Communication-Efficient Federated Low-Rank Update Algorithm and its Connection to Implicit Regularization

## Abstract

Abstract Federated Learning (FL) faces significant challenges related to communication efficiency and heterogeneity. To address these issues, we explore the potential of using low-rank updates. Our theoretical analysis reveals that client’s loss exhibits a higher rank structure (gradients span higher rank subspace of Hessian) compared to the server’s loss. Based on this insight, we hypothesize that constraining client-side optimization to a low-rank subspace could provide an implicit regularization effect. Consequently, we propose FedLoRU, a general low-rank update framework for federated learning. Our framework enforces low-rank client-side updates and accumulates these updates to form a higher-rank model. Additionally, variants of FedLoRU can adapt to environments with statistical and model heterogeneity by employing multiple or hierarchical low-rank updates. Experimental results demonstrate that FedLoRU performs comparably to full-rank algorithms and exhibits robustness to heterogeneous and large numbers of clients.

## 1 Introduction

Federated learning (McMahan et al.,, ) is a collaborative learning framework designed to enhance privacy preservation in machine learning applications. This approach has gained importance due to rising concerns over data privacy, as it allows multiple participants to train a model collectively without sharing raw data.

While federated learning offers privacy benefits, it trades off some performance compared to centralized learning. Two primary factors contributing to this trade-off are communication overhead and heterogeneity. Despite improvements in computation and memory capacities, communication speeds have only slightly improved, making communication overhead a major factor in slowing down federated learning (Zheng et al.,, ). Additionally, various forms of heterogeneity—statistical, system, and device—further complicate FL (Ye et al.,, ; Kairouz et al.,, ). These issues are especially pronounced with a large number of clients, where frequent, less impactful updates slow down training and reduce performance.

Addressing these challenges is becoming increasingly critical for training large language models (LLMs) in a federated learning framework. By low rank herein we refer to gradients spanning a low rank subspace of Hessian at any weights or the weight matrix being of the form $\bm{A}\bm{B}$ where the number of columns of $\bm{A}$ is low. Utilizing private datasets on edge devices for LLM training is promising due to the limited availability of public data (Ye et al.,, ). However, this approach presents significant issues, notably in terms of communication overhead, as edge devices possess heterogeneous resources and data. Additionally, the need for effective regularization across clients is required. Consequently, the development of algorithms to tackle these challenges is an essential problem to bridge the gap between practical and conceptual federated learning applications.

There has been substantial research focusing on the low-rank characteristics in centralized learning. Utilizing low-rank factorized update models such as LoRA (Hu et al.,, ), DyLoRA (Valipour et al.,, ), and QLoRA (Dettmers et al.,, ) can significantly reduce the number of trainable parameters, which helps to conserve memory and computational resources. Further observations (Huh et al.,, ; Ji and Telgarsky,, ) indicate that over-parameterized models tend to find low-rank solutions, which provide implicit regularization effects.

However, the rank properties of the loss landscape in federated learning remain under-explored. We first analyze the difference in the stable rank—defined as the squared ratio of the Frobenius norm to the spectral norm—between client Hessians and the server Hessian of any weights, discovering that client exhibits a higher rank structure. Based on this theoretical insight, we hypothesize that the higher rank structure of client’s loss contributes to increased client discrepancy and that restricting client-side updates could provide an implicit regularization effect across clients. This leads us to the research question:

Can we use low-rank updates to achieve both communication overhead reduction and regularization effects across clients?

We propose the Federated Low-Rank Updates (FedLoRU) algorithm, which addresses communication overhead and the challenges posed by a large number of clients by employing client-side low-rank updates and server-side accumulation of low-rank updates. FedLoRU factorizes client-side update matrices $\bm{A}$ and $\bm{B}$ and applies iterative optimization to these low-rank factorized matrices. Clients and the server share the factorized matrices, which the server then aggregates. Matrices $\bm{A}$ and $\bm{B}$ are communicated between the clients and server, rather than the much larger matrix $\bm{AB}$. To make the model’s weight rank high, the server successively accumulates low-rank matrices. We also generalize the low-rank update strategy within federated learning for various heterogeneous settings.

Figure: Figure 1: Figure LABEL:fig:figure1a provides a flowchart representing the FedLoRU algorithm. In this algorithm, the model training is conducted solely using rank-$r$ matrices, with communication between the server and clients being confined to these matrices. Clients incrementally add low-rank update matrices to their local base model $\bm{W}^{(k)}$ every $\tau$ rounds, resulting in a higher-rank model. Figure LABEL:fig:figure1b depicts the utilization of low-rank factorization within the pFedLoRU and mFedLoRU algorithms. For clarity, the equations assume all $\alpha$ parameters are set to 1.

Our comprehensive approach underscores the potential of low-rank updates not only to enhance communication efficiency but also to impose implicit regularization and harmonize the optimization process across heterogeneous federated learning settings. Our contributions can be summarized as follows. 1) We propose FedLoRU, the first algorithm using successive low-rank updates for both pre-training and fine-tuning in federated learning, and introduce variants of FedLoRU for personalization and model heterogeneity settings; 2) We investigate the rank properties of client and server losses, analytically showing that under stochastic sampling, the rank of the Hessian of the loss function increases with smaller sample sizes; 3) We provide empirical evidence of the higher rank structure of client losses and demonstrate that restricting the rank of local updates aids in implicit regularization; 4) On average, FedLoRU improves state-of-the-art communication-efficient federated learning algorithms on a variety of datasets, including LLM fine-tuning, and exhibits superior performance as the number of clients increases.

## 2 Related work

#### Communication-Efficient Federated Learning

Extensive research has been conducted to address the communication challenges in federated learning (Shahid et al.,, ). FedPAQ (Reisizadeh et al.,, ) and AdaQuantFL (Jhunjhunwala et al.,, ) employ quantization techniques that decrease the precision of weights and activations in neural networks. Besides, Fed-Dropout (Caldas et al.,, ) and FedMP (Jiang et al.,, ) utilize pruning to eliminate less important neurons or connections within models. Since quantization and sparsification techniques do not alter the fundamental network architecture and can be universally applied across any models, they are perceived as additional steps aimed at reducing communication overhead.

In contrast to these methods, model compression techniques in federated learning alter the model structure itself by transforming the original model into a smaller model update before communication, then restoring it before local training or aggregation. FedDLR (Qiao et al.,, ) employs low-rank approximation to compress the model during both server-to-client and client-to-server communications, reverting to the full-rank model during local training. On the other hand, FedHM (Yao et al.,, ) compresses the model solely during server-to-client communication, where clients train factorized low-rank models without reverting them and the server restores the local models before aggregation. While both FedDLR and FedHM effectively reduce communication overheads, their server-side compression approaches can lead to performance degradation. To mitigate potential information loss during server-side compression, we focus on client-side factorization, avoiding compression processes.

#### Low-rank nature of centralized and federated learning

Despite the over-parameterization of current deep learning models, numerous studies (Gur-Ari et al.,, ; Li et al.,, ; Sagun et al.,, ) assert that the training process in deep learning inherently possesses a low-rank nature. Low-Rank Adaptation (LoRA, Hu et al., ()) is a representative algorithm that leverages this low-rank characteristic, particularly for fine-tuning tasks. However, effectively utilizing the low-rank structure during the pre-training phase remains challenging. This difficulty is elucidated by some papers (Yu and Wu,, ; Zhao et al.,, ), which attribute the issue to the weights not exhibiting a low-rank.

Existing research in federated learning has attempted to exploit the low-rank nature observed in centralized learning. LBGM (Azam et al.,, ) and FedLRGD (Jadbabaie et al.,, ) approximate gradients by utilizing old or sampled gradients under the assumption that gradients live in a low-rank subspace. Nonetheless, there is a noticeable gap in their assumption and analysis regarding the rank characteristics specific to federated learning. In the context of federated learning, there is a complex loss landscape involving multiple client-side and a single server-side optimization, and leveraging a low-rank structure needs to consider their respective rank structures. To our knowledge, no prior work has examined the rank structure in federated learning contexts without making very stringent assumptions. Our study is pioneering in addressing this gap, using analytical results and insights to develop a novel algorithm.

#### Low-Rank Adaptation

The fundamental concept of LoRA involves freezing the pre-trained weights and adapting them to new tasks by introducing and optimizing an update through a two-layer low-rank decomposition. This is represented mathematically as $\bm{W}=\bm{W}_{0}+\bm{A}\bm{B}$ where $\bm{W}\in\mathbb{R}^{m\times n}$, $\bm{A}\in\mathbb{R}^{m\times r}$, $\bm{B}\in\mathbb{R}^{r\times n}$, $r\ll m,n$. By constraining the update matrix to a low-rank factorization, the number of parameters in the update matrix is reduced. However, this method demonstrates suboptimal performance during pre-training because the trained weight matrix does not exhibit a low-rank structure. To address this, ReLoRA (Lialin et al.,, ) seeks to achieve a higher-rank model by accumulating multiple low-rank updates, expressed as $\bm{W}=\bm{W}_{0}+\sum_{i=1}^{M}\bm{A}_{i}\bm{B}_{i}$ where $\bm{A}_{i}\in\mathbb{R}^{m\times r}$, $\bm{B}_{i}\in\mathbb{R}^{r\times n}$.

#### Low-Rank Adaptation in Federated Learning

Recent studies have studied the application of LoRA within federated learning frameworks. Notable algorithms, such as FedLoRA (Wu et al.,, ; Yi et al.,, ), FFALoRA (Sun et al.,, ), and Hyperflora (Lu et al.,, ), employ LoRA adapters to facilitate personalization. These methods apply low-rank adaptation to a pre-trained model during the local personalization training phase. On the other hand, other works such as Kuo et al., () and Cho et al., () apply LoRA for fine-tuning within federated learning environments.

These approaches use only one low-rank matrices that restricts the model to lie in low-rank subspace, on the other hand, our work utilizes multiple accumulated layers of low-rank matrices allowing the model to achieve higher rank. Specifically, we extends the concept of LoRA by incorporating client-side low-rank updates and server-side accumulation, aiming to address the low-rank limitation of LoRA as well as the challenges posed by communication and client-server rank disparity. We also generalize the low-rank strategy within federated learning for both pre-training and fine-tuning, and for heterogeneous environments.

## 3 Low-rank updates in federated learning

In centralized learning, neural network losses exhibit a low-rank structure, indicating that the gradient lies within the subspace spanned by the $k$ eigenvectors of the Hessian (Gur-Ari et al.,, ) during training. While efforts have been made to utilize this low-rank structure to enhance federated learning algorithms, there is a lack of studies analyzing the rank structure of federated learning. In federated learning, the clients and server have distinct losses, resulting in different rank structures. Understanding these differing rank structures of client and server losses is crucial for developing low-rank-inspired algorithms tailored for federated learning.

In this section, we conduct a theoretical analysis of the rank structure of federated learning, with a particular emphasis on comparing the rank of client and server Hessians. Drawing from this theoretical analysis, we propose a novel federated learning algorithm FedLoRU designed to address communication efficiency and performance degradation issues associated with a large number of clients. Additionally, we introduce an algorithm that modifies FedLoRU to address challenges related to model and statistical heterogeneity in federated learning.

