Title: On the Convergence of Stochastic Smoothed Multi-Level Compositional Gradient Descent Ascent
Venue: NeurIPS 2025
Accepted: yes
Keywords: Compositional Optimization

Minimax 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

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 descentascent algorithm, incorporating a smoothing technique under the nonconvex-PL condition. We establish a convergence rate to an ( ϵ, ϵ/ √ κ ) -stationary point with improved dependence on the condition number at O ( κ 3 / 2 ) , where ϵ denotes the solution accuracy and κ 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 ( ϵ, ϵ/ √ κ ) -stationary point to an ϵ -stationary point. Finally, extensive experiments validate the effectiveness of our algorithms.

1 Introduction

This paper investigates the stochastic multi-level compositional minimax optimization problem:

where f ( G ( x ) , y ) = E [ f ( G ( x ) , y ; ζ )] and ζ denotes a random variable. The function G ( x ) ≜ g ( K ) ( · · · ( g (1) ( x ))) is a K -level compositional function with K > 1 , where each inner-level function g ( k ) ( · ) = E [ g ( k ) ( · ; ξ ( k ) )] depends on the random sample ξ ( k ) for k ∈ { 1 , · · · , K } . The function f ( · , · ) is referred to as the outer-level objective. In this paper, we consider the general nonconvex-PL setting, where f ( G ( x ) , y ) is nonconvex in the primal variable x and satisfies the Polyak-Lojasiewicz (PL) condition with respect to the dual variable y .

Multi-level compositional optimization has emerged as a 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 [3, 20]; and in reinforcement learning, it 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 as 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

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 a 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 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.

In addition, although [17, 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 a smoothed alternating gradient method for general nonconvex-concave problems, which achieves superior performance compared to conventional approaches. Building on this, [30] further applied this technique to the nonconvex-PL setting-a milder condition than strong concavity-and showed that stochastic smoothed techniques yield improved complexity bounds with better dependence on the condition number κ . However, despite the clear benefits of smoothed techniques in traditional minimax optimization, its application to 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. (1) introduces uncertainty regarding the effectiveness of applying such techniques. It remains unclear whether their use may lead to additional convergence issues. Therefore, it is essential to develop new algorithms that can accommodate biased gradient estimators and guarantee convergence when using smoothing techniques for Eq. (1). On the other hand, as demonstrated in [30], smoothed algorithms typically guarantee an ( ϵ 1 , ϵ 2 ) -stationary point, rather than a standard ϵ -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 [30], 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 a multi-level compositional structure. Therefore, these challenges motivate us to address the problem through the following contributions:

We develop a novel smoothed multi-level compositional minimax optimization algorithm for Eq. (1) by leveraging the variance reduction technique to mitigate the biased gradient estimator issue, and establish a convergence rate of O ( κ 3 / 2 /ϵ 3 ) to an ( ϵ, ϵ/ √ κ ) -stationary point. Compared to existing algorithms, our method achieves a better dependence on the

condition number κ : improving over the O ( κ 3 ) rate of standard two-level compositional minimax algorithms. √

To bridge the gap between an ( ϵ, ϵ/ κ ) -stationary point and a standard ϵ -stationary point, we further propose a stage-wise variance-reduced algorithm for Eq. (1) under the two-sided PL condition. We show that the algorithm achieves a convergence rate of O (1 /ϵ 2 ) to an ϵ -stationary point. As a result, the iteration complexity from the translation is dominated by the complexity of finding an ( ϵ, ϵ/ √ κ ) -stationary point.

Meanwhile, 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-PL assumptions without using the smooth technique, as summarized in Table 1.

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 a 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 batch-normalization [25], etc. Consequently, a series of stochastic multi-level compositional minimization algorithms [31, 2, 5, 35, 23, 13, 14] have been developed to solve this important problem. Notably, [31] introduced the first stochastic multilevel 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 a convergence rate of O (1 /ϵ 4 ) . Later, [35] improved the sample complexity to O (1 /ϵ 3 ) by applying the SPIDER variance-reduction technique [11, 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 to 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 /ϵ 3 ) for the stochastic multi-level compositional problem with a mini-batch size of O (1) . More recently, for the first time, [13] showed that the variance-reduction estimator is not necessary for the Jacobian matrix in each level to achieve a convergence rate of O (1 /ϵ 3 ) . However, these stochastic multilevel 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, 33, 9, 26, 37, 38] has attracted increasing attention due to its important applications in machine learning. To solve the two-level compositional minimax problem, [17] developed the first compositional minimax algorithm based on the mini-batch compositional gradient, achieving a convergence rate of O ( κ 4 /ϵ 4 ) for nonconvex-strongly-concave loss functions. [33] incorporated the momentum technique to reduce the mini-batch size to O (1) while achieving the same convergence rate as [17]. Similarly, [9] used a variance-reduced estimator for the inner-level function, also reducing the mini-batch size to O (1) and achieving the same convergence rate as [17]. [26] introduced the STORM technique for estimating the inner-level function and gradient, achieving a convergence rate of O ( κ 3 /ϵ 3 ) . Recently, [9] claimed to achieve a convergence rate of O ( κ 2 /ϵ 3 ) . 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 a result, it corresponds to the convergence rate for a strongly-convex-strongly-concave loss function, rather than for nonconvex-strongly-concave or nonconvex-PL loss functions. More recently, [37] developed the first stochastic multi-level compositional minimax algorithm for nonconvex-stronglyconcave loss functions in the federated learning setting. However, its convergence rate O (1 /ϵ 4 ) is suboptimal compared to the multi-level compositional minimization algorithm. On the other hand, the smoothed technique was first introduced for nonconvex-concave minimax problems in [36], where the convergence rate of a full alternating gradient descent ascent method was established. Later, [30]

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.

