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.

Adaptive gradient methods are among the standard tools for training machine learning models. Their appeal is that they reduce the need to tune a fixed learning rate by adjusting the effective stepsize using information observed along the optimization trajectory. AdaGrad, introduced by Duchi et al. [2011], is a prototypical example: it rescales the update by the square root of the cumulative sum of past squared subgradients, coordinate by coordinate. The method was originally proposed for nonsmooth Lipschitz-continuous composite convex optimization, achieving the optimal rate O(1/ n) in the objective gap. Later works considered the smooth setting and asked whether AdaGrad can adapt to the unknown smoothness level of the objective, while attaining the corresponding standard rate.

The global minimum-variance portfolio (GMVP) is the canonical decision built from an estimated covariance matrix, yet covariance estimators are universally evaluated by matrix-norm loss, which is not the object the decision depends on. We characterise exactly how covariance-estimation error maps into GMVP suboptimality. We prove an exact regret identity and a non-asymptotic bound showing decision regret depends on the estimation error only through its action on the portfolio weights, scaled by portfolio concentration and the conditioning of the true covariance. From this we derive the decision geometry: GMVP regret is invariant to a (p-1)-dimensional projection of the p^2-dimensional error matrix, with invariance to the covariance-scale direction as an exact special case. We then apply the framework to heavy-tailed returns (tail index kappa in (2,4)), establishing the regret convergence rate implied by the centred operator-norm rate, and confirm the theory on a skew-t/t-copula simulation design with pre-registered analysis. The decision-focused advantage is a sharper constant and a concentration discount rather than a faster rate; we report an honest high-conditioning boundary of the rate prediction. The results complement recent decision-focused learning approaches by supplying the exact estimation geometry and consistency theory they lack.

Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the kth most frequent word πk is proportional to 1/k, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law πk 1/kα parameterized by the exponent α > 0. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the exponent α. Existing theoretical investigations in scaling laws assume that the eigenvalues of the data decay as a power law with exponent α > 1. This assumption effectively makes the problem "finite dimensional" as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case α = 1 as found in language is "worst-case" for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement for large vocabularies.

We study the fundamental problem of calibrating a linear binary classifier of the form σ(ˆw x), where the feature vector xis Gaussian, σis a link function, and ˆw is an estimator of the true linear weight w . By interpolating with a noninformative chance classifier, we construct a well-calibrated predictor whose interpolation weight depends on the angle (ˆw,w) between the estimator ˆw and the true linear weight w . We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate.

We study the dynamics of gradient flow with small weight decay on general training losses F: Rd R. Under mild regularity assumptions and assuming convergence of the unregularised gradient flow, we show that the trajectory with weight decay λ exhibits a two-phase behaviour as λ 0. During the initial fast phase, the trajectory follows the unregularised gradient flow and converges to a manifold of critical points of F. Then, at time of order 1/λ, the trajectory enters a slow drift phase and follows a Riemannian gradient flow minimising the ℓ2-norm of the parameters. This purely optimisation-based phenomenon offers a natural explanation for the grokking effect observed in deep learning, where the training loss rapidly reaches zero while the test loss plateaus for an extended period before suddenly improving. We argue that this generalisation jump can be attributed to the slow norm reduction induced by weight decay, as explained by our analysis.

Interpretable machine learning is essential in high-stakes domains like healthcare. Rule lists are a popular choice due to their transparency and accuracy, but learning them effectively remains a challenge. Existing methods require feature pre-discretization, constrain rule complexity or ordering, or struggle to scale. We present NEURULES, a novel end-to-end framework that overcomes these limitations. At its core, NEURULES transforms the inherently combinatorial task of rule list learning into a differentiable optimization problem, enabling gradient-based learning. It simultaneously discovers feature conditions, assembles them into conjunctive rules, and determines their order--without pre-processing or manual constraints. A key contribution here is a gradient shaping technique that steers learning toward sparse rules with strong predictive performance. To produce ordered lists, we introduce a differentiable relaxation that, through simulated annealing, converges to a strict rule list. Extensive experiments show that NEURULES consistently outperforms combinatorial and neural baselines on binary as well as multi-class classification tasks across a wide range of datasets.

The next frontier of online advertising is revenue generation from LLM-generated content. We consider a setting where advertisers aim to influence the responses of an LLM, while platforms seek to maximize advertiser value and ensure user satisfaction. The challenge is that advertisers' preferences generally conflict with those of the user, and advertisers may misreport their preferences. To address this, we introduce MOSAIC, an auction mechanism that ensures that truthful reporting is a dominant strategy for advertisers and that aligns the utility of each advertiser with their contribution to social welfare. Importantly, the mechanism operates without LLM fine-tuning or access to model weights and provably converges to the output of the optimally fine-tuned LLM as computational resources increase. Additionally, it can incorporate contextual information about advertisers, which significantly improves social welfare. Via experiments with publicly available LLMs, we show that MOSAIC leads to high advertiser value and platform revenue with low computational costs. While our motivating application is online advertising, our mechanism can be applied in any setting with monetary transfers, making it a general-purpose solution for truthfully aggregating the preferences of selfinterested agents over LLM-generated replies.

Multi-arm bandit experimental designs are increasingly being adopted over standard randomized trials due to their potential to improve outcomes for study participants, enable faster identification of the best-performing options, and/or enhance the precision of estimating key parameters. Current approaches for inference after adaptive sampling either rely on asymptotic normality under restricted experiment designs or underpowered martingale concentration inequalities that lead to weak power in practice. To bypass these limitations, we propose a simulation-based approach for conducting hypothesis tests and constructing confidence intervals for arm specific means and their differences. Our simulation-based approach uses positively biased nuisances to generate additional trajectories of the experiment, which we call simulation with optimism. Using these simulations, we characterize the distribution potentially non-normal sample mean test statistic to conduct inference. We provide guarantees for (i) asymptotic type I error control, (ii) convergence of our confidence intervals, and (iii) asymptotic strong consistency of our estimator over a wide variety of common bandit designs. Our empirical results show that our approach achieves the desired coverage while reducing confidence interval widths by up to 50%, with drastic improvements for arms not targeted by the design.

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussiandistributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.