We study minimax optimization problems defined over infinite-dimensional function classes. In particular, we restrict the functions to the class of overparameterized two-layer neural networks and study (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural network. As an initial step, we consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space. For this problem, we establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $\mathcal{O}(T^{-1} + \alpha^{-1} ) $ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural network. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $\mathcal{O}(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, and asset pricing.
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This manuscript presents a novel method for discovering effective connectivity between specified pairs of nodes in a high-dimensional network of time series. To accurately perform Granger causality analysis from the first node to the second node, it is essential to eliminate the influence of all other nodes within the network. The approach proposed is to create a low-dimensional representation of all other nodes in the network using frequency-domain-based dynamic principal component analysis (spectral DPCA). The resulting scores are subsequently removed from the first and second nodes of interest, thus eliminating the confounding effect of other nodes within the high-dimensional network. To conduct hypothesis testing on Granger causality, we propose a permutation-based causality test. This test enhances the accuracy of our findings when the error structures are non-Gaussian. The approach has been validated in extensive simulation studies, which demonstrate the efficacy of the methodology as a tool for causality analysis in complex time series networks. The proposed methodology has also been demonstrated to be both expedient and viable on real datasets, with particular success observed on multichannel EEG networks.
In this work, we introduce LazyBoE, a multi-query method for kinodynamic motion planning with forward propagation. This algorithm allows for the simultaneous exploration of a robot's state and control spaces, thereby enabling a wider suite of dynamic tasks in real-world applications. Our contributions are three-fold: i) a method for discretizing the state and control spaces to amortize planning times across multiple queries; ii) lazy approaches to collision checking and propagation of control sequences that decrease the cost of physics-based simulation; and iii) LazyBoE, a robust kinodynamic planner that leverages these two contributions to produce dynamically-feasible trajectories. The proposed framework not only reduces planning time but also increases success rate in comparison to previous approaches.
In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay $-\omega<0$ for any $\omega>0$. The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay $-\omega+\epsilon$ for any $\epsilon>0$. The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.
Distributed and federated learning algorithms and techniques associated primarily with minimization problems. However, with the increase of minimax optimization and variational inequality problems in machine learning, the necessity of designing efficient distributed/federated learning approaches for these problems is becoming more apparent. In this paper, we provide a unified convergence analysis of communication-efficient local training methods for distributed variational inequality problems (VIPs). Our approach is based on a general key assumption on the stochastic estimates that allows us to propose and analyze several novel local training algorithms under a single framework for solving a class of structured non-monotone VIPs. We present the first local gradient descent-accent algorithms with provable improved communication complexity for solving distributed variational inequalities on heterogeneous data. The general algorithmic framework recovers state-of-the-art algorithms and their sharp convergence guarantees when the setting is specialized to minimization or minimax optimization problems. Finally, we demonstrate the strong performance of the proposed algorithms compared to state-of-the-art methods when solving federated minimax optimization problems.
We study offline reinforcement learning (RL) with linear MDPs under the infinite-horizon discounted setting which aims to learn a policy that maximizes the expected discounted cumulative reward using a pre-collected dataset. Existing algorithms for this setting either require a uniform data coverage assumptions or are computationally inefficient for finding an $\epsilon$-optimal policy with $O(\epsilon^{-2})$ sample complexity. In this paper, we propose a primal dual algorithm for offline RL with linear MDPs in the infinite-horizon discounted setting. Our algorithm is the first computationally efficient algorithm in this setting that achieves sample complexity of $O(\epsilon^{-2})$ with partial data coverage assumption. Our work is an improvement upon a recent work that requires $O(\epsilon^{-4})$ samples. Moreover, we extend our algorithm to work in the offline constrained RL setting that enforces constraints on additional reward signals.
Safe deployment of AI models requires proactive detection of failures to prevent costly errors. To this end, we study the important problem of detecting failures in deep regression models. Existing approaches rely on epistemic uncertainty estimates or inconsistency w.r.t the training data to identify failure. Interestingly, we find that while uncertainties are necessary they are insufficient to accurately characterize failure in practice. Hence, we introduce PAGER (Principled Analysis of Generalization Errors in Regressors), a framework to systematically detect and characterize failures in deep regressors. Built upon the principle of anchored training in deep models, PAGER unifies both epistemic uncertainty and complementary manifold non-conformity scores to accurately organize samples into different risk regimes.
