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Stochastic time-varying optimization is an integral part of learning in whichthe shape of the function changes over time in a non-deterministic manner. Thispaper considers multiple models of stochastic time variation and analyzes thecorresponding notion of hitting time for each model, i.e., the period afterwhich optimizing the stochastic time-varying function reveals informativestatistics on the optimization of the target function. The studied models oftime variation are motivated by adversarial attacks on the computation of valueiteration in Markov decision processes. In this application, the hitting timequantifies the extent that the computation is robust to adversarialdisturbance. We develop upper bounds on the hitting time by analyzing thecontraction-expansion transformation appeared in the time-variation models. Weprove that the hitting time of the value function in the value iteration with aprobabilistic contraction-expansion transformation is logarithmic in terms ofthe inverse of a desired precision. In addition, the hitting time is analyzedfor optimization of unknown continuous or discrete time-varying functions whosenoisy evaluations are revealed over time. The upper bound for a continuousfunction is super-quadratic (but sub-cubic) in terms of the inverse of adesired precision and the upper bound for a discrete function is logarithmic interms of the cardinality of the function domain. Improved bounds for convexfunctions are obtained and we show that such functions are learned faster thannon-convex functions. Finally, we study a time-varying linear model withadditive noise, where hitting time is bounded with the notion of shapedominance.

A Hitting Time Analysis for Stochastic Time-Varying Functions with Applications to Adversarial Attacks on Computation of Markov Decision Processes