On the Convergence of FedProx with Extrapolation and Inexact Prox

## Abstract

Abstract Enhancing the FedProx federated learning algorithm (Li et al., 2020 ) with server-side extrapolation, Li et al. ( 2024a ) recently introduced the FedExProx method.
Their theoretical analysis, however, relies on the assumption that each client computes a certain proximal operator exactly, which is impractical since this is virtually never possible to do in real settings.
In this paper, we investigate the behavior of FedExProx without this exactness assumption in the smooth and globally strongly convex setting.
We establish a general convergence result, showing that inexactness leads to convergence to a neighborhood of the solution.
Additionally, we demonstrate that, with careful control, the adverse effects of this inexactness can be mitigated.
By linking inexactness to biased compression (Beznosikov et al., 2023 ) , we refine our analysis, highlighting robustness of extrapolation to inexact proximal updates.
We also examine the local iteration complexity required by each client to achieved the required level of inexactness using various local optimizers.
Our theoretical insights are validated through comprehensive numerical experiments.

## 1 Introduction

Distributed optimization is becoming increasingly essential in modern machine learning, especially as models grow more complex.
Federated learning (FL), a decentralized approach where multiple clients collaboratively train a shared model while keeping their data locally to preserve privacy, is a key example of this trend (Konečný et al., ; McMahan et al., ).
Often, a central server coordinates the process by aggregating the locally trained models from each client to update the global model without accessing the raw data.
The federated average algorithm ( FedAvg), introduced by McMahan et al. () and Mangasarian & Solodov (), is one of the most popular strategies for tackling federated learning problems.
The algorithm comprises three essential components: client sampling, data sampling, and local training.
During its execution, the server first samples a subset of clients to participate in the training process for a given round.
Each selected client then performs local training using stochastic gradient descent ( SGD), with or without random reshuffling, to enhance communication efficiency, as documented by Bubeck et al. (); Gower et al. (); Moulines & Bach (); Sadiev et al. ().
FedAvg has proven to be highly successful in practice, nevertheless it suffers from client drift when data is heterogeneous (Karimireddy et al., ).

Various techniques have been proposed to address the challenges of data heterogeneity, with FedProx, introduced by Li et al. (), being one notable example.
Rather than having each client perform local SGD rounds, FedProx requires each client to compute a proximal operator locally.
Computing the proximal operator can be regarded as an optimization problem that each client can solve locally.
Proximal algorithms are advantageous when the proximal operators can be evaluated relatively easily (Parikh et al., ).
Algorithms based on proximal operators, such as the proximal point method ( PPM) (Rockafellar, ; Parikh et al., ) and its extension to the stochastic setting ( SPPM) (Bertsekas, ; Asi & Duchi, ; Khaled & Jin, ; Richtárik & Takác, ; Patrascu & Necoara, ), offer greater stability against inaccurately specified step sizes, unlike gradient-based methods.
PPM was introduced by Martinet () and expanded by Rockafellar ().
Its extension into the stochastic setting are often used in federated optimization.
The stability mentioned is particularly useful when problem-specific parameters, such as the smoothness constant of the objective function, are unknown which renders determining the step size for SGD becomes challenging.
Indeed, an excessively large step size in SGD leads to divergence, while a small step size ensures convergence but significantly slows down the training process.

Another approach to mitigating the slowdown caused by heterogeneity is the use of a server step size.
Specifically, in FedAvg, a local step size is employed by each client to minimize their individual objectives, while a server step size is used to aggregate the ‘pseudo-gradients’ obtained from each client (Karimireddy et al., ; Reddi et al., ).
The local step size is set relatively small to mitigate client drift, while the server step size is set larger to avoid slowdowns.
However, the small step sizes result in a slowdown during the initial phase of training, which cannot be fully compensated by the large server step size (Jhunjhunwala et al., ).
Building on the extrapolation technique employed in parallel projection methods to solve the convex feasibility problem (Censor et al., ; Combettes, ; Necoara et al., ), Jhunjhunwala et al. () introduced FedExP as an extension of FedAvg, incorporating adaptive extrapolation as the server step size.
Extrapolation involves moving further along the line connecting the most recent iterate, $x_{k}$, and the average of the projections of $x_{k}$ onto different convex sets, $\mathcal{X}_{i}$, in the parallel projection method, which accelerates the algorithm.
Extrapolation is also known as over-relaxation (Rechardson, ) in fixed point theory.
It is a common technique to effectively accelerate the convergence of fixed point methods including gradient based algorithms and proximal splitting algorithms (Condat et al., ; Iutzeler & Hendrickx, ).
Recently, Li et al. () shows that the combination of extrapolation with FedProx also results in better complexity bounds.
The analysis of the resulting algorithm FedExProx reveals the relationship between the extrapolation parameter and the step size of gradient-based methods with respect to the Moreau envelope associated with the original objective function.
However, it relies on the assumption that each proximal operator is solved accurately, which makes it impractical and less advantageous compared to gradient-based algorithms.

### 1.1 Contributions

Our paper makes the following contributions, please refer to Appendix A for notation details.

- •
We provide a new analysis of FedExProx based on Li et al. (), focusing on the case where the proximal operators are evaluated inexactly in the globally strongly convex setting, removing the need for the assumption of exact proximal operator evaluations.
By properly defining the notion of approximation, we establish a general convergence guarantee of the algorithm to a neighborhood of the solution utilizing the theory of biased SGD (Demidovich et al., ).
Specifically, our algorithm achieves a linear convergence rate of $\mathcal{O}\left(\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\right)$ to a neighborhood of the solution, matching the rate presented by Li et al. ().
- •
Building on our understanding of how the neighborhood arises, we propose a new method of approximation.
This alternative characterization of inexactness eliminates the neighborhood from the previous convergence guarantee, provided that the inexactness is properly bounded, and the extrapolation parameter is chosen to be sufficiently small.
- •
By leveraging the similarity between the definitions of inexactness and compression, we enhance our analysis using the theory of biased compression (Beznosikov et al., ).
The improved analysis offers a faster rate of $\mathcal{O}\left(\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu-4
\varepsilon_{2}L_{\max}}\right)$(^1^11The parameter $\varepsilon_{2}$ is the parameter associated with accuracy of relative approximation as defined in Definition 4. We use the notation $\mathcal{O}\left(\cdot\right)$ to ignore constant factors and $\tilde{\mathcal{O}}\left(\cdot\right)$ when logarithmic factors are also omitted.), leading to convergence to the exact solution, provided that the inexactness is bounded in a more permissive manner.
More importantly, the optimal extrapolation $\nicefrac{{1}}{{\gamma L_{\gamma}}}$ matches the exact case.
This shows that extrapolation aids convergence as long as sufficient accuracy is reached, even with inexact proximal evaluations.
- •
We then analyze how the aforementioned approximations can be obtained by each client.
As examples, we provide the local iteration complexity when the client employs gradient descent ( GD) or Nesterov’s accelerated gradient descent ( AGD), demonstrating that these approximations are readily achievable.
Specifically, for the $i$-th client, the local iteration complexity is $\tilde{\mathcal{O}}\left(1+\gamma L_{i}\right)$ when using GD, and $\tilde{\mathcal{O}}\left(\sqrt{1+\gamma L_{i}}\right)$ when using AGD.
See Table 1 and Table 2 for a detailed comparison of complexities of all relevant quantities.
- •
Finally, we validate our theoretical findings through numerical experiments.
Our numerical results suggest that the proposed technique of relative approximation effectively eliminates bias.
In some cases, the algorithm even outperforms FedProx with exact updates, further validating the effectiveness of server extrapolation, even when proximal updates are inexact.

**Table 1: Comparison of FedExProx (Li et al., ) and our proposed inexact versions of the algorithms using different approximations. In the convergence column, we present the rate at which each algorithm converges to either the solution or a neighborhood in the globally strongly convex setting. Here, $L_{\gamma}$ represents the smoothness constant of $M^{\gamma}$ as defined before Theorem 1. The neighborhood column indicates the size of the neighborhood, while the optimal extrapolation column suggests the best choice of $\alpha$ for each algorithm. The final column outlines the conditions on the inexactness. All quantities are presented with constant factors omitted, $K$ is the number of total iterations, $\gamma$ is the local step size for the proximal operator, $S\left(\varepsilon_{2}\right)$ defined in Theorem 2 is a factor of slowing down due to inexactness in $(0,1]$. For relative approximation, we first present the original theory in the third row and then place the sharper analysis in the following row for comparison.**
| Algorithm | Convergence | Neighborhood | Optimal<br>Extrapolation | Bound on<br>Inexactness |
| --- | --- | --- | --- | --- |
|  |  |  |  |  |
| FedExProx | $\exp\left(-\frac{K\mu}{L_{\gamma}\left(1+\gamma L_{\max}\right)}\right)$ | 0 0 | $\frac{1}{\gamma L_{\gamma}}$ | NA |
| (NEW) FedExProx with<br>$\varepsilon_{1}$ approximation | $\exp\left(-\frac{K\mu}{L_{\gamma}\left(1+\gamma L_{\max}\right)}\right)$ | $\varepsilon_{1}\left(\frac{\frac{1}{\gamma}+L_{\max}}{\mu}\right)^{2}$^(a) | $\frac{1}{4\gamma L_{\gamma}}$ | NA ^(b) |
| (NEW) FedExProx with<br>$\varepsilon_{2}$ relative approximation<br>by biased SGD | $\exp\left(-\frac{K\mu S\left(\varepsilon_{2}\right)}{L_{\gamma}\left(1+\gamma L<br>_{\max}\right)}\right)$^(c) | 0 0 | $<\frac{1}{\gamma L_{\gamma}}$ | $<\frac{\mu^{2}}{4L_{\max}^{2}}$ |
| (NEW) FedExProx with<br>$\varepsilon_{2}$ relative approximation<br>by biased compression | $\exp\left(-\frac{K\left(\mu-4\varepsilon_{2}L_{\max}\right)}{L_{\gamma}\left(1<br>+\gamma L_{\max}\right)}\right)$ | 0 0 | $\frac{1}{\gamma L_{\gamma}}$^(d) | $<\frac{\mu}{4L_{\max}}$ |

**Table 2: Comparison of local iteration complexities of each client in order to obtain an approximation using either GD or AGD (Nesterov, ). We use the $i$-th client as an example, where the local objective $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ is $L_{i}$-smooth and convex, $i\in\left\{1,2,\ldots,n\right\}$.**
| Algorithm | $\varepsilon_{1}$ absolute approximation | $\varepsilon_{2}$ relative approximation |
| --- | --- | --- |
| Gradient descent | $\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\frac{\left\|x_{k}-{\rm<br>prox<br>}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}}{\varepsilon_{1}}\right)\right)$ ^(a) | $\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\frac{1}{\varepsilon_{2}<br>}\right)\right)$ |
| Accelerate gradient descent | $\mathcal{O}\left(\sqrt{1+\gamma L_{i}}\log\left(\frac{\left\|x_{k}-{\rm prox}_<br>{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}}{\varepsilon_{1}}\right)\right)$ | $\mathcal{O}\left(\sqrt{1+\gamma L_{i}}\log\left(\frac{1}{\varepsilon_{2}}<br>\right)\right)$ |

### 1.2 Related work

Arguably, stochastic gradient descent ( SGD) (Robbins & Monro, ; Ghadimi & Lan, ; Gower et al., ; Gorbunov et al., ) remains one of the foundational algorithm in the field of machine learning.
One can simply formulate it as

$$ $x_{k+1}=x_{k}-\eta\cdot g(x_{k}),$ $$

where $\eta>0$ is a scalar step size, $g(x_{k})$ is a possibly stochastic estimator of the true gradient $\nabla f(x_{k})$.
In the case when $g(x_{k})=\nabla f(x_{k})$, SGD becomes GD.
Various extensions of SGD have been proposed since its introduction, examples include compressed gradient descent ( CGD) (Alistarh et al., ; Khirirat et al., ), SGD with momentum (Loizou & Richtárik, ; Liu et al., ), SGD with matrix step size (Li et al., ) and variance reduction (Gower et al., ; Johnson & Zhang, ; Gorbunov et al., ; Tyurin & Richtárik, ; Li et al., ).
Gower et al. () presented a framework for analyzing SGD with unbiased gradient estimator in the convex case based on expected smoothness.
However, in practice, sometimes the gradient estimator could be biased, examples include SGD with sparsified or delayed update (Alistarh et al., ; Recht et al., ).
Beznosikov et al. () examined biased updates in the context of compressed gradient descent.
Demidovich et al. () provides a framework for analyzing SGD with biased gradient estimators in the non-convex setting.

Proximal point method ( PPM) was originally introduced as a method to solve variational inequalities (Martinet, ; Rockafellar, ).
The transition to the stochastic case, driven by the need to efficiently address large-scale optimization problems, leads to the development of SPPM.
Due to its stability and advantage over the gradient based methods, it has been extensively studied, as documented by (Patrascu & Necoara, ; Bianchi, ; Bertsekas, ).
For proximal algorithms to be practical, it is commonly assumed that the proximal operator can be solved efficiently, such as in cases where a closed-form solution is available.
However, in large-scale machine learning models, it is rarely possible to find such a solution in closed form.
To address this issue, most proximal algorithms assume that only an approximate solution is obtained, achieving a certain level of accuracy (Khaled & Jin, ; Sadiev et al., ; Karagulyan et al., ).
Various notions of inexactness are employed, depending on the assumptions made, the properties of the objective, and the availability of algorithms capable of efficiently finding such approximations.

Moreau envelope was first introduced to handle non-smooth functions by Moreau ().
It is also known as the Moreau-Yosida regularization.
The use of the Moreau envelope as an analytical tool to analyze proximal algorithms is not novel.
Ryu & Boyd () noted that running a proximal algorithm on the objective is equivalent to applying gradient methods to its Moreau envelope.
Davis & Drusvyatskiy () analyzed stochastic proximal point method ( SPPM) for weakly convex and Lipschitz functions based on this finding.
Recently, Li et al. () provided an analysis of FedProx with server-side step size in the convex case, based on the reformulation of the problem using the Moreau envelope.
The role of the Moreau envelope extends beyond analyzing proximal algorithms; it has also been applied in the contexts of personalized federated learning (T Dinh et al., ) and meta-learning (Mishchenko et al., ).
The mathematical properties of the Moreau envelope are relatively well understood, as documented by
Jourani et al. (); Planiden & Wang (; ).

Projection methods initially emerged as an effective tool for solving systems of linear equations or inequalities (Kaczmarz, ) and were later generalized to solve the convex feasibility problem (Combettes, ).
The parallel version of this approach involves averaging the projections of the current iterates onto all existing convex sets $\mathcal{X}_{i}$ to obtain the next iterate, a process that is empirically known to be accelerated by extrapolation.
Numerous heuristic rules have been proposed to adaptively set the extrapolation parameter, such as those by Bauschke et al. () and Pierra ().
Only recently, the mechanism behind constant extrapolation was uncovered by Necoara et al. (), who developed the corresponding theoretical framework.
Additionally, Li et al. () provides explanations for the effectiveness of adaptive rules, revealing the connection between the extrapolation parameter and the step size of SGD when using the Moreau envelope as the global objective.

## 2 Mathematical background

In this work, we are interested in the distributed optimization problem which is formulated in the following finite-sum form

$$ $\min_{x\in\mathbb{R}^{d}}\left\{f(x):=\frac{1}{n}\sum_{i=1}^{n}f_{i}(x)\right\},$ (1) $$

where $x\in\mathbb{R}^{d}$ is the model, $n$ is the number of devices/clients, $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ is global objective, each $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ is the empirical risk of model $x$ associated with the $i$-th client.
Each $f_{i}\left(x\right)$ often has the form

$$ $f_{i}(x):={\mathbb{E}}_{\xi\sim\mathcal{D}_{i}}\left[l\left(x,\xi\right)\right],$ (2) $$

where the loss function $l\left(x,\xi\right)$ represents the loss of model $x$ on data point $\xi$ over the training data $\mathcal{D}_{i}$ owned by client $i\in[n]:=\left\{1,2,\ldots,n\right\}$.
We first give the definitions for the proximal operator and Moreau envelope, which we will be using in our analysis.

###### Definition 1 (Proximal operator) .

The proximal operator of an extended real-valued function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ with step size $\gamma>0$ and center $x\in\mathbb{R}^{d}$ is defined as

$$ $\displaystyle{\rm prox}_{\gamma\phi}\left(x\right):=\arg\min_{z\in\mathbb{R}^{ d}}\left\{\phi\left\{z\right\}+\frac{1}{2\gamma}\left\|z-x\right\|^{2}\right\}.$ $$

It is well-known that for any proper, closed, and convex function $\phi$, the proximal operator with any $\gamma>0$ returns a singleton.

###### Definition 2 (Moreau envelope) .

The Moreau envelope of an extended real-valued function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ with step size $\gamma>0$ and center $x\in\mathbb{R}^{d}$ is defined as

$$ $\displaystyle{M}^{\gamma}_{\phi}\left(x\right):=\min_{z\in\mathbb{R}^{d}}\left \{\phi\left(z\right)+\frac{1}{2\gamma}\left\|z-x\right\|^{2}\right\}.$ $$

By the definition of Moreau envelope, it is easy to see that

$$ $\displaystyle{M}^{\gamma}_{\phi}\left(x\right)=\phi\left({\rm prox}_{\gamma \phi}\left(x\right)\right)+\frac{1}{2\gamma}\left\|x-{\rm prox}_{\gamma\phi} \left(x\right)\right\|^{2}.$ (3) $$

Not only are their function values related, but for any proper, closed, and convex function $\phi$, the Moreau envelope is differentiable, specifically, we have:

$$ $\displaystyle\nabla{M}^{\gamma}_{\phi}\left(x\right)=\frac{1}{\gamma}\left(x-{ \rm prox}_{\gamma f}\left(x\right)\right).$ (4) $$

The above identity indicates that $\phi$ and ${M}^{\gamma}_{\phi}$ are intrinsically related.
This relationship plays a key role in our analysis.
We also need the following assumptions on $f$ and $f_{i}$ to carry out our analysis.

