CSE 559A: Computer Vision

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Fall 2018: T-R: 11:30-1pm @ Lopata 101

Instructor: Ayan Chakrabarti (ayan@wustl.edu).
Course Staff: Zhihao Xia, Charlie Wu, Han Liu

November 1, 2018

General

• Look at Proposal Feedback
• Important: This Friday, Office Hours will be shorter.
• Only from 10:30AM - 11 AM (Lopata 103)
• Recitation Next Friday
• Colloquium of Potential Interest
• "Visualizing Scalar Data with Computational Topology and Machine Learning" - Josh Levine from UA
• 11 AM - Noon, Friday (Lopata 101)
• CSE 659A: Advances in Computer Vision

Machine Learning

$w = \arg \min_w \frac{1}{T} \sum_t C_t(w)$

$C_t(w) = y_t \log \left[1 + \exp(-w^T\tilde{x}_t)\right] + (1-y_t) \log \left[1 + \exp(w^T\tilde{x}_t)\right]$

• Defined linear classifier on augmented vector $$\tilde{x}$$
• Used gradient descent to learn $$w$$.
• Looked at behavior of gradients.
• Simplified computation with stochasticity.
• At test time, sign of $$w^T\tilde{x}$$ gives us our label.

This is for binary classification. What about the multi-class case ? $$y \in \{1,2,3,\ldots C\}$$

Machine Learning

Multi-Class Classification

• Want to map an input $$x$$ to a class label $$y \in \{1,2,3,\ldots C\}$$
• Binary case: $$f$$ outputs a single number between 0,1 that represents $$P(y=1)$$.
• Multi-class case: $$f$$ outputs a $$C$$ dimensional vector that represents a probability distribution over $$C$$ classes.

$f(x; W) = \text{SoftMax}(W^T\tilde{x}) = [p_1, p_2, p_3,\ldots p_C]^T$

• Here our learnable parameter is now the $$N\times C$$ matrix $$W$$ ($$N$$ is length of feature vector $$\tilde{x}$$).
• $$p_i$$ represents the probability of class $$i$$
• Each $$p_i > 0$$, and $$\sum_i p_i = 1$$
• SoftMax is a generalization of Sigmoid

$[p_1,p_2,\ldots]^T = \text{SoftMax}([l_1,l_2,\ldots]^T) \rightarrow p_i = \frac{\exp(l_i)}{\sum_{i'} \exp(l_{i'})}$

• At Test Time: $$y = \arg \max_i p_i$$
• $$y = \arg \max_i l_i$$

Machine Learning

Multi-Class Classification

$f(x; W) = \text{SoftMax}(W^T\tilde{x}) = [p_1, p_2, p_3,\ldots p_C]^T$ $[p_1,p_2,\ldots]^T = \text{SoftMax}([l_1,l_2,\ldots]^T) \rightarrow p_i = \frac{\exp(l_i)}{\sum_{i'} \exp(l_{i'})}$

Multi-Class Cross Entropy Loss

$L(y, f(x)) = L(y, [p_1,p_2,\ldots]^T) = - \log p_y$

• Another way to write it:
• $$y^1 = [\delta_1, \delta_2, \ldots]$$, where $$\delta_i = 1$$ if $$y=i$$ and $$0$$ otherwise.
• Called a 1-Hot encoding of the class
• $$y^1$$ also represents a "probability distribution", where the right class has probability 1.
• In some cases, if you have uncertainty in your training data, $$y^1$$ could be a distribution too.

$L(y^1=[\delta_1,\delta_2,\ldots], [p_1,p_2,\ldots]^T) = - \sum_i \delta_i \log p_i$

Machine Learning

Multi-Class Classification

$[l_1,l_2,\ldots]^T = W^T\tilde{x}$ $p_i = \frac{\exp(l_i)}{\sum_{i'} \exp(l_{i'})}$ $L([\delta_1,\delta_2,\ldots], [p_1,p_2,\ldots]^T) = - \sum_i \delta_i \log p_i$

• We're going to use gradient descent to learn $$W$$. What is $$\nabla_W L$$ ?
• First, what is $$\frac{\partial L}{\partial l_i}$$ ? Take 5 mins.
• Derivative is $$p_i - \delta_i$$
• This means that you'll get gradients for all classes (not just the true class)
• Negative gradient wants you to increase probability for right class, and decrease for other classes
• What is $$\nabla_W L$$ ? Take a few minutes!

