Lower Bounds and Optimal Algorithms for Non-Smooth Convex Decentralized Optimization over Time-Varying Networks
Authors: Dmitry Kovalev
Ekaterina Borodich
Alexander Gasnikov
Dmitrii Feoktistov
Venue: NeurIPS 2024 poster (2024)
Keywords: decentralized optimization
convex optimization
distributed optimization
non-smooth optimization
time-varying networks
Primary area: optimization

Abstract: We consider the task of minimizing the sum of convex functions stored in a decentralized manner across the nodes of a communication network. This problem is relatively well-studied in the scenario when the objective functions are smooth, or the links of the network are fixed in time, or both. In particular, lower bounds on the number of decentralized communications and (sub)gradient computations required to solve the problem have been established, along with matching optimal algorithms. However, the remaining and most challenging setting of non-smooth decentralized optimization over time-varying networks is largely underexplored, as neither lower bounds nor optimal algorithms are known in the literature. We resolve this fundamental gap with the following contributions: (i) we establish the first lower bounds on the communication and subgradient computation complexities of solving non-smooth convex decentralized optimization problems over time-varying networks; (ii) we develop the first optimal algorithm that matches these lower bounds and offers substantially improved theoretical performance compared to the existing state of the art.

Lower Bounds and Optimal Algorithms for Non-Smooth Convex Decentralized Optimization over Time- Varying Networks
Dmitry Kovalev
Ekaterina Borodich MIPT borodich ed@phystech.co edu
Yandex Researh Takkovalev@@gmal.... com
Alexander Gasnikov Innopolis University, MIPT, Skoltech gasnikov@yandex.
Dmitrii Feoktistov Innopolis University MSU feoktistovdddmy... msu ru
Abstract
We consider the task of minimizing the sum of convex functions stored in decentralized manner across the nodes of a communication network. This problem is relatively well-studied in the scenario when the objective functions smooth, or the links of the network are fixed in time, or both. In particular, lower bounds the number of decentralized communications and (sub)gradient computations required to solve the problem have been established, along with matching optimal algorithms. However, the remaining and most challenging setting of non-smooth decentralized optimization over time-varying networks is largely underexplored, as neither lower bounds nor optimal algorithms known in the literature. We resolve this fundamental gap with the following contributions: (i) we establish the first lower bounds on the communication and subgradient computation complexities of solving non- non-smooth convex decentralized optimization problems over time-varying networks; (ii) we develop the first optimal algorithm that matches these lower bounds and offers substantially improved theoretical performance compared to the existing state of the art.
Introduction
In this paper, we study the decentralized optimization problem. Specifically, given set of n compute nodes connected through a communication network, our goal is to solve the following finite-sum optimization problem with quadratic regularization:
min
p(x) Σ +
(I)
ZER
where > is regularization parameter, and each function f((x): Rd - IR is stored the corresponding node E (1,...,,1) n... Each node can perform computations based on its local state and data, and can directly communicate with other nodes through the links in the communication network.
"Msccow Institute of Physics and Technology
i Skolkovo Institute of Science and Technology
Research Center for Artificial Intelligence, Innopolis University, Innopolis, Russia Moscow State University
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Decentralized optimization problems find applications in wide variety of fields. These include network resource allocation (Beck al. 2014). distributed model predictive control (Giselsson et al. 2013), power system control (Gan et 2012), distributed spectrum sensing (Bazerque and Giannakis 2009), and optimization in sensor networks (Rabbat and Nowak 2004). In addition, such problems cover the supervised training of machine learning models through empirical risk minimization, thus attracting significant interest from the machine learning community (Lian et al.,
2017
Ryabinin et al. 2021
Ryabinin and Gusev 2020
1.1
Time-varying Networks
In our paper, we focus on the setting in which the links in the communication network are allowed to change over time. Such time-varying networks (Zadeh 1961; Kolar et al. 2010) hold significant relevance to many practical applications. For instance, in sensor networks, changes in the link structure can be caused by the motion of sensors and disturbances in the wireless signal connecting pairs of sensors. Similarly, in distributed machine learning. connections between compute nodes can intermittently appear and disappear due to network unreliability (Ryabinin and Gusev 2020). Lastly, we anticipate that the time-varying setting will be supported by future-generation federated learning systems (Konecný et al. (2066] McMahan et al.) (017)) where communication between pairs of mobile devices or between mobile devices and servers will be affected by their physical proximity, which naturally changes over time.
1.2 Convex Setting
In this work, consider the decentralized optimization problem in the case when the objective function is convex (or strongly convex). At first glance, it may seem that the convexity assumption is restrictive and should not be considered. However, as we will see further, even in this fundamental setting, the existing algorithmic developments are limited and have significant gaps that need to be closed. Moreover, considering the convex optimization setting offers important benefits compared to general non-convex functions. One such benefit is that convex optimization often serves as source of inspiration for the development of algorithms that turn out to be highly effective in solving practical problems, even non-convex ones.
For example, state-of-ther ofeth-rr----rr optimization algorithms such as Adam (Kingma and Ba 2014) and RMSProp (Hinton et al. 2012)) employ the momentum trick, which is observed to be efficient for numerous tasks, including the training of deep neural networks. However, from the perspective of non-convex optimization theory, momentum is useless because, for non-convex problems, the iteration complexity of the standard gradient method cannot be improved (Carmon al. 2020). On the other hand, it was theoretically proven that momentum substantially boosts the convergence speed of the gradient method when applied to convex functions (Nesterov) 1983). In other words, convex optimization theory suggests that the momentum trick should be used, while non-convex theory suggests that it should not, and the former aligns much more closely with practical observations. A similar situation can be seen with other state-of of-te-of-the-art optimization methods, including distributed local gradient methods (Mishchenko et al., 2022, Sadiev et al. 2022 Karimireddy et al. 2020), adap- tive gradient methods (Duchi et al. 2011). 1). etc. Such inconsistency between non-convex theoretical convergence guarantees for optimization algorithms and their actual performance in practice can be attributed to the fact that the class of non-convex functions is far too broad. This is why many optimization research papers try to narrow down this class by considering additional assumptions such as
condition (Karimi et al. (2016), bounded non-convexity (Carmon et al. 2018 Allen-Zhu 2018), quasi-strong convexity (Necoara et al. 2019), etc. However, these assumptions can be seen relaxations of the standard convexity property. Therefore, we naturally opt focus on the convex decentralized optimization problem, leaving potential generalizations for future work.
1.3 Related Work and Main Contributions
Decentralized optimization has been attracting a lot of attention for more than a decade. Plenty of algorithms have been developed, including EXTRA (Shi et al. 2015). DIGing Nedic et al., 2017). SONATA (Scutari and Sun 2019), NIDS (Li et al. 2019). APM-C (Li et al. 2018 Rogozin et al. 202T). ) and many others. recent years, the focus of the research community has shifted towards the more complex task of finding, in some sense, the best possible algorithms for solving decentralized

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Table 1: Summary of the existing state-of-the-ar results in decentralized convex optimization. Multiple paper references are provided for each problem setting: papers marked with provide lower complexity bounds, and papers marked with provide optimal algorithms that match the corresponding lower bounds.
Smooth Setting
Non-Smooth Setting
Fixed Networks
Kovalev et al.
2020
Lan et al. (020))
Scaman et al.
2017)
Scaman et al. 2018
Time- Varring Networks
Kovalev al.
(022a
Algorithm (this paper)
Li and Lin
2021
Theorems
and 2 (this paper)*
optimization problems (Scaman et al. 2017 2018 Lan et al. 2020 Kovalev et al. 2020
202
2022, Hendrikx et al. [021] Li et al. 2022 and Lin 2021 Metelev et al. 2024). This task consists of finding lower bound on the complexity of solving given subclass of decentralized problems and finding an algorithm whose complexity matches this lower bound. Such algorithms are called optimal because their complexity cannot be improved for given problem class due to the established lower bounds.
We discuss the four main classes of decentralized optimization problems that cover smooth and non- smooth objective functions, and fixed and time-varying communication networks. We reference existing state-of-the-ar research papers that collectively solve the task of finding optimal algorithms for these classes. These papers are summarized in Table the case of smooth and strongly convex objective functions and fixed communication networks, Scaman et al. (2017) established the lower bounds on the number of communication rounds and the number of local gradient computations required to find the solution. These lower bounds were matched by OPAPC algorithm of Kovalev et al. 2020). In the case of smooth and strongly convex problems over time-vary marying networks, lower complexity bounds were provided by Kovalev et al. [2021a), and two optimal algorithms were developed: ADOM+ (Kovalev et al.) 2021a) and AccGT (Li and Lin 2021). In the case of non- non-shooth convex problems over fixed networks, lower bounds were established by Scamma al. (2018), , and two optimal algorithms were proposed: DCS (Lan et al.) 2020) and MSPD (Scaman et al. 2018)
Our paper primarily focuses the remaining and most challenging setting of non-smooth convex decentralized optimization problems over time-varying networks. Only a few algorithms have been developed for this setting, including the distributed subgradient method (D-SubGD) by Nedic and method (SubGD-Push) by Nedié and Olshevsky] (2014), and ZOSADOM by Lobanov et al. (2023). Moreover, to the best of our knowledge, neither lower complexity bounds nor optimal al algorithms have been proposed in this setting. Consequently. in this work, close this significant gap with the following key contributions:
Ozdaglar (2009), the subgradient-pee
(i) We establish the first lower bounds on the number of decentralized communications and local subgradient computations required to solve problem in the non-smooth convex setting over time-varying networks.
(ii) We show that our lower bounds are tight by developing the first optimal algorithm that matches these lower bounds. The proposed algorithm has state-of-the-arr theoretical com- munication complexity, which outclasses the existing methods described the literature.
2 Notation and Assumptions
In this paper, we are going to use the following notations: & denotes the Kronecker matrix product, Ip denotes a p x p identity matrix, 1, = (1, (I...., 1)T IR" e" E ER for j (1,.... .l....,p p} denotes the j-th unit basis vector, where p E {1. ...... ,. In addition, || denotes the standard Euclidean norm of a vector, and (.,-) denotes the standard scalar product of two vectors.
'y complexity, we mean, depending on the context, the number of subgradient computations decentralized communications required to solve the problem.
A function is called smooth if it is continuously differentiable and has Lipschitz-continuuus gradient.
3

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2.1
Objective Function
Further, we describe the assumptions that we impose on problem I As discussed Section 1.2 we
assume the convexity of the objective function in problem (I). In particular, we assume that functions
fi..)...., fn(x)) are convex, which formally described Assumption [
Assumption 1. Each function f((x)) is convex. That is, for all x', IR and T E [0, 1). the following inequality holds:
fa(TZ + +1 = (1-
(2)
In addition, we assume that the objective functions
are Lipschitz continuous, which is formalized in Assumption 2 This property is widely used in the theoretical analysis of non-smooth
optimization algorithms,, such as the subgradient method (Nesterov) 2013). dual extrapolation method (Nesterov) 2009), etc.
Assumption 2. Each function fz(z) is M -Lpiscchitz continuous M 2 0. That is. for all x', x E RA.
the following inequality holds:
(1(())) fx(x/) S M =/11-
We also need the following Assumption which ensures the existence of solution to problem I Note that in the strongly convex case (r 0), the solution always exists and is unique. However, in the convex case (r 0), need explicitly assume the existence of solution.
Assumption 3. There exists a solution x² E IRd to problem (T) and a distance >>> 0 such that
|| S.
2.2 Decentralized Communication
Next, we formally describe the decentralized communication setting. The communication network typically represented by a graph E(V,E), where V (l....,77, n. is the set of compute nodes and E. C V x V is the set of links in the network. As mentioned earlier, allow the communication links to change over time. Thus, we introduce the continuous time parameter т 0 and a set-valued function E()): R, + v which represents the time-varying set of edges Our time-varying network is then denoted as (T) = ET)).
Decentralized communication is typically represented via a matrix-vector multiplication with the so-called gossip matrix associated with the communication network (Scaman et 2017) Kovalev al. 2021a) In the varying setting, represent the gossip matrix by a matrix-valued function W(T): R. Raxx, which satisfies the following Assumption 4
Assumption 4. For all T > 0, the gossip matrix W(r) E _g= associated with the time-varying communication network G(V, E(T)) satisfies the following properties:
(i) W(T); 0 if # j and (j.i) ((T).
(ii) W(T),, = 0 and W(T) W1, 00
also define the so-called condition number of the network x > which indicates how well the network ((r) is connected (Scaman et al. 2017 Kovalev et al. 202 In particular, the communica- tion complexity of most decentralized optimization algorithms depends on X. Assumption provides the formal definition of this quantity.
Assumption 5. There exists a constant 2 such that the following inequality holds for all 5 2 0:
[[(x)> - x (1 1/x) | for all E {(II,,..., In) en)) E R² >E= Ti
=0] (5)
Lower Complexity Bounds
3.1 Decentralized Subgradient Optimization Algorithms
In this section, we present the lower bounds on the number decentralized communications and the number of local subgradient computations required to solve problem (I). These lower bounds
By vxv {E: 3 VV V} we denote the of all subsets of V V.

