### Session

## OPT: Non-Convex

##### Room 309

Moderator: Lam Nguyen

**Neural Network Weights Do Not Converge to Stationary Points: An Invariant Measure Perspective**

Jingzhao Zhang · Haochuan Li · Suvrit Sra · Ali Jadbabaie

This work examines the deep disconnect between existing theoretical analyses of gradient-based algorithms and the practice of training deep neural networks. Specifically, we provide numerical evidence that in large-scale neural network training (e.g., ImageNet + ResNet101, and WT103 + TransformerXL models), the neural network's weights do not converge to stationary points where the gradient of the loss is zero. Remarkably, however, we observe that even though the weights do not converge to stationary points, the progress in minimizing the loss function halts and training loss stabilizes. Inspired by this observation, we propose a new perspective based on ergodic theory of dynamical systems to explain it. Rather than studying the evolution of weights, we study the evolution of the distribution of weights. We prove convergence of the distribution of weights to an approximate invariant measure, thereby explaining how the training loss can stabilize without weights necessarily converging to stationary points. We further discuss how this perspective can better align optimization theory with empirical observations in machine learning practice.

**Convergence and Recovery Guarantees of the K-Subspaces Method for Subspace Clustering**

Peng Wang · Huikang Liu · Anthony Man-Cho So · Laura Balzano

The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where $N$ points are randomly sampled from $K \ge 2$ overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within $\Theta(\log\log N)$ iterations with high probability. Moreover, we propose a thresholding inner-product based spectral method for initialization and prove that it produces a point in this neighborhood. We also present numerical results of the studied method to support our theoretical developments.

**Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the $O(\epsilon^{-7/4})$ Complexity**

Huan Li · Zhouchen Lin

This paper studies the accelerated gradient descent for general nonconvex problems under the gradient Lipschitz and Hessian Lipschitz assumptions. We establish that a simple restarted accelerated gradient descent (AGD) finds an $\epsilon$-approximate first-order stationary point in $O(\epsilon^{-7/4})$ gradient computations with simple proofs. Our complexity does not hide any polylogarithmic factors, and thus it improves over the state-of-the-art one by the $O(\log\frac{1}{\epsilon})$ factor. Our simple algorithm only consists of Nesterov's classical AGD and a restart mechanism, and it does not need the negative curvature exploitation or the optimization of regularized surrogate functions. Technically, our simple proof does not invoke the analysis for the strongly convex AGD, which is crucial to remove the $O(\log\frac{1}{\epsilon})$ factor.

**Understanding the unstable convergence of gradient descent**

Kwangjun Ahn · Jingzhao Zhang · Suvrit Sra

Most existing analyses of (stochastic) gradient descent rely on the condition that for $L$-smooth costs, the step size is less than $2/L$. However, many works have observed that in machine learning applications step sizes often do not fulfill this condition, yet (stochastic) gradient descent still converges, albeit in an unstable manner. We investigate this unstable convergence phenomenon from first principles, and discuss key causes behind it. We also identify its main characteristics, and how they interrelate based on both theory and experiments, offering a principled view toward understanding the phenomenon.

**Federated Minimax Optimization: Improved Convergence Analyses and Algorithms**

PRANAY SHARMA · Rohan Panda · Gauri Joshi · Pramod K Varshney

In this paper, we consider nonconvex minimax optimization, which is gaining prominence in many modern machine learning applications, such as GANs. Large-scale edge-based collection of training data in these applications calls for communication-efficient distributed optimization algorithms, such as those used in federated learning, to process the data. In this paper, we analyze local stochastic gradient descent ascent (SGDA), the local-update version of the SGDA algorithm. SGDA is the core algorithm used in minimax optimization, but it is not well-understood in a distributed setting. We prove that Local SGDA has \textit{order-optimal} sample complexity for several classes of nonconvex-concave and nonconvex-nonconcave minimax problems, and also enjoys \textit{linear speedup} with respect to the number of clients. We provide a novel and tighter analysis, which improves the convergence and communication guarantees in the existing literature. For nonconvex-PL and nonconvex-one-point-concave functions, we improve the existing complexity results for centralized minimax problems. Furthermore, we propose a momentum-based local-update algorithm, which has the same convergence guarantees, but outperforms Local SGDA as demonstrated in our experiments.

**Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm**

Pini Zilber · Boaz Nadler

The inductive matrix completion (IMC) problem is to recover a low rank matrix from few observed entries while incorporating prior knowledge about its row and column subspaces. In this work, we make three contributions to the IMC problem: (i) we prove that under suitable conditions, the IMC optimization landscape has no bad local minima; (ii) we derive a simple scheme with theoretical guarantees to estimate the rank of the unknown matrix; and (iii) we propose GNIMC, a simple Gauss-Newton based method to solve the IMC problem, analyze its runtime and derive for it strong recovery guarantees. The guarantees for GNIMC are sharper in several aspects than those available for other methods, including a quadratic convergence rate, fewer required observed entries and stability to errors or deviations from low-rank. Empirically, given entries observed uniformly at random, GNIMC recovers the underlying matrix substantially faster than several competing methods.

**FedNest: Federated Bilevel, Minimax, and Compositional Optimization**

Davoud Ataee Tarzanagh · Mingchen Li · Christos Thrampoulidis · Samet Oymak

Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems - including adversarial robustness, hyperparameter tuning, actor-critic - fall undernested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose FedNest: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for FedNest in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. FedNest introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter & hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice.

**AdaGrad Avoids Saddle Points**

Kimon Antonakopoulos · Panayotis Mertikopoulos · Georgios Piliouras · Xiao Wang

Adaptive first-order methods in optimization have widespread ML applications due to their ability to adapt to non-convex landscapes. However, their convergence guarantees are typically stated in terms of vanishing gradient norms, which leaves open the issue of converging to undesirable saddle points (or even local maxima). In this paper, we focus on the AdaGrad family of algorithms - from scalar to full-matrix preconditioning - and we examine the question of whether the method's trajectories avoid saddle points. A major challenge that arises here is that AdaGrad's step-size (or, more accurately, the method's preconditioner) evolves over time in a filtration-dependent way, i.e., as a function of all gradients observed in earlier iterations; as a result, avoidance results for methods with a constant or vanishing step-size do not apply. We resolve this challenge by combining a series of step-size stabilization arguments with a recursive representation of the AdaGrad preconditioner that allows us to employ center-stable techniques and ultimately show that the induced trajectories avoid saddle points from almost any initial condition.

**Fast and Provable Nonconvex Tensor RPCA**

Haiquan Qiu · Yao Wang · Shaojie Tang · Deyu Meng · QUANMING YAO

In this paper, we study nonconvex tensor robust principal component analysis (RPCA) based on the $t$-SVD. We first propose an alternating projection method, i.e., APT, which converges linearly to the ground-truth under the incoherence conditions of tensors. However, as the projection to the low-rank tensor space in APT can be slow, we further propose to speedup such a process by utilizing the property of the tangent space of low-rank. The resulting algorithm, i.e., EAPT, is not only more efficient than APT but also keeps the linear convergence. Compared with existing tensor RPCA works, the proposed method, especially EAPT, is not only more effective due to the recovery guarantee and adaption in the transformed (frequency) domain but also more efficient due to faster convergence rate and lower iteration complexity. These benefits are also empirically verified both on synthetic data, and real applications, e.g., hyperspectral image denoising and video background subtraction.

