A simple strategy for improving LLM accuracy, especially in math and reasoning problems, is to sample multiple responses and submit the answer most consistently reached. In this paper we leverage Bayesian prior information to save on sampling costs, stopping once sufficient consistency is reached. Although the exact posterior is computationally intractable, we further introduce an efficient "L-aggregated" stopping policy that tracks only the L-1 most frequent answer counts. Theoretically, we prove that L=3 is all you need: this coarse approximation is sufficient to achieve asymptotic optimality, and strictly dominates prior-free baselines, while having a fast posterior computation. Empirically, this identifies the most consistent (i.e., mode) LLM answer using fewer samples, and can achieve similar answer accuracy while cutting the number of LLM calls (i.e., saving on LLM inference costs) by up to 50%.

Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation $\widehat W$ in place of true $W$ ("weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of $W$ depends on the (second order) statistics of $X$ and requires a careful alignment of vector quantization codebook with PCA directions of $X$ (a process known as "waterfilling allocation''). Dependence of the codebook on statistics of $X$, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of $X$, in the sense of being at least as good as an $X$-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in $\mathbb{R}^n$ that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.

Tree-boosting is a widely used machine learning technique for tabular data. However, its out-of-sample accuracy is critically dependent on multiple hyperparameters. In this article, we empirically compare several popular methods for hyperparameter optimization for tree-boosting including random grid search, the tree-structured Parzen estimator (TPE), Gaussian-process-based Bayesian optimization (GP-BO), Hyperband, the sequential model-based algorithm configuration (SMAC) method, and deterministic full grid search using $59$ regression and classification data sets. We find that the SMAC method clearly outperforms all the other considered methods. We further observe that (i) a relatively large number of trials larger than $100$ is required for accurate tuning, (ii) using default values for hyperparameters yields very inaccurate models, (iii) all considered hyperparameters can have a material effect on the accuracy of tree-boosting, i.e., there is no small set of hyperparameters that is more important than others, and (iv) choosing the number of boosting iterations using early stopping yields more accurate results compared to including it in the search space for regression tasks.

In recent years, explaining decisions made by complex machine learning models has become essential in high-stakes domains such as energy systems, healthcare, finance, and autonomous systems. However, the reliability of these explanations, namely, whether they remain stable and consistent under realistic, non-adversarial changes, remains largely unmeasured. Widely used methods such as SHAP and Integrated Gradients (IG) are well-motivated by axiomatic notions of attribution, yet their explanations can vary substantially even under system-level conditions, including small input perturbations, correlated representations, and minor model updates. Such variability undermines explanation reliability, as reliable explanations should remain consistent across equivalent input representations and small, performance-preserving model changes. We introduce the Explanation Reliability Index (ERI), a family of metrics that quantifies explanation stability under four reliability axioms: robustness to small input perturbations, consistency under feature redundancy, smoothness across model evolution, and resilience to mild distributional shifts. For each axiom, we derive formal guarantees, including Lipschitz-type bounds and temporal stability results. We further propose ERI-T, a dedicated measure of temporal reliability for sequential models, and introduce ERI-Bench, a benchmark designed to systematically stress-test explanation reliability across synthetic and real-world datasets. Experimental results reveal widespread reliability failures in popular explanation methods, showing that explanations can be unstable under realistic deployment conditions. By exposing and quantifying these instabilities, ERI enables principled assessment of explanation reliability and supports more trustworthy explainable AI (XAI) systems.

Matrix completion studies the problem of reconstructing a matrix from a (typically random) subset of its entries by exploiting prior structural assumptions such as low rank and incoherence. Roughly speaking, when the underlying n n matrix has low rank and its eigenvectors are sufficiently incoherent, observing Ω(n log n) entries sampled uniformly at random is sufficient for exact recovery via efficient optimization methods [Keshavan et al., 2009, 2010, Candes and Tao, 2010, Candes and Plan, 2010, Recht, 2011, Candes and Recht, 2012, Jain et al., 2013]. This sample complexity is nearly optimal, since specifying a rank-r matrix requires only O(n) degrees of freedom. Tensor completion generalizes this problem to higher-order arrays, aiming to recover a low-rank tensor from a limited set of observed entries, for example, under uniform random sampling. As a natural higher-order analogue of matrix completion, tensor completion has found broad applications in areas such as recommendation systems [Frolov and Oseledets, 2017], signal and image processing [Govindu, 2005, Liu et al., 2012], and data science [Song et al., 2019]. Despite this close analogy, tensor completion behaves fundamentally differently from its matrix counterpart. In contrast to the classical matrix setting, tensor completion exhibits a pronounced trade-off between computational and statistical complexity: while information-theoretic considerations suggest that relatively few samples suffice for recovery, all currently known polynomial-time algorithms require substantially more observations than this optimal limit. Polynomial-time methods A widely used polynomial-time approach to tensor completion is to reduce the problem to matrix completion via matricization.

