Title: Solving a Class of Non-Convex  Minimax Optimization in Federated Learning
Venue: NeurIPS 2023
Accepted: yes
Keywords: Federated Learning

Non-Convex  Optimization

Minimax Optimization
Abstract: The minimax problems arise throughout machine learning applications, ranging from adversarial training and policy evaluation in reinforcement learning to AUROC maximization. To address the large-scale distributed data challenges across multiple clients with communication-efficient distributed training, federated learning (FL) is gaining popularity. Many optimization algorithms for minimax problems have been developed in the centralized setting (\emph{i.e.}, single-machine). Nonetheless, the algorithm for minimax problems under FL is still underexplored. In this paper, we study a class of federated nonconvex minimax optimization problems. We propose FL algorithms (FedSGDA+ and FedSGDA-M) and reduce existing complexity results for the most common minimax problems. For nonconvex-concave problems, we propose FedSGDA+ and reduce the communication complexity to $O(\varepsilon^{-6})$. Under nonconvex-strongly-concave and nonconvex-PL minimax settings, we prove that FedSGDA-M has the best-known sample complexity of $O(\kappa^{3} N^{-1}\varepsilon^{-3})$ and the best-known communication complexity of $O(\kappa^{2}\varepsilon^{-2})$. FedSGDA-M is the first algorithm to match the best sample complexity $O(\varepsilon^{-3})$ achieved by the single-machine method under the nonconvex-strongly-concave setting. Extensive experimental results on fair classification and AUROC maximization show the efficiency of our algorithms.

Solving a Class of Non-Convex Minimax Optimization in Federated Learning

Abstract

The minimax problems arise throughout machine learning applications, ranging from adversarial training and policy evaluation in reinforcement learning to AUROC maximization. To address the large-scale distributed data challenges across multiple clients with communication-efficient distributed training, federated learning (FL) is gaining popularity. Many optimization algorithms for minimax problems have been developed in the centralized setting ( i.e. , single-machine). Nonetheless, the algorithm for minimax problems under FL is still underexplored. In this paper, we study a class of federated nonconvex minimax optimization problems. We propose FL algorithms (FedSGDA+ and FedSGDA-M) and reduce existing complexity results for the most common minimax problems. For nonconvex-concave problems, we propose FedSGDA+ and reduce the communication complexity to O ( ε -6 ) . Under nonconvex-strongly-concave and nonconvex-PL minimax settings, we prove that FedSGDA-M has the best-known sample complexity of O ( κ 3 N -1 ε -3 ) and the best-known communication complexity of O ( κ 2 ε -2 ) . FedSGDA-M is the first algorithm to match the best sample complexity O ( ε -3 ) achieved by the single-machine method under the nonconvex-strongly-concave setting. Extensive experimental results on fair classification and AUROC maximization show the efficiency of our algorithms.

1 Introduction

The nonconvex minimax optimization has been actively applied to solve enormous machine learning problems, such as adversarial training [56, 61], generative adversarial networks (GANs) [16, 18], policy evaluation in reinforcement learning [55, 22, 25], robust optimization [11, 60], AUROC (area under the ROC curve) maximization [39], etc . Many single-machine minimax optimization algorithms have been proposed to address these problems.

Heng Huang ∗

Machine learning tasks with large-scale distributed datasets call for distributed training [3] because of its ability to shorten the calculation time and train models with data from various locations. At the same time, communication overhead has emerged as the most restrictive bottleneck of distributed training due to the increasing model and data size. To tackle these issues, federated learning (FL) [44] has emerged as a promising technique since it makes use of distributed data from different clients, avoids frequent transmission between clients and the central server, and preserves data privacy. In FL, clients train and update their models locally, and the server aggregates and averages the model parameters from all clients periodically. Only models are shared among clients and the training data are stored locally, which provides a certain level of data privacy. In addition, FL also enhances computation power since it utilizes many clients to train models.

Although federated learning has gained popularity, most existing works focus on the standard stochastic minimization problem [33, 48, 34, 36, 71, 58, 70, 72]. Recently, some algorithms for nonminimization optimization in FL are proposed [53, 21, 15, 50, 59]. However, existing FL minimax algorithms have not achieved the complexity level reached by single-machine algorithms. To bridge this gap, we consider the federated nonconvex minimax optimization problem as follows:

̸

where the function f i ( x, y ) = E ξ i ∼D i [ f i ( x, y ; ξ i )] : R d 1 × R d 2 → R is the loss function of the i th client. We restrict our focus to the non-convex minimax problem, where f i ( x, y ) is nonconvex over x ∈ R d 1 and concave or nonconcave over y ∈ R d 2 . N is the total number of clients. ξ i = ( x i , y i ) ∼ D i denotes data point ξ i is sampled from the local data distribution D i on machine i . In this paper, heterogeneous datasets are considered, namely, D i and D j ( i = j ) are not identical.

