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.) 2 Course organization Evaluation: Project teams of two students implementation of a selected model + analysis of given data implementation either in C, C++, or in Java 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 including some proofs that occur only on the whiteboard! Application of any deep learning toolset on given (difficult) data. We prefer TensorFlow but you may use another library (CNTK, Caffe, DeepLearning4j, ...) The goal is to get the best results on increasingly more difficult datasets. The team with the best result on the hardest dataset will automatically get > F at the exam. 3 FAQ Q: Why English? A: Couple of reasons. First, all resources about modern neural nets are in English, it is rather cumbersome to translate everything to Czech (combination of Czech and English is ugly). Second, to attract non-Czech speaking students to the course. 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 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 5 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! 6 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/ 7 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! 8 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 finance - stock prices, risk analysis medicine - diagnosis, signal processing (EKG, EEG, ...), image processing (MRI, roentgen, ...) text and speech processing - automatic translation, text generation, speech recognition other signal processing - filtering, radar tracking, noise reduction · · · I will concentrate on this area! 9 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 a picture 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 10 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, recurent network (LSTM), Hopfield and Boltzmann machines and their use in pre-training of deep nets) Standard applications of these models (image processing, 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, CNTK) 11 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). 12 Biological neural network Zdroj: N. Campbell and J. Reece; Biology, 7th Edition; ISBN: 080537146X 13 Biological neuron Zdroj: http://www.web-books.com/eLibrary/Medicine/Physiology/Nervous/Nervous.htm 14 Synaptic connections 15 Action potential 16 Summation 17 Biological and Mathematical neurons 18 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. 19 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.) 20 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 · · · 21 Neuron and linear separation σ x1 xn · · · x0 = 1 1/0 by 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). 22 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) 23 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. 24 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. 25 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.) 26 Architecture – Cycles A network is cyclic (recurrent) if its architecture contains a directed cycle. Otherwise it is acyclic (feed-forward) 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) 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. 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. 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. 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. 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 33 Activity – XOR 1010 1001 σ 01000110111 σ0001011011 1 σ 0101 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 34 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 35 Activity – example x1 1 σ 010 1 σ01 1 σ 01 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 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 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.) 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 39 Unsupervised learning – illustration X X X X A A A A B B B we search for two centres of clusters red crosses correspond to potential centres before application of the learning algorithm, green ones after the application 40 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 41 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. 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 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. 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. 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. 46 Non-linear separation - sigmoid Theorem (Cybenko 1989 - informal version) Let σ be a continuous function which is sigmoidal, i.e. satisfies σ(x) =    1 pro x → +∞ 0 pro 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 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 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. 49 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 σ(ξ) = 1 1+e−ξ ζ(ξ) = ξ 50 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 . 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 ≥ 0 ; 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). 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. 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. 54 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 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.) 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 History in waves ... Figure: The figure shows two of the three historical waves of articial 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). 60 Current hardware – What do we face? Increasing dataset size ... 61 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) 62 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). 63 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. 64 Current hardware – NVIDIA DGX Station 4x GPU (Tesla V100) TFLOPS = 480 GPU memory 64GB total NVIDIA Tensor Cores: 2,560 NVIDIA CUDA Cores: 20,480 System memory: 256 GB Network: Dual 10 Gb LAN NVIDIA Deep Learning SDK 65 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 library on top of TensorFlow that allows easy description of most modern neural networks CNTK (Microsoft) functionality similar to TensorFlow special input language called BrainScript Theano: 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, Torch (Facebook), Deeplearning4j, ... 