In this work, we address the following question: What minimal structural assumptions are needed to prevent the degradation of statistical learning bounds with increasing dimensionality? We investigate this question in the classical statistical setting of signal estimation from $n$ independent linear observations $Y_i = X_i^{\top}θ+ ε_i$. Our focus is on the generalization properties of a broad family of predictors that can be expressed as linear combinations of the training labels, $f(X) = \sum_{i=1}^{n} l_{i}(X) Y_i$. This class -- commonly referred to as linear prediction rules -- encompasses a wide range of popular parametric and non-parametric estimators, including ridge regression, gradient descent, and kernel methods. Our contributions are twofold. First, we derive non-asymptotic upper and lower bounds on the generalization error for this class under the assumption that the Bayes predictor $θ$ lies in an ellipsoid. Second, we establish a lower bound for the subclass of rotationally invariant linear prediction rules when the Bayes predictor is fixed. Our analysis highlights two fundamental contributions to the risk: (a) a variance-like term that captures the intrinsic dimensionality of the data; (b) the noiseless error, a term that arises specifically in the high-dimensional regime. These findings shed light on the role of structural assumptions in mitigating the curse of dimensionality.

Modern deep learning practice is governed by the surprising predictability of performance improvement with increases in the scale of data, model size, and compute [17]. Often, the scaling of performance as a function of these quantities exhibits remarkably regular power law behavior, termed a neural scaling law [2, 6, 12, 13, 15, 16, 18, 19, 22, 32]. Here, performance is usually measured by some differentiable loss on the predictions of the model on a held out test set representative of the population. Given the relatively universal behavior of the exponents across architectures and optimizers [11, 18, 19], one might hope that relatively simple models of information processing systems might be able to recover the same types of scaling laws. The (stochastic) gradient descent (SGD) dynamics in random feature models were analyzed in recent works [7, 20, 26] which exhibits a surprising breadth of scaling behavior and captures several interesting phenomena in deep network training. Each of the above works has isolated various effects that can hurt performance compared to the idealized infinite data and infinite model size limits. The model was first studied in [7], where the bottlenecks due to finite width and finite dataset size were computed and, for certain data structure, resulted in a Chinchilla-type scaling result as in [18].

Given a budget on total model size, one must decide whether to train a single, large neural network or to combine the predictions of many smaller networks. We study this trade-off for ensembles of random-feature ridge regression models. We prove that when a fixed number of trainable parameters are partitioned among K independently trained models, K = 1 achieves optimal performance, provided the ridge parameter is optimally tuned. We then derive scaling laws which describe how the test risk of an ensemble of regression models decays with its total size. We identify conditions on the kernel and task eigenstructure under which ensembles can achieve near-optimal scaling laws. Training ensembles of deep convolutional neural networks on CIFAR-10 and a transformer architecture on C4, we find that a single large network outperforms any ensemble of networks with the same total number of parameters, provided the weight decay and feature-learning strength are tuned to their optimal values. Ensembling methods are a well-established tool in machine learning for reducing the variance of learned predictors. While traditional ensemble approaches like random forests (Breiman, 2001) and XGBoost (Chen & Guestrin, 2016) combine many weak predictors, the advent of deep neural networks has shifted the state of the art toward training a single large predictor (LeCun et al., 2015). However, deep neural networks still suffer from various sources of variance, such as finite datasets and random initialization (Atanasov et al., 2022; Adlam & Pennington, 2020; Lin & Dobriban, 2021; Atanasov et al., 2024).

Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently handling the nested structure. This paper introduces a novel gradient-based approach for multilevel optimization that overcomes these limitations by leveraging a hierarchically structured decomposition of the full gradient and employing advanced propagation techniques. Extending to n-level scenarios, our method significantly reduces computational complexity while improving both solution accuracy and convergence speed. We demonstrate the effectiveness of our approach through numerical experiments, comparing it with existing methods across several benchmarks. The results show a notable improvement in solution accuracy. To the best of our knowledge, this is one of the first algorithms to provide a general version of implicit differentiation with both theoretical guarantees and superior empirical performance.

Recent years have seen substantial advances in our understanding of high-dimensional ridge regression, but existing theories assume that training examples are independent. By leveraging recent techniques from random matrix theory and free probability, we provide sharp asymptotics for the in- and out-of-sample risks of ridge regression when the data points have arbitrary correlations. We demonstrate that in this setting, the generalized cross validation estimator (GCV) fails to correctly predict the out-of-sample risk. However, in the case where the noise residuals have the same correlations as the data points, one can modify the GCV to yield an efficiently-computable unbiased estimator that concentrates in the high-dimensional limit, which we dub CorrGCV. We further extend our asymptotic analysis to the case where the test point has nontrivial correlations with the training set, a setting often encountered in time series forecasting. Assuming knowledge of the correlation structure of the time series, this again yields an extension of the GCV estimator, and sharply characterizes the degree to which such test points yield an overly optimistic prediction of long-time risk. We validate the predictions of our theory across a variety of high dimensional data.

