On the Convergence of Stochastic Smoothed Multi-Level Compositional Gradient Descent Ascent
Authors: Xinwen Zhang
Hongchang Gao
Venue: NeurIPS 2025 poster ()
Keywords: Compositional Optimization
Minimax Optimization
Primary area: optimization

Abstract: Multi-level compositional optimization is a fundamental framework in machine learning with broad applications. While recent advances have addressed compositional minimization problems, the stochastic multi-level compositional minimax problem introduces significant new challenges—most notably, the biased nature of stochastic gradients for both the primal and dual variables. In this work, we address this gap by proposing a novel stochastic multi-level compositional gradient descent-ascent algorithm, incorporating a smoothing technique under the nonconvex-PL condition. We establish a convergence rate to an $(\epsilon, \epsilon/\sqrt{\kappa})$-stationary point with improved dependence on the condition number at $O(\kappa^{3/2})$, where $\epsilon$ denotes the solution accuracy and $\kappa$ represents the condition number. Moreover,  we design a novel stage-wise algorithm with variance reduction to address the  biased gradient issue under the two-sided PL condition. This algorithm successfully enables a translation from and $(\epsilon, \epsilon/\sqrt{\kappa})$-stationary point to an $\epsilon$-stationary point. Finally, extensive experiments validate the effectiveness of our algorithms.

On the Convergence of Stochastic Smoothed Multi-Level Compositional Gradient Descent Ascent
Xinwen Zhang Temple University Philadelphia, PA, USA el @lLenz@temple.edi edu
Hongchang Gao Temple University Philadelphia, PA, USA hongchang, gao@temple.cr edu
Abstract
Multi-level compositional optimization is fundamental framework in machine learning with broad applications. While recent advances have addressed composi- tional minimization problems, the stochastic multi-level compositional minimax problem introduces significant new challenges-mos notably, the biased nature stochastic gradients for both the primal and dual variables. In this work, we address this gap by proposing novel stochastic multi-level compositional gradient descent- ascent algorithm, incorporating smoothing technique under the nonconvex-Pl PL condition. We establish a convergence rate to an (e, E/ //)R/sssttaiooo point with improved dependence on the condition number at (x23/2), where denotes the solution accuracy and represents the condition number. Moreover, design novel stage-wise algorithm with variance reduction to address the biased gradient issue under the two-sided PL condition. This algorithm successfully enables translation from and (e, (c,e//VR)-staionnaa point to an e-stationary point. Finally, extensive experiments validate the effectiveness of our algorithms.
1 Introduction
This paper investigates the stochastic multi-level compositional minimax optimization problem:
min max f(G(z).)) ZERi yERdy
where f(G(x).)) y) [[f(G(x), y: {)] and Ç denotes random variable. The function G(x) A g(K) (x))) is K- compositional function with K where each inner-level function
g( Eig (k) (+)] depends the random sample g(k) for k f1,... K). The function f(..) is referred to as the outer-level objective. In this paper, we consider the general nonconvex-Pi
setting, where (G(x), is nonconvex in the primal variable and satisfies the folyak-Lojasiewin (PL) condition with respect to the dual variable y.
Multi-level compositional optimization has emerged as vital framework in machine learning, with broad applications across numerous domains. In meta-learning, it enhances model adaptability across tasks [12 [22; in finance, it supports risk-averse portfolio optimization under uncertainty
[ 20] and in reinforcement learning, aids policy evaluation and decision refinement [8] 24]. The widespread
impact of the multi-level compositional structure highlights its importance in handling complex and structured optimization problems. Moreover, the scope of multi-level compositional optimization extends naturally to the minimax setting, with applications in areas such deep AUC maximization (33), multi-instance learning [40), and multi-objective learning [19], etc. Despite the importance of these applications, the stochastic multi-level compositional minimax optimization problem remains
"ooresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Table 1: The comparison of convergence rate between algorithms and existing stochastic compositional minimax algorithms.
Algorithms
Convergence Rate
Assumption
Level
SCGDA 17
O(x"/e))
Nonconnex--stronglyycoom
Two-Ievel
SCGDAM 33
O
Nonconvex-sttrnggyycooom
Two-level
CODA- Primml
(
Vonconnee--tronggyy-onnee
Two-level
NSTORM [26)
O
setnvee-stronglycconee Nonconvex
Two-level
Smoothed-SMCGDA--VV VR (Thm.
(xx
Nonconvex-PP
Multi-level
(Thm.
O (e)
Nonconvex-PP PL
Multi-level
Sngewise-SMGGDA.. (Thm.
O(x//e)
Two-sided-PL
Multi-level
largely underexplored. This gap in the literature motivates our study, which aims to develop an effective algorithmic solution for this challenging class of problems.
Solving multi-level compositional problems is challenging, even in the minimization setting. In particular, when the inner-level functions are nonlinear, the stochastic gradient is no longer an unbiased estimator of the full gradient. Recent research has proposed new algorithms to address this issue. Notably, (31) introduced K-level stochastic compositional gradient descent algorithm, and subsequent efforts 5 23 35) have developed algorithms specifically tailored to address the biased characteristics inherent in the stochastic multi-level compositional framework. Unfortunately, these cattmizzaaiinttaagtteee algorithms cannot directly address the stochastic multi-level compositional minimax optimization problem in Eq. [ the stochastic gradients for both primal and dual variables are biased estimators in stochastic multi-level compositional minimax problems paggems-pposigg greater algorithmic and theoretical challenges.
In addition, although [7 26 9 investigate the two-level compositional minimax problem, it remains within the classical minimax framework and does not consider more advanced techniques that can improve convergence. In contrast, recent progress in classical minimax optimization has demonstrated that smoothed techniques can significantly improve convergence. For instance, (36) proposed
smoothed alternating gradient method for general nonconvex-concavy problems. which achieves superior performance compared to conventional approaches. Building on this, [30] further applied
this technique to the nonconvex-Pi P setting- milder condition than strong concavity- and showed that stochastic smoothed techniques yield improved complexity bounds with better dependence on the condition number K. However, despite the clear benefits of smoothed techniques in traditional minimax optimization, its application multi-level compositional minimax problems remains unexplored. This observation motivates a key question: Can smoothed techniques be effectively integrated into the multi-level compositional framework to improve convergence performance?
Addressing this question is not straightforward and presents substantial algorithmic and theoretical challenges. On the one hand, while existing studies (36]] 30 demonstrate that smoothing techniques are effective for unbiased stochastic gradient estimators, the biased nature of stochastic gradients for both the primal and dual variables in Eq. ( introduces uncertainty regarding the effectiveness of applying such techniques. It remains unclear whether their use may lead to additional convergence issues. Therefore, it essential to develop new algorithms that can accommodate biased gradient estimators and guarantee convergence when using smoothing techniques for Eq. (L. On the other hand, as demonstrated in (30,, smoothed algorithms typically guarantee an (c), Cg sstttionay point,
rather than a standard e-stationary point, which are defined in Definition 3.5 This necessitates a translation between the two measures. While such a translation introduces negligible iteration complexity in classical minimax problems when using an unbiased gradient estimator as shown in 300. there is no known algorithm capable of performing this translation in the context of multi-level compositional minimax optimization. In particular, it remains unclear how to design a translation algorithm without degrading the iteration complexity of the smoothed algorithm in the presence of multi-level compositional structure. Therefore, these challenges motivate us to address the problem through the following contributions:
We develop novel smoothed multi-level compositional minimax optimization algorithm for Eq. [T] by leveraging the variance reduction technique to mitigate the biased gradient estimator issue, and establish a convergence rate of ((x3/3/e3) to an (ssatoa))s point. Compared to existing algorithms, our method achieves a better dependence the

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condition number <: improving over the O(x2) rate of standard two-level compositional minimax algorithms.
To bridge the gap between an -tttttioay point and a standard e-stationary point, we further propose stage-wise variance-reduced algorithm for Eq. under the two-sided PL condition. We show that the algorithm achieves a convergence rate of ((1/c²) to an e-stationary point. As result, the iteration complexity from the translation is dominated by the complexity of finding an (e, (c.e/V/R)-tttionna) point.
Mcanwhile, we obtain two additional results, which may be of independent interest: the convergence rates for the multi-level compositional minimax problem under the nonconvex-PL and two-sided-P PL assumptions without using the smooth technique, as summarized in Table [
We conduct extensive experiments to validate the effectiveness of our proposed algorithms, demonstrating superior performance compared to existing baselines.
2 Related Work
2.1 Stochastic Compositional Minimization Optimization
Recently, a general class of stochastic compositional gradient descent methods 29
18
32
15
16
was developed for two-level compositional minimization problems and established convergence rates for nonconvex loss functions. Aiming to address practical problems with more general stochastic compositional structure, the stochastic two-level compositional problem has been extended to the stochastic multi-level compositional problem. Stochastic multi-level compositional learning has various applications, including multi-step model-agnostic meta-learning [12], the stochastic training of graph neural networks (6), the neural networks with aitcc-normalization [25] etc. Consequently,
series of stochastic multi-level compositional minimization algorithms [I 2 5 35 23
13 14] have
been developed to solve this important problem. Notably, 31] introduced the first stochastic multi- level compositional gradient descent algorithm. Then, [2] employed a moving-average estimator, and (5] used the STORM variance-reduction estimator 7 for each inner-level function, achieving convergence rate of O Later, (35 improved the sample complexity to ((1/c²) by applying the SPIDER variance-reduutionn technique [ [27] to both the inner-level function and Jacobian matrix at each level. Nevertheless, the large batch size required by this method makes it impractical for large-scale models, and the learning rate must be sufficiently small maintain Lipschitz continuity of the variance-reduced gradient. By applying the STORM variance-reduction approach to both the function value and its Jacobian matrix at each level. (23] developed a convergence rate of O(1/e²) for the stochastic multi-level compositional problem with a mini- size of O(1). More recently. for the first time, [13] showed that the puriatce-rddcction estimator is not necessary for the Jacobian matrix in each level to achieve convergence rate of O(1/e².. However, these stochastic multi- level compositional algorithms focus exclusively on minimization problems and therefore cannot be directly applied to multi-level compositional minimax problems.
2.2 Stochastic Compositional Minimax Optimization
Stochastic compositional minimax optimization 17 9
26
37
38
has attracted increasing
attention due to its important applications machine learning. To solve the two-level compositional minimax problem, [17] developed the first compositional minimax algorithm based on the batch
compositional gradient, achieving a convergence rate of O11²* ((1) for wwwroonwee-stoolyyyoooor loss functions. [33] incorporated the momentum technique reduce the mini-batch size to ((1) while achieving the same convergence rate as [77. Similarly, 9 used a variance-reduced estimator for the inner-level function, also reducing the mini-batch size to (1)) and achieving the same convergence rate as [17]. 26 introduced the STORM technique for estimating the inner-level function and gradient, achieving convergence rate of O(x²/ /ea). Recently,
g claimed to achieve a convergence rate of O(x²/e²) However, this convergence rate is established with respect to the stationary point of the Moreau envelope of the primal function, rather than that of the original primal function. As result, it corresponds to the convergence rate for strongly-convex convex-strongly---eeoooooo-----e strongly-concave loss function, or nonconvex-PL. loss functions. More recently, [37]
rather than for nonconvex saarcoovee-stronlyyyyeee strongly-conenn developed the first stochastic multi-level compositional minimax algorithm for nonconvex strongll... concave loss functions in the federated learning setting. However, its convergence rate ((1/e is suboptimal compared to the multi-level compositional minimization algorithm. On the other hand, the nonconvex-concave concave minimax problems in [36), where
smoothed technique was first introduced for nonconvex
the convergence rate of full alternating gradient descent ascent method was established. Later, (30]
3

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extended this technique to the nonconvex PL setting and further investigated the relationship between two stationarity measures. However, none of these algorithms are equipped to handle the challenges posed by multi-level compositional minimax problems, which remain largely unexplored.
Preliminaries
3.1 Notations
We begin by simplifying the complex formulation in Eq.
1 to facilitate analysis:
(x) g (G(
V (x) (xV
(2)
where & E f1.... KK, G( z, and G(z) G (x).
The partial gradients of the objective function can then be expressed as follows:
Vz.(((()).)) =
(z).))
Vyf(G(a).v) = (2f(G(()().v))
(3)
Following prior works 36 30 we introduce auxiliary variable = alongside the primal variable as part of the smoothed technique, and define the smoothed loss function as:
fo (G(x), y: f(G((). + -
211
(4)
where w > 0 is a constant and fo (G(x), y: 2) is strongly convex with respect to x by selecting an appropriate Using the smoothed loss, can derive stochastic estimators of the compositional gradients with respect the primal and dual variables at the I-th iteration:
(x t0 (m ef " " (
Vo
(a eloss") ))(l))) ...
(::51 (kk ): 11)
Vggh (o ( (ril "f (x) (:E(()) )(((())) w(xt - =21)
Vofa(1333., Vuf (10 (3 (:),,
where & = (5e (2) (K)
3.2 Assumptions
We next introduce the following standard assumptions, which are commonly used in stochastic compositional optimization 17 26
9
38
37 30
Assumption 3.1. (Smoothness))
For any k E (1, K}. of ()) and g (x; E) are Cy-Lipschitz continuous, gg (.) and (xEE) are L--Lipschitz continuous, where Cg > O and Lg > 0; f(.-)) and (()) are Ci-Lischhitz continuous, Vf(.)) and v((()( Ly- Lyppsshhiitz continuous, where C > 0 and Ly > O.
Assumption 3.2. (Variance):
K), (K,...,K), the stochastic gradients (xg(() and Vf((s() have upper For any l {1...., bounded variance o², where a > O.
Assumption 3.3. (PL Condition):
For any fixed E TR,,, the maximization problem y max,ert, f(G(x), y) has non-empty solution set and a finite optimal value. Moreover, for all x E Rd= there exists a constant value 0 such that (Vf(((z)))))] 2 -
Here, we define l max(L,, C2K L + CC EK-1 C -I and k denotes the condition number. Then, when w > f, fu (G(x), y: 2) is strongly convex with respect to x. We also introduce the following definitions.
Definition 3.4. (Two-sided PL Condition):
f(x, y) satisfies the two-sided PL condition, if there exist constants μz > 0 and Byy > 0 such that f(-y)) is ub L any E RR,, and ff(x,) :) is py PL any x E Rd.,

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Definition 3.5. (Stationarity measures):
(I,y) is an (E1, (2) st)sttioorryy point of f(.)), if [Vaf(G(z).)) Σ E1 and
[Vf(G(()), v) I ED.
an e-stationary point of ((), if [D(2)]l S 6, where x(x) f(G(z), and y² arg yax y()())
3.3 Challenges
From the algorithmic design perspective, one of the primary challenges in incorporating smoothed techniques is managing the intrinsic bias of stochastic gradients for both the primal and dual variables. Specifically, as shown Eq. 3 the partial gradient regarding the dual variable y relies the stochastic estimator of K-lleel function (-))(), while that regarding the primal variable depends the stochastic estimator of both (+)) and (.) In the stochastic setting, however, computing the stochastic estimator for both the k-th level function and its corresponding gradient introduces bias, as illustrated below:
Flg (8) (1; (())] # (.))
(6) As result, the stochastic gradients with respect to both primal and dual variables are biased estimators of the full gradient. Moreover, as shown in Eq. (6), the estimation biases accumulate across all compositional levels when estimating both the inner-level functions and their gradient. This accumulation of bias introduces greater complexity compared to the two-level case and raises concerns about whether the deeper compositional structure might undermine the effectiveness of smoothed techniques, as all existing smoothed minimux methods handle deterministic grudients or unbiased stochastic gradients.
EIV 2g (A (A- (1i VV gtu (h)( (;:(tt))) ((0)) + V.
From the theoretical analysis perspective, a major challenge arises from the gap between different stationarity measures induced by smoothed techniques. As demonstrated in 300 translation is required from an (c1, -tt-sttion point to an e-stationary point. In standard minimax settings, this can be achieved by applying a stochastic gradient descent-ascent algorithm to the auxiliary problem min maX (zx,y) + f||z - 2 where 2 is the output of the smoothed algorithm. Owing to the fact that this formulation satisfies the the PL condition both x and y, with an iteration complexity of Ó(1/e). Therefore, if the cost of this translation remains lower than that of the smoothed algorithm itself, it does not affect the overall complexity. However, for multi-level compositional minimax problems, there do not exist algorithms for handling the two-sided PL condition to complete the translation, and it is unclear whether the iteration complexity of the translation is smaller than that of the smoothed algorithm or not. In particular, the existing study [3I showed that the standard compositional gradient descent algorithm can only achieve convergence rate with an exponential dependence on the number of levels, even for strongly convex loss functions. As result, the complexity of the translation phase could dominate the overall complexity. Therefore, it remains unclear whether there exists an efficient algorithm to translate from an (61, (20,,22)sttaa -ttationayy point to e-stationary point for multi-level compositional minimax problems.
4
Algorithm T: Smoothed-SMMGDDA-W
4.1
Algorithmic Design
To address the smoothed loss in Eq. (4) design novel algorithm, named stochastic smoothed multi-level compositional gradient descent ascent with variance reduction Mrmoothed-SMCGDA-- as presented in Algorithm [ To mitigate the accumulation of bias at each compositional level, our method incorporates a STORM-like variance-reduced estimator. Specifically, for each inner-level function g (*)) where E (1,..., K). we apply a recursive step that updates the estimator J (k) while controlling variance. This variance reduction technique is also employed for the stochastic gradients: f. Er, Ge) and Vy i Gt).
More concretely, the muriance-reducee estimator for each level-k function is computed as:
(k) h(2) = (k) (1 ((---)--;---) e ) + (1 - ann?) (k)) (1 (k-1), 1), (R)
( )),
where h1 T++1 when ls 0, and a >> is hyperparameter such that ani (0, 1).
5