66 Current software – Keras 67 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. 68 SyNAPSE (USA) Big research project, partially funded by DARPA Among the main subjects IBM a HRL, collaboraton with top US universities, e.g. Boston, Stanford The project started in 2008 Invested tens of millions USD. The goal Develop a neural network comparable with a real brain of a mammal The resulting hw chip should simulate 10 billion neurons, 100 trillion synaptic connections, consume 1 kilowatt (∼ a small heater), size 2 dm3 Oriented towards development of a new parallel computer architecture rather than neuroscience. 69 SyNAPSE (USA) – some results A cat brain simulation (2009) A simulation of a network with 109 neurons, 1013 synapses Simulated on a supercomputer Dawn (Blue Gene/P), 147,450 CPU, 144 tB of memory 643 times slower than the real brain The network modelled according to the real-brain structure (hierarchical model of a visual cortex, 4 layers) The authors claim that they observed some behaviour similar to the behaviour of the real brain (signal propagation, α, γ waves) ... simulation was heavily criticised (see latter) ... in 2012 the number of neurons increased to 530 bn neurons a 100 tn synapses 70 SyNAPSE (USA) – TrueNorth A chip with 5.4 billion elements 4096 neurosynaptic cores connected by a network, implementing 1 million programmable "spiking" neurons, 256 million programmable synaptic connections global frequency 1-kHz low energy consumption, approx. 63mW Offline learning, implemented some known algorithms (convolutional networks, RBM etc.) Applied to simple image recognition tasks. 71 Human Brain Project, HBP (Europe) Funded by EU, budget 109 EUR for 10 years Successor of Blue Brain Project at EPFL Lausanne Blue Brain started in 2005, ended in 2012, Human Brain Project started in 2013 The original goal: Deeper understanding of human brain networking in neuroscience diagnosis of brain diseases thinking machine The approach: study of brain tissue using current technology modelling of biological neurons simulation of the models (program NEURON) 72 HBP (Europe Blue brain project (2008) Model of a part of the brain cortex of a rat (approx. 10,000 neurons), much more complex model of neurons than in SyNAPSE Simulated on a supercomputer of the type Blue Gene/P (provided by IBM on discount), 16,384 CPU, 56 teraflops, 16 terabyt˚u pamˇeti, 1 PB disk space Simulation 300x slower than the real brain Human brain project (2015): Simplified model of the nervous system of a rat (approx. 200 000 neurons) 73 SyNAPSE vs HBP 2011: IBM Simulates 4.5 percent of the Human Brain, and All of the Cat Brain (Scientific American) “... performed the first near real-time cortical simulation of the brain that exceeds the scale of a cat cortex” (IBM) This announcement has been heavily criticised by Dr. Markram (head of HBP) “This is a mega public relations stunt – a clear case of scientific deception of the public” “Their so called “neurons” are the tiniest of points you can imagine, a microscopic dot” “Neurons contain 10’s of thousands of proteins that form a network with 10’s of millions of interactions. These interactions are incredibly complex and will require solving millions of differential equations. They have none of that.” 74 SyNAPSE vs HBP “Eugene Izhikevik himself already in 2005 ran a simulation with 100 billion such points interacting just for the fun of it: (over 60 times larger than Modha’s simulation)” Why did they get the Gordon Bell Prize? “They seem to have been very successful in influencing the committee with their claim, which technically is not peer-reviewed by the respective community and is neuroscientifically outrageous.” But is there any innovation here? “The only innovation here is that IBM has built a large supercomputer.” But did Mohda not collaborate with neuroscientists? “I would be very surprised if any neuroscientists that he may have had in his DARPA consortium realized he was going to make such an outrageous claim. I can’t imagine that the San Fransisco neuroscientists knew he was going to make such a stupid claim. Modha himself is a software engineer with no knowledge of the brain.” 75 ... and in the meantime in Europe In 2014, the European Commission received an open letter signed by more than 130 heads of laboratories demanding a substantial change in the management of the whole project. Peter Dayan, director of the computational neuroscience unit at UCL: “The main apparent goal of building the capacity to construct a larger-scale simulation of the human brain is radically premature.” “We are left with a project that can’t but fail from a scientific perspective. It is a waste of money, it will suck out funds from valuable neuroscience research, and would leave the public, who fund this work, justifiably upset.” 76 ... and in 2016 The European Commission and the Human Brain Project Coordinator, the École Polytechnique Fédérale de Lausanne (EPFL), have signed the first Specific Grant Agreement (SGA1), releasing EUR 89 million in funding retroactively from 1st April 2016 until the end of March 2018. The signature of SGA1 means that the HBP and the European Commision have agreed on the HBP Work Plan for this two year period. The SGA1 work plan will move the Project closer to achieving its aim of establishing a cutting-edge, ICT-based scientific Research Infrastructure for brain research, cognitive neuroscience and brain-inspired computing. 77 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 78 ADALINE Learning: Given a training set 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. 79 Oaks in Wisconsin 80 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). 81 Error function 82 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. 83 Gradient - illustration Caution! This picture just illustrates the notion of gradient ... it is not the convex paraboloid E(w) ! 84 Gradient of the error function (ADALINE) ∂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 85 ADALINE - 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 k = (t mod p) + 1 and 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). 