Modified Policy Iteration for Exponential Cost Risk Sensitive MDPs

Modified policy iteration (MPI) also known as optimistic policy iteration is at the core of many reinforcement learning algorithms. It works by combining elements of policy iteration and value iteration. The convergence of MPI has been well studied in the case of discounted and average-cost MDPs. In this work, we consider the exponential cost risk-sensitive MDP formulation, which is known to provide some robustness to model parameters. Although policy iteration and value iteration have been well studied in the context of risk sensitive MDPs, modified policy iteration is relatively unexplored. We provide the first proof that MPI also converges for the risk-sensitive problem in the case of finite state and action spaces. Since the exponential cost formulation deals with the multiplicative Bellman equation, our main contribution is a convergence proof which is quite different than existing results for discounted and risk-neutral average-cost problems. The proof of approximate modified policy iteration for risk sensitive MDPs is also provided in the appendix.

Performance Bounds for Policy-Based Average Reward Reinforcement Learning Algorithms

Many policy-based reinforcement learning (RL) algorithms can be viewed as instantiations of approximate policy iteration (PI), i.e., where policy improvement and policy evaluation are both performed approximately. In applications where the average reward objective is the meaningful performance metric, often discounted reward formulations are used with the discount factor being close to 1, which is equivalent to making the expected horizon very large. However, the corresponding theoretical bounds for error performance scale with the square of the horizon. Thus, even after dividing the total reward by the length of the horizon, the corresponding performance bounds for average reward problems go to infinity. Therefore, an open problem has been to obtain meaningful performance bounds for approximate PI and RL algorithms for the average-reward setting. In this paper, we solve this open problem by obtaining the first non-trivial error bounds for average-reward MDPs which go to zero in the limit where when policy evaluation and policy improvement errors go to zero.

On The Convergence Of Policy Iteration-Based Reinforcement Learning With Monte Carlo Policy Evaluation

A common technique in reinforcement learning is to evaluate the value function from Monte Carlo simulations of a given policy, and use the estimated value function to obtain a new policy which is greedy with respect to the estimated value function. A well-known longstanding open problem in this context is to prove the convergence of such a scheme when the value function of a policy is estimated from data collected from a single sample path obtained from implementing the policy (see page 99 of [Sutton and Barto, 2018], page 8 of [Tsitsiklis, 2002]). We present a solution to the open problem by showing that a first-visit version of such a policy iteration scheme indeed converges to the optimal policy provided that the policy improvement step uses lookahead [Silver et al., 2016, Mnih et al., 2016, Silver et al., 2017b] rather than a simple greedy policy improvement. We provide results both for the original open problem in the tabular setting and also present extensions to the function approximation setting, where we show that the policy resulting from the algorithm performs close to the optimal policy within a function approximation error.

Reinforcement Learning with Unbiased Policy Evaluation and Linear Function Approximation

We provide performance guarantees for a variant of simulation-based policy iteration for controlling Markov decision processes that involves the use of stochastic approximation algorithms along with state-of-the-art techniques that are useful for very large MDPs, including lookahead, function approximation, and gradient descent. Specifically, we analyze two algorithms; the first algorithm involves a least squares approach where a new set of weights associated with feature vectors is obtained via least squares minimization at each iteration and the second algorithm involves a two-time-scale stochastic approximation algorithm taking several steps of gradient descent towards the least squares solution before obtaining the next iterate using a stochastic approximation algorithm.

MaxWeight With Discounted UCB: A Provably Stable Scheduling Policy for Nonstationary Multi-Server Systems With Unknown Statistics

Multi-server queueing systems are widely used models for job scheduling in machine learning, wireless networks, and crowdsourcing. This paper considers a multi-server system with multiple servers and multiple types of jobs. The system maintains a separate queue for each type of jobs. For each time slot, each available server picks a job from a queue and then serves the job until it is complete. The arrival rates of the queues and the mean service times are unknown and even nonstationary. We propose the MaxWeight with discounted upper confidence bound (UCB) algorithm, which simultaneously learns the statistics and schedules jobs to servers. We prove that the proposed algorithm can stabilize the queues when the arrival rates are strictly within the service capacity region. Specifically, we prove that the queue lengths are bounded in the mean under the assumption that the mean service times change relatively slowly over time and the arrival rates are bounded away from the capacity region by a constant whose value depends on the discount factor used in the discounted UCB. Simulation results confirm that the proposed algorithm can stabilize the queues and that it outperforms MaxWeight with empirical mean and MaxWeight with discounted empirical mean. The proposed algorithm is also better than MaxWeight with UCB in the nonstationary setting.