### 3.1 Higher rank nature of clients in federated learning

#### Notation and problem setup

Suppose $\psi(\bm{x},\bm{y})$ is a data generating distribution for an input-output pair $(\bm{x},\bm{y})\in\mathbb{R}^{d_{x}}\times\mathbb{R}^{d_{y}}$. We consider the problem of finding a prediction function $h^{R}(\cdot;\cdot):\mathbb{R}^{d_{x}}\times\mathbb{R}^{R}\to\mathbb{R}^{d_{y}}$ parameterized by a $R$-dim weight vector $\omega^{R}\in\mathbb{R}^{R}$. Given a loss function $\ell(\cdot,\cdot):\mathbb{R}^{d_{y}}\times\mathbb{R}^{d_{y}}\to\mathbb{R}$, the true or population risk is defined as the loss over the data-generating distribution $\psi(\bm{x},\bm{y})$

$$ $\mathcal{L}_{\text{true}}(h^{R},\omega^{R})=\int\ell(h^{R}(\bm{x};\omega^{R}), \bm{y})d\psi(\bm{x},\bm{y}).$ (1) $$

The corresponding true Hessian is $\bm{H}_{\text{true}}(h^{R},\omega^{R})=\nabla^{2}\mathcal{L}_{\text{true}}(h^{
R},\omega^{R})$. If $\mathcal{D}_{N}=\left\{(\bm{x}_{1},\bm{y}_{1}),\cdots,(\bm{x}_{N},\bm{y}_{N})\right\}$ is a dataset generated from the distribution $\psi$, then the loss and Hessian corresponding to the dataset $\mathcal{D}_{N}$ are given by:

$$ $\displaystyle f_{N}(h^{R},\omega^{R})=\sum_{(x,y)\in\mathcal{D}_{N}}\frac{1}{N }\ell(h^{R}(x;\omega^{R}),y),\quad\bm{H}_{N}(h^{R},\omega^{R})=\sum_{(x,y)\in \mathcal{D}_{N}}\frac{1}{N}\frac{\partial^{2}}{\partial(\omega^{R})^{2}}\ell(h ^{R}(x;\omega^{R}),y).$ (2) $$

We consider random selection of $M$ samples without replacement from $\mathcal{D}_{N}$ to form a sub-dataset $\mathcal{D}_{M}\subseteq\mathcal{D}_{N}$. Let $f_{M}(h^{R},\omega^{R})$ and $\bm{H}_{M}(h^{R},\omega^{R})$ denote the loss and Hessian for the sub-dataset $\mathcal{D}_{M}$. For simplicity, we omit the explicit dependency on $h^{R}$ and $\omega^{R}$ when contextually clear. In federated learning, $f_{N}$ can be considered as the loss that the server optimizes, while $f_{M}$ represents the loss of a local client assuming the homogeneous setting.

For non-zero real numbers $\theta_{1},\cdots,\theta_{k}$, define $\Omega^{R}(\theta_{1},\cdots,\theta_{k})$ as the family of pairs $(h^{R},\omega^{R})$, where $h^{R}$ is an $R$-dimensional prediction function and $\omega^{R}$ is a weight vector, such that the true Hessian has eigenvalues $\theta_{1},\cdots,\theta_{k}$. Specifically, $\Omega^{R}(\theta_{1},\cdots,\theta_{k})=\{(h^{R},\omega^{R}):\bm{H}_{\text{
true}}(h^{R},\omega^{R})\;\text{has eigenvalues}\;\theta_{1},\cdots,\theta_{k}\}$. Let $\Omega(\theta_{1},\cdots,\theta_{k})=\bigcup_{R}\Omega^{R}(\theta_{1},\cdots,
\theta_{k})$, representing the union of $\Omega^{R}(\theta_{1},\cdots,\theta_{k})$ over all dimensions $R$. We aim to show that the difference in stable rank between the Hessians of a server and a client eventually becomes positive as dimension $R$ approaches infinity within the space of $\Omega(\theta_{1},\cdots,\theta_{k})$.

In fact, the set of all possible pairs $(h^{R},\omega^{R})$ is represented by the union over all dimensions $R$, integers $k\leq R$, and non-zero real values $\theta_{1},\cdots,\theta_{k}$ as follows:

$$ $\{(h^{R},\omega^{R}):\text{dimension}\;R<\infty\}=\bigcup_{R=1}^{\infty} \bigcup_{k=1}^{R}\bigcup_{(\theta_{1},\cdots,\theta_{k})\in\mathbb{R}^{k}} \Omega^{R}(\theta_{1},\cdots,\theta_{k}).$ $$

Thus, for any given pair $(h^{R},\omega^{R})$, there exist $\theta_{1},\cdots,\theta_{k}$ such that $(h^{R},\omega^{R})\in\Omega^{R}(\theta_{1},\cdots,\theta_{k})$. According to the following proposition, the proof of which is provided in Appendix A.1, either the set $\Omega(\theta_{1},\cdots,\theta_{k})$ is empty or there exist infinitely many values of $R$ for which $\Omega^{R}(\theta_{1},\cdots,\theta_{k})\neq\emptyset$.

###### Proposition 3.1 .

Let $\theta_{1},\cdots,\theta_{k}$ be fixed non-zero real numbers, and suppose there exists $\tilde{R}>k$ such that $\Omega^{\tilde{R}}(\theta_{1},\cdots,\theta_{k})$ is non-empty. Then there are infinitely many $R$ such that $\Omega^{R}(\theta_{1},\cdots,\theta_{k})$ is non-empty. In particular, $\Omega^{R}(\theta_{1},\cdots,\theta_{k})$ is non-empty for all $R\geq\tilde{R}$.

###### Proposition 3.1 .

#### Comparing the stable rank of the client and server Hessians

Now, we will focus on comparing the stable rank of the client and server Hessians. For given $p,q\in\mathbb{N}$, let $\theta_{1}>\cdots>\theta_{p}>0>\theta_{p+1}>\cdots>\theta_{p+q}$ be deterministic non-zero real numbers, and let $(h^{R},\omega^{R})\in\Omega(\theta_{1},\cdots,\theta_{p+q})$ for some $R$. To compare the stable rank of Hessians for two datasets $\mathcal{D}_{N}$ and $\mathcal{D}_{N}$, we consider the additive perturbed model of the true Hessian as described by Baskerville et al., ():

$$ $\displaystyle\bm{H}_{N}(h^{R},\omega^{R})$ $\displaystyle=\bm{H}_{\text{true}}(h^{R},\omega^{R})+\epsilon^{R}(N)$ (3) $\displaystyle\bm{H}_{M}(h^{R},\omega^{R})$ $\displaystyle=\bm{H}_{\text{true}}(h^{R},\omega^{R})+\epsilon^{R}(M).$ (4) $$

Here, $\epsilon^{R}(N),\epsilon^{R}(M)\in\mathbb{R}^{R\times R}$ are defined as random error matrices associated with each Hessian. These matrices are assumed to be scaled according to $\epsilon^{R}(N)=s(N)X^{R}$, where $X^{R}\in\mathbb{R}^{R\times R}$ is a random real symmetric matrix and $s:\mathbb{N}\to(0,1)$ is a decreasing function.

Another study (Granziol et al.,, ) employs the model $\bm{H}_{M}(h^{R},\omega^{R})=\bm{H}_{N}(h^{R},\omega^{R})+\epsilon^{R}$, implying a dependency structure between $\bm{H}_{M}$ and $\bm{H}_{N}$. However, their analysis assumes independence between these matrices, which is problematic given the underlying model and practical considerations. In contrast, we address this issue by introducing two decoupled additive perturbed models. Additionally, while Granziol et al., () investigates outlier eigenvalues, our focus is on the difference in the rank of the Hessians.

We seek to determine the limiting eigenvalues of the Hessians $\bm{H}_{N}(h^{R},\omega^{R})$ and $\bm{H}_{M}(h^{R},\omega^{R})$ in relation to the eigenvalues of $\bm{H}_{\text{true}}(h^{R},\omega^{R})$. Since $(h^{R},\omega^{R})\in\Omega^{R}(\theta_{1},\cdots,\theta_{p+q})$, the eigenvalues of $\bm{H}_{\text{true}}(h^{R},\omega^{R})$ are $\theta_{1},\cdots,\theta_{p+q}$. Next, we need to make some assumptions about the random error matrix $X^{R}$. Assume $X^{R}$ is a random real symmetric matrix with eigenvalues $\lambda_{1}(X^{R}),\cdots,\lambda_{R}(X^{R})$ and a limiting spectral density $\mu$, such that $\frac{1}{R}\sum_{i=1}^{R}\delta(\lambda-\lambda_{i}(X^{R}))\to\mu(\lambda)$, with convergence in the weak almost sure sense. Examples of matrices exhibiting a well-defined limiting spectral density include Wigner matrices, Wishart matrices, and Gaussian ensembles. We assume $\mu$ is a compactly supported probability measure on $[l_{\mu},r_{\mu}]$ which admits a smooth density with respect to the Lebesque measure and the eigenvectors of $X^{R}$ obey quantum unique ergodicity (QUE). For more detail about the QUE condition, we refer to Baskerville et al., (). We can now find the limiting eigenvalues of $\bm{H}_{N}$ and $\bm{H}_{M}$.

###### Proposition 3.2 (Limiting eigenvalues of $\bm{H}_{N}$ (modified from Baskerville et al., ( 2022 ) )) .

Let $R$ be any integer such that $R\geq\bar{R}$ where $\bar{R}$ is the smallest integer such that $\Omega^{\bar{R}}(\theta_{1},\cdots,\theta_{p+q})$ is non-empty. For any pair $(h^{R},\omega^{R})\in\Omega^{R}(\theta_{1},\cdots,\theta_{p+q})$, consider the Hessian additive error model given by $\bm{H}_{N}(h^{R},\omega^{R})=\bm{H}_{true}(h^{R},\omega^{R})+\epsilon^{R}(N)$. If $\lambda_{i}(\bm{H}_{N}(h^{R},\omega^{R}))$ denotes the $i$-th eigenvalue of $\bm{H}_{N}(h^{R},\omega^{R})$, then for $i=1,\cdots,p$, the following holds:

$$ $\lambda_{i}(\bm{H}_{N}(h^{R},\omega^{R}))\to\begin{cases}g^{-1}_{N}(\theta_{i} )&\text{if}\;g^{-1}_{N}(\theta_{i})>U_{N}\\ U_{N}&\text{otherwise}\end{cases}$ (5) $$

as $R\to\infty$, while for each fixed $i>p$, $\lambda_{i}(H_{N}(h^{R},\omega^{R}))$ that converges to a limit in $\mathbb{R}\backslash[0,U_{N}]$ converges to $U_{N}$, i.e., $\lambda_{i}(H_{N}(h^{R},\omega^{R}))\to U_{N}$. Similarly, for $i=0,\cdots,q-1$, we have

$$ $\lambda_{R-i}(\bm{H}_{N}(h^{R},\omega^{R}))\to\begin{cases}g^{-1}_{N}(\theta_{ p+q-i})&\text{if}\;g^{-1}_{N}(\theta_{p+q-i})<L_{N}\\ L_{N}&\text{otherwise}\end{cases}$ (6) $$

while for each fixed $i\geq q$, $\lambda_{R-i}(\bm{H}_{N}(h^{R},\omega^{R}))$ that converges to a limit in $\mathbb{R}\backslash[L_{N},0]$ converges to $L_{N}$, i.e., $\lambda_{R-i}(\bm{H}_{N}(h^{R},\omega^{R}))\to L_{N}$. Here,

$$ $g^{-1}_{N}(\theta)=\theta+s(N)\mathcal{R}_{\mu}(s(N)\theta^{-1})$ (7) $$

and $U_{N}$ and $L_{N}$ are lower and upper bounds of the limiting distribution $\mu_{N}$ of $\epsilon^{R}(N)$.

Convergence in Proposition 3.2 is weak almost sure convergence and $\mathcal{R}_{\mu}(\omega)$, known as the $\mathcal{R}$-transform, is defined by $\mathcal{R}_{\mu}(\omega)=S^{-1}_{\mu}(\omega)-\frac{1}{\omega}$ where $S_{\mu}(\omega)$ is the Stieltjes transform. Compared to Baskerville et al., (), which focuses solely on outlier eigenvalues, we extend the analysis to bulk eigenvalues and adopt a simpler form of $\mu$. Within the proposition, the $i$-th largest or smallest limiting eigenvalues of $\bm{H}_{N}$ are determined by the values of $g^{-1}(\theta_{i})$. If $g^{-1}(\theta_{i})$ falls within the support of the limiting distribution $\mu_{N}$, the corresponding limiting eigenvalues converge to the bounds. If $g^{-1}(\theta_{i})$ does not lie within this support, it converges to $g^{-1}(\theta_{i})$ itself; these eigenvalues are typically referred to as outlier eigenvalues in the literature. The detailed proof is provided in Appendix A.2 and is similar to the proof in Baskerville et al., ().

###### Proposition 3.2 (Limiting eigenvalues of 𝑯 N subscript 𝑯 𝑁 \bm{H}_{N} bold_italic_H start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT (modified from Baskerville et al., ( 2022 ) )) .