86 ADALINE – Animation 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. 88 ADALINE - classification How to use the ADALINE for classification? The training set is T = x1, d1 , x2, d2 , . . . , xp, dp kde xk = (xk0, xk1, . . . , xkn) ∈ Rn+1 a dk ∈ {1, −1}. Here dk determines a class. Train the network using the ADALINE algorithm. We may expect the following: if dk = 1, then w · xk ≥ 0 if dk = −1, then w · xk < 0 This does not have to be always true but if the training set is reasonably linearly separable, then the algorithm typically gives satisfactory results. 89 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) 90 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) 91 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. 92 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 93 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) ). 94 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 )). 95 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) 96 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 . 97 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.) 98 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 ... 99 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). 100 Illustration of the gradient descent – XOR Source: Pattern Classification (2nd Edition); Richard O. Duda, Peter E. Hart, David G. Stork 101 Animation (sin(x)), network 1-5-1) One iteration: 10 iterations: 102 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) 103 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) 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 2 105 MLP – batch learning 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) = w(t) + ∆w(t) Here ∆w(t) = −ε(t) · E(w(t) ) = −ε(t) · p k=1 Ek (w(t) ) 0 < ε(t) ≤ 1 is a learning rate in step t + 1 E(w(t)) is the gradient of the error function Ek (w(t)) is the gradient of the error function for the training example k 106 MLP – error functions square error: E(w) = p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 mean square error (mse): E(w) = 1 p p k=1 Ek (w) I will use mse throughout the rest of this lecture. 107 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 ∂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 )). 108 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. 109 Issues in gradient descent 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. 110 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. 111 Issues in gradient descent – too fast descent steep cliffs: the gradient is extremely large, descent skips important weight vectors 112 Issues in gradient descent – local vs global structure What if we initialize on the left? 113 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.) 114 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. Some kinds of hardware achieve better runtime with specific sizes of arrays. Especially when using GPUs, it is common 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. 115 Moment 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. 116 Momentum – illustration 117 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 moment 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 Learning rate Generic rules for adaptation of ε(t) Start with a larger learning rate (e.g. ε = 0.1). Later decrease as the descent is supposed to settle in a minimum of E. Some tools allow to set a list of learning rates, each rate for one epoch of the descent. In case you may observe the error evolving: If the error decreases, increase slightly the rate. If the error increases, decrease the rate. Note that the error may increase for the short period without any harm to convergence of the learning process. 119 AdaGrad So far we have considered a uniform learning rate. 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. 120 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) j + δ · k∈T ∂Ek ∂wji (w(t) ) and r (t) j = r (t−1) j +   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. 121 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. 122 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) j + δ · k∈T ∂Ek ∂wji (w(t) ) and r (t) j = ρr (t−1) j + (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. 123 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. 124 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. 125 Activation functions – tanh σ(ξ) = 1.7159 · tanh(2 3 · ξ), we have limξ→∞ σ(ξ) = 1.7159 and limξ→−∞ σ(ξ) = −1.7159 126 Activation functions – tanh σ(ξ) = 1.7159 · tanh(2 3 · ξ) is almost linear on [−1, 1] 127 Activation functions – tanh first derivative: σ(ξ) = 1.7159 · tanh(2 3 · ξ) 128 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) 129 Input preprocessing Individual inputs should not be correlated. Correlated inputs can be removed as a part of dimensionality reduction. (Dimensionality reduction and decorrelation can be implemented using neural networks. There are also standard methods such as PCA.) 130 Initial weights (for tanh) Typically, the weights are chosen randomly 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], extreme values of σ are close to −1 and 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]. 131 Initial weights (for tanh) Standardization gives mean = 0 and variance = 1 of the input data. Assume that individual inputs are (almost) uncorrelated. Consider a neuron j from the first layer with d 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 = d 3 · w. Thus we put w = √ 3√ d . The same works for higher layers, d corresponds to the number of neurons in the layer one level lower. 132 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   Here m is the number of inputs to the neuron, m is the number of outputs of the neuron. 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. 133 Target values (tanh) Target values dkj should be chosen in the range of the output activation functions, in our case [−1.716, 1.716]. Target values too close to extrema of the output activations, in our case ±1.716, may cause that the weights will grow indefinitely (slows down learning). Thus it is good to choose target values from the interval [−1.716 + δ, 1.716 − δ]. As before, ideally [−1.716 + δ, 1.716 − δ] should span the interval on which the activation function is linear, i.e. dkj should be taken from [−1, 1]. 