NnUmg1chLL/NnUmg1chLL.md

Federated Minimax Optimization With Client Heterogeneity

Pranay Sharma pranaysh@andrew.cmu.edu Department of Electrical and Computer Engineering Carnegie Mellon University Rohan Panda rohanpan@andrew.cmu.edu Department of Electrical and Computer Engineering Carnegie Mellon University Gauri Joshi gaurij@andrew.cmu.edu Department of Electrical and Computer Engineering Carnegie Mellon University Reviewed on OpenReview: https: // openreview. net/ forum? id= NnUmg1chLL

Abstract

Minimax optimization has seen a surge in interest with the advent of modern applications such as GANs, and it is inherently more challenging than simple minimization. The difficulty is exacerbated by the training data residing at multiple edge devices or clients, especially when these clients can have heterogeneous datasets and heterogeneous local computation capabilities. We propose a general federated minimax optimization framework that subsumes such settings and several existing methods like Local SGDA. We show that naive aggregation of model updates made by clients running unequal number of local steps can result in optimizing a mismatched objective function - a phenomenon previously observed in standard federated minimization. To fix this problem, we propose normalizing the client updates by the number of local steps. We analyze the convergence of the proposed algorithm for classes of nonconvex-concave and nonconvex-nonconcave functions and characterize the impact of heterogeneous client data, partial client participation, and heterogeneous local computations.

For all the function classes considered, we significantly improve the existing computation and communication complexity results. Experimental results support our theoretical claims.

1 Introduction

The massive surge in machine learning (ML) research in the past decade has brought forth new applications that cannot be modeled as simple minimization problems. Many of these problems, including generative adversarial networks (GANs) Goodfellow et al. (2014); Arjovsky et al. (2017); Sanjabi et al. (2018), adversarial neural network training Madry et al. (2018), robust optimization Namkoong & Duchi (2016); Mohajerin Esfahani & Kuhn (2018), and fair machine learning Madras et al. (2018); Mohri et al. (2019), have an underlying min-max structure. However, the underlying problem is often nonconvex, while classical minimax theory deals almost exclusively with convex-concave problems.

Another feature of modern ML applications is the inherently distributed nature of the training data Xing et al. (2016). The data collection is often outsourced to edge devices or clients. However, the clients may then be unable (due to resource constraints) or unwilling (due to privacy concerns) to share their data with a central server. Federated Learning (FL) Konečn`y et al. (2016); Kairouz et al. (2019) was proposed to alleviate this problem. In exchange for retaining control of their data, the clients shoulder some of the computational load, and run part of the training process locally, using only their own data. The communication with the server is infrequent, leading to further resource savings. Since its introduction, FL has been an active area of research, with some remarkable successes Li et al. (2020); Wang et al. (2021). Research has shown practical Table 1: Comparison of (per client) stochastic gradient complexity and the number of communication rounds needed to reach an ϵ-stationary solution (Definition 1), for different classes of nonconvex minimax problems. Here, n is the total number of clients. For a fair comparison with existing works, our results in this table are specialized to the case when all clients (i) have equal weights (pi = 1/n), (ii) perform equal number of local updates (τi = τ ), and (iii) use the same local update algorithm SGDA. See Table 2 for comparison under more general settings, when (i)-(iii) do not hold.

Setting and Assumptions	Full Client Participation (FCP)
Work	System	Partial Client	Stochastic Gradient	Communication
Heterogeneitya	Participation	Complexity	Rounds
Nonconvex-Strongly-concave (NC-SC)/Nonconvex-Polyak-Łojasiewicz (NC-PL): Theorem 1
(n = 1) Lin et al. (2020a)	-	-	O(1/ϵ4 )	-
Sharma et al. (2022)	✗	✗	O(1/(nϵ4 ))	O(1/ϵ3 )
Yang et al. (2022a)	✗	✓	O(1/(nϵ4 ))	O(1/ϵ2 )
Ours: (Corollary 1.2, Remark 3)	✓	✓	O 1/(nϵ4 )	O 1/ϵ2
Nonconvex-Concave (NC-C): Theorem 2
(n = 1) Lin et al. (2020a)	-	-	O(1/ϵ8 )	-
Sharma et al. (2022)	✗	✗	O(1/(nϵ8 ))	O(1/ϵ7 )
Ours: (Corollary 2.2)	✓	✓	O 1/(nϵ8 )	O 1/ϵ4
Nonconvex-One-point-concave (NC-1PC): Theorem 2
Deng & Mahdavi (2021)	✗	✗	O(1/ϵ12)	O(n 1/6 /ϵ8 )
Sharma et al. (2022)	✗	✗	O(1/ϵ8 )	O(1/ϵ7 )
Ours: (Remark 5)	✓	✓	O 1/(nϵ8 )	O 1/ϵ4
a Individual clients can run an unequal number of local iterations, using different local optimizers (see Section 4)	.

benefits of, and provided theoretical justifications for commonly used practical techniques, such as, multiple local updates at the clients Stich (2018); Khaled et al. (2020); Koloskova et al. (2020); Wang & Joshi (2021), partial client participation Yang et al. (2021), communication compression Hamer et al. (2020); Chen et al.

(2021). Further, impact of heterogeneity in the clients' local data Zhao et al. (2018); Sattler et al. (2019), as well as their system capabilities Wang et al. (2020); Mitra et al. (2021) has been studied. However, all this research has been focused almost solely on simple minimization problems.

With its increasing usage in large-scale applications, FL systems must adapt to a wide range of clients. Data heterogeneity has received significant attention from the community. However, system-level heterogeneity remains relatively unexplored. The effect of client variability or heterogeneity can be controlled by forcing all the clients to carry out an equal number of local updates and utilize the same local optimizer Yu et al. (2019); Haddadpour et al. (2019). However, this approach is inefficient if the client dataset sizes are widely different.

Also, it would entail faster clients sitting idle for long durations Reisizadeh et al. (2022); Tziotis et al. (2022), waiting for stragglers to finish. Additionally, using the same optimizer might be inefficient or expensive for clients, depending on their system capabilities. Therefore, adapting to system-level heterogeneity forms a desideratum for real-world FL schemes. Contributions. We consider a general federated minimax optimization framework, in the presence of both inter-client data and system heterogeneity. System heterogeneity means the participating clients can run an unequal number of local steps, and utilize different local solvers. We consider the problem

$$\underset{\mathbf{x}\in {\mathbb{R}}^{dx}}{\min }\underset{\mathbf{y}\in \mathcal{Y}}{\max }\{F(\mathbf{x},\mathbf{y}):=\sum _{i=1}^n{p}_i{f}_i(\mathbf{x},\mathbf{y})\},\backslash operatorname*\{min\}\_\{\backslash mathbf\{x\}\backslash in\backslash mathbb\{R\}^\{d\; x\}\}\backslash operatorname*\{max\}\_\{\backslash mathbf\{y\}\backslash in\{\backslash mathcal\{Y\}\}\}\backslash left\backslash \{F(\backslash mathbf\{x\},\backslash mathbf\{y\}):=\backslash sum\_\{i=1\}^\{n\}p\_\{i\}f\_\{i\}(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash right\backslash \},$$ i=1 pifi(x, y)} , (1) where fiis the local loss of client i, piis the weight assigned to client i (e.g., the relative sample size), and n is the total number of clients. We study several classes of nonconvex minimax problems (see Table 1). Further,

• In our proposed algorithm, the participating clients may each perform different number of local steps, with different local optimizers. In this setting, naive aggregation of local model updates (as done in existing methods like Local Stochastic Gradient Descent Ascent) may lead to convergence in terms of a mismatched global objective (Corollaries 1.1, 2.1). We propose a simple normalization strategy to fix this problem.

$$(1)(1)$$

• We achieve order-optimal or state-of-the-art computation complexity and significantly improve the communication complexity of existing methods (Corollaries 1.2, 2.2).

• Under the special case where all the clients (i) are assigned equal weights pi = 1/n in (1), (ii) carry out equal number of local updates (τi = τ for all i), and (iii) utilize the same local-update algorithm, our results become directly comparable with existing work (see Table 1) and improve upon them as follows.

1. For nonconvex-strongly-concave (NC-SC - Corollary 1.2) and nonconvex-PL (NC-PL - Remark 3) problems, our method has the order-optimal gradient complexity O(1/(nϵ4)). Further, we improve the communication from O(1/ϵ3) in Sharma et al. (2022) to O(1/ϵ2).

1 2. For nonconvex-concave (NC-C - Corollary 2.2) and nonconvex-one-point-concave (NC-1PC - Remark 5) problems, we achieve state-of-the-art gradient complexity, while significantly improving the communication costs from O(1/ϵ7) in Sharma et al. (2022) to O(1/ϵ4). For NC-1PC functions, we prove the linear speedup in gradient complexity with n that was conjectured in Sharma et al. (2022).

3. As an intermediate result in our proof, we prove the theoretical convergence of Local SGD for one-point-convex function minimization (see Lemma C.5 in Appendix C.4). The achieved convergence rate is the same as that shown for convex minimization in the existing literature Khaled et al. (2020).

It is worth pointing out that our proof technique is different from existing minimax literature (e.g., Sharma et al. (2022); Yang et al. (2022b)). With all the clients carrying out the same number of local steps, the existing federated analyses rely on virtual sequences of average iterates, to mimic the proof steps in centralized settings Lin et al. (2020a); Yang et al. (2022c). In our case, since different clients run different number of local steps, this strategy is no longer viable (see Remark 9).

2 Related Work 2.1 Single-Client Minimax

Nonconvex-Strongly-concave (NC-SC). To our knowledge, Lin et al. (2020a) is the first work to analyze a single-loop algorithm for stochastic (and deterministic) NC-SC problems. Although the O(κ 3/ϵ4) complexity shown is optimal in ϵ, the algorithm required O(ϵ −2) batch-size. Qiu et al. (2020) utilized momentum to achieve O(ϵ −4) convergence with O(1) batch-size. Recent works Yang et al. (2022c); Sharma et al. (2022) achieve the same rate without momentum. Yang et al. (2022c) also improved the dependence on the condition number κ. Second-order stationarity for NC-SC has been recently studied in Luo & Chen (2021). Lower bounds for this problem class have appeared in Luo et al. (2020); Li et al. (2021); Zhang et al. (2021). Nonconvex-Concave (NC-C). Again, Lin et al. (2020a) was the first to analyze a single-loop algorithm for stochastic NC-C problems, proving O(ϵ −8) complexity. In deterministic problems, this has been improved using nested Nouiehed et al. (2019); Thekumparampil et al. (2019) as well as single-loop Xu et al. (2020); Zhang et al. (2020) algorithms. For stochastic problems, Rafique et al. (2021) and the recent work Zhang et al. (2022) improved the complexity to O(ϵ −6). However, both the algorithms have a nested structure, which at every step, solve a simpler problem iteratively. Achieving O(ϵ −6) complexity with a single-loop algorithm has so far proved elusive.

2.2 Distributed/Federated Minimax

Recent years have also seen an increasing body of work in distributed minimax optimization. Some of this work is focused on decentralized settings, as in Rogozin et al. (2021); Beznosikov et al. (2021b,c); Metelev et al. (2022).

Of immediate relevance to us is the federated setting, where clients carry out multiple local updates between successive communication rounds. The relevant works which focused on convex-concave problems include Reisizadeh et al. (2020); Hou et al. (2021); Liao et al. (2021); Sun & Wei (2022). Special classes of nonconvex 1The recent work Yang et al. (2022a) proposes FSGDA algorithm and also achieves O(1/ϵ2) communication cost for NC-PL functions. However, our work is more general since we allow different number of local steps and different local solvers at the clients.

minimax problems in the federated setting have been studied in recent works, such as, nonconvex-linear Deng et al. (2020), nonconvex-PL Deng & Mahdavi (2021); Xie et al. (2021), and nonconvex-one-point-concave Deng & Mahdavi (2021). The complexity guarantees for several function classes considered in Deng & Mahdavi (2021) were further improved in Sharma et al. (2022). However, all these works consider specialized federated settings, either assuming full-client participation, or system-wise identical clients, each carrying out equal number of local updates. As we see in this paper, partial client participation is the most source of error in simple FL algorithms. Also, system-level heterogeneity can have crucial implications on the algorithm performance.

Comparison with Wang et al. (2020); Sharma et al. (2022); Yang et al. (2022a). Wang et al.

(2022a) was, to our knowledge, the first work to consider the problem of system heterogeneity in simple minimization problems, and proposed a normalized averaging scheme to avoid optimizing an inconsistent objective. Compared to Wang et al. (2020), we consider a more challenging problem and achieve higher communication savings (Table 1) 2. Sharma et al. (2022) studied minimax problems in the federated setting but assumed an equal number of SGDA-like local updates, with full client participation. The recent work Yang et al. (2022a) considers NC-SC problem with full and partial client participation and achieves similar communication savings as ours. In comparison, our work considers a more general minimax FL framework with partial client participation, clients running an unequal number of local updates, and using different local solvers. Further, we analyze multiple classes of nonconvex-concave and nonconvex-nonconcave functions, improving the communication and computation complexity of existing minimax methods.

3 Preliminaries

Notations. We let ∥·∥ denote the Euclidean norm ∥·∥2. Given a positive integer m, the set {1, 2*, . . . , m*} is denoted by [m]. Vectors at client i are denoted with subscript i, e.g., xi, while iteration indices are denoted using superscripts, e.g., y (t) or y (t,k). Given a function g, we define its gradient vector as -∇xg(x, y) ⊤, ∇yg(x, y) ⊤⊤, and its stochastic gradient as ∇g(x, y; ξ), where ξ denotes the randomness.

Convergence Metrics. In the presence of nonconvexity, we can only prove convergence to an approximate stationary point, which is defined next.

Definition 1 (ϵ-Stationarity). A point x is an ϵ-stationary point of a differentiable function g if ∥∇g(x)∥ ≤ ϵ. Definition 2. Stochastic Gradient (SG) complexity is the total number of gradients computed by all the clients during the course of the algorithm.

In special cases, where all the clients are weighted equally (pi = 1/n, for all i ∈ [n]) and carry out equal number of local steps τ , we state the per-client gradient complexity for comparison with existing work. See Table 1 and Corollaries 1.2 and 2.2.

Definition 3 (Communication Rounds). During a single communication round, the server sends its global model to a set of clients, which carry out multiple local updates starting from the same model, and return their local vectors to the server. The server then aggregates these local vectors to arrive at a new global model. Throughout this paper, we denote the number of communication rounds by T.

Next, we discuss some assumptions used in the paper.

Assumption 1 (Smoothness). Each local function fiis differentiable and has Lipschitz continuous gradients.

That is, there exists a constant Lf > 0 such that at each client i ∈ [n], for all x, x ′ ∈ R d1 and y, y ′ ∈ Y, ∥∇fi(x, y) − ∇fi(x ′, y ′)∥ ≤ Lf ∥(x, y) − (x ′, y ′)∥ .

Assumption 2 (Bounded Diameter). The constraint set Y is convex and bounded.

2Under the conditions pi = 1*/n, τ*i = τ for all i, for smooth minimization problems, Wang et al. (2020) requires O(1/ϵ3) communication rounds. For NC-SC problems (a harder problem class), we show an improved O(1/ϵ2) communication rounds. Assumption 3 (Local Variance). The stochastic gradient oracle at each client is unbiased. Also, there exist constants σL, βL ≥ 0 such that at each client i ∈ [n], for all x, y,

$${\mathbb{E}}_{{\xi }_i}[\mathrm{\nabla }{f}_i(\mathbf{x},\mathbf{y};{\xi }_i)]=\mathrm{\nabla }{f}_i(\mathbf{x},\mathbf{y}),\backslash mathbb\{E\}\_\{\backslash xi\_\{i\}\}[\backslash nabla\; f\_\{i\}(\backslash mathbf\{x\},\backslash mathbf\{y\};\backslash xi\_\{i\})]=\backslash nabla\; f\_\{i\}(\backslash mathbf\{x\},\backslash mathbf\{y\}),$$ $\mathbb{E}{\xi{i}}\left|\nabla f_{i}(\mathbf{x},\mathbf{y};\xi_{i})-\nabla f_{i}(\mathbf{x},\mathbf{y})\right|^{2}\leq\beta_{L}^{2}\left|\nabla f_{i}(\mathbf{x},\mathbf{y})\right|^{2}+\sigma_{L}^{2}$.
Assumption 4 (Global Heterogeneity). For any set of non-negative weights {wi} n i=1 such that Pn i=1 wi = 1, there exist constants βG ≥ 1, σG ≥ 0 such that for all x, y,

$$\sum _{i=1}^n{w}_i{\parallel {\mathrm{\nabla }}_x{f}_i(\mathbf{x},\mathbf{y})\parallel }^2\le {\beta }_G^2{\parallel \sum _{i=1}^n{w}_i{\mathrm{\nabla }}_x{f}_i(\mathbf{x},\mathbf{y})\parallel }^2+{\sigma }_G^2,\backslash sum\_\{i=1\}^\{n\}w\_\{i\}\backslash left\backslash |\backslash nabla\_\{x\}f\_\{i\}\backslash left(\backslash mathbf\{x\},\backslash mathbf\{y\}\backslash right)\backslash right\backslash |^\{2\}\backslash leq\backslash beta\_\{G\}^\{2\}\backslash left\backslash |\backslash sum\_\{i=1\}^\{n\}w\_\{i\}\backslash nabla\_\{x\}f\_\{i\}\backslash left(\backslash mathbf\{x\},\backslash mathbf\{y\}\backslash right)\backslash right\backslash |^\{2\}+\backslash sigma\_\{G\}^\{2\},$$ $$\sum_{i=1}^{n}w_{i}\left|\nabla_{y}f_{i}\left(\mathbf{x},\mathbf{y}\right)\right|^{2}\leq\beta_{G}^{2}\left|\sum_{i=1}^{n}w_{i}\nabla_{y}f_{i}\left(\mathbf{x},\mathbf{y}\right)\right|^{2}+\sigma_{G}^{2}.$$

If all fi's are identical, we have βG = 1, and σG = 0. Most existing work uses simplified versions of Assumptions 3, 4, assuming βL = 0 and/or βG = 0.

4 Algorithm For Heterogeneous Federated Minimax Optimization

In this section, we propose a federated minimax algorithm to handle system heterogeneity across clients.

4.1 Limitations Of Local Sgda

Following the success of FedAvg McMahan et al. (2017) in FL, Deng & Mahdavi (2021) was the first to explore a simple extension Local stochastic gradient descent-ascent (SGDA) in minimax problems. Between successive communication rounds, clients take multiple simultaneous descent/ascent steps to respectively update the min-variable x and max-variable y. Subsequent work in Sharma et al. (2022) improved the convergence results and showed that LocalSGDA achieves optimal gradient complexity for several classes of nonconvex minimax problems. However, existing work on LocalSGDA also assumes the participation of all n clients in every communication round. More crucially, as observed with simple minimization problems Wang et al. (2020), if clients carry out an unequal number of local updates, or if their local optimizers are not all the same, LocalSGDA (like FedAvg) might converge to the stationary point of a different objective.

This is further discussed in Sections 5.1 and 5.2, and illustrated in Figure 1, where the learning process gets disproportionately skewed towards the clients carrying out more local updates.

Figure 1: FedAvg with heterogeneous local updates. The green (red) triangle represents the local optimizer of f1(f2), while (x ∗, y ∗) is the global optimizer. The number of local updates at the clients is τ1 = 2, τ2 = 5. Generalized Local SGDA Update Rule. To understand this mismatched convergence phenomenon with naive aggregation in local SGDA, recall that Local SGDA updates are of the form

$${\mathbf{x}}^{(t+1)}={\mathbf{x}}^{(t)}+{\gamma }_x^s\sum _{i=1}^n{p}_i{\mathrm{\Delta }}_{\mathbf{x},i}^{(t)},{\mathbf{y}}^{(t+1)}={\mathbf{y}}^{(t)}+{\gamma }_y^s\sum _{i=1}^n{p}_i{\mathrm{\Delta }}_{\mathbf{y},i}^{(t)},\{\backslash bf\; x\}^\{(t+1)\}=\{\backslash bf\; x\}^\{(t)\}+\backslash gamma\_\{x\}^\{s\}\backslash sum\_\{i=1\}^\{n\}p\_\{i\}\backslash Delta\_\{\{\backslash bf\; x\},i\}^\{(t)\},\backslash qquad\{\backslash bf\; y\}^\{(t+1)\}=\{\backslash bf\; y\}^\{(t)\}+\backslash gamma\_\{y\}^\{s\}\backslash sum\_\{i=1\}^\{n\}p\_\{i\}\backslash Delta\_\{\{\backslash bf\; y\},i\}^\{(t)\},$$

where γ s x , γs y are the server learning rates, ∆ (t) x,i = 1 ηcx x (t,τ (t) i) i − x (t), ∆ (t) y,i = 1 ηcy y (t,τ (t) i) i − y (t)are the scaled local updates. x (t,τ (t) i) iis the iterate at client i after taking τ (t) ilocal steps, and η c x , ηcy are the client learning rates. Let us consider a generalized version of this update rule where ∆ (t) x,i, ∆ (t) y,i are linear combinations of local stochastic gradients computed by client i, as ∆ (t) y,i =Pτ (t) i −1 k=0 a (t,k) i ∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i), where a (t,k) i ≥ 0. Commonly used client optimizers, such as, SGD, local momentum, variable local learning rates can be accommodated in this general form (see Appendix A.1 for some examples). For this more general form, we can rewrite the x, y updates at the server as follows

x (t+1) = x (t) − γ s x Pn i=1 piG (t) x,i a¯ (t) i ∥a¯ (t) i ∥1 ∥a¯ (t) i∥1 = x (t) − Xn j=1 pj∥a¯ (t) j ∥1 γ s x Xn i=1 pi∥a¯ (t) i ∥ P 1 n j=1 pj∥a¯ (t) j ∥1 | {z } wi G (t) x,ia¯ (t) i , ∥a¯ (t) i ∥1 | {z } g (t) x,i | {z } τ (t) eff y (t+1) = y (t) + τ (t) eff γ s y Pn i=1 wig (t) y,i, (2) $\frac{1}{2}$ . where G (t) x,i = [∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i)]τ (t) i k=0 ∈ R dx×τ (t) i contains the τ (t) istochastic gradients stacked columnwise, a¯ (t) i = [a t,0 i, a t,1 i*, . . . , a* t,τ (t) i −1 i] ⊤, g (t) x,i, g (t) y,i are the normalized aggregates of the stochastic gradients and τ (t) eff is the effective number of local steps. Note that for simplicity, we assume that the constraint set Y has a large diameter. However, our algorithm can be easily modified to accommodate projection steps. Similar to

Figure 2: Generalized update rule in (2). Note that (g (t) x,i, g (t) y,i) = 1 τi (∆(t) x,i, ∆ (t) y,i). Also, at the server, the weighted sum Pn i=1 wig (t) x,i gets scaled by τ (t) eff .

the observation for simple minimization problems in Wang et al. (2020), we see in Theorems 1, 2 that the resulting iterates of this general algorithm end up converging to the stationary point of a different objective Fe =Pn i=1 wifi. Further, in Corollary 1.1, we observe that this mismatch is a result of using weights wiin (2) to weigh the clients' contribution.

4.2 Proposed Normalized Federated Minimax Algorithm

From the generalized update rule, we can see that setting the weights wi equal to pi will ensure that the surrogate objective F˜ matches with the original global objective F. Setting wi = pi results in normalization Algorithm 1 Fed-Norm-SGDA and Fed-Norm-SGDA+ 1: Input: initialization x (0), y (0), Number of communication rounds T, learning rates: client {η c x , ηcy}, server {γ s x , γs y}, #local-updates {τ (t) i}i,t, S, s = −1 2: for t = 0 to T − 1 do 3: Server selects client set C (t); sends them (x (t), y (t)) 4: if t mod S = 0 then 5: s ← s + 1 6: Server sends xb (s) = x (t)to clients in C (t) 7: end if 8: x (t,0) i = x (t), y (t,0) i = y (t)for i ∈ C(t) 9: for k = 0*, . . . , τ* (t) i − 1 do 10: x (t,k+1) i = x (t,k) i − η c xa (t,k) i ∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i) 11: y (t,k+1) i = y (t,k) i + η c ya (t,k) i ∇yfi(xb (s), y (t,k) i; ξ (t,k) i) # y-update for Fed-Norm-SGDA+ 16: g (t) y,i =Pτ (t) i −1 k=0 a (t,k) i ∥a¯ (t) i ∥1 ∇yfi(xb (s), y (t,k) i; ξ (t,k) i) 17: g (t) y,i =Pτ (t) i −1 k=0 a (t,k) i ∥a¯ (t) i ∥1 ∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i) 12: y (t,k+1) i = y (t,k) i + η c ya (t,k) i ∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i) # y-update for Fed-Norm-SGDA 13: end for 14: Client i aggregates its gradients to compute g (t) x,i, g (t) y,i 15: g (t) x,i =Pτ (t) i −1 k=0 a (t,k) i ∥a¯ (t) i ∥1 ∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i) 18: Clients i ∈ C(t)communicate {g (t) x,i, g (t) y,i} to the server 19: Server computes aggregate vectors {g (t) x , g (t) y } using (3) 20: Server step: nx (t+1) = x (t) − τ (t) eff γ s xg (t) x , y (t+1) = y (t) + τ (t) eff γ s yg (t) y 21: end for 22: Return: x¯ (T) drawn uniformly at random from {x (t)} T t=1 of the local progress at each client before their aggregation at the server. As a result, we can preserve convergence to a stationary point of the original objective function F, even with heterogeneous {τ (t) i}, as we see in Theorem 1 and Theorem 2.

The algorithm follows the steps given in Algorithm 1. In each communication round t, the server selects a client set C (t) and communicates its model parameters (x (t), y (t)) to these clients. The selected clients then run multiple local stochastic gradient steps. The number of local steps {τ (t) i} can vary across clients and across rounds. At the end of τ (t) ilocal steps, client i aggregates its local stochastic gradients into {g (t) x,i, g (t) y,i}, which are then sent to the server. Note that the gradients at client i, {∇fi(·, ·; ξ (t,k) i)} τ (t) i k=0, are normalized by ∥a¯ (t) i∥1 , where a¯ (t) i = [a t,0 i, a t,1 i*, . . . , a* t,τ (t) i −1 i] ⊤ is the vector of weights assigned to individual stochastic gradients in the local updates.3 The server aggregates these local vectors to compute global direction estimates g (t) x , g (t) y , which are then used to update the server model parameters (x (t), y (t)).

3For LocalSGDA Deng & Mahdavi (2021); Sharma et al. (2022), a (t,k) i = 1 for all i ∈ [n], t ∈ [T], k ∈ [τ (t) i] and ∥a¯ (t) i ∥1 = τ (t) i.