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apply to particular class of algorithms, which we refer to as the class of decentralized subgradient optimization algorithms. This class can be seen as an adaptation of black-box optimization procedures (Scaman et al. 2018) to the time-varying network setting, or adaptation of first-order decentralized optimization algorithms (Kovalev et al. 2021a) to the non-smooth optimization setting.
Non-smooth optimization algorithms typically perform incremental updates by computing the sub- gradient of given objective function. The of all subgradients of convex function, called the subdifferential can be multivalued in general. Thus, it is necessary to select the specific subgradient that the algorithm will use. This done by the subgradient oracle, which described by Definition
Definition 1. For each E V, a fanction fi(a): R - I is called a subgradient oracle associated with the function fi(x) if. for all x E IRd, it satisfies V fi(x)) E 0f((z). That is, for each i E V and for all x, x' E R., the following inequality holds:
fi(x') 2 f((x) (Vi(()).'' -r)).
Further, we provide the formal description of the class of decentralized subgradient optimization algorithms in the following Definition
Definition 2. An algorithm is called a decentralized subgradient optimization algorithm with the
subgradient compulation time Twb O and decentralized communication time Teom if satisfies the following constraints:
ii) Internal memory, any time Z each node i V maintains an internal memory, which is represented by a set-valued function M,(T): 1 - 2²".. The internal memory can be updated by subgradient computation or decentralized communication, which is formally represented by the following inclusion:
Mi(T) C. M, mith (T) (M.
where set-valued functions M MQT"M R. + 2 are defined below.
(ii) Subgradient computation. At any time t 2 O, each node i E V can update its internal memory M,(T) by computing the subgradient f((x)) of the function fi(z), which takes time Tsab- That is, for all r 2 O, the set M (T) r) is defined as follows:
M (T) = [span((z, Vl(()) x E EM(T -Taub)))
T 2 Taub
(8)
< Tsub
(iii) Decentralized communication. At any time T 2 O, each node E V can update its internal memory Mi(T) by performing decentralized communication across the communication network, which takes time Teom. That for all r 2 0, the set Manm (T) is defined as follows:
span (pannUUesEEEt)))MM M3(T Teom))
M (T)
7
Teom
Tcom
(iv) Initialization and output. At time T = O. each node V must initialize its internal memory with the zero vector, that is, M,(0) = {0}. At any time T 2 O, each node e V must specify single output vector from its internal memory, Fa,(T) ) E Mi(T).
3.2 Lower Bounds
Now, we are ready to present the lower bounds on the execution time 7 > 0 required to find an e-approximate solution to problem (T) by any algorithm satisfying Definition Theorem provides the lower bound in the strongly convex case (r > 0), and Theorem 22 provides the lower bound in the convex case (r 0). These lower bounds naturally depend on the precision e > 0, the parameters of the problem, including the Lipschitz constant M > 0, the regularization parameter 2 0. the distance R. > O, and the parameters of the network, including the condition number x 2 1, communication time Teom O, and subgradient computation time Tuub 0.
'A vector E IRd is called c-approximate solution to problem D if p(x') S
5

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Table 2: Lower bounds on the communication complexity of solving problem I in the centralized (Arjevani and Shamir, 2015), decentralized fixed network (Scaman al. 2018). and decentralized settings.
vare-ig network (Theorems) and 2
Setting
Centralized
Fixed networks
Time-varying networks
Strongly convex
Ω (M/vrr)
22 (VXXM//re)
(xM//Tr))
Convex
Ω (MR/e)
l (VXMR\/))
((MMR/e)
Theorem 1. For arbitrary parameters M, c, Team Tsub 0 and x there exists an optimization problem of the form ① satisfying Assumptions
by Definition 7 time varying network (T) 22 and corresponding subgradient oracles given E(T)), and corresponding time-varying gossip matrix W (T) satisfying Assumptions and 5 such that at least the following time is required to reach precision p( (T)) p(xx)) S E by any decentralized subgradient optimization algorithm satisfying Definition
XM
M TE
2 S2
Tcom
+
Tzub
(10)
VTe
Theorem 2. For arbitrary parameters M, R., Teom Taub 0 and 1. there exists an optimiza-
tion problem of the form ①T with zero regularization (r = 0) satisfying Assumptions 7 2 and 3
corresponding subgradient oracles given by Definition 7 time varying network ((T) (V. {Tr
and a corresponding time-varying gossip matrix ((T) satisfying Assumptions { Aand such that at
least the following time is required to reach precision p( (T)),((T) p(x") S by any decentralized
subgradient optimization algorithm satisfying Definition 2
XMR
M 22
Ω
Teom
Taub"
(11)
The proofs of Theorems Janieemmsann222 can be found in Appendix B Further, we provide a brief and informal description of the main theoretical ideas that underlie these proofs:
(i) We select a specific "hard" instance of problem ① In particular, we choose the objective function of the form p(x) >
ex x) a(ei, z²) + |||=r|/, which was used by (Arjevani and Shamir (0155); Scaman et al. (2018) in the proof of lower bounds on the communication complexity centralized and netw-kk settings. One can show that the gap p(x) p(x") is lower-bounded by positive constant as long the last component of the vector æ is zero, and it takes M(Taub d) time to break this bound due to the constraint on the subgradient updates (8).
(e [
(ii) We split the objective function between two nodes of a star-topology network with time- varying central node, which was previously utilized by Kovalev et al. (2021a) in the proof of lower bounds for optimizing smooth functions. One can show that it takes 22()) = ?(x) communications exchange information between the two selected nodes due the time- varying center. This contrasts with the fixed path-topology network used by |caaman al. 20177 2018), where such an exchange would take Ω(n) = R(vx) communications. Moreover, using the constraint (8), we can show that it takes Q(Teom nd) time to make the last component of the vector r nonzero and break the lower bound on the gap p(z) p(z*), thanks to the way we split the objective function.
(iii)
Based on the above considerations, we show that the total execution time required to solve the problem is lower-bounded by Ω (Teom nd Tsub d). Thus, obtain the desired results by making a specific choice of the dimension d, network size n, and other parameters of problem
3.3 Comparison with the Lower Bounds in Centralized and Fixed Network Settings
We compare the lower complexity bounds for solving non-smooth convex optimization problems in the three main distributed optimization settings: centralized, decentralized fixed network, and
Scaaaa et al. 2018) do not provide any lower complexity bounds in the strongly convex setting. However, the desired lower bound the communication complexity can be obtained by extending their analysis.

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Algorithm
Theorems I and 2 limply that the communication complexity in the decentralized marying network setting is proportional to the network condition number X. In contrast, the communication complexity in the fixed network setting is proportional to vx. which reflects the fact that time-varying networks are more difficult to deal with compared to fixed networks. In particular, there was a long-standing conjecture that the "upgrade" from the factor x to the factor vx in communication complexity is impossible in the time-varying network setting. Only recently, this conjecture was proved for smooth resolve this open question in the non-smooth case as 1: input: 20 = x- 0 E (R.))
= E (R")", )r, = E C m E (Rd)
2: parameters: K, T (1,2.... .. {(ax 3kr yki ok: Ak. x Wu By V/ 1 c R
3: for k ===00 0. 0.1,...; K do
4:
y Okl + (1- 0x)y k. =
=x))2
5:
f2 V) J V.
where function G(v: 2) is defined in eq. 12
6:
-h (Wx @ La)g - = (W @ La) +m²)
where We denotes the gossip matrix W(T) at the current time
7:
x+1 yf L + sttl),
=lt = zº ~r/ge. ak+1 = x + Th (5t - xbh11)
8:
gh y + ta ty - y²), ghtl = 2 - Ogg m = (vt² / + (mn + + - 95)
k. L²
9:
10:
for 1 = 0, 0...... ...., T - do
ghtt (V (x) kit ),... .....: (n(
11:
12:
x k.4+1 = xhi 1// Igs + Boothtt y 1 + (attttt1 xxt))
13:
xk+1 0xz k.' + (1- (1-00h (1-oa))
Iktl = +E=1 ahi zbtl = agittt + + (1 ork)
14:
(x y *k) (Σ Ax)= E
k=1 (= , b²
15:
output: 2 n Di=1 Fas E Rdd where (x (2 2 xx E (Rd)"
decentralized time-varying network. The lower subgradient computation complexity bounds coincide in these cases (Nesterov) (213)) Scammnn et al. Theorems [nd However, the situation with the communication complexity is different. See Table 2 for a summary.
functions by Kovalev et al. (2021a), and well.
Optimal Algorithm
In this section, we develop optimal algorithm for solving the non-smooth convex decentralized optimization problem ( over time-varying networks. The design of our algorithm relies on a specific saddle-point reformulation of the problem, which we describe in the following section.
4.1 Saddle-Point Reformulation
Let functions F(x): (Red)" - .R. and G(y, 2): (IRd)" (Rdd)² - R. be defined as follows:
F(x)= = Σ f((x)) + f
2 || and G(y. 2) = F9z ly 2
(12)
2
where r (Z1, (#1...., In) E (Rd)", and Tas Tys are some constants that satisfy
rx + 1/T =.
(13)
Consider the following saddle-point problem:
min max max [Q(z, v) F(x) (y.x) G(y, s.t. EE (E(R") VE(R")" =E(Rd)=
(14)
where L C ( is the orthogonal complement to the so-called consensus space L C (R.dy",
defined as follows:
L [( Tx)) 21 =.. = =x,] ). L ((
ー 0}.
(15)
One can show that the saddle-point problem [1) equivalent to the minimization problem 11. This
justified by the following Lemmal) [ The proof of the lemma can be found in the Appendix A
7

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Lemma 1. Problem 14) is equivalent to problem ① in the following sense:
min THE (R (E(R SEL
max max Q(I, v, 2) = nt min p(x).
(16)
The auddle-point reformulation the form (4) was first introduced by Kovalev et al. (2020) 2021a) to develop optimal decentralized algorithms for optimizing smooth functions. However, these are not applicable to the non-smooth case. To the best of our knowledge, the only attempt to adapt the reformulation (14) to the non-smooth setting made by Lobanov et al. (002)). However, their results have sigmificant downsides, which we discuss in Section 4.3)
4.2
New Algorithm and its Convergence
Now, we present Algorithm D for solving problem (1) We provide upper bounds the number of decentralized communications K and the number of subgradient computations 7 required to find an e-approximate solution to the problem. Th [ the upper bounds in the strongly convex (r > 0) and convex (r = 0) cases, respectively. The proofs can be found in Appendix [ The total execution time of Algorithm
= (Tcom K + Tsuh K x T), where the communication time Toom > O and the subgradient computation time Tsub >> O are described Definition 2 This upper-bound on the execution time cannot be improved because of the lower bounds established in the previous Section 3 Therefore, Algorithm is an optimal algorithm for solving problem (I.
[ iss upper-bounded
Theorem 3. Under Assumptions
and ler (strongly convex case). Then Algorithm 7 requires K = O XM decentralized communications (line 6 of Algorithml and K XT = O
M²
subgradient computations (line) of Algorithm
7 reach precision p p( p(x*) S..
Theorem 4. Under Assumptions 7
2 and let r 0 (convex case). Then Algorithm 7 requires
K =
XMM
decentralized communications (line of Algorithm and K x
O M R²
subgradient computations (line 17 of Algorithm to reach precision p(=
p(x*) Se
The design of Algorithm is based the fundamental Forward-Backward algorithm (Bauschke and Combettes (2011). Let E (Rd)" x (Rd)" x L be a Euclidean space, and consider monotone
operator A(u): E E and a maximally- multivalued operator B(u): + 2 defined as
follows:
(FF()) V I
A(u) = V 9 G(y, G(y,
B(u)= =
(17)
PV
where u = (x. 2. E E, and P = (I, - (1/n)1,,1 @ I E Rnndxnd is the orthogonal projection matrix onto L- Then problem (14) is equivalent to the following monotone inclusion problem:
find u such that A(w) + B(u).
(18)
The basic -Bckwarrd algorithm iterates - (id + B)- A(u where id is
the identity operator and (id + B) 11 denotes the inverse of the operator id(u) + B(u), which is
called resolvent. Algorithm T can be obtained by making the following major modifications to these iterations:
ii) We accelerate the convergence of the forward-Backweer algorithm using Nesterov accelera- tion (Nesterov) 1983). Although this mechanism cannot be applied to the general monotone inclusion problem (18), Kovalev al. (2020) showed that it can be used when the operator A(u) is equal to the gradient of a smooth convex function, which is true in our case.
(ii) Computation of the operator A(u) requires multiplication with the matrix P. This, turn, requires an exact averaging of a vector, which is difficult to do over the time-varying network. Kovalev et al. (2021b) showed that this obstacle can tackled with the Error-Feedback mechanism for decentralized communication. which we also utilize.
(iii) At each iteration of the algorithm, we have to compute the resolvent, which requires solving an auxiliary subproblem min,, max, I- || * + F(x) (y, x) Il 1 This problem cannot be solved exactly, so have to find approximate solution using

