Theoretical Convergence Guarantees Research Articles

Scalable Clustering: Large Scale Unsupervised Learning of Gaussian Mixture Models with Outliers

Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. This paper introduces a provably robust clustering algorithm based on loss minimization that performs well on Gaussian mixture models with outliers. It provides theoretical guarantees that the algorithm obtains high accuracy with high probability under certain assumptions. Moreover, it can also be used as an initialization strategy for k-means clustering. Experiments on real-world large-scale datasets demonstrate the effectiveness of the algorithm when clustering a large number of clusters, and a k-means algorithm initialized by the algorithm outperforms many of the classic clustering methods in both speed and accuracy, while scaling well to large datasets such as ImageNet.

Distributed Nonconvex Optimization for Control of Water Networks with Time-coupling Constraints

AbstractIn this paper, we present a new control model for optimizing pressure and water quality operations in water distribution networks. Our formulation imposes a set of time-coupling constraints to manage temporal pressure variations, which are exacerbated by the transition between pressure and water quality controls. The resulting optimization problem is a nonconvex, nonlinear program with nonseparable structure across time steps. This problem proves challenging for state-of-the-art nonlinear solvers, often precluding their direct use for near real-time control in large-scale networks. To overcome this computational burden, we investigate a distributed optimization approach based on the alternating direction method of multipliers (ADMM). In particular, we implement and evaluate two algorithms: a standard ADMM scheme and a two-level variant that provides theoretical convergence guarantees for our nonconvex problem. We use a benchmarking water network and a large-scale operational network in the UK for our numerical experiments. The results demonstrate good convergence behavior across all problem instances for the two-level algorithm, whereas the standard ADMM approach struggles to converge in some instances. With an appropriately tuned penalty parameter, however, both distributed algorithms yield good quality solutions and computational times compatible with near real-time (e.g. hourly) control requirements for large-scale water networks.

Decentralized Counterfactual Value with Threat Detection for Multi-Agent Reinforcement Learning in mixed cooperative and competitive environments

This paper proposes a fully decentralized approach to address the challenge of general mixed cooperation and competition within the domain of Multi-Agent Reinforcement Learning (MARL). Conventional MARL approaches do not achieve full decentralization as they necessitate either the communication of implicit information or the retention of a centralized critic, rendering them impractical in mixed cooperative and competitive environments. To address these challenges, this paper proposes a Decentralized Counterfactual Value (DCV) to model the behaviors of other agents and mitigate the non-stationary problem, accompanied by a Threat Detection (TD) mechanism to discern latent competitive or cooperative relationships. In addition, DCVTD is incorporated into both value-based and policy-based RL paradigms with theoretical convergence guarantee. Finally, empirical validation across four representative environments demonstrates the superior performance of DCVTD in terms of collective returns, computational efficiency, and agent scalability over other fully decentralized approaches, centralized training with decentralized execution approaches, and alternative approaches involving agent modeling or reward shaping in comprehensive experiments.

Cube is a good form: Hyperspectral band selection via multi-dimensional and high-order structure preserved clustering

As an effective strategy for reducing the noisy and redundant information for hyperspectral imagery (HSI), hyperspectral band selection intends to select a subset of original hyperspectral bands, which boosts the subsequent different tasks. In this paper, we introduce a multi-dimensional high-order structure preserved clustering method for hyperspectral band selection, referred to as MHSPC briefly. By regarding original hyperspectral images as a tensor cube, we apply the tensor CP (CANDECOMP/PARAFAC) decomposition on it to exploit the multi-dimensional structural information as well as generate a low-dimensional latent feature representation. In order to capture the local geometrical structure along the spectral dimension, a graph regularizer is imposed on the new feature representation in the lower dimensional space. In addition, since the low rankness of HSIs is an important global property, we utilize a nuclear norm constraint on the latent feature representation matrix to capture the global data structure information. Different to most of previous clustering based hyperspectral band selection methods which vectorize each band as a vector without considering the 2-D spatial information, the proposed MHSPC can effectively capture the spatial structure as well as the spectral correlation of original hyperspectral cube in both local and global perspectives. An efficient alternatively updating algorithm with theoretical convergence guarantee is designed to solve the resultant optimization problem, and extensive experimental results on four benchmark datasets validate the effectiveness of the proposed MHSPC over other state-of-the-arts.

