PV021: Neural networks Tomáš Brázdil 1 Course organization Course materials: Main: The lecture Neural Networks and Deep Learning by Michael Nielsen http://neuralnetworksanddeeplearning.com/ (Extremely well written modern online textbook.) Deep learning by Ian Goodfellow, Yoshua Bengio and Aaron Courville http://www.deeplearningbook.org/ (A very good overview of the state-of-the-art in neural networks.) Suggested: deeplearning.ai courses by Andrew Ng 2 Course organization Evaluation: Project teams of two students implementation of a selected model + analysis of given data implementation either in C, C++ without use of any specialized libraries for data analysis and machine learning need to get over a given accuracy threshold (a gentle one, just to eliminate non-functional implementations) 3 Course organization Evaluation: Project teams of two students implementation of a selected model + analysis of given data implementation either in C, C++ without use of any specialized libraries for data analysis and machine learning need to get over a given accuracy threshold (a gentle one, just to eliminate non-functional implementations) Oral exam I may ask about anything from the lecture! 3 FAQ Q: Why we cannot use specialized libraries in projects? 4 FAQ Q: Why we cannot use specialized libraries in projects? A: In order to "touch" the low level implementation details of the algorithms. You should not even use libraries for linear algebra and numerical methods, so that you will be confronted with rounding errors and numerical instabilities. 4 FAQ Q: Why we cannot use specialized libraries in projects? A: In order to "touch" the low level implementation details of the algorithms. You should not even use libraries for linear algebra and numerical methods, so that you will be confronted with rounding errors and numerical instabilities. Q: Why should you attend this course when there are infinitely many great reasources elsewhere? A: There are at least two reasons: You may discuss issues with me, my colleagues and other students. I will make you truly learn fundamentals by heart. 4 Notable features of the course Use of mathematical notation and reasoning (contains several proofs that are mandatory for the exam) Sometimes goes deeper into statistical underpinnings of neural networks learning The project demands a complete working solution which must satisfy a prescribed performance specification 5 Notable features of the course Use of mathematical notation and reasoning (contains several proofs that are mandatory for the exam) Sometimes goes deeper into statistical underpinnings of neural networks learning The project demands a complete working solution which must satisfy a prescribed performance specification An unusual exam system! You can repeat the oral exam as many times as needed (only the best grade goes into IS). 5 Notable features of the course Use of mathematical notation and reasoning (contains several proofs that are mandatory for the exam) Sometimes goes deeper into statistical underpinnings of neural networks learning The project demands a complete working solution which must satisfy a prescribed performance specification An unusual exam system! You can repeat the oral exam as many times as needed (only the best grade goes into IS). An example of an instruction email (from another course with the same system): It is typically not sufficient to devote a single afternoon to the preparation for the exam. You have to know _everything_ (which means every single thing) starting with the slide 42 and ending with the slide 245 with notable exceptions of slides: 121 - 123, 137 - 140, 165, 167. Proofs presented on the whiteboard are also mandatory. 5 Machine learning in general Machine learning = construction of systems that may learn their functionality from data (... and thus do not need to be programmed.) 6 Machine learning in general Machine learning = construction of systems that may learn their functionality from data (... and thus do not need to be programmed.) spam filter learns to recognize spam from a database of "labelled" emails consequently is able to distinguish spam from ham 6 Machine learning in general Machine learning = construction of systems that may learn their functionality from data (... and thus do not need to be programmed.) spam filter learns to recognize spam from a database of "labelled" emails consequently is able to distinguish spam from ham handwritten text reader learns from a database of handwritten letters (or text) labelled by their correct meaning consequently is able to recognize text 6 Machine learning in general Machine learning = construction of systems that may learn their functionality from data (... and thus do not need to be programmed.) spam filter learns to recognize spam from a database of "labelled" emails consequently is able to distinguish spam from ham handwritten text reader learns from a database of handwritten letters (or text) labelled by their correct meaning consequently is able to recognize text · · · and lots of much much more sophisticated applications ... 6 Machine learning in general Machine learning = construction of systems that may learn their functionality from data (... and thus do not need to be programmed.) spam filter learns to recognize spam from a database of "labelled" emails consequently is able to distinguish spam from ham handwritten text reader learns from a database of handwritten letters (or text) labelled by their correct meaning consequently is able to recognize text · · · and lots of much much more sophisticated applications ... Basic attributes of learning algorithms: representation: ability to capture the inner structure of training data generalization: ability to work properly on new data 6 Machine learning in general Machine learning algorithms typically construct mathematical models of given data. The models may be subsequently applied to fresh data. 7 Machine learning in general Machine learning algorithms typically construct mathematical models of given data. The models may be subsequently applied to fresh data. There are many types of models: decision trees support vector machines hidden Markov models Bayes networks and other graphical models neural networks · · · Neural networks, based on models of a (human) brain, form a natural basis for learning algorithms! 7 Artificial neural networks Artificial neuron is a rough mathematical approximation of a biological neuron. (Aritificial) neural network (NN) consists of a number of interconnected artificial neurons. "Behavior" of the network is encoded in connections between neurons. σ ξ x1 x2 xn y Zdroj obrázku: http://tulane.edu/sse/cmb/people/schrader/ 8 Why artificial neural networks? Modelling of biological neural networks (computational neuroscience). simplified mathematical models help to identify important mechanisms How a brain receives information? How the information is stored? How a brain develops? · · · 9 Why artificial neural networks? Modelling of biological neural networks (computational neuroscience). simplified mathematical models help to identify important mechanisms How a brain receives information? How the information is stored? How a brain develops? · · · neuroscience is strongly multidisciplinary; precise mathematical descriptions help in communication among experts and in design of new experiments. I will not spend much time on this area! 9 Why artificial neural networks? Neural networks in machine learning. Typically primitive models, far from their biological counterparts (but often inspired by biology). 10 Why artificial neural networks? Neural networks in machine learning. Typically primitive models, far from their biological counterparts (but often inspired by biology). Strongly oriented towards concrete application domains: decision making and control - autonomous vehicles, manufacturing processes, control of natural resources games - backgammon, poker, GO, Starcraft, ... finance - stock prices, risk analysis medicine - diagnosis, signal processing (EKG, EEG, ...), image processing (MRI, roentgen, WSI ...) text and speech processing - automatic translation, text generation, speech recognition other signal processing - filtering, radar tracking, noise reduction · · · I will concentrate on this area! 10 Important features of neural networks Massive parallelism many slow (and "dumb") computational elements work in parallel on several levels of abstraction 11 Important features of neural networks Massive parallelism many slow (and "dumb") computational elements work in parallel on several levels of abstraction Learning a kid learns to recognize a rabbit after seeing several rabbits 11 Important features of neural networks Massive parallelism many slow (and "dumb") computational elements work in parallel on several levels of abstraction Learning a kid learns to recognize a rabbit after seeing several rabbits Generalization a kid is able to recognize a new rabbit after seeing several (old) rabbits 11 Important features of neural networks Massive parallelism many slow (and "dumb") computational elements work in parallel on several levels of abstraction Learning a kid learns to recognize a rabbit after seeing several rabbits Generalization a kid is able to recognize a new rabbit after seeing several (old) rabbits Robustness a blurred photo of a rabbit may still be classified as an image of a rabbit 11 Important features of neural networks Massive parallelism many slow (and "dumb") computational elements work in parallel on several levels of abstraction Learning a kid learns to recognize a rabbit after seeing several rabbits Generalization a kid is able to recognize a new rabbit after seeing several (old) rabbits Robustness a blurred photo of a rabbit may still be classified as an image of a rabbit Graceful degradation Experiments have shown that damaged neural network is still able to work quite well Damaged network may re-adapt, remaining neurons may take on functionality of the damaged ones 11 The aim of the course We will concentrate on basic techniques and principles of neural networks, fundamental models of neural networks and their applications. You should learn basic models (multilayer perceptron, convolutional networks, recurrent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets, autoencoders and generative adversarial networks) 12 The aim of the course We will concentrate on basic techniques and principles of neural networks, fundamental models of neural networks and their applications. You should learn basic models (multilayer perceptron, convolutional networks, recurrent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets, autoencoders and generative adversarial networks) Standard applications of these models (image processing, a little bit of speech and text processing) 12 The aim of the course We will concentrate on basic techniques and principles of neural networks, fundamental models of neural networks and their applications. You should learn basic models (multilayer perceptron, convolutional networks, recurrent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets, autoencoders and generative adversarial networks) Standard applications of these models (image processing, a little bit of speech and text processing) Basic learning algorithms (gradient descent & backpropagation, Hebb’s rule) 12 The aim of the course We will concentrate on basic techniques and principles of neural networks, fundamental models of neural networks and their applications. You should learn basic models (multilayer perceptron, convolutional networks, recurrent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets, autoencoders and generative adversarial networks) Standard applications of these models (image processing, a little bit of speech and text processing) Basic learning algorithms (gradient descent & backpropagation, Hebb’s rule) Basic practical training techniques (data preparation, setting various parameters, control of learning) 12 The aim of the course We will concentrate on basic techniques and principles of neural networks, fundamental models of neural networks and their applications. You should learn basic models (multilayer perceptron, convolutional networks, recurrent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets, autoencoders and generative adversarial networks) Standard applications of these models (image processing, a little bit of speech and text processing) Basic learning algorithms (gradient descent & backpropagation, Hebb’s rule) Basic practical training techniques (data preparation, setting various parameters, control of learning) Basic information about current implementations (TensorFlow, Keras) 12 Biological neural network Human neural network consists of approximately 1011 (100 billion on the short scale) neurons; a single cubic centimeter of a human brain contains almost 50 million neurons. Each neuron is connected with approx. 104 neurons. Neurons themselves are very complex systems. 13 Biological neural network Human neural network consists of approximately 1011 (100 billion on the short scale) neurons; a single cubic centimeter of a human brain contains almost 50 million neurons. Each neuron is connected with approx. 104 neurons. Neurons themselves are very complex systems. Rough description of nervous system: External stimulus is received by sensory receptors (e.g. eye cells). 13 Biological neural network Human neural network consists of approximately 1011 (100 billion on the short scale) neurons; a single cubic centimeter of a human brain contains almost 50 million neurons. Each neuron is connected with approx. 104 neurons. Neurons themselves are very complex systems. Rough description of nervous system: External stimulus is received by sensory receptors (e.g. eye cells). Information is futher transfered via peripheral nervous system (PNS) to the central nervous systems (CNS) where it is processed (integrated), and subseqently, an output signal is produced. 13 Biological neural network Human neural network consists of approximately 1011 (100 billion on the short scale) neurons; a single cubic centimeter of a human brain contains almost 50 million neurons. Each neuron is connected with approx. 104 neurons. Neurons themselves are very complex systems. Rough description of nervous system: External stimulus is received by sensory receptors (e.g. eye cells). Information is futher transfered via peripheral nervous system (PNS) to the central nervous systems (CNS) where it is processed (integrated), and subseqently, an output signal is produced. Afterwards, the output signal is transfered via PNS to effectors (e.g. muscle cells). 13 Biological neural network Zdroj: N. Campbell and J. Reece; Biology, 7th Edition; ISBN: 080537146X 14 Summation 15 Biological and Mathematical neurons 16 Formal neuron (without bias) σ ξ x1 x2 xn y w1 w2 · · · wn x1, . . . , xn ∈ R are inputs 17 Formal neuron (without bias) σ ξ x1 x2 xn y w1 w2 · · · wn x1, . . . , xn ∈ R are inputs w1, . . . , wn ∈ R are weights 17 Formal neuron (without bias) σ ξ x1 x2 xn y w1 w2 · · · wn x1, . . . , xn ∈ R are inputs w1, . . . , wn ∈ R are weights ξ is an inner potential; almost always ξ = n i=1 wixi 17 Formal neuron (without bias) σ ξ x1 x2 xn y w1 w2 · · · wn x1, . . . , xn ∈ R are inputs w1, . . . , wn ∈ R are weights ξ is an inner potential; almost always ξ = n i=1 wixi y is an output given by y = σ(ξ) where σ is an activation function; e.g. a unit step function σ(ξ) =    1 ξ ≥ h ; 0 ξ < h. where h ∈ R is a threshold. 17 Formal neuron (with bias) σ ξ x1 x2 xn x0 = 1 bias threshold y w1 w2 · · · wn w0 = −h x0 = 1, x1, . . . , xn ∈ R are inputs 18 Formal neuron (with bias) σ ξ x1 x2 xn x0 = 1 bias threshold y w1 w2 · · · wn w0 = −h x0 = 1, x1, . . . , xn ∈ R are inputs w0, w1, . . . , wn ∈ R are weights 18 Formal neuron (with bias) σ ξ x1 x2 xn x0 = 1 bias threshold y w1 w2 · · · wn w0 = −h x0 = 1, x1, . . . , xn ∈ R are inputs w0, w1, . . . , wn ∈ R are weights ξ is an inner potential; almost always ξ = w0 + n i=1 wixi 18 Formal neuron (with bias) σ ξ x1 x2 xn x0 = 1 bias threshold y w1 w2 · · · wn w0 = −h x0 = 1, x1, . . . , xn ∈ R are inputs w0, w1, . . . , wn ∈ R are weights ξ is an inner potential; almost always ξ = w0 + n i=1 wixi y is an output given by y = σ(ξ) where σ is an activation function; e.g. a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. (The threshold h has been substituted with the new input x0 = 1 and the weight w0 = −h.) 18 Neuron and linear separation ξ = 0 ξ > 0 ξ > 0 ξ < 0 ξ < 0 inner potential ξ = w0 + n i=1 wixi determines a separation hyperplane in the n-dimensional input space in 2d line in 3d plane · · · 19 Neuron and linear separation σ σ( wixi) x1 xn · · · 1/0 by A/B w1 wn n = 8 · 8, i.e. the number of pixels in the images. Inputs are binary vectors of dimension n (black pixel ≈ 1, white pixel ≈ 0). 20 Neuron and linear separation σ x1 xn · · · x0 = 1 1/0 pro A/B w1 wn w0 n = 8 · 8, i.e. the number of pixels in the images. Inputs are binary vectors of dimension n (black pixel ≈ 1, white pixel ≈ 0). 