We consider a decentralized optimization problem for networks affected by communication delays. Examples of such networks include collaborative machine learning, sensor networks, and multi-agent systems. To mimic communication delays, we add virtual non-computing nodes to the network, resulting in directed graphs. This motivates investigating decentralized optimization solutions on directed graphs. Existing solutions assume nodes know their out-degrees, resulting in limited applicability. To overcome this limitation, we introduce a novel gossip-based algorithm, called DT-GO, that does not need to know the out-degrees. The algorithm is applicable in general directed networks, for example networks with delays or limited acknowledgment capabilities. We derive convergence rates for both convex and non-convex objectives, showing that our algorithm achieves the same complexity order as centralized Stochastic Gradient Descent. In other words, the effects of the graph topology and delays are confined to higher-order terms. Additionally, we extend our analysis to accommodate time-varying network topologies. Numerical simulations are provided to support our theoretical findings.

Optimal transport has been very successful for various machine learning tasks; however, it is known to suffer from the curse of dimensionality. Hence, dimensionality reduction is desirable when applied to high-dimensional data with low-dimensional structures. The kernel max-sliced (KMS) Wasserstein distance is developed for this purpose by finding an optimal nonlinear mapping that reduces data into $1$ dimensions before computing the Wasserstein distance. However, its theoretical properties have not yet been fully developed. In this paper, we provide sharp finite-sample guarantees under milder technical assumptions compared with state-of-the-art for the KMS $p$-Wasserstein distance between two empirical distributions with $n$ samples for general $p\in[1,\infty)$. Algorithm-wise, we show that computing the KMS $2$-Wasserstein distance is NP-hard, and then we further propose a semidefinite relaxation (SDR) formulation (which can be solved efficiently in polynomial time) and provide a relaxation gap for the SDP solution. We provide numerical examples to demonstrate the good performance of our scheme for high-dimensional two-sample testing.

In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in Y}\phi(x, y) - \max_{z\in Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i.e., difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.

Semi-implicit variational inference (SIVI) extends traditional variational families with semi-implicit distributions defined in a hierarchical manner. Due to the intractable densities of semi-implicit distributions, classical SIVI often resorts to surrogates of evidence lower bound (ELBO) that would introduce biases for training. A recent advancement in SIVI, named SIVI-SM, utilizes an alternative score matching objective made tractable via a minimax formulation, albeit requiring an additional lower-level optimization. In this paper, we propose kernel SIVI (KSIVI), a variant of SIVI-SM that eliminates the need for lower-level optimization through kernel tricks. Specifically, we show that when optimizing over a reproducing kernel Hilbert space (RKHS), the lower-level problem has an explicit solution. This way, the upper-level objective becomes the kernel Stein discrepancy (KSD), which is readily computable for stochastic gradient descent due to the hierarchical structure of semi-implicit variational distributions. An upper bound for the variance of the Monte Carlo gradient estimators of the KSD objective is derived, which allows us to establish novel convergence guarantees of KSIVI. We demonstrate the effectiveness and efficiency of KSIVI on both synthetic distributions and a variety of real data Bayesian inference tasks.

We develop and analyze algorithms for instrumental variable regression by viewing the problem as a conditional stochastic optimization problem. In the context of least-squares instrumental variable regression, our algorithms neither require matrix inversions nor mini-batches and provides a fully online approach for performing instrumental variable regression with streaming data. When the true model is linear, we derive rates of convergence in expectation, that are of order $\mathcal{O}(\log T/T)$ and $\mathcal{O}(1/T^{1-\iota})$ for any $\iota>0$, respectively under the availability of two-sample and one-sample oracles, respectively, where $T$ is the number of iterations. Importantly, under the availability of the two-sample oracle, our procedure avoids explicitly modeling and estimating the relationship between confounder and the instrumental variables, demonstrating the benefit of the proposed approach over recent works based on reformulating the problem as minimax optimization problems. Numerical experiments are provided to corroborate the theoretical results.

