Arsenii Mustafin

Arsenii Mustafin, Alex Olshevsky, Ioannis Ch. Paschalidis

Temporal difference (TD) learning is a policy evaluation in reinforcement learning whose performance can be enhanced by variance reduction methods. Recently, multiple works have sought to fuse TD learning with Stochastic Variance Reduced Gradient (SVRG) method to achieve a geometric rate of convergence. However, the resulting convergence rate is significantly weaker than what is achieved by SVRG in the setting of convex optimization. In this work we utilize a recent interpretation of TD-learning as the splitting of the gradient of an appropriately chosen function, thus simplifying the algorithm and fusing TD with SVRG. Our main result is a geometric convergence bound with predetermined learning rate of $1/8$, which is identical to the convergence bound available for SVRG in the convex setting. Our theoretical findings are supported by a set of experiments.

Arsenii Mustafin, Aleksei Pakharev, Alex Olshevsky et al.

We present a new geometric interpretation of Markov Decision Processes (MDPs) with a natural normalization procedure that allows us to adjust the value function at each state without altering the advantage of any action with respect to any policy. This advantage-preserving transformation of the MDP motivates a class of algorithms which we call Reward Balancing, which solve MDPs by iterating through these transformations, until an approximately optimal policy can be trivially found. We provide a convergence analysis of several algorithms in this class, in particular showing that for MDPs for unknown transition probabilities we can improve upon state-of-the-art sample complexity results.

Dominik Baumann, Erfaun Noorani, Arsenii Mustafin et al.

In reinforcement learning, we typically aim to optimize the expected value of the sum of rewards an agent collects over a trajectory. However, if the process generating these rewards is non-ergodic, the expected value, i.e., the average over infinitely many trajectories with a given policy, is uninformative for the average over a single, but infinitely long trajectory. Thus, if we care about how the individual agent performs during deployment, the expected value is not a good optimization objective. In this paper, we discuss the impact of non-ergodic reward processes on reinforcement learning agents through an instructive example, relate the notion of ergodic reward processes to more widely used notions of ergodic Markov chains, and present existing solutions that optimize long-term performance of individual trajectories under non-ergodic reward dynamics.

Arsenii Mustafin, Aleksei Pakharev, Alex Olshevsky et al.

We build on a recently introduced geometric interpretation of Markov Decision Processes (MDPs) to analyze classical MDP-solving algorithms: Value Iteration (VI) and Policy Iteration (PI). First, we develop a geometry-based analytical apparatus, including a transformation that modifies the discount factor $γ$, to improve convergence guarantees for these algorithms in several settings. In particular, one of our results identifies a rotation component in the VI method, and as a consequence shows that when a Markov Reward Process (MRP) induced by the optimal policy is irreducible and aperiodic, the asymptotic convergence rate of value iteration is strictly smaller than $γ$.

Arsenii Mustafin, Xinyi Sheng, Dominik Baumann

The theoretical analysis of Markov Decision Processes (MDPs) is commonly split into two cases - the average-reward case and the discounted-reward case - which, while sharing similarities, are typically analyzed separately. In this work, we extend a recently introduced geometric interpretation of MDPs for the discounted-reward case to the average-reward case, thereby unifying both. This allows us to extend a major result known for the discounted-reward case to the average-reward case: under a unique and ergodic optimal policy, the Value Iteration algorithm achieves a geometric convergence rate.

Arsenii Mustafin, Sebastien Colla, Alex Olshevsky et al.

Value Iteration is a widely used algorithm for solving Markov Decision Processes (MDPs). While previous studies have extensively analyzed its convergence properties, they primarily focus on convergence with respect to the infinity norm. In this work, we use absolute probability sequences to develop a new line of analysis and examine the algorithm's convergence in terms of the $L^2$ norm, offering a new perspective on its behavior and performance.

Arsenii Mustafin, Alex Olshevsky, Ioannis Ch. Paschalidis

This paper establishes that an MDP with a unique optimal policy and ergodic associated transition matrix ensures the convergence of various versions of the Value Iteration algorithm at a geometric rate that exceeds the discount factor γ for both discounted and average-reward criteria.