Betting Across Kernels: Anytime-Valid Two-Sample Testing over Kernel Families

Sequential kernel two-sample tests rely on a fixed RKHS chosen in advance, but the right kernel depends on the unknown alternative. We give the first anytime-valid two-sample test that adapts over a continuous family of RKHSs, by composing Shekhar-Ramdas online betting within each RKHS with method-of-mixtures aggregation across kernel parameters. At a point-mass prior the test recovers single-kernel Shekhar-Ramdas exactly. For a discrete family of K kernels, its stopping time is (log K + log(1/α) + C) / MMD² at the best kernel k*; continuous priors achieve the same leading rate plus a burn-in correction. We obtain the first time-uniform confidence sequence on the best-in-family divergence sup over λ of MMD²(k_λ), and a matching lower bound from a bandpass-Gaussian construction shows the log K price is information-theoretically unavoidable. Experiments cover three settings: across six LLM-auditing scenarios the optimal bandwidth varies but the mixture stays competitive throughout (cross-provider Claude vs GPT detected in 18 samples at α = 0.05; the mixture beats a post-hoc oracle by 1.5x on regime-uncertain streams); on capability-rate evaluation it detects an 88% vs 49% accuracy gap between Sonnet 4.6 and Haiku 4.5 in 22 samples; and on a classical A/B test it preserves Type I control under continuous monitoring while a naive peeking z-test inflates to a 34% false-positive rate.