In this work, we unify several expected generalization error bounds based on random subsets using the framework developed by Hellstr\"om and Durisi [1]. First, we recover the bounds based on the individual sample mutual information from Bu et al. [2] and on a random subset of the dataset from Negrea et al. [3]. Then, we introduce their new, analogous bounds in the randomized subsample setting from Steinke and Zakynthinou [4], and we identify some limitations of the framework. Finally, we extend the bounds from Haghifam et al. [5] for Langevin dynamics to stochastic gradient Langevin dynamics and we refine them for loss functions with potentially large gradient norms.

(Gradient) Expectation Maximization (EM) is a widely used algorithm for estimating the maximum likelihood of mixture models or incomplete data problems. A major challenge facing this popular technique is how to effectively preserve the privacy of sensitive data. Previous research on this problem has already lead to the discovery of some Differentially Private (DP) algorithms for (Gradient) EM. However, unlike in the non-private case, existing techniques are not yet able to provide finite sample statistical guarantees. To address this issue, we propose in this paper the first DP version of (Gradient) EM algorithm with statistical guarantees. Moreover, we apply our general framework to three canonical models: Gaussian Mixture Model (GMM), Mixture of Regressions Model (MRM) and Linear Regression with Missing Covariates (RMC). Specifically, for GMM in the DP model, our estimation error is near optimal in some cases. For the other two models, we provide the first finite sample statistical guarantees. Our theory is supported by thorough numerical experiments.

Histogram-based template fits are the main technique used for estimating parameters of high energy physics Monte Carlo generators. Parameterized neural network reweighting can be used to extend this fitting procedure to many dimensions and does not require binning. If the fit is to be performed using reconstructed data, then expensive detector simulations must be used for training the neural networks. We introduce a new two-level fitting approach that only requires one dataset with detector simulation and then a set of additional generation-level datasets without detector effects included. This Simulation-level fit based on Reweighting Generator-level events with Neural networks (SRGN) is demonstrated using simulated datasets for a variety of examples including a simple Gaussian random variable, parton shower tuning, and the top quark mass extraction.

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.

Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{\otimes l}$ of rank $r$ (where $r\ll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = \Omega(d^{l-1})$, while a variant of gradient descent can find an approximate tensor when $m = O^*(r^{2.5l}\log d)$. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.

In this work, we reframe the problem of balanced treatment assignment as optimization of a two-sample test between test and control units. Using this lens we provide an assignment algorithm that is optimal with respect to the minimum spanning tree test of Friedman and Rafsky (1979). This assignment to treatment groups may be performed exactly in polynomial time. We provide a probabilistic interpretation of this process in terms of the most probable element of designs drawn from a determinantal point process which admits a probabilistic interpretation of the design. We provide a novel formulation of estimation as transductive inference and show how the tree structures used in design can also be used in an adjustment estimator. We conclude with a simulation study demonstrating the improved efficacy of our method.

A decision tree is commonly restricted to use a single hyperplane to split the covariate space at each of its internal nodes. It often requires a large number of nodes to achieve high accuracy, hurting its interpretability. In this paper, we propose convex polytope trees (CPT) to expand the family of decision trees by an interpretable generalization of their decision boundary. The splitting function at each node of CPT is based on the logical disjunction of a community of differently weighted probabilistic linear decision-makers, which also geometrically corresponds to a convex polytope in the covariate space. We use a nonparametric Bayesian prior at each node to infer the community's size, encouraging simpler decision boundaries by shrinking the number of polytope facets. We develop a greedy method to efficiently construct CPT and scalable end-to-end training algorithms for the tree parameters when the tree structure is given. We empirically demonstrate the efficiency of CPT over existing state-of-the-art decision trees in several real-world classification and regression tasks from diverse domains.

Despite the wide empirical success of modern machine learning algorithms and models in a multitude of applications, they are known to be highly susceptible to seemingly small indiscernible perturbations to the input data known as adversarial attacks. A variety of recent adversarial training procedures have been proposed to remedy this issue. Despite the success of such procedures at increasing accuracy on adversarially perturbed inputs or robust accuracy, these techniques often reduce accuracy on natural unperturbed inputs or standard accuracy. Complicating matters further the effect and trend of adversarial training procedures on standard and robust accuracy is rather counter intuitive and radically dependent on a variety of factors including the perceived form of the perturbation during training, size/quality of data, model overparameterization, etc. In this paper we focus on binary classification problems where the data is generated according to the mixture of two Gaussians with general anisotropic covariance matrices and derive a precise characterization of the standard and robust accuracy for a class of minimax adversarially trained models. We consider a general norm-based adversarial model, where the adversary can add perturbations of bounded $\ell_p$ norm to each input data, for an arbitrary $p\ge 1$. Our comprehensive analysis allows us to theoretically explain several intriguing empirical phenomena and provide a precise understanding of the role of different problem parameters on standard and robust accuracies.

We present a theoretical framework recasting data augmentation as stochastic optimization for a sequence of time-varying proxy losses. This provides a unified approach to understanding techniques commonly thought of as data augmentation, including synthetic noise and label-preserving transformations, as well as more traditional ideas in stochastic optimization such as learning rate and batch size scheduling. We prove a time-varying Monro-Robbins theorem with rates of convergence which gives conditions on the learning rate and augmentation schedule under which augmented gradient descent converges. Special cases give provably good joint schedules for augmentation with additive noise, minibatch SGD, and minibatch SGD with noise. Implementing gradient-based optimization in practice requires many choices. These include hyperparameters such as learning rate and batch size as well as data augmentation, a popular set of techniques in which data is augmented (i.e.

In this paper, we consider the problem of designing Differentially Private (DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data. The irregularity of such data violates some key assumptions used in almost all existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP guarantees. To better understand this type of challenges, we provide in this paper a comprehensive study of DP-SCO under various settings. First, we consider the case where the loss function is strongly convex and smooth. For this case, we propose a method based on the sample-and-aggregate framework, which has an excess population risk of $\tilde{O}(\frac{d^3}{n\epsilon^4})$ (after omitting other factors), where $n$ is the sample size and $d$ is the dimensionality of the data. Then, we show that with some additional assumptions on the loss functions, it is possible to reduce the \textit{expected} excess population risk to $\tilde{O}(\frac{ d^2}{ n\epsilon^2 })$. To lift these additional conditions, we also provide a gradient smoothing and trimming based scheme to achieve excess population risks of $\tilde{O}(\frac{ d^2}{n\epsilon^2})$ and $\tilde{O}(\frac{d^\frac{2}{3}}{(n\epsilon^2)^\frac{1}{3}})$ for strongly convex and general convex loss functions, respectively, \textit{with high probability}. Experiments suggest that our algorithms can effectively deal with the challenges caused by data irregularity.