Richard Y. Zhang

Gavin Zhang, Hong-Ming Chiu, Richard Y. Zhang

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.]

Gavin Zhang, Salar Fattahi, Richard Y. Zhang

We consider using gradient descent to minimize the nonconvex function $f(X)=φ(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $φ$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally rank deficient, then its rank deficiency certifies it as being globally optimal. This way of certifying global optimality necessarily requires the search rank $r$ of the current iterate $X$ to be overparameterized with respect to the rank $r^{\star}$ of the global minimizer $X^{\star}$. Unfortunately, overparameterization significantly slows down the convergence of gradient descent, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$, even when $φ$ is strongly convex. In this paper, we propose an inexpensive preconditioner that restores the convergence rate of gradient descent back to linear in the overparameterized case, while also making it agnostic to possible ill-conditioning in the global minimizer $X^{\star}$.

Geyu Liang, Gavin Zhang, Salar Fattahi et al.

This paper focuses on the noiseless complete dictionary learning problem, where the goal is to represent a set of given signals as linear combinations of a small number of atoms from a learned dictionary. There are two main challenges faced by theoretical and practical studies of dictionary learning: the lack of theoretical guarantees for practically-used heuristic algorithms and their poor scalability when dealing with huge-scale datasets. Towards addressing these issues, we propose a simple and efficient algorithm that provably recovers the ground truth when applied to the nonconvex and discrete formulation of the problem in the noiseless setting. We also extend our proposed method to mini-batch and online settings where the data is huge-scale or arrives continuously over time. At the core of our proposed method lies an efficient preconditioning technique that transforms the unknown dictionary to a near-orthonormal one, for which we prove a simple alternating minimization technique converges linearly to the ground truth under minimal conditions. Our numerical experiments on synthetic and real datasets showcase the superiority of our method compared with the existing techniques.

Richard Y. Zhang

We consider minimizing a twice-differentiable, $L$-smooth, and $μ$-strongly convex objective $φ$ over an $n\times n$ positive semidefinite matrix $M\succeq0$, under the assumption that the minimizer $M^{\star}$ has low rank $r^{\star}\ll n$. Following the Burer--Monteiro approach, we instead minimize the nonconvex objective $f(X)=φ(XX^{T})$ over a factor matrix $X$ of size $n\times r$. This substantially reduces the number of variables from $O(n^{2})$ to as few as $O(n)$ and also enforces positive semidefiniteness for free, but at the cost of giving up the convexity of the original problem. In this paper, we prove that if the search rank $r\ge r^{\star}$ is overparameterized by a \emph{constant factor} with respect to the true rank $r^{\star}$, namely as in $r>\frac{1}{4}(L/μ-1)^{2}r^{\star}$, then despite nonconvexity, local optimization is guaranteed to globally converge from any initial point to the global optimum. This significantly improves upon a previous rank overparameterization threshold of $r\ge n$, which we show is sharp in the absence of smoothness and strong convexity, but would increase the number of variables back up to $O(n^{2})$. Conversely, without rank overparameterization, we prove that such a global guarantee is possible if and only if $φ$ is almost perfectly conditioned, with a condition number of $L/μ<3$. Therefore, we conclude that a small amount of overparameterization can lead to large improvements in theoretical guarantees for the nonconvex Burer--Monteiro factorization.

Hong-Ming Chiu, Richard Y. Zhang

Adversarial training is well-known to produce high-quality neural network models that are empirically robust against adversarial perturbations. Nevertheless, once a model has been adversarially trained, one often desires a certification that the model is truly robust against all future attacks. Unfortunately, when faced with adversarially trained models, all existing approaches have significant trouble making certifications that are strong enough to be practically useful. Linear programming (LP) techniques in particular face a "convex relaxation barrier" that prevent them from making high-quality certifications, even after refinement with mixed-integer linear programming (MILP) and branch-and-bound (BnB) techniques. In this paper, we propose a nonconvex certification technique, based on a low-rank restriction of a semidefinite programming (SDP) relaxation. The nonconvex relaxation makes strong certifications comparable to much more expensive SDP methods, while optimizing over dramatically fewer variables comparable to much weaker LP methods. Despite nonconvexity, we show how off-the-shelf local optimization algorithms can be used to achieve and to certify global optimality in polynomial time. Our experiments find that the nonconvex relaxation almost completely closes the gap towards exact certification of adversarially trained models.