###### Assumption 1 (Differentiability) .

The function $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ in (1) is differentiable and bounded from below for all $i\in[n]$.

###### Assumption 2 (Interpolation regime) .

There exists $x_{\star}\in\mathbb{R}^{d}$ such that $\nabla f_{i}(x_{\star})=0$ for all $i\in[n]$.

The same as Li et al. (), we assume that we are in the interpolation regime.
This situation arises in modern deep learning scenarios where the number of parameters, $d$, significantly exceeds the number of data points.
For justifications, we refer the readers to Arora et al. (); Montanari & Zhong ().
The motivation for this assumption stems from the parallel projection methods (5) used to solve convex feasibility problems, where the intersection of all convex sets $\mathcal{X}_{i}$ is assumed to be non-empty, which is precisely the interpolation assumption of each $f_{i}$ being the indicator function of $\mathcal{X}_{i}$.

$$ $\displaystyle x_{k+1}=\frac{1}{n}\sum_{i=1}^{n}\Pi_{\mathcal{X}_{i}}\left(x_{k }\right).$ (5) $$

It is known that for (5), the use of extrapolation would enhance its performance both in theory and practice (Necoara et al., ).
Since ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ can be viewed as projection to some level set of $f_{i}$, it is analogous to $\Pi_{\mathcal{X}_{i}}\left(x_{k}\right)$.
Therefore, it is reasonable to assume that extrapolation would be effective under the same assumption.

###### Assumption 3 (Individual convexity) .

The function $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ is convex for all $i\in[n]$.
This means that for each $f_{i}$,

$$ $0\leq f_{i}(x)-f_{i}(y)-\left\langle\nabla f_{i}(y),x-y\right\rangle,\quad \forall x,y\in\mathbb{R}^{d}.$ (6) $$

###### Assumption 4 (Smoothness) .

The function $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ is $L_{i}$-smooth, $L_{i}>0$ for all $i\in[n]$.
This means that for each $f_{i}$,

$$ $f_{i}(x)-f_{i}(y)-\left\langle\nabla f_{i}(y),x-y\right\rangle\leq\frac{L_{i}} {2}\left\|x-y\right\|^{2},\quad\forall x,y\in\mathbb{R}^{d}.$ (7) $$

We will use $L_{\max}$ to denote $\max_{i\in[n]}L_{i}$.

###### Assumption 5 (Global strong convexity) .

The function $f$ is $\mu$-strongly convex, $\mu>0$. That is

$$ $f(x)-f(y)-\left\langle\nabla f(y),x-y\right\rangle\geq\frac{\mu}{2}\left\|x-y \right\|^{2},\quad\forall x,y\in\mathbb{R}^{d}.$ $$

These are all standard assumptions commonly used in convex optimization.
We first present our algorithm as Algorithm 1.
In the following sections, we provide the analysis of this algorithm under different definitions of inexactness, respectively in Section 3 and Section 4.
Details on how these inexactness levels can be achieved by each client are provided in Section 5.
Finally, numerical experiments validating our results are presented in Appendix G.

Figure: Algorithm 1 Inexact FedExProx

###### Definition 1 (Proximal operator) .

###### Definition 2 (Moreau envelope) .

###### Assumption 1 (Differentiability) .

###### Assumption 2 (Interpolation regime) .

###### Assumption 3 (Individual convexity) .

###### Assumption 4 (Smoothness) .

###### Assumption 5 (Global strong convexity) .

## 3 Absolute approximation in distance

As previously suggested, we assume that each proximal operator is solved inexactly, and we need to quantify this inexactness in some way.
Notice that client $i$ is required to solve the following minimization problem.

$$ $\min_{z\in\mathbb{R}^{d}}A^{\gamma}_{k,i}\left(z\right):=f_{i}\left(z\right)+ \frac{1}{2\gamma}\left\|z-x_{k}\right\|^{2},$ (9) $$

where $x_{k}$ is the current iterate and $\gamma>0$ is a constant.
Since we have assumed each function $f_{i}$ is convex, $A^{\gamma}_{k,i}\left(z\right)$ is $\frac{1}{\gamma}$-strongly convex with ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ being its unique minimizer.
One of the most straightforward ways to measure inexactness in this case is through the squared distance to the minimizer, leading to the following definition.

###### Definition 3 (Absolute approximation) .

Given a proper, closed and convex function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$, and a step size $\gamma>0$, we say that a point $y\in\mathbb{R}^{d}$ is an $\varepsilon_{1}$-approximation of ${\rm prox}_{\gamma\phi}\left(x\right)$, if for some $\varepsilon_{1}\geq 0$,

$$ $\left\|y-{\rm prox}_{\gamma f}\left(x\right)\right\|^{2}\leq\varepsilon_{1}.$ (10) $$

In order to analyze Algorithm 1, we first transform the update rule given in (8) in the following way,

$$ $\displaystyle x_{k+1}$ $\displaystyle=$ $\displaystyle x_{k}+\alpha_{k}\left(\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{ i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)+\frac{1}{n}\sum_{i=1 }^{n}{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)-x_{k}\right)$ (11) $\displaystyle\overset{(\ref{eq:ppt-2-gradient})}{=}$ $\displaystyle x_{k}-\alpha_{k}\cdot g(x_{k}),$ $$

where

$$ $g(x_{k}):=\underbrace{\frac{1}{n}\sum_{i=1}^{n}\gamma\nabla{M}^{\gamma}_{f_{i} }\left(x_{k}\right)}_{\text{Gradient}}-\underbrace{\frac{1}{n}\sum_{i=1}^{n} \left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)}_{ \text{Bias}}.$ (12) $$

The above reformulation suggests that Algorithm 1 is in fact, SGD with respect to global objective $\gamma M^{\gamma}\left(x\right):=\frac{1}{n}\sum_{i=1}^{n}\gamma{M}^{\gamma}_{
f_{i}}\left(x\right)$ with a biased gradient estimator.
Compared to SGD with an unbiased gradient estimator, its biased counterpart is less well understood.
However, we are still able to obtain the following convergence guarantee using theories for biased SGD from Demidovich et al. ().

###### Theorem 1 .

Assume 1 (Differentiability), 2 (Interpolation Regime), 3 (Individual convexity), 4 (Smoothness) and 5 (Global strong convexity) hold.
If each client only computes a $\varepsilon_{1}$-absolute approximation $\tilde{x}_{i,k+1}$ in squared distance of ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ at every iteration, such that $\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{
2}\leq\varepsilon_{1}$.
Then we have the following convergence guarantee or Algorithm 1: For a constant extrapolation parameter satisfying $0<\alpha\leq\frac{1}{4}\cdot\frac{1}{\gamma L_{\gamma}}$, where $\gamma$ is the step size of the proximity operator, $\alpha_{k}=\alpha$ is a constant extrapolation parameter, $L_{\gamma}$ is the smoothness constant of $M^{\gamma}$.
The last iterate $x_{K}$ satisfy

$$ $\displaystyle\mathcal{E}_{K}$ $\displaystyle\leq\left(1-\frac{\alpha\gamma\mu}{8\left(1+\gamma L_{\max}\right )}\right)^{K}\mathcal{E}_{0}+\frac{4\varepsilon_{1}\left(1+\gamma L_{\max} \right)}{\mu}\cdot\left(2\alpha L_{\gamma}+\frac{1}{\gamma}\right),$ $$

where $\mathcal{E}_{k}=\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}$.
Specifically, when we choose $\alpha=\frac{1}{4}\cdot\frac{1}{\gamma L_{\gamma}}$, we have

$$ $\displaystyle\Delta_{K}\leq\left(1-\frac{\mu}{32L_{\gamma}\left(1+\gamma L_{ \max}\right)}\right)^{K}\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu} \cdot\Delta_{0}+12\varepsilon_{1}\cdot\left(\frac{\nicefrac{{1}}{{\gamma}}+L_{ \max}}{\mu}\right)^{2},$ $$

where $\Delta_{K}=\left\|x_{K}-x_{\star}\right\|^{2}$, $x_{\star}$ is a minimizer of $f$.

For the sake of brevity in the following discussion, we will use the notation $\mathcal{E}_{k}=\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}$, where $M^{\gamma}_{\inf}$ denotes the infimum of $M^{\gamma}$, $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$, where $x_{\star}$ is a minimizer of $M^{\gamma}$.
Notice that since we are in the interpolation regime, according to 7, the minimizer of $M^{\gamma}$ is also a minimizer of $f$.
Note that instead of converging to the exact minimizer $x_{\star}$, the algorithm converges to a neighborhood whose size depends on both $\varepsilon_{1}$ and $\gamma$; the smaller $\gamma$ is, the larger the neighborhood becomes.
This can be understood intuitively: A smaller $\gamma$ means less progress is made per iteration, leading to a larger accumulated error as the total number of iterations increases.
The parameter $\varepsilon_{1}$ can be arbitrarily large, and the convergence guarantee still holds, indicating that the theory presented is quite general.
However, as $\varepsilon_{1}$ increases, the size of the neighborhood grows proportionally, which limits the practical significance of the result.
When $\varepsilon_{1}=0$, the neighborhood diminishes, and we obtain an iteration complexity of $\tilde{\mathcal{O}}\left(\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\right)$(^2^22We leave out the log factor in $\tilde{\mathcal{O}}\left(\cdot\right)$ notation.), which recovers the result of Li et al. () up to a constant factor.
The optimal constant extrapolation parameter is now given by $\alpha_{\star}=\frac{1}{4}\cdot\frac{1}{\gamma L_{\gamma}}$ which is $4$ times smaller than that of Li et al. ().

###### Definition 3 (Absolute approximation) .

###### Theorem 1 .

## 4 Relative approximation in distance

Theorem 1 offers a general theoretical framework for understanding the behavior of Algorithm 1.
However, a key challenge with Algorithm 1 which utilizes inexact proximal solutions that satisfy Definition 3, is that, unless the proximal operators are solved exactly, convergence will always be limited to a neighborhood of the solution.
The underlying reason is that, as the algorithm progresses, the gradient term in the gradient estimator $g(x_{k})$ diminishes, whereas the bias term remains unchanged.
Building on this observation, we propose employing a different type of approximation, specifically an approximation in relative distance, as defined below.

###### Definition 4 (Relative approximation) .

Given a convex function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$ and a stepsize $\gamma>0$, we say that a point $y\in\mathbb{R}^{d}$ is a $\varepsilon_{2}$-relative approximation of ${\rm prox}_{\gamma\phi}\left(x\right)$, if for some $\varepsilon_{2}\in[0,1)$,

$$ $\left\|y-{\rm prox}_{\gamma\phi}\left(x\right)\right\|^{2}\leq\varepsilon_{2} \cdot\left\|x-{\rm prox}_{\gamma\phi}\left(x\right)\right\|^{2}.$ (13) $$

We impose the requirement that the coefficient $\varepsilon_{2}$ be less than $1$ to ensure that the next iterate is no worse than the current one.
As we can observe, if the approximation of the solution for each proximal operator satisfies Definition 4, both the gradient term and the bias term diminish as the algorithm progresses, ensuring convergence to the exact solution.
Using the theory of biased SGD, we can obtain the following theorem.

###### Theorem 2 .

Assume all the assumptions mentioned in Theorem 1 also hold here.
If each client only computes a $\varepsilon_{2}$-relative approximation $\tilde{x}_{i,k+1}$ in distance with $\varepsilon_{2}<\nicefrac{{\mu^{2}}}{{4L^{2}_{\max}}}$, such that $\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{
2}\leq\varepsilon_{2}\cdot\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}
\right)\right\|^{2}$.
If we are running Algorithm 1 with $\alpha_{k}=\alpha$ satisfying

$$ $\displaystyle 0<\alpha\leq\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu-2\sqrt{ \varepsilon_{2}}L_{\max}}{\mu+4\sqrt{\varepsilon_{2}}L_{\max}+4\varepsilon_{2} L_{\max}}.$ $$

Then the iterates generated by Algorithm 1 satisfies

$$ $\mathcal{E}_{K}\leq\left(1-\alpha\cdot\frac{\gamma\left(\mu-2\sqrt{\varepsilon _{2}}L_{\max}\right)}{4\left(1+\gamma L_{\max}\right)}\right)^{K}\mathcal{E}_{ 0}.$ $$

Specifically, if we choose the largest $\alpha$ possible, we have

$$ $\displaystyle\Delta_{K}\leq\left(1-\frac{\mu}{4L_{\gamma}\left(1+\gamma L_{ \max}\right)}\cdot S\left(\varepsilon_{2}\right)\right)^{K}\cdot\frac{L{\gamma }\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0},$ $$

where $S(\varepsilon_{2}):=\frac{\left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)
\left(1-2\sqrt{\varepsilon_{2}}\frac{L_{\max}}{\mu}\right)}{\mu+4\sqrt{
\varepsilon_{2}}L_{\max}+4\varepsilon_{2}L_{\max}}$ satisfies $0<S(\varepsilon_{2})\leq 1$ is the factor of slowing down due to inexact proximal operator evaluation.

Observe that when $\varepsilon_{2}=0$, meaning the proximal operators are solved exactly, the optimal extrapolation is $\alpha=\frac{1}{\gamma L_{\gamma}}$ and the iteration complexity is $\tilde{\mathcal{O}}\left(\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\right)$.
This recovers the exact result from Li et al. ().
In the case of an inexact solution, as $\varepsilon_{2}$ increases, both $\alpha$ and $S(\varepsilon_{2})$ decrease, leading to a slower rate of convergence.
Note that arbitrary rough approximations are not permissible in this case, as $\varepsilon_{2}$ must satisfy $\varepsilon_{2}<\frac{\mu^{2}}{4L_{\max}^{2}}$.

It is worthwhile noting that Definition 4 is connected to the concept of compression.
Indeed, in our case we have $x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)=\gamma\nabla{M}^{\gamma}_{f_
{i}}\left(x_{k}\right)$, while $\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ can be interpreted as the gradient after compression, that is, $\mathcal{C}(\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right))$.
This indicates that Algorithm 1 with approximation satisfying Definition 4 can be viewed as compressed gradient descent with biased compressor.
We obtain the following convergence guarantee based on theory provided by Beznosikov et al. ().

###### Theorem 3 .

Assume all assumptions of Theorem 1 hold.
Let the approximation $\tilde{x}_{i,k+1}$ all satisfies Definition 4 with $\varepsilon_{2}<\nicefrac{{\mu}}{{4L_{\max}}}$, that is $\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{
2}\leq\varepsilon_{2}\cdot\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}
\right)\right\|^{2}$.
If we are running Algorithm 1 with $\alpha_{k}=\alpha\in(0,\frac{1}{\gamma L_{\gamma}}]$, we have the iterates produced by it satisfying

$$ $\mathcal{E}_{K}\leq\left(1-\left(1-\frac{4\varepsilon_{2}L_{\max}}{\mu}\right) \cdot\frac{\gamma\mu}{4\left(1+\gamma L_{\max}\right)}\cdot\alpha\right)^{K} \mathcal{E}_{0}.$ $$

specifically, if we take the largest extrapolation ($\alpha=\frac{1}{\gamma L_{\gamma}}>1$) possible, we have

$$ $\displaystyle\Delta_{K}\leq\left(1-\left(1-\frac{4\varepsilon_{2}L_{\max}}{\mu }\right)\cdot\frac{\mu}{4L_{\gamma}\left(1+\gamma L_{\max}\right)}\right)^{K} \cdot\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0}.$ $$

The convergence guarantee obtained in this way is sharper, indeed, Theorem 3 suggests that as long as $\varepsilon_{2}<\nicefrac{{\mu}}{{4L}}$, we are able to pick $\alpha=\nicefrac{{1}}{{\gamma L_{\gamma}}}$(^3^33It is shown in Li et al. () that $\nicefrac{{1}}{{\gamma L_{\gamma}}}>1$, which justifies why $\alpha$ is called the extrapolation parameter.)which is the optimal extrapolation for exact proximal computation given in Li et al. ().
Notably, this implies that extrapolation is an effective technique for accelerating the algorithm in this setting, regardless of inexact proximal operator evaluations.
Same as Theorem 2, the convergence is slowed down by the approximation, and in the case of $\varepsilon_{2}=0$, we recover the result in Li et al. ()

###### Definition 4 (Relative approximation) .

###### Theorem 2 .

###### Theorem 3 .

## 5 Achieving the level of inexactness

To fully comprehend the overall complexity of Algorithm 1, it is essential to examine whether the inexactness in evaluating the proximal operators can be effectively achieved.
Since each ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ is computed locally by the corresponding client, the client has access to all the necessary data points for the computation.
Thus, the most straightforward approach is to have each client perform GD.

###### Theorem 4 (Local computation via GD ) .

Assume 1 (Differentiability), 3 (Individual convexity) and 4 (Smoothness) hold.
The iteration complexity for the $i$-th client to provide an approximation using GD in the $k$-th iteration with local step size $\eta_{i}=\frac{\gamma}{1+\gamma L_{i}}$, satisfying Definition 3 is
$\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\nicefrac{{\left\|x_{k}-
{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}}}{{\varepsilon_{1}}}
\right)\right),$
and for Definition 4, it is
$\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\nicefrac{{1}}{{
\varepsilon_{2}}}\right)\right).$

Note that there are no constraints on $\varepsilon_{1}$, and since $\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}\leq\left
\|\gamma\nabla f(x_{k})\right\|^{2}$ by (F.5), it is straightforward to adjust GD to optimize the approximation.
However, for $\varepsilon_{2}$, we require $\varepsilon_{2}<\frac{\mu}{4L_{\max}}$.
In practice, $\varepsilon_{2}$ can be set to a sufficiently small value to satisfy this condition, though this will increase the number of local iterations performed by each client.
The complexity bounds also indicate that as the local step size $\gamma$ increases, it becomes more challenging to compute the approximation.
Alternatively, other algorithms can be employed to find such an approximation.
For instance, by leveraging the structure in (2), SGD can be used as a local solver for the proximal operator when computational resources are limited.
We can use the accelerated gradient descent ( AGD) of Nesterov () to obtain a better iteration complexity for each client.