3 Preliminaries

3.1 Notations

We begin by simplifying the complex formulation in Eq. (1) to facilitate analysis:

The partial gradients of the objective function can then be expressed as follows:

Following prior works [36, 30], we introduce an auxiliary variable z alongside the primal variable x as part of the smoothed technique, and define the smoothed loss function as:

where ω > 0 is a constant and f ω ( G ( x ) , y ; z ) is strongly convex with respect to x by selecting an appropriate ω . Using the smoothed loss, we can derive stochastic estimators of the compositional gradients with respect to the primal and dual variables at the t -th iteration:

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 ∈ { 1 , 2 , · · · , K } , g ( k ) ( · ) and g ( k ) ( · ; ξ ) are C g -Lipschitz continuous, ∇ g ( k ) ( · ) and ∇ g ( k ) ( · ; ξ ) are L g -Lipschitz continuous,where C g > 0 and L g > 0 ;

f ( · , · ) and f ( · , · ; ζ ) are C f -Lipschitz continuous, ∇ f ( · , · ) and ∇ f ( · , · ; ζ ) are L f -Lipschitz continuous, where C f > 0 and L f > 0 .

Assumption 3.2. (Variance):

For any k ∈ { 1 , · · · , K } , the stochastic gradients ∇ g ( k ) ( · ; ξ ( k ) ) and ∇ f ( · , · ; ζ ) have upper bounded variance σ 2 , where σ > 0 .

Assumption 3.3. (PL Condition):

For any fixed x ∈ R d x , the maximization problem y ∗ = max y ∈ R dy f ( G ( x ) , y ) has a non-empty solution set and a finite optimal value. Moreover, for all x ∈ R d x , there exists a constant value µ > 0 such that ∥∇ y f ( G ( x ) , y ) ∥ 2 ≥ 2 µ ( f ( G ( x ) , y ∗ ) -f ( G ( x ) , y )) .

Here, we define ℓ = max { L f , C 2 g K L f + C f ∑ K -1 k =0 L f C K -1+ k g } , and κ = ℓ µ denotes the condition number. Then, when ω > ℓ , f ω ( G ( x ) , y ; z ) 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 µ x > 0 and µ y > 0 such that f ( · , y ) is µ x -PL for any y ∈ R d y , and -f ( x, · ) is µ y -PL for any x ∈ R d x .

Definition 3.5. (Stationarity measures):

( x, y ) is an ( ϵ 1 , ϵ 2 ) -stationary point of f ( · , · ) , if ∥∇ x f ( G ( x ) , y ) ∥ ≤ ϵ 1 and ∥∇ y f ( G ( x ) , y ) ∥ ≤ ϵ 2 .

x is an ϵ -stationary point of Φ( · ) , if ∥∇ Φ( x ) ∥ ≤ ϵ , where Φ( x ) = f ( G ( x ) , y ∗ ) and y ∗ = arg max y ∈ R dy f ( G ( x ) , 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 in Eq.(3), the partial gradient regarding the dual variable y relies on the stochastic estimator of K -level function G ( K ) ( · ) , while that regarding the primal variable x depends on the stochastic estimator of both G ( K ) ( · ) and ∇ G ( K ) ( · ) . 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:

̸

̸

As a 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 minimax methods handle deterministic gradients or unbiased stochastic gradients .

From the theoretical analysis perspective, a major challenge arises from the gap between different stationarity measures induced by smoothed techniques . As demonstrated in [30], a translation is required from an ( ϵ 1 , ϵ 2 ) -stationary point to an ϵ -stationary point. In standard minimax settings, this can be achieved by applying a stochastic gradient descent-ascent algorithm to the auxiliary problem min x ∈ R dx max y ∈ R dy f ( x, y ) + ℓ ∥ x -˜ z ∥ 2 , where ˜ z is the output of the smoothed algorithm. Owing to the fact that this formulation satisfies the the PL condition in both x and y , with an iteration complexity of ˜ O (1 /ϵ 2 ) . 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 [31] showed that the standard compositional gradient descent algorithm can only achieve a convergence rate with an exponential dependence on the number of levels, even for strongly convex loss functions. As a 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 ( ϵ 1 , ϵ 2 ) -stationary point to an ϵ -stationary point for multi-level compositional minimax problems.

4 Algorithm 1: Smoothed-SMCGDA-VR

4.1 Algorithmic Design

To address the smoothed loss in Eq. (4), we design a novel algorithm, named stochastic smoothed multi-level compositional gradient descent ascent with variance reduction (Smoothed-SMCGDA-VR), as presented in Algorithm 1. 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 ( k ) ( · ) , where k ∈ { 1 , . . . , K } , we apply a recursive step that updates the estimator h ( k ) while controlling variance. This variance reduction technique is also employed for the stochastic gradients: ∇ x f ω ( · , · ; · ; ˆ ξ t , ζ t ) and ∇ y f ω ( · , · ; · ; ˆ ξ t , ζ t ) .

More concretely, the variance-reduced estimator for each levelk function is computed as:

where h (0) t +1 = x t +1 when k = 0 , and α > 0 is a hyperparameter such that αη 2 ∈ (0 , 1) .

