Efficient Mirror Descent Ascent Methods for Nonsmooth Minimax Problems
Authors: Feihu Huang
Xidong Wu
Heng Huang
Venue: NeurIPS 2021 Poster (2021)
Keywords: Mirror Descent
Minimax Optimization
Nonconvex-Strongly-Concave
Nonsmooth
Fair Classifier
Robust Neural Network Training

Abstract: In the paper, we propose a class of  efficient mirror descent ascent methods to solve the nonsmooth nonconvex-strongly-concave minimax problems by using dynamic mirror functions, and introduce a convergence analysis framework to conduct rigorous theoretical analysis for our mirror descent ascent methods. For our stochastic algorithms, we first prove that the mini-batch stochastic mirror descent ascent (SMDA) method obtains a gradient  complexity of $O(\kappa^3\epsilon^{-4})$ for finding an $\epsilon$-stationary point, where $\kappa$ denotes the condition number. Further, we propose an accelerated stochastic mirror descent ascent (VR-SMDA) method based on  the variance reduced technique. We prove that our VR-SMDA method achieves a lower gradient complexity of  $O(\kappa^3\epsilon^{-3})$. For our deterministic algorithm, we prove that our deterministic mirror descent ascent (MDA) achieves a lower gradient complexity of $O(\sqrt{\kappa}\epsilon^{-2})$ 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.

Efficient Mirror Descent Ascent Methods for Nonsmooth Minimax Problems
Feihu Huang, Xidong Wu, Heng Huang
Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, USA sanggffihu211@@gmaaa com, xidong wwwpgggwwwppttteeeee leng.huang@pic
Abstract
In the paper, we propose class of efficient mirror descent ascent methods minimax problems by using
solve the nonsmooth vonconnex-stronglyy-ooneee dynamic mirror functions, and introduce a convergence analysis framework to conduct rigorous theoretical analysis for our mirror descent ascent methods. For our stochastic algorithms, we first prove that the mini-bated stochastic mirror descent ascent (SMDA) method obtains gradient complexity of ( for finding an e-stationary point, where x denotes the condition number. Further, we propose accelerated stochastic mirror descent ascent (VR-SMDA) method based on the variance reduced technique. We prove that our VR-SMDA method achieves lower gradient complexity of O For our deterministic algorithm, prove that our deterministic mirror descent ascent (MDA) achieves a lower gradient complexity of O(yie" ²² under mild conditions, which matches the best known complexity in solving smooth nonconvex wwwpenewe--tttogy--ooneeeo
minimax optimization. We conduct the experiments fair classifier and robust neural network training tasks to demonstrate the efficiency of our new algorithms.
Introduction
Minimax optimization recently has attracted increased interest largely due to advance in many
machine learning applications such as generative adversarial networks (GANs) [14] 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-eeeeee
minimax problem:
min max F F(x,y) = (f v) + g(x) - h(3)),
(1)
xEX yEY
where the function f(x, X R smooth and possibly nonconvex in x E X and 4-strongly concave in y E )/ and the functions g(z) and h(y) convex and possibly nonsmooth. Here X' C Rd and )/ C IRP are compact and convex constraint sets, or X' = I and y RP. In many machine learning problems, [xx y) generally represents loss function and a stochastic form, i.e. f(z, y) Eg [f y: {),, where the random variable & follows unknown data distribution. Here both g(z) and ((y)) frequently denote the nonsmooth regularization terms such as g(x) = vi /r||=r/i and hvy) 12 /2]1y/11 with V1,
O. In fact, the above problem ( comes from many machine learning problems, such as fair classifier, robust training, nonlinear temporal-dine differrnce learning in reinforcement learning 4 and robust federated learning [10].
When (xx) ) and h
the problem (T). the classic gradient descent ascent (GDA) methods 40 27 can effectively solve this problem, which alternatively conducts a gradient descent update gradient ascent update on the variable y at each iteration. the same time,
on variable
30 18 46 17] have been proposed to solve the stochastic minimax problem [T]. where f(z, v) Eg [f (x, y {) More recently, some works 2 4 g focus on
some stochastic GDA methods [40] 27
more general minimax problem where both g(z) and h(y) are possibly nonsmooth. Meanwhile,
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Table 1: Gradient complexity of the representative first-order methods for obtaining an c-stationary
point of the nonsmooth nonconvex minimax problem (T). Note that these comparison methods don't t
rely specific strong assumptions the problem Since the convergence properties of the
deterministic proximal gradient descent ascent (PGDA) g build on the Hurdyka-Lojjsiewin
(KL)
geometry assumption, it be excluded. Here к denotes condition number of objective function / [xx y)
in variable y, Since HiBSA algorithm 29] does not provide explicit dependence on K, we use ((K).
