arXiv subjects

Richard Y. Zhang

Publications and source records attributed to Richard Y. Zhang.

31 records · Page 2Linked to original sources

arXiv2022

Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion

The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with $d$ so large that even the simplest full-dimension vector operations with $O(d)$ time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down when the underlying ground truth is ill-conditioned; it requires at least $O(κ\log(1/ε))$ iterations to get $ε$-close to ground truth matrix with condition number $κ$. In this paper, we propose a preconditioned version of SGD that preserves all the favorable practical qualities of SGD for huge-scale online optimization while also making it agnostic to $κ$. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to $ε$-accuracy in $O(\log(1/ε))$ iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with $κ=1$. In our experiments, we observe a similar acceleration for item-item collaborative filtering on the MovieLens25M dataset via a pair-wise ranking loss, with 100 million training pairs and 10 million testing pairs. [See supporting code at https://github.com/Hong-Ming/ScaledSGD.]

cs.LG↗

arXiv2020

How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?

Given a sufficiently large amount of labeled data, the non-convex low-rank matrix recovery problem contains no spurious local minima, so a local optimization algorithm is guaranteed to converge to a global minimum starting from any initial guess. However, the actual amount of data needed by this theoretical guarantee is very pessimistic, as it must prevent spurious local minima from existing anywhere, including at adversarial locations. In contrast, prior work based on good initial guesses have more realistic data requirements, because they allow spurious local minima to exist outside of a neighborhood of the solution. In this paper, we quantify the relationship between the quality of the initial guess and the corresponding reduction in data requirements. Using the restricted isometry constant as a surrogate for sample complexity, we compute a sharp threshold number of samples needed to prevent each specific point on the optimization landscape from becoming a spurious local minimum. Optimizing the threshold over regions of the landscape, we see that for initial points around the ground truth, a linear improvement in the quality of the initial guess amounts to a constant factor improvement in the sample complexity.

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arXiv2020

On the Tightness of Semidefinite Relaxations for Certifying Robustness to Adversarial Examples

The robustness of a neural network to adversarial examples can be provably certified by solving a convex relaxation. If the relaxation is loose, however, then the resulting certificate can be too conservative to be practically useful. Recently, a less conservative robustness certificate was proposed, based on a semidefinite programming (SDP) relaxation of the ReLU activation function. In this paper, we describe a geometric technique that determines whether this SDP certificate is exact, meaning whether it provides both a lower-bound on the size of the smallest adversarial perturbation, as well as a globally optimal perturbation that attains the lower-bound. Concretely, we show, for a least-squares restriction of the usual adversarial attack problem, that the SDP relaxation amounts to the nonconvex projection of a point onto a hyperbola. The resulting SDP certificate is exact if and only if the projection of the point lies on the major axis of the hyperbola. Using this geometric technique, we prove that the certificate is exact over a single hidden layer under mild assumptions, and explain why it is usually conservative for several hidden layers. We experimentally confirm our theoretical insights using a general-purpose interior-point method and a custom rank-2 Burer-Monteiro algorithm.

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arXiv2020

Sparse Semidefinite Programs with Guaranteed Near-Linear Time Complexity via Dualized Clique Tree Conversion

Clique tree conversion solves large-scale semidefinite programs by splitting an $n\times n$ matrix variable into up to $n$ smaller matrix variables, each representing a principal submatrix of up to $ω\timesω$. Its fundamental weakness is the need to introduce overlap constraints that enforce agreement between different matrix variables, because these can result in dense coupling. In this paper, we show that by dualizing the clique tree conversion, the coupling due to the overlap constraints is guaranteed to be sparse over dense blocks, with a block sparsity pattern that coincides with the adjacency matrix of a tree. We consider two classes of semidefinite programs with favorable sparsity patterns that encompass the MAXCUT and MAX $k$-CUT relaxations, the Lovasz Theta problem, and the AC optimal power flow relaxation. Assuming that $ω\ll n$, we prove that the per-iteration cost of an interior-point method is linear $O(n)$ time and memory, so an $ε$-accurate and $ε$-feasible iterate is obtained after $O(\sqrt{n}\log(1/ε))$ iterations in near-linear $O(n^{1.5}\log(1/ε))$ time. We confirm our theoretical insights with numerical results on semidefinite programs as large as $n=13659$. (Supporting code at https://github.com/ryz-codes/dual_ctc )

math.OC↗

arXiv2019

Large-Scale Traffic Signal Offset Optimization

The offset optimization problem seeks to coordinate and synchronize the timing of traffic signals throughout a network in order to enhance traffic flow and reduce stops and delays. Recently, offset optimization was formulated into a continuous optimization problem without integer variables by modeling traffic flow as sinusoidal. In this paper, we present a novel algorithm to solve this new formulation to near-global optimality on a large-scale. Specifically, we solve a convex relaxation of the nonconvex problem using a tree decomposition reduction, and use randomized rounding to recover a near-global solution. We prove that the algorithm always delivers solutions of expected value at least 0.785 times the globally optimal value. Moreover, assuming that the topology of the traffic network is "tree-like", we prove that the algorithm has near-linear time complexity with respect to the number of intersections. These theoretical guarantees are experimentally validated on the Berkeley, Manhattan, and Los Angeles traffic networks. In our numerical results, the empirical time complexity of the algorithm is linear, and the solutions have objectives within 0.99 times the globally optimal value.

