Problem-Dependent Dynamic Regret over a Predictor Class with One-Gradient Feedback

Many online learning systems now have access to several gradient predictors at once, such as simulators, replay buffers, neural surrogates, or time-series forecasters, and the quality of these predictors varies from round to round. We study online convex optimization under one-gradient feedback against an arbitrary moving comparator, with a user-supplied finite class $\Pi$ of gradient-field predictors, and ask whether a single algorithm can compete with the best predictor in $\Pi$ at a logarithmic cost, fall back to a small-loss bound when none of the predictors is useful, and still retain a path-length-adaptive worst-case guarantee when the predictors are adversarial. We give a positive answer through PH-Sword (Predictable-Hybrid Sword.optimism), which combines an exp-concave Hedge over $\Pi$, optimistic projected-gradient base learners on a geometric learning-rate grid, and a correction-term optimistic-entropy meta learner with prediction-error doubling. For any comparator sequence $\mathbf{u}$, PH-Sword attains dynamic regret $\tilde{O}(R\_0\sqrt{(1+P\_T/D)(1+P\_T/D+\min \{\mathcal{E}\_\text{*}/G^2,2LF\_T(\mathbf{u})/G^2\}+\log |\Pi |)})$ up to logarithmic factors, where $R\_0=DG+L{D}^2$, $P\_T$ is the path length, $F\_T(\mathbf{u})$ the cumulative comparator loss, and $\mathcal{E}\_\text{*}$ the best-in-class field error. The bound merges the field-controlled, small-loss, and worst-case regimes into one deterministic guarantee that improves over prior one-gradient dynamic-regret algorithms. On six controlled benchmarks the five predicted scaling laws hold simultaneously: PH-Sword reaches 17% of OGD's final dynamic regret and 54% of Ader's on a slow-drift benchmark, and we observe that a single noisy predictor without model selection can actually do worse than using no predictor at all.