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Algorithm 1 Stochastic Smoothed Multi-Level Compositional Gradient Descent Ascent with
Variance Reduced (Smoothed -SMCGDA-VR)
Input: " 7> 0, > O, Ps. 0. Py 0, 7r > 0. 7o
0, 7/2
0, p211 1, Pyd < 1, an) <
Y2 <1
Initialization: 11 no. no (k) 9 (h (k-1); (k)) for k E .1.... -K.
Po = Vg (z (()) ... Vg (K) (ho (k (K-1), 1) [k)))) VV (h1 (K)
yoi Co) + w0 - %o)
do = 22f(h yo: So). un Po. uo 90-
(K)
1: for =0, 0, ... T do
2: Update I and y. 1++1 Et PPt
Vt+1 y/8 79t
3: Update 2: 2t+1 = 2 + 7= 7 0++1 - 24)
4: 1441 I(+1
5:
for k = 1, 1..... K do
6: Compute k-th inner-level function:
(k) h+41 = ( (h(tz -1 ((+1) + (1 (h) ( ((-11)
7: end for
8: Compute stochastic compositional gradient u and c141:
U1+1a+1 = (g ( .... Vg(K-1) (1(44 ()
(1++1 ; /g (1)
c(K-
(K-1), c(K)
i5t+1) x
Vif(h (K) B(+1: B0-11 + w - 2411)
U141:41 = (e
(K) 31+1:
9:
Compute variance-reduced gradient p++1 and 4++1:
Pt-1 = 1++++11 + -p27 (p: - (1s+1) , qe+1 04 + +(1 -Py??) -02++1)
10: end for
For the luter-level update, compute the stochastic gradient of the smoothed loss defined in Eq. ( 1K
based on the variance-reduced estimator {h++1] k= of the inner-level function, as presented in Step 8. Here, 14+114+ denotes the stochastic compositional gradient regarding primal variable, where the first index indicates the t + 1-th iteration of the variable, and the second reflects the sample indices Č ((g/-) (k) G Similarly, we compute the stochastic gradient with respect to the dual
(R)
variable based on the variance-reduced estimator h of the inner-level function. The algorithm then performs STORM-like updates on p++1 and gr-11, presented in Step 9, where Ps and Py two hyperparameters such that p= E (0, 1) and pyn? 10,
4.2 Theoretical Analysis
We derive the convergence rate of Algorithm I in the following theorem
Theorem 4.1. Given Assumptions 3.7 3.3 when Ps > 0, Py. > 0, a 0, O(f), and the
hyperparameter conditions in Eq. (44) are satisfied, Algorithm achieves the following convergence upper bound:
[BIIV. (EE +
T
KPP
( KO²
+ 0 Ro²
+ o kO?
00
PTPPTS)
PT,PTT))
0 a²1j²0²"
+ +0 (Kpp)tt'0)) + 0(1 a²1j²0²
+ +0 (Kppn''0') + O (xan"to)).
(8)
P#
where Pn= (G( No; No; =0) 2lu,d(uo: =0) + 2g(20), with the definitions of the involved terms provided in Eg. (25).
Corollary 4.2. Given Assumptions 3.1 3.3 by setting 7a 0()), Ty 0(1.. 7. (1/k),
17 = -0 (EExxt/2) Pz (1), Pv (l) 0 0 0 0
"Due to space limitations, the theorem with the full hyperparameter conditions is provided in the Ap-
pendix B.2

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Algorithm can achieve the ( A.e.ee/vv-statiiiioii -sttttoioy solution, where > 0 denotes the solution accuracy, and S is the batch size in the initial iteration.
Note that our Theorem 4.1 provides the convergence rate in terms of the stationary point of the original loss function (z+), Mr). rather than that of the smoothed loss function (G((), y: 2). Therefore, this result corresponds to the convergence rate for nonconvex-Pll loss function, rather than for a two-sided-PL. loss function. As result, the comparison of convergence rates with existing methods in Table is fair and consistent with the comparison made in the context of classical smoothed minimax optimization in [30].
Proof Sketch. To establish the convergence rate Algorithm we propose a novel potential function follows:
H4 = fw (G(z)), yo: 21) 2h (Wi 2)) + 2h(2))
tve Ellpp - Vu.fw(H( yr: 2, 2
Optimizaiion Error: Lemmur R. R.30..4
Gradient Ervor regurding x: Lenmna B.G
my 11l1 V vu (H
)
A
Elll 9ihh (1)
Gradient Ervor regarding y: Lenamu B.7
Inner-level Estimation Evror: Lemona
where the coefficient Va, Vs, and ) are positive, where the notations of V fu ve: and V (H( yr: z,) can be found in Eq. (33
To analyze the descent of the potential function, we decompose and bound each term through sequence lemmas. First, we bound:
P. fo (G(z)) 26)) 2aw,a(yi 2x) 2f(2))
(10)
which characterizes the optimization error introduced by the smoothed technique. Each component
of P, depends on y and x, and the compositional gradient introduces additional bias:
We first derive upper bounds for each component in Pe.
We then combine these bounds in Appendix B.2.1 analyze and quantify their dependence, providing a clear characterization of how the three terms interact.
Second, three additional terms Eq. arise from the gradient errors regarding æ and y, and the inner-level estimation етrоr in the multi-level compositional loss.
Third, the four terms in H4 interdependent. We analyze these dependencies in Appendix B.2.2 and show that 74 satisfies a sufficient descent property, i.e., Me+1 Hi can be bounded under suitable hyperparameter conditions, ensuring convergence to (c, c/ /R))-sationan point. The complete proof is provided in Appendix B
Algorithm 2: Stagewis-SMMGGDA-V
5.1 Algorithmic Design
Algorithm 2 Stagewise-SM SMCGDA-VR
Input: Ps > 0, Py > 0, 0, M2,r > fg.r >
1: for Stage = ..... RR R-1 do
2: Zr,0 2,, Ur.0 Gr. (k)
(( for l {0..... ...KK 1}.
Pr.0 Pr. Or = qr.
3: for f 0.... 1, do
1,
4: Perform one iteration of SMCGDA-VE update
5: Randomly select (5,+1, 1 1 Pr+1 1r-1) from {(Fra (rats Brats (i Pe, dr 1
(k)
6: end for
7: end for
However, to facilitate a fair comparison between the convergence rate of Algorithm I and existing stochastic two-level compositional minimax methods, which establish the rate in terms of e-stationary point instead of (61, <2) sta-stationary point, is necessary convert the ( -tt////oastationaryy solution into e-stationary solution. As discussed in Section 3.3 making this translation is challenging for
7

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multi-level compositional minimax optimization problems. Specifically, in the classical minimax setting, [30] showed that the standard stochastic gradient descent ascent (SGDA) algorithm is problem, since its convergence rate is only Ó(1/c2,, which is dominated by that of the smoothed algorithm. However, this approach does not work for the multi-level compositional minimax optimization problem. Specifically, even for the multi-level compositional minimization optimization problem, the classical stochastic compositional gradient descent algorithm can only achieve a convergence rate with an exponential dependence on the number of levels for strongly convex loss functions, as shown in [I.
sufficient for the translation by solving
The aforementioned challenge motivates the development of a new algorithm handle the translation from an (c, e/ - solution into an e-stationary solution. To this end, we aim to develop a new algorithm to solve the multi-level compositional minimax optimization problem that satisfies the two-sided PL. condition. Specifically, assume Σ is the output of Algorithm then we complete the translation by solving the following problem.
min max EER VERdy
= = + -
E||2
(1I)
Note that ă is the output r from Algorithm| D and it is fixed when solving this problem. Moreover, since w is selected such that f(((x),)) is strongly convex with respect to f(G(x)) y) naturally satisfies the two-sided PL condition. Then, our next goal is to develop an efficient algorithm to solve Eq. II such that its iteration complexity is better than that of Algorithm i.c., the translation does not hurt the overall convergence rate.
To this end, we propose a novel stage-wise algorithm, named Soagewise-SMMGGDA in Algorithm[](NN that more general algorithm is presented in Algorithm for the multi-level compositional minimax optimization problem satisfying the two-sided PL condition. This algorithm may be of independent interest, beyond its use for the translation phase.). The overall optimization divided into R stages, and in each stage, we run the SMCGDA-VR algorithm without updating (i.e., removing the component highlighted in blue) and replacing 2 with in Step 8. At the end of each (k) stage r, the algorithm randomly selects a tuple from the set { Fr.t, Ry,t Pr.t: (r where
as shown
k E {1...., K} }. to be used as the initialization for the next stage + 1. A complete description of the algorithm given in the Appendix
5.2 Theoretical Analysis
We establish the convergence rate of Algorithm 2 in the following theorem. More general results for the extended Algorithm 3] which may be of independent interest, presented in Theorems C.1 C.2
25L2 Theorem 5.1. Given Assumption 3.13.4 by setting co = n Ps = 64400gLL3, Py = 640L
161g 640cgL3, 70.0 = 1 To = max|225, Lgoz }, and 2 1, M2 = (22/ 1 mur 20L.'
0(1((V2r-Lgg), Tr = Ocea/( x 2r) after running Algorithm for the total number of iterations (not stages) ((1/e), we can get [E|| (ā E
Remark 5.2. From Theorem 5.1 it can be observed that the iteration complexity 0(1/²) of the translation phase is much smaller than that of Algorithm 7 Therefore, the translation does not hurt the overall convergence rate.
Remark 5.3. Since the overall iteration complexity is determined by Algorithm 7 we can conclude that our algorithm achieves an iteration complexity of T O (x3/2/e /e)). improving upon the O (k)) complexity of the two-level compositional minimax problem in [77 Ø by offering better dependence on both x and e and the O (w²3 (2) complexity [26] by better dependence on K. To the best of our knowledge, this is the first algorithm to achieve an (x3/22) dependence for (multi-level compositional) minimax problems under the nonconvex-PL. setting.
Proof Sketch. To prove Theorem 5.1
we use an induction approach to handle the stage-wise We introduce two metrics to facilitate convergence analysis:
structure of Algorithms 2
E[V.] EE. 4xx"
CoMl.r (Z.) ((G 3-)]
Lommu
Lamme
ET4- El| v.J G-) Pr
).0)) 11
8

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56
1a (8) (1) - 11
(12)
where Vi denotes the optimization error, and Ue is similar to the last three terms of Eq. (9) is positive constant such that com... = 1
Nu.t To Importantly, following Lemma C.6 and C.7 we establish
how V. and Ur affect each other across stages and derive the following inequalities in Appendix
20co E[V++ + 320d, BR4-] 23,
ERU++1] -
T,
Po
2000
320ca
E[V++] <
Dy E[V.J +
B8.] 33
(13)
Pon T,
These bounds differ only by a factor of 1/u. Using induction, at the r-th stage, we assume
E S c
ERL] = per
(14)
where > 0 is constant. Finally, by selecting appropriate hyperparameters, we prove that
Gr 2
E E cr+1
E S p6,+1
(15)
As such, establish the desired convergence rate. The complete proof is provided in Appendix C
6 Experiment
6.1 Deep AUC Maximization
In the deep AUC maximization problem, applying K-step gradient descent to minimize the cross- entropy loss function results a K-level inner function G(-) in Eq. (1). with a detailed discussion provided in Appendix A We compare smoothed method with three baselines: SCGDA (17), SCGDAM [ and NSTORM [26] across three datasets: CATvsDOG, CIFARIO and STL10. Imbalanced binary datasets generated following the approach described in [33 with imbalance ratio 0.05. ResNet20 is employed as the model. For all algorithms, we set both the learning rate and the momentum variance reduction coefficient to 0.1. In proposed method, employ smoothed techniques during the first 90 epochs, followed by stage-wise updates for the remaining 10 epochs.
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Figure 1: The test AUC score versus the number of stochastic first-order gradient evaluations.
We conduct experiments using our smoothed method for both K = and K 5. with results presented in Figure [ Notably, NSTORM applies STORM-like updates to the two-level(K compositional minimax problem without smoothed techniques. Our results show that the smoothed approach consistently outperforms baselines. Moreover, as the number of levels increases, the smoothed method does not degrade the performance, demonstrating its robustness to increased compositional levels. This improvement observed consistently across all datasets, highlighting the effectiveness of incorporating deeper compositional structures. Additional experiments with varying K are provided the Appendix(A A
6.2 Multi-Instance Learning
Following [39], multi- instance learning can be reformulated as multi-level compositional minimax problem as shown in Eq. []] with details provided in Appendix A For multi-instance learning tasks, our proposed approach utilizes two types of stochastic pooling operations: log-sum-exp (smx) pooling and attention-base based (att) pooling. We compare the performance our smoothed methods against six baseline methods: MIDAM(smx) and MIDAM(att) [40), both utilizing stochastic pooling

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Figure 2: The test AUC score versus the number of epochs for Tabular Datasets.
operations; DAM(mean), DAM(max), DAM(smx), and DAM(att), all of which update the AUC loss with traditional PESG optimizer 34].
We conduct experiments on five commonly used tabular benchmark datasets [0] for MIL tasks MUSKI, MUSK2, Fox, Tiger, and Elephant Breast Cancer, For the tabular datasets, we use a two-layer feed-forward neural network with tanh activation and sigmoid output for AUC loss normalization. For the Breast Cancer dataset, each image is divided into 32 x 32 patches and treated as bag of 672 local patches to enable efficient multi-instance processing, using ResNet20 as the model. All datasets are randomly split into training and testing sets with 0.9/0.1 ratio.
well one histopathological image dataset, namely
For the tabular datasets, perform 5- fold cross-validaie vooss-valiation,, repeating each run with three random seeds. For the image dataset, we use two random seeds. The learning rate for the primal variables is tuned within the set [1e-1, le-2, le- 3), while the learning rate for the dual variables is fixed at 1. We vary the value of from to 5 and ultimately 3 to achieve more stable performance. We present the experimental results on the tabular datasets in Figure 2 and on
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Figure 3: The test AUC score versus the number of epochs for Breast Cancer Dataset.
the image dataset in Figure For the tabular datasets, to ensure clearer visualizations, we omit error bars in the plots and instead report both the mean and standard deviation of the results in Table 2 as shown in Appendix A For the image dataset, we focus our comparison on the softmax- based and attention- based methods. In both experimental settings, our proposed algorithms consistently outperform all baseline methods, demonstrating superior optimization behavior and generalization performance across range of tasks and datasets.
7 Conclusion
In this work, we addressed the challenging problem of stochastic multi-level compositional mini- max optimization by proposing smoothed variance-reduced algorithm. Our theoretical analysis demonstrates that the proposed smoothed method achieves convergence rate of O (1:3/2/e2 to an (e, (t.ee/)k)-stle -tattioaa point. Furthermore, to bridge gap between different stationarity measures, we developed stage-wise algorithm under the two-sided PI. condition, enabling a translation to an e-stationary point. Extensive experiments on deep AUC maximization and multi-instance learning tasks validate the superior performance of our approach.
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Acknowledcements
We thank anonymous reviewers for constructive comments. X. Zhang and H. Gao partially supported by U.S. NSF CAREER 2339545, NSF IIS 2416607, NSF CNS 2107014.
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Contents
Introduction
2
Related Work
2.1
Stochastic Compositional Minimization Optimization
2.2
Stochastic Compositional Minimax Optimization
3
Preliminaries
3.1
Notations
3.2
Assumptions
3.3
Challenges
Algorithm Smootted-SMMCGDA-VV
4.1
Algorithmic Design
4.2
Theoretical Analysis
5
Algorithm 2: Stagewis-SMCCGDA-VR
5.1
Algorithmic Design
5.2
Theoretical Analysis
6
Experiment
6.1
Deep AUC Maximization
6.2
Multi-Instanee Learning
7
Conclusion
10
Applications and Experiments
A.I Deep AUC Maximization
21
A.2 Multi-Instance Learning
A.3 More Experimental Results
22
B Appendix: Smoothed-SSMCGDA.-V
22
Useful Lemmas
23
B.2 Proof of the Theorem 4.1
28
................................
B.2.1
Bound P.+1 P,
29
B.2.2 Bound H4+1 H4
32
Appendix: Stagewis-SMCCGDA-VV
38
C.I Useful Lemmas
38
C22 Proof of the Theorem C.I
47
C.3 Proof of the Theorem C.2
47
C4 Proof of the Theorem 5.1
53
20