Therefore, g (t) x,i, g (t) y,i are simply the average of the stochastic gradients computed in the t-th round. Client Selection. In each round t, the server samples |C(t)| clients uniformly at random without replacement (WOR). While aggregating client updates at the server, client i update is weighed by w˜i = win/|C(t)|, i.e.,

$$\begin{array}{cccc}& {\mathbf{g}}_{\mathbf{x}}^{(t)}=\sum _{i\in {\mathcal{C}}^{(t)}}{\tilde{w}}_i{\mathbf{g}}_{\mathbf{x},i}^{(t)},{\mathbf{g}}_{\mathbf{y}}^{(t)}=\sum _{i\in {\mathcal{C}}^{(t)}}{\tilde{w}}_i{\mathbf{g}}_{\mathbf{y},i}^{(t)}\mathrm{.}& & \text{(1)}\end{array}\backslash mathbf\{g\}\_\{\backslash mathbf\{x\}\}^\{(t)\}=\backslash sum\_\{i\backslash in\backslash mathcal\{C\}^\{(t)\}\}\backslash tilde\{w\}\_\{i\}\backslash mathbf\{g\}\_\{\backslash mathbf\{x\},i\}^\{(t)\},\backslash qquad\backslash mathbf\{g\}\_\{\backslash mathbf\{y\}\}^\{(t)\}=\backslash sum\_\{i\backslash in\backslash mathcal\{C\}^\{(t)\}\}\backslash tilde\{w\}\_\{i\}\backslash mathbf\{g\}\_\{\backslash mathbf\{y\},i\}^\{(t)\}.\backslash tag\{1\}$$ $\quad(3)^{\frac{1}{2}}$ . Note that EC(t) [g $\mathbf{g}^{(t)}=\sum_{i=1}^{n}w_{i}\mathbf{g}^{(t)}{\mathbf{x},i},\mathbf{E}{\mathcal{C}^{(t)}}[\mathbf{g}^{(t)}{\mathbf{y}}]=\sum{i=1}^{n}w_{i}\mathbf{g}^{(t)}_{\mathbf{y},i}$.

5 Convergence Results

Next, we present the convergence results for different classes of nonconvex minimax problems. For simplicity, throughout this section we assume the parameters utilized in Algorithm 1 to be fixed across t. Therefore, a (t,k) i ≡ a (k) i, a¯ (t) i ≡ ai, τ (t) i ≡ τi, τ (t) eff ≡ τeff and |C(t)| = P, for all t.

5.1 Non-Convex-Strongly-Concave (Nc-Sc) Case

Assumption 5 (µ-Strong-concavity (SC) in y). A function f is µ-strong concave (µ > 0) in y if

$-f({\bf x},\bar{\bf y})\geq-f({\bf x},\bar{\bf y})-\langle\nabla_{y}f({\bf x},\bar{\bf y}),\bar{\bf y}-\bar{\bf y}\rangle+\frac{\mu}{2}\big{|}\bar{\bf y}-\bar{\bf y}\big{|}^{2},\qquad\mbox{for all${\bf x}\in\mathbb{R}^{d_{x}}$,and$\bar{\bf y},\bar{\bf y}\in\mathbb{R}^{d_{y}}$.}$ General Convergence Result. We first show that the iterates of Algorithm 1 converge to the stationary point of a surrogate objective Fe, where Fe(x, y) ≜Pn i=1 wifi(x, y). {wi} n i=1 are the aggregation weights used by the server (Line 19). See Appendix B for the full statement and proof.

Theorem 1. Suppose the local loss functions {fi}i satisfy Assumptions 1, 2, 3, 4, 5. Suppose the server selects |C(t)| = P clients in each round t*. Given appropriate choices of client and server learning rates,* (η c x , ηcy ) and (γ s x , γs y ) respectively (see Appendix B.2), the iterates generated by Fed-Norm-SGDA satisfy

$$\begin{array}{cccc}& \underset{t\in [T]}{\min }\mathbb{E}\mathrm{\parallel }\mathrm{\nabla }\tilde{\mathrm{\Phi }}({\mathbf{x}}^{(t)}){\mathrm{\parallel }}^2\le \underset{\text{Parallel transformation}}{\underbrace{\mathcal{O}\left({\kappa }^2{\sigma }_G\sqrt{\frac{n-P}{n-1}\frac{E_{\mathrm{\infty }}}{PT}}\right)}}+\underset{\text{Error with full approximation}}{\underbrace{\mathcal{O}\left({\kappa }^2\sqrt{\frac{{\mathrm{\Delta }}_{\tilde{\mathbf{x}}}+A_{\mathbf{x}}{\sigma }_{\mathbf{x}}^2+B_{\mathbf{x}}{\beta }_{\mathbf{x}}^2{\sigma }_G^2}{P{\tau }_{G}T}}\right)}}+\underset{\text{Local applications}}{\underbrace{\mathcal{O}\left({\kappa }^2\frac{C_{\mathbf{w}}{\sigma }_{\mathbf{x}}^2+D{\sigma }_{\mathbf{w}}^2}{P{T}^{2}T}\right)}}\mathrm{.}& & \text{(4)}\end{array}\backslash min\_\{t\backslash in[T]\}\backslash mathbb\{E\}\backslash |\backslash nabla\backslash tilde\{\backslash Phi\}(\backslash mathbf\{x\}^\{(t)\})\backslash |^\{2\}\backslash leq\backslash underbrace\{\backslash mathcal\{O\}\backslash left(\backslash kappa^\{2\}\backslash sigma\_\{G\}\backslash sqrt\{\backslash frac\{n-P\}\{n-1\}\backslash frac\{E\_\{\backslash infty\}\}\{PT\}\}\backslash right)\}\_\{\backslash text\{Parallel\; transformation\}\}+\backslash underbrace\{\backslash mathcal\{O\}\backslash left(\backslash kappa^\{2\}\backslash sqrt\{\backslash frac\{\backslash Delta\_\{\backslash tilde\{\backslash mathbf\{x\}\}\}+A\_\{\backslash mathbf\{x\}\}\backslash sigma\_\{\backslash mathbf\{x\}\}^\{2\}+B\_\{\backslash mathbf\{x\}\}\backslash beta\_\{\backslash mathbf\{x\}\}^\{2\}\backslash sigma\_\{G\}^\{2\}\}\{P\backslash tau\_\{G\}T\}\}\backslash right)\}\_\{\backslash text\{Error\; with\; full\; approximation\}\}+\backslash underbrace\{\backslash mathcal\{O\}\backslash left(\backslash kappa^\{2\}\backslash frac\{C\_\{\backslash mathbf\{w\}\}\backslash sigma\_\{\backslash mathbf\{x\}\}^\{2\}+D\backslash sigma\_\{\backslash mathbf\{w\}\}^\{2\}\}\{PT^\{2\}T\}\backslash right)\}\_\{\backslash text\{Local\; applications\}\}.\backslash tag\{4\}$$ $$\left(4\right)\backslash left(4\backslash right)$$

where, κ = Lf /µ is the condition number, Φe(x) ≜ maxy Fe(x, y) is the envelope function, ∆Φe ≜ Φe(x (0)) − minx Φe(x), τ¯ = 1 n Pn i=1 τi, τeff =Pn i=1 pi∥ai∥1 , Aw ≜ nτeff Pn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 , Bw ≜ nτeff maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 , Cw ≜ Pn i=1 wi(∥ai∥ 2 2 − [α (t,τi−1) i] 2), D ≜ maxi(β 2 L ∥ai,−1∥ 2 2 + ∥ai,−1∥ 2 1 ), where ai,−1 ≜ [a (0) i, a (1) i*, . . . , a* (τi−2) i] ⊤ and Ew ≜ n maxi wi.

Remark 1. The first term in (4) results from client subsampling (P < n). This explains its dependence on the data heterogeneity σG. The second term represents the optimization error for a centralized algorithm (see Appendix C.3 in Lin et al. (2020a)). The last term represents client-drift, the error if the client(s) run multiple local updates.

Theorem 1 states convergence for a surrogate objective Fe. Next, we see convergence for the true objective F.

Corollary 1.1 (Convergence in terms of F). Given Φ(x) ≜ maxy F(x, y), under the conditions of Theorem 1,

$$\begin{array}{cccc}& \underset{t\in [H]}{\min }{\parallel \mathrm{\nabla }\mathrm{\Phi }({\mathbf{x}}^{(t)})\parallel }^2\le 2(2{\chi }_{\mathbf{p}\mathrm{\mid }\mathbf{w}}^2{\sigma }_H^2+1){ϵ}_{opt}+4{\chi }_{\mathbf{p}\mathrm{\mid }\mathbf{w}}^2{\sigma }_G^2+\frac{4L_t^2}{T}\sum _{t=0}^{T-1}{\parallel {\mathbf{y}}^{\ast }({\mathbf{x}}^{(t)})-{\tilde{\mathbf{y}}}^{\ast }({\mathbf{x}}^{(t)})\parallel }^2\mathrm{.}& & \text{(5)}\end{array}\backslash min\_\{t\backslash in[H]\}\backslash left\backslash |\backslash nabla\backslash Phi(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq2\backslash left(2\backslash chi\_\{\backslash mathbf\{p\}|\backslash mathbf\{w\}\}^\{2\}\backslash sigma\_\{H\}^\{2\}+1\backslash right)\backslash epsilon\_\{opt\}+4\backslash chi\_\{\backslash mathbf\{p\}|\backslash mathbf\{w\}\}^\{2\}\backslash sigma\_\{G\}^\{2\}+\backslash frac\{4L\_\{t\}^\{2\}\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash left\backslash |\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\}^\{(t)\})-\backslash widetilde\{\backslash mathbf\{y\}\}^\{*\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}.\backslash tag\{5\}$$

where χ 2 p∥w ≜Pn i=1 (pi−wi) 2 wi, ϵopt ≜ 1 T PT −1 t=0 ∇Φ( e x (t))

2 denotes the optimization error in (4). If pi = wi for all i ∈ [n], then χ 2 p∥w = 0. Also, then Fe(x, y) ≡ F(x, y). Therefore, y ∗(x) = arg maxy F(x, y) and ye ∗(x) = arg maxy Fe(x, y) are identical, for all x*. Hence,* (5) yields mint∈[T] ∇Φ(x (t)) 2≤ 2ϵopt.

It follows from Corollary 1.1 that if in Algorithm 1, the server aggregation weights {wi} (Line 19) are the same as {pi}, we get convergence to a stationary point of the true objective F. For the rest of this subsection, we assume wi = pi for all i ∈ [n].

Table 2: Comparison of convergence rates of Fed-Norm-SGDA (Theorem 1) and Fed-Norm-SGDA+ (Theorem 2), if all the clients run SGDA/SGDA+ based local updates, i.e., a (t,k) i = 1, for all i, k, t. The results are stated for (i) (pi = 1*/n, τ*i = τ, ∀ i ∈ [n]); and (ii) (pi ̸= pj ), (τi ≠ τj ). The additional factors in (ii) relative to (i) are highlighted in blue. We state the results under partial-client participation (PCP). FCP results follow by choosing P = n. For simplicity, we assume uniformly bounded local variance (βL = 0 in Assumption 3).

Nonconvex-Strongly-concave (NC-SC)/Nonconvex-Polyak-Łojasiewicz (NC-PL): (Theorem 1, Remark 3) System Setting Convergence Rate κ 2σ 2 2nτ Sharma et al. (2022) with P = n: O √ L + κ T σ L + σ 2 2 nτT G 1 pi = n , ∀ i ∈ [n] τi = τ, ∀ i ∈ [n] 2 2 σ P 1 L 2 i Yang et al. (2022a) with P < n: O √ G 1 − + √ P T h 1 + σ τ + σ G P T n 2 2 Ours with P < n: O κ 2σG q (n−P ) √ σL κ κ h σ L 2 i (n−1)P T + P τT + T τ + σ G Ours: κ q(n−P )n maxi pi 2 p 2 κ σ 2 Pn piτi 2 τ 2 pi ̸= pj , τi ̸= τj O 2σG (n−1)P T + κ √ σL q nτeff Pn i + L i=1 τ¯ + σ Gmaxi i 2 P τeffT i=1 τi T τ¯ τ¯ τ¯ = 1 Pn n i=1 τi Nonconvex-Concave (NC-C)/Nonconvex-One-Point-Concave (NC-1PC): (Theorem 2, Remark 5) System Setting Convergence Rate 3/2 1 + O (nτ) Sharma et al. (2022) with P = n: O 1/4 √ T (τP T) pi = 1 , ∀ i ∈ [n] n τi = τ, ∀ i ∈ [n]   Ours with P < n: √ σG n−P √ σL O n−1 1 P T 1/4 + 1 τP T 1/4 + (τP ) 1/4 2 1 + τ n−P −1/4 + O 1 h σ L 2 i O τ + (G2 x + σ G) T 3/4 n−1 T 3/4  2 √ σG n−P n maxi pi P T 1/4 + √ σL nτeff Pn p i 1/4 8 q 1 + n 2 O n−1 i=1 P ∥p∥ 2 τeffP T τi (τeffP ) 1/4 2 −1/4 +O 1 + n ∥p∥ 2 nτeff Pn p i + τeff n−P n maxi pi T 3/4 P 2 i=1 τi n−1 Ours: pi ̸= pj , τi ̸= τj 1 Pn τ¯ = i=1 τi n 1 σ Pn piτi τ 2 +O 2 i=1 2 + (G2 x + σ G)maxi 2 i T 3/4 L τ¯ τ¯ 2

In Table 2, we specialize the bound in (4) to SGDA-based local updates. We compare the bound under two cases: Case 1: equally-weighted clients (pi = 1/n, for all i), all running τi ≡ τ local updates; and Case 2: unequally weighted clients (pi ̸= pj ), running unequal local updates (τi ̸= τj ). The setting in Case 1 has previously been considered in Sharma et al. (2022) (under full participation) and Yang et al. (2022a) 4.

Compared to Sharma et al. (2022), our bound has a smaller local-updates error term. This results in improved communication cost (see Corollary 1.2). The additional factors going from Case 1 to the more general Case 2 are highlighted in blue. The following insights can be drawn from Table 2.

• Partial Client Participation Error: O σG √P T is the most significant component of convergence error.

Unlike the other two errors, it does not decrease with increasing local updates τeff. Consequently, we do not observe communication savings by performing multiple local updates at the clients. It remains an open problem to achieve speedup in terms of local updates in partial participation settings.

• Unequal client weights: if the clients are weighted disparately, we observe an increase in the stochastic gradient complexity. To see this, let τi ≡ τ . The resulting bound is O σG q(n−P )n∥p∥∞ (n−1)P T + σL √n∥p∥ √ 2 P τeffT+ 1 T [ σ 2 L τ¯ +σ 2 G] . Since ∥p∥∞ , ∥p∥2 ≤ 1, in the worst case (when only one of the clients has all the weight), the

4The condition number κ dependence is not explicitly stated in the results in Yang et al. (2022a). complexity is worse by a factor of n. This happens because the client sampling is not done in proportion to their weights. Rather, the server first samples the clients uniformly, and then scales their updates to get an unbiased estimator (3). We leave exploring non-uniform WOR sampling further as a future direction.

Corollary 1.2 (Improved Communication Cost). *Suppose all the clients are weighted equally (*pi = 1/n for all i), with each carrying out τ local steps of SGDA. Further, assume Φ is bounded from below. Then, to reach x such that E∥∇Φ(x)∥ ≤ ϵ,

• Under full participation, the per-client gradient complexity of Fed-Norm-SGDA is T τ = O(κ 4/(nϵ4)). The number of communication rounds required is T = O(κ 2/ϵ2).

• Under partial participation, the per-client gradient complexity of Fed-Norm-SGDA is O(κ 4/(P ϵ4)). In general, running multiple local updates does not yield any communication savings. However, in the special case when inter-client data heterogeneity σG = 0, the communication cost is O(κ 2/ϵ2).

Remark 2. The gradient complexity in Corollary 1.2 is optimal in ϵ, and achieves linear speedup in the number of participating clients. The communication complexity improves the corresponding results in Deng & Mahdavi (2021); Sharma et al. (2022). We match the communication cost in the recent work Yang et al.

(2022a). In addition, our work considers a more general FL setting with unequally weighted clients (pi ̸= pj ), running unequal local updates (τi ̸= τj ), using distinct local solvers (ai ̸= aj ).

Extending The Results To Nonconvex-Pl Functions

Assumption 6. A function f satisfies µ-PL condition in y (µ > 0), if for any fixed x: 1) maxy′ f(x, y ′) has a nonempty solution set; and 2) for all y

$$\mathrm{\parallel }{\mathrm{\nabla }}_{y}f(\mathbf{x},\mathbf{y}){\mathrm{\parallel }}^2\ge 2\mu (\underset{{\mathbf{y}}^{\mathrm{\prime }}}{\max }f(\mathbf{x},{\mathbf{y}}^{\mathrm{\prime }})-f(\mathbf{x},\mathbf{y}))\mathrm{.}\backslash |\backslash nabla\_\{y\}f(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash |^\{2\}\backslash geq2\backslash mu(\backslash operatorname*\{max\}\_\{\backslash mathbf\{y\}^\{\backslash prime\}\}f(\backslash mathbf\{x\},\backslash mathbf\{y\}^\{\backslash prime\})-f(\backslash mathbf\{x\},\backslash mathbf\{y\})).$$

Remark 3. If Assumptions 1, 2, 3, 4 hold, and the global function F satisfies Assumption 6, then for appropriately chosen learning rates (Appendix B.5), the bound in Theorem 1 holds.

5.2 Non-Convex-Concave (Nc-C) Case

In this subsection, we consider smooth nonconvex functions which satisfy the following assumptions.

Assumption 7 (Concavity). The function f is concave in y if for a fixed x ∈ R d1, for all y, y ′ ∈ R d2, f(x, y) ≤ f(x, y ′) + ⟨∇yf(x, y ′), y − y ′⟩.

Assumption 8 (Lipschitz continuity in x). Given a function f, there exists a constant Gx, such that for each y ∈ R d2, and all x, x ′ ∈ R d1,

$$\mathrm{\parallel }f(\mathbf{x},\mathbf{y})-f({\mathbf{x}}^{\mathrm{\prime }},\mathbf{y})\mathrm{\parallel }\le G_{\mathbf{x}}\mathrm{\parallel }\mathbf{x}-{\mathbf{x}}^{\mathrm{\prime }}\mathrm{\parallel }\mathrm{.}\backslash |f(\backslash mathbf\{x\},\backslash mathbf\{y\})-f(\backslash mathbf\{x\}^\{\backslash prime\},\backslash mathbf\{y\})\backslash |\backslash leq\; G\_\{\backslash mathbf\{x\}\}\backslash ,\backslash |\backslash mathbf\{x\}-\backslash mathbf\{x\}^\{\backslash prime\}\backslash |\backslash ,.$$

The envelope function Φ(x) = maxy f(x, y) used so far, may no longer be smooth in the absence of a unique maximizer. However, Φ(·) is weakly convex (Lin et al., 2020a, Lemma 4.7). Therefore, we use the alternate definition of stationarity, proposed in Davis & Drusvyatskiy (2019), utilizing the Moreau envelope of Φ.

Definition 4 (Moreau Envelope). The function ϕλ is the λ-Moreau envelope of ϕ, for λ > 0, if for all x ∈ R dx ,

$${\varphi }_{\lambda }(\mathbf{x})=\underset{{\mathbf{x}}^{\mathrm{\prime }}}{\min }\varphi ({\mathbf{x}}^{\mathrm{\prime }})+\frac{1}{2\lambda }{\parallel {\mathbf{x}}^{\mathrm{\prime }}-\mathbf{x}\parallel }^2\mathrm{.}\backslash phi\_\{\backslash lambda\}(\{\backslash bf\; x\})=\backslash operatorname*\{min\}\_\{\{\backslash bf\; x\}^\{\backslash prime\}\}\backslash phi(\{\backslash bf\; x\}^\{\backslash prime\})+\backslash frac\{1\}\{2\backslash lambda\}\backslash left\backslash |\{\backslash bf\; x\}^\{\backslash prime\}-\{\backslash bf\; x\}\backslash right\backslash |^\{2\}.$$

Drusvyatskiy & Paquette (2019) showed that a small ∥∇ϕλ(x)∥ indicates the existence of some point xe in the vicinity of x, that is nearly stationary for ϕ. Hence, in our case, we focus on minimizing ∥∇Φλ(x)∥.

Proposed Algorithm. For nonconvex-concave functions, we use Fed-Norm-SGDA+. The x-updates are identical to Fed-Norm-SGDA. For the y updates however, the clients compute stochastic gradients ∇yfi(xb (s), y (t,k) i; ξ (t,k) i) keeping the x-component fixed at xb (s)for S communication rounds. This trick, originally proposed in Deng & Mahdavi (2021), gives the analytical benefit of a double-loop algorithm (which updates y several times before updating x once) while also updating x simultaneously. Theorem 2. Suppose the local loss functions {fi} satisfy Assumptions 1, 2, 3, 4, 7, 8, the y iterates are bounded, and the server selects |C(t)| = P clients for all t*. With appropriate client and server learning rates,* (η c x , ηcy ) and (γ s x y min t∈[T] E ∇Φe1/2Lf (x (t)) 2≤ O σ 2 G n − P n − 1 ∆¯Φe Ew P T p1 + Fw 1/4 + O Cwσ 2 L + D(G2x + σ 2 G ) τ¯ 2T3/4 | {z } Partial participation error | {z } Local updates error + O ∆¯eΦ σ 2 LAw τeffP T p1 + Fw 1/4+ ∆¯ eΦ (1 + Fw) T3/4 τeffP Aw + τeff n−P n−1 Ew 1/4 | {z } Error with ful l synchronization , , γs ) respectively (see Appendix C.2), the iterates of Fed-Norm-SGDA+ satisfy $$(6)\backslash quad(6)$$ where Φ1/2Lfis the Moreau envelope of Φ*, and* ∆¯Φe ≜ Φe1/2Lf (x0) − minx Φe1/2Lf (x). The constants Aw, Cw, D, Ew, τ, τ ¯ eff are defined in Theorem 1, and Fw ≜ n(n−P ) P (n−1) Pn i=1 w 2 i .

See Appendix C for the proof. Theorem 2 states convergence for a surrogate objective Fe. Next, we see convergence for the true objective F.

Corollary 2.1 (Convergence in terms of F). Given envelope functions Φ(x) ≜ maxy F(x, y), Φe(x) ≜ maxy Fe(x, y), under the conditions of Theorem 2,

$$\underset{t\in [T]}{\min }{\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\le {ϵ}_{opt}^{\mathrm{\prime }}+\frac{8L_f^2}{T}\sum _{t=0}^{T-1}{\parallel {\tilde{\mathbf{x}}}^{(t)}-{\bar{\mathbf{x}}}^{(t)}\parallel }^2,\backslash operatorname*\{min\}\_\{t\backslash in[T]\}\backslash left\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash epsilon\_\{o\; p\; t\}^\{\backslash prime\}+\backslash frac\{8L\_\{f\}^\{2\}\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash left\backslash |\{\backslash tilde\{\backslash mathbf\{x\}\}\}^\{(t)\}-\{\backslash bar\{\backslash mathbf\{x\}\}\}^\{(t)\}\backslash right\backslash |^\{2\},$$

where Φ1/2Lf is the Moreau envelope of Φ, xe (t) ≜ arg minx′{Φe(x ′)+Lf x ′ −x (t) 2}, x¯ (t) ≜ arg minx′{Φ(x ′)+ Lf x ′ − x (t) 2}, for all t, ϵ ′ opt is the error bound in (6).

Similar to Corollary 1.1, if we replace {wi} with {pi} for all i ∈ [n] in the server updates in Algorithm 1, then Fe ≡ F, and xe (t) and x¯ (t) are identical for all t. Consequently, Theorem 2 gives us convergence in terms of the true objective F. For the rest of this subsection, we assume wi = pi for all i ∈ [n].

Remark 4. Some existing works do not require Assumption 8 for NC-C functions, and also improve the convergence rate. However, these methods either have a double-loop structure Rafique et al. (2021); Zhang et al. (2022), or work with deterministic problems Xu et al. (2020); Zhang et al. (2020). Proposing a single-loop method for stochastic NC-C problems with the same advantages is an open problem.

Again, in Table 2, we specialize the bound in (6) to SGDA+ based local updates. As in the last section

• Partial client participation is the most significant source of convergence error.

• Unequal client weights (pi ̸= pj ) can increase the stochastic gradient complexity, due to the presence of n ∥p∥∞ , n ∥p∥ 2 2 factors.

Corollary 2.2 (Improved Communication Cost). *Suppose all the clients are weighted equally (*pi = 1/n for all i), with each carrying out τ local steps of SGDA+. Further, assume that Φ1/2Lf is bounded from below.

Then, to reach x such that E∥∇Φ1/2Lf (x)∥ ≤ ϵ,

• Under full participation, the per-client gradient complexity of Fed-Norm-SGDA+ is T τ = O(1/(nϵ8)). The number of communication rounds required is T = O(1/ϵ4).

• Under partial participation, the per-client gradient complexity of Fed-Norm-SGDA+ is O(1/(P ϵ8)). In general, running multiple local updates does not yield any communication savings. However, in the special case when inter-client data heterogeneity σG = 0*, the communication cost is* O(1/ϵ4).

In terms of communication requirements, we achieve massive savings (compared to O(1/ϵ7) in Sharma et al.

(2022)). Our gradient complexity results achieve linear speedup in the number of participating clients. Further, as stated earlier, our work considers a more general FL setting with unequally weighted clients (pi ̸= pj ), running unequal local updates (τi ̸= τj ), using distinct local solvers (ai ̸= aj ).

Extending the Results to Nonconvex-One-Point-Concave Functions. One-point-convexity has been observed in SGD dynamics during neural network training Li & Yuan (2017); Kleinberg et al. (2018).

Assumption 9 (One-point-Concavity in y). The function f is said to be one-point-concave in y if fixing x ∈ R d1, for all y ∈ R d2,

$$⟨{\mathrm{\nabla }}_{y}f(\mathbf{x},{\mathbf{y}}^{\mathrm{\prime }}),\mathbf{y}-{\mathbf{y}}^{\ast }(\mathbf{x})⟩\le f(\mathbf{x},\mathbf{y})-f(\mathbf{x},{\mathbf{y}}^{\ast }(\mathbf{x})),\backslash langle\backslash nabla\_\{y\}f(\backslash mathbf\{x\},\backslash mathbf\{y\}^\{\backslash prime\}),\backslash mathbf\{y\}-\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\})\backslash rangle\backslash leq\; f(\backslash mathbf\{x\},\backslash mathbf\{y\})-f(\backslash mathbf\{x\},\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\})),$$

where y ∗(x) ∈ arg maxy f(x, y).

It turns out, Theorem 2 holds for the more general class of nonconvex-one-point-concave (NC-1PC) functions.