###### Theorem 5 (Local computation via AGD ) .

Assume all assumptions mentioned in Theorem 4 hold.
The iteration complexities for the $i$-th client to provide an approximation in the $k$-the iteration using AGD with local step size $\eta_{i}=\frac{\gamma}{1+\gamma L_{i}}$ and momentum parameter $\alpha_{i}=\frac{\sqrt{1+\gamma L_{i}}-1}{\sqrt{1+\gamma L_{i}}+1}$, satisfying Definition 3, Definition 4 are

$$ $\displaystyle\mathcal{O}\left(\sqrt{1+\gamma L_{i}}\log\left(\frac{\left(1+ \gamma L_{i}\right)\cdot\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k} \right)\right\|^{2}}{\varepsilon_{1}}\right)\right);\quad\mathcal{O}\left( \sqrt{1+\gamma L_{i}}\log\left(\frac{1+\gamma L_{i}}{\varepsilon_{2}}\right) \right),$ $$

respectively.

Finally, we provide numerical evidence to support our theoretical findings.
We refer the readers to Appendix G for the details of the settings and the corresponding experiments.

###### Theorem 4 (Local computation via GD ) .

###### Theorem 5 (Local computation via AGD ) .

## 6 Conclusions

### 6.1 Limitations

Despite achieving satisfactory results in the full-batch setting, the client sampling setting did not yield similar outcomes. This may be attributed to the nature of biased compression, which likely requires adjustments to the algorithm itself for resolution.
Nonetheless, we provide the analysis in Appendix E for reference.
Unlike Li et al. (), the presence of bias makes it unclear how to incorporate adaptive step-size rules such as gradient diversity in our case.
The only permissible inexactness for gradient diversity arises from client sub-sampling in the interpolation regime.

### 6.2 Future work

There are still open problems to be addressed.
For example, can Algorithm 1 be modified to incorporate the benefits of error feedback?
Is it possible to eliminate the interpolation regime assumption while still demonstrating that extrapolation is theoretically beneficial for FedExProx?
Another direction that may be of independent interest is to develop adaptive rules of determining the step size for SGD with biased update.

## Appendix A Notations

Throughout the paper, we use the notation $\left\|\cdot\right\|$ to denote the standard Euclidean norm defined on $\mathbb{R}^{d}$ and $\left\langle\cdot,\cdot\right\rangle$ to denote the standard Euclidean inner product.
Given a differentiable function $f:\mathbb{R}^{d}\mapsto\mathbb{R}$, its gradient is denoted as $\nabla f(x)$.
We use the notation ${D}_{f}\left(x,y\right)$ to denote the Bregman divergence associated with a function $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ between $x$ and $y$.
The notation $\inf f$ is used to denote the minimum of a function $f:\mathbb{R}^{d}\mapsto\mathbb{R}$.
We use ${\rm prox}_{\gamma\phi}\left(x\right)$ to denote the proximity operator of function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$ with $\gamma>0$ at $x\in\mathbb{R}^{d}$, and ${M}^{\gamma}_{\phi}\left(x\right)$ to denote the corresponding Moreau Envelope.
We denote the average of the Moreau envelope of each local objective $f_{i}$ by the notation ${M^{\gamma}}:\mathbb{R}^{d}\mapsto\mathbb{R}$.
Specifically, we define ${M}^{\gamma}\left(x\right)=\frac{1}{n}\sum_{i=1}^{n}{M}^{\gamma}_{f}\left(x\right)$.
Note that ${M}^{\gamma}\left(x\right)$ has an implicit dependence on $\gamma$, its smoothness constant is denoted by $L_{\gamma}$.
We say an extended real-valued function $f:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ is proper if there exists $x\in\mathbb{R}^{d}$ such that $f(x)<+\infty$.
We say an extended real-valued function $f:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ is closed if its epigraph is a closed set.
We use the notation $\mathcal{E}_{k}=\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}$ to denote the function value suboptimality of $\gamma M^{\gamma}$ at $x_{k}$, and $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$ to denote the squared distance.
The notation $\mathcal{O}\left(\cdot\right)$ is used to describe complexity while omitting constant factors, whereas $\tilde{\mathcal{O}}\left(\cdot\right)$ is used when logarithmic factors are also omitted.

## Appendix B Facts and lemmas

###### Fact 1 (Young’s inequality) .

For any two vectors $x,y\in\mathbb{R}^{d}$, the following inequality holds,

$$ $\left\|x+y\right\|^{2}\leq 2\left\|x\right\|^{2}+2\left\|y\right\|^{2}.$ (14) $$

###### Fact 2 (Property of convex smooth functions) .

Let $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$ be differentiable.
The following statements are equivalent:

- 1.
$\phi$ is convex and $L$-smooth.
- 2.
$0\leq 2{D}_{\phi}\left(x,y\right)\leq L\left\|x-y\right\|^{2}$ for all $x,y\in\mathbb{R}^{d}$.
- 3.
$\frac{1}{L}\left\|\nabla\phi(x)-\nabla\phi(y)\right\|^{2}\leq 2{D}_{\phi}\left
(x,y\right)$ for all $x,y\in\mathbb{R}^{d}$.

The notation ${D}_{\phi}\left(x,y\right)$ denotes the Bregman divergence associate with $\phi$ at $x,y\in R^{d}$, defined as

$$ ${D}_{\phi}\left(x,y\right)=\phi\left(x\right)-\phi\left(y\right)-\left\langle \nabla\phi(y),x-y\right\rangle.$ $$

The following two facts establish that the convexity and smoothness of a function $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$ ensure the convexity and smoothness of its Moreau envelope.

###### Fact 3 (Convexity of Moreau envelope) .

(Beck, , Theorem 6.55) Let $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ be a proper and convex function.
Then ${M}^{\gamma}_{\phi}$ is a convex function.

###### Fact 4 (Smoothness of Moreau envelope) .

(Li et al., , Lemma 4) Let $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}$ be a convex and $L$-smooth function. Then ${M}^{\gamma}_{\phi}$ is $\frac{L}{1+\gamma L}$-smooth.

The following fact illustrates the relationship between the minimizer of a function $\phi$ and its Moreau envelope ${M}^{\gamma}_{\phi}$.

###### Fact 5 (Minimizer equivalence) .

(Li et al., , Lemma 5) Let $\phi:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ be a proper, closed and convex function.
Then for any $\gamma>0$, $\phi$ and ${M}^{\gamma}_{\phi}$ has the same set of minimizers.

In our case, we assume each $f_{i}$ from (1) is convex and $L_{i}$-smooth.
Therefore by 3 and 4, we know that each ${M}^{\gamma}_{f_{i}}$ is also convex and $\frac{L_{i}}{1+\gamma L_{i}}$-smooth.
This means that $M_{\gamma}=\frac{1}{n}\sum_{i=1}^{n}{M}^{\gamma}_{f_{i}}$ is also convex and smooth.
We denote its smoothness constant as $L_{\gamma}$, and the following fact provides a range for this constant.

###### Fact 6 (Global convexity and smoothness) .

(Li et al., , Lemma 7) Let each $f_{i}$ be proper, closed convex and $L_{i}$-smooth.
Then $M^{\gamma}$ is convex and $L_{\gamma}$-smooth with

$$ $\frac{1}{n^{2}}\sum_{i=1}^{n}\frac{L_{i}}{1+\gamma L_{i}}\leq L_{\gamma}\leq \frac{1}{n}\sum_{i=1}^{n}\frac{L_{i}}{1+\gamma L_{i}}.$ $$

The following fact establishes that the minimizer of $f$ and $M^{\gamma}$ are the same.

###### Fact 7 (Global minimizer equivalence) .

(Li et al., , Lemma 8) If we let every $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}\cup\left\{+\infty\right\}$ be proper, closed and convex, then $f(x)=\frac{1}{n}\sum_{i=1}^{n}f_{i}(x)$ has the same set of minimizers and minimum as

$$ ${M}^{\gamma}\left(x\right)=\frac{1}{n}\sum_{i=1}^{n}{M}^{\gamma}_{f_{i}}\left( x\right),$ $$

if we are in the interpolation regime and $0<\gamma<\infty$.

The above fact demonstrates that running SGD on the objective ${M}^{\gamma}$ will lead us to the correct destination, as the minimizers of ${M}^{\gamma}$ and $f$ are identical in our setting.
In problem (1), if we assume that $f$ is strongly convex, then we have $M^{\gamma}$ satisfies the following star strong convexity inequality.

###### Fact 8 (Star strong convexity) .

(Li et al., , Lemma 11)
Assume 1 (Differentiability), 2 (Interpolation Regime), 3 (Individual convexity), 4 (Smoothness) and 5 (Global strong convexity) hold, then the convex function ${M}^{\gamma}\left(x\right)$ satisfies the following inequality,

$$ ${M}^{\gamma}\left(x\right)-M^{\gamma}_{\inf}\geq\frac{\mu}{1+\gamma L_{\max}} \cdot\frac{1}{2}\left\|x-x_{\star}\right\|^{2},$ $$

for any $x\in\mathbb{R}^{d}$ and a minimizer $x_{\star}$ of ${M}^{\gamma}\left(x\right)$.

The above fact implies that the strong convexity of $f$ translates to the star strong convexity of $M^{\gamma}$.
Star strong convexity is also known as quadratic growth (QG) condition (Anitescu, ).
In the case of a convex function, it is also known as optimal strong convexity (Liu & Wright, ) and semi-strong convexity (Gong & Ye, ).
It is known that for a convex function satisfying quadratic growth condition, it also satisfies the Polyak-Lojasiewicz inequality (Polyak, ) which is described by the following lemma.
Notice that since Algorithm 1 can be viewed as running SGD with objective $\gamma M^{\gamma}$ and a fixed step size $\alpha_{k}=\alpha$, we describe the inequality based on $\gamma M^{\gamma}$ in the following lemma.

###### Lemma 1 (PL-inequality) .

Let 1 (Differentiability), 2 (Interpolation Regime), 3 (Individual convexity), 4 (Smoothness) and 5 (Global strong convexity) hold, then $\gamma M^{\gamma}\left(x\right)$ satisfies the following Polyak-Lojasiewicz inequality,

$$ $\left\|\gamma\nabla M^{\gamma}\left(x\right)\right\|^{2}\geq 2\cdot\frac{ \gamma\mu}{4\left(1+\gamma L_{\max}\right)}\left(\gamma M^{\gamma}\left(x \right)-\gamma M^{\gamma}_{\inf}\right),$ (15) $$

where $x\in\mathbb{R}^{d}$ is an arbitrary vector and $x_{\star}$ is a minimizer of $M^{\gamma}\left(x\right)$.

###### Fact 1 (Young’s inequality) .

###### Fact 2 (Property of convex smooth functions) .

###### Fact 3 (Convexity of Moreau envelope) .

###### Fact 4 (Smoothness of Moreau envelope) .

###### Fact 5 (Minimizer equivalence) .

###### Fact 6 (Global convexity and smoothness) .

###### Fact 7 (Global minimizer equivalence) .

###### Fact 8 (Star strong convexity) .

###### Lemma 1 (PL-inequality) .

## Appendix C Theory of biased SGD

For completeness, we provide the theory of biased SGD we used to analyze our algorithm in this paper.
It is adapted from Demidovich et al. (), which offers a comprehensive study of various assumptions employed in the analysis of SGD with biased gradient updates.
In addition, the authors introduced a new set of assumptions, referred to as the Biased ABC assumption, which are less restrictive than all previous assumptions.
The authors provided convergence guarantees for SGD with biased gradient updates in the non-convex and convex setting.
Specifically, they considered the case of minimizing a function $f:\mathbb{R}^{d}\mapsto\mathbb{R}$,

$$ $\min_{x\in\mathbb{R}^{d}}f(x),$ $$

with

$$ $x_{k+1}=x_{k}-\eta g(x_{k}),$ (biased SGD) $$

where $\eta>0$ is the stepsize, $g(x_{k})$ is a possibly stochastic and biased gradient estimator.
They introduced the biased ABC assumption,

###### Assumption 6 (Biased-ABC) .

(Demidovich et al., , Assumption 9) There exists constants $A,B,C,b,c\geq 0$ such that the gradient estimator $g(x)$ for every $x\in\mathbb{R}^{d}$ satisfies

$$ $\displaystyle\left\langle\nabla f(x),{\mathbb{E}}\left[g(x)\right]\right\rangle$ $\displaystyle\geq$ $\displaystyle b\left\|\nabla f(x)\right\|^{2}-c$ $\displaystyle{\mathbb{E}}\left[\left\|g(x)\right\|^{2}\right]$ $\displaystyle\leq$ $\displaystyle 2A\left(f(x)-f_{\inf}\right)+B\left\|\nabla f(x)\right\|^{2}+C.$ $$

A convergence guarantee was provided for biased SGD under 6 given that $f$ is $\widehat{L}$-smooth and $\widehat{\mu}$-PL, that is, there exists $\widehat{\mu}>0$, such that

$$ $\displaystyle\left\|\nabla f(x)\right\|^{2}\geq 2\widehat{\mu}\left(f(x)-f_{ \inf}\right),$ $$

for all $x\in\mathbb{R}^{d}$.

###### Theorem 6 (Theory of biased SGD ) .

(Demidovich et al., , Theorem 4)
Let $f$ be $\widehat{L}$-smooth and $\widehat{\mu}$-PL and 6 hold.
If we choose a step size $\eta$ satisfying

$$ $\displaystyle 0<\eta<\min\left\{\frac{\widehat{\mu}b}{\widehat{L}\left(A+ \widehat{\mu}B\right)},\frac{1}{\widehat{\mu}b}\right\}.$ (16) $$

Then we have

$$ $\displaystyle{\mathbb{E}}\left[f(x_{k})-f_{\inf}\right]\leq\left(1-\eta \widehat{\mu}b\right)^{k}\left(f(x_{0})-f_{\inf}\right)+\frac{LC\eta}{2 \widehat{\mu}b}+\frac{c}{\widehat{\mu}b}.$ $$

Under the special case of

$$ $\frac{\widehat{\mu}b}{\widehat{L}\left(A+\widehat{\mu}B\right)}<\frac{1}{ \widehat{\mu}b},$ $$

The range of the step size can be simplified to

$$ $\displaystyle 0<\eta\leq\frac{\widehat{\mu}b}{\widehat{L}\left(A+\widehat{\mu} B\right)},$ $$

and if we take the largest possible step size, we have

$$ $\displaystyle{\mathbb{E}}\left[f(x_{k})-f_{\inf}\right]\leq\left(1-\frac{ \widehat{\mu}^{2}b^{2}}{\widehat{L}\left(A+\widehat{\mu}B\right)}\right)^{k} \left(f(x_{0})-f_{\inf}\right)+\frac{LC}{2\widehat{L}\left(A+\widehat{\mu}B \right)}+\frac{c}{\widehat{\mu}b}.$ $$

The constants $C,c$ determine whether the algorithm is converging to the exact solution or just a neighborhood.
For $g(x)=\nabla f(x)$, clearly we have $A=0,B=1,b=1,C=0,c=0$, and there is no neighborhood.
This is expected because the algorithm reduces to standard GD
The iteration complexity is give by $\tilde{\mathcal{O}}\left(\frac{\widehat{L}}{\widehat{\mu}}\right)$, which is also expected for GD.

###### Assumption 6 (Biased-ABC) .

###### Theorem 6 (Theory of biased SGD ) .

## Appendix D Theory of biased compression

In this section, we present the theory of SGD with biased compression.
The theory is adapted from Beznosikov et al. ().
The authors introduced theory for analyzing compressed gradient descent ( CGD) with biased compressor, both in the single node case and in the distributed case when the objective function is assumed to be strongly convex.
Here, we are only concerned with the single node case because distributed compressed gradient descent ( DCGD) with biased compressor may fail to converge.
To address this issue, error feedback mechanism (Seide et al., ; Karimireddy et al., ; Richtárik et al., ) is needed.
In the single node case, the authors considered solving

$$ $\displaystyle\min_{x\in\mathbb{R}^{d}}f(x),$ $$

where $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ is $\widehat{L}$-smooth and $\widehat{\mu}$-strongly convex, with the following compressed gradient descent algorithm

$$ $\displaystyle x_{k+1}=x_{k}-\eta\mathcal{C}\left(\nabla f(x_{k})\right),$ (CGD) $$

where $\mathcal{C}:\mathbb{R}^{d}\mapsto\mathbb{R}$ are potentially biased compression operators, $\eta>0$ is a step size.
The author proved that if certain conditions on $\mathcal{C}$ is satisfied, a corresponding convergence guarantee can then be established.
Three classes of compressor/mapping were introduced.