Graph-based semi-supervised learning with non-convex graph total variation regularization

Graph total variation (GTV) is a widely employed regularization in graph-based semi-supervised learning (GSSL), which enforce the piece-wise smoothness of the label values concerning the underlying graph structure. However, due to its inherently biased estimation, GTV regularization tends to cause an underestimation of label values on boundary nodes . In this paper, we propose a novel GSSL model with non-convex GTV regularization. Specifically, we construct the non-convex GTV regularization by subtracting the generalized Moreau envelope from the original GTV regularization term, which can reduce the estimation bias and thus effectively estimate the label values of the boundary nodes. We demonstrate that under certain conditions of the non-convex control parameter, our proposed GSSL model maintains global convexity, thereby providing a theoretical guarantee for algorithm convergence. Additionally, we present an efficient alternating direction multiplier method (ADMM) to solve the proposed model. Finally, we validate the effectiveness on both synthetic data and background subtraction. The proposed GSSL model outperforms state-of-the-art model with a 17.97% improvement in SNR value on synthetic data, and increases average recall, average precision, and average F-measure by 4.92%, 1.19%, and 1.57% respectively on the pan–tilt-zoom(PTZ) challenge of background subtraction.

A novel framework for online supervised learning with feature selection

Current online learning methods suffer issues such as lower convergence rates and limited capability to select important features compared to their offline counterparts. In this paper, a novel framework for online learning based on running averages is proposed. Many popular offline regularised methods such as Lasso, Elastic Net, Minimax Concave Penalty (MCP), and Feature Selection with Annealing (FSA) have their online versions introduced in this framework. The equivalence between the proposed online methods and their offline counterparts is proved, and then novel theoretical true support recovery and convergence guarantees are provided for some of the methods in this framework. Numerical experiments indicate that the proposed methods enjoy high true support recovery accuracy and a faster convergence rate compared with conventional online and offline algorithms. Finally, applications to large datasets are presented, where again the proposed framework shows competitive results compared to popular online and offline algorithms.

Constrained Bayesian Optimization with Lower Confidence Bound

In this article, we present a hybrid Bayesian optimization (BO) framework to solve constrained optimization problems by adopting a state-of-the-art acquisition function from the unconstrained BO literature, the well-known lower confidence bound acquisition function and propose a novel variant that analyzes the feasible and infeasible regions which ensure the theoretical convergence guarantee. The proposed variant is compared with the existing state-of-the-art approaches in the constrained BO literature via implementing these approaches on six different problems, including black-box, classical engineering, and hyperparameter tuning problems. Further, we demonstrate the effectiveness of our approach through graphical and statistical testing.

A Generalized Neural Diffusion Framework on Graphs

Recent studies reveal the connection between GNNs and the diffusion process, which motivates many diffusion based GNNs to be proposed. However, since these two mechanisms are closely related, one fundamental question naturally arises: Is there a general diffusion framework that can formally unify these GNNs? The answer to this question can not only deepen our understanding of the learning process of GNNs, but also may open a new door to design a broad new class of GNNs. In this paper, we propose a general diffusion equation framework with the fidelity term, which formally establishes the relationship between the diffusion process with more GNNs. Meanwhile, with this framework, we identify one characteristic of graph diffusion networks, i.e., the current neural diffusion process only corresponds to the first-order diffusion equation. However, by an experimental investigation, we show that the labels of high-order neighbors actually appear monophily property, which induces the similarity based on labels among high-order neighbors without requiring the similarity among first-order neighbors. This discovery motives to design a new high-order neighbor-aware diffusion equation, and derive a new type of graph diffusion network (HiD-Net) based on the framework. With the high-order diffusion equation, HiD-Net is more robust against attacks and works on both homophily and heterophily graphs. We not only theoretically analyze the relation between HiD-Net with high-order random walk, but also provide a theoretical convergence guarantee. Extensive experimental results well demonstrate the effectiveness of HiD-Net over state-of-the-art graph diffusion networks.