**On Convergence of Gradient Descent Ascent: A Tight Local Analysis**

Haochuan Li · Farzan Farnia · Subhro Das · Ali Jadbabaie

Gradient Descent Ascent (GDA) methods are the mainstream algorithms for minimax optimization in generative adversarial networks (GANs). Convergence properties of GDA have drawn significant interest in the recent literature. Specifically, for $\min_{x} \max_{y} f(x;y)$ where $f$ is strongly-concave in $y$ and possibly nonconvex in $x$, (Lin et al., 2020) proved the convergence of GDA with a stepsize ratio $\eta_y/\eta_x=\Theta(\kappa^2)$ where $\eta_x$ and $\eta_y$ are the stepsizes for $x$ and $y$ and $\kappa$ is the condition number for $y$. While this stepsize ratio suggests a slow training of the min player, practical GAN algorithms typically adopt similar stepsizes for both variables, indicating a wide gap between theoretical and empirical results. In this paper, we aim to bridge this gap by analyzing the \emph{local convergence} of general \emph{nonconvex-nonconcave} minimax problems. We demonstrate that a stepsize ratio of $\Theta(\kappa)$ is necessary and sufficient for local convergence of GDA to a Stackelberg Equilibrium, where $\kappa$ is the local condition number for $y$. We prove a nearly tight convergence rate with a matching lower bound. We further extend the convergence guarantees to stochastic GDA and extra-gradient methods (EG). Finally, we conduct several numerical experiments to support our theoretical findings.

**Convergence Rates of Non-Convex Stochastic Gradient Descent Under a Generic Lojasiewicz Condition and Local Smoothness**

Kevin Scaman · Cedric Malherbe · Ludovic DOS SANTOS

Training over-parameterized neural networks involves the empirical minimization of highly non-convex objective functions. Recently, a large body of works provided theoretical evidence that, despite this non-convexity, properly initialized over-parameterized networks can converge to a zero training loss through the introduction of the Polyak-Lojasiewicz condition. However, these analyses are restricted to quadratic losses such as mean square error, and tend to indicate fast exponential convergence rates that are seldom observed in practice. In this work, we propose to extend these results by analyzing stochastic gradient descent under more generic Lojasiewicz conditions that are applicable to any convex loss function, thus extending the current theory to a larger panel of losses commonly used in practice such as cross-entropy. Moreover, our analysis provides high-probability bounds on the approximation error under sub-Gaussian gradient noise and only requires the local smoothness of the objective function, thus making it applicable to deep neural networks in realistic settings.

**A Single-Loop Gradient Descent and Perturbed Ascent Algorithm for Nonconvex Functional Constrained Optimization**

Songtao Lu

Nonconvex constrained optimization problems can be used to model a number of machine learning problems, such as multi-class Neyman-Pearson classification and constrained Markov decision processes. However, such kinds of problems are challenging because both the objective and constraints are possibly nonconvex, so it is difficult to balance the reduction of the loss value and reduction of constraint violation. Although there are a few methods that solve this class of problems, all of them are double-loop or triple-loop algorithms, and they require oracles to solve some subproblems up to certain accuracy by tuning multiple hyperparameters at each iteration. In this paper, we propose a novel gradient descent and perturbed ascent (GDPA) algorithm to solve a class of smooth nonconvex inequality constrained problems. The GDPA is a primal-dual algorithm, which only exploits the first-order information of both the objective and constraint functions to update the primal and dual variables in an alternating way. The key feature of the proposed algorithm is that it is a single-loop algorithm, where only two step-sizes need to be tuned. We show that under a mild regularity condition GDPA is able to find Karush-Kuhn-Tucker (KKT) points of nonconvex functional constrained problems with convergence rate guarantees. To the best of our knowledge, it is the first single-loop algorithm that can solve the general nonconvex smooth problems with nonconvex inequality constraints. Numerical results also showcase the superiority of GDPA compared with the best-known algorithms (in terms of both stationarity measure and feasibility of the obtained solutions).

**Anticorrelated Noise Injection for Improved Generalization**

Antonio Orvieto · Hans Kersting · Frank Proske · Francis Bach · Aurelien Lucchi

Injecting artificial noise into gradient descent (GD) is commonly employed to improve the performance of machine learning models. Usually, uncorrelated noise is used in such perturbed gradient descent (PGD) methods. It is, however, not known if this is optimal or whether other types of noise could provide better generalization performance. In this paper, we zoom in on the problem of correlating the perturbations of consecutive PGD steps. We consider a variety of objective functions for which we find that GD with anticorrelated perturbations ("Anti-PGD") generalizes significantly better than GD and standard (uncorrelated) PGD. To support these experimental findings, we also derive a theoretical analysis that demonstrates that Anti-PGD moves to wider minima, while GD and PGD remain stuck in suboptimal regions or even diverge. This new connection between anticorrelated noise and generalization opens the field to novel ways to exploit noise for training machine learning models.