Some recent works have attempted to solve federated minimax optimization for convex-concave setting [11, 26, 52]. Due to the popularity of deep neural networks, nonconvex minimax has wider applications. More recently, some works [50, 53, 10, 19, 67] extend single-machine algorithms, such as SGDA, to federated learning settings for nonconvex minimax optimization. However, theoretical understandings of federated minimax optimization remain limited in the literature. In the context of stochastic smooth nonconvex minimax problems, their analysis either relies on strict assumptions [19] or achieves suboptimal convergence results [50]. For example, single-machine methods [42, 28] achieve O ( κ 3 ε -3 ) under nonconvex-strongly-concave setting, which is much better than O ( κ 4 ε -4 ) achieved by the best FL minimax algorithm [50, 63] in existing literature. Therefore, a natural question arises:

Can we design stochastic gradient decent ascent methods with better sample and communication complexities to match the convergence rate of single-machine counterparts for solving the problem (1) ?

In this paper, we provide an affirmative answer to the aforementioned question and propose a class of algorithms to solve the problem (1) under different settings. In particular, we consider three most common classes of nonconvex minimax optimization problems: 1) NC-C: NonConvex in x , Concave in y ; 2) NC-SC: NonConvex in x , Strongly-Concave in y ; 3) NC- PL: NonConvex in x , PL-condition in y . For each of these problems, we propose a new algorithm with provably better convergence rate (please see Table 1) and provide a theoretical analysis. Our main contributions are four-fold:

NC-C setting. We propose FedSGDA+, and prove it has sample complexity of O ( N -1 ε -8 ) and communication complexity of O ( ε -6 ) . FedSGDA+ takes advantage of the structure of FL by adding the global learning rate at the server and reduces communication complexity to O ( ε -6 ) from O ( ε -7 ) in [50]. It also achieves a linear speedup to the number of clients.

NC-PL setting. We propose a federated stochastic gradient ascent (FedSGDA-M) algorithm with the momentum-based variance reduction technique. It has the best sample complexity of O ( κ 3 N -1 ε -3 ) and the best communication complexity of O ( κ 2 ε -2 ) . Compared with existing momentum-based variance reduction algorithms, our result employs a novel theoretical analysis framework that produces a tighter convergence rate (i.e., our rate gets rid of a logarithmic term appearing in existing works).

NC-SC setting. FedSGDA-M can be directly applied to the NC-SC setting since the PL condition is weaker than strong-concavity. Our algorithm is the first work to reach sample

complexity of O ( ε -3 ) in federated learning. In addition, FedSGDA-M does not rely on a large batch size to reach optimal sample complexity compared with single-machine minimax algorithms [29, 42].

Extensive experimental results on fair classification and AUROC maximization confirm the effectiveness of our proposed algorithm.

2 Related Work

2.1 Single-Machine Minimax

Nonconvex-Concave (NC-C) setting . [31, 46, 45, 54, 35] proposed various deterministic and stochastic algorithms to solve the NC-C minimax problems. All of these algorithms, however, have a double-loop structure and are thus relatively complicated to implement. They decouple the minimax problem into a minimization problem and a maximization problem and use a nested loop to update variable y while keeping variable x constant. Subsequently, [38] studied the complexity result of the single-loop algorithm (SGDA) for the NC-C minimax problem and proves the stochastic algorithm achieves O ( ε -8 ) complexity. SGDA is a direct extension of SGD from minimization optimization to minimax optimization problems. Recently, [43] providing a unified analysis for the convergence of OGDA and EG methods in the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings.

Nonconvex-Strongly-Concave (NC-SC) setting . [38] analyzed the stochastic gradient descent ascent (SGDA) algorithm and proved that SGDA has O ( κ 3 ε -4 ) stochastic gradient complexity. To reduce the convergence rate, [42] proposed a stochastic GDA algorithm (i.e. SREDA) with a double-loop structure based on the variance reduction technique of SPIDER [13] and reduce the complexity to O ( κ 3 ε -3 ) . [29] used momentum-based variance reduction technique of STORM [7] and proposed Acc-MDA. Acc-MDA is a single-loop algorithm, which gets the same convergence result as SREDA. Furthermore, adaptive minimax algorithms are introduced [28, 30] to solve the nonsmooth nonconvex-strongly-concave minimax problems based on dynamic mirror functions. [65] used a nested adaptive framework to design parameter-agnostic nonconvex minimax algorithm. [69] proves that VR-based SAPD+ has the complexity of O ( κ 2 ε -3 ) . However, whether the best convergence result of O ( ε -3 ) in single-machine methods can be achieved in the federated setting is an open question. In addition, [23] conducts an in-depth investigation of limitations of GDA algorithm (e.g., smaller learning rate, cycling/divergence issue) and gives a systematic analysis of how to improve GDA dynamics.