Dual auto-weighted multi-view clustering via autoencoder-like nonnegative matrix factorization

Multi-view clustering (MVC) can exploit the complementary information among multi-view data to achieve the satisfactory performance, thus having extensive potentials for practical applications. Although Nonnegative Matrix Factorization (NMF) has emerged as an effective technique for MVC, the existing NMF-based methods still have two main limitations: 1) They solely focus on the reconstruction of original data, which can be regarded as the decoder of an autoencoder, while neglecting the low-dimensional representation learning. 2) They lack the ability to effectively capture both linear and nonlinear structures of data. To solve these problems, in this paper, we propose a Dual Auto-weighted multi-view clustering model based on Autoencoder-like NMF (DA2NMF), which enables a comprehensive exploration of both linear and nonlinear structures. Specifically, we establish an autoencoder-like NMF model that learns linear low-dimensional representations by integrating data reconstruction and representation learning within a unified framework. Moreover, the adaptive graph learning is introduced to explore the nonlinear structures in data. We further design a dual auto-weighted strategy to adaptively compute weights for different views and low-dimensional representations, thereby obtaining an enhanced consistent graph. An effective algorithm based on Multiplicative Update Rule (MUR) is developed to solve the DA2NMF with the theoretical convergence guarantee. Experimental results show that the proposed DA2NMF can effectively improve the clustering performance compared with the state-of-the-art MVC algorithms.

Unsupervised Multitarget Domain Adaptation With Dictionary-Bridged Knowledge Exploitation.

Unsupervised domain adaptation (UDA) is an emerging learning paradigm that models on unlabeled datasets by leveraging model knowledge built on other labeled datasets, in which the statistical distributions of these datasets are usually not identical. Formally, UDA is to leverage knowledge from a labeled source domain to promote an unlabeled target domain. Although there have been a variety of methods proposed to address the UDA problem, most of them are dedicated to single-source-to-single-target domain, while the works on single-source-to-multitarget domain are relatively rare. Compared to the single-source domain with single-target domain scenario, the UDA from single-source domain to multitarget domain is more challenging since it needs to consider not only the relationships between the source and the target domains but also those among the target domains. To this end, this article proposes a kind of dictionary learning-based unsupervised multitarget domain adaptation method (DL-UMTDA). In DL-UMTDA, a common dictionary is constructed to correlate the single-source and multitarget domains, while individual dictionaries are designed to exploit the private knowledge for the target domains. Through learning the corresponding dictionary representation coefficients in the UDA process, the correlations from the source to the target domains as well as these potential relationships between the target domains can be effectively exploited. In addition, we design an alternating algorithm to solve the DL-UMTDA model with theoretical convergence guarantee. Finally, extensive experiments on benchmark (Office + Caltech) and real datasets (AgeDB, Morph, and CACD) validate the superiority of the proposed method.

Variance Reduced Domain Randomization for Reinforcement Learning With Policy Gradient.

By introducing randomness on the environments, domain randomization (DR) imposes diversity to the policy training of deep reinforcement learning, and thus improves its capability of generalization. The randomization of environments, however, introduces another source of variability for the estimate of policy gradients, in addition to the already high variance incurred by trajectory sampling. Therefore, with standard state-dependent baselines, the policy gradient methods may still suffer high variance, causing a low sample efficiency during the training of DR. In this paper, we theoretically derive a bias-free and state/environment-dependent optimal baseline for DR, and analytically show its ability to achieve further variance reduction over the standard constant and state-dependent baselines for DR. Based on our theory, we further propose a variance reduced domain randomization (VRDR) approach for policy gradient methods, to strike a tradeoff between the variance reduction and computational complexity for the practical implementation. By dividing the entire space of environments into some subspaces and then estimating the state/subspace-dependent baseline, VRDR enjoys a theoretical guarantee of variance reduction and faster convergence than the state-dependent baselines. Empirical evaluations on six robot control tasks with randomized dynamics demonstrate that VRDR not only accelerates the convergence of policy training, but can consistently achieve a better eventual policy with improved training stability.

