# Towards Efficient Communication Federated Recommendation System via Low-rank
Training

Ngoc-Hieu Nguyen<EMAIL_ADDRESS>0009-0009-5873-6532 College of
Engineering & Computer Science, VinUniversityHanoiVietnam ,  Tuan-Anh Nguyen
<EMAIL_ADDRESS>College of Engineering & Computer Science,
VinUniversityHanoiVietnam ,  Tuan Nguyen<EMAIL_ADDRESS>College of
Engineering & Computer Science, VinUniversityHanoiVietnam ,  Vu Tien Hoang
<EMAIL_ADDRESS>College of Engineering & Computer Science,
VinUniversityHanoiVietnam ,  Dung D. Le<EMAIL_ADDRESS>College of
Engineering & Computer Science, VinUniversityHanoiVietnam and  Kok-Seng Wong
<EMAIL_ADDRESS>College of Engineering & Computer Science,
VinUniversityHanoiVietnam

###### Abstract.

In Federated Recommendation (FedRec) systems, communication costs are a
critical bottleneck that arises from the need to transmit neural network
models between user devices and a central server. Prior approaches to these
challenges often lead to issues such as computational overheads, model
specificity constraints, and compatibility issues with secure aggregation
protocols. In response, we propose a novel framework, called Correlated Low-
rank Structure (CoLR), which leverages the concept of adjusting lightweight
trainable parameters while keeping most parameters frozen. Our approach
substantially reduces communication overheads without introducing additional
computational burdens. Critically, our framework remains fully compatible with
secure aggregation protocols, including the robust use of Homomorphic
Encryption. Our approach resulted in a reduction of up to 93.75% in payload
size, with only an approximate 8% decrease in recommendation performance
across datasets. Code for reproducing our experiments can be found at
https://github.com/NNHieu/CoLR-FedRec.

Recommendation System, Federated Learning, Communication efficiency

##  1\. Introduction

In a centralized recommendation system, all user behavior data is collected on
a central server for training. However, this method can potentially expose
private information that users may be hesitant to share with others. As a
result, various regulations such as the General Data Protection Regulation
(GDPR)(Voigt and Von dem Bussche, 2017) and the California Consumer Privacy
Act (CCPA)(Pardau, 2018) have been implemented to limit the centralized
collection of users’ personal data. In response to this challenge, and in
light of the increasing prevalence of edge devices, federated recommendation
(FedRec) systems have gained significant attention for their ability to uphold
user privacy (Ammad-Ud-Din et al., 2019; Lin et al., 2020a; Chai et al., 2020;
Lin et al., 2020b; Wang et al., 2021c, b; Wu et al., 2022; Perifanis and
Efraimidis, 2022; Liu et al., 2022).

The training of FedRec systems is often in a cross-device setting which
involves transferring recommendation models between a central server and
numerous edge devices, such as mobile phones, laptops, and PCs. It is
increasingly challenging to transfer these models due to the growing model
complexity and parameters in modern recommendation systems (Naumov et al.,
2019). In addition, clients participating in FedRec systems often exhibit
differences in their computational processing speeds and communication
bandwidth capabilities, primarily stemming from variations in their hardware
and infrastructure (Li et al., 2020a). These discrepancies can give rise to
stragglers and decrease the number of participants involved in training,
potentially leading to diminished system performance.

Practical FedRec systems require the implementation of mechanisms that reduce
the amount of communication costs. Three commonly used approaches to reduce
communication costs include (i) reducing the frequency of communication by
allowing local updates, (ii) minimizing the size of the message through
message compression, and (iii) reducing the server-side communication traffic
by restricting the number of participating clients per round (Wang et al.,
2021a). Importantly, these three methods are independent and can be combined
for enhanced efficiency.

In this study, we address the challenge of communication efficiency in
federated recommendations by introducing an alternative to compression
methods. Many existing compression methods involve encoding and decoding steps
that can introduce significant delays, potentially outweighing the gains
achieved in per-bit communication time (Vogels et al., 2019). Another crucial
consideration is the compatibility with aggregation protocols. For example,
compression techniques that do not align with all-reduce aggregation may yield
reduced communication efficiency in systems employing these aggregation
techniques (Vogels et al., 2019). This is also necessary for many secure
aggregation protocols such as Homomorphic Encryption (HE) (Bonawitz et al.,
2017). Moreover, many algorithms assume that clients have the same
computational power, but this may induce stragglers due to computational
heterogeneity and can increase the runtime of algorithms.

Based on our observation that the update transferred between clients and the
central server in FedRec systems has a low-rank bias (Section 4.1), we propose
Correlated Low-rank Structure update (CoLR). CoLR increases communication
efficiency by adjusting lightweight trainable parameters while keeping most
parameters frozen. Under this training scheme, only a small amount of
trainable parameters will be shared between the server and clients. Compared
with other compression techniques, our methods offer the following benefits.
(i) Reduce both up-link and down-link communication cost: CoLR avoid the need
of unrolling the low-rank message in the aggregation step by using a
correlated projection, (ii) Low computational overheads: Our method enforces a
low-rank structure in the local update during the local optimization stage so
eliminates the need to perform a compression step. Moreover, CoLR can be
integrated into common aggregation methods such as FedAvg and does not require
additional computation. (iii) Compatible with secure aggregation protocols:
the aggregation step on CoLR can be carried by simple additive operations,
this simplicity makes it compatible with strong secure aggregation methods
such as HE, (iv) Bandwidth heterogeneity awareness: Allowing adaptive rank for
clients based on computational/communication budget.

Our contributions can be summarized as following:

  * •

We propose a novel framework, CoLR, designed to tackle the communication
challenge in training FedRec systems.

  * •

We conducted experiments to showcase the effectiveness of CoLR. Notably, even
with an update size equates to 6.25% of the baseline model, CoLR demonstrates
remarkable efficiency by retaining 93.65% accuracy (in terms of HR) compared
to the much larger baseline.

  * •

We show that CoLR is compatible with HE-based FedRec systems and, hence,
reinforces the security of the overall recommendation systems. Our framework
demonstrates a capability to provide a strong foundation for building a secure
and practical recommendation system.

##  2\. Related Work

#### Federated Recommendation (FedRec) Systems.

In recent years, FedRec systems have risen to prominence as a key area of
research in both machine learning and recommendation systems. FCF (Ammad-Ud-
Din et al., 2019) and FedRec (Lin et al., 2020a) are the pioneering FL-based
methods for collaborative filtering based on matrix factorization. The former
is designed for implicit feedback, while the latter is for explicit feedback.
To enhance user privacy, FedMF (Chai et al., 2020) applies distributed matrix
factorization within the FL framework and introduces the HE technique for
securing gradients before they are transmitted to the server. MetaMF (Lin et
al., 2020b) is a distributed matrix factorization framework using a meta-
network to generate rating prediction models and private item embedding. (Wu
et al., 2022) presents FedPerGNN, where each user maintains a GNN model to
incorporate high-order user-item information. FedNCF (Perifanis and
Efraimidis, 2022) adapts Neural Collaborative Filtering (NCF) (He et al.,
2017) to the federated setting, incorporating neural networks to learn user-
item interaction functions and thus enhancing the model’s learning
capabilities.

#### Communication Efficient Federated Recommendation.

Communication efficiency is of the utmost importance in FL (Konečný et al.,
2016). Some works explore reducing the entire item latent matrix payload by
meta-learning techniques (Lin et al., 2020b; Wang et al., 2021c). For example,
MetaMF (Lin et al., 2020b) adopts the meta recommender to deploy smaller
models on the client to reduce memory consumption. LightFR (Zhang et al.,
2023) proposes a framework to reduce communication costs by exploiting the
learning-to-hash technique under federated settings and enjoys both fast
online inference and reduced memory consumption.

#### Low-rank Structured Update

Konečný et al. (2016) propose to enforce every update to local model
$\Delta_{u}$ to have a low rank structure by express
$\Delta_{u}=A_{u}^{(t)}B_{u}^{(t)}$ where
$A_{u}^{(t)}\in\mathbb{R}^{d_{1}\times k}$ and
$B_{u}^{(t)}\in\mathbb{R}^{k\times d_{2}}$. In subsequent computation,
$A_{u}^{(t)}\in\mathbb{R}^{d\times k}$ is generated randomly and frozen during
a local training procedure. In each round, the method generates the matrix
$A_{u}^{(t)}$ afresh for each client independently. This approach saves a
factor of $d_{1}/k$. They interpret $B_{u}^{(t)}$ as a projection matrix, and
$A_{u}^{(t)}$ as a reconstruction matrix. Hyeon-Woo et al. (2022) proposes a
method that re-parameterizes weight parameters of layers using low-rank
weights followed by the Hadamard product.

$W=W_{1}\odot
W_{2}=\left(X_{1}Y_{1}^{\top}\right)\odot\left(X_{2}Y_{2}^{\top}\right)$

Given this parameterization, the rank of $W$ is upper bound by
$\operatorname{rank}(W_{1})\operatorname{rank}(W_{2})$ which is less
constrained than a conventional low-rank parameterization $W=XY^{\top}$. The
authors show that FedPara can achieve comparable performance to the original
model with 3 to 10 times lower communication costs on various tasks, such as
image classification, and natural language processing.

#### Secure FedRec.

Sending updates directly to the server without implementing privacy-preserving
mechanisms can lead to security vulnerabilities. Chai et al. (2020)
demonstrated that in the case of the Matrix Factorization (MF) model using the
FedAvg learning algorithm, if adversaries gain access to a user’s gradients in
two consecutive steps, they can deduce the user’s rating information.
Therefore, it is crucial to incorporate privacy-preserving mechanisms for the
update parameters transmitted from clients to the server. One approach, as
proposed by Chai et al. (2020), involves leveraging HE to encrypt intermediate
parameters before transmitting them to the server. This method effectively
safeguards user ratings while maintaining recommendation accuracy. However, it
introduces significant computational overhead, including encryption and
decryption steps on the client side, as well as aggregation on the server
side. Approximately 95% of the time consumed by the system is dedicated to
server updates, where all computations are carried out on the ciphertext Chai
et al. (2020).

##  3\. Preliminaries

In this section, we present the preliminaries and the setting that the paper
is working with. Also, this part will discuss the challenges in applying
compression methods.