We study differentially private (DP) mean estimation in the case where each person holds multiple samples. Commonly referred to as the "user-level" setting, DP here requires the usual notion of distributional stability when all of a person's datapoints can be modified. Informally, if $n$ people each have $m$ samples from an unknown $d$-dimensional distribution with bounded $k$-th moments, we show that \[n = \tilde \Theta\left(\frac{d}{\alpha^2 m} + \frac{d }{ \alpha m^{1/2} \varepsilon} + \frac{d}{\alpha^{k/(k-1)} m \varepsilon} + \frac{d}{\varepsilon}\right)\] people are necessary and sufficient to estimate the mean up to distance $\alpha$ in $\ell_2$-norm under $\varepsilon$-differential privacy (and its common relaxations). In the multivariate setting, we give computationally efficient algorithms under approximate DP (with slightly degraded sample complexity) and computationally inefficient algorithms under pure DP, and our nearly matching lower bounds hold for the most permissive case of approximate DP. Our computationally efficient estimators are based on the well known noisy-clipped-mean approach, but the analysis for our setting requires new bounds on the tails of sums of independent, vector-valued, bounded-moments random variables, and a new argument for bounding the bias introduced by clipping.
In the post-deep learning era, the Transformer architecture has demonstrated its powerful performance across pre-trained big models and various downstream tasks. However, the enormous computational demands of this architecture have deterred many researchers. To further reduce the complexity of attention models, numerous efforts have been made to design more efficient methods. Among them, the State Space Model (SSM), as a possible replacement for the self-attention based Transformer model, has drawn more and more attention in recent years. In this paper, we give the first comprehensive review of these works and also provide experimental comparisons and analysis to better demonstrate the features and advantages of SSM. Specifically, we first give a detailed description of principles to help the readers quickly capture the key ideas of SSM. After that, we dive into the reviews of existing SSMs and their various applications, including natural language processing, computer vision, graph, multi-modal and multi-media, point cloud/event stream, time series data, and other domains. In addition, we give statistical comparisons and analysis of these models and hope it helps the readers to understand the effectiveness of different structures on various tasks. Then, we propose possible research points in this direction to better promote the development of the theoretical model and application of SSM. More related works will be continuously updated on the following GitHub: //github.com/Event-AHU/Mamba_State_Space_Model_Paper_List.
In pace with developments in the research field of artificial intelligence, knowledge graphs (KGs) have attracted a surge of interest from both academia and industry. As a representation of semantic relations between entities, KGs have proven to be particularly relevant for natural language processing (NLP), experiencing a rapid spread and wide adoption within recent years. Given the increasing amount of research work in this area, several KG-related approaches have been surveyed in the NLP research community. However, a comprehensive study that categorizes established topics and reviews the maturity of individual research streams remains absent to this day. Contributing to closing this gap, we systematically analyzed 507 papers from the literature on KGs in NLP. Our survey encompasses a multifaceted review of tasks, research types, and contributions. As a result, we present a structured overview of the research landscape, provide a taxonomy of tasks, summarize our findings, and highlight directions for future work.
We study the problem of efficient semantic segmentation for large-scale 3D point clouds. By relying on expensive sampling techniques or computationally heavy pre/post-processing steps, most existing approaches are only able to be trained and operate over small-scale point clouds. In this paper, we introduce RandLA-Net, an efficient and lightweight neural architecture to directly infer per-point semantics for large-scale point clouds. The key to our approach is to use random point sampling instead of more complex point selection approaches. Although remarkably computation and memory efficient, random sampling can discard key features by chance. To overcome this, we introduce a novel local feature aggregation module to progressively increase the receptive field for each 3D point, thereby effectively preserving geometric details. Extensive experiments show that our RandLA-Net can process 1 million points in a single pass with up to 200X faster than existing approaches. Moreover, our RandLA-Net clearly surpasses state-of-the-art approaches for semantic segmentation on two large-scale benchmarks Semantic3D and SemanticKITTI.
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