Finite-Time Analysis of Entropy-Regularized Neural Natural Actor-Critic Algorithm

Natural actor-critic (NAC) and its variants, equipped with the representation power of neural networks, have demonstrated impressive empirical success in solving Markov decision problems with large state spaces. In this paper, we present a finite-time analysis of NAC with neural network approximation, and identify the roles of neural networks, regularization and optimization techniques (e.g., gradient clipping and averaging) to achieve provably good performance in terms of sample complexity, iteration complexity and overparametrization bounds for the actor and the critic. In particular, we prove that (i) entropy regularization and averaging ensure stability by providing sufficient exploration to avoid near-deterministic and strictly suboptimal policies and (ii) regularization leads to sharp sample complexity and network width bounds in the regularized MDPs, yielding a favorable bias-variance tradeoff in policy optimization. In the process, we identify the importance of uniform approximation power of the actor neural network to achieve global optimality in policy optimization due to distributional shift.

Minimax Regret for Cascading Bandits

Cascading bandits model the task of learning to rank $K$ out of $L$ items over $n$ rounds of partial feedback. For this model, the minimax (i.e., gap-free) regret is poorly understood; in particular, the best known lower and upper bounds are $\Omega(\sqrt{nL/K})$ and $\tilde{O}(\sqrt{nLK})$, respectively. We improve the lower bound to $\Omega(\sqrt{nL})$ and show CascadeKL-UCB (which ranks items by their KL-UCB indices) attains it up to log terms. Surprisingly, we also show CascadeUCB1 (which ranks via UCB1) can suffer suboptimal $\Omega(\sqrt{nLK})$ regret. This sharply contrasts with standard $L$-armed bandits, where the corresponding algorithms both achieve the minimax regret $\sqrt{nL}$ (up to log terms), and the main advantage of KL-UCB is only to improve constants in the gap-dependent bounds. In essence, this contrast occurs because Pinsker's inequality is tight for hard problems in the $L$-armed case but loose (by a factor of $K$) in the cascading case.

Robust Multi-Agent Bandits Over Undirected Graphs

We consider a multi-agent multi-armed bandit setting in which $n$ honest agents collaborate over a network to minimize regret but $m$ malicious agents can disrupt learning arbitrarily. Assuming the network is the complete graph, existing algorithms incur $O( (m + K/n) \log (T) / \Delta )$ regret in this setting, where $K$ is the number of arms and $\Delta$ is the arm gap. For $m \ll K$, this improves over the single-agent baseline regret of $O(K\log(T)/\Delta)$. In this work, we show the situation is murkier beyond the case of a complete graph. In particular, we prove that if the state-of-the-art algorithm is used on the undirected line graph, honest agents can suffer (nearly) linear regret until time is doubly exponential in $K$ and $n$. In light of this negative result, we propose a new algorithm for which the $i$-th agent has regret $O( ( d_{\text{mal}}(i) + K/n) \log(T)/\Delta)$ on any connected and undirected graph, where $d_{\text{mal}}(i)$ is the number of $i$'s neighbors who are malicious. Thus, we generalize existing regret bounds beyond the complete graph (where $d_{\text{mal}}(i) = m$), and show the effect of malicious agents is entirely local (in the sense that only the $d_{\text{mal}}(i)$ malicious agents directly connected to $i$ affect its long-term regret).

Learning to Control Partially Observed Systems with Finite Memory

We consider the reinforcement learning problem for partially observed Markov decision processes (POMDPs) with large or even countably infinite state spaces, where the controller has access to only noisy observations of the underlying controlled Markov chain. We consider a natural actor-critic method that employs a finite internal memory for policy parameterization, and a multi-step temporal difference learning algorithm for policy evaluation. We establish, to the best of our knowledge, the first non-asymptotic global convergence of actor-critic methods for partially observed systems under function approximation. In particular, in addition to the function approximation and statistical errors that also arise in MDPs, we explicitly characterize the error due to the use of finite-state controllers. This additional error is stated in terms of the total variation distance between the traditional belief state in POMDPs and the posterior distribution of the hidden state when using a finite-state controller. Further, we show that this error can be made small in the case of sliding-block controllers by using larger block sizes.

The Role of Lookahead and Approximate Policy Evaluation in Policy Iteration with Linear Value Function Approximation

When the sizes of the state and action spaces are large, solving MDPs can be computationally prohibitive even if the probability transition matrix is known. So in practice, a number of techniques are used to approximately solve the dynamic programming problem, including lookahead, approximate policy evaluation using an m-step return, and function approximation. In a recent paper, (Efroni et al. 2019) studied the impact of lookahead on the convergence rate of approximate dynamic programming. In this paper, we show that these convergence results change dramatically when function approximation is used in conjunction with lookout and approximate policy evaluation using an m-step return. Specifically, we show that when linear function approximation is used to represent the value function, a certain minimum amount of lookahead and multi-step return is needed for the algorithm to even converge. And when this condition is met, we characterize the finite-time performance of policies obtained using such approximate policy iteration. Our results are presented for two different procedures to compute the function approximation: linear least-squares regression and gradient descent.