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

TL;DR

This paper establishes almost sure regret bounds for a sub-Gaussian mixture model in unbounded data settings, bridging stochastic and adversarial online learning.

Contribution

It provides path-wise regret bounds for a classic sub-Gaussian mixture, connecting stochastic assumptions with adversarial online learning frameworks.

Findings

Regret bounded by ^2(1/)/V_T + (1/) +    V_T on Ville event _.

On the Ville event _0 of probability one, regret is eventually bounded by    V_T.

The work links stochastic assumptions with adversarial online learning, enabling regret bounds for 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'' $\mathcal E_\alpha$, this regret till time $T$ is bounded by $\ln^2(1/\alpha)/V_T + \ln (1/\alpha) + \ln \ln V_T$ up to universal constants, where $V_T$ is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to $\ln(1/\alpha) + \ln \ln V_T$ if $V_T \geq \ln(1/\alpha)$.) If the data were stochastic, then one can show that $\mathcal E_\alpha$ has probability at least $1-\alpha$ under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event $\mathcal E_0$ of probability one, the regret on every path in $\mathcal E_0$ is eventually bounded by $\ln \ln V_T$ (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.