###  3.1. Federated Learning for Recommendation

In the typical settings of item-based FedRec systems (Lin et al., 2020a),
there are $M$ users and $N$ items where each user $u$ has a private
interaction set denoted as $O_{u}=\\{(i,r_{iu})\\}\subset[N]\times\mathbb{R}$.
These users want to jointly build a recommendation system based on local
computations without violating participants’ privacy. This scenario naturally
aligns with the horizontal federated setting (McMahan et al., 2017), as it
allows us to treat each user as an active participant. In this work, we also
use the terms user and client interchangeably. The primary goal of such a
system is to generate a ranked list of top-K items that a given user has not
interacted with and are relevant to the user’s preferences. Mathematically, we
can formalize the problem as finding a global model parameterized by $\theta$
that minimizes the following global loss function $\mathcal{L}(\cdot)$:

(1)
$\mathcal{L}(\theta)\triangleq\sum_{u=1}^{M}w_{u}\mathcal{L}_{u}(\bm{\theta})$

where $\theta$ is the global parameter, $w_{u}$ is the relative weight of user
$u$. And $\mathcal{L}_{u}(\theta):=\sum_{(i,r_{ui})\in
O_{u}}\ell_{u}(\theta,(i,r_{ui}))$ is the local loss function at user $u$’s
device. Here $(i,r_{ui})$ represents a data sample from the user’s private
dataset, and $\ell_{u}$ is the loss function defined by the learning
algorithm. Setting $w_{u}=N_{u}/N$ where $N_{u}=|O_{u}|$ and
$N=\sum_{u=1}^{M}N_{u}$ makes the objective function $\mathcal{L}(\theta)$
equivalent to the empirical risk minimization objective function of the union
of all the users’ dataset. Once the global model is learned, it can be used
for user prediction tasks.

In terms of learning algorithms, Federated Averaging (FedAvg) (McMahan et al.,
2017) is one of the most popular algorithms in FL. FedAvg divides the training
process into rounds. At the beginning of the $t$-th round ($t\geq 0$), the
server broadcasts the current global model $\theta^{(t)}$ to a subset of users
$\mathcal{S}^{(t)}$ which is often uniformly sampled without replacement in
simulation (Wang et al., 2021a; Lin et al., 2020a). Then each sampled client
in the round’s cohort performs $\tau_{u}$ local SGD updates on its own local
dataset and sends the local model changes
$\Delta_{u}^{(t)}=\theta_{u}^{(t,\tau_{u})}-\theta^{(t)}$ to the server.
Finally, the server performs an aggregation step to update the global model:

(2)
$\theta^{(t+1)}=\theta^{(t)}+\frac{\sum_{u\in\mathcal{S}^{(t)}}w_{u}\Delta_{u}^{(t)}}{\sum_{u\in\mathcal{S}^{(t)}}w_{u}}$

The above procedure will repeat until the algorithm converges.

###  3.2. Limitation of current compression methods

Communication is one of the main bottlenecks in FedRec systems and can be a
serious constraint for both servers and clients. For example, when stragglers
with limited network connections exist, the central server must decide whether
to wait for them to finish or perform the aggregation step with only available
participants. Conversely, on the client side, sending the model updates back
to the server can be challenging, as uplink is typically much slower than
downlink. The download and upload bandwidths in a real cross-device FL system
are estimated at 0.75MB/s and 0.25MB/s, respectively (Wang et al., 2021a).
Although diverse optimization techniques exist to enhance communication
efficiency, such methods may not preserve privacy. Moreover, tackling privacy
and communication efficiency as separate concerns can result in suboptimal
solutions.

#### Top-K compression

The process of allocating memory for copying the gradient (which can grow to a
large size, often in the millions) and then sorting this copied data to
identify the top-K threshold during each iteration is costly enough that it
negates any potential enhancements in overall training time when applied to
real-world systems. As a result, employing these gradient compression methods
in their simplest form does not yield the expected improvements in training
efficiency. As observed in Gupta et al. (2021), employing the Top-K
compression for training large-scale recommendation models takes 11% more time
than the baseline with no compression.

#### SVD compression

After obtaining factorization results $U_{u}$ and $V_{u}$, the aggregation
step requires performing decompression and computing
$\sum_{u\in\mathcal{S}}\frac{N_{u}}{N}U_{u}V_{u}$ and this sum is not
necessarily low-rank so there is no readily reducing cost in the downlink
communication without additional compression-decompression step. The need to
perform matrix multiplication makes this method incompatible with HE.
Moreover, performing SVD decomposition on an encrypted matrix by known schemes
remains an open problem.

##  4\. Proposed method

Figure 1. Illustration of CoLR at training round $t$. At first, the server
conducts aggregation over the local model $A_{u}^{(t-1,\tau_{u})}$ to obtain
the global model update $A^{(t)}$. Subsequently, $A^{(t)}$ are transmitted to
the clients. The client will update their $Q_{u}^{(t)}$ using this $A^{(t)}$,
then initilizes a new matrix $A_{u}^{(t,0)}$ and download the matrix $B^{(t)}$
which is sampled at the server and shared between clients. Finally, the client
carries out local training and then sends the local model update
$A_{u}^{(t,\tau_{u})}$ to the server for the next training round.

###  4.1. Motivation

Our method is motivated by analyzing the optimization process at each user’s
local device. We consider an effective federated matrix factorization (FedMF)
as the backbone model. This model represents each item and user by a vector
with the size of $d$ denoted $\mathbf{q}_{i}$ and $\mathbf{p}_{u}$
respectively. And the predicted ratings $r_{ui}$ are given by
$\hat{r}_{ui}=\mathbf{q}_{i}^{\top}\mathbf{p}_{u}$. Then the user-wise local
parameter $\theta_{u}$ consists of the user $u$’s embedding $\mathbf{p}_{u}$
and the item embedding matrix $Q$, where $\mathbf{q}_{i}$ is the $i$th column
of $Q$. The loss function $\mathcal{L}_{u}$ at user $u$’s device is given in
the following.

$\displaystyle\mathcal{L}_{u}(\mathbf{p}_{u},Q)=$
$\displaystyle\sum_{(i,r_{ui})\in
O_{u}}\ell\left(r_{ui},({Q^{\top}\mathbf{p}_{u}})_{i}\right)+\lambda\|\mathbf{p}_{u}\|_{2}+\lambda\|Q\|_{2}$

Let $\eta$ be the learning rate, the update on the user embedding
$\mathbf{p}_{u}$ at each local optimization step is given by:

(3) $\displaystyle\mathbf{p}_{u}^{(t+1)}=$
$\displaystyle\mathbf{p}_{u}^{(t)}(1-\eta\lambda)+\eta
Q^{(t)\top}(\mathbf{r}-\hat{\mathbf{r}}^{(t)}).$

Let $\mathbf{m}\in\mathbb{R}^{N}$ be a binary vector where $\mathbf{m}_{i}=1$
if $i\in O_{u}$, then the item embedding’s $\mathbf{q}_{i}$ is

(4) $Q^{(t+1)}=Q^{(t)}-\eta(\lambda
Q^{(t)}-\left(\mathbf{m}*(\mathbf{r}_{u}-\hat{\mathbf{r}}_{u})\right)\mathbf{p}_{u}^{(t)\top})$

The update that is sent to the central server has the following formula,

(5)
$\displaystyle\Delta_{Q}^{(t)}=Q^{(t+1)}-Q^{(t)}=\eta\left[\left(\mathbf{m}*(\mathbf{r}_{u}-\hat{\mathbf{r}}_{u})\right)\mathbf{p}_{u}^{(t)\top}-\lambda
Q^{(t)}\right]$

As we can see from equation 4, since each client only stores the presentation
of only one user $\mathbf{p}_{u}$, the update on the item embedding matrix on
each user’s device at each local step are just sum of a rank-1 matrix and a
regularization component. Given that $\lambda$ is typically small, the low-
rank component contributes most to the update $\Delta_{Q}^{(t)}$. And if the
direction of $\mathbf{p}_{u}$ does not change much during the local
optimization phase, the update $\Delta_{Q}^{(t)}$ can stay low-rank. From this
observation, we first assume that the update of the item embedding matrix in
training FedRec systems $\Delta_{Q}^{(t)}$ can be well approximated by a low-
rank matrix. We empirically verify this assumption by monitoring the effective
rank of $\Delta_{Q}^{(t)}$ at each training round for different datasets. The
result is plotted in figure 2 where we plot the mean and standard deviation
averaged over a set of participants in each round.

Figure 2. PCA components progression. The figures show the number of
components that account for 99% (N99-PCA in green) and 90% (N95-PCA in blue)
explained variance of all transfer item embedding matrix across communication
rounds on MovieLens-1M (left) and Pinterest (right) datasets.

This analysis suggests that restricting the update to be low-rank can get away
with aggressive communication reduction without sacrificing much performance.
In the next section, we will propose an efficient communication framework
based on this motivation. Since most of the transferred parameters in
recommendation models are from the item embedding layers, we will focus on
applying the proposed method for embedding layers in this work. Note that our
framework can still be applied to different types of layers that are commonly
used in recommendation models, such as fully connected layers, convolution
layers, and self-attention layers.

###  4.2. Low-rank Structure

We propose explicitly enforcing a low-rank structure on the local update of
the item embedding matrix $Q$. In particular, the local update
$(\Delta_{Q})_{u}^{(t)}$ is parameterized by a matrix product

$(\Delta_{Q})_{u}^{(t)}=B_{u}^{(t)}A_{u}^{(t)}$

where $B_{u}^{(t)}\in\mathbb{R}^{d\times r}$ and
$A_{u}^{(t)}\in\mathbb{R}^{r\times N}$. Given this parameterization, the
embedding $\mathbf{q}_{i}$ of an item with index $i$ is given by

$\mathbf{q}_{i}^{(t)}=\left(Q^{(t)}+B_{u}^{(t)}A_{u}^{(t)}\right)\mathbf{e}_{i}$

where $\mathbf{e}_{i}\in\mathbb{R}^{N}$ is a one-hot vector whose value at
$i$-th is 1. This approach effectively saves a factor of $\frac{N\times
d}{N\times r+d\times r}$ in communication since clients only need to send the
much smaller matrices $A_{u}$ and $B_{u}$ to the central server.