134 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. 135 Output neurons The choice of activation functions for output units depends on the concrete applications. For regression (function approximation) the output is typically linear (or sigmoidal). For classification, the current activation functions of choice are logistic sigmoid or tanh – binary classification softmax: σj(ξj) = eξj i∈Y eξi for multi-class classification. For some reasons 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) + (1 − dkj) ln(1 − yj) ... which somewhat corresponds to the maximum likelihood principle. 136 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. 137 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. 138 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?? 139 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. 140 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 cross-validation etc. 141 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 – bad generalization. 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 using a rule of thumb: the number of neurons ≈ ten times less than the number of training instances. experiment, experiment, experiment. 142 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: 143 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 of the same size by sampling from T uniformly with replacement. If |Ti| = |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. 144 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). update weights of neurons in N (in a standard way), 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. 145 Weight decay 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) + 2ζ ε (w · w) Here 2ζ ε (w · w) penalizes large weights. 146 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. 147 ALVINN (history) 148 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. 149 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.) 150 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. 151 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). 152 ALVINN - learning pure backpropagation constant learning speed momentum, slowly increasing. We used a learning rate of 0.015, a momentum term of 0.9, and we ramped up the learning rate and momentum using a rate term of 0.05. This means that the learning rate and momentum increase linearly over 20 epochs until they reach their maximum value (0.015 and 0.9, respectively). We also used a weight decay term of 0.0001. 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. 153 ALVINN - weight development round 0 round 10 round 20 round 50 h1 h2 h3 h4 h5 Here h1, . . . , h5 are hidden neurons. 154 ALVINN - comments Compare ALVINN with explicit system development: For driving you need to find key features for driving (ALVINN finds automatically) detect the features (ALVINN creates its own detectors) implement driving algorithm (ALVINN learns form the driver) ALVINN was rather limited (but keep in mind that the net is small): just one type of road, no obstacles no higher level control 155 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. 156 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) Abstrakt: 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 the first convolutional network LeNet-1 in 1998. 157 MNIST – LeNet1 158 MNIST – LeNet1 Activity: activation function: hyperbolic tangents 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 Training: modified backpropagation (conjugate gradients), online weights initialized uniformly from [−2.4, 2.4], divided by the number of inputs to a given neuron 159 MNIST – 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) 160 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 161 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. 162 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. 163 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! 164 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. 165 Trained feature maps (20 feature maps, receptive fields 5 × 5) 166 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 · · · 167 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 backprop, which can be easily adapted to convolutional networks. 168 Convolutional network 169 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. 170 Simple convolutional network in Theano 171 MNIST Out of 10 000 images in the test set, only these 33 have been incorrectly classified: 172 More complex convolutional networks Convolutional networks have been used for classification of images from the ImageNet database (16 million color images, 20 thousand classes) 173 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). 174 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 175 ILSVRC 2014 The same set as in 2012, top-5 criterion. GoogLeNet: deep convolutional network, 22 layers Results: Accuracy 93.33% top-5 176 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. 177 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. 178 Convolutional network 179 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. 180 Convolutional layers Neurons grouped into feature maps sharing weights. 181 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. 182 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 · · · 183 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. 184 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) jshare is a set of neurons sharing weights with j i.e. neurons that belong to the same feature map as j 185 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. 186 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 square error (for example): E(w) = 1 p p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 187 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. 188 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, that is shares wji with neurons of jshare: ∂Ek ∂wji = r∈jshare ∂Ek ∂yr · σr (ξr ) · yr Neurons of pooling layers do not have weights. 189 Backprop Now compute derivatives w.r.t. yj: for every j ∈ Y: ∂Ek ∂yj = yj − dkj This holds for the mean-square 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. 