arXiv subjects

Gordei Verbii

Publications and source records attributed to Gordei Verbii.

3 recordsLinked to original sources

arXiv2026

Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off

Worst-case Optimal Recovery (OR) and Bayesian learning describe the same Gaussian-quadratic-Hilbert problems in two vocabularies. We sharpen the correspondence - the radius of information equals a nugget-optimized GP posterior variance and is attained by the posterior mean at a closed-form balance nugget - and measure, inside three published Bayesian systems, where the worst-case side pays. The ledger is two-sided: the losses instruct as much as the wins. Morozov calibration tracks a test-access oracle within $1.00$-$1.19\times$ where $σ$-blind rules fail, is $4.9$-$6.3\times$ more reproducible across noise draws ($p=0.002$-$0.004$), and is the only deployable rule whose selection survives a change of backend ($1.36\times$ against $12$-$30\times$ for the released weight, ML-II and GCV); tight certificates cover at the information-theoretic floor with no numerical slack. But on exchangeable data split-conformal beats the OR head on interval score, a water-filling prior adds nothing without an oracle noise hint, and under covariate shift the OR band keeps coverage on every dataset yet loses interval score to split-conformal, and to a feature-free constant band, on most cells; what pays is not shift but shift on a learnable target, which a training-free audit statistic predicts before any model is fitted. In Bayesian optimization the certified width is a validity floor whose scalar inflation we prove inert under a checkable margin condition and check at every step. Inertness is graded, not binary, and in the size of the inflation as much as in the objective: $κ{=}2$ is inert wherever $κ{=}5$ is and on more cells besides, while $κ{=}5$ moves half the Ackley seeds and every Griewank seed. Exploration is a shape problem, not a scale one. The design rule: match the guarantee tool to the data regime, and audit the regime first.

stat.ML↗

arXiv2026

One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion

One implicit DDIM inversion step is the cheapest probe of whether a pretrained diffusion model encodes local manifold geometry. It is the stationarity condition of an explicit potential, $x-G(x)=\nablaΨ_t(x)$, strongly convex at the Bayes limit with modulus exactly $e^{-h_t}$ for the step's log-SNR gap $h_t$ $-$ for every data law, schedule and point, with no manifold, reach or unimodality hypothesis. Three consequences must be kept apart. (i) The solution is unique at the Bayes limit; a second one requires the trained score to violate the posterior-covariance bound by $1/(1-e^{-h_t})$, a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, $ρ_g^{\star}=1-e^{-h_t}<0.326$ throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on $Ψ_t$, unstable wherever $λ_{\max}(\nabla^2Ψ_t)>2$, so oscillation certifies nothing; damping below $2/λ_{\max}$ cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth $w=rκ_{\max}$ the oscillation shell sits at $w=\tfrac12$, schedule-free, and the divergence shell at $w=1/(1+ρ_g^{\star})$, with a measured finite-noise correction in $\|\mathrm{II}\|^2$. Exact scores reproduce both to within $0.54\%$ on three classes; no trained score we probe shows a shell $-$ a derived limitation, not a null result: the Fermi window conflicts with the model's own training support by $3.6$-$5.6\times$, and the trained Hessian-Lipschitz constant is $2$-$12\%$ of the curvature the law reads, $0$ on a ReLU net. Finally the unconditional ceiling $σ_tλ_{\max}(\mathrm{sym}\,J)\le1$, from $\mathrm{Cov}(x_0\mid x_t)\succeq0$ alone, holds for the exact score to $3\times10^{-7}$ but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by $1.26$-$4.66\times$.

stat.ML↗

arXiv2026

The Distributional View of Knowledge Distillation

Token-level knowledge distillation (KD) matches two conditional distributions per position, yet the standard objectives compare them pointwise: a Kullback-Leibler gradient is blind to which wrong token receives probability mass. We develop a distributional view in which the teacher is represented not by a single softened output but by a family of multi-temperature views - marginals of the annealing path of its logits - and the student is trained against a geometry-aware aggregate of these views under an embedding-based ground cost. We formalize the resulting design space (mixtures, log-linear pooling, entropic Wasserstein barycenters, and a debiased Sinkhorn-divergence flagship in hub and path forms), prove an exact collapse result showing log-linear pooling of tempered views is equivalent to a single temperature, and give a multi-marginal Schrodinger-bridge reading that yields falsifiable predictions. On instruction-tuned Pythia pairs, experiments yield three empirical laws: (i) dispersion law - the benefit of multi-temperature aggregation grows monotonically with the effective temperature dispersion of the views, not with their number; (ii) dispersed views unlock the aggregation operator - the barycenter separates from the arithmetic mixture exactly when transport-based aggregation starts to beat averaging; and (iii) two-regime picture governed by the ceiling gap $Γ=\mathrm{PPL}_{\mathrm{SFT}}-\mathrm{PPL}_{T}$: when the fine-tuned teacher barely beats a supervised student the gentle transport objective is the best KD loss but no KD beats supervised fine-tuning, whereas at a real ceiling the ranking inverts - and the sign of the fidelity-generalization correlation flips. We argue that "which distillation loss is the best" is not a fixed property of the loss but a function of $Γ$.

stat.ML↗