Federated Sampling with Langevin Algorithm under Isoperimetry
Authors: Lukang Sun
Adil Salim
Peter Richtárik
Venue: Accepted by TMLR ()

Abstract: Federated learning uses a set of techniques to efficiently distribute the training of a machine learning algorithm across several devices, who own the training data. These techniques critically rely on reducing the communication cost---the main bottleneck---between the devices and a central server. Federated learning algorithms usually take an optimization approach: they are algorithms for minimizing the training loss subject to communication (and other) constraints. In this work, we instead take a Bayesian approach for the training task, and propose a communication-efficient variant of the Langevin algorithm to sample \textit{a posteriori}. The latter approach is more robust and provides more knowledge of the \textit{a posteriori} distribution than its optimization counterpart. We analyze our algorithm without assuming that the target distribution is strongly log-concave. Instead, we assume the weaker log Sobolev inequality, which allows for nonconvexity.

Published in Transactions on Machine Learning Research (01/2024)
Federated Sampling with Langevin Algorithm under Isoperimetry
Lukang Sun
lukang.sun@koust.eddie edw.
King Abdullah University of Science and Technology
Adil Salim
odilsalimm&mmicrroofff..oom
Microsoft Research
Peter Richtárik
petern.ichttrrikkkauss..ee
King Abdullah University of Science and Technology
Reviewed on OpenReview: https: // http://eepervveeeeeeomee net/ farum? id= Sy7FFPeR6
Abstract
Federated learning uses a set of techniques to efficiently distribute the training of a machine learning algorithm across several devices, who own the training data. These techniques crit- ically rely on reducing the communication cost. the main bottleneck betteeeeeebeeeeen between the devices and central server. Foderated learning algorithms usually take an optimization approach: they are algorithms for minimizing the training loss subject to communication (and other) constraints. In this work, we instead take Bayesian approach for the training task, and propose a communictionn-ffiiinn variant of the Langevin algorithm to sample posteriori. The latter approach is more robust and provides more knowledge of the posteriori distri- bution than its optimization counterpart. We analyze our algorithm without assuming that the target distribution is strongly log-concave. Instead, we assume the weaker log Sobolev inoquality, which allows for nonconvexity.
1 Introduction
Federated learning uses a set of techniques to efficiently distribute the training of a machine learning algorithm across several devicos, who own the training data. In a fedorated learning environment, several considerations are considered, such as privacy or communication between the devices (Kairouz al., 2021). ). In icuticular, the communication cost between the devices and central server (Konečný al. 2016 McMahan et al. 2017)) is one of the main bottlenecks in federated learning environment. Indeed, classical training algorithms such the Stochastic Gradient Descent (SGD) can formally be distributed between the server and the devices. But this requires communications between the devices and the server, raising communication concerns. Therefore, most federated learning algorithms rely on compressing the messages that are exchanged between the server and the devicos. For instance, variants SGD involving compression (Karimireddy et al. 2020:
Horváth et. al. 2019b; Haddadpour al.] 2021 Das et al., 2020 Gorbunov el al. 2021) have successfully
been applied in several engineering setups [Kirrouz al. 2021 Dimitriadis et al. 2022
Sampling in federated learning environment. Most of the literature the training of federated learning learning algorithm focused the optimization paradigm, i.e. the training algorithm is actually communication efficient optimization algorithm to minimize the training loss F. But, optimization is not the only paradigm in the training of learning algorithms. The Bayesian approach relies on sampling from a distribution π proportional to exp(-F) rather than minimizing F. particular, the Bayesian approach provides a number of samples from exp(-F) rather than single minimizer of F. The distribution æ is often the the posterior distribution of some Bayesian model. In this case, sampling from æ allows perform some Bayesian computations such as computing confidence intervals.