Algorithm 1 Stochastic Smoothed Multi-Level Compositional Gradient Descent Ascent with Variance Reduced ( Smoothed -SMCGDA-VR)

1: for t = 0 , · · · , T -1 do

2: Update x and y : x t +1 = x t -γ x ηp t , y t +1 = y t + γ y ηq t ,

3: Update z : z t +1 = z t + γ z η ( x t +1 -z t ) ,

4: h (0) t +1 = x t +1 ,

5: for k = 1 , · · · , K do

6: Compute k -th inner-level function: h ( k ) t +1 = g ( k ) ( h ( k -1) t +1 ; ξ ( k ) t +1 ) + (1 -αη 2 )( h ( k ) t -g ( k ) ( h ( k -1) t ; ξ ( k ) t +1 ))

7: end for

8: Compute stochastic compositional gradient u t +1 and v t +1 : u t +1; t +1 = ∇ g (1) ( h (0) t +1 ; ξ (1) t +1 ) · · · ∇ g ( K -1) ( h ( K -2) t +1 ; ξ ( K -1) t +1 ) ∇ g ( K ) ( h ( K -1) t +1 ; ξ ( K ) t +1 ) × ∇ 1 f ( h ( K ) t +1 , y t +1 ; ζ t +1 ) + ω ( x t +1 -z t +1 ) , ( K )

t

v

+1;

t

+1

=

∇

2

f

(

h

t

+1

, y

t

+1

;

ζ

t

+1

)

,

9: Compute variance-reduced gradient p t +1 and q t +1 :

For the outer-level update, we compute the stochastic gradient of the smoothed loss defined in Eq. (5), based on the variance-reduced estimator { h ( k ) t +1 } K k =1 of the inner-level function, as presented in Step 8. Here, u t +1; t +1 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 ˆ ξ t +1 = {{ ξ ( k ) t +1 } K k =1 , ζ t +1 } . Similarly, we compute the stochastic gradient with respect to the dual variable based on the variance-reduced estimator h ( K ) t +1 of the inner-level function. The algorithm then performs STORM-like updates on p t +1 and q t +1 , as presented in Step 9, where ρ x > 0 and ρ y > 0 are two hyperparameters such that ρ x η 2 ∈ (0 , 1) and ρ y η 2 ∈ (0 , 1) .

4.2 Theoretical Analysis

We derive the convergence rate of Algorithm 1 in the following theorem 2 .

Theorem 4.1. Given Assumptions 3.1-3.3, when ρ x > 0 , ρ y > 0 , α > 0 , ω = O ( ℓ ) , and the hyperparameter conditions in Eq. (94) are satisfied, Algorithm 1 achieves the following convergence upper bound:

where P 0 = f ω ( G ( x 0 ) , y 0 ; z 0 ) -2 f ω,d ( y 0 ; z 0 ) + 2 g ( z 0 ) , with the definitions of the involved terms provided in Eq. (25).

Corollary 4.2. Given Assumptions 3.1-3.3, by setting γ x = O (1) , γ y = O (1) , γ z = O (1 /κ ) , η = O ( ϵ/κ 1 / 2 ) , ρ x = O (1) , ρ y = O (1) , α = O (1) , S = O ( κ 1 / 2 /ϵ ) , T = O ( κ 3 / 2 /ϵ 3 ) ,

Algorithm 1 can achieve the O ( ϵ, ϵ/ √ κ ) -stationary solution, where ϵ > 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 f ( G ( x t ) , y t ) , rather than that of the smoothed loss function f ω ( G ( x ) , y ; z ) . Therefore, this result corresponds to the convergence rate for a nonconvex-PL loss function, rather than for a two-sided-PL loss function. As a result, the comparison of convergence rates with existing methods in Table 1 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 of Algorithm 1, we propose a novel potential function as follows:

where the coefficient ν a , ν b and { λ k } K k =1 are positive, where the notations of ∇ x f ω ( H ( x t ) , y t ; z t ) and ∇ y f ω ( H ( x t ) , y t ; z t ) can be found in Eq. (23).

To analyze the descent of the potential function, we decompose and bound each term through a sequence of lemmas. First, we bound:

which characterizes the optimization error introduced by the smoothed technique. Each component of P t depends on x , y and z , and the compositional gradient introduces additional bias:

We first derive upper bounds for each component in P t .

We then combine these bounds in Appendix B.2.1 to analyze and quantify their dependence, providing a clear characterization of how the three terms interact.

Second, three additional terms in Eq. (9) arise from the gradient errors regarding x and y , and the inner-level estimation error in the multi-level compositional loss.

Third, the four terms in H t are interdependent. We analyze these dependencies in Appendix B.2.2 and show that H t satisfies a sufficient descent property, i.e. , H t +1 -H t can be bounded under suitable hyperparameter conditions, ensuring convergence to an ( ϵ, ϵ/ √ κ ) -stationary point. The complete proof is provided in Appendix B.

5 Algorithm 2: Stagewise-SMCGDA-VR

5.1 Algorithmic Design

However, to facilitate a fair comparison between the convergence rate of Algorithm 1 and existing stochastic two-level compositional minimax methods, which establish the rate in terms of ϵ -stationary point instead of ( ϵ 1 , ϵ 2 ) -stationary point, it is necessary to convert the ( ϵ, ϵ/ √ κ ) -stationary solution into an ϵ -stationary solution. As discussed in Section 3.3, making this translation is challenging for

multi-level compositional minimax optimization problems. Specifically, in the classical minimax setting, [30] showed that the standard stochastic gradient descent ascent (SGDA) algorithm is sufficient for the translation by solving a strongly-convex-strongly-concave problem, since its convergence rate is only ˜ O (1 /ϵ 2 ) , 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 [31].

The aforementioned challenge motivates the development of a new algorithm to handle the translation from an ( ϵ, ϵ/ √ κ ) -stationary solution into an ϵ -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 ˜ z is the output of Algorithm 1, then we complete the translation by solving the following problem.

Note that ˜ z is the output x from Algorithm 1 and it is fixed when solving this problem. Moreover, since ω is selected such that ˆ f ( G ( x ) , y ) is strongly convex with respect to x , ˆ f ( G ( x ) , y ) naturally satisfies the two-sided PL condition. Then, our next goal is to develop an efficient algorithm to solve Eq. (11) such that its iteration complexity is better than that of Algorithm 1, i.e., the translation does not hurt the overall convergence rate .