Disturbance feedback-based model predictive control in uncertain dynamic environments

This paper presents a robust MPC scheme for linear systems subject to time-varying, uncertain constraints that arise from uncertain environments. The predicted input sequence is parameterized over future environment states to guarantee constraint satisfaction despite an imprecise environment prediction and unknown evolution of the future constraints. We provide theoretical guarantees for recursive feasibility and asymptotic convergence. Finally a brief simulation example showcases our results.

Multiple Riemannian Kernel Hashing for Large-Scale Image Set Classification and Retrieval.

Conventional image set methods typically learn from small to medium-sized image set datasets. However, when applied to large-scale image set applications such as classification and retrieval, they face two primary challenges: 1) effectively modeling complex image sets; and 2) efficiently performing tasks. To address the above issues, we propose a novel Multiple Riemannian Kernel Hashing (MRKH) method that leverages the powerful capabilities of Riemannian manifold and Hashing on effective and efficient image set representation. MRKH considers multiple heterogeneous Riemannian manifolds to represent each image set. It introduces a multiple kernel learning framework designed to effectively combine statistics from multiple manifolds, and constructs kernels by selecting a small set of anchor points, enabling efficient scalability for large-scale applications. In addition, MRKH further exploits inter- and intra-modal semantic structure to enhance discrimination. Instead of employing continuous feature to represent each image set, MRKH suggests learning hash code for each image set, thereby achieving efficient computation and storage. We present an iterative algorithm with theoretical convergence guarantee to optimize MRKH, and the computational complexity is linear with the size of dataset. Extensive experiments on five image set benchmark datasets including three large-scale ones demonstrate the proposed method outperforms state-of-the-arts in accuracy and efficiency particularly in large-scale image set classification and retrieval.

Feedback linearization control for uncertain nonlinear systems via generative adversarial networks

This article presents a novel approach to leverage generative adversarial networks(GANs) techniques to learn a feedback linearization controller(FLC) for a class of uncertain nonlinear systems. By estimating uncertainty through the adversarial process, where ground truth samples are exclusively obtained from a predefined integral model, the feedback linearization controller, learned through a minimax two-player optimization framework, enhances the reference tracking performance of the input-output uncertain nonlinear system. Furthermore, we provide theoretical guarantee of convergence and stability, demonstrating the safe recovery of robust FLC. We also address the common challenge of mode collapse in GANs training through the strict convexity of our synthesized generator structure and an enhanced adversarial loss. Comprehensive simulations and practical experiments are conducted to underscore the superiority and efficacy of our proposed approach.

Randomized low rank approximation for nonnegative pure quaternion matrices

Pure quaternion matrix has been widely employed to represent data in real-world applications, such as colour images. Additionally, the imaginary part of quaternion matrix is usually nonnegative due to the natural nonnegativity of real-world data. In this paper, we propose an alternating projection-based algorithm for low rank nonnegative pure quaternion matrix approximation, which can exactly calculate the optimal fixed rank approximation while preserve the pure and nonnegative properties from the given data. More concretely, the proposed algorithm alternatively projects the given quaternion matrix onto the fixed rank quaternion matrix set and nonnegative pure quaternion matrix set in an iterative fashion. Moreover, we establish the theoretical convergence guarantee of the proposed algorithm. To extend the proposed algorithm to large-scaled data, we further propose a randomized algorithm with significant lower computational complexity and comparable accuracy. Numerical experiments on colour images show that our algorithms outperform the other state-of-the-art algorithms.

Image reconstruction through compressive sampling matching pursuit and curvelet transform

An interesting area of research is image reconstruction, which uses algorithms and techniques to transform a degraded image into a good one. The quality of the reconstructed image plays a vital role in the field of image processing. Compressive Sampling is an innovative and rapidly growing method for reconstructing signals. It is extensively used in image reconstruction. The literature uses a variety of matching pursuits for image reconstruction. In this paper, we propose a modified method named compressive sampling matching pursuit (CoSaMP) for image reconstruction that promises to sample sparse signals from far fewer observations than the signal’s dimension. The main advantage of CoSaMP is that it has an excellent theoretical guarantee for convergence. The proposed technique combines CoSaMP with curvelet transform for better reconstruction of image. Experiments are carried out to evaluate the proposed technique on different test images. The results indicate that qualitative and quantitative performance is better compared to existing methods.