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Published in Transactions on Machine Learning Research (01/2024)
In this paper, take a Bayesian approach to federated learning, and similarly to Vono et al (002)) Deng et al. (2021) El Mekkaoui et al. (2021), we focus on the communication complexity (we ignore privacy concerns). We introduce a communication efficient variant of the Langevin algorithm, a widely used algorithm to sample from a target distribution π x cxp(-F) where F is a smooth nonconvex function but т satisfics the log-Sobolev inequality (Villani 2009). We study the complexity of our communication efficient. algorithm sample from æ by exploiting the connections between optimization and sampling. More precisely, our approach has roots in the recent literature on viewing Langevin algorithm an optimization algorithm over space of probability distributions called the Wasserstein space (Wibisono 2018 Ambrosio et al. 2005
Durmus et al. 2019
1.1 Related Work
Most approaches to train federated learning algorithms rely minimizing the training loss with communi-
cation efficient variant of SGD. Several training algorithms have been proposed, but we mention Karimireddy
ct al. (020);; Horváth et al. (2019b); Haddadpour et al. (2021); : Das ct al. (2020); Gorbunov et al. (2021)
because these papers contain the state of the art results in terms of minimization of the training loss. In
this work we shall specifically use the MARINA gradient estimator introduced in Gorbunov et al. 20011 and inspired from Nguyen al. (2017).
The literature on sampling with Langevin algorithm is also large. In recent years, the machine learning research community has specifically been interested in the complexity of Langevin algorithm (Dalalyan
2017 Durmus & Moulines 2017). In the case where F is not convex, recent works include Vempala &
Wibisono )
Wibisono (2019); Ma et al. (2019): ): Chewi et al. (202)) which use an isoperimetric-yyp
inequality (Villani 2009 Chapter 21) such as the log Sobolev inequality to prove convergence in Kullback-
Leibler divergence.
Other works on Langevin in the nonconvex case include Balasubramanian et al. (0022)
Cheng et al.
20188:
Majka et al. (2020): Mattingly al. (2002).
work Vono et al. (2022 Deng et al. (2021). Like work, these papers study the convergence of an efficient variant of Langevin algorithm for federated learning, The key difference between these papers and our work is that they assume F strongly convex, whereas we only assume that the target distribution & exp(-F) satisfies the log Sobolev inequality (LSI), which allows F to be nonconvex. More preciscly, the strong convexity of F implics LSI (Villani 2009 Chapter 21), but LSI allows F to be nonconvex. We compare our complexity results to their Table
The closest papers
Paper
Algorithm
Communication
Assumption
Vono et al.
2022)
QLSD ++
Reduced
Strong convexity
Deng et. al.
2021
(FA-LD)
Reduced
Strong convexity
Wibisono 2019
Langevin
Expensive
log Sobolev inequality
This paper
Langevin-Marina
Reduced
log Sobolev inequality
Table 1: Lanievin-Marina is the only algorithm in the list that works under LSI only, with reduced communication complexily. The four algorithms in the list have the same complexity Ô (d) to achieve 8 accuracy in 2- 2-sassesttein distance in dimension d.
Our contributions summarized below.
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1.2 Contributions
We consider the problem of sampling from exp(-F), where и satislies LSI, in a federated learning setup. We make the following contributions:
We propose a communication efficient variant of Langevin algorithm, called Langevin-MARINg that can be distributively implemented between the central server and the devices. Langevin-MARIN, relics on compressed communications. More preciscly, we borrow the MARINA estimator used in optimization to perform Sampling.
We analyze the complexity of our sampling algorithm in terms of the Kullback-Leibler divergence, the Total Variation distance and the 2- Wassseesttein distance. Our approach relies on viewing our sampling problem as an optimization problem over a space of probability measures, and allows F to be nonconvex.
Our sampling algorithm, Langevin-MARINA, is inspired from an optimization algorithm called MARINA (Gorbunov et al. 2021) for which we give new convergence proof in Appendix B This new proof draws connections between optimization (MARINA) and sampling Llangevin-MARIn
1.3 Paper organization
In Section 2
we review some background material on sampling and optimization. Next, we introduce our federated learning setup in Section 3 In Section 4 give our main algorithm, Langevin-MARINA, and main complexity results. We conclude in Section The proofs, including the new proofs of existing results for MARINA and the experiment results are deferred to the Appendix.
2 Preliminaries
2.1
Mathematical problem
Throughout this paper, we consider a nonconvex function F; Rd + R which is assumed to be L-smooth, i.e., differentiable with an L-Lipschitz continuous gradient: [[V((x) FF()) s L || yll- As often in machino
learning, can be seen as a training loss and lakes the form of a finite sum
F
(1)
where each F; can be seen the loss associated to the dataset stored in the device i. Assuming that f ('oxp(FF(r)ddr E (0, oo), we denote by
x exp(-F),
(2)
the probability distribution whose density is proportional exp(-F). We take a Bayesian approach: in- stead of minimizing F, our goal is to generate random samples from म, which can be seen as the posterior distribution of some Bayesian model.
2.2 Sampling and Optimal Transport
We denote by (2(Rd) the set of Borel measures a on Rd with finite second moment, that is f || do +oo. For every O.V (2( I(a,v) is the set of all the coupling measures between and Rd x Rd that is y T( w) if and only if (dxr) = y(dx, Rd) and v(dy) = >RRd ,dy). The Wasserstein distance bet a and y is defined by
inf / ||x - ull <dxx, dy). (e)())))
Wz(a.v) = V
(3)
The Wasserstein distance W2(..) is metric on (R(RR) and the metric space (P2(R)), W2) is called the Wasserstein space, see Ambrosio et al. (0055)
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The Kullback-Leibler (KL) divergence w.r.t. can be seen the map from the Wasserstein space to (0, oo] defined for every E P22 (R2) by
[ x
[oog()) if H else,
DEL (a ④) =
(4)
where a < π means that a absolutely continuous w.r.t. π and & denotes the density of a w.r.t. n. The KL divergence is always nonnegative and it equal to zero if and only o= x. Therefore, assuming that (R)))) can be seen the solution to the optimization problem
min DKL. (a |T). OEP:(B4)
(5)
Viewing as minimizer of the KL divergence is the cornerstone of our approach. Indeed, we shall view the proposed algorithm as optimization algorithm to solve Problem equalion 5 In particular, follow- ing [iibisono (2018): Durmus et al. (2019), we shall view the Langevin algorithm as first order algorithm over the Wasserstein space. In particular, the relative Fisher information, defined as
V log( (z)) (a(x)) Jx (o) 00,
i
(6)
else,
will play the role of the squared norm of the gradient of the objective function. Indeed, by defining differential structure over the Wasserstein space (Ambrosio et al. 2005), the relative Fisher information is the squared norm of the gradient of the KL divergence.
The analysis of nonconvex optimization algorithm minimizing F often relies on gradient domination con- dition relating the squared norm of the gradient to the function values, such as the following Lojasiewicz condition (Blanchet & Bolte 2018: Karimi et al. 2016)
F(x) - min F s 1 [VF(
2μ
The latter condition has a well-known analogue over the Wasserstein space for the KL divergence, called logarithmic Sobolev inequality.
Assumption 2.1 (Logarithmic Sobolev Incquality (LSI)). The distribution a satisfies the logarithmic Sobolev inequality with constant jc: for all a E (2(
DKL (a m) m) S Jz (o).
(8)
24
The log Sobolev inequality has been studied in the optimal transport community [Villani 2009 Chapter 21) and holds for several target distributions т. First, if is p-strongly convex (we say that и is strongly log-concave), then LSI holds with constant μ. Besides, LSI is preserved under bounded perturbations and Lipschitz mappings (Vempala & Wibisono 2019 Lemma 16, 19). Therefore, small perturbations of strongly- log concave distributions satisfy LSI. For instance, if we add to Gaussian distribution another Gaussian distribution with small weight, then the resulting mixture of Gaussian distributions is not log-concave but it satisfies LSI. One can also find compactly supported examples, see Wibisono 2019 Introduction).
LSI is a condition that has been used in the analysis of the Langevin algorithm
(9) where > 0 is a step size and (Zx) sequence of i.i.d standard Gaussian vectors over Rd, Langevin algorithm can be seen as a gradient descent algorithm for F to which a Gaussian noise is added at each step. For instance, Wibisono (2019) showed that the distributions Pk: of m converges rapidly towards the target distribution æ in terms of the KL divergence under LSI. Other metrics have also been considered, such a the Total Variation distance
Tk-1 Ih - (VV(())))))))))) + V2h7*+1,
||o -||TV
sup AEB(Rd)
(o(A) x(A),,
(10)
where B(RRd) denotes the Borel sigma field of Rd.
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Algorithm MARINA (Gorbunov et al. 2021
1: Input: Starting point To, step-size h, number of iterations K
for y= 1, 2, .... , n in parallel do
Device computes MARINA estimator gó Device i uploads 3 to the central server
end for
6: Server aggregales 90 E %o
7: for A: = =, 0, 2, ... K - do
8: Server broadcasts Ok to all devices i
9: for = 1,2, ............ n in parallel do
10: Device i performs JĄ-1 = Iy hgg
11: Device i computes MARINA estiamtor g
12: Device i uploads g6+1 to the central server
13: end for
14: Server aggregates 0k+1 E gh+1
15: end for
16: Return: g chosen uniformly at random from (ax) or last point æk
3 Federated learning
3.1 Example of federated learning algorithm: MARINA
We now describe our federated learning setup. A central server and number of devices are required to training algorithm in a distributed manner. Each device i E [l.....n} owns a dataset. A meta federated learning algorithm as follows: at each iteration, each device performs some local computation (for instance, a gradient computation) using its own dataset, then each device sends the result of that computation the central server. The central server aggregates the results and sends that aggregation back to the devices. Algorithm | provides an example of federated learning algorithm. This algorithm, called MARINA, was introduced and analyzed in Gorbunov al. 2021 under the assumption that the sequence (gk) is MARINA estimator of the gradient as delined below.
Definition 3.1. Given sequence produced by an algorithm (Ix)x, MARINA estimator of the gradient is a random sequence (gk) satisfying [ggl] = EVVFF(())) and
G S GR + (1 T.2 [|02411 11:1--x-0 =x12] +
(11)
where G2 = E [19x VF(x))] and 0 <<pspp a, a,000
MARINA iterations can be rewritten
Tk+1 = Th hgg.
(12)
Since E [gg] E (**)),, MARINA can be seen as a SGD distributed between the central server and the devices, to minimize the nonconvex function F.
Recall that in federated learning, the main bottleneck is the communication cost between server and devices. The key aspect of MARINA is that il achieves the state of the art in terms of communication complexity among nonconvex optimization algorithms. The reason is the following: there are practical examples of MARINA estimators (gk) that rely on a compression step before the communication and are therefore cheap to communicate
Note that we did specify how of computed Algorithm For the moment we only require ge to MARINA estimator

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3.2 Compression operator
MARINA estimators are lypically cheaper to communicate than full gradient because they involve a com- pression step, making the communication light. A compression operator is a random map from R to F with the following properties.
Definition 3.2 (Compression). A stochastic mapping Q : Rd - Rd is a compression operator if there exists w > 0 such that for any x E R2,
E [[O(r)] = E [1[((a) ==l? s w||bo|??
(13)
The first equation states that is unbiased and the second equation states that the variance of the com- pression operator has a quadratic growth. There are lots of compression operators satisfying equation [3 explicit examples based on quantization and/or sparsification are given in Alistarh et al. (2017) Horváth et al. (019a). In general, compressed quantitics are cheap communicate.
We now fix compression operator and give examples of gradient estimators involving compression, which are provably MARINA estimators.
3.3 MARINA gradient estimators
In every examples presented below, gle := Ei= ghe where g is meant to approximate (..) i.c., the gradient at the device i, at step k. We consider any random sequence (Ix) generated by a training algorithm.
3.3.1 Vanilla MARINA gradient estimator
The vanilla MARINA gradient estimator is defined as the average Ib. == Ei where for every i e {1....,n) n1, Φ6 = (zo) and
prob. p >>6 9h + ((P(((14) - VP(())) prob. 1-p
(F((1+11)
(14)
Node randomly (and independently of 2, 11) computes and uploads to the server the gradient of F; at point 7&+1 with probability p. Else, node i computes and uploads the compressed difference of the gradients (x(()))) Note that the server already knows O from need to send Ok to the server. In average, device computes full gradient
of F; between points # and ry: (VF, the previous iteration, every 1/p steps (p is very small, see Remark 3.6 below).
Proposition 3.3. Assume that F; is -m-sohhth for all i E { [I....,n), i.e.,
(VF()) - VF( L,||x- - yll. Vx, y E Rd.
(15)
Then, Is- Ei= gh. where o is defined by equation 14 is MARINA estimator in the sense of Defini- tion 3.1 with > 12 0 =0 0.
This proposition means that the MARINA estimator reduces the variance induced by the compression oper- ator. Note that, if F; is L--smooth, then I. S E Ti-
3.3.2 Finite sum case
Consider the finite sum case where cach F; is sum over the data points stored in the device i: F; = Eド Fij. where F.y is the loss associated to the data point j in the device i. The finite sum MARINA gradient estimator is analogue to the vanilla estimator, but with subsampling. Tt. is defined as the average 9k == 1 - Dk.
where for every i E (I,..., n), gb = =F)(zo) and
VF((IK+1) Ih + Q @ (VFi)(Ea1)) - -VF)(()))
prob. p > 0
Ik+1
(16)
prob. 1 -

