The local convergence of alternating optimization methods with overrelaxation for low-rank matrix and tensor problems is established. The analysis is based on the linearization of the method which takes the form of an SOR iteration for a positive semidefinite Hessian and can be studied in the corresponding quotient geometry of equivalent low-rank representations. In the matrix case, the optimal relaxation parameter for accelerating the local convergence can be determined from the convergence rate of the standard method. This result relies on a version of Young's SOR theorem for positive semidefinite $2 \times 2$ block systems.
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Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated by power law distributions. In this paper we perform a systematic study of a range of classical single-step and multi-step first order optimization algorithms, with adaptive and non-adaptive, constant and non-constant learning rates: vanilla Gradient Descent, Steepest Descent, Heavy Ball, and Conjugate Gradients. For each of these, we prove that a power law spectral assumption entails a power law for convergence rate of the algorithm, with the convergence rate exponent given by a specific multiple of the spectral exponent. We establish both upper and lower bounds, showing that the results are tight. Finally, we demonstrate applications of these results to kernel learning and training of neural networks in the NTK regime.
In this paper, we aim at unifying, simplifying and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we then introduce a versatile generic scheme, which allows for the design and analysis of Faster LAGrangian (FLAG) methods with new provably sublinear rate of convergence expressed in terms of function values and feasibility violation of the original (non-ergodic) generated sequence. To demonstrate the power and versatility of our approach and results, we show that most well-known iconic Lagrangian-based schemes admit a nice primal algorithmic map, and hence share the new faster rate of convergence results within their corresponding FLAG.
LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation roundoff errors, which can lead solvers to return feasibility and optimality certificates that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are formally established through the derivation of theoretical insights. Their computational advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the proposed methodology is that the length of any coefficient calculated via the associated algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches.
Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train the denoiser as a gradient descent step on a functional parameterized by a deep neural network. Using such a denoiser guarantees the convergence of the PnP version of the Half-Quadratic-Splitting (PnP-HQS) iterative algorithm. In this paper, we show that this gradient denoiser can actually correspond to the proximal operator of another scalar function. Given this new result, we exploit the convergence theory of proximal algorithms in the nonconvex setting to obtain convergence results for PnP-PGD (Proximal Gradient Descent) and PnP-ADMM (Alternating Direction Method of Multipliers). When built on top of a smooth gradient denoiser, we show that PnP-PGD and PnP-ADMM are convergent and target stationary points of an explicit functional. These convergence results are confirmed with numerical experiments on deblurring, super-resolution and inpainting.
The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Mat\'ern covariance functions are essential. However, the Mat\'ern covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Mat\'ern covariance function instead. First, we show that this circular Mat\'ern function is the covariance of the stationary solution to the stochastic differential equation on the circle with a formally defined white noise space measure. Then, for the corresponding conditional autoregressive model, we derive a closed form formula for its covariance function. Together with a closed form formula for the circular Mat\'ern covariance function, the link between these two random fields can be established explicitly. Additionally, it is known that the estimator of the mean is not consistent on circles, we provide an equivalent Gaussian measure explanation for this non-ergodicity issue.
We propose the homotopic policy mirror descent (HPMD) method for solving discounted, infinite horizon MDPs with finite state and action space, and study its policy convergence. We report three properties that seem to be new in the literature of policy gradient methods: (1) HPMD exhibits global linear convergence of the value optimality gap, and local superlinear convergence of the policy to the set of optimal policies with order $\gamma^{-2}$. The superlinear convergence of the policy takes effect after no more than $\mathcal{O}(\log(1/\Delta^*))$ number of iterations, where $\Delta^*$ is defined via a gap quantity associated with the optimal state-action value function; (2) HPMD also exhibits last-iterate convergence of the policy, with the limiting policy corresponding exactly to the optimal policy with the maximal entropy for every state. No regularization is added to the optimization objective and hence the second observation arises solely as an algorithmic property of the homotopic policy gradient method. (3) For the stochastic HPMD method, we further demonstrate a better than $\mathcal{O}(|\mathcal{S}| |\mathcal{A}| / \epsilon^2)$ sample complexity for small optimality gap $\epsilon$, when assuming a generative model for policy evaluation.
Second-order methods have the capability of accelerating optimization by using much richer curvature information than first-order methods. However, most are impractical for deep learning, where the number of training parameters is huge. In Goldfarb et al. (2020), practical quasi-Newton methods were proposed that approximate the Hessian of a multilayer perceptron (MLP) model by a layer-wise block diagonal matrix where each layer's block is further approximated by a Kronecker product corresponding to the structure of the Hessian restricted to that layer. Here, we extend these methods to enable them to be applied to convolutional neural networks (CNNs), by analyzing the Kronecker-factored structure of the Hessian matrix of convolutional layers. Several improvements to the methods in Goldfarb et al. (2020) are also proposed that can be applied to both MLPs and CNNs. These new methods have memory requirements comparable to first-order methods and much less per-iteration time complexity than those in Goldfarb et al. (2020). Moreover, convergence results are proved for a variant under relatively mild conditions. Finally, we compared the performance of our new methods against several state-of-the-art (SOTA) methods on MLP autoencoder and CNN problems, and found that they outperformed the first-order SOTA methods and performed comparably to the second-order SOTA methods.
Second-order optimization methods are among the most widely used optimization approaches for convex optimization problems, and have recently been used to optimize non-convex optimization problems such as deep learning models. The widely used second-order optimization methods such as quasi-Newton methods generally provide curvature information by approximating the Hessian using the secant equation. However, the secant equation becomes insipid in approximating the Newton step owing to its use of the first-order derivatives. In this study, we propose an approximate Newton sketch-based stochastic optimization algorithm for large-scale empirical risk minimization. Specifically, we compute a partial column Hessian of size ($d\times m$) with $m\ll d$ randomly selected variables, then use the \emph{Nystr\"om method} to better approximate the full Hessian matrix. To further reduce the computational complexity per iteration, we directly compute the update step ($\Delta\boldsymbol{w}$) without computing and storing the full Hessian or its inverse. We then integrate our approximated Hessian with stochastic gradient descent and stochastic variance-reduced gradient methods. The results of numerical experiments on both convex and non-convex functions show that the proposed approach was able to obtain a better approximation of Newton\textquotesingle s method, exhibiting performance competitive with that of state-of-the-art first-order and stochastic quasi-Newton methods. Furthermore, we provide a theoretical convergence analysis for convex functions.
In this paper we provide a rigorous convergence analysis for the renowned Particle Swarm Optimization method using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish the convergence to a global minimizer in two steps. First, we prove the consensus formation of the dynamics by analyzing the time-evolution of the variance of the particle distribution. Consecutively, we show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. Our results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum are satisfied. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of the gradient flow for the MSE loss. Second, we provide a technical result (working also for general approximators) relating solutions of the aforementioned differential inclusion to the (discrete) stochastic gradient descent sequences, hence establishing linear convergence towards zero loss for the stochastic gradient descent iterations.
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