This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.
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We employ neural networks to tackle inverse partial differential equations on discretized Riemann surfaces with boundary. To this end, we introduce the concept of a graph with boundary which models these surfaces in a natural way. Our method uses a message passing technique to keep track of an unknown differential operator while using neural ODE solvers through the method of lines to capture the evolution in time. As training data, we use noisy and incomplete observations of sheaves on graphs at various timestamps. The novelty of this approach is in working with manifolds with nontrivial topology and utilizing the data on the graph boundary through a teacher forcing technique. Despite the increasing interest in learning dynamical systems from finite observations, many current methods are limited in two general ways: first, they work with topologically trivial spaces, and second, they fail to handle the boundary data on the ground space in a systematic way. The present work is an attempt at addressing these limitations. We run experiments with synthetic data of linear and nonlinear diffusion systems on orientable surfaces with positive genus and boundary, and moreover, provide evidences for improvements upon the existing paradigms.
Off-policy learning is a framework for optimizing policies without deploying them, using data collected by another policy. In recommender systems, this is especially challenging due to the imbalance in logged data: some items are recommended and thus logged much more frequently than others. This is further perpetuated when recommending a list of items, as the action space is combinatorial. To address this challenge, we study pessimistic off-policy optimization for learning to rank. The key idea is to compute lower confidence bounds on parameters of click models and then return the list with the highest pessimistic estimate of its value. This approach is computationally efficient and we analyze it. We study its Bayesian and frequentist variants, and overcome the limitation of unknown prior by incorporating empirical Bayes. To show the empirical effectiveness of our approach, we compare it to off-policy optimizers that use inverse propensity scores or neglect uncertainty. Our approach outperforms all baselines, is robust, and is also general.
This paper investigates the problem of regret minimization in linear time-varying (LTV) dynamical systems. Due to the simultaneous presence of uncertainty and non-stationarity, designing online control algorithms for unknown LTV systems remains a challenging task. At a cost of NP-hard offline planning, prior works have introduced online convex optimization algorithms, although they suffer from nonparametric rate of regret. In this paper, we propose the first computationally tractable online algorithm with regret guarantees that avoids offline planning over the state linear feedback policies. Our algorithm is based on the optimism in the face of uncertainty (OFU) principle in which we optimistically select the best model in a high confidence region. Our algorithm is then more explorative when compared to previous approaches. To overcome non-stationarity, we propose either a restarting strategy (R-OFU) or a sliding window (SW-OFU) strategy. With proper configuration, our algorithm is attains sublinear regret $O(T^{2/3})$. These algorithms utilize data from the current phase for tracking variations on the system dynamics. We corroborate our theoretical findings with numerical experiments, which highlight the effectiveness of our methods. To the best of our knowledge, our study establishes the first model-based online algorithm with regret guarantees under LTV dynamical systems.
With the rapid development of connecting massive devices to the Internet, especially for remote areas without cellular network infrastructures, space-air-ground integrated networks (SAGINs) emerge and offload computation-intensive tasks. In this paper, we consider a SAGIN, where multiple low-earth-orbit (LEO) satellites providing connections to the cloud server, an unmanned aerial vehicle (UAV), and nearby base stations (BSs) providing edge computing services are included. The UAV flies along a fixed trajectory to collect tasks generated by Internet of Things (IoT) devices, and forwards these tasks to a BS or the cloud server for further processing. To facilitate efficient processing, the UAV needs to decide where to offload as well as the proportion of offloaded tasks. However, in practice, due to the variability of environment and actual demand, the amount of arrival tasks is uncertain. If the deterministic optimization is utilized to develop offloading strategy, unnecessary system overhead or higher task drop rate may occur, which severely damages the system robustness. To address this issue, we characterize the uncertainty with a data-driven approach, and formulate a distributionally robust optimization problem to minimize the expected energy-constrained system latency under the worst-case probability distribution. Furthermore, the distributionally robust latency optimization algorithm is proposed to reach the suboptimal solution. Finally, we perform simulations on the realworld data set, and compare with other benchmark schemes to verify the efficiency and robustness of our proposed algorithm.