Nonconvex-Nonconcave (NC-NC) setting . There is extensive research on NC-NC problems [12] and the Nonconvex-PL condition is a special class of functions that interests us the most. PolyakŁojasiewicz (PL) condition does not require the objective to be concave and recent works show that the PL condition could hold in the training process of overparameterized neural networks with random initialization [1, 5]. Recently, many deterministic methods [45, 64, 14] are proposed for NC-NC problems under the NC-PL setting. [20] proposed a PDAda method for Nonconvex-PL minimax optimization with the restriction of the concavity condition, and [6] focuses on the finite-sum Nonconvex-PL minimax optimization. Stochastic alternating GDA and stochastic smoothed GDA proposed in [66] achieve complexity of O ( κ 4 ε -4 ) and O ( κ 2 ε -4 ) , respectively.

2.2 Distributed/Federated Minimax

Distributed training has rapid development in minimax optimization in recent years, driven by the need to train large-scale datasets [40]. Under the serverless decentralized setting, algorithms for nonconvex minimax have been studied extensively in nonconvex-strongly-concave setting [4, 62, 68, 41, 57] and nonconvex-PL setting [27].

In the FL setting, some works analyzed algorithms for convex-concave problems [11, 37, 26, 52]. However, as nonconvex models (e.g., deep neural networks) being more and more prevalent, there is a growing need for federated nonconvex minimax optimization, such as federated adversarial training [49], federated deep AUROC maximization [19] and federated GAN [47]. [19] and [67] focus on imbalanced data tasks. They reformulated the AUROC maximization problem as the minmax problem under the FL setting. But their analysis relies on strict assumptions that deep models satisfy the PL condition and only focuses on PL-strong-concave minimax. [49] converted the robust federated learning into the minimax problem, where only model parameters, namely min variables, are exchanged among clients via the server. [10] proposed Local SGDA and Local SGDA+. Local SGDA is the local-update version of the SGDA algorithm in FL. Different from local SGDA, in local SGDA+, max variable y is updated with a constant min variable ˜ x and the snapshot ˜ x updates every S iteration. Afterward, [50] improves sample complexity and communication complexity of Local SGDA for NC-SC and NC-PL settings, and Local SGDA+ for NC-C setting. [50] also proposes a Momentum Local SGDA, which achieves the same theoretical results as Local SGDA for NC-PL and NC-SC settings. In addition, [53] designs FEDNEST with two nested loops. Although FEDNEST is composed of FEDINN (a federated stochastic variance reduction algorithm ), their convergence complexity is not improved over vanilla Local SGDA. Afterward, [63] proposes SAGDA under NC-PL setting, which yields a better communication complexity (i.e., O ( ε -2 ) ). However, its analysis does not consider the effect of condition number κ . More recently, [51] considers federated minimax optimization with Client Heterogeneity in nonconvex concave and nonconvex strongly-concave settings.

Relation to Existing Works . We propose FedSGDA+ for NC-C setting and FedSGDA-M for NC-SC and NC-PL settings. In NC-C setting, we discover that the addition of a global step size leads to better communication complexity. Under this circumstance, theoretical analysis is more challenging as we not only need to consider the complicated structure of the minimax problem but also need to handle the local update and global update separately. For FedSGDA-M, we relax the requirement of step size (specifically designed unnatural step size is often required in STORM-like approaches [7, 34]), which requires novel proof techniques to obtain. Thus our result does not contain a logarithmic term and provides a tighter convergence rate (Seen in contributions 1 in [34]). In addition, with different theoretical frameworks, our better sample complexity doesn't rely on a big batch size, while the single-machine minimax algorithm with variance reduction (Acc-MDA) in [29] (Table 2) needs a large batch to achieve the same sample complexity.

3 Algorithms and Convergence Analysis

Notation: ∥ · ∥ denotes the ℓ 2 norm for vectors. a = O ( b ) denotes that a ≤ Cb for some constant C > 0 . Given the mini-batch samples B = { ξ j } b j =1 , we let ∇ f i ( x, y ; B ) = 1 b ∑ b j =1 ∇ f i ( x, y ; ξ i ) .

Assumption 3.1. (i) Unbiased Gradient. The gradient of each component function f i ( x, y ; ξ ) computed at each client is unbiased for all ξ ( i ) ∼ D i , i ∈ [ N ] :

Algorithm 1 FedSGDA+ Algorithm

(ii) Variance Bound. The following inequalities hold for all ξ ( i ) ∼ D i , i, j ∈ [ N ] :

The Assumption 3.1 is a standard assumption in stochastic optimization, which will be used throughout the rest of the paper. In FL algorithms, the Assumption 3.1 (ii) is frequently used to bound the variance and data heterogeneity. The heterogeneity parameter, ζ , denotes the level of data heterogeneity. In the homogeneous data configuration, ζ = 0 .