---

A Applications and Experiments
A.1 Deep AUC Maximization
AUC(Area under the ROC curve) is widely used to evaluate the classifiers for binary classification
with imbalanced data. [33 reformulated the AUC maximization problem the following two-level compositional minimax problem:
min max LAu 0;, v.abb a
(16)
s.t. i = 1fV (w
where w Ed are model parameters while (a, a) are parameters for AUC loss, (x, y) represents feature and label of a sample.
Here, Lce indicates the standard cross cross-entropy loss func- tion, w vjV L.E denotes using the gradient descent ap- proach on cross-entropy loss to update the model parameters, where îj > 0 is the learning rate. Then, the obtained model parameter can be optimized through the AUC loss. The following serves as a generic representation of Eq. (16) as two-level compositional minimax optimization problem:
0,025
0.800
Huraae We
0.775
0.750
0.700
Durax: 11
DuraX 2)
0,675
OuraXi3i
OursXedi (urgK-5)
min max J (g(x),,), #g yER"
(17)
4000
8000
12000
1ehon
2010c
#SFO
where g denotes the inner-level function with one-step gradi- descent and f denotes the outer-level function. Inspired Figure 4: Different on CIFARIO.
by the achievements in addressing the multi-level compositional minimization problem, we extend the one-step gradient descent for the inner-level function to a multi-step update. In detail, for k E (1.... ......K), K, the k-th inner-level function is defined as:
Elo (1; {()) Elz (")), ==1, E0 (3: E (()) Blg ni ())),,
(+)
(18)
k #+
where 9 refers to g(*-1) (.) when k E (2, (,), level function. The learning rate for the inner-level functions is denoted by rj. Consequently, Eq. [17) can be reformulated as multi-level compositional minimax optimization problem exactly as the Eq. (T)
represents the data distribution for the k-th
A.2 Multi-Instance Learning
Multi-instance learning 10] is designed for tasks with training data structured into bags containing many instances, with only bag-level labels known. The symmetric function, also known the pooling operation, a critical component of multi-instance learning. Diverse pooling strategies have been investigated, including mean pooling, pooling (21). Then, to address memory concerns, [40] provided a class of pooling approaches by reformulating the AUC loss function with the pooled prediction as three-level compositional minimax fincttin as follows:
pooling, and softmax pooling (28) and saantion-basedd
stochastic
min max F(w, a, b, b, =EeD [h(w;; - a)²1 + EiED_ [(h(w; bb) w,abb a
a
+ +a(c BIED_ [h(w; X.)1 - ExE, [h(w; Xx)))
(19)
where X; = {xt x.. denotes bag of data instances, D,, represents only containing positive bags with label y4 = 1, D_ represents only containing negative bags with label y = The pooled prediction h(w; X,) fe(fi (w; X.)) denotes the predicted score of the bag i over all its instance, which is a two-level compositional function. For example, for the have:
pooling, we
[w X2) = X
Σ x/EX
exp(o(1;=2))/1
f2(#s) = T log(s:).
(20)
For the attention-barn based (att) pooling, have:
21

---

[ explg(w; w
=(wce=z)) (e f2(8;)
(w) -
8y1
(21)
Exeex, exp(g(w;=)))
822
Similarly, the three-level compositional minimax problem in Eq. 19 can be reformulated stochas- tie multi-level compositional minimax problem by integrating it with cross-entropy loss minimization, as in Eq. (16), after computing the predicted score h(w: X.). In particular, applying K inner gradient steps to optimize the cross-entropy loss results in a K-level inner function. Consequently. (q.(19) can be expressed in the unified form of Eq. 1 which corresponds stochastic (K + -evel compositional minimax problem.
A.3 More Experimental Results
Here, we provide additional empirical results. Specifically, for the deep AUC maximization task, we perform experiments to evaluate the impact of the number of levels K on performance. As shown in Figure E increasing the number of inner levels leads to further improvements in testing performance. For the multi-instance learning task, we report both the mean and standard deviation of the results on tabular datasets in Table 2
Table 2: The test AUC score of different methods all Tabular Datasets.
Methods
MUSKI
MUSK2
Fox
Tiger
Elephant
Ours(att)
0.942(0.039)
0.965(0.029)
0.738(0.018) 8)
0.942(0.017)
(.931(0.034)
Ours(smx)
(0.21))
0.939(0.025)
0.770(0.034)
0.928(0.026)
0.942(0.026)
MIDAM(att)
0.841 1422
0.868(0.087)
0.718(0.078)
0.918(0.030)
0.919(0.029)
MIDAM(
(0.42)
0.905(0.117)
0.702(0.056)
0.909(0.031)
0.903(0.039)
DAM(att)
0.770(0.143)
0.782(0.075)
0.686(0.050)
0.870(0.027)
0.861(0.022)
DAM(smx)
0.802(0.175)
0.847(0.116)
0.684(0.049)
0.889(0.014)
0.908(0.025)
DAM(max)
0.745(0.112)
0.822(0.123)
0.591(0.082)
0.895(0.047)
0.875(0.028)
DAM(mean)
0.795(0.138)
0.826(0.072)
0.653(0.103)
0.855(0.021)
0.895(0.020)
B Appendix: Smoothed-SMCGDA--V
To begin with, we introduce the following terminology to simplify the complex expressions, which will be useful in the subsequent analysis:
V=)(G())),.))) V (") -g)))
V (G(()( vi) = (x((
V.f(H(x))..) = (g (1)
(1) vi (K) vt)
(1)
(z.)) (2/(1 .,)) ...)
(K)
(22)
Therefore, for the smoothed loss, we have
(H (H( = (H((H(
(z)), w(xx - 2x)
(23)
Moreover, we introduce C/ as follows:
= {(K +
(K +16e).
(24)
Following 30 introduce the following auxiliary functions for convergence analysis:
hw.d 2) = min (G(z), 2) dual function =
hw (x) 2) max fa (G(r), VER'v
primal function
22

---

h(2) = min max fu (G(())),
=ER*> yER*>
x" <"( =) = arg min ()
=ER4a
x² x²(2) = arg min (x;;)
#ER"r
y" (=) arg max h ( yER
(25)
Our proof consists of two key components. The first component, including Lemma B. and Lemma addresses the smoothing technique. The second component, comprising Lemma B.5 Lemma B.6] and Lemma B.7 deals with the multi-level compositional structure. In Section B 2 we complete the proof by carefully combining these two components while addressing
Proof Structure.
their interdependence.
B.1 Useful Lemmas
Lemma B.1. Given Assumptions 3.3 we can know
1. (z) is C2 -ippschhit continuous for k {1.... K 1} and G(x) is Ca-Lipschitz continuous where Cg = C
VG(x) is La-Lipschitz continuous where LG= EØ=1 LyCK-iti,
3. V.J(G(r)) v) is L-Lipschitz continuous where L = + Cj
20
4. 4(x) A yax f(G(z),)) ((x) is Li-Lisschitz continuous where La P C,La.
Proof. The first three properties follow from Lemma B.1. in 37 The last property based
on Lemma A.3. in 30 and can be established by showing that [1(())) (x1))l s 208
(22 zill.
Lemma B.2.
30 Given Assumptions 3.1 3.3 the following inequality holds:
|x- x" (y2, 2) - Caay 11 y2ll
||~ (y, 21) x² (y. 22)1 < C,2 ||| 22ll
|| (z1) x² (+2)l - Cz, || z2ll
(26)
Wit w- C23, = بي and Cz, بير and w >> Lemma B.3. Given Assumptions
where Cz2.
3.1 3.3 and " S L, the following inequality holds:
1. The smoothed function fu (G(zz) Mes 2)) satisfies:
(G( ((41; (G Wet1i 2411)
|
2212
(27)
2%277
2. The dual function E( (y6: 2:)
satisfiees:
El a (Bt+ 2
B[h d 2r] + Vf (yr 24), (6)
2/ Las d
Elll + 2
(#++1 - Rz, Ze+1 + z 2r (11+1;
(28)
2
3. The function l((zz) satisfies:
h(*+1)) h(22)) - (2++1 24, 26+1 + 2, 2x" (2041);22)
(29)
23

---

Proof. (1). From the update rule 21+1 = + +:7 - 22), we obtain
fw (G( 1/++122)) - (G(( 2+1))
[1 - =x|l? || Ze+1 113)
2
1
2c+1 - zell² - -I((1 - (=7)( -za)13)
2
1
(1 - -/)))) 1| zel13)
1++1 2011
?/11?
1
1
+ 2y= 7212
| z0l12
?217
w
Yo!
|| - zell?
2 Y2l
2
|2+41 - 2, 1|
(30)
2
where the last step holds uses the fact that 7/2 S
L
Proof. (2). Since the dual function (u 2x) is Lu at it satisfies that
Lund
/ ze) 2 El 2x]] + B
2t), Me+1 - 3111
Bilbb++1
3ell
2
Iw
=ai 20) + To E yfw (x (Ut, 20). 9i)]
Elll 12
E
(31)
2
On the other hand, we have
Mula(Mt-1i444) - -
fa (yt-1i 2111), y 20+1) fw (x" (yt+1i 2++1), Me+1),M(11i2 2x)
2 fz("" (y6-1) 2111) 3t+11 20+1) - fw (x" (y(+1) 24), (14112))
2 (112411 x² (1/+11 2e4 2 1/ 2141111))11
(2e+1 Ez, Ze41 + 2t - 2x (M++15 1(41))
(32)
where we use the fact (a b, + b) = ||a|1 ||61 in the second-to-last step.
By combining the above two inequalities, the proof is complete.
Proof. (3). From the definition of h(2)), we obtain h(2+1)) - h(2z)
hwoall" (2441):241) hw 22)
I hv (2441); - hwedl (ze+ 2t)
fa (y" 2441), (2411;
fo (=" (2441):2) 2)), y" (2(+1))22) 2))
I fx (y" (26+1): 21) (2441); 2(41)) (a (v" (2441):2) (2041))
W [1|2411 x x2"" (2(+1) 22 )| - || - x² (v" (2(+1)): 2x)
(2++1 2++1 + 2t 2x (3" (z
(33)
where we use the fact (a - b, + b) = ||ا||2 ||b1 in the last step.
Lemma B.4. Given Assumptions 3.1 3.3 when v S Pua(w4y and 7=77 $ 1, the following inequali-
ties hold:
We:1; (G ye: 2+)1 - Yol F V fw (G y:: ) w
2
+ 7v7
Elll - 2x [1 + 7 V.f. Me; 2x) ) 1ell
2
2-
24

---

K
A (1) (1ba-1)) h4 + (4Wm12 - 2n" -2=" 4 11112:119
+ Yonk
l ")
B1 [1]
(34)
4
2
Proof. First, from Lemma B.1 follows that the smoothed loss function fw (G ye: 2() is (w ww+- E)- smooth with respect to x. Therefore, we have
(G =x)
< (lL,( 315 =)) B (.) 26)) - -x>i| + w + E
B11l - xz xx11
2
+
= (Gf.)
Yo V
12 A1
=ELL(G( M15 (652)))))))))))))))) 2))
T=
E [=
Y>l/
(L - -pal"]]
2
2
7> 7x" Blllpo11
2,
2
2
< (ILL(G( M15
E (G ye (6: + | (Hr.),, - -Pel1
2
+ V (Gt),,
z()
V (H)
Yzm IE
4
S (Gfu( yo5 (6)))
B vf (z) 263 V 8(i)))) -Pel
2
+oank Σ A (8) (11 (k- 1)) h1 (8) 1| Yoll E [1
(35)
k=1
where the second-to loctad-to-last step holds due to 17 S (yx(w++)
Similarly, since (G(r)), M6 2)) is (w f)-smooth with respect to y, obtain
Bif (G
vo71 + w +
Elllut+1 - vell
2
xx) (() + of (C (G
V..u(G(( Mi (i))))))))
+ 2 [1
3
+
2
Tv
Elfu(G((I ue: 21)
(G
- ( (G))) v6 + 4
+ 1 Elll V vy (G
||
Yo
B gell 2|
w + l ? E[]
2
2
(G(I ye 21)1 +
7/
ff (G wi
+ ?
2
37y w + ?3 11
(36)
2
By combining the inequalities above with Lemmal B.3 the proof is complete.
Lemma B.5. Given Assumptions 3.1 3.3 the following inequality holds:
1. The estimation error between V v.ff((())) yi) and (. v.) satisfies:
(e) K where A₁ = -(03 -022(k-1)323 (
A Alg (1 - 11 11), (37)
A=1
EK-1 kg-tk² 2 + C²2 17
25

---

2. The bounded error between V (H (2t), ye: ye: 21; (4+1) and Vafu ( ye: 2)) satisfies:
Fll (Z)), ye: 21 E441) Vafu (H (z)), yei 24 11 C702
(38)
3. For any k E -1, ... K}. the descent error between h and h k satisfies:
B|lb/41 (1) - 1 - 20"n" (20g)
BIIn ( 90)) 110
k
+ (202))7211 Billp 1 11 20221 0²
Σ (209)-1
(39)
4. For any d > 0 where K E (1,... , K}. the estimation error between g (1g )) and h (2) satisfies:
(
(lgg (1 - 113 S (1 an)²)
B (1) (1e (11(--11) - -1,4001
K
K
K
+ 2/
Σ
2イ(20g)
(h)
h₂ | + 1 Σ
jm
K
+ 2u 2 02
A
(20g))
(40)
1=1
y=0
Proof. First, we have
[B[|| V. (G(z)) - VaJa(H =1]1
= =GIIV v1) + wwf 22) V=f(H( we) - w(x: 2
(G( v1) V=
K
<K> < A (1( - (((()(2))
(41)
where the last step follows from Lemma B.2, Eq. (25) in 37
Similarly,
E V 2 Lo (H..) 2i Val (H() wi: =
E 1)
Vg (K) (K( ( Mei (441) w(x - 22)
(K)
(1
Vg (K
(1) (K-2) / (11 (K- v S (K) .ya)) wlxt zz) )13]
= - (1)) ... Vg [K (1
Vg (1)
(K
(h yy S4+1)
(K)
(K Vg (k) (he
(K- ( (K) (.)
SK (KO3 + cg)o2
(42)
where the last step follows from Lemma B.2, Eq. (28) in 37). From the definition of Cg. the proof is complete.
Subsequently, the remaining inequalities follow from Lemma B.4 and Lemma B.5 in 37
Lemma B.6. Given Assumptions 3.3 we derive
E[[|pe+1 Vafu (1l (Z++1), (t-,)) 24411111111)00000 1 S (1 2 V (H (z:), Mei 2x |
K
2C32021' 21
(20g)=
E (k) - (x) (n (x-1)) + 202
(2C2)
(2 2 [1p..
+ 202,3121111e 12] + 4C 2n
(20 02
(43)
26

---

Proof.
Blllp++1 (aLL((((
= =B[I[(1 - -Pott - VxJ.L Ui ) (F44)) 2411:
= E pan)) (. + (aa
+ -
Vafau V y6r22)
$ (1 - P ) Vg Vg V (h yoi 21)
(K- +,
+ 2 (1)
[K)
(K)
Me41: S(+1)
V0 (0) (11 (1)) 51+1
(K-1), (K)
3e: Get 1
(K)
+
(z)) =]] 27: 2/ C2202
V2.fu H ( y6:
$ (1 - p-- -
(44)
where the last step holds due to Lemma B.5 and third step holds due to the following incquality:
E[|
(H ye zei Ee+11
+ +
+ -1) ((ll)))+([((44))) w
-2+1) -441) V - w(xe - 22)
21) ( w( 26+1)11
< E ( (1:2111)111
= (h .. (1
(10 (K(K1)) (6141) (
-g (()) (1-1))) ((( 1i (K)
Vg
(K
Me (41)111)
(45)
Next, we bound Tj as follows:
T = |||l (1)
Vg (K) (1 Vi " M1tiiit
(M ((4 (M ((( (h (K)
Mr: 1 11
SKK + 1)02 1.7
110 1)02² 1. T.3 ||+ 1=
+ +(K + -)) -1, (x-)) -1 1 11
(K-1)
(K 19 n.
+ (K +K++)) 09 -
2
(k) h 139 C3El1 21 0BEillu41-99 E [1]
1
K
K (2C 1 - + C
I 22 17
(202)*
k
2,
17
9
j=b
k=0
CB7G1" 12 + a2
(20
(46)
Combining this with the previous inequalities completes the proof.
Lemma B.7. Given Assumption 3.7 3.3 we derive:
Elllae+1 Vy (Ze++) M41i | < (1 - p(1--,r))) 1B (H.
Mii =
+ 4a²1)² 1
(2C K g (k)) (+ (202)
k=1
27

---

K
2 L330 Σ (202))-1 + +2ppn'00
(47)
k=1
Proof.
- ( (
(1 111-,""))Bller Vgl Mes 21
2 (411; (25(h
(K)
( U1s Se+1 1113]
+ +225n Bill (1) (K)) . (+41 (4 (K)
M0i 2
$ (1 1- - (H
(R) -4 211
1 - 3x
+ +2p
(48)
by applying Lemma B.5 the proof is complete.
B.2 Proof of the Theorem 4.1
Theorem B.8. (Restatement of Theorem 4.1 Given Assumptions 3.1 3.3
when > 0, Py > 0,
0, w O()) and the hyperparameter conditions are satisfied:
64
1/ S min
(illw + 8)2 (w- 072 022
V
C2 +1 ya
48wC,, Czz. 1 1
166,, (2Lw + w + +)
VPr
Vpu
Va
vPz
4C,, /Cpv 2 o
8V Ex= K
de (20g)
C8
Vpz
VPy
VPu
16C / ( > =K=1 Ez= (2C 16 be,, (203) (203)) 4V100,
Yv Yo (w f)2
(w 1
7/2 Ta 98304w62
64
CTty
1
1
d
S min
(49)
VPa
Py
Va 7/
27 + +
2 a
Ej= )z(20))i-
Algorithm 7 achieves the following convergence upper bound:
[E||]] V.l (GG()))) RBE|| (G
kPo
0
RO?
0 RO?
) 0
NO?
Tx7/T)
PTIPTS))
PyTPTS)
2112TS
o 0227002 + 0 (K)P))102)
0(11 a211022) + (KPN)PP7) + (kanpo²)
(50)
Pr.
Pv.
where Po = fa yo: 20) 2fw.d(V0i 20) + 2g(z)), with the definitions of the involved terms
provided in Eq. (25).
To establish the convergence rate of Algorithm | follows:
Proof.
we propose a novel potential function as
H. fu(G(It) vi)) 2h a0 +2h(2z)
-v.Blllp: Vefu Mi 2) mm vyfu yes 21
28