Type
Algorithm
Reference
Loop(s)
Gradient Complexity
HiBSA
129
Single
O(p(k)e --
MAPGDA
21
Double
0
Deterninistic
PAGDA
4
Single
O(Kte
MDA
Ours
Single
OlVKie
PASGDA
Single
Stochastic
SMDA
Ours
Single
O
VR-SMDA
Ours
Double
some (stochastic) proximal gradient descent ascent (PGDA) methods g have been presented to solve the problem (T). However, they still suffer from the large sample complexities for finding an stationary point of the minimax problem (T) without some specific strong assumptions such as KL.
geometry (Please see [..
In this paper, thus, we propose 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 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 (zx) = 1 |||||| and ((v) = 1 | our methods will include the classic (proximal) gradient descent ascent algorithms. Our main contributions are summarized as follows:
1) We propose class of novel mirror descent ascent methods to solve the minimax problem by using dynamic mirror functions. Moreover, we provide 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() than the existing nonsmooth nonconvex minimax methods. Meanwhile, we propose a fast stochastic mirror descent ascent (SMDA) method, which requires O stochastic gradient evaluations to obtain an c-stationary point of the problem II.
3) We further propose an accelerated stochastic mirror descent ascent "R-SMDA) method by using the variance reduced technique of SARAH/SNVR//SPIDER 37 we prove that our VR-SMDA reaches lower gradient complexity of O
Moreover,
In fact, when our methods solve the minimax problem without nonsmooth regularization terms, ie., 9(x) = 0 and lu(y) 0, our theoretical results also can apply these minimax problems without nonsmooth regularization terms studied (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
g [6] have been proposed for the nonconvex- (stronnly concave minimax optimization. Specifically, Lin et al. (27] studied the convergence properties of both deterministic and stochastic GDA methods. Subsequently, I Luo et al. (30) proposed class of faster stochastic GDA methods based on variance reduced technique of SPIDER [1]. Huang et al. [18] proposed
17
18

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an accelerated single-loop stochastic GDA method based on momentum-basee variance reduced technique of STORM 99. Recently, the manconvve--noconcae minimax optimization problems
with some specific structures also have been studied in 34 48
Specifically, Yang et al. 48 and Diakonikolas al. [1] have studied a class of specific nonconvex -onoonxnx-nnnncccveee minimax
33
inequality and stampacchia variational inequality, respectively. More recently, Zhang et al. [50] and Li al. [24] studied the lower bound of minimax optimization. Xian et al. [45] studied
problems satisfying a so-called two-sided Volyak-Lojasieww
sample complexity for noneonnve--stronglyycooo the decentralized ppnconvee-stronnly-connee minimax optimization. Meanwhile, some research began study the nonsmooth nonconvex minimax problem
works 29
4 8
2.2
Mirror Descent Methods
Mirror descent (a.k.a., Bregman gradient) method 73 is powerful optimization tool in machine learning, since it can fit the geometry of optimization problems by choosing proper Bregman functions 6] The mirror descent methods for convex optimization have been studied in 7 []. 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 al. 400 proposed 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 nonconvex- nonconcave minimax problems with 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.
variance-redueed
3 Preliminaries
3.1 Notations
For two vectors x and y. (r, denotes their inner product. || denotes the lg norm for vectors and spectral norm for matrices, respectively. V V=f(2, y) and V a y) denote the partial derivatives w.r.t. variables r and y respectively, and let ff 3) = f 8). f v)). 8g(x) denotes the subgradient set of function g Given a convex closed set X, we define a projection operation ( ) = arg min || - alla. Given the mini-batch samples = (E*) ) we let V/s(2) =
E 11 V Vf(x;()) Define an increasing o-algebras F. == (Bi, B2, BB for all t 2 2, then let E E Fi).
3.2 Standard Mirror Descent Method
Given p-strongly convex and continuously -ir.ouull--diffeenttabllel function (x)x), i.e. (2) - T2, Ve((z)) V(((z)) 2 pl - I2ll?, we define a Bregman divergence (i.e., Bregman distance) for any E X::
(= (==) ((x) (Vr()),; - -x)-
To solve the problem mingex fz)) the mirror descent method 7 uses the following form to update the variable x, for all 2
1 Do(,
It- arg min ZEX
f(x:) + (Vf(m)), (()))+((V(()...) -x) +
(3)
where >> 0 is stepsize. In the above subproblem 5 the first two terms of its objective function is linear function approximated the function f(x), and the last term is a Bregman distance between x and z₁, Note that the constant terms fzx) and fZ (. xr) can be omitted in the above subproblem
(3). When choosing (z) = 5 ||~ we have D xx) = l| - x; 12 Then the mirror descent method will reduce to the standard projected gradient descent method.
