Swap-agnostic learning compares a forecaster with a different hypothesis on each level set of its predictions. A recent finite-class second-order approachability algorithm attains the sharp $\widetilde O(T^{1/3})$ time exponent. Extending it to infinite classes and optimizing its offset with an oracle were left open. We give a black-box reduction to online square-loss regression. The algorithm maintains one regressor per prediction bucket, locates an adjacent sign crossing with logarithmically many oracle queries, and updates only the sampled bucket. A square-loss identity preserves the comparator's negative quadratic activity. This yields, for any base regret $R$, $\operatorname{SwapReg}_T=O \left(NR(T/N)+T/N^2\right)$ up to loss and confidence factors. Logarithmic-regret regression therefore retains the $T^{1/3}$ exponent for infinite classes. For half-Brier loss and bounded $d$-parameter linear predictors, constrained Online Newton Step gives $\widetilde O(T^{1/3}d^{2/3})$ contextual swap regret. A direct sum of recalibration instances gives a matching $\Omega(T^{1/3}d^{2/3})$ lower bound for a fixed class of pseudodimension $d$. Kernel ridge regression further gives a spectrum-adaptive log-determinant bound for bounded RKHS predictors. For every sequential-entropy exponent $p>0$, fixed convex classes prove that both nonparametric rates are minimax optimal up to logarithmic factors. We also obtain fast canonical-link models and an oracle-efficient offline conversion with excess $\widetilde O((d/m)^{2/3})$.
Keywords: swap regret, agnostic learning, regression, omniprediction, proper losses, calibration