---

Billi
(K (8) - (h (1b >|
(51)
where the coefficient Va, Vb and (Ae) positive.
B.2.1 Bound Po+1 - Pe
First, we aim to derive an upper bound for Pr+1 Pt. To this end, we begin by applying Lemmas B.3 and B.4 from which we obtain
Pe+ - Pt
S [u(((()++1) - (L(())) Mizz)) 2 (M au at)
+ -2(((2++)) - h(2)))
S Yoll Bill V V V.ff (c:
Twml
V yf b112)111 27
B B1[ - -2ell²]
2
K
1 V Tf M622)) Pell + Y
Ax (1( - h2 11
(1)
-M Yall |11121111 + (37'') + +
2 22-2²
2 (x² (yt, z1); ye z)), [t)] + 7/ 1Ell
w(EE+) 2,, 1 + # 2x (ye+15 z+11) )} + w ( - 2x, zt-1 + 2n
2x 2x" (z++1) );z)
B/l [ f. G (24) 21 + 7oml E V (G 21)
E B[[ - 24 ||
=
2
27
K
B V V Mei ve) - P.ll' 112] + Y K
Σ
A
Ellg (b) (1( - -122 (1
h=1
(4num - Yell 1 B11ppe11 + 3 +
l
1 + 1/'1 Lw 1111
4
2
(V (ye, zx), Me5 21), 9t)] + 2w(22+1 - 2,, x² (g++1: z 1) - x (y" (= 21)) (52)
Next, we derive
2(2+11 - 21, x²' 10+11:2411 - x² (y" (#++1)22))
-((1+11 21, (y++1 R0+1) x0 (1' (2+1)) z++))
+ +2(2241 - 2,, x= (v" (2 2 - (2++1);
1
|zc+1 = 6 6 (M4414441) x (y" (2441);
67271
+ 2141
=e|l|>=* (v" (*++1): ); =4)) x (z=
1
<
||2411 - zel12 + 6% (1: - x (y" ((44)) 2C,g. 11.11114111111 - zel1
67
= Gy + 2O22g )| 2 6-=n|l=" (1441: x" (2((2))))
(53)
where the third step follows from Lemma B.2
Additionally, we have
El y (vr, (x) v;))) q:)l
[E V f (M,, 2.) V, Vy Me (V ())))))))))))))...)))))))))))))) 90)
-E of (V,, v)))) - (G(z)) 21; 2.)
- V. 21] 7,7BBl1 + (C 9cll
29

---

S w." |ill V fo (v) (6))
vr: 22 (G
11lal/- o (2:4 2) B/ q: 2 - (H))) ye; 2x9
+ +1yyBBll| V y. (2x) () q:1l²]
m." [1 V fL (we, 21), (6:2)) 2x) -
+ 22yyL33 K
[12]
(1 -
(54)
where c > 0 is a constant, and the last step follows from
yn 2 = D - (H))
E
be: =)) - (H
(1t) (K
hi
||a] x
(k)
k
s L k Σ
c2(K (l (11(t=1))) - n/6 (k)) 11]
(55)
k=1
Setting c = 1. we obtain
P P+11 - Pt
V (G(.)) 21 + 20"
Blll v. fo (G(os)
yi 21)1 20
B[[|241 - =ell²]
2
2
27211
K Yank
+ (.
+ -Pel1
A&Elll9 (2) (11 (1( - 111 [19
k=
+ (tmmai) 2x11 E 1f 172 37/y1
w w+
1/y1 + "220" Lund 5 |/07) E |/
4
+
(ye, 21), U1i - V yo (G.
ye: )
yf ye = )
K
+ 2. K (+)
(1, 1)) (k) |:3| + 2 11 f (H.)) U1; 1=
k=1
+ +w( 6y:
+ 2C |[| - =1|3] + 6.n x: ((+++(++)))))))) - -x (v= (x(+) 2++))111
Elll V fu (G (IT), 2122 70
Elll E (G |11 w(2Oaz
E3 2 - =e
2
2
3
+
=1) - -p + 27
V f H7 (3:), ve) 1el
(NonKA 27y KC
Ellgg (1) ()=- h1 119
+ (11mmi2122 - 7z91
4 Blllpx1 11 + ///01 + w + 3/1 + nnn Lund vv7) Billdd1
7
+ 8 (Mr, 21) x; [1 + 6wy 1 x (y14112441) x (y" (241)) 11
(56)
Furthermore, due to the strong convexity of fa (G(t)) ye: 2)) regarding x. we obtain
1
E [llaa" 2x) - 1tl13] < (w- 0y
B [ ue: = =)]
(57)
In addition, by introducing
xt(2)) = Ue + 7 V fa (x" (ve, 2x), yei 22)
(58)
30

---

we obtain
(1]l= =² (2++1)
= Ellla" -
I 4B(1I=((2(11) - (+(+)=""())))=
+ 4 =='(=))] + + (z1) - xy (z))
(M++1; 4 (M++1i -x (ap))=="((eee (1++1==+1))) 1)1
I 402, |11 21 + 4B[l|=((z)) x" (y (ze); 21) + +4 Blly" (x))=((a) ye+1 [13
+ +40 Blll= - 1 ||
= 4(02, + C22 |B|| = 2e [1] 4011==((2))
x (v +=)
+ +47, 5.
El| V fe (ve. 3,, -
S $4(C2, + C. E 4+11 - 24 41| - x= (v
+ 8 ". Ell
(x (z) - (G (z)); yei =
+ +87 2 C2 EI| fo
21
3
S <4(C2, + 22 E 11+1 4 B (,, -2el13)
(y (2) 201l
+ +877,1 1/. x
(16, 24) 11;
+16721 Eill v. [1
S <4(C%, + +C -| (00 ||
- - (11, - xz 1 (()
+ +8
GE||
)0 -
K
2 x/. + +16
13K
C2(K E|ll
h (8) 1p]
(59)
1=1
Moreover, we derive
| (xx( x (x)()))))
l (hwpl (y' (z)) (x))==(())))))
(y (z)) (1
fu (z)) =)), yn (z))=z))
w
(z)) han (z()=
1
(xx) 21)1
(w 0g
2
(x
(2)) V Vyfu (y:. 31) 21)1
(w 0g
2
Vy (We, 2t), mG
( f)g
28 C2
2
2
11y (2x) -v1l (w f)p (z+) 12 (w- E)p
V (x ye: 21
(w 0p
2(1 7/ PC2 ayy
+ {2)
(I (ye, (1)
(w - -E)u
4(1 P n/y + 222)
(w l)p
4(1 + 1607 2 22 C2, 2,
+ 22
[ (G.z)
(w f)
31

---

4(1 T5 24 4is
+ )2
(|,, 21) - xx11²
(w - 1
4(1 180 P
+ ()
x/ 3/4
|V (1
(w t
4(1 + P/
x/. + 2
| (G 21;
(w pp
4(1 + P
+ 12122
Ruz
(60)
(w t)
Therefore, we obtain the following upper bound for Pr+1 P.:
P.+1
7=1 - (
"y-"" B1[[ 2x11
+
- pr|l 27 2 (4: 9:11
TznkKK Ax 27 21 KC?(K
Elllg "0 (1( 11 1 21w7: + 1 -2211
(410m2222 - 7o
37 w w 2
+ +1217
Lun
Lund nwm) B(ll2|l1)
4
2
8rume + 1448y C. &
96wysm(1 + 72 2
+ +
+
uu
(w- 8)²
(w 2)31
1(1 + 13/ C2
+ ())
+ 24w7."1
B BIIV ([IVV,L((((())... MG))))
(w E)j
+ 96w7
11 0 B[[[ yH ( 2o) - 2e
K
2 C. 1 K + 96w7
"1 20
22 (4) (1) 11) - 11
Am1
8/ + 18 "" Zzyy
96 + 1/7 C + +
(w
-1)sp
4(1 + 721l 2 CC +
+ 24w7yn
7 2
B|ll (G(2))
(w-8))
+ (21 + Ozz.) 20x 34
E Elll =
+ / v)))) 2)) pell + 27 96w.. " C
Elll VII ve - 1=1
(T 2 K + 96w7. 7 C L K2 (n
(1((()1))) - -1
+ (Armria
7a7 4 []]
33 w w run + 7 Lund 7 E[lle
(61)
B.2.2 Bound Hes1 Hr In the following, aim to derive upper bound for Het1 Hei
Hetl Hr
Sy + 18
21 C22
+ *")
7o7 " (G(o.).
(w- E)2
(w -
4(1 + 73m CC2, 2y.
+1))
+ 24wyy.
(f
(w f)1r
2
32

---

+ w (2tran(C2, + -2g.)) + +200g 37,71 E11 - ==al
+ (nan - P.n (((aa ue 10--ppll pell"]
+ (2717 + 96W7s O Ci Byj vn) (()
-9al9
PonKAL + KCg(K 96 / 3 =b. 13 + +2222a2 11 un ( (20g)=+))
+
5=b
K
+ 4a? 1j
+ +
Σ
11(02))-))
ani² (1) - 119
8=441
K [(20g) 200 Vo (20g) K 2212 + Roo1
+ (Home) + 22,
Σ
(2
Y2ll
B ppl |p]
4
k-0
hm1
+
(3rul
4
+
w w+e
2
l
7/'l
2,2
Lu + 20 3 vo 2 Vn. + L
g
7
7/'1 El||e|1 $
K
k
+ 4C2aaa (0,a01e oj Vo
ΣΣ
(2 2p'n vn C + 4a 11 vo
Σ (20g)
k=1j=1
k=1
+ -2pgn" 1oo" + +2a3n
(2 3=0
(62)
We consider the following choice for bounding (GL())) we
Bi] Vll Mei (6+
and
48w7) 121
C2 e
81 (w 8)2
7x
7z7
SO,
S
S0,
8
(w 6)2
512
96w7=1(11
+ C
+ ?2
< 0,
(w 1
512
4(1 + 7/ CC
+ +2))?()
24
ayy
7v
< 0.
(63)
(w - t)μ
8
Since 727/ 1 and Ozy.
with w-
we set
7t = 7x (w E)z
(w
6)3 4
7/2 = Tor 98304w82
64
Ciy
C+z
e2
64f
1/ minn
(64)
6w(w + 8)2
(w - 0)2
VC2 +1
Additionally, we consider the following choice for bounding
|E||24+1
2x 112 we set
w (2C2z. + +247e1(C2, + +Ozg))
1 ( 5
(65)
372
4-
Specifically, we enforce
2wC2g < W
2Aw7an(22 + +02g)5
w
(66)
2472111 Then, based on Eq. 64 from Cz. Czg and v < we obtain
24'="²
1
7/2 5 48wc,y Cag
(67)
To remove the term 26)))) - pell we impose
7x4 pwo7 2oo I 0.
(68)
33

---

From this, we obtain the parameter choice
7/2
(69)
Put
Similarly, to remove the term B (3+)); y/ 21) 2el 3. we impose
2ny + 2 2 C2 uzz
- Pv'l ub S 0.
(70)
From the second inequality Eq. 663 and definition of Co.y we have
2 (w f)2
1 2Tall 64cm, S 7701
96w7= C2
(71)
512
e
512
As result, require
5
2 Nッ S Py
(72)
which leads to the parameter choice
5'18
(73)
2py
(* (1( - -n,401111 we set
Then, for any & E f1,... K}. remove the term E
K
KA + 2 + 96w7s1
1, + +4a21
jmn
K
K
+ + 20
Σ
Σ
11(02)5-))
-017> $0.
(74)
k=1
j=k+1
Plugging the value of and ub, we obtain
K
Jn + KC 96 22 2
13 + +4227'Cp
Ya
(2Cg)--8)
P
j=k
4a²n1² 57y 1 12 + 20211
Σ
))(20g)-))) - -nn2 Ae 50.
(75)
2pp"
j=++1
To analyze this, we first simplify the expression:
K
T A + K + 30 12KK2 + 422
|Eec (2C2))1
j=k
1 S TanKAr + k
402n" 5yy 13 12 An
2p
K
1
3
40217"02
To
Pzn
j=k
57y 17(102)
+ +1
>m
2pp7
2
K
< ani
Yo KA +
15
13 + 4anc2 2
(Eeoz)i-
Lam
2
Pz jmk
It 17(20,) K-k
1
+ +10am
A
(76)
Py
2
Due to anj² & 1, we enforce the following to be non-positive:
K
d
2K As + 5c,, L3KC + Yan
88 Σ
1
(22))-))
+ 20 ST (2C K K-
om)
Pm
Pg
b=
34

---

K
2 or)
2K An + 5cy, 13
a²-12
2a on
8C
1
Σ
(20g)i-h)
20 120
Pr
Py
j=k
(77)
Therefore, obtain the parameter choice for any A where f1,...,
K
2 + 50 1 + +802 a
((ecgy--t
+ +.20 0C L 12(
A
Pv
j=k
Tx Ave
(78)
Moreover, we enforce
2a²1 1)
Σ
X1(202)0t))
-onp2e S = d
(79)
==++
for k E .1, K}. which leads to
1
d
7 S 2\ a (Efratt {2(233)0
(80)
To guarantee that [i]lp:119 cancels out, enforce
K
K
2 + 20, (2 + " + 22m Σ \x()
4
k==
(81)
This is equivalent to enforce
K
K
+ 2 2 To
(20gg
+ 22n 2p
5 21 220
+ +120
7a
\ (202)*
Yall
<0.
PaJ] k=0
00
4
k=
(82)
Specifically, we enforce
K
2n s
7x9
Tr
Σ (2C2)*
?=
16
Pz k=0
16
2212 5% 21 2 (20
<
T Σ
Au (2022
<
Yum
(83)
2p,""
16
aT
16
To solve the first third inequalities, we obtain
VP=
VPa
7s & 1C, 2 45/50,, (2C3)
(84)
For the last inequality, it is equivalent to enforce
802 a
[(2gg)-R
20 aC,, 1 1302))) K-k
(20
(85)
am
Pa
Py.
16
1=1
where
dy = 2K 2K + 5c -2CK
(86)
35

---

Specifically, we enforce
K
2
Ye
dr (20g)*
S Yo!l
[1]
64
k=1
K
8C a
15 1
2 2
Yo
2 (20g))
7x7
[T)
Pre
64
km1
j=
K
(2
2 2
Th
20ac
Yall
(87)
CUIT
Py
64
1=1
For the first inequality, since dę is independent of hyperparameters, we obtain
Va
1/2
(88)
8
k= dx(2g))
For the remaining inequalities, we obtain
VPy
Yr 1
(89)
16C 2 Ej=x(2()) (2C)) 16 5c,, /5, (203)
As for [ we enforce
37 l 2 1/r1 + 7/' Lund 2C +1.77g1 u.
7 877" 50.
(90)
2
This is equivalent to enforce
37
w + 1211² + 7/21 T +2C223'?
22
5% 2p
7
2 SO.
(91)
4
Specifically, we enforce
7/77 (w +)
2/12 Lusd +
Yy"
2
32
7x 2C22211 1/
<
Yy
2L 2117
2n1
57y
74
(92)
P2T
32
2p
32
To solve these inequalities, we obtain
Ps 8 Op 44/10 Lg
PP
Ya S
(93)
16c,, (2Lw + w +)
In summary, by setting
64
Ya S min
(wlww + 8)2
(w- 8)2
V
C2 1/.
+1
48wc,, O2g. / l 16c, (2L + w + E)
VPa
Va
vPz
4C, 4Cpv 2 k k (20g)A V (202)
EK de (2(2))
VPz
VPu
VPu
16C 2 2 K
(203)K ΣK (2C,)) 16/5cm,
4VI0c,, (2C3)
( 2)2
(w 3
Yv = Yo 64
%/ = 7a 98304we2
CTuy
36

