Escaping Saddle Points in Nonconvex Minimax Optimization via Cubic-Regularized Gradient Descent-Ascent

1 INTRODUCTION . Nonconvex minimax optimization is a popular optimization framework that has broad applications in modern machine learning , including game theory ( Ferreira et al. , 2012 ) , generative adversarial networks ( Goodfellow et al. , 2014 ) , adversarial training ( Sinha et al. , 2017 ) , reinforcement learning ( Qiu et al. , 2020 ; Ho and Ermon , 2016 ; Song et al. , 2018 ) , etc . A standard nonconvex minimax optimization problem is shown below , where f is a smooth nonconvex function in x. min x∈Rm max y∈Rn f ( x , y ) . ( P ) In the existing literature , many optimization algorithms have been developed to solve different types of minimax problems . Among them , a simple and popular algorithm is the gradient descent-ascent ( GDA ) , which alternates between a gradient descent update on x and a gradient ascent update on y in each iteration . Specifically , the global convergence of GDA has been established for minimax problems under various types of global geometries , such as convex-concave-type geometry ( f is convex in x and concave in y ) ( Nedić and Ozdaglar , 2009 ; Du and Hu , 2019 ; Mokhtari et al. , 2020 ; Zhang and Wang , 2021 ) , bi-linear geometry ( Neumann , 1928 ; Robinson , 1951 ) and PolyakŁojasiewicz geometry ( Nouiehed et al. , 2019 ; Yang et al. , 2020 ) , yet these geometries are not satisfied by general nonconvex minimax problems . Recently , many studies proved the convergence of GDA in nonconvex minimax optimization for both nonconvex-concave problems ( Lin et al. , 2020 ; Nouiehed et al. , 2019 ; Xu et al. , 2020d ) and nonconvex-strongly-concave problems ( Lin et al. , 2020 ; Xu et al. , 2020d ; Chen et al. , 2021 ) . In these studies , it has been shown that GDA converges sublinearly to a stationary point where the gradient of a certain envelope function of the minimax problem vanishes . Although GDA can find first-order stationary points of nonconvex minimax problems , such a type of convergence guarantee does not rule out the possibility that GDA may get stuck at suboptimal saddle points of the envelope function , which are well known to be the major challenge for training high-dimensional machine learning models ( Dauphin et al. , 2014 ; Jin et al. , 2017 ; Zhou and Liang , 2018 ) . On the other hand , while numerous algorithms have been developed for escaping saddle points in conventional nonconvex optimization , e.g. , first-order algorithms ( Ge et al. , 2015 ; Jin et al. , 2017 ; Carmon and Duchi , 2016 ; Liu and Yang , 2017 ) and second-order algorithms ( Nesterov and Polyak , 2006 ; Agarwal et al. , 2017 ; Yue et al. , 2019 ; Zhou et al. , 2018 ) , such a type of algorithm has not been developed for escaping saddle points in nonconvex minimax optimization . Therefore , we want to ask the following fundamental questions . • Q : How to develop a provably convergent Newton-type GDA algorithm that can effectively escape saddle points in nonconvex minimax optimization ? How fast it converges ? Developing and analyzing such an algorithm is nontrivial due to the following reasons : 1 ) we need to have a good understanding and characterization of both the first-order and second-order information of nonconvex minimax problems ; 2 ) we need to develop a computationally feasible and efficient GDA algorithm that can leverage the local curvature of the function to escape saddle points ; 3 ) we aim to develop a unified analysis framework that can characterize the convergence rate of this algorithm under different types of nonconvex geometry of the minimax problem . In this paper , we provide comprehensive answers to these questions . We develop the first Newton-type GDA algorithm that escapes strict saddle points and converges to second-order stationary points in nonconvex-strongly-concave minimax optimization . We also characterize the global and local convergence rates of this algorithm under various types of nonconvex geometry . We summarize our contributions as follows . 