While the optimal transport (OT) problem was originally formulated as a linear program, regularizing it with an entropic penalty has been favored by practitioners in many recent applications, where that regularization is seen as beneficial from both computational and statistical perspectives. The Sinkhorn fixed-point algorithm isthe most popular approach to solve that regularized problem, and, as a result, multiple attempts have been made to reduce its runtime using, e.g., annealing in the regularization parameter, momentum or acceleration in the iterates. The premiseof this work is that initialization of the Sinkhorn algorithm has received comparatively little attention, possibly due to two preconceptions: since the regularized OT problem is convex, it may not be worth crafting a good initialization, since any isguaranteed to work; secondly, because the outputs of the Sinkhorn algorithm are often differentiated in end-to-end pipelines, a data-dependent initialization would bias Jacobian estimates obtained when unrolling iterations. We challenge this conventional wisdom, and show that data-dependent initializers result in dramatic speed-ups, without affecting the correctness of Jacobian maps, as long as those are recovered using implicit differentiation. Our initializations rely on simple closed-forms for exact or approximate OT solutions, using known results in the 1D, Gaussianor GMM settings. These initializations …

Generative adversarial networks (GANs) represent a game between two neural network machines designed to learn the distribution of data. It is commonly observed that different GAN formulations and divergence/distance measures used could lead to considerably different performance results, especially when the data distribution is multi-modal. In this work, we give a theoretical characterization of the mode-seeking behavior of general f-divergences and Wasserstein distances, and prove a performance guarantee for the setting where the underlying model is a mixture of multiple symmetric quasiconcave distributions. This can help us understand the trade-off between the quality and diversity of the trained GANs' output samples. Our theoretical results show the mode-seeking nature of the Jensen-Shannon (JS) divergence over standard KL-divergence and Wasserstein distance measures. We subsequently demonstrate that a hybrid of JS-divergence and Wasserstein distance measures minimized by Lipschitz GANs mimics the mode-seeking behavior of the JS-divergence. We present numerical results showing the mode-seeking nature of the JS-divergence and its hybrid with the Wasserstein distance while highlighting the mode-covering properties of KL-divergence and Wasserstein distance measures. Our numerical experiments indicate the different behavior of several standard GAN formulations in application to benchmark Gaussian mixture and image datasets.

The "Propose-Test-Release" (PTR) framework is a classic recipe for designing differentially private (DP) algorithms that are data-adaptive, i.e. those that add less noise when the input dataset is "nice". We extend PTR to a more general setting by privately testing data-dependent privacy losses rather than local sensitivity, hence making it applicable beyond the standard noise-adding mechanisms, e.g. to queries with unbounded or undefined sensitivity. We demonstrate the versatility of generalized PTR using private linear regression as a case study. Additionally, we apply our algorithm to solve an open problem from “Private Aggregation of Teacher Ensembles (PATE)” --- privately releasing the entire model with a delicate data-dependent analysis.

Data valuation has wide use cases in machine learning, including improving data quality and creating economic incentives for data sharing. This paper studies the robustness of data valuation to noisy model performance scores. Particularly, we find that the inherent randomness of the widely used stochastic gradient descent can cause existing data value notions (e.g., the Shapley value and the Leave-one-out error) to produce inconsistent data value rankings across different runs. To address this challenge, we introduce the concept of safety margin, which measures the robustness of a data value notion. We show that the Banzhaf value, a famous value notion that originated from cooperative game theory literature, achieves the largest safety margin among all semivalues (a class of value notions that satisfy crucial properties entailed by ML applications and include the famous Shapley value and Leave-one-out error). We propose an algorithm to efficiently estimate the Banzhaf value based on the Maximum Sample Reuse (MSR) principle. Our evaluation demonstrates that the Banzhaf value outperforms the existing semivalue-based data value notions on several ML tasks such as learning with weighted samples and noisy label detection. Overall, our study suggests that when the underlying ML algorithm is stochastic, the Banzhaf value is a …

To achieve a graph representation, most Graph Neural Networks (GNNs) follow two steps: first, each graph is decomposed into a number of subgraphs (which we call the recursion step), and then the collection of subgraphs is encoded by several iterative pooling steps. While recently proposed higher-order networks show a remarkable increase in the expressive power through a single recursion on larger neighborhoods followed by iterative pooling, the power of deeper recursion in GNNs without any iterative pooling is still not fully understood. To make it concrete, we consider a pure recursion-based GNN which we call Recursive Neighborhood Pooling GNN (RNP-GNN). The expressive power of an RNP-GNN and its computational cost quantifies the power of (pure) recursion for a graph representation network. We quantify the power by means of counting substructures, which is one main limitation of the Message Passing graph Neural Networks (MPNNs), and show how RNP-GNN can exploit the sparsity of the underlying graph to achieve low-cost powerful representations. We also compare the recent lower bounds on the time complexity and show how recursion-based networks are near optimal.

