Convergence Rates of Accelerated Markov Gradient Descent with Applications in Reinforcement Learning (2002.02873v3)

Abstract: Motivated by broad applications in machine learning, we study the popular accelerated stochastic gradient descent (ASGD) algorithm for solving (possibly nonconvex) optimization problems. We characterize the finite-time performance of this method when the gradients are sampled from Markov processes, and hence biased and dependent from time step to time step; in contrast, the analysis in existing work relies heavily on the stochastic gradients being independent and sometimes unbiased. Our main contributions show that under certain (standard) assumptions on the underlying Markov chain generating the gradients, ASGD converges at the nearly the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain. One of the key motivations for this study are complicated control problems that can be modeled by a Markov decision process and solved using reinforcement learning. We apply the accelerated method to several challenging problems in the OpenAI Gym and Mujoco, and show that acceleration can significantly improve the performance of the classic temporal difference learning and REINFORCE algorithms.

• The paper demonstrates that accelerated gradient descent maintains near-optimal convergence under Markov-dependent gradients, differing only by a logarithmic factor.
• The paper applies AMGD to reinforcement learning, showing improved performance in temporal difference and policy gradient methods with fewer samples.
• The paper’s analysis in convex, strongly convex, and nonconvex scenarios proves the method’s versatility and practical efficiency in stochastic environments.

Convergence Rates of Accelerated Markov Gradient Descent with Applications in Reinforcement Learning

The paper presents an insight into the accelerated stochastic gradient descent (ASGD) algorithm, particularly addressing its application where gradients are sampled from Markov processes instead of being independent and identically distributed (i.i.d). The authors aim to extend the understanding of ASGD, typically effective under i.i.d. assumptions, to scenarios modeled by Markov decision processes, often observed in reinforcement learning (RL).

Key Contributions

The research pivots around the modification of ASGD under conditions where the gradient samples are dependent and potentially biased due to the underlying Markov processes. The authors propose the Accelerated Markov Gradient Descent (AMGD) framework and provide comprehensive convergence analysis across multiple scenarios: convex, strongly convex, and nonconvex objective functions.

1. Convergence Analysis: The paper successfully demonstrates that the convergence rate for ASGD holds predominantly unchanged when transitioning from i.i.d. gradient samples to those obtained through Markov processes. The primary divergence is a logarithmic factor that contributes due to the mixing time of the Markov chain—a crucial theoretical advancement in stochastic optimization.
2. Reinforcement Learning Applications: The practical implications are examined through reinforcement learning problems where ASGD's application with Markov samples shows significant performance improvements in temporal difference methods and policy gradient algorithms. Using environments like OpenAI Gym and Mujoco, experiments suggest the AMGD's advantage in requiring fewer samples compared to traditional methods.

Theoretical Insights

The theoretical underpinnings include deploying ASGD in nonconvex and convex optimization settings where the gradients are derived through state-dependent Markov processes. The authors discuss standard assumptions about the ergodicity of the chains and leverage the geometric mixing time to alleviate issues of gradient bias and dependence.

• Nonconvex Optimization: A key result indicates that the convergence rate for nonconvex problems approaches that of the independent setting, within a logarithmic factor related to the Markovian nature of the gradient sampling.
• Convex and Strongly Convex Cases: Under these conditions, AMGD shows substantial initial performance improvements and maintains a convergence rate closely aligned with conventional theoretical expectations, illustrating its broader applicability in structured machine learning tasks.

Numerical Results

The paper provides extensive numerical results, delineating the convergence and sample efficiency of the accelerated methods in standard reinforcement learning benchmarks. Importantly, it highlights potential improvements in RL tasks, underpinning the broader applicability of ASGD beyond the confines of traditional settings.

• Policy Evaluation and Control Tasks: Through experiments on GridWorld and various Mujoco environments, the empirical outcomes underscore AMGD's effectiveness in learning robust policies with reduced computational complexity.
• Data Efficiency: Significantly fewer samples are required to attain comparable or superior performance to non-accelerated variants, underscoring the practical benefits of the proposed methodology.

Implications and Future Work

This contribution paves the way for further exploration of accelerated methods in environments characterized by sequential decision-making and complex, stochastic dependencies. Future studies might explore relaxing the ergodicity assumptions or advancing other acceleration techniques within reinforcement learning and broader economic or operational realms characterized by similar stochastic structures.

The implications suggest a paradigm shift in how Markov-based gradient methods can be optimized for machine learning tasks, challenging the prevalent reliance on i.i.d. assumptions. Future work could involve extending this analysis to other areas of artificial intelligence where data is inherently sequential and dependent, such as LLMing or real-time decision systems.