190 Convolutional networks – conclusions 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 backpropagation. Due to the weight sharing allow (very) deep architectures. Typically extended with more adjustments and tricks in their topologies. 191 Recurrent networks – Hopfield network Auto-associative network: Given an input, the network outputs a training example (encoded in its weights) "similar" to the given input. 192 Hopfield network Architecture: complete topology, i.e. output of each neuron is input to all neurons all neurons are both input and output denote by ξ1, . . . , ξn inner potentials and by y1, . . . , yn outputs (states) of individual neurons denote by wji the weight of connection from a neuron i ∈ {1, . . . , n} to a neuron j ∈ {1, . . . , n} assume wjj = 0 for every j = 1, . . . , n For now: no neuron has a bias 193 Hopfield network Learning: Training set T = {xk | xk = (xk1, . . . , xkn) ∈ {−1, 1}n , k = 1, . . . , p} The goal is to "store" the training examples of T so that the network is able to associate similar examples. Hebb’s learning rule: If the inputs to a system cause the same pattern of activity to occur repeatedly, the set of active elements constituting that pattern will become increasingly strongly interassociated. That is, each element will tend to turn on every other element and (with negative weights) to turn off the elements that do not form part of the pattern. To put it another way, the pattern as a whole will become "auto-associated". Mathematically speaking: wji = p k=1 xkjxki 1 ≤ j i ≤ n Intuition: "Neurons that fire together, wire together". 194 Hopfield network Learning: Training set T = {xk | xk = (xk1, . . . , xkn) ∈ {−1, 1}n , k = 1, . . . , p} Hebb’s rule: wji = p k=1 xkjxki 1 ≤ j i ≤ n Note that wji = wij, i.e. the weight matrix is symmetric. Learning can be seen as poll about equality of inputs: If xkj = xki, then the training example votes for "i equals j" by adding one to wji. If xkj xki, then the training example votes for "i does not equal j" by subtracting one from wji. 195 Hopfield network Activity: Initially, neurons set to the network input x = (x1, . . . , xn), thus y (0) j = xj for every j = 1, . . . , n. Cyclically update states of neurons, i.e. in step t + 1 compute the value of a neuron j such that j = (t mod p) + 1, as follows: Compute the inner potential: ξ (t) j = n i=1 wjiy (t) i then y (t+1) j =    1 ξ (t) j > 0 y (t) j ξ (t) j = 0 −1 ξ (t) j < 0 196 Hopfield network – activity The computation stops in a step t∗ if the network is for the first time in a stable state, i.e. y (t∗+n) j = y (t∗) j (j = 1, . . . , n) Theorem Assuming symmetric weights, computation of a Hopfiled network always stops for every input. This implies that a given Hopfiled network computes a function from {−1, 1}n to {−1, 1}n (determined by its weights). Denote by y(W, x) = y (t∗) 1 , . . . , y (t∗) n the value of the network for a given input x and a weight matrix W. Denote by yj(W, x) = y (t∗) j the component of the value of the network corresponding to the neuron j. If W is clear from the context, we write only y(x) a yj(x). 197 Ising model – an analogy Simple models of magnetic materials resemble Hopfield network. atomic magnets organized into square-lattice each magnet may have only one of two possible orientations (in the Hopfield network +1 a −1) orientation of each magnet is influenced by an external magnetic field (input of the network) as well as orientation of the other magnets weights in the Hopfiled net model determine interaction among magnets 198 Energy function Energy function E assigns to every state y ∈ {−1, 1}n a (potential) energy: E(y) = − 1 2 n j=1 n i=1 wjiyjyi states with low energy are stable (few neurons "want to" change their states), states with high energy are not stable i.e. large (positive) wjiyjyi is stable and small (negative) wjiyjyi is not stable The energy does not increase during computation: E(y(t)) ≥ E(y(t+1)), stable states y(t∗) correspond to local minima of E. 199 Energy landscape 200 Hopfield – example 1 2 3 −1 −11 y1 y2 y3 E 1 1 1 1 1 1 −1 1 1 −1 1 −3 1 −1 −1 1 −1 1 1 1 −1 1 −1 −3 −1 −1 1 1 −1 −1 −1 1 Hopfield network with three neurons trained on a single training example (1, −1, 1) using Hebb’s rule (note that (−1, 1, −1) has also been "stored" into the network) 201 Hopfield network – convergence Observe that the energy does not increase during computation: E(y(t)) ≥ E(y(t+1)) if the state is updated in a step t + 1, then E(y(t)) > E(y(t+1)) there are only finitely many states, and thus, eventually, a local minimum of E is reached. This proves that computation of a Hopfield network always stops. 202 Hopfield network – phantoms The energy function E may have local minima that do not correspond to training examples (so called phantoms). Phantoms can be "unlearned" e.g. using the following rule: Given a phantom (x1, . . . , xn) ∈ {−1, 1}n and weights wji, then new weights wji are computed by wji = wji − xixj (i.e. similar to Hebb’s rule but with the opposite sign) 203 Reproduction – statistical analysis Capacity of Hopfield network is defined as the ratio p/n of number of training examples the net is able to learn over the number of neurons. Assume that training examples are chosen randomly: each component of xk is set to 1 with probability 1/2 and to −1 with probability 1/2. Consider a configuration W obtained by learning using the Hebb’s rule. Denote β = P xk = y(W, xk ) pro k = 1, . . . , p Then for n → ∞ and p ≤ n/(4 log n) we have β → 1. I.e. the maximum number of examples that can be effectively stored in Hopfield net is proportional to n/(4 log n). 204 Hopfield network – example figures 12 × 10 (120 neurons, −1 is white and 1 is black) learned 8 figures input generated with 25% noise image shows the activity of the Hopfield network 205 Hopfield network – example 206 Hopfield network – example 207 Hopfield network – local minima We look for "deep" minima of E .... We may get suck in a shallow minimum. Solution: In every state we allow transition to states with higher energy. This transition has a small probability (which will be higher at the beginning and decrease throughout computation). 