3.1 Nonconvex Concave (NC-C) Problems

Assumption 3.2. (Concavity). The nonconvex function f ( · , · ) is concave in y if for a fixed x ∈ R d 1 , ∀ y, y ′ ∈ R d 2 , we have

To solve the problem (1) under the NC-C setting and reduce the complexity, we propose FedSGDA+ (Seen in algorithm 1). Although local SGDA+ [50] achieves the same sample complexity as the singlemachine method (SGDA), its communication complexity is not optimal. This unsatisfactory result is due to the fact that local SGDA+ simply extends the single-machine method into the distributed setting, and does not consider the complicated local-server structure in FL.

In FedSGDA+, Line 5-10 are conducted in the local clients. The updates of variable x are similar to the standard stochastic algorithm for minimization problems, such as FedAvg. We sample data points and update the x locally with the current variable x i t,q and y i t,q . However, for the y -updates, stochastic gradients are calculated with the latest snapshot of x ( i.e., ˜ x k ) and in each local iteration, y updates with the constant ˜ x k as seen in Line 9 in Algorithm 1. The ˜ x k is updated every S rounds ( Line 14-16 in Algorithm 1).

In addition, we make use of the advantage of FL and introduce the global step size η x , η y , which provides the flexibility of FL training (Seen in Line 12-13). Local SGDA+ could be regarded as a special case of FedSGDA+. We now provide the convergence analysis of FedSGDA+ and introduce the necessary assumptions. The details of the proofs are provided in the supplementary materials.

Assumption 3.3. (Smoothness). Each local function f i ( x, y ) has a L f -Lipschitz continuous gradients, i.e., ∀ x 1 , x 2 and y 1 , y 2 , we have

The Assumption 3.3 on the smoothness is a standard assumption in stochastic optimization [2, 17]. Assumption 3.4. (Lipschitz continuity in x). For the function F , there exists a constant G x , such that for each y ∈ R d 2 , and ∀ x, x ′ ∈ R d 1 , we have

Under the NC-C setting, the function F ( · , · ) is concave in y. Following [9], we define Φ( x ) = max y F ( x, y ) and the Moreau envelope of Φ( · ) is defined below:

Definition 3.5. (Moreau Envelope) A function Φ λ ( · ) is the λ -Moreau envelope of Φ( · ) , for λ > 0 , if ∀ x ∈ R d 1 ,

From [38], we know that when we have a point x that is an ε -stationary point of Φ λ ( x ) , then x is close to a point x ′ which is stationary for Φ( x ) . Hence, we focus on minimizing ∥∇ Φ λ ( x ) ∥ under the NC-C setting.

Theorem 3.6. Suppose Assumptions 3.1, 3.2, 3.3, 3.4 hold and the sequences { x t , y t } are generated by Algorithm 1, max { cη y , c } ≤ 1 10 QL f and let ∥ ¯ y t ∥ 2 ≤ D following [10, 50],

Corollary 3.7. By setting c = ˆ c = 1 10 L f QT 1 / 3 , Q = T 1 / 3 N , ˆ cη x = N 10 L f T , cη y = 1 10 L f Q = N 10 L f T 1 / 3 , S = T 1 / 3 , FedSGDA+ has the following convergence rate:

Remark 3.8 . (Complexity) Based on Corollary 3.7, to make 1 T ∑ T -1 t =0 E ∥ ∥ ∇ Φ 1 / 2 L f (¯ x t ) ∥ ∥ 2 ≤ ε 2 , communication complexity T = O ( ε -6 ) . We choose b = O (1) , then we have sample complexity bQT = N -1 ε -8 . It also denotes that FedSGDA+ has linear speedup with respect to the number of clients.

3.2 Nonconvex-PL (NC-PL) Problems

Assumption 3.9. (Polyak Łojasiewicz (PL) Condition in y). The function F ( x, y ) is µ -PL condition in y ( µ > 0) , if ∀ x : 1) y ∗ ( x ) = arg max y F ( x, y ) has a nonempty solution set; 2) ∥∇ y F ( x, y ) ∥ 2 ≥ 2 µ ( F ( x, y ∗ ( x )) -F ( x, y )) , ∀ y .

To solve problem (1) with better convergence complexity under nonconvex-PL, we propose federated stochastic gradient ascent (FedSGDA-M) algorithm with the momentum-based variance reduction technique (Seen in in algorithm 2).

Algorithm 2 FedSGDA-M Algorithm

In FedSGDA-M, each client initializes the gradient estimators { u i 1 , v i 1 } with stochastic gradient as seen in line 2 of Algorithm 2. Following that, each client updates the model variables { x i t , y i t } locally as standard stochastic gradient descent ascent method (lines 12-13 of Algorithm 2). Compared with local momentum SGDA [50], the key difference is that clients utilize variance reduction gradient estimators { u i t , v i t } , which are constructed in lines 15-17 of Algorithm 2. For the update step of { u t , v t } , the coefficients should satisfy 0 < α < 1 and 0 < β < 1 . In every Q iteration, clients transmit model parameters and gradient estimators to the server, which computes { ¯ x t , ¯ y t , ¯ u t , ¯ v t } . Then the server sends averaged model variables and gradient estimators to each client to update the local variables, as shown in lines 5-10 of Algorithm 2.