See Appendix C.4 for more details.

Remark 5. Suppose Assumptions 1, 3, 2, 4, 8 hold. Suppose for all x, all the fi's satisfy Assumption 9 at a common global minimizer y ∗(x). Then, the bound in Theorem 2 holds.

Remark 6. Hence, we settle the conjecture posed in Sharma et al. (2022) that linear speedup can be achieved for NC-1PC functions. As an intermediate step in our proof, we show convergence of Local SGD for one-point-convex functions. This extends the convex result for Local SGD to a larger class of functions.

6 Experiments

In this section, we evaluate the empirical performance of the proposed algorithms. We consider a robust neural training problem Sinha et al. (2017); Nouiehed et al. (2019), and a fair classification problem Mohri et al. (2019); Deng et al. (2020). Due to space constraints, additional details of our experiments, and some additional results are included in Appendix D. Our experiments were run on a network of n = 15 clients, each equipped with an NVIDIA TitanX GPU. We model data heterogeneity across clients using Dirichlet distribution Wang et al. (2019) with parameter α, Dirn(α). Small α ⇒ higher heterogeneity across clients.

Robust NN training. We consider the following robust neural network (NN) training problem.

$$\underset{\mathbf{x}}{\min }\underset{\mathrm{\parallel }\mathbf{y}{\mathrm{\parallel }}^2\le 1}{\max }\sum _{j=1}^N\mathrm{\ell }(h_{\mathbf{x}}({\mathbf{a}}_i+\mathbf{y}),b_i),\backslash operatorname*\{min\}\_\{\backslash mathbf\{x\}\}\backslash ;\backslash operatorname*\{max\}\_\{\backslash |\backslash mathbf\{y\}\backslash |^\{2\}\backslash leq1\}\backslash sum\_\{j=1\}^\{N\}\backslash ell\backslash left(h\_\{\backslash mathbf\{x\}\}(\backslash mathbf\{a\}\_\{i\}+\backslash mathbf\{y\}),b\_\{i\}\backslash right),$$

$$\left(7\right)\backslash left(7\backslash right)$$ ℓ (hx(ai + y), bi), (7) where x denotes the NN parameters, (ai, bi) denote the feature and label of the i-th sample, y denotes the adversarially added feature perturbation, and hx denotes the NN output.

Impact of system heterogeneity. In Figure 3, we compare the effect of heterogeneous number of local updates across clients, on the performance of our proposed Fed-Norm-SGDA+. We compare with Local SGDA+ Deng & Mahdavi (2021), and Local SGDA+ with momentum Sharma et al. (2022). Clients sample the number of epochs they run locally via uniform sampling over the set {2 . . . , E}, i.e., τi ∼ Unif[2 : E]. We observe that Fed-Norm-SGDA+ adapts well to system heterogeneity and outperforms both existing methods.

Impact of partial participation and heterogeneity. Next, we compare the impact of different levels of partial client participation on performance. We compare the full participation setting (n = 15) with P = 5, 10. Clients sample the number of epochs they run locally via τi ∼ Unif[2, 5]. We plot the results for two different values of the data heterogeneity parameter α = 0.1, 1.0. As seen in all our theoretical results where partial participation was the most significant component of convergence error, smaller values of P result in performance loss. Further, higher inter-client heterogeneity (modeled by smaller values of α) results in worse performance. We further explore the impact of α on performance in Appendix D.

Figure 3: Comparison of the effect of heterogeneous number of local updates {τi} on the performance of Fed-Norm-SGDA+ (Algorithm 1), Local SGDA+, and Local SGDA+ with momentum, while solving (7) on CIFAR10 dataset, with VGG11 model. The solid (dashed) curves are for E = 5 (E = 7), and α = 0.1.

Figure 4: Comparison of the effects of partial client participation (PCP) on the performance of Fed-NormSGDA+, for the robust NN training problem on the CIFAR10 dataset, with the VGG11 model. The figure shows the robust test accuracy. The solid (dashed) curves are for α = 0.1 (α = 1.0).

$$(\mathit{\delta })(\{\backslash boldsymbol\{\backslash delta\}\})$$

Figure 5: Comparison of Local SGDA, Local SGDA with momentum, and Fed-Norm-SGDA, for the fair classification task on the CIFAR10 dataset, with the VGG11 model. The solid (dashed) curves are for E = 5 (E = 7), α = 0.1. Fair Classification. We consider minimax formulation of the fair classification problem Mohri et al. (2019); Nouiehed et al. (2019).

$$\underset{\mathbf{x}}{\min }\underset{\mathbf{y}\in {\mathrm{\Delta }}_C}{\max }\sum _{c=1}^C{y}_c{F}_c(\mathbf{x})-\frac{\lambda }{2}{\parallel \mathbf{y}\parallel }^2,\backslash operatorname*\{min\}\_\{\backslash mathbf\{x\}\}\backslash operatorname*\{max\}\_\{\backslash mathbf\{y\}\backslash in\backslash Delta\_\{C\}\}\backslash sum\_\{c=1\}^\{C\}y\_\{c\}F\_\{c\}(\backslash mathbf\{x\})-\{\backslash frac\{\backslash lambda\}\{2\}\}\backslash left\backslash |\backslash mathbf\{y\}\backslash right\backslash |^\{2\},$$ 2, (8)

where x denotes the parameters of the NN, {Fc} C c=1 denote the loss corresponding to class c, and ∆C is the C-dimensional probability simplex. In Figure 5, we plot the worst distribution test accuracy achieved by Fed-Norm-SGDA, Local SGDA Deng & Mahdavi (2021) and Local SGDA with momentum Sharma et al.

(2022). As in Figure 3, clients sample τi ∼ Unif[2, E]. We plot the test accuracy on the worst distribution in each case. Again, Fed-Norm-SGDA outperforms existing methods.

7 Conclusion

In this work, we considered nonconvex minimax problems in the federated setting, where in addition to inter-client data heterogeneity and partial client participation, there is system heterogeneity as well. Clients may run unequal number of local update steps, using different local solvers. In such settings, we observed that existing methods, such as Local SGDA, might converge to the stationary point of an objective quite different from the original intended objective. We showed that normalizing individual client contributions solves this problem. Using our generalized framework, we analyzed several classes of nonconvex minimax functions and significantly improved existing computation and communication complexity results. Potential future directions include analyzing federated systems with unpredictable client presence Yang et al. (2022b).

Acknowledgments

This work was supported in part by NSF grants CCF 2045694, CNS-2112471, CPS-2111751, and ONR N00014-23-1- 2149. Jiarui Li helped with plotting figures for some experiments.

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Contents

1 Introduction 1 2 Related Work 3 2.1 Single-client minimax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Distributed/Federated Minimax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Preliminaries 4 4 Algorithm for Heterogeneous Federated Minimax Optimization 5 4.1 Limitations of Local SGDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Proposed Normalized Federated Minimax Algorithm . . . . . . . . . . . . . . . . . . . . . . . 6 5 Convergence Results 8 5.1 Non-convex-Strongly-Concave (NC-SC) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 Non-convex-Concave (NC-C) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Experiments 12 7 Conclusion 14 A Background 21 A.1 Gradient Aggregation with Different Solvers at Clients . . . . . . . . . . . . . . . . . . . . . . 21 A.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 B Convergence of Fed-Norm-SGDA for Nonconvex-Strongly-Concave Functions (Theorem 1) 23 B.1 Intermediate Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 B.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 B.3 Proofs of the Intermediate Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 B.4 Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 B.5 Convergence under Polyak Łojasiewicz (PL) Condition . . . . . . . . . . . . . . . . . . . . . . 39 C Convergence of Fed-Norm-SGDA+ for Nonconvex Concave Functions (Theorem 2) 39 C.1 Intermediate Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 C.3 Proofs of the Intermediate Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 C.4 Extending the result for Nonconvex One-Point-Concave (NC-1PC) Functions . . . . . . . . . 51 D Additional Experiments 51

Appendix

The appendix are organized as follows. In Section A we mention some basic mathematical results and inequalities which are used throughout the paper. In Appendix B we prove the non-asymptotic convergence of Fed-Norm-SGDA (Algorithm 1) for smooth nonconvex-strongly-concave (and nonconvex-PŁ) functions, and derive gradient complexity and communication cost of the algorithm to achieve an ϵ-stationary point.

In Appendix C, we prove the non-asymptotic convergence of Fed-Norm-SGDA+ (Algorithm 1) for smooth nonconvex-concave and nonconvex-one-point-concave functions. Finally, in Appendix D we provide the details of the additional experiments we performed.

A Background A.1 Gradient Aggregation With Different Solvers At Clients

Local SGDA. Suppose τ (t) i = τ (t) eff = τ for all i ∈ [n], t ∈ [T]. Also, a (t,k) i = 1 for all k ∈ [τ ], t. Then, the local iterate updates in Algorithm 1-Fed-Norm-SGDA reduce to (the updates for Fed-Norm-SGDA+ are analogous)

$$\begin{array}{c}{\mathbf{x}}_i^{(t,k+1)}={\mathbf{x}}_i^{(t,k)}-{\eta }_x^c{\mathrm{\nabla }}_x{f}_i({\mathbf{x}}_i^{(t,k)},{\mathbf{y}}_i^{(t,k)};{\xi }_i^{(t,k)}),\\ {\mathbf{y}}_i^{(t,k+1)}={\mathbf{y}}_i^{(t,k)}+{\eta }_y^c{\mathrm{\nabla }}_y{f}_i({\mathbf{x}}_i^{(t,k)},{\mathbf{y}}_i^{(t,k)};{\xi }_i^{(t,k)}),\end{array}\backslash begin\{array\}\{l\}\{\{\{\backslash bf\; x\}\_\{i\}^\{(t,k+1)\}=\{\backslash bf\; x\}\_\{i\}^\{(t,k)\}-\backslash eta\_\{x\}^\{c\}\backslash nabla\_\{x\}f\_\{i\}(\{\backslash bf\; x\}\_\{i\}^\{(t,k)\},\{\backslash bf\; y\}\_\{i\}^\{(t,k)\};\backslash xi\_\{i\}^\{(t,k)\}),\}\}\backslash \backslash \; \{\{\{\backslash bf\; y\}\_\{i\}^\{(t,k+1)\}=\{\backslash bf\; y\}\_\{i\}^\{(t,k)\}+\backslash eta\_\{y\}^\{c\}\backslash nabla\_\{y\}f\_\{i\}(\{\backslash bf\; x\}\_\{i\}^\{(t,k)\},\{\backslash bf\; y\}\_\{i\}^\{(t,k)\};\backslash xi\_\{i\}^\{(t,k)\}),\}\}\backslash end\{array\}$$

for k ∈ {0*, . . . , τ* − 1} and the gradient aggregate vectors (g (t) x,i, g (t) y,i) are simply the average of individual gradients

$${\mathbf{g}}_{\mathbf{x},i}^{(t)}=\frac{1}{\tau }\sum _{k=0}^{\tau -1}{\mathrm{\nabla }}_x{f}_i({\mathbf{x}}_i^{(t,k)},{\mathbf{y}}_i^{(t,k)};{\xi }_i^{(t,k)}),{\mathbf{g}}_{\mathbf{y},i}^{(t)}=\frac{1}{\tau }\sum _{k=0}^{\tau -1}{\mathrm{\nabla }}_y{f}_i({\mathbf{x}}_i^{(t,k)},{\mathbf{y}}_i^{(t,k)};{\xi }_i^{(t,k)})\{\backslash bf\; g\}\_\{\{\backslash bf\; x\},i\}^\{(t)\}=\backslash frac\{1\}\{\backslash tau\}\backslash sum\_\{k=0\}^\{\backslash tau-1\}\backslash nabla\_\{x\}f\_\{i\}(\{\backslash bf\; x\}\_\{i\}^\{(t,k)\},\{\backslash bf\; y\}\_\{i\}^\{(t,k)\};\backslash xi\_\{i\}^\{(t,k)\}),\backslash quad\{\backslash bf\; g\}\_\{\{\backslash bf\; y\},i\}^\{(t)\}=\backslash frac\{1\}\{\backslash tau\}\backslash sum\_\{k=0\}^\{\backslash tau-1\}\backslash nabla\_\{y\}f\_\{i\}(\{\backslash bf\; x\}\_\{i\}^\{(t,k)\},\{\backslash bf\; y\}\_\{i\}^\{(t,k)\};\backslash xi\_\{i\}^\{(t,k)\})$$

Note that these are precisely the iterates of LocalSGDA proposed in Deng & Mahdavi (2021); Sharma et al.

(2022), with the only difference that in LocalSGDA, the clients communicate the iterates {x (t,τ) i, y (t,τ) i} to the server, which averages them to compute {x (t+1), y (t+1)}. While here, the clients communicate {g (t) x,i, g (t) y,i}.

Also, in Fed-Norm-SGDA, the clients and server use separate learning rates, which results in tighter bounds on the local-updates error. With Momentum in Local Updates. Suppose each local client uses a momentum buffer with momentum scale ρ. Then, for k ∈ {0*, . . . , τ* (t) i − 1}

d t,k+1 x,i = ρd t,k x,i + ∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i), x (t,k+1) i = x (t,k) i − η c xd t,k+1 x,i d t,k+1 y,i = ρd t,k y,i + ∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i), y (t,k+1) i = y (t,k) i + η c yd t,k+1 y,i , Simple calculations show that the coefficient of ∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i) and ∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i) in the gradient aggregate vectors (g (t) x,i, g (t) y,i) is

$$\sum _{j\ge k}^{{\tau }_i^{(t)}-1}=1+\rho +\cdots +{\rho }^{{\tau }_i^{(t)}-1-k}=\frac{1-{\rho }^{{\tau }_i^{(t)}-k}}{1-\rho }\mathrm{.}\backslash sum\_\{j\backslash geq\; k\}^\{\backslash tau\_\{i\}^\{(t)\}-1\}=1+\backslash rho+\backslash cdots+\backslash rho^\{\backslash tau\_\{i\}^\{(t)\}-1-k\}=\{\backslash frac\{1-\backslash rho^\{\backslash tau\_\{i\}^\{(t)\}-k\}\}\{1-\backslash rho\}\}.$$

Therefore, the aggregation vector is a¯ (t) i =1 1−ρ [1 − ρ τ (t) i , 1 − ρ τ (t) i −1*, . . . ,* 1 − ρ], and

$$\mathrm{\parallel }{\bar{\mathbf{a}}}_i^{(t)}{\mathrm{\parallel }}_1=\sum _{k=0}^{{\tau }_i^{(t)}-1}\frac{1-{\rho }^{{\tau }_i^{(t)}-k}}{1-\rho }=\frac{1}{1-\rho }[{\tau }_i^{(t)}-\frac{\rho (1-{\rho }^{{\tau }_i^{(t)}})}{1-\rho }]\mathrm{.}\backslash |\backslash bar\{\backslash mathbf\{a\}\}\_\{i\}^\{(t)\}\backslash |\_\{1\}=\backslash sum\_\{k=0\}^\{\backslash tau\_\{i\}^\{(t)\}-1\}\backslash frac\{1-\backslash rho^\{\backslash tau\_\{i\}^\{(t)\}-k\}\}\{1-\backslash rho\}=\backslash frac\{1\}\{1-\backslash rho\}\backslash left[\backslash tau\_\{i\}^\{(t)\}-\backslash frac\{\backslash rho(1-\backslash rho^\{\backslash tau\_\{i\}^\{(t)\}\})\}\{1-\backslash rho\}\backslash right].$$

A.2 Auxiliary Results

Remark 7 (Impact of heterogeneity σG even with τ = 1). Consider two simple minimization problems:

(P1):$\min\limits_{\bf x}\frac{1}{n}\sum\limits_{i=1}^{n}f_{i}({\bf x})$ and (P2):$\min\limits_{\bf x}f(x)$.
(P1) is a simple distributed minimization problem, with n clients, which we solve using synchronous distributed SGD. At iteration t, each client i computes stochastic gradient ∇fi(x (t); ξ (t) i), and sends it to the server, which averages these, and takes a step in the direction 1n Pn i=1 ∇fi(x (t); ξ (t) i). On the other hand, (P1) is a centralized minimization problem, where at each iteration t, the agent computes a stochastic gradient estimator with batch-size n, 1 n Pn i=1 ∇f(x (t); ξ (t) i). We compare the variance of the two global gradient estimators as follows.

\begin{array}{ll}&\mbox{(\bf P1)}\\ \mathbb{E}\Big{\|}\frac{1}{n}\sum_{i=1}^{n}\nabla f_{i}({\bf x}^{(t)};\xi_{i}^{(t)})-\nabla f({\bf x}^{(t)})\Big{\|}^{2}\\ &\leq\frac{1}{n^{2}}\sum_{i=1}^{n}\left[\rho_{\bf x}^{2}+\beta_{i}^{2}\mathbb{E}\big{|}\nabla f_{i}({\bf x}^{(t)})\big{|}\right]^{2}\\ &=\frac{\sigma_{\bf x}^{2}}{n}+\frac{\beta_{i}^{2}}{n}\left[\beta_{i}^{2}\mathbb{E}\big{|}\nabla f({\bf x}^{(t)})\big{|}^{2}+\sigma_{\bf c}^{2}\right].\end{array} ()

Since almost all the existing works consider the local variance bound (Assumption 3) with βL = 0, the global gradient estimator in both synchronous distributed SGD (P1) and single-agent minibatch SGD (P2) have the same σ 2 L nvariance bound. Therefore, in most existing federated works on minimization Wang et al. (2020); Yang et al. (2021) and minimax problems Sharma et al. (2022), the full synchronization error only depends on the local variance σ 2 L . However, as seen above, for βL > 0, this apparent equivalence breaks down. Koloskova et al. (2020), which considers similar local variance assumption as ours for minimization problems, also show similar dependence on heterogeneity σG.

Lemma A.1 (Young's inequality). Given two same-dimensional vectors u, v ∈ R d, the Euclidean inner product can be bounded as follows:

$$⟨\mathbf{u},\mathbf{v}⟩\le \frac{{\parallel \mathbf{u}\parallel }^2}{2\gamma }+\frac{\gamma {\parallel \mathbf{v}\parallel }^2}{2}\backslash langle\backslash mathbf\{u\},\backslash mathbf\{v\}\backslash rangle\backslash leq\{\backslash frac\{\backslash left\backslash |\backslash mathbf\{u\}\backslash right\backslash |^\{2\}\}\{2\backslash gamma\}\}+\{\backslash frac\{\backslash gamma\backslash left\backslash |\backslash mathbf\{v\}\backslash right\backslash |^\{2\}\}\{2\}\}$$

for every constant γ > 0.

Lemma A.2 (Strong Concavity). A function g : X × Y is strongly concave in y*, if there exists a constant* µ > 0, such that for all x ∈ X , and for all y, y ′ ∈ Y*, the following inequality holds.*

$$g(\mathbf{x},\mathbf{y})\le g(\mathbf{x},{\mathbf{y}}^{\mathbf{\prime }})+⟨{\mathrm{\nabla }}_{y}g(\mathbf{x},{\mathbf{y}}^{\mathbf{\prime }}),{\mathbf{y}}^{\mathbf{\prime }}-\mathbf{y}⟩-\frac{\mu }{2}{\parallel \mathbf{y}-{\mathbf{y}}^{\mathbf{\prime }}\parallel }^2\mathrm{.}g(\{\backslash bf\; x\},\{\backslash bf\; y\})\backslash leq\; g(\{\backslash bf\; x\},\{\backslash bf\; y^\{\backslash prime\}\})+\backslash langle\backslash nabla\_\{y\}g(\{\backslash bf\; x\},\{\backslash bf\; y^\{\backslash prime\}\}),\{\backslash bf\; y^\{\backslash prime\}\}-\{\backslash bf\; y\}\backslash rangle-\backslash frac\{\backslash mu\}\{2\}\backslash left\backslash |\{\backslash bf\; y\}-\{\backslash bf\; y^\{\backslash prime\}\}\backslash right\backslash |^\{2\}.$$

Lemma A.3 (Jensen's inequality). Given a convex function f and a random variable X*, the following holds.*

$$f(\mathbb{E}[X])\le \mathbb{E}[f(X)]\mathrm{.}f\backslash left(\backslash mathbb\{E\}[X]\backslash right)\backslash leq\backslash mathbb\{E\}\backslash left[f(X)\backslash right].$$

Lemma A.4 (Sum of squares). For a positive integer K, and a set of vectors x1, . . . , xK*, the following holds:*

$${\parallel \sum _{k=1}^K{\mathbf{x}}_k\parallel }^2\le K\sum _{k=1}^K{\parallel {\mathbf{x}}_k\parallel }^2\mathrm{.}\backslash left\backslash |\backslash sum\_\{k=1\}^\{K\}\backslash mathbf\{x\}\_\{k\}\backslash right\backslash |^\{2\}\backslash leq\; K\backslash sum\_\{k=1\}^\{K\}\backslash left\backslash |\backslash mathbf\{x\}\_\{k\}\backslash right\backslash |^\{2\}.$$

Lemma A.5 (Quadratic growth condition Karimi et al. (2016)). If function g satisfies Assumptions 1, 5, then for all x*, the following conditions holds*

$$\begin{array}{c}g(\mathbf{x})-{\min }_{\mathbf{z}}g(\mathbf{z})\ge \frac{\mu }{2}{\parallel {\mathbf{x}}_p-\mathbf{x}\parallel }^2,\\ \mathrm{\parallel }\mathrm{\nabla }g(\mathbf{x}){\mathrm{\parallel }}^2\ge 2\mu (g(\mathbf{x})-{\min }_{\mathbf{z}}g(\mathbf{z}))\mathrm{.}\end{array}\backslash begin\{array\}\{c\}\{\{g(\{\backslash bf\; x\})-\backslash operatorname*\{min\}\_\{\{\backslash bf\; z\}\}g(\{\backslash bf\; z\})\backslash geq\backslash frac\{\backslash mu\}\{2\}\backslash left\backslash |\{\backslash bf\; x\}\_\{p\}-\{\backslash bf\; x\}\backslash right\backslash |^\{2\},\}\}\backslash \backslash \; \{\{\backslash qquad\backslash qquad\backslash |\backslash nabla\; g(\{\backslash bf\; x\})\backslash |^\{2\}\backslash geq2\backslash mu\backslash left(g(\{\backslash bf\; x\})-\backslash operatorname*\{min\}\_\{\{\backslash bf\; z\}\}g(\{\backslash bf\; z\})\backslash right).\}\}\backslash end\{array\}$$

Lemma A.6. For L-smooth, convex function g*, the following inequality holds*

$$\begin{array}{cccc}& \mathbb{E}{\parallel \mathrm{\nabla }g(\mathbf{y})-\mathrm{\nabla }g(\mathbf{x})\parallel }^2\le 2L[g(\mathbf{y})-g(\mathbf{x})-\mathrm{\nabla }g(\mathbf{x}{)}^{\mathrm{\top }}(\mathbf{y}-\mathbf{x})]\mathrm{.}& & \text{(9)}\end{array}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\; g(\backslash mathbf\{y\})-\backslash nabla\; g(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq2L\backslash left[g(\backslash mathbf\{y\})-g(\backslash mathbf\{x\})-\backslash nabla\; g(\backslash mathbf\{x\})^\{\backslash top\}(\backslash mathbf\{y\}-\backslash mathbf\{x\})\backslash right].\backslash tag\{9\}$$

Lemma A.7 (Proposition 6 in Cho & Yun (2022)). For L-smooth function g which is bounded below by g ∗, the following inequality holds for all x

$$\mathbb{E}{\parallel \mathrm{\nabla }g(\mathbf{x})\parallel }^2\le 2L[g(\mathbf{x})-g^{\ast }]\mathrm{.}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\; g(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq2L\backslash left[g(\backslash mathbf\{x\})-g^\{*\}\backslash right].$$ ∗] . (10)

B Convergence Of Fed-Norm-Sgda For Nonconvex-Strongly-Concave Functions (Theorem 1)

We organize this section as follows. First, in Appendix B.1 we present some intermediate results, which we use to prove the main theorem. Next, in Appendix B.2, we present the proof of Theorem 1, which is followed by the proofs of the intermediate results in Appendix B.3. Appendix B.4 contains some auxiliary results.

Finally, in Appendix B.5 we discuss the convergence result for nonconvex-PL functions.

The problem we solve is

$$\underset{\mathbf{x}}{\min }\underset{\mathbf{y}}{\max }\{\tilde{F}(\mathbf{x},\mathbf{y})\triangleq \sum _{i=1}^n{w}_i{f}_i(\mathbf{x},\mathbf{y})\}\mathrm{.}\backslash operatorname*\{min\}\_\{\backslash mathbf\{x\}\}\backslash operatorname*\{max\}\_\{\backslash mathbf\{y\}\}\backslash left\backslash \{\{\backslash widetilde\{F\}\}(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash triangleq\backslash sum\_\{i=1\}^\{n\}w\_\{i\}f\_\{i\}(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash right\backslash \}.$$

We define Φe(x) ≜ maxy Fe(x, y) and ye ∗(x) ∈ arg maxy Fe(x, y). Since Fe(x, ·) is µ-strongly concave, ye ∗(x) is unique. In Fed-Norm-SGDA (Algorithm 1), the client updates are given by

$$(10{)}^{\frac{1}{2}}(10)^\{\backslash frac\{1\}\{2\}\}$$

where 1 ≤ k ≤ τi. These client updates are then aggregated to compute {g (t) x,i, g (t) y,i}

g (t) x,i = 1 ∥ai∥ τXi−1 k=0 a (k) i(τi)∇xfi x (t,k) i, y (t,k) i; ξ (t,k) i ; h (t) x,i = 1 ∥ai∥ τXi−1 k=0 a (k) i(τi)∇xfi x (t,k) i, y (t,k) i g (t) y,i = 1 ∥ai∥ τXi−1 k=0 a (k) i(τi)∇yfi x (t,k) i, y (t,k) i; ξ (t,k) i ; h (t) y,i = 1 ∥ai∥ τXi−1 k=0 a (k) i(τi)∇yfi x (t,k) i, y (t,k) i . Remark 8. Note that we have made explicit, the dependence on k in a (j) i(k) above. This was omitted in the main paper to avoid tedious notation. However, for some local optimizers, such as local momentum at the clients (Appendix A.1), the coefficients a (j) i(k) change with k. We assume in our subsequent analysis that a (j) i(k) ≤ α for all j ∈ {0, 1*, . . . , k* −1} and for all k ∈ {1, 2*, . . . , τ*i}. We also use the notation ∥ai∥ ≜ai(τi).

