Eventually LIL Regret: Almost Sure lnln T Regret for a sub-Gaussian Mixture on Unbounded Data

Abstract

We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural “Ville event” ℰ_α, this regret till time T is bounded by ln^2(1/α)/V_T + ln (1/α) + lnln V_T up to universal constants, where V_T is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to ln(1/α) + lnln V_T if V_T ≥ln(1/α).) If the data were stochastic, then one can show that ℰ_α has probability at least 1-α under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event ℰ_0 of probability one, the regret on every path in ℰ_0 is eventually bounded by lnln V_T (up to constants). We explain how this work helps bridge the world of adversarial online learning (which usually deals with regret bounds for bounded data), with game-theoretic statistics (which can handle unbounded data, albeit using stochastic assumptions). In short, conditional regret bounds serve as a bridge between stochastic and adversarial betting.