###### Definition 5 (Class $\mathbb{B}^{1}$ ) .

We say a mapping $\mathcal{C}\in\mathbb{B}^{1}\left(\alpha,\beta\right)$ for some $\alpha,\beta>0$ if

$$ $\displaystyle\alpha\left\|x\right\|^{2}\leq{\mathbb{E}}\left[\left\|\mathcal{C }\left(x\right)\right\|^{2}\right]\leq\beta\left\langle{\mathbb{E}}\left[ \mathcal{C}\left(x\right)\right],x\right\rangle,\qquad\forall x\in\mathbb{R}^{ d}.$ $$

###### Definition 6 (Class $\mathbb{B}^{2}$ ) .

We say a mapping $\mathcal{C}\in\mathbb{B}^{2}\left(\xi,\beta\right)$ for some $\xi,\beta>0$ if

$$ $\displaystyle\max\left\{\xi\left\|x\right\|^{2},\frac{1}{\beta}{\mathbb{E}} \left[\left\|\mathcal{C}\left(x\right)\right\|^{2}\right]\right\}\leq\left \langle{\mathbb{E}}\left[\mathcal{C}\left(x\right)\right],x\right\rangle, \qquad\forall x\in\mathbb{R}^{d}.$ $$

###### Definition 7 (Class $\mathbb{B}^{3}$ ) .

We say a mapping $\mathcal{C}\in\mathbb{B}^{3}\left(\delta\right)$ for some $\delta>0$, if

$$ $\displaystyle{\mathbb{E}}\left[\left\|\mathcal{C}\left(x\right)-x\right\|^{2} \right]\leq\left(1-\frac{1}{\delta}\right)\left\|x\right\|^{2}.$ $$

The authors proved the following theorem about the convergence of the algorithm, the notation $\mathcal{F}_{k}$ is used to denote ${\mathbb{E}}\left[f(x_{k})\right]-f_{\inf}$, with $\mathcal{F}_{0}=f(x_{0})-f_{\inf}$,

###### Theorem 7 .

Let $\mathcal{C}\in\mathbb{B}^{1}\left(\alpha,\beta\right)$.
Then we have $\mathcal{F}_{k}\leq\left(1-\nicefrac{{\alpha}}{{\beta}}\eta\widehat{\mu}\left(
2-\eta\beta\widehat{L}\right)\right)\mathcal{F}_{k-1}$, as long as $0\leq\eta\leq\frac{2}{\beta\widehat{L}}$.
If we choose $\eta=\frac{1}{\beta\widehat{L}}$, we have

$$ $\mathcal{F}_{k}\leq\left(1-\frac{\alpha}{\beta^{2}}\cdot\frac{\widehat{\mu}}{ \widehat{L}}\right)^{K}\mathcal{F}_{0}.$ (17) $$

Let $\mathcal{C}\in\mathbb{B}^{2}\left(\xi,\beta\right)$.
Then we have $\mathcal{F}_{k}\leq\left(1-\xi\eta\left(2-\eta\beta\right)\widehat{L}\right)
\mathcal{F}_{k-1}$, as long as $0\leq\eta\leq\frac{2}{\beta\widehat{L}}$.
If we choose $\eta=\frac{1}{\beta\widehat{L}}$, we have

$$ $\mathcal{F}_{k}\leq\left(1-\frac{\xi}{\beta}\cdot\frac{\widehat{\mu}}{\widehat {L}}\right)^{k}\mathcal{F}_{0}.$ (18) $$

Let $\mathcal{C}\in\mathbb{B}^{3}\left(\delta\right)$.
Then we have $\mathcal{F}_{k}\leq\left(1-\frac{1}{\delta}\eta\widehat{\mu}\right)\mathcal{F}
_{k-1}$, as long as $0\leq\eta\leq\frac{1}{\widehat{L}}$.
If we choose $\eta=\frac{1}{\widehat{L}}$, we have

$$ $\mathcal{F}_{k}\leq\left(1-\frac{1}{\delta}\cdot\frac{\widehat{\mu}}{\widehat{ L}}\right)^{k}\mathcal{F}_{0}.$ (19) $$

Notice that when $\mathcal{C}\left(x\right)=x$, that is, when no compression happens, we have $\alpha=\beta=\xi=\delta=1$.
In this case, the iteration complexity of CGD is given by $\tilde{\mathcal{O}}\left(\frac{\widehat{L}}{\widehat{\mu}}\right)$ and we recover the result of GD.
It is worth noting that Theorem 7 remains valid if the condition of $f$ being $\widehat{\mu}$-strongly convex is replaced with $f$ being $\widehat{\mu}$-PL.

###### Definition 5 (Class 𝔹 1 superscript 𝔹 1 \mathbb{B}^{1} blackboard_B start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) .

###### Definition 6 (Class 𝔹 2 superscript 𝔹 2 \mathbb{B}^{2} blackboard_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .

###### Definition 7 (Class 𝔹 3 superscript 𝔹 3 \mathbb{B}^{3} blackboard_B start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) .

###### Theorem 7 .

## Appendix E Analysis of inexact FedExProx in the client sampling setting

In this section, we will discuss the case where we do client sampling in algorithm 1, we first formulate the algorithm as below.
For the sake of simplicity, we use $\tau$-nice sampling as an example.

Figure: Algorithm 2 Inexact FedExProx with $\tau$-nice sampling

### E.1 Relative approximation in distance

#### The failure of biased compression theory:

Similar to Theorem 7, we initially apply the theory from Beznosikov et al. (), as it provides improved results in the full-batch scenario.
We first define the compressing mapping $\mathcal{C}_{\tau}$ in this case,

$$ $\mathcal{C}_{\tau}\left(\gamma\nabla M^{\gamma}\left(x_{k}\right)\right)=\frac {1}{\tau}\sum_{i\in S_{k}}\left(\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k} \right)-\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right) \right)\right).$ (21) $$

One can verify for every $x_{k}$ and $\varepsilon_{2}$-approximation ${\tilde{x}_{i,k+1}}$ of ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$, we have

$$ $\mathcal{C}_{\tau}\in\mathbb{B}^{3}\left(\delta=\frac{\mu}{\mu-4\varepsilon_{2 }L_{\max}-\frac{n-\tau}{\tau\left(n-1\right)}\left[4\left(2+\varepsilon_{2} \right)L_{\max}-2\mu\right]}\right)$ $$

In the case of $\tau=n$, we have $\mathcal{C}_{n}\in\mathbb{B}^{3}\left(\frac{\mu}{\mu-4\varepsilon_{2}L_{\max}}\right)$, which recovers the result of (42).
When $\tau=1,\varepsilon_{2}=0$, however, this is problematic, as $\mathcal{C}_{1}\in\mathbb{B}^{3}\left(\delta=\frac{\mu}{3\mu-8L_{\max}}\right)$.
Notice that we require $\delta>0$, so we require $3\mu>8L_{\max}$ which only holds in a very restrictive setting.
This is due to the stochasticity contained in (21), which arises from client sampling.

#### Theory of biased SGD :

The algorithm does converge, however, and one can use the theory of Demidovich et al. () to obtain a convergence guarantee.

###### Theorem 8 .

Assume 1 (Differentiability), 2 (Interpolation regime), 3 (Individual convexity), 4 (Smoothness) and 5 (Global strong convexity) hold.
Let the approximation $\tilde{x}_{i,k+1}$ all satisfies Definition 4 with $\varepsilon_{2}<\frac{\mu^{2}}{4L^{2}_{\max}}$, that is

$$ $\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{ 2}\leq\varepsilon_{2}\cdot\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k} \right)\right\|^{2},$ $$

holds for all client $i$ at iteration $k$.
If we are running Algorithm 2 with minibatch size $\tau$ and extrapolation parameter $\alpha_{k}=\alpha>0$ satisfying

$$ $\displaystyle\alpha\leq\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu-2\sqrt{ \varepsilon_{2}}L_{\max}}{\mu+4\varepsilon_{2}L_{\max}+4\sqrt{\varepsilon_{2}} L_{\max}+\frac{n-\tau}{\tau\left(n-1\right)}\cdot\left(4L_{\max}+4\sqrt{ \varepsilon_{2}}L_{\max}-\mu\right)}$ $$

Then the iterates generated by Algorithm 2 satisfies

$$ ${\mathbb{E}}\left[\mathcal{E}_{K}\right]\leq\left(1-\alpha\cdot\frac{\gamma \left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)}{4\left(1+\gamma L_{\max} \right)}\right)^{K}\mathcal{E}_{0}.$ (22) $$

Specifically, if we choose the largest $\alpha$ possible, we have

$$ $\displaystyle{\mathbb{E}}\left[\Delta_{K}\right]\leq\left(1-\frac{\mu}{4L_{ \gamma}\left(1+\gamma L_{\max}\right)}\cdot S\left(\varepsilon_{2},\tau\right) \right)^{K}\cdot\frac{L{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0},$ $$

where $S\left(\varepsilon_{2},\tau\right)$ is defined as

$$ $\displaystyle S\left(\varepsilon_{2},\tau\right):=\frac{\left(\mu-2\sqrt{ \varepsilon_{2}}L_{\max}\right)\left(1-2\sqrt{\varepsilon_{2}}\frac{L_{\max}}{ \mu}\right)}{\mu+4\varepsilon_{2}L_{\max}+4\sqrt{\varepsilon_{2}}L_{\max}+ \frac{n-\tau}{\tau\left(n-1\right)}\cdot\left(4L_{\max}+4\sqrt{\varepsilon_{2} }L_{\max}-\mu\right)},$ $$

satisfying

$$ $\displaystyle 0<S\left(\varepsilon_{2},\tau\right)\leq 1.$ $$

Notice that we have $S\left(\varepsilon_{2},\tau=n\right)=S\left(\varepsilon_{2}\right)$, which appears in Theorem 2.
For the special case when $\varepsilon_{2}=0$, every proximal operator is solved exactly.
The range of $\alpha$ becomes,

$$ $0<\alpha\leq\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu}{\frac{n-\tau}{\tau\left (n-1\right)}\cdot 4L_{\max}+\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)} \mu}.$ $$

According to Li et al. (),

$$ $0<\alpha\leq\frac{1}{\gamma L_{\gamma}}\cdot\frac{L_{\gamma}\left(1+\gamma L_{ \max}\right)}{\frac{n-\tau}{\tau\left(n-1\right)}L_{\max}+\frac{n\left(\tau-1 \right)}{\tau\left(n-1\right)}\cdot L_{\gamma}\left(1+\gamma L_{\max}\right)}.$ $$

Clearly the bound we obtain here is suboptimal, since we have $\mu\leq L_{\gamma}\left(1+\gamma L_{\max}\right)$ according to (27).
This is due to the previously mentioned issue: the nature of biased compression.
When client sampling is used together with biased compressors, it does not necessarily guarantee any benefits.
To solve this, the modification of the algorithm itself may be needed, which we consider as a future work direction.

###### Theorem 8 .

### E.2 Absolute approximation in distance

Similarly to Theorem 8, by applying the theory of biased SGD (Demidovich et al., ), we can derive a convergence guarantee for the minibatch case, though with a suboptimal convergence rate. For brevity and clarity, we do not include the details here.

## Appendix F Proof of theorems and lemmas

### F.1 Proof of Lemma 1

Using 8, we have

$$ $M^{\gamma}\left(x\right)-M^{\gamma}_{\inf}\geq\frac{\mu}{1+\gamma L_{\max}} \cdot\frac{1}{2}\left\|x-x_{\star}\right\|^{2},$ (23) $$

where $x\in\mathbb{R}^{d}$ is any vector, $x_{\star}$ is a minimizer of $M^{\gamma}$, by 5, it is also a minimizer of $f$.
Since we assume each function $f_{i}$ is convex, by 3, we know that ${M}^{\gamma}_{f_{i}}$ is also convex.
As a result, the average of ${M}^{\gamma}_{f_{i}}$, $M^{\gamma}$ is also a convex function.
Utilizing the convexity of $M^{\gamma}$, we have,

$$ $M^{\gamma}_{\inf}\geq M^{\gamma}\left(x\right)+\left\langle\nabla M^{\gamma} \left(x\right),x_{\star}-x\right\rangle.$ $$

Rearranging terms we get,

$$ $\left\langle\nabla M^{\gamma}\left(x\right),x-x_{\star}\right\rangle\geq M^{ \gamma}\left(x\right)-M^{\gamma}_{\inf}.$ (24) $$

As a result, we have

$$ $\left\langle\nabla M^{\gamma}\left(x\right),x-x_{\star}\right\rangle\overset{( \ref{eq:ineq-pr1-1})+(\ref{eq:ineq-pr1-2})}{\geq}\frac{\mu}{1+\gamma L_{\max}} \cdot\frac{1}{2}\left\|x-x_{\star}\right\|^{2}.$ $$

Using Cauchy-Schwarz inequality, we have

$$ $\left\|\nabla M^{\gamma}\left(x\right)\right\|\left\|x-x_{\star}\right\|\geq \left\langle\nabla M^{\gamma}\left(x\right),x-x_{\star}\right\rangle\geq\frac{ \mu}{1+\gamma L_{\max}}\cdot\frac{1}{2}\left\|x-x_{\star}\right\|^{2}.$ $$

When $\left\|x-x_{\star}\right\|>0$, the above inequality leads to

$$ $\left\|\nabla M^{\gamma}\left(x\right)\right\|\geq\frac{\mu}{2\left(1+\gamma L _{\max}\right)}\cdot\left\|x-x_{\star}\right\|,$ (25) $$

which also holds when $\left\|x-x_{\star}\right\|=0$.
Now using (24) and (25), we obtain

$$ $\displaystyle M^{\gamma}\left(x\right)-M^{\gamma}_{\inf}$ $\displaystyle\overset{(\ref{eq:ineq-pr1-2})}{\leq}$ $\displaystyle\left\langle\nabla M^{\gamma}\left(x\right),x-x_{\star}\right\rangle$ $\displaystyle\leq$ $\displaystyle\left\|\nabla M^{\gamma}\left(x\right)\right\|\left\|x-x_{\star}\right\|$ $\displaystyle\overset{(\ref{eq:ineq-pr1-4})}{\leq}$ $\displaystyle\frac{2\left(1+\gamma L_{\max}\right)}{\mu}\left\|\nabla M^{ \gamma}\left(x\right)\right\|^{2}.$ $$

A simple rearranging of terms result in

$$ $\left\|\gamma\nabla M^{\gamma}\left(x\right)\right\|^{2}\geq 2\cdot\frac{ \gamma\mu}{4\left(1+\gamma L_{\max}\right)}\left(\gamma M^{\gamma}\left(x \right)-\gamma M^{\gamma}_{\inf}\right).$ $$

Up till here we have already proved the statement in the lemma, but we want to look at the strongly constant $\mu$ of $f$ a little bit.
In order to provide an upper bound of $\mu$, we notice that due to 4, each ${M}^{\gamma}_{f_{i}}$ is $\frac{L_{i}}{1+\gamma L_{i}}$-smooth and therefore $M^{\gamma}$ is smooth.
We use the notation $L_{\gamma}$ to denote its smoothness constant.
Applying the smoothness of $M^{\gamma}\left(x\right)$, we have

$$ $\displaystyle M^{\gamma}\left(x\right)\leq M^{\gamma}\left(x_{\star}\right)+ \left\langle\nabla M^{\gamma}\left(x_{\star}\right),x-x_{\star}\right\rangle+ \frac{L_{\gamma}}{2}\left\|x-x^{\star}\right\|^{2}.$ $$

Utilizing the fact that $\nabla M^{\gamma}\left(x_{\star}\right)=0$, we have

$$ $\displaystyle M^{\gamma}\left(x\right)-M^{\gamma}_{\inf}$ $\displaystyle\leq\frac{L_{\gamma}}{2}\left\|x-x_{\star}\right\|^{2}$ (26) $$

Combining (26) and (23), we can deduce that

$$ $\frac{\mu}{1+\gamma L_{\max}}\cdot\frac{1}{2}\left\|x-x_{\star}\right\|^{2} \leq M^{\gamma}\left(x\right)-M^{\gamma}_{\inf}\leq\frac{L_{\gamma}}{2}\left\| x-x_{\star}\right\|^{2}.$ $$

which results in the estimate that

$$ $\mu\leq L_{\gamma}\left(1+\gamma L_{\max}\right).$ (27) $$

### F.2 Proof of Theorem 1

Let us first recall that after reformulation, Algorithm 1 can be written as

$$ $\displaystyle x_{k+1}=x_{k}-\alpha\cdot g(x_{k}),$ $$

where $g(x_{k})$ is defined as

$$ $\displaystyle g(x_{k}):=\frac{1}{n}\sum_{i=1}^{n}\gamma\nabla{M}^{\gamma}_{f_{ i}}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right).$ $$

We view this as running full batch biased SGD with stepsize $\alpha$ and global objective $\gamma M^{\gamma}\left(x\right)$.
We first examine if 6 (Biased-ABC) holds for arbitrary $x_{k}$.
Since we are in the full batch case, it is easy to see that

$$ $\displaystyle{\mathbb{E}}\left[g(x_{k})\right]=g(x_{k}).$ $$

Since our objective now is $\gamma M^{\gamma}\left(x\right)$, we have that

$$ $\displaystyle\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),g(x_{k})\right\rangle$ $\displaystyle=\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),\gamma \nabla M^{\gamma}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{ i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right\rangle$ $\displaystyle=\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}- \underbrace{\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),\frac{1}{n} \sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k} \right)\right)\right\rangle}_{:=P_{1}}.$ $$

Now let us focus on $P_{1}$, we have the following upper bound,

$$ $\displaystyle P_{1}$ $\displaystyle\leq$ $\displaystyle\frac{1}{2}\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right \|^{2}+\frac{1}{2}\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|^{2}$ $\displaystyle\overset{(\ref{eq:approx-dist})}{\leq}$ $\displaystyle\frac{1}{2}\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right \|^{2}+\frac{\varepsilon_{1}}{2}.$ $$

As a result, we have

$$ $\displaystyle\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),g(x_{k}) \right\rangle\geq\frac{1}{2}\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right) \right\|-\frac{\varepsilon_{1}}{2},$ $$

which holds for arbitrary $x_{k}$.
This suggests that $b=\frac{1}{2},c=\frac{\varepsilon_{1}}{2}$.
On the other hand,

$$ $\displaystyle{\mathbb{E}}\left[\left\|g(x_{k})\right\|^{2}\right]$ $\displaystyle=$ $\displaystyle\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)+\frac{1}{n}\sum_ {i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right) \right)\right\|^{2}$ $\displaystyle\overset{(\ref{eq:youngs})}{\leq}$ $\displaystyle 2\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}+2 \left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{ i}}\left(x_{k}\right)\right)\right\|^{2}$ $\displaystyle\overset{(\ref{eq:approx-dist})}{\leq}$ $\displaystyle 2\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}+2 \varepsilon_{1}.$ $$

