Learning Deep Architectures for AI. Part II - Vijay Chakilam

Transcription

1 Learning Deep Architectures for AI - Yoshua Bengio Part II - Vijay Chakilam

2 Limitations of Perceptron x1 W, b 0,1 1,1 y x2 weight plane output =1 output =0 There is no value for W and b such that the model results in right target for every example 0,0 1,0 A graphical view of the XOR problem

3 Learning representations x1 W, b φ u1 h1 V, c 2,1 y x2 u2 h2 W = 11 11, b = 0 1, V = 1 2 and c = 0 0,0 1,0

4 y j j z j Backpropagation E = 1 2 (t j y j ) 2 j output E y j = (t j y j ) y i i E z j = dy j dz j E y j = y j (1 y j ) E y j E w ij = z j w ij E z j = y i E z j E y i = dz j E dy i z = w E ij j z j j j

5 Deep vs Shallow Pascanu et al. (2013) compared deep rectifier networks with their shallow counterparts. For a deep model with n 0 inputs and k hidden layers of width n, the maximal number of response regions per parameter behaves as Ω n n 0 k 1 n n 0 2 For a shallow model with inputs and nk hidden units, the maximal number of response regions per parameter behaves as n 0 k O( k n0 1 n n 0 1 )

6 Deep vs Shallow Pascanu et al. (2013) showed that the deep model can generate exponentially more regions per parameter in terms of the number of hidden layers, and at least order polynomially more regions per parameter in terms of layer width n. Montufar et al. (2014) came up with significantly improved lower bound on the maximal linear regions.

7 Unsupervised Pre-training Scarcity of labeled examples and availability of many unlabeled examples. Unknown future tasks. Once a good high-level representation is learned, other learning tasks could be much easier. For example, kernel machines can be very powerful if using an appropriate kernel, i.e. an appropriate feature space.

8 W 1 DNN with unsupervised pre-training using RBMs W 2 X Z1 Z2 W 3 Z3 Z1 Z2 W 1 W 2 W 3 W 4 X Y Z1 Z2 Z3

9 W W T 1 1 ^X DNN with unsupervised pre-training using Autoencoders X Z1 W W T 2 2 Z2 Z3 ^Z 2 W W T 3 3 Z1 Z2 ^Z 1 W 1 W 2 W 3 W 4 X Y Z1 Z2 Z3

10 Restricted Boltzmann Machines

11 Restricted Boltzmann Machine Energy function: E(x, h) = h T Wx c T x b T h = W j,k h j x k c k x k b j h j j k k j Distribution: p(x, h) = exp( E(x, h)) / Z W X h

12 x h - E e -E P(x, h ) P(x) An example of how weights define a distribution h1 h x1-1 x2

13 Log likelihood To train an RBM, we d like to minimize the average negative log-likelihood 1 T t l( f (x (t) )) = 1 T t log p(x (t) ) log p(x (t) ) θ E(x (t), h) = E h θ x (t) E(x, h) E x,h θ

14 Contrastive Divergence Idea: Replace the expectation by a point estimate at x Obtain the point x by Gibbs sampling Start sampling chain at x(t)... ~ p(h x) ~ p(x h) x (t) x 1 x k = x ~

15 Conditional probabilities p(h x) = p(h j x) p(h j =1 x) = j 1 1+ exp( (b j +W j. x)) = sigm(b j +W j. x) p(x h) = p(x k h) k p(x k =1 h) = 1 1+ exp( (c k + h T W.k )) = sigm(c k + h T W.k )

16 Conditional probabilities

17 Contrastive Divergence Parameter Updates Derivation of E(x, h) θ for θ = W jk E(x, h) = W jk h j x k c k x k W jk W b j h j jk jk k j = W jk jk W jk h j x k = h j x k W E(x, h) = hx T

18 Contrastive Divergence Parameter Updates Derivation of E(x, h) Ε h θ x for θ = W jk E(x, h) Ε h W jk x = Ε h h x x j k = h j x k p(h j x) h j {0,1} = x k p(h j =1 x) Ε h [ W E(x, h) x] = h(x)x T

19 Contrastive Divergence Parameter Updates x (t) x ~ θ = W jk Given and the learning rule for becomes W W α ( W log p(x (t) )) ( [ ]) W α Ε h W E(x (t), h) x (t) Ε x,h W E(x, h) W α Ε h W E(x (t), h) x (t) Ε h W E(x ~, h) x ~ W +α h(x (t) )x (t)t h(x ~ ) x ~ T

20 Contrastive Divergence - Algorithm For each training example Generate a negative sample sampling, starting at Update parameters x (t) x (t) x ~ W W +α h(x (t) )x (t)t h(x ~ ) x ~ T b b +α h(x (t) ) h(x ~ ) c c +α x (t) x ~ Repeat until stopping criterion using k steps of Gibbs

