Simultaneously addressing multiple objectives is becoming increasingly important in modern machine learning. At the same time, data is often high-dimensional and costly to label. For a single objective such as prediction risk, conventional regularization techniques are known to improve generalization when the data exhibits low-dimensional structure like sparsity. However, it is largely unexplored how to leverage this structure in the context of multi-objective learning (MOL) with multiple competing objectives. In this work, we discuss how the application of vanilla regularization approaches can fail, and propose a two-stage MOL framework that can successfully leverage low-dimensional structure. We demonstrate its effectiveness experimentally for multi-distribution learning and fairness-risk trade-offs.

The risk of language models reproducing copyrighted material from their training data has led to the development of various protective measures. Among these, inference-time strategies that impose constraints via post-processing have shown promise in addressing the complexities of copyright regulation. However, they often incur prohibitive computational costs or suffer from performance trade-offs. To overcome these limitations, we introduce Copyright-Protecting Model Fusion (CP-Fuse), a novel approach that combines models trained on disjoint sets of copyrighted material during inference. In particular, CP-Fuse adaptively aggregates the model outputs to minimize the reproduction of copyrighted content, adhering to a crucial balancing property to prevent the regurgitation of memorized data. Through extensive experiments, we show that CP-Fuse significantly reduces the reproduction of protected material without compromising the quality of text and code generation. Moreover, its post-hoc nature allows seamless integration with other protective measures, further enhancing copyright safeguards. Lastly, we show that CP-Fuse is robust against common techniques for extracting training data.

Practical and ethical constraints often require the use of observational data for causal inference, particularly in medicine and social sciences. Yet, observational datasets are prone to confounding, potentially compromising the validity of causal conclusions. While it is possible to correct for biases if the underlying causal graph is known, this is rarely a feasible ask in practical scenarios. A common strategy is to adjust for all available covariates, yet this approach can yield biased treatment effect estimates, especially when post-treatment or unobserved variables are present. We propose RAMEN, an algorithm that produces unbiased treatment effect estimates by leveraging the heterogeneity of multiple data sources without the need to know or learn the underlying causal graph. Notably, RAMEN achieves *doubly robust identification*: it can identify the treatment effect whenever the causal parents of the treatment or those of the outcome are observed, and the node whose parents are observed satisfies an invariance assumption. Empirical evaluations across synthetic, semi-synthetic, and real-world datasets show that our approach significantly outperforms existing methods.

Randomized experiments are the preferred approach for evaluating the effects of interventions, but they are costly and often yield estimates with substantial uncertainty. On the other hand, in silico experiments leveraging foundation models offer a cost-effective alternative that can potentially attain higher statistical precision. However, the benefits of in silico experiments come with a significant risk: statistical inferences are not valid if the models fail to accurately predict experimental responses to interventions. In this paper, we propose a novel approach that integrates the predictions from multiple foundation models with experimental data while preserving valid statistical inference. Our estimator is consistent and asymptotically normal, with asymptotic variance no larger than the standard estimator based on experimental data alone. Importantly, these statistical properties hold even when model predictions are arbitrarily biased. Empirical results across several randomized experiments show that our estimator offers substantial precision gains, equivalent to a reduction of up to 20\% in the sample size needed to match the same precision as the standard estimator based on experimental data alone.

In multi-objective learning (MOL), several possibly competing prediction tasks must be solved jointly by a single model. Achieving good trade-offs may require a model class $\mathcal{G}$ with larger capacity than what is necessary for solving the individual tasks. This, in turn, increases the statistical cost, as reflected in known MOL bounds that depend on the complexity of $\mathcal{G}$. We show that this cost is unavoidable for some losses, even in an idealized semi-supervised setting, where the learner has access to the Bayes-optimal solutions for the individual tasks as well as the marginal distributions over the covariates. On the other hand, for objectives defined with Bregman losses, we prove that the complexity of $\mathcal{G}$ may come into play only in terms of unlabeled data. Concretely, we establish sample complexity upper bounds, showing precisely when and how unlabeled data can significantly alleviate the need for labeled data. This is achieved by a simple pseudo-labeling algorithm.

In the era of fast-paced precision medicine, observational studies play a major role in properly evaluating new treatments in clinical practice. Yet, unobserved confounding can significantly compromise causal conclusions drawn from non-randomized data. We propose a novel strategy that leverages randomized trials to quantify unobserved confounding. First, we design a statistical test to detect unobserved confounding above a certain strength. Then, we use the test to estimate an asymptotically valid lower bound on the unobserved confounding strength. We evaluate the power and validity of our statistical test on several synthetic and semi-synthetic datasets. Further, we show how our lower bound can correctly identify the absence and presence of unobserved confounding in a real-world example.

