Gradient-Free Warm-Start Library Recovery: an Amortized-Regret Separation

Continual learning that is gradient-free, local, online, and append-only is attractive for edge and streaming deployment, but its value is usually argued informally. We give a provable account on recurring-regime streams. Given segmentation, a warm-start library learner attains amortized recovery cost O!big(KD/varepsilon2+(R-K)logK/Delta2big) versus a memoryless re-estimator's Theta(RD/varepsilon2), an advantage (R-K),Theta(D/varepsilon2) growing with dimension D and recurrence density. The mechanism is a decoupling: recognizing which of K seen regimes is active costs O(log K/Delta2), independent of D, whereas estimating a regime costs Theta(D/varepsilon2). We prove this is tight: matching lower bounds give recognition Theta(log K/Delta2) and a memoryless-class bound Omega(RD/varepsilon2), so each term is individually minimax-tight (the joint statement is conditional). The separation is born-immune (a memoryless learner's advantage is identically zero) and paradigm-level: it matches, and does not beat, a fair spawn-capable Bayesian baseline; the contribution is attaining this cost structure without end-to-end backprop and with zero forgetting by construction. A count-calibrated variant ties the baseline's leading constant up to a bounded, never-negative per-recurrence overshoot, hyperparameter-free and with no per-step transcendentals. We bound the scope: recognizable regimes are capped by simplex packing (walls eTheta(D)); autonomous segmentation is impossible at the packing wall (no detector escapes the false-alarm/delay frontier as regimes overlap); the advantage vanishes under overlap. The dimension-dependent separation is corroborated on synthetic streams and real k-mer genome distributions (memoryless cost propto D1.04, recognition D-independent); the one real sequential stream sits in the D=1 near-null corner.