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

• The paper establishes that gradient-free, warm-start library recovery decouples regime recognition from estimation, achieving minimax optimality with zero forgetting.
• It demonstrates that recognition cost is dimension-independent, leading to significant sample savings compared to memoryless learners.
• The study shows that local, append-only updates rival Bayesian methods, enabling efficient continual learning on resource-constrained devices.

Gradient-Free Library Recovery: Amortized-Regret Separation in Continual Learning

Problem Formulation and Motivation

The paper "Gradient-Free Warm-Start Library Recovery: an Amortized-Regret Separation" (2606.21253) presents a formal analysis of gradient-free, local, online, append-only continual learners in the context of recurring-regime streams. The primary object is recovery cost following regime changes: the time or number of samples required for a learner to re-adapt after a switch in latent distribution. Memoryless learners always re-estimate from scratch, while warm-start library learners recognize previously-seen regimes and initialize from stored prototypes, only appending new entries on genuine novelty.

The study precisely characterizes the advantage conferred by the library mechanism, which is especially relevant for edge devices and streaming scenarios with strict computational and memory constraints. The retention of identity-clean, append-only memory enables zero-forgetting by construction, distinguishing this approach from gradient-based continual learning methods employing regularization or replay.

Recognition–Estimation Decoupling and Cost Separation

The core of the analysis is the decoupling between recognition and estimation. Estimating a $D$-dimensional categorical regime to total variation $\le\varepsilon$ requires $\Theta(D/\varepsilon^2)$ samples, whereas recognizing which of $K$ previously-seen, separated regimes is active costs only $O(\log K/\Delta^2)$ samples, independent of $D$, for separation parameter $\Delta$ (minimum pairwise total variation). The decoupling is rigorously proven to be tight: lower bounds for recognition ($=\Theta(\log K/\Delta^2)$) and estimation ($=\Omega(D/\varepsilon^2)$) are established in the minimax sense.

The theoretical result is an amortized recovery cost separation:

$$\text{Cost(W)} \le K\,\Theta(D/\varepsilon^2) + (R-K)\,O(\log K/\Delta^2)$$

where $\le\varepsilon$0 is the number of blocks (regime instances). In contrast, memoryless learners incur

$\le\varepsilon$1

yielding an advantage that grows with both regime dimension $\le\varepsilon$2 and recurrence density $\le\varepsilon$3.

Paradigm Reconciliation: Rate Ties with Bayesian Baselines

The paper advances a contradictory claim: gradient-free warm-start library recovery does not surpass fair Bayesian online learners (e.g., BOCPD, online HMMs with state birth, Dirichlet-process mixtures) that also maintain per-regime libraries. Both attain the same amortized cost structure. The significance is thus in realization: the gradient-free learner matches the recovery performance using only local, append-only updates, zero forgetting, and no end-to-end backpropagation.

A count-calibrated variant, which stores each prototype as accumulated counts and uses the prototype's own precision for warm start, further sharpens this tie. It achieves the same leading constant as the Bayesian baseline up to a bounded overshoot per recurrence, $\le\varepsilon$4, where $\le\varepsilon$5 is the minimum category probability. Under interior support ($\le\varepsilon$6), this overshoot is $\le\varepsilon$7, never negative or surpassing the Bayesian baseline, and eliminates hyperparameter tuning.

At equal sample rate and memory order ($\le\varepsilon$8), the count-calibrated learner is transcendental-free per step, contrasting with Bayesian baselines which require $\le\varepsilon$9 transcendental evaluations (log-gamma/digamma) per symbol.

Limits and Geometric Boundaries

The recognizer’s capacity is bounded by geometric packing laws for the simplex. For randomly drawn regimes, typical pairwise separation collapses past a random-model wall $\Theta(D/\varepsilon^2)$0, and optimal packings obey a capacity wall $\Theta(D/\varepsilon^2)$1. The separation advantage disappears when regimes overlap ($\Theta(D/\varepsilon^2)$2), or when block boundaries are not supplied: autonomous segmentation is impossible at the packing wall due to the false-alarm/detection delay frontier. No detector (fixed, adaptive, or Bayesian) can avoid divergent delay as the separation vanishes.

Empirical Corroboration

Empirical evaluations on synthetic streams and real regime distributions (e.g., $\Theta(D/\varepsilon^2)$3-mer distributions from genomic data, varying $\Theta(D/\varepsilon^2)$4 and GC content $\Theta(D/\varepsilon^2)$5–$\Theta(D/\varepsilon^2)$6) consistently corroborate the theory. For synthetic grids, the cold estimation cost rises linearly in $\Theta(D/\varepsilon^2)$7 ($\Theta(D/\varepsilon^2)$8), while recurrence recognition remains flat, dimension-independent. On real genomic regimes, memoryless recovery cost scales as $\Theta(D/\varepsilon^2)$9, with warm-start library advantage growing substantially with dimension—the gap widens from $K$0 to $K$1 samples for $K$2 to $K$3. Controls confirm that the advantage is genuinely recognition-driven.

On a block-structured neuroscience task (IBL biased-choice), the theory predicts a near-null advantage ($K$4 case); empirical results concur. Across Bayesian baseline variants (conjugate-Dirichlet model-averaging, BOCPD, DPMM), the count-calibrated learner consistently ties on recurrence recovery.

Differentiation from Existing Architectures

The analyzed learner differs from comparable gradient-free approaches (e.g., HiCL (2508.16651)) by its strict avoidance of backpropagation, reliance on hard sequential hypothesis testing for recognition, and identity-clean, append-only memory construction. It is also distinct from mechanisms like equilibrium propagation [scellier2017equilibrium], which target energy-based supervised gradients.

Compared to meta-learning and prediction-with-experts theory, the recurrence term here formalizes the sample-complexity reduction possible by reusing prior-regime prototypes within recurring categorical streams.

Practical and Theoretical Implications

This work establishes the validity and limitation of gradient-free library-based continual learners on recurring-regime streams. Strong claims: Library-based recognition produces a dimension-independent recovery term and attains minimax optimality, with zero forgetting by construction.

The practical implication is the possibility of deploying continual learners on edge devices or in real-time streaming applications without the computational cost of backpropagation, with precise guarantees about recovery efficiency. Theoretical implications include explicit geometric boundaries on recognition capacity and an impossibility result for autonomous detection once regime separation falls below the packing threshold.

Future directions could extend the approach to richer regime structures, partial observability, and real-world online learning benchmarks. Integration with neuromorphic architectures or fixed-point arithmetic platforms is facilitated by the complete elimination of transcendental operations.

Conclusion

This paper provides a rigorous cost-separation theorem for gradient-free, append-only continual learning on recurring-regime streams. The recovery advantage is provably dimension-dependent and minimax tight, with paradigm-level ties to Bayesian gold-standard learners. Recognition-based decoupling is constrained by geometric packing, and full autonomy is unattainable at the overlap wall. Empirical validation across synthetic and real tasks confirms the theory, establishing gradient-free library recovery as a technically robust, deployment-ready alternative for continual learning under regime recurrence (2606.21253).