21 Neuron and linear separation ¯w0 + n i=1 ¯wixi = 0 w0 + n i=1 wixi = 0 A A A A B B B Red line classifies incorrectly Green line classifies correctly (may be a result of a correction by a learning algorithm) 22 Neuron and linear separation (XOR) 0 (0, 0) 1 (0, 1) 1 (0, 1) 0 (1, 1) x1 x2 No line separates ones from zeros. 23 Neural networks Neural network consists of formal neurons interconnected in such a way that the output of one neuron is an input of several other neurons. In order to describe a particular type of neural networks we need to specify: Architecture How the neurons are connected. Activity How the network transforms inputs to outputs. Learning How the weights are changed during training. 24 Architecture Network architecture is given as a digraph whose nodes are neurons and edges are connections. We distinguish several categories of neurons: Output neurons Hidden neurons Input neurons (In general, a neuron may be both input and output; a neuron is hidden if it is neither input, nor output.) 25 Architecture – Cycles A network is cyclic (recurrent) if its architecture contains a directed cycle. 26 Architecture – Cycles A network is cyclic (recurrent) if its architecture contains a directed cycle. Otherwise it is acyclic (feed-forward) 26 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers 27 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers layers numbered from 0; the input layer has number 0 E.g. three-layer network has two hidden layers and one output layer 27 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers layers numbered from 0; the input layer has number 0 E.g. three-layer network has two hidden layers and one output layer Neurons in the i-th layer are connected with all neurons in the i + 1-st layer 27 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers layers numbered from 0; the input layer has number 0 E.g. three-layer network has two hidden layers and one output layer Neurons in the i-th layer are connected with all neurons in the i + 1-st layer Architecture of a MLP is typically described by numbers of neurons in individual layers (e.g. 2-4-3-2) 27 Activity Consider a network with n neurons, k input and output. 28 Activity Consider a network with n neurons, k input and output. State of a network is a vector of output values of all neurons. (States of a network with n neurons are vectors of Rn ) State-space of a network is a set of all states. 28 Activity Consider a network with n neurons, k input and output. State of a network is a vector of output values of all neurons. (States of a network with n neurons are vectors of Rn ) State-space of a network is a set of all states. Network input is a vector of k real numbers, i.e. an element of Rk . Network input space is a set of all network inputs. (sometimes we restrict ourselves to a proper subset of Rk ) 28 Activity Consider a network with n neurons, k input and output. State of a network is a vector of output values of all neurons. (States of a network with n neurons are vectors of Rn ) State-space of a network is a set of all states. Network input is a vector of k real numbers, i.e. an element of Rk . Network input space is a set of all network inputs. (sometimes we restrict ourselves to a proper subset of Rk ) Initial state Input neurons set to values from the network input (each component of the network input corresponds to an input neuron) Values of the remaining neurons set to 0. 28 Activity – computation of a network Computation (typically) proceeds in discrete steps. 29 Activity – computation of a network Computation (typically) proceeds in discrete steps. In every step the following happens: 29 Activity – computation of a network Computation (typically) proceeds in discrete steps. In every step the following happens: 1. A set of neurons is selected according to some rule. 2. The selected neurons change their states according to their inputs (they are simply evaluated). (If a neuron does not have any inputs, its value remains constant.) 29 Activity – computation of a network Computation (typically) proceeds in discrete steps. In every step the following happens: 1. A set of neurons is selected according to some rule. 2. The selected neurons change their states according to their inputs (they are simply evaluated). (If a neuron does not have any inputs, its value remains constant.) A computation is finite on a network input x if the state changes only finitely many times (i.e. there is a moment in time after which the state of the network never changes). We also say that the network stops on x. 29 Activity – computation of a network Computation (typically) proceeds in discrete steps. In every step the following happens: 1. A set of neurons is selected according to some rule. 2. The selected neurons change their states according to their inputs (they are simply evaluated). (If a neuron does not have any inputs, its value remains constant.) A computation is finite on a network input x if the state changes only finitely many times (i.e. there is a moment in time after which the state of the network never changes). We also say that the network stops on x. Network output is a vector of values of all output neurons in the network (i.e. an element of R ). Note that the network output keeps changing throughout the computation! 29 Activity – computation of a network Computation (typically) proceeds in discrete steps. In every step the following happens: 1. A set of neurons is selected according to some rule. 2. The selected neurons change their states according to their inputs (they are simply evaluated). (If a neuron does not have any inputs, its value remains constant.) A computation is finite on a network input x if the state changes only finitely many times (i.e. there is a moment in time after which the state of the network never changes). We also say that the network stops on x. Network output is a vector of values of all output neurons in the network (i.e. an element of R ). Note that the network output keeps changing throughout the computation! MLP uses the following selection rule: In the i-th step evaluate all neurons in the i-th layer. 29 Activity – semantics of a network Definition Consider a network with n neurons, k input, output. Let A ⊆ Rk and B ⊆ R . Suppose that the network stops on every input of A. Then we say that the network computes a function F : A → B if for every network input x the vector F(x) ∈ B is the output of the network after the computation on x stops. 30 Activity – semantics of a network Definition Consider a network with n neurons, k input, output. Let A ⊆ Rk and B ⊆ R . Suppose that the network stops on every input of A. Then we say that the network computes a function F : A → B if for every network input x the vector F(x) ∈ B is the output of the network after the computation on x stops. 30 Activity – semantics of a network Definition Consider a network with n neurons, k input, output. Let A ⊆ Rk and B ⊆ R . Suppose that the network stops on every input of A. Then we say that the network computes a function F : A → B if for every network input x the vector F(x) ∈ B is the output of the network after the computation on x stops. Example 1 This network computes a function from R2 to R. 30 Activity – inner potential and activation functions In order to specify activity of the network, we need to specify how the inner potentials ξ are computed and what are the activation functions σ. 31 Activity – inner potential and activation functions In order to specify activity of the network, we need to specify how the inner potentials ξ are computed and what are the activation functions σ. We assume (unless otherwise specified) that ξ = w0 + n i=1 wi · xi here x = (x1, . . . , xn) are inputs of the neuron and w = (w1, . . . , wn) are weights. 31 Activity – inner potential and activation functions In order to specify activity of the network, we need to specify how the inner potentials ξ are computed and what are the activation functions σ. We assume (unless otherwise specified) that ξ = w0 + n i=1 wi · xi here x = (x1, . . . , xn) are inputs of the neuron and w = (w1, . . . , wn) are weights. There are special types of neural network where the inner potential is computed differently, e.g. as a "distance" of an input from the weight vector: ξ = x − w here ||·|| is a vector norm, typically Euclidean. 31 Activity – inner potential and activation functions There are many activation functions, typical examples: Unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. 32 Activity – inner potential and activation functions There are many activation functions, typical examples: Unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. (Logistic) sigmoid σ(ξ) = 1 1 + e−λ·ξ here λ ∈ R is a steepness parameter. Hyperbolic tangens σ(ξ) = 1 − e−ξ 1 + e−ξ ReLU σ(ξ) = max(ξ, 0) 32 Activity – XOR 1 1 σ 01 σ0 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 1 1 σ 11 σ0 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 0 0 σ 01 σ0 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 0 0 σ 01 σ 1 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 1 0 σ 01 σ0 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 1 0 σ 11 σ 1 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 1 0 σ 11 σ 1 1 σ 1 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 0 1 σ 01 σ0 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 0 1 σ 11 σ 1 1 σ 0 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – XOR 0 1 σ 11 σ 1 1 σ 1 1 −22 2 −2 1 −1 1 3 −2 Activation function is a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The network computes XOR(x1, x2) x1 x2 y 1 1 0 1 0 1 0 1 1 0 0 0 33 Activity – MLP and linear separation 0 (0, 0) 1 (0, 1) 1 (0, 1) 0 (1, 1) P1 P2 x1 x2 σ1 σ 1 σ1 −22 2 −2 1 −1 1 3 −2 The line P1 is given by −1 + 2x1 + 2x2 = 0 The line P2 is given by 3 − 2x1 − 2x2 = 0 34 Activity – example x1 1 σ 0 1 σ0 1 σ 0 1 1 2 −5 1 −2 11 −2 −1 The activation function is the unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The input is equal to 1 35 Activity – example x1 1 σ 1 1 σ0 1 σ 0 1 1 2 −5 1 −2 11 −2 −1 The activation function is the unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The input is equal to 1 35 Activity – example x1 1 σ 1 1 σ 1 1 σ 0 1 1 2 −5 1 −2 11 −2 −1 The activation function is the unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The input is equal to 1 35 Activity – example x1 1 σ 1 1 σ 1 1 σ 1 1 1 2 −5 1 −2 11 −2 −1 The activation function is the unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The input is equal to 1 35 Activity – example x1 1 σ 0 1 σ 1 1 σ 1 1 1 2 −5 1 −2 11 −2 −1 The activation function is the unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. The input is equal to 1 35 Learning Consider a network with n neurons, k input and output. 36 Learning Consider a network with n neurons, k input and output. Configuration of a network is a vector of all values of weights. (Configurations of a network with m connections are elements of Rm ) Weight-space of a network is a set of all configurations. 36 Learning Consider a network with n neurons, k input and output. Configuration of a network is a vector of all values of weights. (Configurations of a network with m connections are elements of Rm ) Weight-space of a network is a set of all configurations. initial configuration weights can be initialized randomly or using some sophisticated algorithm 36 Learning algorithms Learning rule for weight adaptation. (the goal is to find a configuration in which the network computes a desired function) 37 Learning algorithms Learning rule for weight adaptation. (the goal is to find a configuration in which the network computes a desired function) Supervised learning The desired function is described using training examples that are pairs of the form (input, output). Learning algorithm searches for a configuration which "corresponds" to the training examples, typically by minimizing an error function. 37 Learning algorithms Learning rule for weight adaptation. (the goal is to find a configuration in which the network computes a desired function) Supervised learning The desired function is described using training examples that are pairs of the form (input, output). Learning algorithm searches for a configuration which "corresponds" to the training examples, typically by minimizing an error function. Unsupervised learning The training set contains only inputs. The goal is to determine distribution of the inputs (clustering, deep belief networks, etc.) 37 Supervised learning – illustration A A A A B B B classification in the plane using a single neuron 38 Supervised learning – illustration A A A A B B B classification in the plane using a single neuron training examples are of the form (point, value) where the value is either 1, or 0 depending on whether the point is either A, or B 38 Supervised learning – illustration A A A A B B B classification in the plane using a single neuron training examples are of the form (point, value) where the value is either 1, or 0 depending on whether the point is either A, or B the algorithm considers examples one after another whenever an incorrectly classified point is considered, the learning algorithm turns the line in the direction of the point 38 Summary – Advantages of neural networks Massive parallelism neurons can be evaluated in parallel 39 Summary – Advantages of neural networks Massive parallelism neurons can be evaluated in parallel Learning many sophisticated learning algorithms used to "program" neural networks 39 Summary – Advantages of neural networks Massive parallelism neurons can be evaluated in parallel Learning many sophisticated learning algorithms used to "program" neural networks generalization and robustness information is encoded in a distributed manned in weights "close" inputs typicaly get similar values 39 Summary – Advantages of neural networks Massive parallelism neurons can be evaluated in parallel Learning many sophisticated learning algorithms used to "program" neural networks generalization and robustness information is encoded in a distributed manned in weights "close" inputs typicaly get similar values Graceful degradation damage typically causes only a decrease in precision of results 39 Expressive power of neural networks 40 Formal neuron (with bias) σ ξ x1 x2 xn x0 = 1 bias threshold y w1 w2 · · · wn w0 = −h x0 = 1, x1, . . . , xn ∈ R are inputs w0, w1, . . . , wn ∈ R are weights ξ is an inner potential; almost always ξ = w0 + n i=1 wixi y is an output given by y = σ(ξ) where σ is an activation function; e.g. a unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. 41 Boolean functions Activation function: unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. 42 Boolean functions Activation function: unit step function σ(ξ) =    1 ξ ≥ 0 ; 0 ξ < 0. σ x1 x2 xn x0 = 1 y = AND(x1, . . . , xn) 1 1 · · · 1 −n σ x1 x2 xn x0 = 1 y = OR(x1, . . . , xn) 1 1 · · · 1 −1 σ x1 x0 = 1 y = NOT(x1) −1 0 42 Boolean functions Theorem Let σ be the unit step function. Two layer MLPs, where each neuron has σ as the activation function, are able to compute all functions of the form F : {0, 1}n → {0, 1}. 43 Boolean functions Theorem Let σ be the unit step function. Two layer MLPs, where each neuron has σ as the activation function, are able to compute all functions of the form F : {0, 1}n → {0, 1}. Proof. Given a vector v = (v1, . . . , vn) ∈ {0, 1}n, consider a neuron Nv whose output is 1 iff the input is v: σ y x1 xi xn x0 = 1 w1 wi · · ·· · · wn w0 w0 = − n i=1 vi wi =    1 vi = 1 −1 vi = 0 Now let us connect all outputs of all neurons Nv satisfying F(v) = 1 using a neuron implementing OR. 43 Non-linear separation x1 x2 y Consider a three layer network; each neuron has the unit step activation function. The network divides the input space in two subspaces according to the output (0 or 1). 44 Non-linear separation x1 x2 y Consider a three layer network; each neuron has the unit step activation function. The network divides the input space in two subspaces according to the output (0 or 1). The first (hidden) layer divides the input space into half-spaces. 44 Non-linear separation x1 x2 y Consider a three layer network; each neuron has the unit step activation function. The network divides the input space in two subspaces according to the output (0 or 1). The first (hidden) layer divides the input space into half-spaces. The second layer may e.g. make intersections of the half-spaces ⇒ convex sets. 44 Non-linear separation x1 x2 y Consider a three layer network; each neuron has the unit step activation function. The network divides the input space in two subspaces according to the output (0 or 1). The first (hidden) layer divides the input space into half-spaces. The second layer may e.g. make intersections of the half-spaces ⇒ convex sets. The third layer may e.g. make unions of some convex sets. 44 Non-linear separation – illustration x1 xk · · · · · · · · · y Consider three layer networks; each neuron has the unit step activation function. Three layer nets are capable of "approximating" any "reasonable" subset A of the input space Rk . 