As machine learning has become more relevant for everyday applications, a natural requirement is the protection of the privacy of the training data. When the relevant learning questions are unknown in advance, or hyper-parameter tuning plays a central role, one solution is to release a differentially private synthetic data set that leads to similar conclusions as the original training data. In this work, we introduce an algorithm that enjoys fast rates for the utility loss for sparse Lipschitz queries. Furthermore, we show how to obtain a certificate for the utility loss for a large class of algorithms.

We present a novel approach for differentially private data synthesis of protected tabular datasets, a relevant task in highly sensitive domains such as healthcare and government. Current state-of-the-art methods predominantly use marginal-based approaches, where a dataset is generated from private estimates of the marginals. In this paper, we introduce PrivPGD, a new generation method for marginal-based private data synthesis, leveraging tools from optimal transport and particle gradient descent. Our algorithm outperforms existing methods on a large range of datasets while being highly scalable and offering the flexibility to incorporate additional domain-specific constraints.

Minimum-norm interpolators have recently gained attention primarily as an analyzable model to shed light on the double descent phenomenon observed for neural networks. The majority of the work has focused on analyzing interpolators in Hilbert spaces, where typically an effectively low-rank structure of the feature covariance prevents a large bias. More recently, tight vanishing bounds have also been shown for isotropic high-dimensional data for $\ell_p$-spaces with $p\in[1,2)$, leveraging sparse structure of the ground truth. However, these proofs are tailored to specific settings and hard to generalize. This paper takes a first step towards establishing a general framework that connects generalization properties of the interpolators to well-known concepts from high-dimensional geometry, specifically, from the local theory of Banach spaces. In particular, we show that under $2$-uniform convexity, the bias of the minimal norm solution is bounded by the Gaussian complexity of the class. We then prove a ``reverse'' Efron-Stein lower bound on the expected conditional variance of the minimal norm solution under cotype $2$. Finally, we prove that this bound is sharp for $\ell_p$-linear regression under sub-Gaussian covariates.

We study the problem of estimating the means of well-separated mixtures when an adversary may add arbitrary outliers. While strong guarantees are available when the outlier fraction is significantly smaller than the minimum mixing weight, much less is known when outliers may crowd out low-weight clusters – a setting we refer to as list-decodable mixture learning (LD-ML). In this case, adversarial outliers can simulate additional spurious mixture components. Hence, if all means of the mixture must be recovered up to a small error in the output list, the list size needs to be larger than the number of (true) components. We propose an algorithm that obtains order-optimal error guarantees for each mixture mean with a minimal list-size overhead, significantly improving upon list-decodable mean estimation, the only existing method that is applicable for LD-ML. Although improvements are observed even when the mixture is non-separated, our algorithm achieves particularly strong guarantees when the mixture is separated: it can leverage the mixture structure to partially cluster the samples before carefully iterating a base learner for list-decodable mean estimation at different scales.

In safety-critical applications, machine learning models should generalize well under worst-case distribution shifts, that is, have a small robust risk. Invariance-based algorithms can provably take advantage of structural assumptions on the shifts when the training distributions are heterogeneous enough to identify the robust risk. However, in practice, such identifiability conditions are rarely satisfied – a scenario so far underexplored in the theoretical literature. In this paper, we aim to fill the gap and propose to study the more general setting of partially identifiable robustness. In particular, we define a new risk measure, the identifiable robust risk, and its corresponding (population) minimax quantity that is an algorithm-independent measure for the best achievable robustness under partial identifiability. We introduce these concepts broadly, and then study them within the framework of linear structural causal models for concreteness of the presentation. We use the introduced minimax quantity to show how previous approaches provably achieve suboptimal robustness in the partially identifiable case. We confirm our findings through empirical simulations and real-world experiments and demonstrate how the test error of existing robustness methods grows increasingly suboptimal as the proportion of previously unseen test directions increases.

Randomized trials are considered the gold standard for making informed decisions in medicine. However, they are often not representative of the patient population in clinical practice. Observational studies, on the other hand, cover a broader patient population but are prone to various biases. Thus, before using observational data for any downstream task, it is crucial to benchmark its treatment effect estimates against a randomized trial. We propose a novel strategy to benchmark observational studies on a subgroup level. First, we design a statistical test for the null hypothesis that the treatment effects – conditioned on a subset of relevant features – differ up to some tolerance value. Our test allows us to estimate an asymptotically valid lower bound on the maximum bias strength for any subgroup. We validate our lower bound in a real-world setting and show that it leads to conclusions that align with established medical knowledge.

