Chapter 6: Multilayer Neural Networks (Sections 6.1-6.3)

• Introduction

• Feedforward Operation and Classification

• Algorithm Pattern Recognition

Two main challenges

• Representation

• Matching

Jain CSE 802, Spring 2013 Representation and Matching How Good is Face Representation?

Courtesy: Pete Langenfeld, MSP

Driver License Information

Gallery: 34 million (30M DMV photos, 4M mugshots) 2009 driver license photo How Good is Face Representation? Smile makes a difference!

Top-10 retrievals

1 2 3 4 5

6 7 8 9 10 Gallery: 34 million (30M DMV photos, 4M mugshots)

Courtesy: Pete Langenfeld, MSP State of the Art in FR: Verification FRGC v2.0 (2006) MBGC (2010) LFW (2007) IJB-A (2015)

LFW Standard Protocol 99.77% (Accuracy) 3,000 genuine & 3,000 imposter pairs; 10-fold CV

LFW BLUFR Protocol 88% TAR @ 0.1% FAR 156,915 genuine, ~46M imposter pairs; 10-fold CV

D. Wang, C. Otto and A. K. Jain, "Face Search at Scale: 80 Million Gallery", arXiv, July 28, 2015 Neural Networks n Massive parallelism is essential for complex recognition tasks (speech & image recognition)

n Humans take only ~200ms for most cognitive tasks; this suggests parallel computation in human brain n Biological networks achieve excellent recognition performance via dense interconnection of simple computational elements (neurons) 10 12 n Number of neurons » 10 – 10 3 4 n Number of interconnections/neuron » 10 – 10 14 n Total number of interconnections » 10

n Damage to a few neurons or synapse (links) does not appear to impair performance (robustness) Neuron

n Nodes are nonlinear, typically analog

x 1 w1 x2 Y (output) xd wd where is internal threshold or offset Neural Networks n Feed-forward networks with one or more layers (hidden) between input & output nodes n How many nodes & hidden layers?

...... c outputs d inputs First hidden layer Second hidden layer

NH1 input units NH2 input units n Network training? Form of the Discriminant Function

• Linear: Hyperplane decision boundaries

• Non-Linear: Arbitrary decision boundaries

• Adopt a model and then use the resulting decision boundary

• Specify the desired decision boundary Linear Discriminant Function • For a 2-class problem, discriminant function that is a linear combination of input features can be written as

Sign of the function value gives Bias or Threshold the class Weight weight label vector Quadratic Discriminant Function

• Quadratic Discriminant Function • Obtained by adding pair-wise products of features

Linear Part Quadratic part, d(d+1)/2 (d+1) parameters additional parameters • g(x) positive implies class 1; g(x) negative implies class 2

• g(x) = 0 represents a hyperquadric, as opposed to hyperplanes in linear discriminant case

• Adding more terms such as wijkxixjxk results in polynomial discriminant functions Generalized Discriminant Function • A generalized linear discriminant function, where y= f(x) can be written as

Dimensionality of the Setting yi(x) to be augmented feature monomials results in space. polynomial discriminant functions

Weights in the augmented feature space. Note that the function is linear in a. • Equivalently, a [a , a ,..., a ]t y = [y (x), y (x),..., y (x)]t = 1 2 dˆ 1 2 dˆ

also called the augmented feature vector. 13 • Perceptron is a linear classifier; it makes predictions based on a linear predictor function combining a set of weights with feature vector

• The perceptron algorithm was invented by Rosenblatt in the late 1950s; its first implementation, in custom hardware, was one of the first artificial neural networks to be produced

• The algorithm allows for online learning; it processes training samples one at a time Two-category Linearly Separable Case

• Let y1,y2,…,yn be n training samples in augmented feature space, which are linearly separable • We need to find a weight vector a such that • aty > 0 for examples from the positive class • aty < 0 for examples from the negative class • “Normalizing” the input examples by multiplying them with their class label (replace all samples from class 2 by their negatives), find a weight vector a such that • aty > 0 for all the examples (here y is multiplied with class label) • The resulting weight vector is called a separating vector or a solution vector The Perceptron Criterion Function

• Goal: Find weight vector a such that aty > 0 for all the samples (assuming it exists) • Mathematically, this can be expressed as finding a weight vector a that minimizes the no. of samples misclassified • Function is piecewise constant (discontinuous, and hence non- differentiable) and is difficult to optimize • Perceptron Criterion Function:

