Understanding the limitations of gradient methods, and stochastic gradient descent (SGD) in particular, is a central challenge in learning theory. To that end, a commonly used tool is the Statistical Queries (SQ) framework, which studies performance limits of algorithms based on noisy interaction with the data. However, it is known that the formal connection between the SQ framework and SGD is tenuous: Existing results typically rely on adversarial or specially-structured gradient noise that does not reflect the noise in standard SGD, and (as we point out here) can sometimes lead to incorrect predictions. Moreover, many analyses of SGD for challenging problems rely on non-trivial algorithmic modifications, such as restricting the SGD trajectory to the sphere or using very small learning rates. To address these shortcomings, we develop a new, non-SQ framework to study the limitations of standard vanilla SGD, for single-index and multi-index models (namely, when the target function depends on a low-dimensional projection of the inputs). Our results apply to a broad class of settings and architectures, including (potentially deep) neural networks.

Linear attention methods offer Transformers $O(N)$ complexity but typically underperform standard softmax attention. We identify two fundamental limitations affecting these approaches: the restriction to convex combinations that only permits additive information blending, and uniform accumulated weight bias that dilutes attention in long contexts. We propose Zero-Sum Linear Attention (ZeroS), which addresses these limitations by removing the constant zero-order term $1/t$ and reweighting the remaining zero-sum softmax residuals. This modification creates mathematically stable weights, enabling both positive and negative values and allowing a single attention layer to perform contrastive operations. While maintaining $O(N)$ complexity, ZeroS theoretically expands the set of representable functions compared to convex combinations. Empirically, it matches or exceeds standard softmax attention across various sequence modeling benchmarks.

A large body of theory and empirical work hypothesizes a connection between the flatness of a neural network's loss landscape during training and its performance. However, there have been conceptually opposite pieces of evidence regarding when SGD prefers flatter or sharper solutions during training. In this work, we partially but causally clarify the flatness-seeking behavior of SGD by identifying and exactly solving an analytically solvable model that exhibits both flattening and sharpening behavior during training. In this model, the SGD training has no \textit{a priori} preference for flatness, but only a preference for minimal gradient fluctuations. This leads to the insight that, at least within this model, it is data distribution that uniquely determines the sharpness at convergence, and that a flat minimum is preferred if and only if the noise in the labels is isotropic across all output dimensions. When the noise in the labels is anisotropic, the model instead prefers sharpness and can converge to an arbitrarily sharp solution, depending on the imbalance in the noise in the labels spectrum. We reproduce this key insight in controlled settings with different model architectures such as MLP, RNN, and transformers.

Weighted empirical risk minimization is a common approach to prediction under distribution drift. This article studies its out-of-sample prediction error under nonstationarity. We provide a general decomposition of the excess risk into a learning term and an error term associated with distribution drift, and prove oracle inequalities for the learning error under mixing conditions. The learning bound holds uniformly over arbitrary weight classes and accounts for the effective sample size induced by the weight vector, the complexity of the weight and hypothesis classes, and potential data dependence. We illustrate the applicability and sharpness of our results in (auto-) regression problems with linear models, basis approximations, and neural networks, recovering minimax-optimal rates (up to logarithmic factors) when specialized to unweighted and stationary settings.

We consider the sequential experimental design problem in the predict-then-optimize paradigm. In this paradigm, the outputs of the prediction model are used as coefficient vectors in a downstream linear optimization problem. Traditional sequential experimental design aims to control the input variables (features) so that the improvement in prediction accuracy from each experimental outcome (label) is maximized. However, in the predict-then-optimize setting, performance is ultimately evaluated based on the decision loss induced by the downstream optimization, rather than by prediction error. This mismatch between prediction accuracy and decision loss renders traditional decision-blind designs inefficient. To address this issue, we propose a directional-based metric to quantify predictive uncertainty. This metric does not require solving an optimization oracle and is therefore computationally tractable. We show that the resulting sequential design criterion enjoys strong consistency and convergence guarantees. Under a broad class of distributions, we demonstrate that our directional uncertainty-based design attains an earlier stopping time than decision-blind designs. This advantage is further supported by real-world experiments on an LLM job allocation problem.

Normative modeling estimates reference distributions of biological measures conditional on covariates, enabling centiles and clinically interpretable deviation scores to be derived. Most neuroimaging pipelines fit one model per imaging-derived phenotype (IDP), which scales well but discards multivariate dependence that may encode coordinated patterns. We propose denoising diffusion probabilistic models (DDPMs) as a unified conditional density estimator for tabular IDPs, from which univariate centiles and deviation scores are derived by sampling. We utilise two denoiser backbones: (i) a feature-wise linear modulation (FiLM) conditioned multilayer perceptron (MLP) and (ii) a tabular transformer with feature self-attention and intersample attention (SAINT), conditioning covariates through learned embeddings. We evaluate on a synthetic benchmark with heteroscedastic and multimodal age effects and on UK Biobank FreeSurfer phenotypes, scaling from dimension of 2 to 200. Our evaluation suite includes centile calibration (absolute centile error, empirical coverage, and the probability integral transform), distributional fidelity (Kolmogorov-Smirnov tests), multivariate dependence diagnostics, and nearest-neighbour memorisation analysis. For low dimensions, diffusion models deliver well-calibrated per-IDP outputs comparable to traditional baselines while jointly modeling realistic dependence structure. At higher dimensions, the transformer backbone remains substantially better calibrated than the MLP and better preserves higher-order dependence, enabling scalable joint normative models that remain compatible with standard per-IDP pipelines. These results support diffusion-based normative modeling as a practical route to calibrated multivariate deviation profiles in neuroimaging.

We study privacy amplification by synthetic data release, a phenomenon in which differential privacy guarantees are improved by releasing only synthetic data rather than the private generative model itself. Recent work by Pierquin et al. (2025) established the first formal amplification guarantees for a linear generator, but they apply only in asymptotic regimes where the model dimension far exceeds the number of released synthetic records, limiting their practical relevance. In this work, we show a surprising result: under a bounded-parameter assumption, privacy amplification persists even when releasing an unbounded number of synthetic records, thereby improving upon the bounds of Pierquin et al. (2025). Our analysis provides structural insights that may guide the development of tighter privacy guarantees for more complex release mechanisms.

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.