208 Boltzmann activity Activity: States of neurons initially set to values of {−1, 1}, i.e., y (0) j ∈ {−1, 1} for j ∈ {1, . . . , n}. In the step t + 1 update value of a randomly chosen neuron j ∈ {1, . . . , n} as follows: Compute the inner potential ξ (t) j = n i=1 wjiy (t) i choose y (t+1) j ∈ {−1, 1} randomly so that P y (t+1) j = 1 = σ ξ (t) j where σ(ξ) = 1 1 + e−2ξ/T(t) The parameter T(t) is called temperature in time t. 209 Temperature and energy High temperature T(t) implies that P y (t+1) j = 1 ≈ 1 2 and thus the network behaves almost randomly. Very low temperature T(t) implies that either P y (t+1) j = 1 ≈ 1 or P y (t+1) j = 1 ≈ 0 depending on whether ξ (t) j > 0 or ξ (t) j < 0. Thus the network behaves almost deterministically (as in the original activity of Hopfield network). Notes: Boltzmann activity = Hopfield activity + random noise, energy E(y) = −1 2 n j=1 n i=1 wjiyjyi may jump to higher levels (with probability depending on the temperature), the probability of transition to higher energy decreases exponentially with the size of the "energy jump". 210 Simulated annealing The following approach may help to reach deep minima of E: Start with higher temperature T(t) Gradually reduce the temperature, e.g. as follows: T(t) = ηt · T(0) where η < 1 is close to 1 or T(t) = T(0)/ log(1 + t) This process resembles annealing used in metallurgy that alters the physical and sometimes chemical properties of a material to increase its ductility and reduce its hardness. It also extends physical motivation of Hopfield networks: magnet orientation is now, in addition, influenced by thermal fluctuations. ... and it gets us close to Boltzmann machines. 211 Boltzmann machine Architecture: Neural network with cycles and symmetric connections (i.e. arbitrary graph) N is a set of all neurons. Denote by ξj the inner potential and by yj the output (i.e. state) of neuron j. State of the machine: y ∈ {−1, 1}|N|. Denote by wji ∈ R the weight of the connection from i to j (and thus also from j to i). No bias and assume wjj = 0 for all j ∈ N. 212 Boltzmann machine Activity: States of neurons initially set to values of {−1, 1}, i.e. y (0) j ∈ {−1, 1} for j ∈ N. In the step t + 1 do the following: Choose a neuron j ∈ N randomly with the uniform probability. Compute the inner potential of j: ξ (t) j = n i∈j← wjiy (t) i Choose y (t+1) j ∈ {−1, 1} randomly so that P y (t+1) j = 1 = σ(ξ (t) j ) where σ(ξ) = 1 1 + e−2ξ/T(t) (T(t) is a temperature at time t.) 213 Boltzmann machine High temperature T(t) implies that P y (t+1) j = 1 ≈ 1 2 and thus the machine behaves almost randomly. Low temperature T(t) means that either P y (t+1) j = 1 ≈ 1 or P y (t+1) j = 1 ≈ 0 depending on whether ξ (t) j > 0 or ξ (t) j < 0. Then the machine behaves almost deterministically (as the Hopfield network). 214 Boltzmann machine represents probability Goal: Construct a network representing a distribution on a set of vectors {−1, 1}|N|. Rough idea: Boltzmann machine has states in {−1, 1}|N|, moves randomly from state to state during computation. If we let the machine run for sufficiently long time (with a fixed temperature), the relative frequencies of visits to states will be independent of the initial state. We consider these frequencies as probabilities of the states. This gives a probability distribution on {−1, 1}|N| represented by the machine. During learning, a probability distribution on states of {−1, 1}|N| will be given, and we adapt weights so that the frequencies match the given probabilities. 215 Equilibrium Fix a temperature T (i.e. T(t) = T for t = 1, 2, . . .). Theorem For every γ∗ ∈ {−1, 1}|N| we have that lim t→∞ P y(t) = γ∗ = 1 Z e−E(γ∗)/T where Z = γ∈{−1,1}|N| e−E(γ)/T E(γ) = − 1 2 i,j wijy γ i y γ j the Boltzmann distribution. Define pN(γ∗) := limt→∞ P y(t) = γ∗ for every γ∗ ∈ {−1, 1}|N|. 216 Equilibrium probabilities Note that pN is a probability distribution on {−1, 1}|N| represented by the machine, for a state γ∗, we have that pN(γ∗) is the probability of γ∗ in the thermal equilibrium, pN(γ∗) can be estimated by P y(t∗) = γ∗ for sufficiently large t∗ That is, in order to compute pN(γ∗ ) it is sufficient to simulate a computation several times for t∗ steps and then compute the relative frequency of stopping in γ∗ . By Markov chains theory, pN(γ∗) is the long-run frequency of visits to γ∗. This gives an alternative procedure for estimating pN(γ∗ ): Execute the machine for very long time, compute the relative frequency of visits to γ∗ along the computation. 217 Boltzmann machine – learning To be able to capture more probability distributions, we introduce hidden neurons. Divide N into two disjoint sets: visible neurons V hidden neurons H For α ∈ {−1, 1}|V| denote pV (α) = β∈{−1,1}|H| pN(α, β) the probability that the state of visible neurons in the thermal equilibrium is α. Our goal is to adapt weights so that pV corresponds to a given probability distribution on {−1, 1}|V|. 218 Boltzmann machine – learning Learning: Let pd be a probability distribution on the states of visible neurons, i.e. on {−1, 1}|V|. The distribution pd can be determined by a sequence of training examples: T = x1, x2, . . . , xm then pd(α) = #(α, T )/m here #(α, T ) is the number of occurrences of α in T . Our goal is to find a configuration of the network W such that pV ≈ pd. 219 Boltzmann machine – learning A suitable measure of difference between probability distributions pV and pd is relative entropy weighted by probabilities of states (Kullback-Leibler divergence): E(W) = α∈{−1,1}|V| pd(α) ln pd(α) pV (α) For pd given by a training set T = x1, x2, . . . , xm we have that minimizing E(W) is equivalent to maximizing likelihood of T . 220 Boltzmann machine – learning Minimize E(w) using gradient descent, i.e. compute a sequence of weight matrices: W(0), W(1), . . . initialise W(0) randomly, close to 0 in step t + 1 compute W(t+1) as follows: W (t+1) ji = W (t) ji + ∆W (t) ji where ∆W (t) ji = −ε(t) · ∂E ∂wji (W(t) ) is the update of the weight wji in the step t + 1 and 0 < ε(t) ≤ 1 is the learning rate in the step t + 1. It remains to compute ∂E ∂wji (W). 