arXiv subjects

Gunner Levi Howe

Publications and source records attributed to Gunner Levi Howe.

8 recordsLinked to original sources

arXiv2026

A Noise Optimum in Rehearsal-Free Continual Learning: Isolation, Mechanism, and Scope

Injecting stochastic noise into a consolidation rule can improve a network's retention of earlier tasks up to an optimal level, then degrade it -- an inverted-U in retention vs. noise. This paper isolates what produces that optimum and maps where it holds, entirely in simulation. (1) Phenomenon: the retention inverted-U appears on several related-task continual-learning benchmarks (Split-MNIST, FashionMNIST, continual Yin-Yang). (2) Isolation: a magnitude-matched ladder shows the effect requires coherent restoring toward the consolidated weights -- a random-direction force of identical magnitude produces no optimum, and a coherent force toward the wrong target actively hurts. (3) Active ingredient: most of the optimum is recovered by coupling the anchor gain to the injected-noise variance sigma^2 -- a one-line rule that neither Ornstein-Uhlenbeck Adaptation (fixed gain) nor MESU (posterior-variance gain) implements. A forced Ornstein-Uhlenbeck calculation derives the rising flank and predicts that the optimal noise rises with per-task interference g -- confirmed out-of-sample in direction against pre-existing measurements (the exponent is unresolved at our grid). The barrier-conditioning of the originating Doob h-transform is a low-sigma safety net that bounds forgetting where the coupled gain is too weak. (4) Scope: the optimum requires shared task structure -- it is absent on permuted-MNIST, and a controlled rotated-vs-permuted comparison localizes the boundary to task structure; the precise governing quantity is left open. (5) Length: at matched severity the advantage persists but attenuates with task count, and we show no rotation family can attribute the trend (a compact-group identity). A single-seed BrainScaleS-2 demonstration of the originating rule is reported separately (Howe, arXiv:2607.06924); this paper makes no hardware claim.

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arXiv2026

Capability Emergence Can Be Forecast: Per-Seed, In Advance, With Calibrated Intervals, Certified False Alarms, and a Blind Pre-Registered Gate

Emergent capabilities are widely treated as unpredictable: loss improves smoothly while abilities appear abruptly. Prior work offers early-warning indicators but never scores them as forecasts: no lead time at controlled false-alarm rate, no calibration, no negatives, no blind tests. We supply that discipline and show that, in grokking model systems and small language models, emergence timing is forecastable per run, in advance, with calibrated uncertainty. Across 30 transformers at identical configuration, the formation time of the previous-token head forecasts each seed's induction-head emergence at Spearman rho=0.977 with median lead 975 steps (~15% of training); a best-case loss rule ties the ranking with 50-step lead (a nowcast). Conformal intervals covered 15/15 held-out seeds, and the frozen rule passed blind pre-registered gates on TWO never-seen configurations (10/10 and 9/10 coverage). A trap-language rung then attacked our own rule as pre-registered: where previous-token context pays for the task itself, the bare precursor false-alarms on 10/10 capability-blocked runs, while the mechanism-composed conjunction is certified in both language classes (0 false alarms) and times emergence at rho=1.000. Finally, a gap-origin study broke the fixed offset (both lr and batch move the gap ~2.3x; no external clock owns it) and revealed the law beneath: across 80 valid-anchor runs the anchor fires at 0.843 of time-to-emergence -- t_event ~= 1.19 x t_anchor -- and this multiplicative rule passed its own blind gate (5/5) at a third unseen configuration. False alarms are certified against 33 manufactured negatives. The precursor leads across 3 public model families (Pythia, OLMo, OLMo-2; 7 suites), with OLMo-2 at 1B tokens showing precursor formed, capability absent. Four pre-registered kill criteria fired and are reported. Every freeze precedes its data in a public commit chain.

