Exponential Concentration in Stochastic Approximation
This provides theoretical guarantees for convergence in stochastic optimization, which is incremental as it extends existing Markov chain results to stochastic approximation algorithms.
The paper tackles the problem of analyzing stochastic approximation algorithms by proving exponential concentration bounds for iterates, contrasting with asymptotic normality results, and applies these to algorithms like Projected Stochastic Gradient Descent, showing faster O(1/t) and linear convergence rates in certain cases.
We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results, which are more frequently associated with stochastic approximation. The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains due to Hajek (1982) to the area of stochastic approximation algorithms. We apply our results to several different Stochastic Approximation algorithms, specifically Projected Stochastic Gradient Descent, Kiefer-Wolfowitz and Stochastic Frank-Wolfe algorithms. When applicable, our results prove faster $O(1/t)$ and linear convergence rates for Projected Stochastic Gradient Descent with a non-vanishing gradient.