#### Stable rank

To compare the rank properties of Hessians of a client and a server, we introduce the concept of stable rank, defined as the square of the ratio between the Frobenius norm and the spectral norm of a matrix $\bm{A}$:

$$ $\text{srank}(\bm{A})=\frac{\|\bm{A}\|_{F}^{2}}{\|\bm{A}\|_{2}^{2}}=\frac{\sum_ {i=1}^{n}\sigma_{i}^{2}(\bm{A})}{\sigma_{1}^{2}(\bm{A})}$ (8) $$

where $n$ is the rank of the matrix $\bm{A}$ and $\sigma_{i}(\bm{A})$ is the $i$-th singular value of the matrix $\bm{A}$. Stable rank serves as a continuous proxy for $\text{rank}(\bm{A})$ and is known for its robustness against small perturbations. In fact, stable rank—which emphasizes eigenvalues near the top eigenvalue—can be considered a more accurate surrogate of the rank structure of the Hessian considering the empirical evidences that gradients are highly influenced by the top Hessian eigenvector, i.e., the eigenvectors corresponding to the largest eigenvalues. Additionally, bounds on the stable rank of a weight provide control over the model’s complexity (Georgiev et al.,, ).

In the following theorem, we demonstrate that smaller datasets result in a higher limiting stable rank. Furthermore, given that modern neural network models typically possess a very large number of parameters, this finding is likely applicable to contemporary models.

###### Theorem 3.3 (Higher rank nature of Hessian of smaller dataset) .

Let $N>M>\bar{R}$ be any integers where $\bar{R}$ is the smallest integer such that $\Omega^{\bar{R}}(\theta_{1},\cdots,\theta_{p+q})$ is non-empty. For any pair $(h^{R},\omega^{R})\in\Omega^{R}(\theta_{1},\cdots,\theta_{p+q})$, let $\bm{H}_{N}(h^{R},\omega^{R})$ and $\bm{H}_{M}(h^{R},\omega^{R})$ be the Hessians as defined previously. The difference in stable rank between $\bm{H}_{N}(h^{R},\omega^{R})$ and $\bm{H}_{M}(h^{R},\omega^{R})$ converges weakly almost surely to positive value $S(\theta_{1},\cdots,\theta_{p+q},\mu)>0$ as $R\to\infty$, i.e.

$$ $\mathrm{srank}(\bm{H}_{M}(h^{R},\omega^{R}))-\mathrm{srank}(\bm{H}_{N}(h^{R}, \omega^{R}))\to S(\theta_{1},\cdots,\theta_{p+q},\mu)>0.$ (9) $$

Here, the value $S(\theta_{1},\cdots,\theta_{p+q},\mu)$ does not dependent on the sequence $(h^{R},\omega^{R})$.

Figure: Figure 2: Figure LABEL:fig:figure2a presents a comparison of the estimated stable rank of the Hessian for dataset sizes of 50 and 500. The estimated spectral density was computed using the pyhessian (Yao et al.,, ) library, and the stable rank was subsequently derived from these estimates. The stable rank of the Hessian for the dataset size of 50 consistently exceeds that of the dataset size of 500. Figure LABEL:fig:figure2b illustrates the test accuracy of FedAvg and FedLoRU across varying numbers of clients. Notably, as the number of clients increases, FedLoRU demonstrates better accuracy compared to FedAvg.

In federated learning, Theorem 3.3 implies that individual clients, often working with smaller, heterogeneous datasets, inherently possess a higher rank structure in their local Hessians compared to the Hessian of the server loss. This higher rank structure can lead to larger discrepancies across clients, as the complexity and variability in the local training landscapes are greater. Consequently, the optimization paths taken by different clients may diverge more significantly, complicating the aggregation process and potentially degrading the overall performance of the global model. This theoretical insight from Theorem 3.3 is further supported by our empirical findings, as shown in Figure 2.

Understanding this phenomenon is crucial for developing more effective federated learning algorithms. By acknowledging the higher rank structure of client’s Hessian, constraining the rank of client-side optimization can mitigate the discrepancies, especially when local dataset sizes are very small. In the subsequent section, we propose an algorithm that exploits this insight.

###### Theorem 3.3 (Higher rank nature of Hessian of smaller dataset) .

### 3.2 Federated Low-Rank Update(FedLoRU) Algorithm

In this section, we introduce the Federated Low-Rank Updates (FedLoRU) algorithm, along with its variants designed to adapt to statistical and model heterogeneity.

Consider a federated learning system with $K$ clients, where each client $k$ has its own loss function $f^{(k)}:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}$. In conventional federated learning algorithms such as FedAvg (McMahan et al.,, ), the server aims to find a global model $\bm{W}\in\mathbb{R}^{m\times n}$ that minimizes the aggregated loss function $f(\bm{W})=\sum_{k=1}^{K}p^{(k)}f^{(k)}(\bm{W})$, where $p^{(k)}$ is the weight of client $k$. During each round $t$, the server selects a set of $M$ clients $\mathcal{K}_{M}$ to participate in local training. These clients receive the global model $\bm{W}_{t-1}$ from the server and perform local training without altering the network architecture. The server then receives the locally updated models $\bm{W}_{t}^{(k)}$ from the clients and aggregates them into a global model by averaging: $\bm{W}t=\sum{k\in\mathcal{K}_{M}}p^{(k)}\bm{W}_{t}^{(k)}$.

#### Fedeated low-rank update algorithm

To enhance communication efficiency, FedLoRU constraints clients’ updates to low-rank. Analogous to LoRA (Hu et al.,, ) approach, upon receiving the global model $\bm{W}\in\mathbb{R}^{m\times n}$, each client $k$ freezes this model and performs local training through low-rank matrices $\bm{A}^{(k)}\in\mathbb{R}^{m\times r}$ and $\bm{B}^{(k)}\in\mathbb{R}^{r\times n}$ by solving:

$$ $\bm{A}^{(k)},\;\bm{B}^{(k)}=\operatorname*{arg\,min}_{\bm{A},\;\bm{B}}f^{(k)}( \bm{W}^{(k)}+\alpha\bm{A}\bm{B})$ (10) $$

where $\alpha$ is a fixed scaling hyperparameter. In FedLoRU, each client solves (10) for $E$ epochs in each round. The server then collects $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$ from the clients and aggregates them by averaging: $\bm{A}=\sum_{k\in\mathcal{K}_{M}}p^{(k)}\bm{A}^{(k)}$, $\bm{B}=\sum_{k\in\mathcal{K}_{M}}p^{(k)}\bm{B}^{(k)}$. After aggregation, the server broadcasts the averaged $\bm{A}$ and $\bm{B}$ to the clients, who continue local training using these matrices.

Occasionally, the server accumulates local updates into the global model after aggregation to achieve a higher-rank global model. Clients subsequently update their local global models by $\bm{W}^{(k)}\leftarrow\bm{W}^{(k)}+\alpha\bm{A}\bm{B}$ and reset their low-rank matrices. When we accumulate low-rank updates every $\tau$ rounds from the initial global model $\bm{W}_{0}$, the final global model at $T$ is

$$ $\bm{W}_{T}=\bm{W}_{0}+\sum_{\begin{subarray}{c}t=1\\ t\;\text{mod}\;\tau=0\end{subarray}}^{T}\bm{A}_{t}\bm{B}_{t}$ (11) $$

We employ two strategies for initializing the low-rank update matrices in FedLoRU. For random initialization, we initialize $\bm{A}$ with a random Gaussian distribution and set $\bm{B}$ to zero, ensuring that $\bm{AB}$ is zero at the start. Alternatively, for momentum initialization, we retain the existing weights of the matrices, continuing to use the previous low-rank update matrices. This approach leverages momentum effects as described in the ReLoRA(Lialin et al.,, ). The scheduling of accumulations is also critical due to the varying nature of the training phases across different rounds; in this study, we employ periodic accumulation, though this area warrants further investigation.

#### Communication overhead

FedLoRU reduces communication overhead from $Kmn$ to $Kmn$ when $r\ll m\;\text{or}\;n$. While we use a low-rank factorized model here, other options like LoKr or LoHa can be employed, differing only in the factorization scheme but maintaining identical fundamental principles. Additionally, without a compression process, there is no extra computation compared to conventional compression-based communication-efficient federated learning algorithms.

#### Low-rank local training, higher-rank global training, and implicit regularization

Moreover, FedLoRU facilitates the training of a higher-rank global model concurrently with low-rank local updates. With each accumulation of low-rank update matrices, the global model’s rank is incrementally enhanced, enabling the initiation of new learning phases. This process is evident in the training curve, which exhibits noticeable pattern shifts following each accumulation. Moreover, constraining the rank of local training introduces a regularization effect, thereby diminishing the discrepancy between updated local models.

Figure: Algorithm 1 FedLoRU. $\bm{W}$ is a model, $\bm{A}_{0},\bm{B}_{0}$ are initial low-rank update matrices, $\alpha$ is a scaling factor, $\tau$ is an accumulation cycle, $T$ is the total training round

#### Federated low-rank update for statistical heterogeneous setting

We develop the personalized FedLoRU (pFedLoRU) algorithm to address statistical heterogeneity in federated learning using low-rank updates idea. In pFedLoRU, each client $k$ maintains a global model $\bm{W}$, global low-rank matrices $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$, and personal low-rank matrices $\bm{L}^{(k)}$ and $\bm{U}^{(k)}$. The matrices $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$ are shared between the server and clients to update common global model, while $\bm{L}^{(k)}$ and $\bm{U}^{(k)}$ are tailored to adapt to the local distribution. In each round $t$, client $k$ optimizes the personal matrices for $E_{\text{per}}$ epochs and the global matrices for $E_{\text{global}}$ by solving:

$$ $\displaystyle\bm{L}^{(k)}_{t},\;\bm{U}^{(k)}_{t}$ $\displaystyle=\operatorname*{arg\,min}_{\bm{L},\;\bm{U}}f^{(k)}(\bm{W}^{(k)}+ \alpha_{\text{global}}\bm{A}^{(k)}_{t-1}\bm{B}^{(k)}_{t-1}+\alpha_{\text{per}} \bm{L}\bm{U})$ (12) $\displaystyle\bm{A}^{(k)}_{t},\;\bm{B}^{(k)}_{t}$ $\displaystyle=\operatorname*{arg\,min}_{\bm{A},\;\bm{B}}f^{(k)}(\bm{W}^{(k)}+ \alpha_{\text{global}}\bm{A}\bm{B}+\alpha_{\text{per}}\bm{L}_{t}^{(k)}\bm{U}_{ t}^{(k)})$ (13) $$

Subsequently, the server collects the global update matrices from the clients, performs aggregation, and updates the global model accordingly. A detailed description of the pFedLoRU algorithm can be found in Appendix B.

#### Federated low-rank update for model heterogeneous setting

When local clients possess varying hardware resources and communication speeds, it becomes impractical to use uniform low-rank matrices across all clients. To address this issue, we develop the model-heterogeneous FedLoRU (mFedLoRU) algorithm, which employs hierarchical low-rank updates. Specifically, to update the low-rank matrices $\bm{A}$ and $\bm{B}$, we apply low-rank updates recursively, enabling model adaptation through nested low-rank updates:

$$ $\bm{A}\bm{B}\leftarrow(\bm{A}+\alpha_{A}\bm{A}_{\text{d}}\bm{A}_{\text{u}})( \bm{B}+\alpha_{B}\bm{B}_{\text{d}}\bm{B}_{\text{u}})$ (14) $$

Here, $\bm{A}\bm{B}$ is the rank-$r$ factorization of a matrix $\bm{W}\in\mathbb{R}^{m\times n}$, and $\bm{A}_{\text{d}}\bm{A}_{\text{u}}$ and $\bm{B}_{\text{d}}\bm{B}_{\text{u}}$ are rank-$r_{A}$ and rank-$r_{B}$ factorizations of $\bm{A}$ and $\bm{B}$, respectively. In mFedLoRU, each client $k$ decides whether to use nested low-rank updates and, if so, determines the locally adapted rank $r^{(k)}$ based on its resources. If a client opts out of nested low-rank updates, it updates its low-rank modules like in FedLoRU. However, if it chooses nested low-rank updates, it generates and optimizes the nested low-rank matrices $\bm{A}_{\text{d}}\bm{A}_{\text{u}}$ and $\bm{B}_{\text{d}}\bm{B}_{\text{u}}$ by solving:

$$ $\bm{A}^{(k)}_{\text{d}},\bm{A}^{(k)}_{\text{u}},\bm{B}^{(k)}_{\text{d}},\bm{B} ^{(k)}_{\text{u}}=\operatorname*{arg\,min}_{\bm{A}_{\text{d}},\bm{A}_{\text{u} },\bm{B}_{\text{d}},\bm{B}_{\text{u}}}f^{(k)}(\bm{W}^{(k)}+\alpha(\bm{A}+ \alpha^{(k)}_{A}\bm{A}_{\text{d}}\bm{A}_{\text{u}})(\bm{B}+\alpha^{(k)}_{B}\bm {B}_{\text{d}}\bm{B}_{\text{u}}))$ (15) $$

After local training, client $k$ sends its nested low-rank matrices to the server, which recovers them into rank-$r$ low-rank matrices $\bm{A}^{(k)}\leftarrow\bm{A}+\alpha^{(k)}_{A}\bm{A}_{\text{d}}^{(k)}\bm{A}_{
\text{u}}^{(k)}$, and $\bm{B}^{(k)}\leftarrow\bm{B}+\alpha^{(k)}_{B}\bm{B}_{\text{d}}^{(k)}\bm{B}_{
\text{u}}^{(k)}$, and then performs aggregation using these rank-$r$ low-rank matrices as in FedLoRU. A detailed description of the mFedLoRU algorithm can be found in Appendix B.