###  4.3. Correlated Low-rank Structure Update

Even though enforcing a low-rank structure on the update can greatly reduce
the uplink communication size, doing aggregation and performing privacy-
preserving is not trivial and faces the following three challenges: (1) the
server needs to multiply out all the pairs $A_{u}^{(t)}$ and $B_{u}^{(t)}$
before performing the aggregation step; (2) the sum of low-rank solutions
would typically leads to a larger rank update so there is no reducing
footprint in the downlink communication; (3) secure aggregation method such as
HE cannot directly apply to $A_{u}^{(t)}$ and $B_{u}^{(t)}$ since it will
require to perform the multiplication between two encrypted matrices, which is
much more costly than simple additive operation.

To reduce the downlink communication cost, we observe that if either
$A_{u}^{(t)}$ or $B_{u}^{(t)}$ is identical between users and is fixed during
the local training process, then the result of the aggregation step can be
represented by a low-rank matrix with the following formulation:

$\Delta_{Q}^{(t)}=B^{(t)}\left(\sum_{u\in S}A_{u}^{(t)}\right).$

Notice that this aggregation is also compatible with HE since it only requires
additive operations on a set of $A_{u}^{(t)}$ and clients can decrypt this
result and then compute the global update $\Delta_{Q}^{(t)}$ at their local
device.

Based on the above observation, we propose the Correlated Low-rank Structure
Update (CoLR) framework. In this framework, the server randomly initializes a
matrix $B^{(t)}$ at the beginning of each training round and shares it among
all participants. Participants then set $B_{u}^{(t)}=B^{(t)}$ and freeze this
matrix during the local training phase and only optimize for $A_{u}^{(t)}$.
The framework is presented in Algorithm 1 and illustrated in Figure 1. Note
that the communication cost can be further reduced by sending only the random
seed of the matrix $B^{(t)}$. A concurrent work (Anonymous, 2023) proposes
FFA-LoRA which also fixes the randomly initialized non-zero matrices and only
finetunes the zero-initialized matrices. They study FFA-LoRA in the context of
federated fine-tuning LLMs and using differential privacy (Dwork et al., 2006)
to provide privacy guarantees.

Input: Initial model $Q^{(0)}$; update rank $r$, a distribution
$\mathcal{D}_{B}$ for initializing $B$; ClientOpt, ServerOpt with learning
rates $\eta,\eta_{s}$;

1

2for  _$t\in\\{0,1,2,\dots,T\\}$ _ do

3 Sample a subset $\mathcal{S}^{(t)}$ of clients

4 Sample $B^{(t)}\sim\mathcal{D}_{B}$

5 for  _client $u\in\mathcal{S}^{(t)}$ in parallel_ do

6 if  _$t >0$_ then

7 Download $A^{(t)}$

8 Merge $Q_{u}^{(t,0)}=Q^{(t-1)}+B^{(t-1)}A^{(t)}$

9

10 end if

11 Initialize $Q_{u}^{(t,0)}=Q^{(t)}$

12 Download $B^{(t)}$ and Initialize $A_{u}^{(t,0)}=\bm{0}$

13 Set trainable parameters
$\theta_{u}^{(t,0)}=\\{A_{u}^{(t,0)},\mathbf{p}_{u}^{(t,0)}\\}$

14 for  _$k=0,\dots,\tau_{u}-1$_ do

15

16 Compute local stochastic gradient
$\nabla\mathcal{L}_{u}(\theta_{u}^{(t,k)})$

17 Perform local update
$\theta_{u}^{(t,k+1)}=\textsc{ClientOpt}\left(\theta_{u}^{(t,k)},\nabla_{\theta_{u}}\mathcal{L}_{u}(\theta_{u}^{(t,k)}),\eta\right)$

18

19 end for

20 $\mathbf{p}_{u}^{(t+1)}=\mathbf{p}_{u}^{(t,\tau_{u})}$

21 Upload $\\{A_{u}^{(t,\tau_{u})}\\}$ to the central server

22

23 end for

24 Aggregate local changes
$A^{(t+1)}=\sum_{u\in\mathcal{S}^{(t)}}\dfrac{N_{u}}{N}A_{u}^{(t,\tau_{u})};$

25 end for

Algorithm 1 Correlated Low-rank Structure Update Matrix Factorization

#### Differences w.r.t. SVD compression.

We compare our method with SVD since it also uses a low-rank structure. The
difference is that in CoLR, participants directly optimize these models on the
low-rank parameterization, while SVD only compresses the result from the local
training step.

###  4.4. Subsampling Correlated Low-rank Structure update (SCoLR)

While CoLR offers its merits, there is a potential drawback to consider - it
may impact recommendation performance as it confines the global update within
a randomly generated low-rank subspace. In the following section, we introduce
a modification to this base algorithm, considering a practical reality:
downlink bandwidth often surpasses uplink capacity, as observed in cross-
device scenarios (Wang et al., 2021a). In these settings, edge devices
establish communication with a central server using network connections that
vary in quality. Practical implementations have demonstrated significant
differences in network bandwidth between download and upload capabilities. We
propose a variant of CoLR termed Subsampling Correlated Low-rank Structure
update (SCoLR) to address this. SCoLR strategically harnesses the more
abundant downlink bandwidth while maintaining communication efficiency and HE
compatibility.

We denote $r_{g}$ as the rank of global update, which is sent from the server
to participants through downlink connections, and $r_{u}$ as the rank of local
update, which is sent from clients to the central server for aggregation
through uplink connections. In practice, we can set $r_{g}$ to be larger than
$r_{u}$, reflecting that downlink bandwidth is often higher than uplink. Given
these rank parameters, at the start of each training round, the central server
first initializes a matrix $B$ with the shape of $\mathbb{R}^{d\times r_{g}}$.
Then, participants in that round will download this matrix to their local
devices and select a subset of columns of $B$ to perform the local
optimization step. In particular, we demonstrate this process through the
following formulation:

(6) $(\Delta_{Q})_{u}^{(t)}=BS_{u}A_{u},$

where $B$ is a matrix with the shape of $\mathbb{R}^{d\times r_{g}}$, $S_{u}$
is a binary matrix with the shape of $\mathbb{R}^{r_{g}\times r_{u}}$ and
$A_{u}$ is a matrix with the shape of $\mathbb{R}^{r_{u}\times N}$.
Specifically, $S_{u}$ is a binary matrix with $r_{u}$ rows and $r_{g}$
columns, where each row has exactly one non-zero element. The non-zero element
in the $i$-th row is at the $j$-th column, where $j$ is the $i$-th element of
a randomly shuffled array of integers from 1 to $r_{g}$. The detail is
presented in Algorithm 2.

Moreover, in scenarios where clients have diverse computational resources,
each can choose a unique local rank, denoted as $r_{u}$, aligning with their
specific computational capacities and individual preferences throughout the
training process. Importantly, sharing the matrix $S_{u}$ does not divulge
sensitive user information. Multiplying this matrix with $S_{u}$ is
essentially a row reordering operation on the matrix $A_{u}$. As a result, we
can effectively perform additive HE between pairs of rows from $A_{u_{1}}$ and
$A_{u_{2}}$. This approach ensures privacy while accommodating varying
computational capabilities among clients.

##  5\. Experiments

Figure 3. HR and NDCG on MovieLens-1M dataset (Top) and Pinterest Dataset
(Bottom). We plot the utilities versus the communication size and compare
CoLRwith the base model with the same transfer size. The x-axis is plotted
using a log scale.

###  5.1. Experimental Setup

Table 1. Statistics of the datasets used in evaluation. Datasets | # Users | # Items | # Ratings | Data Density
---|---|---|---|---
MovieLens-1M (Harper and Konstan, 2015) | $6,040$ | $3,706$ | $1,000,209$ | $4.47\%$
Pinterest (Geng et al., 2015) | $55,187$ | $9,916$ | $1,500,809$ | $0.27\%$

#### Datasets

We experiment with two publicly available datasets, which are MovieLens-1M
(Harper and Konstan, 2015) and Pinterest (Geng et al., 2015). Table 1
summarizes the statistics of our datasets. We follow common practice in
recommendation systems for preprocessing by retaining users with at least 20
interactions and converting numerical ratings into implicit feedback (Ammad-
Ud-Din et al., 2019; He et al., 2017).

#### Evaluation Protocols

We employ the standard leave-one-out evaluation to set up our test set (He et
al., 2017). For each user, we use all their interactions for training while
holding out their last interaction for testing. During the testing phase, we
randomly sampled 99 non-interacted items for each user and ranked the test
item amongst these sampled items.

To evaluate the performance and verify the effectiveness of our model, we
utilize two commonly used evaluation metrics, i.e., Hit Ratio (HR) and
Normalized Discounted Cumulative Gain (NDCG), which are widely adopted for
item ranking tasks. The above two metrics are usually truncated at a
particular rank level (e.g. the first $k$ ranked items) to emphasize the
importance of the first retrieved items. Intuitively, the HR metric measures
whether the test item is present on the top-$k$ ranked list, and the NDCG
metric measures the ranking quality, which comprehensively considers both the
positions of ratings and the ranking precision.

#### Models and Optimization.

For the base models, we adopt Matrix Factorization with the FedAvg learning
algorithm, also used in Chai et al. (2020). In our experiments, the dimension
of user and item embedding $d$ is set to 64 for the MovieLens-1M dataset and
16 for the Pinterest dataset. This is based on our observation that increasing
the embedding size on the Pinterest dataset leads to overfitting and decreased
performance on the test set. The result is also consistent with (He et al.,
2017). We use the simple SGD optimizer for local training at edge devices.

#### Federated settings.

In each round, we sample $M$ clients uniformly randomly, without replacement
in a given round and across rounds. Instead of performing $\tau_{i}$ steps of
ClientOpt, we perform $E$ epochs of training over each client’s dataset. This
is because, in practical settings, clients have heterogeneous datasets of
varying sizes. Thus, specifying a fixed number of steps can cause some clients
to repeatedly train on the same examples, while certain clients only see a
small fraction of their data.

#### Baselines.

We compare our framework with two compression methods: SVD and TopK
compression. The first method is based on singular value decomposition, which
returns a compressed update with a low-rank structure. The second method is
based on sparsification, which represents updates as sparse matrices to reduce
the transfer size.

#### Hyper-parameter settings.

To determine hyper-parameters, we create a validation set from the training
set by extracting the second last interaction of each user and tuning hyper-
parameters on it. We tested the batch size of [32, 64, 128, 256], the learning
rate of [0.5, 0.1, 0.05, 0.01], and weight decay in $[5e-4,1e-4]$. For each
dataset, we set the number of clients participating in each round to be equal
to 1%. The number of aggregation epochs is set at 1000 for MovieLens-1M and
2000 for Pinterest as the training process is converged at these epochs.

