Efficient high-dimensional universal swap-regret forecasting

Determine whether there exists an algorithm that, for d-dimensional prediction spaces in adversarial online settings, produces forecasts guaranteeing every downstream agent swap regret diminishing at a rate of O(T^{O(1)}) while achieving per-round running time that scales polynomially with d.

Background

The paper provides algorithms that guarantee diminishing swap regret for all downstream agents by ensuring forecasts are unbiased over carefully chosen events rather than fully calibrated. In low dimensions (d = 1, 2), they obtain strong rates and computationally feasible procedures, and in higher dimensions they achieve dimension-independent exponents under behavior assumptions but with algorithms whose complexity scales poorly with dimension.

The authors explicitly identify computational efficiency in high dimensions as the central unresolved issue: their current methods, like calibration-based approaches, have complexity that grows badly with d. They pose a concrete open question asking for a polynomial-in-d per-round algorithm that maintains sublinear (polynomial-rate) swap regret guarantees for all downstream agents.