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Vyacheslav Kungurtsev

Publications and source records attributed to Vyacheslav Kungurtsev.

90 records · Page 5Linked to original sources
arXiv2020

Decentralized Langevin Dynamics

Langevin MCMC gradient optimization is a class of increasingly popular methods for estimating a posterior distribution. This paper addresses the algorithm as applied in a decentralized setting, wherein data is distributed across a network of agents which act to cooperatively solve the problem using peer-to-peer gossip communication. We show, theoretically, results in 1) the time-complexity to $ε$-consensus for the continuous time stochastic differential equation, 2) convergence rate in $L^2$ norm to consensus for the discrete implementation as defined by the Euler-Maruyama discretization and 3) convergence rate in the Wasserstein metric to the optimal stationary distribution for the discretized dynamics.

math.OC↗
arXiv2020

A Two-Step Pre-Processing for Semidefinite Programming

In semidefinite programming (SDP), a number of pre-processing techniques have been developed including chordal-completion procedures, which reduce the dimension of individual constraints by exploiting sparsity therein, and facial reduction, which reduces the dimension of the problem by removing redundant rows and columns. This paper suggest that these work in a complementary manner and that facial reduction should be used after chordal-completion procedures. In computational experiments on SDP instances from the SDPLib, a benchmark, and structured instances from polynomial and binary quadratic optimisation, we show that such two-step pre-processing with a standard interior-point method outperforms the interior point method, with or without the traditional pre-processing.

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arXiv2020

Asynchronous Optimization Methods for Efficient Training of Deep Neural Networks with Guarantees

Asynchronous distributed algorithms are a popular way to reduce synchronization costs in large-scale optimization, and in particular for neural network training. However, for nonsmooth and nonconvex objectives, few convergence guarantees exist beyond cases where closed-form proximal operator solutions are available. As most popular contemporary deep neural networks lead to nonsmooth and nonconvex objectives, there is now a pressing need for such convergence guarantees. In this paper, we analyze for the first time the convergence of stochastic asynchronous optimization for this general class of objectives. In particular, we focus on stochastic subgradient methods allowing for block variable partitioning, where the shared-memory-based model is asynchronously updated by concurrent processes. To this end, we first introduce a probabilistic model which captures key features of real asynchronous scheduling between concurrent processes; under this model, we establish convergence with probability one to an invariant set for stochastic subgradient methods with momentum. From the practical perspective, one issue with the family of methods we consider is that it is not efficiently supported by machine learning frameworks, as they mostly focus on distributed data-parallel strategies. To address this, we propose a new implementation strategy for shared-memory based training of deep neural networks, whereby concurrent parameter servers are utilized to train a partitioned but shared model in single- and multi-GPU settings. Based on this implementation, we achieve on average 1.2x speed-up in comparison to state-of-the-art training methods for popular image classification tasks without compromising accuracy.

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arXiv2020

Elastic Consistency: A General Consistency Model for Distributed Stochastic Gradient Descent

Machine learning has made tremendous progress in recent years, with models matching or even surpassing humans on a series of specialized tasks. One key element behind the progress of machine learning in recent years has been the ability to train machine learning models in large-scale distributed shared-memory and message-passing environments. Many of these models are trained employing variants of stochastic gradient descent (SGD) based optimization. In this paper, we introduce a general consistency condition covering communication-reduced and asynchronous distributed SGD implementations. Our framework, called elastic consistency enables us to derive convergence bounds for a variety of distributed SGD methods used in practice to train large-scale machine learning models. The proposed framework de-clutters the implementation-specific convergence analysis and provides an abstraction to derive convergence bounds. We utilize the framework to analyze a sparsification scheme for distributed SGD methods in an asynchronous setting for convex and non-convex objectives. We implement the distributed SGD variant to train deep CNN models in an asynchronous shared-memory setting. Empirical results show that error-feedback may not necessarily help in improving the convergence of sparsified asynchronous distributed SGD, which corroborates an insight suggested by our convergence analysis.