3.3 Some Mild Assumptions
In the subsection, introduce some mild assumptions for the problem
3

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Assumption 1. (Smootiness) For the deterministic and mathi-batch stochastic algorithms (MDA and SMDA), we assume that the function (a. y) has an L-pischittz gradient, i.e., for all 2 22 X and yr, /2 Y, we have
[V yn) - (2 v2)l S 1y1((11,81) - (x2, y2)ll-
(4)
For our variance-red reduced stochastic algorithm (VR-SMDA), we assume that each component fæz Vi has an L Lipischchttz gradient, i.e., for all 11, E X' and y1 V2 3, we have
[2 yn; E) f22, U2: E) & Lfl v1) - (x2, v2) VV.
(5)
2 While the inequality (5) is frequently used in the variance-reducee stochastic optimization 30
In Assumption 1, the inequality 4 is commonly used in the minimax optimization 227
4
g
18
Assumption 2. Each component function [(, y) has an unbiased stochastic gradient with bounded variance o², Le.,
[V v) = Vf(x, v),
E V - Vf(r. v) 5
(6)
Assumption 3. The function fa, y) p--strongly concave w.n. y. i.e., for all X and yı, Y2 E we have V fx. vzf 02)1 2 pllvi 3/2l- Then the following inequality holds
V.
((r S (xx, + (V JyJ(a, (2),))) v1 y2) 2
12112
Assumption 4. The functions g(x) and (yy) are convex but possibly nonsmooth.
Assumption 3 shows that the function f(z y) is u-strongly concave w.r.t y. Assumption shows
that the function h(y) is convex. Thus, the function {f(x, y) h(y)} is strongly concave in y e,
there exists a unique solution to the problem max y) h(v)} for any x. Let (x) (x)= = arg max (f(x,))
h(y)), and ((x) = f(z, y² (z)) h(y" (x)) max (f(z y) h(y))-
Assumption 5. For any E R, the sub-level set {x (xx) + g(x) < a) is compact. The function (xx) + g(x) is bounded below in X, i.e., F infaex ((() + g(x)} -00.
Assumption 5 is frequently used in nonsmooth minimax optimization g In fact, when h(y) where e is a constant, we can only assume the function (xx) + g(x) is bounded below in X' instead of Assumption 5.
4 Mirror Descent Ascent Methods
In the section, we propose a class of novel mirror descent ascent methods to solve the problem I Specifically, we first propose a deterministic mirror descent ascent (MDA) method, and stochastic mirror descent ascent (SMDA) method. Then we further present an accelerated stochastic mirror descent ascent (VR-SMDA) using variance reduced technique of SPIDER 12
4.1 MDA and SMDA algorithms
When (x, y) is deterministic function, we propose a deterministic mirror descent ascent (MDA)
method to solve the deterministic problem [T]. When f(x. v) [e y; {)] stochastic function, we propose a stochastic mirror descent ascent (SMDA) to solve the stochastic problem (I)
Specifically, A ssoow the algorithmic framework of the MDA and SMDA algorithms.
In Algorithm [
we use (stochastic) mirror decent to update variable x, and simultancously use
(stochastic) mirror ascent to update variable y. Specifically, step 7 of Algorithm I mirror descent to update x,
we use the
It+1 arg min ((v,, x) Do, (I, +((x)) xE
(8)
7t.
sEX (f We) + (v,, - rz) + 1 D (x, + )(x)}-
arg min
(9)
7t
In fact, we omit the constant terms f(x, y/) and (ve, Nx.) the above subproblem 88) In the above subproblem (9) the first two terms of its objective function is linear function approximated the function [xx y) based on (stochastic) derivative estimator vi, and the third term is Bregman distance between and x; based on Bregman function th. Since the function g(x) is possibly nonsmooth, we

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Algorithm 1 (Stochastic) Mirror Descent Ascent Algorithm
1: Input: T, stepsizes (7: 0, 0, (0, 1) mini-batch size b;
2: initialize: 21 E X and y1 E J/
3: for t = 1,2...., do
4: MDA: Compute partial derivatives U, = Vxf(x,, yr) and wi Vy Vy,f (2x, 3t);
5: SMDA: Generate randomly mini-batch samples B, = {&i)-- with [B,] = b, and compute
stochastic partial derivatives 1h af (2t, De) and we V Vyfe, (z);
6: Given the mirror functions 1 and dz;
7: 2++1 arg mingex { (wr, z) 1 De (a, vz) + g(x)};
8: 1 5/- + 1/e (Wt++1 We) where Ø++1 arg maxyey (wr,,) v)
Xe Do. (v. h(v)):
9: end for
10: Output: xę and i/2 chosen uniformly random from {fe, Ut y1
Output: (for theoretical) zg and y/ chosen uniformly random from (ae, ye
12: Output: (for practical) 2t and yr.
keep it in the above subproblem (9) as the standard proximal descent algorithm 39 Similarly, at we use the mirror ascent to update y,
step 8 of Algorithm D
J = arg max yEy
{ww,, v)
Doo (y. - [(v)}
(10)
Xx
arg max {f(x v1) + (wi,y y vo) 1 Do. h(y)). Ae
(11)
yEy
In the above subproblem ( the first two terms of its objective function is linear function approximated the function [(x,)) y) based on (stochastic) derivative estimator Wi, and the third term a Bregman distance between y and yt based on Bregman function d,. Moreover, the step 8 of
Algorithm we further use a momentum iteration to update y.
