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 1 σ times that of the uniform distribution; nature then samples an input from this distribution. 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:

Smoothed Online Learning is as Easy as Statistical Learning

“…In this paper, we answer both questions. With regard to the first question, the natural extension of the covering-based argument in [HRS21] would yield suboptimal dependence on σ in the nonparametric regime; instead, we obtain a nonconstructive proof through careful application of combinatorial inequalities and an adaptation of the coupling lemma of [HRS20].…”

“…Smoothed analysis in the setting of online learning was first introduced in [RST11], where the authors showed non-constructively that thresholds again become learnable in this setting. More recently, a series of papers [HRS20,HRS21] has demonstrated that the stochastic perturbation has beneficial effects in far greater generality than the class of thresholds; in fact, any classification task that is possible in the batch setting is also statistically tractable in the smoothed online setting.…”

“…In the more modern formulation of the smoothed paradigm studied in [HRS20,HRS21], instead of choosing an input that is then perturbed, the adversary chooses a distribution that is restricted to be sufficiently anti-concentrated so as to not put too much mass on the set of "hard" instances. This anti-concentration (referred to as σ-smoothness; Definition 1) is quantified by a parameter σ ≤ 1 that governs how far from independent the adversary can be.…”

“…Given a loss function, the objective of the learner is to minimize regret to the best predictor in some function class F. When F is binary-valued and has finite VC dimension d (i.e. is learnable in the batch setting), [HRS21] proved that O dT log (T /σ) regret is achievable, albeit with an inefficient algorithm. Two natural questions arise: can we extend these results to nonparametric, real-valued classes?…”

“…For the question of practical algorithms, we need to more carefully consider what we mean by efficiency. Certainly the algorithm of [HRS21] is not efficient as it requires constructing an ε-net of F as a first step, which is exponential in the VC dimension of F. A natural choice is to look to the batch setting and try to leverage the success of ERM-based approaches to achieve reasonable runtime, as is done in [KV05,HK16]; this approach is supported further by the empirical success of various heuristics for ERM [GBC16]. As such, we suppose the learner has access to an ERM oracle (Definition 2) that can efficiently optimize some loss over our function class F given as input a dataset; we analyze the time complexity of our algorithms in terms of the number of calls to this oracle.…”