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where is the minibatch size and 1 the set of the indices in the minibatch, 4 = b'. Here, (Fix)e.
are Lid random sets consisting in b ii.d samples from the uniform distribution over [I,...,NN.
Proposition 3.4. Assume that Fij is Lay-smooth for all E (1,....n) and all j (I.....N).
Then, gs. + Ei= ghe where si. is defined by equation 16 is a MARINA estimalor in the sense of Defini-
Eng of
tion 3.1 with a w " 1[+(++
where C, = maX Lij and = O.
This proposition means that the MARINA estimator reduces the variance induced by the compression operalor and the variance induced by the subsampling.
3.3.3 Online case
Consider the online case where each F; is an expectation over the randomness of a stream of data arriving online: (z( Eg,DD [Fg, (z)], where Fç. is the loss associated to data point &, device i. The online MARINA gradient estimator analogue the finite sum estimator, but with subsampling from D, instead of the uniform over {1, (1..... N). Moreover, the full gradient V,; is never computed (because it is intractable in online learning). The online MARINA gradient estimator is defined as the average gh == 4/2 Σx gh. where for every i E {I..... ,n), d6 EeEL. VFg (zo) and
b Eeeny VFg( gk + Q Eeen, (xk+1)) Fg (I+)))
prob. p p>0 prob. 1-p
gh+1 1
(17)
where Hi and I are the set of the indices in a minibatch, 11. = b' and |4, = b. Here, (I,,), (resp. (K))),)) is an i.id random set consisting in b (resp. U/) i.dd samples from D,. Besides, in the online case only, we make bounded gradient assumption.
Proposition 3.5. Assume that for all i E fI.....n] there crists o; 2 0 such that for all x E Rd.
BENDD [V (x) VF((z),
BEIND: IV [=(())) S S
Moreover, assume that Fş, is -smoohhth Dj almost surely- Then, gk - gk. where gh is defined
3.1. with " ".. 1i+(1+w)
2iz
by equation 17 is MARINA eslimator in the sense of Definition where C, is the L norm of Lg. (where & ~ Di) and 0 P. of Rab
n=
Note that in the online case 0 o in general, because the online MARINA estimator does not reduce to zero the variance induced by the subsampling. That is because the loss F; is expectation (and not finite sum priorí), for which variance reduction to zero is impossible in general. However, 0 can be made small by taking a large minibatch size b or a small probability p. Moreover, in the online case, Go * 0 unlike in the two other cases.
Remark 3.6. Typically, p very small, for instance, Gorbunov et al. (2021) chooses p = (o /d tion 14 p min (So d, (/N + 6)} in equation 16 and p min {Se/d, 8/ [b)} in equation 17 where Çe supxeRa E [1))))) [lu] and μиılо is the number of non-zero components of y Rd Therefore, with high probability compressed quantities has a low communication cost (Alistarh et al. 2017 Horváth et al. 2019a).
equa-
p. the compressed difference Q FVF((T)+1) - (xg)) is sent to the server. Sending
Remark 3.7. Propositions 3.3 3.4 and 3.5 can be found in Gorbunov et al. (2021 Equation 21, 33, 46) in the case where (xx) is a stochastic gradient descent algorithm (i.c., 1x+1 Fy. hg*)) In the general case where (æx) is produced by any algorithm, we found out that the proofs of these Propositions are the same, but reproduce these proofs in the Appendix for the sake of completeness.
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4 Langevin-MARINA
In this section, we give our main algorithm, Langevin-MARINA, and study its convergence for sampling from & x exp(-F). We prove convergence bounds KL divergence, Wassersteei distance and Total Variation distance under I.SI.
4.1 Proposed algorithm
Our main algorithm, Langevin- MARINA can be seen as a Langevin variant of MARINA which adds Gaussian Langevin-MARINA
noise at each step of MARINA (Algorithm D Our motivation is to obtain a Langevin algorithm whose communication complexity is similar to that of MARINA. Alternatively, one see Langevin-MARIN
MARINA variant of Langevin algorithm which uses MARINA estimator of the gradient 9k. - 9 instead of the full gradient V (zx)) in the Langevin algorithm:
Tk+1
hig + 22h
(18)
Langevin-MARINA is presented in Algorithm 2
Algorithm 2 Langevin-MARINA (proposed algorithm)
Input: Starting point To ~ Po. step-size h, number of iterations K
for i 1.2. n in parallel do
3: Device i computes MARINA estimator dó
4: Device i uploads d to the central server
5: end for
6: Server aggregates 9u = Ei=19h g
for h: = 0, 1,2, .... K do Server broadcasts gt all devices i
Server draws a Gaussian vector Zu+1 NO Ia)
10: Server performs Ik+1 2g hgg + 22h2++1
11:
Server broadcasts 2*+1 to all devices
12:
for i= 1,2, in parallel do
13:
Device i computes MARINA estimator g.+1 14: Device i uploads gh+1 to the central server
15: end for
16: Server aggregates gk+1 n Ein
17: end for
18: Return: JK
4.2 Communication complexity of Langevin-MARINA
Compared to MARINA where n equivalent stochastic gradient descent steps are performed by the devices, here the Langevin step is performed only once, by the server. This allows for computation savings (n times less computations)
the cost of more communication: the iterates Ty need to be broadeast by the server to the devices. Therefore, the commuiccttin complexity of the sampling algorithm Langevin-MARINA is higher than the communication complexity of the optimization algorithm MARINA.
However, compared to the concurrent sampling algorithm of Vono et al. 2022), the communication complex- ity per iteration of LAAReV equivalent. Comparing the communication complexity of Langevin- MARINA to that of FA-LD (Deng et al. 2021) more difficult because FA-LD makes communication savings by performing communication rounds only after number T of local updates, instead of using a compression operator after each local update like Langevin-MARINA.
4.3 Convergence theory for Langevin-Man MARINA
We first prove the convergence of Langevin-MARINA in KT-divergence.
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Theorem 4.1. Assume that LSI (Assumption 2. 2.1 holds and that 9h E g is MARINA estimator in the sense of Definition 3.1. Tj
p p
0 min
(19)
14L V 1 a
6p
then
- c K-uh Kph
DEL (pk Tt) S S--Khiy
T,
(20)
11
where 3/3
DKL (pol T) -- CG. 812129428 1-(1- (1(-p)(4hhaaa 138 B = clthi and is the distribution of a for every k.
(2L2 + 0(1 C(1-p) (SLId 4dhh) + CO, C
In porticular, 0 = 0, set h = o (T(Iaadd): k= (
[ log (-=)) then DKL (px (T) se. 1 + O, p'pe
there will always be an extra residual term I-e Kuh CO in the right hand side of equation 20 which cannot be
diminissed 0 by setting K and h. However the online case (Section 3.3.3),
P we can set
nab
(Eig to make the residual term small Lee-Kuh C0 = O
JURE
From this theorem, we can obtain complexity results in Total Variation distance and 2- Wasserstein distance. Indeed, Pinsker incquality states that
1 ||o π lrv S DKL. (o l Vo, P2(R*),
(21)
and LSI implies Talagrand's T2 inequality
W3 R) s "
DKL (T), Yo P2(R"),
(22)
see Villani 2009
Chapter 21).
Corollary 4.2. Let assumptions and parameters be as in Theorem 4.1 Then,
e CuKKnng + 1 Kyh
|px - n 1122v S
(23)
and
2 (pk, S
e -uKKiig +
1 - e- KKk
(24)
11
In particular, if 0 = 0, set h = 0 upe2 (=(1+a)dd
K = 12 (
(1(( log ( then Ilpx x|rr Sc. c. If 0 * 0. p=p="
there will always be an extra residual term I-c Kuh CO in the right hand side of equation 23 which cannot
be diminished to 0 by setting K and h. However in the ouline case (Section 3.3.3 0 >> we can set b = Ei un=
of
to make the residual term small I-e-Kut C0 O(s?)
Finally, if = 0, set h O (1
p²pe² (2((+a))))
== = S
L2(1-add log | then We(KK, T)) S.. If 0, there p=pp=
will always be an extra residual term 1-e-Kat Kuh 4² C0 in the right hand side of equation 24 which cannot be
diminished by setting K and h. However in the online case, 0 >" of
EE n=
we can = Ω m²b
to make the residual term small --e-Kah C8 O(E2).
We can compare our results to Vono et al. (2022); Deng al. (2021) which also obtain bounds in Wasserstein distance under the stronger assumption that F is jesttoongly convex, see Table In Vono et al.
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20220; they need h = o
where they compute the full gradient every step (we do not include the dependence in p. L in their result for the sake of simplicity). One can think of as 1/p. In Deng et al. (2021), 1, they require h = O
a and K = Q
d log [2
[a =²
n² (<>((>H+1)) and K 52 ( dLL (1= log (2) to get
W2(PK; r) e, where T denotes the number of local updates.
Remark 4.3. Here, we provide insights into determining the optimal choice for the parameter p and sub- soquently evaluate the associated communication cost. For the compression operator, we choose the Tops compression operator from Stich et al. (2018), , for this compression operator,
have = 1 8, where : 2 E [0, 1]. We only consider the vanilla MARINA gradient estimator case (thus 0 = 0), the other case can be considered similarly. Remember the formula for T, C7 in Theorem 4.1 and a in Proposition 33. we can
n² estimate that 1-3, where my:= >. 17. L (1+ (1-p)(1-a) ( d + 4dh), so to have 1-e E
pri
/4
we need/ h = 1 2/
PI or where we omit the dependence on Je, L5 to have - < 6, we need
d(14 (--)
K- = 2
where we omit the log term and dependence on μ. L. With K, the communication
complexity is propotional to d(I+ f ex1
(p + (1 p)s), and from it we can derive the optimal p equals
Pk: := V mits and thus the optimal communication cost is of order 92( A(14 (-ppe--))
e Dpps (pk + (1 p4)ss) for
