arXiv subjects

Song Mei

Publications and source records attributed to Song Mei.

60 records · Page 4Linked to original sources

arXiv2019

Analysis of Sequential Quadratic Programming through the Lens of Riemannian Optimization

We prove that a "first-order" Sequential Quadratic Programming (SQP) algorithm for equality constrained optimization has local linear convergence with rate $(1-1/κ_R)^k$, where $κ_R$ is the condition number of the Riemannian Hessian, and global convergence with rate $k^{-1/4}$. Our analysis builds on insights from Riemannian optimization -- we show that the SQP and Riemannian gradient methods have nearly identical behavior near the constraint manifold, which could be of broader interest for understanding constrained optimization.

math.OC↗

arXiv2018

A Mean Field View of the Landscape of Two-Layers Neural Networks

Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this paper we consider a simple case, namely two-layers neural networks, and prove that -in a suitable scaling limit- SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearly ideal generalization error. This description allows to 'average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD.

stat.ML↗

arXiv2018

The landscape of the spiked tensor model

We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $λ_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $λ\ge C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1\ll λ\ll n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $λ$ larger than a constant) critical points are either very close to the unknown vector ${\boldsymbol u}$, or are confined in a band of width $Θ(λ^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${\boldsymbol u}$. For local maxima, this band shrinks to be of size $Θ(λ^{-1/(k-2)})$. These `uninformative' local maxima are likely to cause the failure of optimization algorithms.

math.ST↗

arXiv2017

Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality

A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional problems. In order to cope with this problem, Burer and Monteiro proposed a non-convex rank-constrained formulation, which has good performance in practice but is still poorly understood theoretically. In this paper we study the rank-constrained version of SDPs arising in MaxCut and in synchronization problems. We establish a Grothendieck-type inequality that proves that all the local maxima and dangerous saddle points are within a small multiplicative gap from the global maximum. We use this structural information to prove that SDPs can be solved within a known accuracy, by applying the Riemannian trust-region method to this non-convex problem, while constraining the rank to be of order one. For the MaxCut problem, our inequality implies that any local maximizer of the rank-constrained SDP provides a $ (1 - 1/(k-1)) \times 0.878$ approximation of the MaxCut, when the rank is fixed to $k$. We then apply our results to data matrices generated according to the Gaussian ${\mathbb Z}_2$ synchronization problem, and the two-groups stochastic block model with large bounded degree. We prove that the error achieved by local maximizers undergoes a phase transition at the same threshold as for information-theoretically optimal methods.

math.OC↗

arXiv2017

The Landscape of Empirical Risk for Non-convex Losses

Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process theory implies uniform convergence of the empirical risk to the population risk. While uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. In order to capture the complexity of computing M-estimators, we propose to study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we can establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as non-convex binary classification, robust regression, and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provide a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the unknown parameters vector (modulo logarithmic factors), then a suitable uniform convergence result takes place. We apply this result to non-convex binary classification and robust regression in very high-dimension.

stat.ML↗

arXiv2015

On a molecular based Q-tensor model for liquid crystals with density variations

In this article, we study the new Q-tensor model previously derived from Onsager's molecular theory by Han \textit{et al.} [Arch. Rational Mech. Anal., 215.3 (2014), pp. 741-809] for static liquid crystal modeling. Taking density and Q-tensor as order parameters, the new Q-tensor model not only characterizes important phases while capturing density variation effects, but also remains computationally tractable and efficient. We report the results of two numerical applications of the model, namely the isotropic--nematic--smectic-A--smectic-C phase transitions and the isotropic--nematic interface problem, in which density variations are indispensable. Meanwhile, we show the connections of the new Q-tensor model with classical models including generalized Landau-de Gennes models, generalized McMillan models, and the Chen-Lubensky model. The new Q-tensor model is the pivot and an appropriate trade-off between the classical models in three scales.

cond-mat.soft↗