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.