In this paper, we propose a novel uniform generalization bound on the time and inverse temperature for stochastic gradient Langevin dynamics (SGLD) in a non-convex setting. While previous works derive their generalization bounds by uniform stability, we use Rademacher complexity to make our generalization bound independent of the time and inverse temperature. Using Rademacher complexity, we can reduce the problem to derive a generalization bound on the whole space to that on a bounded region and therefore can remove the effect of the time and inverse temperature from our generalization bound. As an application of our generalization bound, an evaluation on the effectiveness of the simulated annealing in a non-convex setting is also described. For the sample size $n$ and time $s$, we derive evaluations with orders $\sqrt{n^{-1} \log (n+1)}$ and $|(\log)^4(s)|^{-1}$, respectively. Here, $(\log)^4$ denotes the $4$ times composition of the logarithmic function.
This paper studies an intriguing phenomenon related to the good generalization performance of estimators obtained by using large learning rates within gradient descent algorithms. First observed in the deep learning literature, we show that a phenomenon can be precisely characterized in the context of kernel methods, even though the resulting optimization problem is convex. Specifically, we consider the minimization of a quadratic objective in a separable Hilbert space, and show that with early stopping, the choice of learning rate influences the spectral decomposition of the obtained solution on the Hessian's eigenvectors. This extends an intuition described by Nakkiran (2020) on a two-dimensional toy problem to realistic learning scenarios such as kernel ridge regression. While large learning rates may be proven beneficial as soon as there is a mismatch between the train and test objectives, we further explain why it already occurs in classification tasks without assuming any particular mismatch between train and test data distributions.
In federated learning, participating clients typically possess non-i.i.d. data, posing a significant challenge to generalization to unseen distributions. To address this, we propose a Wasserstein distributionally robust optimization scheme called WAFL. Leveraging its duality, we frame WAFL as an empirical surrogate risk minimization problem, and solve it using a local SGD-based algorithm with convergence guarantees. We show that the robustness of WAFL is more general than related approaches, and the generalization bound is robust to all adversarial distributions inside the Wasserstein ball (ambiguity set). Since the center location and radius of the Wasserstein ball can be suitably modified, WAFL shows its applicability not only in robustness but also in domain adaptation. Through empirical evaluation, we demonstrate that WAFL generalizes better than the vanilla FedAvg in non-i.i.d. settings, and is more robust than other related methods in distribution shift settings. Further, using benchmark datasets we show that WAFL is capable of generalizing to unseen target domains.
We consider the problem of controlling an unknown linear dynamical system under adversarially changing convex costs and full feedback of both the state and cost function. We present the first computationally-efficient algorithm that attains an optimal $\smash{\sqrt{T}}$-regret rate compared to the best stabilizing linear controller in hindsight, while avoiding stringent assumptions on the costs such as strong convexity. Our approach is based on a careful design of non-convex lower confidence bounds for the online costs, and uses a novel technique for computationally-efficient regret minimization of these bounds that leverages their particular non-convex structure.
Trajectory Distribution Control for Model Predictive Path Integral Control using Covariance Steering
This paper presents a novel control approach for autonomous systems operating under uncertainty. We combine Model Predictive Path Integral (MPPI) control with Covariance Steering (CS) theory to obtain a robust controller for general nonlinear systems. The proposed Covariance-Controlled Model Predictive Path Integral (CC-MPPI) controller addresses the performance degradation observed in some MPPI implementations owing to unexpected disturbances and uncertainties. Namely, in cases where the environment changes too fast or the simulated dynamics during the MPPI rollouts do not capture the noise and uncertainty in the actual dynamics, the baseline MPPI implementation may lead to divergence. The proposed CC-MPPI controller avoids divergence by controlling the dispersion of the rollout trajectories at the end of the prediction horizon. Furthermore, the CC-MPPI has adjustable trajectory sampling distributions that can be changed according to the environment to achieve efficient sampling. Numerical examples using a ground vehicle navigating in challenging environments demonstrate the proposed approach.
In this paper, we study a sequential decision-making problem, called Adaptive Sampling for Discovery (ASD). Starting with a large unlabeled dataset, algorithms for ASD adaptively label the points with the goal to maximize the sum of responses. This problem has wide applications to real-world discovery problems, for example drug discovery with the help of machine learning models. ASD algorithms face the well-known exploration-exploitation dilemma. The algorithm needs to choose points that yield information to improve model estimates but it also needs to exploit the model. We rigorously formulate the problem and propose a general information-directed sampling (IDS) algorithm. We provide theoretical guarantees for the performance of IDS in linear, graph and low-rank models. The benefits of IDS are shown in both simulation experiments and real-data experiments for discovering chemical reaction conditions.
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