This paper investigates sparse high-dimensional linear regression, particularly examining the properties of the posterior under conditions of random design and unknown error variance. We provide consistency results for the posterior and analyze its concentration rates, demonstrating adaptiveness to the unknown sparsity level of the regression coefficient vector. Furthermore, we extend our investigation to establish concentration outcomes for parameter estimation using specific distance measures. These findings are in line with recent discoveries in frequentist studies. Additionally, by employing techniques to address model misspecification through a fractional posterior, we broaden our analysis through oracle inequalities to encompass the critical aspect of model misspecification for the regular posterior. Our novel findings are demonstrated using two different types of sparsity priors: a shrinkage prior and a spike-and-slab prior.

Clustering is a critical challenge in network science, pivotal for detecting underlying patterns and structures in unlabeled data. To explore the boundaries of this challenge, stochastic block models (SBMs) have been effectively utilized as a mathematical framework to assess the performance of clustering algorithms. Specifically, an SBM is a statistical model developed to reveal the structural dynamics of networks or graphs, where nodes represent individual entities and edges symbolize the connections between them. In a typical SBM, nodes are categorized into blocks or communities according to their connectivity patterns, with the probability of an edge existing between any two nodes depending on the blocks to which they belong [3]. For example, in a social network using an SBM, nodes might be organized by attributes such as age, gender, or geographic location, with friendship probabilities determined by their block memberships [1, 6]. The Bipartite Stochastic Block Model(B-SBM)[2] extends the conventional SBM to accommodate networks comprising two distinct node types, forming a bipartite graph structure. This adaptation is particularly beneficial in contexts such as recommendation systems, where nodes represent users and products, or in particular social networks, where nodes might denote individuals and the groups or events they participate in. In B-SBMs, the connections between nodes from different sets are governed by an "affinity matrix" that specifies the likelihood of linkage based on group affiliations. This matrix is integral to capturing interaction patterns within the network, allowing for a sophisticated estimation of model parameters from observed connections.

Graph neural networks (GNNs) are powerful black-box models which have shown impressive empirical performance. However, without any form of uncertainty quantification, it can be difficult to trust such models in high-risk scenarios. Conformal prediction aims to address this problem, however, an assumption of exchangeability is required for its validity which has limited its applicability to static graphs and transductive regimes. We propose to use unfolding, which allows any existing static GNN to output a dynamic graph embedding with exchangeability properties. Using this, we extend the validity of conformal prediction to dynamic GNNs in both transductive and semi-inductive regimes. We provide a theoretical guarantee of valid conformal prediction in these cases and demonstrate the empirical validity, as well as the performance gains, of unfolded GNNs against standard GNN architectures on both simulated and real datasets.

In nonsmooth, nonconvex stochastic optimization, understanding the uniform convergence of subdifferential mappings is crucial for analyzing stationary points of sample average approximations of risk as they approach the population risk. Yet, characterizing this convergence remains a fundamental challenge. This work introduces a novel perspective by connecting the uniform convergence of subdifferential mappings to that of subgradient mappings as empirical risk converges to the population risk. We prove that, for stochastic weakly-convex objectives, and within any open set, a uniform bound on the convergence of subgradients -- chosen arbitrarily from the corresponding subdifferential sets -- translates to a uniform bound on the convergence of the subdifferential sets itself, measured by the Hausdorff metric. Using this technique, we derive uniform convergence rates for subdifferential sets of stochastic convex-composite objectives. Our results do not rely on key distributional assumptions in the literature, which require the population and finite sample subdifferentials to be continuous in the Hausdorff metric, yet still provide tight convergence rates. These guarantees lead to new insights into the nonsmooth landscapes of such objectives within finite samples.

Recent works have extended notions of feature importance to \emph{semantic concepts} that are inherently interpretable to the users interacting with a black-box predictive model. Yet, precise statistical guarantees, such as false positive rate control, are needed to communicate findings transparently and to avoid unintended consequences in real-world scenarios. In this paper, we formalize the global (i.e., over a population) and local (i.e., for a sample) statistical importance of semantic concepts for the predictions of opaque models, by means of conditional independence, which allows for rigorous testing. We use recent ideas of sequential kernelized testing (SKIT) to induce a rank of importance across concepts, and showcase the effectiveness and flexibility of our framework on synthetic datasets as well as on image classification tasks using vision-language models such as CLIP.