This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.
We extend recent work establishing an equivalence between one-layer transformers and nearest-neighbor classifiers in the binary setting to the multiclass case. By leveraging the simplex encoding, we show that one-layer transformers with an argmax classification head behave identically to a one-nearest-neighbor classifier in the multiclass setting. This closes a gap left by prior work, whose multiclass result relied on a non-standard rounding-based approach rather than the typical argmax head used in practice.
We prove that a multi-head scaled dot product attention can be viewed as a parameter identification strategy. The ratio of unidentified parameters to the total number of parameters scales like the reciprocal of the number of heads ($1/2 \to 1/(2H)$), meaning models with more heads are structurally more identified. A subtle side effect of the mathematics observation that attention can never be fully identified. Similarly we also show that some bias terms can have no effect on softmax-based attention layers in both the single- and multiple-head settings, though this is mostly a curiosity that should have a marginal effect on model size and model training/prediction efficiency. We also touch on modern improvements to transformers including RoPE and GQA from this perspective, illustrating how those as well can improve the ratio of ``meaningful'' parameters to all parameters. Simple numerical examples demonstrate that training can indeed involve updates that overlap model-invariant subspaces that arise from a lack of identification. As part of our experiments we use a ``rebalancing'' approach that can ``fix'' updates that overlap unindentified subspaces but do not try to present evidence this should actually be adopted. Instead we simply view our numerical results as exploring and confirming the theoretical results. As a whole we discuss a purely mathematical/statistical explanation, identification, for why specific architectural choices in transformers may have improved performance.
The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as $f = g + \varepsilon h$; in the limit $\varepsilon \to 0$, the dynamics of gradient descent with learning rate $\eta$ converge to the gradient flow of $h$ constrained to the minimizers of $g$ of sharpness at most $2/\eta$. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in $\varepsilon$. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of $g$-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in $\varepsilon$: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.
Rapha\"el Berthier· 0 citations
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Sampling from distributions conditioned on desired semantic properties is an emerging challenge in modern generative modeling. Metropolis-Hastings (MH) provides a principled route to conditional sampling, but requires access to exact pointwise target-density evaluations, which are not available in generative settings. Meanwhile, pairwise comparisons by humans or model "judge" are highly accessible and have proved valuable across diverse applications. We introduce Pref-MH, a general exact MH sampler for judge-induced conditional distributions using only stochastic binary pairwise comparisons. Our key observation is that the MH unnormalized density ratio matches the preference odds of the Bradley-Terry (BT) choice model. The central challenge is that while MH requires precise ratio computation, BT judges provide only sampled binary feedback. To this end, we develop a valid accept/reject rule whose resulting Markov chain provably converges to the target distribution. We further show that, for a fixed proposal kernel and budget, Pref-MH is optimal in the Peskun-Tierney sense among this class of exact reversible acceptance rules. Experiments on text generation and molecular design with LLM judges, as well as image generation with VLM judges, demonstrate that Pref-MH provides a practical and flexible approach to conditional sampling when comparative feedback is relatively easy to obtain.
Post-hoc out-of-distribution detectors are fitted on a finite reference set, so every score they produce is an estimate. If we had chosen a different set, some verdicts would have moved. We measure that movement by resampling the reference set and recording the bootstrap standard deviation of the score, which we call verdict instability. It admits a closed form with no fitted parameters. The instability of a verdict is the within-class dispersion of the assigned class along the query's direction, divided by the square root of that class's reference count. That count is what separates verdict instability from the geometry of the score distribution, and it is identifiable only under class imbalance. Instability grows with the local dispersion. Far-OOD queries lie along the low-variance directions of an anisotropic embedding, so every distance-based score we test assigns its highest values to the verdicts that are most reproducible. Only estimators of local dispersion carry the sign a practitioner expects. We give a rule that predicts this sign for any score from a single label-free correlation, and abstention driven by a wrong-signed score turns out worse than abstention at random on every dataset we test.
We combine neural network means with exact Mat\'ern kernel regressions of their residuals and of their learned features, and evaluate the pairing on two public emulation problems with published baselines: the structural-mechanics benchmark of de Hoop et al. and the OCO-2 radiative-transfer emulator of Lamminp\"a\"a et al. On structural mechanics the combination reaches 4.55% test error, matching the best published architecture, and 5.38% against a published 6.49% in the low-data regime. On OCO-2 it improves on the published Gaussian-process emulator on that problem's own test points, outright on two of the three spectral bands; the same kernel that trails the network tenfold on the raw state overtakes it on the network's features, and we measure why (the target's squared native-space norm drops about fortyfold at fixed effective dimension) and prove the mechanism. Where the two families tie instead, the residuals of every architecture we train correlate above 0.86 and their shared component is flat in diversity and sample size, which reads the published plateau as a property of the data. Supporting results include a second-moment identity that predicts stacking outcomes from measured correlations, an optimal-recovery certificate, and a distribution-free coverage band, the only uncertainty signal that survives our tests.
