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Neural Networks and Kernel Methods

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Neural Networks and Kernel Methods

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

y(x,w)= w0+ w1 f1(x)+ w2 f2(x)+ … + wM fM(x)

- xcan 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

…

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 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

- 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):

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) = 2x=-∞a N(x|0,1)dx - 1

- This one has a lighter tail

Normalized to have same range and slope ata=0

As above, but h is on a log-scale

a

- Only weights corresponding to the feed-forward topology are instantiated
- The sum is over those values of j with instantiated weights wkj

From now on, we’ll denote all activation functions by h

- As for regression, we consider a squared error cost function:
E(w)= ½SnSk ( 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…

E(w)= ½SnSk ( 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

Weight

Local error signal

Local input signal

- 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

- Forward propagation gives all the a’s and z’s

- Back-propagation gives all the d ’s

t2

t1

- To compute En/aj (dj), note that aj appears in all those expressions ak = Siwkih(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 = wkjh’(aj)
- Noting that En/ak=dk, we get the back-propagation rule:
- For output units: -

- 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

(M = # hidden units)

- Because of random weight initialization, each training run will find a different solution

Validation error

M

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

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

- 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

- 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)

x = entire scan line

- 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

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

- 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

x2

f2

x1

f1

- 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

- k(x1,x2) = x1T x2
- k(x1,x2) = x1T S-1x2
(S-1is symmetric positive definite)

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

- k(x1,x2) = p(x1)p(x2)

Example

- 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

- 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

- Idea: Identify a small number of training cases, called support vectors, which are used to make predictions
- See textbook for details

Support vector

Questions?

Same set of weights used for all hidden units

- We can now automatically learn the features needed to track both people

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

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