Stochastic Approximation with Delayed Updates: Finite-Time Rates under Markovian Sampling (2402.11800v3)

Abstract: Motivated by applications in large-scale and multi-agent reinforcement learning, we study the non-asymptotic performance of stochastic approximation (SA) schemes with delayed updates under Markovian sampling. While the effect of delays has been extensively studied for optimization, the manner in which they interact with the underlying Markov process to shape the finite-time performance of SA remains poorly understood. In this context, our first main contribution is to show that under time-varying bounded delays, the delayed SA update rule guarantees exponentially fast convergence of the \emph{last iterate} to a ball around the SA operator's fixed point. Notably, our bound is \emph{tight} in its dependence on both the maximum delay $\tau_{max}$, and the mixing time $\tau_{mix}$. To achieve this tight bound, we develop a novel inductive proof technique that, unlike various existing delayed-optimization analyses, relies on establishing uniform boundedness of the iterates. As such, our proof may be of independent interest. Next, to mitigate the impact of the maximum delay on the convergence rate, we provide the first finite-time analysis of a delay-adaptive SA scheme under Markovian sampling. In particular, we show that the exponent of convergence of this scheme gets scaled down by $\tau_{avg}$, as opposed to $\tau_{max}$ for the vanilla delayed SA rule; here, $\tau_{avg}$ denotes the average delay across all iterations. Moreover, the adaptive scheme requires no prior knowledge of the delay sequence for step-size tuning. Our theoretical findings shed light on the finite-time effects of delays for a broad class of algorithms, including TD learning, Q-learning, and stochastic gradient descent under Markovian sampling.

• The paper presents a finite-time convergence analysis for delayed stochastic approximation under Markovian sampling, showing exponential convergence to a noise- and delay-dependent error ball.
• It introduces a delay-adaptive scheme that adjusts step sizes based on the average delay, significantly enhancing convergence compared to non-adaptive methods without prior delay knowledge.
• The study offers profound implications for distributed reinforcement learning by addressing how communication delays and temporal correlations impact learning performance.

Exploring Delay-Adaptive Stochastic Approximation under Markovian Sampling

In the field of reinforcement learning and optimization, the timely processing of information and updates is crucial for the efficiency and effectiveness of learning algorithms. A foundational aspect of stochastic approximation (SA) schemes, including temporal-difference (TD) learning and Q-learning, involves iteratively updating policies or value functions based on newly acquired samples. However, in distributed and asynchronous systems, updates are often subject to delays, which can significantly hamper the convergence rate and overall performance of these algorithms. Understanding the impact of such delays, especially under Markovian sampling, is vital for advancing learning algorithms in realistic, large-scale applications.

Analytical Contributions

The paper of stochastic approximation schemes with delayed updates under Markovian sampling brings forth several analytical contributions, which are both novel and critical for advancing the field. The first notable contribution is the comprehensive finite-time convergence analysis for delayed SA schemes under Markovian sampling, a problem formulation that captures the essence of many practical reinforcement learning setups, especially those involving distributed systems. The analysis reveals that for a class of algorithms employing constant or time-varying bounded delays, the convergence to the fixed point of the SA operator is exponentially fast, albeit to a larger error ball, the size of which is dictated by the noise level and the maximum delay in the system.

A key insight from this analysis is that for systems with time-varying delays, the convergence rate is affected by the maximum possible delay, highlighting the impact of worst-case scenarios on learning performance. This insight propels the investigation into delay-adaptive strategies, where the delay's impact on the convergence rate is mitigated by adapting the step size based on the staleness of updates. Such an adaptive scheme, as proposed in this paper, depends on the average delay, demonstrating a significant improvement over non-adaptive methods in terms of convergence rate, without necessitating prior knowledge of the delay sequence for tuning.

Practical Implications

Beyond theoretical advancements, this research holds profound implications for implementing reinforcement learning algorithms in asynchronous and distributed systems, such as those encountered in federated learning and multi-agent setups. Specifically, the delay-adaptive stochastic approximation scheme enables a learner to adjust update frequencies dynamically, based on the delay's current state, thereby optimizing the learning process in the face of inevitable communication lags. This adaptiveness ensures that the algorithm remains robust, maintaining a faster convergence rate without the need for conservative step-size settings.

Furthermore, the findings underscore the importance of accounting for the temporal correlations induced by Markovian sampling, adding a layer of complexity to the handling of outdated information. The proofs and techniques developed in this paper could serve as a foundation for exploring similar delay-adaptive mechanisms across a broader spectrum of learning algorithms, potentially enhancing their robustness and efficiency in real-world scenarios where delays cannot be entirely avoided.

Future Directions

The exploration of stochastic approximation with delayed updates under Markovian sampling paves the way for several future research avenues. An immediate extension would involve adapting the delay-adaptive scheme to multi-agent reinforcement learning environments, where delays are not merely a computational artifact but also a byproduct of the strategic interactions among agents. Additionally, investigating the interplay between delay-adaptiveness and other types of perturbations or constraints, such as communication bandwidth and privacy requirements, could yield further insights into designing resilient distributed learning algorithms.

In summary, this research makes significant strides in understanding and mitigating the effects of delayed updates in stochastic approximation schemes under Markovian sampling. The proposed delay-adaptive scheme represents a pertinent advance toward more robust and efficient reinforcement learning algorithms, essential for tackling the challenges posed by today's increasingly complex and interconnected world.