Natural Language Processing
with Deep Learning
CS224N/Ling284
Christopher Manning
Lecture 3: Neural net learning: Gradients by hand (matrix calculus)
and algorithmically (the backpropagation algorithm)
1. Introduction
2
Assignment 2 is all about making sure you really understand the math of neural networks
… then we’ll let the software do it!
We’ll go through it all quickly today, but this is the week of quarter to most work through
the readings!
This will be a tough week for some! à
Make sure to get help if you need it
Visit office hours
Read tutorial materials given in the syllabus
Thursday will be mainly linguistics! Some people find that tough too 😉
Named Entity Recognition (NER)
• The task: find and classify names in text, for example:
Last night , Paris Hilton wowed in a sequin gown .
PER PER
Samuel Quinn was arrested in the Hilton Hotel in Paris in April 1989 .
PER PER LOC LOC LOC DATE DATE
• Possible uses:
• Tracking mentions of particular entities in documents
• For question answering, answers are usually named entities
• Often followed by Named Entity Linking/Canonicalization into Knowledge Base
3
Simple NER: Window classification using binary logistic classifier
• Idea: classify each word in its context window of neighboring words
• Train logistic classifier on hand-labeled data to classify center word {yes/no} for each
class based on a concatenation of word vectors in a window
• Really, we usually use multi-class softmax, but trying to keep it simple J
• Example: Classify “Paris” as +/– location in context of sentence with window length 2:
the museums in Paris are amazing to see .
Xwindow = [ xmuseums xin xParis xare xamazing ]T
• Resulting vector xwindow = x ∈ R5d , a column vector!
• To classify all words: run classifier for each class on the vector centered on each word
in the sentence
4
NER: Binary classification for center word being location
• We do supervised training and want high score if it’s a location
𝐽! 𝜃 = 𝜎 𝑠 =
1
1 + 𝑒"#
5
x = [ xmuseums xin xParis xare xamazing ]
predicted model
probability of class
Remember: Stochastic Gradient Descent
Update equation:
i.e., for each parameter: 𝜃$
%&'
= 𝜃$
()*
− 𝛼
+, -
+-!
"#$
In deep learning, we update the data representation (e.g., word vectors) too!
How can we compute ∇- 𝐽(𝜃)?
1. By hand
2. Algorithmically: the backpropagation algorithm
𝛼 = step size or learning rate
6
Lecture Plan
Lecture 4: Gradients by hand and algorithmically
1. Introduction (5 mins)
2. Matrix calculus (40 mins)
3. Backpropagation (35 mins)
7
Computing Gradients by Hand
8
• Matrix calculus: Fully vectorized gradients
• “Multivariable calculus is just like single-variable calculus if you use matrices”
• Much faster and more useful than non-vectorized gradients
• But doing a non-vectorized gradient can be good for intuition; recall the first
lecture for an example
• Lecture notes and matrix calculus notes cover this material in more detail
• You might also review Math 51, which has a new online textbook:
http://web.stanford.edu/class/math51/textbook.html
or maybe you’re luckier if you did Engr 108
Gradients
9
• Given a function with 1 output and 1 input
𝑓 𝑥 = 𝑥.
• It’s gradient (slope) is its derivative
*/
*0
= 3𝑥1
“How much will the output change if we change the input a bit?”
At x = 1 it changes about 3 times as much: 1.013 = 1.03
At x = 4 it changes about 48 times as much: 4.013 = 64.48
Gradients
• Given a function with 1 output and n inputs
• Its gradient is a vector of partial derivatives with
respect to each input
10
Jacobian Matrix: Generalization of the Gradient
• Given a function with m outputs and n inputs
• It’s Jacobian is an m x n matrix of partial derivatives
11
Chain Rule
• For composition of one-variable functions: multiply derivatives
• For multiple variables at once: multiply Jacobians
12
Example Jacobian: Elementwise activation Function
13
Example Jacobian: Elementwise activation Function
Function has n outputs and n inputs → n by n Jacobian
14
Example Jacobian: Elementwise activation Function
15
Example Jacobian: Elementwise activation Function
16
Example Jacobian: Elementwise activation Function
17
Other Jacobians
• Compute these at home for practice!
• Check your answers with the lecture notes
18
Other Jacobians
• Compute these at home for practice!
• Check your answers with the lecture notes
19
Other Jacobians
• Compute these at home for practice!
• Check your answers with the lecture notes
20
Fine print: This is the correct Jacobian.
Later we discuss the “shape convention”;
using it the answer would be h.
Other Jacobians
• Compute these at home for practice!
• Check your answers with the lecture notes
21
Back to our Neural Net!
x = [ xmuseums xin xParis xare xamazing ]
22
Back to our Neural Net!