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Table 3: The execution time required to find an e-approximate solution to the decentralized optimization problem T by the following algorithms: D-SubGD (Nedic and Ozdaglar (2009). SubGD-Puse (Nedić and Olshevsky 2014), ZO-SADOM (Lobanov et 2202), and Algorithm (this paper). Decentralized communication and subgradient computation complexities are marked with green and yellow colors, respectively. For D-SubGD, the complexity is not provided because the algorithm converges only to neighborhood of the solution. For SubGD-Push, poly(M, M, R, d) denotes a certain polynomial in M, R, For ZO-SADOM, the differences from the optimal complexities are highlighted in red color.
Algorithm
Strongly-convex case complexity
Convex case complexity
D-SubGD
N/A
poly(M, R. d)
2 log"?
poly(M, R, d)
n log"
SubGD-Push
Teom
22
+ Taub
xM (MM4/4 log
M d log
xM Rd/44 log
M²Reddog
ZO-SADOM
Teom
Vre
+ Tash
re
Team
+ Tinb
e²
XM
M
XMR
MR
Algorithm
Teom
Tuub
Teom
Tsub
e²
XM
M
XMR
M²
Lower Bounds
+ Tuub
+ Tsub
/rc
rc
additional "inner" algorithm based on the subgradient method (Nesterov 2013) and the Chambolle-Pock operator splitting (Chambolle and Pock [201T). We also have to conduct a careful analysis to find an efficient way to combine the inner and the "outer" Forward- Backward algorithms and avoid unnecessary waste of subgradient calls.
The design of Algorithm I shares some similarities with the algorithm of Kovalev et al. (2021a) such as (i) and (ii) above. However, Kovalev al. (2021a) simply add the gradient VF(x) to the operator A(u) and use the accelerated version of the Forward- Bacrrdd-Bawwrdd algorithm, which we obviously cannot do as the function ((x) is not smooth. Instead, we have to put the subdifferential FF into the operator (vu) and follow (iii) above. Part (iii), in turn, shares some similarities with the algorithm of Lan (2020). However, Lan et al.) (2020) simply have a zero operator A(u) = O, which makes (i) and (ii) above unnecessary in their case. In contrast, we cannot make such simplifications because work in the much more complicated setting time-varying networks.
4.3 Comparison with the Existing Results
One could naturally expect that the existing optimal algorithms, originally developed for fixed networks, such as DCS (Lan et al.) (020) and MSPD (Scaman et al.] 2018), could be applied to solve problem ( over tamevirrying networks. However, this is not the case, which justified by the lack of corresponding theoretical guarantees and was shown empirically by Kovalev et al. (20216). Therefore, we have consider only those algorithms that were specifically developed for the time-varying network setting.
We provide comparison of our Algorithm|] T with the existing state-of-the- decentralized methods for solving convex non-smooth optimization problems over time-varying networks in Table 3 10
These include D-SubGD (Nedic and Ozdaglar 2009), SubGD-Push (Nedić and Olshevsky 2014). 4h,
and ZO-SADOM (Lobanov et al. 2023). The first two algorithms have poor performance: D-bubD
converges only to limited precision, and SubGD-Puse converges at slow rate of O(log" (1 (1/() /))) which does not match even the iteration complexity of the standard centralized subgradient method,
let alone the improved complexity of Algorithm [ The complexity of ZO-SADOM is also worse
than the lower bounds. Moreover, the theoretical results Lobanov et (2023) have substantial drawbacks compared to ours:
We ignore universal constants in Table like in the O(-) and 12(-) notation.

---

(i) Lobanov et al. 2023) do not provide any theoretical insights or innovations in the analysis their algorithm. In particular, they use the randomized smoothing technique (Duchi et al. 2012 to obtain a smooth approximation of the objective p(æ), and apply the existing algorithm Kovalev et al. (2021a) minimize this approximation. In contrast, we develop new algorithm that directly works with the original non-smooth objective p(x).
(ii) ZO-SADOM has extra factors d'/4 log(1/c) and (llo((1/) in the decentralized communi- cation and subgradient computation complexities, respectively, compared to the optimal complexity of our Algorithm Thus, the performance of ZO-SADOM can be poor when applied, for instance, to large-scale machine learning problems in which the dimension d can be huge.
Acknowledgments and Disclosure of F Funding
This research has been financially supported by The Analytical Center for the Government of the Russian Federation (Agreement No. 70-2021-00143 01. IGK 0000000D730324454000))
References
Allen-Zhu, Z. (2018). Natasha 2: Faster non-convex optimization than sgd. Advances in neural information processing systems, 31.
Arjevani, Y. and Shamir, O. (2015). Communication complexity of distributed convex learning and optimization. Advances in neural information processing systems, 28.
Bauschke, H. H. and Combettes, P. L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer Science Business Media.
Bazerque, J. A. and Giannakis, G. B. (2009). Distributed spectrum sensing for cognitive radio networks by exploiting sparsity. IEEE Transactions on Signal Processing 5867
Beck, A., Nedié, A., Ozdaglar, A., and Teboulle, M. (2014). An O(1/k) gradient method for network resource allocation problems. IEEE Transactions on Control of Network Systems, 1(1):64-33.
Carmon, Y., Duchi, J. C.,, Hinder, O., and Sidford, A. (2018). Accelerated methods for nonconvex optimization. SIAM Journal on Optimization, 28(2):1751-1772.
Carmon, Y., Duchi, J. C., Hinder, O., and Sidford, A. (2020). Lower bounds for finding stationary points i. Mathematical Programming, 184(1):71-120.
Chambolle, A. and Pock, T. (2011). A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of mathematical imaging and vision, 40: 120-155
Duchi, J., Hazan, E.,, and Singer, Y. (2011). Adaptive subgradient methods for online learning and stochastic optimization. Journal of machine learning research, 12(7).
Duchi. J. C., Bartlett, P. L.. and Wainwright, M. J. (2012). Randomized smoothing for stochastic optimization. SIAM Journal on Optimization, 22(2):674-701.
Gan, I.., Topcu, U., and Low, S. H. (2012). Optimal decentralized protocol for electric vehicle charging. IEEE Transactions on Power Systems, 28(2):900-951.
Giselsson, P., Doan, M. D., Keviczky, T., De Schutter, B., and Rantzer, A. (2013). Accelerated gradicnt methods and dual decomposition in distributed model predictive control. Automatica, 49(3):829-833,
Hendrikx, H., Bach, F., and Massoulie, I.. (2021). An optimal algorithm for decentralized finite-sum optimization. SIAM Journal Optimization, 31(4):2753-2783,
Hinton, G., Srivastava, N., and Swersky, K. (2012). Neural networks for machine learning lecture 6a overview of mini-batch gradient descent. Cited on, 14(8):2.
10

---

Karimi, H., Nutini, J., and Schmidt, M. (2016), Linear convergence of gradient and meoximal-gradien methods under the polyak-lojasiewezz condition. In Machine Learning and Knowledge Discovery in Databases: European Conference, ECML. PKDD 2016, Riva del Garda, Italy, September 19-23, 2016, Proceedings, Part 16, pages 795-811. Springer.
Karimireddy, S. P., Kale, S., Mohri, M., Reddi, S., Stich, S., and Suresh, A. T. (2020). Scaffold: Stochastic controlled averaging for federated learning. In International conference on machine learning, pages 5132-5143, PMLR.
Kingma, and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.
Kolar, M., Song, L.,, Ahmed, A., and Xing, E. P. (2010), Estimating time-varying networks. The Annals of Applied Statistics, pages 94-123.
Konecný, J., McMahan, H. B., Yu, F. X., Richtárik, P., Suresh, A. T., and Bacon, D. (2016). Federated learning: Strategies for improving communication efficiency. arXiv preprint arXIv:1610.05492,, 8.
Kovalev, D., Beznosikov, A., Sadiev, A., Persiianov, M., Richtárik, P., and Gasnikov, A. (2022). Optimal algorithms for decentralized stochastic variational incqualities. Advances in Neural Information Processing Systems, 35:31073-31088.
Kovalev, D., Gasanov, E., Gasnikov, A., and Richtarik, P. (2021a). Lower bounds and optimal algorithms for smooth and strongly convex decentralized optimization over tire-narying networks. Advances in Neural Information Processing Systems, 34:22325-22333
Kovalev, D., Salim, A., and Richtárik, P. (2020). Optimal and practical algorithms for smooth and strongly convex decentralized optimization. Advances in Neural Information Processing Systems, 13:18342-18352.
Kovalev, D., Shulgin, E., Richtárik, P., Rogozin, A. V., and Gasnikov, A. (2021b). Adom: Accelerated decentralized optimization method for time-varying networks. In International Conference on Machine Learning, pages 5784-5793. PMLR.
Lan, G., Lee, S., and Zhou, Y. (2020). Communicttion-eficieent algorithms for decentralized and stochastic optimization. Mathematical Programming, 180(1):237-284.
Li, H., Fang, C., Yin, W., and Lin, Z. (2018). A sharp convergence rate analysis for distributed accelerated gradient methods. arXiv preprint 1rXiv:1810.0100:
Li, H. and Lin, Z. (2021). Accelerated gradient tracking over time-varying graphs for decentralized optimization. arXiv preprint 104v:2200.002566
Li, H., Lin, Z., and Fang, Y. (2022). Variance reduced extra and diging and their optimal acceleration for strongly convex decentralized optimization. Journal of Machine Learning Research, 23((22): 1- 41.
Li, Z., Shi, W., and Yan, M. (2019). A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates. IEEE Transactions on Signal Processing. 67(17):4494 4506.
Lian, X., Zhang, C., Zhang, H., Hsieh, C.-J., Zhang, W., and Liu, J. (2017). Can decentralized algorithms outperform centralized algorithms? case study for decentralized parallel stochastic gradient descent. Advances in neural information processing systems, 30.
Lobanov, A., Veprikov, A., Konin, G., Beznosikov, A., Gasnikov, A., and Kovalev, D. (2023). Non- smooth setting of stochastic decentralized convex optimization problem over time-varying graphs. Computational Management Science, 20(1):48.
McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. (2017). Communication- efficient learning of deep networks from decentralized data. Artificial intelligence and statistics, pages 1273-1282. PMLR.
11

---

Metelev, D., Chezhegov, S., Rogozin, A., Kovalev, D., Beznosikov, A., Sholokhov, A., and Gasnikov, A. (2024). Decentralized finite-sum optimization over vire-ng networks. arXiv preprint arXiv:2402.0249..
Mishchenko, K., Malinovsky, G., Stich, S., and Richtárik, P. (2022). Proxskip: Yes! local gradient steps provably lead to communication acceleration! finally! In International Conference on Machine Learning, pages 15750-15769. PMLR.
Necoara, I., Nesterov, Y., and Glineur, F. (2019). Linear convergence of first order methods for non-strongly convex optimization. Mathematical Programming, 175:69-107.
Nedić, A. and Olshevsky, A. (2014). Distributed optimization over time-varying directed graphs. IEEE Transactions on Automatic Control, 60(3):60--615
Nedic, A., Olshevsky, A., and Shi, W. (2017). Achieving geometric convergence for distributed optimization over varying graphs. SIAM Journal on Optimization, 27(4):2597-2333.
Nedic, A. and Ozdaglar, A. (2009). Distributed subgradient methods for multi-agen optimization. IEEE Transactions on Automatic Control, 54(1):48-6) 61.
Nesterov, Y. (1983). A method for unconstrained convex minimization problem with the rate of convergence O(1/k²). In Dokl. Akad. Nauk. SSSR, volume 269, page 543.
Nesterov, Y. (2009), Primal-dual subgradient methods for convex problems. Mathematical program- ming, 120(1):221-259,
Nesterov, Y. (2013). Introductory lectures on convex optimization: A basic course, volume 87. Springer Science Business Media.
Rabbat, M. and Nowak, R. (2004). Distributed optimization sensor networks. In Proceedings of the 3rd international symposium Information processing in sensor networks, pages 20-27.
Rogozin, A., Lukoshkin, V., Gasnikov, A., Kovalev, D., and Shulgin, E. (2021). Towards accelerated rates for distributed optimization over time-varying networks. In Optimization and Applications: 12th International Conference, OPTIMA 2021, Petrovac, Montenegro, September 27-October 1, 2021, Proceedings 12, pages 258-272. Springer.
Ryabinin, M., Gorbunov, E., Plokhotnyuk, V., and Pekhimenko, G. (2021). Moshpit sgd: Communication-effiieee decentralized training on heterogeneous unreliable devices. Advances in Neural Information Processing Systems, 34:18195-18211
Ryabinin, M. and Gusev, A. (2020). Towards crowdsourced training of large neural networks using decentralized mixuure-of-exeert.. Advances in Neural Information Processing Systems, 33:3659-3672.
Sadiev, A., Kovalev, D., and Richtárik, P. (2022). Communication acceleration of local gradient methods via an accelerated primal-dual algorithm with an inexact prox. Advances in Neural Information Processing Systems, 35:21777-2111 -17911.
Scaman, K., Bach, F., Bubeck, S.. Lee, Y. T., and Massoulié, L. (2017). Optimal algorithms for smooth and strongly convex distributed optimization in networks. In international conference on machine learning, pages 3027-3036. PMLR.
Scaman, K., Bach, F., Bubeck, S., Massoulié, I.., and Lee, (2018). Optimal algorithms for non-smooth distributed optimization in networks. Advances in Neural Information Processing Systems, 31.
Scutari, G. and Sun, Y. (2019). Distributed nonconvex constrained optimization over time-varying digraphs. Mathematical Programming, 176:497-544,
Shi. W. Ling, Q.. Wu, G., and Yin, W. (2015). Extra: An exact first-order algorithm for decentralized consensus optimization. SIAM Journal on Optimization, 25(2):944-966.
Zadeh, L. A. (1961). Time-varying networks, i. Proceedings of the IRE, 49(10))1488-11
12

