People overthink; language models over-sample, and the extra effort can talk both into a worse answer. Reasoning systems answer a hard question by sampling it many times (test-time scaling), and the more they draw, the more often a correct answer turns up somewhere, so coverage, the fraction of problems with at least one correct try, climbs and appears to be progress. But a deployed system must return one answer, and choosing it, not knowing which try is right, is selection; selection is capped, and past a point extra samples only make the model surer of a confident mistake, even as every draw adds cost. The gap between climbing coverage and stalled selection, the identifiability gap, is the answer a model can produce but not pick. So the real question is not whether to sample but how far, and the answer is: not far. For picking an answer, the vote has already settled within a few dozen draws, the modal ceiling; for scoring a benchmark, sooner still, the correlation ceiling. Beyond that, extra draws cost compute and add nothing, and can even make the answer worse. This paper turns the cutoff into a single number, the effective number of samples, that any sampling run already reveals. The bottleneck is recognizing a right answer, not generating one.

Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(η^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $τ=ηk$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(η)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(η^2)$; a bare fixed-$β$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$β$ momentum, and 4.00 for SGD; matching the memory clock to $τ$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation. Subject to an abstention budget $α$, we analyze the probability of undetected error--the risk of recommending a suboptimal arm without abstaining. Our central finding is that abstention induces a phase transition: without abstention, the error probability decays polynomially in the sampling budget $T$; in contrast, introducing any small positive abstention budget shifts this to an exponential decay. For Gaussian priors and rewards, in the regime $T\to\infty$ followed by $α\downarrow0$, we establish exact matching information-theoretic lower bounds and algorithmic upper bounds on the optimal error exponent, which takes the form $\exp(-\frac{α^{2}T}{8κ_ν^{2}})$. The hardness parameter $κ_ν$ represents the prior density of the top-two gap at zero, highlighting that nearly tied instances drive the fundamental error. We introduce an adaptive algorithm, PGWS, that successfully achieves this optimal exponent by expending its abstention budget on statistically ambiguous instances. We further demonstrate that this polynomial-to-exponential improvement is exclusively a Bayesian phenomenon--in the frequentist setting, abstention only affects lower-order exponent terms. We also extend our results beyond the Gaussian model.

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

This paper introduces the R package spca, which provides a computational framework for least squares sparse principal component analysis (LS-SPCA). Unlike other SPCA methods, LS-SPCA generates uncorrelated sparse principal components (sPCs) that effectively maximize the explained variance while maintaining strong correlations with standard principal components (PCs). The framework also includes more computationally efficient variants that produce mildly correlated sPCs, which often have lower cardinality while explaining equal or greater variance than the LS-SPCA optimal sPCs. The spca package is built on an efficient C++ backend for matrix computations, with distinct engines for tall and fat matrices, and a flexible R frontend. The user interface offers several options for computing sPCs, such as deciding whether sparsification should stop when a threshold for cumulative variance explained or R2 with the PCs is reached, and choosing between simple forward selection, stepwise forward selection, or backward elimination for variable selection. In addition to the print(), summary(), and plot() methods, the package includes tools for comparing different "spca" solutions, grouping sparse loadings, and representing foreign SPCA solutions as "spca" objects. This article demonstrates with real datasets the use of the package in a typical LS-SPCA workflow and briefly contrasts LS-SPCA with conventional SPCA solutions . Then it compares different LS-SPCA solutions obtained from the dataset. Finally, the performance of spca on large tall and fat matrices is discussed, showing that spca offers a computationally efficient alternative for computing interpretable sPCs.