• Let’s find
• Really, we care about the gradient of the loss Jt but we
will compute the gradient of the score for simplicity
23
x = [ xmuseums xin xParis xare xamazing ]
1. Break up equations into simple pieces
24
Carefully define your variables and keep track of their dimensionality!
2. Apply the chain rule
25
2. Apply the chain rule
26
2. Apply the chain rule
27
2. Apply the chain rule
28
3. Write out the Jacobians
Useful Jacobians from previous slide
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3. Write out the Jacobians
30
𝒖!
Useful Jacobians from previous slide
3. Write out the Jacobians
31
𝒖!
Useful Jacobians from previous slide
3. Write out the Jacobians
32
𝒖!
Useful Jacobians from previous slide
3. Write out the Jacobians
33
𝒖!
𝒖!
Useful Jacobians from previous slide
Re-using Computation
• Suppose we now want to compute
• Using the chain rule again:
34
Re-using Computation
• Suppose we now want to compute
• Using the chain rule again:
The same! Let’s avoid duplicated computation …
35
Re-using Computation
• Suppose we now want to compute
• Using the chain rule again:
36
𝛿 is the local error signal
𝒖!
Derivative with respect to Matrix: Output shape
• What does look like?
• 1 output, nm inputs: 1 by nm Jacobian?
• Inconvenient to then do
37
Derivative with respect to Matrix: Output shape
• What does look like?
• 1 output, nm inputs: 1 by nm Jacobian?
• Inconvenient to then do
• Instead, we leave pure math and use the shape convention:
the shape of the gradient is the shape of the parameters!
• So is n by m:
38
Derivative with respect to Matrix
• What is
• is going to be in our answer
• The other term should be because
• Answer is:
39
𝛿 is local error signal at 𝑧
𝑥 is local input signal
Deriving local input gradient in backprop
• For
"𝒛
"𝑾
in our equation:
• Let’s consider the derivative of a single weight Wij
• Wij only contributes to zi
• For example: W23 is only
used to compute z2 not z1
40
x1 x2 x3 +1
f(z1)= h1 h2 =f(z2)
s u2
W23
b2
𝜕𝑠
𝜕𝑾
= 𝜹
𝜕𝒛
𝜕𝑾
= 𝜹
𝜕
𝜕𝑾
(𝑾𝒙 + 𝒃)
𝜕𝑧2
𝜕𝑊2$
=
𝜕
𝜕𝑊2$
𝑾23 𝒙 + 𝑏2
=
+
+4%!
∑567
*
𝑊25 𝑥5 = 𝑥$
Why the Transposes?
41
• Hacky answer: this makes the dimensions work out!
• Useful trick for checking your work!
• Full explanation in the lecture notes
• Each input goes to each output – you want to get outer product
What shape should derivatives be?
• Similarly, is a row vector
• But shape convention says our gradient should be a column vector because b is
a column vector …
• Disagreement between Jacobian form (which makes the chain rule
easy) and the shape convention (which makes implementing SGD easy)
• We expect answers in the assignment to follow the shape convention
• But Jacobian form is useful for computing the answers
42
What shape should derivatives be?
Two options:
1. Use Jacobian form as much as possible, reshape to
follow the shape convention at the end:
• What we just did. But at the end transpose to make the
derivative a column vector, resulting in
2. Always follow the shape convention
• Look at dimensions to figure out when to transpose and/or
reorder terms
• The error message 𝜹 that arrives at a hidden layer has the
same dimensionality as that hidden layer
43
3. Backpropagation
We’ve almost shown you backpropagation
It’s taking derivatives and using the (generalized, multivariate, or matrix)
chain rule
Other trick:
We re-use derivatives computed for higher layers in computing
derivatives for lower layers to minimize computation
44
Computation Graphs and Backpropagation
Ÿ + Ÿ
• Software represents our neural
net equations as a graph
• Source nodes: inputs
• Interior nodes: operations
45
Computation Graphs and Backpropagation
Ÿ + Ÿ
• Software represents our neural
net equations as a graph
• Source nodes: inputs
• Interior nodes: operations
• Edges pass along result of the
operation
46
Computation Graphs and Backpropagation
Ÿ + Ÿ
• Software represents our neural
net equations as a graph
• Source nodes: inputs
• Interior nodes: operations
• Edges pass along result of the
operation
“Forward Propagation”
47
Backpropagation
Ÿ + Ÿ
• Then go backwards along edges
• Pass along gradients
48
Backpropagation: Single Node
• Node receives an “upstream gradient”
• Goal is to pass on the correct
“downstream gradient”
Upstream
gradient49
Downstream
gradient
Backpropagation: Single Node
Downstream
gradient
Upstream
gradient
• Each node has a local gradient
• The gradient of its output with
respect to its input
Local
gradient50
Backpropagation: Single Node
Downstream
gradient
Upstream
gradient
• Each node has a local gradient
• The gradient of its output with
respect to its input
Local
gradient51
Chain
rule!