## 4 Experiments

In this section, we extensively evaluate FedLoRU on pre-training and fine-tuning on different heterogeneous settings. We first provide the experiment setup such as baselines and heterogeneous settings, then move on to the performance evaluation.

### 4.1 Experiment setup

#### Datasets and Models

To evaluate the performance of our proposed algorithms, we employ three commonly used datasets in machine learning: CIFAR-10, CIFAR-100 (Krizhevsky et al.,, ), and Alpaca (Taori et al.,, ). For the image datasets, we employ ResNet-18 (He et al.,, ) as the model architecture, and for the language dataset, we use LLaMa2-3B (Touvron et al.,, ). In pre-training experiments on the image datasets, we vary the number of clients between 20 and 400 to evaluate the algorithms under diverse conditions and to investigate the effect of client numbers. At each round, 50% of the clients are randomly sampled to train models over 5 local epochs. For fine-tuning the language model, we configure 10 clients with a 50% participation rate and conduct training for 1 local epoch. The learning rate is selected through grid search, and various rank configurations are tested for FedHM and FedLoRU.

#### Baseline Algorithms

We conduct a comparative analysis of FedLoRU against several benchmarks: FedAvg, the standard federated learning algorithm that trains full-rank models; FedLoRA, which trains low-rank modules without accumulating low-rank updates; and FedHM, the state-of-the-art in communication-efficient federated learning. For evaluating pFedLoRU, we compare it with pFedLoRA, and for mFedLoRU, we use the system-heterogeneous version of FedHM as the comparison baseline.

#### Heterogeneous Settings

In the statistically heterogeneous setting, we generate disjoint Non-IID client data using a Dirichlet distribution, Dir($\alpha$), with a concentration parameter $\alpha$ set to 0.5, as described in Hsu et al., (). For the system heterogeneous setting, we simulate virtual environments where each client is assigned a different level of computational complexity, thereby restricting them to use low-rank update matrices of varying ranks. The specific configurations for these settings are detailed in Appendix C.

**Table 1: Top-1 test accuracy comparison with different communication-efficient federated learning methods under various FL settings.**
| Dataset | Data setting | #clients | FedAvg | FedLoRA | FedHM | FedLoRU |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| #params | acc | #params | acc | #params | acc | #params | acc |  |  |  |
| CIFAR-10 | IID | K=20 | 11.17M | 93.48 | 4.59M | 91.65 | 4.59M | 90.76 | 4.59M | 92.43 |
|  |  |  |  | 3.44M | 90.32 | 3.44M | 90.71 |  |  |  |
|  |  |  |  | 2.30M | 90.77 | 2.30M | 90.85 |  |  |  |
| K=100 | 11.17M | 85.14 | 4.59M | 79.48 | 4.59M | 81.41 | 4.59M | 81.36 |  |  |
|  |  |  |  | 3.44M | 81.58 | 3.44M | 86.01 |  |  |  |
|  |  |  |  | 2.30M | 82.12 | 2.30M | 86.10 |  |  |  |
| Non-IID | K=20 | 11.17M | 79.65 | 4.59M | 69.60 | 4.59M | 70.55 | 4.59M | 75.19 |  |
|  |  |  |  | 3.44M | 66.39 | 3.44M | 69.71 |  |  |  |
|  |  |  |  | 2.30M | 65.48 | 2.30M | 67.88 |  |  |  |
| CIFAR-100 | IID | K=20 | 11.22M | 69.97 | 4.63M | 65.53 | 4.63M | 59.43 | 4.63M | 66.81 |
|  |  |  |  | 3.49M | 58.40 | 3.49M | 60.78 |  |  |  |
|  |  |  |  | 2.35M | 58.52 | 2.35M | 61.42 |  |  |  |
| K=100 | 11.22M | 55.14 | 4.63M | 39.44 | 4.63M | 40.88 | 4.63M | 57.76 |  |  |
|  |  |  |  | 3.49M | 40.04 | 3.49M | 53.25 |  |  |  |
|  |  |  |  | 2.35M | 40.82 | 2.35M | 53.53 |  |  |  |
| Non-IID | K=20 | 11.22M | 19.18 | 4.63M | 14.41 | 4.63M | 16.88 | 4.63M | 16.46 |  |
|  |  |  |  | 3.49M | 15.04 | 3.49M | 13.70 |  |  |  |
|  |  |  |  | 2.35M | 15.13 | 2.35M | 14.52 |  |  |  |

### 4.2 Performance Evaluation

We evaluate the Top-1 accuracy of models with varying parameter sizes in both IID and Non-IID scenarios across different federated learning configurations.

#### Performance of Pre-training

Table 1 shows the performance of FedLoRU and baseline algorithms. In our experimental evaluation, FedLoRU consistently achieves competitive or superior accuracy compared to other algorithms, demonstrating its effectiveness in federated learning environments. While FedLoRU exhibits slightly lower performance compared to FedAvg in most settings, the performance degradation is minimal when considering the significantly reduced number of parameters. This indicates that FedLoRU’s low-rank update and accumulation strategy effectively reduces communication overhead without substantially sacrificing model performance. Moreover, FedLoRU surpasses other communication-efficient federated learning algorithms.

The client regularization effect of FedLoRU, as predicted by our theoretical analysis, suggests that using client-side low-rank updates is particularly beneficial in environments with a large number of clients. This benefit is evident in experiments under IID conditions with 100 clients, where FedLoRU attains the highest accuracy among the tested methods. Furthermore, under Non-IID settings, all methods experience considerable performance degradation compared to the IID setting; however, FedAvg still exhibits the best performance. However, as we will discuss in subsequent sections, the challenge of statistical heterogeneity can be addressed by utilizing personalized federated learning algorithms, and pFedLoRU surpasses the state-of-the-art personalized FL algorithms.

#### Performance of LLM Fine-tuning

Figure 3 presents the loss curves of FedLoRA and FedLoRU during the fine-tuning of a LLaMA2-3B model on the Alpaca dataset. The train loss curves show that both algorithms achieve similar convergence rates, with minimal differences in training optimization. However, a notable distinction emerges in the test loss results, where FedLoRU consistently outperforms FedLoRA after the 25th communication round.

Figure: Figure 3: Loss curve comparison of FedLoRU and FedLoRA for fine-tuning LLaMa2-3B on Alpaca dataset in rank=8 setting.

In this fine-tuning experiment, we accumulate the results every 15 communication rounds. Notably, despite FedLoRU performing an additional accumulation at round 30, the test loss does not show any further improvement. This suggests that beyond a certain point, further accumulation may not necessarily enhance the model’s generalization performance.

#### Scalability and Performance of FedLoRU in Large-Client Federated Learning

As the number of clients in federated learning increases, the scalability and performance of algorithms become critical. Our experiments reveal that FedAvg’s performance drops drastically as the number of clients grows—from a Top-1 accuracy of 69.97% at $K=20$ to just 21.44% at $K=400$. This sharp decline suggests that FedAvg struggles to maintain model accuracy in large-scale federated learning environments.

**Table 2: Comparison of FedAvg and FedLoRU with different K (number of clients) values and corresponding ratios.**
| K | FedAvg | FedLoRU | Ratio |
| --- | --- | --- | --- |
| 20 | 69.97 | 66.81 | -0.046 |
| 50 | 64.68 | 62.45 | -0.034 |
| 100 | 55.14 | 57.76 | 0.048 |
| 200 | 38.85 | 42.55 | 0.095 |
| 300 | 24.94 | 33.81 | 0.356 |
| 400 | 21.44 | 33.25 | 0.551 |

In contrast, FedLoRU outperforms FedAvg as the number of clients increases. While FedAvg slightly outperforms FedLoRU with smaller numbers of clients ($K=20$ and $K=50$), FedLoRU consistently surpasses FedAvg when the client count ranges from $K=100$ to $K=400$. Moreover, the performance gap widens with larger client numbers, indicating that FedLoRU scales more effectively with a higher number of clients—an essential feature for large–scale federated learning scenarios. To quantify the performance difference, we define the ratio:

$$ $\text{Ratio}=\frac{\text{FedLoRU}-\text{FedAvg}}{\text{FedLoRU}}$ $$

This ratio represents the relative decrease in accuracy when using FedAvg instead of FedLoRU, expressed as a proportion of FedLoRU’s accuracy. Higher ratio values with increasing $K$ highlight FedLoRU’s superior scalability and effectiveness in environments with extensive client participation.

#### Performance of pFedLoRU and mFedLoRU

In our experiments, we evaluate the performance of pFedLoRU and mFedLoRU on statistical heterogeneous and system heterogeneous FL environments. Table 3 shows the performance of pFedLoRU and pFedLoRA. Under both non-IID levels ($\alpha=0.1$ and $\alpha=0.5$), pFedLoRU shows a clear advantage in terms of accuracy compared to pFedLoRA. In addition, despite having less than half the number of parameters, pFedLoRU consistently achieves higher accuracy.

**Table 3: Comparison of pFedLoRA and pFedLoRU with varying non-iidness ($\alpha$) on CIFAR100 dataset.**
| Dataset | $\alpha$ | pFedLoRA(1) | pFedLoRA(2) | pFedLoRU |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| #params | acc | #params | acc | #params | acc |  |  |
| CIFAR100 | 0.1 | 11.22M | 45.36 | 11.22M | 47.45 | 4.63M | 49.65 |
| 0.5 | 11.22M | 42.14 | 11.22M | 42.28 | 4.63M | 46.50 |  |

**Table 4: Comparison of FedHM and mFedLoRU in two system-heterogeneous setting.**
| Dataset | Heterogeneous Setting | FedHM | mFedLoRU |
| --- | --- | --- | --- |
| CIFAR-10 | setting 1 | 88.09 | 84.81 |
| setting 2 | 88.68 | 84.36 |  |
| CIFAR-100 | setting 1 | 49.84 | 51.16 |
| setting 2 | 50.52 | 50.89 |  |

On the other hands, Table 4 shows the performanec of mFedLoRU and FedHM. FedHM outperforms mFedLoRU in both heterogeneous settings (setting 1 and setting 2) for the CIFAR-10 dataset, indicating that FedHM handles model heterogeneity more effectively for simpler tasks. This suggests that FedHM is better suited for less complex datasets such as CIFAR-10, where its approach proves more efficient. However, mFedLoRU outperforms FedHM in both heterogeneous settings for the more complex CIFAR-100 dataset, demonstrating its potential in addressing the model-heterogeneous problem in federated learning. A key advantage of mFedLoRU is that it does not require additional computational steps, such as the weight factorization used in FedHM, making it a more efficient solution in scenarios involving more challenging tasks.

## 5 Conclusion

In conclusion, FedLoRU demonstrates comparable performance to FedAvg while utilizing significantly fewer trainable and communicated parameters. This highlights its efficiency in reducing communication overhead without sacrificing accuracy. Furthermore, as the number of clients increases, FedLoRU consistently outperforms FedAvg, even though it operates with a much smaller parameter set. This scalability makes FedLoRU an effective solution for federated learning in large-scale environments, maintaining strong performance under more challenging conditions.

