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: The paper proposes a second-order cubic regularization type method for an unconstrained min-max problem in which the function is twice continuously differential, strongly concave w.r.t the max decision variable, with Lipschitz continuous first and second derivatives. The proposed method comprises a nested gradient ascent process of the max variable, with a dynamic number of iterations that guarantee the desired properties to support the convergence resutls, and a cubic proximal update on the min variable.  The method is proved to converge to second-order stationary point, under standard assumptions.