Backpropagation: Single Node
Downstream
gradient
Upstream
gradient
• Each node has a local gradient
• The gradient of its output with
respect to its input
Local
gradient
• [downstream gradient] = [upstream gradient] x [local gradient]
52
Backpropagation: Single Node
*
• What about nodes with multiple inputs?
53
Backpropagation: Single Node
Downstream
gradients
Upstream
gradient
Local
gradients
*
• Multiple inputs → multiple local gradients
54
An Example
55
An Example
+
*
max
56
Forward prop steps
An Example
+
*
max
57
Forward prop steps
6
3
2
1
2
2
0
An Example
+
*
max
58
Forward prop steps
6
3
2
1
2
2
0
Local gradients
An Example
+
*
max
59
Forward prop steps
6
3
2
1
2
2
0
Local gradients
An Example
+
*
max
60
Forward prop steps
6
3
2
1
2
2
0
Local gradients
An Example
+
*
max
61
Forward prop steps
6
3
2
1
2
2
0
Local gradients
An Example
+
*
max
62
Forward prop steps
6
3
2
1
2
2
0
Local gradients
upstream * local = downstream
1
1*3 = 3
1*2 = 2
An Example
+
*
max
63
Forward prop steps
6
3
2
1
2
2
0
Local gradients
upstream * local = downstream
1
3
2
3*1 = 3
3*0 = 0
An Example
+
*
max
64
Forward prop steps
6
3
2
1
2
2
0
Local gradients
upstream * local = downstream
1
3
2
3
0
2*1 = 2
2*1 = 2
An Example
+
*
max
65
Forward prop steps
6
3
2
1
2
2
0
Local gradients
1
3
2
3
0
2
2
Gradients sum at outward branches
66
+
Gradients sum at outward branches
67
+
Node Intuitions
+
*
max
68
6
3
2
1
2
2
0
1
2
2
2
• + “distributes” the upstream gradient to each summand
Node Intuitions
+
*
max
69
6
3
2
1
2
2
0
1
3
3
0
• + “distributes” the upstream gradient to each summand
• max “routes” the upstream gradient
Node Intuitions
+
*
max
70
6
3
2
1
2
2
0
1
3
2
• + “distributes” the upstream gradient
• max “routes” the upstream gradient
• * “switches” the upstream gradient
Efficiency: compute all gradients at once
* + Ÿ
• Incorrect way of doing backprop:
• First compute
71
Efficiency: compute all gradients at once
* + Ÿ
• Incorrect way of doing backprop:
• First compute
• Then independently compute
• Duplicated computation!
72
Efficiency: compute all gradients at once
* + Ÿ
• Correct way:
• Compute all the gradients at once
• Analogous to using 𝜹 when we
computed gradients by hand
73
1. Fprop: visit nodes in topological sort order
- Compute value of node given predecessors
2. Bprop:
- initialize output gradient = 1
- visit nodes in reverse order:
Compute gradient wrt each node using
gradient wrt successors
Done correctly, big O() complexity of fprop and
bprop is the same
In general, our nets have regular layer-structure
and so we can use matrices and Jacobians…
Back-Prop in General Computation Graph
…
…
…
= successors of
Single scalar output
74
Automatic Differentiation
• The gradient computation can be
automatically inferred from the symbolic
expression of the fprop
• Each node type needs to know how to
compute its output and how to compute
the gradient wrt its inputs given the
gradient wrt its output
• Modern DL frameworks (Tensorflow,
PyTorch, etc.) do backpropagation for
you but mainly leave layer/node writer
to hand-calculate the local derivative
75
Backprop Implementations
76
Implementation: forward/backward API
77
Implementation: forward/backward API
78
Manual Gradient checking: Numeric Gradient
• For small h (≈ 1e-4),
• Easy to implement correctly
• But approximate and very slow:
• You have to recompute f for every parameter of our model
• Useful for checking your implementation
• In the old days, we hand-wrote everything, doing this everywhere was the key test
• Now much less needed; you can use it to check layers are correctly implemented
79
Summary
80
We’ve mastered the core technology of neural nets! 🎉
• Backpropagation: recursively (and hence efficiently) apply the chain rule
along computation graph
• [downstream gradient] = [upstream gradient] x [local gradient]
• Forward pass: compute results of operations and save intermediate
values
• Backward pass: apply chain rule to compute gradients
Why learn all these details about gradients?
81
• Modern deep learning frameworks compute gradients for you!
• Come to the PyTorch introduction this Friday!
• But why take a class on compilers or systems when they are implemented for you?
• Understanding what is going on under the hood is useful!
• Backpropagation doesn’t always work perfectly
• Understanding why is crucial for debugging and improving models
• See Karpathy article (in syllabus):
• https://medium.com/@karpathy/yes-you-should-understand-backprop-e2f06eab496b
• Example in future lecture: exploding and vanishing gradients