To this end, we propose a novel stage-wise algorithm, named Stagewise-SMCGDA-VR, as shown in Algorithm 2 ( Note that a more general algorithm is presented in Algorithm 3 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 is divided into R stages, and in each stage, we run the SMCGDA-VR algorithm without updating z (i.e., removing the component highlighted in blue) and replacing z with ˜ z in Step 8. At the end of each stage r , the algorithm randomly selects a tuple from the set { ( x r,t , y r,t , h ( k ) r,t , p r,t , q r,t ) } T r -1 t =0 , where k ∈ { 1 , . . . , K } , to be used as the initialization for the next stage r +1 . A complete description of the algorithm is given in the Appendix C.

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, are presented in Theorems C.1-C.2.

Theorem 5.1. Given Assumption 3.1-3.4, by setting c 0 = 25 L 2 f µ 2 , ρ x = 6400 c 0 L 2 β , ρ y = 640 L 2 β , α = 640 c 0 L 2 β , η y, 0 = 1 20 L β , T 0 = max { 225 , 16 V 0 L β σ 2 } , and for r ≥ 1 , η x,r = O ( µ 2 / ( √ 2 r -1 L β )) , η y,r = O (1 / ( √ 2 r -1 L β )) , T r = O ( c 0 / ( µ × 2 r -1 )) , after running Algorithm 2 for the total number of iterations (not stages) O (1 /ϵ 2 ) , we can get E [ ∥∇ Φ(˜ x R ) ∥ 2 ] ≤ ϵ 2 .

Remark 5.2. From Theorem 5.1, it can be observed that the iteration complexity O (1 /ϵ 2 ) of the translation phase is much smaller than that of Algorithm 1. Therefore, the translation does not hurt the overall convergence rate.

Remark 5.3. Since the overall iteration complexity is determined by Algorithm 1, we can conclude that our algorithm achieves an iteration complexity of T = O ( κ 3 / 2 /ϵ 3 ) , improving upon the O ( κ 4 /ϵ 4 ) complexity of the two-level compositional minimax problem in [17, 9] by offering better dependence on both κ and ϵ and the O ( κ 3 /ϵ 3 ) complexity in [26] by a better dependence on κ . To the best of our knowledge, this is the first algorithm to achieve an O ( κ 3 / 2 ) 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 structure of Algorithms 2. We introduce two metrics to facilitate convergence analysis:

where V t denotes the optimization error, and U t is similar to the last three terms of Eq. (9), c 0 is a positive constant such that c 0 η x,t η y,t = 1 10 . Importantly, following Lemma C.6 and C.7, we establish how V t and U t affect each other across stages and derive the following inequalities in Appendix C.3:

These bounds differ only by a factor of 1 /µ . Using induction, at the r -th stage, we assume

where ϵ r > 0 is a constant. Finally, by selecting appropriate hyperparameters, we prove that

As such, we 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 crossentropy loss function results in a K -level inner function G ( · ) in Eq. (1), with a detailed discussion provided in Appendix A. We compare our smoothed method with three baselines: SCGDA [17], SCGDAM [33], and NSTORM [26] across three datasets: CATvsDOG, CIFAR10 and STL10. Imbalanced binary datasets are generated following the approach described in [33], with an imbalance ratio of 0.05. ResNet20 is employed as the model. For all algorithms, we set both the learning rate and the momentum or variance reduction coefficient to 0.1. In our proposed method, we employ smoothed techniques during the first 90 epochs, followed by stage-wise updates for the remaining 10 epochs.

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(c) STL10

We conduct experiments using our smoothed method for both K = 1 and K = 5 , with results presented in Figure 1. Notably, NSTORM applies STORM-like updates to the two-level( K = 1 ) compositional minimax problem without smoothed techniques. Our results show that the smoothed approach consistently outperforms all 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 is observed consistently across all datasets, highlighting the effectiveness of incorporating deeper compositional structures. Additional experiments with varying K are provided in the Appendix A.

6.2 Multi-Instance Learning

Following [39], multi-instance learning can be reformulated as a multi-level compositional minimax problem as shown in Eq. 1, 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-based (att) pooling. We compare the performance of our smoothed methods against six baseline methods: MIDAM(smx) and MIDAM(att) [40], both utilizing stochastic pooling

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We conduct experiments on five commonly used tabular benchmark datasets [10, 1] for MIL tasks MUSK1, MUSK2, Fox, Tiger, and Elephant - as well as one histopathological image dataset, namely Breast Cancer. For the tabular datasets, we use a two-layer feed-forward neural network with tanh activation and a sigmoid output for AUC loss normalization. For the Breast Cancer dataset, each image is divided into 32 × 32 patches and treated as a 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 a 0.9/0.1 ratio.

For the tabular datasets, we perform 5fold cross-validation, 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, 1e-2, 1e3}, while the learning rate for the dual variables is fixed at 1. We vary the value of K from 1 to 5 and ultimately fix it at 3 to achieve more stable performance. We present the experimental results on the tabular datasets in Figure 2, and on

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the image dataset in Figure 3. 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 a range of tasks and datasets.

7 Conclusion

In this work, we addressed the challenging problem of stochastic multi-level compositional minimax optimization by proposing a smoothed variance-reduced algorithm. Our theoretical analysis demonstrates that the proposed smoothed method achieves a convergence rate of O ( κ 3 / 2 /ϵ 3 ) to an ( ϵ, ϵ/ √ κ ) -stationary point. Furthermore, to bridge the gap between different stationarity measures, we developed a stage-wise algorithm under the two-sided PL condition, enabling a translation to an ϵ -stationary point. Extensive experiments on deep AUC maximization and multi-instance learning tasks validate the superior performance of our approach.

Acknowledcements

We thank anonymous reviewers for constructive comments. X. Zhang and H. Gao were partially supported by U.S. NSF CAREER 2339545, NSF IIS 2416607, NSF CNS 2107014.