---

1
1
1
1
d
S min
(94)
VPz
VPy
va
7z
275 (w + +0)
21 a
2j= 23 ((C2))))-
we obtain
Ho+1 H,
7x [01l V (G 21)]
Y (G
18 2141
zel17]
4
47/21
K
k
5%
+ 4C a 1) 0220002000000000
220 + 22
02
4a²"1"
Li
252>(20 (20g))-1
P>71 k=1
Pa g
2py
k=1
+ +2p2n*
57g 0² + 222211 10000
K
Th &
(202)
2py'
[2)
k=1
1=0
7x
[|| V
Mii 24) )11 - C%l
- EE
V (x - Cy, w
=1]1
FBll,0x - zzz =+
4
4
K
K
+ +40 21 0²
Po
2022r=1 +
2p
4
"
C20² - 4o 0² 57 12
> (20
P= km1
pz
2py
k=
5o 0² + 2a 20"11'0" 11
K
k-1
+ 22p 2p
A
202y
(95)
km
j=0
From
W.f((G())).)))) [. ( Mei (G)))) 200 zell?
[ =²)l== (G(z)) Me; =)]3
(96)
by summing over from to T and reformulate we obtain
Wei =1) 2x V
Bl/ - ()
T 1=0
K
8K
Swi
Ho - Hr
(20 0²+ + 22 C302
< maxx
T
Pr.
k=1j=1
a²n² L3 Σ (202))
K
K
k-1
+ 10c,, Py
1 25p5ppp77 /poppcyn 0² + 2am2 o²
Σ
M (202)3)
(97)
km
j=U
When t = 0, we derive
5% Elloo (H yo: 20)
Ho = Po +
E Ellpo - Vzfu(H(EO), yo: +1
PuTl
2p,"1
K
E (k) 9 (1) (1(60 112
am
k=
Ye σ²
57cc, o²
K
< Pa
>
(98)
pz7
2py
an
s
Finally, we have
T-1
T [IIV ((t))
37

---

kPo 0
0 ko²
+ +0
ko²
0 RO²
Tont
Pan
PyTRT55
(x72TT))
+ 0 021202 + +((PPttPP0))) +0(1: (1101000) + O + (o07)++((ma)) 00
0
(99)
Pr
Pg
C
Appendix: StagewiseSMMCGDA-VV
Note that in this section, we provide general algorithm for the multi-level compositional minimax optimization problem satisfying the two-sided PL condition. Specifically, we aim to solve the following problem:
min max f(G(z). ZER" yER
(100)
where f(::-) satisfies the two-sided PL condition. To simplify the analysis, we further assume that the loss function satisfies the same continuity, smoothness, and bounded variance assumptions as stated in the main text.
In this general setting, we can obtain two results which may be of independent interest beyond their use for the translation phase.
First, when the number of stages is just one, i.., R = 1, multi-level compositional minimax optimization problem satisfying the nonconvex-PL. assumption in Theorem CA1
obtain the convergence rate for the
Theorem C.1. Given Assumption
3.1
3.3
when R = 1, by setting 7l.1 O(E/k). na.1 O(E/k²),
and the initial batch size as O c), after running Algorithm for Ti = (x (c2) total iterations,
have Ti Ei' (
2
we do not use the PL condition with respect z. Therefore. the result provides convergence rate for the nonconvex-PL. minimax problem. In addition, this convergence rate corresponds the standard compositional minimax algorithm without the use of the smoothing technique. Therefore, in Table[C] we compare the convergence rate and learning rate with and without the use of the smoothing technique. It can be seen that we should use smaller learning rate for compared to y when not using the smoothing technique, the condition number k > 1.
Note that in the proof of Theorem C.I
Table 3: The comparison of the convergence rate and learning rate with and without the use of the smoothing technique. LR-r denotes the learning rate for I. and LR-y denotes that for y.
Algorithms
Convergence Rate
LR-a (le/2122) OlE/K²)
LR-y
LR-a/L.Ryy
Smmottedd-SMCGGA--V (Thm. 4.1
O ((33/2T7)
0(e/x-72)
0(1)
Onestage-SMCGDA-VVE (Thm.
O
O(E/R)
0(1/x²)
Second, when the number of stages is greater than one, i.e., R can obtain the convergence rate for the multi-level compositional minimax optimization problem satisfying the two-sided PL condition in Theorem
Theorem C.2. Given Assumption 3.1 3.4 by setting co 2513 μ²² Pr. 6400co L2, Pv 640L
640cg L3 79.0 = 1 ((1/(v2r- Lg)), T. O(co/ (n 2r) after running Algorithm 3 for Okk0/ /e) total iterations, we can get E (5²)) - ((x*) Sc.
20L.'' To max max
T.goz
16V }. and for r 2 1, 11..r (( Ng.
C.1 Useful Lemmas
Lemma C.3. Given Assumptions 3./ 3.3 and Na,r
S
2.. we know
1.
E E (2) & E E(((x..))
nr.
(1,,,
ne. E Ellp [
2
4
38

---

Algorithm 3 Stagewise Stochastic Multi-level Compositional Gradient Descent Ascent with Variance Reduced Algorithm (Stagewse-SMMGGD.) SMCGDA-VR)
Input: Pr > 0. Py > 0, a > O, 12,r > O, Ay. > 0.
0 xo and (,00) g (hg (k-1), -( for & E {I,..., K). po gg (zro c(t) (g(k)(())) (K-1), ((K) Vi f1 yo: Co.0),
(K)
do V 22 yo: Co.0).
1: for = 0, .R do
2: Fr E. 1r.,0 Dr h 17.
(dc)
(b)
for k E {0, K 1}.
Pr.0 = Pr. 9r,0 = gr.
3:
fort 0,... T. 1, do
4:
2r,41 = Ir,t Pp.t.
Vr.++1 -r,tt + 1r.
(0) 1r
5:
6:
for k = -,... K do
7:
Compute k-th inner-level function:
(, = (*) (h (,.41) (1 on (1 (n (.,t1)))
8:
end for
9:
Compute stochastic compositional gradient ur,4+1 and Un,+11 (h
Wr,++1:11= (M (R) Vif(h "1 Br.41i
cl441) ( (K- -2),
(1/141 c( ( (n ( (S0tt)))) i5:441)x
Ur,++1;0-1 (2f 1.11 Nr,e+1: Gr.))
(K)
10:
Compute variance-reduced gradient Pr,+1 and gra+11
Pr.+1 Ur,++1:+11 + P= ( (Pr. - ura8+1),
Gr.++1 = G.1+1:11 + (1 - py ) (grz. Urses+)),
11:
end for
Randomly select (#r+1, 1r+1, (k) Pr+1, (r+1) from {(Tr,s Uras by Pr.t (r,))==0
12:
13: end for
+ 1/. E| V
(r:) 2n, K Σ
( (k-11)
1/ (12]
+ 2m,, El (H) -pr,e1
(101)
Proof.
Lo
B[ < (x,,) E (r,41 x,,,] +
E B1l 1:::4------0 x1,,el
22
I.
= =E)((z,,)) - 75 E (Ir.)).
Pr..) +
2
n Lux
E(((F,,))
Je
F
||3
nax
E
[lllpr,el
Pr.el11
2
2
2
n L.
< E[4(zr,))
Ba.r E
|
Na.
[E
2
2
2
+ El
V
(Ix.)) (G(r,))) Vr. + n1, Me, ( pr.cl
< (()()r)))
Na.5
(zx
Norr
Billpr.el
+ nr. B http:///ccccnooooo..........o...
2
+ 2m,,
V Vn.))
(z,, V2f(H
Vr. | + 2m { (1.
< E((()r)))
ne E V
(
|| Te.r
nn. -.
2
K
+ 21 K
A (1)
(1=1))
- 2n V
(102) (100))
A=1
39

---

where the second-to lo-last step holds due to Mr S
1
2h4
Lemma C.4. Given Assumption 3.1 3.3
Ny.
we have
(lf(G(Zra))
3/,
E. (r.))))))
Fy. 22.5 Blll (G (r.)))1
2
K
K
+ Mar K
Agll (1) (h/ (1(k-1)) -h 11 + 2y. L K Σ
2 E|
(h, ( n/ 1/2111 11?1
km1
km
7/. [1]
x (H (Zr,)) yr.,) Pr.e 1 27, Ell V V y (Zr,), Vr.e) 9r..11
2 (Ir,1 ||3]
My. Blll [1
(103)
Proof. First, from Lemma B.1 we obtain
Eff(G(Z,))) Vr.))
$ (C0 (n.), -2r,)) l
B11 - Fr,,
n
E Esllpree
2
+ *1, Dr )
E B(Ilp,,e11
2
Te.r
E|] V V (G(rn)) Pr.el
2
2
2 Blllpr.e11
2
3m (G..)
2
E
B[lppr,el
2
2
Th,, Ell. My.e)
Vz. (H w..)1 1., V .. J [ (Hr.)) -Pr..)
n
B(llpr.e11
2
K
+ #1,5 K
(U Pr.el]].
(104)
Moreover, obtain
E[f(G(a,.42))
- vr.1)l :1.) + Ellbb..-1 vr.ell
2
-y (,,_ + n E [|ll
!u.r 1
ny. E V (G Ur.e) gr.ell (r.e
Tu. 111lll..
2
< E (G(
ny. Eil (G..))
vr.1
n/.
2
2
2
[El V -,
(G( Vr.e) gr.ell
12
17..
) gr. 11
2
40

---

3
2 " ((g. (o. (n
vr.a)1 ( gr,e112)
ng Tu
(I) Ju. -tt
BBllgna
2
3
2+,111 " - -rr.el² 2 "
< ([J(G((rr4+))
Nu. (
Ny. [1 [1']
2 Vin [ ||te| [1]
3
+ Bll.
v
Ur )l1] 2n,, B11 V / (Ir,), vr,1) gr
Bl (r,
7/. B1| /./ (G 11 T/.r B3 Bl11 11 + 2 Vun EE| Pr.a | 17
+ 2 - 1,,, (K) [| + 2 E1l / (,,,),,,,.1 0r. [|
$ ())((((r)4)))))))
Ty. Blll (G vo.,)1
Te.r Bl.BBllyr: [12] + 2 Uest EB| [B[1pr.:.11
K
+ 2g I2
ca
B)
22(K
(h/ n 27 E V (2r,)) g..1
k=1
(105)
where the sixth step holds due to myrr S t. the fourth step follows from the following inequality:
2 2 Vvf(G((r vr,e)1 + of (G(rr.t))) vr,)) V (.
(106)
By combining these two inequalities, the proof is complete.
Lemma C.5. Given Assumption 3.1 3.3 Ms, S 162 we have
E[D(Ir,(+1) f(G((z,,+1) (r,+1)) (z)() - f(G((r,)) vra))
5/ E[I ]
B, (G( (G vr. + B BlIVo((rr.) V=((((()))))
4
K
Bllgg (k-1))
(1)00 1| + 2gg L3
(*)
n 11/11191
-33 K
+ +312, B V ff (I,,)) Vr - Pr. + 2n, B ( - 1..1
Ne. B[llpr. [| Ng. E 1ll.. 1|3]
(107)
Proof. In terms of Lemma and Lemma 4 we obtain
E ([ (G(( Mr.e 1] -E ( 2. - ((G(aa)))))))..
na [ (T.
Na E pellpr.:11 na, ( - vr.)))
2
4
22, K
B |, Pr.ell
km
+ 2
V Bra))
Tuu Thr (G(,,)
4
1/ K
A (1) (1
-1
(k) [12] + 21 T K
[ (1) (k-11) 002213
T/
( (H vr.))
Pr.e 11 + 21 F V (H 11)
2m2
Blllgr:el!
4
Bll
Ne. Elll pr. 11 Nn. E[|
-V vr.))
2
41

---

K
K
+ +37 K
A (1) - //0 1| + 20y Li L Σ
c2(K Ellg (0)
(1) (hr,e
- 11/111991
k=1
k=1
(z yr 1r2) !v.r B1l (C
+ 37,,
vx (z), vr - Pr.ell | + 2my Blll V (H - qr,1199
2m 2IE [1pr1 1|] - Tur Blllgnell [1=]
5/2..
[l[1 (zr )1 Tyr B
AMz, A BlI (=((((())))
K
K
+ +3m K
Blll (8) (1 (1)) || +2 L K)
-2ik-N (h - n,
km
+ 3122 B
f (Ir,)) Vr.,)) Pr.ell 2 (H - -
Marr Blllpr,e ||
Blll [1
(108)
4
where the last step follows from || (G vn.)) (2V (1( and Yarr S I6c
Lemma C.6. Given Assumption 3.7 3.3
by setting
1
d
Mur I min {1]
22. S min 124 16 2\
a o(Dj=k+ X (2C3)5-
K
22
25
Py = =4002 P= = 6400cg 640cgL3 co
(109)
where do is defined in Eg.
116,, Ly is defined in Eq. 132 such that us
Du.r
have
Iden
B1 (I(x,)))
Collerr
Bll V (C
T t=
1ly.
40cgVrs T,
160co
(o + oho 560 + 330cg
(110)
Pg Pym
Proof. We first propose novel Lyapunov function follows:
Hl,e+1 E(D(zr,++1) ((x.)+ Co (E(D(2r,++1) EEf(GG(z, (.)
4
(H(D,,4))) - Prattll31
Palar
PyTu.
Arll (k) (1-11)
11,41119),
(111)
where Na
Yu.r I0co
Then, from Lemma C.5
B.5
B.6
and
B.7
we
obtain
Hr.111 Hr. <
Na.
Cn
E
V
V
(G vr
4
4
4co
+ (Ma,r +
[LI - - v..))
Ny.r
3cother
+ (2mar
Am BlIl vr.e) pr.el
1/..
(2c07h,r
4nly
E -
Sma, c [2cg) + 2202
K
4 nar
Colerr Ner
E Elllp,,ell
Pr.
Pyyarr
4 np
k=0
42

---

81y.r
Cg + 8yyr 1.3
Cola.r
Blllar, [2]
Pulleir
Py
4
300 KAs 200 13 +
16a2
KAp +
Cz
(202)--1
Mar
Poller
5=
K
17 + +2a
Σ
2)(202)-b))
am dR
Ellle g
(
(1((t-11) )
Pyller
$=
16a2
K
K
CZO²
(202)-11 + Ta
20 20'
0²
A
Σ (20gg)
k=1j=1
1=0
16021/5
1.300
E(02
8p
(112)
Pylbuir
h=1
To begin with, for any E (1,... K), remove the term Elll (0) (1(t-))))) set
30o
16a2
K op
27,, KA +
KAs + 2007e K2K
((20g)=0)
Ty.r
Puller
jmn
1602n
+
(202 20
Σ
22(20g)-k)
an 50. (113)
Pyl
s=
Since Trr
Ayr 10cg
we enforce
16a2mhr
3 K Av + K3
+
C 1220g)+
Patler
j=k jam 50.
16a2 ne. 1, 13((2 k-
1
(114)
By solving this, obtain
A
SKAr
800 K
((K-K)
32
C2 (2
32
12 1,((22)) K-k
0fler
Pallar
Py
8A
ScoR1.2 + 32\x.3
327a,A
(115)
am7,r
ameer
Pedler
Pyulur
where & E (1,....
K} and X₁ maxa(xx Ak.2, Ak.3, )k.)).
Given that d can be organized as:
8AR.1 + 880 +
32
32Ak,4
(116)
10copy
Nar
Pr
nu.r
Moreover, we enforce
2a Merr
2 4
Σ
2)(2
am 5 -
1
(117)
2
j=k+1
for k ... K}. which leads to
1
d at a (2(23)0-
ne.r
2\
(118)
Additionally, by plugging the value of Ae, we obtain
3con2.r
21s.. K
Au +
Σ Au + 20u K
c x2(--)) + 2x 2 nr.r
Σ 22(2)))--))
1mm1 By.r
5m
43

---

+
16a 16. Nar Cx
Σ
((20g)=th)
K
+
16a
Warr 1,,
K
(22gg) K-b
-mminr
K
A
Pet
Pollor
h=1
1
h=1
A=1
-
23
Ba K
Au + 200me.
c2(K-k)+
16.
C
Σ
(20g)-4) +
16a Taar L7 > (20g) K-k
10
Pelar
Pyllair
5=
K
KO ())
K
8KAr
Σ
(20g)=**+
32
1 (20%))
2 0ff..r
Pollar
Patuur
k=
j=k
23
Marr
K
A 2007s T.f K
o2(K-K)
160miar
Σ
(20g)5-t)
16ami
K T.j
Σ [200))K
10
Pel
Pylloir
k=1
R=1
h=1
==
k=1
K
16amper
16amhr (200)))
AKAupar + Acota L3 K3
Cp
1(20g)i=*
Patarr
Pyuue
K
K Ak 200 13
10 Ta.
K)
Σ
0g
o2(K-R)
(119)
To guarantee that Billpr,e|1 cancels out, we enforce
K E(0g)) + +223(202))
4n2,r
Mer
(2
O
ya
coller nar
S 0
Pz
Pylly.
4
My.r
8
k=0
(120)
This can be done by setting
K
An (202)*
Bar
S 0,
16
k=1
K
8Mar
C2 Ps
(20g))
U2. < 0,
16
km0
4m2.
213 (2C
ya.r
1 0.
(121)
Pyfly.
16
For the first inequality, we enforce
K
8Ak.1
K Sco (202)*
1
(202)*
< 0,
00
64
64
k=
32\k.3
(202)
1
<
0,
32\x,4
(202)* g)
1
00
(122)
Pr
64
10coP
64
is easy to obtain
K
K
Ox > max
512 1 (20g)*
Σ
51 31200 AR. 2 (2C2)/).
k=1
k=1
K
1024A (2C2)*.
P2
2048A k. 3 (202)*,
Py
(123)
5co
km
For the second and third inequalities, obtain
K (20g))
64
Pr 2 128C 2 p k=
Pv Z 5co
13 13(23
(124)
Similarly, to guarantee that [r[ll || cancels out, we enforce
81 O2 +
81y. 1.
Coller
0.
(125)
Pa7arr
Py
4
44

