Smoothed Analysis with Adaptive Adversaries

Abstract

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by a multiplicative factor from the uniform distribution; nature then samples an input from this distribution. This interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm's current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: Online learning, Online discrepancy and Dispersion in online optimization. We show that in these setting, we can get bounds that match bounds we can get for non-adaptive adversaries.