A Contraction Framework for Stochastic Operators with Bootstrapping: Application to TD Learning

arXiv:2609.29961v1 Announce Type: new Abstract: Many iterative algorithms rely on bootstrapping. A variable is updated using a second, frozen copy as a target, which is periodically replaced with the updated variable. Majorize-minimize and inexact proximal-point methods share this structure, as does temporal-difference (TD) learning. However, existing convergence guarantees for scenarios that combine sampled updates with targets refreshed only every $K$ steps rely on the specific structure of the update, such as linear approximation or gradient-based inner steps, and on uniformly bounded sampling error. We instead model the sampled update as a stochastic operator on the parameter space, which reduces the analysis to a contraction argument that needs no gradient structure and allows the sampling error to grow with the iterates. Within this framework, we derive a finite-time bound for i.i.d. samples and any target-update period $K$. We show that the iterates converge geometrically in root mean square to a ball around the fixed point, provided the sensitivity to the frozen target is smaller than the contraction slack of the inner map. Existing deterministic frozen-target contraction and stochastic-gradient-type bounds follow as special cases of our framework, and simulations of TD learning reproduce the predicted contraction rate and scaling of the error floor with the step size.

Abstract:Many iterative algorithms rely on bootstrapping. A variable is updated using a second, frozen copy as a target, which is periodically replaced with the updated variable. Majorize-minimize and inexact proximal-point methods share this structure, as does temporal-difference (TD) learning. However, existing convergence guarantees for scenarios that combine sampled updates with targets refreshed only every $K$ steps rely on the specific structure of the update, such as linear approximation or gradient-based inner steps, and on uniformly bounded sampling error. We instead model the sampled update as a stochastic operator on the parameter space, which reduces the analysis to a contraction argument that needs no gradient structure and allows the sampling error to grow with the iterates. Within this framework, we derive a finite-time bound for i.i.d. samples and any target-update period $K$. We show that the iterates converge geometrically in root mean square to a ball around the fixed point, provided the sensitivity to the frozen target is smaller than the contraction slack of the inner map. Existing deterministic frozen-target contraction and stochastic-gradient-type bounds follow as special cases of our framework, and simulations of TD learning reproduce the predicted contraction rate and scaling of the error floor with the step size.

Machine Learning (cs.LG); Signal Processing (eess.SP); Optimization and Control (math.OC)

arXiv:2609.29961 [cs.LG]

(or arXiv:2609.29961v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.29961 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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A Contraction Framework for Stochastic Operators with Bootstrapping: Application to TD Learning

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