Top2 compression operator. It is hard to derive the optimal & from this complicated formula, however once we have estimate of nt, we can approximate the optimal k numerically by optimization algorithm. Since = _ e [0, and ny if the variance of {Li}i=1 is not big); so we have pk Re V nd
J. and so the commu-
nication cost is 52 (K+ X mi!
(so the best & = 1 and the corresponding p 2 V (d)). For the algorithm in Vono et al.) (2022), since the algorithm use compression operator in each step, so the communication complexity is of order Q2( ( 5) b())
) (so the best h 1) and similarly the algorithm in Deng ct al. (2021)
has communication complexity Q( di.." ddd" +)) (so the best T W since only communicate with the 1
central server without compression after every T local updates.
Regarding the original MARINA algorithm for optimization, our proofs for Langevin-MARINA employ a dynamical point of view: we consider a path { that goes from Tgg to Fk+1 and we bound the derivative of F(xz) along this path, this challenge is addressed in novel manner by combining Lemma C.2 in the appendix and the properties of MARINA estimator, later integrate along this path to bound the objective gap. Compared with existing federated sampling approaches, Langevin-MARINA MARINA does not rely on the strong convexity of F, and because of this lack of convexity we cannot hope to bound the distance between the iterate and the solution in Wasserstein distance directly. Therefore, we make a detour to first bound the KL divergence, which plays the role of an objective gap.
5 Conclusion
We introduced a communication efficient variant of Langevin algorithm for federated learning called Langevin- MARINA. We studied the complexity of this algorithm in terms of KT. divergence, Total Variation distance and 2- Wassscrsteinn distance. Unlike existing works on sampling for foderated learning, we only require the target distribution to satisfy LSI, which allows the target distribution to not be log-concave.
LARReVIA is inspired by optimization algorithm for federated learning called MARINA. MARINA achieves the state of the art in communication complexity among nonconvex optimization algorithms. How- ever, MARReV requires more communication rounds than MARINA. The fundamental reason for needs to communicate the Gaussian noise. To solve this issue, one approach
this is that Sangevin-MARING is to design a Langevin algorithm allowing for the compression of the Gaussian noisc. Another approach is to put the same random seed in each device in order to ensure that they all generate the same Z at step k. This approach does not require the communication of the Gaussian noise.
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Finally, simple experiments have shown similar performance for Langevin-Marina and QLSD ++ Vono et al. (2022), although the convergence of QLSD+ under T.ST has not been established. Therefore, we wonder if a convergence theory of QLSD++ under LSI would be possible. We leave this question for future work.
References
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Herbert Amann. Ordinary differential equations: an introduction to nonlinear analysis, volume 13. Walter de Gruyter, 2011.
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric spaces and in the space of probabilily mensures. Springer Science & Business Media, 2005.
Krishnakumar Balasubramanian, Sinho Chewi, Murat A Erdogdu, Adil Salim, and Matthew Zhang, Towards a theory of non-log-concave sampling: first-order stationarity guarantees for Langevin Monte Carlo. arXiv preprint Xiv:2200 05214, 2022.
Adrien Blanchet and Jérôme Bolte. A family of functional incqualities: Lojasiewicz incqualitics and dis- placement convex functions. Journal of Punctional Analysis, 275(7):1650-167) 2018.
Xiang Cheng, Niladri S Chatterji, Yasin Abbasi-Yadkort Peter I, Bartlett, and Michael I Jordan. Sharp convergence rates for Langevin dynamics in the nonconvex setting. ar preprint arXvv18055.0165 2018.
Sinho Chewi, Murat A Erdogdu, Mufan Bill Li, Ruoqi Shen, and Matthew Zhang. Analysis of Langevin Monte Carlo from Poincaré to log-Sobolev. arXiv preprint arXiv:2112, 12662, 2021.
Arnak Dalalyan. Theoretical guarantees for approximate sampling from smooth and log-concave densities. Journal of the Royal Statisticnl Society: Series B (Statistical Methodology), 79(3):651-66 1-661, 2017.
Rudrajit Das, Anish Acharya, Abolfazl Hashemi, Sujay Sanghavi, Inderjit S Dhillon, and Ufuk Topeu. Faster non-convex federated learning via global and local momentum. arXiv preprint arXi::002.04061, 2020.
Wei Deng, Yi-An Ma, Zhao Song, Qian Zhang, and Guang Lin. On convergence of federated averaging Langevin dynamics. ar preprinl or 1t::2112.05100 2021.
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Alain Durmus and Eric Moulines. Nonasymptotic convergence analysis for the unadjusted Langevin algo- rithm. The Annals of Applied Probability, 2753)1155111887, 2017.
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Eduard Gorbunov, Konstantin P Burlachenko, Zhize Li, and Peter Richtárik. Marina: faster non-convex distributed learning with compression. In International Conference on Machine Learning, pp. 3788-3798.
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Samuel Horváth, Yu Ho, Ludovit Horvath, Atal Narayan Sahu, Marco Canini, and Peter Richtárik. Natural compression for distributed deep learning. orXiv preprint Xrv: 1905.900.10 10988, 2019a.
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Maxime Vono, Vincent Plassier, Alain Durmus, Aymeric Dieuleveut, and Eric Moulines. Qlsd: quantised
Langevin stochastic dynamics for Bayesian foderated learning. In International Conference Artificial Intelligence and Slatislics, pp. 6459 6500. PMLR, 2022.
Andre Wibisono. Sampling as optimization in the space measures: The Langevin dynamics as a composite optimization problem. In Conference on Learning Theory (COLT), pp. 2093 1193-3027, 2018.
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arXiv.1911.01469, 2019.
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Donsity the Target Distribution
01.00
11.15
01.00
0.00
01.0)
Figure Density n.
A Experiments
We experiment our algorithm and compare it to QLSD++ (Vono et al.) 2022 Since this paper is theoretical, we chose simple one dimensional example, in order to be able to visualize the performance of the algorithms on histograms.
The target distribution π x exp(-F), where
(1+
2
10
Z
I
F(x)
cos(x)
+
+1+
-I,
(25)
10
(a-11)² m1)
x2
2
and where II is the area of the disc with unit radius. The density of the target distribution is plotted in Figure [ We can obscrve that F is L-smooth but not convex. Morcover, we observe that æ is a bounded perturbation of a strongly log concave distribution. Therefore, π satisfies LSI (with an unknown constant).
Experimental setup. We use n = 5 clients, and client has N(2) subfunctions Fgy. The mumber Ni) is randomly chosen between 10 and 20. The subfunctions Fig are written Poy(x) = F(x) + Eú sin(x), where Súj is noise factor such that Fi(x) == No. Σ (z) We compare Langevin- to QLSD++ Vono
let al. 2022). In Langevin-Marina we set = 0.001, ie., a full gradient is computing every 1000 iterations expectation. In QLSD+ +++ (Vono al.) 2022) we set
0.2, the initial memory term r7) 0, [5] and 1 1000, i.e., full gradient is computing every 1000 iterations. We the compression operator from Vono et al. 2022) with quantization level = 28. For the ease of comparison, we also compress the p- p-branch in Langevin-Marne Marina, so the communication complexity of Langovin-Marina and QLSD* are the same. To compute the stochastic gradient we subsample one index from {1,..., (.....,((i)) NNii) in each iteration. We run 400000 steps of both algorithms and collect all the points generated by each algorithm to plot their histogram. The results are listed in Figure
On Figure 2, the performance of the algorithms can be seen visually the histogram (in blue) fits the curve (in red). Langevin-Marina and QLS+++ have similar performances in this one-dimensional experiment, although the convergence of QLSD++ under LSI has not been established, unlike Langevin-Marina. Therefore, we wonder if a convergence theory of QLSD++ under LSI would be possible.
B New proofs of existing optimization results for MARINA (Algorithm 1
The goal of this section to we show that the approach that we used study Langevin-MARINA can be used study MARINA. This section is, with the proof section Appendix [C.6. the ONLY section of the paper that deals with the problem of minimizing F rather than sampling exp(-F).
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Langevin-Marina Intial point=0 stepsize:0.0om 002
QUSD+ intia point=0 stepsize=0.00 002
0.30
0.30
Target distibution
Tarpet dis butoon
Hettgramm the samples
Hastggam the aumples
0.35
0.25
0.15
0.16
0.00
0.05
0.00
00000
Langevin-Marina
intiaa paint-3
stepsize-O.000
QLSD++ initial point-3
stepsize-O..o 0002
Target distribution
Target distriutioo
Histogram of the camples
Hästogram ammggammma of the samples
0.10
0.10
0.05
0.00
0.00
0.00
Figure 2: Langevin- Marrina vs QLSD++ with the same step-size 7 = 0.002, but different initial point 2g = 0 and 2g 3.
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In particular, we provide analysis of MARINA (Algorithm 1 establishing the convergence of the whole continuous trajectories (24)120 generated by the algorithm, unlike Gorbunov et al. (2021) which focuses on the discrete iterates (==) Compared to Gorbunov et al. (2021). but with a different method.
