Ensemble Kalman inversion (EKI) is a sequential Monte Carlo method used to solve inverse problems within a Bayesian framework. Unlike backpropagation, EKI is a gradient-free optimization method that only necessitates the evaluation of artificial neural networks in forward passes. In this study, we examine the effectiveness of EKI in training neural ordinary differential equations (neural ODEs) for system identification and control tasks. To apply EKI to optimal control problems, we formulate inverse problems that incorporate a Tikhonov-type regularization term. Our numerical results demonstrate that EKI is an efficient method for training neural ODEs in system identification and optimal control problems, with runtime and quality of solutions that are competitive with commonly used gradient-based optimizers.

Phase association groups seismic wave arrivals according to their originating earthquakes. It is a fundamental task in a seismic data processing pipeline, but challenging to perform for smaller, high-rate seismic events which carry fundamental information about earthquake dynamics, especially with a commonly assumed inaccurate wave speed model. As a consequence, most association methods focus on larger events that occur at a lower rate and are thus easier to associate, even though microseismicity provides a valuable description of the elastic medium properties in the subsurface. In this paper, we show that association is possible at rates much higher than previously reported even when the wave speed is unknown. We propose Harpa, a high-rate seismic phase association method which leverages deep neural fields to build generative models of wave speeds and associated travel times, and first solves a joint spatio--temporal source localization and wave speed recovery problem, followed by association. We obviate the need for associated phases by interpreting arrival time data as probability measures and using an optimal transport loss to enforce data fidelity. The joint recovery problem is known to admit a unique solution under certain conditions but due to the non-convexity of the corresponding loss a simple gradient scheme converges to poor local minima. We show that this is effectively mitigated by stochastic gradient Langevin dynamics (SGLD). Numerical experiments show that \harpa~efficiently associates high-rate seismicity clouds over complex, unknown wave speeds and graciously handles noisy and missing picks.

Two widely considered decentralized learning algorithms are Gossip and random walk-based learning. Gossip algorithms (both synchronous and asynchronous versions) suffer from high communication cost, while random-walk based learning experiences increased convergence time. In this paper, we design a fast and communication-efficient asynchronous decentralized learning mechanism DIGEST by taking advantage of both Gossip and random-walk ideas, and focusing on stochastic gradient descent (SGD). DIGEST is an asynchronous decentralized algorithm building on local-SGD algorithms, which are originally designed for communication efficient centralized learning. We design both single-stream and multi-stream DIGEST, where the communication overhead may increase when the number of streams increases, and there is a convergence and communication overhead trade-off which can be leveraged. We analyze the convergence of single- and multi-stream DIGEST, and prove that both algorithms approach to the optimal solution asymptotically for both iid and non-iid data distributions. We evaluate the performance of single- and multi-stream DIGEST for logistic regression and a deep neural network ResNet20. The simulation results confirm that multi-stream DIGEST has nice convergence properties; i.e., its convergence time is better than or comparable to the baselines in iid setting, and outperforms the baselines in non-iid setting.

Addressing the interpretability problem of NMF on Boolean data, Boolean Matrix Factorization (BMF) uses Boolean algebra to decompose the input into low-rank Boolean factor matrices. These matrices are highly interpretable and very useful in practice, but they come at the high computational cost of solving an NP-hard combinatorial optimization problem. To reduce the computational burden, we propose to relax BMF continuously using a novel elastic-binary regularizer, from which we derive a proximal gradient algorithm. Through an extensive set of experiments, we demonstrate that our method works well in practice: On synthetic data, we show that it converges quickly, recovers the ground truth precisely, and estimates the simulated rank exactly. On real-world data, we improve upon the state of the art in recall, loss, and runtime, and a case study from the medical domain confirms that our results are easily interpretable and semantically meaningful.