Definition 3.10. According to Assumption 3.9, there exists at least one solution to the problem max y F ( x, y ) for any x . Here we define Φ( x ) = F ( x, y ∗ ( x )) = max y F ( x, y ) . We use ε -stationary point of Φ( x ) , i.e. ∥∇ Φ( x ) ∥ ≤ ε as the convergence metric.

We know Φ( x ) is differentiable and ( L + κL ) -smooth and y ∗ ( · ) is κ -Lipschitz from [50]. Given that ∇ y F (¯ x t , y ∗ ( x t )) = 0 , we have ∇ Φ(¯ x t ) = ∇ x F (¯ x t , y ∗ ( x t )) + ∇ y F (¯ x t , y ∗ ( x t )) · ∂y ∗ (¯ x t ) = ∇ x F (¯ x t , y ∗ ( x t )) which is widely used in the analysis of nonconvex-PL [10] and nonconvex-stronglyconcave minimax optimization [62, 38]. Then we discuss the convergence analysis of FedSGDA-M. The proofs are provided in the supplementary materials.

Assumption 3.11. (Lipschitz Smoothness) Each component function f i ( x, y ; ξ ) has a L f -Lipschitz gradient, i.e., ∀ x 1 , x 2 and y 1 , y 2 , we have

F ( x, y ) has an L f -Lipschitz gradient based on the convexity of norm and Assumption 3.11. We have

In optimization analysis, it is standard to use the Assumption 3.11. Several widely used singlemachine stochastic algorithms, such as SPIDER [13] and STORM [7], make use of this assumption. It is also used by numerous FL algorithms, including MIME [32] Fed-GLOMO [8], STEM [34] and FAFED [58].

Theorem 3.12. Suppose that sequence { ¯ x t , ¯ y t } T t =0 is generated from Algorithm 2. Under the above Assumptions (3.1,3.9,3.11), given η = 1 20 QL , α = c 1 η 2 , β = c 2 η 2 , c 1 = 30 L 2 bNκ 1 -ν , c 2 = 30 L 2 bNκ 2 -2 ν , c = 1 6 κ 1 -ν , ˆ c = 1 54 κ 3 -ν where ν ∈ [0 , 1] we have

Corollary 3.13. By setting b = O ( κ ν ) for ν ∈ [0 , 1] , c 1 = 30 L 2 bNκ 1 -ν , c 2 = 30 L 2 bNκ 2 -2 ν , c = 1 6 κ 1 -ν , ˆ c = 1 54 κ 3 -ν , T = κ 3 -ν T 0 , Q = T 1 / 3 0 N 2 / 3 , η = 1 20 QL = N 2 / 3 20 LT 1 / 3 0 , B = T 1 / 3 0 bκ 1 -ν N 2 / 3 , we have α = c 1 η 2 = 3 N 1 / 3 40 T 2 / 3 0 bκ 1 -ν , β = c 2 η 2 = 3 N 1 / 3 40 T 2 / 3 0 bκ 2 -2 ν .

where Φ ∗ is the optimal.

Remark 3.14 . (Complexity) To make the 1 T ∑ T -1 t =0 E ∥∇ Φ(¯ x t ) ∥ 2 ≤ ε 2 , we get T 0 = O ( N -1 ε -3 ) and T = O ( κ 3 -ν N -1 ε -3 ) . Considering the b = κ ν , we have communication complexity T Q = κ 3 -ν ( NT 0 ) 2 / 3 = κ 3 -ν ε -2 and sample complexity bT = O ( κ 3 N -1 ε -3 ) . When ν = 1 , b = κ , communication Complexity T Q = κ 2 ε -2 for finding an ε -stationary point. Sample complexity bT = O ( κ 3 N -1 ε -3 ) matches the complexity result achieved by the single-machine algorithms, such as SREDA and Acc-MDA in [42, 29] but we do not require a large batch size b compared with these algorithms. And O ( κ 3 N -1 ε -3 ) also exhibits a linear speed-up compared with the aforementioned single-machine algorithms.

3.3 Nonconvex-Strongly-Concave (NC-SC) Problems

Assumption 3.15. Each local function function f i ( x, y ) is µ -strongly concave in y ∈ Y , i.e., ∀ x ∈ X and y 1 , y 2 ∈ Y , we have

When the function F ( x, y ) is strongly concave in y ∈ Y , there exists a unique solution to the problem max y ∈Y f ( x, y ) for any x . Since PL condition is weaker than strong concavity, the convergence result of FedSGDA-M in Algorithm 2 under NC-PL also apply to NC-SC problem and FedSGDA-M has the sample complexity of O ( κ 3 n -1 ε -3 ) and the communication complexity of O ( κ 2 ε -2 ) under nonconvex-strongly-concave setting.

