Statistical Learning Theory

In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated.

This paper uses classical high-rate quantization theory to provide new insights into mixture-of-experts (MoE) models for regression tasks. Our MoE is defined by a segmentation of the input space to regions, each with a single-parameter expert that acts as a constant predictor with zero-compute at inference. Motivated by high-rate quantization theory assumptions, we assume that the number of experts is sufficiently large to make their input-space regions very small. This lets us study the approximation error of our MoE model class: (i) for one-dimensional inputs, we formulate the test error and its minimizing segmentation and experts; (ii) for multidimensional inputs, we formulate an upper bound for the test error and study its minimization. Moreover, we consider the learning of the expert parameters from a training dataset, given an input-space segmentation, and formulate their statistical learning properties. This leads us to theoretically and empirically show how the tradeoff between approximation and estimation errors in MoE learning depends on the number of experts.

Deep neural networks excel in high-dimensional problems, outperforming models such as kernel methods, which suffer from the curse of dimensionality. However, the theoretical foundations of this success remain poorly understood. We follow the idea that the compositional structure of the learning task is the key factor determining when deep networks outperform other approaches. Taking a step towards formalizing this idea, we consider a simple compositional model, namely the multi-index model (MIM). In this context, we introduce and study hyper-kernel ridge regression (HKRR), an approach blending neural networks and kernel methods. Our main contribution is a sample complexity result demonstrating that HKRR can adaptively learn MIM, overcoming the curse of dimensionality. Further, we exploit the kernel nature of the estimator to develop ad hoc optimization approaches. Indeed, we contrast alternating minimization and alternating gradient methods both theoretically and numerically. These numerical results complement and reinforce our theoretical findings.

Test-time training (TTT) enhances model performance by explicitly updating designated parameters prior to each prediction to adapt to the test data. While TTT has demonstrated considerable empirical success, its theoretical underpinnings remain limited, particularly for nonlinear models. In this paper, we investigate the combination of TTT with in-context learning (ICL), where the model is given a few examples from the target distribution at inference time. We analyze this framework in the setting of single-index models, where the feature vector is drawn from a hidden low-dimensional subspace. For single-layer transformers trained with gradient-based algorithms and adopting TTT, we establish an upper bound on the prediction risk. Our theory reveals that TTT enables the single-layer transformers to adapt to both the feature vector and the link function, which vary across tasks. This creates a sharp contrast with ICL alone, which is theoretically difficult to adapt to shifts in the link function. Moreover, we provide the convergence rate with respect to the data length, showing the predictive error can be driven arbitrarily close to the noise level as the context size and the network width grow.

We study multivariate linear regression under Gaussian covariates in two settings, where data may be erased or corrupted by an adversary under a coordinate-wise budget. In the incomplete data setting, an adversary may inspect the dataset and delete entries in up to an $η$-fraction of samples per coordinate; a strong form of the Missing Not At Random model. In the corrupted data setting, the adversary instead replaces values arbitrarily, and the corruption locations are unknown to the learner. Despite substantial work on missing data, linear regression under such adversarial missingness remains poorly understood, even information-theoretically. Unlike the clean setting, where estimation error vanishes with more samples, here the optimal error remains a positive function of the problem parameters. Our main contribution is to characterize this error up to constant factors across essentially the entire parameter range. Specifically, we establish novel information-theoretic lower bounds on the achievable error that match the error of (computationally efficient) algorithms. A key implication is that, perhaps surprisingly, the optimal error in the missing data setting matches that in the corruption setting-so knowing the corruption locations offers no general advantage.

Meta-learning aims to leverage information across related tasks to improve prediction on unlabeled data for new tasks when only a small number of labeled observations are available ("few-shot" learning). Increased task diversity is often believed to enhance meta-learning by providing richer information across tasks. However, recent work by Kumar et al. (2022) shows that increasing task diversity, quantified through the overall geometric spread of task representations, can in fact degrade meta-learning prediction performance across a range of models and datasets. In this work, we build on this observation by showing that meta-learning performance is affected not only by the overall geometric variability of task parameters, but also by how this variability is allocated relative to an underlying low-dimensional structure. Similar to Pimonova et al. (2025), we decompose task-specific regression effects into a structurally informative component and an orthogonal, non-informative component. We show theoretically and through simulation that meta-learning prediction degrades when a larger fraction of between-task variability lies in orthogonal, non-informative directions, even when the overall geometric variability of tasks is held fixed.