45 Non-linear separation – illustration x1 xk · · · · · · · · · y Consider three layer networks; each neuron has the unit step activation function. Three layer nets are capable of "approximating" any "reasonable" subset A of the input space Rk . Cover A with hypercubes (in 2D squares, in 3D cubes, ...) 45 Non-linear separation – illustration x1 xk · · · · · · · · · y Consider three layer networks; each neuron has the unit step activation function. Three layer nets are capable of "approximating" any "reasonable" subset A of the input space Rk . Cover A with hypercubes (in 2D squares, in 3D cubes, ...) Each hypercube K can be separated using a two layer network NK (i.e. a function computed by NK gives 1 for points in K and 0 for the rest). 45 Non-linear separation – illustration x1 xk · · · · · · · · · y Consider three layer networks; each neuron has the unit step activation function. Three layer nets are capable of "approximating" any "reasonable" subset A of the input space Rk . Cover A with hypercubes (in 2D squares, in 3D cubes, ...) Each hypercube K can be separated using a two layer network NK (i.e. a function computed by NK gives 1 for points in K and 0 for the rest). Finally, connect outputs of the nets NK satisfying K ∩ A ∅ using a neuron implementing OR. 45 Non-linear separation - sigmoid Theorem (Cybenko 1989 - informal version) Let σ be a continuous function which is sigmoidal, i.e. satisfies σ(x) =    1 for x → +∞ 0 for x → −∞ For every "reasonable" set A ⊆ [0, 1]n, there is a two layer network where each hidden neuron has the activation function σ (output neurons are linear), that satisfies the following: For "most" vectors v ∈ [0, 1]n we have that v ∈ A iff the network output is > 0 for the input v. For mathematically oriented: "reasonable" means Lebesgue measurable "most" means that the set of incorrectly classified vectors has the Lebesgue measure smaller than a given ε > 0 46 Non-linear separation - practical illustration ALVINN drives a car 47 Non-linear separation - practical illustration ALVINN drives a car The net has 30 × 32 = 960 inputs (the input space is thus R960 ) 47 Non-linear separation - practical illustration ALVINN drives a car The net has 30 × 32 = 960 inputs (the input space is thus R960 ) Input values correspond to shades of gray of pixels. 47 Non-linear separation - practical illustration ALVINN drives a car The net has 30 × 32 = 960 inputs (the input space is thus R960 ) Input values correspond to shades of gray of pixels. Output neurons "classify" images of the road based on their "curvature". Zdroj obrázku: http://jmvidal.cse.sc.edu/talks/ann/alvin.html 47 Function approximation - three layers Let σ be a logistic sigmoid, i.e. σ(ξ) = 1 1 + e−ξ For every continuous function f : [0, 1]n → [0, 1] and ε > 0 there is a three-layer network computing a function F : [0, 1]n → [0, 1] such that there is a linear activation in the output layer, i.e. the value of the output neuron is its inner potential ξ, 48 Function approximation - three layers Let σ be a logistic sigmoid, i.e. σ(ξ) = 1 1 + e−ξ For every continuous function f : [0, 1]n → [0, 1] and ε > 0 there is a three-layer network computing a function F : [0, 1]n → [0, 1] such that there is a linear activation in the output layer, i.e. the value of the output neuron is its inner potential ξ, the remaining neurons have the logistic sigmoid σ as their activation, 48 Function approximation - three layers Let σ be a logistic sigmoid, i.e. σ(ξ) = 1 1 + e−ξ For every continuous function f : [0, 1]n → [0, 1] and ε > 0 there is a three-layer network computing a function F : [0, 1]n → [0, 1] such that there is a linear activation in the output layer, i.e. the value of the output neuron is its inner potential ξ, the remaining neurons have the logistic sigmoid σ as their activation, for every v ∈ [0, 1]n we have that |F(v) − f(v)| < ε. 48 Function approximation – three layer networks x1 x2 σ σ σ σ σ· · · · · · · · · ζ y weighted sum of "spikes" ... + the other two 90 degree rotations a "spike" inner potential the value of the neuron 49 Function approximation - two-layer networks Theorem (Cybenko 1989) Let σ be a continuous function which is sigmoidal, i.e. is increasing and satisfies σ(x) =    1 pro x → +∞ 0 pro x → −∞ For every continuous function f : [0, 1]n → [0, 1] and every ε > 0 there is a function F : [0, 1]n → [0, 1] computed by a two layer network where each hidden neuron has the activation function σ (output neurons are linear), that satisfies the following |f(v) − F(v)| < ε pro každé v ∈ [0, 1]n . 50 Neural networks and computability Consider recurrent networks (i.e. containing cycles) 51 Neural networks and computability Consider recurrent networks (i.e. containing cycles) with real weights (in general); 51 Neural networks and computability Consider recurrent networks (i.e. containing cycles) with real weights (in general); one input neuron and one output neuron (the network computes a function F : A → R where A ⊆ R contains all inputs on which the network stops); 51 Neural networks and computability Consider recurrent networks (i.e. containing cycles) with real weights (in general); one input neuron and one output neuron (the network computes a function F : A → R where A ⊆ R contains all inputs on which the network stops); parallel activity rule (output values of all neurons are recomputed in every step); 51 Neural networks and computability Consider recurrent networks (i.e. containing cycles) with real weights (in general); one input neuron and one output neuron (the network computes a function F : A → R where A ⊆ R contains all inputs on which the network stops); parallel activity rule (output values of all neurons are recomputed in every step); activation function σ(ξ) =    1 ξ ≥ 1 ; ξ 0 ≤ ξ ≤ 1 ; 0 ξ < 0. 51 Neural networks and computability Consider recurrent networks (i.e. containing cycles) with real weights (in general); one input neuron and one output neuron (the network computes a function F : A → R where A ⊆ R contains all inputs on which the network stops); parallel activity rule (output values of all neurons are recomputed in every step); activation function σ(ξ) =    1 ξ ≥ 1 ; ξ 0 ≤ ξ ≤ 1 ; 0 ξ < 0. We encode words ω ∈ {0, 1}+ into numbers as follows: δ(ω) = |ω| i=1 ω(i) 2i + 1 2|ω|+1 E.g. ω = 11001 gives δ(ω) = 1 2 + 1 22 + 1 25 + 1 26 (= 0.110011 in binary form). 51 Neural networks and computability A network recognizes a language L ⊆ {0, 1}+ if it computes a function F : A → R (A ⊆ R) such that ω ∈ L iff δ(ω) ∈ A and F(δ(ω)) > 0. 52 Neural networks and computability A network recognizes a language L ⊆ {0, 1}+ if it computes a function F : A → R (A ⊆ R) such that ω ∈ L iff δ(ω) ∈ A and F(δ(ω)) > 0. Recurrent networks with rational weights are equivalent to Turing machines For every recursively enumerable language L ⊆ {0, 1}+ there is a recurrent network with rational weights and less than 1000 neurons, which recognizes L. The halting problem is undecidable for networks with at least 25 neurons and rational weights. There is "universal" network (equivalent of the universal Turing machine) 52 Neural networks and computability A network recognizes a language L ⊆ {0, 1}+ if it computes a function F : A → R (A ⊆ R) such that ω ∈ L iff δ(ω) ∈ A and F(δ(ω)) > 0. Recurrent networks with rational weights are equivalent to Turing machines For every recursively enumerable language L ⊆ {0, 1}+ there is a recurrent network with rational weights and less than 1000 neurons, which recognizes L. The halting problem is undecidable for networks with at least 25 neurons and rational weights. There is "universal" network (equivalent of the universal Turing machine) Recurrent networks are super-Turing powerful 52 Neural networks and computability A network recognizes a language L ⊆ {0, 1}+ if it computes a function F : A → R (A ⊆ R) such that ω ∈ L iff δ(ω) ∈ A and F(δ(ω)) > 0. Recurrent networks with rational weights are equivalent to Turing machines For every recursively enumerable language L ⊆ {0, 1}+ there is a recurrent network with rational weights and less than 1000 neurons, which recognizes L. The halting problem is undecidable for networks with at least 25 neurons and rational weights. There is "universal" network (equivalent of the universal Turing machine) Recurrent networks are super-Turing powerful For every language L ⊆ {0, 1}+ there is a recurrent network with less than 1000 nerons which recognizes L. 52 Summary of theoretical results Neural networks are very strong from the point of view of theory: All Boolean functions can be expressed using two-layer networks. Two-layer networks may approximate any continuous function. Recurrent networks are at least as strong as Turing machines. 53 Summary of theoretical results Neural networks are very strong from the point of view of theory: All Boolean functions can be expressed using two-layer networks. Two-layer networks may approximate any continuous function. Recurrent networks are at least as strong as Turing machines. These results are purely theoretical! "Theoretical" networks are extremely huge. It is very difficult to handcraft them even for simplest problems. From practical point of view, the most important advantage of neural networks are: learning, generalization, robustness. 53 Neural networks vs classical computers Neural networks "Classical" computers Data implicitly in weights explicitly Computation naturally parallel sequential, localized Robustness robust w.r.t. input corruption & damage changing one bit may completely crash the computation Precision imprecise, network recalls a training example "similar" to the input (typically) precise Programming learning manual 54 History & implementations 55 History of neurocomputers 1951: SNARC (Minski et al) the first implementation of neural network a rat strives to exit a maze 40 artificial neurons (300 vacuum tubes, engines, etc.) 56 History of neurocomputers 1957: Mark I Perceptron (Rosenblatt et al) - the first successful network for image recognition single layer network image represented by 20 × 20 photocells intensity of pixels was treated as the input to a perceptron (basically the formal neuron), which recognized figures weights were implemented using potentiometers, each set by its own engine it was possible to arbitrarily reconnect inputs to neurons to demonstrate adaptability 57 History of neurocomputers 1960: ADALINE (Widrow & Hof) single layer neural network weights stored in a newly invented electronic component memistor, which remembers history of electric current in the form of resistance. Widrow founded a company Memistor Corporation, which sold implementations of neural networks. 1960-66: several companies concerned with neural networks were founded. 58 History of neurocomputers 1967-82: dead still after publication of a book by Minski & Papert (published 1969, title Perceptrons) 1983-end of 90s: revival of neural networks many attempts at hardware implementations application specific chips (ASIC) programmable hardware (FPGA) hw implementations typically not better than "software" implementations on universal computers (problems with weight storage, size, speed, cost of production etc.) 59 History of neurocomputers 1967-82: dead still after publication of a book by Minski & Papert (published 1969, title Perceptrons) 1983-end of 90s: revival of neural networks many attempts at hardware implementations application specific chips (ASIC) programmable hardware (FPGA) hw implementations typically not better than "software" implementations on universal computers (problems with weight storage, size, speed, cost of production etc.) end of 90s-cca 2005: NN suppressed by other machine learning methods (support vector machines (SVM)) 2006-now: The boom of neural networks! deep networks – often better than any other method GPU implementations ... some specialized hw implementations (Google’s TPU) 59 Some highlights Breakthrough in image recognition. Accuracy of image recognition improved by an order of magnitude in 5 years. Breakthrough in game playing. Superhuman results in Go and Chess almost without any human intervention. Master level in Starcraft, poker, etc. Breakthrough in machine translation. Switching to deep learning produced a 60% increase in translation accuracy compared to the phrase-based approach previously used in Google Translate (in human evaluation) Breakthrough in speech processing. 60 History in waves ... Figure: The figure shows two of the three historical waves of artificial neural nets research, as measured by the frequency of the phrases "cybernetics" and "connectionism" or "neural networks" according to Google Books (the third wave is too recent to appear). 61 Current hardware – What do we face? Increasing dataset size ... 62 Current hardware – What do we face? ... and thus increasing size of neural networks ... 2. ADALINE 4. Early back-propagation network (Rumelhart et al., 1986b) 8. Image recognition: LeNet-5 (LeCun et al., 1998b) 10. Dimensionality reduction: Deep belief network (Hinton et al., 2006) ... here the third "wave" of neural networks started 15. Digit recognition: GPU-accelerated multilayer perceptron (Ciresan et al., 2010) 18. Image recognition (AlexNet): Multi-GPU convolutional network (Krizhevsky et al., 2012) 20. Image recognition: GoogLeNet (Szegedy et al., 2014a) 63 Current hardware – What do we face? ... as a reward we get this ... Figure: Since deep networks reached the scale necessary to compete in the ImageNetLarge Scale Visual Recognition Challenge, they have consistently won the competition every year, and yielded lower and lower error rates each time. Data from Russakovsky et al. (2014b) and He et al. (2015). 64 Current hardware In 2012, Google trained a large network of 1.7 billion weights and 9 layers The task was image recognition (10 million youtube video frames) The hw comprised a 1000 computer network (16 000 cores), computation took three days. 65 Current hardware In 2012, Google trained a large network of 1.7 billion weights and 9 layers The task was image recognition (10 million youtube video frames) The hw comprised a 1000 computer network (16 000 cores), computation took three days. In 2014, similar task performed on Commodity Off-The-Shelf High Performance Computing (COTS HPC) technology: a cluster of GPU servers with Infiniband interconnects and MPI. Able to train 1 billion parameter networks on just 3 machines in a couple of days. Able to scale to 11 billion weights (approx. 6.5 times larger than the Google model) on 16 GPUs. 65 Current hardware – NVIDIA DGX-1 Station 8x GPU (Tesla V100) TFLOPS = 1000 GPU memory 256GB total NVIDIA Tensor Cores: 5,120 NVIDIA CUDA Cores: 40,960 System memory: 512 GB Network: Dual 10 Gb LAN NVIDIA Deep Learning SDK 66 Deep learning in clouds Several companies offer cloud services for deep learning: Amazon Web Services Google Cloud Deep Cognition ... Advantages: Do not have to care (too much) about technical problems. Do not have to buy and optimize highend hw/sw, networks etc. Scaling & virtually limitless storage. Disadvatages: Do not have full control. Performance can vary, connectivity problems. Have to pay for services. Privacy issues. 67 Current software TensorFlow (Google) open source software library for numerical computation using data flow graphs allows implementation of most current neural networks allows computation on multiple devices (CPUs, GPUs, ...) Python API Keras: a part of TensorFlow that allows easy description of most modern neural networks PyTorch (Facebook) similar to TensorFlow object oriented Theano (dead): The "academic" grand-daddy of deep-learning frameworks, written in Python. Strongly inspired TensorFlow (some people developing Theano moved on to develop TensorFlow). There are others: Caffe, Deeplearning4j, ... 68 Current software – Keras 69 Current software – Keras functional API 70 Current software – TensorFlow 71 Current software – TensorFlow 72 Current software – PyTorch 73 Other software implementations Most "mathematical" software packages contain some support of neural networks: MATLAB R STATISTICA Weka ... The implementations are typically not on par with the previously mentioned dedicated deep-learning libraries. 74 Training linear models 75 Linear regression (ADALINE) Architecture: x1 x2 xn · · · y x0 = 1 w0 w1 w2 wn w = (w0, w1, . . . , wn) and x = (x0, x1, . . . , xn) where x0 = 1. Activity: inner potential: ξ = w0 + n i=1 wixi = n i=0 wixi = w · x activation function: σ(ξ) = ξ network function: y[w](x) = σ(ξ) = w · x 76 Linear regression (ADALINE) Learning: Given a training dataset T = x1, d1 , x2, d2 , . . . , xp, dp Here xk = (xk0, xk1 . . . , xkn) ∈ Rn+1, xk0 = 1, is the k-th input, and dk ∈ R is the expected output. Intuition: The network is supposed to compute an affine approximation of the function (some of) whose values are given in the training set. 77 Oaks in Wisconsin 78 Linear regression (ADALINE) Error function: E(w) = 1 2 p k=1 w · xk − dk 2 = 1 2 p k=1   n i=0 wixki − dk   2 The goal is to find w which minimizes E(w). 79 Error function 80 Gradient of the error function Consider gradient of the error function: E(w) = ∂E ∂w0 (w), . . . , ∂E ∂wn (w) Intuition: E(w) is a vector in the weight space which points in the direction of the steepest ascent of the error function. Note that the vectors xk are just parameters of the function E, and are thus fixed! 