CS7643-Problem Set 1 Gradient Descent and Automatic Differentiation Solved

1         Gradient Descent
1. We often use iterative optimization algorithms such as Gradient Descent to find w that minimizes a loss function f(w). Recall that in gradient descent, we start with an initial value of w (say w(1)) and iteratively take a step in the direction of the negative of the gradient of the objective function i.e.

                                                                                 w(t+1) = w(t) − η∇f(w(t))                                           (1)

for learning rate η 0.

In this question, we will develop a slightly deeper understanding of this update rule, in particular for minimizing a convex function f(w). Note: this analysis will not directly carry over to training neural networks since loss functions for training neural networks are typically not convex, but this will (a) develop intuition and (b) provide a starting point for research in non-convex optimization (which is beyond the scope of this class).

Recall the first-order Taylor approximation of f at w(t):

                                                                                             (2)

When f is convex, this approximation forms a lower bound of f, i.e.
 
 w
(3)
affine lower bound to f(·)

Since this approximation is a ‘simpler’ function than f(·), we could consider minimizing the approximation instead of f(·). Two immediate problems: (1) the approximation is affine (thus unbounded from below) and (2) the approximation is faithful for w close to w(t). To solve both problems, we add a squared `2 proximity term to the approximation minimization:

argmin(4)

w

 affine lower bound to f(·)

trade-off proximity term

Notice that the optimization problem above is an unconstrained quadratic programming problem, meaning that it can be solved in closed form (hint: gradients).

What is the solution w∗ of the above optimization? What does that tell you about the gradient descent update rule? What is the relationship between λ and η?

2.     Let’s prove a lemma that will initially seem devoid of the rest of the analysis but will come in handy in the next sub-question when we start combining things. Specifically, the analysis in this sub-question holds for any w?, but in the next sub-question we will use it for w? that minimizes f(w).

Consider a sequence of vectors v1,v2,...,vT, and an update equation of the form w(t+1) = w(t) − ηvt with w(1) = 0. Show that:

                                                                                        (5)

3.    Now let’s start putting things together and analyze the convergence rate of gradient descent i.e. how fast it converges to w?. First, show that for w¯ 

                                                                                  (6)

Next, use the result from part 2, with upper bounds B and ρ for ||w?|| and  respectively and show that for fixed , the convergence rate of gradient descent is

 √

O(1/ T) i.e. the upper bound for .

4.    Consider an objective function f(w) := f1(w) + f2(w) comprised of N = 2 terms:

                                       and                                                                   (7)

Now consider using SGD (with a batch-size B = 1) to minimize f(w). Specifically, in each iteration, we will pick one of the two terms (uniformly at random), and take a step in the direction of the negative gradient, with a constant step-size of η. You can assume η is small enough that every update does result in improvement (aka descent) on the sampled term. Is SGD guaranteed to decrease the overall loss function in every iteration? If yes, provide a proof. If no, provide a counter-example.

2         Automatic Differentiation
5. (In practice, writing the closed-form expression of the derivative of a loss function f

w.r.t. the parameters of a deep neural network is hard (and mostly unnecessary) as f becomes complex. Instead, we define computation graphs and use the automatic differentiation algorithms (typically backpropagation) to compute gradients using the chain rule. For example, consider the expression

f(x,y) = (x + y)(y + 1)

Let’s define intermediate variables a and b such that
(8)
a = x + y
(9)
b = y + 1
(10)
f = a × b
(11)
A computation graph for the “forward pass” through f is shown in Fig. 1.

 

Figure 1

We can then work backwards and compute the derivative of f w.r.t. each intermediate variable (  and  ) and chain them together to get   and .

Let σ(·) denote the standard sigmoid function. Now, for the following vector function:

f1(w1,w2) = eew1+e2w2 + σ(ew1 + e2w2)
(12)
f2(w1,w2) = w1w2 + max(w1,w2)
(13)
(a)     Draw the computation graph. Compute the value of f at w~ = (1,−1).

(b)    At this w~, compute the Jacobian  using numerical differentiation (using ∆w = 0.01).

(c)     At this w~, compute the Jacobian using forward mode auto-differentiation.

(d)    At this w~, compute the Jacobian using backward mode auto-differentiation.

(e)     Don’t you love that software exists to do this for us?

3         Paper Review
The first of our paper reviews for this course comes from a much acclaimed spotlight presentation at NeurIPS 2019 on the topic ‘Weight Agnostic Neural Networks’ by Adam Gaier and David Ha from Google Brain.

The paper presents a very interesting proposition that, through a series of experiments, re-examines some fundamental notions about neural networks - in particular, the comparative importance of architectures and weights in a network’s predictive performance.

The paper can be viewed here. The authors have also written a blog post with intuitive visualizations to help understand its key concepts better.

Guidelines: Please restrict your reviews to no more than 350 words. The evaluation rubric for this section is as follows :

6.    What is the main contribution of this paper? Briefly summarize its key insights, strengths and weaknesses.

7.    What is your personal takeaway from this paper? This could be expressed either in terms of relating the approaches adopted in this paper to your traditional understanding of learning parameterized models, or potential future directions of research in the area which the authors haven’t addressed, or anything else that struck you as being noteworthy.

4         Implement and train a network on CIFAR-10
Setup Instructions: Before attempting this question, look at setup instructions at here.

8. Now, we will learn how to implement a softmax classifier, vanilla neural networks (or Multi-Layer Perceptrons), and ConvNets. You will begin by writing the forward and backward passes for different types of layers (including convolution and pooling), and then go on to train a shallow ConvNet on the CIFAR-10 dataset in Python. Next you will learn to use PyTorch, a popular open-source deep learning framework, and use it to replicate the experiments from before. Follow the instructions provided