#### Machine

The experiments were conducted on a machine equipped with an Intel(R) Xeon(R)
W-1250 CPU @ 3.30GHz and a Quadro RTX 4000 GPU.

###  5.2. Experimental Results

#### (1) CoLR can achieve comparable performances with the base models.

Given our primary focus is on recommendation performance within communication-
limited environments, we commence our investigation by comparing the
recommendation performance between CoLR and the base model FedMF given the
same communication budget. On the ML-1M dataset, we adjust the dimensions of
user and item embeddings across the set [1, 2, 4, 8, 16, 32, 64] for FedMF
while fixing the embedding size of CoLR to 64, with different rank settings
within [1, 2, 4, 8, 16, 32]. Similarly, for Pinterest, the embedding range for
FedMF is [1, 2, 4, 8, 16], while CoLR has an embedding size of 16 and ranks in
the range of [1, 2, 4, 8]. Our settings lead to approximately equivalent
transfer sizes for both methods in each dataset.

In Figure 3, we present the HR and NDCG metrics across a range of different
transfer sizes. With equal communication budgets, CoLR consistently
outperforms their counterparts on both datasets. To illustrate, on the
Pinterest dataset, even with an update size equates to 6.25% of the largest
model, CoLR achieves a notable performance (81.03% HR and 48.50% NDCG)
compared to the base model (84.74% HR and 51.79% NDCG) while attaining a much
larger reduction in terms of communication cost (16x). In contrast, the FedMF
models with corresponding embedding sizes achieve much lower accuracies. On
the MovieLens-1M dataset, we also observe a similar pattern where CoLR
consistently demonstrates higher recommendation performance when compared to
their counterparts.

The result from this experiment highlights that CoLR can achieve competitive
performance when compared to the fully-trained model, FedMF while greatly
reducing the cost of communication.

#### (2) Comparison between CoLR and other compression-based methods.

Table 2. Communication times and Training times for MovieLens-1M dataset. Method |  Communication time (minutes) |  Computation time (minutes) |  Total Training Time (minutes)
---|---|---|---
MF-64 |  80.43 |  169.07 |  249.50
CoLR@1 |  1.26 |  169.18 |  170.43
CoLR@2 |  2.51 |  169.21 |  171.72
CoLR@4 |  5.03 |  169.27 |  174.30
CoLR@8 |  10.05 |  169.29 |  179.34
CoLR@16 |  20.11 |  169.30 |  189.41
CoLR@32 |  40.21 |  169.38 |  209.60
SVD@1 |  1.26 |  169.49 |  170.75
SVD@2 |  2.51 |  169.50 |  172.02
SVD@4 |  5.03 |  169.53 |  174.55
SVD@8 |  10.05 |  169.59 |  179.65
SVD@16 |  20.11 |  169.64 |  189.74
SVD@32 |  40.21 |  169.60 |  209.82
TopK@1 |  2.51 |  169.76 |  172.28
TopK@2 |  5.03 |  169.79 |  174.81
TopK@4 |  10.05 |  169.82 |  179.87
TopK@8 |  20.11 |  169.92 |  190.03
TopK@16 |  40.21 |  170.14 |  210.35
Figure 4. HR and NDCG on MovieLens-1M dataset (Top) and Pinterest Dataset
(Bottom). We plot the utilities versus the communication size and compare
CoLRwith the base model with the same transfer size. The x-axis is plotted
using a log scale. The dashed black line is the base model’s performance with
different embedding sizes. For fair comparisons, we set the compression ratio
of each method to be equal.

We run the above experiment with two compression methods, SVD and TopK, using
equal compression ratios as CoLR. For a fair comparison, we compress both the
upload and download messages with the same compression ratio. The result in
terms of transfer size is presented in Figure 4. In the case with the same
communication budget, CoLR achieves better performance across the range of
communication budgets.

In the previous results, the evaluation of techniques focuses on the overall
number of transmitted bits. Although this serves as a broad indicator, it
fails to consider the time consumed by encoding/decoding processes and the
fixed network latency within the system. When these time delays significantly
exceed the per-bit communication time, employing compression techniques may
offer limited or minimal benefits. In the following, we do an analysis to
understand the effects of using CoLR and compression methods in training
FedRec models.

We follow the model from (Wang et al., 2021a) to estimate the communication
efficiency of deploying methods to real-world systems. The execution time per
round when deploying an optimization algorithm in a cross-device FL system is
estimated as follows,

$\displaystyle T_{\text{round }}(\mathcal{A})=T_{\text{comm
}}(\mathcal{A})+T_{\text{comp }}(\mathcal{A}),$ $\displaystyle T_{\text{comm
}}(\mathcal{A})=\frac{S_{\text{down }}(\mathcal{A})}{B_{\text{down
}}}+\frac{S_{\text{up }}(\mathcal{A})}{B_{\text{up }}}$ $\displaystyle
T_{\text{comp }}(\mathcal{A})=\max_{j\in\mathcal{D}_{\text{round
}}}T_{\text{client }}^{j}+T_{\text{server }}(\mathcal{A}),$ $\displaystyle
T_{\text{client }}^{j}(\mathcal{A})=R_{\text{comp }}T_{\text{sim
}}^{j}(\mathcal{A})+C_{\text{comp }}$

where client download size $S_{\text{d}own}(\mathcal{A})$, upload size
$S_{\text{u}p}(\mathcal{A})$, server computation time $T_{\text{server }}$,
and client computation time $T_{\text{client }}^{j}$ depend on model and
algorithm $\mathcal{A}$. Simulation time $T_{\text{server }}$ and
$T_{\text{client }}^{j}$ can be estimated from FL simulation in our machine.
We get the estimation of parameters $(B_{\text{down }},B_{\text{up
}}),R_{\text{comp }},C_{\text{comp }}$ from Wang et al. (2021a).

$\displaystyle B_{\text{down }}\sim
0.75\mathrm{MB}/\mathrm{secs},B_{\mathrm{up}}\sim
0.25\mathrm{MB}/\mathrm{secs},$ $\displaystyle R_{\mathrm{comp}}\sim 7,\text{
and }C_{\text{comp }}\sim 10\text{ secs. }$

Table 2 presents our estimation in terms of communication times and
computation time. Notice that CoLR adds smaller overheads to the computation
time while still greatly reducing the communication cost.

#### (3) CoLR is compatible with HE

Table 3. Overheads, and Communication ratios for MovieLens-1M dataset; Comm Ratio is calculated by file sizes of Ciphertext over file sizes of Plaintext. Method | Client overheads | Server overheads | Ciphertext size | Plaintext size | Comm Ratio
---|---|---|---|---|---
FedMF | 0.93 s | 2.39 s | 24,587 KB | 927 KB | 26.52
FedMF w/ TopK@1/64 | 88.20 s | 88.06 s | 3,028 KB | 29 KB | 103.09
FedMF w/ TopK@2/64 | 182.02 | 185.59 | 6,056 KB | 58 KB | 103.83
FedMF w/ TopK@4/64 | 353.25 | 364.67 | 12,112 KB | 116 KB | 104.20
FedMF w/ TopK@8/64 | 723.45 | 750.98 | 24,225 KB | 232 KB | 104.40
FedMF w/ TopK@16/64 | 1449.90 | 1483.91 | 48,448 KB | 464 KB | 104.49
FedMF w/ CoLR@1 | 0.07 | 0.24 | 3,073 KB | 15 KB | 206.31
FedMF w/ CoLR@2 | 0.07 | 0.25 | 3,073 KB | 29 KB | 104.63
FedMF w/ CoLR@4 | 0.07 | 0.25 | 3,073 KB | 58 KB | 52.69
FedMF w/ CoLR@8 | 0.08 | 0.25 | 3,073 KB | 116 KB | 26.44
FedMF w/ CoLR@16 | 0.15 | 0.51 | 6,147 KB | 232 KB | 26.49
FedMF w/ CoLR@32 | 0.30 | 1.03 | 12,293 KB | 464 KB | 26.51

In this section, we argue that tackling privacy and communication efficiency
as separate concerns can result in suboptimal solutions and point out the
limitation in applying SVD and TopK compression on HE-based FedRec systems.

Limitation of HE with SVD Compression: The SVD method necessitates the
decomposition of a matrix $M=U*S*V$, requiring matrix multiplication on
encrypted matrices U, S, and V derived from local clients’ updates. There are
several research endeavors aimed at providing efficient algorithms for
applying homomorphic encryption in this context, specifically using the CKKS
cryptosystem. However, a limitation of this method is that the dimension of
the matrix must be in the form of $2^{n}$, often necessitating additional
padding on the original matrices to achieve this form, particularly in the
case of larger dimension matrices. Additionally, to reduce the size of global
updates sent to the server, additional SVD decompositions are required.
Performing SVD decomposition on an encrypted matrix by known schemes remains
an open problem, making SVD compression an operation that can result in high
bandwidth consumption when transmitting large encrypted matrices in their raw
form.

Limitation of HE with TopK Compression: The TopK compression method
necessitates in-place homomorphic operations, a characteristic not compatible
with the CKKS scheme, designed to perform homomorphic encryption on tensors.
As an alternative, we have employed the Paillier cryptosystem, a partially
homomorphic encryption scheme capable of encrypting individual numbers. While
Paillier allows for the implementation of the FedAvg aggregation strategy for
TopK compression, it requires both encryption and decryption operations on
local clients, and the secured aggregation process on the server must be
executed on each number in the TopK vector. Consequently, increasing the value
of K results in higher operational costs for encryption, decryption, and
secured aggregation.

In our experiment, we conduct tests using two different compression methods:
CoLR and TopK. These tests are carried out under identical configurations,
encompassing local updates and the number of clients involved in training
rounds. The current setup entails the utilization of the CKKS Cryptosystem for
our CoLR method, while the TopK method employs the Paillier cryptosystem for
encryption, decryption, and aggregation in place of the TopK vector.

Regarding the runtime aspect, our CoLR compression method leverages the
inherent efficiency of the CKKS cryptosystem, which can execute operations on
multiple values as a vector. In our approach, for a flattened vector of size
$n$, both clients and the server need to perform operations on at most
$\lceil\frac{n}{8096}\rceil$ blocks, which incurs minimal computational time.
On the other hand, the TopK method necessitates operations to be executed on
every value within the TopK vector, leading us to opt for the Paillier
cryptosystem as it is partially homomorphic and can accommodate the
requirements of both our schemes and the TopK method. In our experimental
setup, when the value of $k$ doubles (i.e., doubling the TopK vector’s size),
the operation time for both client-side and server-side operations also
doubles, as it mandates operations on each value within the vector. Throughout
the experiment, our CoLR consistently outperforms the TopK method across
various $k$ values, exhibiting lower time overheads on both the client and
server sides.