Find a that minimizes this criterion The criterion is proportional to the sum of distances from the misclassified samples to the decision boundary Now, the minimization is mathematically tractable, and hence it is a better criterion fn. than no. of misclassifications. Fixed-increment Single Sample Perceptron • Also called perceptron learning in an online setting • For large datasets, this is more efficient compared to batch mode

n = no. of training samples; a = weight vector; k = iteration # Chapter 5, page 230 Perceptron Convergence Theorem

If training samples are linearly separable, then the sequence of weight vectors given by Fixed- increment single-sample Perceptron will terminate at a solution vector

What happens if the patterns are non-linearly separable? 18 Multilayer Perceptron

Can we learn the nonlinearity at the same time as the linear discriminant? This is the goal of multilayer neural networks or multilayer

Pattern Classification, Chapter 6 19

Pattern Classification, Chapter 6 20 Feedforward Operation and Classification

• A three-layer neural network consists of an input layer, a hidden layer and an output layer interconnected by modifiable (learned) weights represented by links between layers • Multilayer neural network implements linear discriminants, but in a space where the inputs have been mapped nonlinearly • Figure 6.1 shows a simple three-layer network

Pattern Classification, Chapter 6 21

NNo training here

No training involved here, since we are implementing a known input/output mapping 22

Pattern Classification, Chapter 6 23

• A single “bias unit” is connected to each unit in addition to the input units d d t • Net activation: net j = å xi w ji + w j0 = å xi w ji º w j .x, i=1 i=0

where the subscript i indexes units in the input layer, j in the hidden layer; wji denotes the input-to-hidden layer weights at the hidden unit j. (In neurobiology, such weights or connections are called “synapses”)

• Each hidden unit emits an output that is a nonlinear function of its activation, that is: yj = f(netj)

Pattern Classification, Chapter 6 24 Figure 6.1 shows a simple threshold function ì1 if net ³ 0 f ( net ) = sgn( net ) º í î- 1 if net < 0 • The function f(.) is also called the activation function or “nonlinearity” of a unit. There are more general activation functions with desirables properties

• Each output unit similarly computes its net activation based on the hidden unit signals as:

nH nH t netk = å y j wkj + wk0 = å y j wkj = wk .y, j=1 j=0 where the subscript k indexes units in the ouput layer and nH denotes the number of hidden units

Pattern Classification, Chapter 6 25

• The output units are referred as zk. An output unit computes the nonlinear function of its net input, emitting

zk = f(netk)

• In the case of c outputs (classes), we can view the network as computing c discriminant functions

zk = gk(x); the input x is classified according to the largest discriminant function gk(x) " k = 1, …,c

• The three-layer network with the weights listed in fig. 6.1 solves the XOR problem

Pattern Classification, Chapter 6 26

• The hidden unit y1 computes the boundary:

³ 0 Þ y1 = +1

x1 + x2 + 0.5 = 0

< 0 Þ y1 = -1

• The hidden unit y2 computes the boundary:

£ 0 Þ y2 = +1

x1 + x2 -1.5 = 0

< 0 Þ y2 = -1

• Output unit emits z1 = +1 if and only if y1 = +1 and y2 = +1 Using the terminology of computer logic, the units are behaving like gates, where the first hidden unit is an OR gate, the second hidden unit is an AND gate, and the output unit implements

zk = y1 AND NOT y2 = (x1 OR x2) and NOT(x1 AND x2) =

x1 XOR x2

which provides the nonlinear decision of fig. 6.1Pattern Classification, Chapter 6 27 • General Feedforward Operation – case of c output units

æ nH æ d ö ö g ( x ) º z = f ç w f ç w x + w ÷ + w ÷ (1) k k çå kj çå ji i j0 ÷ k0 ÷ è j=1 è i=1 ø ø (k = 1,...,c) • Hidden units enable us to express more complicated nonlinear functions and extend classification capability • Activation function does not have to be a sign function, it is often required to be continuous and differentiable • We can allow the activation in the output layer to be different from the activation function in the hidden layer or have different activation for each individual unit • Assume for now that all activation functions are identical

Pattern Classification, Chapter 6 28 • Expressive Power of multi-layer Networks

Question: Can every decision boundary be implemented by a three-layer network?

