Uncertainty Score

Diffusion models generate samples by iteratively querying learned score estimates. A rapidly growing literature focuses on accelerating sampling by minimizing the number of score evaluations, yet the information-theoretic limits of such acceleration remain unclear. In this work, we establish the first score query lower bounds for diffusion sampling. We prove that for $d$-dimensional distributions, given access to score estimates with polynomial accuracy $\varepsilon=d^{-O(1)}$ (in any $L^p$ sense), any sampling algorithm requires $\widetildeΩ(\sqrt{d})$ adaptive score queries. In particular, our proof shows that, within any polynomial total-query budget, successful sampling requires searching over $\widetildeΩ(\sqrt{d})$ distinct noise levels, providing a formal explanation for why multiscale noise schedules are necessary in practice.

Large language model (LLM) evaluation platforms increasingly rely on pairwise human judgments. These data are noisy, sparse, and non-uniform, yet leaderboards are reported with limited uncertainty quantification. We study this as semiparametric inference for a low-rank latent score tensor observed through pairwise comparisons under Bradley-Terry-Luce-type models. This places LLM evaluation in a new tensor completion setting with structured observations, non-uniform sampling, and pairwise contrasts. Our target is a smooth functional $ψ(T^\star)$, including linear estimands such as ability gaps and nonlinear ones such as win probabilities. We derive the information operator on the low-rank tangent space, the efficient influence function, and the semiparametric efficiency bound, then construct a one-step debiased estimator with asymptotic normality. A central challenge is that the information operator is anisotropic and does not commute with the tangent-space projection, creating a bottleneck absent from isotropic models. We introduce a score-whitening method that equalizes local Fisher information and restores stable inference at the optimal sample-complexity scale. Our results provide a principled framework for uncertainty quantification in LLM evaluation and more broadly for inference on low-rank structures from pairwise data.

Converting a pretrained Transformer into a more efficient hybrid model through distillation offers a promising approach to reducing inference costs. However, achieving high-quality generation in distilled models requires careful joint design of both the student architecture and the distillation process. Many prior distillation works evaluate downstream multiple-choice benchmarks by ranking candidate answers with log-likelihood rather than requiring autoregressive generation, which can obscure important differences in model quality. For example, on overlapping benchmarks, we show that a 7B distilled model that nearly matches its teacher to within 0.2 pp under log-likelihood scoring falls behind by 20.8 pp when it must generate answers autoregressively. We investigate this phenomenon with GenDistill, a multi-stage pipeline we designed for distilling a pretrained Transformer into an efficient Hybrid Kimi Delta Attention (Hybrid-KDA) student. Using it as a controlled testbed on Qwen3-0.6B, we systematically ablate six design axes (training objective, loss masking, training duration, dataset selection, parameter freezing, and architecture choice) and evaluate every choice under both log-likelihood and generation-based protocols. We find that log-likelihood-based evaluation consistently underestimates the gap between teacher and student, and can in some cases reverse the ranking of design choices, so conclusions drawn from perplexity-only evaluation may be misleading. Among the factors we study, dataset selection, completion-only masking, and freezing attention layers during post-training have the largest impact on generation quality. Our best distillation recipe, using a Hybrid-KDA model as the student, retains 86-90% of teacher accuracy on knowledge benchmarks while reducing KV cache memory by up to 75% and improving time-to-first-token by 2-4x at 128K-token contexts.

Proper scoring rules are essential for evaluating probabilistic forecasts. We propose a simple algebraic rearrangement of the Yates covariance decomposition of the Brier score into three independently non-negative terms: a variance mismatch term, a correlation deficit term, and a calibration-in-the-large term. This rearrangement makes the optimality conditions for perfect forecasting transparent: the optimal forecast must simultaneously match the variance of outcomes, achieve perfect positive correlation with outcomes, and match the mean of outcomes. Any deviation from these conditions results in a positive contribution to the Brier score.

Tokenization is a hardcoded compression step which remains in the training pipeline of Large Language Models (LLMs), despite a general trend towards architectures becoming increasingly end-to-end. Prior work has shown promising results at scale in bringing this compression step inside the LLMs' architecture with heuristics to draw token boundaries, and also attempts to learn these token boundaries with straight-through estimates, which treat the problem of drawing discrete token boundaries as a continuous one. We show that these token boundaries can instead be learned using score function estimates, which have tighter theoretical guarantees due to directly optimizing the problem of drawing discrete token boundaries to minimize loss. We observe that techniques from reinforcement learning, such as time discounting, are necessary to reduce the variance of this score function sufficiently to make it practicable. We demonstrate that the resultant method outperforms prior proposed straight-through estimates, both qualitatively and quantitatively at the $100$ million parameter scale.

Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We show how in some generality, the availability of scores allows the general problem to be ``reduced'' to sampling from an adaptively constructed sequence of $K$ strongly log-concave (SLC) sub-problems. The reduction is simple, constructive and algorithm-independent, so that any SLC sampler can be used as a subroutine. Various bounds on score-based sampling complexity follow directly: for instance, high-accuracy SLC samplers yield $\tilde{\mathcal{O}}(K \sqrt{d} \operatorname{polylog}(1/\varepsilon))$ guarantees for accuracy $\varepsilon$ in dimension $d$, where randomized midpoint SLC schemes yield $\tilde{\mathcal{O}}(K d^{1/3} \operatorname{poly}(1/\varepsilon))$ guarantees. When the original distribution itself is SLC, we prove that $K \leq 1 + \log_2(κ)$, thereby obtaining the first efficient procedure with logarithmic dependence on condition number $κ$; for general distributions, the quantity $K$ depends on the geometry of score Hessian across the trajectory. Our analysis is direct and simple, involving techniques and insights complementary to those in standard analyses of discretized diffusions.

Standard language model evaluation assigns scores to single predicted answers, rewarding high-confidence responses regardless of how residual probability mass is distributed over alternative options. This creates a systematic pressure toward overconfident guessing: under accuracy-based schemes, a model maximises its expected score by always committing to an answer rather than abstaining, even when its uncertainty is high. While penalty-based approaches partially address this by raising the confidence threshold for strategic guessing, they still treat all sub-threshold responses identically, ignoring a fundamental distinction in how models can express uncertainty - for example between hedging toward incorrect answers versus hedging toward "I don't know" responses. This paper introduces a novel evaluation metric to solve this problem of not considering a model's entire probability distribution over answer choices. The metric naturally distinguishes between harmful overconfidence in wrong answers and uncertainty expressed through abstention, providing scores in an interpretable default range. Through theoretical analysis and illustrative examples, the metric is shown to offer a more nuanced and aligned evaluation paradigm that incentivises models to express genuine uncertainty rather than guessing. Adapting 12 existing evaluation benchmarks to the metric's variants and measuring performance on six language models shows that for half of the tested benchmarks scores are negative across all tested models, indicating significant tendencies towards hallucination.

Robust medical image segmentation across modalities remains challenging due to severe domain shifts and the lack of target-domain labels. While diffusion models have been explored for cross-domain generation and augmentation, target-domain conditional diffusion training typically relies on highly noisy pseudo masks; naively conditioning on a single Arg-Max pseudo-label can corrupt diffusion training and downstream segmentation. We propose UPDiff-UDA, a unified UDA framework whose core is an uncertainty-guided training objective for target-domain conditional diffusion. Given an imperfect source-trained segmenter, we use its per-pixel softmax distribution to form ranked pseudo-label maps (Arg-Max, Arg-2nd, Arg-3rd, ...). Each map yields a conditional score estimate, and we aggregate them via pixel-wise confidence weighting to obtain an uncertainty-reweighted score for score matching, improving robustness to pseudo-label noise while leveraging alternative plausible labels in uncertain regions. We further provide a theoretical justification showing that confidence-weighted aggregation follows a minimum-MSE convex-combination principle under the segmenter-induced surrogate label distribution. To improve pseudo-condition quality, we also introduce a feature-guided, low-degree-of-freedom Bézier curve adaptation to reduce appearance gaps. Experiments on multiple public datasets and modality shifts show that UPDiff-UDA generates high-fidelity labeled target-style samples for augmentation and consistently outperforms strong UDA baselines. The code for this project is available at: https://github.com/superlc1995/Multi-Channel-Uncertainty-Diffusion-UDA

Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, a task that in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality with dimension-independent error bounds, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Locality structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.

Diffusion models achieve remarkable generation quality, yet face a fundamental challenge known as memorization, where generated samples can replicate training samples exactly. We develop a theoretical framework to explain this phenomenon by showing that the empirical score function (the score function corresponding to the empirical distribution) is a weighted sum of the score functions of Gaussian distributions, in which the weights are sharp softmax functions. This structure causes individual training samples to dominate the score function, resulting in sampling collapse. In practice, approximating the empirical score function with a neural network can partially alleviate this issue and improve generalization. Our theoretical framework explains why: In training, the neural network learns a smoother approximation of the weighted sum, allowing the sampling process to be influenced by local manifolds rather than single points. Leveraging this insight, we propose two novel methods to further enhance generalization: (1) Noise Unconditioning enables each training sample to adaptively determine its score function weight to increase the effect of more training samples, thereby preventing single-point dominance and mitigating collapse. (2) Temperature Smoothing introduces an explicit parameter to control the smoothness. By increasing the temperature in the softmax weights, we naturally reduce the dominance of any single training sample and mitigate memorization. Experiments across multiple datasets validate our theoretical analysis and demonstrate the effectiveness of the proposed methods in improving generalization while maintaining high generation quality.