Fast Nonlinear Two-Time-Scale Stochastic Approximation: Achieving O(1/k) Finite-Sample Complexity

Conceptos Básicos

Proposing a new variant of two-time-scale stochastic approximation to achieve O(1/k) convergence rate in mean-squared errors.

Resumen

This paper introduces a novel approach to improve the convergence rate of two-time-scale stochastic approximation methods. By leveraging Ruppert-Polyak averaging techniques, the proposed method achieves an optimal convergence rate of O(1/k) in the mean-squared regime, surpassing the existing rate of O(1/k2/3). The key idea involves dynamically estimating operators through their samples before updating main iterates, reducing the impact of sampling noise on convergence towards desired solutions. Theoretical results and simulations demonstrate the effectiveness of this approach in reinforcement learning algorithms and linear quadratic control problems.
Introduction:

Proposes a new variant of two-time-scale stochastic approximation.
Leverages Ruppert-Polyak averaging techniques for dynamic operator estimation.
Improves convergence rate to O(1/k) in mean-squared errors.
Main Results:

Theoretical analysis shows improved finite-time convergence rates.
Simulations confirm faster convergence compared to existing methods.
Applications:

Applied to reinforcement learning algorithms with linear function approximation.
Utilized in online actor-critic methods for solving LQR problems.

Estadísticas

Underlying nonlinear operators converge at an optimal rate O(1/k).
Best known finite-time convergence rate is O(1/k2/3).

Citas

"Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate O(1/k)." - Thinh T. Doan

Ideas clave extraídas de

Fast Nonlinear Two-Time-Scale Stochastic Approximation

by Thinh T. Doa... a las arxiv.org 03-25-2024

https://arxiv.org/pdf/2401.12764.pdf

Consultas más profundas

How can this new variant be applied to other optimization or control problems?

The new variant of the two-time-scale stochastic approximation, utilizing Ruppert-Polyak averaging techniques, can be applied to a wide range of optimization and control problems. One application could be in distributed optimization scenarios where communication constraints exist. By leveraging the proposed method, one can potentially improve convergence rates in distributed settings with limited communication capabilities. Additionally, the technique could also be beneficial in game theory applications, particularly in zero-sum games where finding Nash equilibria is crucial. The improved convergence rate offered by this approach could enhance the performance of algorithms designed for solving such games.

What are potential limitations or drawbacks of using Ruppert-Polyak averaging techniques?

While Ruppert-Polyak averaging techniques offer benefits such as improved convergence rates and stability in iterative algorithms, there are some limitations and drawbacks to consider. One limitation is that these techniques may introduce additional computational complexity due to the need for maintaining averaged estimates alongside main iterates. This increased computational burden could impact real-time applications or scenarios with strict resource constraints.
Another drawback is related to sensitivity to hyperparameters such as step sizes and constants used in the averaging process. Improper tuning of these parameters may lead to suboptimal performance or even divergence of the algorithm. Additionally, implementing Ruppert-Polyak averaging requires careful consideration of noise levels and sampling characteristics since noisy estimates can affect the effectiveness of the technique.

How does this research contribute to advancements in reinforcement learning algorithms?

This research significantly advances reinforcement learning algorithms by introducing a novel variant of two-time-scale stochastic approximation with optimal finite-time convergence rates (O(1/k)). By incorporating Ruppert-Polyak averaging techniques into traditional SA methods, this approach enhances algorithmic performance by reducing noise impacts on iterate updates.
The improved convergence rates provided by this research have direct implications for reinforcement learning tasks such as policy evaluation and online actor-critic methods. These advancements enable more efficient training processes, faster model convergence, and enhanced overall performance in complex decision-making environments typically encountered in reinforcement learning applications like game theory or control systems design.