Answer: Yes (due to A. Kolmogorov) “Any continuous function from input to output can be implemented in a three-layer net, given sufficient number of hidden units nH, proper nonlinearities, and weights.” Any continuous function g(x) defined on the unit cube can be represented in the following form 2n+1 n g( x ) = åd j (Sbij ( xi )) "x Î I ( I = [0,1];n ³ 2 ) j=1 for properly chosen functions dj and bij

Pattern Classification, Chapter 6 29

Pattern Classification, Chapter 6 30 • Network has two modes of operation:

• Learning The consists of presenting an input pattern and modifying the network parameters (weights) to bring the actual outputs closer to the desired target values • Feedforward The feedforward operations consists of presenting a pattern to the input units and passing (or feeding) the signals through the network in order to yield a decision from the outputs units

Pattern Classification, Chapter 6 Network Learning: 31 Backpropagation Algorithm • Goal is to learn the interconnection weights based on the training patterns and the desired outputs

• In a three-layer network, it is a straightforward matter to understand how the output, and thus the error, depends on the hidden-to-output layer weights

• The power of backpropagation is that it enables us to compute an effective error for each hidden unit, and thus derive a learning rule for the input-to-hidden weights. This is known as: The credit assignment problem

Pattern Classification, Chapter 6 32

Pattern Classification, Chapter 6 33 Network Learning • Start with an untrained network, present a training pattern to the input layer, pass the signal through the network and determine the output. • Let tk be the k-th target (or desired) output and zk be the k-th computed output with k = 1, …, c. Let w represent all the weights of the network

c 1 1 2 The training error: 2 • J( w ) = å( tk - zk ) = t - z 2 k=1 2 • The backpropagation learning rule is based on • The weights are initialized with random values and are changed in a direction that will reduce the error: ¶J Dw = -h ¶w Pattern Classification, Chapter 6 34 where h is the learning rate which indicates the relative size of the change in weights w(m +1) = w(m) + Dw(m) where m is the m-th training pattern presented

• Error on the hidden–to-output weights

¶J ¶J ¶netk ¶netk = . = -d k ¶wkj ¶netk ¶wkj ¶wkj where the sensitivity of unit k is defined as: ¶J d k = - ¶netk and describes how the overall error changes with the activation of the unit’s net activation

¶J ¶J ¶zk d k = - = - . = ( tk - zk ) f' ( netk ) ¶netk ¶zk ¶netk Pattern Classification, Chapter 6 35

t Since netk = wk .y, therefore: ¶netk = y j ¶wkj

Conclusion: the weight update (or learning rule) for the hidden-to-output weights is:

Dwkj = hdkyj = h(tk – zk) f’ (netk)yj

• Learning rule for the input-to-hiden units is more subtle and is the crux of the credit assignment problem • Error on the input-to-hidden units: Using the chain rule ¶J ¶J ¶y ¶net = . j . j ¶w ji ¶y j ¶net j ¶w ji Pattern Classification, Chapter 6 c c ¶J ¶ é 1 2 ù ¶z 36 = ( t - z ) = - ( t - z ) k However, ê å k k ú å k k ¶y j ¶y j ë2 k=1 û k=1 ¶y j c c ¶zk ¶netk = -å( tk - zk ) . = -å( tk - zk ) f' ( netk )wkj k=1 ¶netk ¶y j k=1 Similarly as in the preceding case, we define the sensitivity of a hidden unit: c d j º f' ( net j )å wkjd k k=1 Above equation is the core of the “credit assignment” problem: “The sensitivity at a hidden unit is simply the sum of the individual sensitivities at the output units weighted by the hidden-to-output weights wkj, all multiplied by f’(netj)”; see fig 6.5 Conclusion: Learning rule for the input-to-hidden weights:

Dw = hx d = h[Sw d ] f' ( net )x ji i j $!kj!!k #!!!"j i Pattern Classification, Chapter 6 d j Sensitivity at Hidden Node 38 Backpropagation Algorithm • More specifically, the “backpropagation of errors” algorithm • During training, an error must be propagated from the output layer back to the hidden layer to learn the input-to-hidden weights • It is gradient descent in a layered network • Exact behavior of the learning algorithm depends on the starting point • Start the process with random values of weights; in practice you learn many networks with different initializations

Pattern Classification, Chapter 6 • Training protocols: 39 • Stochastic: patterns are chosen randomly from training set; network weights are updated for each pattern • Batch: Present all patterns before updating weights • On-line: present each pattern once & only once (no memory for storing patterns) • Stochastic backpropagation algorithm:

Begin initialize nH; w, criterion q, h, m ¬ 0 do m ¬ m + 1 xm ¬ randomly chosen pattern

wji ¬ wji + hdjxi; wkj ¬ wkj + hdkyj until ||ÑJ(w)|| < q return w

End Pattern Classification, Chapter 6 40 • Stopping criterion

• The algorithm terminates when the change in the criterion function J(w) is smaller than some preset value q; other stopping criteria that lead to better performance than this one

• A weight update may reduce the error on the single pattern being presented but can increase the error on the full training set • In stochastic backpropgation and batch propagation, we must make several passes through the training data