Thus, we can choose $A=0,B=2,C=2\varepsilon_{1}$.
Since we have assumed 3 (Individual convexity) and 4 (Smoothness), it is easy to see that $M^{\gamma}$ is smooth, and we denote its smoothness constant as $L_{\gamma}$.
It is therefore straightforward to see that our global objective $\gamma M^{\gamma}$ is $\gamma L_{\gamma}$-smooth.
We also assume $f$ is $\mu$-strongly convex, which by 8 indicates that $M^{\gamma}$ is $\frac{\mu}{1+\gamma L_{\max}}$ star strongly convex.
We immediately obtain using Lemma 1 that $\gamma M^{\gamma}$ is $\frac{\gamma\mu}{4\left(1+\gamma L_{\max}\right)}$-PL.
Now, we have validated all the assumptions for using Theorem 6.
Applying Theorem 6, we obtain that when the extrapolation parameter satisfies

$$ $0<\alpha<\frac{1}{4}\cdot\min\left\{\frac{1}{\gamma L_{\gamma}},\frac{2\left(1 +\gamma L_{\max}\right)}{\gamma\mu}\right\},$ $$

the last iterate $x_{K}$ of Algorithm 1 with each proximal operator solved inexactly according to Definition 1 satisfies

$$ $\displaystyle\mathcal{E}_{K}$ $\displaystyle\leq\left(1-\frac{\alpha\gamma\mu}{8\left(1+\gamma L_{\max}\right )}\right)^{K}\mathcal{E}_{0}+\frac{8\varepsilon_{1}\alpha L_{\gamma}\left(1+ \gamma L_{\max}\right)}{\mu}+\frac{4\varepsilon_{1}\left(1+\gamma L_{\max} \right)}{\gamma\mu},$ $$

where $\mathcal{E}_{k}=\gamma M^{\gamma}\left(x_{k}\right)-M^{\gamma}_{\inf}$.
Let us now prove that

$$ $\displaystyle\frac{1}{\gamma L_{\gamma}}<\frac{2\left(1+\gamma L_{\max}\right) }{\gamma\mu}.$ $$

This is equivalent to prove

$$ $\displaystyle\mu<2L_{\gamma}\left(1+\gamma L_{\max}\right),$ $$

which is always true since (27) holds.
As a result, we can simplify the range of the extrapolation parameter to

$$ $\displaystyle 0<\alpha\leq\frac{1}{4\gamma L_{\gamma}}.$ $$

If we pick the largest possible $\alpha$, we have

$$ $\displaystyle\mathcal{E}_{K}$ $\displaystyle\leq\left(1-\frac{\mu}{32L_{\gamma}\left(1+\gamma L_{\max}\right) }\right)^{K}\mathcal{E}_{0}+\frac{6\varepsilon_{1}\left(1+\gamma L_{\max} \right)}{\gamma\mu}.$ $$

This result is not directly comparable to that of Li et al. ().
However, using smoothness of $\gamma L_{\gamma}$, if we denote $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$ where $x_{\star}$ is a minimizer of both $M^{\gamma}$ and $f$ since we assume we are in the interpolation regime (2), we have

$$ $\displaystyle\mathcal{E}_{0}\leq\frac{\gamma L_{\gamma}}{2}\Delta_{0}.$ $$

Using star strong convexity, we have

$$ $\displaystyle\mathcal{E}_{K}\geq\frac{\gamma\mu}{2\left(1+\gamma L_{\max} \right)}\Delta_{K}.$ $$

As a result, we can transform the above convergence guarantee into

$$ $\displaystyle\Delta_{K}\leq\left(1-\frac{\mu}{32L_{\gamma}\left(1+\gamma L_{ \max}\right)}\right)^{K}\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu} \cdot\Delta_{0}+12\varepsilon_{1}\cdot\left(\frac{\nicefrac{{1}}{{\gamma}}+L_{ \max}}{\mu}\right)^{2}.$ $$

This completes the proof.

### F.3 Proof of Theorem 2

Since we based our analysis on the theory of biased SGD, we first verify the validity of 6.

#### Finding b 𝑏 b italic_b and c 𝑐 c italic_c :

Let us start with finding a lower bound on $\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),{\mathbb{E}}\left[g(x_{k
})\right]\right\rangle$.
We have

$$ $\displaystyle\left\langle\gamma M^{\gamma}\left(x_{k}\right),{\mathbb{E}}\left [g(x_{k})\right]\right\rangle$ $\displaystyle=\left\langle\gamma M^{\gamma}\left(x_{k}\right),\gamma M^{\gamma }\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\rangle$ $\displaystyle=\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}-\left \langle\gamma M^{\gamma}\left(x_{k}\right),\frac{1}{n}\sum_{i=1}^{n}\left( \tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right\rangle$ $\displaystyle\geq\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}-\left \|\gamma M^{\gamma}\left(x_{k}\right)\right\|\cdot\left\|\frac{1}{n}\sum_{i=1} ^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right) \right\|,$ $$

where the last inequality is obtained using Cauchy-Schwarz inequality.
We then utilize the convexity of $\left\|\cdot\right\|$ and obtain,

$$ $\displaystyle\left\langle\gamma M^{\gamma}\left(x_{k}\right),{\mathbb{E}}\left [g(x_{k})\right]\right\rangle$ $\displaystyle\geq$ $\displaystyle\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}-\left\| \gamma M^{\gamma}\left(x_{k}\right)\right\|\cdot\frac{1}{n}\sum_{i=1}^{n}\left \|\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|$ $\displaystyle\overset{(\ref{eq:index2-approx})}{\geq}$ $\displaystyle\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}-\sqrt{ \varepsilon_{2}}\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|\cdot\frac{1 }{n}\sum_{i=1}^{n}\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle=$ $\displaystyle\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}-\sqrt{ \varepsilon_{2}}\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|\cdot\frac{1 }{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right) \right\|.$ $$

Notice that

$$ $\displaystyle\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\| =\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma\nabla{M}^{ \gamma}_{f_{i}}\left(x_{\star}\right)\right\|,$ $$

holds for any $x_{\star}$ that is a minimizer of $M^{\gamma}\left(x\right)$ due to interpolation regime assumption.
As a result, we can provide an upper bound based on smoothness of each individual $\gamma{M}^{\gamma}_{f_{i}}\left(x\right)$ using 2,

$$ $\displaystyle\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma \nabla{M}^{\gamma}_{f_{i}}\left(x_{\star}\right)\right\|\leq\frac{\gamma L_{i} }{1+\gamma L_{i}}\left\|x_{k}-x_{\star}\right\|.$ (28) $$

Thus,

$$ $\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f_{i}} \left(x_{k}\right)\right\|$ $\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\frac{\gamma L_{i}}{1+\gamma L_{i}} \left\|x_{k}-x_{\star}\right\|\leq\frac{\gamma L_{\max}}{1+\gamma L_{\max}} \cdot\left\|x_{k}-x_{\star}\right\|.$ $$

In addition, we have due to Cauchy-Schwarz inequality and the convexity of $M^{\gamma}\left(x\right)$

$$ $\left\|\nabla M^{\gamma}\left(x_{k}\right)\right\|\cdot\left\|x_{k}-x_{\star} \right\|\geq\left\langle\nabla M^{\gamma}\left(x_{k}\right),x_{k}-x_{\star} \right\rangle\geq M^{\gamma}\left(x_{k}\right)-M^{\gamma}_{\inf},$ (29) $$

and due to quadratic growth condition that

$$ $M^{\gamma}\left(x_{k}\right)-M^{\gamma}_{\inf}\geq\frac{\mu}{1+\gamma L_{\max} }\cdot\frac{1}{2}\left\|x_{k}-x_{\star}\right\|^{2}.$ (30) $$

Combining (29) and (30), we have

$$ $\displaystyle\frac{\mu}{2\left(1+\gamma L_{\max}\right)}\cdot\left\|x_{k}-x_{ \star}\right\|^{2}\overset{(\ref{eq:lemma-5t1})+(\ref{eq:lemma-5t2})}{\leq} \left\|\nabla M^{\gamma}\left(x_{k}\right)\right\|\cdot\left\|x_{k}-x_{\star} \right\|.$ $$

This indicates that

$$ $\displaystyle\left\|x_{k}-x_{\star}\right\|\leq\frac{2\left(1+\gamma L_{\max} \right)}{\mu}\left\|\nabla M^{\gamma}\left(x_{k}\right)\right\|.$ (31) $$

Combining (28) and (31), we generate the following lower bound

$$ $\displaystyle\left\langle\gamma M^{\gamma}\left(x_{k}\right),{\mathbb{E}}\left [g(x_{k})\right]\right\rangle$ $\displaystyle\overset{(\ref{eq:lemma-5t0})}{\geq}\left\|\gamma M^{\gamma}\left (x_{k}\right)\right\|^{2}-\sqrt{\varepsilon_{2}}\left\|\gamma M^{\gamma}\left( x_{k}\right)\right\|\cdot\frac{\gamma L_{\max}}{1+\gamma L_{\max}}\left\|x_{k} -x_{\star}\right\|$ $\displaystyle\overset{(\ref{eq:lemma-5t3})}{\geq}\left\|\gamma M^{\gamma}\left (x_{k}\right)\right\|^{2}-\sqrt{\varepsilon_{2}}\cdot\frac{L_{\max}}{1+\gamma L _{\max}}\cdot\frac{2\left(1+\gamma L_{\max}\right)}{\mu}\left\|\gamma M^{ \gamma}\left(x_{k}\right)\right\|^{2}$ $\displaystyle=\left(1-\sqrt{\varepsilon_{2}}\cdot\frac{2L_{\max}}{\mu}\right) \cdot\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}.$ $$

Thus, as long as $\varepsilon_{2}<\frac{\mu^{2}}{4L_{\max}^{2}}$, we have $b=1-\sqrt{\varepsilon_{2}}\cdot\frac{2L_{\max}}{\mu}$, and $c=0$.

#### Finding A , B 𝐴 𝐵 A,B italic_A , italic_B and C 𝐶 C italic_C :

We start with expanding $\left\|g(x_{k})\right\|^{2}$,

$$ $\displaystyle{\mathbb{E}}\left[\left\|g(x_{k})\right\|^{2}\right]$ $\displaystyle=\left\|\gamma M^{\gamma}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1} ^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right) \right\|^{2}$ $\displaystyle=\left\|\gamma M^{\gamma}\left(x_{k}\right)\right\|^{2}+ \underbrace{\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox} _{\gamma f_{i}}\left(x_{k}\right)\right)\right\|^{2}}_{:=T_{2}}$ $\displaystyle\quad\underbrace{-2\left\langle\gamma M^{\gamma}\left(x_{k}\right ),\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}} \left(x_{k}\right)\right)\right\rangle}_{:=T_{3}}.$ (32) $$

It is easy to bound $T_{2}$ utilizing the convexity of $\left\|\cdot\right\|^{2}$,

$$ $\displaystyle T_{2}$ $\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left\|\tilde{x}_{i,k+1}-{\rm prox}_ {\gamma f_{i}}\left(x_{k}\right)\right\|^{2}$ $\displaystyle\overset{(\ref{eq:index2-approx})}{\leq}\frac{\varepsilon_{2}}{n} \sum_{i=1}^{n}\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\| ^{2}=\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|\gamma{M}^{\gamma}_{f_{i}} \left(x_{k}\right)\right\|^{2}.$ $$

Let $x_{\star}$ be a minimizer of $M^{\gamma}$, since we assume 2 holds, it is also a minimizer of each ${M}^{\gamma}_{f_{i}}$.
As a result,

$$ $\displaystyle T_{2}$ $\displaystyle\leq\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|\gamma{M}^{ \gamma}_{f_{i}}\left(x_{k}\right)-\gamma{M}^{\gamma}_{f_{i}}\left(x_{\star} \right)\right\|^{2}$ $\displaystyle\leq\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\frac{2\gamma L_{i}}{1 +\gamma L_{i}}\left(\gamma M^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma M^{ \gamma}_{f_{i}}\left(x_{\star}\right)\right)\leq\frac{2\varepsilon_{2}\gamma L _{\max}}{1+\gamma L_{\max}}\cdot\left(\gamma M^{\gamma}\left(x_{k}\right)- \gamma M^{\gamma}_{\inf}\right).$ (33) $$

We then consider $T_{3}$, and start with applying Cauchy-Schwarz inequality

$$ $\displaystyle T_{3}\leq 2\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right) \right\|\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{ \gamma f_{i}}\left(x_{k}\right)\right)\right\|.$ (34) $$

Using the convexity of $\left\|\cdot\right\|$, we have

$$ $\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|$ $\displaystyle\leq$ $\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left\|\tilde{x}_{i,k+1}-{\rm prox}_{ \gamma f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle\overset{(\ref{eq:index2-approx})}{\leq}$ $\displaystyle\frac{\sqrt{\varepsilon_{2}}}{n}\sum_{i=1}^{n}\left\|x_{k}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle\overset{(\ref{eq:ppt-2-gradient})}{=}$ $\displaystyle\frac{\sqrt{\varepsilon_{2}}}{n}\sum_{i=1}^{n}\left\|\gamma\nabla {M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x _{\star}\right)\right\|$ $\displaystyle\overset{\text{\lx@cref{creftypecap~refnum}{tech-lemma:smooth}}}{\leq}$ $\displaystyle\frac{\sqrt{\varepsilon_{2}}}{n}\sum_{i=1}^{n}\frac{\gamma L_{i}} {1+\gamma L_{i}}\left\|x_{k}-x_{\star}\right\|$ $\displaystyle\leq$ $\displaystyle\frac{\sqrt{\varepsilon_{2}}\gamma L_{\max}}{1+\gamma L_{\max}} \cdot\left\|x_{k}-x_{\star}\right\|.$ $$

Utilizing (31), we have

$$ $\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|$ $\displaystyle\leq\frac{\sqrt{\varepsilon_{2}}\gamma L_{\max}}{1+\gamma L_{\max }}\cdot\frac{2\left(1+\gamma L_{\max}\right)}{\mu}\left\|\nabla M^{\gamma} \left(x_{k}\right)\right\|$ $\displaystyle=\frac{2\sqrt{\varepsilon_{2}}L_{\max}}{\mu}\cdot\left\|\gamma \nabla M^{\gamma}\left(x_{k}\right)\right\|$ (35) $$

Plug the above inequality into (34), we have

$$ $\displaystyle T_{3}\leq\frac{4\sqrt{\varepsilon_{2}}L_{\max}}{\mu}\cdot\left\| \gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}.$ (36) $$

Combining (36) and (F.3), plug them into (F.3), we have

$$ $\displaystyle{\mathbb{E}}\left[\left\|g\left(x_{k}\right)\right\|^{2}\right] \leq\frac{2\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}}\cdot\left(\gamma M ^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}\right)+\left(1+\frac{4 \sqrt{\varepsilon_{2}}L_{\max}}{\mu}\right)\cdot\left\|\gamma\nabla M^{\gamma} \left(x_{k}\right)\right\|^{2}.$ $$

Thus, we have

$$ $\displaystyle A=\frac{\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}},\quad B =\frac{\mu+4\sqrt{\varepsilon_{2}}L_{\max}}{\mu},\quad C=0.$ $$

#### Applying Theorem 6 :

First, we list our the values appeared respectively,

$$ $\displaystyle A=\frac{\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}},\quad B =\frac{\mu+4\sqrt{\varepsilon_{2}}L_{\max}}{\mu},\quad b=\frac{\mu-2\sqrt{ \varepsilon_{2}}L_{\max}}{\mu},$ $\displaystyle C=c=0.$ $$

We know that the PL constant of $\gamma M^{\gamma}$ is given by $\frac{\gamma\mu}{4\left(1+\gamma L_{\max}\right)}$ and the corresponding smoothness constant is $\gamma L_{\gamma}$.
Applying Theorem 6, the range of $\alpha$ is given by

$$ $\displaystyle 0<\alpha<\min\left\{\underbrace{\frac{1}{\gamma L_{\gamma}}\cdot \frac{\mu-2\sqrt{\varepsilon_{2}}L_{\max}}{\mu+4\sqrt{\varepsilon_{2}}L_{\max} +4\varepsilon_{2}L_{\max}}}_{:=B_{1}},\underbrace{\frac{4\left(1+\gamma L_{ \max}\right)}{\gamma\left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)}}_{:=B_{2 }}\right\}.$ (37) $$

Now notice that actually we can prove that for $\varepsilon_{2}<\frac{\mu^{2}}{4L_{\max}^{2}}$, we have $B_{2}>B_{1}$, and we can simplify the range of $\alpha$ to

$$ $\displaystyle 0<\alpha\leq\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu-2\sqrt{ \varepsilon_{2}}L_{\max}}{\mu+4\sqrt{\varepsilon_{2}}L_{\max}+4\varepsilon_{2} L_{\max}}.$ $$

#### Proof of B 2 > B 1 subscript 𝐵 2 subscript 𝐵 1 B_{2}>B_{1} italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

:
It is easy to verify that the above inequality ($B_{2}>B_{1}$) can be equivalently written as

$$ $\displaystyle 4L_{\gamma}\left(1+\gamma L_{\max}\right)\left(\mu+4\sqrt{ \varepsilon_{2}}L_{\max}+4\varepsilon_{2}L_{\max}\right)>\left(\mu-2\sqrt{ \varepsilon_{2}}L_{\max}\right)^{2},$ $$

since when $\sqrt{\varepsilon_{2}}<\frac{\mu}{2L_{\max}}$, we have $\mu-2\sqrt{\varepsilon_{2}}L_{\max}>0$.
We expand the right-hand side and obtain:

$$ $\displaystyle\left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)^{2}$ $\displaystyle=\mu^{2}-4\sqrt{\varepsilon_{2}}L_{\max}+4\varepsilon_{2}L_{\max} ^{2}<2\mu^{2}-4\sqrt{\varepsilon_{2}}L_{\max}<2\mu^{2}.$ $$

For the left-hand side, as we have already shown in 27, we have

$$ $\displaystyle 4L_{\gamma}\left(1+\gamma L_{\max}\right)\left(\mu+4\sqrt{ \varepsilon_{2}}L_{\max}+4\varepsilon_{2}L_{\max}\right)$ $\displaystyle\geq 4\mu\left(\mu+4\sqrt{\varepsilon_{2}}L_{\max}+2\varepsilon_{ 2}L_{\max}\right)>4\mu^{2}.$ $$

Combining the above inequality we arrive at $B_{2}>B_{1}$.