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arXiv2020

Stochastic Gradient Langevin with Delayed Gradients

Stochastic Gradient Langevin Dynamics (SGLD) ensures strong guarantees with regards to convergence in measure for sampling log-concave posterior distributions by adding noise to stochastic gradient iterates. Given the size of many practical problems, parallelizing across several asynchronously running processors is a popular strategy for reducing the end-to-end computation time of stochastic optimization algorithms. In this paper, we are the first to investigate the effect of asynchronous computation, in particular, the evaluation of stochastic Langevin gradients at delayed iterates, on the convergence in measure. For this, we exploit recent results modeling Langevin dynamics as solving a convex optimization problem on the space of measures. We show that the rate of convergence in measure is not significantly affected by the error caused by the delayed gradient information used for computation, suggesting significant potential for speedup in wall clock time. We confirm our theoretical results with numerical experiments on some practical problems.

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arXiv2020

Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity

We consider nonconvex constrained optimization problems and propose a new approach to the convergence analysis based on penalty functions. We make use of classical penalty functions in an unconventional way, in that penalty functions only enter in the theoretical analysis of convergence while the algorithm itself is penalty-free. Based on this idea, we are able to establish several new results, including the first general analysis for diminishing stepsize methods in nonconvex, constrained optimization, showing convergence to generalized stationary points, and a complexity study for SQP-type algorithms.

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arXiv2020

Second-order Guarantees of Distributed Gradient Algorithms

We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i) the renowned Distributed Gradient Descent (DGD) algorithm likely converges to a neighborhood of a Second-order Stationary (SoS) solution; and (ii) the more recent class of distributed algorithms based on gradient tracking--implementable also over digraphs--likely converges to exact SoS solutions, thus avoiding (strict) saddle-points. Furthermore, new convergence rate results to first-order critical points is established for the latter class of algorithms.

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arXiv2020

Reinforcement Learning Based on Real-Time Iteration NMPC

Reinforcement Learning (RL) has proven a stunning ability to learn optimal policies from data without any prior knowledge on the process. The main drawback of RL is that it is typically very difficult to guarantee stability and safety. On the other hand, Nonlinear Model Predictive Control (NMPC) is an advanced model-based control technique which does guarantee safety and stability, but only yields optimality for the nominal model. Therefore, it has been recently proposed to use NMPC as a function approximator within RL. While the ability of this approach to yield good performance has been demonstrated, the main drawback hindering its applicability is related to the computational burden of NMPC, which has to be solved to full convergence. In practice, however, computationally efficient algorithms such as the Real-Time Iteration (RTI) scheme are deployed in order to return an approximate NMPC solution in very short time. In this paper we bridge this gap by extending the existing theoretical framework to also cover RL based on RTI NMPC. We demonstrate the effectiveness of this new RL approach with a nontrivial example modeling a challenging nonlinear system subject to stochastic perturbations with the objective of optimizing an economic cost.

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arXiv2019

Distributed Stochastic Nonsmooth Nonconvex Optimization

Distributed consensus optimization has received considerable attention in recent years; several distributed consensus-based algorithms have been proposed for (nonsmooth) convex and (smooth) nonconvex objective functions. However, the behavior of these distributed algorithms on {\it nonconvex, nonsmooth and stochastic} objective functions is not understood. This class of functions and distributed setting are motivated by several applications, including problems in machine learning and signal processing. This paper presents the first convergence analysis of the decentralized stochastic subgradient method for such classes of problems, over networks modeled as undirected, fixed, graphs.

math.OC↗
arXiv2019

Lifted Weight Learning of Markov Logic Networks Revisited

We study lifted weight learning of Markov logic networks. We show that there is an algorithm for maximum-likelihood learning of 2-variable Markov logic networks which runs in time polynomial in the domain size. Our results are based on existing lifted-inference algorithms and recent algorithmic results on computing maximum entropy distributions.

cs.AI↗
arXiv2018

A Comparison of Nonsmooth, Nonconvex, Constrained Optimization Solvers for the Design of Time-Delay Compensators

We present a detailed set of performance comparisons of two state-of-the-art solvers for the application of designing time-delay compensators, an important problem in the field of robust control. Formulating such robust control mechanics as constrained optimization problems often involves objective and constraint functions that are both nonconvex and nonsmooth, both of which present significant challenges to many solvers and their end-users hoping to obtain good solutions to these problems. In our particular engineering task, the main difficulty in the optimization arises in a nonsmooth and nonconvex stability constraint, which states that the infinite spectrum of zeros of the so-called shaper should remain in the open left half-plane. To perform our evaluation, we make use $β$-relative minimization profiles, recently introduced visualization tools that are particularly suited for benchmarking solvers on nonsmooth, nonconvex, constrained optimization problems. Furthermore, we also introduce new visualization tools, called Global-Local Profiles, which for a given problem and a fixed computational budget, assess the tradeoffs of distributing the budget over few or many starting points, with the former getting more budget per point and latter less.