Table 3 shows that CoLR can reduce client and server overheads by up to 10x.
When comparing ciphertext to plaintext sizes, the TopK compression method with
Paillier encryption demands encryption for each value within the TopK vector.
Consequently, whenever the size of the TopK vector doubles, the ciphertext
size also doubles. In contrast, as previously explained, our scheme produced
at most $\lceil\frac{n}{8096}\rceil$ blocks of ciphertext, with the ciphertext
size not doubling each time $k$ doubles. This phenomenon illustrates why, in
several cases, the ciphertext size remains consistent even as the plaintext
size increases. With higher $k$ values aimed at achieving greater precision,
our scheme demonstrates smaller ciphertext sizes, offering a reduction in
bandwidth consumption.

###  5.3. Dynamic local rank

In this section, we evaluate our proposed method SCoLR, where we explore the
scenario where each client can dynamically select a random $r_{u}$ value
during each training round $t$. This scenario reflects real-world federated
learning, where clients often showcase differences in computing capabilities
and communication capacities due to hardware discrepancies, as exemplified in
(Lai et al., 2021; Li et al., 2020b). It becomes inefficient to impose a
uniform training model on all clients within this heterogeneous context, as
some devices may not be able to harness their computational resources fully.

For this experiment, we set the global rank $r$ in range $\\{4,8,16,32\\}$ and
uniformly sample the local rank $r_{u}$ such that $1\leq r_{u}\leq r$. It’s
crucial to emphasize that $r_{u}$ is independently sampled for each user and
may differ from one round to the next. This configuration mirrors a practical
scenario where the available resources of a specific user may undergo
substantial variations at different time points during the training phase. We
present the result on the MovieLens-1M dataset in Table 4. This result
demonstrates that SCoLR is effective under the device heterogeneity setting.

Table 4. HR, NDCG of SCoLR algorithm on the MovieLens-1M dataset under computation/device heterogeneity settings. Global low-rank |  Local-rank | HR | NDCG
---|---|---|---
32 |  $1-32$ | 64.59 | 37.56
16 |  $1-16$ | 60.25 | 34.33
8 |  $1-8$ | 52.75 | 28.41
4 |  $1-4$ | 45.99 | 23.70

##  6\. Conclusion

In this work, we propose Correlated Low-rank Structure update (CoLR), a
framework that enhances communication efficiency and privacy preservation in
FedRec by leveraging the inherent low-rank structure in updating transfers,
our method reduces communication overheads. CoLR also benefits from the CKKS
cryptosystem, which allows the implementation of a secured aggregation
strategy within FedRec. With minimal computational overheads and bandwidth-
heterogeneity awareness, it offers a flexible and efficient means to address
the challenges of federated learning. For future research, we see several
exciting directions. First, our framework still involves a central server, we
would like to test how our methods can be effectively adapted to a fully
decentralized, peer-2-peer communication setting. Secondly, investigating
methods to handle dynamic network conditions and straggler mitigation in real-
world settings will be crucial. Lastly, expanding our approach to accommodate
more advanced secure aggregation techniques for reduced server-side
computational costs and extending its compatibility with various encryption
protocols can further enhance its utility in privacy-sensitive applications.

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##  Appendix A An analysis on the initialization of the matrix B

If each client performs only one GD step locally then $B$ can be seen as the
projection matrix and $B\mathbf{a}_{i}$ is the projection of the update of
item $i$ on the subspace spanned by columns of $B$. We denoite the error of
the update on each item embedding $i$ by $\epsilon_{i}$ which has the
following formulation:

(7)
$\epsilon_{i}=\mathbb{E}_{B}\left[\left\|\bar{\Delta}_{Q}-\frac{1}{|S|}\left(\sum_{u\in
S}B_{u}\mathbf{a}_{u}\right)\right\|_{2}^{2}\right].$

We analyze the effect of different initialization of $B$ on this error. First,
we state the proposition A.1 which gives an upper bound on the error
$\epsilon_{i}$.

###### Proposition A.1 (Upper bound the error).

If $B_{u}$ is independently generated between users and are chosen from a
distribution $\mathcal{B}$ that satisfies:

  1. (1)

Bounded operator norm: $\mathbb{E}\left[\|B\|^{2}\right]\leq L_{\mathcal{B}}$

  2. (2)

Bounded bias:
$\|\mathbb{E}B_{u}B_{u}^{\top}\bar{\mathbf{p}}_{u}-\bar{\mathbf{p}}_{u}\|_{2}\leq\sqrt{\delta_{\mathcal{B}}}$

Then,

(8) $\displaystyle\epsilon_{i}$
$\displaystyle=\mathbb{E}_{B}\left[\left\|\bar{\Delta}_{Q}-\frac{1}{|S|}\left(\sum_{u\in
S}B_{u}\mathbf{a}_{u}\right)\right\|_{2}^{2}\right]$ (9)
$\displaystyle\leq\frac{1}{|S|}C_{\mathbf{p}}^{2}\delta_{\mathcal{B}}+\frac{1}{|S|}\max_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}\left(L_{B}^{2}+1\right).$

###### Proof.

Assume $B_{u}$ is independently generated between users, we have

$\displaystyle\epsilon_{i}$
$\displaystyle=\frac{1}{|S|^{2}}\mathbb{E}_{B}\left[\left\|\sum_{u\in
S}(r_{ui}-\hat{r}_{ui})\left(B_{u}B_{u}^{\top}\mathbf{p}_{u}-\mathbf{p}_{u}\right)\right\|_{2}^{2}\right]$
$\displaystyle=\frac{1}{|S|^{2}}\mathbb{E}_{B}\left[\left\|\sum_{u\in
S}\alpha_{u}\left(B_{u}B_{u}^{\top}\mathbf{p}_{u}-\mathbf{p}_{u}\right)\right\|_{2}^{2}\right]$
$\displaystyle=\frac{1}{|S|^{2}}\sum_{u_{1}\in S}\sum_{u_{2}\neq
u_{1}}\alpha_{u_{1}}\alpha_{u_{2}}\mathbb{E}_{B}\left<B_{u_{1}}B_{u_{1}}^{\top}\mathbf{p}_{u_{1}}-\mathbf{p}_{u_{1}},B_{u_{2}}B_{u_{2}}^{\top}\mathbf{p}_{u_{2}}-\mathbf{p}_{u_{2}}\right>$
$\displaystyle+\frac{1}{|S|^{2}}\sum_{u\in
S}\alpha^{2}_{u}\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\mathbf{p}_{u}-\mathbf{p}_{u}\right\|_{2}^{2}\right]$

If $B_{u}$ are independently chosen from a distribution $\mathcal{B}$ that
satisfies:

  1. (1)

Bounded operator norm: $\mathbb{E}\left[\|B\|^{2}\right]\leq L_{\mathcal{B}}$

  2. (2)

Bounded bias:
$\|\mathbb{E}B_{u}B_{u}^{\top}\bar{\mathbf{p}}_{u}-\bar{\mathbf{p}}_{u}\|_{2}\leq\sqrt{\delta_{\mathcal{B}}}$

We have

$\displaystyle\mathbb{E}_{B}\left<B_{u_{1}}B_{u_{1}}^{\top}\mathbf{p}_{u_{1}}-\mathbf{p}_{u_{1}},B_{u_{2}}B_{u_{2}}^{\top}\mathbf{p}_{u_{2}}-\mathbf{p}_{u_{2}}\right>$
$\displaystyle=\left\|\mathbf{p}_{u_{1}}\right\|\left\|\mathbf{p}_{u_{2}}\right\|\mathbb{E}_{B}\left<B_{u_{1}}B_{u_{1}}^{\top}\bar{\mathbf{p}}_{u_{1}}-\bar{\mathbf{p}}_{u_{1}},B_{u_{2}}B_{u_{2}}^{\top}\bar{\mathbf{p}}_{u_{2}}-\bar{\mathbf{p}}_{u_{2}}\right>$
(10)
$\displaystyle=\left\|\mathbf{p}_{u_{1}}\right\|\left\|\mathbf{p}_{u_{2}}\right\|\left<\mathbb{E}B_{u_{1}}B_{u_{1}}^{\top}\bar{\mathbf{p}}_{u_{1}}-\bar{\mathbf{p}}_{u_{1}},\mathbb{E}B_{u_{2}}B_{u_{2}}^{\top}\bar{\mathbf{p}}_{u_{2}}-\bar{\mathbf{p}}_{u_{2}}\right>$
$\displaystyle\leq\left\|\mathbf{p}_{u_{1}}\right\|\left\|\mathbf{p}_{u_{2}}\right\|\|\mathbb{E}B_{u_{1}}B_{u_{1}}^{\top}\bar{\mathbf{p}}_{u_{1}}-\bar{\mathbf{p}}_{u_{1}}\|_{2}\|\mathbb{E}B_{u_{2}}B_{u_{2}}^{\top}\bar{\mathbf{p}}_{u_{2}}-\bar{\mathbf{p}}_{u_{2}}\|_{2}$
(11) $\displaystyle\leq C_{\mathbf{p}}^{2}\delta_{\mathcal{B}}$

where (A) follows since $B_{u}$ are independently sampled between users. The
second term is

$\displaystyle\frac{1}{|S|^{2}}\sum_{u\in
S}\alpha^{2}_{u}\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\mathbf{p}_{u}-\mathbf{p}_{u}\right\|_{2}^{2}\right]$
$\displaystyle=\frac{1}{|S|^{2}}\sum_{u\in
S}\alpha^{2}_{u}\|\mathbf{p}_{u}\|_{2}^{2}\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\bar{\mathbf{p}}_{u}-\bar{\mathbf{p}}_{u}\right\|_{2}^{2}\right]$
$\displaystyle=\frac{1}{|S|^{2}}\sum_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\bar{\mathbf{p}}_{u}\|_{2}^{2}+\|\bar{\mathbf{p}}_{u}\right\|_{2}^{2}-2\bar{\mathbf{p}}_{u}^{\top}B_{u}B_{u}^{\top}\bar{\mathbf{p}}_{u}\right]$
$\displaystyle=\frac{1}{|S|^{2}}\sum_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\bar{\mathbf{p}}\right\|_{2}^{2}+1-2\left\|B_{u}^{\top}\bar{\mathbf{p}}\right\|_{2}^{2}\right]$
$\displaystyle\leq\frac{1}{|S|}\max_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}\left(\mathbb{E}_{B}\left[\left\|B_{u}B_{u}^{\top}\right\|_{2}^{2}\right]+1\right)$
$\displaystyle\leq\frac{1}{|S|}\max_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}\left(L_{B}^{2}+1\right)$

∎

Next, we bound the bias and the operator norm of $B_{u}$ if it is sampled from
a Gaussian distribution in the lemma A.2.