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.

A membership-inference attack gets the output of a learning algorithm, and a target individual, and tries to determine whether this individual is a member of the training data or an independent sample from the same distribution. A successful membership-inference attack typically requires the attacker to have some knowledge about the distribution that the training data was sampled from, and this knowledge is often captured through a set of independent reference samples from that distribution. In this work we study how much information the attacker needs for membership inference by investigating the sample complexity-the minimum number of reference samples required-for a successful attack. We study this question in the fundamental setting of Gaussian mean estimation where the learning algorithm is given $n$ samples from a Gaussian distribution $\mathcal{N}(μ,Σ)$ in $d$ dimensions, and tries to estimate $\hatμ$ up to some error $\mathbb{E}[\|\hat μ- μ\|^2_Σ]\leq ρ^2 d$. Our result shows that for membership inference in this setting, $Ω(n + n^2 ρ^2)$ samples can be necessary to carry out any attack that competes with a fully informed attacker. Our result is the first to show that the attacker sometimes needs many more samples than the training algorithm uses to train the model. This result has significant implications for practice, as all attacks used in practice have a restricted form that uses $O(n)$ samples and cannot benefit from $ω(n)$ samples. Thus, these attacks may be underestimating the possibility of membership inference, and better attacks may be possible when information about the distribution is easy to obtain.

We study fairness-accuracy tradeoffs when a single predictive model must serve multiple demographic groups. A useful tool for understanding this tradeoff is the fairness-accuracy (FA) Pareto frontier, which characterizes the set of models that cannot be improved in either fairness or accuracy without worsening the other. While characterizing the FA frontier requires full knowledge of the data distribution, we focus on the finite-sample regime, quantifying how well a designer can approximate any point on the frontier from limited data and bounding the worst-case gap. In particular, we derive worst-case-optimal estimators that depend on the designer's knowledge of the covariate distribution. For each estimator, we characterize how finite-sample effects asymmetrically impact each group's welfare and identify optimal sample allocation strategies. Finally, we provide uniform finite-sample bounds along the entire FA frontier, quantifying finite-sample deviations across alternative fairness-accuracy tradeoffs.

Causal identification asks whether a causal target can be recovered from the observational distribution in an idealized infinite-data setting. In practice, only finite samples are available, and identifiability alone does not determine how much data are needed for reliable estimation. We introduce Probably Approximately Correct Causal (PACC) learnability, which formalizes uniform accuracy-confidence guarantees over a causal model class. From a PAC-learning perspective, we ask how many samples are sufficient to estimate a causal target to accuracy $ε$ with probability at least $1-δ$. We show that PACC sample complexity is shaped by three recurring factors: causal identification strength, target complexity, and nuisance complexity, and derive a general sufficient bound linking them to finite-sample causal learning. We illustrate the framework through self-controlled case series, instrumental variables, and propensity-score-based estimation, showing how distinct causal methods can be analyzed within a common framework.

Learning is a complex dynamical process shaped by a range of interconnected decisions. Careful design of hyperparameter schedules for artificial neural networks or efficient allocation of cognitive resources by biological learners can dramatically affect performance. Yet, theoretical understanding of optimal learning strategies remains sparse, especially due to the intricate interplay between evolving metaparameters and nonlinear learning dynamics. The search for optimal protocols is further hindered by the high dimensionality of the learning space, often resulting in predominantly heuristic, difficult to interpret, and computationally demanding solutions. Here, we combine statistical physics with control theory in a unified theoretical framework to identify optimal learning protocols in prototypical neural network models. In the high-dimensional limit, we derive closed-form ordinary differential equations that track online stochastic gradient descent through low-dimensional order parameters. We formulate the design of learning protocols as an optimal control problem directly on the dynamics of the order parameters with the goal of minimizing the generalization error. This formulation encompasses a variety of learning scenarios, optimization constraints, and control budgets. We apply it to representative cases, including optimal curricula, adaptive dropout regularization and noise schedules in denoising autoencoders. We find nontrivial yet interpretable strategies highlighting how optimal protocols mediate learning trade-offs. Our results establish a principled foundation for understanding and designing optimal protocols and suggest a path toward a theory of meta-learning grounded in statistical physics.