We also introduce two variants of the FedLoRU algorithm: pFedLoRU and mFedLoRU. These variants perform particularly well in heterogeneous environments, demonstrating their potential for broader applicability across various federated learning settings. However, a key limitation of FedLoRU lies in determining the optimal accumulation schedule for low-rank updates. The performance of the algorithm is highly dependent on the timing of these accumulations, and finding the best schedule can be challenging. Testing multiple accumulation schedules is not efficient, and thus, future research should focus on developing methods to optimize or automate this process.

## Appendix A Proof of the main theorem

In this section, we provide proofs of Proposition 3.1 Proposition 3.2 and Theorem 3.3. We begin by presenting some lemmas that will be required for our analysis, then proceed to prove the propositions and the theorem.

###### Lemma A.1 (Theorem 2.2 from Pielaszkiewicz and Singull, ( 2015 ) ) .

Let $\mu_{n}$ be a sequence of probability measures on $\mathbb{R}$ and let $g_{\mu_{n}}$ denote the Stieltjes transform of $\mu_{n}$. Then

- a)
if $\mu_{n}\to\mu$ weakly, where $\mu$ is a measure on $\mathbb{R}$, then $g_{\mu_{n}}(z)\to g_{\mu}(z)$ pointwise for any $z\in\{z:z\in\mathbb{C},\;\mathfrak{I}(z)>0\}$
- b)
if $g_{\mu_{n}}(z)\to g(z)$ pointwise, for all $z\in\{z:z\in\mathbb{C},\;\mathfrak{I}(z)>0\}$, then there exists a unique non-negative and finite measure such that $g=g_{\mu}$ and $\mu_{n}\to\mu$ weakly

###### Lemma A.2 (Theorem 3.4 from Baskerville et al., ( 2022 ) ) .

Let $X$ be an $N\times N$ real symmetric random matrix and let $D$ be an $N\times N$ symmetric matrix(deterministic or random). Let $\widehat{\mu}_{X}$, $\widehat{\mu}_{D}$ be the empirical spectral measures of the sequence of matrices $X,D$ and assume there exist deterministic limit measures $\mu_{X}$, $\mu_{D}$. Assume that $X$ has QUE and $\widehat{\mu}_{X}$ concentrates in the sense that

$$ $\mathbb{P}(W_{1}(\widehat{\mu}_{X},\mu_{X})>\delta)\leq e^{-N^{\tau}f(\delta)}$ $$

where $\tau>0$ and $f$ is some positive increasing function. Then $H=X+D$ has a limiting spectral measure and it is given by the free convolution $\mu_{X}\boxplus\mu_{D}$

###### Lemma A.3 (Weyl’s inequality) .

For Hermitian matrices $\bm{A},\bm{B}\in\mathbb{C}^{R\times n}$ and $i,j\in\{1,c\dots,n\}$,

$$ $\displaystyle\lambda_{i+j-1}(\bm{A}+\bm{B})\leq$ $\displaystyle\;\lambda_{i}(\bm{A})+\lambda_{j}(\bm{B}),\quad i+j\leq n+1,$ (16) $\displaystyle\lambda_{i+j-n}(\bm{A}+\bm{B})\geq$ $\displaystyle\;\lambda_{i}(\bm{A})+\lambda_{j}(\bm{B}),\quad i+j\geq n+1,$ (17) $$

where $\lambda_{i}(\bm{A})$ is $i$-th eigenvalue of $\bm{A}$.

###### Lemma A.1 (Theorem 2.2 from Pielaszkiewicz and Singull, ( 2015 ) ) .

###### Lemma A.2 (Theorem 3.4 from Baskerville et al., ( 2022 ) ) .

###### Lemma A.3 (Weyl’s inequality) .

### A.1 Proof of Proposition 3.1

Suppose $(h^{\tilde{R}},\omega^{\tilde{R}})\in\Omega^{\tilde{R}}(\theta_{1},\cdots,
\theta_{k})$, i.e., $\bm{H}_{\text{true}}(h^{\tilde{R}},\omega^{\tilde{R}})$ has non-zero eigenvalues $\theta_{1},\cdots,\theta_{k}$. To construct a prediction function $h^{\tilde{R}+1}$ and a weight $\omega^{\tilde{R}+1}$ of dimension $\tilde{R}+1$ such that the true Hessian retains the same non-zero eigenvalues, define:

$$ $\displaystyle h^{\tilde{R}+1}(\omega^{\tilde{R}},z)$ $\displaystyle=h^{\tilde{R}}(\omega^{\tilde{R}})+g^{\tilde{R}+1}(\omega^{\tilde {R}},z)$ (18) $\displaystyle\omega^{\tilde{R}+1}$ $\displaystyle=(\omega^{\tilde{R}},z)$ (19) $$

where $\nabla^{2}\int\ell(g^{\tilde{R}+1}(\bm{x};(\omega^{\tilde{R}},z)),\bm{y})d\psi
(\bm{x},\bm{y})=0$ ensures that the second derivative with respect to the new function $g^{\tilde{R}+1}$ vanishes. Thus, since $h^{\tilde{R}+1}$ and $h^{\tilde{R}}$ share the same true Hessian, it follows that $(h^{\tilde{R}+1},\omega^{\tilde{R}+1})\in\Omega^{\tilde{R}+1}(\theta_{1},
\cdots,\theta_{k})$. Specifically, if we consider feed-forward neural networks as prediction functions, one can easily construct a larger neural network that maintains the same non-zero eigenvalues by adding an additional neuron with a single connection to a neuron in the previous layer. This additional neuron does not affect the final output, thereby preserving the desired eigenvalue properties.

### A.2 Proof of Proposition 3.2

To prove Proposition 3.2, we decompose the eigenvalue analysis into two distinct parts. First, we demonstrate that the $i$-th eigenvalues, where $i\in\{p+1,\cdots,P-q-1\}$, converge to the upper or lower bounds of the spectral density of $\mu_{N}$. This portion of the proof parallels the approach employed by Benaych-Georges and Nadakuditi, (). Second, we show that the remaining eigenvalues converge to the Stieltjes transformation. This part of the proof follows the methodology outlined by Baskerville et al., ().

###### Proof.

In the proof, we drop dependency on $(h^{R},\omega^{R})$ since it is clear. First, consider $\lambda_{i}(\bm{H}_{N})$ where $p<i<R-q$. By using Lemma A.3, we have

$$ $\displaystyle\lambda_{i}(\bm{H}_{N})$ $\displaystyle\leq\lambda_{1+i-j}(\bm{H}_{\text{true}})+\lambda_{1+i-k}( \epsilon(N)),\;\quad i\leq R,\;i=j+k-1,\;j,k\in\{1,\cdots,R\}$ (20) $\displaystyle\lambda_{i}(\bm{H}_{N})$ $\displaystyle\leq\lambda_{R+i-j}(\bm{H}_{\text{true}})+\lambda_{R+i-k}( \epsilon(N)),\quad i\geq 1,\;i=j+k-R,\;j,k\in\{1,\cdots,R\}$ (21) $$

If we put $k=1+p$ on (20) and $k=R-q$ on (21), since $\lambda_{1+p}(\bm{H}_{\text{true}})=0$ and $\lambda_{R-q}(\bm{H}_{\text{true}})=0$, we deduce

$$ $\lambda_{i+q}(\epsilon(N))\leq\lambda_{i}(H_{N})\leq\lambda_{i-p}(\epsilon(N)) ,\quad\forall i\in\{1,\cdots,R\}$ (22) $$

where $\lambda_{k}(\epsilon(N))=-\infty$ if $k>R$ and $+\infty$ if $k\leq 0$. Additionally, since $\epsilon(N)$ has the limiting spectral density $\mu_{N}$ and $L_{N},U_{N}$ are lower and upper bound of $\mu_{N}$, we have for all $1\leq i\leq R$,

$$ $\displaystyle\liminf_{R\to\infty}\lambda_{i}(\epsilon(N))\geq U_{N}\;\;\;$ $\displaystyle\text{and}\;\;\;\limsup_{R\to\infty}\lambda_{R+1-i}(\epsilon(N)) \leq L_{N}$ (23) $\displaystyle\lambda_{1}(\epsilon(N))\to U_{N}\;\;\;$ $\displaystyle\text{and}\;\;\;\lambda_{P}(\epsilon(N))\to L_{N}$ (24) $$

From the above relations, it follows that for all fixed $1\leq i\leq R$, $\lambda_{i}(\epsilon(N))\to U_{N}$ and $\lambda_{R+1-i}(\epsilon(N))\to L_{N}$. By (22), we have

$$ $\liminf_{n\to\infty}\lambda_{i}(H_{N})\geq U_{N}\;\;\;\text{and}\;\;\;\limsup_ {n\to\infty}\lambda_{i}(H_{N})\leq L_{N}$ (25) $$

and for all $i>p$ (resp. $i\geq q)$ fixed, we have

$$ $\lambda_{i}(H_{N})\to U_{N}\;\;\;\;(\text{resp.}\;\lambda_{R-i}(H_{N})\to L_{N})$ (26) $$

Now, we are going to prove the remaining eigenvalues $\lambda_{i}(\bm{H}_{N})$, where $i\in\{1,\cdots,p,R-q+1,\cdots R\}$. Note that, since $p+q\ll R$ when $R$ is large enough, the limiting spectral density of $H_{true}$ converges to $\nu=\delta_{0}$.

Consider $\lambda_{i}(H_{N})$ where $i\leq p$ or $i\geq R-q$. By the Lemma A.2, the limiting spectral density $\mu_{H_{N}}$ of $H_{N}$ is $\mu_{N}\boxplus\nu$ where $\mu_{N}$ is the limiting spectral density of $\epsilon(N)$. Then by the Lemma A.1, the Stieltjes transform $g_{\mu_{H_{N}}}(z)$ converges pointwise to $g_{\nu\boxplus\mu_{N}}(z)$ for any $z\in\{z:z\in\mathbb{C},\mathfrak{I}(z)>0\}$. Therefore, we have

$$ $\displaystyle\begin{split}\widehat{g}_{H_{N}(h^{R},\omega^{R})}(z)&=g_{\mu_{H_ {N}}(h^{R},\omega^{R})}(z)+o(1)\\ &=g_{\mu_{N}(h^{R},\omega^{R})\boxplus\nu(h^{R},\omega^{R})}(z)+o(1)\\ &=g_{\nu(h^{R},\omega^{R})}(k(z))+o(1)\\ &=\widehat{g}_{H_{\text{true}}(h^{R},\omega^{R})}(k(z))+o(1)\end{split}$ (27) $$

where $k$ is the subordination function such that $g_{\mu_{N}\boxplus\nu}(z)=g_{\nu}(k(z))$.

Let $\lambda\in\mathbb{R}\backslash\text{supp}(\mu_{N}\boxplus\nu)$ be an eigenvalue of $H_{N}$. Then $\widehat{g}_{H_{N}}$ has a singularity at $\lambda$, thus $\widehat{g}_{H_{\text{true}}}$ has a singularity at $k(\lambda)$, thus, for any $R$, this singularity should persist and $k(\lambda)$ must coincide with one of the outliers of $H_{N}$, i.e., $\theta_{i}$ is an outlier eigenvalue of $H_{true}$ if and only if there exists an eigenvalue $\lambda$ of $H_{N}$ contained in $\mathbb{R}\backslash\text{supp}(\mu_{N}\boxplus\nu)$ such that $k(\lambda)=\theta_{i}$. Thus, we can write the outliers of $H_{N}$ as

$$ $\{k^{-1}(\theta_{j}):k^{-1}(\theta_{j})\in\mathbb{R}\backslash\text{supp}(\mu_ {N}\boxplus\nu)\}$ (28) $$

Note that $\text{supp}(\mu_{N}\boxplus\nu)=\text{supp}(\mu_{N}\boxplus\delta_{0})=\text{
supp}(\mu_{N})$. Now, we want to find the form of $k^{-1}(\theta_{j})$. From the subordination function relation, we have

$$ $\displaystyle\begin{split}k^{-1}(\theta)&=g^{-1}_{\mu_{N}\boxplus\nu}(g_{\nu}( \theta))\\ &=\mathcal{R}_{\mu_{N}}(g_{\nu}(\theta)+g_{\nu}^{-1}(g_{\nu}(\theta))\\ &=\mathcal{R}_{\mu_{N}}\left(1/\theta\right)+\theta\end{split}$ (29) $$

Note that by the definition of Stieltjes transformation and $\mathcal{R}$-transform $g_{\nu}(\theta)=g_{\delta_{0}}(\theta)=1/\theta$.