This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.
Daniel Nicorici, Olli Yli-Harja, Jaakko Astola· 0 citations
Persistent entropy is a scalar summary of persistence barcodes widely used to detect regime changes, yet there is no account of when a structural change in a barcode must produce a detectable change in entropy. We establish a model-agnostic theorem supplying such conditions. Treating persistence diagrams as random objects indexed by a control parameter, we identify a dispersion-condensation mechanism in the normalized persistence weights and derive an explicit lower bound on the entropy difference between the two regimes, valid with high probability at finite sample size and insensitive to the absolute scale of bar lifetimes. We also give a procedure for verifying the hypotheses on empirical barcodes. Applied to convolutional networks, the criterion shows that the circular organization of learned filters reported by Gabrielsson and Carlsson emerges through a sharp topological phase transition, and locates its onset: within a few hundred iterations on MNIST, but an order of magnitude later on CIFAR-10. The same criterion detects the Kuramoto synchronization and Vicsek order-disorder transitions.
This paper considers a novel variant of the online fair division problem involving multiple agents in which a learner sequentially observes an indivisible item that must be irrevocably allocated to one of the agents to achieve a desired balance between fairness and efficiency. Existing algorithms assume a small number of items with a sufficiently large number of copies, which ensures a good utility estimation for all item-agent pairs from noisy observed utilities. However, this assumption may not hold in many real-life applications, e.g., an online platform with a large number of users (items) who use the platform's service providers (agents) only a few times (a few copies of items), making it difficult to accurately estimate utilities for all item-agent pairs. To address this limitation, we assume utility is an unknown function of item-agent features. We propose algorithms that model online fair division as a contextual bandit problem and achieve provable sublinear regret. Our experimental results further validate the effectiveness of the proposed algorithms.
Arun Verma, Indrajit Saha, Makoto Yokoo et al.· 0 citations
The Rashomon effect is a machine learning phenomenon where equally accurate models produce different predictions for the same inputs (predictive multiplicity). Existing work primarily focuses on multiplicity within individual models, but in more complex decision systems, the impact of the Rashomon effect is less well understood. In this work, we study multiplicity from the perspective of auditing incorrect ensemble predictions, where the decision to divert an instance for human review is based on a consistency criterion that combines the ensemble margin with a measure of local prediction variability for each constituent model. With mild assumptions about stability and smoothness, we show that the consistency scores of finite ensembles converge to the corresponding consistency score of the expected model from the Rashomon set as the ensemble size and the number of samples used to measure local prediction variability increase. To demonstrate the efficacy of the proposed criterion, we evaluate the framework with respect to transformer models applied to natural language understanding tasks and parameter-efficient fine-tuning of large language models used for tabular data classification tasks. Our experiments show that ensembling models from the Rashomon set substantially reduces the risk of incorrect predictions going unchecked compared with auditing a single model, while incurring only a moderate increase in the number of diversions. Moreover, the auditing behavior of the full Rashomon set can be closely approximated by finite ensembles of relatively modest size, with the risk approaching zero for some datasets. We further demonstrate that the proposed measure exhibits stronger agreement with established predictive multiplicity metrics than existing consistency measures, providing a more reliable way to capture multiplicity in the Rashomon set.
Sinjini Banerjee, Tim Marrinan, Anand D. Sarwate· 0 citations
Uniform-state discrete diffusion models update all tokens in parallel while keeping every position revisable. Even when the commonly used top-$p$ rule leaves only one candidate at a position, that choice affects only the current reverse step and can be revised at the next sampling step. We ask what changes when selected hypotheses instead become persistent context for later predictions. We therefore propose committed reveal sampling (CRS), a training-free sampler that stores selected argmax tokens and inserts them into subsequent model inputs. Our analysis gives a rationale for selecting later and for keeping selected tokens visible. Under the exact forward process, the Bayes error of selecting a clean token cannot increase as noise decreases, while in a simple latent-mode model, keeping the selected token visible helps later parallel predictions agree on the same sequence-level choice. Empirically, paired experiments on Duo-distilled then separate this persistent effect from single-step top-$p$ restriction and scalar temperature scaling. Under the same finalization rule, CRS without top-$p$ truncation reaches lower generative perplexity (GenPPL) than fixed $p=0.95$ and $p=0.9$ baselines across budgets of 8--64 function evaluations (NFE). At 64 NFE, the comparison at matched unigram entropy also gives lower GenPPL for CRS, yielding a more favorable GenPPL--entropy tradeoff. Base Duo shows the same direction in a descriptive comparison, while other diversity and continuation metrics can rank these operating points differently. These results identify support restriction and persistent context as distinct controls of that tradeoff.