Modern sensing technologies routinely generate multi-dimensional functional data, often accompanied by scalar covariates, motivating statistical methods that can jointly accommodate nonlinear functional effects, complex cross-component dependencies, and irregular sampling patterns. Existing functional logistic regression approaches rely on basis expansions whose performance is critically sensitive to the choice of basis, smoothing parameters, and sampling design; meanwhile, current signature-based methods require heuristic selection of the signature truncation order. We propose Adaptively Truncated Signature-based Logistic Regression (ATSLR), a semi-parametric classification framework that integrates path-signature representations with a data-driven procedure for selecting the signature truncation order via penalized empirical risk minimization. Our method is supported by rigorous theoretical guarantees, including the existence of an optimal truncation order, consistency of its adaptive estimator, convergence rates for the classifier risk, a finite and computable search bound, and a general error-propagation analysis for irregular sampling. Experiments on synthetic and real-world datasets demonstrate that our framework consistently outperforms traditional functional classifiers and fixed-order signature baselines in accuracy, robustness, and interpretability. Our results highlight the practical and theoretical value of integrating rough path theory with adaptive model complexity control.

What fundamentally distinguishes an adversarial attack from a misclassification due to limited model expressivity or finite data? In this work, we investigate this question in the setting of high-dimensional binary classification, where statistical effects due to limited data availability play a central role. We introduce a new error metric that precisely capture this distinction, quantifying model vulnerability to consistent adversarial attacks -- perturbations that preserve the ground-truth labels. Our main technical contribution is an exact and rigorous asymptotic characterization of these metrics in both well-specified models and latent space models, revealing different vulnerability patterns compared to standard robust error measures. The theoretical results demonstrate that as models become more overparameterized, their vulnerability to label-preserving perturbations grows, offering theoretical insight into the mechanisms underlying model sensitivity to adversarial attacks.

Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic \enquote{free lunch} for PPI++, an adaptive form of PPI, showing that the asymptotic variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds regardless of the quality of the pseudo-labels. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a \enquote{no free lunch} result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made empirical advances by incorporating symmetry and geometry priors, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while providing well-calibrated uncertainty estimates.

We revisit the foundations of fairness and its interplay with utility and efficiency in settings where the training data contain richer labels, such as individual types, rankings, or risk estimates, rather than just binary outcomes. In this context, we propose algorithms that achieve stronger notions of evidence-based fairness than are possible in standard supervised learning. Our methods support classification and ranking techniques that preserve accurate subpopulation classification rates, as suggested by the underlying data distributions, across a broad class of classification rules and downstream applications. Furthermore, our predictors enable loss minimization, whether aimed at maximizing utility or in the service of fair treatment. Complementing our algorithmic contributions, we present impossibility results demonstrating that simultaneously achieving accurate classification rates and optimal loss minimization is, in some cases, computationally infeasible. Unlike prior impossibility results, our notions are not inherently in conflict and are simultaneously satisfied by the Bayes-optimal predictor. Furthermore, we show that each notion can be satisfied individually via efficient learning. Our separation thus stems from the computational hardness of learning a sufficiently good approximation of the Bayes-optimal predictor. These computational impossibilities present a choice between two natural and attainable notions of accuracy that could both be motivated by fairness.

Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop a formal framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between ``full-data'' and stereotypical predictors, (ii) the dominance of a region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.

We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.

We initiate the study of structured Stackelberg games, a novel form of strategic interaction between a leader and a follower where contextual information can be predictive of the follower's (unknown) type. Motivated by applications such as security games and AI safety, we show how this additional structure can help the leader learn a utility-maximizing policy in both the online and distributional settings. In the online setting, we first prove that standard learning-theoretic measures of complexity do not characterize the difficulty of the leader's learning task. Notably, we find that there exists a learning-theoretic measure of complexity, analogous to the Littlestone dimension in online classification, that tightly characterizes the leader's instance-optimal regret. We term this the Stackelberg-Littlestone dimension, and leverage it to provide a provably optimal online learning algorithm. In the distributional setting, we provide analogous results by showing that two new dimensions control the sample complexity upper- and lower-bound.