$\nabla_W L = \tilde{x}~~~[p_1-\delta_1, p_2-\delta_2, \ldots]$

This is a matrix multiply or outer-product of an $$N\times 1$$ vector with an $$1 \times C$$ vector.

Machine Learning

• For regression and both binary and multi-class classification:
• Defined linear classifier on augmented vector $$\tilde{x}$$
• Run optimization to learn parameters

The problem is:

• The definition of augmented vector $$\tilde{x}$$ is hand-crafted
• We have manually engineered our features.
• The only thing we're learning is a linear classifier on top.

Want to learn the features themselves !

Given that SGD works, what's stopping us from learning a function $$g$$ such that $$g(x)=\tilde{x}$$ ?

Classification

• Learn $$\tilde{x} = g(x;\theta)$$ and do binary classification on its output.

$w = \arg \min_{w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^T\tilde{x}_t)\right] + (1-y_t) \log \left[1 + \exp(w^T\tilde{x}_t)\right]$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$

• Again, use (stochastic) gradient descent.
• But this time, the cost is no longer convex.

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$w = \arg \min_{w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^T\tilde{x}_t)\right] + (1-y_t) \log \left[1 + \exp(w^T\tilde{x}_t)\right]$ $\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$

• Again, use (stochastic) gradient descent.
• But this time, the cost is no longer convex.
• Turns out .. doesn't matter (sort of).

Recall in the previous case: (where $$C_t$$ is the cost of one sample)

$\nabla_w C_t = \tilde{x}_t~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

Exactly the same, with $$\tilde{x} = g(x;\theta)$$ for the current value of $$\theta$$.

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$

$\nabla_w C_t = \tilde{x}_t~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

What about $$\nabla_\theta C_t$$ ?

First, what is the $$\nabla_{\tilde{x}_t} C_t$$ ?

Take 5 mins

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$

$\nabla_w C_t = \tilde{x}_t~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

What about $$\nabla_\theta C_t$$ ?

First, what is the $$\nabla_{\tilde{x}_t} C_t$$ ?

$\nabla_{\tilde{x}_t} C_t = ~~~~\color{red}{?}~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$

$\nabla_w C_t = \tilde{x}_t~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

What about $$\nabla_\theta C_t$$ ?

First, what is the $$\nabla_{\tilde{x}_t} C_t$$ ?

$\nabla_{\tilde{x}_t} C_t = ~~~~w~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$ $\nabla_{\tilde{x}_t} C_t = ~~~~w~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

• Now, let's say $$\theta$$ was an $$M\times N$$ matrix, and $$g(x;\theta) = \theta x$$.
• $$N$$ is the length of the vector $$x$$
• $$M$$ is the length of the encoded vector $$\tilde{x}$$

What is $$\nabla_{\theta} C_t$$ ?

Take 5 mins!

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$ $\nabla_{\tilde{x}_t} C_t = ~~~~w~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

• Now, let's say $$\theta$$ was an $$M\times N$$ matrix, and $$g(x;\theta) = \theta~x$$.
• $$N$$ is the length of the vector $$x$$
• $$M$$ is the length of the encoded vector $$\tilde{x}$$

What is $$\nabla_{\theta} C_t$$ ?