Let $m_{n}^{(\mu)}$ be the $n$-th moment of a distribution $\mu$ and $C_{n}^{(\mu)}$ be the $n$-th cumulant of $\mu$. Then we have the relationship between $m_{n}^{(\mu)}$ and $C_{n}^{(\mu)}$ ([3]) as

$$ $m_{n}^{(\mu)}=\sum_{r=1}^{n}\sum_{\begin{subarray}{c}0\leq i_{1},\cdots,i_{r} \leq n-r\\ i_{1}+\cdots+i_{r}=n-r\end{subarray}}C_{r}^{(\mu)}\;\left[\Pi_{j=1}^{r}m_{i_{j }}^{(\mu)}\right]$ (30) $$

Therefore, from the moment’s scaling property, $m_{n}^{\mu_{N}}=s(N)^{n}m_{n}^{\mu}$, we can deduce the scaling relation property of the cumulants, $C_{n}^{(\mu_{N})}=s(N)^{n}C_{n}^{(\mu)}$, therefore we have the scaling property of $\mathcal{R}$-transform:

$$ $\mathcal{R}_{\mu_{N}}(\theta)=s(N)\mathcal{R}_{\mu}(s(N)\theta)$ (31) $$

Finally, we have a expression for the outliers of $H_{N}$ as

$$ $k^{-1}(\theta)=s(N)\mathcal{R}_{\mu}(s(N)/\theta)+\theta$ (32) $$

∎

###### Proof.

### A.3 Proof of Theorem 3.3

###### Proof.

Suppose $\alpha$ and $\beta$ be the size of sets $\{i>p:\lambda_{i}(H_{N})\to U_{N}\}$ and $\{i\geq q:\lambda_{R-i}(H_{N})\to L_{N}\}$ respectively, and $a_{N}$ and $b_{N}$ be integers such that

$$ $\displaystyle g^{-1}_{N}(\theta_{a_{N}})>$ $\displaystyle U_{N}>g^{-1}_{N}(\theta_{a_{N}+1})$ $\displaystyle g^{-1}_{N}(\theta_{p+q-b_{N}})>$ $\displaystyle L_{N}>g^{-1}_{N}(\theta_{p+q-b_{N}+1})$ $$

and we can define $a_{M}$ and $b_{M}$ in the similar manner. Then we have

$$ $\displaystyle\overbrace{\theta_{1}>\cdots>\theta_{a_{N}}}^{a_{N}}>\overbrace{ \theta_{a_{N}+1}>\cdots>\theta_{p}}^{p-a_{N}}>0>\overbrace{\theta_{p+1}>\cdots >\theta_{p+q-b_{N}}}^{q-b_{N}}>\overbrace{\theta_{p+q-b_{N}+1}>\cdots>\theta_{ p+q}}^{b_{N}}$ (33) $\displaystyle\overbrace{\theta_{1}>\cdots>\theta_{a_{M}}}^{a_{M}}>\overbrace{ \theta_{a_{M}+1}>\cdots>\theta_{p}}^{p-a_{M}}>0>\overbrace{\theta_{p+1}>\cdots >\theta_{p+q-b_{M}}}^{q-b_{M}}>\overbrace{\theta_{p+q-b_{M}+1}>\cdots>\theta_{ p+q}}^{b_{M}}$ (34) $$

WLOG, we can assume $\|\theta_{1}\|>\|\theta_{p+q}\|$ and define $g^{-1}_{M}(\theta_{j})=\theta_{j}+s(M)\mathcal{R}_{\mu}(s(M)\theta_{j}^{-1})$. We will consider the limiting stable rank of the Hessians. From Proposition 3.2, the stable ranks of Hessians converges to the limiting stable ranks $\hat{\text{srank}}(H_{N})$ and $\hat{\text{srank}}(H_{M})$. Since $a_{N}>a_{M}$ and $b_{N}<b_{M}$, we can express the difference of the limiting stable rank of $H_{N}$ and $H_{M}$ as

$$ $\displaystyle\begin{split}&\hat{\text{srank}}(H_{M})-\hat{\text{srank}}(H_{N}) \\ =&\;\sum_{j=2}^{a_{M}}\left\{\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M}( \theta_{1})}\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}(\theta_ {1})}\right)^{2}\right\}+\sum_{j=a_{M}+1}^{a_{N}}\left\{\left(\frac{U_{M}}{g^{ -1}_{M}(\theta_{1})}\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N} (\theta_{1})}\right)^{2}\right\}+\\ &\sum_{j=a_{N}+1}^{p+\alpha}\left\{\left(\frac{U_{M}}{g^{-1}_{M}(\theta_{1})} \right)^{2}-\left(\frac{U_{N}}{g^{-1}_{N}(\theta_{1})}\right)^{2}\right\}+\sum _{j=1}^{b_{M}}\left\{\left(\frac{g^{-1}_{M}(\theta_{p+q+1-j})}{g^{-1}_{M}( \theta_{1})}\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{p+q+1-j})}{g^{-1}_{N}( \theta_{1})}\right)^{2}\right\}+\\ &\sum_{j=b_{M}+1}^{b_{N}}\left\{\left(\frac{L_{M}}{g^{-1}_{M}(\theta_{1})} \right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{p+q+1-j})}{g^{-1}_{N}(\theta_{1})} \right)^{2}\right\}+\sum_{j=b_{N}+1}^{q+\beta}\left\{\left(\frac{L_{M}}{g^{-1} _{M}(\theta_{1})}\right)^{2}-\left(\frac{L_{N}}{g^{-1}_{N}(\theta_{1})}\right) ^{2}\right\}\end{split}$ (35) $$

We have six summation terms in (35) and will show each term is negative.

(i) Consider the first term in (35):

$$ $S_{1}=\sum_{j=2}^{a_{M}}\left\{\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M}( \theta_{1})}\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}(\theta_ {1})}\right)^{2}\right\}$ $$

We will show each term $F_{j}=\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M}(\theta_{1})}\right)^{2}-
\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}(\theta_{1})}\right)^{2}$ in the summation is negative. We can expand $F_{j}$ as follow:

$$ $\displaystyle\begin{split}F_{j}&=\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M }(\theta_{1})}\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}( \theta_{1})}\right)^{2}\\ &=\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M}(\theta_{1})}+\frac{g^{-1}_{N} (\theta_{j})}{g^{-1}_{N}(\theta_{1})}\right)\left(\frac{g^{-1}_{M}(\theta_{j}) }{g^{-1}_{M}(\theta_{1})}-\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}(\theta_{1}) }\right)\\ &=\left(\frac{g^{-1}_{M}(\theta_{j})}{g^{-1}_{M}(\theta_{1})}+\frac{g^{-1}_{N} (\theta_{j})}{g^{-1}_{N}(\theta_{1})}\right)\left(\frac{g^{-1}_{M}(\theta_{j}) g^{-1}_{N}(\theta_{1})-g^{-1}_{N}(\theta_{j})g^{-1}_{M}(\theta_{1})}{g^{-1}_{M }(\theta_{1})g^{-1}_{N}(\theta_{1})}\right)\end{split}$ (36) $$

To verify the sign of $F_{j}$, we have to focus on the numerator of the second part of multiplicative term. We can simplify the numerator part as

$$ $\displaystyle g^{-1}_{M}(\theta_{j})g^{-1}_{N}(\theta_{1})-g^{-1}_{N}(\theta_{ j})g^{-1}_{M}(\theta_{1})$ $\displaystyle\;\;\;\;\;\;\;=\theta_{1}\left\{s(N)\mathcal{R}_{\mu}(s(N)\theta_ {j}^{-1})-s(M)\mathcal{R}_{\mu}(s(M)\theta_{j}^{-1})\right\}+\theta_{j}\left\{ s(M)\mathcal{R}_{\mu}(s(M)\theta_{1}^{-1})-s(N)\mathcal{R}_{\mu}(s(N)\theta_{1 }^{-1})\right\}$ $\displaystyle\quad\quad\;\;\;\;+s(N)s(M)\left\{\mathcal{R}_{\mu}(s(M)\theta_{1 }^{-1})\mathcal{R}_{\mu}(s(N)\theta_{j}^{-1})-\mathcal{R}_{\mu}(s(N)\theta_{1} ^{-1})\mathcal{R}_{\mu}(s(M)\theta_{j}^{-1})\right\}$ $$

Consider the term:

$$ $F_{j,1}=\theta_{1}\left\{s(N)\mathcal{R}_{\mu}(s(N)\theta_{j}^{-1})-s(M) \mathcal{R}_{\mu}(s(M)\theta_{j}^{-1})\right\}+\theta_{j}\left\{s(M)\mathcal{R }_{\mu}(s(M)\theta_{1}^{-1})-s(N)\mathcal{R}_{\mu}(s(N)\theta_{1}^{-1})\right\}$ $$

We know the $R$-transform can be expressed as power series as

$$ $\mathcal{R}_{\mu}(s(N)\theta^{-1})=\sum_{n=1}^{\infty}C_{n}^{(\mu)}\left(\frac {s(N)}{\theta}\right)^{n-1}$ $$

where $C_{n}^{(\mu)}$ is the $n$-th cumulant of $\mu$. Then we can calculate $F_{j,1}$ as

$$ $\displaystyle F_{j,1}$ $\displaystyle=\sum_{n=1}^{\infty}C_{n}^{(\mu)}(s(N)^{n}-s(M)^{n})\;\theta_{1} \cdot(1/\theta_{j})^{n-1}-\sum_{n=1}^{\infty}C_{n}^{(\mu)}(s(N)^{n}-s(M)^{n}) \;\theta_{j}\cdot(1/\theta_{1})^{n-1}$ $\displaystyle=\sum_{n=1}^{\infty}C_{n}^{(\mu)}(s(N)^{n}-s(M)^{n})\left(\theta_ {1}\cdot\left(\frac{1}{\theta_{j}}\right)^{n-1}-\theta_{j}\cdot\left(\frac{1}{ \theta_{1}}\right)^{n-1}\right)$ $$

Since $\theta_{1}>\theta_{j}$ and $s(N)<s(M)$, we can easily show that $F_{j,1}$ is negative.
Next, consider the term:

$$ $F_{j,2}=\mathcal{R}_{\mu}(s(N)\theta_{j}^{-1})\mathcal{R}_{\mu}(s(M)\theta_{1} ^{-1})-\mathcal{R}_{\mu}(s(M)\theta_{j}^{-1})\mathcal{R}_{\mu}(s(N)\theta_{1}^ {-1})$ $$

By using the power series expression of $\mathcal{R}$-Transform, we have

$$ $F_{j,2}=\sum_{n=1}^{\infty}C_{n}^{(\mu)}\left(\frac{s(M)}{\theta_{1}}\right)^{ n-1}\sum_{n=1}^{\infty}C_{n}^{(\mu)}\left(\frac{s(N)}{\theta_{j}}\right)^{n-1} -\sum_{n=1}^{\infty}C_{n}^{(\mu)}\left(\frac{s(N)}{\theta_{1}}\right)^{n-1} \sum_{n=1}^{\infty}C_{n}^{(\mu)}\left(\frac{s(M)}{\theta_{j}}\right)^{n-1}$ $$

We know that when $\sum_{1}^{\infty}a_{n}$ and $\sum_{1}^{\infty}b_{N}$ converges, then

$$ $\sum_{1}^{\infty}a_{n}\sum_{1}^{\infty}b_{n}=\sum_{n=1}^{\infty}\sum_{k=1}^{n} a_{k}b_{n-1+1}$ $$

Therefore, we have

$$ $\displaystyle F_{j,2}$ $\displaystyle=\sum_{n=1}^{\infty}\sum_{k=1}^{n}C_{k}^{(\mu)}C_{n-k+1}^{(\mu)} \left(\frac{s(N)}{\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}}\right) ^{n-k}$ $\displaystyle-\sum_{n=1}^{\infty}\sum_{k=1}^{n}C_{k}^{(\mu)}C_{n-k+1}^{(\mu)} \left(\frac{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{\theta_{1}}\right) ^{n-k}$ $\displaystyle=\sum_{n=1}^{\infty}\sum_{k=1}^{n}C_{k}^{(\mu)}C_{n-k+1}^{(\mu)} \left\{\left(\frac{s(N)}{\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}} \right)^{n-k}-\left(\frac{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{ \theta_{1}}\right)^{n-k}\right\}$ $$