At iteration t, the server samples |C(t)| clients without replacement (WOR) uniformly at random. While aggregating at the server, client i update is weighed by w˜i = win/|C(t)|. The aggregates (g (t) x , g (t) y ) computed at the server are of the form

g (t) x = X i∈C(t) w˜ig (t) x,i, such that EC(t) [g (t) x ] = EC(t) hXn i=1 I(i ∈ C(t)) ˜wig (t) x,ii= Xn i=1 wig (t) x,i (12) g (t) y = X i∈C(t) w˜ig (t) y,i, such that EC(t) [g (t) y ] = EC(t) hXn i=1 I(i ∈ C(t)) ˜wig (t) y,ii= Xn i=1 wig (t) y,i For simplicity of analysis, unless stated otherwise, we assume that |C(t)| = P for all t. Finally, server updates the x, y variables as

$${\mathbf{x}}^{(t+1)}={\mathbf{x}}^{(t)}-{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_x^s{\mathbf{g}}_{\mathbf{x}}^{(t)},{\mathbf{y}}^{(t+1)}={\mathbf{y}}^{(t)}+{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_y^s{\mathbf{g}}_{\mathbf{y}}^{(t)}\mathrm{.}\{\backslash bf\; x\}^\{(t+1)\}=\{\backslash bf\; x\}^\{(t)\}-\backslash tau\_\{\backslash mathrm\{eff\}\}\backslash gamma\_\{x\}^\{s\}\{\backslash bf\; g\}\_\{\backslash bf\; x\}^\{(t)\},\backslash qquad\{\backslash bf\; y\}^\{(t+1)\}=\{\backslash bf\; y\}^\{(t)\}+\backslash tau\_\{\backslash mathrm\{eff\}\}\backslash gamma\_\{y\}^\{s\}\{\backslash bf\; g\}\_\{\backslash bf\; y\}^\{(t)\}.$$

We define by F(t ′) the σ-algebra generated by {{x (t,k) i, y (t,k) i}i,k} t ′−1 t=0 . Throughout, we denote the conditional expectation E[·|F(t)] by the shorthand Et[·].

B.1 Intermediate Lemmas

We begin with the following result from Nouiehed et al. (2019) about the smoothness of Φ( e ·).

Lemma B.1. If a function f(·, ·) satisfies Assumptions 1, 5 (Lf -smoothness and µ-strong concavity in y), then ϕ(·) ≜ maxy f(·, y) is LΦ*-smooth with* LΦ = κLf /2 + Lf , where κ = Lf /µ is the condition number. Lemma B.2. Suppose the local client loss functions {fi} satisfy Assumptions 1, 4, and the stochastic oracles for the local functions satisfy Assumption 3. Suppose the server selects P clients in each round. Then the iterates generated by Fed-Norm-SGDA (Algorithm 1) satisfy

Et g (t) x 2 = Et X i∈C(t) w˜ig (t) x,i 2 ≤ n P P − 1 n − 1 Et Xn i=1 wih (t) x,i 2 + n P Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2 σ 2 L + β 2 LEt ∇xfi(x (t,k) i, y (t,k) i) 2 (13) + n(n − P) n − 1 "2L 2 f P Xn i=1 w 2 i ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + (max iwi) 2 P β 2 G ∇xFe(x (t), y (t)) 2 + σ 2 G #, where, ∆ (t,k) x,y (i) ≜ Et h∥x (t,k) i − x (t)∥ 2 + ∥y (t,k) i − y (t)∥ 2iis the iterate drift for client i*, at local iteration* k in the t*-th communication round.* Lemma B.3. Suppose the local client loss functions {fi} satisfy Assumptions 1, 4, 5, and the stochastic oracles for the local functions satisfy Assumption 3. Also, the server learning rate γ s xsatisfies 64τeffγ s xLΦβ 2 Lβ 2 G n P (maxi wi∥ai∥ 2 2 /∥ai∥ 2 1 ) ≤ 1, 8τeffγ s xLΦ(maxi wi) n P n−P n−1 max{8β 2 G, 1} ≤ 1*, and* 8τeffγ s xLΦβ 2 L n P (maxi,k wia (k) i(τi)/∥ai∥1 ) ≤ 1. Then the iterates generated by Algorithm 1 satisfy

Et hΦ( e x (t+1)) − Φ( e x (t)) i≤ − 7τeffγ s x 16 ∇Φ( e x (t)) 2 − τeffγ s x 2 1 − n(P − 1) P(n − 1) τeffγ s xLΦ Et Xn i=1 wih (t) x,i 2 + 5 4 τeffγ s xL 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 9τeffγ s xL 2 f 4µ hΦ( e x (t)) − Fe(x (t), y (t)) i (14) + τ 2 eff[γ s x ] 2LΦ 2 n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + σ 2 G 2(max iwi) n − P n − 1 + 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# . Remark. The bound in Equation (14) looks very similar to the corresponding one-step decay bound for simple smooth minimization problems. The major difference is the presence of hΦ( e x (t)) − Fe(x (t), y (t)) i, which quantifies the inaccuracy of y (t)in solving the max problem maxy Fe(x (t), y). The term Pn i=1wi ∥ai∥1 Pτi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) is the client drift and is bounded in Lemma B.4 below.

Lemma B.4. Suppose the local loss functions {fi} satisfy Assumptions 1, 4, 5, and the stochastic oracles for the local functions satisfy Assumption 3. Further, in Algorithm 1, we choose learning rates η c x , ηcy such that max{η c x , ηcy} ≤ 1 2Lf (maxi ∥ai∥1 ) √2(1+β 2 L ) . Then, the iterates {x (t) i, y (t) i} generated by Fed-Norm-SGDA (Algorithm 1) satisfy

L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ 2[η c x ] 2 + [η c y ] 2L 2 fσ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 4L 2 fMa−1 [η c x ] 2 + [η c y ] 2σ 2 G + 8L 2 fMa−1 β 2 G[η c x ] 2∇Φ( e x (t)) 2 + 8L 3 fMa−1 β 2 G 2κ[η c x ] 2 + [η c y ] 2hΦ( e x (t)) − Fe(x (t), y (t)) i, where $M{\mathbf{a}{-1}}\triangleq\max{i}\left(\left|\mathbf{a}{i,-1}\right|{1}^{2}+\beta_{L}^{2}\left|\mathbf{a}{i,-1}\right|{2}^{2}\right)$._ Lemma B.5. Suppose the local loss functions {fi} satisfy Assumptions 1, 4, 5, and the stochastic oracles for the local functions satisfy Assumption 3. The server learning rates γ s x , γs y satisfy the following conditions:

$$\begin{array}{c}{\tau }_{eff}{\gamma }_y^s\kappa L_f{\beta }_G^2\frac{n}{P}\max \{{\beta }_L^2{\max }_i\frac{w_i\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_2^2}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1^2},\frac{n-P}{n-1}{\max }_i{w}_i\}\le \frac{1}{64},{\gamma }_x^s\le \frac{{\gamma }_y^s}{81{\kappa }^2},\\ \\ \\ 8{\tau }_{ef}{L}_f{\gamma }_y^s\frac{n}{P}\max \{\frac{n-P}{n-1}{\max }_i{w}_i,{\beta }_L^2{\max }_{i,k}\frac{w_i{a}_i^{(k)}(r_i)}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1}\}\le 1\end{array}\backslash begin\{array\}\{c\}\{\{\backslash tau\_\{e\; f\; f\}\backslash gamma\_\{y\}^\{s\}\backslash kappa\; L\_\{f\}\backslash beta\_\{G\}^\{2\}\backslash frac\{n\}\{P\}\backslash max\backslash left\backslash \{\backslash beta\_\{L\}^\{2\}\backslash max\_\{i\}\backslash frac\{w\_\{i\}\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{2\}^\{2\}\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}^\{2\}\},\backslash frac\{n-P\}\{n-1\}\backslash max\_\{i\}w\_\{i\}\backslash right\backslash \}\backslash leq\backslash frac\{1\}\{64\},\backslash gamma\_\{x\}^\{s\}\backslash leq\backslash frac\{\backslash gamma\_\{y\}^\{s\}\}\{81\backslash kappa^\{2\}\},\}\}\backslash \backslash \; \{\{\{\}\}\}\backslash \backslash \; \{\{\{\}\}\}\backslash \backslash \; \{\{8\backslash tau\_\{e\; f\}L\_\{f\}\backslash gamma\_\{y\}^\{s\}\backslash frac\{n\}\{P\}\backslash max\backslash left\backslash \{\backslash frac\{n-P\}\{n-1\}\backslash max\_\{i\}w\_\{i\},\backslash beta\_\{L\}^\{2\}\backslash max\_\{i,k\}\backslash frac\{w\_\{i\}a\_\{i\}^\{(k)\}(r\_\{i\})\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}\}\backslash right\backslash \}\backslash leq1\}\}\backslash end\{array\}$$

The client learning rates η c x , ηcy satisfy η c yLfβG ≤1 16√κMa−1 and η c x : η c xLfβG ≤1 64κ √Ma−1 , respectively. Then the iterates generated by Fed-Norm-SGDA (Algorithm 1) satisfy

1 T T X−1 t=0 E hΦ( e x (t)) − Fe(x (t), y (t)) i ≤ 4 hΦ( e x (0)) − Fe(x (0), y (0)) i τeffγ s yµT +1 12µκ2 1 T T X−1 t=0 E ∇Φ( e x (t)) 2 + 4τeff[γ s x ] 2LΦ γ s yµ n(P − 1) P(n − 1)E Xn i=1 wih (t) x,i 2 + 8τeffγ s yκ n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# + 8κLf [η c x ] 2 + [η c y ] 2"σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2σ 2 GMa−1 # . (15)

Remark 9. The proof of Lemma B.5 differs from similar results in the existing literature Sharma et al. (2022); Yang et al. (2022a). As in these works, if all the clients are running the same number of local steps (τi = τ , for all i), we can define virtual sequences of average iterates x (t,k) = 1 P Pi∈C(t) x (t,k) i, y (t,k) = 1 P Pi∈C(t) y (t,k) i, for all k ∈ [0, τ − 1], t. Define F ′(t, k) as the σ-algebra generated by

$${\mathcal{F}}^{\mathrm{\prime }}(t,k)\triangleq \sigma \{\{\{{\mathbf{x}}_i^{(s,k)},{\mathbf{x}}_i^{(s,k)}{\}}_{i,k}{\}}_{s=0}^{t-1}\bigcup {\{\{{\mathbf{x}}_i^{(t,j)},{\mathbf{x}}_i^{(t,j)}{\}}_i\}}_{j=0}^{k-1}\}\mathrm{.}\{\backslash mathcal\{F\}\}^\{\backslash prime\}(t,k)\backslash triangleq\backslash sigma\backslash left\backslash \{\backslash \{\backslash \{\{\backslash mathbf\{x\}\}\_\{i\}^\{(s,k)\},\{\backslash mathbf\{x\}\}\_\{i\}^\{(s,k)\}\backslash \}\_\{i,k\}\backslash \}\_\{s=0\}^\{t-1\}\backslash ,\backslash bigcup\backslash left\backslash \{\backslash \{\{\backslash mathbf\{x\}\}\_\{i\}^\{(t,j)\},\{\backslash mathbf\{x\}\}\_\{i\}^\{(t,j)\}\backslash \}\_\{i\}\backslash right\backslash \}\_\{j=0\}^\{k-1\}\backslash right\backslash \}.$$

Since, conditioned on F ′(t, k), x t,k+1 ⊥ {∇yfi(x (t,k) i, y (t,k) i; ξ (t,k) i)} n i=1, using {x (t,k), y (t,k)} considerably simplifies the analysis. However, with τi ̸= τj , the virtual sequences {x (t,k), y (t,k)} can no longer be defined for all k. Hence, we need an alternate proof strategy.

B.2 Proof Of Theorem 1

For the sake of completeness, we first state the full statement of Theorem 1 here.

Theorem. Suppose the local loss functions {fi}i satisfy Assumptions 1, 3, 4, 5. Suppose the server selects clients using without-replacement sampling scheme (WOR). Also, the server learning rates γ s x , γs y and the client learning rates η c x , ηcy satisfy the conditions specified in Lemma B.5. Then the iterates generated by

Fed-Norm-SGDA (Algorithm 1) satisfy min t∈[0:T −1] E ∇Φ( e x (t)) 2 ≤ 1 T T X−1 t=0 E ∇Φ( e x (t)) 2 ≤ O κ 2 ∆Φe τeffγ s yT + γ s yLf P Awσ 2 L + Bwβ 2 Lσ 2 G | {z } Error with ful l synchronization + O κ 2 n − P n − 1 γ s yLfEwτeffσ 2 G P ! + Oκ 2[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + Dσ2G | {z } Error due to local updates , | {z } Partial Participation Error

where κ = Lf /µ is the condition number, Φe(x) ≜ maxy Fe(x, y) is the envelope function, ∆Φe ≜ Φe(x (0)) − minx Φe(x), Aw ≜ nτeff Pn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 , Bw ≜ nτeff maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 , Cw ≜Pn i=1 wi(∥ai∥ 2 2 − [α (t,τi−1) i] 2), D ≜ maxi(β 2 L ∥ai,−1∥ 2 2 + ∥ai,−1∥ 2 1 ), where ai,−1 ≜ [a (0) i, a (1) i*, . . . , a* (τi−2) i] ⊤ for all i and Ew ≜ n maxi wi. Using γ s y = Θ r P τeffLf T-∆eΦ +Awσ 2 L+(Bwβ 2 L+ n−P n−1 Ewτeff)σ 2 G

! and η c x ≤ η c y = Θ1 Lf τ¯ √T , where τ¯ = 1 n Pn i=1 τi in the bounds above, we get min t∈[T] E

∇Φ( e x (t))

2 ≤ O

• O

κ 2 s∆Φe + Awσ 2 L + Bwβ 2 L σ 2 G P τeffT κ 2 sn − P n − 1 · Ewσ 2 G P T

• O κ 2 Cwσ 2 L + Dσ2G τ¯ 2T

.

| {z } Local updates error | {z } Error with ful l synchronization | {z } Partial participation error Proof. Using Lemma B.3, and substituting in the bound on iterates' drift from Lemma B.4, we can bound

Et hΦ( e x (t+1)) − Φ( e x (t)) i≤ − 7τeffγ s x 16 ∇Φ( e x (t))

2 − τeffγ s x 2 1 − n(P − 1) P(n − 1) τeffγ s xLΦ Et

Xn i=1 wih (t) x,i

2 + 9τeffγ s xL 2 f 4µ hΦ( e x (t)) − Fe(x (t), y (t)) i + τ 2 eff[γ s x ] 2LΦ 2 n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1

• σ 2 G

2(max iwi) n − P n − 1

• 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#
• 5 2 τeffγ s x [η c x ] 2 + [η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2σ 2 GMa−1

• 10τeffγ s xL 2 fMa−1 β 2 G [η c x ] 2∇Φ( e x (t))

2

• Lf 2κ[η c x ] 2 + [η c y ] 2hΦ( e x (t)) − Fe(x (t), y (t)) i. (16) Summing (16) over t = 0*, . . . , T* − 1, substituting the bound on E hΦ( e x (t)) − Fe(x (t), y (t)) ifrom Lemma B.5, and rearranging the terms, we get 1 T T X−1 t=0 E ∇Φ( e x (t)) 2 = O κ 2∆Φe τeffγ s yT + τeffγ s yLfκ 2 n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#! + O κ 2[η c x ] 2 + [η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai∥ 2 2 − [a (t,τi−1) i] 2+ σ 2 G max i ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 #! (17) Consequently, using constants Aw, Bw, Cw, D, Ew, (17) can be simplified to 1 T T X−1 t=0 E ∇Φ( e x (t)) 2 ≤O κ 2 ∆Φe τeffγ s yT + γ s yLf P Awσ 2 L + Bwβ 2 L + n − P n − 1 Ewτeffσ 2 G 26

$$+\mathcal{O}\left({\kappa }^2([{\eta }_x^c{]}^2+[{\eta }_y^c{]}^2)L_f^2[C_w{\sigma }_L^2+D{\sigma }_G^2]\right),+\backslash ,\{\backslash mathcal\{O\}\}\backslash left(\backslash kappa^\{2\}\backslash left([\backslash eta\_\{x\}^\{c\}]^\{2\}+[\backslash eta\_\{y\}^\{c\}]^\{2\}\backslash right)L\_\{f\}^\{2\}\backslash left[C\_\{w\}\backslash sigma\_\{L\}^\{2\}+D\backslash sigma\_\{G\}^\{2\}\backslash right]\backslash right),$$ which completes the proof.

Convergence In Terms Of F

Proof of Corollary 1.1. According to the definition of F(x) and Fe(x), we have

∇Φ(x) − ∇Φ( e x) = Xn i=1 [pi∇xfi(x, y ∗(x)) − wi∇xfi(x, ye ∗(x))] (y ∗(x) ∈ arg maxy F(x, y)) = Xn i=1 pi[∇xfi(x, y ∗(x)) − ∇xfi(x, ye ∗(x))] +Xn i=1 (pi − wi) ∇xfi(x, ye ∗(x)) = [∇xF(x, y ∗(x)) − ∇xF(x, ye ∗(x))] +Xn i=1 pi − wi √wi· √wi∇xfi(x, ye ∗(x)). Taking norm, using Lf -smoothness and applying Cauchy–Schwarz inequality, we get

$${\parallel \mathrm{\nabla }\mathrm{\Phi }(\mathbf{x})-\mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x})\parallel }^2\le 2L_f^2{\parallel {\mathbf{y}}^{\ast }(\mathbf{x})-{\tilde{\mathbf{y}}}^{\ast }(\mathbf{x})\parallel }^2+2\left[\sum _{i=1}^n\frac{{(p_i-w_i)}^2}{w_i}\right]\left[\sum _{i=1}^n{w}_i{\parallel {\mathrm{\nabla }}_x{f}_i(\mathbf{x},{\tilde{\mathbf{y}}}^{\ast }(\mathbf{x}))\parallel }^2\right]\backslash left\backslash |\backslash nabla\backslash Phi(\backslash mathbf\{x\})-\backslash nabla\backslash widetilde\{\backslash Phi\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq2L\_\{f\}^\{2\}\backslash left\backslash |\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\})-\backslash widetilde\{\backslash mathbf\{y\}\}^\{*\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}+2\backslash left[\backslash sum\_\{i=1\}^\{n\}\backslash frac\{\backslash left(p\_\{i\}-w\_\{i\}\backslash right)^\{2\}\}\{w\_\{i\}\}\backslash right]\backslash left[\backslash sum\_\{i=1\}^\{n\}w\_\{i\}\backslash left\backslash |\backslash nabla\_\{x\}f\_\{i\}\backslash left(\backslash mathbf\{x\},\backslash widetilde\{\backslash mathbf\{y\}\}^\{*\}(\backslash mathbf\{x\})\backslash right)\backslash right\backslash |^\{2\}\backslash right]$$ $$\leq2L_{f}^{2}\left|\mathbf{y}^{}(\mathbf{x})-\widetilde{\mathbf{y}}^{}(\mathbf{x})\right|^{2}+2\chi_{\mathbf{p}|\mathbf{w}}^{2}\left[\left.\partial_{G}^{2}\right|\nabla\widetilde{\Phi}(\mathbf{x})\right|^{2}+\sigma_{G}^{2}\right],$$ $\square$ $\square$ where the last inequality uses Assumption 4. Next, note that

$${\parallel \mathrm{\nabla }\mathrm{\Phi }(\mathbf{x})\parallel }^2\le 2{\parallel \mathrm{\nabla }\mathrm{\Phi }(\mathbf{x})-\mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x})\parallel }^2+2{\parallel \mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x})\parallel }^2\mathrm{.}\backslash left\backslash |\backslash nabla\backslash Phi(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq2\backslash left\backslash |\backslash nabla\backslash Phi(\backslash mathbf\{x\})-\backslash nabla\backslash widetilde\{\backslash Phi\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}+2\backslash left\backslash |\backslash nabla\backslash widetilde\{\backslash Phi\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}.$$

Therefore, we obtain

min t∈[T] ∇Φ(x (t)) 2 ≤ 1 T T X−1 t=0 ∇Φ(x (t)) 2 ≤ 2 h2χ 2 p∥wβ 2 G + 1i1 T T X−1 t=0 ∇Φ( e x (t)) 2 + 4 "χ 2 p∥wσ 2 G + L 2 f 1 T T X−1 t=0 y ∗(x (t)) − ye ∗(x (t)) 2# = 2 h2χ 2 p∥wβ 2 G + 1iϵopt + 4 "χ 2 p∥wσ 2 G + L 2 f 1 T T X−1 t=0 y ∗(x (t)) − ye ∗(x (t)) 2# . where ϵopt denotes the optimization error in the right hand side of (4) in Theorem 1.

Proof of Corollary 1.2. If clients are weighted equally (wi = pi = 1/n for all i), with each carrying out τ steps of local SGDA, from (4) we get

$$\underset{t\in [T]}{\min }{\parallel \mathrm{\nabla }\mathrm{\Phi }({\mathbf{x}}^{(t)})\parallel }^2\le \mathcal{O}(\sqrt{\frac{n-P}{n-1}}\frac{{\kappa }^2{\sigma }_G}{\sqrt{PT}}+{\kappa }^2(\frac{{\sigma }_L+{\beta }_L{\sigma }_G}{\sqrt{P\tau T}}+\frac{{\sigma }_L^2+\tau {\sigma }_G^2}{\tau T}\left)\right)\mathrm{.}\backslash operatorname*\{min\}\_\{t\backslash in[T]\}\backslash left\backslash |\backslash nabla\backslash Phi(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash mathcal\{O\}\backslash left(\{\backslash sqrt\{\backslash frac\{n-P\}\{n-1\}\}\}\{\backslash frac\{\backslash kappa^\{2\}\backslash sigma\_\{G\}\}\{\backslash sqrt\{P\; T\}\}\}+\backslash kappa^\{2\}\backslash Big(\{\backslash frac\{\backslash sigma\_\{L\}+\backslash beta\_\{L\}\backslash sigma\_\{G\}\}\{\backslash sqrt\{P\backslash tau\; T\}\}\}+\{\backslash frac\{\backslash sigma\_\{L\}^\{2\}+\backslash tau\backslash sigma\_\{G\}^\{2\}\}\{\backslash tau\; T\}\}\backslash Big)\backslash right).$$

• For full client participation, this reduces to

$$\underset{t\in [T]}{\min }\mathbb{E}{\parallel \mathrm{\nabla }\tilde{\mathrm{\Phi }}({\mathbf{x}}^{(t)})\parallel }^2\le \mathcal{O}(\frac{1}{\sqrt{n\tau T}}+\frac{1}{T})\mathrm{.}\backslash operatorname*\{min\}\_\{t\backslash in[T]\}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\backslash tilde\{\backslash Phi\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\{\backslash mathcal\{O\}\}\backslash left(\{\backslash frac\{1\}\{\backslash sqrt\{n\backslash tau\; T\}\}\}+\{\backslash frac\{1\}\{T\}\}\backslash right).$$

To reach an ϵ-stationary point, assuming nτ ≤ T, the per-client gradient complexity is T τ = O κ 4 nϵ4 .

Since τ ≤ T /n, the minimum number of communication rounds required is T = O κ 2 ϵ 2 .

• For partial participation, O n − P n − 1 σ 2 G q τ P T is the dominant term, and we do not get any convergence benefit of multiple local updates. Consequently, per-gradient client complexity and number of communication rounds are both T τ = O κ 4 P ϵ4 , for τ = O(1). However, if the data across clients comes from identical distributions (σG = 0), then we recover per-client gradient complexity of O κ 4 P ϵ4 , and number of communication rounds = O κ 2 ϵ 2 .

B.3 Proofs Of The Intermediate Lemmas

Proof of Lemma B.2. Et

X i∈C(t) w˜ig (t) x,i

2 = Et

X i∈C(t) w˜i g (t) x,i − h (t) x,i + h (t) x,i

2 (a) = Et

X i∈C(t) w˜i g (t) x,i − h (t) x,i

2

• Et

X i∈C(t) w˜ih (t) x,i

2

• Et

X i∈C(t) w˜ih (t) x,i

2 (∵ Et[g (t) x,i] = h (t) x,i for all clients i ∈ C(t)) = Et

X i∈C(t) w˜ 2 i g (t) x,i − h (t) x,i

2

= n P Xn i=1 w 2 i Et g (t) x,i − h (t) x,i

2

• Et

X i∈C(t) w˜ih (t) x,i

2 (∵ w˜i = win/P and P(i ∈ C(t)) = P/n) k=0 [a (k) i(τi)]2 σ 2 L + β 2 L ∇xfi(x (t,k) i, y (t,k) i)

2+ Et

X i∈C(t) w˜ih (t) x,i

2 . (18) ≤ n P Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 Here, (a) follows from the following reasoning. Et

X i,j∈C(t) w˜iw˜j Dh (t) x,i, g (t) x,j − h (t) x,jE = Et

X i∈C(t) w˜ 2 i E hDh (t) x,i, g (t) x,i − h (t) x,iE| F(t), C (t)i+ X i̸=j w˜iw˜j E hDh (t) x,i, g (t) x,j − h (t) x,jE| F(t), C (t)i

| {z } =0 (Assumption 3; independence of stochastic gradients across clients) = Et

X i∈C(t) w˜ 2 i E hDh (t) x,i, g (t) x,i − h (t) x,iE| F(t), C (t)i w˜ 2 i ∥ai∥ 2 1 τXi−1 k=0 τXi−1 j=0 a (k) i(τi)a (j) i(τi) E h D∇xfi x (t,k) i, y (t,k) i; ξ (t,k) i − ∇xfi x (t,k) i, y (t,k) i , = Et " X i∈C(t) ∇xfi x (t,j) i, y (t,j) i E | F(t), C (t)i# w˜ 2 i ∥ai∥ 2 1 "τXi−1 = Et " X k=0 [a (k) i(τi)]2 E hD E

h∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i) − ∇xfi(x (t,k) i, y (t,k) i) x (t,k) i, y (t,k) i i , i∈C(t) | {z } =0 ∇xfi(x (t,k) i, y (t,k) i) EF(t), C (t)i# + 2X j<k a (k) i(τi)a (j) i(τi)E hD E h∇xfi(x (t,k) i, y (t,k) i; ξ (t,k) i) − ∇xfi(x (t,k) i, y (t,k) i) x (t,j) i, y (t,j) i i , | {z } =0 ∇xfi(x (t,j) i, y (t,j) i) EF(t), C (t)i# = 0. (∵ Assumption 3) The last inequality in (18) follows from Assumption 1 and 3. Further, we can bound the second term in (18)

as follows.