###### Lemma A.2 (Gaussian Initialization).

Let $r<d$. Consider $B\in\mathbb{R}^{d\times r}$ be sampled from the Gaussian
distribution where $B$ has i.i.d. $\mathcal{N}(0,1/k)$ entries and a fixed
unit vector $\mathbf{v}\in\mathbf{R}^{d}$. Then

  1. (1)

Bounded operator norm:

$\mathbb{E}\|B\|^{2}\leq\frac{d}{r}\left(1+O\left(\sqrt{\frac{r}{d}}\right)\right)$

  2. (2)

Unbias: for every unit vector $\mathbf{v}\in\mathbb{R}^{d}$

$\left\|\mathbb{E}BB^{\top}\mathbf{v}-\mathbf{v}\right\|=0$

###### Proof.

Let $B^{\prime}=PB$ where $P\in\mathbb{R}^{d\times d}$ is the rotation matrix
such that $P\mathbf{v}=\mathbf{e}_{1}$. Due to the rotational symmetry of the
normal distribution, $B^{\prime}$ is a random matrix with i.i.d.
$\mathcal{N}(0,1/r)$ entries. Note that $B=P^{\top}B^{\prime}$.

$\displaystyle\mathbb{E}_{B}\left[BB^{\top}\mathbf{v}\right]$
$\displaystyle=\mathbb{E}_{B}\left[P^{\top}PBB^{\top}P^{\top}P\mathbf{v}\right]$
$\displaystyle=P^{\top}\mathbb{E}_{B}\left[B^{\prime}B^{\prime\top}\mathbf{e}_{1}\right]$

Let $\mathbf{z}=B^{\prime}B^{\prime\top}\mathbf{e}_{1}$. Notice that
$\mathbf{z}_{j}=\left<B^{\prime\top}\mathbf{e}_{j},B^{\prime\top}\mathbf{e}_{1}\right>$.
Because $B^{\prime}$ has i.i.d. $\mathcal{N}(0,1/r)$ entries,
$\mathbf{z}_{1}=\|B^{\prime\top}\mathbf{e}_{1}\|_{2}^{2}=\sum_{k=1}^{r}(B^{\prime}_{1k})^{2}$
is $1/r$ times a Chi-square random variable with $r$ degrees of freedom. So
$\mathbb{E}[\mathbf{z}_{1}]=\frac{1}{r}r=1$ and
$\mathbb{E}[\mathbf{z}_{j}]=0\forall j>1$. Thus,
$\mathbb{E}[\mathbf{z}]=\mathbf{e}_{1}$. Therefore,
$\left\|\mathbb{E}_{B}\left[BB^{\top}\mathbf{v}\right]\right\|=\left\|P^{\top}\mathbf{e}_{1}-\mathbf{v}\right\|=0.$
∎

From Proposition A.1 and Lemma A.2, we can directly get the following theorem
which bould the error of restricting the local update in a low-rank subspace
which is randomly sampled from a normal distribution.

###### Theorem A.3.

Assume $B_{u}$ is independently generated between users and are chosen from
the normal distribution $\mathcal{N}(0,1/r)$. Then,

$\displaystyle\epsilon_{i}$
$\displaystyle=\mathbb{E}_{B}\left[\left\|\bar{\Delta}_{Q}-\frac{1}{|S|}\left(\sum_{u\in
S}B_{u}\mathbf{a}_{u}\right)\right\|_{2}^{2}\right]$
$\displaystyle\leq\frac{1}{|S|}\max_{u\in
S}\alpha_{u}\|\mathbf{p}_{u}\|_{2}^{2}~{}O\left(\frac{d}{r}\right).$

This result demonstrates that the square error can increase for lower values
of the local rank $r$. Building on this insight, we suggest scaling the
learning rate of the low-rank components by $\sqrt{\frac{r}{d}}$ to counter
the error.

#### Experementing with different initialization strategies for $B$

The result in the main text is reported using $B$ sampled from a normal
distribution $\mathcal{N}(0,1/r)$. In this section, we experiment with
$\mathcal{D}_{B}$ chosen from a normal distribution, distribution of
orthonormal matrix, and Gaussian distribution on a unit sphere. The result is
s

Figure 5. Result with different initialization strategy for $B$ on
MovieLens-1M dataset

##  Appendix B Algorithm Details

In Section 4.4, we presented SCoLR to address the bandwidth heterogeneity
problem. We provide the detail of this method in Algorithm 2.

Input: Initial model $Q^{(0)}$; global update rank $r_{g}$, local update rank
$\\{r_{u}\\}$, a distribution $\mathcal{D}_{B}$ for initializing $B$;
ClientOpt, ServerOpt with learning rates $\eta,\eta_{s}$;

1

2for  _$t\in\\{0,1,2,\dots,T\\}$ _ do

3 Sample a subset $\mathcal{S}^{(t)}$ of clients

4 Sample $B^{(t)}\sim\mathcal{D}_{B}$

5 for  _client $u\in\mathcal{S}^{(t)}$ in parallel_ do

6 if  _$t >0$_ then

7 Download $A^{(t)}$

8 Merge $Q_{u}^{(t,0)}=Q^{(t-1)}+B^{(t-1)}A^{(t)}$

9

10 end if

11 Initialize $Q_{u}^{(t,0)}=Q^{(t)}$

12 Download $B^{(t)}$, Initialize $A_{u}^{(t,0)}=\bm{0}$, and Randomly sample
$S_{u}^{(t)}$

13 Set trainable parameters
$\theta_{u}^{(t,0)}=\\{A_{u}^{(t,0)},\mathbf{p}_{u}^{(t,0)}\\}$

14 for  _$k=0,\dots,\tau_{u}-1$_ do

15

16 Compute local stochastic gradient
$\nabla\mathcal{L}_{u}(\theta_{u}^{(t,k)})$

17 Perform local update
$\theta_{u}^{(t,k+1)}=\textsc{ClientOpt}\left(\theta_{u}^{(t,k)},\nabla_{\theta_{u}}\mathcal{L}_{u}(\theta_{u}^{(t,k)}),\eta\right)$

18

19 end for

20 $\mathbf{p}_{u}^{(t+1)}=\mathbf{p}_{u}^{(t,\tau_{u})}$

21 Upload $\\{S_{u}^{(t)},A_{u}^{(t,\tau_{u})}\\}$ to the central server

22

23 end for

24 Aggregate local changes
$A^{(t+1)}=\sum_{u\in\mathcal{S}^{(t)}}\dfrac{N_{u}}{N}S_{u}^{(t)}A_{u}^{(t,\tau_{u})};$

25 end for

Algorithm 2 Subsampling Correlated Low-rank Structure update (SCoLR)


# Swampland Conjectures for an Almost Topological Gravity Theory

Rafael Álvarez-García  II. Institut für Theoretische Physik, Universität
Hamburg, Luruper Chaussee 149, 22607 Hamburg, Germany  Ralph Blumenhagen  Max-
Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany
Christian Kneißl  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany  Ludwig-Maximilians-Universität München, Fakultät für Physik,
Theresienstr. 37, 80333 München, Germany  Andriana Makridou  Max-Planck-
Institut für Physik, Föhringer Ring 6, 80805 München, Germany  Lorenz
Schlechter  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München,
Germany

###### Abstract

We analyze $AdS$ and $dS$ swampland conjectures in a three-dimensional higher
spin theory with self-interacting matter, which contains conformal gravity and
is almost topological. A theory of a similar type was proposed as the
effective theory in the high energy phase of non-critical M-theory in 3D. With
some details differing from the usual string theory story, it is found that
the resulting effective theory, namely topologically massive gravity, fits
well into the web of the proposed swampland conjectures. Supporting a recent
proposal, in particular we find that this 3D theory gives rise to a quantum
break time that scales like $t_{Q}\sim H^{-1}$.

††preprint: MPP-2021-110

##  I Introduction

The swampland program [1] implies a paradigm shift in the field of string
phenomenology. Instead of analyzing the string landscape for certain well
understood classes of string compactifications, one tries to mark the boundary
between the landscape and the swampland. This is quantified in the form of a
number of swampland conjectures, which are supported by concrete string models
and/or by more general quantum gravity arguments (see [2, 3, 4] for recent
reviews). In this endeavor, our intuition is led by experience from critical
string theory or M-theory and their low-energy effective actions. As such, we
have a universal gravity sector that is governed by an Einstein-Hilbert action
that upon quantization leads to gravitons interacting with a strength set by
the Planck mass.

The purpose of this letter is to consider effective gravity theories that are
of a different kind but have still appeared in the string theory literature,
hence having a good chance to be in the string landscape. We want to challenge
such theories with some of the swampland conjectures, in particular the $AdS$
and $dS$ swampland conjectures, as well as the proposal of quantum breaking.

One instance where such gravity theories of a different kind are expected to
appear is in the high energy or high temperature regime of string theory.
Unfortunately, for temperatures beyond the Hagedorn transition these theories
are not completely understood, but there are indications, like the quadratic
scaling of the free-energy with temperature [5], that the number of degrees of
freedom gets reduced. This letter is also motivated by the recent proposal [6]
that in the high temperature regime of string theory, the quantum break time
should scale like $t_{Q}\sim H^{-1}$ irrespective of the number of dimensions.
Recall that, as shown in [7], for Einstein gravity in $d$ space-time
dimensions this scaling is $t_{Q}\sim M_{\rm pl}^{d-2}/H^{d-1}$.