Graphons are general and powerful models for generating graphs of varying size. In this paper, we propose to directly model graphons using neural networks, obtaining Implicit Graphon Neural Representation (IGNR). Existing work in modeling and reconstructing graphons often approximates a target graphon by a fixed resolution piece-wise constant representation. Our IGNR has the benefit that it can represent graphons up to arbitrary resolutions, and enables natural and efficient generation of arbitrary sized graphs with desired structure once the model is learned. Furthermore, we allow the input graph data to be unaligned and have different sizes by leveraging the Gromov-Wasserstein distance. We first demonstrate the effectiveness of our model by showing its superior performance on a graphon learning task. We then propose an extension of IGNR that can be incorporated into an auto-encoder framework, and demonstrate its good performance under a more general setting of graphon learning. We also show that our model is suitable for graph representation learning and graph generation.

Recent work has shown that sparse graphs containing many triangles cannot be reproduced using a finite-dimensional representation of the nodes, in which link probabilities are inner products. Here, we show that such graphs can be reproduced using an infinite-dimensional inner product model, where the node representations lie on a low-dimensional manifold. Recovering a global representation of the manifold is impossible in a sparse regime. However, we can zoom in on local neighbourhoods, where a lower-dimensional representation is possible. As our constructions allow the points to be uniformly distributed on the manifold, we find evidence against the common perception that triangles imply community structure.

Statistical modeling of random networks has been a widely used approach to uncovering interaction mechanisms of complex systems and predicting unobserved links in real-world networks. In many applications, network connections are collected via egocentric sampling: a subset of nodes was sampled first, after which all links involving this subset of nodes were recorded; all other information was missing. Compared with the typical assumption of uniformly missing at random, the egocentrically sampled partial networks requires specially designed modeling strategies. The previous available statistical methods are either computationally infeasible or based on intuitive designs without theoretical justification. We propose a method to fit general low-rank models for egocentrically sampled networks, which include several popular network models. The method is based on spectral properties and is computationally efficient for large-scale networks. The proposed method gives a consistent recovery of the missing subnetwork due to egocentric sampling for sparse networks. To our knowledge, this is the first available theoretical guarantee for egocentric partial network estimation in the scope of low-rank models. We evaluate the method on several synthetic and real-world networks and show that it delivers competitive performance in link prediction tasks.

We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In particular, partially stochastic networks with only n stochastic biases are universal probabilistic predictors for n-dimensional predictive problems. In empirical investigations, we find no systematic benefit of full stochasticity across four different inference modalities and eight datasets; partially stochastic networks can match and sometimes even outperform fully stochastic networks, despite their reduced memory costs.

Most modern probabilistic generative models, such as the variational autoencoder (VAE), have certain indeterminacies that are unresolvable even with an infinite amount of data. Different tasks tolerate different indeterminacies, however recent applications have indicated the need for strongly identifiable models, in which an observation corresponds to a unique latent code. Progress has been made towards reducing model indeterminacies while maintaining flexibility, and recent work excludes many---but not all---indeterminacies. In this work, we motivate model-identifiability in terms of task-identifiability, then construct a theoretical framework for analyzing the indeterminacies of latent variable models, which enables their precise characterization in terms of the generator function and prior distribution spaces. We reveal that strong identifiability is possible even with highly flexible nonlinear generators, and give two such examples. One is a straightforward modification of iVAE (Khemakhem et al., 2020); the other uses triangular monotonic maps, leading to novel connections between optimal transport and identifiability.

Constrained learning is prevalent in countless statistical tasks. Recent work proposes distance-to-set penalties to derive estimators under general constraints that can be specified as sets, but focuses on obtaining point estimates that do not come with corresponding measures of uncertainty. To remedy this, we approach distance-to-set regularization from a Bayesian lens. We consider a class of smooth distance-to-set priors, showing that they yield well-defined posteriors toward quantifying uncertainty for constrained learning problems. We discuss relationships and advantages over prior work on Bayesian constraint relaxation. Moreover, we prove that our approach is optimal in an information geometric-sense for finite penalty parameters ρ, and enjoys favorable statistical properties when ρ → ∞. The method is designed to perform effectively within gradient-based MCMC samplers, as illustrated on a suite of simulated and real data applications.

