We develop a new class of fast Krasnosel’skiǐ-Mann methods with variance reduction to solve a finite-sum cocoercive equation, [Formula: see text]. Our algorithm is single loop and leverages a new family of unbiased variance-reduced estimators specifically designed for a wider class of root-finding algorithms. Our method achieves both [Formula: see text] and [Formula: see text] last-iterate convergence rates in terms of [Formula: see text], where k is the iteration counter, and [Formula: see text] is the total expectation. We also establish almost-sure [Formula: see text] convergence rates and the almost-sure convergence of iterates [Formula: see text] to a solution of [Formula: see text]. We instantiate our framework for two prominent estimators: SVRG and SAGA. By an appropriate choice of parameters, both variants attain an oracle complexity of [Formula: see text] to reach an [Formula: see text]-solution, where n denotes the number of summands in the finite-sum operator G. Furthermore, under [Formula: see text]-strong quasimonotonicity, our method achieves a linear convergence rate and an oracle complexity of [Formula: see text], where [Formula: see text]. We extend our approach to solve a class of finite-sum inclusions (possibly nonmonotone), demonstrating that our schemes retain the same theoretical guarantees as in the equation setting. Finally, numerical experiments demonstrate that our methods exhibit promising performance compared with state-of-the-art methods. Funding: This work was partially supported by the U.S. National Science Foundation [Grant RTG-DMS-2134107] and the Office of Navel Research [Grant N00014-23-1-2588].
The classical Euclidean subgradient algorithm extends, via tangent constructions and exponential maps, to geodesically convex optimization on manifolds. General complexity analysis for manifolds with an upper curvature bound of zero, as developed by Zhang and Sra in 2016 [Zhang H, Sra S (2016) First-order methods for geodesically convex optimization. Feldman V, Rakhlin A, Shamir O, eds. Proc. 29th Conf. Learn. Theory, vol. 49 (PMLR, New York), 1617-1638] depends unavoidably on an additional lower curvature bound. We present a fresh approach to subgradient-type methods, suitable for objectives with "horospherically convex" level sets. Our method avoids both tangential constructions in its description and lower curvature bounds in its complexity analysis. Furthermore, it applies beyond manifolds to general geodesic metric spaces with curvature nonpositive but possibly unbounded below. As applications in such spaces, which include CAT(0) cubical complexes such as the Billera-Holmes-Vogtmann space of phylogenetic trees, we consider previously inaccessible problems such as recognizing weighted Frechet means and computing minimal enclosing balls.
We consider the problem of estimating a multivariate distribution based on multiple data sources, which include both joint and marginal-only data. Specifically, we introduce a nonparametric approach called Ensemble Copula Coupling (ECC), which can be viewed as a data fusion approach that combines joint and marginal information. The particular setting of interest is input modeling for output analysis of simulated stochastic systems. We apply an ECC-based input model to address uncertainty quantification with correlated inputs. We prove the consistency of ECC and provide limit theorems that can be used to construct confidence intervals for the estimators. Resampling schemes are proposed to improve the accuracy of the confidence intervals. Simulation experiments on queueing and finance examples illustrate the advantages of using the proposed approach and demonstrate that the ECC-based input model can be applied to higher-dimensional problems at lower computational cost compared with existing approaches.
Travel times in road networks are subject to stochastic uncertainty resulting from various parameters. A benevolent mobility service provider observing the actual travel times from data may use its informational advantage to steer the traffic equilibrium in a favorable direction through a processs known as signaling or Bayesian persuasion. Previous work has shown that the underlying signaling problem can be NP-hard to approximate better than [Formula: see text], even for static Wardrop flows with affine cost functions with stochastic offsets. In contrast, we show that in this case, the signaling problem is easy for many networks. First, we tightly characterize the class of single-commodity networks in which full information revelation is always an optimal signaling strategy. Second, we construct a reduction from optimal signaling to computing an optimal collection of active edges for the Wardrop equilibrium. For two states, this allows us to efficiently compute an optimal signaling scheme whenever the number of different active edge sets is bounded by a polynomial in the input size. Using a cell decomposition technique, we extend the approach to a polynomial-time algorithm for multicommodity parallel-edge networks provided that both the number of commodities and states are constant. Funding: This work was supported by Deutsche Forschungsgemeinschaft EXC-2046/1 [Grant 390685689] and Ho 3831/9-1 [Grant 514505843].
The restarted primal-dual hybrid gradient method (rPDHG) is a first-order method recently known for its computational effectiveness in solving linear programming (LP) problems. Despite its impressive practical performance, the theoretical iteration bounds for rPDHG can be exponentially poor. To investigate this gap from a probabilistic perspective, we show that rPDHG achieves polynomial-time complexity in a high-probability sense, under assumptions on the probability distribution from which the data instance is generated. We consider both Gaussian and more general sub-Gaussian distribution models. For standard-form LP instances with m constraints and n variables, our bounds take a particularly simple form when m is not too close to n: rPDHG iterates settle on the optimal basis in [Formula: see text] iterations, followed by [Formula: see text] iterations to compute an [Formula: see text]-optimal solution, with probability at least [Formula: see text] for [Formula: see text] that is not exponentially small. The Stage I bound further improves to [Formula: see text] in the Gaussian distribution model. Experimental results confirm the tail behavior and the polynomial-time dependence on problem dimensions of the iteration counts. As an application of our probabilistic analysis, we explore how the disparity among the components of the optimal solution bears on the performance of rPDHG, and we provide guidelines for generating challenging LP instances. Funding: Part of this work was performed at the Massachusetts Institute of Technology and was supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0356]. Another part of this work was performed at the Georgia Institute of Technology and was supported by the Office of Naval Research [Award N00014-25-1-2088].