Clearly, for making progress we need a concrete and well treatable starting
ground. For our purposes, this is provided by the non-critical M-theory in 3D
proposed by Hořava/Keeler [9, 10, 11] which is exactly solvable in terms of a
non-relativistic Fermi liquid in 2 +1 dimensions. This is motivated by the
known tachyon condensation from ten-dimensional type 0 theories, which
represent the infinite temperature limit of type II theories, to the two-
dimensional type 0 theories [8]. The non-critical M-theory arises from the
thermal version of this two-dimensional theory when D0-branes are included [9,
10, 11]. In the conformal limit this theory was suggested to be described by
an effective higher spin generalization of conformal gravity equipped with a
higher spin matter field. The ground state of non-critical M-theory
corresponds to an $AdS_{2}\times S^{1}$ solution of this theory plus a
massless fermion. Without the matter field, the theory would be topological
with no propagating degrees of freedom. Note that such topological theories
were also invoked in the recent proposal [12] of a topological phase in the
early universe.

Here we will also include a conformally coupled, scalar matter field that is
expected to decouple in the strict conformal limit, but is still present at
large but finite energies. By also including a conformal self-interaction of
the scalar field one can find non-trivial $AdS_{3}$ and $dS_{3}$ solutions for
a non-vanishing vacuum expectation value of the scalar. The resulting
effective theory is known as topologically massive gravity (TMG) [13],
carrying the wrong sign of the Einstein-Hilbert term. In this letter we
consider this almost topological theory as a toy model for the high energy
phase of string theory and analyze whether it does still satisfy in particular
those swampland conjectures dealing with $dS$ and $AdS$ solutions.

##  II Non-critical M-theory in 3D

Three-dimensional non-critical M-theory was proposed in [9, 10] as the strong
coupling limit of non-critical type 0A theory in 2D. In [9] a non-perturbative
definition of this theory was formulated as a double-scaled non-relativistic
Fermi liquid in $2+1$ dimensions. This theory is closely related to the matrix
model description of 2D type 0A theory by summing over all D0-brane charges.
Moreover, it has an infinite dimensional symmetry described by a bosonic
higher-spin Lie algebra, called ${\cal W}_{0}$ in [10]. Important in the
following is that the conformal algebra $SO(3,2)$ is a finite dimensional
subalgebra of ${\cal W}_{0}$. For more details we refer the reader to the
original literature.

In [10] it was found that in the high-temperature regime the non-perturbative
partition function is closely related to the one of the topological A-model on
the resolved conifold. In the conformal limit
($\alpha^{\prime}\rightarrow\infty$) it was argued [11] that the ground state
of non-critical M-theory should be an $AdS_{2}\times S^{1}$ space-time
accompanied by a massless free fermion as the only propagating degree of
freedom.

As proposed in [11], a higher spin theory admitting such a solution is given
by the higher spin Chern-Simons theory [14]

$\begin{split}S_{\rm HCS}={1\over 4}\int_{M}{\rm Tr}\left({\cal A}\wedge
d{\cal A}+{2\over 3}{\cal A}\wedge{\cal A}\wedge{\cal A}\right)\end{split}$
(1)

with the gauge field one-form ${\cal A}$ taking values in the infinite-
dimensional higher-spin Lie-algebra ${\cal W}_{0}$. As shown in [15], matter
fields can be coupled to this topological theory at least at the level of the
matter field equations of motion in a CS background field ${\cal A}$ via

$\begin{split}d|\Phi\rangle+{\cal A}\star|\Phi\rangle=0\,,\end{split}$ (2)

where $|\Phi\rangle$ is a higher spin matter field that contains a scalar and
a fermion as the lowest components. In an $AdS_{2}\times S^{1}$ (or also
$AdS_{3}$) background with all higher spin fields set to zero, all these
equations reduce to two equations of motion. One is the Klein-Gordon equation
for a conformally coupled massless scalar field and the other is the Dirac
equation for a massless fermion. Because of the absence of a propagating
scalar degree of freedom in the conformal limit, the scalar field was set to
zero by hand in [11]. The scalar is related to the “tachyon” in the effective
2D type 0A theory and is only expected to completely decouple in the strict
$\alpha^{\prime}\to\infty$ limit, which could be made manifest by setting
$\phi=\varphi/(\alpha^{\prime})^{1/4}$ in the following.

###  II.1 Conformal gravity with matter

The theory of interest in this paper is a truncation of this higher spin
theory where we consider just the spin-2 mode and the scalar and fermionic
matter fields. Actually, the fermion will not play any role in the following.
Recall that restricting the higher spin CS theory (1) to just the spin-2 mode
one gets conformal gravity in 2+1 dimensions [16], a topological gravity
theory governed by the Chern-Simons action

$\begin{split}S^{SO(3,2)}_{\rm CS}={1\over 4}\int_{M}{\rm Tr}\left(A\wedge
dA+{2\over 3}A\wedge A\wedge A\right)\end{split}$ (3)

with the one-form $A$ taking values in the 3D conformal algebra $SO(3,2)$. The
resulting equation of motion says that the Cotton tensor is vanishing

$\begin{split}C_{\mu\nu}=\epsilon_{\mu}{}^{\alpha\beta}\,\nabla_{\alpha}\Big{(}R_{\beta\nu}-{1\over
4}g_{\beta\nu}R\Big{)}=0\,,\end{split}$ (4)

which is satisfied for a conformally flat metric. Note that being topological,
this theory does not admit propagating degrees of freedom. Moreover, by being
conformal it does not have a dimensionful gravitational coupling like the
Planck mass.

Now, one can couple this gravity theory to fermionic and bosonic matter
fields. Here we focus on a scalar field, whose action takes the following form

$\begin{split}S_{\rm eff}=S^{SO(3,2)}_{\rm CS}+S_{\rm ferm}-\int
d^{3}x&\sqrt{-g}\Big{(}{1\over
2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\\\ &+{\xi_{3}\over
2}R\,\phi^{2}+{\kappa\over 6}\phi^{6}\Big{)}\,\end{split}$ (5)

with $\xi_{3}=1/8$ and where we also added a conformal, sixth order self-
coupling of the field $\phi$ with a dimensionless coupling constant $\kappa$.
From now on we suppress the contribution from the fermionic matter field. One
can show that this action is indeed invariant under a local Weyl-
transformation

$\begin{split}g_{\mu\nu}\to e^{2\Lambda(x)}g_{\mu\nu}\,,\qquad\phi\to
e^{-\Lambda(x)/2}\phi\,.\end{split}$ (6)

It was argued in [17] that also for the higher spin generalization, such an
interaction term can be consistently introduced. Varying now $S_{\rm eff}$
with respect to the metric, one arrives at the gravity equation of motion
$C_{\mu\nu}-T^{\rm bos}_{\mu\nu}=0$, that in detail takes the form

$\begin{split}&C_{\mu\nu}-{\phi^{2}\over 8}\Big{(}R_{\mu\nu}-{1\over
2}g_{\mu\nu}R\Big{)}+{\kappa\over 6}\phi^{6}g_{\mu\nu}\\\\[2.84544pt]
&-\Big{(}\partial_{\mu}\phi\partial_{\nu}\phi-{1\over
2}g_{\mu\nu}(\partial\phi)^{2}\Big{)}-{1\over
4}\Big{(}\phi\Box\phi+(\partial\phi)^{2}\Big{)}g_{\mu\nu}\\\ &+{1\over
4}\Big{(}\phi\nabla_{\mu}\nabla_{\nu}\phi+\partial_{\mu}\phi\partial_{\nu}\phi\Big{)}=0\,.\end{split}$
(7)

It is remarkable that the Einstein tensor reappears in the equation of motion,
though not from varying the Chern-Simons gravity action but from varying the
action for the conformally coupled scalar. However, the sign in front of the
Einstein tensor is opposite so that effectively gravity acts repulsively. If
$\phi$ carries a non-vanishing vacuum expectation value $\phi_{0}$, then the
prefactor of the Einstein tensor can be thought of as an effective Planck
scale

$\begin{split}\widetilde{M}_{\rm pl}=-{\phi_{0}^{2}\over 8}\,.\end{split}$ (8)

Taking the trace of (LABEL:eommetric) one finds an expression that vanishes if
the equation of motion for the scalar field is obeyed

$\begin{split}\Box\phi-{1\over 8}R\phi-\kappa\phi^{5}=0\,.\end{split}$ (9)

This is of course a consequence of the underlying Weyl symmetry.

###  II.2 AdS and dS solutions

Let us consider potential $AdS_{3}$ and $dS_{3}$ solutions of radius $\ell$.
These are conformally flat and have a constant Ricci scalar, so the equation
of motion for $\phi$ admits the non-trivial solution

$\begin{split}\phi_{0}^{4}={3\over 4|\kappa|\ell^{2}}\,.\end{split}$ (10)

Here $AdS_{3}/dS_{3}$ has curvature $R=\mp 6/\ell^{2}$ so that one needs
$\kappa>0$ for $AdS_{3}$ and $\kappa<0$ for $dS_{3}$. One can show that also
the gravity equation of motion (LABEL:eommetric) is satisfied.

Since $\ell$ sets a length scale, this can be thought of as breaking the Weyl
symmetry spontaneously, with the flat direction in the $(\phi,\ell)$ plane
giving a Goldstone mode. Note that conformal gravity equipped with an
additional scalar field with non-vanishing vacuum expectation value
$\phi_{0}\neq 0$ is nothing else than the well known theory of topologically
massive gravity (TMG), whose action is usually written as

$\begin{split}S_{\rm TMG}&={\widehat{M}_{\rm pl}\over 2}\int
d^{3}x\Big{[}\sqrt{-g}\big{(}R-2\Lambda\big{)}\\\ &+{1\over
2\mu}\epsilon^{\mu\nu\rho}\Big{(}\Gamma^{\alpha}_{\mu\beta}\partial_{\nu}\Gamma^{\beta}_{\rho\alpha}+{2\over
3}\Gamma^{\alpha}_{\mu\gamma}\Gamma^{\gamma}_{\nu\beta}\Gamma^{\beta}_{\rho\alpha}\Big{)}\Big{]}\,.\end{split}$
(11)

Note that the Weyl symmetry can be used to eliminate fluctuations of the field
$\phi$. By comparison, one finds a negative Planck scale $\widehat{M}_{\rm
pl}=\widetilde{M}_{\rm pl}$ and

$\begin{split}\mu={\phi_{0}^{2}\over 8}={1\over
16\ell}\sqrt{3\over|\kappa|}\,,\qquad\Lambda=\mp{1\over\ell^{2}}\,.\end{split}$
(12)

This theory was first introduced in [13], where it was shown that around a
Minkowski solution it has one massive propagating spin-2 degree of freedom of
mass $m_{(2)}=\mu$. Thus, as expected for a local symmetry, the Goldstone mode
does not survive as a massless mode.

