arXiv subjects

Wotao Yin

Publications and source records attributed to Wotao Yin.

At least 127 records · Page 7Linked to original sources

arXiv2018

Straggler Mitigation in Distributed Optimization Through Data Encoding

Slow running or straggler tasks can significantly reduce computation speed in distributed computation. Recently, coding-theory-inspired approaches have been applied to mitigate the effect of straggling, through embedding redundancy in certain linear computational steps of the optimization algorithm, thus completing the computation without waiting for the stragglers. In this paper, we propose an alternate approach where we embed the redundancy directly in the data itself, and allow the computation to proceed completely oblivious to encoding. We propose several encoding schemes, and demonstrate that popular batch algorithms, such as gradient descent and L-BFGS, applied in a coding-oblivious manner, deterministically achieve sample path linear convergence to an approximate solution of the original problem, using an arbitrarily varying subset of the nodes at each iteration. Moreover, this approximation can be controlled by the amount of redundancy and the number of nodes used in each iteration. We provide experimental results demonstrating the advantage of the approach over uncoded and data replication strategies.

stat.ML↗

arXiv2017

On the Convergence of Asynchronous Parallel Iteration with Unbounded Delays

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call delay, is the number of times it has been updated since its creation. Almost all recent works prove convergence under the assumption of a finite maximum delay and set their stepsize parameters accordingly. However, the maximum delay is practically unknown. This paper presents convergence analysis of an async-parallel method from a probabilistic viewpoint, and it allows for large unbounded delays. An explicit formula of stepsize that guarantees convergence is given depending on delays' statistics. With $p+1$ identical processors, we empirically measured that delays closely follow the Poisson distribution with parameter $p$, matching our theoretical model, and thus the stepsize can be set accordingly. Simulations on both convex and nonconvex optimization problems demonstrate the validness of our analysis and also show that the existing maximum-delay induced stepsize is too conservative, often slowing down the convergence of the algorithm.

math.OC↗

arXiv2017

Proximal-Proximal-Gradient Method

In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to minimization problems written as a sum of many differentiable and many non-differentiable convex functions. The non-differentiable functions can be coupled. We furthermore present a related stochastic variation, which we call stochastic PPG (S-PPG). S-PPG can be interpreted as a generalization of Finito and MISO over to the sum of many coupled non-differentiable convex functions. We present many applications that can benefit from PPG and S-PPG and prove convergence for both methods. A key strength of PPG and S-PPG is, compared to existing methods, its ability to directly handle a large sum of non-differentiable non-separable functions with a constant stepsize independent of the number of functions. Such non-diminishing stepsizes allows them to be fast.

math.OC↗

arXiv2017

A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs

In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge. Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. As a first-order method, the proposed algorithm relies on simple subroutines, and therefore is simple to implement and has low per-iteration cost.

math.OC↗

arXiv2017

Fast non-coplanar beam orientation optimization based on group sparsity

The selection of beam orientations, which is a key step in radiation treatment planning, is particularly challenging for non-coplanar radiotherapy systems due to the large number of candidate beams. In this paper, we report progress on the group sparsity approach to beam orientation optimization, wherein beam angles are selected by solving a large scale fluence map optimization problem with an additional group sparsity penalty term that encourages most candidate beams to be inactive. The optimization problem is solved using an accelerated proximal gradient method, the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). We derive a closed-form expression for a relevant proximal operator which enables the application of FISTA. The proposed algorithm is used to create non-coplanar treatment plans for four cases (including head and neck, lung, and prostate cases), and the resulting plans are compared with clinical plans. The dosimetric quality of the group sparsity treatment plans is superior to that of the clinical plans. Moreover, the runtime for the group sparsity approach is typically about 5 minutes. Problems of this size could not be handled using the previous group sparsity method for beam orientation optimization, which was slow to solve much smaller coplanar cases. This work demonstrates for the first time that the group sparsity approach, when combined with an accelerated proximal gradient method such as FISTA, works effectively for non-coplanar cases with 500-800 candidate beams.

physics.med-ph↗

arXiv2017

Online Convolutional Dictionary Learning

While a number of different algorithms have recently been proposed for convolutional dictionary learning, this remains an expensive problem. The single biggest impediment to learning from large training sets is the memory requirements, which grow at least linearly with the size of the training set since all existing methods are batch algorithms. The work reported here addresses this limitation by extending online dictionary learning ideas to the convolutional context.

