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.

Continue reading “Stochastic Finite Arm Bandits”

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


\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.

Continue reading “Perturbation of Markov Matrices”

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.]

Continue reading “A Short Discussion on Policy Gradients in Bandits”