81 Gradient of the error function Consider gradient of the error function: E(w) = ∂E ∂w0 (w), . . . , ∂E ∂wn (w) Intuition: E(w) is a vector in the weight space which points in the direction of the steepest ascent of the error function. Note that the vectors xk are just parameters of the function E, and are thus fixed! Fact If E(w) = 0 = (0, . . . , 0), then w is a global minimum of E. For ADALINE, the error function E(w) is a convex paraboloid and thus has the unique global minimum. 81 Gradient - illustration Caution! This picture just illustrates the notion of gradient ... it is not the convex paraboloid E(w) ! 82 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) E(w0, w1) = 1 2 [(w0+w1·2−1)2+(w0+w1·3−2)2+(w0+w1·4−5)2] 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) E(w0, w1) = 1 2 [(w0+w1·2−1)2+(w0+w1·3−2)2+(w0+w1·4−5)2] δE δw0 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) E(w0, w1) = 1 2 [(w0+w1·2−1)2+(w0+w1·3−2)2+(w0+w1·4−5)2] δE δw0 = (w0 +w1 ·2−1)·1+(w0 +w1 ·3−2)·1+(w0 +w1 ·4−5)·1 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) E(w0, w1) = 1 2 [(w0+w1·2−1)2+(w0+w1·3−2)2+(w0+w1·4−5)2] δE δw0 = (w0 +w1 ·2−1)·1+(w0 +w1 ·3−2)·1+(w0 +w1 ·4−5)·1 δE δw1 83 Gradient of the error function First, consider n = 1. Then the model is y = w0 + w1 · x. Consider a concrete training set: T = {((1, 2), 1), ((1, 3), 2), ((1, 4), 5)} = ((x10, x11), d1), ((x20, x21), d2), ((x30, x31), d3) E(w0, w1) = 1 2 [(w0+w1·2−1)2+(w0+w1·3−2)2+(w0+w1·4−5)2] δE δw0 = (w0 +w1 ·2−1)·1+(w0 +w1 ·3−2)·1+(w0 +w1 ·4−5)·1 δE δw1 = (w0 +w1 ·2−1)·2+(w0 +w1 ·3−2)·3+(w0 +w1 ·4−5)·4 83 Gradient of the error function ∂E ∂w (w) = 1 2 p k=1 δ δw   n i=0 wixki − dk   2 84 Gradient of the error function ∂E ∂w (w) = 1 2 p k=1 δ δw   n i=0 wixki − dk   2 = 1 2 p k=1 2   n i=0 wixki − dk   δ δw   n i=0 wixki − dk   84 Gradient of the error function ∂E ∂w (w) = 1 2 p k=1 δ δw   n i=0 wixki − dk   2 = 1 2 p k=1 2   n i=0 wixki − dk   δ δw   n i=0 wixki − dk   = 1 2 p k=1 2   n i=0 wixki − dk     n i=0 δ δw wixki − δE δw dk   84 Gradient of the error function ∂E ∂w (w) = 1 2 p k=1 δ δw   n i=0 wixki − dk   2 = 1 2 p k=1 2   n i=0 wixki − dk   δ δw   n i=0 wixki − dk   = 1 2 p k=1 2   n i=0 wixki − dk     n i=0 δ δw wixki − δE δw dk   = p k=1 w · xk − dk xk 84 Gradient of the error function ∂E ∂w (w) = 1 2 p k=1 δ δw   n i=0 wixki − dk   2 = 1 2 p k=1 2   n i=0 wixki − dk   δ δw   n i=0 wixki − dk   = 1 2 p k=1 2   n i=0 wixki − dk     n i=0 δ δw wixki − δE δw dk   = p k=1 w · xk − dk xk Thus E(w) = ∂E ∂w0 (w), . . . , ∂E ∂wn (w) = p k=1 w · xk − dk xk 84 Linear regression - learning Batch algorithm (gradient descent): Idea: In every step "move" the weights in the direction opposite to the gradient. 85 Linear regression - learning Batch algorithm (gradient descent): Idea: In every step "move" the weights in the direction opposite to the gradient. The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 85 Linear regression - learning Batch algorithm (gradient descent): Idea: In every step "move" the weights in the direction opposite to the gradient. The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1, weights w(t+1) are computed as follows: w(t+1) = w(t) − ε · E(w(t) ) = w(t) − ε · p k=1 w(t) · xk − dk · xk Here 0 < ε ≤ 1 is a learning rate. 85 Linear regression - learning Batch algorithm (gradient descent): Idea: In every step "move" the weights in the direction opposite to the gradient. The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1, weights w(t+1) are computed as follows: w(t+1) = w(t) − ε · E(w(t) ) = w(t) − ε · p k=1 w(t) · xk − dk · xk Here 0 < ε ≤ 1 is a learning rate. Proposition For sufficiently small ε > 0 the sequence w(0), w(1), w(2), . . . converges (componentwise) to the global minimum of E (i.e. to the vector w satisfying E(w) = 0). 85 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 Linear regression - animation 86 ADALINE - learning Online algorithm (Delta-rule, Widrow-Hoff rule): weights in w(0) initialized randomly close to 0 in the step t + 1, weights w(t+1) are computed as follows: w(t+1) = w(t) − ε(t) · w(t) · xk − dk · xk Here k = t mod p + 1 and 0 < ε(t) ≤ 1 is a learning rate in the step t + 1. Note that the algorithm does not work with the complete gradient but only with its part determined by the currently considered training example. 87 ADALINE - learning Online algorithm (Delta-rule, Widrow-Hoff rule): weights in w(0) initialized randomly close to 0 in the step t + 1, weights w(t+1) are computed as follows: w(t+1) = w(t) − ε(t) · w(t) · xk − dk · xk Here k = t mod p + 1 and 0 < ε(t) ≤ 1 is a learning rate in the step t + 1. Note that the algorithm does not work with the complete gradient but only with its part determined by the currently considered training example. Theorem (Widrow & Hoff) If ε(t) = 1 t , then w(0), w(1), w(2), . . . converges to the global minimum of E. 87 What about classification? Binary classification: Desired outputs 0 and 1. Ideally, capture the probability distribution of classes. 88 What about classification? Binary classification: Desired outputs 0 and 1. ... does not capture probability well (it is not a probability at all) 88 What about classification? Binary classification: Desired outputs 0 and 1. ... logistic sigmoid 1 1+e−(w·x) is much better! 88 Logistic regression x1 x2 xn · · · y x0 = 1 w0 w1 w2 wn w = (w0, w1, . . . , wn) and x = (x0, x1, . . . , xn) where x0 = 1. Activity: inner potential: ξ = w0 + n i=1 wixi = n i=0 wixi = w · x activation function: σ(ξ) = 1 1+e−ξ network function: y[w](x) = σ(ξ) = 1 1+e−(w·x) Intuition: The output y is now interpreted as the probability of the class 1 given the input x. 89 But what is the meaning of the sigmoid? The model gives a probability y of the class 1 given an input x. But why we model such a probability using 1/(1 + e−w·x) ?? 90 But what is the meaning of the sigmoid? The model gives a probability y of the class 1 given an input x. But why we model such a probability using 1/(1 + e−w·x) ?? Let ˆy be the "true" probability of the class 1 to be modeled. What about odds of the class 1? odds(ˆy) = ˆy/(1 − ˆy) 90 But what is the meaning of the sigmoid? The model gives a probability y of the class 1 given an input x. But why we model such a probability using 1/(1 + e−w·x) ?? Let ˆy be the "true" probability of the class 1 to be modeled. What about log odds (aka logit) of the class 1? logit(ˆy) = log(ˆy/(1 − ˆy)) Looks almost linear ... 90 But what is the meaning of the sigmoid? Assume that ˆy is the probability of the class 1. Put log(ˆy/(1 − ˆy)) = w · x 91 But what is the meaning of the sigmoid? Assume that ˆy is the probability of the class 1. Put log(ˆy/(1 − ˆy)) = w · x Then log((1 − ˆy)/ˆy) = −w · x 91 But what is the meaning of the sigmoid? Assume that ˆy is the probability of the class 1. Put log(ˆy/(1 − ˆy)) = w · x Then log((1 − ˆy)/ˆy) = −w · x and (1 − ˆy)/ˆy = e−w·x 91 But what is the meaning of the sigmoid? Assume that ˆy is the probability of the class 1. Put log(ˆy/(1 − ˆy)) = w · x Then log((1 − ˆy)/ˆy) = −w · x and (1 − ˆy)/ˆy = e−w·x and ˆy = 1 1 + e−w·x That is, if we model log odds using a linear function, the probability is obtained by applying the logistic sigmoid on the result of the linear function. 91 Logistic regression Learning: Given a training dataset T = x1, d1 , x2, d2 , . . . , xp, dp Here xk = (xk0, xk1 . . . , xkn) ∈ Rn+1, xk0 = 1, is the k-th input, and dk ∈ {0, 1} is the expected output. 92 Logistic regression Learning: Given a training dataset T = x1, d1 , x2, d2 , . . . , xp, dp Here xk = (xk0, xk1 . . . , xkn) ∈ Rn+1, xk0 = 1, is the k-th input, and dk ∈ {0, 1} is the expected output. What error function? (Binary) cross-entropy: E(w) = p k=1 −(dk log(yk ) + (1 − dk ) log(1 − yk )) What?!? 92 Log likelihood is your friend! Let’s have a "coin" (sides 0 and 1). 93 Log likelihood is your friend! Let’s have a "coin" (sides 0 and 1). The probability of 1 is ˆy and is unknown! 93 Log likelihood is your friend! Let’s have a "coin" (sides 0 and 1). The probability of 1 is ˆy and is unknown! You have tossed the coin 5 times and got a training dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Consider this to be a very special case where the input dimension is 0 93 Log likelihood is your friend! Let’s have a "coin" (sides 0 and 1). The probability of 1 is ˆy and is unknown! You have tossed the coin 5 times and got a training dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Consider this to be a very special case where the input dimension is 0 What is the best model y of ˆy based on the data? 93 Log likelihood is your friend! Let’s have a "coin" (sides 0 and 1). The probability of 1 is ˆy and is unknown! You have tossed the coin 5 times and got a training dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Consider this to be a very special case where the input dimension is 0 What is the best model y of ˆy based on the data? Answer: The one that generates the data with maximum probability! 93 Log likelihood is your friend! Keep in mind our dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} 94 Log likelihood is your friend! Keep in mind our dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Assume that the data was generated by independent trials, then the probability of getting exactly T from our model is L = y · y · (1 − y) · (1 − y) · y How to maximize this w.r.t. y? 94 Log likelihood is your friend! Keep in mind our dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Assume that the data was generated by independent trials, then the probability of getting exactly T from our model is L = y · y · (1 − y) · (1 − y) · y How to maximize this w.r.t. y? Maximize LL = log(L) = log(y)+log(y)+log(1−y)+log(1−y)+log(y) 94 Log likelihood is your friend! Keep in mind our dataset: T = {1, 1, 0, 0, 1} = {d1, . . . , d5} Assume that the data was generated by independent trials, then the probability of getting exactly T from our model is L = y · y · (1 − y) · (1 − y) · y How to maximize this w.r.t. y? Maximize LL = log(L) = log(y)+log(y)+log(1−y)+log(1−y)+log(y) But then −LL = −1·log(y)−1·log(y)−(1−0)·log(1−y)−(1−0)·log(1−y)−1·log(y) i.e. −LL is the cross-entropy. 94 Let the coin depend on the input Consider our model: y = 1 1 + e−(w·x) 95 Let the coin depend on the input Consider our model: y = 1 1 + e−(w·x) The training dataset is now standard: T = x1, d1 , x2, d2 , . . . , xp, dp Here xk = (xk0, xk1 . . . , xkn) ∈ Rn+1, xk0 = 1, is the k-th input, and dk ∈ {0, 1} is the expected output. 95 Let the coin depend on the input Consider our model: y = 1 1 + e−(w·x) The training dataset is now standard: T = x1, d1 , x2, d2 , . . . , xp, dp Here xk = (xk0, xk1 . . . , xkn) ∈ Rn+1, xk0 = 1, is the k-th input, and dk ∈ {0, 1} is the expected output. The likelihood: L = p k=1 ydk k · (1 − yk )(1−dk ) and LL = log(L) = p k=1 (dk log(yk ) + (1 − dk ) log(1 − yk )) and thus −LL = the cross-entropy. Minimizing the cross-entropy maximizes the log-likelihood (and vice versa). 95 Normal Distribution Distribution of continuous random variables. Density (one dimensional, that is over R): p(x) = 1 σ √ 2π exp − (x − µ)2 2σ2 =: N[µ, σ2 ](x) µ is the expected value (the mean), σ2 is the variance. 96 Maximum Likelihood vs Least Squares (Dim 1) Fix a training set D = (x1, d1) , (x2, d2) , . . . , xp, dp 97 Maximum Likelihood vs Least Squares (Dim 1) Fix a training set D = (x1, d1) , (x2, d2) , . . . , xp, dp Assume that each dk has been generated randomly by dk = (w0 + w1 · xk ) + k w0, w1 are unknown numbers k are normally distributed with mean 0 and an unknown variance σ2 97 Maximum Likelihood vs Least Squares (Dim 1) Keep in mind: dk = (w0 + w1 · xk ) + k Assume that 1, . . . , p were generated independently. 98 Maximum Likelihood vs Least Squares (Dim 1) Keep in mind: dk = (w0 + w1 · xk ) + k Assume that 1, . . . , p were generated independently. Denote by p(d1, . . . , dp | w0, w1, σ2) the probability density according to which the values d1, . . . , dn were generated assuming fixed w0, w1, σ2, x1, . . . , xp. 98 Maximum Likelihood vs Least Squares (Dim 1) Keep in mind: dk = (w0 + w1 · xk ) + k Assume that 1, . . . , p were generated independently. Denote by p(d1, . . . , dp | w0, w1, σ2) the probability density according to which the values d1, . . . , dn were generated assuming fixed w0, w1, σ2, x1, . . . , xp. The independence and normality imply p(d1, . . . , dp | w0, w1, σ2 ) = p k=1 N[w0 + w1xk , σ2 ](dk ) = p k=1 1 σ √ 2π exp − (dk − w0 − w1xk )2 2σ2 98 Maximum Likelihood vs Least Squares Our goal is to find (w0, w1) that maximizes the likelihood that the training set D with fixed values d1, . . . , dn has been generated: L(w0, w1, σ2 ) := p(d1, . . . , dp | w0, w1, σ2 ) 99 Maximum Likelihood vs Least Squares Our goal is to find (w0, w1) that maximizes the likelihood that the training set D with fixed values d1, . . . , dn has been generated: L(w0, w1, σ2 ) := p(d1, . . . , dp | w0, w1, σ2 ) Theorem (w0, w1) maximizes L(w0, w1, σ2) for arbitrary σ2 iff (w0, w1) minimizes squared error E(w0, w1) = p k=1 (dk − w0 − w1xk )2. 99 Maximum Likelihood vs Least Squares Our goal is to find (w0, w1) that maximizes the likelihood that the training set D with fixed values d1, . . . , dn has been generated: L(w0, w1, σ2 ) := p(d1, . . . , dp | w0, w1, σ2 ) Theorem (w0, w1) maximizes L(w0, w1, σ2) for arbitrary σ2 iff (w0, w1) minimizes squared error E(w0, w1) = p k=1 (dk − w0 − w1xk )2. Note that maximizing L(w0, w1, σ2) w.r.t. (w0, w1) does not depend on σ2. 99 Maximum Likelihood vs Least Squares Our goal is to find (w0, w1) that maximizes the likelihood that the training set D with fixed values d1, . . . , dn has been generated: L(w0, w1, σ2 ) := p(d1, . . . , dp | w0, w1, σ2 ) Theorem (w0, w1) maximizes L(w0, w1, σ2) for arbitrary σ2 iff (w0, w1) minimizes squared error E(w0, w1) = p k=1 (dk − w0 − w1xk )2. Note that maximizing L(w0, w1, σ2) w.r.t. (w0, w1) does not depend on σ2. Maximizing σ2 satisfies σ2 = 1 p p k=1 (dk − w0 − w1 · xk )2. 99 MLP training – theory 100 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers layers numbered from 0; the input layer has number 0 E.g. three-layer network has two hidden layers and one output layer Neurons in the i-th layer are connected with all neurons in the i + 1-st layer Architecture of a MLP is typically described by numbers of neurons in individual layers (e.g. 2-4-3-2) 101 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) 102 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops 102 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) 102 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) wji is the weight of the connection from i to j (in particular, wj0 is the weight of the connection from the formal unit input, i.e. wj0 = −bj where bj is the bias of the neuron j) 102 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) wji is the weight of the connection from i to j (in particular, wj0 is the weight of the connection from the formal unit input, i.e. wj0 = −bj where bj is the bias of the neuron j) j← is a set of all i such that j is adjacent from i (i.e. there is an arc to j from i) 102 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) wji is the weight of the connection from i to j (in particular, wj0 is the weight of the connection from the formal unit input, i.e. wj0 = −bj where bj is the bias of the neuron j) j← is a set of all i such that j is adjacent from i (i.e. there is an arc to j from i) j→ is a set of all i such that j is adjacent to i (i.e. there is an arc from j to i) 102 MLP – activity Activity: inner potential of neuron j: ξj = i∈j← wjiyi 103 MLP – activity Activity: inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable) [ e.g. logistic sigmoid σj(ξ) = 1 1+e −λjξ ] 103 MLP – activity Activity: inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable) [ e.g. logistic sigmoid σj(ξ) = 1 1+e −λjξ ] State of non-input neuron j ∈ Z \ X after the computation stops: yj = σj(ξj) (yj depends on the configuration w and the input x, so we sometimes write yj(w, x) ) 103 MLP – activity Activity: inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable) [ e.g. logistic sigmoid σj(ξ) = 1 1+e −λjξ ] State of non-input neuron j ∈ Z \ X after the computation stops: yj = σj(ξj) (yj depends on the configuration w and the input x, so we sometimes write yj(w, x) ) The network computes a function R|X| do R|Y| . Layer-wise computation: First, all input neurons are assigned values of the input. In the -th step, all neurons of the -th layer are evaluated. 103 MLP – learning Learning: Given a training set T of the form xk , dk k = 1, . . . , p Here, every xk ∈ R|X| is an input vector end every dk ∈ R|Y| is the desired network output. For every j ∈ Y, denote by dkj the desired output of the neuron j for a given network input xk (the vector dk can be written as dkj j∈Y ). 