cs.LG↗

arXiv2017

On Unbounded Delays in Asynchronous Parallel Fixed-Point Algorithms

The need for scalable numerical solutions has motivated the development of asynchronous parallel algorithms, where a set of nodes run in parallel with little or no synchronization, thus computing with delayed information. This paper studies the convergence of the asynchronous parallel algorithm ARock under potentially unbounded delays. ARock is a general asynchronous algorithm that has many applications. It parallelizes fixed-point iterations by letting a set of nodes randomly choose solution coordinates and update them in an asynchronous parallel fashion. ARock takes some recent asynchronous coordinate descent algorithms as special cases and gives rise to new asynchronous operator-splitting algorithms. Existing analysis of ARock assumes the delays to be bounded, and uses this bound to set a step size that is important to both convergence and efficiency. Other work, though allowing unbounded delays, imposes strict conditions on the underlying fixed-point operator, resulting in limited applications. In this paper, convergence is established under unbounded delays, which can be either stochastic or deterministic. The proposed step sizes are more practical and generally larger than those in the existing work. The step size adapts to the delay distribution or the current delay being experienced in the system. New Lyapunov functions, which are the key to analyzing asynchronous algorithms, are generated to obtain our results. A set of applicable optimization algorithms with large-scale applications are given, including machine learning and scientific computing algorithms.

math.OC↗

arXiv2017

More Iterations per Second, Same Quality -- Why Asynchronous Algorithms may Drastically Outperform Traditional Ones

In this paper, we consider the convergence of a very general asynchronous-parallel algorithm called ARock, that takes many well-known asynchronous algorithms as special cases (gradient descent, proximal gradient, Douglas Rachford, ADMM, etc.). In asynchronous-parallel algorithms, the computing nodes simply use the most recent information that they have access to, instead of waiting for a full update from all nodes in the system. This means that nodes do not have to waste time waiting for information, which can be a major bottleneck, especially in distributed systems. When the system has $p$ nodes, asynchronous algorithms may complete $Θ(\ln(p))$ more iterations than synchronous algorithms in a given time period ("more iterations per second"). Although asynchronous algorithms may compute more iterations per second, there is error associated with using outdated information. How many more iterations in total are needed to compensate for this error is still an open question. The main results of this paper aim to answer this question. We prove, loosely, that as the size of the problem becomes large, the number of additional iterations that asynchronous algorithms need becomes negligible compared to the total number ("same quality" of the iterations). Taking these facts together, our results provide solid evidence of the potential of asynchronous algorithms to vastly speed up certain distributed computations.

math.OC↗

arXiv2017

Asynchronous Coordinate Descent under More Realistic Assumptions

Asynchronous-parallel algorithms have the potential to vastly speed up algorithms by eliminating costly synchronization. However, our understanding to these algorithms is limited because the current convergence of asynchronous (block) coordinate descent algorithms are based on somewhat unrealistic assumptions. In particular, the age of the shared optimization variables being used to update a block is assumed to be independent of the block being updated. Also, it is assumed that the updates are applied to randomly chosen blocks. In this paper, we argue that these assumptions either fail to hold or will imply less efficient implementations. We then prove the convergence of asynchronous-parallel block coordinate descent under more realistic assumptions, in particular, always without the independence assumption. The analysis permits both the deterministic (essentially) cyclic and random rules for block choices. Because a bound on the asynchronous delays may or may not be available, we establish convergence for both bounded delays and unbounded delays. The analysis also covers nonconvex, weakly convex, and strongly convex functions. We construct Lyapunov functions that directly model both objective progress and delays, so delays are not treated errors or noise. A continuous-time ODE is provided to explain the construction at a high level.

math.OC↗

arXiv2017

Cyclic Coordinate Update Algorithms for Fixed-Point Problems: Analysis and Applications

Many problems reduce to the fixed-point problem of solving $x=T(x)$. To this problem, we apply the coordinate-update algorithms, which update only one or a few components of $x$ at each step. When each update is cheap, these algorithms are faster than the full fixed-point iteration (which updates all the components). In this paper, we focus on the coordinate-update algorithms based on the cyclic selection rules, where the ordering of coordinates in each cycle is arbitrary. These algorithms are fast, but their convergence is unknown in the fixed-point setting. When $T$ is a nonexpansive operator and has a fixed point, we show that the sequence of coordinate-update iterates converges to a fixed point under proper step sizes. This result applies to the primal-dual coordinate-update algorithms, which have applications to optimization problems with nonseparable nonsmooth objectives, as well as global linear constraints. Numerically, we apply coordinate-update algorithms with the cyclic, shuffled cyclic, and random selection rules to $\ell_1$ robust least squares, a CT image reconstruction problem, as well as nonnegative matrix factorization. They converge much faster than the standard fixed-point iteration. Among the three rules, cyclic and shuffled cyclic rules are overall faster than the random rule.

math.OC↗

arXiv2017

Decentralized Consensus Optimization with Asynchrony and Delays

We propose an asynchronous, decentralized algorithm for consensus optimization. The algorithm runs over a network in which the agents communicate with their neighbors and perform local computation. In the proposed algorithm, each agent can compute and communicate independently at different times, for different durations, with the information it has even if the latest information from its neighbors is not yet available. Such an asynchronous algorithm reduces the time that agents would otherwise waste idle because of communication delays or because their neighbors are slower. It also eliminates the need for a global clock for synchronization. Mathematically, the algorithm involves both primal and dual variables, uses fixed step-size parameters, and provably converges to the exact solution under a bounded delay assumption and a random agent assumption. When running synchronously, the algorithm performs just as well as existing competitive synchronous algorithms such as PG-EXTRA, which diverges without synchronization. Numerical experiments confirm the theoretical findings and illustrate the performance of the proposed algorithm.

