Due to the communication bottleneck in distributed and federated learning applications, algorithms using communication compression have attracted significant attention and are widely used in practice. Moreover, the huge number, high heterogeneity and limited availability of clients result in high client-variance. This paper addresses these two issues together by proposing compressed and client-variance reduced methods COFIG and FRECON. We prove an $O(\frac{(1+\omega)^{3/2}\sqrt{N}}{S\epsilon^2}+\frac{(1+\omega)N^{2/3}}{S\epsilon^2})$ bound on the number of communication rounds of COFIG in the nonconvex setting, where $N$ is the total number of clients, $S$ is the number of clients participating in each round, $\epsilon$ is the convergence error, and $\omega$ is the variance parameter associated with the compression operator. In case of FRECON, we prove an $O(\frac{(1+\omega)\sqrt{N}}{S\epsilon^2})$ bound on the number of communication rounds. In the convex setting, COFIG converges within $O(\frac{(1+\omega)\sqrt{N}}{S\epsilon})$ communication rounds, which, to the best of our knowledge, is also the first convergence result for compression schemes that do not communicate with all the clients in each round. We stress that neither COFIG nor FRECON needs to communicate with all the clients, and they enjoy the first or faster convergence results for convex and nonconvex federated learning in the regimes considered. Experimental results point to an empirical superiority of COFIG and FRECON over existing baselines.

Survival analysis is a statistical method employed to scrutinize the duration until a specific event of interest transpires, known as time-to-event information characterized by censorship. Recently, deep learning-based methods have dominated this field due to their representational capacity and state-of-the-art performance. However, the black-box nature of the deep neural network hinders its interpretability, which is desired in real-world survival applications but has been largely neglected by previous works. In contrast, conventional tree-based methods are advantageous with respect to interpretability, while consistently grappling with an inability to approximate the global optima due to greedy expansion. In this paper, we leverage the strengths of both neural networks and tree-based methods, capitalizing on their ability to approximate intricate functions while maintaining interpretability. To this end, we propose a Neural Survival Oblique Tree (NSOTree) for survival analysis. Specifically, the NSOTree was derived from the ReLU network and can be easily incorporated into existing survival models in a plug-and-play fashion. Evaluations on both simulated and real survival datasets demonstrated the effectiveness of the proposed method in terms of performance and interpretability.

Explicit finite-sample statistical guarantees on model performance are an important ingredient in responsible machine learning. Previous work has focused mainly on bounding either the expected loss of a predictor or the probability that an individual prediction will incur a loss value in a specified range. However, for many high-stakes applications, it is crucial to understand and control the dispersion of a loss distribution, or the extent to which different members of a population experience unequal effects of algorithmic decisions. We initiate the study of distribution-free control of statistical dispersion measures with societal implications and propose a simple yet flexible framework that allows us to handle a much richer class of statistical functionals beyond previous work. Our methods are verified through experiments in toxic comment detection, medical imaging, and film recommendation.

The quintessential learning algorithm of empirical risk minimization (ERM) is known to fail in various settings for which uniform convergence does not characterize learning. It is therefore unsurprising that the practice of machine learning is rife with considerably richer algorithmic techniques for successfully controlling model capacity. Nevertheless, no such technique or principle has broken away from the pack to characterize optimal learning in these more general settings. The purpose of this work is to characterize the role of regularization in perhaps the simplest setting for which ERM fails: multiclass learning with arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit optimal learning algorithms that dovetail with tried-and-true algorithmic principles: Occam's Razor as embodied by structural risk minimization (SRM), the principle of maximum entropy, and Bayesian reasoning. Most notably, we introduce an optimal learner which relaxes structural risk minimization on two dimensions: it allows the regularization function to be "local" to datapoints, and uses an unsupervised learning stage to learn this regularizer at the outset. We justify these relaxations by showing that they are necessary: removing either dimension fails to yield a near-optimal learner. We also extract from OIGs a combinatorial sequence we term the Hall complexity, which is the first to characterize a problem's transductive error rate exactly. Lastly, we introduce a generalization of OIGs and the transductive learning setting to the agnostic case, where we show that optimal orientations of Hamming graphs -- judged using nodes' outdegrees minus a system of node-dependent credits -- characterize optimal learners exactly. We demonstrate that an agnostic version of the Hall complexity again characterizes error rates exactly, and exhibit an optimal learner using maximum entropy programs.

