Multi-armed bandit (MAB) is a widely adopted framework for sequential decision-making under uncertainty. Traditional bandit algorithms rely solely on online data, which tends to be scarce as it must be gathered during the online phase when the arms are actively pulled. However, in many practical settings, rich auxiliary data, such as covariates of past users, is available prior to deploying any arms. We introduce a new setting for MAB where pre-trained machine learning (ML) models are applied to convert side information and historical data into \emph{surrogate rewards}. A prominent feature of this setting is that the surrogate rewards may exhibit substantial bias, as true reward data is typically unavailable in the offline phase, forcing ML predictions to heavily rely on extrapolation. To address the issue, we propose the Machine Learning-Assisted Upper Confidence Bound (MLA-UCB) algorithm, which can be applied to any reward prediction model and any form of auxiliary data. When the predicted and true rewards are jointly Gaussian, it provably improves the cumulative regret, provided that the correlation is non-zero -- even in cases where the mean surrogate reward completely misaligns with the true mean rewards. Notably, our method requires no prior knowledge of the covariance matrix between true and surrogate rewards. We compare MLA-UCB with the standard UCB on a range of numerical studies and show a sizable efficiency gain even when the size of the offline data and the correlation between predicted and true rewards are moderate.

Most modern learning problems are over-parameterized, where the number of learnable parameters is much greater than the number of training data points. In this over-parameterized regime, the training loss typically has infinitely many global optima that completely interpolate the data with varying generalization performance. The particular global optimum we converge to depends on the implicit bias of the optimization algorithm. The question we address in this paper is, ``What is the implicit bias that leads to the best generalization performance?". To find the optimal implicit bias, we provide a precise asymptotic analysis of the generalization performance of interpolators obtained from the minimization of convex functions/potentials for over-parameterized linear regression with non-isotropic Gaussian data. In particular, we obtain a tight lower bound on the best generalization error possible among this class of interpolators in terms of the over-parameterization ratio, the variance of the noise in the labels, the eigenspectrum of the data covariance, and the underlying distribution of the parameter to be estimated. Finally, we find the optimal convex implicit bias that achieves this lower bound under certain sufficient conditions involving the log-concavity of the distribution of a Gaussian convolved with the prior of the true underlying parameter.

Robust estimation of heterogeneous treatment effects is a fundamental challenge for optimal decision-making in domains ranging from personalized medicine to educational policy. In recent years, predictive machine learning has emerged as a valuable toolbox for causal estimation, enabling more flexible effect estimation. However, accurately estimating conditional average treatment effects (CATE) remains a major challenge, particularly in the presence of many covariates. In this article, we propose pretraining strategies that leverage a phenomenon in real-world applications: factors that are prognostic of the outcome are frequently also predictive of treatment effect heterogeneity. In medicine, for example, components of the same biological signaling pathways frequently influence both baseline risk and treatment response. Specifically, we demonstrate our approach within the R-learner framework, which estimates the CATE by solving individual prediction problems based on a residualized loss. We use this structure to incorporate side information and develop models that can exploit synergies between risk prediction and causal effect estimation. In settings where these synergies are present, this cross-task learning enables more accurate signal detection, yields lower estimation error, reduced false discovery rates, and higher power for detecting heterogeneity.

The Median of Means (MoM) is a mean estimator that has gained popularity in the context of heavy-tailed data. In this work, we analyze its performance in the task of simultaneously estimating the mean of each function in a class $\mathcal{F}$ when the data distribution possesses only the first $p$ moments for $p \in (1,2]$. We prove a new sample complexity bound using a novel symmetrization technique that may be of independent interest. Additionally, we present applications of our result to $k$-means clustering with unbounded inputs and linear regression with general losses, improving upon existing works.

Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.

Synthetic Data Generation has become essential for scalable, privacy-preserving statistical analysis. While standard approaches based on generative models, such as Normalizing Flows, have been widely used, they often suffer from slow convergence in high-dimensional settings, frequently converging more slowly than the canonical $1/\sqrt{n}$ rate when approximating the true data distribution. To overcome these limitations, we propose a Latent Noise Injection method using Masked Autoregressive Flows (MAF). Instead of directly sampling from the trained model, our method perturbs each data point in the latent space and maps it back to the data domain. This construction preserves a one to one correspondence between observed and synthetic data, enabling synthetic outputs that closely reflect the underlying distribution, particularly in challenging high-dimensional regimes where traditional sampling struggles. Our procedure satisfies local $(ε, δ)$-differential privacy and introduces a single perturbation parameter to control the privacy-utility trade-off. Although estimators based on individual synthetic datasets may converge slowly, we show both theoretically and empirically that aggregating across $K$ studies in a meta analysis framework restores classical efficiency and yields consistent, reliable inference. We demonstrate that with a well-calibrated perturbation parameter, Latent Noise Injection achieves strong statistical alignment with the original data and robustness against membership inference attacks. These results position our method as a compelling alternative to conventional flow-based sampling for synthetic data sharing in decentralized and privacy-sensitive domains, such as biomedical research.

