Reliable Scale-Free Bandits via Empirical Likelihood

We study stochastic bandits without knowledge of reward scales or tail parameters. Almost-sure regret guarantees describe eventual learning but do not quantify expected performance or the risk of large regret in a finite run. We introduce EL-F, an anytime empirical-likelihood (EL) extension of the g-Forcing policy of Cowan and Katehakis (2020), and characterize its performance at three levels. First, for integrable rewards and non-decreasing, diverging exploration schedules g with vanishing increments, EL-F preserves the g-Forcing asymptotics for pseudo-regret R(T): R(T)/g(T) to sum i:Δ i>0Δ i almost surely as T to infinity, where Δ i is the suboptimality gap of arm i. Second, on every sub-exponential instance, for non-decreasing g(t)=ω(t) satisfying a mild feasibility condition, expected pseudo-regret is at most g(T) sum i:Δ i>0Δ i+C, where C is an instance-dependent constant independent of T. Moreover, we also establish finite-time tail bounds with super-polynomial decay in regret beyond the forced-exploration budget. For slowly varying g, we determine the exact exponent of linear regret deviations on two-armed Gaussian instances with common variance and establish a matching lower bound among policies satisfying the same expected-regret budget on every such Gaussian instance. These quantitative guarantees also cover the greedy variant obtained at zero EL radius. A fixed positive radius introduces local optimism while preserving the leading asymptotic regret. We establish coupled pathwise dominance over the greedy variant in two-armed bandits with a deterministic suboptimal arm. Experiments demonstrate reductions in prolonged suboptimal allocation compared with g-Forcing on challenging instances.