A second order regret bound for NormalHedge

We consider the problem of prediction with expert advice for ``easy’’ sequences. We show that a variant of NormalHedge enjoys a second-order $ε$-quantile regret bound of $O\big(\sqrt{V_T \log(V_T/ε)}\big) $ when $V_T > \log N$, where $V_T$ is the cumulative second moment of instantaneous per-expert regret averaged with respect to a natural distribution determined by the algorithm. The algorithm is motivated by a continuous time limit using Stochastic Differential Equations. The discrete time analysis uses self-concordance techniques.

💡 Research Summary

The paper studies online prediction with expert advice under the classic setting where a learner must combine the advice of N experts, each round incurring a loss bounded by B. The goal is to bound the ε‑quantile regret, i.e., the loss of the ⌊N ε⌋‑th best expert after T rounds. While many algorithms adapt to “easy” sequences via variance‑based first‑order or second‑order bounds, and others achieve quantile‑based bounds, no prior method simultaneously achieved a second‑order quantile bound. Freund (2016) conjectured that NormalHedge could fill this gap.

The authors introduce a general “Constant Potential” (CP) framework. A potential function ϕ(y,t) depends on a regret coordinate y and a continuous time variable t. A “good” potential must be jointly strictly convex, three‑times differentiable, monotone in y, non‑increasing in t, converge as t → ∞, be projection‑compatible, and satisfy the backward heat equation ∂ₜϕ = −½ ∂²_{yy}ϕ. Two concrete potentials are examined:

1. Exponential potential ϕ_exp(y,t)=exp(√2 η y − η² t) (η>0, domain ℝ).
2. Normal‑Hedge‑BH potential ϕ_NH(y,t)=t^{‑½} exp(y²/(2t)) (domain