---

Appendix
A Proof of Lemma I
The orthogonal complement _ to the consensus space L is given as follows:
C { Tn) E (I²)" : I1 + +... + In = 0}
(19)
Let us perform the maximization of Qz,, 2) in the variable y E (I=
max wE(Rd))
Qzx yr
(a)
max NE(Ra)"
F(x) (v.x) G(y. 2)
F(x) + max
((,xx)
Ts
|1 =113] 2
yE(3d)n
F(x) + yryz
|r,||
(x, 2)
where (a) uses the definition of (z 2) eq. [14; (b) uses the definition of G(y, 2) in eq. 12). Next,
we perform maximization in the variable E
max max Q( 2) = max ZEC- VE(Re)n
F(x)= + 2ryz
|| (xr, 2)
ZEC-
=((x) +
||~||/
max ZEC-
| (ax, 2)1
2r
F(x) +
|| + Lc(x),
2ryz
where (x)): (RRd)= R is the indicator function, which is defined as follows:
0
x 2
Tc(x) = max
[- (x,,
(20)
=EC-
+o0
otherwise
Now, we can rewrite the saddle-point problem
14 as follows
(a)
min max max Q(r,
min max max Q(x, y, 2) =E(Ra)n ZEL BE(Rdyn
=E(Rd)" yE 2E
1
min #E(Rd)>
F(x) + 2r
|||| + Ic(x)
[((x.)) + + 11/Tue |11:113) + [c(x)
min #E(Rd))
(d) min @E(R
(Ii(z:) + "Ilosllp) + Tc(x)
min p(x).
ZER
where (a) uses the fact that we can exhange the order of the two consecutive maximizations; (b) uses
the previous equation; (c) uses the definition of F(x) in eq. [2): (d) uses eq. (13); (e) uses the definition of p(z) in eq. and the definition of I (x).
13

---

B Proof of Theorems andd 2
B.1 The Hard Instance of Problem
Compute nodes. this proof, we consider the case when 2 3. The case < can be proven using the fixed-network argument of Scaman et al. (2018). We choose
3[x/3,, which implies
that n 2 and n mod 3 = 0. We also divide the set of nodes V = (1..... n) into the following
three disjoint subsets: V1 (.,.....,....................... n/33. V22 {n/3 L.... , and V/3 (2n/3- ,,...,
Objective functions. We fix an arbitrary odd integer d E f3,5, .,... and define functions
f1(x), ((()): Rd IR as follows:
h2g 12)) a(x, o4) i V
fx(2) 1-12/2 1//21122( (22((2)
i V/2
(21)
i V/3
where a is an arbitrary constant and functions hx)), hd- (x)): Rd + R are defined as follows:
h1(x)) = (xx e - e
(22)
Consequently, the objective function p(z) in problem (T) given follows:
d-1
a (e(.z) x) + 2 ||l.
p(r) = 3 (x())
(23)
3
We also define the subgradient oracles (z)))), ... vz((x): Rd Rd as follows:
d 1)/2 / (1) ae i V1 1)/2 (1)
fi())
i E V2 i V
(24)
where (2), ..., ( Rd + nd are the subgradient oracles associated with functions
h1(x), ..... 1 defined follows:
j+1
og
(e
1=) > (ed x)) 1. 2) = (ed
(h,(x)
0
(25)
j+1
++110)
(ed
Time-varying network. We choose the tire-varying network ((T) (V,E(T)) to be a star- topology undirected graph with the time- tine-varying center node ic(T) E V. Formally, define the edges of the time-varying network ((T C V V as follows:
E(T) = U
((1,
(26)
iEV.i7ie(T))
We also specify the center node (T) at a given time T 2 0 as follows:
(T) 2n/3 + 1 (IT/Tcoml mod n//)..
(27)
We choose the time-varying gossip matrix W(T) E Rax to be the Laplacian matrix of the graph (()). Formally, W(r) is defined as follows:
0
* and (i,j) $ E(r)
W(T)
i and (i.j) {(T),
(28)
(eg, i=
where deg, (T) denotes the degree of the node E V in the graph ((T), i.e.
drg, H[ (i.j) E(T))
(29)
One can observe, that the varying gossip matrix W(T) satisfies Assumption in particular, ker W(T) ker W sp Moreover,
show that W(r) is symmetric matrix, and Amax W(r) and - (W(()) 1/n 2 1/x. Hence, W(T) satisfies Assumption 5
14

---

B.2
Auxiliary Lemmas
Further, we define linear spaces Ku...., Kud C R2 as follows:
Ko (0) and K, = span (fot....,,f)) .ff. for jE (1......d))
(30)
In order prove Theorems D and we will use the following auxiliary lemmas. The proofs of these lemmas can be found in Appendix Appendix [3.]] and the proof of Theorem 2
C
Furthermore, the proof of Theorem is contained in
contained in Appendix B34
Lemma 2. For all t 2 O, the following statements hold:
ii Leri E V2 Then, for all j E (I....,(d (d-..,/d1)))))))
Mi(T) C K2s implies MM (T + Taub) C K2s.
(31)
(ii) Leti E V2. Then, for all E (O,...,(dd (d-))))
Mi(T) C K2j+1 implies M (r + Taub) K2j+1.
(32)
(ii) Let i E V3. Then, for all E (O....,d], d}.
Mz(T) Cj
implies M. (T Taub) Kj-
(33)
The proof of Lemma 2 is contained in Appendix C.1
Lemma 3. Let E (O,...,n(4. 1)/6 - 13. Then, for all < (k + Tcom the following inclusion holds:
M((T) C (Kpp2 i V1 or (i E V3 and < 2n/3 + 1-+440 1)) (Kppt1 i V/2 or (i E V/ and > 2n/3 + + +1)
(34)
where [3k/n] and q = k mod (n/3).
The proof of Lemma is contained in Appendix C.2
Lemma 4. Let functions f1,..., (x) be defined by eq. solution x² E Rdd, which is given follows:
21
Then problem eq.
has unique
a
1a.
(35)
3rd
Moreover, for all x E K8-1, the following inequality holds:
a
p(x) p(xx)) 2 18rd'
(36)
The proof of Lemma is contained in Appendix C.3
B.3 Proof of Theorem
Decentralized communication. Lemma 3 implies that M2(T) C. Ka-1 as long as 7 < Toom nndd
1)/6. Hence, Lemma 4 iimplis eq. (36) for all x M2(r) as long as 7 < Teom n(d 1)/6. Let the
constant a >> 0 be chosen as follows:
M
(37)
2vd
Then, each function fa(x) defined by eq. (11 is M-Lipschitz. Indeed, the case E V3 is trivial. In the case when i V1, we can prove the -i-sschhitz continuity of f((x) follows:
(d 1)/2
f1)')
Σ
(1,z,
e
e
(x', - a(x x'.
1=1
(d- 1)/2
(x x' e - + al(r - x', efil
15

---

(d- 1)/2
(1 - n', e + 1 - x', )))1) + a (x'
<
d-1
Σ
(0 - x' eg)l + a (0 ea)
j=1
<
Σ
(or x' , eggl
2 2avd| x' S M| - r'|l.
j=1
In the case when i E V2, prove the M M-Lipschitz continuity of f((z) similarly,
Without loss of generality, we assume S M² /576r) and define d {3, 5, ...} follows:
M
d=
(38)
12 re
Using eqs. (37 and 38). for T < Teum n2d /) and x E M,(r). we obtain
M
p(x) - p(x*) z
36r E.
Hence, to reach precision p(z) - p(x*)====-))))))>)))) S for some E M;(T), it is necessary that т satisfies
n(d 1)
2 Toom
M
12 -1)
Teom
3
(39)
M
Teom
3
12vrE
Mx
Ω
Teom
Vrr
Subgradient computation. We also need to prove that to reach precision p(æ) - -((x)) S e for some x E M,(T), it is necessary that satisfies
M
Tsuh
(40)
P6
We can do this by providing extended version of our hard problem instance, described Ap- pendix B. In particular, we consider the following instance of problem
min 1 n (n,a"))Rd xRd n (fAe))+
|||| 1111
(41)
where functions fi( (x): IRd + R are defined in Appendix B. by eq. (11 and functions
f(x'),..., (xr): Rdd + IR are defined as follows:
[x(') = b max
(e x'),
(42)
JE(L....d')
where some constant. Then, by choosing an appropriate subgradient oracle V associated with each function (z') (see Section 3.2.1 of Nestcro (2013)) we can obtain both lower bounds (39) and (40). which concludes the proof.
16

---

B.4 Proof of Theorem 2
Our proof of Theorem 2 is very similar to the proof of Theorem with the following differences. Let function hz(x): R R be the Huber function, which is defined as follows:
1 S 8
hx(x) Σ (g(lef, a)), where k1((1)
(43)
88/1 2 82
1 8
Note that function hz(z) is continuously differentiable and (v (Vdo)-ipsshhiz continuous.
In the proof of Theorem 3 we used functions fi())..... fn (z) defined eq. (21) of Appendix B.1
Here we use a slightly different choice, that is, functions fi (n,) are defined as follows:
(d-1)/2 1)/2 h axx, of) 11/2 h2)(2))
i E V1
= = h1(x)) +
i V/2
(44)
i V/3
Consequently, our hard instance of problem following:
T which is described in Appendix B.I
turns into the
d-1 a
min ZERd
h5(() ((x) = Σ
3 (e9,2) + cho(x)
(45)
j=1
where O is some constant, and functions h1 (x). h are defined in eq. 22
One show that Lemmas 2 and 3 still hold true. We can also replace Lemma 4 with the following
Lemma 5 The proof of this lemma is a trivial extension of the proof of Lemma 4 which uses the fact
that V 2 (x+* Vha(2" (x*) long 8 and x² defined by eq. (46) and eq. (47). respectively.
Lemma 5. Let defined as follows:
(46)
3ed
Problem eq.
45 has a solution x² E IRd, which is given as follows:
a
(47)
3cd
Moreover, for all a E Ka--1, the following inequality holds:
a²
p(x) p(x')((==))
(48)
18ed'
One can also show that each function f((x) defined in eq. 44
M--Lischtttz continuous, where
My is defined as follows:
My 2avd+ civa= 2avd + a/33Vd) s 3avd.
(49)
Let us choose and follows:
M
M
== 9Rd 3
and
(50)
This choice of and e implies M, s M and ||="' < R. Moreover, eq. 48 implies
MR
p(z)
p(x') N
(51)
18d
as long as x" Ka-1- Next, without loss of generality we can assume < (MR)/72 and choose d E (3,5....) as follows:
MR
d = 2
1,
(52)
36c
which, for all M:(7), implies
pxx) - p(xx)
as long as 7 satisfies
n(d 1) Q
M MRX
T 2 Toom
Toom
(53)
which concludes the proof.
17

---

C Proofs of Lemmas from Section B.2
C.1 Proof of Lemma
Statement (i). Let i E V and x C K2j for E {I..... ......., (d 1)/2}. Then for / 2 2j + we obtain
(of+ ef,x) 0, which implies (hr(x) = 0 due to eq. 25 Hence, obtain the following:
144 (T) <2)(2) + Vha-2(x) eñ
Vh1 (z) Vh3(a) + Vh2j-1 12)) eç
3 span (fe) e U... 1 2
K2j
where (a) uses eq. (24); (b) uses the fact that Vh((r) 0 for / > 2j+1: uses eq. 25 (d) uses the definition of K2j in eq. 30.. Hence, Mi(r) C K2j implies M (T + Tsuh) K2j by definition of M! (.) eq.
Statement (ii). Let i E V/2 and x C K2jt1 for j E {O,.... ... (d 1)/2). Then for 1 2 2j + 2 obtain (ef x) O, which implies Vhy(x) due eq. (55). Hence, obtain the
following:
fi(r)
Vh2(r) + (ha(2) + ... (x))
Vhz(s) + Vha(x) + Vh2y
e span ((e), e U d c 0
K2js
where (a) uses eq. 24 (b) uses the fact that (x) 0 for / 2 2j + 2: (c) uses eq. 25 (d) uses
the definition of K2j+1 in eq. (30). Hence, Mi(r) C K22++ implies M (r + Taub) C Kaj+1 by the
definition of Ma sub in eq. 8
Statement (iii). This statement is trivially implied by the definition of V (z) in eq. 24 and the definition of Mbth( (-) eq. (8).
C.2 Proof of Lemma 3
We prove the lemma using the induction on
Base case: k = 0. In this case, assume < (k 1)Tam Toom- Hence, for all E V. obtain (T) = Ø and M,(T) C M,bb(T). Using Lemma 2 and the fact that M (7) = Ø, we easily obtain
K₂ i V
Mi(T) C (T)
K1 i 1/2
Kg i V/3
which implies the desired eq. (34) for = =.
Induction hypothesis. Let K'' E 10,1,2.... We assume that 1) Fooms that is,
34 holds for all + < (K' +
M.(T) c (K2 i Vi or (i E V/ and < 2n/3 3 + of + 1) K2p441
(54)
i 122 or (i E V/3 and > 2n/3 +
where p' =33// /a] and g' k' mod (n/3).
18