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Contents

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 as the following two-level compositional minimax problem:

where w ∈ R d are model parameters while ( a, b, α ) are parameters for AUC loss, ( x, y ) represents feature and label of a sample.

Here, L CE indicates the standard cross-entropy loss function, w -˜ η ∇ w L CE denotes using the gradient descent approach on cross-entropy loss to update the model parameters, where ˜ η > 0 is the learning rate. Then, the obtained model parameter ˜ w can be optimized through the AUC loss. The following serves as a generic representation of Eq. (16) as a two-level compositional minimax optimization problem:

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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 ∈ { 1 , · · · , K } , the k -th inner-level function is defined as:

̸

where ˜ g refers to g ( k -1) ( · ) when k ∈ { 2 , · · · , K } , ξ ( k ) represents the data distribution for the k -th level function. The learning rate for the inner-level functions is denoted by ˜ η . Consequently, Eq. (17) can be reformulated as a multi-level compositional minimax optimization problem exactly as the Eq. (1).

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 as the pooling operation, is a critical component of multi-instance learning. Diverse pooling strategies have been investigated, including mean pooling, max pooling, and softmax pooling [28] and attention-based pooling [21]. Then, to address memory concerns, [40] provided a class of variance-reduced stochastic pooling approaches by reformulating the AUC loss function with the pooled prediction as a three-level compositional minimax function as follows:

where X i = { x 1 i , · · · , x n i i } denotes a bag of data instances, D + represents only containing positive bags with label y i = 1 , D -represents only containing negative bags with label y i = 0 . The pooled prediction h ( w ; X i ) = f 2 ( f 1 ( w ; X i )) denotes the predicted score of the bag i over all its instance, which is a two-level compositional function. For example, for the log-sum-exp(smx) pooling, we have:

For the attention-based (att) pooling, we have:

Similarly, the three-level compositional minimax problem in Eq. (19) can be reformulated as a stochastic 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 i ) . In particular, applying K inner gradient steps to optimize the cross-entropy loss results in a K -level inner function. Consequently, Eq.(19) can be expressed in the unified form of Eq. (1), which corresponds to a stochastic ( K +3) -level 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 4, 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.

B Appendix: Smoothed-SMCGDA-VR

To begin with, we introduce the following terminology to simplify the complex expressions, which will be useful in the subsequent analysis:

Therefore, for the smoothed loss, we have

Moreover, we introduce C 2 p as follows:

Following [30], we introduce the following auxiliary functions for convergence analysis:

Proof Structure. Our proof consists of two key components. The first component, including Lemma B.3 and Lemma B.4, 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 their interdependence.

B.1 Useful Lemmas

Lemma B.1. Given Assumptions 3.1-3.3, we can know

G ( k ) ( x ) is C k g -Lipschitz continuous for k ∈ { 1 , · · · , K -1 } and G ( x ) is C G -Lipschitz continuous where C G = C K g ;

∇ G ( x ) is L G -Lipschitz continuous where L G = ∑ K -1 j =0 L g C K -1+ j g ;

∇ x f ( G ( x ) , y ) is ˆ L -Lipschitz continuous where ˆ L = C 2 G L f + C f L G ;

Φ( x ) ≜ max y ∈ R dy f ( G ( x ) , y ) , Φ( x ) is L Φ -Lipschitz continuous where L Φ = 2 C 2 G L 2 f µ + C f L G .

Proof. The first three properties follow from Lemma B.1. in [37]. The last property is based on Lemma A.3. in [30] and can be established by showing that ∥ Φ( x 2 ) -Φ( x 1 ) ∥ ≤ ( 2 C 2 G L 2 f µ + C f L G ) ∥ x 2 -x 1 ∥ .

Lemma B.2. [30] Given Assumptions 3.1-3.3, the following inequality holds:

where C x 1 yz = ω + ℓ ω -ℓ , C x 2 yz = ω ω -ℓ , and C x z = ω ω -ℓ and ω > ℓ .

Lemma B.3. Given Assumptions 3.1-3.3, and γ z η ≤ 1 , the following inequality holds:

The smoothed function f ω ( G ( x t ) , y t ; z t ) satisfies:

The dual function E [ h ω,d ( y t ; z t )] satisfies:

The function h ( z t ) satisfies:

Proof. (1). From the update rule z t +1 = z t + γ z η ( x t +1 -z t ) , we obtain

where the last step holds uses the fact that γ z η ≤ 1 .

Proof. (2). Since the dual function h ω,d ( y t ; z t ) is L ω,d -smooth, it satisfies that

On the other hand, we have

where we use the fact ⟨ a -b, a + b ⟩ = ∥ a ∥ 2 -∥ b ∥ 2 in the second-to-last step. By combining the above two inequalities, the proof is complete.

Proof. (3). From the definition of h ( z t ) , we obtain

where we use the fact ⟨ a -b, a + b ⟩ = ∥ a ∥ 2 -∥ b ∥ 2 in the last step.

Lemma B.4. Given Assumptions 3.1-3.3, when η ≤ 1 2 γ x ( ω + ℓ ) , and γ z η ≤ 1 , the following inequalities hold:

Proof. First, from Lemma B.1, it follows that the smoothed loss function f ω ( G ( x t ) , y t ; z t ) is ( ω + ℓ ) -smooth with respect to x . Therefore, we have

where the second-to-last step holds due to η ≤ 1 2 γ x ( ω + ℓ ) .

Similarly, since f ω ( G ( x t ) , y t ; z t ) is ( ω + ℓ ) -smooth with respect to y , we obtain

By combining the inequalities above with Lemma B.3, the proof is complete.

