Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization

( √ κ −2 ) for deterministic oracles , where defines the level of approximate stationarity and κ is the condition number . Our lower bound matches the best existing upper bound in the and κ dependence up to logarithmic factors . For stochastic oracles , we provide a lower bound of Ω ( √ κ −2 + κ1/3 −4 ) . It suggests that there is a gap between the best existing upper boundO ( κ3 −4 ) and our lower bound in the condition number dependence . 1 Introduction . In this paper , we study the oracle complexity lower bound of the following min-max optimization problem : min x∈X max y∈Y f ( x ; y ) , ( 1 ) where X ⊂ Rm and Y ⊂ Rn are nonempty closed convex sets . Such a problem arises in a wide range of applications , e.g. , two-player zero-sum games [ 38 ] , Generative Adversarial Networks [ 17 ] , robust optimization [ 4 ] , including defending adversarial attacks [ 18 , 29 ] . The research in min-max problems ( 1 ) has a long history . If f is linear in both x and y , the problem is known as bilinear min-max optimization , for which von Neumann ’ s min-max theorem [ 38 ] guarantees the existence of a saddle point , a point ( x∗ , y∗ ) that satisfies strong duality ( i.e . min-max equals max-min ) . Later , Sion [ 44 ] generalized the statement to convex-concave functions under certain regularity conditions , which led to the theory of KKT conditions . Aside from studying the existence of strong duality , Nemirovski [ 33 ] , Nesterov [ 36 ] proposed algorithms that can find approximate saddle points for convex-concave objectives with an O ( 1/ ) convergence rate . The rate was proven to be worst-case optimal even for min-max problems with a bilinear cross-term [ 41 ] . Going beyond the assumption of convex-concavity poses a big challenge . In general nonconvexnonconcave min-max optimization problems , a saddle-point may not exist [ 22 ] . Defining a notion of min-max points that is simultaneously nontrivial and tractable is still an open question . If one uses the 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . standard definition of saddle points as in the strong duality problems , then determining its existence is known to be NP-hard [ 9 ] , and finding an approximate local saddle point is PPAD-complete [ 9 ] . Alternatively , a relaxed definition of min-max point known as Stackelberg equilibrium is guaranteed to exist [ 22 ] , yet only local asymptotic convergence is known [ 22 , 47 ] . Due to the lack of a well-formulated suboptimality measure for general nonconvex-nonconcave problems , recent research has considered problems with special structures . For example , Lin et al . [ 26 , 27 ] studied nonconvex-concave problems , Yang et al . [ 50 ] studied problems satisfying the Polyak-Lojasiewicz inequality , Diakonikolas et al . [ 11 ] studied problems with weak Minty variational inequality solutions , Mangoubi and Vishnoi [ 30 ] considered an algorithm specific equilibrium . Our work focuses on the analysis of gradient oracle complexity in the nonconvex-strongly-concave min-max setting . In the standard convex-concave setup , Lin et al . [ 27 ] designed a set of algorithms that achieve near-optimal gradient complexity for all the three variants of convex-concave min-max optimization . However , in the nonconvex-concave setting , it is unclear whether their algorithm is optimal given that no matching lower bound exists . In this work , our main contribution is as follows : • We construct an explicit example on which no first-order zero-respecting algorithm can achieve less than Ω ( √ κ −2 ) oracle complexity for smooth nonconvex-strongly-concave problems , matching the Õ ( √ κ −2 ) complexity in [ 27 ] . • We further extend our results to the stochastic ( -oracle ) setting , achieving the complexity lower bound Ω ( √ κ −2 +κ1/3 −4 ) . Compared against the results in [ 26 ] , our result suggests that the dependency on the condition number may not be optimal . 