We prove that single-parameter natural exponential families with subexponential tails are self-concordant with polynomial-sized parameters. For subgaussian natural exponential families we establish an exact characterization of the growth rate of the self-concordance parameter. Applying these findings to bandits allows us to fill gaps in the literature: We show that optimistic algorithms for generalized linear bandits enjoy regret bounds that are both second-order (scale with the variance of the optimal arm's reward distribution) and free of an exponential dependence on the bound of the problem parameter in the leading term. To the best of our knowledge, ours is the first regret bound for generalized linear bandits with subexponential tails, broadening the class of problems to include Poisson, exponential and gamma bandits.

The recent cohort of publicly available generative models can produce human expert level content across a variety of topics and domains. Given a model in this cohort as a base model, methods such as parameter efficient fine-tuning, in-context learning, and constrained decoding have further increased generative capabilities and improved both computational and data efficiency. Entire collections of derivative models have emerged as a byproduct of these methods and each of these models has a set of associated covariates such as a score on a benchmark, an indicator for if the model has (or had) access to sensitive information, etc. that may or may not be available to the user. For some model-level covariates, it is possible to use "similar" models to predict an unknown covariate. In this paper we extend recent results related to embedding-based representations of generative models - the data kernel perspective space - to classical statistical inference settings. We demonstrate that using the perspective space as the basis of a notion of "similar" is effective for multiple model-level inference tasks. Generative models have recently met or surpassed human-level standards on benchmarks across a range of tasks (Nori et al., 2023; Katz et al., 2024; Dubey et al., 2024). While these claims warrant skepticism and further robustness evaluation (Ness et al., 2024), the impressive capabilities have created a competitive environment for training state-of-the-art models and have inspired the development of complementary methods to adapt models to particular use cases.

We study high-dimensional, ridge-regularized logistic regression in a setting in which the covariates may be missing or corrupted by additive noise. When both the covariates and the additive corruptions are independent and normally distributed, we provide exact characterizations of both the prediction error as well as the estimation error. Moreover, we show that these characterizations are universal: as long as the entries of the data matrix satisfy a set of independence and moment conditions, our guarantees continue to hold. Universality, in turn, enables the detailed study of several imputation-based strategies when the covariates are missing completely at random. We ground our study by comparing the performance of these strategies with the conjectured performance -- stemming from replica theory in statistical physics -- of the Bayes optimal procedure. Our analysis yields several insights including: (i) a distinction between single imputation and a simple variant of multiple imputation and (ii) that adding a simple ridge regularization term to single-imputed logistic regression can yield an estimator whose prediction error is nearly indistinguishable from the Bayes optimal prediction error. We supplement our findings with extensive numerical experiments.

Many applications involve reasoning about time durations before a critical event happens--also called time-to-event outcomes. When will a customer cancel a subscription, a coma patient wake up, or a convicted criminal reoffend? Time-to-event outcomes have been studied extensively within the field of survival analysis primarily by the statistical, medical, and reliability engineering communities, with textbooks already available in the 1970s and '80s. This monograph aims to provide a reasonably self-contained modern introduction to survival analysis. We focus on predicting time-to-event outcomes at the individual data point level with the help of neural networks. Our goal is to provide the reader with a working understanding of precisely what the basic time-to-event prediction problem is, how it differs from standard regression and classification, and how key "design patterns" have been used time after time to derive new time-to-event prediction models, from classical methods like the Cox proportional hazards model to modern deep learning approaches such as deep kernel Kaplan-Meier estimators and neural ordinary differential equation models. We further delve into two extensions of the basic time-to-event prediction setup: predicting which of several critical events will happen first along with the time until this earliest event happens (the competing risks setting), and predicting time-to-event outcomes given a time series that grows in length over time (the dynamic setting). We conclude with a discussion of a variety of topics such as fairness, causal reasoning, interpretability, and statistical guarantees. Our monograph comes with an accompanying code repository that implements every model and evaluation metric that we cover in detail.

In recent years, there has been growing interest in the field of functional neural networks. They have been proposed and studied with the aim of approximating continuous functionals defined on sets of functions on Euclidean domains. In this paper, we consider functionals defined on sets of functions on spheres. The approximation ability of deep ReLU neural networks is investigated by novel spherical analysis using an encoder-decoder framework. An encoder comes up first to accommodate the infinite-dimensional nature of the domain of functionals. It utilizes spherical harmonics to help us extract the latent finite-dimensional information of functions, which in turn facilitates in the next step of approximation analysis using fully connected neural networks. Moreover, real-world objects are frequently sampled discretely and are often corrupted by noise. Therefore, encoders with discrete input and those with discrete and random noise input are constructed, respectively. The approximation rates with different encoder structures are provided therein.