Neal and Hinton (1998) recast maximum likelihood estimation of any given latent variable model as the minimization of a free energy functional F, and the EM algorithm as coordinate descent applied to F. Here, we explore alternative ways to optimize the functional. In particular, we identify various gradient flows associated with F and show that their limits coincide with F's stationary points. By discretizing the flows, we obtain practical particle-based algorithms for maximum likelihood estimation in broad classes of latent variable models. The novel algorithms scale to high-dimensional settings and perform well in numerical experiments.

Inferring causal structures from experimentation is a central task in many domains. For example, in biology, recent advances allow us to obtain single-cell expression data under multiple interventions such as drugs or gene knockouts. However, the targets of the interventions are often uncertain or unknown and the number of observations limited. As a result, standard causal discovery methods can no longer be reliably used.To fill this gap, we propose a Bayesian framework (BaCaDI) for discovering and reasoning about the causal structure that underlies data generated under various unknown experimental or interventional conditions.BaCaDI is fully differentiable, which allows us to infer the complex joint posteriorover the intervention targets and the causal structure via efficient gradient-based variational inference.In experiments on synthetic causal discovery tasks and simulated gene-expression data, BaCaDI outperforms related methods in identifying causal structures and intervention targets.

Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.

Bayesian optimization (BO) is a popular approach to sample-efficient optimization of black-box objective functions. While BO has been successfully applied to a wide range of scientific applications, traditional approaches to single-objective BO only seek to find a single best solution. This can be a significant limitation in situations where solutions may later turn out to be intractable, e.g., a designed molecule may turn out to later violate constraints that can only be evaluated after the optimization process has concluded. To combat this issue, we propose Rank-Ordered Bayesian Optimization with Trust-regions (ROBOT) which aims to find a portfolio of high-performing solutions that are diverse according to a user-specified diversity metric. We evaluate ROBOT on several real-world applications and show that it can discover large sets of high-performing diverse solutions while requiring few additional function evaluations compared to finding a single best solution.

Sparse Gaussian processes are a key component of high-throughput Bayesian optimisation (BO) loops; however, we show that existing methods for allocating their inducing points severely hamper optimisation performance. By exploiting the quality-diversity decomposition of determinantal point processes, we propose the first inducing point allocation strategy designed specifically for use in BO. Unlike existing methods which seek only to reduce global uncertainty in the objective function, our approach provides the local high-fidelity modelling of promising regions required for precise optimisation. More generally, we demonstrate that our proposed framework provides a flexible way to allocate modelling capacity in sparse models and so is suitable for a broad range of downstream sequential decision making tasks.

Confidence sequences are confidence intervals that can be sequentially tracked, and are valid at arbitrary data-dependent stopping times. This paper presents confidence sequences for a univariate mean of an unknown distribution with a known upper bound on the p-th central moment (p > 1), but allowing for (at most) ε fraction of arbitrary distribution corruption, as in Huber's contamination model. We do this by designing new robust exponential supermartingales, and show that the resulting confidence sequences attain the optimal width achieved in the nonsequential setting. Perhaps surprisingly, the constant margin between our sequential result and the lower bound is smaller than even fixed-time robust confidence intervals based on the trimmed mean, for example. Since confidence sequences are a common tool used within A/B/n testing and bandits, these results open the door to sequential experimentation that is robust to outliers and adversarial corruptions.

Random Fourier Features (RFF) is among the most popular and broadly applicable approaches for scaling up kernel methods. In essence, RFF allows the user to avoid costly computations with a large kernel matrix via a fast randomized approximation. However, a pervasive difficulty in applying RFF is that the user does not know the actual error of the approximation, or how this error will propagate into downstream learning tasks. Up to now, the RFF literature has primarily dealt with these uncertainties using theoretical error bounds, but from a user's standpoint, such results are typically impractical---either because they are highly conservative or involve unknown quantities. To tackle these general issues in a data-driven way, this paper develops a bootstrap approach to numerically estimate the errors of RFF approximations. Three key advantages of this approach are: (1) The error estimates are specific to the problem at hand, avoiding the pessimism of worst-case bounds. (2) The approach is flexible with respect to different uses of RFF, and can even estimate errors in downstream learning tasks. (3) The approach enables adaptive computation, in the sense that the user can quickly inspect the error of a rough initial kernel approximation and then predict how much extra …

The most relevant problems in discounted reinforcement learning involve estimating the mean of a function under the stationary distribution of a Markov reward process, such as the expected return in policy evaluation, or the policy gradient in policy optimization. In practice, these estimates are produced through a finite-horizon episodic sampling, which neglects the mixing properties of the Markov process. It is mostly unclear how this mismatch between the practical and the ideal setting affects the estimation, and the literature lacks a formal study on the pitfalls of episodic sampling, and how to do it optimally. In this paper, we present a minimax lower bound on the discounted mean estimation problem that explicitly connects the estimation error with the mixing properties of the Markov process and the discount factor. Then, we provide a statistical analysis on a set of notable estimators and the corresponding sampling procedures, which includes the finite-horizon estimators often used in practice. Crucially, we show that estimating the mean by directly sampling from the discounted kernel of the Markov process brings compelling statistical properties w.r.t. the alternative estimators, as it matches the lower bound without requiring a careful tuning of the episode horizon.

