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

Abstract

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 $O\!\big(KD/\varepsilon^2+(R-K)\logK/Δ^2\big)$ versus a memoryless re-estimator's $\Theta (RD/{\epsilon }^2)$, an advantage $(R-K)\Theta (D/{\epsilon }^2)$ growing with dimension $D$ and recurrence density. The mechanism is a decoupling: recognizing which of $K$ seen regimes is active costs $O(\log K/{\Delta }^2)$, independent of $D$, whereas estimating a regime costs $\Theta (D/{\epsilon }^2)$. We prove this is tight: matching lower bounds give recognition $\Theta (\log K/{\Delta }^2)$ and a memoryless-class bound $\Omega (RD/{\epsilon }^2)$, 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 $e^{\Theta (D)}$); 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 $\propto D^{1.04}$, recognition $D$-independent); the one real sequential stream sits in the $D=1$ near-null corner.

21 pages, 2 figures. v2: expanded related work (switching-regret with memory; universal coding for repeating statistics; concurrent TTA-learnability work); two clarifying remarks. No changes to results