Lemma B.5. Given Assumptions 3.1-3.3, the following inequality holds:

The estimation error between ∇ x f ( G ( x t ) , y t ) and ∇ x f ( H ( x t ) , y t ) satisfies:

The bounded error between ∇ x f ω ( H ( x t ) , y t ; z t ; ˆ ξ t +1 ) and ∇ x f ω ( H ( x t ) , y t ; z t ) satisfies: E [ ∥∇ x f ω ( H ( x t ) , y t ; z t ; ˆ ξ t +1 ) -∇ x f ω ( H ( x t ) , y t ; z t ) ∥ 2 ] ≤ C 2 p σ 2 . (38)

For any k ∈ { 1 , · · · , K } , the descent error between h ( k ) t +1 and h ( k ) t satisfies:

For any λ k > 0 where k ∈ { 1 , · · · , K } , the estimation error between g ( k ) ( h ( k -1) t ) and h ( k ) t satisfies:

Proof. First, we have

where the last step follows from Lemma B.2, Eq. (25) in [37]. Similarly,

where the last step follows from Lemma B.2, Eq. (28) in [37]. From the definition of C 2 p , the proof is complete.

Subsequently, the remaining inequalities follow from Lemma B.4 and Lemma B.5 in [37].

Lemma B.6. Given Assumptions 3.1-3.3, we derive

Proof.

where the last step holds due to Lemma B.5 and third step holds due to the following inequality:

Next, we bound T 1 as follows:

Combining this with the previous inequalities completes the proof.

Lemma B.7. Given Assumption 3.1-3.3, we derive:

Proof.

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 ρ x > 0 , ρ y > 0 , α > 0 , ω = O ( ℓ ) , and the hyperparameter conditions are satisfied:

Algorithm 1 achieves the following convergence upper bound:

where P 0 = f ω ( G ( x 0 ) , y 0 ; z 0 ) -2 f ω,d ( y 0 ; z 0 ) + 2 g ( z 0 ) , with the definitions of the involved terms provided in Eq. (25).

Proof. To establish the convergence rate of Algorithm 1, we propose a novel potential function as follows:

where the coefficient ν a , ν b and { λ k } K k =1 are positive.

B.2.1 Bound P t +1 -P t

First, we aim to derive an upper bound for P t +1 -P t . To this end, we begin by applying Lemmas B.3 and B.4, from which we obtain

Next, we derive

where the third step follows from Lemma B.2.

Additionally, we have

where c > 0 is a constant, and the last step follows from

Setting c = 1 8 , we obtain

Furthermore, due to the strong convexity of f ω ( G ( x t ) , y t ; z t ) regarding x , we obtain

In addition, by introducing

we obtain

Moreover, we derive

Therefore, we obtain the following upper bound for P t +1 -P t :

B.2.2 Bound H t +1 -H t

In the following, we aim to derive an upper bound for H t +1 -H t :

We consider the following choice for bounding E [ ∥∇ x f ω ( G ( x t ) , y t ; z t ) ∥ 2 ] and E [ ∥∇ y f ω ( G ( x t ) , y t ; z t ) ∥ 2 ] :

Since γ z η ≤ 1 and C x 1 yz = ω + ℓ ω -ℓ , we set

Additionally, we consider the following choice for bounding E [ ∥ z t +1 -z t ∥ 2 ] , we set

Specifically, we enforce

Then, based on Eq. (64), from C x z = C x 2 yz and η < 1 , we obtain

To remove the term E [ ∥∇ x f ω ( H ( x t ) , y t ; z t ) -p t ∥ 2 ] , we impose

From this, we obtain the parameter choice

Similarly, to remove the term E [ ∥∇ y f ω ( H ( x t ) , y t ; z t ) -q t ∥ 2 ] , we impose

From the second inequality in Eq. (63) and definition of c γ y , we have

As a result, we require which leads to the parameter choice

Plugging the value of ν a and ν b , we obtain

To analyze this, we first simplify the expression:

Due to αη 2 ≤ 1 , we enforce the following to be non-positive:

Therefore, we obtain the parameter choice for any λ k where k ∈ { 1 , · · · , K } :

Moreover, we enforce

for k ∈ { 1 , · · · , K } , which leads to

To guarantee that E [ ∥ p t ∥ 2 ] cancels out, we enforce

This is equivalent to enforce

Specifically, we enforce

To solve the first third inequalities, we obtain

For the last inequality, it is equivalent to enforce

where

Specifically, we enforce

For the first inequality, since d k is independent of hyperparameters, we obtain

For the remaining inequalities, we obtain

As for E [ ∥ q t ∥ 2 ] , we enforce

This is equivalent to enforce

Specifically, we enforce

To solve these inequalities, we obtain

In summary, by setting

we obtain

From

by summing over t from 0 to T -1 and reformulate it, we obtain

When t = 0 , we derive

Finally, we have

C Appendix: Stagewise-SMCGDA-VR

Note that in this section, we provide a general algorithm for the multi-level compositional minimax optimization problem satisfying the two-sided PL condition. Specifically, we aim to solve the following problem:

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.e., R = 1 , we can obtain the convergence rate for the multi-level compositional minimax optimization problem satisfying the nonconvex-PL assumption in Theorem C.1.

Theorem C.1. Given Assumption 3.1-3.3, when R = 1 , by setting η y, 1 = O ( ϵ/κ ) , η x, 1 = O ( ϵ/κ 3 ) , and the initial batch size as O ( κ/ϵ ) , after running Algorithm 3 for T 1 = O ( κ 3 /ϵ 3 ) total iterations, we have 1 T 1 ∑ T 1 -1 t =0 E [ ∥∇ Φ( x 1 ,t ) ∥ 2 ] ≤ ϵ 2 .

Note that in the proof of Theorem C.1, we do not use the PL condition with respect to x . Therefore, the result provides a convergence rate for the nonconvex-PL minimax problem. In addition, this convergence rate corresponds to 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 a smaller learning rate for x compared to y when not using the smoothing technique, as the condition number κ > 1 .