Et

X i∈C(t) w˜ih (t) x,i − Xn i=1 wih (t) x,i + Xn i=1 wih (t) x,i

2 = Et

Xn i=1 wih (t) x,i

2

• Et

Xn i=1 I(i ∈ C(t)) ˜wih (t) x,i − Xn i=1 wih (t) x,i

2 ((WOR) sampling) = Et

Xn i=1 wih (t) x,i

2 + Xn i=1 Et (I(i ∈ C(t)))2w˜ 2 i + w 2 i − 2I(i ∈ C(t)) ˜wiwi h (t) x,i

2 i̸=j Et D(I(i ∈ C(t)) ˜wi − wi)h (t) x,i,(I(j ∈ C(t)) ˜wj − wj )h (t) x,jE + X = Et

Xn i=1 wih (t) x,i

2 + Xn i=1 Et w 2 i n P − 1 h (t) x,i

2 i̸=j Et hI(i ∈ C(t)) · I(j ∈ C(t)) ˜wiw˜j − I(j ∈ C(t)) ˜wjwi − I(i ∈ C(t)) ˜wiwj + wiwj Dh (t) x,i, h (t) x,jEi + X = Et

Xn i=1 wih (t) x,i

2 + n P − 1 Xn i=1 Et w 2 i h (t) x,i

2+ X i̸=j Et wiwj n P P − 1 n − 1 − 1 Dh (t) x,i, h (t) x,jE

n P P − 1 n − 1 Et

Xn i=1 wih (t) x,i

2 + n P n − P n − 1 Xn i=1 w 2 i Et h (t) x,i

2 , (19) Next, we bound the second term in (19). Xn i=1 w 2 i Et h (t) x,i − ∇xfi(x (t), y (t)) + ∇xfi(x (t), y (t)) 2 ≤ 2L 2 f Xn i=1 w 2 i ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 2(max iwi) β 2 G ∇xFe(x (t), y (t)) 2 + σ 2 G , (20) using Assumption 4. ∆ (t,k) x,y (i) ≜ Et h∥x (t,k) i − x (t)∥ 2 + ∥y (t,k) i − y (t)∥ 2i. Substituting (20) in (19), and using the resulting bound in (18) we get the bound in (13). Proof of Lemma B.3. Since the local functions {fi} satisfy Assumption 5, F(x, ·) is µ-SC for any x. In the proof, we use the quadratic growth property of µ-SC function F(x, ·), i.e., for any given x

$$\frac{\mu }{2}{\parallel \mathbf{y}-{\mathbf{y}}^{\ast }(\mathbf{x})\parallel }^2\le F(\mathbf{x},{\mathbf{y}}^{\ast }(\mathbf{x}))-F(\mathbf{x},\mathbf{y}),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathbf{y},\{\backslash frac\{\backslash mu\}\{2\}\}\backslash left\backslash |\backslash mathbf\{y\}-\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq\; F(\backslash mathbf\{x\},\backslash mathbf\{y\}^\{*\}(\backslash mathbf\{x\}))-F(\backslash mathbf\{x\},\backslash mathbf\{y\}),\backslash quad\{\backslash mathrm\{~for~all~\}\}\backslash mathbf\{y\},$$ $$(21{)}^{\frac{1}{2}}(21)^\{\backslash frac\{1\}\{2\}\}$$

where y ∗(x) = arg maxy′ F(x, y ′). Using LΦ-smoothness of Φ( e ·),

Et hΦ( e x (t+1)) i≤ EtΦ( e x (t)) − Et * ∇Φ( e x (t)), τeffγ s x X i∈C(t) w˜ig (t) x,i++ τ 2 eff[γ s x ] 2LΦ 2 Et g (t) x 2 = Φ( e x (t)) − τeffγ s xEt * ∇Φ( e x (t)), Xn i=1 wih (t) x,i++ τ 2 eff[γ s x ] 2LΦ 2 Et g (t) x 2 (using Assumption 3 and (3)) ∇Φ( e x (t)) 2 + Et Xn i=1 wih (t) x,i 2 + τeffγ s x 2 Et ∇Φ( e x (t)) − Xn i=1 wih (t) x,i 2 = Φ( e x (t)) − τeffγ s x 2 + τ 2 eff[γ s x ] 2LΦ 2 Et g (t) x 2 . (22) The last term is bounded in Lemma B.2. Next, we bound the third term above. Et

∇Φ( e x (t)) − Xn i=1 wih (t) x,i

2 = Et

Xn i=1 wi ∇xfi(x (t), ye ∗(x (t))) − ∇xfi(x (t), y (t)) + ∇xfi(x (t), y (t)) − h (t) x,i 2 (since ye ∗(x) = arg maxy′ Fe(x, y ′)) k=0 a (k) i(τi)∇xfi x (t,k) i, y (t,k) i ! 2 ≤ 2L 2 fEt ye ∗(x (t)) − y (t) 2

• 2Et

Xn i=1 wi

∇xfi(x (t), y (t)) −1 ∥ai∥1 τXi−1 (Lf -smoothness; Young's inequality) ≤ 4L 2 f µ hΦ( e x (t)) − Fe(x (t), y (t)) i+ 2Xn i=1 wi1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)Et ∇xfi(x (t), y (t)) − ∇xfi(x (t,k) i, y (t,k) i)

2 (Quadratic growth of µ-SC functions (21); Jensen's inequality) ≤ 4L 2 f µ hΦ( e x (t)) − Fe(x (t), y (t)) i+ 2L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)Et x (t,k) i − x (t) 2 + y (t,k) i − y (t) 2 (Lf -smoothness)

4L 2 f µ hΦ( e x (t)) − Fe(x (t), y (t)) i+ 2L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i). (23) where ∆ (t,k) x,y (i) ≜ Et x (t,k) i − x (t) 2 + y (t,k) i − y (t) 2. Further, the term containing ∇xfi(x (t,k) i, y (t,k) i)

2 in (13) is bounded in Lemma B.7. Substituting the bounds from (23), (13) and Lemma B.7 into (22), we get ∇Φ( e x (t)) 2 + Et Xn i=1 wih (t) x,i 2 Et hΦ( e x (t+1)) i≤ Φ( e x (t)) − τeffγ s x 2 + τeffγ s x 2 "4L 2 f µ hΦ( e x (t)) − Fe(x (t), y (t)) i+ 2L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) # k=0 a (k) i(τi)∆(t,k) x,y (i) n P P − 1 n − 1 Et Xn i=1 wih (t) x,i 2 + n P n − P n − 1 2L 2 f Xn i=1 w 2 i ∥ai∥1 τXi−1 + τ 2 eff[γ s x ] 2LΦ 2 + τ 2 eff[γ s x ] 2LΦ 2 nσ2L P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + τ 2 eff[γ s x ] 2LΦ 2 n P n − P n − 1 2(max iwi) β 2 G ∇xFe(x (t), y (t)) 2 + σ 2 G w 2 i ∥ai∥ 2 1 τXi−1 $$+,\frac{\tau_{\mathrm{eff}}^2[\gamma_x^s]^2}{2}$$ $$+,\frac{\tau_{\mathrm{eff}}^2[\gamma_x^s]^2}{2}$$ $$\leq\tilde{\Phi}(\mathbf{x}^{(t)})-$$ $$+,\frac{9\tau_{\mathrm{eff}}\gamma_x^s\mathbf{I}}{4\mu}$$ $$+,\frac{\tau_{\mathrm{eff}}^2[\gamma_x^s]^2}{2}$$. 2LΦ n P β 2 L " 2L 2 f Xn i=1 k=0 [a (k) i(τi)]2∆(t,k) x,y (i) + 2σ 2 G

max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# 2LΦ n P β 2 L4β 2 G

max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! "2L 2 f µ Φ( e x (t)) − Fe(x (t), y (t)) + ∇xΦ( e x (t))

2# 7τeffγ s x 16 ∇Φ( e x (t))

2 − τeffγ s x 2 1 − n P P − 1 n − 1 τeffγ s xLΦ Et

Xn i=1 wih (t) x,i

2 xL 2 f hΦ( e x (t)) − Fe(x (t), y (t)) i+ 5 4 τeffγ s xL 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) 2LΦ n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1

• σ 2 G

2(max iwi) n − P n − 1

• 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# , where, the coefficients are simplified, using assumptions on the learning rate γ s x

$$\begin{array}{c}{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_x^s{L}_{\mathrm{\Phi }}[\left({\max }_i{w}_i\right)\left(\frac{n-P}{n-1}\right)+{\beta }_L^2{\max }_{i,k}\frac{w_i{a}_i^{(k)}({\tau }_i)}{\mathrm{\parallel }{a}_i{\mathrm{\parallel }}_1}]\le \frac{P}{4n}\\ {\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_x^s{L}_{\mathrm{\Phi }}[\left({\max }_i{w}_i\right)\left(\frac{n-P}{n-1}\right)+{\beta }_L^2{\max }_i\frac{w_i\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_2^2}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1^2}]\le \frac{P}{32{\beta }_G^{2}n}\mathrm{.}\end{array}\backslash begin\{array\}\{c\}\{\{\backslash tau\_\{\backslash mathrm\{eff\}\}\backslash gamma\_\{x\}^\{s\}L\_\{\backslash Phi\}\backslash left[\backslash left(\backslash operatorname*\{max\}\_\{i\}w\_\{i\}\backslash right)\backslash left(\backslash frac\{n-P\}\{n-1\}\backslash right)+\backslash beta\_\{L\}^\{2\}\backslash operatorname*\{max\}\_\{i,k\}\backslash frac\{w\_\{i\}a\_\{i\}^\{(k)\}(\backslash tau\_\{i\})\}\{\backslash |a\_\{i\}\backslash |\_\{1\}\}\backslash right]\backslash leq\backslash frac\{P\}\{4n\}\}\}\backslash \backslash \; \{\{\backslash tau\_\{\backslash mathrm\{eff\}\}\backslash gamma\_\{x\}^\{s\}L\_\{\backslash Phi\}\backslash left[\backslash left(\backslash operatorname*\{max\}\_\{i\}w\_\{i\}\backslash right)\backslash left(\backslash frac\{n-P\}\{n-1\}\backslash right)+\backslash beta\_\{L\}^\{2\}\backslash operatorname*\{max\}\_\{i\}\backslash frac\{w\_\{i\}\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{2\}^\{2\}\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}^\{2\}\}\backslash right]\backslash leq\backslash frac\{P\}\{32\backslash beta\_\{G\}^\{2\}n\}.\}\}\backslash end\{array\}$$

This finishes the proof.

Proof of Lemma B.4. We use the client update equations for individual iterates (11). To bound ∆ (t,k) x,y (i), first we bound a single component term Et x (t,k) i − x (t) 2 . For 1 ≤ k ≤ τi,

Et x (t,k) i − x (t) 2 = [η c x ] 2Et

k X−1 j=0 a (j) i(k) ∇xfi x (t,j) i, y (t,j) i; ξ (t,j) i − ∇xfi x (t,j) i, y (t,j) i

• ∇xfi x (t,j) i, y (t,j) i

2 = [η c x ] 2

Et

k X−1 j=0 a (j) i(k) ∇xfi x (t,j) i, y (t,j) i; ξ (t,j) i − ∇xfi x (t,j) i, y (t,j) i

2

• Et

k X−1 j=0 a (j) i(k)∇xfi x (t,j) i, y (t,j) i

2

(using unbiasedness in Assumption 3) = [η c x ] 2

k X−1 j=0 [a (j) i(k)]2Et ∇xfi x (t,j) i, y (t,j) i; ξ (t,j) i − ∇xfi x (t,j) i, y (t,j) i

2

• Et

k X−1 j=0 a (j) i(k)∇xfi x (t,j) i, y (t,j) i

2

≤ [η c x ] 2

k X−1 j=0 [a (j) i(k)]2 σ 2 L + β 2 LEt ∇xfi(x (t,j) i, y (t,j) i)

2+

k X−1 j=0 a (j) i(k)

k X−1 j=0 a (j) i(k)Et ∇xfi x (t,j) i, y (t,j) i

2

. (24) where the last inequality follows from Jensen's inequality (Lemma A.3). Next, note that

1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) k X−1 j=0 [a (j) i(k)]2 ≤1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) τXi−2 k=0 [a (j) i(k)]2 = τXi−2 k=0 [a (j) i(k)]2 ≤ ∥ai∥ 2 2 − [a (t,τi−1) i(τi)]2, 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) k X−1 j=0 a (j) i(k) ≤ 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) τXi−2 k=0 a (j) i(k) = τXi−2 k=0 a (j) i(k) ≤ ∥ai∥1 − [a (t,τi−1) i(τi)], 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) k X−1 j=0 [a (j) i(k)]2 ≤1 ∥ai∥1 τXi−1 k=0 a (k) i(τi) τXi−2 j=0 [a (j) i(k)]2 ≤ α · τXi−2 k=0 a (k) i(τi), (25) where a (j) i(k) ≤ α for all j, k. We define ∥ai,−1∥ 2 2 ≜ ∥ai∥ 2 2 − [a (t,τi−1) i(τi)]2, ∥ai,−1∥1 ≜ ∥ai∥1 − [a (t,τi−1) i(τi)] for the sake of brevity. Using (25), we bound the individual terms in (24). $$\frac{1}{\mathrm{\parallel }{\mathbf{a}}_i\mathrm{\parallel }}\sum _{k=0}^{{\tau }_i-1}{a}_i^{(k)}({\tau }_i)\sum _{j=0}^{k-1}[a_i^{(j)}(k){]}^2{\beta }_L^2{\parallel {\mathrm{\nabla }}_x{f}_i({\mathbf{x}}_i^{(t,j)},{\mathbf{y}}_i^{(t,j)})\parallel }^2\backslash frac\{1\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\}\backslash sum\_\{k=0\}^\{\backslash tau\_\{i\}-1\}a\_\{i\}^\{(k)\}(\backslash tau\_\{i\})\backslash sum\_\{j=0\}^\{k-1\}[a\_\{i\}^\{(j)\}(k)]^\{2\}\backslash beta\_\{L\}^\{2\}\backslash left\backslash |\backslash nabla\_\{x\}f\_\{i\}(\backslash mathbf\{x\}\_\{i\}^\{(t,j)\},\backslash mathbf\{y\}\_\{i\}^\{(t,j)\})\backslash right\backslash |^\{2\}$$ $$\leq2\beta_{L}^{2}\alpha L_{f}^{2}\sum_{j=0}^{\tau_{i}-2}[a_{i}^{(j)}(k)]\Delta_{\mathbf{x},\mathbf{y}}^{(t,j)}+2\beta_{L}^{2}\left|\mathbf{a}{i,-1}\right|{2}^{2}\left|\nabla_{x}f_{i}(\mathbf{x}^{(t)},\mathbf{y}^{(t)})\right|^{2}.\tag{26}$$ Similarly,

k=0 a (k) i(τi) k X−1 j=0 a (j) i(k) 1 ∥ai∥1 τXi−1 k X−1 j=0 a (j) i(k)Et ∇xfi x (t,j) i, y (t,j) i 2 k=0 a (k) i(τi) τXi−2 j=0 a (j) i(k) ≤2 ∥ai∥1 τXi−1 τXi−2 j=0 a (j) i(k) L 2 f∆(t,j) x,y + ∇xfi(x (t), y (t)) 2 ≤ 2 ∥ai,−1∥1 L 2 f τXi−2 j=0 a (j) i(k)∆(t,j) x,y + 2 ∥ai,−1∥ 2 1 ∇xfi(x (t), y (t)) 2 . (27) Substituting (26), (27) in (24), we get

1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)Et x (t,k) i − x (t) 2 ≤ [η c x ] 2σ 2 L ∥ai,−1∥ 2 2 + 2[η c x ] 2L 2 f ∥ai,−1∥1 + β 2 Lα τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 2[η c x ] 2∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 ∇xfi(x (t), y (t)) 2 . (28) $$(27)(27)$$ $${\mathbf{\Sigma }}^0(i)\backslash mathbf\{\backslash Sigma\}^\{0\}(i)$$ Similarly, we can bound the y error

1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)Et y (t,k) i − y (t) 2 ≤ [η c y ] 2σ 2 L ∥ai,−1∥ 2 2 + 2[η c y ] 2L 2 f ∥ai,−1∥1 + β 2 Lα τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 2[η c y ] 2∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 ∇yfi(x (t), y (t)) 2 . (29) $${\mathbf{\Sigma }}^{)}(i)\backslash mathbf\{\backslash Sigma\}^\{)\}(i)$$ Combining the two bounds in (28) and (29), we get

1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)Et x (t,k) i − x (t) 2 + y (t,k) i − y (t) 2≤[η c x ] 2 + [η c y ] 2σ 2 L ∥ai,−1∥ 2 2 + 2 [η c x ] 2 + [η c y ] 2L 2 f ∥ai∥1 ∥ai,−1∥1 + β 2 Lα 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i)

• 2 ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 [η c x ] 2∇xfi x (t), y (t) 2 + [η c y ] 2∇yfi x (t), y (t) 2. (30) Define Am ≜ 2L 2 f [η c x ] 2 + [η c y ] 2maxi ∥ai∥1 ∥ai,−1∥1 + β 2 Lα. Rearranging the terms in (30), and taking the weighted sum over agents, we get

wi ∥ai∥1 τXi−1 L 2 f Xn i=1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ [η c x ] 2 + [η c y ] 2L 2 fσ 2 L 1 − Am Xn i=1 wi ∥ai,−1∥ 2 2 + 2L 2 f 1 − Am Xn i=1 wi ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 [η c x ] 2∇xfi x (t), y (t) 2 + [η c y ] 2∇yfi x (t), y (t) 2 ≤ [η c x ] 2 + [η c y ] 2L 2 f 1 − Amσ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2L 2 fMa−1 1 − Am [η c x ] 2 β 2 G Xn i=1 wi∇xfi x (t), y (t) 2 + σ 2 G + 2L 2 fMa−1 1 − Am [η c y ] 2 β 2 G Xn i=1 wi∇yfi x (t), y (t) 2 + σ 2 G . (31) where (31) follows from Assumption 4, and we define Ma−1 ≜ maxi ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 . We bounded ∇xFx (t), y (t) 2in Lemma B.6. Similarly, we can bound∇yFx (t), y (t) 2as follows.

\left\|\nabla_{\mathbf{y}}F\left(\mathbf{x}^{(t)},\mathbf{y}^{(t)}\right)\right\|^{2}=\left\|\nabla_{\mathbf{y}}F\left(\mathbf{x}^{(t)},\mathbf{y}^{(t)}\right)-\nabla_{\mathbf{y}}F\left(\mathbf{x}^{(t)},\mathbf{y}^{\prime}(\mathbf{x}^{(t)})\right)\right\|^{2}\qquad\left(\cdot\cdot\mathbf{y}^{\prime}(\mathbf{x})=\arg\max_{\mathbf{y}^{\prime}}F(\mathbf{x},\mathbf{y}^{\prime})\right)\right\| $$\leq2L_{f}\left[\widehat{\Phi}(\mathbf{x}^{(t)})-\widehat{F}(\mathbf{x}^{(t)},\mathbf{y}^{(t)})\right].\tag{32}$$

using Lf -smoothness and concavity of F(x, ·) (Lemma A.6). Also, for the choice of η c x , ηcy , we get Am ≤ 1/2.

Consequently, substituting the two bounds in (31), we complete the proof.

Proof of Lemma B.5. First we see that

Et hΦ( e x (t+1)) − Fe(x (t+1), y (t+1)) i = Et hΦ( e x (t+1)) − Φ( e x (t)) i+ hΦ( e x (t)) − Fe(x (t), y (t)) i+ Et hFe(x (t), y (t)) − Fe(x (t+1), y (t+1)) i. (33) Et hΦ( e x (t+1)) − Φ( e x (t)) iis already bounded in Lemma B.3. We bound Et hFe(x (t), y (t)) − Fe(x (t+1), y (t+1)) i as follows. Using the notation z (t) = (x (t), y (t)) and using Lf -smoothness (Assumption 1) of Fe(·, ·), −Fe(z (t+1)) ≤ −Fe(z (t)) − D∇zFe(z (t)), z (t+1) − z (t)E+ Lf 2 z (t+1) − z (t) 2 = −Fe(z (t)) − D∇yFe(x (t), y (t)), y (t+1) − y (t)E+ Lf 2 y (t+1) − y (t) 2 − D∇xFe(x (t), y (t)), x (t+1) − x (t)E+ Lf 2 x (t+1) − x (t) 2 ⇒ −Et hFe(x (t+1), y (t+1)) i≤ −Fe(x (t), y (t)) − τeffγ s yEt * ∇yFe(x (t), y (t)), Xn i=1 wih (t) y,i++ τ 2 eff[γ s y ] 2Lf 2 Et g (t) y 2 + τeffγ s xEt * ∇xFe(x (t), y (t)), Xn i=1 wih (t) x,i++ τ 2 eff[γ s x ] 2Lf 2 Et g (t) x 2 . Rearranging the terms a bit, we get Et hFe(x (t), y (t)) − Fe(x (t+1), y (t+1)) i ≤ − τeffγ s y 2 Et ∇yFe(x (t), y (t)) 2 + Xn i=1 wih (t) y,i 2 − ∇yFe(x (t), y (t)) − Xn i=1 wih (t) y,i 2 + τ 2 eff[γ s y ] 2Lf 2 Et g (t) y 2 + τeffγ s x 2 Et ∇xFe(x (t), y (t)) 2 + Xn i=1 wih (t) x,i 2 − ∇xFe(x (t), y (t)) − Xn i=1 wih (t) x,i 2 + τ 2 eff[γ s x ] 2Lf 2 Et g (t) x 2 . (34) Next, we bound the individual terms in (34). Using quadratic growth of µ-SC functions (Lemma A.5),

$${\parallel {\mathrm{\nabla }}_y\tilde{F}({\mathbf{x}}^{(t)},{\mathbf{y}}^{(t)})\parallel }^2\ge 2\mu [\tilde{\mathrm{\Phi }}({\mathbf{x}}^{(t)})-\tilde{F}({\mathbf{x}}^{(t)},{\mathbf{y}}^{(t)})]\mathrm{.}\backslash left\backslash |\backslash nabla\_\{y\}\backslash widetilde\{F\}(\{\backslash bf\; x\}^\{(t)\},\{\backslash bf\; y\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash geq2\backslash mu\backslash left[\backslash widetilde\{\backslash Phi\}(\{\backslash bf\; x\}^\{(t)\})-\backslash widetilde\{F\}(\{\backslash bf\; x\}^\{(t)\},\{\backslash bf\; y\}^\{(t)\})\backslash right].$$ $\mathfrak{c}^{(t)},{\bf y}^{(t)})-\sum_{i=1}^{n}w_{i}{\bf h}_{{\bf x},i}^{(t)}\right|^{2}$, we use similar reasoning as in (23).
To bound ∇yFe(x (t), y i=1 wih y,i , we use similar reasoning as in (23). Et ∇yFe(x (t), y (t)) − Xn i=1 wih (t) y,i 2 = Xn i=1 wi ∇yfi(x (t), y (t)) − h (t) y,i 2 ≤ L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i), (35) Substituting the bounds in (35) and Lemma B.6 into (34) we get Substituting the bounds in (35) and Lemma B.6 into (34) we get Et hFe(x (t), y (t)) − Fe(x (t+1), y (t+1)) i hΦ( e x (t)) − Fe(x (t), y (t)) i− τeffγ s y 2 Et Xn i=1 wih (t) y,i 2 ≤ −τeffµγs y 1 − 2κ 2γ s x γ s y + τeffγ s x ∇Φ( e x (t)) 2 + τeffγ s x 2 Et Xn i=1 wih (t) x,i 2 + τ 2 effLf 2 [γ s y ] 2Et g (t) y 2 + [γ s x ] 2Et g (t) x 2 + τeffγ s yL 2 f 2 Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i). (36) (t) x 2is already bounded in Lemma B.2. We further substitute the bound on Et ∇xfi(x (t,k) i, y (t,k) i) 2 Et g from Lemma B.7 to get

Et g (t) x

2 ≤ n(P − 1) P(n − 1)Et

Xn i=1 wih (t) x,i

2 + σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G P n(n − P) n − 1max iwi

• β 2 Ln P " 2L 2 f Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2∆(t,k) x,y (i) + 4β 2 G

max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! ∇xΦ( e x (t))

2# + β 2 Ln P2 max i wi∥ai∥ 2 2 ∥ai∥ 2 1 hσ 2 G + 4β 2 GLfκ hΦ( e x (t)) − Fe(x (t), y (t)) ii w 2 i ∥ai∥1 τXi−1 + n(n − P) n − 1 2L 2 f P Xn i=1 k=0 a (k) i(τi)∆(t,k) x,y (i) + n(n − P) n − 1max iwi 2β 2 G P 4Lfκ hΦ( e x (t)) − Fe(x (t), y (t)) i+ 2 ∇xΦ( e x (t))

2 34

≤ n(P − 1) P(n − 1)Et Xn i=1 wih (t) x,i 2 + σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2nσ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! wi ∥ai∥1 τXi−1 + n − P n − 1 max iwi + β 2 L max i,k wia (k) i(τi) ∥ai∥1 !2nL2 f P Xn i=1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 4nβ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! ∇xΦ( e x (t)) 2 + 8nβ2GLfκ P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !hΦ( e x (t)) − Fe(x (t), y (t)) i. (37) Similarly, we can bound Et (t) y 2to get g

Et g (t) y

2 ≤ n(P − 1) P(n − 1)Et

Xn i=1 wih (t) y,i

2 + σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 Gn P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! + n − P n − 1 max iwi + β 2 L max i,k wia (k) i(τi) ∥ai∥1 !2nL2 f P Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 4β 2 GLfn P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !hΦ( e x (t)) − Fe(x (t), y (t)) i. (38) Substituting (37), (38) in (36), and again substituting the resulting bound in (33), we get Et hΦ( e x (t+1)) − Fe(x (t+1), y (t+1)) i (a) ≤

1 − τeffµγs y

1 − κ 2γ s x γ s y 2 + 9 4 − τeffκγs y 4nβ2GLf P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !!! hΦ( e x (t)) − Fe(x (t), y (t)) i

• τeffγ s x 9 16
• τeffLf γ s x 2nβ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !! ∇Φ( e x (t))

2 + τ 2 eff[γ s x ] 2LΦ 2 n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1

• σ 2 G

2(max iwi) n − P n − 1

• 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#
• τ 2 effLf 2 [γ s y ] 2 + [γ s x ] 2"σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1
• 2nσ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#

− τeffγ s x 2 1 − n(P − 1) P(n − 1) τeffγ s xLΦ − 1 − n(P − 1) P(n − 1) τeffγ s xLf Et

Xn i=1 wih (t) x,i

2 − τeffγ s y 2 1 − n(P − 1) P(n − 1) τeffγ s yLf Et

Xn i=1 wih (t) y,i

2

• τeff 3γ s x 2+ 3γ s y 4 L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) (b) ≤

1 − τeffµγs y

1 − κ 2γ s x γ s y 2 + 9 4 − τeffκγs y 4nβ2GLf P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !!! hΦ( e x (t)) − Fe(x (t), y (t)) i

• τeffγ s x 9 16
• τeffLf γ s x 2nβ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !! ∇Φ( e x (t))

2 + τ 2 eff[γ s x ] 2 2 n(P − 1) P(n − 1)(LΦ + Lf )Et

Xn i=1 wih (t) x,i

2 35 − τeffγ s x 2

1 − n(P − 1) P(n − 1) τeff [γ s y ] 2 γ sx Lf !Et

Xn i=1 wih (t) y,i

2 + τ 2 eff[γ s x ] 2LΦ 2 n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1

• σ 2 G

2(max iwi) n − P n − 1

• 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#
• τ 2 effLf 2 [γ s y ] 2 + [γ s x ] 2"σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1
• 2nσ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#
• τeffγ s y " 2[η c x ] 2 + [η c y ] 2L 2 fσ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 4L 2 fMa−1 [η c x ] 2 + [η c y ] 2σ 2 G

• τeffγ s y 8L 2 fMa−1 β 2 G[η c x ] 2∇Φ( e x (t))

2

• 8L 3 fMa−1 β 2 G 2κ[η c x ] 2 + [η c y ] 2hΦ( e x (t)) − Fe(x (t), y (t)) i, (39) where (a) follows since $${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{L}_f{\gamma }_y^s(\frac{n-P}{n-1}\underset{i}{\max }{w}_i+{\beta }_L^2\underset{i,k}{\max }\frac{w_i{a}_i^{(k)}({\tau }_i)}{\mathrm{\parallel }{a}_i{\mathrm{\parallel }}_1})\frac{n}{P}\le \frac{1}{4},\backslash tau\_\{\backslash mathrm\{eff\}\}L\_\{f\}\backslash gamma\_\{y\}^\{s\}\backslash left(\backslash frac\{n-P\}\{n-1\}\backslash operatorname*\{max\}\_\{i\}w\_\{i\}+\backslash beta\_\{L\}^\{2\}\backslash operatorname*\{max\}\_\{i,k\}\backslash frac\{w\_\{i\}a\_\{i\}^\{(k)\}(\backslash tau\_\{i\})\}\{\backslash |\backslash mathbf\{\backslash mathit\{a\}\}\_\{i\}\backslash |\_\{1\}\}\backslash right)\backslash frac\{n\}\{P\}\backslash leq\backslash frac\{1\}\{4\},$$ and (b) follows by using the bound in Lemma B.4. Next, we simplify the coefficients of different terms in (39).

