Neural Networks and Kernel Methods

How are we doing on the pass sequence? Generally, this will take a lot longer than 24 hours… We need to avoid doing this by hand! We can now track both men, provided with Hand-labeled coordinates of both men in 30 frames Hand-extracted features (stripe detector, white blob detector) Hand-labeled classes for the white-shirt tracker We have a framework for how to optimally make decisions and track the men

Recall: Multi-input linear regression y(x,w) = w0 + w1 f1(x) + w2 f2(x) + … + wM fM(x) x can be an entire scan-line or image! We could try to uniformly distribute basis functions in the input space: This is futile, because of the curse of dimensionality x = entire scan line …

Neural networks and kernel methods Two main approaches to avoiding the curse of dimensionality: “Neural networks” Parameterize the basis functions and learn their locations Can be nested to create a hierarchy Regularize the parameters or use Bayesian learning “Kernel methods” The basis functions are associated with data points, limiting complexity A subset of data points may be selected to further limit complexity

Neural networks and kernel methods Two main approaches to avoiding the curse of dimensionality: “Neural networks” Parameterize the basis functions and learn their locations Can be nested to create a hierarchy Regularize the parameters or use Bayesian learning “Kernel methods” The basis functions are associated with data points, limiting complexity A subset of data points may be selected to further limit complexity

Two-layer neural networks Before, we used Replace each fj with a variable zj, where and h() is a fixed activation function The outputs are obtained from where s() is another fixed function In all, we have (simplifying biases):

Typical activation functions h(a) Logistic sigmoid, aka logit: h(a) = s(a) = 1/(1+e-a) Hyperbolic tangent: h(a) = tanh(a) = (ea-e-a)/(ea+e-a) Cumulative Gaussian (error function): h(a) = 2x=-∞a N(x|0,1)dx - 1 This one has a lighter tail Normalized to have same range and slope at a=0 As above, but h is on a log-scale a

Examples of functions learned by a neural network (3 tanh hidden units, one linear output unit)

Multi-layer neural networks From now on, we’ll denote all activation functions by h Only weights corresponding to the feed-forward topology are instantiated The sum is over those values of j with instantiated weights wkj

Learning neural networks As for regression, we consider a squared error cost function: E(w) = ½ Sn Sk ( tnk – yk(xn,w) )2 which corresponds to a Gaussian density p(t|x) We can substitute and use a general purpose optimizer to estimate w, but it is illustrative and useful to study the derivatives of E…

Learning neural networks E(w) = ½ Sn Sk ( tnk – yk(xn,w) )2 Recall that for linear regression: E(w)/wm = -Sn ( tn - yn ) xnm We’ll use the chain rule of differentiation to derive a similar-looking expression, where Local input signals are forward-propagated from the input Local error signals are back-propagated from the output Weight in-between error signal and input signal Error signal Input signal

Local signals needed for learning For clarity, consider the error for one training case: To compute En/wji, note that wji appears in only one term of the overall expression, namely Using the chain rule of differentiation, we have where if wji is in the 1st layer, zi is actually input xi Weight Local error signal Local input signal

Forward-propagating local input signals Forward propagation gives all the a’s and z’s

Back-propagating local error signals Back-propagation gives all the d ’s

Back-propagating error signals To compute En/aj (dj), note that aj appears in all those expressions ak = Si wki h(ai) that depend on aj Using the chain rule, we have The sum is over k s.t. unit j is connected to unit k and for each such term, ak/aj = wkj h’(aj) Noting that En/ak = dk, we get the back-propagation rule: For output units: -

Putting the propagations together For each training case n, apply forward propagation and back-propagation to compute for each weight wji Sum these over training cases to compute Use these derivatives for steepest descent learning or as input to a conjugate gradients optimizer, etc On-line learning: After each pattern presentation, use the above gradient to update the weights

The number of hidden units determines the complexity of the learned function (M = # hidden units)

The effect of local minima Because of random weight initialization, each training run will find a different solution Validation error M

Regularizing neural networks Demonstration of over-fitting (M = # hidden units)

