We study gradient descent (GD) with a constant stepsize for ℓ2-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in eO(κ) steps with κ being the condition number. Surprisingly, we show that this can be accelerated to eO( κ)by simply using a large stepsize--for which the objective evolves nonmonotonically. The acceleration brought by large stepsizes extends to minimizing the population risk for separable distributions, improving on the best-known upper bounds on the number of steps to reach a nearoptimum. Finally, we characterize the largest stepsize for the local convergence of GD, which also determines the global convergence in special scenarios. Our results extend the analysis of Wu et al. (2024) from convex settings with minimizers at infinity to strongly convex cases with finite minimizers.

Most existing transfer learning algorithms for high-dimensional models employ a two-step regularization framework, whose success heavily hinges on the assumption that the pre-trained model closely resembles the target. To relax this assumption, we propose a co-regularization process to directly exploit beneficial knowledge from the source domain for high-dimensional generalized linear models. The proposed method learns the target parameter by constraining the source parameters to be close to the target one, thereby preventing fine-tuning failures caused by significantly deviated pre-trained parameters. Our theoretical analysis demonstrates that the proposed method accommodates a broader range of sources than existing two-step frameworks, thus being more robust to less similar sources. Its effectiveness is validated through extensive empirical studies.

Direct preference optimization (DPO) is widely used as a simple and stable method for aligning large language models (LLMs) with human preferences. This paper investigates a generalized DPO loss that enables a policy model to match the target policy from a likelihood ratio estimation perspective. The ratio of the target policy provides a unique identification of the policy distribution without relying on reward models or partition functions. This allows the generalized loss to retain both simplicity and theoretical guarantees, which prior work such as f-PO fails to achieve simultaneously. We propose Bregman preference optimization (BPO), a generalized framework for ratio matching that provides a family of objective functions achieving target policy optimality.

Gaussian sketching, which consists of pre-multiplying the data with a random Gaussian matrix, is a widely used technique in data science and machine learning. Beyond computational benefits, this operation also provides differential privacy guarantees due to its inherent randomness. In this work, we revisit this operation through the lens of Rényi Differential Privacy (RDP), providing a refined privacy analysis that yields significantly tighter bounds than prior results. We then demonstrate how this improved analysis leads to performance improvement in different linear regression settings, establishing theoretical utility guarantees. Empirically, our methods improve performance across multiple datasets and, in several cases, reduce runtime.

Background Childhood Anemia affects an estimated 40% of children aged 6-59 months globally and arises from heterogeneous nutritional, infectious, and socioeconomic factors that vary substantially across settings. This variability challenges the generalizability of predictive machine learning models, which often degrade under cross-population or temporal shifts. We investigated the utility a modern transformer-based tabular foundation model (TabPFN) as a complementatry framework with respect to supervised classical machine learning methods across diverse country contexts, with particular attention to data-scarce settings where surveillance capacity is most limited. Methods We conducted a multi-country prediction study using Demographic and Health Surveys (DHS) children's recode data from 16 countries spanning Africa, Asia, Latin America, the Caucasus, and the Middle East. The harmonized analytic cohort comprised of (n = 68,856)children aged 6-59 months with valid hemoglobin measurements. Anemia was defined using WHO age and altitude-adjusted thresholds and treated as a binary outcome. We trained Logistic Regression, XGBoost, and LightGBM models using standard supervised learning, and evaluated TabPFN v2.6 in an in-context learning setting. Performance was assessed using Area Under the Receiver Operating Characteristic Curve (AUC-ROC) and other standard classification metrics, with calibration evaluated via Brier score and expected calibration error (ECE). Uncertainty in performance estimates was quantified using bootstrap resampling to derive 95% confidence intervals. Robustness was assessed in a few-shot learning setting. Cross-population generalization was examined using leave-one-country-out (LOCO) validation and reverse-LOCO experiments to assess directional transferability. Subgroup analyses were conducted across five demographic strata: child age group, sex, maternal education, residence type, and household wealth quintile. Feature importance was assessed using standard linear and tree-based explainer SHAP values for the three supervised models and an adapted version of SHAP for TabPFN, aggregated across countries and examined at the country level. TabPFN also yielded the best probabilistic calibration across all 16 countries, achieving the lowest mean Brier score (0.203) and Expected Calibration Error (ECE = 0.042) of all models evaluated; LightGBM and Logistic Regression exhibited the greatest miscalibration, particularly at higher predicted probabilities. Under full-data conditions, within-country discrimination was moderate across all models (AUC-ROC 0.59-0.76) Under LOCO validation, performance declined modestly (AUC-ROC 0.58-0.69) Reverse-LOCO analyses revealed asymmetric and directional transferability, with epidemiologically diverse populations serving as more informative training sources and certain target populations remaining persistently difficult to predict regardless of model or training data.

