Asymptotic Regularity of a Generalised Stochastic Halpern Scheme

Area of Science:

• Optimization Theory
• Stochastic Analysis
• Machine Learning

Background:

• Stochastic optimization methods are crucial for large-scale problems.
• Existing iterations like Halpern and Krasnoselskii-Mann have limitations in stochastic settings.
• Tikhonov regularization is used to stabilize optimization problems.

Purpose of the Study:

• To develop a generalized stochastic iteration with abstract incorporation of stochasticity.
• To analyze asymptotic regularity rates for this new iteration.
• To explore applications in optimization and reinforcement learning.

Main Methods:

• A generalized stochastic Halpern-style iteration incorporating a Krasnoselskii-Mann style mapping.
• Abstract formulation of stochasticity to accommodate various sampling methods.
• Analysis of asymptotic regularity rates, including linear and quadratic bounds.

Main Results:

• Achieved abstract, general, and uniform rates of asymptotic regularity.
• Obtained linear rates for specific cases (stochastic Halpern and Krasnoselskii-Mann with Tikhonov regularization).
• Derived quadratic rates in inner product spaces for the general iteration.

Conclusions:

• The generalized iteration unifies and improves existing stochastic optimization schemes.
• Rates can be adapted for oracle complexity bounds and practical variance management (e.g., minibatching).
• Potential for novel Q-learning methods in reinforcement learning through instantiation.