#### The convergence guarantee

:
Given that we select $\alpha$ properly, we have

$$ $\displaystyle\mathcal{E}_{K}\leq\left(1-\alpha\cdot\frac{\gamma\left(\mu-2 \sqrt{\varepsilon_{2}}L_{\max}\right)}{4\left(1+\gamma L_{\max}\right)}\right) ^{K}\mathcal{E}_{0},$ $$

where $\mathcal{E}_{k}=\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}$.
We do not have expectation here since we are in the full batch case.
Specifically, if we choose the largest $\alpha$ possible, we have

$$ $\mathcal{E}_{K}\leq\left(1-\frac{\mu}{4L_{\gamma}\left(1+\gamma L_{\max}\right )}\cdot S\left(\varepsilon_{2}\right)\right)^{k}\mathcal{E}_{0},$ $$

where

$$ $S(\varepsilon_{2})=\frac{\left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)\left (1-2\sqrt{\varepsilon_{2}}\frac{L_{\max}}{\mu}\right)}{\mu+4\sqrt{\varepsilon_ {2}}L_{\max}+4\varepsilon_{2}L_{\max}},$ $$

satisfies $0<S(\varepsilon_{2})\leq 1$ is the factor of slowing down due to inexact proximity operator evaluation.
Using smoothness of $\gamma L_{\gamma}$, if we denote $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$ where $x_{\star}$ is a minimizer of both $M^{\gamma}$ and $f$ since we assume we are in the interpolation regime (2), we have

$$ $\displaystyle\mathcal{E}_{0}\leq\frac{\gamma L_{\gamma}}{2}\Delta_{0}.$ $$

Using star strong convexity (quadratic growth property), we have

$$ $\displaystyle\mathcal{E}_{K}\geq\frac{\gamma\mu}{2\left(1+\gamma L_{\max} \right)}\Delta_{K}.$ $$

As a result, we can transform the above convergence guarantee into

$$ $\displaystyle\Delta_{K}\leq\left(1-\frac{\mu}{4L_{\gamma}\left(1+\gamma L_{ \max}\right)}\cdot S\left(\varepsilon_{2}\right)\right)^{K}\cdot\frac{L{\gamma }\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0}.$ $$

This completes the proof.

### F.4 Proof of Theorem 3

We start with formalizing the problem.
Using (11) and (12), we can write the update rule of Algorithm 1 as

$$ $x_{k+1}=x_{k}-\alpha\cdot\left(\frac{1}{n}\sum_{i=1}^{n}\gamma\nabla{M}^{ \gamma}_{f_{i}}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i, k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right).$ (38) $$

Since by Definition 4, we have $\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{
2}\leq\varepsilon_{2}\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)
\right\|^{2}$, we can view the left hand side as a compressed version of the true gradient.
Specifically, there are two possible perspectives:

- (I).
Let $\mathcal{C}_{i}\left(\cdot\right)$ be the compressing mapping with the $i$-th client, $i\in\left\{1,2,\ldots,n\right\}$, defined as
$\displaystyle\mathcal{C}_{i}\left(\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}
\right)\right):=\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\left(
\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right).$
In this way, we reformulate (38) as
$\displaystyle x_{k+1}=x_{k}-\alpha\cdot\frac{1}{n}\sum_{i=1}^{n}\mathcal{C}_{i
}\left(\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right).$
(39)
(39) is exactly DCGD with biased compression.
We can easily prove that
$\displaystyle\mathcal{C}_{i}$
$\displaystyle\in\mathbb{B}^{1}\left(\alpha=1-2\sqrt{\varepsilon_{2}},\beta=
\frac{1-\sqrt{\varepsilon_{2}}}{1+\varepsilon_{2}}\right)$
$\displaystyle\mathcal{C}_{i}$
$\displaystyle\in\mathbb{B}^{2}\left(\xi=1-\sqrt{\varepsilon_{2}},\beta=\frac{1
-\sqrt{\varepsilon_{2}}}{1+\varepsilon_{2}}\right)$
$\displaystyle\mathcal{C}_{i}$
$\displaystyle\in\mathbb{B}^{3}\left(\delta=\frac{1}{1-\varepsilon_{2}}\right).$
However, DCGD with biased compression may fail to converge even if the above formulation of compression mapping seems quite nice.
For an example of such failure, we refer the readers to Beznosikov et al. (, Example 1).
This limitation can be circumvented by employing an error feedback mechanism; however, this approach requires modifications to the original algorithm.
We therefore leave it as a future research direction.
- (II).
We can also view it as if we are in the single node case.
Let $\mathcal{C}\left(\cdot\right)$ be the compressing mapping defined as
$\displaystyle\mathcal{C}\left(\nabla M^{\gamma}\left(x_{k}\right)\right)$
$\displaystyle:=\frac{1}{n}\sum_{i=1}^{n}\gamma\nabla{M}^{\gamma}_{f_{i}}\left(
x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{
\gamma f_{i}}\left(x_{k}\right)\right)$
$\displaystyle=\gamma\nabla M^{\gamma}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^
{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right).$
(40)
This formulation leads us to the convergence guarantee appeared in Theorem 3, as we illustrate below.

Let us first analyze $\mathcal{C}$ defined in ((II).).
We will verify it belongs to $\mathbb{B}^{3}\left(\delta\right)$.
The inequality we want to prove can be written equivalently as

$$ $\displaystyle\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)-\frac{1}{n}\sum_ {i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right) \right)-\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}\leq\left(1-\frac {1}{\delta}\right)\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2},$ (41) $$

which is exactly

$$ $\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|^{2}\leq\left\|\gamma\nabla M ^{\gamma}\left(x_{k}\right)\right\|^{2}$ $$

For the left-hand side, using the convexity of $\left\|\cdot\right\|^{2}$ in combination with Definition 4, we obtain

$$ $\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|^{2}$ $\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left\|\tilde{x}_{i,k+1}-{\rm prox}_ {\gamma f_{i}}\left(x_{k}\right)\right\|^{2}$ $\displaystyle\leq\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|x_{k}-{\rm prox }_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}.$ $$

Let $x_{\star}$ be a minimizer of $f$, since we assume 2 holds, by 7, it is also a minimizer of $\gamma M^{\gamma}$,

$$ $\displaystyle\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|x_{k}-{\rm prox}_{ \gamma f_{i}}\left(x_{k}\right)\right\|^{2}$ $\displaystyle\overset{(\ref{eq:ppt-2-gradient})}{=}$ $\displaystyle\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{ \gamma}_{f_{i}}\left(x_{k}\right)\right\|^{2}$ $\displaystyle=$ $\displaystyle\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{ \gamma}_{f_{i}}\left(x_{k}\right)-\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{ \star}\right)\right\|^{2}$ $\displaystyle\overset{\text{\lx@cref{creftypecap~refnum}{tech-lemma:smooth}}}{\leq}$ $\displaystyle\frac{2\varepsilon_{2}}{n}\sum_{i=1}^{n}\frac{\gamma L_{i}}{1+ \gamma L_{i}}\left(\gamma{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma{M}^{ \gamma}_{f_{i}}\left(x_{\star}\right)\right)$ $\displaystyle\leq$ $\displaystyle\frac{2\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}}\left( \gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}\left(x_{\star}\right) \right).$ $$

We then notice that as it is illustrated by Lemma 1, we have

$$ $\displaystyle\left(1-\frac{1}{\delta}\right)\left\|\gamma\nabla M^{\gamma} \left(x_{k}\right)\right\|^{2}\geq\left(1-\frac{1}{\delta}\right)\frac{\gamma \mu}{2\left(1+\gamma L_{\max}\right)}\left(\gamma M^{\gamma}\left(x_{k}\right) -\gamma M^{\gamma}\left(x_{\star}\right)\right).$ $$

Combining the above two inequalities, we know that the following inequality is a sufficient condition for (41),

$$ $\displaystyle\frac{2\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}}\left( \gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}\left(x_{\star}\right) \right)\leq\left(1-\frac{1}{\delta}\right)\frac{\gamma\mu}{2\left(1+\gamma L_{ \max}\right)}\left(\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}\left( x_{\star}\right)\right).$ $$

It is easy to check that if we pick

$$ $\delta=\frac{\mu}{\mu-4\varepsilon_{2}L_{\max}}>0,$ (42) $$

the condition is met.
However, for this to hold, we must ensure that $\varepsilon_{2}<\frac{\mu}{4L_{\max}}$.

As we mentioned in Appendix D, Beznosikov et al. () provided the theory of CGD with biased compressor belongs to $\mathbb{B}^{3}\left(\delta\right)$.
We have already shown that $\mathcal{C}\in\mathbb{B}^{3}\left(\delta=\frac{\mu}{\mu-4\varepsilon_{2}L_{
\max}}\right)$, when $\varepsilon_{2}<\frac{4L_{\max}}{\mu}$.
Notice that our objective $\gamma M^{\gamma}$ is $\gamma L_{\gamma}$-smooth and $\frac{\gamma\mu}{1+\gamma L_{\max}}$-PL.(^4^44Theorem 7 remains valid if we replace $f$ being strongly convex with PL.)
Therefore, as long as $0<\alpha\leq\frac{1}{\gamma L_{\gamma}}$ and $\varepsilon_{2}<\frac{\mu}{4L_{\max}}$, we have

$$ $\displaystyle\mathcal{E}_{K}\leq\left(1-\frac{\mu-4\varepsilon_{2}L_{\max}}{ \mu}\cdot\frac{\gamma\mu}{4\left(1+\gamma L_{\max}\right)}\cdot\alpha\right)^{ K}\mathcal{E}_{0},$ $$

Taking $\alpha=\frac{1}{\gamma L_{\gamma}}$, which is the largest step size possible, we can further simplify the above convergence into

$$ $\displaystyle M^{\gamma}\left(x_{k}\right)-M^{\gamma}_{\star}\leq\left(1-\left (1-\frac{4\varepsilon_{2}L_{\max}}{\mu}\right)\cdot\frac{\mu}{4L_{\gamma}\left (1+\gamma L_{\max}\right)}\right)^{K}\left(M^{\gamma}\left(x_{0}\right)-M^{ \gamma_{\star}}\right).$ $$

Using smoothness of $\gamma L_{\gamma}$, if we denote $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$ where $x_{\star}$ is a minimizer of both $M^{\gamma}$ and $f$ since we assume we are in the interpolation regime (2), we have

$$ $\displaystyle\mathcal{E}_{0}\leq\frac{\gamma L_{\gamma}}{2}\Delta_{0}.$ $$

Using star strong convexity (quadratic growth property), we have

$$ $\displaystyle\mathcal{E}_{K}\geq\frac{\gamma\mu}{2\left(1+\gamma L_{\max} \right)}\Delta_{K}.$ $$

As a result, we can transform the above convergence guarantee into

$$ $\displaystyle\Delta_{K}\leq\left(1-\left(1-\frac{4\varepsilon_{2}L_{\max}}{\mu }\right)\cdot\frac{\mu}{4L_{\gamma}\left(1+\gamma L_{\max}\right)}\right)^{K} \cdot\frac{L_{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0}.$ $$

This completes the proof.

### F.5 Proof of Theorem 4

Notice that we assume each $f_{i}$ is $L_{i}$-smooth and convex.
The local optimization of each client can be written as

$$ $\min_{z\in\mathbb{R}^{d}}\left\{A^{\gamma}_{k,i}\left(z\right)=f_{i}\left(z \right)+\frac{1}{2\gamma}\left\|z-x_{k}\right\|^{2}\right\},$ $$

It is easy to see that $A^{\gamma}_{k,i}\left(z\right)$ is $L_{i}+\frac{1}{\gamma}$-smooth and $\frac{1}{\gamma}$-strongly convex.
We first provide the convergence theory of GD for reference.

#### Theory of GD :

For a $\widehat{\mu}$-strongly convex, $\widehat{L}$-smooth function $\phi$, the algorithm can be formulated as

$$ $z_{t+1}=z_{t}-\eta\nabla\phi(z_{t}),$ (GD) $$

where $z_{t}$ is the iterate in the $t$-th iteration, and $\eta>0$ is the step size.
GD with step size $\eta\in(0,\frac{1}{\widehat{L}}]$ generates iterates that satisfy

$$ $\left\|z_{t}-z_{\star}\right\|^{2}\leq\left(1-\eta\widehat{\mu}\right)^{t} \left\|z_{0}-z_{\star}\right\|^{2},$ $$

where $z_{\star}$ is a minimizer of $\phi$, $t$ is the number of iterations (number of gradient evaluations).

#### Approximation satisfying Definition 3 :

Notice that ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ is the minimizer of $A^{\gamma}_{k,i}\left(z\right)$ and $z_{0}=x_{k}$.
As a result, if we run GD with the largest step size $\frac{\gamma}{1+\gamma L_{i}}$,

$$ $\displaystyle\left\|z_{t}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^ {2}\leq\left(1-\frac{1}{1+\gamma L_{i}}\right)^{t}\left\|x_{k}-{\rm prox}_{ \gamma f_{i}}\left(x_{k}\right)\right\|^{2}$ (43) $$

We have

$$ $t=\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\frac{\left\|x_{k}-{ \rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}}{\varepsilon_{1}}\right )\right).$ $$

The unknown term $\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}$ within the log can be bounded by

$$ $\displaystyle\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^ {2}$ $\displaystyle=\left\|z_{0}-z_{\star}\right\|^{2}$ $\displaystyle\leq\gamma^{2}\left\|\nabla A^{\gamma}_{k,i}\left(z_{0}\right)- \nabla A^{\gamma}_{k,i}\left(z_{\star}\right)\right\|^{2}=\left\|\gamma\nabla f _{i}\left(x_{k}\right)\right\|^{2},$ (44) $$

which can be easily calculated.

#### Approximation satisfying Definition 4 :

According to (43), we have

$$ $t=\mathcal{O}\left(\left(1+\gamma L_{i}\right)\log\left(\frac{1}{\varepsilon_{ 2}}\right)\right).$ $$

This completes the proof.

### F.6 Proof of Theorem 5

We first provide the theory of AGD (Nesterov, ).

#### Theory of AGD :

For a $\widehat{\mu}$-strongly convex, $\widehat{L}$-smooth function $\phi$, the algorithm can be formulated as

$$ $\displaystyle y_{t+1}$ $\displaystyle=z_{t}+\alpha\left(z_{t}-z_{t-1}\right)$ $\displaystyle z_{t+1}$ $\displaystyle=y_{t+1}-\eta\nabla\phi\left(y_{t+1}\right),$ (AGD) $$

where $z_{t},y_{t}$ are iterates, $\eta>0$ is the step size, $\alpha>0$ is the momentum parameter.
AGD with step size $\eta=\frac{1}{\widehat{L}}$, momentum $\alpha=\frac{\sqrt{\widehat{L}}-\sqrt{\widehat{\mu}}}{\sqrt{\widehat{L}}+\sqrt
{\widehat{\mu}}}$ generates iterates that satisfy

$$ $\displaystyle\left\|z_{t}-z_{\star}\right\|^{2}\leq\frac{2\widehat{L}}{ \widehat{\mu}}\cdot\left(1-\sqrt{\frac{\widehat{\mu}}{\widehat{L}}}\right)^{t} \left\|z_{0}-z_{\star}\right\|^{2},$ $$

where $z_{\star}$ is a minimizer of $\phi$, $t$ is the number of iterations (number of gradient evaluations).

#### Approximation satisfying Definition 3 :

Notice that ${\rm prox}_{\gamma f_{i}}\left(x_{k}\right)$ is the minimizer of $A^{\gamma}_{k,i}\left(z\right)$ and $z_{0}=x_{k}$.
As a result, if we run AGD with the step size $\frac{\gamma}{1+\gamma L_{i}}$ and momentum $\alpha=\frac{\sqrt{1+\gamma L_{i}}-1}{\sqrt{1+\gamma L_{i}}+1}$,

$$ $\displaystyle\left\|z_{t}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^ {2}\leq 2\cdot\left(1+\gamma L_{i}\right)\left(1-\frac{1}{\sqrt{1+\gamma L_{i} }}\right)^{t}\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^ {2}.$ (45) $$

We have

$$ $t=\mathcal{O}\left(\sqrt{1+\gamma L_{i}}\log\left(\frac{\left(1+\gamma L_{i} \right)\cdot\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{ 2}}{\varepsilon_{1}}\right)\right)$ $$

Similar to the proof of Theorem 4, since we have according to (F.5),

$$ $\left\|x_{k}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}\leq\left \|\gamma\nabla f_{i}\left(x_{k}\right)\right\|^{2},$ $$

it is straightforward to determine the number of local iterations needed.