we obtain the same convergence rate,
The approach we take to study MARINA similar to the approach we took to shudy Langevin-MARIN: we establish the convergence of the whole continuous trajectories. The key difference that MARINA is optmization algorithm in the Euclidean space Rd so the underlying space in the convergence proofs of MARINA is R.. On the contrary, we viewed Langevin-MARINA as optimization algorithm in the Wasserstein space, so the underlying space in the convergence proofs of Langevin-MARINA is the Wasscrstein space.
In this section, we focus on the minimization, by MARINA (Algorithm D of the empirical risk:
72
min #ER
F(x)
Σ B(x).
(26)
One can see MARINA as a variant of the gradient descent algorithm which uses a MARINA estimator of the gradient Ok n E g instoad of the full gradient (F(x)) in the gradient descent algorithm:
Th+1 = Ib - hgk.
(27)
MARINA is presented in Algorithm 1
We now provide the convergence results MARINA. Theorem B.1. Let (ow) be MARINA estimalors. If the step-size sulisfies
The proofs are provided later in the Appendix.
1 p 10T
0 h S
(28)
T+a
then
1 Kh
di 2 F((*))
E [VP((())
+20,
(29)
Kh
Kh
where 1 = F(xo) + hC GCG C' I- 1-(1-p(aL"hhatt)
81,22242
l + (1 (1- [HJL)))
Let h T0Z V Pha (hence C' = O (1) and ft == TT, where T is a uniform random variable over [0, Kh)
independent of (zt). Then, E [[IVF(&))]]] d S [F(+))] dt. To achieve E [|VF(())) 2
2
when 0 = (for instance for the MARINA estimators equation 14 and equation 161, F(z= +0
g
iterations suffice. If f * 0 (for instance for the MARINA estimator equation 17 we choose the batch-size of then 2C0 O(s2), so Ω (#) -F=")
Ω (Ein
[tt T.)..) iterations suffice.
If further satisfics the Lojasiewicz condition 7 we can obtain the following stronger result.
K-1 Theorem B.2. Let (==) be MARINA estimators. Assume that the gradient domination condition holds. If the step-size satisfies
0 h < min
(30)
14L V 1 1 a 6μ
we will have
1 1 c- Kuh
(((xx)))F((+") - - ukk (V22 F(x"))
CO,
(11)
1
where 122 = F(xo) -he
CGo, C= 1-(1-)](aAh~a++ 3 euh.
$1.222
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Let minf 14Z / afa )n) (hence ( To achieve E [[((2)) - F(x*) I E when = (for
instance for the MARINA estimators equation 14 and equation 160. 2(max 1ta L
1s 1, log ()-=((=2))
iterations suffice. If 0 o 0 (for instance for the MARINA estimator equation of then term
17
if we choose the batch-size
-- Ruch C0 is of order O(e), so Q mmax
1+
log( "2-F(=")) iterations
suffice.
Since we recover the convergence rates of Gorbunov et al. 2021 we refer to the latter paper for discussion of these results.
C Proofs
C.1 Proof of Proposition 3.3
For gradient estimator
wc have
F. [198+11
F ( )1 11
2
(11- (1-pp) -
gk 2(VF.((0+1)) -VP((x)) v F 20
(32)
(1 P)
Q(VF((I++1) VF((=))) VF(()))) +
12241,
+ (1 E) [1.h VF(xx)]1 23
Since QVVFi (Ix+1) V (...))..., @ (VF. (ER+)
VFF (zx)) are independent random vectors, now we
have
E |1xkt1 - VF(Ix+1)1 2x41,
-V(())) + - + 1I_11
= (1 -1-20 P)E
+ +11 E)E
110x VF(*))1 I
(33)
1- 22
F. [1 (F(()))) (F((x+1) + VF(x))] #K41,
n²
+ (1 E)E [19x
VF(x))1 ar
(1 - p)w
[VF(((011) - VE(WW! 1841 xK +(1 p)B [11.k VF(xx))1 xk
E
n²
Use Equation
15 and the tower property, we obtain
E |1kt11 - VF(I+-1)11
= |1ox+1 VF | 26
(1 p)w
n
E || - 261l -x61
(1 p)E [119x V=(xx)11]
(34)
<
n2
2 Σn= I?
= (1 p)1
E ||
- sullp]
+ (1 - p)E 11ox VF(x))]?
n² L
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Σ Li L 0 O.
S0
C.2 Proof of Proposition 3.4
For gradient estimator
16 (finite sum case, that is for each i [n], F := 1. Ei= Fty), we have
E 190+1 V P (xxx+1 (1)+
1 2t
= (1 - P)F
9k. +
Σ
- VF((++1)) 1k
1=1
jET/
(35)
(1 - =11-pFF
Σ o
Σ ( - -VF(*++) + VF(m))))+(((()))
1k
==1
JE
+ - (1-
|19x
Fx(x))] x
Next, we use the notation:
A & Eyer,, (V (T++1) - Viy (rx)) and A VF, (#1+1) VF, (rx).. A for all E [n]. Moreover, 2 e(A)) ,...., Q (Ah) are independent random
They satisfy E A 1k+1, Tb vectors, now we have
E |1 VF( 2842.
= -[-
n ト
Σ
(((A))) - =t)|/ | 16
+ (1 (1-p)E [198 VF(xx)] mx]
p n
Σ E
||
A + A 1
1K4+1, Ik
+ (1 - (1- [1su VF(xx)) Irk
1- P "
((())))
+ ||| | 1
2241, Ix ) + (1-p) [1.9x VF(xx))1 |
#K+1, IT
n²
1
WE
A
[ 2k
E A
1224 1) + (p- [loak VF(xx)]1 2
p)E
n
| - 1/ 112211.28 1) + (1 pp)p [lox VF(*+)]1 x
WE
|
[
+ (1 W|E
2
(36)
Next we need to calculate E A - | 1x41,
For convenience, we will denote aij = VF
(Fi((z))) and a; VF((2++1) Ve((x.). Define
with prob. & with prob. n.
Xa
(37)
N
with prob. 1
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(x.)% independent with each other.
Let 1,,
U
Xs. so
A
Σ
( - (sx))
D
JEL
(38)
j=
then
E
A
A
E
x==
(ars
-a)
Ik
2
8
N
1
Σ
E
1x,=((a5 aai)
1k+1, 2b
Σ
E
Σ
11x==(005 -a), E
\x a:) 1
1/2
j=1
V/
N
2
1
E
Σ
xx= (ass - aai
[kk+1
1/2
N Σ 1=1
N
x. x (ai - ass) 1kt1, I.h 1)
x. (ai ai)
1K41, It
E
Σ
E
Σ
X1 0zy
Ik+1, 2k
- -1lail?
1=1
2
<
Σ
1
x1 Ris
|I2+1. 3k
Σ E
||1 ||
||a;)]| 111 + Σ [((xx= xx=rr)) (air, aij)
j48
1
|ayyl 2
N
Ø=1
C3 B
| xx
(39)
Use Equation
15 and the tower property, we get
|1aht1 - (F(+++1)11)
= E E ||lu-11 - VF(I++1)11 IK+1,
p n
Wb? + (1 W)+
E || 2x
+ (1 -p)E [19xx vF(x))13
(40)
< n
i=!
E c
$ (1 (1-p) Ei=
T2? (1 w)
|100-1- - + (1- p)E [lo0 VF(x+)]]]
n² 2
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Dis 1.2 +7++0 )w) 8 = 0. n 1²
C.3 Proof of Theorem 3,5
For gradient estimator 17 (online case, that is for each i E [n]], F; == EEIND [Fg.)), we first have
11x+1 - VF (#x+1)|1 xx+1, 2x
=(1-p)E
9k + 71
Σ
v
1
Σ EET
(VFe (2++1) - Vlfe (FF(())))
F (x2+1)
1x Tk
P
Σ
(VFs (Ix+1)
VF (Ix+2))
ax+1
n2
EET.
-(1 E)
Q
Σ
(VFg (#K+1) - Ve (xx))
FFF VF (xx)
| 1
B'
EE
+ (1 - E
[1.
FF (ax))² )1
xR
+
E ( - VF (2x++)))1 x
n2
EEL,
<1 F
Σ
(VFe (*K+1)
VF (F(41))) VF V (oxa))
1 T6
EE
p EE o²
+ F) [11.
VF (xæ)l1 æ
2b
(41)
here 11, consists of b' elements i.i.d. sampled from distribution D, In the following, we use the notation:
A = Ecer, (VFg (Ix+1) - VFg. (æx)) and A = VF, (#x+1) V., (xx). They satisfy A 2641 Tx =
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Af for all i E [n]. Moreover, (Aa) ...... Q (Aa) are independent random vectors, then we have
E |1att1 - VF(+++1)]l" 26411
-
Σ
n
(o()))) 4+)
+ (1 p)E [1oa. VF(xx)]] Ix + P p. o?
(11-p)E
n²b
1-p 2
| A + - - 1
+ - (1- [lean [F(xx)] P o7
E
n2b
i=1
1- P-P- Σ
3t1|
+ | - =11l (24411 27h 1) + (1 (1- |19. - FF(xx))] x
E
2k+1, Jt
2
1=1
P > of
nb
1- P
wE ||
+E |ل - 1.1
104t1, 20h 1) + (1 (1-p)BB [1..0. VF(xx)1 z
18+1, IK
2
1=1
b
A - 1812 1 22 1) + (1 p)E [19x VF(x>)1 a
p
wE
| 2x + (1 + W/E
n
1=1
Σ (=1 o,
bbbb
(42)
|| - 1tIl 1 1 I Let et. ~ DD,s = 1,2,..., be V i.dd random
Now we need to calculate E
variables, Me = U sh. and denote agt, = VFet (IR+1) - VFet (In) == (F((E)+1) - VF((x), then
A
Σ
ag
1
aek
(43)
D'
SEl
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So
1011²
E
8'
E
ueh
ai)
V
k=1
b'
1/22
13
12
26 Jt
1/2
B
{
(uet.
us Uxi 0z
xk+ 2t
8'
2
8/2
Oct
a:
#k++1, Ik
(44)
8'
1
ass 2
F
Mgt.
1k 37k
8/2
F
|aet.
|2
1 8k
- -Ila:ll
| 12
$
E
2
$ ÿ
| xx
Combine with the tower property, we finally get
E |lokt1 [=)
= E
E
|11n-+1 VF(x+++1)] )|
Σ
WL? + (1 + W
E
|| =x11? + (1 (1-p))
p)E [1oan
vF(=))1?
+ p PΣis o² n²b
n
E
2) Em= L + (1 + w)
+ p PI of²
||ER+1 2xll7] + (1 (1--)D [19x VF(WW)]?)
E
2212
n²b
(45)
+1++a) > S0
p >'
8
n=b
C.4 Proof of Theorem
The following lemma is an integral form of Grönwall inequality from Amann 2001 Chapter II.), which plays important role in the proof of Theorem 4.1 and B.2
Lemma C.1 (Grönwall Inequality). Assume 6, B [0, + R are bounded non-negative measurable function and C:: [) + I is non-negative integrable function with the property that
@(t) < B(t)+ ' C(T))(T)dr for all t E [0,
(46)
Then
((t) < B(t)+ + ( B(s)C(s) exp
ds for all E [].
(47)
We also need the following lemma from Chewi al. 2021
Lemma 16.).
Lemma C.2. Assume that VF is L-Lipschitz. For any probability measure μ,, it holds that
E | < F
Vlogg H
+ 2dl. = Jx (1)) + 2dI.
(48)
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Remind you the definition of KL divergence and Fisher information:
= log( Pt (z)dpe Jn (pt) f | V log G)| dp.
DKL (pe | x)
(49)
We follow the proof of Vempala & Wibisono (2019 Lemma 3). Consider the following SDE dæ, - fe(oo)dt + vdd,,
(50)
where f() R x Rd is of some probability space (E,,, let Poe (1o. (, denote the joint distribution of ( which write in terms of the conditionals and marginals
Por (1o. m.) po (2o, k) Pejo (It zo. Ps (ax) Poje (zo, E 2t).
Conditioning on (3r, 5), the drift vector field (e( is a constant, so the Fokker-Planck formula for the conditional density Prjo (xre 2o) is
pppj (x, Zo. E)
V (pejo (zt To. E) fe (zo)) + Aprio (æt xo,E)
0t
To derive the evolution of Pt, we take expectation over (ro, {) ~ Pu, we obtain
@p((x)
Op:i (x vo. E) po (z0, {) dxodg
Öt
R2x=
0t
(V (Pejo (ara xo, fe (Ra)) Apdo (oa Io, E)) po (2o {) doodg
Rdx=
(51)
(V (pox (z, wo. E) fe (mo)) + Apor (Ir, xo, {)) drodg
R4X3
(me()) { /RAX3 Poje (oro E x) fe (zo) drode) Ap(())
V (p)( Ife (xo) ot x1) Ap.(x),
so we have
dDRt. (pe ②²
(V. (pr [Se (mr) x4 = =:)) + Ap((x)) log( Pt (*) (x)dda
Epooi life (xo) xt ====]] V V log( (xx) pa (x))x
V log Mt (x) Vog(( Pt (xx + Epo [ie (zo)
x = x + (log(p)) v logg (o)) (x()) I
π
( (V log( PC (xx Epona [ife (zr) It = x]| VF(x), V log( 2 (xx) P (r(x)do
π
Jn (pt) [ (Bpole lffe (20) xe - =2 = v log( (a)) pz(z)dz
S Jr (p:) + 7 (pt) + [ (Epon [e (mr) xt = x] V((x), Eooc [/e (20) xr = x) -VF(x)) pe pc(x)de
3 Jn (p.) +E | (fe VF(xz) =1]1
4
3
S Ja (p.) +E E [1.fe( VF(x:)11 xx
3
J (p)) + (1) VF(0))11
(52)
22