---

Induction step. We assume that the induction hypothesis 54 is true. Our goal is to prove that eq. (34) holds for k' + When T cTomm the desired eq. (34 is implied by the induction hypothesis (54). Thus, we can assume kToom S T < (I: + 1) com. Further, we consider two cases: q / 0 and 4 0.
Induction step, case + O. In this case, p p' and of + 1.
Part (i), First, we consider the case
kTocom I < min (I: + T)mm R:Tcom + Tsub }-
(55)
Equation 55 implies 5 Tsub < (k' + Temm and Toom (K' + 1)Tcom Using the induction hypothesis (54) and the fact that p' = p and q' = q we get
K2p+2 i Vi or (i E V/3 and i S 2n/3 + v) K2pt1
M Tsub), M,(T Toom]
(56)
i E V/ (i V/3 and i 2n/3 9)
Hence, using Lemma
we obtain
K2p+2 K2pp1
i E Vi or (i E V/ and i < 2n/3 + 4) i V/2 or (i E V/3 and i > 2n/3 + 9)
M C
(57)
Equations 27 and (55 imply ic(T) = 2n/3 + 1. Hence, using eq. 56). we get
Mia(r)(T Tcom) C K2p+1.
For i + ie(T), using eqs. and 56 we get
M f.om (T) = span (Mis - Trom))
K2p-1.
(58)
For i = ic(T) = 2n/3 + + 1, using eqs.
(9) and (56). we get
com
(c( (G))(()))))))))))))))(()) (T) = span
Ms(r Tcom)
c K2p42
(59)
(ti(
Hence, using eqs.
58 and (59.. for all E V, we obtain
K2p42 i 2n/3 3 + q
M (T) C
(60)
K2p+1 i 2n/3 9
Now, we combine eqs. (57 and 660, and obtain
Mi(T) C. (T) M (T) c K2p42
E V1 or (i E V/3 and - 2n/3 + +
i V22 or (i E V/3 and > 2n/3+q+1 (61)
K2p+1
Thus, we were able to prove eq. for r satisfying (55)
Part (ii). We can prove the general case
krTuu I r < min (R: + +1)))aanaa Toums KTecm + 1T
for arbitrary 1 E ... using the induction on I. The only difference compared to the proof in the previous part is in eq. (56), which will change to
Mi(T Tsub) {
[2ptz i E Vi (i E V/3 and i < 2n/3 + y + 1) Kkppt
e V2 (i V/ and > 2n/3 + q 1)
and eq. (57 will change follows due Lemma 2
(2ppt K2pt1
i V or (i E V/ and < +
M,a c
i V2 or (i V23 and
However, the rest of the proof, including eq. 61 will remain unchanged.
19

---

Induction step, case 9 = 0. In this case p = p' + and q = =n/3 1. Part (i). First, consider the
cToc S T < min (1: + Toom oToom + Tsub }.
(62)
Equation 62) implies T Tanb < (k' + 1) and T - Teom < (k' + 1) Using the induction
hypothesis (4) and the fact that p' p and o' n1/3 1, we get
K2p
i Pric E V/.
MI(T Tub); Mi(T Tcom) C
(63)
K2p-1
i 1/2
Equations 27 and 662 imply ic(T) = 2n/3 + 1. Using eq.
63 we get
(T. - Tcom) C K2pp
For * ie(T), using eqs. and 63 we get
M (T) span (Mi,(t((T - Teom)) C Kep-
(64)
For i = ic(T) = 2n/3 3 + 1, using eqs. 99 and 63
get
M (e(r) (T) = span
U (+L.(T)
Mj(T Toom)
C K2p
(65)
Hence, using eqs. (64 and (65). for all i E V, we obtain
M (r) K2p-
(66)
Using Lemma from eq. 663 we obtain
{ K2p
i V or i V/3
M
(67)
K2p-1
E 1/2
Hence, using eqs. (66) and (67). for all i E V, we obtain
Mi(r) c CM (7) U M (7) C K2p which implies eq. for satisfying (62)
(68)
Part (ii). Next, we consider the case
kTcom + Tsub S < min (/: + 1)Tcom KToom + 2
(69)
Equation (66) still holds for all E and t satisfying eq. 469). From eqs. (68 and 669 for all E V. obtain
Mi(T- - Tuub) C KCzp
which, due Lemma 2 implies the following:
K2p E V1 or V/3 K2pt1
Mr (T) c
(70)
iE V2
Hence, using eqs. (66) and 70p. we obtain
Kap K2p41
i Vi or V/3
M2(7) c M U?)
(71)
i V/2
which implies eq. for satisfying (69) Part(iii). We can prove the general case
kraa + |Tub S < min((k: + T)Ta kToam + (1 + 1) 1)Tahh)
(72)
for / E (2, 3,...... using the induction L. There will be no differences compared to the proof in the previous part. Indeed, eqs. (66) and (71) will still hold for all i E V and 7 satisfying eq. 722.
20

---

C.3
Proof Lemma 4
One can show, that x* defined eq. (35) is indeed the unique minimizer of the function p(x) defined in eq. (2)). Moreover, we can obtain the following:
a²
p(=*)
18rd'
We can lower-bound function p(z) follows:
d-1
r p(x) 10212110 ||01|² (c (0f, + Σ
(x, o - of11
j=l
3 (of
Σ ((, - (2,, cz+))
a (e x)
3
0
long as x Ka-1- Hence, for all x E Kj,, we obtain
p(z) p(x*)= 2 a²
18rd
which concludes the proof.
21

---

D Proof of Theorems Ba and 4
D.1 Auxiliary Lemmas
In order to prove Theorems and we will use the following auxiliary lemmas. The proof's of
these lemmas can be found in Appendix E. Furthermore, the proof of Theorem 3 is contained in
Appendix D.2, and the proof of Theorem
is contained in Appendix [.3
Lemma 6. Under Assumptions 7
and 3 let > 0 (strongly convex case). Then there exists
solution (w* "0"""")== E C (I.
C to problem T.. which satisfies the following conditions
E 0, ;"),
0 0=
VyQ((w, "OQW"'V'2"" v²,"",, L V=Q(w", ,y" ;²),
(73)
Moreover, the following inequalities hold:
|| 1|2 < nM2 n/2
11v"11 (1 (1+ral (attr//)nn'' 1|= S 4nM2.
(74)
The proof of Lemma 6 is contained in Appendix E.1
Lemma 7. Under Assumptions
T1 K-1 and Th= = 1/(TTT), Bg T:z: ox /2T + Bx) for k E .0....,K 1}.
[ [and?
2 let n/ "2:...,,
be chosen as follows:
(75)
Then, for all x E (I and l E (O....,
...,. K 1}, the following inequality holds:
(T + +b2x)11 x12 T la z112 2n (T+TT
(76)
- (F( - F(x) - kt gbt11 - x) + 1 g
2 2
The proof of Lemma] is contained in Appendix E.2
Lemma 8. Under Assumption 4 for all l E {0, .0......k. K 1}, the iterates of Algorithm 7 satisfy
Px =h, where P E Ranxxn is the orthogonal projection matrix onto Lt, which is given as follows: P (Ia n 1 @ Ia.
p = P2 =",
(77)
(78)
The proof of Lemma & is contained in Appendix
Lemma 9. Under Assumptions and S for all l E {o.... K - 1} the following inequality holds:
|| 1 S 2x I1 - 2x|ln k+ m
++ 1 + 4x 1
(79)
The proof of Lemma 9 is contained in Appendix E..4
Lemma 10. Under Assumptions f and let parameters 0% be chosen as follows:
-1
0 = g2rr for k= =,....., 1-
(80)
Then, for all k E {0,.... K - 1}. the following inequality holds:
5 -ax ((z - -0 (g))) + Tys ry=ll l|z - 2² I?) - (4au 1|
(81)
The proof of Lemma i0 is contained in Appendix E.5
Lemma 11. Under Assumptions D A and under conditions of Lemmas and 10 let parameters .o...., aK-1 1 and 70. 7 be chosen as follows:
Oi = 3/(k + +), 7k- = (k+ + 2)/ + 3) for k= =....., 1. K nh. .........
(82)
Let parameters To
41 K-1 and .e....,
m2 K-
be chosen as follows:
- = Tzax -1 T =
-1 tt 11
-1 for k= =.....,A K 1,
(83)
where Tax Tly and 112 are defined as follows:
T2 = Ty (4rya) Te = (10rgax2)
To T==2/1 Tyz 3/r.
(84)
Let parameters >1...., AK be chosen as follows:
Ak = ax and d = = + a a for k = l,..., K 1.
(85)
Let the input of Algorithm Tbe chosen as follows:
x0 ===0 0, y¹ 0, 00 0, m = 0.
(86)
Then, for all x. y E (IRd)" and E C- the following inequality holds:
2
|||x=|12 1bll + 45x² 1121²) +
18
72nM2
Qx K y, 2) Q(2, V. K <
(87)
K2
rKT
The proof of Lemma II is contained in Appendix E.6
22

---

D.2 Proof of Theorem
3
We can upper-bound
2 |, w*1 where w" is defined in Lemma 6 as follows:
Fz
w 2=11
(4)
- Q( K x²²²²) - Q(w"( ,y,,²")
2
Q
a K
VV, 2²² - -Q(w", :Ya K
2
r||l"|| + r
18
11 1 45x2
11=²1 +
72nM2
k
r
rKT
(d)
2 nn22
18(1 + Ta / 180nx"M²
72nM²
+
K
rKT
where (a) uses Lemma and the strong convexity of Q(x; y, 2) in x; (b) and (d) use Lemma (c) uses Using the definition of r. in eq. (84)
Lemma 11.
51nM²
180nx 22²
72nM2
r||
w01l3 -
+
K2
rKT
1386nx2
2M
72nM²
rk
rKT
Next, we can upper- upper-bound n(p(x/)) - p(xx)) as follows:
n(p(x!)) p(x))) (a)
f(((x) - f((x*) +
2 11 12 2
11 [1(
(F(CE)) - f1(x*) + r _ b
x x 113
1/2 ||=0 1?2)
(F((T)) 1(aa)) + 2 |1 a,i
r
r 2 ||=²1(]
(fi(=!)) - f((x') + 1
|0 k 113
2 ||="11 11
2
(1,(21))) - -1((0) + M 1,
20 =11)) + r
K 112
r
11
2
111
1 F(akk) -F(w") +
|100
K
11?
1
|
1|2
+
M|
M
2r,z
a
2ry
F(a - -F(w") +
| K ||2
1
1|00112
2r
2ryz
E M2 V E ||,KK
(b) F(zk) K F(w") + yr
||
| 112 + VnM llp
2ry==
where (a) uses the definition of p(x:) in eq. [ (b) uses the definition of K on line 15 of Algo-
rithm (c) uses the convexity of |1-11: (d) uses the definition of w in eq. 95 (e) uses Assump-
tion 2 (f) uses the definition of function F(x) in eq. (12) and eq. (13 (g) uses the Cauchy-Schwarz
inequality; (h) uses the definition of P in eq. (88)
Next, for arbitrary E L we define y -r 2 K Q(=:! y, 2) F(x 2ry
2. Then, we get vaQ( v, = 0 and |--kK 112 + (oo 2). Plugging this into the previous upper-bound gives
the following:
1 ||=0
n(p(a.)) - p(x*)) S Q(=** 2) F(w*)
(n 2) + VnM ll
2r
23

---

Q( 2) (('",,;²") (x 2) + vTM ||0 llP
ax v, Q(w", , K 2/
(x + /nM 1|
where (a) uses the definition of y² in eq.
98 and the definition of ~ in eq. 99 (b) uses Lemma 6
Next, we choose E L follows:
+ M
K + 0 La Fa 0
(88)
Then, (x. 2) + /nM ||p and we obtain the following:
n(p(x,) p(x*))
S
Q a y. 2)
Q(
Va
(a)
2
18
| + 45x²
72nM²
r||""|
+
1|
k
rKT
(
K2
2
r|
11
+
18
||
yz
1
K
+ +1?
45x2
1x13
+
72nM2
rKT
2
18
x5x2
+ 72nM²
r||
||
gz (xr
w² + +w") + +1112
11=113
K2
rKT
2
54
54
54
x5x²
72nM2
r||w"||
w 112
||w0| 112 +
||2113
+
11:113
K2
rr
yz
rr yz
7
rKT
2
54
54
99x2
72nMP
r||w"||
|
w" 1|2
+
||=0||
1|2113
K2
rr
rr
rKT
(4)
2
54
54
90mx 2 M2
72nM2
r||=" 1|
||
|x
w01
112
+
| 1| +
K2
rr
mr
r
rKT
2
nM
54nM²
54
99mx2M² 2 r
72nM²
-.
w0
1|2
K2
r3,2
rr
rKT
yz
(
2
+
Gr|1
w 111
+ 99nx2 M2
72nM2
K2
rKT
212nx²
72nM²
12r
w²1l
rK2
rKT
K²
(0)
212mx2M'*
72nM²
12
(1386nx²
222
P22
72nM2
rK2
rKT
K2
rK2
KT
where (a) uses Lemma (b) uses our choice of y; (c) uses the parallelogram rule and Young's
inequality; (d) uses our choice of 2; (e) uses Lemma 6 (f) uses the definition of Tya in eq. (84); (g) uses
the previously obtained upper-bound on r| w ||2. Dividing both sides of the inequality by n gives the following:
M2
72M
12
11386x2 2M²
72M
p p(=))
rK²
rKT
K
rk2
rKT
Hence, choosing the parameters K and T such that
XM
M KxT20
K>O O
and
Vre
rc
implies p( - p(x*)= S, which concludes the proof.
24