---

With Zr.
2xc. 10cm
we obtain
80
c +
81gy,r
1gr
S 0.
(126)
Pz
Py
40
To solve this inequality, we enforce
8000
1
8
C -
1 v
(127)
Pv
80
Py
80
We obtain
P. 2 6400cr
Pv 2 640L3
(128)
In summary, the hyperparameters should be set as follows:
1
1
d a
/he S min {
Tar S min
(8)
2L
166" 2
((C==
k
K
K
P5 2 max
128C 2048 3
> (20g)) 6400coC2).
p
k=1
k==0
K
1024\k
64
Py
2 max
(2
(202) ,,,00))),66011)
5eo
5e0
k=1
K
2 max
512 (2C2)*
512cg (2
(129)
Moreover, by setting Co
25
get
4co
82
Cofler
5
(130)
nu.T
112
16
4cong Ty.
Then from Lemma A.7 in we have (Na,r + coffe V (G((r)))) Bra))
V Vaf(G(e,,) (r.)))
As result, we have
Ho,1+1 Hy,t
Ba.r ( Co EI V yf (G((r.)) 3+
Co
4
16
177z.r
E[l V. (H(o))) Br.)) Pr.cl 197y.
(Ir.), vr.)) -..l1" 1..1
10
-110 Mm K
17
AA 200 LiK
(2(K-N) Elll (1 ( ||
16a 2ma2ma
n2.
K k
K
k-
CB0²
(202)-11 + E. 2a ,
Σ
A > (209))
Pz
k=1s=1
j=0
16a2nd
13002
(202)
3
Pyllur
Ma. Conn.r Elll V. yf (G(z..)) vr.)) Coller
V
E ( (l.)))
my.
16
17 Ell V (H(zr.) pr.cl
197y. B V (Zr.)) vr.)) c...
10
K Σ
-110 n,, K Σ
17
A 200 L3K
Elll (2(K-k)
k=1
45

---

16a m
K
k
K
k-1
O20²
Σ>
0200 + n -
Na o
Σ
As
(20gg
Pz
k=1Ø=1
j=0
1
Nase
13002
(20g)
(131)
Pyllur
where the last step holds due amerr S and
Pour
since Te
10cg
CD
25
/²
Then, set
K 0
13 max (1, Σ 1
(20g)
Σ Σ (22gg)),
1=
1=0
k=1
Py = 64003, P2 = 6400coL a = 640cgL3
(132)
where Ng is defined in Eq. 115 Meanwhile, this indicates Pa 10copy copy.
It is easy to verify that the conditions in Eq. 119h are satisfied.
By summing from to T, - 1, we obtain
T
[1 (
Colle.
(G(as)
T. 1=
Ty.r
4 4(Hr.0 - H.,T.)
64
K
k
cg
(20 +32pz
Ta Tr
Pa
k=1j=1
K
k-1
64a211a
K E
y2 0²
+ 80 12
Au
Σ
(20g))
+
(202)
Ds.r
k=1
jm0
k=1
4Vr0
160
16
40
8d
Scode.2
+ 32
32ARA
Tr
P Ty
Pyl
Nu.
nn T
001
amper
Pa
Pyl
K
6402
K
+
62o2
(2
1- + 3222 ,2p +
1300" >
2202))0-
Px
Pytlerr
k=1ş=1
k==
+
K
SAR,1
ScoAR.2
32>a.3
+
32>x.4
Σ (20g)) +
k-1
32
amTar
Pallar
Pyr
1a.
1=1
1=0
40co Vn.
1600
160cno
8960coo
Tr
Py
Tr
Py
Py
K
k
8
32
Py
C2O²
2
(202)=
+
Py
C2o2
L3,00
Σ
(202)a-1
125
10
p.r
125
88
Σ
K
100 Py.
8
copy
32p
32copy
(202)s
320co
100
1000
1000
k=1
40coVr.
160co
(0% + t 560 + +300c
(133)
Tr
TT
where
Vr.0 B[D(I,,))] - ((xx)) +
Codle (E()(z,,)) Elf(g(z,o), o
Ny.r
E|||V, (HH(v,()) V..0) pr. 0%,0 h.k
[II gr.ol and 0,,0
h.k
E||gg (k) - [1] oAo -E Noro..
(b)
46

---

C.2
Proof of the Theorem C11
Proof. Based on Lemma C.6
we have
T
Co (G am.)))))))))))
T, 100
Mly
40cp V,,0
160co
Py Py T, to + o 560%,) + 33
(134)
ny
Since co O(²),, then is easy to verify that by setting by setting fy.1 O(c/k), M7,11 O(c/n3).
Ti = Oxx0/e/// and the initial batch size (K//e), we have
40coVr.0 T,
S <E(e)),
160co
Py'ii.TT, Tr (0 + 560 S 0(()),
330cg 1000gLL2p00 I2 (((2).
(135)
As a result, we can conclude 1
E V |1 e
πi
C.3 Proof of the Theorem
Lemma C.7. Assumption 3.1 3.4
we have
2.110 + of o + 560 S
Pun or, + o²,0 + 56070) + 20c
320co
+ 338cop 1i002
T.
(136)
Proof. In the following, we will bound 07O0 + 0000 + 566 At first, from Lemma B.5 we get
K
X XEI
(* (h1t
T.
k=1
K
X 11 (1(00
200
K
Σ
K
anh Tr
T
k=1
k 1=++1
A-1
Elllrrell 11 + 20m
N
(20
T.
1=0
2a 2 11,r
BBI (0) (n (1 1?] 11
- - - (1 "1
am2 T
2072 Tr
j=k+1
K h-1
+
(2
Σ Ellle.ell [1 + 20m
X/1
(20
T.
k=1
j=0
100
X (n) (k- 111
50
XB (4)
g (1(tt
Pytti. T
Po
T
L3
Pemger
E| +
(
(137)
S.p T,
50
where the last step holds due am < Then, according the random sampling operation, we have
(*) - (1A
X (1 -1.00) [?]
1.0
T
hm1
47

---

150
K
h.k
+
L 1 1--1
Bllp P]
PuTuer
K
X/2
k-1
(20g))
Py T k=1
copy
Tr
50
t=0
k=1
150
7,00 +
L3
B[Ilpell
1 1
Σ
Bllareel
+
2
K
X
(202))
PT T.
Py
10 T.
50
150g
T
Σ
Bill p. 112]
1
T.-
Σ Bll4.1111)
+ "0%0
Lig0
(138)
P T
640
10 T7
0
where the last step holds due to Py = 640
Then, based on Lemma B.6 we have
E|l (a. prall?]
T.
2nhrr
1 T--1
Elll v af
(M( vn.o) p,
[[[
T
Tr
Pumber
Tr
+
4a
Pr
TT
1
T-
K
( Σ (20g)- j=
K
El| ( th) (1 -,,,"
Pr k==
2
K
T
T
B[Ip
||
t=0
k=1
t=0
40- 2
K
+
Cg0²
(20 2p Cso²
(139)
Pr
$
10
(H - -
243
1
Σ Ellp 11
20L3 1
T.
B B1
P T,
100 T,
Py T,
1
ΣΣ
K
XEBII - -9 (h(t--))))
Putheir C/O0 2 ΣΣ 250
1 ((2gg)' 1- + Po Cho'
+ 250 Tr
j=1
where the last step holds due to the definition of Xx and any S 1. Then, due to the random sampling in each outer iteration, it easy to know
v I V
Pr+10
[H pr.ell
T
10 Pv 1 T,
V (H((0)))) Mr,o) pro +
2001
1
B[lp,, [1
T
Elll [$]
Pr.
T,
I. 1=0
1=0
150
oND +
T--1
Σ
Ellpp.. 1|
1
Ella- + 50 Pon Li00
250
PomR. Tp
640
T.
10 T.
1=
Thu 1000
C
1-
+ 5 Pur CHOO
1
k=1j=1
100 TT T
t B T
201
T--1
Σ
T. E Ellpr.ell [1 10 T Σ
Blla.:19)) + 2 -om (140)
640
T,
t-u
where the last step holds due to By = 6400.3 co >> 1 and
colie.r 1
Th.r TO
Similarly, from Lemma B.7 we get
VIIVv,(H v..) (..
T. D
< TT
E B of (Z..o) - 2 + 4a na
L Tr
T,-1 K
(20g) K K - -00)(h(t-))) 11
T
+ py
2n
1
T--1
Σ
Elll,, 11
2D 1 T
Blll,,, 11
se²
1 4
Lij0"
2 Σ (20g) + i
T-
Py T,
1=1
48

---

T--1 K
(H
1
X
[
B
1P
(141)
Po T
25 T,
L
T+-1
2
T--1
4p
K
El| Pr, 19++
Σ
Li00
Σ (20 2p
50cypy
T.
Py
100
1=
where the second step holds due to the definition of Xy. and the last step holds due to an Then, due to the randomly sampling operation in each outer iteration, is easy to know
S 1.
of +1.00 (( ,..1,
9mell
T,
L3 1
24 T.-1
E|l (H( gn.ol11++
Bllpaell
Σ
P. T
50 T,
Py T.
2
1 T. E 9 (0) (106=-1)110 + 4pp 1270²
(20 h-1 + 2p
25
10000
Elll V v (G(z,)) vr.o) 9,,ol
2003
1
1 1 9++
Σ
Bllly.,
Py p.r
Pv.
T,
10
1
150
1
-.-1
Σ
E
Pll 12]+
T.
Σ
Ellgaa111)
Pom
1.30²
25
Po T,
640
Tr
10 T.
50
1=
4py 10000
K
Lj00"
(20g)--1
(142)
k=1
21
E
Ellped 11 11+
1 Σ
Blgesll
I11) +
11
5 12g00
TT,
T.
T
640
Tr
10 T,
where the last step holds due to By 640L2 /3'
Then, we combine these three inequalities together as follows:
-01,0 + oy 0 + +500
100
201
T
Σ
BlIl 12]
T.
Σ
Ell
2 ( Pa
TT
Tr
BT
T
640
T,
TO
+
1=0
2=
+ 0%
60000
+
21
Bllle 1
1
Bllg.. 111)
11
1.30²
TT
TT
640
10 T.
t=0
1=0
+
$4000
56
T--1
=1 - Σ
Ellgnel
E
+
56
Po L.00
(143)
T.
640
Tr
10 Tr
50
100
84070 T.
1/1 Tx Σ Bllle..111
T--1
1
Tr-1
+
Blgg..ll11 + ni.
Po
10 T
Then, we need to bound /. ET= E Billpr, ||31 +
conv.r Ne.r
T. ET= Billg,, 1P]. In particular, we have
Coller B[
Tu.r
< 2BBl|pr,
(2(x.,) )l (4..))
+ Colleur
2B[l1 - Ho |
Vyrr
+ + +
<
- 11 ht
+ 8Ell (
+ comler
((T - (H u..
Wyrr
49

---

(..)))))))
+
8
ns. B. (
7ar
con...
(144)
4
K
SK B (C (r.))
A& (1 1/22 (.) proll
40v
qn qneel
L3 Σ
(1 ( 112
Myar
1/.r
km
By plugging Eq. 131 we obtain
El] P
ca Elllg, 119 8(Hre - Hnet1)
C0
Ny.r
na.r
2
68
Priell]] 152 (
V ( (Hr,) Urie)
B
grell
5
5mparr
68
K
A4 + 160 Lj Σ o2
C02(x-t))B(11117 - (1, ((((--))))1111
02(x-k))
Bllli
4
E (C 8K Σ AuBllla (( - 1
SE - pr,ell
K x2(K Ellg (1.t n1.2
4colla Elllgrt V..
400llas
thr
qurr
k=1
8(Hr,t Host1) 660cc
(145)
7a.r
where co
258 A
By summing up from T, L, we get
Bllpp,.11]+
conler
B[llg..111) <
8(Hr.0 He.r.)
660cp 12 Pspo
T,
uurr
Tr
80c0
3200
320co0
17920cgo" + 660cs L
T,
TT
Pumh.
P T
80c
320cp
o + 0% + 560%/0 + 660coLa
(146)
T.
T,
By plugging this inequality to Eq.
143
we get
0+1,0 + 0.-100 + 560"+11
100
o.h
8407
T--1
T- Blaa,ell11
Bllp 2
PT
Pr P Tr
2 T,
10 T,
t=0
100
0he
84070".
PT + P,
T Pr
+
1 Vr.o
320co
(ora +o + 500 ( + L20
2 Nu.r Tr P Tr
< 320ca 07,0 + 0% + 5607.0
20cg
+
Mo.T. + 13
(147)
In the following, we prove Theorem
Proof. Under the two-sided PI. condition, we get
2p (x,))))))
(xx.)
Collerr
- (E]
7y.r
50

---

Co [ ur
(148)
My.r
Due to the random sampling in each outer iteration, get
T.
V++1.
[[
(xx)) + collar (E((((r);)] - EE (9( Mr.))))))))))))))))))))))
T
My.
T.
V (Fr.e))
co E [I v ( (rx)))
2p
T,
fl.
1=
1
40co 400
160cg
(0 m0 + 00 + 560 h 330000 12z0²)
2/1
Myrr Tr
Py T
1
20.0Vr.0
320co
700 + o²,0 + 560%)0) 338c0 13302
(149)
1
Pu T.
Therefore, when O, have +0, have
+560
58 Based on Eq. 149 and Lemma C.7
18560cpL2
20coVo.0
i + 0².0 + 560 1
0²
+ 338c0
Po
ny.ooo
1 1 11856000133
1.2
20coVo.
V2.0 S μ Po
+ o0
(150)
Ry,oRo
When 0, by setting N4,0
and Rg max 16Vo.9 Lzo=
), we have
30
18560 x 45cg
20 x 30co Vo.0 0²
i.0 + oooo + +60
<
+ 338rg L3 < 500c 1,302
R2R.
Ro
V1.0 5002 μ
(151)
where the second step holds due to Py 640L
Therefore, we denote 61 500cg o2 / such that
T.. + oo.o- + 56 S HE1, V1.0 S E1
(152)
In the following, 0 + 5h + 560": S pe, and Vr,0 S er, we will prove of + 0,41, + 560 S pe, / and Vr+10 & 4./2. At first, we have
the inductive approach prove the desired result. Specifically, suppose
0+1.0 + o/ 0,+1.0 560
320coL3
20co
jier
3 LZ00
(153)
Py T,
T,
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I 2 we enforce each term to be smaller than /,/6. In
338cop 13302 per
(154)
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338cg640L3311,
L330² < /EE
per
6
My. 1140 /c0
(155)
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51

---

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20co
Cr S
(156)
ny.T, T,
6
we get
120 1140cg 1.3 /co
T, 2
(157)
MVRE
By setting
320cg L2
per
Py T, Kec, S
(158)
6
we get
570 x 1140cg
T, 2
(159)
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VPc.
and T, 0
2. Ryker
V
agig0
)
get
1140,7m130
pe,
71,0 + 560" ++100 0
PC,
<
uCr
(160)
6
2
and
1
320coL
20c N,
V4+10
P T
Me,
Er 33. 5
(161)
11
500c 50Uc,L300 L300 get
Since 6 2
1
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c6LLz02
2"-1 4
= Lăco x 2r-1,
x
coL.302
Per
CoLl//Coa
col3/a0o 22- coLo x 2r-1 <
CoLo /
x 2r-1
(162)
PVT
VCoLBo
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2 = c so that
R = log, 25,, As such, the total number of iterations
(To + Σ Tr) = (mmxxmmmxx2)))
R
16V0.0 Lga2
+
Rf
Co
x 2--1)=
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=
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60
c : (163)
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R
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Co 2000 291
=
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(164)
14 2 1
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UC,
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ny.
= O(1 ((1/22 2011/22-111 11g)
(165)
1140
2 11
and it easy to know Mz.r = 0(121 00122//2-111, 22 Lg))
52

---

C.4 Proof of the Theorem 5.1
Proof. Because f(G(z). v) is strongly convex with respect to x and satisfies the PL condition with respect to y, we have
2 B(((OR) 6(x")),
B Bllaa x= x=1
(166)
where set 28 such that f(G(w), y) is E-strongly convex with respect to æ; and define ((x) max,, f(G(z),)) v²) with 3² = arg yER", (G(()),)) y). Then, according to Proposition
2.1 in 30], to guarantee E. F|| - =01|² z ( such that ( = O we can enforce E ( g) - (x")) ( Then, from Theorem C.2 it is easy to that after running Algorithm for the total number of iterations ( (Note that /E is usually large in practice (3) so that we omit other factors.), we have = - x 11" < O(e2) and then (ix\)( & O(e²).
53