104 MLP – learning Learning: Given a training set T of the form xk , dk k = 1, . . . , p Here, every xk ∈ R|X| is an input vector end every dk ∈ R|Y| is the desired network output. For every j ∈ Y, denote by dkj the desired output of the neuron j for a given network input xk (the vector dk can be written as dkj j∈Y ). Error function: E(w) = p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 104 MLP – learning algorithm Batch algorithm (gradient descent): The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: w (t+1) ji = w (t) ji + ∆w (t) ji 105 MLP – learning algorithm Batch algorithm (gradient descent): The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: w (t+1) ji = w (t) ji + ∆w (t) ji where ∆w (t) ji = −ε(t) · ∂E ∂wji (w(t) ) is a weight update of wji in step t + 1 and 0 < ε(t) ≤ 1 is a learning rate in step t + 1. 105 MLP – learning algorithm Batch algorithm (gradient descent): The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: w (t+1) ji = w (t) ji + ∆w (t) ji where ∆w (t) ji = −ε(t) · ∂E ∂wji (w(t) ) is a weight update of wji in step t + 1 and 0 < ε(t) ≤ 1 is a learning rate in step t + 1. Note that ∂E ∂wji (w(t) ) is a component of the gradient E, i.e. the weight update can be written as w(t+1) = w(t) − ε(t) · E(w(t) ). 105 MLP – error function gradient For every wji we have ∂E ∂wji = p k=1 ∂Ek ∂wji 106 MLP – error function gradient For every wji we have ∂E ∂wji = p k=1 ∂Ek ∂wji where for every k = 1, . . . , p holds ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi 106 MLP – error function gradient For every wji we have ∂E ∂wji = p k=1 ∂Ek ∂wji where for every k = 1, . . . , p holds ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi and for every j ∈ Z X we get ∂Ek ∂yj = yj − dkj for j ∈ Y 106 MLP – error function gradient For every wji we have ∂E ∂wji = p k=1 ∂Ek ∂wji where for every k = 1, . . . , p holds ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi and for every j ∈ Z X we get ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj for j ∈ Z (Y ∪ X) (Here all yj are in fact yj(w, xk )). 106 MLP – error function gradient If σj(ξ) = 1 1+e −λjξ for all j ∈ Z, then σj (ξj) = λjyj(1 − yj) 107 MLP – error function gradient If σj(ξ) = 1 1+e −λjξ for all j ∈ Z, then σj (ξj) = λjyj(1 − yj) and thus for all j ∈ Z X: ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · λr yr (1 − yr ) · wrj for j ∈ Z (Y ∪ X) 107 MLP – error function gradient If σj(ξ) = 1 1+e −λjξ for all j ∈ Z, then σj (ξj) = λjyj(1 − yj) and thus for all j ∈ Z X: ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · λr yr (1 − yr ) · wrj for j ∈ Z (Y ∪ X) If σj(ξ) = a · tanh(b · ξj) for all j ∈ Z, then σj (ξj) = b a (a − yj)(a + yj) 107 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) For every k = 1, . . . , p do: 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) For every k = 1, . . . , p do: 1. forward pass: compute yj = yj(w, xk ) for all j ∈ Z 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) For every k = 1, . . . , p do: 1. forward pass: compute yj = yj(w, xk ) for all j ∈ Z 2. backward pass: compute ∂Ek ∂yj for all j ∈ Z using backpropagation (see the next slide!) 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) For every k = 1, . . . , p do: 1. forward pass: compute yj = yj(w, xk ) for all j ∈ Z 2. backward pass: compute ∂Ek ∂yj for all j ∈ Z using backpropagation (see the next slide!) 3. compute ∂Ek ∂wji for all wji using ∂Ek ∂wji := ∂Ek ∂yj · σj (ξj) · yi 108 MLP – computing the gradient Compute ∂E ∂wji = p k=1 ∂Ek ∂wji as follows: Initialize Eji := 0 (By the end of the computation: Eji = ∂E ∂wji ) For every k = 1, . . . , p do: 1. forward pass: compute yj = yj(w, xk ) for all j ∈ Z 2. backward pass: compute ∂Ek ∂yj for all j ∈ Z using backpropagation (see the next slide!) 3. compute ∂Ek ∂wji for all wji using ∂Ek ∂wji := ∂Ek ∂yj · σj (ξj) · yi 4. Eji := Eji + ∂Ek ∂wji The resulting Eji equals ∂E ∂wji . 108 MLP – backpropagation Compute ∂Ek ∂yj for all j ∈ Z as follows: 109 MLP – backpropagation Compute ∂Ek ∂yj for all j ∈ Z as follows: if j ∈ Y, then ∂Ek ∂yj = yj − dkj 109 MLP – backpropagation Compute ∂Ek ∂yj for all j ∈ Z as follows: if j ∈ Y, then ∂Ek ∂yj = yj − dkj if j ∈ Z Y ∪ X, then assuming that j is in the -th layer and assuming that ∂Ek ∂yr has already been computed for all neurons in the + 1-st layer, compute ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj (This works because all neurons of r ∈ j→ belong to the + 1-st layer.) 109 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 1. forward pass, i.e. computes yj(w, xk ) 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 1. forward pass, i.e. computes yj(w, xk ) 2. backpropagation, i.e. computes ∂Ek ∂yj 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 1. forward pass, i.e. computes yj(w, xk ) 2. backpropagation, i.e. computes ∂Ek ∂yj 3. computes ∂Ek ∂wji and adds it to Eji (a constant time operation in the unit cost framework) 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 1. forward pass, i.e. computes yj(w, xk ) 2. backpropagation, i.e. computes ∂Ek ∂yj 3. computes ∂Ek ∂wji and adds it to Eji (a constant time operation in the unit cost framework) The steps 1. - 3. take linear time. 110 Complexity of the batch algorithm Computation of ∂E ∂wji (w(t−1)) stops in time linear in the size of the network plus the size of the training set. (assuming unit cost of operations including computation of σr (ξr ) for given ξr ) Proof sketch: The algorithm does the following p times: 1. forward pass, i.e. computes yj(w, xk ) 2. backpropagation, i.e. computes ∂Ek ∂yj 3. computes ∂Ek ∂wji and adds it to Eji (a constant time operation in the unit cost framework) The steps 1. - 3. take linear time. Note that the speed of convergence of the gradient descent cannot be estimated ... 110 Illustration of the gradient descent – XOR Source: Pattern Classification (2nd Edition); Richard O. Duda, Peter E. Hart, David G. Stork 111 MLP – learning algorithm Online algorithm: The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: w (t+1) ji = w (t) ji + ∆w (t) ji where ∆w (t) ji = −ε(t) · ∂Ek ∂wji (w (t) ji ) is the weight update of wji in the step t + 1 and 0 < ε(t) ≤ 1 is the learning rate in the step t + 1. There are other variants determined by selection of the training examples used for the error computation (more on this later). 112 SGD weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: Choose (randomly) a set of training examples T ⊆ {1, . . . , p} Compute w(t+1) = w(t) + ∆w(t) where ∆w(t) = −ε(t) · k∈T Ek (w(t) ) 0 < ε(t) ≤ 1 is a learning rate in step t + 1 Ek (w(t) ) is the gradient of the error of the example k Note that the random choice of the minibatch is typically implemented by randomly shuffling all data and then choosing minibatches sequentially. 113 MLP training – practical issues 114 Architecture – Multilayer Perceptron (MLP) Input Hidden Output x1 x2 y1 y2 Neurons partitioned into layers; one input layer, one output layer, possibly several hidden layers layers numbered from 0; the input layer has number 0 E.g. three-layer network has two hidden layers and one output layer Neurons in the i-th layer are connected with all neurons in the i + 1-st layer Architecture of a MLP is typically described by numbers of neurons in individual layers (e.g. 2-4-3-2) 115 MLP – architecture Notation: Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) wji is the weight of the connection from i to j (in particular, wj0 is the weight of the connection from the formal unit input, i.e. wj0 = −bj where bj is the bias of the neuron j) j← is a set of all i such that j is adjacent from i (i.e. there is an arc to j from i) j→ is a set of all i such that j is adjacent to i (i.e. there is an arc from j to i) 116 MLP – learning Learning: Given a training set T of the form xk , dk k = 1, . . . , p Here, every xk ∈ R|X| is an input vector end every dk ∈ R|Y| is the desired network output. For every j ∈ Y, denote by dkj the desired output of the neuron j for a given network input xk (the vector dk can be written as dkj j∈Y ). Error function: E(w) = p k=1 Ek (w) 117 SGD weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: Choose (randomly) a set of training examples T ⊆ {1, . . . , p} Compute w(t+1) = w(t) + ∆w(t) where ∆w(t) = −ε(t) · k∈T Ek (w(t) ) 0 < ε(t) ≤ 1 is a learning rate in step t + 1 Ek (w(t) ) is the gradient of the error of the example k Note that the random choice of the minibatch is typically implemented by randomly shuffling all data and then choosing minibatches sequentially. 118 MLP – mse gradient For every wji we have ∂E ∂wji = 1 p p k=1 ∂Ek ∂wji 119 MLP – mse gradient For every wji we have ∂E ∂wji = 1 p p k=1 ∂Ek ∂wji where for every k = 1, . . . , p holds ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi and for every j ∈ Z X we get (for squared error) ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj for j ∈ Z (Y ∪ X) (Here all yj are in fact yj(w, xk )). 119 (Some) error functions squared error: E(w) = p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 mean squared error (mse): E(w) = 1 p p k=1 Ek (w) (categorical) cross entropy: E(w) = − 1 p p k=1 j∈Y dkj ln(yj) 120 Practical issues of gradient descent Training efficiency: What size of a minibatch? How to choose the learning rate ε(t) and control SGD ? How to pre-process the inputs? How to initialize weights? How to choose desired output values of the network? 121 Practical issues of gradient descent Training efficiency: What size of a minibatch? How to choose the learning rate ε(t) and control SGD ? How to pre-process the inputs? How to initialize weights? How to choose desired output values of the network? Quality of the resulting model: When to stop training? Regularization techniques. How large network? For simplicity, I will illustrate the reasoning on MLP + mse. Later we will see other topologies and error functions with different but always somewhat related issues. 121 Issues in gradient descent Small networks: Lots of local minima where the descent gets stuck. The model identifiability problem: Swapping incoming weights of neurons i and j leaves the same network topology – weight space symmetry. Recent studies show that for sufficiently large networks all local minima have low values of the error function. 122 Issues in gradient descent Small networks: Lots of local minima where the descent gets stuck. The model identifiability problem: Swapping incoming weights of neurons i and j leaves the same network topology – weight space symmetry. Recent studies show that for sufficiently large networks all local minima have low values of the error function. Saddle points One can show (by a combinatorial argument) that larger networks have exponentially more saddle points than local minima. 122 Issues in gradient descent – too slow descent flat regions E.g. if the inner potentials are too large (in abs. value), then their derivative is extremely small. 123 Issues in gradient descent – too fast descent steep cliffs: the gradient is extremely large, descent skips important weight vectors 124 Issues in gradient descent – local vs global structure What if we initialize on the left? 125 Gradient Descent in Large Networks Theorem Assume (roughly), activation functions: "smooth" ReLU (softplus) σ(z) = log(1 + exp(z)) In general: Smooth, non-polynomial, analytic, Lipschitzs. inputs xk of Euclidean norm equal to 1, desired values dk satisfying |dk | ∈ O(1), the number of hidden neurons per layer sufficiently large (polynomial in certain numerical characteristics of inputs roughly measuring their similarity, and exponential in the depth of the network), the learning rate constant and sufficiently small. The gradient descent converges (with high probability) to a global minimum with zero error at linear rate. Later we get to a special type of networks called ResNet where the above result demands only polynomially many neurons per layer (w.r.t. depth). 126 Issues in computing the gradient vanishing and exploding gradients ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj for j ∈ Z (Y ∪ X) 127 Issues in computing the gradient vanishing and exploding gradients ∂Ek ∂yj = yj − dkj for j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj for j ∈ Z (Y ∪ X) inexact gradient computation: Minibatch gradient is only an estimate of the true gradient. Note that the variance of the estimate is (roughly) σ/ √ m where m is the size of the minibatch and σ is the variance of the gradient estimate for a single training example. (E.g. minibatch size 10 000 means 100 times more computation than the size 100 but gives only 10 times less variance.) 127 Minibatch size Larger batches provide a more accurate estimate of the gradient, but with less than linear returns. 128 Minibatch size Larger batches provide a more accurate estimate of the gradient, but with less than linear returns. Multicore architectures are usually underutilized by extremely small batches. 128 Minibatch size Larger batches provide a more accurate estimate of the gradient, but with less than linear returns. Multicore architectures are usually underutilized by extremely small batches. If all examples in the batch are to be processed in parallel (as is the typical case), then the amount of memory scales with the batch size. For many hardware setups this is the limiting factor in batch size. 128 Minibatch size Larger batches provide a more accurate estimate of the gradient, but with less than linear returns. Multicore architectures are usually underutilized by extremely small batches. If all examples in the batch are to be processed in parallel (as is the typical case), then the amount of memory scales with the batch size. For many hardware setups this is the limiting factor in batch size. It is common (especially when using GPUs) for power of 2 batch sizes to offer better runtime. Typical power of 2 batch sizes range from 32 to 256, with 16 sometimes being attempted for large models. 128 Minibatch size Larger batches provide a more accurate estimate of the gradient, but with less than linear returns. Multicore architectures are usually underutilized by extremely small batches. If all examples in the batch are to be processed in parallel (as is the typical case), then the amount of memory scales with the batch size. For many hardware setups this is the limiting factor in batch size. It is common (especially when using GPUs) for power of 2 batch sizes to offer better runtime. Typical power of 2 batch sizes range from 32 to 256, with 16 sometimes being attempted for large models. Small batches can offer a regularizing effect, perhaps due to the noise they add to the learning process. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. ("On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima". Keskar et al, ICLR’17) 128 Momentum Issue in the gradient descent: E(w(t)) constantly changes direction (but the error steadily decreases). 129 Momentum Issue in the gradient descent: E(w(t)) constantly changes direction (but the error steadily decreases). Solution: In every step add the change made in the previous step (weighted by a factor α): ∆w(t) = −ε(t) · k∈T Ek (w(t) ) + α · ∆w (t−1) ji where 0 < α < 1. 129 Momentum – illustration 130 SGD with momentum weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: Choose (randomly) a set of training examples T ⊆ {1, . . . , p} Compute w(t+1) = w(t) + ∆w(t) where ∆w(t) = −ε(t) · k∈T Ek (w(t) ) + α∆w(t−1) 0 < ε(t) ≤ 1 is a learning rate in step t + 1 0 < α < 1 measures the "influence" of the momentum Ek (w(t) ) is the gradient of the error of the example k Note that the random choice of the minibatch is typically implemented by randomly shuffling all data and then choosing minibatches sequentially. 131 Learning rate 132 Adaptive learning rate Power scheduling: Set (t) = 0/(1 + t/s) where 0 is an initial learning rate and s a number of steps (after s steps the learning rate is 0/2, after 2s it is 0/3 etc.) 133 Adaptive learning rate Power scheduling: Set (t) = 0/(1 + t/s) where 0 is an initial learning rate and s a number of steps (after s steps the learning rate is 0/2, after 2s it is 0/3 etc.) Exponential scheduling: Set (t) = 0 · 0.1t/s . (the learning rate decays faster than in the power scheduling) 133 Adaptive learning rate Power scheduling: Set (t) = 0/(1 + t/s) where 0 is an initial learning rate and s a number of steps (after s steps the learning rate is 0/2, after 2s it is 0/3 etc.) Exponential scheduling: Set (t) = 0 · 0.1t/s . (the learning rate decays faster than in the power scheduling) Piecewise constant scheduling: A constant learning rate for a number of steps/epochs, then a smaller learning rate, and so on. 133 Adaptive learning rate Power scheduling: Set (t) = 0/(1 + t/s) where 0 is an initial learning rate and s a number of steps (after s steps the learning rate is 0/2, after 2s it is 0/3 etc.) Exponential scheduling: Set (t) = 0 · 0.1t/s . (the learning rate decays faster than in the power scheduling) Piecewise constant scheduling: A constant learning rate for a number of steps/epochs, then a smaller learning rate, and so on. 1cycle scheduling: Start by increasing the initial learning rate from 0 linearly to 1 (approx. 