Machine learning algorithms typically perform optimization over a class of non-convex functions. In this work, we provide bounds on the fundamental hardness of identifying the global minimizer of a non convex function. Specifically, we design a family of parametrized non-convex functions and employ statistical lower bounds for parameter estimation. We show that the parameter estimation problem is equivalent to the problem of function identification in the given family. We then claim that non convex optimization is at least as hard as function identification. Jointly, we prove that any first order method can take exponential time to converge to a global minimizer.
We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.
We introduce a new concept, data irrecoverability, and show that the well-studied concept of data privacy is sufficient but not necessary for data irrecoverability. We show that there are several regularized loss minimization problems that can use perturbed data with theoretical guarantees of generalization, i.e., loss consistency. Our results quantitatively connect the convergence rates of the learning problems to the impossibility for any adversary for recovering the original data from perturbed observations. In addition, we show several examples where the convergence rates with perturbed da…
Matrix completion has attracted much interest in the past decade in machine learning and computer vision. For low-rank promotion in matrix completion, the nuclear norm penalty is convenient due to its convexity but has a bias problem. Recently, various algorithms using nonconvex penalties have been proposed, among which the proximal gradient descent (PGD) algorithm is one of the most efficient and effective. For the nonconvex PGD algorithm, whether it converges to a local minimizer and its convergence rate are still unclear. This work provides a nontrivial analysis on the PGD algorithm in the …
Non-convergence to global minimizers in data driven supervised deep learning: Adam and stochastic gradient descent optimization provably fail to converge to global minimizers in the training of deep neural networks with ReLU activation
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Vietnam Academy of Science and Technology（越南科学技术研究院）；Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）深圳大数据研究院）
Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networks
Aiming towards the minimizers: fast convergence of SGD for overparametrized problems
Halicioğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校哈利乔格鲁数据科学研究所）；University of Washington（华盛顿大学）；Cornell University（康奈尔大学）
Georgia Institute of Technology（佐治亚理工学院）；Singapore University of Technology & Design（新加坡科技设计大学）
Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.
Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient large-scale learning algorithms. In this study, we consider solving a binary classification problem using random features and stochastic gradient descent. In recent research, an exponential convergence rate of the expected classification error under the strong low-noise condition has been shown. We extend these analyses to a random features setting, analyzing …
Although SGD with random reshuffle has been widely-used in machine learning applications, there is a limited understanding of how model characteristics affect the convergence of the algorithm. In this work, we introduce model incoherence to characterize the diversity of model characteristics and study its impact on convergence of SGD with random reshuffle under weak strong convexity. Specifically, minimizer incoherence measures the discrepancy between the global minimizers of a sample loss and those of the total loss and affects the convergence error of SGD with random reshuffle. In particular…
In overparametrized models, the noise in stochastic gradient descent (SGD) implicitly regularizes the optimization trajectory and determines which local minimum SGD converges to. Motivated by empirical studies that demonstrate that training with noisy labels improves generalization, we study the implicit regularization effect of SGD with label noise. We show that SGD with label noise converges to a stationary point of a regularized loss $L(θ) +λR(θ)$, where $L(θ)$ is the training loss, $λ$ is an effective regularization parameter depending on the step size, strength of the label noise, and the…
We analyze the convergence rate of gradient flows on objective functions induced by Dropout and Dropconnect, when applying them to shallow linear Neural Networks (NNs) - which can also be viewed as doing matrix factorization using a particular regularizer. Dropout algorithms such as these are thus regularization techniques that use 0,1-valued random variables to filter weights during training in order to avoid coadaptation of features. By leveraging a recent result on nonconvex optimization and conducting a careful analysis of the set of minimizers as well as the Hessian of the loss function, …
A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity
Department of Statistics and Data Science, The Wharton School, University of Pennsylvania（统计与数据科学系，沃顿商学院，宾夕法尼亚大学）；Halicioğlu Data Science Institute, University of California, San Diego（Halicioğlu数据科学研究所，加州大学圣地亚哥分校）
Tsinghua University（清华大学）；University of California, Los Angeles（加州大学洛杉矶分校）
Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an "inspect" phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a radius $R$ around the current point. When a sample point yields a sufficient decrease in the objective, we move there and resume an existing algorithm. If no sufficient decrease is found, the current point is called an app…
Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growth
Wharton Department of Statistics and Data Science, University of Pennsylvania（沃顿统计与数据科学系，宾夕法尼亚大学）；Department of Mathematics, U. Washington（数学系，华盛顿大学）；Edwardson School of Industrial Engineering, Purdue University（工业工程学院，普渡大学）
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Academy for Advanced Interdisciplinary Studies, Peking University（北京大学前沿交叉学科研究院）
In this paper, we follow Eftekhari's work to give a non-local convergence analysis of deep linear networks. Specifically, we consider optimizing deep linear networks which have a layer with one neuron under quadratic loss. We describe the convergent point of trajectories with arbitrary starting point under gradient flow, including the paths which converge to one of the saddle points or the original point. We also show specific convergence rates of trajectories that converge to the global minimizer by stages. To achieve these results, this paper mainly extends the machinery in Eftekhari's work …
African Institute for Mathematical Sciences（非洲数学科学研究所）；Stellenbosch University（Stellenbosch大学）；RWTH Aachen University（亚琛工业大学）
We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover…
Technical University of Munich（慕尼黑工业大学）；University of Calgary（卡尔加里大学）；University of Ferrara（费拉拉大学）
We investigate the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a functi…
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a se…
