Core Concepts
We present an efficient randomized algorithm that guarantees O(√T log T) expected maximum swap regret (MSR) in online binary prediction, matching the Ω(√T) lower bound up to a logarithmic factor.
Abstract
The key insights are:
The MSR can often be significantly smaller than the K1 calibration error, despite their linear relationship in the worst-case. This allows us to bypass the Ω(T^0.528) lower bound for the K1 calibration error.
We establish a general lemma (Lemma 5.2) that attributes MSR to bucket-wise biases. This lemma plays a crucial role in our analysis.
We show that the guarantee |b̂qi - qi| ≤ O(1/√ni) can be approximately achieved in the online binary prediction setting, using a refinement of the result from Noarov et al. (2023).
Combining Lemma 5.2 with the bound on |b̂qi - qi|, we obtain the final O(√T log T) expected MSR guarantee.
Our algorithm works in the standard online binary prediction setting. In each round t, the algorithm makes a prediction pt ∈ [0, 1] and the adversary reveals the true state θt ∈ {0, 1}. Both pt and θt can depend on the past history, but they cannot depend on each other. This allows our algorithm to leverage the power of randomization.