Skip connections [He et al., 2016] have become an ubiquitous feature of neural network architectures from facilitating the stable training of deep models. However, despite their success, prior works [Veit et al., 2016, Gromov et al., 2024, Zhang et al., 2024] have raised the concern that the benefits of skip connections, namely ease of training, may be obscuring deeper issues, in terms of representation learning, that skip connections induce. The main point behind these criticisms is that skip connections appear to bias models away from properly utilising the full depth of their architectures. For instance, Ji et al. [2025a] argues that since skip connections continually reintroduce earlier features into deeper layers, they disrupt the learning of hierarchical and progressively more abstract representations, fundamentally harming representation learning. Motivated by this line of reasoning, we explore designing Transformers that are able to be trained stably without skip connections. Previous works [He et al., 2023, Ji et al., 2025a] have tackled this through modifications to Softmax Self-Attention (SSA) [Vaswani et al., 2017] and weight initialisations to improve signal propagation and the conditioning of the Jacobian matrix. However, these works restrict themselves to standard Softmax-based Transformers which appear to be inherently unstable without skip connections [Dong et al., 2021, Ji et al., 2025b] due to SSA.

In this work, we provide a sharp theory of scaling laws for two-layer neural networks trained on a class of hierarchical multi-index targets, in a genuinely representation-limited regime. We derive exact information-theoretic scaling laws for subspace recovery and prediction error, revealing how the hierarchical features of the target are sequentially learned through a cascade of phase transitions. We further show that these optimal rates are achieved by a simple, target-agnostic spectral estimator, which can be interpreted as the small learning-rate limit of gradient descent on the first-layer weights. Once an adapted representation is identified, the readout can be learned statistically optimally, using an efficient procedure. As a consequence, we provide a unified and rigorous explanation of scaling laws, plateau phenomena, and spectral structure in shallow neural networks trained on such hierarchical targets.

We consider the problem of estimating the Fréchet and conditional Fréchet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fréchet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fréchet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.

Differentially private synthetic data enables the sharing and analysis of sensitive datasets while providing rigorous privacy guarantees for individual contributors. A central challenge is to achieve strong utility guarantees for meaningful downstream analysis. Many existing methods ensure uniform accuracy over broad query classes, such as all Lipschitz functions, but this level of generality often leads to suboptimal rates for statistics of practical interest. Since many common data analysis queries exhibit smoothness beyond what worst-case Lipschitz bounds capture, we ask whether exploiting this additional structure can yield improved utility. We study the problem of generating $(\varepsilon,δ)$-differentially private synthetic data from a dataset of size $n$ supported on the hypercube $[-1,1]^d$, with utility guarantees uniformly for all smooth queries having bounded derivatives up to order $k$. We propose a polynomial-time algorithm that achieves a minimax error rate of $n^{-\min \{1, \frac{k}{d}\}}$, up to a $\log(n)$ factor. This characterization uncovers a phase transition at $k=d$. Our results generalize the Chebyshev moment matching framework of (Musco et al., 2025; Wang et al., 2016) and strictly improve the error rates for $k$-smooth queries established in (Wang et al., 2016). Moreover, we establish the first minimax lower bound for the utility of $(\varepsilon,δ)$-differentially private synthetic data with respect to $k$-smooth queries, extending the Wasserstein lower bound for $\varepsilon$-differential privacy in (Boedihardjo et al., 2024).

Graph-structured data is ubiquitous and powerful in representing complex relationships in many online platforms. While graph neural networks (GNNs) are widely used to learn from such data, counterfactual graph learning has emerged as a promising approach to improve model interpretability. Counterfactual explanation research focuses on identifying a counterfactual graph that is similar to the original but leads to different predictions. These explanations optimize two objectives simultaneously: the sparsity of changes in the counterfactual graph and the validity of its predictions. Building on these qualitative optimization goals, this paper introduces CFRecs, a novel framework that transforms counterfactual explanations into actionable insights. CFRecs employs a two-stage architecture consisting of a graph neural network (GNN) and a graph variational auto-encoder (Graph-VAE) to strategically propose minimal yet high-impact changes in graph structure and node attributes to drive desirable outcomes in recommender systems. We apply CFRecs to Zillow's graph-structured data to deliver actionable recommendations for both home buyers and sellers with the goal of helping them navigate the competitive housing market and achieve their homeownership goals. Experimental results on Zillow's user-listing interaction data demonstrate the effectiveness of CFRecs, which also provides a fresh perspective on recommendations using counterfactual reasoning in graphs.