1 = 10 0) halfway through training. Then decrease from 1 linearly to 0. Finish by dropping the learning rate by several orders of magnitude (still linearly). According to a 2018 paper by Leslie Smith this may converge much faster (100 epochs vs 800 epochs on CIFAR10 dataset). For comparison of some methods see: AN EMPIRICAL STUDY OF LEARNING RATES IN DEEP NEURAL NETWORKS FOR SPEECH RECOGNITION, Senior et al 133 AdaGrad So far we have considered fixed schedules for learning rates. It is better to have larger rates for weights with smaller updates, smaller rates for weights with larger updates. AdaGrad uses individually adapting learning rate for each weight. 134 SGD with AdaGrad weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), compute w(t+1) : Choose (randomly) a minibatch T ⊆ {1, . . . , p} Compute w (t+1) ji = w (t) ji + ∆w (t) ji 135 SGD with AdaGrad weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), compute w(t+1) : Choose (randomly) a minibatch T ⊆ {1, . . . , p} Compute w (t+1) ji = w (t) ji + ∆w (t) ji where ∆w (t) ji = − η r (t) ji + δ · k∈T ∂Ek ∂wji (w(t) ) and r (t) ji = r (t−1) ji +   k∈T ∂Ek ∂wji (w(t) )   2 η is a constant expressing the influence of the learning rate, typically 0.01. δ > 0 is a smoothing term (typically 1e-8) avoiding division by 0. 135 RMSProp The main disadvantage of AdaGrad is the accumulation of the gradient throughout the whole learning process. In case the learning needs to get over several "hills" before settling in a deep "valley", the weight updates get far too small before getting to it. RMSProp uses an exponentially decaying average to discard history from the extreme past so that it can converge rapidly after finding a convex bowl, as if it were an instance of the AdaGrad algorithm initialized within that bowl. 136 SGD with RMSProp weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), compute w(t+1) : Choose (randomly) a minibatch T ⊆ {1, . . . , p} Compute w (t+1) ji = w (t) ji + ∆w (t) ji 137 SGD with RMSProp weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), compute w(t+1) : Choose (randomly) a minibatch T ⊆ {1, . . . , p} Compute w (t+1) ji = w (t) ji + ∆w (t) ji where ∆w (t) ji = − η r (t) ji + δ · k∈T ∂Ek ∂wji (w(t) ) and r (t) ji = ρr (t−1) ji + (1 − ρ)   k∈T ∂Ek ∂wji (w(t) )   2 η is a constant expressing the influence of the learning rate (Hinton suggests ρ = 0.9 and η = 0.001). δ > 0 is a smoothing term (typically 1e-8) avoiding division by 0. 137 Other optimization methods There are more methods such as AdaDelta, Adam (roughly RMSProp combined with momentum), etc. A natural question: Which algorithm should one choose? 138 Other optimization methods There are more methods such as AdaDelta, Adam (roughly RMSProp combined with momentum), etc. A natural question: Which algorithm should one choose? Unfortunately, there is currently no consensus on this point. According to a recent study, the family of algorithms with adaptive learning rates (represented by RMSProp and AdaDelta) performed fairly robustly, no single best algorithm has emerged. 138 Other optimization methods There are more methods such as AdaDelta, Adam (roughly RMSProp combined with momentum), etc. A natural question: Which algorithm should one choose? Unfortunately, there is currently no consensus on this point. According to a recent study, the family of algorithms with adaptive learning rates (represented by RMSProp and AdaDelta) performed fairly robustly, no single best algorithm has emerged. Currently, the most popular optimization algorithms actively in use include SGD, SGD with momentum, RMSProp, RMSProp with momentum, AdaDelta and Adam. The choice of which algorithm to use, at this point, seems to depend largely on the user’s familiarity with the algorithm. 138 Choice of (hidden) activations Generic requirements imposed on activation functions: 1. differentiability (to do gradient descent) 2. non-linearity (linear multi-layer networks are equivalent to single-layer) 3. monotonicity (local extrema of activation functions induce local extrema of the error function) 4. "linearity" (i.e. preserve as much linearity as possible; linear models are easiest to fit; find the "minimum" non-linearity needed to solve a given task) The choice of activation functions is closely related to input preprocessing and the initial choice of weights. I will illustrate the reasoning on sigmoidal functions; say few words about other activation functions later. 139 Activation functions – tanh σ(ξ) = 1.7159 · tanh(2 3 · ξ), we have limξ→∞ σ(ξ) = 1.7159 and limξ→−∞ σ(ξ) = −1.7159 140 Activation functions – tanh σ(ξ) = 1.7159 · tanh(2 3 · ξ) is almost linear on [−1, 1] 141 Activation functions – tanh first derivative: σ(ξ) = 1.7159 · tanh(2 3 · ξ) 142 Input preprocessing Some inputs may be much larger than others. E.g..: Height vs weight of a person, maximum speed of a car (in km/h) vs its price (in CZK), etc. 143 Input preprocessing Some inputs may be much larger than others. E.g..: Height vs weight of a person, maximum speed of a car (in km/h) vs its price (in CZK), etc. Large inputs have greater influence on the training than the small ones. In addition, too large inputs may slow down learning (saturation of activation functions). 143 Input preprocessing Some inputs may be much larger than others. E.g..: Height vs weight of a person, maximum speed of a car (in km/h) vs its price (in CZK), etc. Large inputs have greater influence on the training than the small ones. In addition, too large inputs may slow down learning (saturation of activation functions). Typical standardization: average = 0 (subtract the mean) variance = 1 (divide by the standard deviation) Here the mean and standard deviation may be estimated from data (the training set). (illustration of standard deviation) 143 Initial weights (for tanh) Assume weights chosen uniformly in random from an interval [−w, w] where w depends on the number of inputs of a given neuron. 144 Initial weights (for tanh) Assume weights chosen uniformly in random from an interval [−w, w] where w depends on the number of inputs of a given neuron. Consider the activation function σ(ξ) = 1.7159 · tanh(2 3 · ξ) for all neurons. σ is almost linear on [−1, 1], σ saturates out of the interval [−4, 4] (i.e. it is close to its limit values and its derivative is close to 0. 144 Initial weights (for tanh) Assume weights chosen uniformly in random from an interval [−w, w] where w depends on the number of inputs of a given neuron. Consider the activation function σ(ξ) = 1.7159 · tanh(2 3 · ξ) for all neurons. σ is almost linear on [−1, 1], σ saturates out of the interval [−4, 4] (i.e. it is close to its limit values and its derivative is close to 0. Thus for too small w we may get (almost) linear model. for too large w (i.e. much larger than 1) the activations may get saturated and the learning will be very slow. Hence, we want to choose w so that the inner potentials of neurons will be roughly in the interval [−1, 1]. 144 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. 145 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. Consider a neuron j from the first layer with n inputs. Assume that its weights are chosen uniformly from [−w, w]. 145 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. Consider a neuron j from the first layer with n inputs. Assume that its weights are chosen uniformly from [−w, w]. The rule: choose w so that the standard deviation of ξj (denote by oj) is close to the border of the interval on which σj is linear. In our case: oj ≈ 1. 145 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. Consider a neuron j from the first layer with n inputs. Assume that its weights are chosen uniformly from [−w, w]. The rule: choose w so that the standard deviation of ξj (denote by oj) is close to the border of the interval on which σj is linear. In our case: oj ≈ 1. Our assumptions imply: oj = n 3 · w. Thus we put w = √ 3√ n . 145 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. Consider a neuron j from the first layer with n inputs. Assume that its weights are chosen uniformly from [−w, w]. The rule: choose w so that the standard deviation of ξj (denote by oj) is close to the border of the interval on which σj is linear. In our case: oj ≈ 1. Our assumptions imply: oj = n 3 · w. Thus we put w = √ 3√ n . The same works for higher layers, n corresponds to the number of neurons in the layer one level lower. 145 Glorot & Bengio initialization The previous heuristics for weight initialization ignores variance of the gradient (i.e. it is concerned only with the "size" of activations in the forward pass). 146 Glorot & Bengio initialization The previous heuristics for weight initialization ignores variance of the gradient (i.e. it is concerned only with the "size" of activations in the forward pass). Glorot & Bengio (2010) presented a normalized initialization by choosing w uniformly from the interval:  − 6 m + n , 6 m + n   =   − 3 (m + n)/2 , 3 (m + n)/2   Here n is the number of inputs to the layer, m is the number of outputs of the layer (i.e. the number of neurons in the layer). 146 Glorot & Bengio initialization The previous heuristics for weight initialization ignores variance of the gradient (i.e. it is concerned only with the "size" of activations in the forward pass). Glorot & Bengio (2010) presented a normalized initialization by choosing w uniformly from the interval:  − 6 m + n , 6 m + n   =   − 3 (m + n)/2 , 3 (m + n)/2   Here n is the number of inputs to the layer, m is the number of outputs of the layer (i.e. the number of neurons in the layer). This is designed to compromise between the goal of initializing all layers to have the same activation variance and the goal of initializing all layers to have the same gradient variance. The formula is derived using the assumption that the network consists only of a chain of matrix multiplications, with no non-linearities. Real neural networks obviously violate this assumption, but many strategies designed for the linear model perform reasonably well on its non-linear counterparts. 146 Modern activation functions For hidden neurons sigmoidal functions are often substituted with piece-wise linear activations functions. Most prominent is ReLU: σ(ξ) = max{0, ξ} THE default activation function recommended for use with most feedforward neural networks. As close to linear function as possible; very simple; does not saturate for large potentials. Dead for negative potentials. 147 More modern activation functions Leaky ReLU (greenboard): Generalizes ReLU, not dead for negative potentials. Experimentally not much better than ReLU. 148 More modern activation functions Leaky ReLU (greenboard): Generalizes ReLU, not dead for negative potentials. Experimentally not much better than ReLU. ELU: "Smoothed" ReLU: σ(ξ) =    α(exp(ξ) − 1) for ξ < 0 ξ for ξ ≥ 0 Here α is a parameter, ELU converges to −α as ξ → −∞. As opposed to ReLU: Smooth, always non-zero gradient (but saturates), slower to compute. 148 More modern activation functions Leaky ReLU (greenboard): Generalizes ReLU, not dead for negative potentials. Experimentally not much better than ReLU. ELU: "Smoothed" ReLU: σ(ξ) =    α(exp(ξ) − 1) for ξ < 0 ξ for ξ ≥ 0 Here α is a parameter, ELU converges to −α as ξ → −∞. As opposed to ReLU: Smooth, always non-zero gradient (but saturates), slower to compute. SELU: Scaled variant of ELU: : σ(ξ) = λ    α(exp(ξ) − 1) for ξ < 0 ξ for ξ ≥ 0 Self-normalizing, i.e. output of each layer will tend to preserve a mean (close to) 0 and a standard deviation (close to) 1 for λ ≈ 1.050 and α ≈ 1.673, properly initialized weights (see below) and normalized inputs (zero mean, standard deviation 1). 148 Initializing with Normal Distribution Denote by n the number of inputs to the initialized layer, and m the number of neurons in the layer. Glorot & Bengio (2010): Choose weights randomly from the normal distribution with mean 0 and variance 2/(n + m) Suitable for activation functions: None, tanh, logistic, softmax 149 Initializing with Normal Distribution Denote by n the number of inputs to the initialized layer, and m the number of neurons in the layer. Glorot & Bengio (2010): Choose weights randomly from the normal distribution with mean 0 and variance 2/(n + m) Suitable for activation functions: None, tanh, logistic, softmax He (2015): Choose weights randomly from the normal distribution with mean 0 and variance 2/n Designed for ReLU, leaky ReLU 149 Initializing with Normal Distribution Denote by n the number of inputs to the initialized layer, and m the number of neurons in the layer. Glorot & Bengio (2010): Choose weights randomly from the normal distribution with mean 0 and variance 2/(n + m) Suitable for activation functions: None, tanh, logistic, softmax He (2015): Choose weights randomly from the normal distribution with mean 0 and variance 2/n Designed for ReLU, leaky ReLU LeCun (1990): Choose weights randomly from the normal distribution with mean 0 and variance 1/n Suitable for SELU 149 How to choose activation of hidden neurons Default is ReLU. According to Aurélien Géron: SELU > ELU > leakyReLU > ReLU > tanh > logistic For discussion see: Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems, Aurélien Géron 150 Output neurons The choice of activation functions for output units depends on the concrete applications. For regression (function approximation) the output is typically linear. 151 Output neurons The choice of activation functions for output units depends on the concrete applications. For regression (function approximation) the output is typically linear. For classification, the current activation functions of choice are logistic sigmoid – binary classification softmax: Given an output neuron j ∈ Y yj = σj(ξj) = eξj i∈Y eξi for multi-class classification. 151 Output neurons The choice of activation functions for output units depends on the concrete applications. For regression (function approximation) the output is typically linear. For classification, the current activation functions of choice are logistic sigmoid – binary classification softmax: Given an output neuron j ∈ Y yj = σj(ξj) = eξj i∈Y eξi for multi-class classification. The error function used with softmax (assuming that the target values dkj are from {0, 1}) is typically cross-entropy: − 1 p p k=1 j∈Y dkj ln(yj) ... which somewhat corresponds to the maximum likelihood principle. 151 Sigmoidal outputs with cross-entropy – in detail Consider Binary classification, two classes {0, 1} One output neuron j, its activation logistic sigmoid σj(ξj) = 1 1 + e−ξj The output of the network is y = σj(ξj). 152 Sigmoidal outputs with cross-entropy – in detail Consider Binary classification, two classes {0, 1} One output neuron j, its activation logistic sigmoid σj(ξj) = 1 1 + e−ξj The output of the network is y = σj(ξj). For a training set T = xk , dk k = 1, . . . , p (here xk ∈ R|X| and dk ∈ R), the cross-entropy looks like this: Ecross = − 1 p p k=1 [dk ln(yk ) + (1 − dk ) ln(1 − yk )] where yk is the output of the network for the k-th training input xk , and dk is the k-th desired output. 152 Generalization Intuition: Generalization = ability to cope with new unseen instances. Data are mostly noisy, so it is not good idea to fit exactly. In case of function approximation, the network should not return exact results as in the training set. 153 Generalization Intuition: Generalization = ability to cope with new unseen instances. Data are mostly noisy, so it is not good idea to fit exactly. In case of function approximation, the network should not return exact results as in the training set. More formally: It is typically assumed that the training set has been generated as follows: dkj = gj(xk ) + Θkj where gj is the "underlying" function corresponding to the output neuron j ∈ Y and Θkj is random noise. The network should fit gj not the noise. Methods improving generalization are called regularization methods. 153 Regularization Regularization is a big issue in neural networks, as they typically use a huge amount of parameters and thus are very susceptible to overfitting. 154 Regularization Regularization is a big issue in neural networks, as they typically use a huge amount of parameters and thus are very susceptible to overfitting. von Neumann: "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." ... and I ask you prof. Neumann: What can you fit with 40GB of parameters?? 