1.1 OUR CONTRIBUTIONS . We consider the minimax optimization problem ( P ) , where f is a twice-differentiable and nonconvexstrongly-concave function , and its gradient and Jacobian matrices are Lipschitz continuous . Define an envelope function Φ ( x ) : = maxy∈Rn f ( x , y ) . The existing GDA algorithms can only find first-order stationary points that satisfy ∇Φ ( x∗ ) = 0 . In this paper , we develop a Newton-type GDA algorithm that converges to second-order stationary points of the nonconvex minimax problem ( P ) . Specifically , we propose Cubic-GDA – a Newton-type GDA algorithm that leverages the classical cubic regularization technique to escape saddle points . Different from the standard cubic regularization algorithm that uses the Hessian information of the function , the Hessian of Φ ( x ) is not directly available in nonconvex minimax optimization , and hence we develop a rigorous and computationally feasible scheme in Cubic-GDA to estimate the Hessian . We study the global convergence property of Cubic-GDA in general nonconvex-strongly-concave optimization . Specifically , we show that Cubic-GDA admits an intrinsic potential functionH ( x , x′ , y ) ( see Proposition 2 ) , which monotonically decreases along the trajectory of Cubic-GDA . Based on the monotonicity of this potential function , we show that every limit point of the parameter sequence { xt } t generated by Cubic-GDA is a second-order stationary point of the minimax problem . We further analyze the aymptotic convergence rates of Cubic-GDA under a broad spectrum of the local nonconvex Łojasiewicz gradient geometry . In this case , we show that Cubic-GDA converges to a unique limit point , which is a second-order stationary point . Moreover , as the geometry parameter increases ( i.e. , sharper local geometry ) , the convergence rate of Cubic-GDA accelerates from sublinear convergence up to super-linear convergence , as we summarize in Table 1 below . In particular , these convergence rates are orderwise faster than those of the standard GDA under the same type of nonconvex geometry ( Chen et al. , 2021 ) 1 . 1Note that the geometry parameter θ in this paper corresponds to 1 1−θ in ( Chen et al. , 2021 ) . 1.2 RELATED WORK . Deterministic GDA algorithms : Many studies characterized the convergence of GDA in nonconvex minimax optimization . Specifically , Lin et al . ( 2020 ) ; Nouiehed et al . ( 2019 ) ; Xu et al . ( 2020d ) studied the convergence of GDA in the nonconvex-concave setting whereas Lin et al . ( 2020 ) ; Xu et al . ( 2020d ) focused on the nonconvex-strongly-concave setting . In these general nonconvex settings , it is shown that GDA converges to a certain stationary point at a sublinear rate . Recently , Chen et al . ( 2021 ) proved the parameter convergence of proximal-GDA in regularized nonconvex-stronglyconcave optimization under the Kurdyka-Łojasiewicz geometry . The convergence rates obtained there are orderwise slower than that of Cubic-GDA . Yang et al . ( 2020 ) studied an alternating gradient descent-ascent ( AGDA ) algorithm in which the gradient ascent step uses the current variable xt+1 instead of xt . Xu et al . ( 2020d ) studied an alternating gradient projection algorithm which applies ` 2 regularizer to the local objective function of GDA followed by projection onto the constraint sets . Daskalakis and Panageas ( 2018 ) ; Mokhtari et al . ( 2020 ) ; Zhang and Wang ( 2021 ) analyzed optimistic gradient descent-ascent ( OGDA ) which applies negative momentum to accelerate GDA . Mokhtari et al . ( 2020 ) also studied an extra-gradient algorithm which applies two-step GDA in each iteration . Nouiehed et al . ( 2019 ) studied multi-step GDA where multiple gradient ascent steps are performed , and they also studied the momentum-accelerated version . Cherukuri et al . ( 2017 ) ; Daskalakis and Panageas ( 2018 ) ; Jin et al . ( 2020 ) studied GDA in continuous time dynamics using differential equations . Adolphs et al . ( 2019 ) analyzed a second-order variant of the GDA algorithm . In a concurrent work ( Luo and Chen , 2021 ) , the authors proposed and studied the same Cubic-GDA algorithm . They