Scaling laws describe how learning performance improves with data, compute, or training time, and have become a central theme in modern deep learning. We study this phenomenon in a canonical nonlinear model: phase retrieval with anisotropic Gaussian inputs whose covariance spectrum follows a power law. Unlike the isotropic case, where dynamics collapse to a two-dimensional system, anisotropy yields a qualitatively new regime in which an infinite hierarchy of coupled equations governs the evolution of the summary statistics. We develop a tractable reduction that reveals a three-phase trajectory: (i) fast escape from low alignment, (ii) slow convergence of the summary statistics, and (iii) spectral-tail learning in low-variance directions. From this decomposition, we derive explicit scaling laws for the mean-squared error, showing how spectral decay dictates convergence times and error curves. Experiments confirm the predicted phases and exponents. These results provide the first rigorous characterization of scaling laws in nonlinear regression with anisotropic data, highlighting how anisotropy reshapes learning dynamics.

We propose a way of transforming the problem of conditional density estimation into a single nonparametric regression task via the introduction of auxiliary samples. This allows leveraging regression methods that work well in high dimensions, such as neural networks and decision trees. Our main theoretical result characterizes and establishes the convergence of our estimator to the true conditional density in the data limit. We develop condensité, a method that implements this approach. We demonstrate the benefit of the auxiliary samples on synthetic data and showcase that condensité can achieve good out-of-the-box results. We evaluate our method on a large population survey dataset and on a satellite imaging dataset. In both cases, we find that condensité matches or outperforms the state of the art and yields conditional densities in line with established findings in the literature on each dataset. Our contribution opens up new possibilities for regression-based conditional density estimation and the empirical results indicate strong promise for applied research.

We propose and analyze a variant of Sparse Polyak for high dimensional M-estimation problems. Sparse Polyak proposes a novel adaptive step-size rule tailored to suitably estimate the problem's curvature in the high-dimensional setting, guaranteeing that the algorithm's performance does not deteriorate when the ambient dimension increases. However, convergence guarantees can only be obtained by sacrificing solution sparsity and statistical accuracy. In this work, we introduce a variant of Sparse Polyak that retains its desirable scaling properties with respect to the ambient dimension while obtaining sparser and more accurate solutions.

In this work, we propose a set of conformal prediction procedures tailored to compositional responses, where outcomes are proportions that must be positive and sum to one. Building on Dirichlet regression, we introduce a split conformal approach based on quantile residuals and a highest-density region strategy that combines a fast coordinate-floor approximation with an internal grid refinement to restore sharpness. Both constructions are model-agnostic at the conformal layer and guarantee finite-sample marginal coverage under exchangeability, while respecting the geometry of the simplex. A comprehensive Monte Carlo study spanning homoscedastic and heteroscedastic designs shows that the quantile residual and grid-refined HDR methods achieve empirical coverage close to the nominal 90\% level and produce substantially narrower regions than the coordinate-floor approximation, which tends to be conservative. We further demonstrate the methods on household budget shares from the BudgetItaly dataset, using standardized socioeconomic and price covariates with a train, calibration, and test split. In this application, the grid-refined HDR attains coverage closest to the target with the smallest average widths, closely followed by the quantile residual approach, while the simple triangular HDR yields wider, less informative sets. Overall, the results indicate that conformal prediction on the simplex can be both calibrated and efficient, providing practical uncertainty quantification for compositional prediction tasks.

Minimizing loss functions is central to machine-learning training. Although first-order methods dominate practical applications, higher-order techniques such as Newton's method can deliver greater accuracy and faster convergence, yet are often avoided due to their computational cost. This work analyzes the impact of finite-precision arithmetic on Newton steps and establishes a convergence theorem for mixed-precision Newton optimizers, including "quasi" and "inexact" variants. The theorem provides not only convergence guarantees but also a priori estimates of the achievable solution accuracy. Empirical evaluations on standard regression benchmarks demonstrate that the proposed methods outperform Adam on the Australian and MUSH datasets. The second part of the manuscript introduces GN_k, a generalized Gauss-Newton method that enables partial computation of second-order derivatives. GN_k attains performance comparable to full Newton's method on regression tasks while requiring significantly fewer derivative evaluations.