Second, when the number of stages is greater than one, i.e., R > 1 , we can obtain the convergence rate for the multi-level compositional minimax optimization problem satisfying the two-sided PL condition in Theorem C.2.

Theorem C.2. Given Assumption 3.1-3.4, by setting c 0 = 25 L 2 f µ 2 , ρ x = 6400 c 0 L 2 β , ρ y = 640 L 2 β , α = 640 c 0 L 2 β , η y, 0 = 1 20 L β , T 0 = max { 225 , 16 V 0 L β σ 2 } , and for r ≥ 1 , η x,r = O ( µ 2 / ( √ 2 r -1 L β )) , η y,r = O (1 / ( √ 2 r -1 L β )) , T r = O ( c 0 / ( µ × 2 r -1 )) , after running Algorithm 3 for O ( κ 6 /ϵ ) total iterations, we can get E [Φ(˜ x R ) -Φ( x ∗ )] ≤ ϵ .

C.1 Useful Lemmas

Lemma C.3. Given Assumptions 3.1-3.3 and η x,r ≤ 1 2 L Φ , we know

Algorithm 3 Stagewise Stochastic Multi-level Compositional Gradient Descent Ascent with Variance Reduced Algorithm (Stagewise-SMCGDA-VR)

4:

x

r,t

+1

=

x

r,t

-

η

y

r,t

+1

=

y

r,t

+

η

x,r y,r

p

q

r,t r,t

7:

Compute

k

-th inner-level function:

8: end for

9: Compute stochastic compositional gradient u r,t +1 and v r,t +1

v

r,t

+1;

t

+1

=

∇

(

K

)

2

f

(

h

r,t

+1

, y r,t

+1

;

ζ

r,t

)

,

Compute variance-reduced gradient

10:

p

r,t

+1

and

q

r,t

+1

11: end for

13: end for

Proof.

,

.

:

where the second-to-last step holds due to η x,r ≤ 1 2 L Φ .

Lemma C.4. Given Assumption 3.1-3.3 , η y,r ≤ 1 ℓ , we have

Proof. First, from Lemma B.1, we obtain

Moreover, we obtain

where the sixth step holds due to η y,r ≤ 1 ℓ , the fourth step follows from the following inequality:

By combining these two inequalities, the proof is complete.

Lemma C.5. Given Assumption 3.1-3.3 , η x,r ≤ 1 16 ℓ , we have

Proof. In terms of Lemma C.3 and Lemma C.4, we obtain

where the last step follows from ∥∇ x f ( G ( x r,t ) , y r,t ) ∥ 2 ≤ 2 ∥∇ Φ( x r,t ) ∥ 2 +2 ∥∇ x f ( G ( x r,t ) , y r,t ) -∇ Φ( x r,t ) ∥ 2 and η x,r ≤ 1 16 ℓ .

Lemma C.6. Given Assumption 3.1-3.3 , by setting

where ˜ λ k is defined in Eq. (116), L β is defined in Eq. (132), such that η x,r = η y,r 10 c 0 , we have

Proof. We first propose a novel Lyapunov function as follows:

where η x,r = η y,r 10 c 0 . Then, from Lemma C.5, B.5, B.6 and B.7, we obtain

Since η x,r = η y,r 10 c 0 , we enforce

By solving this, we obtain

where k ∈ { 1 , · · · , K } and λ ′ k = max { λ k, 1 , λ k, 2 , λ k, 3 , λ k, 4 } . Given that λ k can be organized as:

Moreover, we enforce

for k ∈ { 1 , · · · , K } , which leads to

Additionally, by plugging the value of λ k , we obtain

To guarantee that E [ ∥ p r,t ∥ 2 ] cancels out, we enforce

This can be done by setting

For the first inequality, we enforce

It is easy to obtain

For the second and third inequalities, we obtain

Similarly, to guarantee that E [ ∥ q r,t ∥ 2 ] cancels out, we enforce

With η x,r = η y,r 10 c 0 , we obtain

We obtain

To solve this inequality, we enforce

In summary, the hyperparameters should be set as follows:

Moreover, by setting c 0 = 25 ℓ 2 µ 2 , we get

Then from Lemma A.7 in [4] , we have ( η x,r + 4 c 0 η 2 x,r η y,r ) E [ ∥∇ Φ( x r,t ) -∇ x f ( G ( x r,t ) , y r,t ) ∥ 2 ] ≤ c 0 η x,r 16 E [ ∥∇ y f ( G ( x r,t ) , y r,t ) ∥ 2 ] .

As a result, we have

where the last step holds due to αη 2 x,r ≤ 1 , and -η x,r 4 ≥ -η y,r 8 since η x,r = η y,r 10 c 0 , c 0 = 25 ℓ 2 µ 2 . Then, we set

where λ ′ k is defined in Eq.(115). It is easy to verify that the conditions in Eq. (129) are satisfied. Meanwhile, this indicates ρ x = 10 c 0 ρ y , α = c 0 ρ y .

By summing t from 0 to T r -1 , we obtain

where V r, 0 = E [Φ( x r, 0 )] -Φ( x ∗ ) + c 0 η x,r η y,r ( E [Φ( x r, 0 )] -E [ f ( g ( x r, 0 ) , y r, 0 )]) , σ x r, 0 = E [ ∥∇ x f ( H ( x r, 0 ) , y r, 0 ) -p r, 0 ∥ 2 ] , σ y r, 0 = E [ ∥∇ y f ( H ( x r, 0 ) , y r, 0 ) -q r, 0 ∥ 2 ] and σ h,k r, 0 = E [ ∥ g ( k ) ( h ( k -1) r, 0 ) -h ( k ) r, 0 ∥ 2 ] , σ h r, 0 = ∑ K k =1 λ ′ k σ h,k r, 0 .

