Contextual Swap Regret via Localized Martingale Absorption

Online forecasts can be assessed through errors that remain correctable in the issued record. Contextual swap regret measures the reduction in cumulative squared loss from contextual replacements and supports calibration guarantees. Distribution-weighted guarantees do not directly control randomized forecasters' realized records. Over $T$ prediction rounds, at fixed dimension and comparator radius, existing methods attain $\tilde{O}(T^{1/3})$ distribution-weighted regret, whereas the prior realized bound is $\tilde{O}(T^{3/5})$ with high probability. For affine replacements, we improve the latter to $\tilde{O}(T^{1/3})$, matching the distribution-weighted horizon exponent. Sampling fluctuations shrink with replacement distance. Our analysis exploits this relationship to let square loss absorb the sampling cost, preserving a finer prediction grid. Lower bounds establish the optimal horizon exponent at fixed confidence. We also characterize expected minimax regret across dimensions and horizons up to logarithmic factors for every fixed $R\ge 1$, where $R$ bounds replacement parameter norms. Experiments assess the grid choice on synthetic streams and census income prediction data. Within a shared ridge implementation, the finer grid reduces median realized regret to $1/15.5$ of the coarse grid value at $T=10^6$ on synthetic data, and improves realized regret and affine calibration error on the census benchmarks.