This paper presents a succinct derivation of the training and generalization performance of a variety of high-dimensional ridge regression models using the basic tools of random matrix theory and free probability. We provide an introduction and review of recent results on these topics, aimed at readers with backgrounds in physics and deep learning. Analytic formulas for the training and generalization errors are obtained in a few lines of algebra directly from the properties of the $S$-transform of free probability. This allows for a straightforward identification of the sources of power-law scaling in model performance. We compute the generalization error of a broad class of random feature models. We find that in all models, the $S$-transform corresponds to the train-test generalization gap, and yields an analogue of the generalized-cross-validation estimator. Using these techniques, we derive fine-grained bias-variance decompositions for a very general class of random feature models with structured covariates. These novel results allow us to discover a scaling regime for random feature models where the variance due to the features limits performance in the overparameterized setting. We also demonstrate how anisotropic weight structure in random feature models can limit performance and lead to nontrivial exponents for finite-width corrections in the overparameterized setting. Our results extend and provide a unifying perspective on earlier models of neural scaling laws.

Recent evaluations of cross-domain text classification models aim to measure the ability of a model to obtain domain-invariant performance in a target domain given labeled samples in a source domain. The primary strategy for this evaluation relies on assumed differences between source domain samples and target domain samples in benchmark datasets. This evaluation strategy fails to account for the similarity between source and target domains, and may mask when models fail to transfer learning to specific target samples which are highly dissimilar from the source domain. We introduce Depth $F_1$, a novel cross-domain text classification performance metric. Designed to be complementary to existing classification metrics such as $F_1$, Depth $F_1$ measures how well a model performs on target samples which are dissimilar from the source domain. We motivate this metric using standard cross-domain text classification datasets and benchmark several recent cross-domain text classification models, with the goal of enabling in-depth evaluation of the semantic generalizability of cross-domain text classification models.

Transformed Gaussian Processes (TGPs) are stochastic processes specified by transforming samples from the joint distribution from a prior process (typically a GP) using an invertible transformation; increasing the flexibility of the base process. Furthermore, they achieve competitive results compared with Deep Gaussian Processes (DGPs), which are another generalization constructed by a hierarchical concatenation of GPs. In this work, we propose a generalization of TGPs named Deep Transformed Gaussian Processes (DTGPs), which follows the trend of concatenating layers of stochastic processes. More precisely, we obtain a multi-layer model in which each layer is a TGP. This generalization implies an increment of flexibility with respect to both TGPs and DGPs. Exact inference in such a model is intractable. However, we show that one can use variational inference to approximate the required computations yielding a straightforward extension of the popular DSVI inference algorithm Salimbeni et al (2017). The experiments conducted evaluate the proposed novel DTGPs in multiple regression datasets, achieving good scalability and performance.

Over-parameterized models estimated with some form of gradient descent come in various forms, such as linear regression with potentially non-linear features, neural networks, or kernel methods. The double descent phenomenon can be seen empirically in several of these models [6, 15]: Given a fixed prediction problem, when the number of parameters of the model is increasing from zero to the number of observations, the generalization performance traditionally goes down and then up, due to overfitting. Once the number of parameters exceeds the number of observations, the generalization error decreases again, as illustrated in Figure 1. The phenomenon has been theoretically analyzed in several settings, such as random features based on neural networks [27], random Fourier features [24], or linear regression [7, 17]. While the analysis of [27, 24] for random features corresponds to a single prediction problem with a sequence of increasingly larger prediction models, most of the analysis of [17] for linear regression does not consider a single problem, but varying problems, which does not actually lead to a double descent curve. Random subsampling on a single prediction problem was analyzed with a simpler model with isotropic covariance matrices in [7] and [17, Section 5.2], but without a proper double descent as the model is too simple to account for a U-shaped curve in the under-parameterized regime. In work related to ours, principal component regression was analyzed by [37] with a double descent curve but with less general assumptions regarding the spectrum of the covariance matrix and the optimal predictor.

Estimating treatment effects is of great importance for many biomedical applications with observational data. Particularly, interpretability of the treatment effects is preferable for many biomedical researchers. In this paper, we first provide a theoretical analysis and derive an upper bound for the bias of average treatment effect (ATE) estimation under the strong ignorability assumption. Derived by leveraging appealing properties of the Weighted Energy Distance, our upper bound is tighter than what has been reported in the literature. Motivated by the theoretical analysis, we propose a novel objective function for estimating the ATE that uses the energy distance balancing score and hence does not require correct specification of the propensity score model. We also leverage recently developed neural additive models to improve interpretability of deep learning models used for potential outcome prediction. We further enhance our proposed model with an energy distance balancing score weighted regularization. The superiority of our proposed model over current state-of-the-art methods is demonstrated in semi-synthetic experiments using two benchmark datasets, namely, IHDP and ACIC.