C.2 Proof of the Theorem C.1

Proof. Based on Lemma C.6, we have

Since c 0 = O ( κ 2 ) , then it is easy to verify that by setting by setting η y, 1 = O ( ϵ/κ ) , η x, 1 = O ( ϵ/κ 3 ) , T 1 = O ( κ 3 /ϵ 3 ) , and the initial batch size as O ( κ/ϵ ) , we have

As a result, we can conclude 1 T 1 ∑ T 1 -1 t =0 E [ ∥∇ Φ( x 1 ,t ) ∥ 2 ] ≤ ϵ 2 .

C.3 Proof of the Theorem C.2

Lemma C.7. Assumption 3.1-3.4 , we have

Proof. In the following, we will bound σ x r, 0 + σ y r, 0 +56 σ h r, 0 . At first, from Lemma B.5, we get

where the last step holds due to αη 2 x,r ≤ 1 . Then, according to the random sampling operation, we have

where the last step holds due to ρ y = 640 L 2 β .

Then, based on Lemma B.6, we have

where the last step holds due to the definition of λ ′ k and αη 2 x,r ≤ 1 . Then, due to the random sampling in each outer iteration, it is easy to know

where the last step holds due to ρ y = 640 L 2 β , c 0 > 1 and 0 x,r η y,r = 1 10 . Similarly, from Lemma B.7, we get

where the second step holds due to the definition of λ ′ k , and the last step holds due to αη 2 x,r ≤ 1 . Then, due to the randomly sampling operation in each outer iteration, it is easy to know

where the last step holds due to ρ y = 640 L 2 β .

Then, we combine these three inequalities together as follows:

Then, we need to bound 1 T r ∑ T r -1 t =0 E [ ∥ p r,t ∥ 2 ] + c 0 η y,r η x,r 1 T r ∑ T r -1 t =0 E [ ∥ q r,t ∥ 2 ] . In particular, we have

By plugging Eq. (131), we obtain

By summing up t from 0 to T r -1 , we get

By plugging this inequality to Eq. (143), we get

In the following, we prove Theorem C.2.

Proof. Under the two-sided PL condition, we get

we set

Due to the random sampling in each outer iteration, we get

Therefore, when r = 0 , we have σ x 0 , 0 + σ y 0 , 0 +56 σ h 0 , 0 = 58 L 2 β σ 2 . Based on Eq. (149) and Lemma C.7, we have

When r = 0 , by setting η y, 0 = 1 30 L β and R 0 = max { 225 , 16 V 0 , 0 L β σ 2 } , we have

where the second step holds due to ρ y = 640 L 2 β .

Therefore, we denote ϵ 1 ≜ 500 c 0 L 2 β σ 2 /µ such that

In the following, we use the inductive approach to prove the desired result. Specifically, suppose σ x r, 0 + σ y r, 0 +56 σ h r, 0 ≤ µϵ r and V r, 0 ≤ ϵ r , we will prove σ x r +1 , 0 + σ y r +1 , 0 +56 σ h r +1 , 0 ≤ µϵ r / 2 and V r +1 , 0 ≤ ϵ r / 2 . At first, we have

To make σ x r +1 , 0 + σ y r +1 , 0 +56 σ h r +1 , 0 ≤ µϵ r / 2 , we enforce each term to be smaller than ϵ r / 6 . In particular, by setting

It is easy to verify that ρ y η 2 y,r < 1 for t ≥ 1 .

By setting we get

By setting we get

Therefore, by setting η y,r = √ µϵ r 1140 √ c 0 L 2 β σ and T r = O ( c 0 L 2 β √ c 0 σ µ √ µϵ r ∨ c 2 0 L 4 β σ 2 µϵ r ) , we get

and

Since ϵ r = ϵ 1 2 r -1 = 500 c 0 L 2 β σ 2 2 r -1 µ , we get

Therefore, we set T r = O ( c 0 µ × 2 r -1 ) . Finally, to achieve V R, 0 ≤ ϵ , we need ϵ 1 2 ( R -1) = ϵ so that R = log 2 2 ϵ 1 ϵ . As such, the total number of iterations is

where the second step holds due to

Moreover, we get

and it is easy to know η x,r = O ( µ 2 / 2 r -1 L β ) .

C.4 Proof of the Theorem 5.1

Proof. Because ˆ f ( G ( x ) , y ) is strongly convex with respect to x and satisfies the PL condition with respect to y , we have

where we set ω = 2 ℓ such that ˆ f ( G ( x ) , y ) is ℓ -strongly convex with respect to x , and we define ˆ Φ( x ) = max y ˆ f ( G ( x ) , y ∗ ) with y ∗ = arg max y ∈ R dy ˆ f ( G ( x ) , y ) . Then, according to Proposition 2.1 in [30], to guarantee E [ ∥ x ˜ R -x ∗ ∥ 2 ] ≤ O ( ϵ 2 ) such that E [ ∥∇ Φ(˜ x R ) ∥ 2 ] ≤ O ( ϵ 2 ) , we can enforce E [ ˆ Φ( x ˜ R ) -ˆ Φ( x ∗ )] ≤ O ( ϵ 2 ) . Then, from Theorem C.2, it is easy to see that after running Algorithm 2 for the total number of iterations O (1 /ϵ 2 ) (Note that 1 /ϵ is usually large in practice [30] so that we omit other factors.), we have E [ ∥ x ˜ R -x ∗ ∥ 2 ] ≤ O ( ϵ 2 ) and then E [ ∥∇ Φ(˜ x R ) ∥ 2 ] ≤ O ( ϵ 2 ) .

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\nQuestions\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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n## 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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?\n\nQUESTIONS:\nN/A - Please comment on the weakness of the paper\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\n1) 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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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.\n\nQUESTIONS:\nBelow 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.\n\nLIMITATIONS:\nSome 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?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}