$\nabla_{\theta} C_t = \nabla_{\tilde{x}_t} C_t~~~~~\color{red}{?}$

Classification

• Learn $$\tilde{x} = g(x;\theta)$$

$\theta,w = \arg \min_{\theta,w} \frac{1}{T} \sum_t y_t \log \left[1 + \exp(-w^Tg(x_t;\theta))\right] + (1-y_t) \log \left[1 + \exp(w^Tg(x_t;\theta))\right]$ $\nabla_{\tilde{x}_t} C_t = ~~~~w~~~~\left[\frac{\exp(w^T\tilde{x}_t)}{1+\exp(w^T\tilde{x}_t)} - y_t\right]$

• Now, let's say $$\theta$$ was an $$M\times N$$ matrix, and $$g(x;\theta) = \theta~x$$.
• $$N$$ is the length of the vector $$x$$
• $$M$$ is the length of the encoded vector $$\tilde{x}$$

What is $$\nabla_{\theta} C_t$$ ?

$\nabla_{\theta} C_t = \left(\nabla_{\tilde{x}_t} C_t\right) x_t^T$

• This is actually a linear classifier on $$x$$
• $$w^T\theta~x = (\theta^T w)^T~x = \tilde{w}^T~x$$
• But because of our factorization, is no longer convex.
• If we want to increase the expressive power of our classifier, $$g$$ has to be non-linear !

Classification

The Multi-Layer Perceptron

$x$

Classification

The Multi-Layer Perceptron

$x~~\overset{h~=~\theta x}{\longrightarrow}~~h$

Classification

The Multi-Layer Perceptron

$x~~\overset{h~=~\theta x}{\longrightarrow}~~h~~\overset{\tilde{h}^j~=~\kappa(h^j)}{\longrightarrow}~~\tilde{h}$

• $$\kappa$$ is an "element-wise" non-linearity.
• For example $$\kappa(x) = \sigma(x)$$. More on this later.
• Has no learnable parameters.

Classification

The Multi-Layer Perceptron

$x~~\overset{h~=~\theta x}{\longrightarrow}~~h~~\overset{\tilde{h}^j~=~\kappa(h^j)}{\longrightarrow}~~\tilde{h}~~\overset{y~=w^T\tilde{h}}{\longrightarrow}~~y$

• $$\kappa$$ is an "element-wise" non-linearity.
• For example $$\kappa(x) = \sigma(x)$$. More on this later.
• Has no learnable parameters.

Classification

The Multi-Layer Perceptron

$x~~\overset{h~=~\theta x}{\longrightarrow}~~h~~\overset{\tilde{h}^j~=~\kappa(h^j)}{\longrightarrow}~~\tilde{h}~~\overset{y~=w^T\tilde{h}}{\longrightarrow}~~y~~\overset{p~=~\sigma(y)}{\longrightarrow}~~p$

• $$\kappa$$ is an "element-wise" non-linearity.
• For example $$\kappa(x) = \sigma(x)$$. More on this later.
• Has no learnable parameters.
• $$\sigma$$ is our sigmoid to convert log-odds to probability. $\sigma(y) = \frac{\exp(y)}{1+\exp(y)}$
• Multiplication by $$\theta$$ and action of $$\kappa$$ is a "layer".
• Called a "hidden" layer, because you're learning a "latent representation".
• Don't have direct access to the true value of its outputs
• Learning a representation that jointly with a learned classifier is optimal

Classification

The Multi-Layer Perceptron

$x~~\overset{h~=~\theta x}{\longrightarrow}~~h~~\overset{\tilde{h}^j~=~\kappa(h^j)}{\longrightarrow}~~\tilde{h}~~\overset{y~=w^T\tilde{h}}{\longrightarrow}~~y~~\overset{p~=~\sigma(y)}{\longrightarrow}~~p$

• This is a neural network:
• A complex function formed by composition of "simple" linear and non-linear functions.
• This network has learnable parameters $$\theta,w$$.
• Train by gradient descent with respect to classification loss.
• What are the gradients ?

Doing this manually is going to get old really fast.

• Express complex function as a composition of simpler functions.
• Store this as nodes in a 'computation graph'
• Use chain rule to automatically back-propagate

Popular Autograd Systems: Tensorflow, Torch, Caffe, MXNet, Theano, ...