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arXiv2026

The Automaton Underneath: The Additive Input Pathway Is a Parasitic Attractor for State Tracking in Householder Linear RNN

Linear RNNs with input-dependent Householder-product transitions (DeltaNet/DeltaProduct-class) can provably represent hard state-tracking automata, yet trained models fail to length-generalize -- a gap recent work attributes to optimization, without a causal account. We give one, in a pre-registered, within-architecture causal ablation: the same model with one term deleted -- the additive input injection $b_t = W_b e_t$. With $b_t$, models fit length 32 and collapse out-of-distribution on parity, $S_4$, $A_5$, and non-solvable $S_5$ word problems (0.20 at position 512 on $S_5$). Without it -- input acting only through the orthogonal transitions -- the same architecture learns the exact automaton: median accuracy 1.00 at 16x the training length, at every width admitted by a representation law we state and test: the minimal number of Householder factors per token equals the maximal reflection length of the task's generators in the format-pinned representation (parity 1, $S_4$ 3, $A_5$ and $S_5$ 4); below it, nothing fits. The contrast with DeltaProduct's $S_4$/$A_5$ at $n_h{=}2$ (group-element classification, $SO(3)$ realization) shows the law is representation-relative: task format is a hidden variable in state-tracking benchmarks. Two pre-registered arms locate the mechanism. (i) Initialized at a verified-exact solution with $W_b{=}0$, Adam grows the additive path and pulls the model off the exact solution; $-b$ controls stay at 1.00. (ii) Our registered prediction that the fit routes through $b_t$ fired its kill criterion: all 49 fitting seeds retain in-domain fit under $W_b{:=}0$ -- and at law-minimal width, zeroing $W_b$ at inference restores exact generalization (parity 5/5, $S_5$ 5/5, $S_4$ 4/5, $A_5$ 4/5). The additive pathway is parasitic: it destabilizes, then conceals, a correctly learned automaton. All 202 runs pre-registered; all numbers regenerate from artifacts.

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arXiv2026

Structure-Specific Representational Priors Causally Control the Grokking Delay

Grokking -- generalization long after training-set interpolation -- has been accelerated by structure-agnostic interventions (gradient filtering, weight-norm clamping, geometric penalties). Whether the delay specifically measures the time to form task-structured representations has remained observational. We test it causally by injecting representational priors of varying content into a one-layer transformer learning modular addition, via a supervised-contrastive loss whose positives encode (i) the task's true structure ($(a+b) \bmod p$), (ii) a coherent-but-wrong sibling ($(a-b) \bmod p$), or (iii) a random partition -- all with identical loss form, strength, class sizes, and geometry. Whether generalization occurs follows a clean gradation: true 22/30 runs, sibling (same periodic features, wrong combination) 14/15, random (only memorizable) 0/20 (Fisher $p=1.3\times10^{-7}$). A weight-norm-matched control replaying the norm trajectory onto plain cross-entropy generalizes 0/15, ruling out the norm as mediator. Probes show structure formation precedes and predicts generalization in all runs. Only the true structure also accelerates grokking (up to $2.75\times$), but this is dose-dependent and bimodal. We then confirm the mechanism by prediction: because the acceleration is gated by a weight-norm side-effect, clamping the norm during training yields a reliable, standalone accelerator with a median $8.6\times$ speedup (up to $22\times$ on the fastest seeds, under 1000 epochs), growing monotonically as the norm is held lower; the residual stalls also vanish, though significant only pooled over the two mitigations run at both strengths ($0/40$ vs $6/20$, $p=7.7\times10^{-4}$), not per method. The grokking delay is, causally, the time to form the right representational structure -- decided at the level of features, not labels.

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arXiv2026

What Makes a Representational Prior Work? Feature Families, Label-Free Invariances, and Critical Windows in Grokking

Companion work showed the grokking delay is causally the time to form task-structured representations, injectable via a contrastive prior. Here we characterize what makes such a prior work, across four axes, in 188 new runs. Content: a coherent, learnable prior built from the wrong feature family (magnitude bands) blocks generalization like a random partition (1/15 vs 0/20 grok; $p=0.43$ between them), confirming the companion's prediction that priors act at the level of the circuit's features. Supervision: a fully label-free invariance prior -- positives are commuted pairs $(a,b)\sim(b,a)$ only -- generalizes in 15/15 runs at a median $2.7\times$ speedup, more reliably than the label-supervised prior itself ($p=0.038$), and combined with a weight-norm clamp yields the strongest method we test (median $17\times$, 5/5) -- strongest meaning reliably fast: plain cross-entropy with a clamp matches this speed only at the exact critical norm, while the prior keeps it fast across the entire clamp range. Timing: the prior is only needed early -- applied solely during the first 2000 epochs (4% of budget) it generalizes 10/10 at $2.7\times$, beating continuous application (8/10, $1.25\times$) and a duration-matched later window ($2.1\times$). Setting: the dissociation replicates on modular multiplication and across depths and normalization variants, and a clamp sweep quantifies the companion's central claim: structure injection flattens the weight-norm delay-law exponent about 17-fold (plain cross-entropy slows $31\times$ per +10 norm units, a lower bound as higher cells are censored, versus $1.22\times$ with the prior). Honest boundary: tasks that generalize before memorizing have no delay to control. Feature-family alignment decides whether a prior permits generalization; invariance content suffices for acceleration without labels; a brief early window captures nearly all of the benefit.

