Efficient Mirror Descent Ascent Methods for Nonsmooth Minimax Problems

√ κ −2 ) under mild conditions , which matches the best known complexity in solving smooth nonconvex-strongly-concave minimax optimization . We conduct the experiments on fair classifier and robust neural network training tasks to demonstrate the efficiency of our new algorithms . 1 Introduction . Minimax optimization recently has attracted increased interest largely due to advance in many machine learning applications such as generative adversarial networks ( GANs ) [ 14 , 41 ] , robust neural networks training [ 32 ] , fair learning [ 31 ] , federated learning [ 10 ] , and policy evaluation [ 43 ] . In the paper , we study the following nonsmooth nonconvex-strongly-concave minimax problem : min x∈X max y∈Y F ( x , y ) = { f ( x , y ) + g ( x ) − h ( y ) } , ( 1 ) where the function f ( x , y ) : X ×Y → R is smooth and possibly nonconvex in x ∈ X and µ-strongly concave in y ∈ Y , and the functions g ( x ) and h ( y ) are convex and possibly nonsmooth . Here X ⊆ Rd and Y ⊆ Rp are compact and convex constraint sets , or X = Rd and Y = Rp . In many machine learning problems , f ( x , y ) generally represents loss function and is a stochastic form , i.e. , f ( x , y ) = Eξ [ f ( x , y ; ξ ) ] , where the random variable ξ follows an unknown data distribution . Here both g ( x ) and h ( y ) frequently denote the nonsmooth regularization terms such as g ( x ) = ν1‖x‖1 and h ( y ) = ν2‖y‖1 with ν1 , ν2 > 0 . In fact , the above problem ( 1 ) comes from many machine learning problems , such as fair classifier , robust training , nonlinear temporal-difference learning in reinforcement learning [ 43 ] and robust federated learning [ 10 ] . When g ( x ) = 0 and h ( y ) = 0 in the problem ( 1 ) , the classic gradient descent ascent ( GDA ) methods [ 40 , 27 ] can effectively solve this problem , which alternatively conducts a gradient descent update on the variable x and a gradient ascent update on the variable y at each iteration . At the same time , some stochastic GDA methods [ 40 , 27 , 30 , 18 , 46 , 17 ] have been proposed to solve the stochastic minimax problem ( 1 ) , where f ( x , y ) = Eξ [ f ( x , y ; ξ ) ] . More recently , some works [ 2 , 4 , 8 ] focus on more general minimax problem ( 1 ) , where both g ( x ) and h ( y ) are possibly nonsmooth . Meanwhile , 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . some ( stochastic ) proximal gradient descent ascent ( PGDA ) methods [ 2 , 4 , 8 ] have been presented to solve the problem ( 1 ) . However , they still suffer from the large sample complexities for finding an stationary point of the minimax problem ( 1 ) without some specific strong assumptions such as KL geometry ( Please see Table 1 ) . In this paper , thus , we propose a class of efficient mirror descent ascent methods by using dynamic mirror function ( i.e. , Bregman function ) . Specifically , our methods perform an adaptive mirror descent update to variable x and an adaptive mirror ascent update to variable y alternatively at each iteration . Our new algorithmic framework can generate many popular methods and their variants by adopting different mirror functions . For example , by adopting the mirror functions ψ ( x ) = 12‖x‖ 2 and φ ( y ) = 12‖y‖ 2 , our methods will include the classic ( proximal ) gradient descent ascent algorithms . Our main contributions are summarized as follows : 1 ) We propose a class of novel mirror descent ascent methods to solve the minimax problem ( 1 ) by using dynamic mirror functions . Moreover , we provide a convergence analysis framework for our mirror descent ascent methods . 2 ) We present a faster deterministic adaptive mirror descent ascent ( MDA ) method , which reaches a lower gradient complexity of O ( √ κ −2 ) than the existing nonsmooth nonconvex minimax methods . Meanwhile , we propose a fast stochastic mirror descent ascent ( SMDA ) method , which requires O ( κ3 −4 ) stochastic gradient evaluations to obtain an -stationary point of the problem ( 1 ) . 