Financial time series forecasting is central to trading, portfolio optimization, and risk management, yet it remains challenging due to noisy, non-stationary, and heterogeneous data. Recent advances in time series foundation models (TSFMs), inspired by large language models, offer a new paradigm for learning generalizable temporal representations from large and diverse datasets. This paper presents the first comprehensive empirical study of TSFMs in global financial markets. Using a large-scale dataset of daily excess returns across diverse markets, we evaluate zero-shot inference, fine-tuning, and pre-training from scratch against strong benchmark models. We find that off-the-shelf pre-trained TSFMs perform poorly in zero-shot and fine-tuning settings, whereas models pre-trained from scratch on financial data achieve substantial forecasting and economic improvements, underscoring the value of domain-specific adaptation. Increasing the dataset size, incorporating synthetic data augmentation, and applying hyperparameter tuning further enhance performance.

Artificial intelligence (AI) and data science are transforming chemical research, yet few formal courses are tailored to synthetic and experimental chemists, who often face steep entry barriers due to limited coding experience and lack of chemistry-specific examples. We present the design and implementation of AI4CHEM, an introductory data-driven chem-istry course created for students on the synthetic chemistry track with no prior programming background. The curricu-lum emphasizes chemical context over abstract algorithms, using an accessible web-based platform to ensure zero-install machine learning (ML) workflow development practice and in-class active learning. Assessment combines code-guided homework, literature-based mini-reviews, and collaborative projects in which students build AI-assisted workflows for real experimental problems. Learning gains include increased confidence with Python, molecular property prediction, reaction optimization, and data mining, and improved skills in evaluating AI tools in chemistry. All course materials are openly available, offering a discipline-specific, beginner-accessible framework for integrating AI into synthetic chemistry training.

Pediatric pancreatitis is a progressive and debilitating inflammatory condition, including acute pancreatitis and chronic pancreatitis, that presents significant clinical diagnostic challenges. Machine learning-based methods also face diagnostic challenges due to limited sample availability and multimodal imaging complexity. To address these challenges, this paper introduces Upstream Probabilistic Meta-Imputation (UPMI), a light-weight augmentation strategy that operates upstream of a meta-learner in a low-dimensional meta-feature space rather than in image space. Modality-specific logistic regressions (T1W and T2W MRI radiomics) produce probability outputs that are transformed into a 7-dimensional meta-feature vector. Class-conditional Gaussian mixture models (GMMs) are then fit within each cross-validation fold to sample synthetic meta-features that, combined with real meta-features, train a Random Forest (RF) meta-classifier. On 67 pediatric subjects with paired T1W/T2W MRIs, UPMI achieves a mean AUC of 0.908 $\pm$ 0.072, a $\sim$5% relative gain over a real-only baseline (AUC 0.864 $\pm$ 0.061).

Online sparse linear regression is the task of applying linear regression analysis to examples arriving sequentially subject to a resource constraint that a limited number of features of examples can be observed. Despite its importance in many practical applications, it has been recently shown that there is no polynomial-time sublinear-regret algorithm unless NP$\subseteq$BPP, and only an exponential-time sublinear-regret algorithm has been found. In this paper, we introduce mild assumptions to solve the problem.

We propose a Bayesian regression method that accounts for multi-way interactions of arbitrary orders among the predictor variables. Our model makes use of a factorization mechanism for representing the regression coefficients of interactions among the predictors, while the interaction selection is guided by a prior distribution on random hypergraphs, a construction which generalizes the Finite Feature Model. We present a posterior inference algorithm based on Gibbs sampling, and establish posterior consistency of our regression model. Our method is evaluated with extensive experiments on simulated data and demonstrated to be able to identify meaningful interactions in applications in genetics and retail demand forecasting.