1.1 Related work . Upper bounds . Since standard gradient descent-ascent diverges for even the simplest case of bilinear zero-sum games , researchers proposed different variations to achieve convergence in the convex-concave setting , with possibly additional structures . One of the variations is known as extragradient method [ 24 , 46 , 33 , 7 , 49 ] , which takes two gradient steps per iteration . Another line studies the optimistic gradient method [ 43 , 8 , 15 , 31 , 20 ] . Mokhtari et al . [ 32 ] proposed a unified analysis through the lens of proximal update . Some works [ 1 , 27 ] focused on accelerating the known rates in terms of the conditional number dependence . Golowich et al . [ 16 ] studied last iterate convergence . Lee et al . [ 25 ] , Daskalakis et al . [ 10 ] , Wei et al . [ 48 ] studied two-player zero-sum stochastic games . Convergence under the nonconvex-concave setup is less studied . Rafique et al . [ 42 ] studied a proximal version of gradient methods . Lin et al . [ 26 ] studied gradient descent ascent with different time scales . Thekumparampil et al . [ 45 ] proposed an implicit algorithm for the nonconvex-concave setting . Many other works also studied different variations of convergence [ 39 , 23 , 28 , 40 ] . Lower bounds . Our work follows the complexity framework introduced by Nemirovski in the 1980s . The most classical lower bounds are due to Nemirovski and Nesterov , and are discussed in their textbooks [ 35 , 37 ] . Recently , much progress was made [ 13 , 5 , 6 , 3 ] by extending the original analysis to the nonconvex setting . Our work builds upon these results . Another line of works studies lower bounds for quadratic problems in min-max setting and utilizes a different framework sometimes known as Stationary Canonical Linear Iterative ( SCLI ) [ 2 , 41 , 51 , 21 ] . It exploits the closed-form updates for linear dynamics and studies convergence properties via the transition matrices . Concurrently , Zhang et al . [ 52 ] and Han et al . [ 19 ] provided lower complexity bounds similar to ours in the deterministic nonconvex-strongly-concave setting . Their constructions are similar to each other but different from ours . Their constructions can be roughly viewed as a simple addition of the quadratic instance in [ 41 ] and a nonconvex component as in [ 6 ] . Informally speaking , their √ κ factor comes from the rate in [ 41 ] . However , our construction is mainly inspired by the chain-like instance in [ 5 ] . To obtain the √ κ factor , we insert quadratic sub-chains into it to increase its length by a factor of √ κ . One possible advantage of our construction is that the minimization and maximization variables are weakly coupled so that we may have more freedom to modify the component functions to extend our results to other settings such as the nonconvex-concave function class . Zhang et al . [ 52 ] and Han et al . [ 19 ] also provided novel analyses with extensions to the upper and lower bounds in the finite-sum setting , whereas we further extend our results to the stochastic setting . 2 Preliminaries . Before introducing our main results , we describe the problem setup and algorithm complexity in this section . First , we introduce the notation . Next , we define the function class and algorithm class . Finally , we provide the formal definition of optimality measure and gradient complexity . Notation . We use bold lower-case letters to denote vectors and use xi to denote the i-th coordinate of vector x . Let supp ( x ) : = { i : xi 6= 0 } denote the support of x . Let ‖x‖2 and ‖x‖∞ denote its ` 2 and ` ∞ norm respectively . For a matrix A , we use Ai , j to denote its ( i , j ) -th entry and A > its transpose . We use ‖A‖2 to denote its spectral norm and det ( A ) its determinant . We use calligraphic upper-case letters to denote sets , as in X and Y . For a nonempty closed convex set X ∈ Rd , let PX denote the Euclidean projection onto X . We also use a semi-comma besides commas to split minimization and maximization variables of a function . For example , when we write f ( x , z ; y ) , it means x and z are the variables to minimize and y is the variable to maximize . Finally , we use the standard O ( · ) and Ω ( · ) notation , with Õ ( · ) and Ω̃ ( · ) further hiding log factors . 