21 Autoencoders

22 3 Autoencoder x h(x) c k b j W * = W T (tied weights) Decoder x = o(a(x)) = sigm(c + W * h(x)) for binary inputs W h(x) Encoder = g(a(x)) = sigm(b + Wx)

23 Autoencoder: Loss Function Loss function: l( f (x)) = log p(x µ) Use as the loss function Example: We get the squared error loss l( f (x)) = 1 ( ˆx k x k ) 2 2 By choosing a Gaussian distribution with mean µ and identity covariance 1 p(x µ) = (2π ) exp( 1 (x D/2 k µ k ) 2 ) 2 k and choosing µ = c +W * h(x) For binary inputs: k k l( f (x)) = (x k log( ˆx k )+ (1 x k )log(1 ˆx k ))

24 Autoencoder: Gradient For both binary and real valued inputs, the gradient has a very simple form â(x (t ) ) l( f (x(t) )) = ˆx (t) x (t) Parameter gradients are obtained by backpropagating the gradient like in a regular network

25 3 Denoising Autoencoder Representation should be robust to introduction of noise Random assignment of subset of inputs to 0, with probability v Gaussian additive noise Reconstruction computed from the corrupted input Loss function compares reconstruction with original input h(x~) X^ X~ x c k b j W * = W T (tied weights) W noise process p(x~ x)

26 Dataset MNIST ("Modified National Institute of Standards and Technology") is the de facto hello world dataset of computer vision. Contains gray-scale images of hand-drawn digits, from zero through nine. Each image is 28 pixels in height and 28 pixels in width, for a total of 784 pixels in total. Each pixel has a single pixel-value associated with it, indicating the lightness or darkness of that pixel, with higher numbers meaning darker. This pixelvalue is an integer between 0 and 255, inclusive.

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29 Softmax The output units in a softmax group use a non-local non-linearity: y i y i = softmax group e z i j group e z j z i y i = y (1 - y ) z i i i

30 Theano Theano is a Python library that allows you to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently. Tight integration with NumPy Transparent use of a GPU. Perform data-intensive computations much faster than on a CPU. Efficient symbolic differentiation. Theano does your derivatives for functions with one or many inputs. Speed and stability optimizations.

31 Theano Installation and more information: Latest reference: Theano: A Python framework for fast computation of mathematical expressions

32 Theano.Tensor Tensor A Theano module for symbolic NumPy types and operations There are many types of symbolic expressions. Basic Tensor functionality includes creation of tensor variables such as scalar, vector, matrix, etc.

33 Create algebraic expressions on symbolic variables: Theano Functions

34 Create a shared variable so we can do gradient descent Define a cost function that has a minimum value Compute the gradient

35 Get data and initialize network

36 Define Theano variables and expressions

37 Define training expressions and functions

38 Loop through the data in batches

39 DNN without unsupervised pretraining

40 DNN with unsupervised pre-training W 1 using RBMs W 2 X Z1 Z2 W 3 Z3 Z1 Z2 W 1 W 2 W 3 W 4 X Y Z1 Z2 Z3

41 DNN unsupervised pre-training with RBMs

42 W W T 1 1 ^X DNN with unsupervised pre-training using Autoencoders X Z1 W W T 2 2 Z2 Z3 ^Z 2 W W T 3 3 Z1 Z2 ^Z 1 W 1 W 2 W 3 W 4 X Y Z1 Z2 Z3

43 DNN unsupervised pre-training with AutoEncoders

44 Unsupervised Pre-training Layer-wise unsupervised learning: The author believes that greedy layer-wise unsupervised pre-training overcomes the challenges of deep learning by introducing a useful prior to the supervised fine-tuning training procedure. Gradients of a criterion defined at the output layer could become less useful as they are propagated backwards to lower layers. The author states it is reasonable to believe unsupervised learning criterion defined at the level of a single layer could be used to move its parameters in a favorable direction.

45 Unsupervised Pre-training The author claims unsupervised pre-training acts as a regularizer by establishing an initialization point of the fine-tuning procedure inside a region of parameter space in which the parameters are henceforth restricted. Another motivation is that greedy layer-wise unsupervised pre-training could be a way to naturally decompose the problem into subproblems associated with different levels of abstraction and extract salient information about the input distribution and capture in a distributed representation.

46 References Yoshua Bengio, Learning deep architecures for AI: Hugo Larochelle, Neural networks: neural_networks/content.html Geoff Hinton, Neural networks for machine learning: Goodfellow et al., Deep Learning: Montufar et al., On the number of linear regions of Deep Networks:

47 References Pascanu et al., On the number of response regions of deep feed forward neural networks with piece-wise linear activations: Perceptron weight update graphics: print.pdf Proof of Theorem (Block, 1962, and Novikoff, 1962). Python and Theano code reference:

48 References Erhan et al., Why does unsupervised pre-training help deep learning: erhan10a.pdf Theano installation and more information: Theano latest reference: Theano: A Python framework for fast computation of mathematical expressions