Regularizing neural networks Over-fitting: Use cross-validation to select the network architecture (number of layers, number of units per layer) Add to E a term (l/2)Sjiwji2 that penalizes large weights, so Use cross-validation to select l Use early-stopping and cross-validation (next slide) Take a Bayesian approach: Put a prior on the w’s and integrate over them to make predictions

Early stopping The weights start at small values and grow Perhaps the number of learning iterations is a surrogate for model complexity? This works for some learning tasks Training error Validation error Number of learning iterations

Can we use a standard neural network to automatically learn the features needed for tracking? x = entire scan line x is 320-dimensional, so the number of parameters would be at least 320 We have only 15 data points (setting aside 15 for cross validation) so over-fitting will be an issue We could try weight decay, Bayesian learning, etc, but a little thinking reveals that our approach is wrong… In fact, we want the weights connecting different positions in the scan line to use the same feature (eg, stripes)

Convolutional neural networks Recall that a short portion of the scan line was sufficient for tracking the striped shirt We can use this idea to build a convolutional network With constrained weights, the number of free parameters is now only ~ one dozen, so… We can use Bayesian/regularized learning to automatically learn the features Same set of weights used for all hidden units

Convolutional neural networks in 2-D (from Le Cun et al, 1989)

Neural networks and kernel methods Two main approaches to avoiding the curse of dimensionality: “Neural networks” Parameterize the basis functions and learn their locations Can be nested to create a hierarchy Regularize the parameters or use Bayesian learning “Kernel methods” The basis functions are associated with data points, limiting complexity A subset of data points may be selected to further limit complexity

Kernel methods Basis functions offer a way to enrich the feature space, making simple methods (such as linear regression and linear classifiers) much more powerful Example: Input x; Features x, x2, x3, sin(x), … There are two problems with this approach Computational efficiency: Generally, the appropriate features are not known, so there is a huge (possibly infinite) number of them to search over Regularization: Even if we could search over the huge number of features, how can we select appropriate features so as to prevent overfitting? The kernel framework enables efficient approaches to both problems

Kernel methods x2 f2 x1 f1

Definition of a kernel f(x1)Tf(x2) = k(x1, x2) Suppose f(x) is a mapping from the D-dimensional input vector x to a high (possibly infinite) dimensional feature space Many simple methods rely on inner products of feature vectors, f(x1)Tf(x2) For certain feature spaces, the “kernel trick” can be used to compute f(x1)Tf(x2) using the input vectors directly: f(x1)Tf(x2) = k(x1, x2) k(x1, x2) is referred to as a kernel If a function satisfies “Mercer’s conditions” (see textbook), it can be used as a kernel

Examples of kernels k(x1, x2) = x1T x2 k(x1, x2) = x1T S-1 x2 (S-1 is symmetric positive definite) k(x1, x2) = exp(-||x1-x2||2/2s2) k(x1, x2) = exp(-½ x1T S-1 x2 ) k(x1, x2) = p(x1)p(x2)

Gaussian processes Recall that for linear regression: Using a design matrix F, our prediction vector is Let’s use a simple prior on w: Then K is called the Gram matrix, where Result: The correlation between two predictions equals the kernel evaluated for the corresponding inputs Example

Gaussian processes: “Learning” and prediction As before, we assume The target vector likelihood is Using , we can obtain the marginal predictive distribution over targets: where Predictions are based on where , = is Gaussian with

Example: Samples from

Example: Learning and prediction

Sparse kernel methods and SVMs Idea: Identify a small number of training cases, called support vectors, which are used to make predictions See textbook for details Support vector

Questions?

How are we doing on the pass sequence? We can now automatically learn the features needed to track both people Same set of weights used for all hidden units

How are we doing on the pass sequence? Same set of weights used for all hidden units Pretty good! We can now automatically learn the features needed to track both people But, it sucks that we need to hand-label the coordinates of both men in 30 frames and hand-label the 2 classes for the white-shirt tracker

Lecture 5 Appendix

Constructing kernels Provided with a kernel or a set of kernels, we can construct new kernels using any of the rules: