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Perfect synchronization in distributed machine learning problems is inefficient and even impossible due to the existence of latency, package losses and stragglers. We propose a Robust Fully-Asynchronous Stochastic Gradient Tracking method (R-FAST), where each device performs local computation and communication at its own pace without any form of synchronization. Different from existing asynchronous distributed algorithms, R-FAST can eliminate the impact of data heterogeneity across devices and allow for packet losses by employing a robust gradient tracking strategy that relies on properly designed auxiliary variables for tracking and buffering the overall gradient vector. More importantly, the proposed method utilizes two spanning-tree graphs for communication so long as both share at least one common root, enabling flexible designs in communication architectures. We show that R-FAST converges in expectation to a neighborhood of the optimum with a geometric rate for smooth and strongly convex objectives; and to a stationary point with a sublinear rate for general non-convex settings. Extensive experiments demonstrate that R-FAST runs 1.5-2 times faster than synchronous benchmark algorithms, such as Ring-AllReduce and D-PSGD, while still achieving comparable accuracy, and outperforms existing asynchronous SOTA algorithms, such as AD-PSGD and OSGP, especially in the presence of stragglers.

A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner through the variational formulation of the problem and by applying the finite element method. To obtain the lower bounds we formulate the dual variational problem and introduce appropriate approximation spaces employing the finite element method as well. We deal with the 3D setting, which has been rarely considered in the literature so far. The theoretical justification of the procedure is presented and supported with illustrative examples.

Modern mainstream financial theory is underpinned by the efficient market hypothesis, which posits the rapid incorporation of relevant information into asset pricing. Limited prior studies in the operational research literature have investigated tests designed for random number generators to check for these informational efficiencies. Treating binary daily returns as a hardware random number generator analogue, tests of overlapping permutations have indicated that these time series feature idiosyncratic recurrent patterns. Contrary to prior studies, we split our analysis into two streams at the annual and company level, and investigate longer-term efficiency over a larger time frame for Nasdaq-listed public companies to diminish the effects of trading noise and allow the market to realistically digest new information. Our results demonstrate that information efficiency varies across years and reflects large-scale market impacts such as financial crises. We also show the proximity to results of a well-tested pseudo-random number generator, discuss the distinction between theoretical and practical market efficiency, and find that the statistical qualification of stock-separated returns in support of the efficient market hypothesis is dependent on the driving factor of small inefficient subsets that skew market assessments.

For any particular class of graphs, algorithms for computational problems restricted to the class often rely on structural properties that depend on the specific problem at hand. This begs the question if a large set of such results can be explained by some common problem conditions. We propose such conditions for $HH$-subgraph-free graphs. For a set of graphs $HH$, a graph $G$ is $HH$-subgraph-free if $G$ does not contain any of graph from $H$ as a subgraph. Our conditions are easy to state. A graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem satisfies all three conditions, then for every finite set $HH$, it is ``efficiently solvable'' on $HH$-subgraph-free graphs if $HH$ contains a disjoint union of one or more paths and subdivided claws, and is ``computationally hard'' otherwise. We illustrate the broad applicability of our meta-classification by obtaining a dichotomy between polynomial-time solvability and NP-completeness for many well-known partitioning, covering and packing problems, network design problems and width parameter problems. For other problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). The proposed framework thus gives a simple pathway to determine the complexity of graph problems on $HH$-subgraph-free graphs. This is confirmed even more by the fact that along the way, we uncover and resolve several open questions from the literature.

We present a novel variant of the multi-level Monte Carlo method that effectively utilizes a reserved computational budget on a high-performance computing system to minimize the mean squared error. Our approach combines concepts of the continuation multi-level Monte Carlo method with dynamic programming techniques following Bellman's optimality principle, and a new parallelization strategy based on a single distributed data structure. Additionally, we establish a theoretical bound on the error reduction on a parallel computing cluster and provide empirical evidence that the proposed method adheres to this bound. We implement, test, and benchmark the approach on computationally demanding problems, focusing on its application to acoustic wave propagation in high-dimensional random media.

This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension d; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself, and propose a simple stopping rule applicable to any optimization method which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.