221 Boltzmann machine – learning For sufficiently large t∗ (i.e. in thermal equilibrium) we have ∂E ∂wji ≈ − 1 T y (t∗) j y (t∗) i fixed − y (t∗) j y (t∗) i free y (t∗) j y (t∗) i fixed is the expected value of y (t∗) j y (t∗) i in the thermal equilibrium assuming that values of visible neurons are fixed at the beginning of computation according to pd. y (t∗) j y (t∗) i free is the expected value of y (t∗) j y (t∗) i in the thermal equilibrium (no values fixed). Thus ∆w ( ) ji = −ε( ) · ∂E ∂wji (W( −1) ) = ε( ) T y (t∗) j y (t∗) i fixed − y (t∗) j y (t∗) i free 222 Boltzmann machine – learning Compute y (t∗) j y (t∗) i fixed as follows: Let Y := 0 and do the following q times: 1. choose α ∈ {−1, 1}|V| randomly according to pd, 2. fix values of visible neurons to α and do not update them throughout the remaining steps 3. and 4., 3. simulate t∗ steps, now the current values of neurons j and i are y (t∗ ) j and y (t∗ ) i , respectively, 4. add y (t∗ ) j y (t∗ ) i to Y. For sufficiently large q, the value Y/q will be a good estimate of y (t∗) j y (t∗) i fixed . y (t∗) j y (t∗) i free can be estimated similarly, the only difference is that the steps 1. and 2. are omitted. 223 Boltzmann machine – learning For completeness, the analytic version: y (t∗) i y (t∗) j fixed = = α∈{−1,1}|V| pd(α) β∈{−1,1}|S| pN(α, β) pV (α) y αβ j y αβ i here y αβ j is the output of the neuron j in the state (α, β). y (t∗) i y (t∗) j free = γ∈{−1,1}|N| pN(γ)y γ j y γ i 224 Restricted Boltzmann machine (RBM) Architecture: Neural network with cycles and symmetric connections, neurons divided into two disjoint sets: V - visible H - hidden Connections: V × S (complete bipartite graph) N is a set of all neurons. Denote by ξj the inner potential and by yj the output (i.e. state) of neuron j. State of the machine: y ∈ {0, 1}|N|. Denote by wji ∈ R the weight of the connection from i to j (and thus also from j to i). Consider bias: wj0 is the weight between j and a neuron 0 whose value y0 is always 1. 225 RBM – activity Activity: States of neurons initially set to values of {0, 1}, i.e. y (0) j ∈ {0, 1} for j ∈ N. In the step t + 1 do the following: t even: randomly choose new values of all hidden neurons, for every j ∈ H P y (t+1) j = 1 = 1  1 + exp  −wj0 − i∈V wjiy (t) i     t odd: randomly choose new values of all visible neurons, for every j ∈ V P y (t+1) j = 1 = 1  1 + exp  −wj0 − i∈H wjiy (t) i     226 Thermal equilibrium Fix a temperature T (i.e. T(t) = T for t = 1, 2, . . .). Theorem For every γ∗ ∈ {0, 1}|N| we have that lim t→∞ P y(t) = γ∗ = 1 Z e−E(γ∗)/T where Z = γ∈{0,1}|N| e−E(γ)/T and E(γ) = − i∈V, j∈H wjiy γ j y γ i − i∈V wi0y γ i − j∈H wj0y γ j Define pN(γ∗) := limt→∞ P y(t) = γ∗ for every γ∗ ∈ {0, 1}|N|. 227 RBM – learning Learning: Let pd be a probability distribution on states of visible neurons, i.e. on {0, 1}|V|. Our goal is to find a configuration of the network W such that pV ≈ pd. A suitable measure of difference between probability distributions pV and pd is relative entropy weighted by probabilities of states (Kullback-Leibler divergence): E(W) = α∈{0,1}|V| pd(α) ln pd(α) pV (α) 228 RBM – learning Minimize E(w) using gradient descent, i.e. compute a sequence of weight matrices: W(0), W(1), . . . initialise W(0) randomly, close to 0 in step t + 1 compute W(t+1) as follows: W (t+1) ji = W (t) ji + ∆W (t) ji where ∆W (t) ji = −ε(t) · ∂E ∂wji (W(t) ) is the update of the weight wji in the step t + 1 and 0 < ε(t) ≤ 1 is the learning rate in the step t + 1. It remains to compute ∂E ∂wji (W). 229 RBM – learning For sufficiently large t∗ (i.e. in thermal equilibrium) we have ∂E ∂wji ≈ − 1 T y (t∗) j y (t∗) i fixed − y (t∗) j y (t∗) i free y (t∗) j y (t∗) i fixed is the expected value of y (t∗) j y (t∗) i in the thermal equilibrium assuming that values of visible neurons are fixed at the beginning of computation according to pd. y (t∗) j y (t∗) i free is the expected value of y (t∗) j y (t∗) i in the thermal equilibrium (no values fixed). Problem: Computation of y (t∗) j y (t∗) i free takes long time. y (t∗) j y (t∗) i free can be estimated with yjyi recon , the expectation of y (3) j y (3) i when values of visible neurons chosen by pd. 230 RBM – learning Thus ∆w (t) ji = ε(t) · yjyi fixed − yjyi recon Compute yjyi fixed as follows: Let Y := 0 and repeat the following q times: fix values of visible neurons randomly by pd simulate one step of computation, add yjyi to Y For a suitable q we have that Y/q estimates yjyi fixed well. Compute yjyi recon as follows: Let Y := 0 and repeat q times: choose initial values of visible neurons by pd simuate three steps, add yjyi to Y (i.e. compute values of hidden neurons, then of visible ones (reconstruction of the input) and then of hidden neurons) For a suitable q we have that Y/q estimates yjyi recon well. 231 Deep 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) 232 Deep MLP 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) 233 Deep MLP – activity inner potential of neuron j: ξj = i∈j← wjiyi activation function σj for neuron j (arbitrary differentiable): yj = σj(ξj) A deep MLP is evaluated layer-wise, 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. 234 Deep MLP – 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 square error (for example): E(w) = 1 p p k=1 Ek (w) where Ek (w) = 1 2 j∈Y yj(w, xk ) − dkj 2 Other errors such as cross-entropy possible. 235 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. 236 Why deep networks ... if one hidden layer is able to represent an arbitrary (reasonable) function? One hidden layer may be very inefficient, i.e. huge amount of neurons may be needed. One can show that the number of hidden neurons may be exponential w.r.t. the dimension of the input, networks with multiple layers may be exponentially more succinct as opposed to single hidden layer. ... ok, so let’s try to teach deep networks ... using backpropagation? Problems: Gradient may vanish/explode when backpropagated through many layers. Deep networks (with many neurons) overfit very easily. 237 Deep MLP - vanishing 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 holds ∂Ek ∂yj = yj − dkj pro j ∈ Y ∂Ek ∂yj = r∈j→ ∂Ek ∂yr · σr (ξr ) · wrj pro j ∈ Z (Y ∪ X) σr (ξr ) · wrj is less than one for standard logistic sigmoid and weights initialized close to 0. 