• Coefficient of $\mathbb{E}\left[\widetilde{\Phi}(\mathbf{x}^{(t)})-\widetilde{F}(\mathbf{x}^{(t)},\mathbf{y}^{(t)})\right]$ can be simplified to . 1 − τeffµγs y 1 − κ 2γ s x γ s y 17 4 − τeffκγs y 4nβ2GLf P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !!! + τeffγ s y8L 3 fMa−1 β 2 G 2κ[η c x ] 2 + [η c y ] 2 ≤ 1 − τeffγ s yµ 4. using γ s x ≤γ s y 17κ2 , τeffγ s yκLfβ 2 G n P max nβ 2 L maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 , n−P n−1 maxi wi

$$\}\le \frac{1}{64},\kappa L_f{\beta }_G{\eta }_x^c\le \frac{1}{16\sqrt{2M_{{\mathbf{n}}_{-1}}}}\backslash Big\backslash \}\backslash leq\{\backslash frac\{1\}\{64\}\},\backslash ;\backslash kappa\; L\_\{f\}\backslash beta\_\{G\}\backslash eta\_\{x\}^\{c\}\backslash leq\{\backslash frac\{1\}\{16\backslash sqrt\{2M\_\{\backslash mathbf\{n\}\_\{-1\}\}\}\}\}$$

and LfβGη c y ≤1 16√κMa−1 .

• Coefficient of E $|\sum_{i=1}^{n}\Phi_{i}\mathbf{n}{\Phi,i}|$ can be simplified to $$\frac{\tau{\text{eff}}^{2}[\gamma_{x}^{s}]^{2}}{2}(L_{\Phi}+L_{f})\frac{n(P-1)}{P(n-1)}\leq\tau_{\text{eff}}^{2}[\gamma_{x}^{s}]^{2}L_{\Phi}\frac{n(P-1)}{P(n-1)}.$$ $\sum_{i=1}^n w_i\mathbf{h}_{\mathbf{x},i}^{(t)}\Big|^2$ can be simplified to .
• Coefficient of E ∇Φ( e x (t))

2 can be simplified to

τeffγ s x 9 16 + τeffLf γ s x 2nβ2G P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !! + τeffγ s y8L 2 fMa−1 β 2 G[η c x ] 2 ≤ τeffγ s x 9 16 + 4γ s x γ s y 1 64κ + τeffγ s y 64κ 2 ≤ τeffγ s y 48κ 2 . ( γ s x γs y ≤1 using γ s x ≤γ s y 81κ2 , η c xLfβG ≤1 64κ √Ma−1 and τeffγ s yκLfβ 2 G n P max nn−P n−1 maxi wi, β2L maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 $$(\frac{{\gamma }_x^s}{{\gamma }_y^s}\le \frac{1}{81{\kappa }^2})(\backslash frac\{\backslash gamma\_\{x\}^\{s\}\}\{\backslash gamma\_\{y\}^\{s\}\}\backslash leq\backslash frac\{1\}\{81\backslash kappa^\{2\}\})$$ o≤ 1 64 .

• Coefficient of σ 2 L can be simplified to

τ 2 eff[γ s x ] 2LΦ 2 n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 # + τ 2 effLf 2 [γ s y ] 2 + [γ s x ] 2"n P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 # + τeffγ s y " 2[η c x ] 2 + [η c y ] 2L 2 f Xn i=1 wi ∥ai,−1∥ 2 2 # ≤ τ 2 eff [γ s y ] 2Lf + [γ s x ] 2LΦ n P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2τeffγ s y [η c x ] 2 + [η c y ] 2L 2 f Xn i=1 wi ∥ai,−1∥ 2 2 .

• Coefficient of σ 2 G can be simplified to

τ 2 eff[γ s x ] 2LΦ 2 n P " 2(max iwi) n − P n − 1 + 2β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 # + τ 2 effLf 2 [γ s y ] 2 + [γ s x ] 2 2n P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! + τeffγ s y4L 2 fMa−1 [η c x ] 2 + [η c y ] 2 ≤ τ 2 eff [γ s y ] 2Lf + [γ s x ] 2LΦ 2n P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! + 4τeffγ s y [η c x ] 2 + [η c y ] 2L 2 fMa−1 . Substituting these simplified coefficients in (39), and rearranging the terms, we get

Et hΦ( e x (t+1)) − Fe(x (t+1), y (t+1)) i ≤ 1 − τeffγ s yµ 4 hΦ( e x (t)) − Fe(x (t), y (t)) i+ τeffγ s y 48κ 2 ∇Φ( e x (t)) 2 + τ 2 eff[γ s x ] 2LΦ n(P − 1) P(n − 1)Et Xn i=1 wih (t) x,i 2 + τ 2 eff [γ s y ] 2Lf + [γ s x ] 2LΦ n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# + 2τeffγ s y [η c x ] 2 + [η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2σ 2 GMa−1 # . (40) Summing both sides of (40) over t = 0, . . . , T − 1 and rearranging the terms, we get 1 T T X−1 t=0 E hΦ( e x (t)) − Fe(x (t), y (t)) i Φ( e x (0)) − Fe(x (0), y (0)) T− E Φ( e x (T)) − Fe(x (T), y (T)) ≤4 τeffγ s yµ + 1 12µκ2 1 T T X−1 t=0 E ∇Φ( e x (t)) 2 T + 4τeff[γ s x ] 2LΦ γ s yµ n(P − 1) P(n − 1)E Xn i=1 wih (t) x,i 2 + 8κLf [η c x ] 2 + [η c y ] 2"σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2σ 2 GMa−1 # + 8τeffγ s yκ n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# , (41) where, we use [γ s y ] 2Lf ≥ [γ s x ] 2LΦ. This concludes the proof.

B.4 Auxiliary Lemmas

Lemma B.6. If the local client function fi(x, ·) satisfy Assumptions 1, 5 (Lf -smoothness and µ-strong concavity in y), then the function F satisfies

$${\parallel {\mathrm{\nabla }}_x\tilde{F}(\mathbf{x},\mathbf{y}\parallel }^2\le 2\parallel \mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x}){\parallel }^2+\frac{4L_f^2}{\mu }[\tilde{\mathrm{\Phi }}(\mathbf{x})-\tilde{F}(\mathbf{x},\mathbf{y})]\mathrm{.}\backslash left\backslash |\backslash nabla\_\{x\}\backslash widetilde\{F\}(\{\backslash bf\; x\},\{\backslash bf\; y\}\backslash right\backslash |^\{2\}\backslash leq2\backslash big\backslash |\backslash nabla\backslash widetilde\{\backslash Phi\}(\{\backslash bf\; x\})\backslash big\backslash |^\{2\}+\backslash frac\{4L\_\{f\}^\{2\}\}\{\backslash mu\}\backslash left[\backslash widetilde\{\backslash Phi\}(\{\backslash bf\; x\})-\backslash widetilde\{F\}(\{\backslash bf\; x\},\{\backslash bf\; y\})\backslash right].$$

Proof. $${\parallel {\mathrm{\nabla }}_x\tilde{F}(\mathbf{x},\mathbf{y}\parallel }^2\backslash left\backslash |\backslash nabla\_\{x\}\{\backslash tilde\{F\}\}(\backslash mathbf\{x\},\backslash mathbf\{y\}\backslash right\backslash |^\{2\}$$ 2 $$\le 2{\parallel \mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x})\parallel }^2+2{\parallel {\mathrm{\nabla }}_{x}F(\mathbf{x},\mathbf{y})-\mathrm{\nabla }\tilde{\mathrm{\Phi }}(\mathbf{x})\parallel }^2\backslash leq2\backslash left\backslash |\backslash nabla\backslash tilde\{\backslash Phi\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}+2\backslash left\backslash |\backslash nabla\_\{x\}F\backslash left(\backslash mathbf\{x\},\backslash mathbf\{y\}\backslash right)-\backslash nabla\backslash tilde\{\backslash Phi\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}$$ $$\leq2\left|\nabla\tilde{\Phi}(\mathbf{x})\right|^{2}+2L_{f}^{2}\left|\tilde{\mathbf{y}}^{*}(\mathbf{x})-\mathbf{y}\right|^{2}\qquad\qquad\qquad\left(L_{f}\text{-smoothness(Assumption1)}\right)\right|$$ $$\leq2\left|\nabla\tilde{\Phi}(\mathbf{x})\right|^{2}+\frac{4L_{f}^{2}}{\mu}\left[\tilde{\Phi}(\mathbf{x})-\widetilde{F}(\mathbf{x},\mathbf{y})\right].\qquad\qquad\qquad\qquad\text{(Assumption5)}$$ Lemma B.7. If the local client function fi(x, ·) satisfy Assumptions 1, 4 and 5, then the iterates {x (t,k) i, y (t,k) i}i,(t,k) generated by Algorithm 1 satisfy

Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2E∇xfi(x (t,k) i, y (t,k) i) 2 ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2σ 2 G max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! + 4β 2 G max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! "2L 2 f µ E Φ( e x (t)) − Fe(x (t), y (t)) + ∇xΦ( e x (t)) 2# .

Proof. Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2E ∇xfi(x (t,k) i, y (t,k) i) ± ∇xfi(x (t), y (t)) 2 ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 fE x (t,k) i − x (t) 2 + y (t,k) i − y (t) 2+ 2Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 E ∇xfi(x (t), y (t)) 2 (Lf -smoothness) ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2 max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! β 2 G ∇xFe(x (t), y (t)) 2 + σ 2 G (Assumption 4) Using Lemma B.6 gives the result.

Lemma B.8. If the local client function fi(x, ·) satisfy Assumptions 1, 4 and 5, then the iterates {x (t,k) i, y (t,k) i}i,(t,k) generated by Algorithm 1 satisfy

Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2E∇yfi(x (t,k) i, y (t,k) i) 2 ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2 max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !hσ 2 G + 2β 2 GLfE Φ( e x (t)) − Fe(x (t), y (t)) i . 38 Proof. Following closely the proof of Lemma B.7,

Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2E ∇yfi(x (t,k) i, y (t,k) i) ± ∇yfi(x (t), y (t))

2 ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 fE x (t,k) i − x (t) 2 + y (t,k) i − y (t) 2+ 2Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 E

∇yfi(x (t), y (t))

2 (Lf -smoothness) ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2 max i wi∥ai∥ 2 2 ∥ai∥ 2 1 ! β 2 G ∇yFe(x (t), y (t))

2

• σ 2 G

(Assumption 4) ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2 max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !hσ 2 G + 2β 2 GLfE Φ( e x (t)) − Fe(x (t), y (t)) i . where the final inequality follows from smoothness and concavity of F in y.

B.5 Convergence Under Polyak Łojasiewicz (Pl) Condition

In case the global function satisfies Assumption 6, the results in this section follow with minor modifications.

The crucial difference is that Lemma A.6 no longer holds. Lemma B.2 and Lemma B.3 follow exactly. The statement of Lemma B.4 needs some modification, since we use Lemma A.6 in the proof.

Lemma B.9. Suppose the local loss functions {fi} satisfy Assumptions 1, 4, 6, and the stochastic oracles for the local functions satisfy Assumption 3. Under the conditions of Lemma B.9, the iterates {x (t) i, y (t) i} generated by Algorithm 1 satisfy

pi ∥ai∥1 τXi−1 L 2 f Xn i=1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ 2[η c x ] 2 + [η c y ] 2L 2 fσ 2 L Xn i=1 pi ∥ai,−1∥ 2 2 + 4L 2 fMa−1 [η c x ] 2 + [η c y ] 2σ 2 G + 8L 2 fMa−1 β 2 G[η c x ] 2E ∇Φ( e x (t)) 2 + 8κL 3 fMa−1 β 2 G 2[η c x ] 2 + [η c y ] 2E hΦ( e x (t)) − Fe(x (t), y (t)) i, where Ma−1 ≜ maxi ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 .

$$M_{{\mathbf{a}}_{-1}}\triangleq \underset{i}{\max }({\parallel a_{i,-1}\parallel }_1^2+{\beta }_L^2{\parallel a_{i,-1}\parallel }_2^2)\mathrm{.}M\_\{\backslash mathbf\{a\}\_\{-1\}\}\backslash triangleq\backslash operatorname*\{max\}\_\{i\}\backslash left(\backslash left\backslash |a\_\{i,-1\}\backslash right\backslash |\_\{1\}^\{2\}+\backslash beta\_\{L\}^\{2\}\backslash left\backslash |a\_\{i,-1\}\backslash right\backslash |\_\{2\}^\{2\}\backslash right).$$ The bound in Lemma B.8 also changes to

Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2E∇yfi(x (t,k) i, y (t,k) i) 2 ≤ 2 Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2L 2 f∆(t,k) x,y (i) + 2 max i pi∥ai∥ 2 2 ∥ai∥ 2 1 !hσ 2 G + 2β 2 GκLfE Φ( e x (t)) − Fe(x (t), y (t)) i . The same bound in Lemma B.5 holds, but with more stringent conditions on learning rates, namely η c yLfβG ≤1 16κ √Ma−1and τeffγ s yκLfβ 2 G 1 P max nκβ2L maxi ∥ai∥ 2 2 ∥ai∥ 2 1 , 1 o≤1 64 . Consequently, the bounds in Theorem 1 hold, under slightly more stringent conditions on the learning rates.

C Convergence Of Fed-Norm-Sgda+ For Nonconvex Concave Functions (Theorem 2)

We organize this section as follows. First, in Appendix C.1 we present some intermediate results, which we use in the proof of Theorem 2. Next, in Appendix C.2, we present the proof of Theorem 2, which is followed by the proofs of the intermediate results in Appendix C.3. Finally, we discuss the extension of our results to nonconvex-one-point-concave functions in Appendix C.4.

The problem we solve is

$$\underset{\mathbf{x}}{\min }\underset{\mathbf{y}}{\max }\{\tilde{F}(\mathbf{x},\mathbf{y})\triangleq \sum _{i=1}^n{w}_i{f}_i(\mathbf{x},\mathbf{y})\}\mathrm{.}\backslash operatorname*\{min\}\_\{\backslash mathbf\{x\}\}\backslash operatorname*\{max\}\_\{\backslash mathbf\{y\}\}\backslash left\backslash \{\{\backslash widetilde\{F\}\}(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash triangleq\backslash sum\_\{i=1\}^\{n\}w\_\{i\}f\_\{i\}(\backslash mathbf\{x\},\backslash mathbf\{y\})\backslash right\backslash \}.$$

We define Φe(x) ≜ maxy Fe(x, y) and ye ∗(x) ∈ arg maxy Fe(x, y). Since Fe(x, ·) is no longer strongly concave, y ∗(x) need not be unique. In Algorithm 1-Fed-Norm-SGDA+ , the client updates are given by

where 1 ≤ k ≤ τi. The server updates are given by

$$\begin{array}{cccc}& {\mathbf{x}}^{(t+1)}={\mathbf{x}}^{(t)}-{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_x^s{\mathbf{g}}_{\mathbf{x}}^{(t)},{\mathbf{y}}^{(t+1)}={\mathbf{y}}^{(t)}+{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_y^s{\mathbf{g}}_{\mathbf{y}}^{(t)},& & \text{(43)}\end{array}\{\backslash bf\; x\}^\{(t+1)\}=\{\backslash bf\; x\}^\{(t)\}-\backslash tau\_\{\backslash rm\; eff\}\backslash gamma\_\{x\}^\{s\}\{\backslash bf\; g\}\_\{\backslash bf\; x\}^\{(t)\},\backslash qquad\{\backslash bf\; y\}^\{(t+1)\}=\{\backslash bf\; y\}^\{(t)\}+\backslash tau\_\{\backslash rm\; eff\}\backslash gamma\_\{y\}^\{s\}\{\backslash bf\; g\}\_\{\backslash bf\; y\}^\{(t)\},\backslash tag\{43\}$$

where g (t) x , g (t) y are defined in (3). The normalized (stochastic) gradient vectors are defined as

g (t) x,i = 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)∇xfi x (t,k) i, y (t,k) i; ξ (t,k) i ; h (t) x,i = 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)∇xfi x (t,k) i, y (t,k) i , k=0 a (k) i(τi)∇yfi xb (s), y (t,k) i . (44) g (t) y,i = 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)∇yfi xb (s), y (t,k) i; ξ (t,k) i ; h (t) y,i = 1 ∥ai∥1 τXi−1

C.1 Intermediate Lemmas

As discussed in Section 5.2, we analyze the convergence of the smoothed envelope function Φe1/2Lf . We begin with a bound on the one-step decay of this function.

Lemma C.1 (One-step decay of Smoothed-Envelope). Suppose the local loss functions {fi} satisfy Assumptions 1, 3, 7, and 8. Then, the iterates generated by Algorithm 1-Fed-Norm-SGDA+ satisfy

E hΦe1/2Lf (x (t+1)) i≤ E hΦe1/2Lf (x (t)) i+ τ 2 eff[γ s x ] 2Lf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x + G 2 x P − 1 n − 1 + n − P n − 1 Xn i=1 w 2 i !# + 2τeffγ s x ( L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + LfE hΦ( e x (t)) − Fe(x (t), y (t)) i)− τeffγ s x 8 E ∇Φe1/2Lf (x (t)) 2 , where ∆ (t,k) x,y (i) = E h∥x (t,k) i − x (t)∥ 2 + ∥y (t,k) i − y (t)∥ 2iis the drift of client i ∈ [n], at the k-th local step of epoch t.

Between two successive synchronization time instants (for example, t, t + 1), the clients drift apart due to local descent/ascent steps, resulting in the {∆ (t,k) x,y (i)}i,k terms. Also, E hΦ( e x (t)) − Fe(x (t), y (t)) iquantifies the error of the inner maximization. In the subsequent lemmas, we bound both these error terms.

Lemma C.2 (Consensus Error). Suppose the local loss functions {fi} satisfy Assumptions 1, 4, 7, and 8. The stochastic oracles for the local functions satisfy Assumption 3. Further, in Algorithm 1-Fed-Norm-SGDA+, we choose the client learning rate η c y such that η c y ≤1 2Lf (maxi ∥ai∥1 ) √2 max{1,β2L } . Then, the iterates {x (t) i, y (t) i} generated by Algorithm 1-Fed-Norm-SGDA+ satisfy

$$L_f^2\sum _{i=1}^n\frac{w_i}{\mathrm{\parallel }{a}_i{\mathrm{\parallel }}_1}\sum _{k=0}^{n-1}{a}_i^{(k)}(r_i){\mathrm{\Delta }}_{\mathbf{x},\mathbf{y}}^{(l,k)}(i)\le 2(\mathrm{\mid }{\eta }_x^e{\mathrm{\mid }}^2+\mathrm{\mid }{\eta }_y^e{\mathrm{\mid }}^2)L_f^2{\sigma }_L^2\sum _{i=1}^n{w}_i\mathrm{\parallel }{\mathbf{a}}_{i,-1}{\mathrm{\parallel }}_2^2+4L_f^2{M}_{\mathbf{a},-1}(\mathrm{\mid }{\eta }_x^e{\mathrm{\mid }}^2{G}_{\mathbf{x}}^2+\mathrm{\mid }{\eta }_y^e{\mathrm{\mid }}^2{\sigma }_G^2),L\_\{f\}^\{2\}\backslash sum\_\{i=1\}^\{n\}\backslash frac\{w\_\{i\}\}\{\backslash |a\_\{i\}\backslash |\_\{1\}\}\backslash sum\_\{k=0\}^\{n-1\}a\_\{i\}^\{(k)\}(r\_\{i\})\backslash Delta\_\{\backslash mathbf\{x\},\backslash mathbf\{y\}\}^\{(l,k)\}(i)\backslash leq2\backslash left(|\backslash eta\_\{x\}^\{e\}|^\{2\}+|\backslash eta\_\{y\}^\{e\}|^\{2\}\backslash right)L\_\{f\}^\{2\}\backslash sigma\_\{L\}^\{2\}\backslash sum\_\{i=1\}^\{n\}w\_\{i\}\backslash ,\backslash |\backslash mathbf\{a\}\_\{i,-1\}\backslash |\_\{2\}^\{2\}+4L\_\{f\}^\{2\}M\_\{\backslash mathbf\{a\},-1\}\backslash left(|\backslash eta\_\{x\}^\{e\}|^\{2\}G\_\{\backslash mathbf\{x\}\}^\{2\}+|\backslash eta\_\{y\}^\{e\}|^\{2\}\backslash sigma\_\{G\}^\{2\}\backslash right),$$ $$+8[{\eta }_y^c{]}^2{L}_f^3{M}_{{\mathbf{a}}_{-1}}{\beta }_G^2\mathbb{E}[\tilde{\mathrm{\Phi }}({\widehat{\mathbf{x}}}^{(s)})-\tilde{F}({\widehat{\mathbf{x}}}^{(s)},{\mathbf{y}}^{(t)})],+\backslash ,8[\backslash eta\_\{y\}^\{c\}]^\{2\}L\_\{f\}^\{3\}M\_\{\{\backslash bf\; a\}\_\{-1\}\}\backslash beta\_\{G\}^\{2\}\backslash mathbb\{E\}\backslash left[\backslash widetilde\{\backslash Phi\}(\backslash widehat\{\{\backslash bf\; x\}\}^\{(s)\})-\backslash widetilde\{F\}(\backslash widehat\{\{\backslash bf\; x\}\}^\{(s)\},\{\backslash bf\; y\}^\{(t)\})\backslash right],$$

where Ma−1 ≜ maxi(∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 ).

Note that consensus error depends on the difference E[Φe(xb (s)) − Fe(xb (s), y (t))]. This is different from the term E[Φe(x (t)) − Fe(x (t), y (t))] in Lemma C.1. Since in Algorithm 1-Fed-Norm-SGDA+ , the x-component stays fixed at xb (s)for S communication rounds while updating y (t,k) i, the difference

$$\sum _{t=kS}^{(k+1)S-1}\mathbb{E}[\tilde{\mathrm{\Phi }}({\widehat{\mathbf{x}}}^{(s)})-\tilde{F}({\widehat{\mathbf{x}}}^{(s)},{\mathbf{y}}^{(t)})]\backslash sum\_\{t=k\; S\}^\{(k+1)S-1\}\backslash mathbb\{E\}\backslash left[\backslash widetilde\{\backslash Phi\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\})-\backslash widetilde\{F\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\},\backslash mathbf\{y\}^\{(t)\})\backslash right]$$

can be interpreted as the optimization error, when maximizing the concave function F(xb (s), ·) over S

communication rounds. Next, we bound this error. The following result essentially extends the analysis of FedNova (Wang et al. (2020)) to concave maximization (analogously, convex minimization) problems. We also generalize the corresponding analyses in Khaled et al. (2020); Koloskova et al. (2020) to heterogeneous local updates. Lemma C.3 (Local SG updates for Concave Maximization). Suppose the local functions satisfy Assumptions 1, 3, 4 and 7. Further, lety (t) 2≤ R for all t. We run Algorithm 1-Fed-Norm-SGDA+ with client step-size η c y y f G y 2τeffγ s yLf n P max{β 2 G, 1} max (β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 , n − P n − 1 max iwi ) ≤ 1 8 , 2τeffγ s yLf n P max (P − 1 n − 1 , β2L max i,k wia (k) i(τi) ∥ai∥1 ) ≤ 1 8 . Then the iterates generated by Algorithm 1-Fed-Norm-SGDA+ satisfy such that 64[η c ] 2Ma−1 L 2 β 2 n P ≤ 1*. Further, the server step-size* γ s satisfies 1 S (s+1) X S−1 t=sS E hΦ( e xb (s)) − Fe(xb (s), y (t)) i ≤4R τeffγ s yS + τeffγ s y n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# + 4Lf ([η c x ] 2 + [η c y ] 2) " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2Ma−1 (G 2 x + σ 2 G) # , where Ma−1 ≜ maxi ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 .