---

Published in Transactions on Machine Learning Research (01/2024)
If we replace fe(2o) by g& (xx))) in equation 52
we will have
dDKL (pil x)
3
1 n (pt) + VF(22) 9xll"]
dt
4
- - Jn (pt) + 2E
[IVF(a)) VF(xx)]1 + 2E [IVF(I>) -x1l²)
(53)
3
4 Jr (pt) +2
[VF(x)) VF(x))]] +2G
We bound term A, denote F. the filtration generated by {B,) (Ba)sn) then
A Ep IVF(xx)) (FF(m))))FFF(M)) |,:]]
(54)
next estimate the inner expectation,
B [IVF((a))) -VFFF()))VV((a)I1
|F < 1
|x Ik 12 Ft:
(gk v2 (Bunte - BEA)) 1|
| 1
1
1.3/2 |gg 1 + 2I,2dt
(55)
S h
lluull + +L
LE
|| 2x.1? [F
from equation 55 and equation
53
we finally have
dD (pe T)
J- (pt) + 2 [|00+1 -aap)) 2Ga
(56)
dl
Next, we use Lemma C.2 to bound E || xx 1261
E ||
2x1
h
11 2 2dh
E
V F (2rx) - gxll + [IVF(==)](] + 2dh
24
|[F
+ +2h 2dh
(57)
4h2
[=I[V/==;)]]++
VF(ax))11 ((VEE=+ [IVF(x.)
2/2 + 2dh
[V== 4 ||0041 - 2211² + 2h 2dh,
44"E
so let h < 11 we have
2/21'
| - 2x12 { ShPE
[VF(x))] [
+ +4 4dh.
(58)
23