#### Approximation satisfying Definition 4 :

Using (45), we have

$$ $\displaystyle t=\mathcal{O}\left(\sqrt{1+\gamma L_{i}}\log\left(\frac{1+\gamma L _{i}}{\varepsilon_{2}}\right)\right).$ $$

### F.7 Proof of Theorem 8

In this case, the gradient estimator is defined as

$$ $g(x_{k})=\frac{1}{\tau}\sum_{i\in S_{k}}\left(\gamma\nabla{M}^{\gamma}_{f_{i}} \left(x_{k}\right)-\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k }\right)\right)\right).$ (46) $$

Notice that we have

$$ $\displaystyle\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),{\mathbb{E} }\left[g(x_{k})\right]\right\rangle$ $\displaystyle=\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),{\mathbb{E }}\left[\frac{1}{\tau}\sum_{i\in S_{k}}\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x _{k}\right)-\frac{1}{\tau}\sum_{i\in S_{k}}\left(\tilde{x}_{i,k+1}-{\rm prox}_ {\gamma f_{i}}\left(x_{k}\right)\right)\right]\right\rangle$ $\displaystyle=\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),\gamma \nabla M^{\gamma}\left(x_{k}\right)-\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{x}_{ i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right\rangle.$ $$

Using the same technique in the proof of Theorem 2, we are able to obtain that

$$ $\displaystyle\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),{\mathbb{E} }\left[g(x_{k})\right]\right\rangle\geq\left(1-\frac{2\sqrt{\varepsilon_{2}}L_ {\max}}{\mu}\right)\cdot\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right \|^{2}.$ $$

Thus, as long as we pick $\varepsilon_{2}<\frac{\mu^{2}}{4L_{\max}^{2}}$, we can pick $b=1-\sqrt{\varepsilon_{2}}\cdot\frac{2L_{\max}}{\mu}$ and $c=0$.
We then compute ${\mathbb{E}}\left[\left\|g(x_{k})\right\|^{2}\right]$,

$$ $\displaystyle{\mathbb{E}}\left[\left\|g(x_{k})\right\|^{2}\right]$ $\displaystyle={\mathbb{E}}\left[\left\|\frac{1}{\tau}\sum_{i\in S_{k}}\gamma \nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\frac{1}{\tau}\sum_{i\in S_{k}} \left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right) \right\|^{2}\right]$ $\displaystyle=\underbrace{{\mathbb{E}}\left[\left\|\frac{1}{\tau}\sum_{i\in S_ {k}}\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\|^{2}\right]}_{:= T_{1}}+\underbrace{{\mathbb{E}}\left[\left\|\frac{1}{\tau}\sum_{i\in S_{k}} \left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right) \right\|^{2}\right]}_{:=T_{2}}$ $\displaystyle\quad\underbrace{-2{\mathbb{E}}\left[\left\langle\frac{1}{\tau} \sum_{i\in S_{k}}\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right),\frac{1}{ \tau}\sum_{i\in S_{k}}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x _{k}\right)\right)\right\rangle\right]}_{:=T_{3}}.$ $$

We try to provide upper bounds for those terms separately.

#### Term T 1 subscript 𝑇 1 T_{1} italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT :

We have

$$ $\displaystyle T_{1}$ $\displaystyle=\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{1}{n}\sum_{i=1}^{n }\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\|^{2}+\frac{n \left(\tau-1\right)}{\tau\left(n-1\right)}\cdot\left\|\gamma\nabla M^{\gamma} \left(x_{k}\right)\right\|^{2}.$ $$

Using smoothness of $\gamma{M}^{\gamma}_{f_{i}}$ and the fact that we are in the interpolation regime, we have

$$ $\displaystyle T_{1}$ $\displaystyle=\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{1}{n}\sum_{i=1}^{n }\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)-\gamma\nabla{M}^{ \gamma}_{f_{i}}\left(x_{\star}\right)\right\|^{2}+\frac{n\left(\tau-1\right)}{ \tau\left(n-1\right)}\cdot\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right) \right\|^{2}$ $\displaystyle\leq\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{1}{n}\sum_{i=1} ^{n}\frac{2\gamma L_{i}}{1+\gamma L_{i}}\cdot\left(\gamma M^{\gamma}_{f_{i}} \left(x_{k}\right)-\gamma\left(M^{\gamma}_{f_{i}}\right)_{\inf}\right)+\frac{n \left(\tau-1\right)}{\tau\left(n-1\right)}\cdot\left\|\gamma\nabla M^{\gamma} \left(x_{k}\right)\right\|^{2}$ $\displaystyle\leq\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{2\gamma L_{\max }}{1+\gamma L_{\max}}\cdot\left(\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{ \gamma}_{\inf}\right)+\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)}\cdot \left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|^{2}.$ (47) $$

#### Term T 2 subscript 𝑇 2 T_{2} italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT :

It is easy to see that using convexity of the squared Euclidean norm, we have

$$ $\displaystyle T_{2}$ $\displaystyle\leq{\mathbb{E}}\left[\frac{1}{\tau}\sum_{i\in S_{k}}\left\| \tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|^{2}\right]$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left\|\tilde{x}_{i,k+1}-{\rm prox}_{ \gamma f_{i}}\left(x_{k}\right)\right\|^{2}\overset{(\ref{eq:index2-approx})}{ \leq}\frac{\varepsilon_{2}}{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f _{i}}\left(x_{k}\right)\right\|^{2}.$ $$

Using smoothness of each individual $\gamma{M}^{\gamma}_{f_{i}}\left(x_{k}\right)$ and the fact we are in the interpolation regime, we have

$$ $\displaystyle T_{2}$ $\displaystyle\leq\frac{2\varepsilon_{2}\gamma L_{\max}}{1+\gamma L_{\max}} \left(\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}\right).$ (48) $$

#### Term T 3 subscript 𝑇 3 T_{3} italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT :

We have

$$ $\displaystyle T_{3}$ $\displaystyle=-2\cdot\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{1}{n}\sum_{ i=1}^{n}\left\langle\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right),\tilde{ x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\rangle$ $\displaystyle\quad-2\cdot\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)} \cdot\left\langle\gamma\nabla M^{\gamma}\left(x_{k}\right),\frac{1}{n}\sum_{i= 1}^{n}\left(\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right) \right)\right\rangle.$ $$

Using Cauchy-Schwarz inequality and convexity, we further obtain

$$ $\displaystyle T_{3}$ $\displaystyle\leq 2\cdot\frac{n-\tau}{\tau\left(n-1\right)}\cdot\frac{1}{n} \sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right \|\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle\quad+2\cdot\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)} \left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|\cdot\frac{1}{n}\sum_{ i=1}^{n}\left\|\tilde{x}_{i,k+1}-{\rm prox}_{\gamma f_{i}}\left(x_{k}\right) \right\|.$ $$

Using similar approaches in the previous paragraphs, we have

$$ $\displaystyle T_{3}$ $\displaystyle\overset{(\ref{eq:index2-approx})}{\leq}\frac{2\left(n-\tau\right )}{\tau\left(n-1\right)}\cdot\frac{\sqrt{\varepsilon_{2}}}{n}\sum_{i=1}^{n} \left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\|^{2}+\frac{2n \left(\tau-1\right)}{\tau\left(n-1\right)}\left\|\gamma M^{\gamma}\left(x_{k} \right)\right\|\frac{\sqrt{\varepsilon_{2}}}{n}\cdot\sum_{i=1}^{n}\left\| \gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle\leq\frac{2\left(n-\tau\right)}{\tau\left(n-1\right)}\cdot\frac{ \sqrt{\varepsilon_{2}}}{n}\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f_{i} }\left(x_{k}\right)-\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{\star}\right) \right\|^{2}$ $\displaystyle\quad+\frac{2n\left(\tau-1\right)}{\tau\left(n-1\right)}\left\| \gamma M^{\gamma}\left(x_{k}\right)\right\|\frac{\sqrt{\varepsilon_{2}}}{n} \cdot\sum_{i=1}^{n}\left\|\gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)- \gamma\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right)\right\|$ $\displaystyle\leq\frac{4\sqrt{\varepsilon_{2}}\left(n-\tau\right)}{\tau\left(n -1\right)}\cdot\frac{\gamma L_{\max}}{1+\gamma L_{\max}}\left(\gamma M^{\gamma }\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}\right)$ $\displaystyle\quad+\frac{4\sqrt{\varepsilon_{2}}n\left(\tau-1\right)}{\tau \left(n-1\right)}\cdot\frac{\gamma L_{\max}}{1+\gamma L_{\max}}\left\|x_{k}-x_ {\star}\right\|\left\|\gamma\nabla M^{\gamma}\left(x_{k}\right)\right\|$ $\displaystyle\overset{(\ref{eq:ineq-pr1-4})}{\leq}\frac{4\sqrt{\varepsilon_{2} }\left(n-\tau\right)}{\tau\left(n-1\right)}\cdot\frac{\gamma L_{\max}}{1+ \gamma L_{\max}}\left(\gamma M^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{ \inf}\right)$ $\displaystyle\quad+\frac{4\sqrt{\varepsilon_{2}}n\left(\tau-1\right)}{\tau \left(n-1\right)}\cdot\frac{L_{\max}}{\mu}\left\|\gamma\nabla M^{\gamma}\left( x_{k}\right)\right\|^{2}.$ (49) $$

Combining (F.7), (48) and (F.7), we have

$$ $\displaystyle\sum_{i=1}^{3}T_{i}$ $\displaystyle\leq 2\left(\varepsilon_{2}+\frac{2\sqrt{\varepsilon_{2}}\left(n- \tau\right)}{\tau\left(n-1\right)}+\frac{\left(n-\tau\right)}{\tau\left(n-1 \right)}\right)\cdot\frac{\gamma L_{\max}}{1+\gamma L_{\max}}\cdot\left(\gamma M ^{\gamma}\left(x_{k}\right)-\gamma M^{\gamma}_{\inf}\right)$ $\displaystyle\quad+\left(\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)}+ \frac{4\sqrt{\varepsilon_{2}}n\left(\tau-1\right)}{\tau\left(n-1\right)}\right )\cdot\frac{L_{\max}}{\mu}\cdot\left\|\gamma M^{\gamma}\left(x_{k}\right) \right\|^{2}.$ (50) $$

Therefore, it is easy to see that we can pick

$$ $\displaystyle A$ $\displaystyle=\left(\varepsilon_{2}+\frac{2\sqrt{\varepsilon_{2}}\left(n-\tau \right)}{\tau\left(n-1\right)}+\frac{\left(n-\tau\right)}{\tau\left(n-1\right) }\right)\cdot\frac{\gamma L_{\max}}{1+\gamma L_{\max}}$ $\displaystyle B$ $\displaystyle=\left(\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)}+\frac{4 \sqrt{\varepsilon_{2}}n\left(\tau-1\right)}{\tau\left(n-1\right)}\right)\cdot \frac{L_{\max}}{\mu},\qquad C=0.$ $$

Applying Theorem 4 of Demidovich et al. (), we list the corresponding values of $A,B,C,b,c\geq 0$ below,

$$ $\displaystyle A$ $\displaystyle=\frac{\gamma L_{\max}}{1+\gamma L_{\max}}\left(\varepsilon_{2}+ \frac{2\sqrt{\varepsilon_{2}}\left(n-\tau\right)}{\tau\left(n-1\right)}+\frac{ \left(n-\tau\right)}{\tau\left(n-1\right)}\right)$ $\displaystyle B$ $\displaystyle=\frac{n\left(\tau-1\right)}{\tau\left(n-1\right)}\left(1+\frac{4 \sqrt{\varepsilon_{2}}L_{\max}}{\mu}\right),\quad C=0$ $\displaystyle b$ $\displaystyle=\frac{\mu-2\sqrt{\varepsilon_{2}}L_{\max}}{\mu},\quad c=0.$ $$

We know that the PL constant of $\gamma M^{\gamma}$ is given by $\frac{\gamma\mu}{4\left(1+\gamma L_{\max}\right)}$ and the corresponding smoothness constant is $\gamma L_{\gamma}$.
As a result, when $\alpha>0$ satisfies

$$ $\displaystyle\alpha<\underbrace{\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu-2 \sqrt{\varepsilon_{2}}L_{\max}}{\mu+4\varepsilon_{2}L_{\max}+4\sqrt{ \varepsilon_{2}}L_{\max}+\frac{n-\tau}{\tau\left(n-1\right)}\cdot\left(4L_{ \max}+4\sqrt{\varepsilon_{2}}L_{\max}-\mu\right)}}_{:=B_{1}^{\prime}},$ $$

and

$$ $\displaystyle\alpha<\underbrace{\frac{4\left(1+\gamma L_{\max}\right)}{\gamma \left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)}}_{=B_{2}},$ $$

we can obtain a convergence guarantee for the algorithm.
Notice that $B_{1}^{\prime}\leq B_{1}<B_{2}$(^5^55The definition of $B_{1}$ is given in (37)), thus we can further simplify the range of $\alpha$ to

$$ $\displaystyle\alpha\leq\underbrace{\frac{1}{\gamma L_{\gamma}}\cdot\frac{\mu-2 \sqrt{\varepsilon_{2}}L_{\max}}{\mu+4\varepsilon_{2}L_{\max}+4\sqrt{ \varepsilon_{2}}L_{\max}+\frac{n-\tau}{\tau\left(n-1\right)}\cdot\left(4L_{ \max}+4\sqrt{\varepsilon_{2}}L_{\max}-\mu\right)}}_{:=B_{1}^{\prime}}.$ $$

Given that we select $\alpha$ properly, we have

$$ $\displaystyle{\mathbb{E}}\left[\mathcal{E}_{K}\right]\leq\left(1-\alpha\cdot \frac{\gamma\left(\mu-2\sqrt{\varepsilon_{2}}L_{\max}\right)}{4\left(1+\gamma L _{\max}\right)}\right)^{K}\mathcal{E}_{0}.$ $$

Specifically, if we choose the largest $\alpha$ possible, we have

$$ $\displaystyle{\mathbb{E}}\left[\mathcal{E}_{K}\right]\leq\left(1-\frac{\mu}{4L _{\gamma}\left(1+\gamma L_{\max}\right)}\cdot S\left(\varepsilon_{2},\tau \right)\right)^{K}\mathcal{E}_{0},$ $$

where $S\left(\varepsilon_{2},\tau\right)$ is defined as

$$ $\displaystyle S\left(\varepsilon_{2},\tau\right)=\frac{\left(\mu-2\sqrt{ \varepsilon_{2}}L_{\max}\right)\left(1-2\sqrt{\varepsilon_{2}}\frac{L_{\max}}{ \mu}\right)}{\mu+4\varepsilon_{2}L_{\max}+4\sqrt{\varepsilon_{2}}L_{\max}+ \frac{n-\tau}{\tau\left(n-1\right)}\cdot\left(4L_{\max}+4\sqrt{\varepsilon_{2} }L_{\max}-\mu\right)},$ $$

satisfying

$$ $\displaystyle 0<S\left(\varepsilon_{2},\tau\right)\leq 1.$ $$

Using smoothness of $\gamma L_{\gamma}$, if we denote $\Delta_{k}=\left\|x_{k}-x_{\star}\right\|^{2}$ where $x_{\star}$ is a minimizer of both $M^{\gamma}$ and $f$ since we assume we are in the interpolation regime (2), we have

$$ $\displaystyle\mathcal{E}_{0}\leq\frac{\gamma L_{\gamma}}{2}\Delta_{0}.$ $$

Using star strong convexity (quadratic growth property), we have

$$ $\displaystyle\mathcal{E}_{K}\geq\frac{\gamma\mu}{2\left(1+\gamma L_{\max} \right)}\Delta_{K}.$ $$

As a result, we can transform the above convergence guarantee into

$$ $\displaystyle{\mathbb{E}}\left[\Delta_{K}\right]\leq\left(1-\frac{\mu}{4L_{ \gamma}\left(1+\gamma L_{\max}\right)}\cdot S\left(\varepsilon_{2},\tau\right) \right)^{K}\cdot\frac{L{\gamma}\left(1+\gamma L_{\max}\right)}{\mu}\Delta_{0}.$ $$

This completes the proof.

## Appendix G Experiments

We describe the settings for the numerical experiments and the corresponding results to validate our theoretical findings.
We are interested in the following optimization problem in the distributed setting,

$$ $\displaystyle\min_{x\in\mathbb{R}^{d}}\left\{f(x)=\frac{1}{n}\sum_{i=1}^{n}f_{ i}\left(x\right)\right\}.$ $$

Here $n$ denotes the number of clients, $d$ is the dimension, each function $f_{i}:\mathbb{R}^{d}\mapsto\mathbb{R}$ has the following form

$$ $f_{i}(x)=\frac{1}{2}x^{\top}{\bm{A}}_{i}x+b_{i}^{\top}x+c_{i},$ $$

where ${\bm{A}}_{i}\in\mathbb{S}_{+}^{d},b_{i}\in\mathbb{R}^{d},c_{i}\in\mathbb{R}$.
Specifically, we pick $n=20$ and $d=300$ for the experiments.
Notice that we have

$$ $\displaystyle\nabla f_{i}(x)={\bm{A}}_{i}x-b_{i};\qquad\nabla^{2}f_{i}(x)={\bm {A}}_{i}\succeq{\bm{O}}_{d},$ $$

which suggests that each $f_{i}$ is convex and smooth.
We can easily compute that in this case, we have

$$ $\displaystyle{\rm prox}_{\gamma f_{i}}\left(x\right)=\left({\bm{A}}_{i}+\frac{ 1}{\gamma}{\bm{I}}_{d}\right)^{-1}\left(\frac{1}{\gamma}x-b_{i}\right).$ $$

All experiment codes were implemented in Python 3.11 using the NumPy and SciPy libraries.
The computations were performed on a system powered by an AMD Ryzen 9 5900HX processor with Radeon Graphics, featuring 8 cores and 16 threads, running at 3.3 GHz.
Code availability: [https://anonymous.4open.science/r/Inexact-FedExProx-code-E783/](https://anonymous.4open.science/r/Inexact-FedExProx-code-E783/)

### G.1 Comparison of FedProx , FedExProx , FedExProx with absolute approximation and relative approximation

In this section, we compare the convergence of FedProx, FedExProx and FedExProx with absolute approximation and relative approximation.
For FedProx, we simply set the server extrapolation to be $1$ while for FedExProx, we set its extrapolation parameter to be $\frac{1}{\gamma L_{\gamma}}$.
We assume exact proximal evaluation for the above two algorithms.
For FedExProx with approximations, we fix $\varepsilon_{1}$ and $\varepsilon_{2}$ to be reasonable values, respectively.
We then set their extrapolation parameter to be the optimal value under the specific setting.
Throughout the experiment, we vary the value of the local step size $\gamma$ to see its effect on all the algorithms.
Specifically, we select $\gamma$ from the set $\left\{1000,100,10,1,0.1.0.01\right\}$, and we fix $\varepsilon_{1}=0.001$, $\varepsilon_{2}=0.01$ first, then we set them to $\varepsilon_{1}=1e-6$, $\varepsilon_{2}=0.001$.

