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Exact-Form Regret for Gradient Descent, Mirror Descent and Follow-the-Regularized-Leader

arXiv · AI, language, vision and robotics · article · Sep 8, 2026 · UTC

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used b

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