3 ) We further propose an accelerated stochastic mirror descent ascent ( VR-SMDA ) method by using the variance reduced technique of SARAH/SNVRG/SPIDER [ 37 , 51 , 12 , 44 ] . Moreover , we prove that our VR-SMDA reaches a lower gradient complexity of O ( κ3 −3 ) . In fact , when our methods solve the minimax problem ( 1 ) without nonsmooth regularization terms , i.e. , g ( x ) = 0 and h ( y ) = 0 , our theoretical results also can apply these minimax problems without nonsmooth regularization terms studied in [ 27 , 28 ] . 2 Related Works . In this section , we review some existing typical minimax optimization methods and stochastic mirror descent methods , respectively . 2.1 Minimax Optimization Methods . Minimax optimization recently has been widely studied in machine learning community . The convergence properties of ( strongly ) convex- ( strongly ) concave minimax optimization have been studied in [ 42 , 26 , 35 , 49 ] . Due to the popularity of nonconvex models in machine learning , many recent studies focused on the nonconvex minimax problems such as robustly deep neural networks training and GANs . For example , some effective gradient descent ascent ( GDA ) methods [ 40 , 38 , 27 , 28 , 46 , 47 , 17 , 18 , 19 , 16 ] have been proposed for the nonconvex- ( strongly ) concave minimax optimization . Specifically , Lin et al . [ 27 ] studied the convergence properties of both deterministic and stochastic GDA methods . Subsequently , Luo et al . [ 30 ] proposed a class of faster stochastic GDA methods based on variance reduced technique of SPIDER [ 12 ] . Huang et al . [ 18 ] proposed an accelerated single-loop stochastic GDA method based on momentum-based variance reduced technique of STORM [ 9 ] . Recently , the nonconvex-nonconcave minimax optimization problems with some specific structures also have been studied in [ 34 , 48 , 11 , 33 ] . Specifically , Yang et al . [ 48 ] and Diakonikolas et al . [ 11 ] have studied a class of specific nonconvex-nonconcave minimax problems satisfying a so-called two-sided Polyak-Lojasiewicz inequality and stampacchia variational inequality , respectively . More recently , Zhang et al . [ 50 ] and Li et al . [ 24 ] studied the lower bound of sample complexity for nonconvex-strongly-concave minimax optimization . Xian et al . [ 45 ] studied the decentralized nonconvex-strongly-concave minimax optimization . Meanwhile , some research works [ 29 , 2 , 4 , 8 ] began to study the nonsmooth nonconvex minimax problem ( 1 ) . 2.2 Mirror Descent Methods . Mirror descent ( a.k.a. , Bregman gradient ) method [ 7 , 3 ] is a powerful optimization tool in machine learning , since it can fit the geometry of optimization problems by choosing proper Bregman functions [ 5 , 6 ] . The mirror descent methods for convex optimization have been studied in [ 7 , 3 ] . Subsequently , Lei et al . [ 23 ] integrated the variance reduced technique to the mirror descent algorithm for stochastic convex optimization . More recently , a variance-reduced adaptive stochastic mirror descent algorithm [ 25 ] has been proposed to solve the nonsmooth nonconvex finite-sum mini optimization . Recently , the mirror descent method also has been used to solve minimax optimization problems . For example , Babanezhad et al . [ 1 ] presented a mirror-type algorithm for convex minimax optimization . Rafique et al . [ 40 ] proposed a class of mirror descent methods for weakly convex minimax optimization . Meanwhile , a new mirror descent-type method [ 34 ] has been proposed to solve a class of nonconvexnonconcave minimax problems with a non-monotone variational inequality structure . To the best of our knowledge , recently few work focuses on explicitly using the mirror-decent-type method to solve the nonsmooth nonconvex minimax problems . 3 Preliminaries . 3.1 Notations . For two vectors x and y , 〈x , y〉 denotes their inner product . ‖ · ‖ denotes the ` 2 norm for vectors and spectral norm for matrices , respectively . ∇xf ( x , y ) and ∇yf ( x , y ) denote the partial derivatives w.r.t . variables x and y respectively , and let ∇f ( x , y ) = ( ∇xf ( x , y ) , ∇yf ( x , y ) ) . ∂g ( x ) denotes the subgradient set of function g ( x ) . Given a convex closed set X , we define a projection operation ProjX ( α ) = arg minx∈X ‖x− α‖2 . Given the mini-batch samples B = { ξi } bi=1 , we let∇fB ( x ) = 1 b ∑b i=1∇f ( x ; ξi ) . Define an increasing σ-algebras Ft : = { B1 , B2 , · · · , Bt−1 } for all t ≥ 2 , then let E [ · ] = E [ ·|Ft ] . 