We address the problem of active logistic regression in the realizable setting. It is well known that active learning can require exponentially fewer label queries compared to passive learning, in some cases using $\log \frac{1}{\eps}$ rather than $\poly(1/\eps)$ labels to get error $\eps$ larger than the optimum. We present the first algorithm that is polynomially competitive with the optimal algorithm on every input instance, up to a factor $d^2$ and factors polylogarithmic in the error and domain size. In particular, on a simple family of logistic threshold instances with a uniform distribution, our algorithm uses $\polylog(1/\eps)$ labels, whereas every passive learner requires $Ω(1/\eps^2)$ labels. Our algorithm is based on sampling and can be extended to learn more general class of functions. We further support our theoretical results with experiments demonstrating performance gains for logistic regression compared to existing active learning algorithms.

Data-driven optimization aims to translate a machine learning model into decision-making by optimizing decisions on estimated costs. Such a pipeline can be conducted by fitting a distributional model which is then plugged into the target optimization problem. While this fitting can utilize traditional methods such as maximum likelihood, a more recent approach uses estimation-optimization integration that minimizes decision error instead of estimation error. Although intuitive, the statistical benefit of the latter approach is not well understood yet is important to guide the prescriptive usage of machine learning. In this paper, we dissect the performance comparisons between these approaches in terms of the amount of model misspecification. In particular, we show how the integrated approach offers a ``universal double benefit'' on the top two dominating terms of regret when the underlying model is misspecified, while the traditional approach can be advantageous when the model is nearly well-specified. Our comparison is powered by finite-sample tail regret bounds that are derived via new higher-order expansions of regrets and the leveraging of a recent Berry-Esseen theorem.

Recent studies identified an intriguing phenomenon in recursive generative model training known as model collapse, where models trained on data generated by previous models exhibit severe performance degradation. Addressing this issue and developing more effective training strategies have become central challenges in generative model research. In this paper, we investigate this phenomenon within a novel framework, where generative models are iteratively trained on a combination of newly collected real data and synthetic data from the previous training step. To develop an optimal training strategy for integrating real and synthetic data, we evaluate the performance of a weighted training scheme in various scenarios, including Gaussian distribution estimation, generalized linear models, and nonparametric estimation. We theoretically characterize the impact of the mixing proportion and weighting scheme of synthetic data on the final model's performance. Our key finding is that, across different settings, the optimal weighting scheme under different proportions of synthetic data asymptotically follows a unified expression, revealing a fundamental trade-off between leveraging synthetic data and model performance. In some cases, the optimal weight assigned to real data corresponds to the reciprocal of the golden ratio. Finally, we validate our theoretical results on extensive simulated datasets and a real tabular dataset.

We study the fundamental problem of calibrating a linear binary classifier of the form $σ(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $σ$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ and the true linear weight $w_\star$. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift. While minimax optimality of misspecified spectral algorithms has been established, existing guarantees are typically restricted to the non-saturation regime. We demonstrate that the infinite smoothness of fixed-bandwidth Gaussian kernels provides universal robustness to model misspecification by showing that this kernel choice enables any spectral algorithm to attain minimax optimal rates, provided the regularization parameter decays exponentially. This result effectively decouples optimality from the algorithm's inherent qualification. Building on this, we then advocate Gaussian spectral algorithms as powerful components in a learning framework for robust and adaptive transfer. Specifically, we derive the adaptive convergence rate of the excess risk for this framework and show that the rates are optimal up to logarithmic factors. Our results also reveal the impact of the magnitude of the concept shift and the sample size on the generalization error.