---

D.3 Proof of Theorem 4
With the original problem
turns into the following problem:
min EE
f(x)
Σ (()))
(89)
Let x E Rd be the solution to problem (89), such that ||~||| S R, which always exists due to Assumption 3 Let > be an arbitrary regularization parameter. We upper-bound function f(a)
using the regularized objective function p(xx) defined in eq. (1) as follows:
f(x) $ f(x) + = p(x).
On the other hand, we can lower-bound f(x*) as follows:
f(x*) = p(x')))))))))))r))) r
- min p(z') ク ||00012
min p(x') x'
TR²
x'E
Hence, we can bound the function suboptimality gap in problem
89 as follows:
min p(x') + rR æ'ER4
f(x) - f(x") S p(x)
2
Let the regularization parameter > be chosen as follows:
e/R².
(90)
Then, obtain the following:
- f(x)-((x") S p(z)) min p(z')
+
(91)
Elzu 2
We can apply Algorithm|B [ to solving the regularized problem with the regularization parameter
defined in eq. (90). Theorem 3 jiplies that, to reach precision
p(
K
min p(x') ZRE
(92)
2
it is sufficient to perform the following number of decentralized communications:
K= O
X
(a)
o
XMR
(93)
Vre
and the following number of subgradient computations:
2
M R²
KxT=0
(94)
re
2²
where (a) and (b) use the definition of in eq. 900)
Using eqs.
91 and 92
we also obtain the
desired precision f(x:5) f(x") S G, which concludes the proof.
25

---

E Proofs of Lemmas from Section D.1
E.1 Proof of Lemma 6
First, we pick the solution x² F to problem (I. which is unique due to Assumption 3 and the fact
that O. Next, we define w² L as follows:
w"
(x x²).
(95)
From Assumptions 1 and 2 it follows that ((op((x) R2 and dom fi(x) Rd for all i U.,, which implies the following:
E Op(x')))=)))))))))))rr))) -xx 1 Σ af((*))
(96)
Hence, there exists a vector A = (A) {1, [I....,n], n.,, and the following relation holds:
A E (RR")" such that A E 0fi(z") for all E
rx"
+
A = 0.
(97)
1=1
Next, we define y² (RR4) as follows:
= A + +Tz"".
(98)
From Assumptions and it follows that dom F(x) = (Idd", which implies y" E OF(w") and 0 E ((F()) - (v" () Q,Q(w", Qw",, y" 2*)
Next, we define z E C- as follows:
rw" A
(99)
Note that the inclusion 2² C- is implied by eq. 97.. Further, we get
V= y" 2*) (a)
- (y" =*) -rzz(z - r)w" E L.
where (a) uses the definition of (z, y, 2) in eq. 14); (b) uses the definition of y and z", and the last inclusion follows from the definition of wt. Moreover, we obtain the following
(r + Tr r)w" (c) 0,
V 3²
w" Tyz (y" +
where (a) uses the definition of Q( y. 2) in eq. 14 : (b) uses the definition of y and 2"; (c) uses eq. 3
M for E fI....,m). Hence, using eq. we get r r| < M, which implies ||= 1 < nM2 /r2. Moreover, we get
From Assumption it follows that | ||
< A
w S Vn(M M/r) (1 (1+T=/T)/MM,
which implies ||y
2 < (1 T
2 nM Finally, we obtain
2 $ r|| || + ||~||| 2
which implies ||=*1|2 < 4m and concludes the proof.
26

---

E.2 Proof of Lemma 7
We start with the following apper-bount on |2
- x112:
2
1
| 1ett1 - x²|2
2/1
1
1
+ 1 (athett
llattre -x|13
x
1
2
21/2112
2t. x x)
2
2
1/22
(6)
1lottie - 20112
| k 21..112
2n/
20
" h.t
+ Mitt - sx+ 1
+ (, - xth) x 1++ x)
1
1la
- xx122
1
|1
112 + (g ky 2x,,+1 2)
21
20
(Bk ka+1
+ Tx
10
M,t+1 x) - gg h
x k,441 - x))
-x0l x0112
1
x
2/. + (y h+1 s. n,et1 xz)
2
2n
T
T 1, x/1²
I|.
11112
+
2
2
2
3x
5x
1
112
1|
xz1??
+
|| 12 (g x - x)
(o 22,,t zk:0)
2
2
2
(2)
1
1
ktt
xx12
11
k,t+1
= 112
+ (g
++1 k =x)
201
21
A
|1
k,t11
- =1-112
2111
x=| 2 +
Ts
||
x:l1
2
2
2
3k
Bx
| 1,11
||
1,tt - =0112
+
||
2
2
2
n
> (f((x)) - 1e(0./1)
(g k.0+1 2. k::1)))
where (o ( = (V (a
fz(aat)))) of E (Rd)", (a) and (c) uses the paral-
lelogram rule; (b) uses line 12 of Algorithm ! (d) uses line Tf of Algorithm Definition !
and
Assumption 1 Further, we obtain
h xx12
2
1 R
2012
kt+1 - k.t 1 ( , k.tt1 x))
2//
2
-
||
kit+1
=1²
T2
11++1 =0112
r ||a xx|?
2
8k
Be 1
Be
1,11111
x0|2 2||2+
+
||
2
2
+
(fe(x))) - lelabeet2)
M ||= 1.041
k.
||
|| ||
k.
k,0+1 xr. (11
1=1
(6)
1
1
Ila - =x122
||
h.tt - 2.2 + ++
k x))
2nt
2
k
- 22112
|
=01l2 +
1| x012
2
2
2
8k
Br
Br
||
x0122 +
||=l|
2
2
2
+
Σ
(Fo(x)) - fo(0
k,241 + 2M
| h.+1
xx 101
27

---

(0)
1
1
hi - -x12-
|| 1ahhet11 - x2.11/2 + (y .hilt1 x)
2
2
22
|| - x22112-
-
_2
||
|ittet =x112 + _
1|2
=x12
2
2
+ F(x) - F(ahhett)
12
||
||.kttt1
- x²1² + Σ
| k.t 2/200112 + 2n 2122
2
2
iml
1
1 - -x/12= +
T
*
|1
1/12 2 + (1/ x
klt+1
x) + 2n1 M²
2
2
2 |1chett1
2
+
|| l x012 F(xtt(11) + F(x);
2
2
where (a) uses Assumption 2 and the Cauchy- Schwwrz inequality; (b) uses the inequality |s.::1| s M, which follows from Assumption 2. (c) uses the definition of 83 in eq. (75), the definition of F(x) in eq. (12) and Young's inequality. After rearranging, we obtain
1 1tt - -x112 + Th 11 - x112 + 2n A Arrt+1
x k.t+1
x012 -
2
2
2
where A k is defined
A k
F(x F(x))
(g x/ =x)
* ra || 1,1t++111111 - 2012 + *k
| - =11112
2
2
Now, we sum these inequalities for & = O.....T- and obtain
1 Io x1|2
1. x=12+
I TT 11a - x112 + 2n0 M2T
T
A h.t.
2
2
2
Dividing both sides of the inequality by T gives
1 R.T x112 1
1
1:0 x11 +
T
1|2 xx| + M
1
a
2
2
2
T
Using the definition of A.t, the definition of t* on line 13 of Algorithm obtain
and Assumption
1
110 - x012 + The
R.T x:11 -
| - x/12 + 2 M²
2
2
2
(F(s+t1) - F(x) - (1 [ x²))
+ + Fx k+l x/12 + The
1 +1 - =001?
2
Using the definition of th and 3r in eq.
75
we obtain
k.r.T 2
2=112 +
* 3k
||
rr|
*k
k. - -1/|
T
|, xr12 + 2n M2
2
2
((( F(x)
(y h+ gk+1 - x) +
|| x² (
2
Using the definition of k.0 on line 9 of Algorithm|| the definition of * on line 13 Algorithm the definition of nt 3x and og in eq. (7)) and the convexity of ||-I|, we obtain
Mn
(T + 1 3r ++1 x012 S - ||
x +
T
F(athtl) - -F(x)= - (2) |+| 1,,1, - x)) +
T || 2²(1p).
2
which concludes the proof.
28

---

E.3 Proof of Lemma g
Using Assumption 4 and the definition of in eq.
78 we obtain
P(W @ Ia) = (W & I,)P - (Wx & Ia).
(100)
Then, the desired relations be trivially obtained by analyzing the lines of Algorithm
!
E.4 Proof of Lemma
9
We can upper-bound
|| + m R+ 1| as follows:
|1
++1
m
++1
In * + + - 3111
Im * + -h - (W @ Ia)(m + +f)))
(Pm(( + + - (Wn @ I2)(mm + g
(1 1/x 1/x)ll + 9
M (1 11--1, 1/x) ((1 1 1/(2x)) || + (1 2x)lm
$ (1 1/(2x))
where (a) uses line & of Algorithm [ (b) uses line of Algorithm I (c) uses eq. 100): (d) uses
Assumption [5] (e) uses the parallelogram rule and Young's 's inequality. Using this, we obtain
|| s 2x "I 2x m 113 + 4x 1n
which concludes the proof.
E.5 Proof of Lemma 10
We can upper bound llo.
11 as follows:
1
- Potl
IIW,, & La)g P
II( L.) P
W
(1 - 1/x0llg1
where (a) uses line of Algorithm I (b) uses eq. (00); (c) uses Assumption 5
On the other hand,
ll - 12 is equal to the following:
Po
(a)
| + |
2 Pgtt)
2 ( Pg 0h
(0:)²
2²
11 1B
+
where (a) uses the parallelogram rule; (b) uses line of Algorithml I Hence, we obtain the following
21 + ok
2h
Pg
g 11 S O.
gh After rearranging and multiplying both sides of the inequality by 2ar
we obtain
gk
0 a 11
( 2 Pgt
+t
z² 12)
20k
2aux
(a) (( - )² + ryz 1| - 11) +
1
11o.11?
0x
4ax yz
b
((g
2h + +r k+l 2
1
4cra
where (a) uses eq.
880 (b) uses Lemma which concludes the proof.
29

---

E.6 Proof of Lemma
We can upper-bount
y/12 follows:
2nd
1
110
A+1
w11
(a)
11
w112-
1
11
- 112
k+
+
1 (e 30, ,0 s - v)
2/1/
2nh
21/y
(0)
1
llye
u12
1²11 (of + ghit1 - - 1)
20g
21/y
1
11
yl1
v/ 111 - (att1 .yttt - 1)
20/
2
(og y h+1 - y* + 3
3d + yr - 1)
( 1
2m 11 th u11
11
k+1 111 (1tt11, - y)
2n
Wy g/
1²)) + (1 oxs)
21 W J/th 1n) + (shy yn
ak
2n 110" y11 - 1
2n llt - (5 , - v)
o (g ght1
+1 - x²) + (1 as) Lo Iz y/) + (g y y/m)
+ (i ght1, / - v)
where (a) uses the parallelogram rule; (b) uses line of Algorithm Algorithm| Further, we can upper-bound the term (7 ik+l
T (c) uses Lines and g of y) as follows:
(ak+1
gk+1 y - y)
(4) (i++1 - x²²² Ty (3* xxt-1), y y)
(x - xhth 3²t - y) - (xt - -k+t1 gh+ - 1) + Th (1 ( zh, y/t gr)
(6) (2 k- ,h, u) fx - gktl , - 1) + An/
11 1²12
+ 2n > 12112 ||aa 2x112.
where (a) uses Line of Algorithm II (b) uses Young's inequality. Plugging this into the previous inequality gives
110
x+1
w/1 -
2n 11 y11 - 4n 11 - 8²11 - (3 - - y)
2
ox 1 ( yb g+ )nd) + (1 - -ass)ar (gy y² - -²²h + (g/) y/ - y
+ (x
tt,y ",* - u) fxx - ghtl , - -) + 2n 2 221 11 11555111 1/111
(a)
111 - y11 - (5*+1, +1 , k+ v) + 2n 2 2 |1ا - 1 - 1112
2ng
(o 1
4 + (1 ax )x)a ( :z 1x) - ax -1 [ J. +++1 y
1
x²1 + 7/ (20 - 5xh y² - v) - (at - xh+1 1/t 4)
4n5af
a
(b)
1
l1u*
-y12 2n 2 x
1 - xx12 112 - (1+t11 ; ,kt - y)
2n5
+ 7m x 11 v² v) - (1x - xht1 . - 4) + (وا 31 - y²
on )x) (9th v² -1²) ox (ogg k+1 - ) rr 11y y (12)
++
where (a) line of Algorithm [ (b) uses eqs. (83) and 84
Let gk be defined for all & E (O..... K} as follows:
= 2 Pmn
(101)
30

---

Using eq. 101 and lines 7 and 8f of Algorithm ! we obtain
gb+1
2 + gh 2t P(n m h+1 n m²
= 2 P( + nls k+t mht ++ 1/ m²
8 - P( + ne
Not1 " (NE/m) k+ "mm" + o -92) - m²)
8thh r/PP""
Hence, we can upper-hound
1. 24/
|| - =112 as follows:
1
1
k
- 2||2
(0)
11
=||² +
|1 - 1/22 + _ (sktt
2k, 2)
211
21
2
115
(6)
1*
=11²
+
1
1p
(Pp z* - =2)
21/
2
T
|
2112
+
(P 2t - 2) + P - -2h)
2n
2
(c)
=||²
p/- (P 2h =) 1/2 p Pm*
2n
where (a) uses the parallelogram rule; (b) uses the update rule for 2h which we previously ob-
tained; (c) uses eq. (11)). Further, we can upper-bound the term v/. (P Pm follows
(a)
1/2 (Pg.
Pm
I
Il
1
(b)
1
(2xl1/
2x |InEm*113
2n
(e)
1
(4x11 gf
Il m
k+1 m ++1 111))
2n
where (a) uses the Cauchy-Schward inequality; (b) uses Young's inequality; (c) uses Lemma Plugging this into the previous inequality gives
1 1 x+1 - 2112 z
_S 11g 11 2
11
=112
+
(Pg -2)
2
2
1
(4x1InE
+ Il 11
11 k+ 1 1111)
2113
(4)
||
=||2
Th + 4x)
1
(0 2 - -** =
27h
(1IT)-
m
13
|| +1 m
k+1 11/
2n1
(8)
||
=11
(1 + 4x)
|
+
1
(In -1mm 1 ++*1 112)
2
2/
(00 2 2/ (1 an )a ( z/
=
(0)
1
|| - 2|1 [1 + 4x)
2 2n1
|/ +
(1lm-m 1 112 -11m
1++1 m 112-)
2n
(g2, 7 2/) + (1 an o (g=, -
+
of 1 (Et+) 2² ( + Fa
|| |z 211
1
l1g
(d)
|| =11? + 2n (1 m +13 -1 + +1 + (g - 2n)
5
2n/
+ (1 ax) )
( zt - 2² ax -1 (124+1 - =zt.00 =² ryz 1| - 2 (12)
31