math.OC↗

arXiv2017

A Primer on Coordinate Descent Algorithms

This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to their effectiveness in solving large-scale optimization problems in machine learning, compressed sensing, image processing, and computational statistics. Coordinate descent algorithms solve optimization problems by successively minimizing along each coordinate or coordinate hyperplane, which is ideal for parallelized and distributed computing. Avoiding detailed technicalities and proofs, this monograph gives relevant theory and examples for practitioners to effectively apply coordinate descent to modern problems in data science and engineering.

math.OC↗

arXiv2016

Expander Graph and Communication-Efficient Decentralized Optimization

In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy. We discover that the so-called expander graphs are near-optimal choices. We propose three approaches to construct expander graphs for different numbers of nodes and node degrees. Our numerical results show that the performance of decentralized optimization is significantly better on expander graphs than other regular graphs.

math.OC↗

arXiv2016

Coordinate Friendly Structures, Algorithms and Applications

This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.

math.OC↗

arXiv2016

TMAC: A Toolbox of Modern Async-Parallel, Coordinate, Splitting, and Stochastic Methods

TMAC is a toolbox written in C++11 that implements algorithms based on a set of modern methods for large-scale optimization. It covers a variety of optimization problems, which can be both smooth and nonsmooth, convex and nonconvex, as well as constrained and unconstrained. The algorithms implemented in TMAC, such as the coordinate up- date method and operator splitting method, are scalable as they decompose a problem into simple subproblems. These algorithms can run in a multi-threaded fashion, either synchronously or asynchronously, to take advantages of all the cores available. TMAC architecture mimics how a scientist writes down an optimization algorithm. Therefore, it is easy for one to obtain a new algorithm by making simple modifications such as adding a new operator and adding a new splitting, while maintaining the multicore parallelism and other features. The package is available at https://github.com/uclaopt/TMAC.

math.OC↗

arXiv2016

ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates

Finding a fixed point to a nonexpansive operator, i.e., $x^*=Tx^*$, abstracts many problems in numerical linear algebra, optimization, and other areas of scientific computing. To solve fixed-point problems, we propose ARock, an algorithmic framework in which multiple agents (machines, processors, or cores) update $x$ in an asynchronous parallel fashion. Asynchrony is crucial to parallel computing since it reduces synchronization wait, relaxes communication bottleneck, and thus speeds up computing significantly. At each step of ARock, an agent updates a randomly selected coordinate $x_i$ based on possibly out-of-date information on $x$. The agents share $x$ through either global memory or communication. If writing $x_i$ is atomic, the agents can read and write $x$ without memory locks. Theoretically, we show that if the nonexpansive operator $T$ has a fixed point, then with probability one, ARock generates a sequence that converges to a fixed points of $T$. Our conditions on $T$ and step sizes are weaker than comparable work. Linear convergence is also obtained. We propose special cases of ARock for linear systems, convex optimization, machine learning, as well as distributed and decentralized consensus problems. Numerical experiments of solving sparse logistic regression problems are presented.

math.OC↗

arXiv2016

Sparse Recovery via Differential Inclusions

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to \emph{Linearized Bregman iteration}, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal $l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.

math.ST↗

arXiv2015

A globally convergent algorithm for nonconvex optimization based on block coordinate update

Nonconvex optimization problems arise in many areas of computational science and engineering and are (approximately) solved by a variety of algorithms. Existing algorithms usually only have local convergence or subsequence convergence of their iterates. We propose an algorithm for a generic nonconvex optimization formulation, establish the convergence of its whole iterate sequence to a critical point along with a rate of convergence, and numerically demonstrate its efficiency. Specially, we consider the problem of minimizing a nonconvex objective function. Its variables can be treated as one block or be partitioned into multiple disjoint blocks. It is assumed that each non-differentiable component of the objective function or each constraint applies to one block of variables. The differentiable components of the objective function, however, can apply to one or multiple blocks of variables together. Our algorithm updates one block of variables at time by minimizing a certain prox-linear surrogate. The order of update can be either deterministic or randomly shuffled in each round. We obtain the convergence of the whole iterate sequence under fairly loose conditions including, in particular, the Kurdyka-Łojasiewicz (KL) condition, which is satisfied by a broad class of nonconvex/nonsmooth applications. We apply our convergence result to the coordinate descent method for non-convex regularized linear regression and also a modified rank-one residue iteration method for nonnegative matrix factorization. We show that both the methods have global convergence. Numerically, we test our algorithm on nonnegative matrix and tensor factorization problems, with random shuffling enable to avoid local solutions.

math.OC↗