Remark 10. It is worth noting that the proof of Lemma C.3 does not require global concavity of local $${\mathbf{\Delta }}_{\mathbf{i}-1}\triangleq \underset{i}{\max }({\parallel {\mathbf{a}}_{i,-1}\parallel }_1^2+{\beta }_L^2{\parallel {\mathbf{a}}_{i,-1}\parallel }_2^2)\mathrm{.}\{\backslash bf\backslash Delta\}\_\{\{\backslash bf\; i\}-1\}\backslash triangleq\backslash operatorname*\{max\}\_\{i\}\backslash left(\backslash left\backslash |\{\backslash bf\; a\}\_\{i,-1\}\backslash right\backslash |\_\{1\}^\{2\}+\backslash beta\_\{L\}^\{2\}\backslash left\backslash |\{\backslash bf\; a\}\_\{i,-1\}\backslash right\backslash |\_\{2\}^\{2\}\backslash right).$$ functions. Rather, given x, we only need concavity of local functions {fi} at some point y ∗(x). This is precisely the one-point-concavity assumption (Assumption 9) discussed earlier in Deng & Mahdavi (2021); Sharma et al. (2022). Therefore, Lemma C.3 for a much larger class of functions. Further, the bound in Lemma C.3 improves the corresponding bounds derived in existing work. As we discuss in Appendix C.4, this helps us achieve improve complexity results for nonconvex-one-point-concave (NC-1PC) functions. Next, we bound the difference E hΦ( e x $${\mathbf{x}}^{(t)})-\tilde{F}({\mathbf{x}}^{(t)},{\mathbf{y}}^{(t)})]\mathrm{.}\backslash mathbf\{x\}^\{(t)\})-\{\backslash widetilde\{F\}\}(\backslash mathbf\{x\}^\{(t)\},\backslash mathbf\{y\}^\{(t)\})\backslash Big].$$ Lemma C.4. Suppose the local functions satisfy Assumptions 1, 3, 4, 8. Then the iterates generated by Algorithm 1-Fed-Norm-SGDA+ satisfy 1 T T X−1 t=0 E hΦ( e x (t)) − Fe(x (t), y (t)) i≤ 1 T T /S X−1 s=0 (s+1) X S−1 t=sS E hΦ( e xb (s)) − Fe(xb (s), y (t)) i + 2τeffγ s xGx(S − 1)rn P vuutXn i=1 w2 i ∥ai∥ 2 2 ∥ai∥ 2 1 (σ 2 L + β 2 LG2x ) + G2x P − 1 n − 1 + n − P n − 1 Xn i=1 w2 i ! .

C.2 Proof Of Theorem 2

For the sake of completeness, we first state the full statement of Theorem 2 here.

Theorem 3. Suppose the local loss functions {fi} satisfy Assumptions 1, 3, 4, 7, 8. Further, let y (t) 2≤ R for all t. We run Algorithm 1-Fed-Norm-SGDA+ with client step-size η c ysuch that 64[η c y ] 2Ma−1 L 2 f max{β 2 G n P , 1} ≤ 1*. Further, the server step-size* γ s y satisfies

$$2{\tau }_{eff}{\gamma }_y^s{L}_f\frac{n}{P}\max \{{\beta }_L^2\underset{i}{\max }\frac{w_i\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_2^2}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1^2},\frac{n-P}{n-1}\underset{i}{\max }{w}_i\}\le \frac{1}{8},2\backslash tau\_\{e\; f\; f\}\backslash gamma\_\{y\}^\{s\}L\_\{f\}\backslash frac\{n\}\{P\}\backslash max\backslash left\backslash \{\backslash beta\_\{L\}^\{2\}\backslash max\_\{i\}\backslash frac\{w\_\{i\}\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{2\}^\{2\}\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}^\{2\}\},\backslash frac\{n-P\}\{n-1\}\backslash max\_\{i\}w\_\{i\}\backslash right\backslash \}\backslash leq\backslash frac\{1\}\{8\},$$ $$2\tau_{e f f}\gamma_{y}^{s}L_{f}\frac{n}{P}\max\left{\frac{P-1}{n-1},\beta_{L}^{2}\max_{i,k}\frac{w_{i}a_{i}^{(k)}(\tau_{i})}{|\mathbf{a}{i}|{1}}\right}\leq\frac{1}{8}.$$

Then the iterates generated by Algorithm 1-Fed-Norm-SGDA+ satisfy

1 T T X−1 t=0 E ∇Φe1/2Lf (x (t)) 2 ≤ O ∆¯Φe τeffγ sxT + τeffγ s xLf Aw P τeff σ 2 L + β 2 LG 2 x + G 2 x n(P − 1) P(n − 1) + Fw ! + O τeffγ s xLfGx(S − 1)sAw P τeff (σ 2 L + β 2 LG2x ) + G2x n(P − 1) P(n − 1) + Fw ! + O LfR τeffγ s yS + γ s yLf P Awσ 2 L + σ 2 G τeff n − P n − 1 Ew + Bwβ 2 L + O[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + DG 2 x + σ 2 G ,

where Φe1/2Lf (x) ≜ minx′ Φe(x ′) + Lf ∥x ′ − x∥ 2is the envelope function, ∆¯Φe ≜ Φe1/2Lf (x0) − minx Φe1/2Lf (x), Aw ≜ nτeff Pn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 , Bw ≜ nτeff maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 , Ew ≜ n maxi wi, Cw ≜Pn i=1 wi(∥ai∥ 2 2 − [α (t,τi−1) i] 2), and D ≜ maxi(∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 ), Fw ≜ n(n−P ) P (n−1) Pn i=1 w 2 i . With the following parameter values:

$${\eta }_x^c={\eta }_y^c=\mathrm{\Theta }\left(\frac{1}{L_f\tau T^{3\mathrm{/}8}}\right),{\gamma }_x^{\ast }=\mathrm{\Theta }\left(\frac{P^{1\mathrm{/}4}}{({\tau }_{cB}T{)}^{3\mathrm{/}4}}\right),{\gamma }_y^{\ast }=\mathrm{\Theta }\left(\frac{P^{3\mathrm{/}4}}{({\tau }_{cq}T{)}^{1\mathrm{/}4}}\right),S=\mathrm{\Theta }\left(\sqrt{\frac{T}{{\tau }_{cq}P}}\right),\backslash eta\_\{x\}^\{c\}=\backslash eta\_\{y\}^\{c\}=\backslash Theta\backslash left(\backslash frac\{1\}\{L\_\{f\}\backslash tau\; T^\{3/8\}\}\backslash right),\backslash gamma\_\{x\}^\{*\}=\backslash Theta\backslash left(\backslash frac\{P^\{1/4\}\}\{(\backslash tau\_\{c\; B\}T)^\{3/4\}\}\backslash right),\backslash quad\backslash gamma\_\{y\}^\{*\}=\backslash Theta\backslash left(\backslash frac\{P^\{3/4\}\}\{(\backslash tau\_\{c\; q\}T)^\{1/4\}\}\backslash right),\backslash quad\; S=\backslash Theta\backslash left(\backslash sqrt\{\backslash frac\{T\}\{\backslash tau\_\{c\; q\}P\}\}\backslash right),$$

where τ¯ = 1 n Pn i=1 τi*, we can further simplify to*

min t∈[T] E ∇Φe1/2Lf (x (t)) 2≤ O σ 2 G n − P n − 1 ∆¯Φe Ew P T p1 + Fw 1/4 + O Cwσ 2 L + D(G2x + σ 2 G ) τ¯ 2T3/4 | {z } Partial participation error | {z } Local updates error + O ∆¯eΦ σ 2 LAw τeffP T p1 + Fw 1/4+ ∆¯Φe (1 + Fw) T3/4 τeffP Aw + τeff n−P n−1 Ew 1/4! , | {z } Error with ful l synchronization Proof. We sum the bound in Lemma C.1 over t = 0 to T − 1 and rearrange the terms to get

1 T T X−1 t=0 E ∇Φe1/2Lf (x (t)) 2 ≤8 τeffγ sxT T X−1 t=0 E hΦe1/2Lf (x (t)) − Φe1/2Lf (x (t+1)) i + 8τeffγ s xLf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x + G 2 x P − 1 n − 1 + n − P n − 1 Xn i=1 w 2 i !# 42

• 16Lf 1 T T X−1 t=0 E

hΦ( e x (t)) − Fe(x (t), y (t)) i+ 16 T T /S X−1 s=0 (s+1) X S−1 t=sS L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ 8 hΦe1/2Lf (x (0)) − Φe1/2Lf (x (T)) i τeffγ sxT+ 8τeffγ s xLf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x

• G 2 x P − 1 n − 1
• n − P n − 1 Xn i=1 w 2 i !#
• 32 [η c x ] 2 + [η c y ] 2L 2 fσ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 64L 2 fMa−1 [η c x ] 2G 2 x + [η c y ] 2σ 2 G (From Lemma C.2)
• 128[η c y ] 2L 3 fMa−1 β 2 G 1 T T /S X−1 s=0 (s+1) X S−1 t=sS E

hΦ( e xb (s)) − Fe(xb (s), y (t)) i+ 16Lf 1 T T X−1 t=0 E

hΦ( e x (t)) − Fe(x (t), y (t)) i ≤ 8∆¯Φe τeffγ sxT

• 8τeffγ s xLf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x
• G 2 x P − 1 n − 1
• n − P n − 1 Xn i=1 w 2 i !#

(where ∆¯Φe ≜ Φe1/2Lf (x0) − minx Φe1/2Lf (x))

• 32 [η c x ] 2 + [η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2Ma−1 G 2 x + σ 2 G

• 32τeffγ s xLfGx(S − 1)rn P vuutXn i=1 w2 i ∥ai∥ 2 2 ∥ai∥ 2 1 (σ 2 L + β 2 LG2x ) + G2x P − 1 n − 1
• n − P n − 1 Xn i=1 w2 i ! (From Lemma C.4)
• 18Lf "4R τeffγ s yS
• τeffγ s y n P

σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1

• 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !!#

(From Lemma C.3; using Am ≤ min{ 1 2 ,1 16β 2 G })

• 72[η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2σ 2 GMa−1

. (45) We can simplify the notation using the constants Aw ≜ nτeff Pn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 , Bw ≜ τeffn maxi wi∥ai∥ 2 2 ∥ai∥ 2 1 , Ew ≜ n maxi wi, Cw ≜Pn i=1 wi ∥ai,−1∥ 2 2 , D ≜ Ma−1 , Fw ≜ n P Pn i=1 w 2 i and drop the numerical constants, for simplicity, to get 1 T T X−1 t=0 E

∇Φe1/2Lf (x (t))

2 ≲

∆¯Φe τeffγ sxT

• τeffγ s xLf Aw P τeff σ 2 L + β 2 LG 2 x

• G 2 x n(P − 1) P(n − 1) + (n − P) (n − 1) Fw

• τeffγ s xLfGx(S − 1)sAw P τeff (σ 2 L + β 2 LG2x ) + G2x n(P − 1) P(n − 1) + (n − P) (n − 1) Fw

+LfR τeffγ s yS + γ s yLf P Awσ 2 L + σ 2 G τeff n − P n − 1 Ew + Bwβ 2 L +[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + DG 2 x + σ 2 G

= ∆¯Φe τeffγ sxT

• τeffγ s xLf I 2 1 + γ s yLf I2 P+ Lf τeffγ s xGx(S − 1)I1 + R τeffγ s yS

+[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + D(G 2 x + σ 2 G), (46) where in (46), to simplify notation, we have defined I1 ≜ rAw P τeff (σ 2 L + β 2 LG2x ) + G2x n(P −1) P (n−1) + (n−P ) (n−1) Fw , I2 ≜ Awσ 2 L + (Bwβ 2 L + τeff n−P n−1 Ew)σ 2 G. Next, we optimize the algorithm parameters S, γsx , γs y , ηcy , ηcy to achieve a tight bound on (46). If R = 0, we let S = 1. Else, let S = q R τ 2 effγsxγs yGxI1 . Substituting this in (46), we get

$$\begin{array}{cccc}& \frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}{\parallel \widehat{\mathbf{v}}{\widehat{\mathbf{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\lesssim \frac{{\widehat{\mathrm{\Delta }}}_{\widehat{\mathbf{\Phi }}}}{{\tau }_{\text{eff}}{\gamma }_{\pi }^{\ast }T}+{\tau }_{\text{eff}}{\gamma }_{\pi }^{\ast }{L}_f{\chi }_1^2+\frac{{\gamma }_{\pi }^{\ast }{L}_f{\chi }_2}{P}+L_f\sqrt{\frac{R{\gamma }_{\pi }^{\ast }{G}_{\pi }{\chi }_1}{{\gamma }_{\text{f}}^{\ast }}}& & \text{(47)}\end{array}\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash mathbb\{E\}\backslash left\backslash |\backslash widehat\{\backslash mathbf\{v\}\}\backslash widehat\{\backslash mathbf\{\backslash Phi\}\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash lesssim\backslash frac\{\backslash widehat\{\backslash Delta\}\_\{\backslash widehat\{\backslash mathbf\{\backslash Phi\}\}\}\}\{\backslash tau\_\{\backslash text\{eff\}\}\backslash gamma\_\{\backslash pi\}^\{*\}T\}+\backslash tau\_\{\backslash text\{eff\}\}\backslash gamma\_\{\backslash pi\}^\{*\}L\_\{f\}\backslash chi\_\{1\}^\{2\}+\backslash frac\{\backslash gamma\_\{\backslash pi\}^\{*\}L\_\{f\}\backslash chi\_\{2\}\}\{P\}+L\_\{f\}\backslash sqrt\{\backslash frac\{R\backslash gamma\_\{\backslash pi\}^\{*\}G\_\{\backslash pi\}\backslash chi\_\{1\}\}\{\backslash gamma\_\{\backslash text\{f\}\}^\{*\}\}\}\backslash tag\{47\}$$ $$+\left([\eta_{\pi}^{e}]^{2}+[\eta_{\text{p}}^{e}]^{2}\right)L_{f}^{2}\left[G_{w}\sigma_{L}^{2}+D(G_{\pi}^{*}+\sigma_{G}^{2})\right],,$$

Next, we focus on the terms in (47) containing γ s y : Lf hγ yI2 P + qRγsxGxI1 γs y i. To optimize these, we choose γ s y = P 2I2 2/3(RγsxGxI1) 1/3. Substituting in (47), we get

$$\begin{array}{cccc}& \frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}{\parallel \mathrm{\nabla }{\widehat{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2⩽\frac{{\widehat{\mathrm{\Delta }}}_{\widehat{\mathbf{x}}}}{{\tau }_{\text{eff}}{\gamma }_{\pi }^{2}T}+{\tau }_{\text{eff}}{\tau }_{\pi }^{\ast }{L}_f{T}_1^2+L_f{\left(\frac{T_2}{P}R{\gamma }_{\pi }^{\ast }{G}_{\mathbf{x}}{T}_1\right)}^{1\mathrm{/}3}& & \text{(48)}\end{array}\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\backslash widehat\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leqslant\backslash frac\{\backslash widehat\{\backslash Delta\}\_\{\backslash widehat\{\backslash mathbf\{x\}\}\}\}\{\backslash tau\_\{\backslash text\{eff\}\}\backslash gamma\_\{\backslash pi\}^\{2\}T\}+\backslash tau\_\{\backslash text\{eff\}\}\backslash tau\_\{\backslash pi\}^\{*\}L\_\{f\}T\_\{1\}^\{2\}+L\_\{f\}\backslash left(\backslash frac\{T\_\{2\}\}\{P\}R\backslash gamma\_\{\backslash pi\}^\{*\}G\_\{\backslash mathbf\{x\}\}T\_\{1\}\backslash right)^\{1/3\}\backslash tag\{48\}$$ $$+\left([\eta_{c}^{\sigma^{2}}]+[\eta_{0}^{\sigma^{2}}]\right)L_{f}^{2}\left[C_{w}\sigma_{L}^{2}+D(G_{\mathbf{x}}^{2}+\sigma_{G}^{2})\right],$$

Finally, we focus on the terms in (48) containing γ s x :∆¯eΦ τeffγsxT + Lf I2 P RγsxGxI1 1/3. We ignore the higher order linear term. With γ s x = 3∆¯ eΦ τeffLf T 3/4I2 P RGxI1 −1/4, and absorbing numerical constants inside O(·) we get,

1 T T X−1 t=0 E ∇Φe1/2Lf (x (t)) 2 ≤ O ∆¯Φe I1I2 1/4 + O (∆¯Φe τeffP) 1/4 T3/4 I 2 1 (I1I2) 1/4 ! + O[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + D(G 2 x + σ 2 G) (τeffP T) 1/4 ≤ O ∆¯Φe pG2x (1 + Fw)Awσ 2 L 1/4 + O ∆¯Φe pG2x (1 + Fw)τeff n−P n−1 Ewσ 2 G 1/4 (τeffP T) 1/4 (τeffP T) 1/4 + O (∆¯Φe τeffP) 1/4 Aw + τeff n−P n−1 Ew 1/4 + O[η c x ] 2 + [η c y ] 2L 2 f -Cwσ 2 L + D(G 2 x + σ 2 G) . (49) G2x 1 + n−P n−1 Fw T3/4 Lastly, we specify the algorithm parameters in terms of n, T, τeff, τ¯.

Lastly, we specify the algorithm parameters in terms of $n,T,\tau_{\rm eff},\dot{\tau}$. $$\gamma_{\rm g}^{}=\Theta\left(\frac{P^{1/4}}{(\tau_{\rm eff}T)^{3/4}}\right),\quad\gamma_{\rm g}^{}=\Theta\left(\frac{P^{3/4}}{(\tau_{\rm eff}T)^{1/4}}\right),\quad S=\Theta\left(\sqrt{\frac{T}{\tau_{\rm eff}P}}\right).$$ Finally, choosing the client learning rates $\eta_{\rm g}^{}=\eta_{\rm g}^{}=\frac{1}{\tau_{\rm eff}T^{3/4}}$, we get the result.

Convergence In Terms Of F

Proof of Corollary 2.1. Following Lin et al. (2020a), we define

Φe1/2Lf (x) ≜ minx′ nΦ( e x ′) + Lf ∥x ′ − x∥ 2o; xe ≜ arg minx′ nΦ( e x ′) + Lf ∥x ′ − x∥ 2o, Φ1/2Lf (x) ≜ minx′ nΦ(x ′) + Lf ∥x ′ − x∥ 2o; x¯ ≜ arg minx′ nΦ(x ′) + Lf ∥x ′ − x∥ 2o.(50) Also, it follows from Lemma 2.2 in Davis & Drusvyatskiy (2019) that ∇Φe1/2Lf (x) = 2Lf (x − xe) and ∇Φ1/2Lf (x) = 2Lf (x − x¯). Therefore,

$${\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}(\mathbf{x})\parallel }^2\le 2{\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}(\mathbf{x})-\mathrm{\nabla }{\tilde{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}(\mathbf{x})\parallel }^2+2{\parallel \mathrm{\nabla }{\tilde{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}(\mathbf{x})\parallel }^2\backslash left\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}\backslash leq2\backslash left\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\})-\backslash nabla\backslash widetilde\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}+2\backslash left\backslash |\backslash nabla\backslash widetilde\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\})\backslash right\backslash |^\{2\}$$ $$=8L_{f}^{2}\left|\widetilde{\mathbf{x}}-\widetilde{\mathbf{x}}\right|^{2}+2\left|\nabla\widetilde{\Phi}{1/2L{f}}(\mathbf{x})\right|^{2}$$

Consequently, we obtain

$$\underset{t\in [T]}{\min }{\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\le \frac{1}{T}\sum _{t=0}^{T-1}{\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\backslash min\_\{t\backslash in[T]\}\backslash left\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash left\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}$$ $$\leq\frac{2}{T}\sum_{t=0}^{T-1}\left[\left|\nabla\tilde{\Phi}{1/2L{f}}(\mathbf{x}^{(t)})\right|^{2}+4L_{f}^{2}\left|\tilde{\mathbf{x}}^{(t)}-\tilde{\mathbf{x}}^{(t)}\right|^{2}\right].\tag{51}$$ $\square$ . where xe (t), x¯ (t)follow the same definition as in (50), with x replaced with x (t).

Proof of Corollary 2.2. If clients are weighted equally (wi = pi = 1/n for all i), with each carrying out τ steps of local SGDA+, then (6) reduces to

\min_{t\in[T]}\mathbb{E}\big{\|}\nabla\Phi_{1/2L_{f}}(\mathbf{x}^{(t)})\big{\|}^{2}\leq\mathcal{O}\Big{(}\frac{1}{(r\,PT)^{1/4}}+\frac{(r\,P)^{1/4}}{T^{3/4}}\Big{)}+\mathcal{O}\left(\frac{\sigma_{L}^{2}+\tau(G_{n}^{2}+\sigma_{D}^{2})}{rT^{3/4}}\right)+\mathcal{O}\Big{(}\Big{(}\frac{n-P}{n-1}\cdot\frac{1}{PT}\Big{)}^{1/4}\Big{)}.

• For full client participation, this reduces to

$$\underset{t\in [T]}{\min }\mathbb{E}\parallel \mathrm{\nabla }{\mathrm{\Phi }}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)}){\parallel }^2\le \mathcal{O}(\frac{1}{(\tau nT{)}^{1\mathrm{/}4}}+\frac{(\tau n{)}^{1\mathrm{/}4}}{T^{3\mathrm{/}4}})\mathrm{.}\backslash operatorname*\{min\}\_\{t\backslash in[T]\}\backslash mathbb\{E\}\backslash big\backslash |\backslash nabla\backslash Phi\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash big\backslash |^\{2\}\backslash leq\backslash mathcal\{O\}\backslash Big(\{\backslash frac\{1\}\{(\backslash tau\; n\; T)^\{1/4\}\}\}+\{\backslash frac\{(\backslash tau\; n)^\{1/4\}\}\{T^\{3/4\}\}\}\backslash Big).$$

To reach an ϵ-stationary point, assuming nτ ≤ T, the per-client gradient complexity is T τ = O1 nϵ8 .

Since τ ≤ T /n, the minimum number of communication rounds required is T = O1 ϵ 4 .

• For partial participation, O n − P n − 1 ·1 P T 1/4is the dominant term, and we do not get any convergence benefit of multiple local updates. Consequently, per-gradient client complexity and number of communication rounds are both T τ = O1 P ϵ8 , for τ = O(1). However, if the data across clients comes from identical distributions (σG = 0), then we recover per-client gradient complexity of O1 P ϵ8 , and number of communication rounds = O1 ϵ 4 .

Special Cases

• Centralized, deterministic case (σL = σG = 0, βG = 1, τeff = n = 1): in this case Aw = Bw = 1, Cw = D = 0. Also, I1 = Gx pβ 2 L + 1, I2 = 0. The bound in (45) reduces to

$$\begin{array}{cccc}& \frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}{\parallel \mathrm{\nabla }{\widehat{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\le \mathcal{O}(\frac{\widehat{{\mathrm{\Delta }}_{\widehat{\mathrm{\Phi }}}}}{{\gamma }_s^{\ast }T}+{\gamma }_s^{\ast }{L}_f{G}_{\mathbf{x}}^2[({\beta }_L^2+1)+(S-1)\sqrt{{\beta }_L^2+1}]+\frac{L_{f}R}{{\gamma }_s^{\ast }S})\mathrm{.}& & \text{(52)}\end{array}\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\backslash widehat\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash mathcal\{O\}\backslash left(\backslash frac\{\backslash widehat\{\backslash Delta\_\{\backslash widehat\{\backslash Phi\}\}\}\}\{\backslash gamma\_\{s\}^\{*\}T\}+\backslash gamma\_\{s\}^\{*\}L\_\{f\}G\_\{\backslash mathbf\{x\}\}^\{2\}\backslash left[(\backslash beta\_\{L\}^\{2\}+1)+(S-1)\backslash sqrt\{\backslash beta\_\{L\}^\{2\}+1\}\backslash right]+\backslash frac\{L\_\{f\}R\}\{\backslash gamma\_\{s\}^\{*\}S\}\backslash right).\backslash tag\{52\}$$

For βL = 0, (52) yields the convergence result in Lin et al. (2020a).

• Single node, stochastic case (σG = 0, βG = 1, τeff = n = 1): in this case Aw = Bw = 1, Cw = D = 0.

Also, I1 =pσ 2 L + (β 2 L + 1)G2x , I2 = σ 2 L . The bound in (45) reduces to

$$\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}{\parallel \mathrm{\nabla }{\tilde{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\le \mathrm{\Theta }(\frac{{\bar{\mathrm{\Delta }}}_{\tilde{\mathrm{\Phi }}}}{{\gamma }_x^{s}T}+{\gamma }_x^s{L}_f({\sigma }_L^2+({\beta }_L^2+1)G_{\mathbf{x}}^2)+{\gamma }_y^s{L}_f{\sigma }_L^2)\mathrm{.}\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\backslash tilde\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash Theta\backslash left(\backslash frac\{\backslash bar\{\backslash Delta\}\_\{\backslash tilde\{\backslash Phi\}\}\}\{\backslash gamma\_\{x\}^\{s\}T\}+\backslash gamma\_\{x\}^\{s\}L\_\{f\}(\backslash sigma\_\{L\}^\{2\}+(\backslash beta\_\{L\}^\{2\}+1)G\_\{\backslash mathbf\{x\}\}^\{2\})+\backslash gamma\_\{y\}^\{s\}L\_\{f\}\backslash sigma\_\{L\}^\{2\}\backslash right).$$

$$\begin{array}{cccc}& +\mathrm{\Theta }\left(L_f[{\gamma }_x^s{G}_{\mathbf{x}}(S-1)\sqrt{{\sigma }_L^2+({\beta }_L^2+1)G_{\mathbf{x}}^2}+\frac{R}{{\gamma }_y^{s}S}]\right)\mathrm{.}& & \text{(53)}\end{array}+\backslash Theta\backslash left(L\_\{f\}\backslash left[\backslash gamma\_\{x\}^\{s\}G\_\{\backslash bf\; x\}(S-1)\backslash sqrt\{\backslash sigma\_\{L\}^\{2\}+(\backslash beta\_\{L\}^\{2\}+1)G\_\{\backslash bf\; x\}^\{2\}\}+\backslash frac\{R\}\{\backslash gamma\_\{y\}^\{s\}S\}\backslash right]\backslash right).\backslash tag\{53\}$$

Again, for βL = 0, (53) yields the convergence result in Lin et al. (2020a).

• Multiple equally weighted (wi = 1/n, ∀ i ∈ [n]) clients, full client participation, stochastic case with synchronous client updates (τeff = 1): in this case Aw = 1, Bw = 1, Cw = D = 0, 0. The bound in (45) reduces to

$$\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}{\parallel \mathrm{\nabla }{\tilde{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t)})\parallel }^2\le \mathrm{\Theta }(\frac{{\tilde{\mathrm{\Delta }}}_{\mathrm{\Phi }}^{-}}{{\gamma }_x^{s}T}+{\gamma }_x^s{L}_f(G_{\mathbf{x}}^2+\frac{{\sigma }_L^2+{\beta }_L^2{G}_{\mathbf{x}}^2}{n}))\backslash frac\{1\}\{T\}\backslash sum\_\{t=0\}^\{T-1\}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\backslash tilde\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\backslash mathbf\{x\}^\{(t)\})\backslash right\backslash |^\{2\}\backslash leq\backslash Theta\backslash left(\backslash frac\{\backslash tilde\{\backslash Delta\}\_\{\backslash Phi\}^\{-\}\}\{\backslash gamma\_\{x\}^\{s\}T\}+\backslash gamma\_\{x\}^\{s\}L\_\{f\}\backslash left(G\_\{\backslash mathbf\{x\}\}^\{2\}+\backslash frac\{\backslash sigma\_\{L\}^\{2\}+\backslash beta\_\{L\}^\{2\}G\_\{\backslash mathbf\{x\}\}^\{2\}\}\{n\}\backslash right)\backslash right)$$ $$\quad+\Theta\left(\frac{\gamma_{y}^{s}L_{f}}{n}(\sigma_{L}^{2}+\sigma_{G}^{2}\beta_{L}^{2})+\gamma_{x}^{s}L_{f}G_{\mathbf{x}}(S-1)\sqrt{G_{\mathbf{x}}^{2}+\frac{\sigma_{L}^{2}+\beta_{L}^{2}G_{\mathbf{x}}^{2}}{n}}+\frac{RL_{f}}{\gamma_{y}^{s}S}\right),\tag{54}$$

Note that unlike existing analyses of synchronous update algorithms Woodworth et al. (2020); Yun et al. (2022); Sharma et al. (2022), the bound in (54) depends on the inter-client heterogeneity σ 2 G.