characterize the computation complexity under a special type of inexactness that approximates the inverse Jacobian using matrix Chebyshev polynomials . As a comparison , this study focuses on analyzing the global and local convergence properties of Cubic-GDA . Stochastic GDA algorithms : Lin et al . ( 2020 ) ; Yang et al . ( 2020 ) analyzed stochastic GDA and stochastic AGDA , which are direct extension of GDA and AGDA to the stochastic setting . Variance reduction techniques have been applied to stochastic minimax optimization , including SVRG-based ( Du and Hu , 2019 ; Yang et al. , 2020 ) , SPIDER-based ( Xu et al. , 2020c ) , SREDA ( Xu et al. , 2020b ) , STORM ( Qiu et al. , 2020 ) and its gradient free version ( Huang et al. , 2020 ) . Xie et al . ( 2020 ) studied the complexity lower bound of first-order stochastic algorithms for finite-sum minimax problem . Cubic regularization ( CR ) : CR algorithm dates back to ( Griewank , 1981 ) , where global convergence of the algorithm is established . In Nesterov and Polyak ( 2006 ) , the authors analyzed the convergence rate of CR to second-order stationary points for nonconvex optimization . In ( Nesterov , 2008 ) , the authors established the sub-linear convergence of CR for solving convex smooth problems , and they further proposed an accelerated version of CR with improved sub-linear convergence . Recently , Yue et al . ( 2019 ) studied the asymptotic convergence properties of CR under the error bound condition , and established the quadratic convergence of the iterates . Recently , Hallak and Teboulle ( 2020 ) proposed a framework of two directional method for finding second-order stationary points in general smooth nonconvex optimization . This main idea of the algorithm is to search for a feasible direction toward the solution and is not based on cubic regularization . Several other works proposed different methods to solve the cubic subproblem of CR , e.g. , ( Agarwal et al. , 2017 ; Carmon and Duchi , 2016 ; Cartis et al. , 2011b ) . Another line of work aimed at improving the computation efficiency of CR by solving the cubic subproblem with inexact gradient and Hessian information . In particular , Ghadimi et al . ( 2017 ) proposed an inexact CR for solving convex problem . Also , Cartis et al . ( 2011a ) proposed an adaptive inexact CR for nonconvex optimization , whereas Jiang et al . ( 2017 ) further studied the accelerated version for convex optimization . Several studies explored subsampling schemes to implement inexact CR algorithms , e.g. , ( Kohler and Lucchi , 2017 ; Xu et al. , 2020a ; Zhou and Liang , 2018 ; Wang et al. , 2018 ) . 2 PROBLEM FORMULATION AND PRELIMINARIES . In this section , we introduce the problem formulation and present some preliminary results that will be used in the analysis . Notation : For notation simplicity , we denote∇1f , ∇2f as the gradients with respect to the first and the second input arguments of f , respectively . We also denote ∇11f , ∇22f as the Jacobian matrices where the second-order derivatives are taken over the first and second arguments of f , respectively . Moreover , we denote ∇12f as the Jacobian matrix where the second-order derivative is taken over the first argument of f and followed by the second argument , and∇21f is defined in a similar way . We consider the minimax optimization problem ( P ) that satisfies the following standard assumptions . Assumption 1 . The minimax optimization problem ( P ) satisfies : 1 . Function f ( · , · ) is L1-smooth and function f ( x , · ) is µ-strongly concave for all fixed x ; 2 . The Jacobian matrices∇11f , ∇12f , ∇21f , ∇22f are L2-Lipschitz ; 3 . Function Φ is bounded below and has compact sub-level sets . To elaborate , item 1 considers the class of nonconvex-strongly-concave functions f that has been widely studied in the minimax optimization literature ( Lin et al. , 2020 ; Jin et al. , 2020 ; Xu et al. , 2020d ; Lu et al. , 2020 ) . Items 2 assumes that the block Jacobian matrices of f are Lipschitz , which is a standard assumption for analyzing many second-order optimization algorithms ( Nesterov