4 Experiments

We conduct experiments on AUROC maximization and fair classification tasks to verify the efficiency of our algorithms under nonconvex-strongly-concave and nonconvex-concave settings. Experiments are completed on the computer cluster with AMD EPYC 7513 Processors and NVIDIA RTX A6000. The code is available ∗

Datasets and Models : We test the performance of algorithms on three typical datasets: FashionMNIST dataset, CIFAR-10 dataset and Tiny-ImageNet. Fashion-MNIST dataset has 70 , 000 28 × 28 gray images ( 10 categories, 60 , 000 training images and 10 , 000 testing images). CIFAR-10 dataset consists of 50 , 000 training images and 10 , 000 testing images. Each image is the 3 × 32 × 32 arrays of color image. Tiny-ImageNet dataset has 200 classes of ( 64 × 64 ) colored images and each class has 500 training images, 50 validation images, and 50 test images. For Fashion MNIST and Cifar10 data sets, we choose convolutional neural network from [28] (The details are shown in the supplementray materials). For Tiny-ImageNet, we choose ResNet-18 [24] as the model.

4.1 Fair Classification

First, we follow [50, 28] and train fair classification networks by minimizing the maximum loss over different categories.

where L i c denotes the cross-entropy loss functions corresponding to the class c in C different classes and x denotes the CNN model parameters. Clearly, the problem in (6) is nonConvex in x (deep model parameters), and Concave in y. We compare our algorithm (FedSGDA+) and local SGDA+ with varying model, datasets, local update numbers, step size. Although the constraint is not considered in the theoretical analysis of local SGDA+ and FedSGDA+, FedSGDA+ still shows better performance compared with local SGDA+.

The network has 20 clients. The datasets are partitioned into disjoint sets across all clients and each client holds part of the data from all the classes [34]. We initialize the renset18 with pretrained weights in PyTorch. In experiments, we run grid search for step size, and choose the step size for primal variable in the set { 0 . 01 , 0 . 03 , 0 . 05 , 0 . 1 , 0 . 3 } and that for dual variable in the set { 0 . 001 , 0 . 01 , 0 . 1 } . We choose the global step size in the set { 0 . 1 , 0 . 5 , 1 , 1 . 5 , 2 } . The batch-size b is in 50 and the inner loop number Q is seleted from { 20 , 50 , 100 } , The outer loop number S is selected from { 1 , 5 , 10 } for FedSGDA+ and { 1 , 5 , 10 , Q } for local SGDA+.

Figure 1 shows that FedSGDA+ has a better convergence rate than local SGDA+. This confirms that our algorithm can effectively accelerate SGDA by using the structure of federated learning. Due to the page limititation, the ablation analysis of step size is presented in the supplementary materials.

(a) Fashion-MNIST

(b) CIFAR-10

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4.2 AUROC Maximization

[39] showed the AUROC maximization problem could be reformulated as the non-convex-stronglyconcave minimax optimization, as below:

where

ξ i = ( x , y ) ∼ D i denotes a random data point and x represents the data features and y ∈ Y = {-1 , +1 } is the label. h ( m ; x ) denotes the prediction score of the data point x calculated by a model with parameter m . p = Pr ( y = 1) = E y [ I [ y =1] ] denotes the prior probability of the positive data.

(a) Fashion-MNIST

(b) CIFAR-10

(c) Tiny ImageNet

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CommunicationRound

Following [19, 67], we constructed the imbalanced binary-class versions of datasets as follow: firstly, the first half of the classes (0 - 4) in the original Fashion-MNIST, CIFAR10 and classes (0 - 99) in Tiny-ImageNet datasets are converted into the negative class, and the rest half of classes are considered to be a positive class. 80% of the negative data points are randomly dropped in each dataset. Then the datasets are evenly divided into disjoint sets across 16 clients. In this case, each clients share completely different imbalanced datasets. In the experiment, we use xavier normal initialization to deep models.

We compare our algorithm (i.e., FedSGDA-M) with local SGDA [10, 50], CODA+ [19, 67], Momentum SGDA [50], CODASCA [67] and SAGDA [63] as baselines in AUROC maximization. In experiments, we carefully tune hyperparameters for all methods. We run a grid search for step size, and choose the step size for the primal variable in the set { 0 . 001 , 0 . 005 , 0 . 01 } and that for dual variable in the set { 0 . 0001 , 0 . 001 , 0 . 01 } . We choose the global step size from { 0 . 9 , 1 , 1 . 5 , 2 } for CODASCA and SAGDA. We choose the momentum parameter in Local Momentum SGDA in the set { 0 . 1 , 0 . 5 , 0 . 9 } . The α and β in FedSGDA-M are chosen from { 0 . 1 , 0 . 5 , 0 . 9 } . The batch-size b is 50 and the inner loop number Q ∈ { 10 , 20 , 50 } .

As shown in Figure 2, we compare the performance of FedSGDA-M and other baseline methods against the number of communication rounds. Figure 2 shows that our algorithms consistently outperform the other baseline algorithms on testing datasets, which validates the efficacy of our algorithms. Due to space limitation, other test results are provided in the supplementary materials.