238 Deep MLP – pretraining Assume k layers. Denote Wi the weight matrix between layers i − 1 and i Fi function computed by the "lower" part of the MLP consisting of layers 0, 1, . . . , i F1 is a function which consists of the input and the first hidden layer (which is now considered as the output layer). Crucial observation: For every i, the layers i − 1 and i together with the matrix Wi can be considered as a RBM (assume T = 1). Denote such a RBM as Bi. 239 Deep MLP – pretraining For now, consider only input vectors x1, . . . , xp where xk ∈ {0, 1}n for all k = 1, . . . , p. unsupervised pretraining: Gradually, for every i = 1, . . . , k, train RBM Bi on randomly selected inputs from the training set: Fi−1(x1), . . . , Fi−1(xp) using the training algorithm for RBM (here F0(xi) = xi). (Thus Bi learns from training samples transformed by the already pretrained layers 0, . . . , i − 1) We obtain a deep belief network D representing a distribution given by x1, . . . , xp. (Recall that in such a distribution the probability of a given x is equal to the relative frequency of x in x1, . . . , xp.) 240 Deep belief network The network D can be used to sample from the distribution as follows: Simulate the topmost RBM for some steps (ideally to thermal equilibrium), this gives values of neurons in the two topmost layers. Propagate the values downwards by always simulating one step of the corresponding RBM. That is, you have already computed values of neurons in layers k and k − 1. To compute values of neurons in the layer k − 2, simulate one step of RBM Bk−1, that is sample values of neurons in the layer k − 2 using RBM dynamics of Bk−1 with values of the layer k − 1 fixed. Similarly, compute values of k − 3 by simulating Bk−2 ... etc. ... finally obtain values of input neurons. Probability with which a concrete input x is sampled by the above procedure is the probability of x in the distribution represented by D. 241 Deep MLP – training with pretraining Now consider supervised learning with a training set: T = xk , dk k = 1, . . . , p . Still assume that xk ∈ {0, 1}n . unsupervised pretraining: Gradually, for every i = 1, . . . , k, train RBM Bi on randomly selected inputs from the training set: Fi−1(x1), . . . , Fi−1(xp) using the training algorithm for RBM (here F0(xi) = xi). (Thus Bi learns from training samples transformed by the already pretrained layers 0, . . . , i − 1) Obtain D. Add one (or more) layer to the top of D and consider the result to be MLP. (i.e. forget the RBM dynamics and start considering the network as MLP with sigmoidal activations). supervised fine-tuning: Train in supervised mode (on the training set T ) using e.g. gradient descent + backprop. 242 Application – dimensionality reduction Dimensionality reduction: A mapping R from Rn to Rm where m < n, for every example x we have that x can be "reconstructed" from R(x). Standard method: PCA (there are many linear as well as non-linear variants) 243 Reconstruction – PCA 1024 pixels compressed to 100 dimensions (i.e. 100 numbers). 244 Autoencoders Dimensionality reduction using MLP. Consider MLP n − m − n where m << n. The same vector on the input as well as output. Dimensionality reduction: Encoding: Compute values of hidden neurons. Reconstruction: Compute values of output neurons given values of hidden neurons. Can also be used for compression (in communication). One can show that if linear neurons are used, the method implements PCA. 245 Autoencoder – historical implementation Architecture: MLP 64 − 16 − 64 Activity: activation function: hyperbolic tangens with limits −1 and 1 Learning: Data: Images 256 × 256, 8 bits per pixel. Samples: input and output is a frame 8 × 8, randomly selected in the image. Inputs normalized to [−1, 1]. Learning: Backpropagation Learning rate: 0.01 for hidden, 0.1 pro output trained in 50 000 - 100 000 iterations The goal was to compress images to smaller data size. 246 Dimensionality reduction – compression A frame 8 × 8 passes through the image 256 × 256 (no overlap) (A) original (B) compression (C) compression + rounding to 6 bits (1.5 bit per pixel) (D) compression + rounding to 4 bits (1 bit per pixel) 247 Dimensionality reduction – compression New image (trained on the previous one): (A) original (B) compression (C) compression + rounding to 6 bits (1.5 bit per pixel) (D) compression + rounding to 4 bits (1 bit per pixel) 248 Deep MLP – dimensionality reduction Hinton, G. E., Osindero, S. and Teh, Y. (2006) A fast learning algorithm for deep belief nets. Neural Computation, 18, pp 1527-1554. Hinton, G. E. and Salakhutdinov, R. R. (2006) Reducing the dimensionality of data with neural networks. Science, Vol. 313. no. 5786, pp. 504 - 507, 28 July 2006. This basically started all the deep learning craze ... 249 Deep MLP – dimensionality reduction 250 Images – pretraining Data: 165 600 black-white images, 25 × 25, mean intensity 0, variance 1. Images obtained from Olivetti Faces database of images 64 × 64 using standard transformations. 103 500 training set, 20 700 validation, 41 400 test Network: 2000-100-500-30, training using layered RBM. Notes: Training of the lowest layer (2000 neurons): Values of pixels distorted using Gaussian noise, low learning rate: 0.001, 200 iterations Training all hidden layers: Values of neurons are binary. Training of output layer: Values computed directly using the sigmoid activation functions + noise. That is, values of output neurons are from the interval [0, 1]. 251 Images – fine-tuning Stochastic activation substituted with deterministic. That is the value of hidden neurons is not chosen randomly but directly computed by application of sigmoid on the inner potential (this gives the mean activation). Backpropagation. Error function: cross-entropy − i pi ln ˆpi − i (1 − pi) ln(1 − ˆpi) here pi is the intensity of i-th pixel of the input and ˆpi of the reconstruction. 252 Results 1. Original 2. Reconstruction using deep networks (reduction to 30-dim) 3. Reconstruction using PCA (reduction to 30-dim) 253