Richard Y. Zhang

The classical low-rank matrix recovery problem is well-known to exhibit \emph{benign nonconvexity} under the restricted isometry property (RIP): local optimization is guaranteed to converge to the global optimum, where the ground truth is recovered. We investigate whether benign nonconvexity continues to hold when the factor matrices are constrained to be elementwise nonnegative -- a common practical requirement. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $δ=0$. Surprisingly, however, this property fails to extend to the partially-observed case with any arbitrarily small RIP constant $δ\to0^{+}$, irrespective of rank overparameterization. This finding exposes a critical theoretical gap: the continuity argument widely used to explain the empirical robustness of low-rank matrix recovery fundamentally breaks down once nonnegative constraints are imposed.

Yubo Zhuang, Xiaohui Chen, Yun Yang et al.

$K$-means clustering is a widely used machine learning method for identifying patterns in large datasets. Recently, semidefinite programming (SDP) relaxations have been proposed for solving the $K$-means optimization problem, which enjoy strong statistical optimality guarantees. However, the prohibitive cost of implementing an SDP solver renders these guarantees inaccessible to practical datasets. In contrast, nonnegative matrix factorization (NMF) is a simple clustering algorithm widely used by machine learning practitioners, but it lacks a solid statistical underpinning and theoretical guarantees. In this paper, we consider an NMF-like algorithm that solves a nonnegative low-rank restriction of the SDP-relaxed $K$-means formulation using a nonconvex Burer--Monteiro factorization approach. The resulting algorithm is as simple and scalable as state-of-the-art NMF algorithms while also enjoying the same strong statistical optimality guarantees as the SDP. In our experiments, we observe that our algorithm achieves significantly smaller mis-clustering errors compared to the existing state-of-the-art while maintaining scalability.

Gavin Zhang, Hong-Ming Chiu, Richard Y. Zhang

Non-convex gradient descent is a common approach for estimating a low-rank $n\times n$ ground truth matrix from noisy measurements, because it has per-iteration costs as low as $O(n)$ time, and is in theory capable of converging to a minimax optimal estimate. However, the practitioner is often constrained to just tens to hundreds of iterations, and the slow and/or inconsistent convergence of non-convex gradient descent can prevent a high-quality estimate from being obtained. Recently, the technique of preconditioning was shown to be highly effective at accelerating the local convergence of non-convex gradient descent when the measurements are noiseless. In this paper, we describe how preconditioning should be done for noisy measurements to accelerate local convergence to minimax optimality. For the symmetric matrix sensing problem, our proposed preconditioned method is guaranteed to locally converge to minimax error at a linear rate that is immune to ill-conditioning and/or over-parameterization. Using our proposed preconditioned method, we perform a 60 megapixel medical image denoising task, and observe significantly reduced noise levels compared to previous approaches.

Richard Y. Zhang

Recent work established that rank overparameterization eliminates spurious local minima in nonconvex low-rank matrix recovery under the restricted isometry property (RIP). But this does not fully explain the practical success of overparameterization, because real algorithms can still become trapped at nonstrict saddle points (approximate second-order points with arbitrarily small negative curvature) even when all local minima are global. Moreover, the result does not accommodate for noisy measurements, but it is unclear whether such an extension is even possible, in view of the many discontinuous and unintuitive behaviors already known for the overparameterized regime. In this paper, we introduce a novel proof technique that unifies, simplifies, and strengthens two previously competing approaches -- one based on escape directions and the other based on the inexistence of counterexample -- to provide sharp global guarantees in the noisy overparameterized regime. We show, once local minima have been converted into global minima through slight overparameterization, that near-second-order points achieve the same minimax-optimal recovery bounds (up to small constant factors) as significantly more expensive convex approaches. Our results are sharp with respect to the noise level and the solution accuracy, and hold for both the symmetric parameterization $XX^{T}$, as well as the asymmetric parameterization $UV^{T}$ under a balancing regularizer; we demonstrate that the balancing regularizer is indeed necessary.

Gavin Zhang, Richard Y. Zhang

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.

Richard Y. Zhang

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.

Richard Y. Zhang, Somayeh Sojoudi, Javad Lavaei

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.

Richard Y. Zhang, Cédric Josz, Somayeh Sojoudi et al.

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.

Richard Y. Zhang, Salar Fattahi, Somayeh Sojoudi

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.

Salar Fattahi, Richard Y. Zhang, Somayeh Sojoudi

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.