3.2 Standard Mirror Descent Method . Given a ρ-strongly convex and continuously-differentiable function ψ ( x ) , i.e. , 〈x1 − x2 , ∇ψ ( x1 ) − ∇ψ ( x2 ) 〉 ≥ ρ‖x1−x2‖2 , we define a Bregman divergence ( i.e. , Bregman distance ) for any x , z ∈ X : Dψ ( z , x ) = ψ ( z ) − ψ ( x ) − 〈∇ψ ( x ) , z − x〉 . ( 2 ) To solve the problem minx∈X f ( x ) , the mirror descent method [ 7 , 3 ] uses the following form to update the variable x , for all t ≥ 1 xt+1 = arg min x∈X { f ( xt ) + 〈∇f ( xt ) , x− xt〉+ 1 α Dψ ( x , xt ) } , ( 3 ) where α > 0 is stepsize . In the above subproblem ( 3 ) , the first two terms of its objective function is a linear function approximated the function f ( x ) , and the last term is a Bregman distance between x and xt . Note that the constant terms f ( xt ) and 〈∇f ( xt ) , xt〉 can be omitted in the above subproblem ( 3 ) . When choosing ψ ( x ) = 12‖x‖ 2 , we have Dψ ( x , xt ) = 12‖x − xt‖ 2 . Then the mirror descent method will reduce to the standard projected gradient descent method . 3.3 Some Mild Assumptions . In the subsection , we introduce some mild assumptions for the problem ( 1 ) . Assumption 1 . ( Smoothness ) For the deterministic and mini-batch stochastic algorithms ( MDA and SMDA ) , we assume that the function f ( x , y ) has an Lf -Lipschitz gradient , i.e. , for all x1 , x2 ∈ X and y1 , y2 ∈ Y , we have ‖∇f ( x1 , y1 ) −∇f ( x2 , y2 ) ‖ ≤ Lf‖ ( x1 , y1 ) − ( x2 , y2 ) ‖ . ( 4 ) For our variance-reduced stochastic algorithm ( VR-SMDA ) , we assume that each component function f ( x , y ; ξ ) has an Lf -Lipschitz gradient , i.e. , for all x1 , x2 ∈ X and y1 , y2 ∈ Y , we have ‖∇f ( x1 , y1 ; ξ ) −∇f ( x2 , y2 ; ξ ) ‖ ≤ Lf‖ ( x1 , y1 ) − ( x2 , y2 ) ‖ , ∀ξ . ( 5 ) In Assumption 1 , the inequality ( 4 ) is commonly used in the minimax optimization [ 27 , 2 , 4 , 8 ] . While the inequality ( 5 ) is frequently used in the variance-reduced stochastic optimization [ 30 , 18 ] . Assumption 2 . Each component function f ( x , y ; ξ ) has an unbiased stochastic gradient with bounded variance σ2 , i.e. , E [ ∇f ( x , y ; ξ ) ] = ∇f ( x , y ) , E‖∇f ( x , y ; ξ ) −∇f ( x , y ) ‖2 ≤ σ2 . ( 6 ) Assumption 3 . The function f ( x , y ) is µ-strongly concave w.r.t y , i.e. , for all x ∈ X and y1 , y2 ∈ Y , we have ‖∇yf ( x , y1 ) −∇yf ( x , y2 ) ‖ ≥ µ‖y1 − y2‖ . Then the following inequality holds f ( x , y1 ) ≤ f ( x , y2 ) + 〈∇yf ( x , y2 ) , y1 − y2〉 − µ 2 ‖y1 − y2‖2 . ( 7 ) Assumption 4 . The functions g ( x ) and h ( y ) are convex but possibly nonsmooth . Assumption 3 shows that the function f ( x , y ) is µ-strongly concave w.r.t y . Assumption 4 shows that the function h ( y ) is convex . Thus , the function { f ( x , y ) − h ( y ) } is strongly concave in y ∈ Y , there exists a unique solution to the problem maxy∈Y { f ( x , y ) − h ( y ) } for any x . Let y∗ ( x ) = arg maxy∈Y { f ( x , y ) − h ( y ) } , and Φ ( x ) = f ( x , y∗ ( x ) ) − h ( y∗ ( x ) ) = maxy∈Y { f ( x , y ) − h ( y ) } . Assumption 5 . For any α ∈ R , the sub-level set { x : Φ ( x ) + g ( x ) ≤ α } is compact . The function Φ ( x ) + g ( x ) is bounded below in X , i.e. , F ∗ = infx∈X { Φ ( x ) + g ( x ) } > −∞ . Assumption 5 is frequently used in nonsmooth minimax optimization [ 8 ] . In fact , when h ( y ) = c where c is a constant , we can only assume the function Φ ( x ) + g ( x ) is bounded below in X instead of Assumption 5 .

Summary: This paper proposes efficient mirror descent ascent (MDA) methods to solve the nonsmooth nonconvex-strongly-concave minimax problems. The SOTA convergence rates are established both for deterministic and stochastic settings. In practice, some experimetal results are provided to demonstrate the efficiencies of the proposed MDA methods. Overall, this is a very good paper.