Smoothed Analysis of Sequential Probability Assignment

TL;DR

This paper explores smoothed analysis for sequential probability assignment, establishing minimax rates and developing algorithms that leverage MLE oracles to achieve sublinear regret.

Contribution

It introduces a general reduction framework linking minimax rates in smoothed adversaries to transductive learning, and develops efficient algorithms with theoretical guarantees.

Findings

Optimal logarithmic rates for parametric classes

Sublinear regret algorithms for general classes

Reduction framework connecting smoothed adversaries to transductive learning

Abstract

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.