Reviews:
[{"review_id": "nIQNvBuuH0", "reviewer": "Reviewer_7Drm", "summary": "The paper provides a new algorithm and results for the stochastic multi-level compositional minmax problem. The approach includes variance reduction and smoothing techniques for which have yet to be applied to the stochastic case. Stationary point convergence is claimed for different metrics. To get an $\\epsilon$-stationary point convergence, from an $(\\epsilon_1,\\epsilon_2)$ stationary point requires solving a two-sided stochastic PL multi-level problem for which the paper proposes a new method and analysis. The new algorithm is also tested empirically in deep AUC maximization and multi-instance learning.", "questions": "Below I include some questions and comments that I think will shed light on my previous concerns as well potentially improve the paper. \n\n\n+ Lemma B.1 proof seems incomplete, where is the reasoning for property (3)? Also the reference for property (4) seems missing?\n+ What are the $\\gamma$ constants in the algorithm input or in the bound?\n+ Why is there an expectation in in Lemma B.3? h is a deterministic function?\nWhy is the dual function smooth? Taking the min over a smooth function can yield a non-differentiable function unless assumptions such as strong convexity are used (e.g. like in Nesterov smoothing). It is smooth if $\\omega$ is sufficiently large, which I think is the implied assumption but this should be included somewhere explicitly in the assumptions.\n+ I think there is a typo on line 486 in appendix, I believe we should have $f_{\\omega}(x^{\\ast}(y_{t+1};z_{t}), y_{t+1};z_t)$.\n+ What are the meaning of the $\\rho$ constants in the main Theorem? Looking at the proofs they seem to be related to different Lipschitz constants but this is not clear at all.\n+ The stepsize condition (97) is very far in the appendix. It should be brough up earlier.\n+ Requiring $g^{(k)}$ to be Lipschitz for all k seems a bit restrictive? I understand that existing works make this assumption but  it seems this would be violated even in the simple convex example $x^2 = x * x$. \n+ What does $\\nabla f(\\cdot,\\cdot)$ mean exactly in Assumption 3.1? Do you mean  the gradient of each player holding the other player fixed is Lipschitz with respect to both players?\n+ second player has noise variable $\\zeta_{t+1}$, is there a reason why we are using a different source of noise here? Also since the dual variable has no multi-level structure why is the gradient estimator for the dual variable biased?\n+ Why would it be bad to say take say $K$ independent samples  so that the gradient can be unbiased. Is this possible?  I think  better discussion on the problem setting might answer this question.", "limitations": "Some of the implied assumptions from the theory can be better stated for example how large should $\\omega$ be? and how easy is it to satisfy condition (97) in practice?", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Quality\nUnfortunately I cannot verify the technical soundness of the paper. The overall paper is very difficult to follow, below I point out several examples on clarity issues. On the technical side, I cannot verify a Lemma (Lemma B.3). More precisely, I don't see how line (31) is true in the appendix. Applying the difference of squares does not explain this line. You would get something like $\\|z'-x\\|^2 - \\|z-x\\|^2 = (\\|z'-x\\|-\\|z-x\\|)(\\|z'-x\\|+\\|z-x\\|)$ which is not equivalent to what is written. Additionally Lemma B.1 seems incomplete (see below for details).\n\n## Clarity\nThe paper significantly lacks clarity. There is a lot of discussion motivating the problem and highlighting different challenges on the theoretical side especially in comparison to related setting (non-stocastic min-max or stochastic minimization) but it is very difficult to understand the problem setting and the theoretical results. Below I list some examples:\n+ The stochastic oracle is never formally defined. I had to look at other papers to understand why the naive gradient estimator in (5) is biased. My understanding it is because in this setting we are incrementally sampling $\\xi_t^{(i)}$ across different $i$, e.g. sample $\\xi_t^{(1)}$ then $\\xi_t^{(2)}$. Additionally terms like $\\hat{\\xi}_t$ are never defined.  And in even in equation (6) where the biased is claimed, it is not obvious at this point without reading about the multilevel setting elsewhere.\n+ It is also difficult to follow the proofs either because of typos or incomplete reasoning or conditions like on the stepsize that are in the middle of a long proof instead of stated outside  (see below for more details)\n+ There are also constants that seem important for the Theory and algorithm that are never discussed such as $\\rho_x, \\rho_y, \\gamma$.\n\n## Significance\nThe authors do a good job placing where the contribution lies in the multilevel literature. I think there is interest in this community for this contribution. But due to the issues mentioned above the paper should be revised due to clarity issues on both the problem setting and the theory.\n\n## Originality\nSee above in significance.", "was_revised": true, "final_justification": "Technical issues were clarified by reviewers, there still remains issues on clarity which was also shared with at least one more reviewer.", "rebuttal": "Thank you for carefully reading our paper. The reviewer's questions can be classified two categories: **writing issues** and **clarification of specific details.** We appreciate your feedback and address your concerns below.\n\n---\n\n## **I. Writing issues**\n\n**Problem setting and biased issue:** We provide a formal stochastic problem formulation in Eq. (1), where different noise sources arise naturally: the outer-level function $f$ involves noise $\\zeta$, and each inner-level function $g^k$ involves noise ${\\xi}^k$. In Eq. (5), the stochastic set  $\\hat{\\xi}\\_{t} = \\\\{ \\xi\\_{t}^{(1)}, \\cdots, \\xi\\_{t}^{(K)}, \\zeta\\_{t}\\\\}$ is defined implicitly through the second equality. For better clarity, we will make this explicit in the revised version. \n\nThe bias issue in Eq.(5) is explained by Eq.(6). Just as shown in Eq. (6), because the $k$-th level function $g^{(k)}$ is a non-linear function, those stochastic estimators in Eq.(6) are biased estimators.  We will revise the paper to clarify this discussion around Eq. (6) for better understanding.\n\n**Follow the proofs:** We have provided the proof sketches for the two theorems in Lines 204–216 and Lines 250–258. These sketches are helpful for understanding the overall structure and intuition behind our proofs.\n\n\n**Other typos or writing issues:** We will carefully revise our writing by following the reviewer's suggestions.\n\n---\n\n## **II. Clarification of specific details**\n\n**Lemma B.1:**  The first **three** properties follow from Lemma B.1 in [37]; the last follows from Lemma 4.3 in [22]. We apologize for the typo and any misunderstanding it may have caused.\n\n**Lemma B.3:** \n* In Eq. (31), the second-to-last step uses the vector version of that equation: $||a||^2-\\|\\|b\\|\\|^2= < a-b, a+b >$. It is easy to verify it is true: $< a-b, a+b >=\\|\\|a\\|\\|^2 +a^Tb-b^Ta-\\|\\|b\\|\\|^2=\\|\\|a\\|\\|^2-\\|\\|b\\|\\|^2$. We will make it clear. \n\n* The dual variable $y$ is updated by _stochastic_ gradient,  so we should take the expectation.\n\n* The smoothness of the dual function can be found in Lemma B.3 of [36].  \n\n* The scale of $\\omega$ is provided in Line 478. We will make it clear. \n\n**Hyperparameters:** $\\gamma_x, \\gamma_y, \\gamma_z$ are coefficients for learning rate updates, $\\rho_x, \\rho_y$ are coefficients for momentum updates. O(1) means they are independent of $\\epsilon$ or $\\kappa$, as can be seen from Eq.(97). In practice, to determine learning rate, we fix the coefficient $\\gamma$ to constants and only tune $\\eta$. Once $\\eta$ is selected, tuning $\\rho$ is straightforward since $\\rho\\eta^2 \\in (0, 1)$. Thus, satisfying Eq.(97) is not difficult, and all of our experiments comply with this condition.\n\n**Assumptions:** \n* Lipschitz continuity of $g^k$ is common and not restrictive in non-convex multi-level optimization [5, 19, 31, 35]. To cover more cases, such as that the reviewer mentioned, we can consider the generalized Lipschitz condition, e.g., [R1], and we will leave it for future work. \n\n\n\n* The smoothness of $\\nabla f(\\cdot, \\cdot)$ in Assumption 3.1 is also a common assumption in compositional minimax optimization [9, 13, 23]. Its formal definition is: $\\|\\nabla\\_1 f(x\\_1, y\\_1)-\\nabla\\_1 f(x\\_2, y\\_2)\\| \\leq l[\\|x\\_1-x\\_2\\|+\\|y\\_1-y\\_2\\|]$, $\\|\\nabla\\_2 f(x\\_1, y\\_1)-\\nabla\\_2 f(x\\_2, y\\_2)\\| \\leq l[\\|x\\_1-x\\_2\\|+\\|y\\_1-y\\_2\\|],$\n\n\n[R1] Li, Haochuan, et al. \"Convex and non-convex optimization under generalized smoothness.\" Advances in Neural Information Processing Systems 36 (2023): 40238-40271.\n\n**Second player has noise variable**: Because each level function has its own source of noise, we use $\\zeta$ to differentiate the noise source of the dual variable from those levels of functions.  For the stochastic gradient with respect to the dual variable: $\\nabla\\_y f(\\hat{G}_t, y\\_t ; \\zeta_t)$ where $\\hat{G}\\_t$ is the biased stochastic estimator of $G(x_t)$ as discussed in Eq.(6),  it is obvious that $\\mathbb{E}[\\nabla\\_y f(\\hat{G}_t, y\\_t ; \\zeta_t)] \\neq \\nabla\\_y f(G(x\\_t), y\\_t )$.\n\n**The gradient can be unbiased. Is this possible?** When each level function $g^{(k)}$ is a linear function, it is possible. But we are not aware of such applications in machine learning, so our focus is to study the more general setting that can benefit real-world machine learning models. \n\n---\nAs shown above, **the reviewer’s questions primarily concern writing issues and clarification of specific details. We believe these points are minor and can be easily addressed, hence not affecting the technical contribution of our paper**. \n\nMoreover, we would like to emphasize that all theoretical results are correct and supported with complete proofs. Additionally, we provide high-level proof sketches for both algorithms in lines 204–216 and 250–258 for better understanding. To further improve clarity, we will add more explanations of the problem setting and key equations for readers. Given that the reviewer has acknowledged the significance and originality of our work, we respectfully ask for a reconsideration of the overall evaluation."}, {"review_id": "uZj8gSgpPt", "reviewer": "Reviewer_bNHb", "summary": "The paper studies \"stochastic multi-level compositional minimax\" optimisation, a setting that arises in deep AUC maximisation and multi-instance learning.\n\nTwo main algorithms are proposed:\n\n1. Smoothed-SMCGDA-VR – a variance-reduced gradient descent ascent method that uses smoothing term and STORM-style recursive estimators at every compositional level. Theorem 4.1 shows it finds an $(\\epsilon,\\;\\epsilon/\\sqrt\\kappa)$-stationary point in $\\tilde O(\\kappa^{3/2}/\\varepsilon^{3})$ stochastic first-order calls, improving the $\\kappa$-dependence over prior two-level results.\n\n2. Stagewise-SMCGDA-VR – a two-phase scheme that, given the first algorithm’s output, solves a smoothed auxiliary game under a two-sided PL condition, converting to a true $\\epsilon$-stationary point with an extra $\\tilde O(1/\\mu^{3}\\log(1/\\epsilon))$ cost.", "questions": "1) Regarding the lines 38 to 42, \"Unfortunately, these minimization-targeted algorithms cannot directly address the stochastic multi-level compositional minimax optimization problem in Eq. (1), as the stochastic gradients for both primal and dual variables are biased estimators in stochastic multi-level compositional minimax problems—posing greater algorithmic and theoretical challenges.\", do you claim that the challenge in the stochastic multi-level compositional minimax optimization problem is different from the generic min-targeted one? If yes, could you elaborate on that?\n\n2) Regarding lines 55 and 56, \"Addressing this question is not straightforward and presents substantial algorithmic and theoretical\nchallenges.\" This sentence is not clear to me. Could you explain it more?\n\n3) Is there any application for two-sided PL?\n\n4) I couldn't follow the lines 163 to 176; what is the smoothed algorithm which outputs \\tilde{x}? What does it mean a function satisfies two-sided PL WITH iteration complexity O(\\eps^{-2})?\n\n5) Have you tried replacing STORM with SPIDER/SARAH updates at each level; if so, how did constants affect performance?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1) Multi-level *minimax* problems are largely unexplored; to me, the bias in both primal and dual stochastic gradients is well motivated.\n2) Their theoretical and experimental contributions in this work seem good to me.\n3) Improved complexity: The $\\kappa$-exponent drops from 3 or 4 in prior work to 3/2, and the conversion phase is provably negligible in cost.\n\n\nWeaknesses:\n\nWhile the paper makes a good theoretical step for stochastic multi-level minimax optimisation, its practical impact is circumscribed by strong structural assumptions, heavy hyper-parameter dependence, and sizeable computational demands. Relaxing the PL requirements, developing adaptive parameter-free variants, and broadening empirical validation would substantially strengthen the work.", "was_revised": true, "final_justification": "The author’s reply resolves my concerns; I’m keeping my current rating.", "rebuttal": "Thank you for carefully reading our paper.  Suggestions such as relaxing PL conditions, developing adaptive parameter-free variants, and exploring more empirical application are insightful direction for future work. Regarding questions, they are mainly about **the clarification of specific details**.  We appreciate your feedback and provide detailed clarification for each of them below.\n\n---\n\n## **I. Clarification of specific details**\n\n**Lines 38 to 42:** The challenge in stochastic multi-level compositional minimax optimization differs from that in the standard minimization setting. In the minimization case, there is only one variable, whereas in the minimax setting, biases arise in both the primal and dual variables, making the problem more difficult to handle.\n\n**Line 55 and 56:** We provide an explanation of this sentence in lines 56–65. The challenges lie in two aspects:\n\n* Algorithmic challenge: the stochastic gradients for both primal and dual variables are biased;\n\n* Theoretical challenge: comparing convergence rates requires translating between two different stationarity measures.\n\nA more detailed discussion of these challenges is given in Section 3.3.\n\n**Application for two-sided PL:** The two-sided PL condition is easy to satisfy. Our experiments, such as deep AUC maximization and multi-instance learning when using over-parameterized deep neural networks, satisfy this condition, because [R1] shows that over-parameterized deep neural networks satisfy the PL condition. \n\n[R1] Chaoyue Liu, Libin Zhu, and Mikhail Belkin. Loss landscapes and optimization in over-parameterized non-linear\nsystems and neural networks. arXiv preprint arXiv:2003.00307, 2020a.\n\n**Line 163 to 176:** $\\tilde{z}$ refers to the updated primal variable $x_{T}$ obtained using a smoothed algorithm. This interpretation follows from Proposition 2.1 in [30], which includes the relevant proof. \n\n**STORM:** STORM is a more practical technique compared to SPIDER/SARAH, as it avoids the need for large batch sizes to compute a checkpoint gradient. This makes STORM more suitable for large-scale applications."}, {"review_id": "CMQGGF4n5Y", "reviewer": "Reviewer_Z4zL", "summary": "## Summary\n\nThis paper addresses the **multi-level stochastic compositional minimax optimization** problem, a challenging framework where the objective involves nested expectations and a min-max structure over decision variables. Such settings introduce significant complexities, particularly due to the biased nature of stochastic gradient estimates when applied to both primal and dual variables.\n\nTo tackle these difficulties, the authors propose a **stochastic gradient ascent algorithm** tailored for multi-level composition. Their method integrates **variance reduction** and a **smoothing strategy** under the **nonconvex Polyak–Łojasiewicz (PL) condition**. They theoretically establish a convergence guarantee to an $(\\epsilon, \\epsilon / \\sqrt{\\kappa})$-stationary point, with iteration complexity scaling as $\\mathcal{O}(\\kappa^{3/2} / \\epsilon^2)$, where $\\kappa$ denotes the condition number. Additionally, they design a stage-wise scheme under a two-sided PL condition that enables translation between different notions of stationarity.\n\nWhile prior work has shown the advantages of smoothing techniques in classical minimax problems, especially in improving convergence under PL-type conditions, their application in **multi-level compositional settings** has remained largely unexplored. This paper fills that gap, showing that such techniques can be effectively adapted to the multi-level setting, achieving both improved theoretical bounds and empirical performance.", "questions": "N/A - Please comment on the weakness of the paper", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- The paper is **clearly written** and well-structured, making the technical contributions accessible even within a complex multi-level stochastic minimax setting.\n- It addresses a **concrete and timely gap** in the literature—namely, the challenge of biased stochastic gradients in compositional min-max problems—by introducing a method that is both novel and theoretically justified.\n- The proposed algorithm achieves an improved complexity bound in terms of the condition number $\\kappa$, specifically $\\mathcal{O}(\\kappa^{3/2} / \\epsilon^2)$, which improves over previous works.\n- The incorporation of smoothed techniques into the multi-level compositional setting is original and motivated by recent advances in classical minimax optimization.\n\n## Weaknesses / Questions\n\n- It is not entirely clear whether the proposed algorithm can be **extended to ReLU networks**, where subgradients rather than gradients are used. Would the variance reduction or convergence guarantees still hold under such non-smooth settings?\n- A natural question is how the **complexity would change** in the case where the problem satisfies the **two-sided PL condition** (i.e., the function is PL in both $x$ and $y$). Can the current framework be adapted, and would it yield better rates?\n- There is some concern regarding **practical relevance**: how far is this multi-level compositional model from real-world applications, such as adversarial training or bilevel learning in neural networks?", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for carefully reading our paper.  Regarding questions, they are mainly about **the clarification of specific details**.  We appreciate your feedback and provide detailed clarification for each of them below.\n\n---\n\n## **I. Clarification of specific details**\n\n**Extended to ReLU networks:** Extending the proposed algorithm to non-smooth settings using subgradients is a promising direction for future work. \n\n**Two-sided PL condition:**  Thank you for the insightful question. We believe the smoothing framework can be applied to the two-sided PL problem; however, it would require new convergence analysis, and the achievable convergence rate remains unclear. We will consider this promising direction for future work.\n\n\n**Real-world applications:** Multi-level compositional models can be applied to practical scenarios such as hyperparameter optimization (bilevel optimization) and deep AUC maximization (bilevel minimax optimization)."}, {"review_id": "BP90RjllYi", "reviewer": "Reviewer_GHtS", "summary": "The paper discusses the problem of multi-level compositional optimization and proposes a smoothed variance-reduced algorithm that finds a $(\\epsilon,\\epsilon/\\sqrt \\kappa)$-stationary point with $O(\\kappa ^{3/2})$ iterations.", "questions": "1. The definition of $\\omega$ is unclear. Is it an algorithm parameter? If so, how is it chosen? If not, the bound should depend on $\\omega$. In addition, in line 582, the bound should be multiplied by $\\omega$, which appears in the third term of the $\\max$. Moreover, the proof assumes $\\omega \\ge L_f$, but choosing $\\omega$ too large would worsen the result because $f_\\omega$ diverges from $f$. How is this fact expressed in the proof or the algorithm?\n\n2. Could the authors elaborate on where the improvement in the result comes from? Is it due to a better algorithm, a sharper analysis, or both?", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1. The paper is well written and presents its motivation clearly.\n\n2. The results improve the bounds achieved by previous work, even for the case $k = 2$.\n\n3. The authors clearly discuss the challenges they tackled while solving the problem.\n\n4. The authors validated their results experimentally.\n\nWeaknesses\n\n1. My major concern is about the correctness of the bound. . In the transition from line 580 to line 582, the term $H_0$ should be multiplied by $\\kappa/\\eta$ (the second term in the $\\max$), shouldn't it? If so, this would weaken the bound stated in the paper. Could the authors clarify?  If I’m mistaken, I will revise my score accordingly.\n\n2.  Although the authors describe the difficulties they faced, it would help readers if they also explained how each challenge was overcome and which part of their algorithm or analysis addressed it.\n\n3.  The proofs are hard to follow; many inequalities need further explanation. A proof sketch outlining the key ideas behind each central lemma would be valuable.\n\n4. The formality of the proofs is somewhat lacking. For example, the authors do not specify concrete values for the algorithm’s constants (they provide only $O$-notation). Because there are many constraints on these constants, it is hard to verify that they are mutually consistent.", "was_revised": true, "final_justification": "My concern regarding the transition from line 580 to line 582 has been addressed. However, I still found the proofs difficult to follow and believe they would benefit from further clarification. Given that the authors intend to include a proof sketch and clarify the transitions, and since I was unable to fully follow or verify the original proofs, I am increasing my score to 4 while maintaining low confidence.", "rebuttal": "Thank you for carefully reading our paper. The reviewer's questions can be classified two categories: **writing issues** and **clarification of specific details.** We appreciate your feedback and address your questions below.\n\n---\n\n## **I. Writing issues**\n\n**Correctness of the bound:** There is a typo in line 580. $\n\\frac{\\kappa(\\mathcal{H}\\_0-\\mathcal{H}\\_T)}{\\gamma\\_x \\eta T}$ should be $\n\\frac{\\mathcal{H}\\_0-\\mathcal{H}\\_T}{T}$. Specifically, eq.(100) is derived from eq.(98). It is easy to know that the coefficient of $\\frac{\\mathcal{H}\\_0-\\mathcal{H}\\_T}{T}$ is $\n\\frac{8}{\\gamma\\_x \\eta}$, which is moved to the max operator. However, this typo does not affect the final result in eq.(102), as it is easy to verify that the coefficient of $\\mathcal{P}_0$ in eq.(102) is correct. Hence, the overall convergence rate remains valid.   Thanks for pointing this out this typo, and we will revise it accordingly. \n\n**Proof:** We provide proof sketch outlining the high-level ideas in line 204-216 for the first algorithm, and line 250-258 for the second. The use of $O$-notation is standard in optimization analysis, and we provide complete proofs alongside it. Additional explanations for each lemma will be included in the appendix to improve clarity. \n\n**The authors do not specify concrete values for the algorithm’s constants:** They have already been provided in our proof. Specifically, Eq.(97) summarizes the upper bound of hyperparameters. \n\n---\n\n## **II. Clarification of specific details**\n\n\n\n**Addressing challenges:** We summarize how the challenges are addressed in line 68-79. For the algorithmic challenge, we propose a variance-reduced algorithm based on smoothed techniques. For the theoretical challenges, after obtaining an $O(\\epsilon, \\sqrt{\\epsilon})$-stationary point, we further develop an algorithm under the two-sided PL condition, successfully performing the translation between two stationary measures.\n\n\n\n\n\n**Hyperparameter $\\omega$:**  $\\omega$ is defined in eq.(4), as a hyperparameter in the smoothing term. In Theorem 4.1, we state that choosing $\\omega = O(L_f)$, e.g., $\\omega=2L_f$,  is a common practice for the smoothing technique.  Moreover, since $\\omega$ is in the same scale as $L_f$, it does not worsen the convergence. In particular, Eq.(97) shows how $\\omega$ affects the hyperparameter. It is easy to see that $\\omega=2L_f$ does not have a significant influence on them, as well as the convergence rate. \n\n**Elaborate on where the improvement in the result comes from:** It is due to both a better algorithm design and sharper analysis. On the one hand, compared to [9, 13], the improvement is due to the variance-reduced estimator, while the improvement over [23] is due to the smoothing technique. On the other hand, we developed a novel Algorithm 2 and provided sharper analysis to translate the $(\\epsilon, \\epsilon/\\sqrt{\\kappa})$-stationary point to $\\epsilon$-stationary point. This has never been investigated for compositional miniax optimization, so this is another important contribution of our paper. \n\n---\n\nIf we have addressed the concerns, particularly regarding the correctness of the bound, we kindly ask the reviewer to reconsider the evaluation."}, {"review_id": "xOVIej0FqO", "reviewer": "Reviewer_g7hY", "summary": "This paper studies the stochastic multi-level compositional minimax optimization problem, where the objective function has a multi-level compositional structure. \n\nThe authors propose a variance-reduced algorithm that incorporates smoothing techniques and provide a convergence analysis under a non-convex PL condition. \n\nTheoretical analysis demonstrates improvements in the dependency on the condition number and the convergence rate, compared to existing methods. \n\nThe experiments cover key applications, such as deep AUC maximization and multi-instance learning.", "questions": "Questions\n1. How would a smaller batch size affect the algorithm's performance? Have the authors considered any techniques to mitigate this issue to make the algorithm more suitable for large-scale training scenarios?\n\n2. Could the authors provide a more detailed comparison of the per-iteration computation time and the overall training time between the Smoothed-SMCGDA-VR algorithm and the baseline methods?\n\n3. Have the authors conducted experiments to investigate how the algorithm's performance scales with the number of compositional levels, K?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n\n1. The proposed Smoothed-SMCGDA-VR algorithm integrates stochastic smoothing with **STORM-like variance reduction techniques**, offering an effective solution to the biased gradient issue in the multi-level compositional minimax optimization problem.\n\n2. The paper presents a thorough convergence analysis for the proposed algorithm, demonstrating that the method improves the dependency on the condition number.\n\n3. Experiments on deep AUC maximization and multi-instance learning are conducted, showcasing the superior performance of the proposed algorithm.\n\n**Weaknesses**:\n\n1. The proposed method is not a mini-batch algorithm. The convergence result in Theorem 4.1 requires the batch size $S$ to satisfy a sufficiently large condition with $B = \\mathcal{O}(1/\\epsilon)$ in order to achieve the claimed convergence rate. This may necessitate a large batch size, which could become a bottleneck in practical applications. \n\n2. The iteration complexity of the proposed algorithm is $O(\\kappa^{1/2}\\epsilon^{-3})$, which shows only a limited improvement over the most representative algorithms, whose iteration complexity is $O(\\kappa^{3}\\epsilon^{-3})$, especially when $\\kappa$ is close to 1. Moreover, the algorithm’s fast convergence seems to rely on a relatively large mini-batch size $B = \\mathcal{O}(1/\\epsilon)$ as a trade-off.\n\n3. The algorithm presented is a stochastic extension of the methods in [R1,R2], where the authors use the component-wise smoothness of the objective function $f(\\cdot,\\cdot)$ and variance-reduction technique used in [R3] to achieve an iteration complexity of $O(\\epsilon^{-3})$. The notation in this paper is highly complex, and the conclusions appear predictable, making the paper less engaging. It is unclear where the core challenge and primary novelty of the work lie.\n\n4. The pseudocode for Algorithm 1 (Smoothed-SMCGDA-VR) involves numerous hyperparameters that require manual tuning. It would be more informative to explicitly present their roles and suggested values to guide implementation.\n\nReferences\n\n[R1] J. Yang, A. Orvieto, A. Lucchi, and N. He. Faster single-loop algorithms for minimax optimization without strong concavity. AISTATS 2022.\n\n[R2] T. Zheng, L. Zhu, A. So, J. Blanchet, J. Li. Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimization, NeurIPS 2023.\n\n[R3] A. Cutkosky and F. Orabona. Momentum-based variance reduction in non-convex sgd, NeurIPS 2019.", "was_revised": true, "final_justification": "My main concerns have been resolved.", "rebuttal": "Thank you for carefully reading our paper.  The reviewer's questions can be classified two categories: **misunderstanding**  and **clarification of specific details.** We appreciate your feedback and address your questions below.\n\n---\n\n## **I. Misunderstanding**\n\n\n\n**Mini-batch:** Our proposed algorithm is a mini-batch algorithm with batch size $S=O(1)$. The larger batch size **$B=O(1/\\epsilon)$ is used only in the initial iteration**, when $t=0$. Throughout the remainder of training ($ t \\in \\\\{1, \\cdots, T-1\\\\}$), the batch size remains $O(1)$, which is consistent with the STORM algorithm[7]. Thus, our algorithm is suitable for large-scale training scenarios. We will make this point clearer in the revised version to avoid potential misunderstanding.\n\n\n\n\n\n\n\n\n---\n\n## **II. Clarification of specific details**\n\n**Iteration complexity:** First, we would like to highlight that $O(\\kappa^2\\epsilon^{-3})$ is obtained for the **two-level** problem, rather than the **multi-level** compositional minimax problem.  The multi-level structure brings more challenges for convergence analysis. Therefore, the improvement in the dependence on $\\kappa$ is non trivial from an analytical perspective, particularly given that our work addresses a new problem class: stochastic **multi-level** compositional minimax optimization. Second, the condition number $\\kappa$ is typically large for practical machine learning applications. Therefore, the improvement in terms of $\\kappa$ is significant. \n\n**Core challenge and primary novelty:** Our algorithm is not a straightforward extension of [R1, R2]. Those works consider only standard minimax problems and do not encounter the gradient bias issue during updates. In contrast, \n\n* the **multi-level structure** and the **compositional structure** in our problem introduce accumulated bias across levels, which poses a significant challenge;\n\n* another challenge lies in **relating two different stationary measures in the multi-level compositional minimax setting**. \n\nThese challenges are discussed in detail in lines 55–67 and Section 3.3.  The novelty of our work lies in how we address these challenges, through both a novel algorithm and improved convergence analysis, as summarized in lines 68–79. In particular, we developed a novel Algorithm 2  with a stage-wise structure for translating different stationary measures, which has never been studied in [R1-R3]. This is an important novelty and contribution of our paper. \n\n**Hyperparameters:**\n\n- **$\\eta$**: Learning rate.\n- **$\\omega$**: Used in the smoothing term.\n- **$\\gamma_x, \\gamma_y, \\gamma_z$**: Coefficients for learning rate updates.\n- **$\\rho_x, \\rho_y$**: Coefficients for momentum updates.\n\nIn practice, we first tune $\\eta$ and $\\omega$. $\\omega$ should not be set too large to avoid deviating too far from the original problem. The coefficient $\\gamma$ can be fixed to constants. Once $\\eta$ is selected, tuning $\\rho$ is straightforward since $\\rho\\eta^2 \\in (0, 1)$. \n\n**Experiments:** We have included experiments with the number of compositional levels $k$, as shown in Figure 4. Additional comparisons on per-iteration computation time and the overall training time can be added in the revised version.\n\n---\nIt appears that **the main concern lies in the mini-batch setting, which may stem from a misunderstanding**. We believe all the reviewer's questions are minor and can be easily addressed, hence not affecting the technical contribution of our paper.  If our clarification resolves reviewer's questions, we kindly ask the reviewer to reconsider the evaluation."}]