Pattern Classification, Chapter 6 41 • Learning Curves • Before training starts, the error on the training set is high; as the learning proceeds, error becomes smaller • Error per pattern depends on the amount of training data and the expressive power (such as the number of weights) in the network • Average error on an independent test set is always higher than on the training set, and it can decrease as well as increase • A validation set is used in order to decide when to stop training ; we do not want to overfit the network and decrease the power of the classifier’s generalization “Stop training when the error on the validation set is minimum”

Pattern Classification, Chapter 6 42

Pattern Classification, Chapter 6 Representation at the Hidden Layer

• What do the learned weights mean? • The weights connecting hidden layer to output layer form a linear discriminant • The weights connecting input layer to hidden layer represent a mapping from the input feature space to a latent feature space • For each hidden unit, the weights from input layer describe the input pattern that leads to the maximum activation of that node Backpropagation as Feature Mapping Input layer to hidden layer weights for a character recognition task Weights at two hidden nodes represented as 8x8 patterns

Left gets activated for F, right gets activated for L, and both get activated for E • 64-2-3 sigmoidal network for classifying three characters (E,F,L) • Non-linear interactions between the features may cause the features of the pattern to not manifest in a single hidden node (in contrary to the example shown above) • It may be difficult to draw similar interpretations in large networks and caution must be exercised while analyzing weights Practical Techniques for Improving Backpropagation • A naïve application of backpropagation procedures may lead to slow convergence and poor performance • Some practical suggestions; no theoretical results • Activation Function f(.) • Must be non-linear (otherwise, 3-layer network is just a linear discriminant) and saturate (have max and min value) to keep weights and activation functions bounded • Activation function and its derivative must be continuous and smooth; optionally monotonic • Choice may depend on the problem. Eg. Gaussian activation if the data comes from a mixture of Gaussians • Eg: sigmoid (most popular), polynomial, tanh, sign function • Parameters of activation function (e.g. Sigmoid) • Centered at 0, odd function f(-net) = -f(net) (anti-symmetric); leads to faster learning • Choice depends on the range of the input values Activation Function

First & second derivative

The anti-symmetric sigmoid function: f(-x) = -f(x). a = 1.716, b = 2/3. Practical Considerations • Scaling inputs (important not just for neural networks) • Large differences in scale of different features due to the choice of units is compensated by normalizing them to be in the same range, [0,1] or [-1,1]; without normalization, error will hardly depend on feature with very small values • Standardization: Shift the inputs to have zero mean and unit variance • Target Values • One-of-C representation for the target vector (C is no. of classes). Better to use +1 and –1 that lie well within the range of sigmoid function saturation values (+1.716, -1.716) • Higher values (e.g. 1.716, saturation point of a sigmoid) may require the weights to go to infinity to minimize the error • Training with Noise • For small training sets, it is better to add noise to the input patterns and generate new “virtual” training patterns Practical Considerations

• Number of Hidden units (nH) • Governs the expressive power of the network • The easier the task, the fewer the nodes needed • Rule of thumb: total no. of weights must be less than the number of training examples (preferably 10 times less); no. of hidden units determines the total no. of weights • A more principled method is to adjust the network complexity in response to training data; e.g. start with a “large” no. of hidden units and “decay”, prune, or eliminate weights • Initializing weights • We cannot initialize weights to zero, otherwise learning cannot take place • Choose initial weights w such that |w| < w’ • w’ too small – slow learning; too large – early saturation and no learning

• w’ is chosen to be 1/Öd for input layer, and 1/ ÖnH for hidden layer Total no. of Weights

Error per pattern with the increase in number of hidden nodes.

• 2-nH-1 network (with bias) trained on 90 2D-Gaussian patterns (n = 180) from each class (sampled from mixture of 3 Gaussians) • Minimum test error occurs at 17-21 weights in total (4-5 hidden nodes). This illustrates the rule of thumb that n/10 weights often gives lowest error Practical Considerations

• Learning Rate • Small learning rate: slow convergence • Large learning rate: high oscillation and slow convergence Practical Considerations

• Momentum • Prevents the algorithm from getting stuck at plateaus and local minima • Weight decay • Avoid overfitting by imposing the condition that weights must be small • After each update, weights are decayed by some factor • Related to regularization (also used in SVM) • Hints • Additional input nodes added to NN that are only used during training. Help learn better feature representation. 52 Practical Considerations

• Training setup • Online, stochastic, batch-mode • Stop training • Halt when validation error reaches (first) minimum • Number of hidden layers • More layers -> more complex • Networks with more hidden layers are more prone to get caught in local minima • Smaller the better (KISS) • Criterion function • We talked about squared error, but there are others Pattern Classification, Chapter 6