When Bregman functions vhe (x) ||= 12 and (3)) llvll for all t > 1, we have De, x2)
2 ||æ 2x and Da, (y. v1) 3/2 |ly 122 Under this case, Algorithm 1 will reduce the standard (stochastic) proximal gradient descent ascent algorithm. When Bregman functions (/((x)) xx² H
and (y) = _ G.y for all 1. 2 1, we have Da, (ar, xz. = 1 (r - x²,)² H, - xx)) and Do, (y: 91)
(y ye) Gr - )t). where H, 2 pla and G. ł plp. For example, given a E (0. 1) and p > 0, we can generate the matrices H, and G, like as in Adam-type algorithms [2] 20) defined
io 0, é a + (1 a V MiE)
He diag p), t t2
(12)
whp 0, w. = a,. + (1 - a) V,J ve: (e: E.).
G. = diag( / W1 + p), t 121
(13)
Under this case, our SMDA algorithm will reduce novel adaptive gradient descent ascent algorithm.
In the problem II. the functions g(x) and h(y) are generally nonsmooth, e.g., g(x) vi ||||||
and 1(()) V2 wil with V1 > 0, 1/2 > O. When Bregman functions (x) = 1x2 He and
(x(y) yy T' G.y for all 2 1, and the matrices H; and G, are diagonal, e.g., generated from the above and 13), we can use the soft thresholding operator S(a, A) = sign max(lal - 0) =
arg min,
+ }|z{} to obtain the closed-form solutions of the following subproblems:
min (1r, x) + ZEX
(a H(( xZ) + m111=/11.
(14)
2x
max yEY
(wwe,a) v)
2x: (y )² G' - w)) - -
(15)
where H; = diag(hi...... ha,e) with hi, > 0 for E [d]. and G. = diag(o1.,. .pp)) with
Ij.1 > 0 ffr E [p]. Without loss of generality, let X' = IRd and y = RR,, we have
8(1
7
7 arg min aigmin { (0x,,, + ReR
has (x+ x2.2) + + i [d]
(16)
hig
his
2%t
S
Xa
Wjst
X142) arg max { (103, Vn)
Ii.t (リı w3.0) m21u, j [p]. (17)
9j.t
gj.t
2\,
5

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Algorithm 2 Accelerated Stochastic Mirror Descent Ascent (VR-SMDA) Algorithm
1: Input: T, q. stepsizes {7/ 0, A, 0, E (0, (0,1))=, mini-batch sizes and b1;
2: initialize: 21 E X and E J/:
3: for = 1, ................................ T do
4: if' mod (t, = 0 then
5: Randomly generate mini-batch samples B, = { with [B|] = b:
6: Compute stochastic partial derivatives v V. V,, (Zez Vt)) and wt VyJis (1i,
7: else
8:
Randomly generate mini-batch samples I- (5)) with [ = bi:
9:
Compute stochastic partial derivatives
Ut Va (x))
V (It- 3/- + 1t
(18)
W2 V yIz (Ie, M1)
yIII, 111-1,13-111 + w6-1
(19)
10:
end if
Given the mirror functions 2h and di;
12: 2++1 arg mingex { (v),, 2) + 7. Dv, (x, x2) + g(x)};
13: Vt+1 = y/ + (Wh+1 - yr) where Ut+1 arg maxyey {(wr, v) x/2 Do. (v. h(v))::
14: end for
15: Output: (for theoretical) 25 and yę chosen uniformly random from (II, Vt ]T=1
16: Output: (for practical) 2T and yr.
4.2 VR-SMDA Algorithm
In this subsection, we propose an accelerated stochastic mirror descent ascent (VR-SMIA) algorithm to solve the stochastic problem ( Algorithm (eattihm@ddsscbbbbe describes the detailed algorithmic framework the VR-SMDA method.