and Polyak , 2006 ; Agarwal et al. , 2017 ; Yue et al. , 2019 ) . Moreover , item 3 guarantees that the minimax problem has at least one solution . By strong concavity of f ( x , · ) , it is clear that the maximizer y∗ ( x ) : = arg maxy∈Rn f ( x , y ) is unique for every x ∈ Rm . In particular , if x∗ is a second-order stationary point of Φ ( x ) , then ( x∗ , y∗ ( x∗ ) ) is the desired solution of the minimax problem ( P ) . Define an envelope function Φ ( x ) : = maxy∈Rn f ( x , y ) . Then the minimax problem ( P ) is equivalent to the minimization problem . minx∈Rm Φ ( x ) , where Φ ( x ) = maxy∈Rn f ( x , y ) . As we show in item 2 of Proposition 1 later , this envelope function Φ ( x ) is smooth and nonconvex . The existing GDA algorithms can only find first-order stationary points of the minimax problem that satisfy∇Φ ( x∗ ) = 0 . In this paper , we aim to develop a provably convergent algorithm that can find second-order stationary points x∗ of the function Φ ( x ) that satisfy the following set of conditions . ( Second-order stationary ) : ∇Φ ( x∗ ) = 0 , ∇2Φ ( x∗ ) 0 . In the existing literature , many optimization algorithms have been developed for finding secondorder stationary points in conventional nonconvex minimization problems . This includes first-order algorithms ( Ge et al. , 2015 ; Jin et al. , 2017 ; Carmon and Duchi , 2016 ; Liu and Yang , 2017 ) and second-order algorithms ( Nesterov and Polyak , 2006 ; Agarwal et al. , 2017 ; Yue et al. , 2019 ; Zhou et al. , 2018 ) . However , these algorithms are not directly applicable to solve the problem ( P ’ ) , as the function Φ ( x ) involves a special maximization structure and hence its specific function form Φ as well as the gradient∇Φ and Hessian∇2Φ are not available in practice . Instead , our algorithm design can only leverage information of the bi-variate function f . Next , we present some important properties regarding the gradient and Jacobian matrices of the functions f ( x , y ) and Φ ( x ) . Throughout , we denote κ = L1/µ as the condition number . Proposition 1 . Let Assumption 1 hold . Then , the following statements hold . 1 . Mapping y∗ ( x ) is κ-Lipschitz continuous ; 2 . Function Φ ( x ) is L1 ( 1 + κ ) -smooth and ∇Φ ( x ) = ∇1f ( x , y∗ ( x ) ) ; 3 . Define G ( x , y ) = ∇11f ( x , y ) − ∇12f ( x , y ) [ ∇22f ( x , y ) ] −1∇21f ( x , y ) . Then , G is a Lipschitz mapping with constant LG = L2 ( 1 + κ ) 2 , i.e. , ‖G ( x′ , y′ ) −G ( x , y ) ‖ ≤ LG‖ ( x′ , y′ ) − ( x , y ) ‖ ; 4 . The Hessian of Φ satisfies∇2Φ ( x ) = G ( x , y∗ ( x ) ) , which is Lipschitz continuous with constant LΦ = LG ( 1 + κ ) = L2 ( 1 + κ ) 3 . The first two items characterize the gradient of the envelope function Φ in terms of the partial gradient of the bi-variate objective function f . They are proved in the previous work ( Lin et al. , 2020 ) and we include them for completeness . On the other hand , the last two items further characterize the Hessian of Φ in terms of the block Jacobian matrices of f . As we present in the next section , the Lipschitz continuous Hessian ∇2Φ ( x ) allows us to develop a cubic-regularization-based algorithm for finding second-order stationary points . We also note that the proof of items 3 & 4 are not trivial . Specifically , we need to first develop bounds for the spectrum norm of the block Jacobian matrices in Lemma 1 ( see the first page of the appendix ) , which helps prove the Lipschitz continuity of the G mapping in item 3 . Moreover , we leverage the optimality condition of f ( x , · ) to derive an expression for the maximizer mapping y∗ ( x ) ( see ( 15 ) in the appendix ) , which is used to further prove item 4 .

Summary:  In this paper, the authors develop a second-order method based on the cubic regularization technique for solving nonconvex-strongly-concave minimax problems. They analyze its convergence rate under the standard smoothness assumptions as well as under the local Lojasiewicz gradient geometry. Notably, this is the first algorithm that provably converges to a second-order stationary point and hence escapes strict saddle points.