Smoothed Analysis with Adaptive Adversaries
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We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by 1σ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for adaptive adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems: -Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d. We bound the regret by Õ(√(T dln(1/σ)) + d√(ln(T/σ))). This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ_2 unit ball. We bound the ℓ_∞ norm of the discrepancy vector by Õ(ln^2( nT/σ) ). -Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is ( σ/√(Tℓ), Õ(√(Tℓ)))-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.
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