In Algorithm [ we only draw a mini-batch samples B, {St) at each iteration. Clearly, the mini-batch samples will take large variances in our SMDA algorithm. Thus, we use the variance reduced technique of SPIDER in our VR-SMDA algorithm to accelerate it. Specifically, when mod (t, g) = O, we draw a relative large batch samples B, = { to estimate our stochastic partial derivatives in and wy; when mod (1, 4) * O, we only draw a mini-batch samples I, = {5t)== (b > 61) to estimate D, and w; in (18) and (19). Since samples Ie are independent to variables (e-1 Me 1/- 1.-1 by using Assumption 2, we have
RI, [ox] = V.f( me) - V=I(
1- + V f Vt);
(20)
ET, [w:] = VyI(xe, ve. VyJ (z+-1, + wt-1 + v (K),
(21)
Thus, the partial derivative estimators v and wy are biased. As Algorithm [ we also use the mirror descent iteration to update x, and use both the mirror ascent and momentum iterations to update in Algorithm
Convergence Analysis
In this section, we study the convergence properties of our algorithms (i.e., MDA, SMDA and VR- SMDA) under some mild conditions. All related proofs are provided in the supplementary materials. We first introduce a useful convergence metric || (or Bl| as in [3 25 measure convergence properties of our algorithms. Given the parameters 2, at f-th iteration by our algorithms, we define
gradient mapping [3 [3 as
1 (Ft
G,
x xtt1),
(22)
Ft
(VD(())) x) + 1 Dv, (x, + +0(x)
x arg min redt
(23)
7t
where S (z)) - hx) = max, - h(y)). When X' Rd and g(x) is a constant, and rix (x) = || we have G. Vo(x,) = V.I Under this case, our convergence metric [ = V.. v² (xx))] is common convergence metric used in (27].

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Since the objective function f(x y) is j--strongly concave over y, the standard (stochastic) proximal gradient ascent can easily obtain the global solution of the subproblem max {f (a. y) h(y)
Without loss of generalization, our theoretical analysis, we give the mirror functions dx = 2 +lyll
for all & 2 1, and all mirror functions { {=) are p-strong convex. Here, the constant p can be seen as lower bound of the strong convexity of all functions (x)( as in [25].
5.1 Convergence Analysis of the SMDA and MDA Algorithms
In the subsection, we provide the convergence properties of our SMDA and MDA algorithms.
Theorem 1. Suppose the sequence (zt, V Y be generated from Algorithm using stochastic partiaal
derivatives (i.e., SMDA algorithm). Let 0 < 1| = 1/2 S 1, <
7/ S min(
3e
9ypud
2mupd and
41 800 252,
1 we have 0 d S EL,'
4v 2 F*)
1
4v2A1
10g
+ 2
Σ
(24)
T
V3TYp
v3ITp
V3bp
3
where K = T.p/u, L 1. + K). F(x) = (1x)) + g(z) and A |l v" (2())))
Remark Without loss of generality, let Ls 2 1.2 Given 0 n S 1, 2 O
y min( "L' 800m?? 2512
9mpue 2npad) and OLI7) (v 2 0), we have 7 = (n: and yp = ( (x22v33)
Thus, SMDA algorithm has convergence rate of O 103-2
33-
When let
b b and = e/3, we have T OK2e ?) and b O(NE Since our SMDA
algorithm requires 2b stochastic gradient evaluations to estimate the stochastic partial directives or and we al each iteration, and needs T iterations, it has a gradient complexity of 2bT O for finding an e-stationary point, the same complexity in [4]. When let v 4/3, b T/x and
1/3 = e/3, we have = 0(x2/3 and b = =OE=2), Thus, our SMDA algorithm has a near optimal gradient complexity of 26T = O(x1/3E-4), which matches a gradient complexity lower bound given in (24) for solving the problem (1) without the nonsmooth regularization terms. 7 using the deter-
Theorem 2. Suppose the sequence {ıx,, T be generated from Algorithm
ministic partial derivatives (i.e., MDA algorithm). Let < 1j = T S 1, < <
7 7t S
min( TI 800k" 25177
3e 9npud 2mpspd ) and O < X d S टl.. we have
T | S
4 2(F2(())-FF - F.)
4v221
+
(25)
T
V3Tmpp
V3T
E=
where K = T.g/11. 1. 1. + k). F(x) = ((z) + g(x) and A = 11ln - y*()))l
Remark 2. Without loss of generality, let Lg 1. Given O 7 s O(H)),
= mmin( 30 InpuA 2nueh and p = OLI (v 2 0). we have TO = ( Since IL S00K" + 2517
our MDA algorithm requires 2 gradient evaluations at each iteration, and needs T iterations, it has a gradient complexity of 2T (x.(2-22) (2(2)))))-2) for finding an e-stationary point. When let v our MDA algorithm has a gradient complexity of 2I =O(k2 the same complexity in M. When let = 1/2, our MDA algorithm has a lower gradient complexity of O(Ke 2) than the complexity in 4 2). When let 3/4, our MDA algorithm has near optimal gradient complexity of O RC 2), which is the same complexity in (28) for solving the problem (1) without the nonsmooth regularization terms.
5.2 Convergence Analysis of the VR-SMDA Algorithm
In the subsection, we provide the convergence properties of the VR-SMDA algorithm.