---

where (a) uses Lemma 8 and the fact that E L'; (b) uses line of Algorithm
(c) uses
Lemma 10 (d) uses eqs. (33 and (84)
Now we combine the upper-bounds bounds for 2n5 11y h+1 (1lb+tlyy1 y/1 and 1 || - =12 and obtain the following:
2n
1
11 b+1
y112+ +
1| - =2112
2n
21
1
111 u11
+
11 - =1|2+ + 2nl 22 21112=111111 5411 (ik+1,, ,1 , -1)) 11)
21/8
2
Th 7-(x0-1 a .y y) -(x² - 5h+l, - 1 1/
1
(1 " 11p -1ln5 mmmmmm" 1n++1111)
2
<oog y - 2²)) + <g 2- - -²- + (1 - ox) )
1 (lg y² - 21/)) + (9 - -14
ok 1 (1o gh - )/ + (2k+1 - , + + Illlltt1 - 2² + r ry=al 11 - 11)
(4) 1
S 2n
x - y/12 + 2
11 =112 + 2 7xl 1 x-1 - 1 (5 , n - v)
1
+ n (x 8h. 4 y) (1x 1t+l -
- y)
+
(I - 11 + 111)
2n
+ +Gly, 2) - Gly 2² + (1 - -ax)az (G(T". (Ct) - Gly ))
ak (G(T ++ 1 ) Glyy 2²)
1 l1o* - yl1 + 2 11 - 211 + 2n 72 11 1| - 5/113 - (iktl .xtt 3)
2nd
+ 7k (I thty 3² y) - fxtt - - , - y) +
2 (1 m k 113 - -11 m k+ 11:)
+ (1 - o -1 (G(T", G(y, =)) 0kk -1 (C(g"~1,
Glyy z)).
where (a) uses the definition of ot and O on line of Algorithm I and the convexity and (2r smoothness of the function G(y, 2). Further, we divide both sides of the inequality by as and, using eq. (83) obtain the following:
1
1
11 6+1 3112 +
|1
- -212++ +
|| k+1 m
A+1 12
27/y
21/²
21/²
1 -11* - -y112 + _111=1 11
S 201y
-211² + 27/²
11 m 113 + 2n -2 222111= - 1t112
201
+ Th a x
-1 ((a--1 - 5², y² - v) - oxx (x xh+l , - y) - 0x
- (itht petl - 2)
+ (a 2 - oxx (G(T" G(y, z)) -o (G(T"+) , - G(y, =))
Next, we divide the inequality in Lemma by ag and, using the definition of - and in eqs. 83 and 84 obtain the following:
(a -2 + ox (=))([=tt x=12 S Tz 2 x0|2 Tz | 112 + 2nM2
2
TxT
ax
(F(*hH)) - F(x)) (w [4tligh1110 - x²)
Combining this inequality with the previous upper-bound gives the following:
Tx -2 + oz ==))|l=,tt 2012
+ 2ny
1
t - v112
+
1
|| - =1|2 + 21/=
I 1+1 m k+ 113
1
21/=
1
1
11 - -212² + 1 In/m= 1/ + 2n.1
I xz 1110 - -x01²² + 2n/y
11y* y11²
+
27:
2n
T
Tz 2
| - 1/1 + 2x 22221 x - 11112 - ox [,x* xh+1 p+1 - y)
2
+ 7600 1
(1 - - x²² 4) -ax -1 (F(a - -xx)) + (v ),),)))))) (5 vv)
32

---

+ (ak 2 - -ax') (G(T², (G(TT**)) - G(y. 2)) - 0k 2 (G(g (e((p+t,hht)) - G(v,2))
( T
21 211E* - =x12²= +
_11b² - -11² +
- -11
1
11 - -211²= + 21/²
1
l1mmm 113 + 2nM2
27/y
21
TxT
T2ak 2
| - 111 + 2 2
N/ 2/2110 k- - -/1 - ox 11tt - _ktl,, g - y)
2
1 (1/-1 - -x
, .x* v) =kk 11 (F - (5tht11 .)) - -G(y, 2))
+ y
+ (ax²² ax 2 (G(T²* x²) (0" x) -(x)) -ax (Gly ++1 xz=11 } + (ight) xx) - F())
( Tx k
2 111* k - xx|² 2 + 27yy
1
11 - -yl?= +
1
1| - =1|² + 21/=
1
|| m
& | TxT
2n
27²=
T 2
| - -//13 - 21/ -2 2221122 - = - ok - (2t x ++1 n - v)
2
+ 2 (a--1 11l--t -#" "²."²" - y) - -lax -2 ox 1(xx J.h ~ヤ + ax
2 Q((, g 1/
- 0x
(c)
1 11 - -y1l= +
1
1
M
-2 *|1*** -xl|2- + 21y
11 - ==12= +
k m 1
k pp +
2n=
21
TxT
Tz 2
|1 - 2//122 + a 2222110 10 -
1111
ax (1( xk2 y ybtz y)
2
+ 7
1 (a k- - -z",a 1² -) - -(aEe
2 - 0x
a= ) Q Fh , + okk
2 20(,,, Iut1, ,
0% (1x
++ ,3,22) + (ak -2 .ak 11((((z) (),)
= - 2110** - -x|13= + 21yy
111* -v1 + 1
11 - 2|12 + 1
1 m/ + 2nM2
211;
2n1-
T.T
2
T2 || - 112 + a 2 222110² - 111
-1 - ax ,/\(1" - ght1 sebtt,yatt , - y)
2
+ n 1 (er - - - v) + (a -2 - -ax (Q(T* .), [(xx, 1²1)
2x (Q(**+1 2) Qax, .. n1 , 1)*
where (a) uses the fact that + = 0x g4+11 - (1 - og) T which follows from lines 4 and
of Algorithm I (b) uses the definition of Q(x 2) in eq. 14 (c) uses line 13 of Algorithm I and Assumption
Further, let a-K = 3/(K + 3). Then from eq.
82 it follows that a + aj 2 a 76
(k + 2)/3 and ag -1 = (k: + 3)/3. Hence, we obtain the following:
1 11 - y/1² + 21
2 a k+1 1| - ==13== + 27y
|| 1++ - =112 + 21/2
1|m
k+ m
k+ 2
1 |11** - 2112 +
1
2nM2
S &Tx -2 211x²* - -0012 + 20/y
|| - =11 + 21/;
|
113 +
2/12
TsT
Tz(k (1k+33 3)2
| -01112 + 4ng (k + 2)2
||
k-1 - 1/112
k + 3
(1t - #tt1 12 - 5)
18
18
3
A: 2
+
(a
- -.h 1² - 1) + (a -2 - 1x
(Q(x* v222) - Q(z J² 211)
3
xxx 2 (Q(T"+1 ,,,,) Q(z ght! , 1)
(4)
0x 2 9110* - xr|1 + 27yy
11 yl1 + 2m:
5| - =||2 +
| m
& + 2nd12
21/²
TzT
21y + 3)2
| - x²1 x11 +
2n (k + 2)2
|| 1 - _1122 + 3
x -
xbt1 , ++1 - v)
9
9
3
++2 2
+
(x
b- 11
8x 1,.5==0 1) + (a 2 - az -1) (Q(=t, v,,)) - Q(z (5,
3
2
(Q( 2) - Q(z, ott ,(t1,
2
33

---

where (a) uses eq. (8
Next, we sum these inequalities for h: =0,...; K 1 and obtain the following:
1
1
Tx 2 |1 K - -x12²+ + 217y
11y* - y112 + 27/=
|| =112 +
||
m
K
pp
27/=
S <Ta00" 2 211200 - -x11²= + 1 11 - -yll² + 27/
- ==11² + 1
2n K
11
113 +
24y
21=
T
8Mg 11500 xx- x=1 + 3 -1 .9.8 y)
+
9
(K 2)2
ji - 1x11 112 1/2 (K + +2)(k K-1 -j , -5- v)
9
K-1
(0k 2 -ax ) (Q(=", 22, - Q(x, 3/ )
K-1
0x 2 (Q(x*+] 2) - Q(x, y g ))+11)
0=0
(a)
1
+ 1
2nM22 K
S 0o 11 - -x1|2+ + 27y
2
1110 - vl1 +
1| - =212+
0
m +
2112
217=
TxT
2ng (K- + 2)²
| xx 1x-1 + (K + 2)2
1
1ko - 5
1|2
+
1
11 -21
9
9
41yy
K-1
(ap
2
ox
(Q(T)) . 22 - Q(x ,", ))
K-1
0k -2 (Q(x+t) -), - Q(x, y ( )
A=
(6)
- -0112
+
+
1
1100 - y1² + 21
||00 - 2|| +
1
mm m1 + 2nM2K
27
27²
T
(K
+ 2222
|
12
+
1
1
1/ ax (Q(x :1,2) Q2x, "0,,""",, )
9
41yy
K
1
- - (Q(x** ys 2) - Q(x, ,/, 20))
1=1
(c)
To
|120
- x11 + 2//y
1188 y1? + 211²
11:0 - 21|
+
1
0 0 11p +
2nM2
21/²
T&T
my (K +
2)2
1 4nyy
K
|
xk 1112 +
2/12
Σ
Au (Q(**, v, Q(z J"h z²1)
9
where (a) uses the Schhhazz inequality, Young's inequality, and the initialization 30 x- on line of Algorithm (b) uses the fact that on = 1, which follows from eq. (82) (c) uses the
definition of A in eq. (85) Further, we obtain the following:
1
1
T
K
2 11
- ==|12= + 21/y
11 - y11 +
21/²
|1
-
-212+
+ 211=
| m
1
(0)
Ta
Tx - =0112- +
1
11100 - yl
+
1
|1
211
+
1
Inmm +
2nM2K
217y
2n
2m:
T.T
(b(K + 2)
||
-112 112
1
11
K
y112 Σ
Ale
(Q(10' K -1,2) Qux v. =k
Afly
k=1
(6)
Ta 1, 2:12
+
1
11
u11
+
1
||
=112
+
11 m 1| +
2n K T.T
27yy
27:
2m
34

---

7 + 222 11 - 111
1 Afly
K-1
+
11 K yl1
Σ k=0
0 (Q(x y,2) Qxx, v =x ))
9
where (a) uses the convexity of (1r, z) in æ (follows from Assumption 1 and the concavity of
Qxr y, =) in (u:z), z), line 14 of Algorithm and the fact that A 2 which follows
follws from eqs. (82)
and (85) (b) use the definition of X in eq. (85) and the fact that 1. which follows from eq. (85).
Next, do rearranging and use eqs. (84 and (86). which gives the following:
K-1
Σ a )
(Q(o v1,)) - Q(x, vo =o )) ||
Ilull +
15x² |=||
1
rT
Next, we divide both sides of the inequality by EkE og which gives the following:
Q1x 2) Qe Va
2)
S
K-1
Σ
G - llvll +
15x² r
||
12nM² K
rT
Further, we can estimate
Σ
K-1 k==
Oik
as follows:
K-1 Σ
K-1
3
K-1
K(K 1)
K(K +
K²
K
k=
3
6
6
6
k=0
k=0
k=0
where (a) uses eq.
82
Plugging this into the previous inequality gives
Q(x v,
Qz Vo
2
<
6
r
|=||
-llll1 +
x5x² 2
2| +
12nM2 K
K²
3
rT
1
2r|1ar|12
+
36
|
90x2 2
||=13)
72n
R²
rKT
which concludes the proof.
35

---

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3. Theory Assumptions and Proofs
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The assumptions are described in Section The proof of Lemma 1 is provided in Appendix The proofs Theorems and are provided in Appendix The proofs of Theorems 3 and 4 are provided in Appendix
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37