##  III Swampland conjectures

In this section we confront this effective almost topological theory with
various swampland conjectures, in particular those dealing with $AdS$ and $dS$
solutions.

###  III.1 AdS swampland conjectures

Thus, we consider the case $\kappa>0$ so that the action allows us to read off
an effective potential

$\begin{split}V(\phi)=-{3\over 8\ell^{2}}\phi^{2}+{\kappa\over
6}\phi^{6}\,.\end{split}$ (13)

The form of this potential is shown in figure 1, where the non-trivial
solution (10) for $\phi_{0}$ is evident.

$\phi$$V$

Figure 1: The potential $V(\phi)$ for $\kappa=1$ and $AdS$ length scale
$\ell={10}$.

It can readily be checked that the $AdS$ minimum satisfies the $AdS$/moduli
scale separation conjecture [18] which in our case says $m_{\phi}\,\ell\leq
c$, for some $O(1)$ constant $c$. Just from $V(\phi)$ we find
$m^{2}_{\phi}=3/\ell^{2}$.

Next, we take a look at the $AdS$ distance conjecture [19]. From (12) we have
for the cosmological constant $|\Lambda_{\rm AdS}|\sim\phi_{0}^{4}$. Therefore
it vanishes for $\phi_{0}=0$ which, opposed to the $AdS$ distance conjecture,
is at finite distance in the moduli space.

The $AdS$ distance conjecture also says that there should be a tower of light
states with scaling $m\sim|\Lambda|^{\alpha},\alpha>0$. From the (refined)
swampland distance conjecture [20, 21], one expects a tower of states with
masses

$\begin{split}m=m_{0}\,e^{-{\Delta\phi/\sqrt{|\widetilde{M}_{\rm
pl}}|}}\sim\phi_{0}^{2}\sim{1\over\ell}\sim\Lambda_{\rm AdS}^{1\over
2}\end{split}$ (14)

where we have used that the only mass-scale in the problem is
$m_{0}\sim\phi_{0}^{2}$ and the effective Planck scale (8).

Since non-critical M-theory is defined in 3D, there is no compactifcation and
hence no tower of Kaluza-Klein modes involved here. A natural candidate for
this tower of states are the infinitely many higher spin fields in the Chern-
Simons action. For the spin-2 field, the mass of the single massive degree of
freedom of TMG with a negative cosmological constant was computed in [22, 23]

$\begin{split}m^{2}_{(2)}=\Big{(}\mu+{2\over\ell}\Big{)}^{2}-{1\over\ell^{2}}={1\over\ell^{2}}(\mu\ell+3)(\mu\ell+1)\end{split}$
(15)

with $\mu\ell=\sqrt{3/(256\kappa)}$. This can be thought of as the
$\mu\ell$-corrected mass of the field $\phi$ that we read off from the
effective scalar potential (13) as $m^{2}_{\phi}=3/\ell^{2}$. Moreover, the
relation (15) correctly reproduces the mass of the graviton in the Minkowski
limit $\ell\to\infty$.

Since we have $\mu\ell>0$, this mass can never vanish. Thus, our case is
different from the chiral model discussed in [24], which in contrast to our
case had a positive Planck mass. Taking some results for higher spin
topologically massive gravity [25, 26, 27] into account, a natural conjecture
for the mass of the higher spin fields would be

$\begin{split}m^{2}_{(s)}={(s-1)\over\ell^{2}}\Big{(}(s-1)\mu\ell+(s+1)\Big{)}\Big{(}\mu\ell+1\Big{)}\,,\end{split}$
(16)

which for $s\gg 1$ indeed gives the desired scaling $m_{(s)}\sim s\mu\sim
s\phi_{0}^{2}$. Of course, a more thorough computation in the framework of the
full ${\cal W}_{0}$ higher spin theory is necessary to really confirm this
relation.

Therefore, consistent with the finite distance in moduli space, there is only
a polynomially and not an exponentially light tower of states. This is
analogous to a similar polynomial scaling found for the finite distance
conifold point in [28]. Similar to that case, applying the emergence proposal
[29, 30, 31] will lead us to a field dependent UV cut-off

$\begin{split}\Lambda_{\rm UV}\sim\phi_{0}^{2}\sim{1\over\ell}\end{split}$
(17)

so that the number of light species is constant and does not depend on the
field $\phi_{0}$. This cut-off is also consistent with the spin-2 swampland
conjecture proposed in [32].

For 3D gravity with negative cosmological constant, it is known that there
exist BTZ black-holes which are locally isometric to $AdS_{3}$ space.
Therefore, they will also be solutions of TMG and also of the full equations
of motion (LABEL:eommetric) and (9). However, with the wrong sign of the
Einstein-Hilbert term they will carry negative energy and therefore signal a
non-perturbative instability of the $AdS_{3}$ background. Let us remark that
this is consistent with the proposed instability [33] of non-supersymmetric
$AdS$-backgrounds.

To summarize, up to the finite distance in moduli space, this almost
topological higher spin theory has all the ingredients to satisfy the $AdS$
swampland conjectures.

###  III.2 dS swampland conjecture

Now let us choose $\kappa<0$ and consider the non-trivial $dS_{3}$ solution.
Writing $\ell=1/H$ the action features a potential

$\begin{split}V(\phi)={3\over 8}H^{2}\phi^{2}-{|\kappa|\over
6}\phi^{6}\,\end{split}$ (18)

which is just the one from figure 1 flipped at the $x$-axis. Therefore, we
have a $dS$ maximum and can check whether the (refined) $dS$ swampland
conjecture [34, 35, 36] in 3D is satisfied, i.e. in particular the relation

$\begin{split}M_{\rm pl}|V^{\prime\prime}|>c^{\prime}\,V\,.\end{split}$ (19)

Computing both sides we find $V(\phi_{0})={\kappa\over 3}\phi_{0}^{6}$ and
$V^{\prime\prime}(\phi_{0})=-4\kappa\phi_{0}^{4}=-3/\ell^{2}$. Using now the
effective Planck scale (8) we get that both sides scale as $\phi_{0}^{6}$ and
that (19) is satisfied if $c^{\prime}<3/2$. The value of $c^{\prime}$ is not
known. One proposal [6] is $c^{\prime}=1/(D-1)(D-2)$ which for $D=3$ yields
$c^{\prime}=1/2$.

###  III.3 Quantum break time

It has been proposed in [7, 37, 38] that $dS$ solutions in quantum gravity
experience the phenomenon of quantum breaking. This is argued to come from the
decoherence of the $dS$ bound state of graviton species.

First we notice that since $dS_{3}$ is conformally flat, it is a solution to
pure conformal gravity and without any additional propagating modes present it
would be eternal and no quantum breaking would occur. However, after
conformally coupling it to a scalar field, one finds a massive spin-2 degree
of freedom so that a priori quantum breaking can occur.

Since the only scale in the problem is set by $H$ or $\phi_{0}^{2}$,
respectively, just by dimensional reasoning the quantum break time should
scale as $t_{Q}\sim H^{-1}$. At first sight this is different from the quantum
break time $t_{Q}\sim{M_{\rm pl}/H^{2}}$ obtained in [7] for a three-
dimensional Einstein gravity theory. However, taking into account the
effective Planck scale (8) of TMG, the two expressions indeed agree.

This finding is consistent with the proposal in [6], namely that in the high
temperature regime of string theory, the quantum break time should scale like
$t_{Q}\sim H^{-1}$ irrespective of the number of dimensions. Due to the
censorship of quantum breaking, there should be a classical mechanism leading
to a faster decay. From the form of the potential in figure 1 it is clear that
there exists a tachyonic instability (in the $\phi$ direction) of mass
$m^{2}=-3H^{2}$ leading indeed to a decay time $t_{\rm dec}\sim H/|m|^{2}\sim
H^{-1}$.

###  III.4 Higuchi bound

Finally, let us make a short comment on the Higuchi bound [39, 40]. In
particular, it says that in a $d$-dimensional $dS$ background there exists a
lower bound for the mass of an helicity $t$ mode of a massive field of spin
$s$

$\begin{split}m_{(s,t)}^{2}\geq H^{2}(s-t-1)(s+t+d-4)\,.\end{split}$ (20)

Since in our case the sign of the Einstein-Hilbert term is reversed, we expect
that ghost freedom in $dS$ is guaranteed in the opposite regime, i.e. for the
helicity $t=s$ mode in 3D we get

$\begin{split}m_{(s,s)}^{2}\leq-H^{2}(2s-1)\,.\end{split}$ (21)

Next we need to know the mass of the spin-2 mode for TMG. Redoing the
computation from [22, 23] for positive cosmological constant
$\Lambda=\ell^{-2}=H^{2}$, we find

$\begin{split}m^{2}_{(2)}=-\Big{(}\mu+{2\over\ell}\Big{)}^{2}+{1\over\ell^{2}}\,,\end{split}$
(22)

which correctly reproduces the mass of the tachyonic mode in the $\mu\ell\to
0$ limit. This calls for the higher spin generalization

$\begin{split}m^{2}_{(s)}=-\Big{(}(s-1)\mu+{s\over\ell}\Big{)}^{2}+{1\over\ell^{2}}\leq(-s^{2}+1)H^{2}\,.\end{split}$
(23)

which for $s\geq 2$ indeed satisfies the (sign reversed) Higuchi bound (21).

##  IV Conclusions

In this letter we have studied a well motivated (toy) model for the high
energy phase of string theory regarding its consistency with some of the
swampland conjectures. We considered the truncation of the higher spin Chern-
Simons theory to just the spin-2 mode equipped with a conformal, self-
interacting scalar field. For this model we found $AdS$ and $dS$ vacua that,
up to some slight modifications, have a good chance to satisfy the known
swampland criteria. Some steps like the determination of the masses of the
higher spin modes were left open. It would also be interesting to consider
similar models of this type, as for instance the four-dimensional Weyl-squared
action [41, 42, 43].

The result of this letter can be interpreted in two ways. First, it
demonstrates that there is a good chance for the existence of non-standard
effective gravity theories in the landscape of string theory. Second, it gives
credence to the proposal that the high energy phase of string theory does also
satisfy the swampland criteria.

Turning the logic around and using the swampland conjectures as a guide to the
high temperature phase, we conclude that a purely topological theory without
any propagating degrees of freedom might be too simple.

_Acknowledgements:_ We thank Dieter Lüst for useful comments.

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