Modern machine learning algorithms aim to extract fine-grained information from data to provide accurate predictions, which often conflicts with the goal of privacy protection. This paper addresses the practical and theoretical importance of developing privacy-preserving machine learning algorithms that ensure good performance while preserving privacy. In this paper, we focus on the privacy and utility (measured by excess risk bounds) performances of differentially private stochastic gradient descent (SGD) algorithms in the setting of stochastic convex optimization. Specifically, we examine the pointwise problem in the low-noise setting for which we derive sharper excess risk bounds for the differentially private SGD algorithm. In the pairwise learning setting, we propose a simple differentially private SGD algorithm based on gradient perturbation. Furthermore, we develop novel utility bounds for the proposed algorithm, proving that it achieves optimal excess risk rates even for non-smooth losses. Notably, we establish fast learning rates for privacy-preserving pairwise learning under the low-noise condition, which is the first of its kind.

The learning of Gaussian Mixture Models (also referred to simply as GMMs) plays an important role in machine learning. Known for their expressiveness and interpretability, Gaussian mixture models have a wide range of applications, from statistics, computer vision to distributional reinforcement learning. However, as of today, few known algorithms can fit or learn these models, some of which include Expectation-Maximization algorithms and Sliced Wasserstein Distance. Even fewer algorithms are compatible with gradient descent, the common learning process for neural networks. In this paper, we derive a closed formula of two GMMs in the univariate, one-dimensional case, then propose a distance function called Sliced Cram\'er 2-distance for learning general multivariate GMMs. Our approach has several advantages over many previous methods. First, it has a closed-form expression for the univariate case and is easy to compute and implement using common machine learning libraries (e.g., PyTorch and TensorFlow). Second, it is compatible with gradient descent, which enables us to integrate GMMs with neural networks seamlessly. Third, it can fit a GMM not only to a set of data points, but also to another GMM directly, without sampling from the target model. And fourth, it has some theoretical guarantees like global gradient boundedness and unbiased sampling gradient. These features are especially useful for distributional reinforcement learning and Deep Q Networks, where the goal is to learn a distribution over future rewards. We will also construct a Gaussian Mixture Distributional Deep Q Network as a toy example to demonstrate its effectiveness. Compared with previous models, this model is parameter efficient in terms of representing a distribution and possesses better interpretability.

The performance of data fusion and tracking algorithms often depends on parameters that not only describe the sensor system, but can also be task-specific. While for the sensor system tuning these variables is time-consuming and mostly requires expert knowledge, intrinsic parameters of targets under track can even be completely unobservable until the system is deployed. With state-of-the-art sensor systems growing more and more complex, the number of parameters naturally increases, necessitating the automatic optimization of the model variables. In this paper, the parameters of an interacting multiple model (IMM) filter are optimized solely using measurements, thus without necessity for any ground-truth data. The resulting method is evaluated through an ablation study on simulated data, where the trained model manages to match the performance of a filter parametrized with ground-truth values.

We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the "sub-optimality" of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines--stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)--which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.

To date, no "information-theoretic" frameworks for reasoning about generalization error have been shown to establish minimax rates for gradient descent in the setting of stochastic convex optimization. In this work, we consider the prospect of establishing such rates via several existing information-theoretic frameworks: input-output mutual information bounds, conditional mutual information bounds and variants, PAC-Bayes bounds, and recent conditional variants thereof. We prove that none of these bounds are able to establish minimax rates. We then consider a common tactic employed in studying gradient methods, whereby the final iterate is corrupted by Gaussian noise, producing a noisy "surrogate" algorithm. We prove that minimax rates cannot be established via the analysis of such surrogates. Our results suggest that new ideas are required to analyze gradient descent using information-theoretic techniques.

We study differentially private (DP) machine learning algorithms as instances of noisy fixed-point iterations, in order to derive privacy and utility results from this well-studied framework. We show that this new perspective recovers popular private gradient-based methods like DP-SGD and provides a principled way to design and analyze new private optimization algorithms in a flexible manner. Focusing on the widely-used Alternating Directions Method of Multipliers (ADMM) method, we use our general framework to derive novel private ADMM algorithms for centralized, federated and fully decentralized learning. For these three algorithms, we establish strong privacy guarantees leveraging privacy amplification by iteration and by subsampling. Finally, we provide utility guarantees using a unified analysis that exploits a recent linear convergence result for noisy fixed-point iterations.