It is widely believed that given the same labeling budget, active learning (AL) algorithms like margin-based active learning achieve better predictive performance than passive learning (PL), albeit at a higher computational cost. Recent empirical evidence suggests that this added cost might be in vain, as margin-based AL can sometimes perform even worse than PL. While existing works offer different explanations in the low-dimensional regime, this paper shows that the underlying mechanism is entirely different in high dimensions: we prove for logistic regression that PL outperforms margin-based AL even for noiseless data and when using the Bayes optimal decision boundary for sampling. Insights from our proof indicate that this high-dimensional phenomenon is exacerbated when the separation between the classes is small. We corroborate this intuition with experiments on 20 high-dimensional datasets spanning a diverse range of applications, from finance and histology to chemistry and computer vision.

Prior theoretical and empirical works have established that semi-supervised learning algorithms can leverage the unlabeled data to improve over the labeled sample complexity of supervised learning (SL) algorithms. However, existing theoretical work focuses on regimes where the unlabeled data is sufficient to learn a good decision boundary using unsupervised learning (UL) alone. This begs the question: Can SSL algorithms simultaneously improve upon both UL and SL? To this end, we derive a tight lower bound for 2-Gaussian mixture models that explicitly depends on the labeled and the unlabeled dataset size as well as the signal-to-noise ratio of the mixture distribution. Surprisingly, our result implies that no SSL algorithm improves upon the minimax-optimal statistical error rates of SL or UL algorithms for these distributions. Nevertheless, in our real-world experiments, SSL algorithms can often outperform UL and SL algorithms. In summary, our work suggests that while it is possible to prove the performance gains of SSL algorithms, this would require careful tracking of constants in the theoretical analysis.

Good generalization performance on high-dimensional data crucially hinges on a simple structure of the ground truth and a corresponding strong inductive bias of the estimator. Even though this intuition is valid for regularized models, in this paper we caution against a strong inductive bias for interpolation in the presence of noise: While a stronger inductive bias encourages a simpler structure that is more aligned with the ground truth, it also increases the detrimental effect of noise. Specifically, for both linear regression and classification with a sparse ground truth, we prove that minimum $\ell_p$-norm and maximum $\ell_p$-margin interpolators achieve fast polynomial rates close to order $1/n$ for $p > 1$ compared to a logarithmic rate for $p = 1$. Finally, we provide preliminary experimental evidence that this trade-off may also play a crucial role in understanding non-linear interpolating models used in practice.

As machine learning algorithms are deployed on sensitive data in critical decision making processes, it is becoming increasingly important that they are also private and fair. In this paper, we show that, when the data has a long-tailed structure, it is not possible to build accurate learning algorithms that are both private and results in higher accuracy on minority subpopulations. We further show that relaxing overall accuracy can lead to good fairness even with strict privacy requirements. To corroborate our theoretical results in practice, we provide an extensive set of experimental results using a variety of synthetic, vision (CIFAR-10 and CelebA), and tabular (Law School) datasets and learning algorithms.

Deep neural networks often predict samples with high confidence even when they come from unseen classes and should instead be flagged for expert evaluation. Current novelty detection algorithms cannot reliably identify such near OOD points unless they have access to labeled data that is similar to these novel samples. In this paper, we develop a new ensemble-based procedure for semi-supervised novelty detection (SSND) that successfully leverages a mixture of unlabeled ID and novel-class samples to achieve good detection performance. In particular, we show how to achieve disagreement only on OOD data using early stopping regularization. While we prove this fact for a simple data distribution, our extensive experiments suggest that it holds true for more complex scenarios: our approach significantly outperforms state-of-the-art SSND methods on standard image data sets (SVHN/CIFAR-10/CIFAR-100) and medical image data sets with only a negligible increase in computation cost.

Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for high-dimensional kernel regression for a number of specific assumptions on the data distribution. In this paper, we show that in high dimensions, the rotational invariance property of commonly studied kernels (such as RBF, inner product kernels and fully-connected NTK of any depth) leads to inconsistent estimation unless the ground truth is a low-degree polynomial. Our lower bound on the generalization error holds for a wide range of distributions and kernels with different eigenvalue decays. This lower bound suggests that consistency results for kernel ridge regression in high dimensions generally require a more refined analysis that depends on the structure of the kernel beyond its eigenvalue decay.

Numerous recent works show that overparameterization implicitly reduces variance for min-norm interpolators and max-margin classifiers. These findings suggest that ridge regularization has vanishing benefits in high dimensions. We challenge this narrative by showing that, even in the absence of noise, avoiding interpolation through ridge regularization can significantly improve generalization. We prove this phenomenon for the robust risk of both linear regression and classification, and hence provide the first theoretical result on \emph{robust overfitting}.