If we let $T(n)=\sum_{k=1}^{n}C_{k}^{(\mu)}C_{n-k+1}^{(\mu)}\left\{\left(\frac{s(N)}{
\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}}\right)^{n-k}-\left(\frac
{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{\theta_{1}}\right)^{n-k}\right\}$, we can write $F_{j,2}=\sum_{n=1}^{\infty}T(n)$. We will show $F_{j,2}$ is negative by showing each $T(n)$ is negative for $n=2m$ and $n=2m+1$, where $m\in\mathbb{N}$. For $n=2m$ ($m\in\mathbb{N}$),

$$ $\displaystyle T(n)=T(2m)$ $\displaystyle=\sum_{k=1}^{2m}C_{k}^{(\mu)}C_{2m-k+1}^{(\mu)}\left\{\left(\frac {s(N)}{\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}}\right)^{2m-k}- \left(\frac{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{\theta_{1}}\right) ^{2m-k}\right\}$ $\displaystyle=\sum_{k=1}^{m}\Biggl{[}C_{k}^{(\mu)}C_{2m-k+1}^{(\mu)}\left\{ \left(\frac{s(N)}{\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}}\right) ^{2m-k}-\left(\frac{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{\theta_{1} }\right)^{2m-k}\right\}$ $\displaystyle\;\;\;\;\;\;\;\;-C_{2m-k+1}^{(\mu)}C_{k}^{(\mu)}\left\{\left( \frac{s(N)}{\theta_{j}}\right)^{2m-k}\left(\frac{s(M)}{\theta_{1}}\right)^{k-1 }-\left(\frac{s(M)}{\theta_{j}}\right)^{2m-k}\left(\frac{s(N)}{\theta_{1}} \right)^{k-1}\right\}\Biggr{]}$ $\displaystyle=\sum_{k=1}^{m}C_{k}^{(\mu)}C_{2m-k+1}^{(\mu)}\left(s(N)^{k-1}s(M )^{2m-k}-s(M)^{k-1}s(N)^{2m-k}\right)\left(\frac{1}{\theta_{j}^{k-1}\theta_{1} ^{2m-k}}-\frac{1}{\theta_{j}^{2m-k}\theta_{1}^{k-1}}\right)$ $$

We can easily show two conditions:

$$ $\displaystyle s(N)^{k-1}s(M)^{2m-k}-s(M)^{k-1}s(N)^{2m-k}>0$ $\displaystyle\iff 2m-2k+1>0$ $\displaystyle\frac{1}{\theta_{j}^{k-1}\theta_{1}^{2m-k}}-\frac{1}{\theta_{j}^{ 2m-k}\theta_{1}^{k-1}}>0$ $\displaystyle\iff 2m-2k+1<0$ $$

Therefore, we can deduce $T(2m)$ is negative.

For $n=2m+1$ ($m\in\mathbb{N}$),

$$ $\displaystyle T(n)=T(2m+1)$ $\displaystyle=\sum_{k=1}^{2m+1}C_{k}^{(\mu)}C_{2m-k+2}^{(\mu)}\left\{\left( \frac{s(N)}{\theta_{j}}\right)^{k-1}\left(\frac{s(M)}{\theta_{1}}\right)^{2m-k +1}-\left(\frac{s(M)}{\theta_{j}}\right)^{k-1}\left(\frac{s(N)}{\theta_{1}} \right)^{2m-k+1}\right\}$ $$

$(m+1)$-th term of $T(2m+1)$ is zero since it is symmetric, thus we can write $T(2m+1)$ as

$$ $T(2m+1)=\sum_{k=1}^{m}C_{k}^{(\mu)}C_{2m-k+2}^{(\mu)}\left(s(N)^{k-1}s(M)^{2m- k+1}-s(M)^{k-1}s(N)^{2m-k+1}\right)\left(\frac{1}{\theta_{j}^{k-1}\theta_{1}^{ 2m-k+1}}-\frac{1}{\theta_{j}^{2m-k+1}\theta_{1}^{k-1}}\right)$ $$

We can show $T(2m+1)$ is negative as similar way of $T(2m)$. Therefore, $F_{j}$ is negative.

(ii) Consider the second term in (35):

$$ $S_{2}=\sum_{j=a_{M}+1}^{a_{N}}\left\{\left(\frac{U_{M}}{g^{-1}_{M}(\theta_{1}) }\right)^{2}-\left(\frac{g^{-1}_{N}(\theta_{j})}{g^{-1}_{N}(\theta_{1})}\right )^{2}\right\}$ $$

We will show $S_{2}$ is negative by showing each term in the summation $G_{j}=\frac{U_{M}^{2}}{\left\{\theta_{1}+s(M)\mathcal{R}_{\mu}(s(M)\theta_{1}^
{-1}\right\}^{2}}-\left\{\frac{\theta_{j}+s(N)\mathcal{R}_{\mu}(s(N)\theta_{j}
^{-1})}{\theta_{1}+s(N)\mathcal{R}_{\mu}(s(N)\theta_{1}^{-1})}\right\}^{2}$ is negative. Since $\theta_{j}+s(M)\mathcal{R}_{\mu}(S(M)\theta_{j}^{-1})>U_{M}$ for all $j\in\{a_{M}+1,\cdots,a_{N}\}$, we have $G_{j}<F_{j}<0$. Therefore, $S_{2}$ is negative.

(iii) Consider the third term in (35):

$$ $\displaystyle S_{3}$ $\displaystyle=\sum_{j=a_{N}+1}^{p+\alpha}\left\{\left(\frac{U_{M}}{g^{-1}_{M}( \theta_{1})}\right)^{2}-\left(\frac{U_{N}}{g^{-1}_{N}(\theta_{1})}\right)^{2}\right\}$ $\displaystyle=(p+\alpha-a_{N}-1)\left(\frac{U_{M}^{2}}{\left\{\theta_{1}+s(M) \mathcal{R}_{\mu}(s(M)\theta_{1}^{-1}\right\}^{2}}-\frac{U_{N}^{2}}{\left\{ \theta_{1}+s(N)\mathcal{R}_{\mu}(s(N)\theta_{1}^{-1}\right\}^{2}}\right)$ $$

By using the fact $U_{N}/s(N)=U_{M}/s(M)$, we can write the above term as

$$ $\displaystyle S_{3}$ $\displaystyle=(p+\alpha-a_{N}-1)\left(\frac{U_{M}^{2}}{\left\{\theta_{1}+s(M) \mathcal{R}_{\mu}(s(M)\theta_{1}^{-1}\right\}^{2}}-\frac{U_{M}^{2}}{\left\{ \theta_{1}+s(N)\mathcal{R}_{\mu}(s(N)\theta_{1}^{-1}\right\}^{2}}\frac{s(N)^{2 }}{s(M)^{2}}\right)$ $$

By using the power series expansion of $\mathcal{R}$-Transform, we can easily show that $Q=0$. For fourth, fifth, and sixth terms in (35), they are negative in similar way of (i), (ii), and (iii). Therefore, $\hat{\text{srank}}(H_{M})-\hat{\text{srank}}(H_{N})$ is negative.

∎

###### Proof.

## Appendix B Detail of the algorithms

### B.1 Personalized Federated Low-Rank Updates (pFedLoRU)

Figure: Algorithm 2 pFedLoRU. $\bm{W}$ is a model, $\bm{A}_{0},\bm{B}_{0}$ are initial global low-rank update matrices, $\bm{L}_{0},\bm{U}_{0}$ are initial personal low-rank update matrices, $\alpha_{\text{global}}$, $\alpha_{\text{per}}$ are the scaling factors, $\tau$ is an accumulation cycle, $T$ is the total training round

The pFedLoRU algorithm enables each client $k$ to train a personalized model adapted to its data distribution. In pFedLoRU, client $k$ retains global low-rank update matrices $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$ for updating the shared model, as well as personalized low-rank update matrices $\bm{L}^{(k)}$ and $\bm{U}^{(k)}$ for learning the personalized model. The communication between the server and clients involves only the low-rank matrices $\bm{A}$ and $\bm{B}$, which substantially reduces communication overhead. These matrices, $\bm{A}$ and $\bm{B}$, are aggregated to update the global model $\bm{W}^{(k)}$. Finally, each client possesses a personalized model of the form $\bm{W}^{(k)}+\bm{L}^{(k)}\bm{U}^{(k)}$.

In practice, since the global model incorporates general knowledge from the all clients’ dataset, and the personalized model is essentially a fine-tuned version of the global model, we typically assign higher ranks to $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$. Additionally, although we use the same rank for $\bm{L}^{(k)}$ and $\bm{U}^{(k)}$ across all clients in our experiments, each client can, in practice, use different ranks based on the complexity and size of their local dataset. It is also noteworthy that different ranks for $\bm{A}^{(k)}$ and $\bm{B}^{(k)}$ can be employed by integrating pFedLoRU and mFedLoRU.

### B.2 Model-Heterogeneous Federated Low-Rank Updates (mFedLoRU)

Figure: Algorithm 3 mFedLoRU. $\bm{W}$ is a model, $\bm{A}_{0},\bm{B}_{0}$ are initial low-rank update matrices, $\alpha,\alpha^{(k)}_{A},\alpha^{(k)}_{B}$ are scaling factors, $\tau$ is an accumulation cycle, $T$ is the total training round

The mFedLoRU algorithm enables each client $k$ to utilize a rank tailored to its resource constraints. Similar to FedLoRU, client $k$ maintains low-rank update matrices $\bm{A}^{(k)}\in\mathbb{R}^{m\times r}$ and $\bm{B}^{(k)}\in\mathbb{R}^{r\times n}$, but employs recursive low-rank updates during training. First, client $k$ determines personal update ranks $r_{A}^{(k)},r_{B}^{(k)}<r$ based on its resources. At each round, it initializes nested low-rank update matrices $\bm{A}_{\text{d}}^{(k)}\in\mathbb{R}^{m\times r_{A}^{(k)}}$, $\bm{A}_{\text{u}}^{(k)}\in\mathbb{R}^{r_{A}^{(k)}\times r}$ and $\bm{B}_{\text{d}}^{(k)}\in\mathbb{R}^{r\times r_{B}^{(k)}}$, $\bm{B}_{\text{u}}^{(k)}\in\mathbb{R}^{r_{B}^{(k)}\times n}$ such that $\bm{A}_{\text{d}}^{(k)}\bm{A}_{\text{u}}^{(k)}=0$ and $\bm{B}_{\text{d}}^{(k)}\bm{B}_{\text{u}}^{(k)}=0$. After local training by solving (15), we can update client $k$’s original low-rank matrices as follows:

$$ $\bm{A}^{(k)}\leftarrow\bm{A}^{(k)}+\alpha_{A}^{(k)}\bm{A}_{\text{d}}^{(k)}\bm{ A}_{\text{u}}^{(k)},\quad\bm{B}^{(k)}\leftarrow\bm{B}^{(k)}+\alpha_{A}^{(k)} \bm{B}_{\text{d}}^{(k)}\bm{B}_{\text{u}}^{(k)}$ (37) $$

In mFedLoRU, to reduce communication overhead, the client does not recover its original low-rank matrices directly. Instead, it sends the nested low-rank matrices to the server, and the server recovers them. By using this strategy, the communication overhead is reduced from $2mn$ to $r(m+n)+r_{A}(m+r)+r_{B}(n+r)$.

### B.3 Personalized Federated Low-Rank Adaptation (pFedLoRA)

Figure: Algorithm 4 pFedLoRA. $\bm{W}$ is a model, $\bm{L}_{0},\bm{U}_{0}$ are initial personal low-rank update matrices, $\alpha_{\text{per}}$ is the scaling factor, $T$ is the total training round.

We outline two variants of the personalized FedLoRA algorithm here. Both versions of pFedLoRA follow a similar framework, where each client maintains a full-rank global model $\bm{W}$ and its own personalization modules $\bm{L}^{(k)}$ and $\bm{U}^{(k)}$.