Theorem 3. Suppose the sequence {xt, Ut \T, be generated from Algorithm
2 Let by 2. < 1 qpdp 3p en 9paud and 0 X min( 6l, 100 L
n $ 1, 0 < = "t S min( 37. 3SL, 19L7m 8 400k3
7

---

we have
1 T
2 =t)
AV201
+ 2v20
Σ E Gill
(26)
T
V3TTP
V3TyP
Vippbb,yj
where K. L1/, L, L,t)) = ((x) g(x) and A1 11 -v"(x1))
Remark 3. Without loss of generality, let L 2 +.- Given < v 1, = 0 (
min 3p quap 3p pn 9pqud
and O(T, (v Muvz 0), we have
= O Thus, our
41 38L77 1923"²
8 400k2
VR-SMDA algorithm has convergence rate of (
121 2nd
2n)
When let 0, T/K
and e/2, we have T = O(P2E Further let b1 = - q== O) and = O(KE 2). Since our VR-SMDA algorithm requires 2b stochastic gradient evaluations estimate the stochastic directives vr and wi al each iteration when mod (1, = 0, otherwise needs 4b, stochastic gradient evaluations, and need T iterations, it has gradient complexity of Ab, - 24T finding an e-stationary point of the problem ( which is the same complexity in (30] for solving the problem ① without the nonsmooth regularization terms.
O )
Remark 4. The above optimal gradient complexities are obtained when given OLLÿ') (v'> >0) 0),
where L is the smooth parameter of objective function f(x y). Although in the objective function [(x y), by may be large, can easily change the original objective function f(a, v) to new function f(x,1) y) = Tf y), < 1. Since V f(( y) = TV f(x, 9). the gradient of function f(x y) is I--Lipschitz continuous (ī. = T.ş)). Thus, we can choose a suitable hyper-parameter T to let this new objective function fxx, y) satisfy the condition (L))
6 Numerrcal Experiments
In this section, perform two tasks (i.e., fair classifier and robust neural network training) to validate efficiency our algorithms. Specifically, we conduct these tasks on the Fashion-MNIST dataset in (38] as well MNIST dataset and CIFAR-10 dataset. Fashion-MNIST dataset and MNIST dataset consist of 28 x 28 arrays of grayscale pixel images classified into 10 categories, and includes 60, 000 training images and 10, 000 testing images. CIFAR- 10 dataset includes 60, 000 32 x 32 colour images (50, 000 training images and 10, 000 testing images). In the experiment, we compare algorithms (MDA, SMDA and VR-SMDA) with the existing proximal gradient descent ascent algorithms (MAPGDA (2), PAGDA 4 and PASGDA 4 for solving these nonsmooth nonconvex minimax problems. Note that both HiBSA algorithm of (29 and Proximal-GDA algorithm of [8] only are non-accelerated version of MAPGDA algorithm (2). so we omit them in the comparison methods. The experiments are run CPU machines with 2.3 GHz Intel Core i9 well as NVIDIA Tesla P40 GPU.
MAPGDA
MAPGDA
MAPGDA
2.8
PAGDA
3.1
PAGDA
MDA
MDA
MDA
Loss
2.9
2.8
1.8
2.6
1.8
2.5
200
400
600
800
1000
200
400
800
000
1000
200
400
800
500
1000
Time
Time
Time
(a) Dashion-MNIST"
(b) MNIST
(c) CIFAR-10
Figure I: Results of different deterministic methods on the fair classifier task.
6.1 Fair Classifier
The first task is to train fair classifier minimize the maximum loss over categories. Here, we use a nonconvex Convolutional Neural Network (CNN) model as classifier. Similar to (38), we limit our experiment to the three categories. To be precise, the MNISST dataset is limited to T-shirt/top.

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3.2
2.5
PPAGDD
SMDA
2.8
-PASGDA
SMDA
3.1
-PASODA
SMDA
VR-SMDA
2.5
Loss
2.4
2.9
2.8
2.2
1,8
28
2
1.6
2.6
30
40
30
40
Epoch
Epoch
Epoch
(a) Fashion-MNIST
(b) MNIST
(c) CIFAR-10
Figure 2: Results of different stochastic methods on the fair classifier task.
Coat and Shirt categories. The MNIST dataset is limited to digital numbers 2,3 and CIFARIO dataset is limited to airplane, automobile and bird. Specifically, we will solve the minimax problem:
Σ v.L,(w) g(w) - noo). s.t. D- E 36 1: O,
3
3
min max yEy
Σ yi = =} (27)
where w denotes the parameters in CNN model, and L1, L2 and L3 are the loss functions correspond- ||w]| and h(y) 2/2 y 2
ing to the samples in three different categories. Here we let g(w) where 1/1 > 0 and 1/2 > O. Clearly the inner maximization problem is strongly concave, and the outer minimization problem is nonconvex nonsmooth. Thus our theory can be applied.
