Shrinking-Tube Concentration for Adaptive Markovian Stochastic Approximation

2026-09-24

AI Summary

This study addresses the lack of convergence stability guarantees for adaptive algorithms in dynamic data reshaping by establishing a concentration bound on the shrinking tube for projected stochastic approximation, ensuring that iterates remain within a time-varying tightening target neighborhood with high probability. Methodologically, it integrates backward kernel substitution, finite-time mean squared error bounds, and block-wise maximal first-exit arguments to precisely characterize the sharp trade-off between tolerance contraction and exit probability decay, while proving the unimprovability of the derived exponential lower bound. The theory is further extended to noisy and biased settings, quantifying how gradient precision affects long-term policy reliability in inventory learning. Ultimately, this work provides a rigorous finite-time theoretical foundation for adaptive learning systems.

Abstract

Adaptive algorithms increasingly make decisions while reshaping the dynamics that generate their future data. We establish a shrinking-tube concentration bound for projected stochastic approximation driven by an adaptive Markov chain. The bound guarantees, with high probability, that every iterate after a chosen time remains within a tolerance around the target that tightens over time. The probability of any exit after the chosen time admits a polynomially decaying upper bound, and a matching lower bound shows that its polynomial exponent cannot be improved in general under finite second moments. The result therefore identifies a sharp tradeoff between how quickly the tolerance shrinks and how rapidly the probability of any future exit decreases. We also extend the analysis to recursions with additional martingale-difference noise and predictable bias, showing how growth in the martingale-difference noise scale slows the decay of the exit-probability bound while predictable bias restricts the admissible tube shrinkage. The proof combines backward kernel replacement, a finite-time mean-squared-error bound, and a blockwise maximal first-exit argument. We apply the theory to inventory learning with stockout-dependent demand and fixed stockout costs, and quantify how numerical gradient accuracy affects the all-future reliability of the resulting policies.

Problem

• Adaptive Markovian stochastic approximation
• Shrinking-tube concentration
• Exit probability
• Martingale-difference noise
• Predictable bias

Innovation

• Shrinking-Tube Concentration
• Adaptive Markov Chain
• Stochastic Approximation
• Backward Kernel Replacement
• Exit Probability Bound