Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.

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.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

Empirical Bayes (EB) estimators can match the first-order asymptotic risk of maximum likelihood (ML) while behaving very differently at second order: recent excess mean squared error (XMSE) analysis shows that kernel-based EB estimation may be worse than ML when the kernel is poorly aligned with the true parameter. This paper turns that diagnostic into a design principle. We propose an XMSE-aware mixed estimator that interpolates between ML and EB shrinkage. Its fixed-weight XMSE is a scalar quadratic, yielding a closed-form oracle mixing weight that is no worse than both ML and the base EB estimator at the XMSE scale. A plug-in implementation based on finite-sample XMSE approximations is proved consistent, with a second-order oracle regret rate for an interior oracle weight. We further establish a transfer of the regret bound to the fixed-weight risk curve evaluated at the selected weight, a thresholded boundary rule, and extensions to compact kernel families and to finite and growing kernel dictionaries with high-probability oracle bounds. Finite impulse response simulations with SURE-tuned, hard-selection, and trace-corrected baselines, together with the public Silverbox and Cascaded Tanks benchmarks, show that the proposed estimator retains most of the benefit of regularization when it is helpful and retreats toward ML under kernel misspecification, with an identified finite-de analyzed on the benchmarks.

We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

Synthetic data augmentation is widely used to mitigate class imbalance, but its theoretical effects on score-based classification remain poorly understood. This paper develops a framework for characterizing when synthetic minority augmentation can improve threshold-integrated and threshold-optimized metrics, including AUROC, AUPRC, best-threshold balanced accuracy, and best-threshold \(\F_1\) score. We separate the effect of augmentation into two components: a change in effective class weighting and a discrepancy between the synthetic and true minority distributions. Under well-specified score models, the raw estimator already targets the likelihood-ratio ordering, which is population-optimal for the metrics considered. Consequently, augmentation cannot provide a fundamental population-level improvement beyond possible finite-sample variance reduction, and may introduce additional bias through synthetic distributional error. We further establish minimax lower bounds showing that the raw estimator already achieves the optimal metric-regret rate in the well-specified regime. Under misspecification, however, augmentation can play a qualitatively different role: by changing the effective class balance, it can alter the restricted-class projection and correct ranking errors induced by the raw imbalanced objective. We provide explicit improvement bounds quantifying the roles of approximation error, finite-sample estimation error, and synthetic distributional error. Simulation studies corroborate the theory, demonstrating limited gains under well-specification and nontrivial but nonmonotone improvements under misspecification.

We study Transformer-type residual layers under cross-entropy training through a continuous-depth mean field control viewpoint. Depth is treated as time, layer parameters as controls, and the residual Transformer recursion as an explicit Euler scheme for a controlled hidden-state flow. For fixed controls, we prove an $O(\varepsilon)$ pathwise approximation of finite-depth trajectories by the continuous flow and combine this with high-probability sampling bounds for the empirical cross-entropy risk. We formulate the limiting population problem as a first-order transport control problem for the law of hidden states and derive a Pontryagin condition whose terminal adjoint contains the softmax residual. We also give finite-class and metric-entropy uniform estimates, compare optimal values, and discuss existence, stability, continuous-to-discrete recovery, initialization, and range estimates for continuous minimizers.

In this paper we prove some concentration inequalities for two types of error probabilities in the Empirical Risk Principle (ERP) in statistical learning, which provide a lower bound and an upper bound for the minimal risk (in terms of the minimal empirical risk) with non-asymptotic high confidence. The usual boundedness condition of the empirical risk function is relaxed to the Gaussian or exponential integrability condition. The confidence of the lower bound of the minimal risk is shown to be independent of the number of training parameters and the dimension of the input vectors, allowing one to detect the deficiency of a learning machine efficiently; and the confidence of the upper bound of the minimal risk is proved to be high provided that the sample size $n$ is much greater than the box dimension of the parameter set $Θ$ in the Orlicz metric $d_{ψ_1}$ associated with the risk functions. Our work is based on Talagrand's concentration inequalities (the sharp versions by Bousquet and Klein-Rio), transport-entropy inequalities and the recent progress in the theory of empirical processes and statistical learning.

We study online decision making problems under resource constraints, where both reward and cost functions are drawn from distributions that may change adversarially over time. We focus on two canonical settings: (i) online resource allocation where rewards and costs are observed before action selection, and (ii)online learning with resource constraints where they are observed after action selection, under full feedback or bandit feedback. It is well known that achieving sublinear regret in these settings is impossible when reward and cost distributions may change arbitrarily over time. To address this challenge, we analyze a framework in which the learner is guided by a spending plan--a sequence prescribing expected resource usage across rounds. We design general (primal-)dual methods that achieve sublinear regret with respect to baselines that follow the spending plan. Crucially, the performance of our algorithms improves when the spending plan ensures a well-balanced distribution of the budget across rounds. We additionally provide a robust variant of our methods to handle worst-case scenarios where the spending plan is highly imbalanced. To conclude, we study the regret of our algorithms when competing against benchmarks that deviate from the prescribed spending plan.

Searching through chemical space is an exceptionally challenging problem because the number of possible molecules grows combinatorially with the number of atoms. Large, autoregressive models trained on databases of chemical compounds have yielded powerful generators, but we still lack robust strategies for generating molecules with desired properties. This molecular search problem closely resembles the "alignment" problem for large language models, though for many chemical tasks we have a specific and easily evaluable reward function. Here, we introduce an algorithm called energy rank alignment (ERA) that leverages an explicit reward function to produce a gradient-based objective that we use to optimize autoregressive policies. We show theoretically that this algorithm is closely related to proximal policy optimization (PPO) and direct preference optimization (DPO), but has a minimizer that converges to an ideal Gibbs-Boltzmann distribution with the reward playing the role of an energy function. Furthermore, this algorithm is highly scalable, does not require reinforcement learning, and performs well relative to DPO when the number of preference observations per pairing is small. We deploy this approach to align molecular transformers and protein language models to generate molecules and protein sequences, respectively, with externally specified properties and find that it does so robustly, searching through diverse parts of chemical space.