Author rebuttal:

Meta-review: This work focuses on stochastic, multi-level compositional (in the language of some earlier work, compositional-on-x) min-max optimization by assuming nonconvexity on one side and PL on the other side. The proposed method has complexity $O(\kappa^{3/2} \epsilon^{-3})$ in the nonconvex-PL case for finding a two sided $(\epsilon, \epsilon/\kappa)$ stationary point where the two sided refers to the norms of gradients w.r.t. $x$ and $y$. Since the earlier work focused on $\epsilon$ stationarity of the objective function which is given in the $\max$ form, the authors specialize to PL-PL case and show $O(\kappa^{3/2} \epsilon^{-3})$ complexity for $\epsilon$-stationarity and then argue that they improve the earlier complexity by $O(\kappa^{3/2})$ whereas the earlier work focused on nonconvex-PL case (a more general case than PL-PL) and showed $\epsilon$-stationarity.

Here, the condition number $\kappa$ is defined as the ratio of the Lipschitz constant and the PL constant. 

The reviewing team found the studied problem and the complexity improvement interesting, however significant concerns remain about the writing of the paper. In particular, the reviewers and the AC shares concerns on the difficulty of following the developments in the paper and the proofs in the appendix. 

Below are some remarks from the private discussion between myself and the reviewers:

- "In my view, the appendices need to be revised, and it could take considerable time and effort to produce fully justified proofs. I agree with the AC that the authors did not make a significant effort to make it easy for readers to verify their arguments."

- "I agree that it is difficult to follow the proofs, I was not able to verify every step. I got stuck on multiple parts, and though the authors addressed the parts for which I got stuck, to be honest, I would most likely have to spend another few days just to go over all the proofs again. Moreover, I am not satisfied in the author's responses defending sufficient clarity. For example, their main arguments were:

- We have a proof sketch

- This other paper does is similarly

- Although a proof sketch is nice, for me it has not helped. I know at least one other reviewer (GHtS) mentioned clarity issues, maybe they also agree.

For the second point, this is in the context of so many parameters and variables that are not well explained to me in the paper. Moreover, important stepsize conditions like (97) in the appendix are quite hidden. The authors' response was that we only give $O$ scale for such parameters. Another example is the  momentum constant $\rho$, which are featured everywhere in the Theorems but I believe there is no discussion on them. They are technically in the pseudocode of Algorithm 1 but they should be atleast discussed in detail if they are main hyperparameters featured in the Theorems. Maybe I missed this discussion somewhere but I can't currently find anything on it."

I also wish to emphasize my agreement with the reviewer that just because another paper from ICML 2025 presents hyperparameters in terms of their orders in the main text and in their detailed form in the appendix **does not justify anything**. The authors should not use such justifications. It seems appropriate to acknowledge that the concern of the reviewer is fair (I also totally agree with this concern) but unfortunately the analysis is extremely technical and hence the conditions at Eq. (97) are not very readable.

- "For example, looking at the estimations in pages 18, 22, 23, 28, 31, 32, 33, 36, 37, 38, 39, 40 (and honestly, most of the pages in the 30 page appendix), it looks like the authors unfortunately did not give much effort to make their proof readable."

As a result, the reviewing team has significant concerns on the lack of clarity in the writing of this paper. Given that the authors have the full responsibility in making their main text and proofs readable and convincing to the readers, I find the lack of clarity a significant problem, especially for a theoretical paper with 30 pages of proofs. As a result **the authors need to spend a considerable amount of time to streamline their proof**. They should avoid long equality chains in the pages referenced above and make sure to have ample justification for each of their proof steps. **To be published, a proof have to be readable.** Otherwise, neither the reviewers nor the readers of the paper can verify or understand the proof: **the authors should do every effort to make their proof much more readable than its current state.**

- Another issue is that the presentation of the contribution is a bit misleading. In particular, in Remark 5.2, the authors say

> To the best of our knowledge, this is the first algorithm to achieve an $O(\kappa^{3/2})$ dependence for (multi-level compositional) minimax problems under the nonconvex-PL setting.

which is technically correct but the authors are unfortunately not acknowledging here that they actually need to assume PL in the primal to get this result. That is, even thought technically a PL-PL problem is still a nonconvex-PL problem, it is a significantly smaller class than nonconvex-PL problems. Indeed the complexity that the authors are comparing do not assume PL in the primal, [23] seems to only assume smoothness in the primal. The authors should make this distinction **crystal clear**. I find the comparison right now unfairly presented with respect to [23]. 

As it stands, it is not clear if there is a strict improvement compared to the state-of-the-art in the 2-level case. The comparison is either with a different stationarity measure (Section 4) and hence not comparable to other works or the comparison is with different assumptions (Section 5), that is the authors assume PL in the primal. On the other hand, this work is more general because they are allowing more than 2-level composition.

As a result, the authors should make this comparison very clear and stop claiming strict improvement. From what I see, their work is complementing the state-of-the-art, generalizing in some senses, but more restrictive in other senses. Hence a direct comparison is not possible.

Yet, the smoothing idea and the result is potentially interesting to the line of research in this direction.
Decision: Accept (poster)