We'll write our own!

• Say we want to minimize a loss $$L$$, that is a function of parameters and training data.
• Let's say for a parameter $$\theta$$ we can write: $L = f(x); x = g(\theta,y)$ where $$y$$ is independent of $$\theta$$, and $$f$$ does not use $$\theta$$ except through $$x$$.
• Now, let's say I gave you the value of $$y$$ and the gradient of $$L$$ with respect to $$x$$.
• $$x$$ is an $$N-$$ dimensional vector
• $$\theta$$ is an $$M-$$ dimensional vector (if its a matrix, just think of each element as a separate paramter)

Express $$\frac{\partial{L}}{\partial{\theta^j}}$$ in terms of $$\frac{\partial{L}}{\partial{x^i}}$$ and $$\frac{\partial{g(\theta,y)^i}}{\partial \theta^j}$$: which is the partial derivative of one of the dimensions of the outputs of $$g$$ with respect to one of the dimensions of its inputs.

For every $$j$$

$\frac{\partial L}{\partial \theta^j} = \sum_i \frac{\partial{L}}{\partial{x^i}}\frac{\partial{g(\theta,y)^i}}{\partial \theta^j}$

We can similarly compute gradients for the "other" input to $$g$$, i.e. y.

$L = f(x,x'); x = g(\theta,y), x' = g'(\theta,y')$

Let's say a specific variable had two "paths" to the loss.

$\frac{\partial L}{\partial \theta^j} = \sum_i \frac{\partial{L}}{\partial{x^i}}\frac{\partial{g(\theta,y)^i}}{\partial \theta^j} + \sum_i \frac{\partial{L}}{\partial{x'^i}}\frac{\partial{g'(\theta,y')^i}}{\partial \theta^j}$

• Build a directed computation graph with a (python) list of nodes
G = [n1,n2,n3 ...]
• Each node is an "object" that is one of three kinds:
• Input
• Parameter
• Operation . . .

We will define the graph by calling functions that define functional relationships.

import edf

x = edf.Input()
theta = edf.Parameter()

y = edf.matmul(theta,x)
y = edf.tanh(y)

w = edf.Parameter()
y = edf.matmul(w,y)

We will define the graph by calling functions that define functional relationships.

import edf

x = edf.Input()
theta = edf.Parameter()

y = edf.matmul(theta,x)
y = edf.tanh(y)

w = edf.Parameter()
y = edf.matmul(w,y)
• Each of these statements adds a node to the list of nodes.
• Operation nodes are added by matmul, tanh, etc., and are linked to previous nodes that appear before it in the list as input.
• Every node object is going to have a member element n.top which will be the value of its "output"
• This can be an arbitrary shaped array.
• For input and parameter nodes, these top values will just be set (or updated by SGD).
• For operation nodes, the top values will be computed from the top values of their inputs.
• Every operation node will be an object of a class that has a function called forward.
• A forward pass will begin with values of all inputs and parameters set.
• Then we will go through the list of nodes in order, and compute the value of all operation nodes.

import edf

x = edf.Input()
theta = edf.Parameter()

y = edf.matmul(theta,x)
y = edf.tanh(y)

w = edf.Parameter()
y = edf.matmul(w,y)
• A forward pass will begin with values of all inputs and parameters set.
• Then we will go through the list of nodes in order, and compute the value of all operation nodes.

• Because nodes were added in order, if we go through them in order,
the tops of our inputs will be available.

import edf

x = edf.Input()
theta = edf.Parameter()

y = edf.matmul(theta,x)
y = edf.tanh(y)

w = edf.Parameter()
y = edf.matmul(w,y)

Somewhere in the training loop, where the values of parameters have been set before.

x.set(...)
edf.Forward()
print(y.top)
• And this will give us the value of the output.
• But now, we want to compute "gradients".
• For each "operation" class, we will also define a function backward.
• All operation and paramter nodes will also have an element called grad.
• We will have to then back-propagate gradients in order.