Despite recent efforts, neural networks still struggle to learn in non-stationary environments, and our understanding of catastrophic forgetting (CF) is far from complete. In this work, we perform a systematic study on the impact of model scale and the degree of feature learning in continual learning. We reconcile existing contradictory observations on scale in the literature, by differentiating between lazy and rich training regimes through a variable parameterization of the architecture. We show that increasing model width is only beneficial when it reduces the amount of feature learning, yielding more laziness. Using the framework of dynamical mean field theory, we then study the infinite width dynamics of the model in the feature learning regime and characterize CF, extending prior theoretical results limited to the lazy regime. We study the intricate relationship between feature learning, task non-stationarity, and forgetting, finding that high feature learning is only beneficial with highly similar tasks. We identify a transition modulated by task similarity where the model exits an effectively lazy regime with low forgetting to enter a rich regime with significant forgetting. Finally, our findings reveal that neural networks achieve optimal performance at a critical level of feature learning, which depends on task non-stationarity and transfers across model scales. This work provides a unified perspective on the role of scale and feature learning in continual learning.

When applicants get rejected by an algorithmic decision system, recourse explanations provide actionable suggestions for how to change their input features to get a positive evaluation. A crucial yet overlooked phenomenon is that recourse explanations are performative: When many applicants act according to their recommendations, their collective behavior may change statistical regularities in the data and, once the model is refitted, also the decision boundary. Consequently, the recourse algorithm may render its own recommendations invalid, such that applicants who make the effort of implementing their recommendations may be rejected again when they reapply. In this work, we formally characterize the conditions under which recourse explanations remain valid under performativity. A key finding is that recourse actions may become invalid if they are influenced by or if they intervene on non-causal variables. Based on our analysis, we caution against the use of standard counterfactual explanations and causal recourse methods, and instead advocate for recourse methods that recommend actions exclusively on causal variables.

Random feature approximation is arguably one of the most widely used techniques for kernel methods in large-scale learning algorithms. In this work, we analyze the generalization properties of random feature methods, extending previous results for Tikhonov regularization to a broad class of spectral regularization techniques. This includes not only explicit methods but also implicit schemes such as gradient descent and accelerated algorithms like the Heavy-Ball and Nesterov method. Through this framework, we enable a theoretical analysis of neural networks and neural operators through the lens of the Neural Tangent Kernel (NTK) approach trained via gradient descent. For our estimators we obtain optimal learning rates over regularity classes (even for classes that are not included in the reproducing kernel Hilbert space), which are defined through appropriate source conditions. This improves or completes previous results obtained in related settings for specific kernel algorithms.

As conversational agents increasingly engage in emotionally supportive dialogue, it is important to understand how closely their interactions resemble those in traditional therapy settings. This study investigates whether the concerns shared with a robot align with those shared in human-to-human (H2H) therapy sessions, and whether robot responses semantically mirror those of human therapists. We analyzed two datasets: one of interactions between users and professional therapists (Hugging Face's NLP Mental Health Conversations), and another involving supportive conversations with a social robot (QTrobot from LuxAI) powered by a large language model (LLM, GPT-3.5). Using sentence embeddings and K-means clustering, we assessed cross-agent thematic alignment by applying a distance-based cluster-fitting method that evaluates whether responses from one agent type map to clusters derived from the other, and validated it using Euclidean distances. Results showed that 90.88% of robot conversation disclosures could be mapped to clusters from the human therapy dataset, suggesting shared topical structure. For matched clusters, we compared the subjects as well as therapist and robot responses using Transformer, Word2Vec, and BERT embeddings, revealing strong semantic overlap in subjects' disclosures in both datasets, as well as in the responses given to similar human disclosure themes across agent types (robot vs. human therapist). These findings highlight both the parallels and boundaries of robot-led support conversations and their potential for augmenting mental health interventions.