Sequential decision making significantly speeds up research and is more cost-effective compared to fixed-n methods. We present a method for sequential decision making for stratified count data that retains Type-I error guarantee or false discovery rate under optional stopping, using e-variables. We invert the method to construct stratified anytime-valid confidence sequences, where cross-talk between subpopulations in the data can be allowed during data collection to improve power. Finally, we combine information collected in separate subpopulations through pseudo-Bayesian averaging and switching to create effective estimates for the minimal, mean and maximal treatment effects in the subpopulations.

Feature attribution methods identify which features of an input most influence a model's output. Most widely-used feature attribution methods (such as SHAP, LIME, and Grad-CAM) are "class-dependent" methods in that they generate a feature attribution vector as a function of class. In this work, we demonstrate that class-dependent methods can "leak" information about the selected class, making that class appear more likely than it is. Thus, an end user runs the risk of drawing false conclusions when interpreting an explanation generated by a class-dependent method. In contrast, we introduce "distribution-aware" methods, which favor explanations that keep the label's distribution close to its distribution given all features of the input. We introduce SHAP-KL and FastSHAP-KL, two baseline distribution-aware methods that compute Shapley values. Finally, we perform a comprehensive evaluation of seven class-dependent and three distribution-aware methods on three clinical datasets of different high-dimensional data types: images, biosignals, and text.

Semi-supervised learning (SSL) promises improved accuracy compared to training classifiers on small labeled datasets by also training on many unlabeled images. In real applications like medical imaging, unlabeled data will be collected for expediency and thus uncurated: possibly different from the labeled set in classes or features. Unfortunately, modern deep SSL often makes accuracy worse when given uncurated unlabeled data. Recent complex remedies try to detect out-of-distribution unlabeled images and then discard or downweight them. Instead, we introduce Fix-A-Step, a simpler procedure that views all uncurated unlabeled images as potentially helpful. Our first insight is that even uncurated images can yield useful augmentations of labeled data. Second, we modify gradient descent updates to prevent optimizing a multi-task SSL loss from hurting labeled-set accuracy. Fix-A-Step can "repair" many common deep SSL methods, improving accuracy on CIFAR benchmarks across all tested methods and levels of artificial class mismatch. On a new medical SSL benchmark called Heart2Heart, Fix-A-Step can learn from 353,500 truly uncurated ultrasound images to deliver gains that generalize across hospitals.

Federated Learning (FL) under distributed concept drift is a largely unexplored area. Although concept drift is itself a well-studied phenomenon, it poses particular challenges for FL, because drifts arise staggered in time and space (across clients). Our work is the first to explicitly study data heterogeneity in both dimensions. We first demonstrate that prior solutions to drift adaptation, with their single global model, are ill-suited to staggered drifts, necessitating multiple-model solutions. We identify the problem of drift adaptation as a time-varying clustering problem, and we propose two new clustering algorithms for reacting to drifts based on local drift detection and hierarchical clustering. Empirical evaluation shows that our solutions achieve significantly higher accuracy than existing baselines, and are comparable to an idealized algorithm with oracle knowledge of the ground-truth clustering of clients to concepts at each time step.

This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than 1/3, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than 1/3. Compared to the existing results …

We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.

Empirical risk minimization (ERM) and distributionally robust optimization (DRO) are popular approaches for solving stochastic optimization problems that appear in operations management and machine learning. Existing generalization error bounds for these methods depend on either the complexity of the cost function or dimension of the uncertain parameters; consequently, the performance of these methods is poor for high-dimensional problems with objective functions under high complexity. We propose a simple approach in which the distribution of uncertain parameters is approximated using a parametric family of distributions. This mitigates both sources of complexity; however, it introduces a model misspecification error. We show that this new source of error can be controlled by suitable DRO formulations. Our proposed parametric DRO approach has significantly improved generalization bounds over existing ERM / DRO methods and parametric ERM for a wide variety of settings. Our method is particularly effective under distribution shifts. We also illustrate the superior performance of our approach on both synthetic and real-data portfolio optimization and regression tasks.