---

Published in Transactions on Machine Learning Research (01/2024)
Add CG&+1 to both sides of inequality equation
56 C' is some constant to be determined later, then use
Theorem 3.1
the right hand side of equation 56 will be
3
RHS -
Ja (p:) + 2
|| -
+ 2G + CG++1
4
3
Ju (pt) + [108+1- - + +2G +C (1-p) (1- || - m211]] + +0)
4
3
J (pt) (2L2 + C(1 C(I-p)Laa) E
|108411 - xx|l? -xx||
(2+ c(1-p)) G) + +C0
4
equation
58
3 Jx (p:) + (212 +c(1 +((1-))(aa (sh*E) ( |VF 44 +4dh)
+
- p)) G. +CO
Lemma
3
S
4 J (pt) + (2L² + C(1 - p) (8h² (Ja (pr) + 2nL) + 4112G, + +dhh)
+ +2+ -0-))) p)) G + CB
3 - 8h (2L2 + C(1 - -p)L2a) Jn (px) + 2) G
=
4
+ (2T.² + C(1 - 1a) (8The + 4dth) + C0
denote as
(pt)+ 3 - 8h2 (2L² + +C(11-) -D) Ja (0t) + (8L3)/ C(1 C(1))+(((1))) (412)/a + 1) + 2) G5 + = =-
(59)
Choose parameter (B = or B = clh)) and let C (8L3h² + C(1 p) (4I + 1) +2) 3, solve this, we
gel
8L2 2
C I- p) ( + 1)
(60)
To make sure C 0, we should require h S 1 21. V (1-p)a when & = the case 3 = cau is bit complicated:
when h small (for example I 1t) we have = eith I 1 + 2uh, insert this into the denominator of and make sure it positive, that is
(1 (AL' + 1) (ph))h) 0,
(61)
which is equivalent to
P 8L² uahs + 1-P PAL"ah" +
1 1-1 2uh < 1,
(62)
p
11
111
one simple solution for equation 62 is to let I < }. 1I < 3, 1,111 < 1, which is
h < min 2212 P
PAL - p) 12L - p) )2/2
1/3
p
p 6p(11 - p)
}.
(63)
Insert equation 60 into the parameter before E
(x and require
81 + 23
1
8h? (2T.2 + p()))) = 81.2 2T. +
1- (1 p) (4L 1) (1 -p)1.a S
(64)
4'
solve this we get
1
1 1- (1 (1-p)B
h <
(65)
2L |
16 (1 p17a 16)59
If 3 = 1, we need
h
10L
I+a a 2L
P
S
17a
p S min(,
p)
2L
16p +17011 - pp)
P
p)'Va(1-) pm
a(1
p
(66)
24

---

Published in Transactions on Machine Learning Research (01/2024)
to guarantee C 2 0 and equation 65
The 3 = e case is complicated: from equation 63
we know h < م
so 8 = elth I 1 < 1+ 3
insert this upper bound of 3 into equation 65 that is
get lower bound of the right hand side of equation 65
1
2p 17a(3 - 2p) + 32p
1
- (1-p)(1 अリー 16 + (1 - p)(17a (1) + 3(1-p)
2L
2L
(67)
1
1- (1-111pp)) (1-p)3
<
2T.
16 + (1 - p17 c 16)/3*
So we need
minf 1
2p
2AL2 p) 11/3
P
p 12L - p) 1122
P
>p)
(68)
2L
V 17a(3
2p) + 32p
66((1 -
p
We can further simplify equation 68 the first and third term in equation 68 will be greater than
14L
1+a
the fourth term is greater than and min by is a lower bound of the second term in equation 68, since
6p
1/3 < 24112 1/3 min(a,b) 6} s 2/3/2/33 (117612((14a)
So finally when 3 = eth, we need
P P
h < min!
(69)
14T. V
1 a
6p
guarantce C 2 and equation 65
Once we have C 2 0 and equation 64 then
dDD. (pe T) + CG Jn (p:) + 8--CGG +r.
(70)
dt
Case I. B 3=1, 1, integrate equation 70 we have then
(k+1)h Ja (pt) dl < 2 DKL. (pun 1) + CG&h - (DKI. (p(++IJ (T) + CG++1h)) + CTh,
Juh
(71)
use equation 71 for = 0,1,2,1... ... K, we finally have
1 Kh
{
2 (DKL (po | T) + OGoh - (DKL (PKK σ)) + CGxh)) +
Jx (px) dl
Kh
Kh.
(72)
s 2(DKL (po T) + hCG.)
+
Kh
Case II. Suppose æ satisfies LSST with parameter 14, that is
DKL (v ②) < 1 Jr (v),
(73)
2p
so with LSI, we have from equation 70 that
dDRL (pe m) + CG S -DDKL (pr (T) + B +
(74)
dt
25

---

Published in Transactions on Machine Learning Research (01/2024)
Change equation DEL (px [T), B()) have
74 into its equivalent integral form, then it satisfies equation 46 with o(t) (B-1CG
CG++1 + +T) 1. + DKL. (PEA x), C(t) -14, then by equation
47
we
DKL (pt | 1x) ext DEL (pkn (T) 1 - e- (B-ICG: CG&+1 +
(75)
1
let t= h and B = elth then we have
DRL. (P(++2) m) + 1 - e
DKr. (pen T) + elth 1 1- E-h
1 - e-Th
CGk-1 S Se-uh
B B-'CGR
1
1
11
(76)
= - DKL (pen I T) e-eum
1- e-u
CGk)
T,
1
use equation 76 for = =0,122.... K - 1, we have finally
DEL (pxh x) DEL (pKa I 1t) + 1 CGK
(77)
S eek DKT (Po x) 1- e CGo) + e -Kph
T,
this proves Theorem 4.1
C.5 Complexity of Langevin-MARINA
I < min T47 T+a J. we wil have C = O()). To achieve DRL (pk T) S e eukkhyz I-e. - =(()),
uKh
we need to bound the residual term 1-e-Kpa Kpa
7 and the contraction term e e-KKy respectively. When = 0,
h 0
O pic
[<(I+aaa)) is enough to make
1pe-Rph Ryh
O(e); when 0 * need
sca pipe Ei to make the left term R-c-Ruh CB O(e).
#
further require b 22
For the contraction term, we need
und
K = Ω
('(+aa p'pe
log( (2) (here we assume DT(+aajd <
MPs
min
P
Gy }} to make e eukkyy = O
IIL
Ita
So for all gradient estimators, need K Ω
(2(1)) pepee
log(
Va
uPE and h O II+taj
to guarantee
Ei Ju
DEL (px T) = O(E), for algorithm based on equation 17 we need further assume b = Ω
The analysis of the complexity under the Total Variation distance and the 2- Wasserstein distance are sim- ilar. We want llpk- - ग 1 S (s22) and (2(pK x) < OE by Theorem 4.2 we only need to guarantee e e-ukhyy l-k-
OE and e euukhy I-g-nka 7 Ouus?) respectively. So we only need to replace
" the e in h, b in the above by e² and ne?, then we will have lle - F Tllzv = O(s) and W2(PK, TT) O(s).
C.6 Proofs of Theorem B.1 and B.2
Define two flows,
Ft Ib -ggk:
(78)
$ s<t
x
(79)
Iz Je (F(a)))
s>
where Zh the gradient estimator of equation is continuous with respect to and when x Zt.
14 equation
16 or equation 17
Zo = Zg. Tb. = 2x+1, note x
Then follow the same procedure in the last section, we have
26

---

Published in Transactions on Machine Learning Research (01/2024)
(F()))
dF())) s== + dF(w)) dF(x)) ls=t
dt
ds
dt
ds
[V 112 (VF(x)), Fz()2) 9k)
- IVFF(=)) 11 + |IVE (VVF(x)) 1x
(80)
S /VF [|VF(.)) - 9x
3 [VF(E))) +2 v2)()( VF(xx)] +2
V2 + g& 12
|VF(x))] + | +2/ 9xll" since [VF((z)) vF(xx)]1 < 1 1 2
2b4 |
| x&
We define G3
E [ - -xxl) then take expectation on both sides of equation 80 we have
dF (F(e)) S -44
B [VF
+ 22L'
[| xx11?]
+ 2G..
(81)
dl
Next, we bound |10841 - -xxll" xxl1?
E |
2x11?]
1
9kl 1|
2
V xx )
+ 2
VF(x)) SKll"]
(82)
$ 4h
VV xz, 2x))) )1(
4h-E [VF((=)) VF(xx))1? 2haGr
S F
|VF
+ 41,2
|| -
malli]]
Gk:
which is equivalent to
(1 (1-417 4L E [xxe+)=xx1]]]
xxll]] < 4h2E [[F(())]) + 2
(83)
If we let h S 1
2V2L' then based on equation 83 we have
E
|| xx 12]
S [VF(()))) 4h2GG.
(84)
Add CG to both sides of equation the right hand side based on Theorem
is some parameter be determined later. Then we calculate
RHS
3 E
VF(x)))
2
|aa41
2x1l]
+ 2G + CG++1
E
(VF(2)) )1?
+ 2 11241-2217 2x11
+ 2G + C (11 - GR (1-p)L'aB) | [ +0)
4
s
E
(VF(2)) (x:) )1²
+ (2L² + -C(1 C(1- ((1-p)L2a)
1110041-22 - - 11 + +22 C(1 - pp) Gx + +C#
4
3
E
[VF
)12
(2L' ((1-p)L3a)
(8/"E |V + G/ +2 C(1 -p) Gr +CO
$
3
8h2 (2I.²
+C(1 ((1-p)Lta)) E [[IVF(=))]]] + (8I + p) (4I + 1) + +)) G +CB
4
(85)
27

