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In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

Non-rigid registration, which deforms a source shape in a non-rigid way to align with a target shape, is a classical problem in computer vision. Such problems can be challenging because of imperfect data (noise, outliers and partial overlap) and high degrees of freedom. Existing methods typically adopt the $\ell_{p}$ type robust norm to measure the alignment error and regularize the smoothness of deformation, and use a proximal algorithm to solve the resulting non-smooth optimization problem. However, the slow convergence of such algorithms limits their wide applications. In this paper, we propose a formulation for robust non-rigid registration based on a globally smooth robust norm for alignment and regularization, which can effectively handle outliers and partial overlaps. The problem is solved using the majorization-minimization algorithm, which reduces each iteration to a convex quadratic problem with a closed-form solution. We further apply Anderson acceleration to speed up the convergence of the solver, enabling the solver to run efficiently on devices with limited compute capability. Extensive experiments demonstrate the effectiveness of our method for non-rigid alignment between two shapes with outliers and partial overlaps, with quantitative evaluation showing that it outperforms state-of-the-art methods in terms of registration accuracy and computational speed. The source code is available at //github.com/yaoyx689/AMM_NRR.

In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leverage the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the distributed setting, this enables smaller rate-distortion terms which are shown to translate into tighter generalization bounds. The bounds are then applied to the distributed support vector machines (SVM), suggesting that the generalization error of the distributed setting decays faster than that of the centralized one with a factor of $\mathcal{O}(\log(K)/\sqrt{K})$. This finding is validated also experimentally. A similar conclusion is obtained for a multiple-round federated learning setup where each client uses stochastic gradient Langevin dynamics (SGLD).

In this paper, we consider distributed optimization problems where $n$ agents, each possessing a local cost function, collaboratively minimize the average of the local cost functions over a connected network. To solve the problem, we propose a distributed random reshuffling (D-RR) algorithm that invokes the random reshuffling (RR) update in each agent. We show that D-RR inherits favorable characteristics of RR for both smooth strongly convex and smooth nonconvex objective functions. In particular, for smooth strongly convex objective functions, D-RR achieves $\mathcal{O}(1/T^2)$ rate of convergence (where $T$ counts epoch number) in terms of the squared distance between the iterate and the global minimizer. When the objective function is assumed to be smooth nonconvex and has Lipschitz continuous component functions, we show that D-RR drives the squared norm of gradient to $0$ at a rate of $\mathcal{O}(1/T^{2/3})$. These convergence results match those of centralized RR (up to constant factors) and outperform the distributed stochastic gradient descent (DSGD) algorithm if we run a relatively large number of epochs. Finally, we conduct a set of numerical experiments to illustrate the efficiency of the proposed D-RR method on both strongly convex and nonconvex distributed optimization problems.

Gradient coding schemes effectively mitigate full stragglers in distributed learning by introducing identical redundancy in coded local partial derivatives corresponding to all model parameters. However, they are no longer effective for partial stragglers as they cannot utilize incomplete computation results from partial stragglers. This paper aims to design a new gradient coding scheme for mitigating partial stragglers in distributed learning. Specifically, we consider a distributed system consisting of one master and N workers, characterized by a general partial straggler model and focuses on solving a general large-scale machine learning problem with L model parameters using gradient coding. First, we propose a coordinate gradient coding scheme with L coding parameters representing L possibly different diversities for the L coordinates, which generates most gradient coding schemes. Then, we consider the minimization of the expected overall runtime and the maximization of the completion probability with respect to the L coding parameters for coordinates, which are challenging discrete optimization problems. To reduce computational complexity, we first transform each to an equivalent but much simpler discrete problem with N\llL variables representing the partition of the L coordinates into N blocks, each with identical redundancy. This indicates an equivalent but more easily implemented block coordinate gradient coding scheme with N coding parameters for blocks. Then, we adopt continuous relaxation to further reduce computational complexity. For the resulting minimization of expected overall runtime, we develop an iterative algorithm of computational complexity O(N^2) to obtain an optimal solution and derive two closed-form approximate solutions both with computational complexity O(N). For the resultant maximization of the completion probability, we develop an iterative algorithm of...

In deep neural networks, better results can often be obtained by increasing the complexity of previously developed basic models. However, it is unclear whether there is a way to boost performance by decreasing the complexity of such models. Intuitively, given a problem, a simpler data structure comes with a simpler algorithm. Here, we investigate the feasibility of improving graph classification performance while simplifying the learning process. Inspired by structural entropy on graphs, we transform the data sample from graphs to coding trees, which is a simpler but essential structure for graph data. Furthermore, we propose a novel message passing scheme, termed hierarchical reporting, in which features are transferred from leaf nodes to root nodes by following the hierarchical structure of coding trees. We then present a tree kernel and a convolutional network to implement our scheme for graph classification. With the designed message passing scheme, the tree kernel and convolutional network have a lower runtime complexity of $O(n)$ than Weisfeiler-Lehman subtree kernel and other graph neural networks of at least $O(hm)$. We empirically validate our methods with several graph classification benchmarks and demonstrate that they achieve better performance and lower computational consumption than competing approaches.

In this paper, a new weighted average estimator (WAVE) is proposed to enhance the performance of the simple-averaging based distributed estimator, under a general loss with a high dimensional parameter. To obtain an efficient estimator, a weighted least-square ensemble framework plus an adaptive $L_1$ penalty is proposed, in which the local estimator is estimated via the adaptive-lasso and the weight is inversely proportional to the variance of local estimators. It can be proved that WAVE enjoys the same asymptotic properties as the global estimator and simultaneously spend a very low communication cost, only requiring the local worker to deliver two vectors to the master. Moreover, it is shown that WAVE is effective even when the samples across local workers have different mean and covariance. In particular, the asymptotic normality is established under such conditions, while other competitors may not own this property. The effectiveness of WAVE is further illustrated by an extensive numerical study and a real data analysis.

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 this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an $\varepsilon$-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose an first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for finding an $\varepsilon$-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are parameter-free or almost parameter-free except that the knowledge on convexity parameter is required. To the best of our knowledge, no prior studies were conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are entirely new.

The aim of this work is to develop a fully-distributed algorithmic framework for training graph convolutional networks (GCNs). The proposed method is able to exploit the meaningful relational structure of the input data, which are collected by a set of agents that communicate over a sparse network topology. After formulating the centralized GCN training problem, we first show how to make inference in a distributed scenario where the underlying data graph is split among different agents. Then, we propose a distributed gradient descent procedure to solve the GCN training problem. The resulting model distributes computation along three lines: during inference, during back-propagation, and during optimization. Convergence to stationary solutions of the GCN training problem is also established under mild conditions. Finally, we propose an optimization criterion to design the communication topology between agents in order to match with the graph describing data relationships. A wide set of numerical results validate our proposal. To the best of our knowledge, this is the first work combining graph convolutional neural networks with distributed optimization.