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arXiv2018

Algorithms for solving optimization problems arising from deep neural net models: smooth problems

Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the empirical risk is, however, highly nonlinear. This presents a challenge to application and development of appropriate optimization algorithms for solving the problem. In this paper, we summarize the primary challenges involved and present the case for a Newton-based method incorporating directions of negative curvature, including promising numerical results on data arising from security anomally deetection.

math.OC↗
arXiv2018

Algorithms for solving optimization problems arising from deep neural net models: nonsmooth problems

Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the empirical risk is, however, highly nonconvex. This alone presents a challenge to application and development of appropriate optimization algorithms for solving the problem. However, in addition, there are a number of interesting problems for which the objective function is non- smooth and nonseparable. In this paper, we summarize the primary challenges involved, the state of the art, and present some numerical results on an interesting and representative class of problems.

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arXiv2018

Asynchronous Parallel Algorithms for Nonconvex Optimization

We propose a new asynchronous parallel block-descent algorithmic framework for the minimization of the sum of a smooth nonconvex function and a nonsmooth convex one, subject to both convex and nonconvex constraints. The proposed framework hinges on successive convex approximation techniques and a novel probabilistic model that captures key elements of modern computational architectures and asynchronous implementations in a more faithful way than current state-of-the-art models. Other key features of the framework are: i) it covers in a unified way several specific solution methods; ii) it accommodates a variety of possible parallel computing architectures; and iii) it can deal with nonconvex constraints. Almost sure convergence to stationary solutions is proved, and theoretical complexity results are provided, showing nearly ideal linear speedup when the number of workers is not too large.

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arXiv2017

Asynchronous Parallel Algorithms for Nonconvex Big-Data Optimization. Part II: Complexity and Numerical Results

We present complexity and numerical results for a new asynchronous parallel algorithmic method for the minimization of the sum of a smooth nonconvex function and a convex nonsmooth regularizer, subject to both convex and nonconvex constraints. The proposed method hinges on successive convex approximation techniques and a novel probabilistic model that captures key elements of modern computational architectures and asynchronous implementations in a more faithful way than state-of-the-art models. In the companion paper we provided a detailed description on the probabilistic model and gave convergence results for a diminishing stepsize version of our method. Here, we provide theoretical complexity results for a fixed stepsize version of the method and report extensive numerical comparisons on both convex and nonconvex problems demonstrating the efficiency of our approach.

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arXiv2014

An Inequality Constrained SL/QP Method for Minimizing the Spectral Abscissa

We consider a problem in eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa - the value of a parameter that results in the smallest value of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control systems. Many systems require the spectra to lie in the left half plane in order for them to be stable. The optimization problem, however, is difficult to solve because the underlying objective function is nonconvex, nonsmooth, and non-Lipschitz. In addition, local minima tend to correspond to points of non-differentiability and locally non-Lipschitz behavior. We present a sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing the surfaces corresponding to the largest eigenvalues. We present numerical results comparing the algorithms to the state of the art.

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arXiv2014

Hybrid Random/Deterministic Parallel Algorithms for Nonconvex Big Data Optimization

We propose a decomposition framework for the parallel optimization of the sum of a differentiable {(possibly nonconvex)} function and a nonsmooth (possibly nonseparable), convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. The main contribution of this work is a novel \emph{parallel, hybrid random/deterministic} decomposition scheme wherein, at each iteration, a subset of (block) variables is updated at the same time by minimizing local convex approximations of the original nonconvex function. To tackle with huge-scale problems, the (block) variables to be updated are chosen according to a \emph{mixed random and deterministic} procedure, which captures the advantages of both pure deterministic and random update-based schemes. Almost sure convergence of the proposed scheme is established. Numerical results show that on huge-scale problems the proposed hybrid random/deterministic algorithm outperforms both random and deterministic schemes.

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