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arXiv2026

Intrinsic-Noise Consolidation: A Doob-Barrier-Conditioned Diffusion Turns Analog Device Noise into a Continual-Learning Resource

On analog neuromorphic hardware, intrinsic device noise is normally an accuracy tax. We ask whether it can instead consolidate memories. We cast per-synapse consolidation as a Doob h-transform: condition each weight's stochastic dynamics on never crossing a memory-critical barrier around its consolidated value. The conditioned diffusion gains an extra drift sigma^2 d/dw log h, a restoring force amplified by the noise variance itself that diverges at the barrier. We are explicit about novelty: the anchored drift -s(w-mu) our rule also contains is not ours (the limit of OUA, MESU, and EWC), and we surrender it. We claim only the conjunction of (a) the Doob barrier-conditioning as a synaptic rule, to our knowledge unclaimed (every h-transform use we found is generative modeling, none synaptic), and (b) a falsifiable prediction: increasing intrinsic noise non-monotonically improves sequential-task retention, an inverted-U that anchored-drift methods cannot produce. We pre-registered this as a go/no-go gate; it passes. On single-head Split-MNIST (8 seeds) the rule lifts retention 10.9 points at an interior optimum (paired Wilcoxon p=0.004), while matched OU/EWC/MESU anchors are monotone. Ablating the conditioning removes the effect; the optimum tracks the barrier; the inverted-U survives a second task stream and the realization where noise enters the forward pass. We then measure the intrinsic noise on real BrainScaleS-2 silicon (additive, trial-to-trial independent, tunable via on-chip averaging) and run the rule on the chip with its noise in the training loop: barrier-conditioning retains a prior task 15.6 points better than the matched control at matched average accuracy, a stability-plasticity shift, not a net-accuracy win (single seed; retention measured, energy modelled). Intrinsic analog noise thus becomes a consolidation dividend a digital accelerator must spend energy to generate.

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arXiv2026

Heckman-Corrected Epistemic Uncertainty: Selection on Unobservables Defeats Importance Weighting

Training data for machine learning is routinely collected by a selection process the model never sees: loans are observed only when granted, outcomes only when a test was ordered. The standard fixes -- importance weighting, covariate-shift correction, MAR imputation -- assume selection is ignorable given observables. Econometrics solved the harder case in 1979: Heckman's two-equation model jointly fits a probit selection equation and an outcome equation linked through correlated errors, and the inverse-Mills-ratio term corrects for selection on unobservables, where importance weighting is structurally helpless. We instantiate this for deep epistemic uncertainty: a deep outcome network, a linear selection head, and a joint bivariate-normal likelihood over all units, ensembled for predictive variance. In a controlled generator where sampling probability depends on an unobservable correlated (rho up to 0.9) with the outcome noise, deep ensembles, MC dropout, and GP baselines are overconfident exactly where data was avoided: coverage of nominal-90% intervals falls to 64.4% at rho=0.9, and importance weighting with oracle propensities does not fix it (43.1%) -- reweighting corrects the covariate distribution, not the conditional bias E[y|x,selected] != E[y|x]. The Heckman correction restores coverage (88.9%) when the selection equation has an instrument -- a variable affecting selection but not the outcome -- and degrades measurably without one (40.3%); we chart this honesty curve rather than hide it. On real tabular data with induced MNAR selection, the corrected intervals are the best-calibrated (lowest region-ECE) non-oracle method in selected-against regions; baselines matching its raw coverage do so only by over-widening everywhere. Our estimators reproduce classic Stata output to seven digits. We state which identification regime a practitioner is in, and release the code.

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arXiv2026

Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.

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