2.1 Function class . Our analysis focuses on the class of smooth nonconvex-strongly-concave functions defined below . Definition 1 . Given L ≥ µ > 0 and ∆ > 0 , we use F ( L , µ , ∆ ) to denote the set of all functions f : X × Y → R for some nonempty closed convex sets X ⊂ Rm and Y ⊂ Rn where m , n ∈ N , which satisfies the following assumptions : 1. f is L-smooth , that is , for every x1 , x2 ∈ X and y1 , y2 ∈ Y , ‖∇f ( x1 ; y1 ) −∇f ( x2 ; y2 ) ‖2 ≤ L ‖ ( x1 , y1 ) − ( x2 , y2 ) ‖2 ; 2 . For any fixed x ∈ X , f is µ-strongly concave in y , that is , for any y1 , y2 ∈ Y , f ( x ; y1 ) ≤ f ( x ; y2 ) +∇yf ( x ; y2 ) · ( y1 − y2 ) − 1 2 µ ‖y1 − y2‖22 ; 3. fm ( 0 ) −minx∈X fm ( x ) ≤ ∆ , where fm ( x ) : = maxy∈Y f ( x ; y ) . For any fixed y ∈ Y , f is potentially nonconvex in x . Note that F ( L , µ , ∆ ) includes functions with domain on Rm × Rn for all m , n ∈ N , following the framework of dimension-free convergence guarantees [ 34 , 37 ] . In Section 5 , we will construct a hard instance with domain dimensions m , n both growing inversely in the required accuracy . 2.2 Algorithm class . In this section , we describe the algorithms of interest for the min-max problems with the function class defined above . In particular , we restrict our analysis to first-order algorithms that optimize objectives using first-order oracles defined below . Definition 2 ( Deterministic first-order oracle ) . The deterministic first-order oracle of a differentiable function f : X → R is a mapping O : x 7→ ( f ( x ) , ∇f ( x ) ) for x ∈ X . Definition 3 ( Stochastic first-order oracle ) . A stochastic first-order oracle with bounded variance σ2 of a differentiable function f : X → R is a mapping O : x 7→ ( f ( x ) , g ( x , ξ ) ) for x ∈ X , where ξ is a random variable satisfying E [ g ( x , ξ ) ] = ∇f ( x ) and E [ ‖g ( x , ξ ) −∇f ( x ) ‖22 ] ≤ σ2 . We say O is a first-order oracle if it is a deterministic or stochastic first-order oracle . Furthermore , we consider first-order algorithms satisfying the zero-respecting assumption . Formally , we define first-order algorithms as follows . Definition 4 ( First-order algorithm ) . A first-order ( zero-respecting ) algorithm is one that for any function f : X → R and its associated first-order oracle Of : x 7→ ( f ( x ) , g ) , the ( t+ 1 ) -th iterate xt+1 satisfies xt+1 ∈ PX ( v ) : supp ( v ) ⊂ ⋃ 0≤i≤t ( supp ( xi ) ∪ supp ( gi ) ) . Definition 4 extends the standard zero-respecting algorithms [ 5 , 3 ] to the constrained setting . It covers most existing first-order methods used in the literature including ( projected ) stochastic gradient descent , adaptive methods , and more importantly , the algorithms used in [ 26 , 27 ] which achieve the upper bounds for nonconvex-strongly-concave min-max optimization in the deterministic and stochastic settings , respectively . There is a standard reduction from a lower bound for zero-respecting algorithms to that for arbitrary deterministic algorithms with deterministic or even stochastic firstorder oracles [ 5 , 3 ] . We defer this extension to future work .

Summary: This paper establishes lower bounds for finding approximate stationary points of min_x max_y f(x;y), where f is nonconvex in x and strongly concave in y. With deterministic gradients, the lower bound matches the convergence rate of the accelerated "Minimax-PPA" algorithm from [Lin et al. '20b], settling the oracle complexity in this setting, combining the bump-function construction of [Carmon et al. '19] with the classic zero-chain construction for convex optimization. A stochastic gradient oracle is constructed, following the same stochastic zero-chain construction as [Arjevani et al. '19], providing a lower bound scaling as \kappa^{1/3} / \eps^4 times the variance.