In the last decade, developments in tropical geometry have provided a number of uses directly applicable to problems in statistical learning. The TML package is the first R package which contains a comprehensive set of tools and methods used for basic computations related to tropical convexity, visualization of tropically convex sets, as well as supervised and unsupervised learning models using the tropical metric under the max-plus algebra over the tropical projective torus. Primarily, the TML package employs a Hit and Run Markov chain Monte Carlo sampler in conjunction with the tropical metric as its main tool for statistical inference. In addition to basic computation and various applications of the tropical HAR sampler, we also focus on several supervised and unsupervised methods incorporated in the TML package including tropical principal component analysis, tropical logistic regression and tropical kernel density estimation.

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with large amounts of experimental data. Unfortunately, data-driven methods often fail to discover valid laws when data is noisy or scarce. Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory. However, the problem of searching over the space of formulae consistent with background theory to find one that fits the data best is not well-solved. We propose a solution to this problem when all axioms and scientific laws are expressible via polynomial equalities and inequalities and argue that our approach is widely applicable. We further model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivestellensatz certificates. Remarkably, the optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory, or non-deterministic polynomial (NP) time with partially correct background theory. We demonstrate that some famous scientific laws, including Kepler's Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from background axioms and experimental data.

In deep geological repositories for high level nuclear waste with close canister spacings, bentonite buffers can experience temperatures higher than 100 {\deg}C. In this range of extreme temperatures, phenomenological constitutive laws face limitations in capturing the thermo-hydro-mechanical (THM) behavior of the bentonite, since the pre-defined functional constitutive laws often lack generality and flexibility to capture a wide range of complex coupling phenomena as well as the effects of stress state and path dependency. In this work, a deep neural network (DNN)-based soil-water retention curve (SWRC) of bentonite is introduced and integrated into a Reproducing Kernel Particle Method (RKPM) for conducting THM simulations of the bentonite buffer. The DNN-SWRC model incorporates temperature as an additional input variable, allowing it to learn the relationship between suction and degree of saturation under the general non-isothermal condition, which is difficult to represent using a phenomenological SWRC. For effective modeling of the tank-scale test, new axisymmetric Reproducing Kernel basis functions enriched with singular Dirichlet enforcement representing heater placement and an effective convective heat transfer coefficient representing thin-layer composite tank construction are developed. The proposed method is demonstrated through the modeling of a tank-scale experiment involving a cylindrical layer of MX-80 bentonite exposed to central heating.

The widely observed 'benign overfitting phenomenon' in the neural network literature raises the challenge to the 'bias-variance trade-off' doctrine in the statistical learning theory. Since the generalization ability of the 'lazy trained' over-parametrized neural network can be well approximated by that of the neural tangent kernel regression, the curve of the excess risk (namely, the learning curve) of kernel ridge regression attracts increasing attention recently. However, most recent arguments on the learning curve are heuristic and are based on the 'Gaussian design' assumption. In this paper, under mild and more realistic assumptions, we rigorously provide a full characterization of the learning curve: elaborating the effect and the interplay of the choice of the regularization parameter, the source condition and the noise. In particular, our results suggest that the 'benign overfitting phenomenon' exists in very wide neural networks only when the noise level is small.

Many clustering algorithms fail when clusters are of arbitrary shapes, of varying densities, or the data classes are unbalanced and close to each other, even in two dimensions. A novel clustering algorithm, DenMune is presented to meet this challenge. It is based on identifying dense regions using mutual nearest neighborhoods of size K, where K is the only parameter required from the user, besides obeying the mutual nearest neighbor consistency principle. The algorithm is stable for a wide range of values of K. Moreover, it is able to automatically detect and remove noise from the clustering process as well as detecting the target clusters. It produces robust results on various low and high-dimensional datasets relative to several known state-of-the-art clustering algorithms.

Utilizing auxiliary outlier datasets to regularize the machine learning model has demonstrated promise for out-of-distribution (OOD) detection and safe prediction. Due to the labor intensity in data collection and cleaning, automating outlier data generation has been a long-desired alternative. Despite the appeal, generating photo-realistic outliers in the high dimensional pixel space has been an open challenge for the field. To tackle the problem, this paper proposes a new framework DREAM-OOD, which enables imagining photo-realistic outliers by way of diffusion models, provided with only the in-distribution (ID) data and classes. Specifically, DREAM-OOD learns a text-conditioned latent space based on ID data, and then samples outliers in the low-likelihood region via the latent, which can be decoded into images by the diffusion model. Different from prior works, DREAM-OOD enables visualizing and understanding the imagined outliers, directly in the pixel space. We conduct comprehensive quantitative and qualitative studies to understand the efficacy of DREAM-OOD, and show that training with the samples generated by DREAM-OOD can benefit OOD detection performance. Code is publicly available at https://github.com/deeplearning-wisc/dream-ood.