154 Early stopping Early stopping means that we stop learning before it reaches a minimum of the error E. When to stop? 155 Early stopping Early stopping means that we stop learning before it reaches a minimum of the error E. When to stop? In many applications the error function is not the main thing we want to optimize. E.g. in the case of a trading system, we typically want to maximize our profit not to minimize (strange) error functions designed to be easily differentiable. Also, as noted before, minimizing E completely is not good for generalization. For start: We may employ standard approach of training on one set and stopping on another one. 155 Early stopping Divide your dataset into several subsets: training set (e.g. 60%) – train the network here validation set (e.g. 20%) – use to stop the training (possibly) test set (e.g. 20%) – use to compare trained models What to use as a stopping rule? 156 Early stopping Divide your dataset into several subsets: training set (e.g. 60%) – train the network here validation set (e.g. 20%) – use to stop the training (possibly) test set (e.g. 20%) – use to compare trained models What to use as a stopping rule? You may observe E (or any other function of interest) on the validation set, if it does not improve for last k steps, stop. Alternatively, you may observe the gradient, if it is small for some time, stop. (recent studies shown that this traditional rule is not too good: it may happen that the gradient is larger close to minimum values; on the other hand, E does not have to be evaluated which saves time. To compare models you may use ML techniques such as various types of cross-validation etc. 156 Size of the network Similar problem as in the case of the training duration: Too small network is not able to capture intrinsic properties of the training set. Large networks overfit faster. Solution: Optimal number of neurons :-) 157 Size of the network Similar problem as in the case of the training duration: Too small network is not able to capture intrinsic properties of the training set. Large networks overfit faster. Solution: Optimal number of neurons :-) there are some (useless) theoretical bounds there are algorithms dynamically adding/removing neurons (not much use nowadays) In practice: start with a model solving similar problem (transfer learning). experiment, experiment, experiment. 157 Feature extraction Consider a two layer network. Hidden neurons are supposed to represent "patterns" in the inputs. Example: Network 64-2-3 for letter classification: 158 Ensemble methods Techniques for reducing generalization error by combining several models. The reason that ensemble methods work is that different models will usually not make all the same errors on the test set. Idea: Train several different models separately, then have all of the models vote on the output for test examples. 159 Ensemble methods Techniques for reducing generalization error by combining several models. The reason that ensemble methods work is that different models will usually not make all the same errors on the test set. Idea: Train several different models separately, then have all of the models vote on the output for test examples. Bagging: Generate k training sets T1, ..., Tk by sampling from T uniformly with replacement. If the number of samples is |T |, then on average |Ti| = (1 − 1/e)|T |. For each i, train a model Mi on Ti. Combine outputs of the models: for regression by averaging, for classification by (majority) voting. 159 Dropout The algorithm: In every step of the gradient descent choose randomly a set N of neurons, each neuron is included in N independently with probability 1/2, (in practice, different probabilities are used as well). do forward and backward propagations only using the selected neurons (i.e. leave weights of the other neurons unchanged) 160 Dropout The algorithm: In every step of the gradient descent choose randomly a set N of neurons, each neuron is included in N independently with probability 1/2, (in practice, different probabilities are used as well). do forward and backward propagations only using the selected neurons (i.e. leave weights of the other neurons unchanged) Dropout resembles bagging: Large ensemble of neural networks is trained "at once" on parts of the data. Dropout is not exactly the same as bagging: The models share parameters, with each model inheriting a different subset of parameters from the parent neural network. This parameter sharing makes it possible to represent an exponential number of models with a tractable amount of memory. In the case of bagging, each model is trained to convergence on its respective training set. This would be infeasible for large networks/training sets. 160 Weight decay and L2 regularization Generalization can be improved by removing "unimportant" weights. Penalising large weights gives stronger indication about their importance. 161 Weight decay and L2 regularization Generalization can be improved by removing "unimportant" weights. Penalising large weights gives stronger indication about their importance. In every step we decrease weights (multiplicatively) as follows: w (t+1) ji = (1 − ζ)w (t) ji + ∆w (t) ji Intuition: Unimportant weights will be pushed to 0, important weights will survive the decay. 161 Weight decay and L2 regularization Generalization can be improved by removing "unimportant" weights. Penalising large weights gives stronger indication about their importance. In every step we decrease weights (multiplicatively) as follows: w (t+1) ji = (1 − ζ)w (t) ji + ∆w (t) ji Intuition: Unimportant weights will be pushed to 0, important weights will survive the decay. Weight decay is equivalent to the gradient descent with a constant learning rate ε and the following error function: E (w) = E(w) + ζ 2ε (w · w) Here ζ 2ε (w · w) is the L2 regularization that penalizes large weights. 161 More optimization, regularization ... There are many more practical tips, optimization methods, regularization methods, etc. For a very nice survey see http://www.deeplearningbook.org/ ... and also all other infinitely many urls concerned with deep learning. 162 Some applications 163 ALVINN (history) 164 ALVINN Architecture: MLP, 960 − 4 − 30 (also 960 − 5 − 30) inputs correspond to pixels 165 ALVINN Architecture: MLP, 960 − 4 − 30 (also 960 − 5 − 30) inputs correspond to pixels Activity: activation functions: logistic sigmoid Steering wheel position determined by "center of mass" of neuron values. 165 ALVINN Learning: Trained during (live) drive. Front window view captured by a camera, 25 images per second. Training samples of the form (xk , dk ) where xk = image of the road dk = corresponding position of the steering wheel position of the steering wheel "blurred" by Gaussian distribution: dki = e−D2 i /10 where Di is the distance of the i-th output from the one which corresponds to the correct position of the wheel. (The authors claim that this was better than the binary output.) 166 ALVINN – Selection of training samples Naive approach: take images directly from the camera and adapt accordingly. 167 ALVINN – Selection of training samples Naive approach: take images directly from the camera and adapt accordingly. Problems: If the driver is gentle enough, the car never learns how to get out of dangerous situations. A solution may be turn off learning for a moment, then suddenly switch on, and let the net catch on, let the driver drive as if being insane (dangerous, possibly expensive). The real view out of the front window is repetitive and boring, the net would overfit on few examples. 167 ALVINN – Selection of training examples Problem with a "good" driver is solved as follows: 168 ALVINN – Selection of training examples Problem with a "good" driver is solved as follows: 15 distorted copies of each image: desired output generated for each copy 168 ALVINN – Selection of training examples Problem with a "good" driver is solved as follows: 15 distorted copies of each image: desired output generated for each copy "Boring" images solved as follows: a buffer of 200 images (including 15 copies of the original), in every step the system trains on the buffer after several updates a new image is captured, 15 copies are made and they will substitute 15 images in the buffer (5 chosen randomly, 10 with the smallest error). 168 ALVINN - learning pure backpropagation constant learning rate momentum, slowly increasing. Results: Trained for 5 minutes, speed 4 miles per hour. ALVINN was able to drive well on a new road it has never seen (in different weather conditions). 169 ALVINN - learning pure backpropagation constant learning rate momentum, slowly increasing. Results: Trained for 5 minutes, speed 4 miles per hour. ALVINN was able to drive well on a new road it has never seen (in different weather conditions). The maximum speed was limited by the hydraulic controller of the steering wheel, not the learning algorithm. 169 ALVINN - weight development round 0 round 10 round 20 round 50 h1 h2 h3 h4 h5 Here h1, . . . , h5 are hidden neurons. 170 MNIST – handwritten digits recognition Database of labelled images of handwritten digits: 60 000 training examples, 10 000 testing. Dimensions: 28 x 28, digits are centered to the "center of gravity" of pixel values and normalized to fixed size. More at http: //yann.lecun.com/exdb/mnist/ The database is used as a standard benchmark in lots of publications. 171 MNIST – handwritten digits recognition Database of labelled images of handwritten digits: 60 000 training examples, 10 000 testing. Dimensions: 28 x 28, digits are centered to the "center of gravity" of pixel values and normalized to fixed size. More at http: //yann.lecun.com/exdb/mnist/ The database is used as a standard benchmark in lots of publications. Allows comparison of various methods. 171 MNIST One of the best "old" results is the following: 6-layer NN 784-2500-2000-1500-1000-500-10 (on GPU) (Ciresan et al. 2010) Abstract: Good old on-line back-propagation for plain multi-layer perceptrons yields a very low 0.35 error rate on the famous MNIST handwritten digits benchmark. All we need to achieve this best result so far are many hidden layers, many neurons per layer, numerous deformed training images, and graphics cards to greatly speed up learning. A famous application of a learning convolutional network LeNet-1 in 1998. 172 MNIST – LeNet1 173 MNIST – LeNet1 Interpretation of output: the output neuron with the highest value identifies the digit. the same, but if the two largest neuron values are too close together, the input is rejected (i.e. no answer). Learning: Inputs: training on 7291 samples, tested on 2007 samples Results: error on test set without rejection: 5% error on test set with rejection: 1% (12% rejected) compare with dense MLP with 40 hidden neurons: error 1% (19.4% rejected) 174 Modern convolutional networks The rest of the lecture is based on the online book Neural Networks and Deep Learning by Michael Nielsen. http://neuralnetworksanddeeplearning.com/index.html Convolutional networks are currently the best networks for image classification. Their common ancestor is LeNet-5 (and other LeNets) from nineties. Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 1998 175 AlexNet In 2012 this network made a breakthrough in ILVSCR competition, taking the classification error from around 28% to 16%: A convolutional network, trained on two GPUs. 176 Convolutional networks - local receptive fields Every neuron is connected with a field of k × k (in this case 5 × 5) neurons in the lower layer (this filed is receptive field). Neuron is "standard": Computes a weighted sum of its inputs, applies an activation function. 177 Convolutional networks - stride length Then we slide the local receptive field over by one pixel to the right (i.e., by one neuron), to connect to a second hidden neuron: The "size" of the slide is called stride length. The group of all such neurons is feature map. all these neurons share weights and biases! 178 Feature maps Each feature map represents a property of the input that is supposed to be spatially invariant. Typically, we consider several feature maps in a single layer. 179 Pooling Neurons in the pooling layer compute functions of their receptive fields: Max-pooling : maximum of inputs L2-pooling : square root of the sum of squres Average-pooling : mean · · · 180 Trained feature maps (20 feature maps, receptive fields 5 × 5) 181 Trained feature maps 182 Simple convolutional network 28 × 28 input image, 3 feature maps, each feature map has its own max-pooling (field 5 × 5, stride = 1), 10 output neurons. Each neuron in the output layer gets input from each neuron in the pooling layer. Trained using the gradient descent with the backprop, which can be easily adapted to convolutional networks. 183 Convolutional network 184 Simple convolutional network vs MNIST two convolutional-pooling layers, one 20, second 40 feature maps, two dense (MLP) layers (1000-1000), outputs (10) Activation functions of the feature maps and dense layers: ReLU max-pooling output layer: soft-max Error function: negative log-likelihood (= cross-entropy) Training: SGD, mini-batch size 10 learning rate 0.03 L2 regularization with "weight" λ = 0.1 + dropout with prob. 1/2 training for 40 epochs (i.e. every training example is considered 40 times) Expanded dataset: displacement by one pixel to an arbitrary direction. Committee voting of 5 networks. 185 MNIST Out of 10 000 images in the test set, only these 33 have been incorrectly classified: 186 More complex convolutional networks Convolutional networks have been used for classification of images from the ImageNet database (16 million color images, 20 thousand classes) 187 ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) Competition in classification over a subset of images from ImageNet. Started in 2010, assisted in breakthrough in image recognition. Training set 1.2 million images, 1000 classes. Validation set: 50 000, test set: 150 000. Many images contain more than one object ⇒ model is allowed to choose five classes, the correct label must be among the five. (top-5 criterion). 188 AlexNet ImageNet classification with deep convolutional neural networks, by Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton (2012). Trained on two GPUs (NVIDIA GeForce GTX 580) Výsledky: accuracy 84.7% in top-5 (second best algorithm at the time 73.8%) 63.3% "perfect" (top-1) classification 189 ILSVRC 2014 The same set as in 2012, top-5 criterion. GoogLeNet: deep convolutional network, 22 layers Results: Accuracy 93.33% top-5 190 ILSVRC 2015 Deep convolutional network Various numbers of layers, the winner has 152 layers Skip connections implementing residual learning Error 3.57% in top-5. 191 ILSVRC 2016 Trimps-Soushen (The Third Research Institute of Ministry of Public Security) There is no new innovative technology or novelty by Trimps-Soushen. Ensemble of the pretrained models from Inception-v3, Inception-v4, Inception-ResNet-v2, Pre-Activation ResNet-200, and Wide ResNet (WRN-68–2). Each of the models are strong at classifying some categories, but also weak at classifying some categories. Test error: 2.99% 192 Top-k accuracy analyzed https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 193 Top-20 typical errors Out of 1458 misclassified images in Top-20: https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 194 Top-k accuracy analyzed https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 195 Top-k accuracy analyzed https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 196 Top-k accuracy analyzed https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 197 Top-k accuracy analyzed https://towardsdatascience.com/review-trimps-soushen-winner-in-ilsvrc-2016-image-classification-dfbc423111dd 198 Superhuman convolutional nets?! Andrej Karpathy: ...the task of labeling images with 5 out of 1000 categories quickly turned out to be extremely challenging, even for some friends in the lab who have been working on ILSVRC and its classes for a while. First we thought we would put it up on [Amazon Mechanical Turk]. Then we thought we could recruit paid undergrads. Then I organized a labeling party of intense labeling effort only among the (expert labelers) in our lab. Then I developed a modified interface that used GoogLeNet predictions to prune the number of categories from 1000 to only about 100. It was still too hard - people kept missing categories and getting up to ranges of 13-15% error rates. In the end I realized that to get anywhere competitively close to GoogLeNet, it was most efficient if I sat down and went through the painfully long training process and the subsequent careful annotation process myself... The labeling happened at a rate of about 1 per minute, but this decreased over time... Some images are easily recognized, while some images (such as those of fine-grained breeds of dogs, birds, or monkeys) can require multiple minutes of concentrated effort. I became very good at identifying breeds of dogs... Based on the sample of images I worked on, the GoogLeNet classification error turned out to be 6.8%... My own error in the end turned out to be 5.1%, approximately 1.7% better. 199 Does it really work? 200 Convolutional networks – theory 201 Convolutional network 202 Convolutional layers Every neuron is connected with a (typically small) receptive field of neurons in the lower layer. Neuron is "standard": Computes a weighted sum of its inputs, applies an activation function. 203 Convolutional layers Neurons grouped into feature maps sharing weights. 204 Convolutional layers Each feature map represents a property of the input that is supposed to be spatially invariant. Typically, we consider several feature maps in a single layer. 