Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise (2002.01268v1)

Abstract: Linear two-timescale stochastic approximation (SA) scheme is an important class of algorithms which has become popular in reinforcement learning (RL), particularly for the policy evaluation problem. Recently, a number of works have been devoted to establishing the finite time analysis of the scheme, especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous in practice. In this paper, we provide a finite-time analysis for linear two timescale SA. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise, only the constants are affected by the mixing time of the Markov chain. With an appropriate step size schedule, the transient term in the expected error bound is $o(1/k^c)$ and the steady-state term is ${\cal O}(1/k)$, where $c>1$ and $k$ is the iteration number. Furthermore, we present an asymptotic expansion of the expected error with a matching lower bound of $\Omega(1/k)$. A simple numerical experiment is presented to support our theory.

Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise

The paper "Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise" addresses a core issue in the field of reinforcement learning (RL)—namely, the efficacy and convergence rate of linear two-timescale stochastic approximation (SA) algorithms under Markovian noise conditions. The authors provide a comprehensive finite-time analysis by deriving non-asymptotic bounds, which are crucial for understanding transient behaviors in practical applications involving SA schemes.

Two-timescale SA is essential in RL algorithms such as Gradient Temporal Difference (GTD) learning, where evaluating policies accurately and efficiently is critical, especially in off-policy paradigms. The paper advances existing literature by establishing rigorous performance bounds without A-priori stability assumptions and eliminating projection steps traditionally used to ensure stability.

Under consideration are both martingale differences and Markovian noise scenarios, which are ubiquitous in RL practice due to the inherent nature of the environment interactions. The authors demonstrate that the convergence rates in these contexts align in terms of asymptotic performance, although the constants involved vary depending on the mixing time of the Markov chain governing the noise.

Key Numerical Findings

• Convergence Rate: With an appropriately chosen step size, the transient error term in the expected error bounds decays as $o(1/k^c)$ and the steady-state error term scales as ${\cal O}(1/k)$, where $c > 1$ and $k$ is the iteration number.
• Lower Bound Analysis: The paper establishes a matching lower bound $\Omega(1/k)$ by conducting an asymptotic expansion of the expected error, reinforcing the tightness of their results.
• Error Term Contributions: The analysis explores the contributions from various noise components and finds faster decay rates for Markovian noise-driven iterates compared to martingale noise-driven iterates amid similar step size schedules.

Theoretical and Practical Implications

From a theoretical standpoint, these results provide significant insight into the convergence dynamics of two-timescale SA schemes. Such bounds clarify under what circumstances optimal convergence is possible or hindered by step size configurations and noise characteristics. Practically, these insights are critical for tuning RL algorithms—particularly GTD methods—to ensure efficient learning in environments modeled by Markov processes.

Moreover, avoiding projection steps allows for genuine practical deployment of these schemes without contriving stability through artificial constraints. This aligns algorithmic practice closely with theoretical behaviors, promising more reliable application in real-world scenarios.

Future Directions

The paper opens several pathways for future exploration. There is potential to extend similar finite-time analyses to nonlinear stochastic approximation or multi-scale reinforcement learning architectures, which are becoming increasingly significant in high-dimensional RL tasks. Another promising direction could involve deepening the analysis of noise mixing times and their influence on the constants in convergence bounds—empowering even more refined tuning techniques for RL implementations.

In conclusion, this paper enriches the deterministic discussion surrounding stochastic approximation schemes with finite-time analyses, paving the way for robust reinforcement learning deployments in naturally noisy environments. The technical rigor and insights it provides lay solid groundwork for subsequent innovations both in methodology and application domains.