Adversarial training augments the training set with perturbations to improve the robust error (over worst-case perturbations), but it often leads to an increase in the standard error (on unperturbed test inputs). Previous explanations for this tradeoff rely on the assumption that no predictor in the hypothesis class has low standard and robust error. In this work, we precisely characterize the effect of augmentation on the standard error in linear regression when the optimal linear predictor has zero standard and robust error. In particular, we show that the standard error could increase even when the augmented perturbations have noiseless observations from the optimal linear predictor. We then prove that the recently proposed robust self-training (RST) estimator improves robust error without sacrificing standard error for noiseless linear regression. Empirically, for neural networks, we find that RST with different adversarial training methods improves both standard and robust error for random and adversarial rotations and adversarial l_infty perturbations in CIFAR-10.

This work provides theoretical and empirical evidence that invariance-inducing regularizers can increase predictive accuracy for worst-case spatial transformations (spatial robustness). Evaluated on these adversarially transformed examples, standard and adversarial training with such regularizers achieves a relative error reduction of 20% for CIFAR-10 with the same computational budget. This even surpasses handcrafted spatial-equivariant networks. Furthermore, we observe for SVHN, known to have inherent variance in orientation, that robust training also improves standard accuracy on the test set. We prove that this no-trade-off phenomenon holds for adversarial examples from transformation groups.

Early stopping of iterative algorithms is a widely used form of regularization in statistics, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including L2-boost, LogitBoost, and AdaBoost, among others), we exhibit a direct connection between the performance of a stopped iterate and the localized Gaussian complexity of the associated function class. This connection allows us to show that the local fixed point analysis of Gaussian or Rademacher complexities, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

In the online multiple testing problem, p-values corresponding to different null hypotheses are presented one by one, and the decision of whether to reject a hypothesis must be made immediately, after which the next p-value is presented. Alpha-investing algorithms to control the false discovery rate were first formulated by Foster and Stine and have since been generalized and applied to various settings, varying from quality-preserving databases for science to multiple A/B tests for internet commerce. This paper improves the class of generalized alpha-investing algorithms (GAI) in four ways : (a) we show how to uniformly improve the power of the entire class of GAI procedures under independence by awarding more alpha-wealth for each rejection, giving a near win-win resolution to a dilemma raised by Javanmard and Montanari, (b) we demonstrate how to incorporate prior weights to indicate domain knowledge of which hypotheses are likely to be null or non-null, (c) we allow for differing penalties for false discoveries to indicate that some hypotheses may be more meaningful/important than others, (d) we define a new quantity called the \emph{decaying memory false discovery rate, or $\memfdr$} that may be more meaningful for applications with an explicit time component, using a discount factor to incrementally forget past decisions and alleviate some potential problems that we describe and name ``piggybacking'' and ``alpha-death''. Our GAI++ algorithms incorporate all four generalizations (a, b, c, d) simulatenously, and reduce to more powerful variants of earlier algorithms when the weights and decay are all set to unity.

Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L^2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

We propose an alternative framework to existing setups for controlling false alarms when multiple A/B tests are run over time. This setup arises in many practical applications, e.g. when pharmaceutical companies test new treatment options against control pills for different diseases, or when internet companies test their default webpages versus various alternatives over time. Our framework proposes to replace a sequence of A/B tests by a sequence of best-arm MAB instances, which can be continuously monitored by the data scientist. When interleaving the MAB tests with an online false discovery rate (FDR) algorithm, we can obtain the best of both worlds: low sample complexity and any time online FDR control. Our main contributions are: (i) to propose reasonable definitions of a null hypothesis for MAB instances; (ii) to demonstrate how one can derive an always-valid sequential p-value that allows continuous monitoring of each MAB test; and (iii) to show that using rejection thresholds of online-FDR algorithms as the confidence levels for the MAB algorithms results in both sample-optimality, high power and low FDR at any point in time. We run extensive simulations to verify our claims, and also report results on real data collected from the New Yorker Cartoon Caption contest.

The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known to be susceptible to local optima. In this paper, we first give a general characterization of the basin of attraction associated with any global optimum of the population likelihood. By exploiting this characterization, we provide non-asymptotic finite sample guarantees on the Baum-Welch updates and show geometric convergence to a small ball of radius on the order of the minimax rate around a global optimum. As a concrete example, we prove a linear rate of convergence for a hidden Markov mixture of two isotropic Gaussians given a suitable mean separation and an initialization within a ball of large radius around (one of) the true parameters. To our knowledge, these are the first rigorous local convergence guarantees to global optima for the Baum-Welch algorithm in a setting where the likelihood function is nonconvex. We complement our theoretical results with thorough numerical simulations studying the convergence of the Baum-Welch algorithm and illustrating the accuracy of our predictions.

This paper considers the problem of signal recovery from magnitude measurements for signals in modulation invariant spaces. It proposes a measurement setup such that almost every signal in such a signal space can be reconstructed from its amplitude measurements up to a global constant phase and with a sampling rate of four times the rate of innovation of the signal space. The applicability of the proposed scheme under noise measurements is demonstrated by computer simulations.