Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated with each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications.

Sampling-based methods promise scalability improvements when paired with stochastic gradient descent in training Graph Convolutional Networks (GCNs). While effective in alleviating the neighborhood explosion, due to bandwidth and memory bottlenecks, these methods lead to computational overheads in preprocessing and loading new samples in heterogeneous systems, which significantly deteriorate the sampling performance. By decoupling the frequency of sampling from the sampling strategy, we propose LazyGCN, a general yet effective framework that can be integrated with any sampling strategy to substantially improve the training time. The basic idea behind LazyGCN is to perform sampling periodically and effectively recycle the sampled nodes to mitigate data preparation overhead. We theoretically analyze the proposed algorithm and show that under a mild condition on the recycling size, by reducing the variance of inner layers, we are able to obtain the same convergence rate as the underlying sampling method.

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With O(r 3 \kappa 2 n \log n) random measurements of a positive semidefinite n\times n matrix of rank r and condition number \kappa, our method is guaranteed to converge linearly to the global optimum.

Decentralized optimization has wide applications in machine learning, signal processing, and control. In this paper, we study the decentralized composite optimization problem with a non-smooth regularization term. Many proximal gradient based decentralized algorithms have been proposed in the past. However, these algorithms do not achieve near optimal computational complexity and communication complexity. In this paper, we propose a new method which establishes the optimal computational complexity and a near optimal communication complexity.

SGD with momentum (SGDM) has been widely applied in many machine learning tasks, and it is often applied with dynamic stepsizes and momentum weights tuned in a stagewise manner. Despite of its empirical advantage over SGD, the role of momentum is still unclear in general since previous analyses on SGDM either provide worse convergence bounds than those of SGD, or assume Lipschitz or quadratic objectives, which fail to hold in practice. Furthermore, the role of dynamic parameters has not been addressed. In this work, we show that SGDM converges as fast as SGD for smooth objectives under both strongly convex and nonconvex settings. We also prove that multistage strategy is beneficial for SGDM compared to using fixed parameters.

This uniform budget represents an overall maximal privacy violation that any user is willing to face by contributing their data to the training set. We argue that this approach is limited because different users may have different privacy expectations. Thus, setting a uniform privacy budget across all points may be overly conservative for some users or, conversely, not sufficiently protective for others. In this paper, we capture these preferences through individualized privacy budgets. To demonstrate their practicality, we introduce a variant of Differentially Private Stochastic Gradient Descent (DP-SGD) which supports such individualized budgets.

To measure the difference between two probability distributions, referred to as the source and target, respectively, we exploit both the chain rule and Bayes' theorem to construct conditional transport (CT), which is constituted by both a forward component and a backward one. The forward CT is the expected cost of moving a source data point to a target one, with their joint distribution defined by the product of the source probability density function (PDF) and a source-dependent conditional distribution, which is related to the target PDF via Bayes' theorem. The backward CT is defined by reversing the direction. The CT cost can be approximated by replacing the source and target PDFs with their discrete empirical distributions supported on mini-batches, making it amenable to implicit distributions and stochastic gradient descent-based optimization. When applied to train a generative model, CT is shown to strike a good balance between mode-covering and mode-seeking behaviors and strongly resist mode collapse.

Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access.

We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ the notion of efficiency ordering', a well-analyzed tool for comparing the performance of Markov Chain Monte Carlo (MCMC) samplers, for SGD algorithms in the form of Loewner ordering of covariance matrices associated with the scaled iterate errors in the long term. Using this ordering, we show that input sequences that are more efficient for MCMC sampling also lead to smaller covariance of the errors for SGD algorithms in the limit. This also suggests that an arbitrarily weighted MSE of SGD iterates in the limit becomes smaller when driven by more efficient chains. Our finding is of particular interest in applications such as decentralized optimization and swarm learning, where SGD is implemented in a random walk fashion on the underlying communication graph for cost issues and/or data privacy.

What is the information leakage of an iterative randomized learning algorithm about its training data, when the internal state of the algorithm is \emph{private}? How much is the contribution of each specific training epoch to the information leakage through the released model? We study this problem for noisy gradient descent algorithms, and model the \emph{dynamics} of R\'enyi differential privacy loss throughout the training process. Our analysis traces a provably \emph{tight} bound on the R\'enyi divergence between the pair of probability distributions over parameters of models trained on neighboring datasets. We prove that the privacy loss converges exponentially fast, for smooth and strongly convex loss functions, which is a significant improvement over composition theorems (which over-estimate the privacy loss by upper-bounding its total value over all intermediate gradient computations).