In the experiment, we let 111 = 0.001 and 1/2 = 0.1 in the above problem (7).. For fair comparison, we use the same step size for all methods. Specifically, step-size for is 0.001 and step-size for y is 0,00001. We apply xavier normal initialization to CNN layer. In our algorithms, we choose the mirror functions t/x (w) 1wu H,w and or (v) f G.yy for all > 1, where H, and G; are generated from (12) and (3 respectively, given a = 0.1 and
0.00005. We set 17 = 7 = 1 in algorithms. We all deterministic algorithms for 1000 seconds and all stochastic algorithms for 50 epochs. Then we record the loss value. For stochastic methods, batch sizes of PASGDA and SMDA are 3000. For our VR-SMDA, set the large batch size b 60000 and the mini-bate maii-bachh size b 3000.
Figure [ shows the loss vs time of different deterministic methods. Figure plots the loss epoch of different stochastic methods. From these results, we can find that our algorithms consistently outperform the other algorithms with great margin. The main reason is that our algorithms use the preconditioned (adaptive) matrices H, and G. in updating æ and y, respectively.
6.2 Robust Neural Network Training
The second task is to train robust Neural Networks (NNs). Although the NNs have been widely used in many applications such as image classification, they are vulnerable to adversarial attacks such as Fast Gradient Sign Method (FGSM) [15] and Projected Gradient Descent (PGD) attack [22]. In other word, small perturbation in the input of NN can significantly change its output. Thus, we try to train a robust NN against these adversarial attacks, which generally reformulate this robust training into the following minimax problem:
min Σ max C(f(a; + us: ()) y = E Rd Huello se, é E [m]} WEY
(28)
where (a,, b.) denotes the i-th data point, and w is the parameter of NN, and y4 E Ed denotes is the perturbation added to the i-th data point. Following [38], we approximate the inner maximization problem of the above minimax problem (28) with the following finite max problem
((F(or)(w))))).) b.), ... ( (()()..(w)::.)..))
min
max
(29)
where (w) is the result of targeted attack on data point a, that is changed the output of NN to label j. Following (8), we can obtain āi, (w) by using the following procedure: the last layer of
the NN architecture for learning classification on MNIST (Fashion-MNISS) we have 10 different

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neurons, each corresponding with one category of classification. For any sample (a,, by) in the dataset and starting from a = a, for any 0, 1.... 9. we rum projected gradient ascent to obtain the following chain of points:
k+1 = Projy [at, v. (Zz(a w) Zb. (a w))], k = 1, K - G
(30)
where > is a stepsize, and Zj is the network logit before softmax corresponding to label j. Finally, we can set dı,((w) = o, in the above minimax problem
PASGDA
PASGDA
SMDA
SMDA VR-SMDA
VR-SMDA
3
3-
Losa
2.5
1.5
150
Epoch
Epoch
Fashion-MNIST
(b) MNIST
Figure 3: Results of different stochastic methods on the robust NN training task at Pashion-MNIS and MNIST datasets.
Next, we can replace the above problem 29 with the following nonconvex nonsmooth problem:
min
max UEU
Σ
uj
((f K w), b.) + +r|l|ull 12 12 2
(31)
1=0
9
s.t. = {u 10 U 2 0, Σ
"
11
1=0
where #1 > 0 and P2 > 0. Clearly, the inner maximization problem in 31 is strongly-concave, and its outer minimization problem is nonconvex and nonsmooth.
In the experiment, we V1 0.0001 and 1/2 0.1 the above problem ( In the above problem (30). we set K = 5. For fair comparison, we use the same step size for all methods. Specifically, step-size for w is 0.0005 and step-size for 0,00001. We set 17 = 7J 1 in our algorithms. For our algorithms, we choose the mirror functions 2
w² H2W and (u) = 1 Gru for all 2 1. where H, and G, are generated from (12) and (3) respectively, given a 0.1 and 0.0005. Here we only conduct experiments with stochastic methods, and batch-sizes of PASGDA and SMDA are 600. For our VR-SMDA, we set b 1200 and
= 600. Following (38, we set 0.4 in the above problem (88). Figure 3 shows the loss vs epoch of different stochastic methods. From these results, we can find that our algorithms outperform the other algorithms, and the VR-SMDA consistently outperforms the SMDA.
Conclusions
In the paper, we proposed class of novel adaptive mirror descent ascent methods to solve the nonconvex-stronnlyy-oonee.
minimax optimization problems with nonsmooth regularization terms. Moreover, we provided a useful convergence analysis framework for our methods. Some experimental results on fair classifier and robust neural network training tasks verify that our new algorithms consistently outperform the related algorithms.
Acknowledgments and Disclosure of Funding
This work was partially supported by NSF IIS 1845666, 1852606, 1838627, 1837956, 1956002, OIA 2040588.