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Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
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The assumptions made should be given (e.g.. Normally distributed errors).
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Reviews:
[{"review_id": "OZjgmdkf85", "reviewer": "Reviewer_7QFG", "summary": "This paper derives lower bounds on the communication and computation complexities of solving non-smooth convex decentralized optimization problems over time-varying networks and designs and algorithm that achieves these lower bounds.", "questions": "None.", "limitations": "Yes.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The problem studied in the paper is interesting.\n\nThe results look to be rigorous and strong (by achieving the derived lower bound).", "weaknesses": "None.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time, effort, and appreciation of the theoretical results of our paper.", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "3w5cBuW9Is", "reviewer": "Reviewer_Eyy5", "summary": "The paper studies non-smooth decentralized optimization with time-varying communication networks. The paper presents execution time lower bounds of subgradient algorithms for strongly-convex and convex cases. Then, the paper develops algorithms achieving matching execution time.", "questions": "Why does the objective in eq (1) have a quadratic regularization?\n\nThe theorem 1 and 2 state lower bounds of execution times but the authors interpret these results as communication and computation bounds. I wonder can you make this argument formal in the classical definition of communication complexity (Yao, 1979) and computation complexity (Papadimitriou, 2003).\n\nReferences:\n\nYao, A. C. C. (1979). Some complexity questions related to distributive computing (preliminary report). In Proceedings of the eleventh annual ACM symposium on Theory of computing (pp. 209-213).\n\nPapadimitriou, C. H. (2003). Computational complexity. In Encyclopedia of computer science (pp. 260-265).", "limitations": "Yes.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I am not very familiar with the field of this paper. As far as I see, the paper makes good theoretical contributions by resolving the open problem concerning the complexity of non-smooth decentralized optimization with time-varying communication networks. This problem is natural and has many potential applications in practice. The auithors draw detailed comparison with existing works and present their results well. There is no major flaw I see in this work, but again I am not very familiar with the field. One minor weakness might be a lack of numerical experiments validating the convergence rate of the proposed algorithm. Nonetheless, I think it is not a big issue for a theory paper.", "weaknesses": "There is no major flaw I see in this work, but again I am not very familiar with the field. One minor weakness might be a lack of numerical experiments validating the convergence rate of the proposed algorithm. Nonetheless, I think it is not a big issue for a theory paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time, effort, and feedback. Below, we provide our detailed response to the review.\n\n### Weaknesses\n\n>One minor weakness might be a lack of numerical experiments validating the convergence rate of the proposed algorithm. Nonetheless, I think it is not a big issue for a theory paper.\n\nThank you for the suggestion. We are currently working on the basic experiments and will try to include them in the revised version of the paper.\n\n### Questions\n\n>Why does the objective in eq (1) have a quadratic regularization?\n\nWe decided to use an explicit regularizer because it allows for a cleaner transition from the original problem (1) to the saddle-point reformulation (14). Note that any strongly convex finite-sum optimization problem can be presented in the form of equation (1). In the case of a non-strongly convex problem, we use the standard regularization technique, which allows the reduction of convex problems to strongly convex ones (see, for instance, [1]). We describe this in Appendix D.3. We will try to add this clarification to the main part of the paper in the revised version.\n \n[1] Allen-Zhu, Zeyuan, and Elad Hazan. \"Optimal black-box reductions between optimization objectives.\" Advances in Neural Information Processing Systems 29 (2016).\n\n>The theorem 1 and 2 state lower bounds of execution times but the authors interpret these results as communication and computation bounds. I wonder can you make this argument formal in the classical definition of communication complexity (Yao, 1979) and computation complexity (Papadimitriou, 2003).\n\nUnfortunately, to the best of our knowledge, the vast majority of results in the optimization literature are provided in terms of limited computation models, for instance, black-box optimization procedures [2] (similar to what we use in the paper), or polynomial time methods [3]. Note that the latter is significantly different from the classical notion of polynomial time algorithms. The important aspect is that we need to obtain very refined results, rather than classical polynomial or exponential complexity, which is, in some sense, similar to the complexity results obtained for sorting algorithms in the comparison tree model, or for matrix multiplication algorithms in terms of algebraic complexity. Another potential issue is that our results depend not on the size of the input, but rather on the internal properties of the problem such as Lipschitz constants, network condition numbers, etc. Thus, making the arguments of Theorems 1 and 2 in terms of classical computation models may be difficult, at least within the limited author response period. However, this would be an interesting direction for future research.\n\n[2] Nesterov, Yu E. \"Introductory Lectures on Convex Optimization. A Basic Course.\" (2004).\n\n[3] Nemirovski, Arkadi. \"Interior point polynomial time methods in convex programming.\" Lecture notes 42.16 (2004): 3215-3224.", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "PkaF4hFtPV", "reviewer": "Reviewer_ap5m", "summary": "The authors introduce an algorithm which optimally bounds the complexity of algorithms for non-smooth convex decentralised optimisation over time varying networks.", "questions": "Based on the above comments, I would ask the authors the following question,\n\n - Would it be possible to make the main paper less dense and focus more on guiding the user through the intuition of the proofs?", "limitations": "The authors have sufficiently addressed any limitations that might arise from the submitted work.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces an algorithm for an unsolved problem setting as there have been several prior works that provide solutions and bounds for the smooth convex case but this is the first work that that provides bounds an optimal algorithms for the non-smooth case in a time varying domain.", "weaknesses": "The paper can be quite dense and while the authors provide slight intuition about the proof sketch in the main paper, most of the actual paper lies in the appendix which can make it quite inaccessible.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time, effort, and feedback. Below, we provide our detailed response to the review.\n\n### Weaknesses and Questions\n\n>The paper can be quite dense and while the authors provide slight intuition about the proof sketch in the main paper, most of the actual paper lies in the appendix which can make it quite inaccessible.\n\n>Would it be possible to make the main paper less dense and focus more on guiding the user through the intuition of the proofs?\n\nThank you for the suggestion. For the proof of lower bounds, the main theoretical ideas are listed on lines 204-223. We will try to enrich this part by adding a more detailed explanation of how to \"assemble\" these ideas into the final proof in the revised version of the paper.\nFor the proof of algorithm convergence, the main algorithmic components are listed on lines 282-296. We will try to provide an informal explanation of how these components work from a theoretical perspective and how to obtain the final proof using them, with appropriate references to the lemmas in the appendix.\nWe will try to obtain additional space for these changes, perhaps by shortening some parts of the introduction (as suggested by Reviewer 3roP), or by using an extra page if the paper is accepted.", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "LfM03mWc8A", "reviewer": "Reviewer_3roP", "summary": "This paper studies the problem of decentralized optimization for non-smooth convex opjectives and time-varying networks. The paper introduces an algorithm to solve this problem together with matching lower bounds on the required communication and subgradient computations, thereby proving that the proposed algorithm is optimal in this sense.", "questions": "Why does (57) not change in Part(ii) (line 591)? I believe the result is correct, but I am not sure whether this statement is true.", "limitations": "I do not fully agree with the authors' answer to Question 2 in the NeurIPS Checklist, as limitations of the presented results and avenues for future research are not clearly discussed in the paper. More specifically, the authors only provide the required assumptions, but do not discuss whether these assumptions are restrictive in certain scenarios (except for Assumption 1) and whether these assumptions could be relaxed in future work.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is very well written and explains the concepts used in this work very clearly. Furthermore, the paper solves a relevant research question by proposing an optimal algorithm together with matching lower bounds. The results and proofs established in this paper seem to be correct.", "weaknesses": "In my experience, optimization algorithms can sometimes yield unsatisfactory performance despite having good theoretical guarantees. Therefore, I believe that adding (even a small) simulation example showcasing the proposed algorithm's performance on a relevant problem would greatly improve the paper. \n\nFurthermore, while the paper is generally very well written, the algorithm could be explained a little bit better. Even just mentioning which lines of the algorithm correspond to which step in lines 282 - 296 would greatly help a reader who is not necessarily familiar with each of the references. In exchange, I believe that the justification for considering convex cost functions (Section 1.2) could be shortened. \n\nFinally, there are a few small typos that should be corrected:\nI think that $\\mathbf{1}_p$ should be defined as $(1,\\dots,1)^\\top$; Algorithm 1 requires an initialization for $\\bar{z}^0$ and $\\bar{y}^0$, and in the proofs (e.g., after line 765) you need to define w.l.o.g. that $\\tilde{x}^0 = x^{-1}$; in line 657, I think you should have $+\\langle x_a^K,z \\rangle$ and you need to flip the sign in (87); in lines 692-693, the function $F(x)$ maps from $(\\mathbb{R}^n)^d$, i.e., the argument should be $w^*$ instead of $x^*$; in Appendix E.3, E.4, and E:5, you previously used the notation $W(k)$ instead of $W_k$; in the first equations of Appendix E.6 (after (b)), I believe it should be $y^k$ instead of $y^{k+1}$, which is corrected in the next equation, and I do not see why you need line 7 of Algorithm 1 for (d); after line 773, there should be $k$ instead of $K$ in the third sum.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time, effort, and feedback. Below, we provide our detailed response to the review.\n\n### Weaknesses\n\n>I believe that adding (even a small) simulation example showcasing the proposed algorithm's performance on a relevant problem would greatly improve the paper.\n\nThank you for the suggestion. We are currently working on the basic experiments and will try to include them in the revised version of the paper.\n\n> Just mentioning which lines of the algorithm correspond to which step in lines 282 - 296 would greatly help a reader who is not necessarily familiar with each of the references.\n\nWe will try to provide an additional explanation of the algorihtm development with references to the corresponding algorithmic lines in the revised version of the paper by shortening Section 1.2, or by using an additional page if the paper is accepted.\n\n>Finally, there are a few small typos that should be corrected.\n\nThank you for finding the typos. We have corrected them as follows:\n- changed $\\mathbb{1}_p = (1,\\ldots,1)^\\top$;\n- added missing initializations on line 1 of Algorithm 1:\n  - added explicit initialization $\\overline{y}^0 = y^0$ and $\\overline{z}^0 = z^0$ (note that $\\overline{y}^0$ and $\\overline{z}^0$ are not used eventually due to line 4 and the fact that we use $\\alpha_0 = 1$);\n  - added explicit initialization $x^{-1} = \\tilde{x}^0 = x^0$ and annotated corresponding transitions after line 765;\n- fixed the sign of $\\langle x_a^K,z \\rangle$ on line 657 and after, and fixed the sign in equation (87);\n- replaced $x^*$ with $w^*$ on lines 692-693;\n- we use the notation $\\mathbf{W}_k$ when $k$ is the iteration number, and $\\mathbf{W}(\\tau)$ when $\\tau$ is the continuous time, which is mentioned on line 6 of Algorithm 1;\n- in the first equation of Appendix E.6:\n  - $y^{k+1}$ after (b) is a typo and is replaced with $y^k$;\n  - (d) indeed does not use line 7 of Algorithm 1, it is just an algebraic transformation;\n- fixed the third sum after line 773.\n\n### Questions\n\n>Why does (57) not change in Part(ii) (line 591)? I believe the result is correct, but I am not sure whether this statement is true.\n\nThere is indeed a small inaccuracy on line 591. Similarly to part(i) and due to Lemma 2 equation (57) will change to the following:\n$$\n\\\\mathcal{M}_i^{\\\\text{sub}}(\\\\tau)\n\\\\subset\n\\\\begin{cases}\n    \\\\mathcal{K}\\_{2p + 2} & i \\\\in \\\\mathcal{V}_1 \\\\text{ or } \\\\left(i \\\\in \\\\mathcal{V}_3 \\\\text{ and } i \\\\leq 2n/3 + q + 1\\\\right) \\\\\\\\\n    \\\\mathcal{K}\\_{2p + 1} & i \\\\in \\\\mathcal{V}_2 \\\\text{ or } \\\\left(i \\\\in \\\\mathcal{V}_3 \\\\text{ and } i > 2n/3 + q + 1\\\\right)\n\\\\end{cases},\n$$\nand the rest of the proof remains unchanged, including the last step:\n$$\n\\\\begin{aligned}\n\\\\mathcal{M}_i(\\\\tau) \n&\\\\subset\n\\\\mathcal{M}_i^{\\\\text{sub}}(\\\\tau) \\\\cup \\\\mathcal{M}_i^{\\\\text{com}}(\\\\tau)\n\\\\\\\\&\\\\subset\n\\\\begin{cases}\n    \\\\mathcal{K}\\_{2p + 2} & i \\\\in \\\\mathcal{V}_1 \\\\text{ or } \\\\left(i \\\\in \\\\mathcal{V}_3 \\\\text{ and } i \\\\leq 2n/3 + q + 1\\\\right) \\\\\\\\\n    \\\\mathcal{K}\\_{2p + 1} & i \\\\in \\\\mathcal{V}_2 \\\\text{ or } \\\\left(i \\\\in \\\\mathcal{V}_3 \\\\text{ and } i > 2n/3 + q + 1\\\\right)\n\\\\end{cases}.\n\\\\end{aligned}\n$$\nWe added this clarification to the proof. Thank you for noticing this.\n\n### Limitations\n\n>More specifically, the authors only provide the required assumptions, but do not discuss whether these assumptions are restrictive in certain scenarios (except for Assumption 1) and whether these assumptions could be relaxed in future work.\n\nWe will try to add an appropriate discussion on the restrictiveness and the possibility of relaxing the assumptions. For instance, Assumption 5 can be relaxed to a more general joint spectrum property (Nedic et al., 2017), and a corresponding extension of our results would be trivial. Assumption 2 can probably be relaxed too, if we replace the subgradient method with some adaptive method like Adagrad [1].\n\n[1] Duchi, John, Elad Hazan, and Yoram Singer. \"Adaptive subgradient methods for online learning and stochastic optimization.\" Journal of machine learning research 12.7 (2011).", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their time, effort, and high evaluation of our work. As far as we understand, there were no major common issues raised by the reviewers. Hence, we are providing our detailed responses to each review in separate messages.

Meta-review: This paper derives lower bounds on the communication and computation complexities of solving non-smooth convex decentralized optimization problems over time-varying networks and designs an algorithm that achieves these lower bounds. All reviewers appreciate the contributions of this work, recognizing its novelty and nontrivial nature. The only minor weakness lies in the somewhat limited simulation setups.

In the final version I recommend the authors to discuss the time-varying assumption. In general, time-varying graphs are assumed to be contractive over a finite length, but this work seem to assume contraction for every graph.
Decision: Accept (poster)