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Computationally solving multi-marginal optimal transport (MOT) with squared Euclidean costs for $N$ discrete probability measures has recently attracted considerable attention, in part because of the correspondence of its solutions with Wasserstein-$2$ barycenters, which have many applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. While entropic regularization has been successfully applied to approximate Wasserstein barycenters, this loses the sparsity of the optimal solution, making it difficult to solve the MOT problem directly in practice because of the curse of dimensionality. Thus, for obtaining barycenters, one usually resorts to fixed-support restrictions to a grid, which is, however, prohibitive in higher ambient dimensions $d$. In this paper, after analyzing the relationship between MOT and barycenters, we present two algorithms to approximate the solution of MOT directly, requiring mainly just $N-1$ standard two-marginal OT computations. Thus, they are fast, memory-efficient and easy to implement and can be used with any sparse OT solver as a black box. Moreover, they produce sparse solutions and show promising numerical results. We analyze these algorithms theoretically, proving upper and lower bounds for the relative approximation error.

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a "martingale loss function", whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions" with test functions. Solving these equations in different ways recovers various classical TD algorithms, such as TD($\lambda$), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero, and we provide the convergence rate. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

The isogeometric approximation of the Stokes problem in a trimmed domain is studied. This setting is characterized by an underlying mesh unfitted with the boundary of the physical domain making the imposition of the essential boundary conditions a challenging problem. A very popular strategy is to rely on the so-called Nitsche method \cite{MR3264337}. We show that the Nitsche method lacks stability in some degenerate trimmed domain configurations, potentially polluting the computed solutions. After extending the stabilization procedure of \cite{MR4155233} to incompressible flow problems, we show that we recover the well-posedness of the formulation and, consequently, optimal a priori error estimates. Numerical experiments illustrating stability and converge rates are included.

In black-box function optimization, we need to consider not only controllable design variables but also uncontrollable stochastic environment variables. In such cases, it is necessary to solve the optimization problem by taking into account the uncertainty of the environmental variables. Chance-constrained (CC) problem, the problem of maximizing the expected value under a certain level of constraint satisfaction probability, is one of the practically important problems in the presence of environmental variables. In this study, we consider distributionally robust CC (DRCC) problem and propose a novel DRCC Bayesian optimization method for the case where the distribution of the environmental variables cannot be precisely specified. We show that the proposed method can find an arbitrary accurate solution with high probability in a finite number of trials, and confirm the usefulness of the proposed method through numerical experiments.

Second-order optimization methods are among the most widely used optimization approaches for convex optimization problems, and have recently been used to optimize non-convex optimization problems such as deep learning models. The widely used second-order optimization methods such as quasi-Newton methods generally provide curvature information by approximating the Hessian using the secant equation. However, the secant equation becomes insipid in approximating the Newton step owing to its use of the first-order derivatives. In this study, we propose an approximate Newton sketch-based stochastic optimization algorithm for large-scale empirical risk minimization. Specifically, we compute a partial column Hessian of size ($d\times m$) with $m\ll d$ randomly selected variables, then use the \emph{Nystr\"om method} to better approximate the full Hessian matrix. To further reduce the computational complexity per iteration, we directly compute the update step ($\Delta\boldsymbol{w}$) without computing and storing the full Hessian or its inverse. We then integrate our approximated Hessian with stochastic gradient descent and stochastic variance-reduced gradient methods. The results of numerical experiments on both convex and non-convex functions show that the proposed approach was able to obtain a better approximation of Newton\textquotesingle s method, exhibiting performance competitive with that of state-of-the-art first-order and stochastic quasi-Newton methods. Furthermore, we provide a theoretical convergence analysis for convex functions.

Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, with probability tending to 1 or almost surely, as the sample size goes to infinity. Under the Bayesian nonparametric framework, a common assumption in the study of PCRs is that the model is dominated for the observations; that is, it is assumed that the posterior can be written through the Bayes formula. In this paper, we consider the problem of establishing PCRs in Bayesian nonparametric models where the posterior distribution is not available through the Bayes formula, and hence models that are non-dominated for the observations. By means of the Wasserstein distance and a suitable sieve construction, our main result establishes PCRs in Bayesian nonparametric models where the posterior is available through a more general disintegration than the Bayes formula. To the best of our knowledge, this is the first general approach to provide PCRs in non-dominated Bayesian nonparametric models, and it relies on minimal modeling assumptions and on a suitable continuity assumption for the posterior distribution. Some refinements of our result are presented under additional assumptions on the prior distribution, and applications are given with respect to the Dirichlet process prior and the normalized extended Gamma process prior.

We present a novel neural-networks-based algorithm to compute optimal transport maps and plans for strong and weak transport costs. To justify the usage of neural networks, we prove that they are universal approximators of transport plans between probability distributions. We evaluate the performance of our optimal transport algorithm on toy examples and on the unpaired image-to-image style translation task.

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.