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arXiv2019

Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery

Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $δ$. If $δ$ is too large, however, then counterexamples containing spurious local minima are known to exist. In this paper, we introduce a proof technique that is capable of establishing sharp thresholds on $δ$ to guarantee the inexistence of spurious local minima. Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $δ<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point). We also prove a local recovery result: given an initial point $x_{0}$ satisfying $f(x_{0})\le(1-δ)^{2}f(0)$, any descent algorithm that converges to second-order optimality guarantees exact recovery.

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arXiv2018

How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i.e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: we prove that every x is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant $δ=1/2$, but causes randomly initialized stochastic gradient descent (SGD) to fail 12% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

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arXiv2018

GMRES-Accelerated ADMM for Quadratic Objectives

We consider the sequence acceleration problem for the alternating direction method-of-multipliers (ADMM) applied to a class of equality-constrained problems with strongly convex quadratic objectives, which frequently arise as the Newton subproblem of interior-point methods. Within this context, the ADMM update equations are linear, the iterates are confined within a Krylov subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its ability to accelerate convergence. The basic ADMM method solves a $κ$-conditioned problem in $O(\sqrtκ)$ iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in $O(κ^{1/4})$ iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of $O(\sqrtκ)$ iterations. The method is shown to be competitive against standard preconditioned Krylov subspace methods for saddle-point problems. The method is embedded within SeDuMi, a popular open-source solver for conic optimization written in MATLAB, and used to solve many large-scale semidefinite programs with error that decreases like $O(1/k^{2})$, instead of $O(1/k)$, where $k$ is the iteration index.

math.OC↗

arXiv2018

Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion

The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as "graphical lasso", but its computational cost becomes prohibitive for large data sets. A recent line of results showed--under mild assumptions--that the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix and solving a maximum determinant matrix completion (MDMC) problem. This paper proves an extension of this result, and describes a Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $ε$-accurate solution in $O(n\log(1/ε))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve the associated MDMC problems with as many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.

stat.ML↗

arXiv2017

Sparse Inverse Covariance Estimation for Chordal Structures

In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have shown that the sparsity pattern of the optimal solution of GL is equivalent to the one obtained from simply thresholding the sample covariance matrix, for sparse graphs under different conditions. We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure. As a major generalization of the previous result, in this paper we derive a closed-form solution for the GL for graphs with chordal structures. We show that the GL and thresholding equivalence conditions can significantly be simplified and are expected to hold for high-dimensional problems if the thresholded sample covariance matrix has a chordal structure. We then show that the GL and thresholding equivalence is enough to reduce the GL to a maximum determinant matrix completion problem and drive a recursive closed-form solution for the GL when the thresholded sample covariance matrix has a chordal structure. For large-scale problems with up to 450 million variables, the proposed method can solve the GL problem in less than 2 minutes, while the state-of-the-art methods converge in more than 2 hours.

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arXiv2017

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.

math.OC↗

arXiv2017

Modified Interior-Point Method for Large-and-Sparse Low-Rank Semidefinite Programs

Semidefinite programs (SDPs) are powerful theoretical tools that have been studied for over two decades, but their practical use remains limited due to computational difficulties in solving large-scale, realistic-sized problems. In this paper, we describe a modified interior-point method for the efficient solution of large-and-sparse low-rank SDPs, which finds applications in graph theory, approximation theory, control theory, sum-of-squares, etc. Given that the problem data is large-and-sparse, conjugate gradients (CG) can be used to avoid forming, storing, and factoring the large and fully-dense interior-point Hessian matrix, but the resulting convergence rate is usually slow due to ill-conditioning. Our central insight is that, for a rank-$k$, size-$n$ SDP, the Hessian matrix is ill-conditioned only due to a rank-$nk$ perturbation, which can be explicitly computed using a size-$n$ eigendecomposition. We construct a preconditioner to "correct" the low-rank perturbation, thereby allowing preconditioned CG to solve the Hessian equation in a few tens of iterations. This modification is incorporated within SeDuMi, and used to reduce the solution time and memory requirements of large-scale matrix-completion problems by several orders of magnitude.

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arXiv2016

Parameter Insensitivity in ADMM-Preconditioned Solution of Saddle-Point Problems

We consider the solution of linear saddle-point problems, using the alternating direction method-of-multipliers (ADMM) as a preconditioner for the generalized minimum residual method (GMRES). We show, using theoretical bounds and empirical results, that ADMM is made remarkably insensitive to the parameter choice with Krylov subspace acceleration. We prove that ADMM-GMRES can consistently converge, irrespective of the exact parameter choice, to an $ε$-accurate solution of a $κ$-conditioned problem in $O(κ^{2/3}\logε^{-1})$ iterations. The accelerated method is applied to randomly generated problems, as well as the Newton direction computation for the interior-point solution of semidefinite programs in the SDPLIB test suite. The empirical results confirm this parameter insensitivity, and suggest a slightly improved iteration bound of $O(\sqrtκ\logε^{-1})$.

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