Applied Probability Notes

Neural Tangent Kernel

The Neural Tangent Kernel is a way of understanding the training performance of Neural Networks by relating them to Kernel methods. Here we overview the results of the paper [Jacot et al. here]. The paper considers a deep neural network with a fixed amount of data and a fixed depth. The weights applied to neurons are initially independent and normally distributed. We take a limit where the width of each layer tends to infinity.

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Stochastic Control Notes Update – 2021

I’ve updated the notes for this year’s stochastic control course, here:

Stochastic_Control_2021_Jan.pdfDownload

Asside from general tidying. New material includes:

Equilibrium distributions of Markov chains
Occupancy measure of infinite time horizon MDPs
Linear programming as an algorithm for solving MDPs
Convergence of Asynchronous Value Iteration
(s,S) Inventory Control
POMDP (though there is still more to add)
Calculus of Variations (though there is still more to add)
Pontyagin’s Maximum Prinicple
Linear-Quadratic Lyapunov functions (Sylvester’s equation and Hurwitz matrices)
(some) Online Convex Optimization
Stochastic Bandits (UCB and Lai-Robbins Lower bound)
Gittins’ Index Theorem.
Sequential/Stochastic Linear Regression (Lai and Wei)
More discussion on TD methods
Discussion on double Q-learning and Dueling/Advantage updating
Convergence proof for SARSA
Policy Gradients (some convergence arguments from Bhanhari and Russo, but still more to do)
Cross Entropy Method (but still more to do)
Several new appendices (but mostly from old notes)

Like last year, I will likely update the notes further (and correct typos) towards the end of the course.

Inventory Control

We consider the problem where there is an amount of stock $x_t$ at time $t$ . You can perform the action to order $a_t$ units of stock. Further the demand at time $t$ is $d_t$ . We assume $d_t$ is independent over $t$ . The change in the amount of stock follows the dynamic:

$\begin{aligned} x_{t+1} = x_t + a_t -d_t \,.\end{aligned}$

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Lai Robbins Lower-Bound

We continue the earlier post on finite-arm stochastic multi-arm bandits. The results so far have suggest that, for independent identically distributed arms, the correct size of the regret is of the order $\log T$ . We now more formally prove this with the Lai and Robbins lower-bound .

Euler-Maruyama

The Euler-Maruyama scheme is a method of approximating an stochastic differential equation. Here we investigate two forms of error the scheme: the weak error and the strong error. The aim is to later we will cover Multi-Level Monte Carlo (MLMC) and related topics.

Stochastic Finite Arm Bandits

We discuss a canonical multi-arm bandit setting. The stochastic multi-arm bandit with finite arms and bounded rewards. We let $a=1,...,N$ index the set of arms. We let $\mathcal A=\{1,...,N\}$ be the set of arms. If you play the arm $a$ at time $t\in\mathbb N$ , you receive rewards $r_t(a)$ which are independent and identically distributed in $t$ . However, the distribution between arms may change.

We let $\bar r(a)$ be the mean of arm $a$ . We want to find the machine with highest mean reward.

Perturbation of Markov Matrices

We consider the impact on the stationary distribution of perturbing a Markov chain’s transition matrix. This argument due to Seneta. We show if $P$ and $\tilde P$ are probability matrices of a irreducible finite state Markov chains with stationary distributions $\pi$ and $\pi$ then we can show that

Theorem.

$\displaystyle || \pi - \tilde \pi ||_{TV} \leq \frac{1}{1- \tau(P)} \sum_{x} \left\| P_{x, \cdot} -\tilde P_{x, \cdot} \right\|_{TV}$

where $\tau(P)$ is the total variation between the rows of $P$ .

Cross Entropy Method

The Cross Entropy Method (CEM) is a generic optimization technique. It is a zero-th order method, i.e. you don’t gradients.¹ So, for instance, it works well on combinatorial optimization problems, as well as reinforcement learning.

Exponential Families

The exponential family of distributions are a particularly tractable, yet broad, class of probability distributions. They are tractable because of a particularly nice [Fenchel] duality relationship between natural parameters and moment parameters. Moment parameters can be estimated by taking the empirical mean of sufficient statistics and the duality relationship can then recover an estimate of the distributions natural parameters.

A Short Discussion on Policy Gradients in Bandits

This is a quick note on policy gradients in bandit problems. There are a number of excellent papers coming out on the convergence of policy gradients [1 ,2, 3]. I wrote this sketch argument a few months ago. Lockdown and homeschooling started and I found I did not have time to pursue it. Then yesterday morning I came across the following paper Mei et al. [1], which uses a different Lyapunov argument to draw an essentially the same conclusion [first for bandits then for general RL]. Although most of the time I don’t talk about my own research on this blog, I wrote this short note here just in case this different Lyapunov argument is of interest.

Further, all these results on policy gradient are [thus far] for deterministic models. I wrote a short paper that deals with a stochastic regret bound for a policy gradient algorithm much closer to projected gradient descent, again in the bandit setting. I’ll summarize this, as it might be possible to use it to derive stochastic regret bounds for policy gradients more generally. At the end, I discuss some issues around getting some proofs to work in the stochastic case and emphasize the point that $1/q^\star$ is a good choice of lyapunov function [where $latex $q^\star$ is the probability of not choosing the optimal action.]