A Concentration Bound for Stochastic Approximation via Alekseev's Formula

Gugan Thoppe; Vivek S. Borkar

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arXiv · 1506.08657

A Concentration Bound for Stochastic Approximation via Alekseev's Formula

Gugan Thoppe·Vivek S. Borkar

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Abstract

Given an ODE and its perturbation, the Alekseev formula expresses the solutions of the latter in terms related to the former. By exploiting this formula and a new concentration inequality for martingale-differences, we develop a novel approach for analyzing nonlinear Stochastic Approximation (SA). This approach is useful for studying a SA's behaviour close to a Locally Asymptotically Stable Equilibrium (LASE) of its limiting ODE; this LASE need not be the limiting ODE's only attractor. As an application, we obtain a new concentration bound for nonlinear SA. That is, given $\epsilon >0$ and that the current iterate is in a neighbourhood of a LASE, we provide an estimate for i.) the time required to hit the $\epsilon-$ball of this LASE, and ii.) the probability that after this time the iterates are indeed within this $\epsilon-$ball and stay there thereafter. The latter estimate can also be viewed as the `lock-in' probability. Compared to related results, our concentration bound is tighter and holds under significantly weaker assumptions. In particular, our bound applies even when the stepsizes are not square-summable. Despite the weaker hypothesis, we show that the celebrated Kushner-Clark lemma continues to hold. %

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Gugan Thoppe, Vivek S. Borkar. 2015-06-26. A Concentration Bound for Stochastic Approximation via Alekseev's Formula. https://arxiv.org/abs/1506.08657

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