10

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13


Reviews:
[{"review_id": "XfJ2Na0KkiG", "reviewer": "Reviewer_SMH6", "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.", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper proposes a class of effective mirror descent ascent (MDA) methods to solve non-smooth non-convex strongly-concave minimax problems. It proves that the proposed MDA has a lower sample complexity than the best known results. At the same time, some experimental results on fair classifier and robust neural network training tasks verify the efficiency of the proposed algorithms. \n\nThe novelty of this paper lies in the use of the Bregman distance to reduce the sample complexity of the minimax optimization methods, which is very interesting in theory. Moreover, it provides a rigorous theoretical analysis. I only have some minor comments as follows. \n \n1) Line 24: “f(x)” should be “h(y)”;\n2) Line 85: “f(x;\\xi^i)” should be “\\nabla f(x;\\xi^i)”;\n3) Line 122: “f(x,y)=\\mathbb{E}[f(x;\\xi)]”  should be “f(x,y)=\\mathbb{E}[f(x,y;\\xi)]”;\n4) Line 149, “mini-batch samples will take large variances in our SMDA algorithm” should be “our SMDA algorithm suffer from large variances due to only using mini-batch stochastic gradients”;\n5) Line 213, “between our works and …” should be “between our methods...”;\n6) In the experiment, how to choose these tuning parameters (such as \\lambda, \\gamma, \\alpha) in the algorithms?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "TpGDyIy9z--", "reviewer": "Reviewer_uFvW", "summary": "This paper propose mirror descent ascent methods for nonsmooth nonconvex-strongly-concave minimax problem. The algorithms requires $O(\\kappa\\epsilon^{-2})$ gradient complexity for deterministic case and $O(\\kappa^3\\epsilon^{-3})$ stochastic gradient complexity for stochastic case.", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper studies mirror descent for nonsmooth nonconvex-strongly-concave minimax problem. The algorithms integrate the mirror ascent iteration with the momentum technique to update $y$. The proposed mirror descent ascent algorithm requires $O(\\kappa\\epsilon^{-2})$ gradient complexity to find $\\epsilon$-first-order stationary point and the complexity of variance-reduced mirror descent ascent algorithm is $O(\\kappa^3\\epsilon^{-3})$ stochastic gradient complexity.\n\nThe theoretical and empirical results of this paper is interesting, but comparison with some related work should be discussed in more details.\n1. In deterministic case, Lin’s [A] algorithm could find $\\epsilon$-stationary point with $\\tilde O(\\sqrt{\\kappa}\\epsilon^{-2})$ complexity, which is better than the algorithm in this paper.\n2. In stochastic case, Luo's [20] algorithm also requires  $O(\\kappa^3\\epsilon^{-3})$ complexity. Although Luo's [20] algorithm does not consider constraint on $x$, it looks not difficult to be extended by introduced additional projection step on $x$.\n\nI hope the authors address the above concerns in revision/feedback, which could make the paper more convincing. \n\n[A] Lin, Tianyi, Chi Jin, and Michael I. Jordan. \"Near-optimal algorithms for minimax optimization.\" Conference on Learning Theory. PMLR, 2020.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "This is a theoretical paper.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}, {"review_id": "HhcWFx1tkQU", "reviewer": "Reviewer_Mhm6", "summary": "This paper proposes a class of mirror descent ascent (MDA) methods for solving nonsmooth and nonconvex-strongly-concave minimax optimization problems by using dynamic mirror functions. It also studies the convergence properties of the proposed algorithms. In particular, the MDA method achieves a lower sample complexity by using the strongly-convex mirror function. The paper also provides some numerical experiments on fair classifiers and robust neural network training tasks to demonstrate the efficiency of the proposed algorithms. This paper first applies the mirror descent algorithm to nonsmooth minimax optimization. It provides a rigorous convergence analysis of the proposed algorithms. Overall, this paper is interesting and has some novelties. Thus I recommend accepting this paper. ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Some comments:\n1) In the paper, problem (1) is a constrained optimization. Can the convergence results in the paper apply to the unconstrained minimax optimization? \n\n2) In the convergence analysis, could the authors provide some intuitions about the defined Lyapunov function $\\Omega_t$? Also, intuitively, why adopting mirror descent can help achieve a lower sample complexity?\n\n3) In the experiment, how to choose the mirror functions used in the proposed algorithms?\n \nSome typos:\n1) In the line 85, “f(x;\\xi^i)”  should be “\\nabla f(x;\\xi^i)”;\n2) In the line 122, “f(x,y)=E[f(x;\\xi)]”  should be  “f(x,y)=E[f(x,y;\\xi)]”.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper considers nonconvex-strongly concave minimax optimization with Bregman divergence setup and provides several new results based on deterministic and stochastic mirror descent based algorithms. All the reviewers agree that the results in this paper are novel and interesting. The reviewers also raise some important concerns such as potential extension to concave but not strongly concave settings, which the authors should properly address in the camera ready version. I suggest to the authors to give serious thought to all other suggestions of the reviewers in the camera ready version, which will help this become an even stronger paper.
Decision: Accept (Poster)