In pFedLoRA(1), the first variant, as suggested by Wu et al., () and other FedLoRA algorithms, the personalization modules are optimized separately from the global model. Specifically, the algorithm first optimizes the personalization modules for $E_{\text{per}}$ and subsequently optimizes the global full-rank model for $E_{\text{global}}$ by solving:

$$ $\displaystyle\bm{L}_{t}^{(k)},\bm{U}_{t}^{(k)}$ $\displaystyle=\operatorname*{arg\,min}_{\bm{L},\bm{U}}f^{(k)}(\bm{W}^{(k)}+ \alpha_{\text{per}}\bm{L}\bm{U})$ (38) $\displaystyle\bm{W}_{t}^{(k)}$ $\displaystyle=\operatorname*{arg\,min}_{\bm{W}}f^{(k)}(\bm{W}+\alpha_{\text{ per}}\bm{L}_{t}^{(k)}\bm{U}_{t}^{(k)})$ (39) $$

However, pFedLoRA(1) has been found to be less effective compared to our modified version pFedLoRA(2). The second variant, pFedLoRA(2), optimizes both the personalization modules and the global full-rank model simultaneously for $E=E_{\text{per}}+E_{\text{global}}$ by solving:

$$ $\bm{W}_{t}^{(k)},\bm{L}_{t}^{(k)},\bm{U}_{t}^{(k)}=\operatorname*{arg\,min}_{ \bm{W},\bm{L},\bm{U}}f^{(k)}(\bm{W}+\alpha_{\text{per}}\bm{L}\bm{U})$ (40) $$

## Appendix C Detail of the experiment setting

In this section, we provide a detailed explanation of the experiments, including the datasets and hyperparameters used. The implementation is based on PyTorch.

### C.1 Detail of the implementation in experiments

#### Detailed implementation of FedLoRU

In FedLoRU, FedHM, and their variant algorithms, we generate factorized low-rank modules for convolutional layers only. We utilize several low-rank environments where the rank of the factorized low-rank modules are 16, 32, 64, 128. Further, we use $\alpha=16$ for all experiments as it turns out to have consistent results for $\alpha\in\{16,32,64\}$.

#### Federated learning setting

The federated learning experiments were conducted using three datasets: CIFAR-10, CIFAR-100, and Alpaca. The client sampling rate, representing the proportion of clients selected per communication round, was set at 0.5 for all datasets. Each client performed 5 local epochs per communication round on CIFAR-10 and CIFAR-100 with a batch size of 32, while client performed 1 local epochs on Alpaca with a batch size of 16. The optimizer used for training was stochastic gradient descent (SGD) with a momentum of 0.9 on CIFAR-10 and CIFAR-100, while AdamW (Loshchilov,, ). We find the learning rate using grid search and use Cosine-Annealing as scheduler, and the cycle step is set to 50 or the total communication round.

#### Model heterogeneous setting

In this subsection, we describe the system heterogeneous settings used in our experiments. To simulate varying client capabilities, we tested two different system heterogeneous configurations in mFedLoRU experiments where the clients had different ranks, denoted as $r$, which reflect the computational resources or constraints of each client. For FedHM, we match the number of trainable parameters corresponding to the model with specific rank in mFedLoRU experiments.

**Table A1: Detailed system heterogeneous settings in our experiments. Both settings include total 20 clients.**
| Rank of a client | $r=128$ | $r=64$ | $r=32$ | $r=16$ |  |
| --- | --- | --- | --- | --- | --- |
| #Clients | setting 1 | 5 | 5 | 5 | 5 |
| setting 2 | - | 6 | 6 | 7 |  |

The motivation behind these settings was to create a challenging system heterogeneous environment. This is particularly important since we observed that FedLoRU with $r=128$ produces similar results to FedAvg with a full-rank model. Hence, these settings were designed to push the algorithm’s adaptability under harsher, more diverse client conditions.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: Q1. The paper presents FedLoRU and its variants by applying low-rank updates in a federated learning setting. However, the novelty of this proposed method is limited. The idea of using low-rank updates in federated learning has been explored before, and the paper does not provide a compelling argument for why the proposed method outperforms existing approaches. 

Q2. While the paper utilizes the stable rank metric to analyze rank properties between local clients and the central server, the discussion around this metric is lacking. The claim that stable rank "serves as a continuous proxy for rank and is robust" is made without sufficient references or supporting literature. Additionally, more discussion is needed on how this concept is adapted from related fields, and why it is appropriate for the federated learning context.

Q3. Figure 2(a) is difficult to interpret. Both the datasets with 50 and 500 samples show a high stable rank at the 15th epoch, which is counterintuitive and requires further explanation. It would strengthen the paper if the authors could repeat the experiment multiple times and provide clearer insights to support the observed trends.

Q4. The experiment shown in Figure 2(b) does not convincingly support the authors' intuition without a more detailed description. A thorough explanation of the experimental setup and its relation to Theorem 3.2 would significantly improve the clarity and impact of the results.
soundness: 2
strengths: FedLoRU uses successive low-rank updates for both pre-training and fine-tuning in federated learning and achieves good performance.
summary: The paper applies FedLoRU and its variants to impose the local update in a low-rank subspace to achieve implicit regularization.
weaknesses: W1. The novelty is not justified sufficiently. 

W2. More discussions and justifications regarding the stable rank metric are needed.

W3. The experiment setup and results are not convincing.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 2
questions: See in weaknesses.
soundness: 2
strengths: 1. The proposed method is well-motivated, the paper investigates the rank properties of client and server losses, analytically showing that under stochastic sampling, the rank of the Hessian of the loss function increases with smaller sample sizes.
2. The empirical results show empirical evidence of the higher rank structure of client losses and demonstrate that restricting the rank of local updates aids in implicit regularization.
summary: To address the issue of communication efficiency and heterogeneity in Federated Learning, this paper proposes the FedLoRU method. This general low-rank update framework enforces low-rank client-side updates and accumulates these updates to form a higher-rank model. The authors provide empirical results to demonstrate that FedLoRU performs better than other algorithms.
weaknesses: 1. In the theorems that are presented, summarizing the main insights of these theorems may be needed since currently they are just written as long paragraphs.
2. In experiments, the least partial client participation ratio is set as 0.5. In more realistic settings, the participation ratio is lower with more clients.
3. The author should consider more baselines, which apply low-rank factorized update models, such as [1].
[1] Nam Hyeon-Woo, Moon Ye-Bin, Tae-Hyun Oh. FedPara: Low-rank Hadamard Product for Communication-Efficient Federated Learning. ICLR 2022.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: Please refer to the limitation.
soundness: 2
strengths: This paper reveals that client loss in federated learning has a higher rank structure (in gradients and Hessian subspaces) than the server's loss.
 Based on this, they propose that restricting client optimization to a low-rank subspace could provide implicit regularization. They then introduce FedLoRU, a framework that enforces low-rank updates on the client side and aggregates them into a higher-rank model. Finally, they add another low-rank module pair to adapt to environments with statistical and model heterogeneity.
summary: This paper reveals that client loss in federated learning has a higher rank structure (in gradients and Hessian subspaces) than the server's loss. Based on this, they propose that restricting client optimization to a low-rank subspace could provide implicit regularization and then introduce FedLoRU, a framework that enforces low-rank updates on the client side and aggregates them into a higher-rank model. Finally, they add another low-rank module pair to adapt to environments with statistical and model heterogeneity.
weaknesses: The novelty is limited, there is no close connectiong between the analysis and the algorithm. I think this algorithm is a federated version of ReLoRA if we consider on the non-personalized version, aggregating low-rank modules for higher rank training. 

There is no theoretical analysis for the algorithm. It's fully heuristic. When we consider the personalized strategy this paper studied, I don't know what kind of solution will this algoritm converge to. Will the introduced L, U fully concel out the A, B modules and make this algorithm fully consider local loss? The author didn't provide the reasonability of their strategy. 

According to my understanding, this algorithm is still a full parameter training algorithm as it initializes W every $\tau$ step. So the comparison to LoRA is unfair. On the other hand, there are numerous algorithms for conventional federated learning. If you want to highlight your algorithm's advantage, you should compare your algorithm with the conventional algorithm, rather than just beatting LoRA. 

You can't accurately solve argmin_{A,B} f. This step is computation-heavy even if you use an \epsilon-approixition. This step is actually one step of full LoRA tuning.  Therefore, this algorithm is not suitable for LLM fine-tuning.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: * The title mentions "its connection to implicit regularization". To my knowledge, implicit regularization refers to the phenomenon that optimizers without explicit regularization, such as SGD, prefer regularized solutions [1]. However, FedLoRU explicitly works in a specific rank-$r$ space. Could the authors please explain in what sense is FedLoRU connected to implicit regularization?
* The theory part analyzes the rank structures of *loss Hessians* at server and client side. At the same time, FedLoRU proposes to perform low-rank updates on the model's *weight matrices*. Could the authors please explain the connection between the rank structure of loss Hessians and weight matrices?

[1] Ziwei Ji and Matus Telgarsky. Gradient descent aligns the layers of deep linear networks. arXiv preprint arXiv:1810.02032, 2018.
soundness: 2
strengths: * The paper provides rigorous theoretical analysis on the Hessian rank structures, establishing interesting asymptotic results within a mathematically general framework.
* The proposed algorithm achieves performance comparable or superior to other known methods in experiments, while significantly reducing the communication overhead by low-rank updates.
* The presentation of this paper is well-organized and the motivation and methodology are clear to follow.
summary: This paper studies communication-efficient low-rank update framework for federated learning. 

It provides theoretical asymptotic analysis for the rank structures of the Hessian at server side and client side, which motivates the design of FedLoRU algorithm. Generalizations of FedLoRU under statistical and model heterogeneity, namely pFedLoRU and mFedLoRU, are also presented. Finally, the authors conduct experiments on computer vision pre-training and language model fine-tuning tasks to demonstrate the performance of FedLoRU and its generalizations.
weaknesses: * Although the authors provide some Hessian rank structure analysis, the design of FedLoRU can be better supported from the theoretical side. For example, some convergence guarantees, since low-rank updates lead to loss of information compared to full-rank updates and may hurt the optimization.
* The title mentions "its connection to implicit regularization", but I was not able to spot sufficient discussion on implicit regularization of FedLoRU; also please see a conceptual question below.
* The design of FedLoRU seems a straightforward extension to federated setting of existing methods for low-rank matrix accumulation such as ReLoRA [1]. 
* This is not a major weakness but more evaluations on LLM fine-tuning could be done, as most of the experiment details are devoted to computer vision tasks on small datasets.

[1] Vladislav Lialin, Sherin Muckatira, Namrata Shivagunde, and Anna Rumshisky. Relora: High- rank training through low-rank updates. In The Twelfth International Conference on Learning Representations, 2023.

Meta-review: The paper proposes the FedLoRU method: a federated optimization method combining low-rank updates on the clients' side (equivalent to running LORA locally on each client), aggregating the updates on the servers, and repeating this process. The authors claim this method is motivated by the needs for communication-efficient federated learning methods, combined with regularization offered by low-rank adaptation.

I do not really observe any particular strengths in this paper. The core of the idea is straightforward, and appeared in some previous papers already (e.g., Kuo et al, 2024). The difference here is that multiple low-rank updates are added to the base model over the algorithm run. However, this idea was explored before, too, in the COLA (Chain of LORA) paper (Xia et al, 2024). 

Some issues:

- There is no convergence analysis of the new method. 
- The aggregation mechanism is problematic from a theoretical point of view: the A updates are aggregated on their own, and the B updates are aggregated on their own, whereas the "mathematically correct" update would be to average the products A*B; i.e., average the updates. This approach does not lead to a low-rank update, however. Thus, the authors resort to a heuristic, which needs justification. Multiple papers were written on this topic before.
- The connection between weights being low-rank and Hessians being of low-rank, which plays a key role in their justification of the method, was questions by the reviewers, and the explanation is not satisfying - it relies on hand-waving arguments instead of on solid mathematical reasoning.
- Experimental results were not found to be convincing enough - a very high bar is needed for an empirical work; and this bar was certainly not achieved.
- The authors seem to be not aware of prior FL literature with very closely connected algorithms which, unlike their work, have strong theoretical backing. For example, literature on federated optimization with contractive compression applied to the updates by the clients (e.g., all work on error feedback, started in 2014 by Seide, with hundreds of follow-up works) is very closely connected. This is because low-rank approximation is known to be a contractive compressor. 
- I doubt the convergence of the presented method can be analyzed mathematically - in fact, I believe simple counterexamples can be found on which this methods fails. 

Finally, no reviewer recommended this paper for acceptance. I've read the reviews, the rebuttals and the discussion, and have briefly looked at the paper as well. I agree with the overall judgement that this paper has many significant weaknesses, and should not be accepted.

AC

Decision: Reject