The Random Forest (RF) algorithm can be applied to a broad spectrum of problems, including time series prediction. However, neither the classical IID (Independent and Identically distributed) bootstrap nor block bootstrapping strategies (as implemented in rangerts) completely account for the nature of the Data Generating Process (DGP) while resampling the observations. We propose the combination of RF with a residual bootstrapping technique where we replace the IID bootstrap with the AR-Sieve Bootstrap (ARSB), which assumes the DGP to be an autoregressive process. To assess the new model's predictive performance, we conduct a simulation study using synthetic data generated from different types of DGPs. It turns out that ARSB provides more variation amongst the trees in the forest. Moreover, RF with ARSB shows greater accuracy compared to RF with other bootstrap strategies. However, these improvements are achieved at some efficiency costs.

The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the assumption that $\pi$ is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of $\pi$ needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If $\pi$ is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed.

Synthetic data generation has been proven successful in improving model performance and robustness in the context of scarce or low-quality data. Using the data valuation framework to statistically identify beneficial and detrimental observations, we introduce a novel augmentation pipeline that generates only highvalue training points based on hardness characterization. We first demonstrate via benchmarks on real data that Shapley-based data valuation methods perform comparably with learning-based methods in hardness characterisation tasks, while offering significant theoretical and computational advantages. Then, we show that synthetic data generators trained on the hardest points outperform non-targeted data augmentation on simulated data and on a large scale credit default prediction task. In particular, our approach improves the quality of out-of-sample predictions and it is computationally more efficient compared to non-targeted methods. Training complex machine learning models requires large amounts of data, but in real-world applications data may be of poor quality, insufficient in amount, or subject to privacy, safety, and regulatory limitations. Such challenges have sparked an interest in synthetic data generation (SDG), representing the practice of using available data to generate realistic synthetic samples (Lu et al., 2024). In this work, we argue that, when the objective is to use synthetic data to make an existing machine learning model better generalize to unseen data, augmenting only the hardest training points is more effective than augmenting the entire training dataset.

Large Language Models (LLMs) based on the pre-trained fine-tuning paradigm have become pivotal in solving natural language processing tasks, consistently achieving state-of-the-art performance. Nevertheless, the theoretical understanding of how model complexity influences fine-tuning performance remains challenging and has not been well explored yet. In this paper, we focus on autoregressive LLMs and propose to employ Hidden Markov Models (HMMs) to model them. Based on the HMM modeling, we investigate the relationship between model complexity and the generalization capability in downstream tasks. Specifically, we consider a popular tuning paradigm for downstream tasks, head tuning, where all pre-trained parameters are frozen and only individual heads are trained atop pre-trained LLMs. Our theoretical analysis reveals that the risk initially increases and then decreases with rising model complexity, showcasing a "double descent" phenomenon. In this case, the initial "descent" is degenerate, signifying that the "sweet spot" where bias and variance are balanced occurs when the model size is zero. Obtaining the presented in this study conclusion confronts several challenges, primarily revolving around effectively modeling autoregressive LLMs and downstream tasks, as well as conducting a comprehensive risk analysis for multivariate regression. Our research is substantiated by experiments conducted on data generated from HMMs, which provided empirical support and alignment with our theoretical insights.

Principal component analysis (PCA) is a widely used technique for dimension reduction. As datasets continue to grow in size, distributed-PCA (DPCA) has become an active research area. A key challenge in DPCA lies in efficiently aggregating results across multiple machines or computing nodes due to computational overhead. Fan et al. (2019) introduced a pioneering DPCA method to estimate the leading rank-$r$ eigenspace, aggregating local rank-$r$ projection matrices by averaging. However, their method does not utilize eigenvalue information. In this article, we propose a novel DPCA method that incorporates eigenvalue information to aggregate local results via the matrix $\beta$-mean, which we call $\beta$-DPCA. The matrix $\beta$-mean offers a flexible and robust aggregation method through the adjustable choice of $\beta$ values. Notably, for $\beta=1$, it corresponds to the arithmetic mean; for $\beta=-1$, the harmonic mean; and as $\beta \to 0$, the geometric mean. Moreover, the matrix $\beta$-mean is shown to associate with the matrix $\beta$-divergence, a subclass of the Bregman matrix divergence, to support the robustness of $\beta$-DPCA. We also study the stability of eigenvector ordering under eigenvalue perturbation for $\beta$-DPCA. The performance of our proposal is evaluated through numerical studies.