GPUs have significantly accelerated first-order methods for large-scale optimization, especially in continuous optimization. However, this success has not transferred cleanly to problems with discrete variables, combinatorial structure, and nonlinear objectives, such as certifying optimal solutions for cardinality-constrained generalized linear models. Major challenges include the sequential processing of heterogeneous nodes in branch and bound (BnB) and frequent data movement between the CPU and GPU. We propose a simple, generic, and modular CPU--GPU framework that processes multiple BnB nodes in batches on GPUs. The framework is built around a small set of GPU-efficient routines and uses padding together with lightweight custom kernels to handle irregular node data structures. Experiments show one to two orders of magnitude speedups and zero optimality gap on challenging instances. The framework can also be extended to collect the entire Rashomon set, enabling downstream statistical analysis such as variable-importance analysis and model selection under secondary user-specific measures (e.g., AUC in classification).

Many clinical risk scores are deployed as additive rules with nonnegative integer points assigned to relevant binary predictive features. These integer weights not only make the score easier to use in practice but also promote sparsity in the resulting prediction model. Such risk scores are often derived by first fitting a regression model and then rounding the estimated coefficients to the nearest integer after appropriate scaling. This approach is computationally fast but does not guarantee optimality of the resulting score. Alternatively, one may search over all possible integer weights to directly optimize a value function by posing the problem as an integer programming task. However, the associated computational burden can be substantial, especially when the value function is nonconcave or even discontinuous. In this paper, we develop new machine learning algorithms that employ a flexible greedy optimization strategy to learn such additive scoring directly under explicit and sensible optimality objectives. We apply the proposed method to a large electronic health record (EHR) cohort in Epic Cosmos to construct an integer-weighted comorbidity score for measuring the risk of post-discharge mortality. We also conduct a simulation study to examine the finite-sample operating characteristics.

Stochastic gradient methods are central to modern large-scale learning, but their use with incomplete covariates remains delicate since imputation schemes generally introduce systematic gradient biases, as shown for linear models. In this work, we prove that all parametric models exhibit similar gradient bias for various imputation procedures and characterize exactly the dependence on the missingness ratio vector $p$, with $O(\|p\|)$ as the leading term. We exploit this analysis to propose a simple debiasing procedure for stochastic gradient descent (SGD) with missing values based on Richardson extrapolation, which leverages the exact expression of the gradient bias. The key idea is to \emph{deliberately add missingness}: from an already incomplete observation, we generate a further-thinned version at a higher, controlled missingness level, and combine the two resulting stochastic gradients to cancel the leading bias term. We prove that one Richardson step reduces the gradient bias from $O(\|p\|)$ to $O(\|p\|^2)$ under several missingness scenarios. Our proposed method is computationally efficient, model-agnostic and applies to any parametric loss whose stochastic gradient can be computed after imputation. Furthermore, when missing indicators are independent, the population gradient bias is a multilinear polynomial in $p$ and depends only on population gradient errors induced by declaring a single coordinate missing. In this case, our method generalizes to a multi-step Richardson procedure which recursively cancels higher-order terms. Empirically, Richardson debiasing improves optimization and estimation across several generalized linear models and combines positively with widely used imputation procedures such as MICE. These results suggest that, somewhat counter-intuitively, adding controlled missingness on top of existing missing data can make stochastic learning from incomplete data more accurate.

We introduce a general framework that extends Bayesian inference by allowing the researcher to explicitly encode confidence in each source of uncertainty within the model. This mechanism provides a new handle for model design and regularisation control. Building on this framework, we develop a general approach for inducing sparsity in statistical models and illustrate its use in linear and logistic regression, as well as in Bayesian neural networks.

This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical largesample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Candès [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.