We investigate the problem of testing the global null in the high-dimensional regression models when the feature dimension p grows proportionally to the number of observations n . Despite a number of prior work studying this problem, whether there exists a test that is model-agnostic, efficient to compute and enjoys a high power, still remains unsettled. In this paper, we answer this question in the affirmative by leveraging the random projection techniques, and propose a testing procedure that blends the classical F -test with a random projection step. When combined with a systematic choice of the projection dimension, the proposed procedure is proved to be minimax optimal and, meanwhile, reduces the computation and data storage requirements. We illustrate our results in various scenarios when the underlying feature matrix exhibits an intrinsic lower dimensional structure (such as approximate low-rank or has exponential/polynomial eigen-decay), and it turns out that the proposed test achieves sharp adaptive rates.

A common challenge faced in practical supervised learning, such as medical image processing and robotic interactions, is that there are plenty of tasks but each task cannot afford to collect enough labeled examples to be learned in isolation. However, by exploiting the similarities across those tasks, one can hope to overcome such data scarcity. Under a canonical scenario where each task is drawn from a mixture of k linear regressions, we study a fundamental question: can abundant small-data tasks compensate for the lack of big-data tasks? Existing second moment based approaches of \cite{2020arXiv200208936K} show that such a trade-off is efficiently achievable, with the help of medium-sized tasks with \Omega(k {1/2}) examples each. However, this algorithm is brittle in two important scenarios.

We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution \pi\propto e {-V} on \R d . In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to \pi, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence. We also provide a convergence result of the finite particle system corresponding to the practical implementation of SVGD to its population version.

Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice, which has been hypothesized to play an important role in the generalization of modern machine learning approaches. In this work, we seek to understand these issues in the simpler setting of linear regression (including both underparameterized and overparameterized regimes), where our goal is to make sharp instance-based comparisons of the implicit regularization afforded by (unregularized) average SGD with the explicit regularization of ridge regression. For a broad class of least squares problem instances (that are natural in high-dimensional settings), we show: (1) for every problem instance and for every ridge parameter, (unregularized) SGD, when provided with \emph{logarithmically} more samples than that provided to the ridge algorithm, generalizes no worse than the ridge solution (provided SGD uses a tuned constant stepsize); (2) conversely, there exist instances (in this wide problem class) where optimally-tuned ridge regression requires \emph{quadratically} more samples than SGD in order to have the same generalization performance. Taken together, our results show that, up to the logarithmic factors, the generalization performance of SGD is always no worse than that of ridge regression in a wide range of overparameterized problems, and, in fact, could be much better for some problem instances. More generally, our results show how algorithmic regularization has important consequences even in simpler (overparameterized) convex settings.

Differentially Private Stochastic Gradient Descent (DP-SGD) forms a fundamental building block in many applications for learning over sensitive data. Two standard approaches, privacy amplification by subsampling, and privacy amplification by shuffling, permit adding lower noise in DP-SGD than via na\"{\i}ve schemes. A key assumption in both these approaches is that the elements in the data set can be uniformly sampled, or be uniformly permuted --- constraints that may become prohibitive when the data is processed in a decentralized or distributed fashion. In this paper, we focus on conducting iterative methods like DP-SGD in the setting of federated learning (FL) wherein the data is distributed among many devices (clients). Our main contribution is the \emph{random check-in} distributed protocol, which crucially relies only on randomized participation decisions made locally and independently by each client.

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires \mathcal{O}(\max\\{1/\epsilon_f {2},1/\epsilon_g {2}\\}) stochastic oracle queries to obtain a solution that is \epsilon_f -optimal for the upper-level and \epsilon_g -optimal for the lower-level. Moreover, for the case that the upper-level function is non-convex, our method requires at most \mathcal{O}(\max\\{1/\epsilon_f {3},1/\epsilon_g {3}\\}) stochastic oracle queries to find an (\epsilon_f, \epsilon_g) -stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are \mathcal{O}(\sqrt{n}/\epsilon) and \mathcal{O}(\sqrt{n}/\epsilon {2}) for the convex and non-convex settings, respectively, where \epsilon \min \\{\epsilon_f,\epsilon_g\\} .

We provide practical, efficient, and nonparametric methods for auditing the fairness of deployed classification and regression models. Whereas previous work relies on a fixed-sample size, our methods are sequential and allow for the continuous monitoring of incoming data, making them highly amenable to tracking the fairness of real-world systems. We also allow the data to be collected by a probabilistic policy as opposed to sampled uniformly from the population. This enables auditing to be conducted on data gathered for another purpose. Moreover, this policy may change over time and different policies may be used on different subpopulations. Finally, our methods can handle distribution shift resulting from either changes to the model or changes in the underlying population.

Inference, estimation, sampling and likelihood evaluation are four primary goals of probabilistic modeling. Practical considerations often force modeling approaches to make compromises between these objectives. We present a novel probabilistic learning framework, called Fenchel Mini-Max Learning (FML), that accommodates all four desiderata in a flexible and scalable manner. Our derivation is rooted in classical maximum likelihood estimation, and it overcomes a longstanding challenge that prevents unbiased estimation of unnormalized statistical models. By reformulating MLE as a mini-max game, FML enjoys an unbiased training objective that (i) does not explicitly involve the intractable normalizing constant and (ii) is directly amendable to stochastic gradient descent optimization.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. [2016] provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on nonsmooth convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses.

Commonly used classification algorithms in machine learning, such as support vector machines, minimize a convex surrogate loss on training examples. In practice, these algorithms are surprisingly robust to errors in the training data. In this work, we identify a set of conditions on the data under which such surrogate loss minimization algorithms provably learn the correct classifier. This allows us to establish, in a unified framework, the robustness of these algorithms under various models on data as well as error. In particular, we show that if the data is linearly classifiable with a slightly non-trivial margin (i.e. a margin at least C\div\sqrt{d} for d -dimensional unit vectors), and the class-conditional distributions are near isotropic and logconcave, then surrogate loss minimization has negligible error on the uncorrupted data even when a constant fraction of examples are adversarially mislabeled.