Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Making

The online learning setting has become the most popular regime for studying sequential decision making with dependent and potentially adversarial data. While this paradigm is attractive due to its great generality and minimal set of assumptions [Cesa-Bianchi and Lugosi, 2006], the worstcase nature of the adversary creates statistical and computational challenges [Rakhlin et al., 2015, Littlestone, 1988, Hazan and Koren, 2016]. In order to mitigate these difficulties, Rakhlin et al. [2011] proposed the smoothed setting, wherein the adversary is constrained to sample data from a distribution whose likelihood ratio is bounded above by 1/σ with respect to a fixed dominating measure, which ensures that the adversary cannot choose worst-case inputs with high probability. As in other online learning settings, performance is measured via regret with respect to a best-inhindsight comparator [Cesa-Bianchi and Lugosi, 2006]. Recent works have demonstrated strong computational-statistical tradeoffs in smoothed online learning: while there are statisticaly efficient algorithms that can enjoy regret logarithmic in 1/σ, oracle-efficient algorithms necessarily suffer regret scaling polynomially in 1/σ [Haghtalab et al., 2022a,b, Block et al., 2022], where the learner is assumed access to an Empirical Risk Minimization (ERM) oracle that is able to efficiently optimize functionals on the parameter space. This gap is significant, because in many applications of interest, the natural scaling of σ is exponential in ambient problem dimension [Block and Simchowitz, 2022]. A natural question remains: under which types of smoothing is it possible to design oracleefficient algorithms with regret that scales polynomially in problem dimension? A partial answer was provided by Block and Simchowitz [2022], who demonstrate an efficient algorithm based on the John Ellipsoid which attains log(T/σ) poly(dimension)-regret for noiseless linear classification, and for a suitable generalization to classification with polynomial features.