Representative Selection (RS) is the problem of finding a small subset of exemplars from a dataset that is representative of the dataset. In this paper, we study RS for attributed graphs, and focus on finding representative nodes that optimize the accuracy of a model trained on the selected representatives. Theoretically, we establish a new hardness result forRS (in the absence of a graph structure) by proving that a particular, highly practical variant of it (RS for Learning) is hard to approximate in polynomial time within any reasonable factor, which implies a significant potential gap between the optimum solution of widely-used surrogate functions and the actual accuracy of the model. We then study the setting where a (homophilous) graph structure is available, or can be constructed, between the data points.We show that with an appropriate modeling approach, the presence of such a structure can turn a hard RS (for learning) problem into one that can be effectively solved. To this end, we develop RS-GNN, a representation learning-based RS model based on Graph Neural Networks. Empirically, we demonstrate the effectiveness of RS-GNN on problems with predefined graph structures as well as problems with graphs induced from node feature similarities, by showing that RS-GNN achieves significant improvements over established baselines on a suite of eight benchmarks.

Frequency analysis is useful for understanding the mechanisms of representation learning in neural networks (NNs). Most research in this area focuses on the learning dynamics of NNs for regression tasks, while little for classification. This study empirically investigates the latter and expands the understanding of frequency shortcuts. First, we perform experiments on synthetic datasets, designed to have a bias in different frequency bands. Our results demonstrate that NNs tend to find simple solutions for classification, and what they learn first during training depends on the most distinctive frequency characteristics, which can be either low- or high-frequencies. Second, we confirm this phenomenon on natural images. We propose a metric to measure class-wise frequency characteristics and a method to identify frequency shortcuts. The results show that frequency shortcuts can be texture-based or shape-based, depending on what best simplifies the objective. Third, we validate the transferability of frequency shortcuts on out-of-distribution (OOD) test sets. Our results suggest that frequency shortcuts can be transferred across datasets and cannot be fully avoided by larger model capacity and data augmentation. We recommend that future research should focus on effective training schemes mitigating frequency shortcut learning.

Decision trees are among the most popular machine learning models and are used routinely in applications ranging from revenue management and medicine to bioinformatics. In this paper, we consider the problem of learning optimal binary classification trees with univariate splits. Literature on the topic has burgeoned in recent years, motivated both by the empirical suboptimality of heuristic approaches and the tremendous improvements in mixed-integer optimization (MIO) technology. Yet, existing MIO-based approaches from the literature do not leverage the power of MIO to its full extent: they rely on weak formulations, resulting in slow convergence and large optimality gaps. To fill this gap in the literature, we propose an intuitive flow-based MIO formulation for learning optimal binary classification trees. Our formulation can accommodate side constraints to enable the design of interpretable and fair decision trees. Moreover, we show that our formulation has a stronger linear optimization relaxation than existing methods in the case of binary data. We exploit the decomposable structure of our formulation and max-flow/min-cut duality to derive a Benders' decomposition method to speed-up computation. We propose a tailored procedure for solving each decomposed subproblem that provably generates facets of the feasible set of the MIO as constraints to add to the main problem. We conduct extensive computational experiments on standard benchmark datasets on which we show that our proposed approaches are 29 times faster than state-of-the-art MIO-based techniques and improve out-of-sample performance by up to 8%.

Link prediction on knowledge graphs (KGs) is a key research topic. Previous work mainly focused on binary relations, paying less attention to higher-arity relations although they are ubiquitous in real-world KGs. This paper considers link prediction upon n-ary relational facts and proposes a graph-based approach to this task. The key to our approach is to represent the n-ary structure of a fact as a small heterogeneous graph, and model this graph with edge-biased fully-connected attention. The fully-connected attention captures universal inter-vertex interactions, while with edge-aware attentive biases to particularly encode the graph structure and its heterogeneity. In this fashion, our approach fully models global and local dependencies in each n-ary fact, and hence can more effectively capture associations therein. Extensive evaluation verifies the effectiveness and superiority of our approach. It performs substantially and consistently better than current state-of-the-art across a variety of n-ary relational benchmarks. Our code is publicly available.