---

Published in Transactions on Machine Learning Research (01/2024)
Choose 3 (we will set 3 = 1 3 elth in the later) and let C' (8L + C(1 p) (4L 1) +2) solve this equation, we get.
8T2123 + 2 C 1 - (1-111) (4L2h2a 1)3"
(86)
We need
2 O,
Sh (2L + C(1 - pp) 2 2
(87)
By similar analysis as Appendix C.4 we need require
5
when B and h S
T0Z
T+a
min
147. V
L P_
140 op
} when 8 = en to guarantee equation 87
Once we have equation 87 then we finally have
by equation 85
dF (F(x))
E [IVF(a)) B CG CO. 2
(88)
dt
Case I. Let 3 = 1, then we have
dE [F(xe] CG S - [iV + CGr + CO, dt
(89)
where C is defined in equation
86 integrating both sides from 0 to h, have
1 -- - E [VF()))] a lCGx hCO,
B[F(#++1)] B [[(()))] hCGx+1
(90)
which is equivalent to
f E [IVF(NN)) dt (B[[(())) (E[F(r +cG 2hC8.
(91)
If we first do the same procodure as above for k: = 0,1,2, ... K then take summation from both sides, will have
Kh /
[IVE<EI)) di ((E + hCGg) -
hCGx)) + +00
Kh.
(92)
< (F(ao) + l.G. - F(x*)) + 200,
Kh
this proves Theorem B.1
Case II.Supoose Lojasiewicz condition holds, that is
|F(2)| > 2p (F(x) min F), Vr Edd
(93)
Combine equation
88 and equation 93 have
-() I - (E(F(x)) + CG + C8,
(E (x.)
(94)
dt
which is equivalent to the intogral form
E(F(x)))=((2) s (8- -CC CG++1 CB) (FF=+++((()))F(((()))) [ -p -p (E(((()))) F(x*)) - dr,
te [0.
(95)
Now we use Grönwall incquality C.1 note equation
95
satisfics equation 46 with @(t) = F(x.) - F(x"),
B(I) = (8-tCG; CG&+1 + C0) 1 F(xx) - F(x C(1)) = - then by equation 47 we have
E(F(x,) - F(x*) < md (E F(x*)) + 1 1 1 (B - CG + C0)
(96)
P
28

---

Published in Transactions on Machine Learning Research (01/2024)
let t= h and B elth then we have
E ([(x++1)]-((2) 1 e CG S Se
[F(F((x))] F(x*) eat 1 - -e
FCG))) 1 1-
CB
1
=e ph (FF(x))]-F(A))) + - e--u CG²)
1 1 1-e-a
+
C8,
(97)
use equation 97 for k 0,1,2, K - 1, we have finally
[[F(xx)] - F(xx)) S (F 1 -e CGK
1
[[F(20)] - F(x*) + 1 - e (Go)
1 Kah
S -ukh
C0
(98)
1- CGo F(x*))
1 1-e e-e-kul Kuk
S Se- (((z))
+
C8,
/
this proves Theorem
B.2
29


Reviews:
[{"review_id": "aSC2WhzWz9", "reviewer": "Reviewer_P7nJ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. It improved the convergence rate of the standard MARINA algorithm. \n\n2. It investigates a new setting: the weaker log Sobolev inequality.\n\nWeakness:\nThe contributions of this submission are not clear. \n1. To improve the convergence rate of the standard MARINA algorithm, what is the unique challenge here? What new techniques are developed to address these challenges?\n\n2. When extending MARINA to Langevin-MARINA, what are the unique challenges here? What new techniques are developed to address these challenges?\n\n3. Compared with existing federated sampling approaches, what are the unique challenges when considering the weaker log Sobolev inequality?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies federated sampling problems. In particular, it extended the standard MARINA algorithm to the sampling problem and provided the convergence rate. \n\nHowever, the current submission does not demonstrate the unique challenges when extending MARINA to Langevin-MARINA, and how these unique challenges are addressed. From the current version, it seems the extension is not straightforward. It would be good if the authors could provide more discussions about this critical point.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The contributions of this submission are not clear. \n1. To improve the convergence rate of the standard MARINA algorithm, what is the unique challenge here? What new techniques are developed to address these challenges?\n\n2. When extending MARINA to Langevin-MARINA, what are the unique challenges here? What new techniques are developed to address these challenges?\n\n3. Compared with existing federated sampling approaches, what are the unique challenges when considering the weaker log Sobolev inequality?", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "6H6Y7hAKt1", "reviewer": "Reviewer_ZmLB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths.\nThe paper is on the whole clearly written and well structured. Furthermore, the math appears correct. The result should be of interest to the sampling community, and is therefore pertinent to TMLR.\n\nWeaknesses.\nI am inclined to recommend acceptance of the paper as it is, but there is one addition that I would like the authors to make. After Corollary 4.2, the authors discuss the communication cost of their protocol and compare it to the communication cost of previous works. However, I found this discussion difficult to read and the comparison to previous works difficult to interpret because certain quantities like $p$ are not made explicit, and are left as a choice to the reader.\n\nI would like the authors to add calculations of the communication cost in the specific case that the compression operator $\\\\mathcal{Q}$ is the operator that rounds each number to $B$ bits. What are the optimal choices of $B$ and $p$ in that case to minimize the communication? And what is the total communication complexity cost in terms of # of bits? And how does that compare to previous works? This would help me (and future readers) immensely to understand the significance of the contribution. (Alternatively, another compression operator $\\\\mathcal{Q}$ could also be analyzed, but the point is that there should be a specific example of the implication of the theorem.)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a federated algorithm for sampling distributions of the form $\\\\mu \\\\propto \\\\exp(-F)$, assuming that they satisfy a log-sobolev inequality. The paper presents an analysis for this algorithm, proving that the procedure samples from a distribution that is close in KL divergence to $\\\\mu$.\n\nThe analysis proceeds by adapting the analysis of MARINA (which was an algorithm previously proposed by Gorbunov et al. (2021) for federated optimization) to the setting of sampling, by adapting a proof of Vempala & Wibisono (2019) for the convergence of Langevin sampling.\n\nOne of the main aspects that differentiates this work from previous works on federated sampling is that this work applies to distributions satisfying an LSI, while previous work along these lines assumed the more restrict condition of strong convexity.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "As well as what I wrote above in the weaknesses section, please address these typos, and questions:\n\n* Typo: \"Proof of Theorem 3.3\" and \"Theorem 3.4\" in the appendix, when these are Proposition 3.3 and Proposition 3.4 in the main text\n\n* Comment: In Section 3.3.1, make it more clear that all nodes coordinate in either uploading $\\\\nabla F_i(x_{k+1})$ or $g_k + \\\\mathcal{Q}(\\\\nabla F_i(x_{k+1}) - \\\\nabla F_i(x_k))$. I.e., the choices are not independent. (unless I am misreading the proof here, this is necessary)\n\n* Typo: some equalities in proof of Prop. 3.4 should be inequalities\n\n* Comment: On page 7, please make it clear that $\\\\mathcal{I}'_{i,k}$ is a multiset with elements chosen with replacement from $[N]$.\n\n* Typo: Equation (39) has a typo with $\\\\rangle$ vs. $|$ on the 5th line.\n\n* Typo: Equation (39), $\\\\chi_s'$ should be $\\\\chi_{s'}$\n\n* Question: \"Therefore the communication complexity is higher...\" on page 8. This is only higher by a factor of 2, no?\n\n* Typo: Equation (21), $H_{\\\\pi}(\\\\sigma)$ is not defined. You meant $D_{KL}$ instead?\n\n* Typo: In (59), $J_{\\\\pi}$ should be $J_{\\\\pi}(\\rho_t)$", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "tKuuZo0CgX", "reviewer": "Reviewer_M5MN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The strength is that analyses are given to the proposed algorithm, showing improvement over the previous works on communication cost.\nWeaknesses are mainly in three folds. \n1. The assumed log Sobolev inequality is a pretty strong assumption due to its strong connection to gradient dominance and strongly concave conditions.\n2. There is no comparison with commonly used FedAvg style algorithms in federated learning, both in terms of communication and convergence guarantee. \n3. No experiments in the paper to validate the actual performance of the proposed algorithm.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a Langevin algorithm for federated learning, along with its convergence analysis under log Sobolev inequality.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The paper would benefit from a thorough comparison with more commonly used federated learning algorithms, both in theory and in experiments.", "broader_impact_concerns": "No concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The submission meets both *claim and evidence* and *audience* criteria. However, the reviewers were generally lukewarm due to the incremental contribution and lack of experiments [see more above]. Only one reviewer engaged in the discussion/rebuttal phase, 2/3 reviewers proposed *leaning accept*.
Decision: Accept (poster)