This is due to the more general noise assumption (Assumption 3). In the existing works, β 2 L is assumed zero, in which case, the bound in (54) is also independent of σ 2 G. See Appendix A.2 for a more detailed explanation.

• Multiple, equally weighted (wi = 1/n, ∀ i ∈ [n]) clients, full client participation, multiple, but equal number of client updates (τi = τeff = τ, ∀ i ∈ [n]). In this case Aw = Bw = 1, Cw = τ − 1, D = (τ − 1)(τ − 1 + β 2 L ). The bound in (45) then reduces to

t=0 E ∇Φe1/2Lf (x (t)) 2 ≤ Θ ∆¯ eΦ τ γsxT + τ γ s xLf G 2 x + σ 2 L + β 2 LG 2 x nτ + γ s yLf (σ 2 L + β 2 Lσ 2 G) n ! (55) 1 T TX−1 + Θ Lf " τ γ s xGx(S − 1)rG2x + σ 2 L + β 2 LG2x nτ+ R τ γs yS # + (τ − 1) [η c x] 2 + [η c y] 2L 2 f -σ 2 L + (τ − 1 + β 2 L)(G 2 x + σ 2 G)!.

For βL = βG = 0, this setting reduces to the one considered in Sharma et al. (2022). However, as stated earlier, our bound on the local update error is tighter.

C.3 Proofs Of The Intermediate Lemmas

Proof of Lemma C.1. Using the definition in (50) x¯ (t) = arg minx Φ( e x) + Lf x − x (t) 2. Also, note that

$$\begin{array}{cccc}& {\tilde{\mathrm{\Phi }}}_{1\mathrm{/}2L_f}({\mathbf{x}}^{(t+1)})\le \tilde{\mathrm{\Phi }}({\bar{\mathbf{x}}}^{(t)})+L_f{\parallel {\bar{\mathbf{x}}}^{(t)}-{\mathbf{x}}^{(t+1)}\parallel }^2\mathrm{.}& & \text{(56)}\end{array}\backslash widetilde\{\backslash Phi\}\_\{1/2L\_\{f\}\}(\{\backslash bf\; x\}^\{(t+1)\})\backslash leq\backslash widetilde\{\backslash Phi\}(\backslash bar\{\backslash bf\; x\}^\{(t)\})+L\_\{f\}\backslash left\backslash |\backslash bar\{\backslash bf\; x\}^\{(t)\}-\{\backslash bf\; x\}^\{(t+1)\}\backslash right\backslash |^\{2\}.\backslash tag\{56\}$$

Using the x updates in (43),

E x¯ (t) − x (t+1) 2 = E x¯ (t) − x (t) + τeffγ s x X i∈C(t) w˜ig (t) x,i 2 = E x¯ (t) − x (t) 2 + τ 2 eff[γ s x ] 2E X i∈C(t) w˜ig (t) x,i 2 + 2τeffγ s xE * x¯ (t) − x (t), Xn i=1 wih (t) x,i+ ≤ E x¯ (t) − x (t) 2 + 2τeffγ s xE Dx¯ (t) − x (t), ∇xFe(x (t), y (t)) E+ τ 2 eff[γ s x ] 2E X i∈C(t) w˜ig (t) x,i 2 + τeffγ s xE k=0 a (k) i(τi) ∇xfi(x (t,k) i, y (t,k) i) − ∇xfi(x (t), y (t)) 2 Lf 2 x¯ (t) − x (t) 2 +2 Lf Xn i=1 wi ∥ai∥1 τXi−1

≤ E x¯ (t) − x (t) 2 + τ 2 eff[γ s x ] 2E X i∈C(t) w˜ig (t) x,i 2 (57) + τeffγ s xE "Lf 2 x¯ (t) − x (t) 2 + 2Lf Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 2 Dx¯ (t) − x (t), ∇xFe(x (t), y (t)) E#, (58)

where (58) follows from Lf -smoothness (Assumption 1) and Jensen's inequality. From (18), (19), we can bound E Pi∈C(t ′) w˜ig (t ′) x,i 2as follows.

E X i∈C(t ′) w˜ig (t ′) x,i 2 ≤ n P Xn i=1 w 2 i ∥ai∥ 2 1 τXi−1 k=0 [a (k) i(τi)]2 σ 2 L + β 2 LE ∇xfi(x (t,k) i, y (t,k) i) 2 + n P P − 1 n − 1 E Xn i=1 wih (t) x,i 2 + n P n − P n − 1 Xn i=1 w 2 i E h (t) x,i 2 ≤ n P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x + n(P − 1) P(n − 1)G 2 x + n(n − P) P(n − 1) G 2 x Xn i=1 w 2 i , (59)

where the final inequality by using Assumption 8. Next, we bound the inner product term in (58). Using Lf -smoothness of F (Assumption 1):

E Dx¯ (t) − x (t), ∇xFe(x (t), y (t)) E≤ E Fe(x¯ (t), y (t)) − Fe(x (t), y (t)) + Lf 2 x¯ (t) − x (t) 2 ≤ E Φ( e x¯ (t)) + Lf x¯ (t) − x (t) 2− EFe(x (t), y (t)) − Lf 2 E x¯ (t) − x (t) 2 ≤ E Φ( e x (t)) + Lf x (t) − x (t) 2− EFe(x (t), y (t)) − Lf 2 E x¯ (t) − x (t) 2 (by definition of x¯ (t)) ≤ E Φ( e x (t)) − Fe(x (t), y (t)) − Lf 2 x¯ (t) − x (t) 2. (60)

Substituting the bounds from (58) and (60) into (56), we get

E

hΦe1/2Lf (x (t+1)) i≤ E Φ( e x¯ (t)) + Lf x¯ (t) − x (t) 2

• τ 2 eff[γ s x ] 2Lf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x
• G 2 x P − 1 n − 1
• n − P n − 1 Xn i=1 w 2 i !#
• 2τeffγ s xL 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + 2τeffγ s xLfE hΦ( e x (t)) − Fe(x (t), y (t)) i− τeffγ s xL 2 f 2 E

x¯ (t) − x (t) 2 ≤ E hΦe1/2Lf (x (t)) i+ τ 2 eff[γ s x ] 2Lf n P "Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 σ 2 L + β 2 LG 2 x

• G 2 x P − 1 n − 1
• n − P n − 1 Xn i=1 w 2 i !#
• 2τeffγ s x ( L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) + LfE hΦ( e x (t)) − Fe(x (t), y (t)) i)− τeffγ s x 8 E

∇Φe1/2Lf (x (t))

2 , where we use ∇Φe1/2Lf (x) = 2Lf (x − x¯) from (50). Proof of Lemma C.2. We use the client update equations for individual iterates in (42). To bound ∆ (t,k) x,y (i), first we bound the x-error E x (t,k) i − x (t) 2 . Starting from (24), using Assumption 8, for 1 ≤ k ≤ τi,

k=0 a (k) i(τi) k X−1 j=0 [a (j) i(k)]2σ 2 L + β 2 LG 2 x + k X−1 j=0 a (j) i(k) k X−1 j=0 a (j) i(k)G 2 x 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)E x (t,k) i − x (t) 2 ≤ [η c x ] 2 ∥ai∥1 τXi−1 ≤ [η c x ] 2hσ 2 L ∥ai,−1∥ 2 2 + G 2 x ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 i , (61) where we use (25). Next, we bound E y (t,k) i − y (t) 2 , using the bound from (29), to get 1 ∥ai∥1 τXi−1 k=0 a (k) i(τi)E y (t,k) i − y (t) 2 ≤ [η c y ] 2σ 2 L ∥ai,−1∥ 2 2 + 2[η c y ] 2L 2 f ∥ai,−1∥1 + β 2 Lα τXi−1 k=0 a (k) i(τi)∆(t,k) y(i) + 2[η c y ] 2∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 E ∇yfi(xb (s), y (t)) 2 . (62) (t,k) (t,k)

Compared to (29), the difference is the presence of ∆ y (i) in (62), rather than ∆ x,y (i). Taking a weighted sum over agents in (62), we get

L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) y(i) ≤ 2[η c y ] 2L 2 f " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2Ma−1 β 2 GE ∇yFe(xb (s), y (t)) 2 + σ 2 G #. (63) where, we choose η c y such that Am ≜ 2L 2 f [η c y ] 2 maxi ∥ai∥1 ∥ai,−1∥1 + β 2 Lα≤12 , and define Ma−1 ≜ $\square$ maxi ∥ai,−1∥ 2 1 + β 2 L ∥ai,−1∥ 2 2 . Next, it follows from Lf -smoothness (Assumption 1) and Lemma A.7 that

$$\mathbb{E}{\parallel {\mathrm{\nabla }}_y\tilde{F}({\widehat{\mathbf{x}}}^{(s)},{\mathbf{y}}^{(t)})\parallel }^2\le 2L_f\mathbb{E}[\tilde{\mathrm{\Phi }}({\widehat{\mathbf{x}}}^{(s)})-\tilde{F}({\widehat{\mathbf{x}}}^{(s)},{\mathbf{y}}^{(t)})]\mathrm{.}\backslash mathbb\{E\}\backslash left\backslash |\backslash nabla\_\{y\}\backslash widetilde\{F\}\backslash left(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\},\backslash mathbf\{y\}^\{(t)\}\backslash right)\backslash right\backslash |^\{2\}\backslash leq2L\_\{f\}\backslash mathbb\{E\}\backslash left[\backslash widetilde\{\backslash Phi\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\})-\backslash widetilde\{F\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\},\backslash mathbf\{y\}^\{(t)\})\backslash right].$$ $(\otimes)=1/(\otimes)$.

Subsequently, combining (61) and (63), we get

L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ 2[η c x ] 2 + [η c y ] 2L 2 fσ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 4L 2 fMa−1 [η c x ] 2G 2 x + [η c y ] 2σ 2 G + 8[η c y ] 2L 3 fMa−1 β 2 GE hΦ( e xb (s)) − Fe(xb (s), y (t)) i. which finishes the proof. Proof of Lemma C.3. We define y ∗(xb (s)) ∈ arg maxy Fe(xb (s), y). Then, E

y (t+1) − y ∗(xb (s))

2 (43) = E y (t) + τeffγ s yg (t) y − y ∗(xb (s))

2 = E y (t) − y ∗(xb (s))

2

• τ 2 eff[γ s y ] 2E

g (t) y

2

• 2τeffγ s yE

• y (t) − y ∗(xb (s)), Xn i=1 wih (t) y,i+. (64) E

g (t) y

2 is bounded in (38). We only need to further bound E Pn i=1 wih (t) y,i

2 . E

Xn i=1 wih (t) y,i

2 ≤ E

Xn i=1 k=0 [a (k) i(τi)] ∇yfi(xb (s), y (t,k) i) − ∇yfi(xb (s), y (t)) + ∇yfi(xb (s), y (t))

2 wi ∥ai∥1 τXi−1 ≤ 2L 2 f Xn i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]∆(t,k) y(i) + 2 ∇yFe(xb (s), y (t))

2 (Jensen's inequality) $$\begin{array}{cccc}& \le 2L_f^2\sum _{i=1}^n\frac{w_i}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1}\sum _{k=0}^{n-1}[a_i^{(k)}({\tau }_i)\mathrm{\mid }{\mathrm{\Delta }}_{\mathbf{y}}^{(L,k)}(i)+4L_f\mathbb{E}[\widehat{\mathrm{\Phi }}({\widehat{\mathbf{x}}}^{(s)})-\tilde{F}({\widehat{\mathbf{x}}}^{(s)},{\mathbf{y}}^{(t)})]\mathrm{.}& & \text{(65)}\end{array}\backslash leq2L\_\{f\}^\{2\}\backslash sum\_\{i=1\}^\{n\}\backslash frac\{w\_\{i\}\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}\}\backslash sum\_\{k=0\}^\{n-1\}[a\_\{i\}^\{(k)\}(\backslash tau\_\{i\})|\backslash Delta\_\{\backslash mathbf\{y\}\}^\{(L,k)\}(i)+4L\_\{f\}\backslash mathbb\{E\}\backslash left[\backslash widehat\{\backslash Phi\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\})-\backslash widetilde\{F\}(\backslash widehat\{\backslash mathbf\{x\}\}^\{(s)\},\backslash mathbf\{y\}^\{(t)\})\backslash right].\backslash tag\{65\}$$ Next, we bound the third term in (6:).
E

• y (t) − y ∗(xb (s)), Xn i=1 wih (t) y,i+= E
• y (t) − y ∗(xb (s)), Xn i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]∇yfi(xb (s), y (t,k) i)

• = Xn i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]E hDy (t) − y (t,k) i, ∇yfi(xb (s), y (t,k) i) E+ Dy (t,k) i − y ∗(xb (s)), ∇yfi(xb (s), y (t,k) i) Ei ≤ Xn i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]E " fi(xb (s), y (t)) − fi(xb (s), y (t,k) i) + Lf 2 y (t) − y (t,k) i

2 (Lf -smoothness) + fi(xb (s), y (t,k) i) − fi(xb (s), y ∗(xb (s)))#(Concavity in y)

Lf 2 Xn i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]∆(t,k) y(i) − E hΦ( e xb (s)) − Fe(xb (s), y (t)) i. (66) Substituting (38), (65), (66) in (64), we get E

y (t+1) − y ∗(xb (s))

2 ≤ E y (t) − y ∗(xb (s))

2

• τ 2 eff[γ s y ] 2 "σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1
• 2σ 2 Gn P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#

− 2τeffγ s y

1 − 2τeffγ s yLf n P "P − 1 n − 1

• β 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !#! E

hΦ( e xb (s)) − Fe(xb (s), y (t)) i

• τeffγ s yLf " 1 + 2τeffγ s yLf n P P − 1 n − 1
• n − P n − 1 max iwi + β 2 L max i,k wia (k) i(τi) ∥ai∥1 !#Xn

i=1 wi ∥ai∥1 τXi−1 k=0 [a (k) i(τi)]∆(t,k) x,y (i), $$(68)(68)$$ (67) since ∆ (t,k) y ≤ ∆ (t,k) x,y . Using the bound on ∆ (t,k) x,y from Lemma C.2, Xn i=1 wi ∥ai∥1 τXi−1 k=0 a (k) i(τi)∆(t,k) x,y (i) ≤ 2[η c x ] 2 + [η c y ] 2σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 4Ma−1 [η c x ] 2G 2 x + [η c y ] 2σ 2 G + 8[η c y ] 2LfMa−1 β 2 GE hΦ( e xb (s)) − Fe(xb (s), y (t)) i. (68) We substitute (68) in (67), and simplify the terms using the choice of γ s y , ηcy to get y E y (t+1) − y ∗(xb (s)) 2 ≤ E y (t) − y ∗(xb (s)) 2 + τ 2 eff[γ s y ] 2 "σ 2 Ln P Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 Gn P n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# − τeffγ s yE hΦ( e xb (s)) − Fe(xb (s), y (t)) i+ 4τeffγ s yLf ([η c x ] 2 + [η c y ] 2) " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2Ma−1 (G 2 x + σ 2 G) # . using γ s y , ηcy that satisfy

$$2{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_y^s{L}_f\frac{n}{P}(\frac{P-1}{n-1}+\frac{n-P}{n-1}\underset{i}{\max }{w}_i+{\beta }_L^2\underset{i,k}{\max }\frac{w_i{a}_i^{(k)}({\tau }_i)}{\mathrm{\parallel }{a}_i{\mathrm{\parallel }}_1})\le 1,2\backslash tau\_\{\backslash mathrm\{eff\}\}\backslash gamma\_\{y\}^\{s\}L\_\{f\}\backslash frac\{n\}\{P\}\backslash left(\backslash frac\{P-1\}\{n-1\}+\backslash frac\{n-P\}\{n-1\}\backslash operatorname*\{max\}\_\{i\}w\_\{i\}+\backslash beta\_\{L\}^\{2\}\backslash operatorname*\{max\}\_\{i,k\}\backslash frac\{w\_\{i\}a\_\{i\}^\{(k)\}(\backslash tau\_\{i\})\}\{\backslash |a\_\{i\}\backslash |\_\{1\}\}\backslash right)\backslash leq1,$$ $$2{\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}{\gamma }_y^s{L}_f\frac{n}{P}[\frac{P-1}{n-1}+{\beta }_G^2(\frac{n-P}{n-1}\underset{i}{\max }{w}_i+{\beta }_L^2\underset{i}{\max }\frac{w_i\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_2^2}{\mathrm{\parallel }{\mathbf{a}}_i{\mathrm{\parallel }}_1^2})]\le 2\backslash tau\_\{\backslash rm\; eff\}\backslash gamma\_\{y\}^\{s\}L\_\{f\}\backslash frac\{n\}\{P\}\backslash left[\backslash frac\{P-1\}\{n-1\}+\backslash beta\_\{G\}^\{2\}\backslash left(\backslash frac\{n-P\}\{n-1\}\backslash max\_\{i\}w\_\{i\}+\backslash beta\_\{L\}^\{2\}\backslash max\_\{i\}\backslash frac\{w\_\{i\}\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{2\}^\{2\}\}\{\backslash |\backslash mathbf\{a\}\_\{i\}\backslash |\_\{1\}^\{2\}\}\backslash right)\backslash right]\backslash leq$$ $$2L_{f}\frac{n}{P}\left[8[\eta_{y}^{c}]^{2}M_{\mathbf{a}{-1}}L{f}\beta_{G}^{2}\right]\leq$$ $$\cdot$$ $$\frac{1}{4},\{\backslash frac\{1\}\{4\}\},$$ $${\frac{1}{4}}$$ Then the coefficient of E hΦ( e xb (s)) − Fe(xb (s), y (t)) ican we bounded by −τeffγ s y . Consequently, by rearranging

the terms and summing over t, we get the result. 1 S (s+1) X S−1 t=sS E hΦ( e xb (s)) − Fe(xb (s), y (t)) i ≤ E y sS − y ∗(xb (s)) 2 τeffγ s yS + τeffγ s y n P " σ 2 L Xn i=1 w 2 i ∥ai∥ 2 2 ∥ai∥ 2 1 + 2σ 2 G n − P n − 1 max iwi + β 2 L max i wi∥ai∥ 2 2 ∥ai∥ 2 1 !# + 4Lf ([η c x ] 2 + [η c y ] 2) " σ 2 L Xn i=1 wi ∥ai,−1∥ 2 2 + 2Ma−1 (G 2 x + σ 2 G) # .

$$\mathrm{\square }\backslash square$$ Proof of Lemma C.4. Let t = sS, sS + 1*, . . . ,*(s + 1)S − 1, where k is a positive integer. Let xb (s)is the latest snapshot iterate for the y-update in Algorithm 1-Fed-Norm-SGDA+ . Then

E hΦ( e x (t)) − Fe(x (t), y (t)) i = E hFe(x (t), y ∗(x (t))) − Fe(xb (s), y ∗(xb (s))) + Fe(xb (s), y ∗(xb (s))) − Fe(xb (s), y (t)) + Fe(xb (s), y (t)) − Fe(x (t), y (t)) i ≤ E hFe(x (t), y ∗(x (t))) − Fe(xb (s), y ∗(x (t)))i+ E hFe(xb (s), y ∗(xb (s))) − Fe(xb (s), y (t)) i+ GxE x (t) − xb (s) ≤ 2GxE x (t) − xb (s) + E hΦ( e xb (s)) − Fe(xb (s), y (t)) i. (69) where, y ∗(·) ∈ arg maxy Fe(·, y) and (69) follows from Gx-Lipschitz continuity of F(·, y) (Assumption 8).

where, y ∗(·) ∈ arg maxy Fe(·, y) and (69) follows from Gx-Lipschitz continuity of F(·, y) (Assumption 8). Next, we see that E xb (s) − x (t) ≤ qE xb (s) − x(t) 2(Jensen's inequality) (43) = vuuutE τeffγ sx Xt−1 t ′=sS X i∈C(t ′) w˜ig (t ′) x,i 2 ≤ τeffγ s x vuuut(S − 1) Xt−1 t ′=sS E X i∈C(t ′) w˜ig (t ′) x,i 2 ≤ τeffγ s x (S − 1)rn P vuutXn i=1 w2 i ∥ai∥ 2 2 ∥ai∥ 2 1 (σ 2 L + β 2 LG2x ) + G2x P − 1 n − 1 + n − P n − 1 Xn i=1 w2 i ! . (from (59)) Using this bound in (69), and summing over t, we get

1 S (s+1) X S−1 t=sS E hΦ( e x (t)) − Fe(x (t), y (t)) i≤ 1 S (s+1) X S−1 t=sS E hΦ( e xb (s)) − Fe(xb (s), y (t)) i + 2τeffγ s xGx(S − 1)rn P vuutXn i=1 w2 i ∥ai∥ 2 2 ∥ai∥ 2 1 (σ 2 L + β 2 LG2x ) + G2x P − 1 n − 1 + n − P n − 1 Xn i=1 w2 i ! . Finally, summing over s = 0 to T /S − 1 we get the result.

C.4 Extending The Result For Nonconvex One-Point-Concave (Nc-1Pc) Functions

Carefully revisting the proof of Theorem 2, we notice that Lemma C.1 and Lemma C.2 do not rely on the concavity assumption. Lemma C.3 does use concavity of local functions {fi}. However, it is only needed to derive (66). Further, this only requires concavity of local functions at a global point y ∗(xb (s)). Therefore, as mentioned earlier in Remark 10, it holds even for NC-1PC functions. This is an independent result in itself, since we have extended the existing convergence result of local stochastic gradient method for convex minimization (concave maximization) problems, to a much more general one-point-convex minimization (or one-point-convex maximization) problem. Therefore, we restate it here for the more general case.

Lemma C.5 (Local SG updates for One-Point-Concave Maximization). Suppose the local loss functions {fi} satisfy Assumptions 1, 3, 4, 8. Suppose for all x, all the fi's satisfy Assumption 9 at a common global minimizer y ∗(x), and thaty (t) 2≤ R for all t*. If we run Fed-Norm-SGDA+ with the same conditions* on the client and server step-sizes η c y , γs y respectively, as in Lemma C.3, then the iterates generated by Fed-Norm-SGDA+ also satisfy the bound in Lemma C.3. Next, Lemma C.4 also holds irrespective of concavity. Therefore, the resulting convergence result in Theorem 2 for nonconvex-concave minimax problems holds for a much larger class of functions. We restate the modified theorem statement briefly.

Theorem. Suppose the local loss functions {fi} satisfy Assumptions 1, 3, 4, 8. Suppose for all x*, all the* fi's satisfy Assumption 9 at a common global minimizer y ∗(x), and thaty (t) 2≤ R for all t*. If we run* Algorithm 1-Fed-Norm-SGDA+ with the same conditions on the client and server step-sizes η c y , γs y respectively, as in Theorem 3, then the iterates generated by Algorithm 1-Fed-Norm-SGDA+ also satisfy the bound in Theorem 3.

Remark 11. Again, choosing client weights {wi} the same as in the original global objective {pi}, we get convergence in terms of the original objective F.

D Additional Experiments

For communicating parameters and related information amongst the clients, ethernet connections were used.

Our algorithm was implemented using parallel training tools in PyTorch 1.0.0 and Python 3.6.3. For both robust NN Training and fair classification experiments, we use batch-size of 32 in all the algorithms. Momentum parameter 0.9 is used only in Momentum Local SGDA(+).

Robust NN Training. Here we further explore performance of Fed-Norm-SGDA+ on the robust NN training problem. We use VGG-11 model to classify CIFAR10 dataset. In Figure 6, we show the training loss curves corresponding to the results in Figure 3 on varying number of local steps. Similarly, in Figure 6, we show the training loss curves corresponding to Figure 4 on the effect of partial participation. In Figure 7, we demonstrate the effect of increasing data heterogeneity across clients, whle in Figure 9 we show the advantage of using multiple clients for the federated minimax problem. With k-fold increase in n, we observe an almost k-fold drop in the number of communication rounds needed to reach a target test accuracy (70% here.).

We use batch-size of 32. Momentum parameter 0.9 is used only in Local SGDA+(M).

Communication rounds	1-100	101-200	>200
c
Client Learning Rate (η y )	0.02	2 × 10−3	2 × 10−4
Client Learning Rate (η c )	0.016	1.6 × 10−3	1.6 × 10−4
x s	s
Server Learning Rate (γ x = γ y )	1	1	1

Table 3: Parameter values for experiments in robust NN training experiments.

Fair Classification We also demonstrate the impact of partial client participation in the fair classification problem. Figure 10 complements Figure 10 in the main paper, evaluating fairness of a VGG11 model on

Figure 6: Comparison of the effect of heterogeneous number of local updates {τi} on the performance of Fed-Norm-SGDA+ (Algorithm 1), Local SGDA+, and Local SGDA+ with momentum, while solving (7) on CIFAR10 dataset, with VGG11 model. The solid (dashed) curves are for E = 5 (E = 7), and α = 0.1.

Figure 7: Comparison of the effects of partial client participation (PCP) on the performance of Fed-NormSGDA+, for the robust NN training problem on the CIFAR10 dataset, with the VGG11 model. The figure shows the robust test accuracy. The solid (dashed) curves are for α = 0.1 (α = 1.0). CIFAR10 dataset. We have plotted the test accuracy of the model over the worst distribution. With an increasing number of participating clients, the performance consistently improves.

Batch-size of 32 is used. Momentum parameter 0.9 is used only in Local SGDA (M).

Table 4: Parameter values for experiments in fair classification experiments.

	c
Client Learning Rate (η y )	0.02
Client Learning Rate (η c )	0.016
	x
Server Learning Rate (γ s	s
	x = γ y )	1

Figure 8: Effect of inter-client data heterogeneity (quantified by α) on the performance of Fed-Norm-SGDA+ in a robust NN training task.

Figure 9: Effect of increasing client-set on the performance of Fed-Norm-SGDA+ in a robust NN training task.

Figure 10: Effect of partial client participation on the performance of Fed-Norm-SGDA in a fair image classification task.