Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning (1902.00923v3)

Abstract: We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm from the equilibrium point of an associated ordinary differential equation (ODE). We obtain finite-time bounds on the mean-square error in the case of constant step-size algorithms by considering the drift of an appropriately chosen Lyapunov function. The Lyapunov function can be interpreted either in terms of Stein's method to obtain bounds on steady-state performance or in terms of Lyapunov stability theory for linear ODEs. We also provide a comprehensive treatment of the moments of the square of the 2-norm of the approximation error. Our analysis yields the following results: (i) for a given step-size, we show that the lower-order moments can be made small as a function of the step-size and can be upper-bounded by the moments of a Gaussian random variable; (ii) we show that the higher-order moments beyond a threshold may be infinite in steady-state; and (iii) we characterize the number of samples needed for the finite-time bounds to be of the same order as the steady-state bounds. As a by-product of our analysis, we also solve the open problem of obtaining finite-time bounds for the performance of temporal difference learning algorithms with linear function approximation and a constant step-size, without requiring a projection step or an i.i.d. noise assumption.

• The paper introduces a Lyapunov function to upper bound the mean-square error in constant step-size stochastic approximation algorithms.
• It derives precise finite-time bounds for TD learning with linear function approximations without relying on i.i.d. samples or projection steps.
• By analyzing Markovian noise effects, the study offers robust numerical and theoretical insights to enhance reinforcement learning applications.

Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning

The paper authored by R. Srikant and Lei Ying provides a detailed analytical framework for understanding the finite-time dynamics of linear stochastic approximation algorithms influenced by Markovian noise, particularly within the context of reinforcement learning. The authors focus on deriving precise finite-time error bounds, emphasizing the behavior of constant step-size algorithms without relying on assumptions of i.i.d. noise or the inclusion of projection steps that other studies often require.

Key Contributions

1. Lyapunov Function and Finite-Time Bounds: The paper introduces the use of a Lyapunov function, adopting either Stein's method or Lyapunov stability theory suitable for linear ODEs. This approach provides a means to upper bound the mean-square error encountered in stochastic approximation algorithms. For constant step-size strategies, finite-time bounds on the moments of error deviation from equilibrium are established. An essential technical innovation is demonstrating that the lower-order moments are upper bounded by their Gaussian counterparts, while higher-order moments can diverge beyond a certain threshold.
2. Temporal Difference Learning: The paper addresses an open problem by deriving finite-time bounds for temporal difference (TD) learning algorithms using a linear function approximation, eliminating the need for a projection step or the assumption of i.i.d. samples. This distinction is significant given the assumption in prior works that these were necessary for analyzing the TD learning algorithms effectively.
3. Markovian Noise Model: Extending the applicability of their analysis, the authors consider a more generalized form of linear stochastic approximation under the influence of Markovian noise. This is particularly relevant for TD learning algorithms, which are a specific case of these approximation methods when viewed with Markovian samples.
4. Theoretical and Numerical Insights: The framework they develop further elucidates the convergence characteristics by proving that the 2-norm of the error does not demonstrate sub-exponential decay, a noted departure from classic assumptions related to stability in stochastic approximations. This is supported by strong numerical results applicable to a wide range of reinforcement learning tasks, ensuring relevance for practical applications.

Implications and Future Considerations

Theoretical insights into the finite-time behavior of stochastic approximation algorithms, as provided in this paper, open new avenues for research into more robust and efficient reinforcement learning algorithms. The removal of standard assumptions regarding independence and stationary noise allows these methods to be applied more broadly, and the establishment of finite-time bounds enables the assessment of algorithm performance within a set operational timeframe.

From a future research perspective, there is potential for extending this analysis to other forms of learning algorithms, perhaps exploring nonlinear stochastic approximations or expanding these bounds to varying step-size algorithms in reinforcement learning. Additionally, the understanding of higher moment behaviors could be refined to provide more nuanced insights into the long-term stability and error distribution characteristics of such systems. This work lays a solid foundation for advancing the understanding of how linear-style approximators behave when applied under real-world stochastic conditions often encountered in AI-driven decision-making processes.