In this paper, the method of gaps, a technique for deriving closed-form expressions in terms of information measures for the generalization error of supervised learning algorithms, is introduced. This method relies on the notion of gaps, which characterize the variation of the expected empirical risk (when either the model or dataset is kept fixed) with respect to changes in the probability measure on the varying parameter. This distinction results in two classes of gaps: algorithm-driven gaps (fixed dataset) and data-driven gaps (fixed model). The method relies on two central observations: (i) the generalization error is the expectation of an algorithm-driven gap or a data-driven gap. In the first case, the expectation is with respect to a measure on the datasets; in the second case, it is with respect to a measure on the models. (ii) Both algorithm-driven gaps and data-driven gaps exhibit closed-form expressions in terms of relative entropies. In particular, algorithm-driven gaps involve a Gibbs probability measure on the set of models, which represents a supervised Gibbs algorithm. Alternatively, data-driven gaps involve a worst-case data-generating (WCDG) probability measure on the set of data points, which is also a Gibbs probability measure. Interestingly, such Gibbs measures, which are exogenous to the analysis of generalization, place the supervised Gibbs algorithm and the WCDG probability measure as natural references for the analysis of supervised learning algorithms. New exact expressions and all existing exact expressions for the generalization error of supervised learning algorithms can be obtained with the proposed method. Such new expressions are intended as structural and conceptual characterizations, not computational shortcuts. Finally, these expressions unveil strong connections among generalization, hypothesis testing, information measures, and Pythagorean identities.

Physically motivated stochastic dynamics are widely used to sample from high-dimensional distributions. However, such samplers often get trapped in metastable states, approximately sampling from a distribution that differs significantly from the desired stationary state. We rigorously show that for multivariable discrete distributions, the true stationary model can nevertheless be recovered from these metastable samples. This relies on a fundamental observation: for distributions satisfying a strong metastability condition, their single-variable conditional probabilities are on average extremely close to those of the true stationary distribution. This remains true even when the two distributions are far apart under global metrics such as Kullback-Leibler divergence. Consequently, we can effectively learn the true model using a conditional-likelihood estimator even when the samples are drawn from a restricted state space. Extending these general results to Ising models, we prove rigorous parameter and structure learning guarantees. Finally, we demonstrate this phenomenon numerically on higher-alphabet spin glass models.

We resolve a fundamental question about the ability to perform a statistical task, such as learning, when an adversary corrupts the sample. Such adversaries are specified by the types of corruption they can make and their level of knowledge about the sample. The latter distinguishes between sample-adaptive adversaries which know the contents of the sample when choosing the corruption, and sample-oblivious adversaries, which do not. We prove that for all types of corruptions, sample-adaptive and sample-oblivious adversaries are equivalent up to polynomial factors in the sample size. This resolves the main open question introduced by [BLMT22] and further explored in [CHL+23]. Specifically, consider any algorithm $A$ that solves a statistical task even when a sample-oblivious adversary corrupts its input. We show that there is an algorithm $A'$ that solves the same task when the corresponding sample-adaptive adversary corrupts its input. The construction of $A'$ is simple and maintains the computational efficiency of $A$: It requests a polynomially larger sample than $A$ uses and then runs $A$ on a uniformly random subsample.

In this paper, we establish a collection of covering number bounds for linear function classes under various norm constraints on the inputs and matrices. We then combine these results with existing covering number bounds to derive improved estimates and, based on these estimates, develop generalization error bounds for single-layer Transformers. The resulting generalization bounds improve upon several existing results in the literature and, in particular, are independent of the input sequence length. Moreover, our generalization error bound decays at the rate $O(1/\sqrt{n})$, where $n$ denotes the sample size, thereby improving upon existing bounds that scale as $O((\log n)/\sqrt{n})$. Furthermore, our covering number analysis explicitly incorporates rank constraints on the underlying matrix classes, allowing us to characterize how low-rank structures affect the metric entropy and, consequently, the resulting generalization bounds for Transformer architectures.

In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learnability. Classical results show that embedding a function space into a reproducing kernel Hilbert space (RKHS) implies a bound on its metric entropy growth. Surprisingly, we prove a converse: a bound on the metric entropy growth of a function space allows its embedding to a $L_p-$type Reproducing Kernel Banach Space (RKBS). This shows that the ${L}_p-$type RKBS provides a broad modeling framework for learnable function classes with controlled metric entropies. Our results shed new light on the power and limitations of kernel methods for learning complex function spaces.

This paper develops theory for deep neural network (DNN) estimators under dependent data. To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical $\mathcal{L}^{2}$-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary $β$-mixing data taking values in unbounded sets. I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights. For both DNN classes, I derive general results when the function to be estimated is Hölder smooth and the data are nonstationary, subgaussian, and $β$-mixing with either exponential or polynomial decay. I then specialize these to nonparametric regression, logistic regression, and quantile regression settings. Under exponential $β$-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.