Limitation . Minimax optimization has many applications and a more comprehensive discussion of our proposed algorithms on these tasks will be a future study because the theoretical analysis is the main contribution of this paper.

5 Conclusion

In this paper, we study a class of federated nonconvex minimax optimization problems (1). We consider the three most common settings (NC-SC, NC-PL, NC-C). Under the NC-C setting, we propose FedSGDA+ and prove it has the best communication complexity of O ( ε -6 ) . It also achieves a linear speedup to the number of clients. Under NC-PL and NC-PL settings, we propose FedSGDAMwith variance reduction technique and we prove that our algorithm (FedSGDA-M ) has the best sample complexity ( O ( κ 3 n -1 ε -3 ) ) and the best sample communication complexity ( O ( κ 2 ε -2 ) ). We prove that FedSGDA also enjoys linear speedup with respect to the number of clients. Therefore, we reduce the existing complexity results for the most common nonconvex minimax optimization problems under the federated learning setting.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper focuses on the minimax optimization problem within the context of the federated learning framework. In order to address this problem, the authors introduced two algorithms named as FedSGDA+ and FedSGDA-M, to handle distinct types of losses, specifically focusing on the concavity of the maximizer component. The study presented a comprehensive analysis encompassing both theoretical and numerical evaluations.\n\nSTRENGTHS:\nThis paper contributed two novel algorithms to solve various types of minimax optimization problems within the federated learning framework, which holds significant impact. As many machine learning problems are closely tied to minimax optimization, the developed algorithms offer valuable solutions for a wide range of applications. The solid theoretical analysis not only validates the effectiveness of their proposed algorithms but also lays a foundation for future investigations in the fields of minimax optimization and federated learning.\n\nWEAKNESSES:\nThe paper needs presentation improvement as well as more clarification in the motivation and algorithm development. \n\nCurrently, I am inclining towards border acceptance with my questions below. I encourage authors to provide responses for my better assessment.\n\n1. Line 27-29: The authors claim the federated learning is proposed to tackle the communication issue. But to my best knowledge, the federated learning has many advantages such as local data privacy preserving, utilizing more clients' data and enhancing computation power etc. Only highlighting communication advantages might be one-sided.\n\n2. Line 145: The second inequality in (2) seems to be a strong assumption. It requires the deviation of the local data from the global average  to be bounded. However, under severe data heterogeneity and large parameters magnitude ,  such bound might not hold. Many federated learning works have studied to avoid such assumption, e.g. [1]. Can we also relax this assumption here?\n\n3. Line 167: This assumption seems to be an addition assumption compared with the existing works in Table 1. Basically, it requires the gradient of $F$ w.r.t. $x$ to be bounded. It might not hold for many applications.\n\n4. Wonder whether linear speed up w.r.t. to clients' number hold for this study. Can the authors provide some numerical validation?\n\n5. For the algorithm presentation, can authors separate the server-side and client-side for better understanding?\n\n6. Is FedSGDA-M motivated by the STORM-type of variance reduction method? Why is the algorithm not applicable for NC-C case?\n\n7. Section 4.1: The loss looks be to a constrained one. How do the proposed algorithm hold the constraint?\n\n8. Figure 2: Is FedSGDA in the figure the proposed FedSGDA+ or FedSGDA-M? \n\n9. Can the authors provide numerical studies on the data heterogeneity, clients' number and computation cost?\n\n[1] Karimireddy, Sai Praneeth, et al. \"Scaffold: Stochastic controlled averaging for federated learning.\" International Conference on Machine Learning. PMLR, 2020.\n\nQUESTIONS:\nsee weakness\n\nLIMITATIONS:\nThe author acknowledged the need for further numerical studies and a more comprehensive analysis as a limitation of the paper. Since the research primarily focuses on theoretical algorithm development, there is no anticipated potential for negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-28"}

comments: SUMMARY:
this paper proposed new algorithms for solving federated minimax problems in both nonconvex concave and nonconvex PL (or strongly concave) settings under the assumption of data heterogeneity. the authors propose a novel way to incorporate the variance reduction technique into the federated setting and provide solid convergence analysis in both settings. numerical experiments on auc maximization and fairness problems are reported to show the advantage of the proposed algorithms.

STRENGTHS:
1. the proposed algorithms achieve the best convergence rate compared with existing works in the same setting. the final rate comes with linear speedup under the heterogeneous setting.
2. the idea of updating y with a fixed x in the concave case helps to improve communication complexity.
3. the numerical experiments are convincing. the authors compare with the majority of existing works.
4. the paper is easy to follow.

WEAKNESSES:
the step size \eta needs to be very small to guarantee the convergence since it's inverse propositional to q, the local updates. this requirement hurts the performance in practice

QUESTIONS:
both algorithms in this paper use the same batch data to compute the x gradient and the y gradient. in that case, those gradients (\nabla_x f and \nabla_y f) are not independent of each other. will this cause problems in the proof? can the author double-check?