205 Pooling layers Neurons in the pooling layer compute simple functions of their receptive fields (the fields are typically disjoint): Max-pooling : maximum of inputs L2-pooling : square root of the sum of squres Average-pooling : mean · · · 206 Convolutional networks – architecture Neurons organized in layers, L0, L1, . . . , Ln, connections (typically) only from Lm to Lm+1. 207 Convolutional networks – architecture Neurons organized in layers, L0, L1, . . . , Ln, connections (typically) only from Lm to Lm+1. Several types of layers: input layer L0 207 Convolutional networks – architecture Neurons organized in layers, L0, L1, . . . , Ln, connections (typically) only from Lm to Lm+1. Several types of layers: input layer L0 dense layer Lm: Each neuron of Lm connected with each neuron of Lm−1. 207 Convolutional networks – architecture Neurons organized in layers, L0, L1, . . . , Ln, connections (typically) only from Lm to Lm+1. Several types of layers: input layer L0 dense layer Lm: Each neuron of Lm connected with each neuron of Lm−1. convolutional & pooling layer Lm: Contains two sub-layers: convolutional layer: Neurons organized into disjoint feature maps, all neurons of a given feature map share weights (but have different inputs) pooling layer: Each (convolutional) feature map F has a corresponding pooling map P. Neurons of P have inputs only from F (typically few of them), compute a simple aggregate function (such as max), have disjoint inputs. 207 Convolutional networks – architecture Denote X a set of input neurons Y a set of output neurons Z a set of all neurons (X, Y ⊆ Z) individual neurons denoted by indices i, j etc. ξj is the inner potential of the neuron j after the computation stops yj is the output of the neuron j after the computation stops (define y0 = 1 is the value of the formal unit input) wji is the weight of the connection from i to j (in particular, wj0 is the weight of the connection from the formal unit input, i.e. wj0 = −bj where bj is the bias of the neuron j) j← is a set of all i such that j is adjacent from i (i.e. there is an arc to j from i) j→ is a set of all i such that j is adjacent to i (i.e. there is an arc from j to i) [ji] is a set of all connections (i.e. pairs of neurons) sharing the weight wji. 208 Convolutional networks – activity neurons of dense and convolutional layers: inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable): yj = σj(ξj) 209 Convolutional networks – activity neurons of dense and convolutional layers: inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable): yj = σj(ξj) Neurons of pooling layers: Apply the "pooling" function: max-pooling: yj = max i∈j← yi avg-pooling: yj = i∈j← yi |j←| A convolutional network is evaluated layer-wise (as MLP), for each j ∈ Y we have that yj(w, x) is the value of the output neuron j after evaluating the network with weights w and input x. 209 Convolutional networks – learning Learning: Given a training set T of the form xk , dk k = 1, . . . , p Here, every xk ∈ R|X| is an input vector end every dk ∈ R|Y| is the desired network output. For every j ∈ Y, denote by dkj the desired output of the neuron j for a given network input xk (the vector dk can be written as dkj j∈Y ). Error function – mean squared error (for example): E(w) = 1 p p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 210 Convolutional networks – SGD The algorithm computes a sequence of weight vectors w(0), w(1), w(2), . . .. weights in w(0) are randomly initialized to values close to 0 in the step t + 1 (here t = 0, 1, 2 . . .), weights w(t+1) are computed as follows: Choose (randomly) a set of training examples T ⊆ {1, . . . , p} Compute w(t+1) = w(t) + ∆w(t) where ∆w(t) = −ε(t) · 1 |T| k∈T Ek (w(t) ) Here T is a minibatch (of a fixed size), 0 < ε(t) ≤ 1 is a learning rate in step t + 1 Ek (w(t)) is the gradient of the error of the example k Note that the random choice of the minibatch is typically implemented by randomly shuffling all data and then choosing minibatches sequentially. Epoch consists of one round through all data. 211 Backprop Recall that Ek (w(t)) is a vector of all partial derivatives of the form ∂Ek ∂wji . How to compute ∂Ek ∂wji ? 212 Backprop Recall that Ek (w(t)) is a vector of all partial derivatives of the form ∂Ek ∂wji . How to compute ∂Ek ∂wji ? First, switch from derivatives w.r.t. wji to derivatives w.r.t. yj: Recall that for every wji where j is in a dense layer, i.e. does not share weights: ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi 212 Backprop Recall that Ek (w(t)) is a vector of all partial derivatives of the form ∂Ek ∂wji . How to compute ∂Ek ∂wji ? First, switch from derivatives w.r.t. wji to derivatives w.r.t. yj: Recall that for every wji where j is in a dense layer, i.e. does not share weights: ∂Ek ∂wji = ∂Ek ∂yj · σj (ξj) · yi Now for every wji where j is in a convolutional layer: ∂Ek ∂wji = r ∈[ji] ∂Ek ∂yr · σr (ξr ) · y Neurons of pooling layers do not have weights. 212 Backprop Now compute derivatives w.r.t. yj: for every j ∈ Y: ∂Ek ∂yj = yj − dkj This holds for the squared error, for other error functions the derivative w.r.t. outputs will be different. 213 Backprop Now compute derivatives w.r.t. yj: for every j ∈ Y: ∂Ek ∂yj = yj − dkj This holds for the squared error, for other error functions the derivative w.r.t. outputs will be different. for every j ∈ Z Y such that j→ is either a dense layer, or a convolutional layer: ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj 213 Backprop Now compute derivatives w.r.t. yj: for every j ∈ Y: ∂Ek ∂yj = yj − dkj This holds for the squared error, for other error functions the derivative w.r.t. outputs will be different. for every j ∈ Z Y such that j→ is either a dense layer, or a convolutional layer: ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj for every j ∈ Z Y such that j→ is max-pooling: Then j→ = {i} for a single "max" neuron and we have ∂Ek ∂yj =    ∂Ek ∂yi if j = arg maxr∈i← yr 0 otherwise I.e. gradient can be propagated from the output layer downwards as in MLP. 213 Convolutional networks – summary Conv. nets. are nowadays the most used networks in image processing (and also in other areas where input has some local, "spatially" invariant properties) Typically trained using the gradient descent with the backpropagation. Due to the weight sharing allow (very) deep architectures. Typically extended with more adjustments and tricks in their topologies. 214 Recurrent Neural Networks - LSTM 215 RNN Input: x = (x1, . . . , xM) Hidden: h = (h1, . . . , hH) Output: y = (y1, . . . , yN) 216 RNN example Activation function: σ(ξ) =    1 ξ ≥ 0 0 ξ < 0 y 1 0 1 h (0, 0) (1, 1) (1, 0) (0, 1) · · · x (0, 0) (1, 0) (1, 1) 217 RNN example Activation function: σ(ξ) =    1 ξ ≥ 0 0 ξ < 0 y y1 = 1 y2 = 0 y3 = 1 h h0 = (0, 0) h1 = (1, 1) h2 = (1, 0) h3 = (0, 1) · · · x x1 = (0, 0) x2 = (1, 0) x3 = (1, 1) 217 RNN example y y1 = 1 y2 = 0 y3 = 1 h h0 = (0, 0) h1 = (1, 1) h2 = (1, 0) h3 = (0, 1) · · · x x1 = (0, 0) x2 = (1, 0) x3 = (1, 1) 217 RNN – formally M inputs: x = (x1, . . . , xM) H hidden neurons: h = (h1, . . . , hH) N output neurons: y = (y1, . . . , yN) Weights: Ukk from input xk to hidden hk Wkk from hidden hk to hidden hk Vkk from hidden hk to output yk 218 RNN – formally Input sequence: x = x1, . . . , xT xt = (xt1, . . . , xtM) 219 RNN – formally Input sequence: x = x1, . . . , xT xt = (xt1, . . . , xtM) Hidden sequence: h = h0, h1, . . . , hT ht = (ht1, . . . , htH) We have h0 = (0, . . . , 0) and htk = σ   M k =1 Ukk xtk + H k =1 Wkk h(t−1)k   219 RNN – formally Input sequence: x = x1, . . . , xT xt = (xt1, . . . , xtM) Hidden sequence: h = h0, h1, . . . , hT ht = (ht1, . . . , htH) We have h0 = (0, . . . , 0) and htk = σ   M k =1 Ukk xtk + H k =1 Wkk h(t−1)k   Output sequence: y = y1, . . . , yT yt = (yt1, . . . , ytN) where ytk = σ H k =1 Vkk htk . 219 RNN – in matrix form Input sequence: x = x1, . . . , xT 220 RNN – in matrix form Input sequence: x = x1, . . . , xT Hidden sequence: h = h0, h1, . . . , hT where h0 = (0, . . . , 0) and ht = σ(Uxt + Wht−1) 220 RNN – in matrix form Input sequence: x = x1, . . . , xT Hidden sequence: h = h0, h1, . . . , hT where h0 = (0, . . . , 0) and ht = σ(Uxt + Wht−1) Output sequence: y = y1, . . . , yT where yt = σ(Vht ) 220 RNN – Comments ht is the memory of the network, captures what happened in all previous steps (with decaying quality). RNN shares weights U, V, W along the sequence. Note the similarity to convolutional networks where the weights were shared spatially over images, here they are shared temporally over sequences. RNN can deal with sequences of variable length. Compare with MLP which accepts only fixed-dimension vectors on input. 221 RNN – training Training set T = (x1, d1), . . . , (xp, yp) here each x = x 1, . . . , x T is an input sequence, each d = d 1, . . . , d T is an expected output sequence. Here each x t = (x t1, . . . , x tM) is an input vector and each d t = (d t1, . . . , d tN) is an expected output vector. 222 Error function In what follows I will consider a training set with a single element (x, d). I.e. drop the index and have x = x1, . . . , xT where xt = (xt1, . . . , xtM) d = d1, . . . , dT where dt = (dt1, . . . , dtN) The squared error of (x, d) is defined by E(x,d) = T t=1 N k=1 1 2 (ytk − dtk )2 Recall that we have a sequence of network outputs y = y1, . . . , yT and thus ytk is the k-th component of yt 223 Gradient descent (single training example) Consider a single training example (x, d). The algorithm computes a sequence of weight matrices as follows: 224 Gradient descent (single training example) Consider a single training example (x, d). The algorithm computes a sequence of weight matrices as follows: Initialize all weights randomly close to 0. 224 Gradient descent (single training example) Consider a single training example (x, d). The algorithm computes a sequence of weight matrices as follows: Initialize all weights randomly close to 0. In the step + 1 (here = 0, 1, 2, . . .) compute "new" weights U( +1), V( +1), W( +1) from the "old" weights U( ), V( ), W( ) as follows: U ( +1) kk = U ( ) kk − ε( ) · δE(x,d) δUkk V ( +1) kk = V ( ) kk − ε( ) · δE(x,d) δVkk W ( +1) kk = W ( ) kk − ε( ) · δE(x,d) δWkk 224 Gradient descent (single training example) Consider a single training example (x, d). The algorithm computes a sequence of weight matrices as follows: Initialize all weights randomly close to 0. In the step + 1 (here = 0, 1, 2, . . .) compute "new" weights U( +1), V( +1), W( +1) from the "old" weights U( ), V( ), W( ) as follows: U ( +1) kk = U ( ) kk − ε( ) · δE(x,d) δUkk V ( +1) kk = V ( ) kk − ε( ) · δE(x,d) δVkk W ( +1) kk = W ( ) kk − ε( ) · δE(x,d) δWkk The above is THE learning algorithm that modifies weights! 224 Backpropagation Computes the derivatives of E, no weights are modified! 225 Backpropagation Computes the derivatives of E, no weights are modified! δE(x,d) δUkk = T t=1 δE(x,d) δhtk · σ · xtk k = 1, . . . , M δE(x,d) δVkk = T t=1 δE(x,d) δytk · σ · htk k = 1, . . . , H δE(x,d) δWkk = T t=1 δE(x,d) δhtk · σ · h(t−1)k k = 1, . . . , H 225 Backpropagation Computes the derivatives of E, no weights are modified! δE(x,d) δUkk = T t=1 δE(x,d) δhtk · σ · xtk k = 1, . . . , M δE(x,d) δVkk = T t=1 δE(x,d) δytk · σ · htk k = 1, . . . , H δE(x,d) δWkk = T t=1 δE(x,d) δhtk · σ · h(t−1)k k = 1, . . . , H Backpropagation: δE(x,d) δytk = ytk − dtk (assuming squared error) δE(x,d) δhtk = N k =1 δE(x,d) δytk · σ · Vk k + H k =1 δE(x,d) δh(t+1)k · σ · Wk k 225 Long-term dependencies δE(x,d) δhtk = N k =1 δE(x,d) δytk · σ · Vk k + H k =1 δE(x,d) δh(t+1)k · σ · Wk k Unless H k =1 σ · Wk k ≈ 1, the gradient either vanishes, or explodes. For a large T (long-term dependency), the gradient "deeper" in the past tends to be too small (large). A solution: LSTM 226 LSTM ht = ot ◦ σh(Ct ) output Ct = ft ◦ Ct−1 + it ◦ ˜Ct memory ˜Ct = σh(WC · ht−1 + UC · xt ) new memory contents ot = σg(Wo · ht−1 + Uo · xt ) output gate ft = σg(Wf · ht−1 + Uf · xt ) forget gate it = σg(Wi · ht−1 + Ui · xt ) input gate ◦ is the component-wise product of vectors · is the matrix-vector product σh hyperbolic tangents (applied component-wise) σg logistic sigmoid (aplied component-wise) 227 RNN vs LSTM Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 228 LSTM ⇒ ht = ot ◦ σh(Ct ) ⇒ Ct = ft ◦ Ct−1 + it ◦ ˜Ct ⇒ ˜Ct = σh(WC · ht−1 + UC · xt ) ⇒ ot = σg(Wo · ht−1 + Uo · xt ) ⇒ ft = σg(Wf · ht−1 + Uf · xt ) ⇒ it = σg(Wi · ht−1 + Ui · xt ) Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 229 LSTM ⇒ ht = ot ◦ σh(Ct ) ⇒ Ct = ft ◦ Ct−1 + it ◦ ˜Ct ⇒ ˜Ct = σh(WC · ht−1 + UC · xt ) ⇒ ot = σg(Wo · ht−1 + Uo · xt ) ⇒ ft = σg(Wf · ht−1 + Uf · xt ) ⇒ it = σg(Wi · ht−1 + Ui · xt ) Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 229 LSTM ⇒ ht = ot ◦ σh(Ct ) ⇒ Ct = ft ◦ Ct−1 + it ◦ ˜Ct ⇒ ˜Ct = σh(WC · ht−1 + UC · xt ) ⇒ ot = σg(Wo · ht−1 + Uo · xt ) ⇒ ft = σg(Wf · ht−1 + Uf · xt ) ⇒ it = σg(Wi · ht−1 + Ui · xt ) Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 229 LSTM ⇒ ht = ot ◦ σh(Ct ) ⇒ Ct = ft ◦ Ct−1 + it ◦ ˜Ct ⇒ ˜Ct = σh(WC · ht−1 + UC · xt ) ⇒ ot = σg(Wo · ht−1 + Uo · xt ) ⇒ ft = σg(Wf · ht−1 + Uf · xt ) ⇒ it = σg(Wi · ht−1 + Ui · xt ) Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 229 LSTM ⇒ ht = ot ◦ σh(Ct ) ⇒ Ct = ft ◦ Ct−1 + it ◦ ˜Ct ⇒ ˜Ct = σh(WC · ht−1 + UC · xt ) ⇒ ot = σg(Wo · ht−1 + Uo · xt ) ⇒ ft = σg(Wf · ht−1 + Uf · xt ) ⇒ it = σg(Wi · ht−1 + Ui · xt ) Source: https://colah.github.io/posts/2015-08-Understanding-LSTMs/ 229 LSTM – summary LSTM (almost) solves the vanishing gradient problem w.r.t. the "internal" state of the network. Learns to control its own memory (via forget gate). Revolution in machine translation and text processing. 230 Convolutions & LSTM in action – cancer research 231 Colorectal cancer outcome prediction The problem: Predict 5-year survival probability from an image of a small region of tumour tissue (1 mm diameter). 232 Colorectal cancer outcome prediction The problem: Predict 5-year survival probability from an image of a small region of tumour tissue (1 mm diameter). Input: Digitized haematoxylin-eosin-stained tumour tissue microarray samples. Output: Estimated survival probability. 232 Colorectal cancer outcome prediction The problem: Predict 5-year survival probability from an image of a small region of tumour tissue (1 mm diameter). Input: Digitized haematoxylin-eosin-stained tumour tissue microarray samples. Output: Estimated survival probability. Data: Training set: 420 patients of Helsinki University Centre Hospital, diagnosed with colorectal cancer, underwent primary surgery. Test set: 182 patients Follow-up time and outcome known for each patient. 232 Colorectal cancer outcome prediction The problem: Predict 5-year survival probability from an image of a small region of tumour tissue (1 mm diameter). Input: Digitized haematoxylin-eosin-stained tumour tissue microarray samples. Output: Estimated survival probability. Data: Training set: 420 patients of Helsinki University Centre Hospital, diagnosed with colorectal cancer, underwent primary surgery. Test set: 182 patients Follow-up time and outcome known for each patient. Human expert comparison: Histological grade assessed at the time of diagnosis. Visual Risk Score: Three pathologists classified to high/low-risk categories (by majority vote). Source: D. Bychkov et al. Deep learning based tissue analysis predicts outcome in colorectal cancer. Scientific Reports, Nature, 2018. 232 Colorectal cancer outcome prediction 233 Colorectal cancer outcome prediction 233 Data & workflow Input images: 3500 px × 3500 px Cut into tiles: 224 px × 224 px ⇒ 256 tiles Each tile pased to a convolutional network (CNN) Ouptut of CNN: 4096 dimensional vector. A "string" of 256 vectors (each of the dimension 4096) pased into a LSTM. LSTM outputs the probability of 5-year survival. 234 Data & workflow Input images: 3500 px × 3500 px Cut into tiles: 224 px × 224 px ⇒ 256 tiles Each tile pased to a convolutional network (CNN) Ouptut of CNN: 4096 dimensional vector. A "string" of 256 vectors (each of the dimension 4096) pased into a LSTM. LSTM outputs the probability of 5-year survival. The authors also tried to substitute the LSTM on top of CNN with logistic regression naive Bayes support vector machines 234 CNN architecture – VGG-16 (Pre)trained on ImageNet (cats, dogs, chairs, etc.) 235 LSTM architecture LSTM has three layers (264, 128, 64 cells) 236 LSTM – training L1 regularization (0.005) at each hidden layer of LSTM i.e. 0.005 times the sum of absolute values of weights added to the error L2 regularization (0.005) at each hidden layer of LSTM i.e. 0.005 times the sum of squared values of weights added to the error Dropout 5% at the input and the last hidden layers of LSTM Datasets: Training: 220 samples, Validation 60 samples, Test 140 samples. 237 Colorectal cancer outcome prediction Source: D. Bychkov et al. Deep learning based tissue analysis predicts outcome in colorectal cancer. Scientific Reports, Nature, 2018. 238 Feed-forward networks summary Architectures: Multi-layer perceptron (MLP): dense connections between layers Convolutional networks (CNN): local receptors, feature maps pooling Recurrent networks (RNN, LSTM): self-loops but still feed-forward through time Training: gradient descent algorithm + heuristics 239