Outlier detection (OD) aims to identify anomalous instances by learning the underlying structure of normal data (inliers), and is particularly challenging in fully unsupervised settings where no information about anomalies is available during training. Recent advances have leveraged the inlier-memorization (IM) effect, a phenomenon in which deep models memorize inlier patterns earlier than those of outliers, as a powerful signal for distinguishing outliers. However, despite its empirical success, the theoretical understanding of the IM effect remains limited. In this work, we present a theoretical study of the IM effect. Focusing on a simple autoencoder, we show that, under mild assumptions, the model can successfully memorize inliers while failing to memorize outliers during certain stages of early training. In particular, we characterize not only the emergence of the IM effect, but also its strength and persistence, and analyze how these properties depend on the data distribution and parameter initialization. In addition, building on these insights, we derive simple yet practical guidelines for enhancing the IM effect, including data preprocessing and parameter initialization schemes, achieving state-of-the-art performance on the ADBench datasets. Our findings provide a theoretical foundation for the IM effect and offer actionable directions for improving IM-based outlier detection methods.

The standard convergence analysis of mini-batch stochastic gradient descent (SGD) models gradient noise using a single variance term that treats all parameter directions equally, ignoring the fact that noise in high-curvature directions has less impact because learning rates are already constrained there. We introduce Curvature-Weighted Gradient Diversity (CWGD), a geometry-aware measure that weights per-sample gradient diversity by the inverse square root of the Hessian, providing a tighter proxy for the effective optimization noise. For strongly convex quadratic objectives with diagonal Hessians and isotropic noise, we prove that a CWGD-modulated cosine learning-rate schedule can reduce the asymptotic optimization error floor by up to a factor of two compared with standard cosine annealing. We implement this idea as CWGD-Cosine using a Hutchinson-based diagonal Hessian estimator that is exact for quadratic objectives. Across a range of condition numbers, batch sizes, and noise structures, CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine annealing while incurring negligible overhead in the quadratic setting. We also identify and correct a degenerate curvature estimator, analyze the robustness of the proposed estimator, and explicitly discuss the limitations of the method, including Hessian staleness in non-convex optimization. These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.

Prediction sets should have high coverage to be useful, but some coverage notions are more practically relevant than others. In the classification setting, class-conditional coverage requires that the prediction set (i.e., the set of candidate labels for a new test point) must achieve the target accuracy level within each class, which may be challenging to satisfy when many classes are rare and have few calibration points. At the other extreme, marginal coverage requires only that coverage holds on average over the distribution of all classes, which can lead to low-probability labels being essentially ignored. To find a middle ground, recent work has introduced macro-coverage, defined as the unweighted average of class-conditional coverages. Macro-coverage offers a compromise between marginal coverage and class-conditional coverage that is particularly appropriate for long-tailed settings. In this work, we show that label-weighted conformal prediction can be used to produce prediction sets with a finite-sample macro-coverage guarantee, and more generally a guarantee on a family of generalized macro-coverage objectives that aggregate coverage at the level of arbitrary class groupings and take a weighted average. We further characterize the form of the smallest prediction sets satisfying a given generalized macro-coverage objective and propose a corresponding conformal score function. We validate our theoretical results on two large-scale image classification datasets.

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.

Speculative decoding accelerates language model inference by using a fast drafter to propose candidate tokens that are then verified by a larger target model. Existing theory largely studies the stochastic, distribution-preserving setting, where the goal is to exactly sample from the target distribution. In contrast, many practical systems use greedy decoding, relaxed acceptance rules, or tree-based candidate sets, where success is governed by local ranking and threshold events rather than exact distributional equality. We develop a theory for these regimes. We identify that many common acceptance criteria have rejection regions that can be characterized as lower level sets of the target distribution. For these, we characterize the exact KL divergence required for rejection yielding exact certificates and sharp margin-based bounds for strict greedy decoding, additive and multiplicative relaxed acceptance, top-(m) relaxed criteria, and entropy-thresholded acceptance. We then extend the framework to greedy tree decoding, deriving exact and margin-only certificates for when the target greedy token remains covered by the drafter's top-(m) candidates. Finally, we evaluate the resulting certificates on Qwen3 models, showing that relaxed and tree-based criteria substantially enlarge the region of certified acceptance, especially on decoding steps with low target model distribution margin. These results complement existing distribution-preserving analyses of speculative decoding by characterizing the deterministic local acceptance events common in practical inference systems.