Siddharth Chandak

Siddharth Chandak, Pratik Shah, Vivek S Borkar et al.

Motivated by the novel paradigm developed by Van Roy and coauthors for reinforcement learning in arbitrary non-Markovian environments, we propose a related formulation and explicitly pin down the error caused by non-Markovianity of observations when the Q-learning algorithm is applied on this formulation. Based on this observation, we propose that the criterion for agent design should be to seek good approximations for certain conditional laws. Inspired by classical stochastic control, we show that our problem reduces to that of recursive computation of approximate sufficient statistics. This leads to an autoencoder-based scheme for agent design which is then numerically tested on partially observed reinforcement learning environments.

Siddharth Chandak, Ilai Bistritz, Nicholas Bambos

Consider a decision-maker that can pick one out of $K$ actions to control an unknown system, for $T$ turns. The actions are interpreted as different configurations or policies. Holding the same action fixed, the system asymptotically converges to a unique equilibrium, as a function of this action. The dynamics of the system are unknown to the decision-maker, which can only observe a noisy reward at the end of every turn. The decision-maker wants to maximize its accumulated reward over the $T$ turns. Learning what equilibria are better results in higher rewards, but waiting for the system to converge to equilibrium costs valuable time. Existing bandit algorithms, either stochastic or adversarial, achieve linear (trivial) regret for this problem. We present a novel algorithm, termed Upper Equilibrium Concentration Bound (UECB), that knows to switch an action quickly if it is not worth it to wait until the equilibrium is reached. This is enabled by employing convergence bounds to determine how far the system is from equilibrium. We prove that UECB achieves a regret of $\mathcal{O}(\log(T)+τ_c\log(τ_c)+τ_c\log\log(T))$ for this equilibrium bandit problem where $τ_c$ is the worst case approximate convergence time to equilibrium. We then show that both epidemic control and game control are special cases of equilibrium bandits, where $τ_c\log τ_c$ typically dominates the regret. We then test UECB numerically for both of these applications.

Siddharth Chandak

Two-time-scale stochastic approximation is an algorithm with coupled iterations which has found broad applications in reinforcement learning, optimization and game control. While several prior works have obtained a mean square error bound of $O(1/k)$ for linear two-time-scale iterations, the best known bound in the non-linear contractive setting has been $O(1/k^{2/3})$. In this work, we obtain an improved bound of $O(1/k)$ for non-linear two-time-scale stochastic approximation. Our result applies to algorithms such as gradient descent-ascent and two-time-scale Lagrangian optimization. The key step in our analysis involves rewriting the original iteration in terms of an averaged noise sequence which decays sufficiently fast. Additionally, we use an induction-based approach to show that the iterates are bounded in expectation.

Siddharth Chandak

Two-time-scale stochastic approximation algorithms are iterative methods used in applications such as optimization, reinforcement learning, and control. Finite-time analysis of these algorithms has primarily focused on fixed point iterations where both time-scales have contractive mappings. In this work, we broaden the scope of such analyses by considering settings where the slower time-scale has a non-expansive mapping. For such algorithms, the slower time-scale can be viewed as a stochastic inexact Krasnoselskii-Mann iteration. We also study a variant where the faster time-scale has a projection step which leads to non-expansiveness in the slower time-scale. We show that the last-iterate mean square residual error for such algorithms decays at a rate $O(1/k^{1/4-ε})$, where $ε>0$ is arbitrarily small. We further establish almost sure convergence of iterates to the set of fixed points. We demonstrate the applicability of our framework by applying our results to minimax optimization, linear stochastic approximation, and Lagrangian optimization.

Siddharth Chandak, Vivek S. Borkar

We derive a concentration bound of the type `for all $n \geq n_0$ for some $n_0$' for TD(0) with linear function approximation. We work with online TD learning with samples from a single sample path of the underlying Markov chain. This makes our analysis significantly different from offline TD learning or TD learning with access to independent samples from the stationary distribution of the Markov chain. We treat TD(0) as a contractive stochastic approximation algorithm, with both martingale and Markov noises. Markov noise is handled using the Poisson equation and the lack of almost sure guarantees on boundedness of iterates is handled using the concept of relaxed concentration inequalities.

Siddharth Chandak, Ilai Bistritz, Nicholas Bambos

Consider a strongly monotone game where the players' utility functions include a reward function and a linear term for each dimension, with coefficients that are controlled by the manager. Gradient play converges to a unique Nash equilibrium (NE) that does not optimize the global objective. The global performance at NE can be improved by imposing linear constraints on the NE, also known as a generalized Nash equilibrium (GNE). We therefore want the manager to control the coefficients such that they impose the desired constraint on the NE. However, this requires knowing the players' rewards and action sets. Obtaining this game information is infeasible in a large-scale network and violates user privacy. To overcome this, we propose a simple algorithm that learns to shift the NE to meet the linear constraints by adjusting the controlled coefficients online. Our algorithm only requires the linear constraints violation as feedback and does not need to know the reward functions or the action sets. We prove that our algorithm converges with probability 1 to the set of GNE given by coupled linear constraints. We then prove an L2 convergence rate of near-$O(t^{-1/4})$.

Siddharth Chandak, Vivek S. Borkar, Harsh Dolhare

The popular LSPE($λ$) algorithm for policy evaluation is revisited to derive a concentration bound that gives high probability performance guarantees from some time on.

Siddharth Chandak, Vivek S. Borkar, Parth Dodhia

Using a martingale concentration inequality, concentration bounds `from time $n_0$ on' are derived for stochastic approximation algorithms with contractive maps and both martingale difference and Markov noises. These are applied to reinforcement learning algorithms, in particular to asynchronous Q-learning and TD(0).