Global and local similarity learning in multi-kernel space for nonnegative matrix factorization

Most of existing nonnegative matrix factorization (NMF) methods do not fully exploit global and local similarity information from data. In this paper, we propose a novel local similarity learning approach in the convex NMF framework, which encourages inter-class separability that is desired for clustering. Thus, the new model is capable of enhancing intra-class similarity and inter-class separability with simultaneous global and local learning. Moreover, the model learns the factor matrices in an augmented kernel space, which is a convex combination of pre-defined kernels with auto-learned weights. Thus, the learnings of cluster structure, representation factor matrix, and the optimal kernel mutually enhance each other in a seamlessly integrated model, which leads to informative representation. Multiplicative updating rules are developed with theoretical convergence guarantee. Extensive experimental results have confirmed the effectiveness of the proposed model.

Deep Reinforcement Learning‐Based Air Defense Decision‐Making Using Potential Games

This study addresses the challenge of intelligent decision‐making for command‐and‐control systems in air defense combat operations. Current autonomous decision‐making systems suffer from limited rationality and insufficient intelligence during operation processes. Recent studies have proposed methods based on deep reinforcement learning (DRL) to address these issues. However, DRL methods typically face challenges related to weak interpretability, lack of convergence guarantees, and high‐computing power requirements. To address these issues, a novel technique for large‐scale air defense decision‐making by combining a DRL technique with game theory is discussed. The proposed method transforms the target assignment problem into a potential game that provides theoretical guarantees for Nash equilibrium (NE) from a distributed perspective. The air‐defense decision problem is decomposed into separate target selection and target assignment problems. A DRL method is used to solve the target selection problem, while the target assignment problem is translated into a target assignment optimization game. This game is proven to be an exact potential game with theoretical convergence guarantees for an NE. Having simulated the proposed decision‐making method using a digital battlefield environment, the effectiveness of the proposed method is demonstrated.

A Derivative-Incorporated Adaptive Gradient Method for Federated Learning

As a new machine learning technique, federated learning has received more attention in recent years, which enables decentralized model training across data silos or edge intelligent devices in the Internet of Things without exchanging local raw data. All kinds of algorithms are proposed to solve the challenges in federated learning. However, most of these methods are based on stochastic gradient descent, which undergoes slow convergence and unstable performance during the training stage. In this paper, we propose a differential adaptive federated optimization method, which incorporates an adaptive learning rate and the gradient difference into the iteration rule of the global model. We further adopt the first-order moment estimation to compute the approximate value of the differential term so as to avoid amplifying the random noise from the input data sample. The theoretical convergence guarantee is established for our proposed method in a stochastic non-convex setting under full client participation and partial client participation cases. Experiments for the image classification task are performed on two standard datasets by training a neural network model, and experiment results on different baselines demonstrate the effectiveness of our proposed method.

Faster Convergence in Deep-Predictive-Coding Networks to Learn Deeper Representations.

Deep-predictive-coding networks (DPCNs) are hierarchical, generative models. They rely on feed-forward and feedback connections to modulate latent feature representations of stimuli in a dynamic and context-sensitive manner. A crucial element of DPCNs is a forward-backward inference procedure to uncover sparse, invariant features. However, this inference is a major computational bottleneck. It severely limits the network depth due to learning stagnation. Here, we prove why this bottleneck occurs. We then propose a new forward-inference strategy based on accelerated proximal gradients. This strategy has faster theoretical convergence guarantees than the one used for DPCNs. It overcomes learning stagnation. We also demonstrate that it permits constructing deep and wide predictive-coding networks. Such convolutional networks implement receptive fields that capture well the entire classes of objects on which the networks are trained. This improves the feature representations compared with our lab's previous nonconvolutional and convolutional DPCNs. It yields unsupervised object recognition that surpass convolutional autoencoders and is on par with convolutional networks trained in a supervised manner.