LIMITATIONS:
the algorithm names in the table are not consistent with the names in the algorithms.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n(1) This paper describes two algorithms, FedSGDA+ and FedSGDA-M, for solving non-convex minimax optimization problems. \n(2) Theoretical guarantees are establised for these two algorithms, under several assumptions.\n(3) Empirical tests are conducted.\n\nSTRENGTHS:\n**Originality:**\n\nThe techniques used in this paper are not novel from an optimization perspective. However, they can indeed be applied to minimax problems and have a theoretical bound.\n \n**Quality:**\n\nThe notations, problem statements, and mathematical proofs appear rigorous, although I have not scrutinized them word by word.\n \n**Clarity:**\n\nThe readability of this article is good.\n \n**Significance:**\n\nThis paper provides agorithms with better convergence rates (although some extra assumptions were made in the paper, ). The techniques (and tricks) may inspire and guide future researchers.\n\nWEAKNESSES:\n1. The algorithms described in \"Federated minimax optimization: Improved convergence analyses and algorithms\" [36] and the ones in this paper are very similar. Specifically, FedSGDA-M and Local SGDA-M in [36] are basically the same, and the difference between FedSGDA+ and Local SGDA in [36] is very small, only in the step size, which may not be the most crucial part of the FedSGDA+ algorithm. Therefore, I believe that there is not much improvement in the algorithm aspect of the paper.\n \n2. On the other hand, the convergence rates established in this paper appear to be better than previous algorithms, but the assumptions used are different from those of the previous. Thus, this comparison seems somewhat unfair and does not necessarily indicate a substantial improvement in convergence rates.\n \n3. As for the experiments, firstly, this paper does not describe whether the experimental settings conform to the theoretical assumptions, so it is unclear whether the empirical studies can support the theoretical results. Secondly, the relationship between the complexity established in this paper and the number of worker nodes is not reflected in the experiments. Lastly, only Local SGDA was put into the experimental baseline set, and some algorithms that perform well in this field, such as FEDNEST and SAGDA, unfortunately did not become the experimental baselines.\n\nQUESTIONS:\nI have detailed the issues I identified as weaknesses in the previous message. Please address these questions I posed to them.\n\nLIMITATIONS:\nYes, they have.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studied a class of federated nonconvex minimax optimization problems. Authors proposed FL algorithms and provided sample complexity under three different settings including nonconvex-concave, nonconvex-strongly conconcave, and nonconvex-PL conditioned. Authors showed that derived rates has the best sample complexity. Experimental results demonstrated the superiority.\n\nSTRENGTHS:\n1. The theoretical analysis is one main contribution of this paper. It's nontrivial to derive sample complexity for different settings. \n2. Experiments also validated the efficacy of proposed algorithms. \n3. Code is available and looks neat.\n\nWEAKNESSES:\nI have a major concern regarding the discussion and comparison of existing works in the paper. Specifically, when addressing local algorithms, the authors solely focus on SGDA (with and without momentum), which is considered a classical and relatively old algorithm. However, it is worth noting that several recent works have demonstrated improvements in convergence rates. For instance, in [1], Section 4.2 provides insights into enhanced rates, while [2] presents Theorems C.9 that showcase accelerated convergence rates for convex-nonconcave minimax problems. Additionally, [3] introduces Theorem 4.1, which is relevant to this discussion. Although these works may not directly address federated learning, their findings on accelerated convergence rates for general minimax problems should be brought into the discussion to avoid confusion among readers in the future.\n\n### reference:\n1. Mahdavinia et.al., Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems\n2. He et.al., GDA-AM: ON THE EFFECTIVENESS OF SOLVING MIN-IMAX OPTIMIZATION VIA ANDERSON MIXING, ICLR 2022\n3  Lee et.al., Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems, Neurips 2021\n\nQUESTIONS:\nSee above\n\nLIMITATIONS:\nAuthors discussed limitations, although generic.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

comments: SUMMARY:
Authors propose federated stochastic gradient descent-ascent method for solving minimax problems in federated learning setting and demonstrate that oracle and communication complexity of their method is significantly better than that of analogues in nonconvex-(concave/non-concave/PL) cases with respect to dependence on desired accuracy.

STRENGTHS:
Theoretical analysis of the proposed method shows that its complexity is the best among alternatives, construction of the algorithm is simple enough, important practical cases of nonconvex and PL functions are considered.

WEAKNESSES:
I guess, within the goals of the paper there are no significant weaknesses: theoretical results are good and well-justified, minimal experiments confirm the advantages of the proposed method.

QUESTIONS:
I suggest authors to reflect deviation of accuracy for runs with different realisations of randomness on their figures in addition to particular realisation of convergence curve — for example, with shadow. Its important for understanding stability of advantage of the proposed method. And maybe the title should be changed to correctly reflect the content of the paper: "solving a class" seems to have unclear meaning, etc.

LIMITATIONS:
Everything is okay.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-12