Figure: Figure 1: Comparison of FedProx, FedExProx with exact proximal evaluations, FedExProx with $\varepsilon_{1}$-absolute approximation and FedExProx with $\varepsilon_{2}$-relative approximation. In this case, we fix $\varepsilon_{1}=0.001$, $\varepsilon_{2}=0.01$ and pick the local step size $\gamma\in\left\{1000,100,10,1,0.1.0.01\right\}$. The $y$-axis is the squared distance to the minimizer of $f$, and the $x$-axis denotes the iterations.
Refer to caption: x1.png

Figure: Figure 2: Comparison of FedProx, FedExProx with exact proximal evaluations, FedExProx with $\varepsilon_{1}$-absolute approximation and FedExProx with $\varepsilon_{2}$-relative approximation. In this case, we fix $\varepsilon_{1}=1e-6$, $\varepsilon_{2}=0.001$ and pick the local step size $\gamma\in\left\{1000,100,10,1,0.1.0.01\right\}$. The $y$-axis is the squared distance to the minimizer of $f$, and the $x$-axis denotes the iterations.
Refer to caption: x7.png

Notably in Figure 1 and Figure 2, in all cases, FedExProx with absolute approximation exhibits the poorest performance and converges only to a neighborhood of the solution.
This is expected, since the bias in this case does not go to zero as the algorithm progresses.
It is worth mentioning that as the local step size $\gamma$ increases, the size of the neighborhood decreases, which supports our claim in Theorem 1.
As anticipated, in all cases, FedExProx outperforms FedProx due to server extrapolation.
However, as $\gamma$ increases, the performance gap between them diminishes.
The performance of FedExProx with relative approximation is surprisingly good, outperforming FedProx in several cases.
This suggests the effectiveness of server extrapolation even when the proximal evaluations are inexact.

### G.2 Comparison of FedExProx with absolute approximation under different inaccuracies

In this section, we compare FedExProx with absolute approximations under different level of inaccuracies.
We fix the local step size $\gamma$ to be a reasonable value, and we vary the level of inexactness for the algorithm.
Specifically, we select $\gamma$ from the set $\left\{0.1,1,10\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{1}$ from the set $\left\{0.001,0.005,0.01,0.05,0.1\right\}$.

Figure: Figure 3: Comparison of FedExProx with $\varepsilon_{1}$-absolute approximation under different level of inexactness. We select $\gamma$ from the set $\left\{0.1,1,10\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{1}$ from the set $\left\{0.001,0.005,0.01,0.05,0.1\right\}$. The $y$-axis denotes the squared distance to the minimizer and the $x$-axis is the number of iterations.
Refer to caption: x13.png

As observed in Figure 3, the size of the neighborhood increases with $\varepsilon_{1}$, further corroborating our theoretical findings in Theorem 1.
Before reaching the neighborhood, the convergence rates of FedExProx with different level of inexactness are similar, which is expected.

### G.3 Comparison of FedExProx with relative approximation under different inaccuracies

In this section, we compare FedExProx with relative approximations under different level of relative inaccuracies.
We fix the local step size $\gamma$ to be a reasonable value, and we vary the level of inexactness for the algorithm.
Specifically, we select $\gamma$ from the set $\left\{0.1,0.05,0.01\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{2}$ from the set $\left\{0.001,0.005,0.01,0.05,0.1\right\}$.

Figure: Figure 4: Comparison of FedExProx with $\varepsilon_{2}$-relative approximation under different level of inexactness. We select $\gamma$ from the set $\left\{0.01,0.05,0.1\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{2}$ from the set $\left\{0.001,0.005,0.01,0.05,0.1\right\}$. The $y$-axis denotes the squared distance to the minimizer and the $x$-axis is the number of iterations.
Refer to caption: x16.png

As observed in Figure 4, in all cases, a smaller $\varepsilon_{2}$ corresponds to faster convergence of the algorithm.
This supports the claim of Theorem 3.
All the tested algorithm converges to the exact solution linearly, which validates the effectiveness of the proposed technique of relative approximation to reduce the bias term.

### G.4 Adaptive extrapolation for inexact proximal evaluations

In this section, we study the possibility of applying adaptive extrapolation to FedExProx with relative approximations.
We do not consider the case of absolute approximation since it converges only to a neighborhood, which causes problems when combined with adaptive step sizes such as gradient diversity and Polyak step size.

We are using the following definition of gradient diversity based extrapolation,

$$ $\displaystyle\alpha_{k}=\alpha_{k,G}:=\frac{1+\gamma L_{\max}}{\gamma L_{\max} }\cdot\frac{\frac{1}{n}\sum_{i=1}^{n}\left\|x_{k}-{\rm prox}_{\gamma f_{i}} \left(x_{k}\right)\right\|^{2}}{\left\|\frac{1}{n}\sum_{i=1}^{n}\left(x_{k}-{ \rm prox}_{\gamma f_{i}}\left(x_{k}\right)\right)\right\|^{2}}.$ $$

for Polyak type extrapolation, we use

$$ $\displaystyle\alpha_{k}=\alpha_{k,S}:=\frac{\frac{1}{n}\sum_{i=1}^{n}\left({M} ^{\gamma}_{f_{i}}\left(x_{k}\right)-\inf{M}^{\gamma}_{f_{i}}\right)}{\gamma \left\|\frac{1}{n}\sum_{i=1}^{n}\nabla{M}^{\gamma}_{f_{i}}\left(x_{k}\right) \right\|^{2}}.$ $$

Figure: Figure 5: Comparison of FedExProx with $\varepsilon_{2}$-relative approximation under different level of inexactness using gradient diversity based extrapolation. we select $\gamma$ from the set $\left\{1,0.1,0.01\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{2}$ from the set $\left\{0.0001,0.05\right\}$. The $y$-axis denotes the squared distance to the minimizer and the $x$-axis is the number of iterations.
Refer to caption: x19.png

Figure: Figure 6: Comparison of FedExProx with $\varepsilon_{2}$-relative approximation under different level of inexactness using Polyak step size based extrapolation. we select $\gamma$ from the set $\left\{10,100,1000\right\}$ and for each choice of $\gamma$, we select $\varepsilon_{2}$ from the set $\left\{1e-4,1e-5\right\}$. The $y$-axis denotes the squared distance to the minimizer and the $x$-axis is the number of iterations.
Refer to caption: x22.png

As it can be observed from Figure 5, in all cases, the use of a gradient diversity based adaptive extrapolation results in faster convergence of the algorithm.
This suggests the possibility of developing an adaptive extrapolation for our methods.
However, as we can see from Figure 6, a direct implementation of Polyak step size type extrapolation results in divergence of the algorithm, indicating that the challenge may be more complex than anticipated.
In our case, this is equivalent to designing adaptive step sizes for SGD with biased updates or CGD with biased compression.
To the best of our knowledge, this field remains open and requires further investigation, as biased updates are quite common in practice.

## Peer Reviews

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: See above
soundness: 2
strengths: The paper addresses a significant gap in existing work on FedExProx by relaxing the exact proximal computation assumption. This makes the analysis more applicable to real-world federated learning systems, where inexact computations are the norm due to resource constraints.
summary: This paper investigates the convergence behavior of FedExProx, a recent extension of the FedProx federated learning algorithm, which includes server-side extrapolation to improve performance in federated settings. A key issue with existing analyses of FedExProx is the assumption that each client can compute the proximal operator exactly, which is unrealistic in practical applications. This paper relaxes this assumption, examining the algorithm’s behavior in cases where the proximal operator is only computed approximately. The authors establish convergence results in smooth, globally strongly convex settings, demonstrating that the algorithm still converges, albeit to a neighborhood around the solution. They also show that careful control can reduce the negative impact of inexact proximal updates and draw connections to biased compression methods. Additionally, they provide an analysis of the local iteration complexity needed for clients to achieve a specific level of inexactness, with empirical validation of their findings through numerical experiments.
weaknesses: - The theoretical analysis is restricted to globally strongly convex problems, which may limit its applicability to a broader range of federated learning applications that involve non-convex objectives. Extending this analysis to non-convex cases would significantly increase the paper’s impact.

- The assumption of smoothness might not always hold in federated learning, particularly when clients have heterogeneous data distributions. A discussion on how the proposed approach might generalize or be adapted for non-smooth settings would strengthen the paper.

- The experimental part is the weakest part of this work...

**The authors have addressed my concerns in rebuttals. I raise my grade to 6**

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: - What is the main challenge and main novelty in your paper?

- Why do you not take stochastic gradients into account? How will the results change assuming this?
soundness: 3
strengths: - The question is indeed natural and relevant to concurrent FL problems

- The authors cleverly define two kind of approximations and show that one is better then the other, allowing us to converge to the true optimum

- The writing is very clear and easy to follow

- Experiments are illustrative, and in a sense validate the theory
summary: This work sets to explore a recent algorithm in FL called FEDPROX. This algorithm leans on exact computation of what is called proximal operator.
The paper asks the following natural question: what if we do not fully solve the operator, but rather solve approximately?
In the smooth+strongly-convex case, this paper explore this questions assuming two kinds of approximations $\epsilon_1$ and $\epsilon_2$.
weaknesses: - While the question is natural and important, the solution is quite straightforward, and does not introduce any novel tools or analysis. Excluding the  $\epsilon_2$ approximation which is nice.

- The paper does not consider stochastic gradients which is the more relevant case in practice, it is important to understand how will the results change in light of this?

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: - Theorem 1 seems to provide convergence guarantees under the natural assumption of absolute approximation. However, the guarantee provided, as mentioned, includes a neighbourhood of convergence which is not optimal. On the other hand, the connection with biased compression provides a refined theorem (Theorem 3), establishing convergence to the exact solution. The amount of inexactness, though, in Theorem 3 is bounded. Do you think that one can achieve the best of both worlds, namely convergence to the exact solution but for arbitrary inexactness.
- How one can compute the relative inexactness $\epsilon_2$ in practice? Are there inherent computational tradeoffs or challenges in the computation of the relative inexactness $\epsilon_2$ in comparison to estimating the constant $\epsilon_1$? It would be nice also if you could comment on ways to approximate $\epsilon_2$ in practical federated learning problems.  
- Is it possible to raise the assumption on interpolation in the strongly convex setting by using a more refined proof technique or do you think that extrapolation might be beneficial only on that regime?
soundness: 3
strengths: - The convergence is established under different notions of inexactness of the proximal operator.
- The paper recovers as a special case the results of the original paper on FedExProx, when the proximal operators are evaluated exactly.
- The connection with the biased compression is interesting.
summary: The paper focuses on a federated learning algorithm, called FedExProx, that requires each client to compute exactly a proximal operator. The authors analyze the FedExProx method when the proximal operators of each client is not computed exactly. Theoretical guarantees are provided in the strongly convex and smooth setting and the convergence rate of the algorithm is established. Moreover, the authors highlight a connection with the bias compression methods, that allows them to obtain more refined convergence guarantees. The iteration complexity of the local updates for gradient descent and accelerated gradient descent are also provided. Experimental results validate the theoretical results and showcase the effect of the different notions of inexactness in the computation of the proximal operator in the convergence of the method.
weaknesses: - Theorems 1, 2, 3 require the notion of interpolation. Even though an explanation of regimes that satisfy this condition is provided, considering that there are previous works [1], [2] that extend beyond that setting, this assumption seems to be an avenue for future work in this field. More specifically, the initial FedProx algorithm [1] is analyzed in the general non-interpolated setting. In addition, the follow-up work regarding the FedExProx algorithm [2] considers in the main paper the interpolated regime. However, the authors provide additionally an illustration of the algorithm's behaviour in the non-interpolated setting (see Appendix F.3 in [2]). In that sense, it would be useful to provide some additional details on the behaviour of the algorithm in the non-interpolated setting or to comment on the main challenges in extending the current proof technique beyond the interpolation framework, offering in that way a more complete picture and direction for future research. 
- Theorems 4, 5 seem to evaluate the inexactness achieved in each client. However, the inexactness is only with respect to the notion of the absolute approximation, for which we know that Theorem 1 is not optimal (since for the same amount of inexactness Theorem 3 gives convergence to the exact solution). Thus, it seems that a characterization of the inexactness in terms of the relative approximation would be also useful. Hence, providing similar theorems for the relative approximation case seems to be a nice addition to the current results. 
- Minor: The statement of Theorem 1 can be made shorter in order to increase the readability of the paper.
- Minor typo: In Figure 1, it is mentioned “Figure (c) demonstrates how varying values of $\epsilon_1$ affect FedExProx with relative approximation.” but as shown the varying values correspond to $\epsilon_2$.

References:  
[1] Federated Optimization in Heterogeneous Networks, T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, V. Smith    
[2] The Power of Extrapolation in Federated Learning, H. Li, K. Acharya, P. Richtárik

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 2
questions: - Fig. 1(a) indicates that inexactness can help whereas the theory predict otherwise. For inexact proximal gradient inexactness have shown to help for certain regimes (see e.g. page 5 of [3]). Is it possible that something analogue can be said in your setting?
- It is not very clear how much having a more stringent requirement on the relative inexactness ($\mu/L_{max}$ as compared with $\mu^2/L_{max}^2$) buys in terms of the global rate. Is it possible to explicitly compare $S(\varepsilon_2)$ with $(1-4\varepsilon_2 L_{max})$?

[3] https://proceedings.neurips.cc/paper_files/paper/2011/file/8f7d807e1f53eff5f9efbe5cb81090fb-Paper.pdf
soundness: 2
strengths: - The writing is very clear and transparent. They state how results are obtained (relative inexactness by using analysis from biased SGD and from compression) and discuss limitations.
- Considering relative inexactness for federated learning seems interesting
summary: The paper considers a finite-sum $\mu$-strongly convex problems for which the interpolation conditions holds, and where each client objective is convex and $L$-smooth.
The work then considers the FedExProx method, which combines proximal client updates with an extrapolated server step, and extends this work to handle inexactness of the client prox computations. Specifically:

- with fixed absolute inexactness they show that the method converges to a neighbourhood of the solution (using a factor $1/4$ smaller extrapolation step).
- with a type of relative inexactness (smaller than order $\mu^2/L^2$) they show exact convergence but for a restrictive extrapolation server stepsize $\alpha$.
- for relative inexactness with a more stringent condition (smaller than order $\mu/L$) they show that the same (large) extrapolation stepsize can be used as in the exact case.
- They provide convergence rate for the local strongly convex and smooth objective with gradient descent and Nesterov acceleration.
weaknesses: - the work requires very strong assumptions: the solution needs to be unique (strong convexity) and shared amongst all clients (interpolation condition)
- There is a large body of work on relative inexactness for proximal methods starting with [1], where it is used to essentially inherent the nice properties of an exact proximal computation. Considering the strong assumptions (strong convexity and interpolation condition) it does not seem very surprising that one can extend to a multi-client setting. It would be good to cite this work and put it into context. 
- The work does not treat adaptive stepsizes and partial participation as in (exact) FedExProx (they do discuss the difficulty of client sampling in the appendix).

Minor:

- The local convergence rates are not new. It would be good to explicitly state this.
- After Theorem 2 when discussing the slowdown due to small $\alpha$, it would be informative to plug in $\varepsilon_2=c\mu^2/L_{max}^2$ for some $c<1$ and simplify the expression.
- It it possible to to get convergence not only to a neighborhood even for absolute inexactness. It might be worth choosing the $\varepsilon_1$ sufficiently small, to make the comparison with relative inexactness more direct (how does the choice effect the client steps and the communication rounds?).
- For absolute inexactness the server stepsize $\alpha$ is a factor ¼  smaller. Maybe stress that this affects the rate explicitly in Table 1.
- It is maybe worth stating how many iterations (e.g. with Nesterov) are needed to make $\varepsilon_2 =\mu/L$ vs $\varepsilon_2 =\mu^2/L^2$ to make the comparison/tradeoff more explicit between the two relative inexactness results.
- It seems like some concurrent work is treating absolute inexactness which might be worth mentioning [2]

Typos:

- Eq. 4 both f and $\phi$ are present

[1] https://arxiv.org/pdf/2410.15368v1

[2] https://www.emis.de/journals/JCA/vol.6_no.1/j149.pdf

Meta-review: The paper considers a recently introduced algorithm FedExProx addressing federated learning settings, where an extrapolation step at the server is combined with the local proximal updates at the clients. The addressed finite sum optimization problem is assumed to be such that each component function is smooth and convex, the sum is strongly convex, and an "interpolation regime" condition applies, meaning that the global minimizer is also a minimizer of each of the component functions. The paper relaxes the assumption from prior work that the proximal updates at the clients are computed exactly and instead studies two settings: with additive and multiplicative approximation error. For additive error, it proves that the algorithm converges to a neighborhood of the minimum, while for the multiplicative error convergence to the exact minimum is possible (asymptotically). 

Although federated learning as a topic is of high interest to the ML community and the paper provides interesting technical contributions, it's scope is somewhat specialized since the problem assumptions are quite strong and the novelty in coming up with the problem and the solution seems somewhat limited. Perhaps adding the results for the convex (but not strongly convex) objectives claimed by the authors would strengthen the paper.

Decision: Reject