Title: Lower Bounds and Optimal Algorithms for Non-Smooth Convex Decentralized Optimization over Time-Varying Networks
Venue: NeurIPS 2024
Accepted: yes
Keywords: decentralized optimization

convex optimization

distributed optimization

non-smooth optimization

time-varying networks
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

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.

1 Introduction

In this paper, we study the decentralized optimization problem. Specifically, given a set of n compute nodes connected through a communication network, our goal is to solve the following finite-sum optimization problem with quadratic regularization:

where r ≥ 0 is a regularization parameter, and each function f i ( x ): R d → R is stored on the corresponding node i ∈ { 1 , . . . , n } . Each node i can perform computations based on its local state and data, and can directly communicate with other nodes through the links in the communication network.

Decentralized optimization problems find applications in a wide variety of fields. These include network resource allocation (Beck et al., 2014), distributed model predictive control (Giselsson et al., 2013), power system control (Gan et al., 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 (Konecn` y et al., 2016; McMahan et al., 2017), 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, we 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 a 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-the-art 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 et 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-the-art optimization methods, including distributed local gradient methods (Mishchenko et al., 2022; Sadiev et al., 2022; Karimireddy et al., 2020), adaptive gradient methods (Duchi et al., 2011), 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 Polyak-Łojasiewicz 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 as relaxations of the standard convexity property. Therefore, we naturally opt to 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., 2021), and many others. In 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

optimization problems (Scaman et al., 2017, 2018; Lan et al., 2020; Kovalev et al., 2020, 2021b,a, 2022; Hendrikx et al., 2021; Li et al., 2022; Li and Lin, 2021; Metelev et al., 2024). This task consists of finding a lower bound on the complexity 5 of solving a 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 a given problem class due to the established lower bounds.

We discuss the four main classes of decentralized optimization problems that cover smooth 6 and nonsmooth objective functions, and fixed and time-varying communication networks. We reference the existing state-of-the-art research papers that collectively solve the task of finding optimal algorithms for these classes. These papers are summarized in Table 1. In 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-varying 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-smooth convex problems over fixed networks, lower bounds were established by Scaman et al. (2018), and two optimal algorithms were proposed: DCS (Lan et al., 2020) and MSPD (Scaman et al., 2018).

Our paper primarily focuses on 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 Ozdaglar (2009), the subgradient-push method ( SubGD-Push ) by Nedi´ c and Olshevsky (2014), and ZOSADOM by Lobanov et al. (2023). Moreover, to the best of our knowledge, neither lower complexity bounds nor optimal algorithms have been proposed in this setting. Consequently, in this work, we close this significant gap with the following key contributions:

(i) We establish the first lower bounds on the number of decentralized communications and local subgradient computations required to solve problem (1) 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-art theoretical communication complexity, which outclasses the existing methods described in the literature.

2 Notation and Assumptions

In this paper, we are going to use the following notations: ⊗ denotes the Kronecker matrix product, I p denotes a p × p identity matrix, 1 p = (1 , . . . , 1) ⊤ ∈ R p , e p j ∈ R p for j ∈ { 1 , . . . , p } denotes the j -th unit basis vector, where p ∈ { 1 , 2 , . . . } . In addition, ∥·∥ denotes the standard Euclidean norm of a vector, and ⟨· , ·⟩ denotes the standard scalar product of two vectors.

2.1 Objective Function

Further, we describe the assumptions that we impose on problem 1. As discussed in Section 1.2, we assume the convexity of the objective function in problem (1). In particular, we assume that functions f 1 ( x ) , . . . , f n ( x ) are convex, which is formally described in Assumption 1.

Assumption 1. Each function f i ( x ) is convex. That is, for all x ′ , x ∈ R d and τ ∈ [0 , 1] , the following inequality holds:

In addition, we assume that the objective functions f 1 ( x ) , . . . , f n ( x ) 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 f i ( x ) is M -Lipschitz continuous for M ≥ 0 . That is, for all x ′ , x ∈ R d , the following inequality holds:

We also need the following Assumption 3, which ensures the existence of a solution to problem (1). Note that in the strongly convex case ( r > 0 ), the solution always exists and is unique. However, in the convex case ( r = 0 ), we need to explicitly assume the existence of a solution.

Assumption 3. There exists a solution x ∗ ∈ R d to problem (1) and a distance R > 0 such that

2.2 Decentralized Communication

Next, we formally describe the decentralized communication setting. The communication network is typically represented by a graph G ( V , E ) , where V = { 1 , . . . , n } is the set of compute nodes and E ⊂ V × V is the set of links in the network. As mentioned earlier, we allow the communication links to change over time. Thus, we introduce the continuous time parameter τ ≥ 0 and a set-valued function E ( τ ): R + → 2 V×V , which represents the time-varying set of edges. 7 Our time-varying network is then denoted as G ( τ ) = ( V , E ( τ )) .

Decentralized communication is typically represented via a matrix-vector multiplication with the so-called gossip matrix associated with the communication network (Scaman et al., 2017; Kovalev et al., 2021a). In the time-varying setting, we represent the gossip matrix by a matrix-valued function W ( τ ): R + → R n × n , which satisfies the following Assumption 4.

Assumption 4. For all τ ≥ 0 , the gossip matrix W ( τ ) ∈ R n × n associated with the time-varying communication network G ( V , E ( τ )) satisfies the following properties:

̸

(i) W ( τ ) ij = 0 if i = j and ( j, i ) / ∈ E ( τ ) ,

We also define the so-called condition number of the network χ ≥ 1 , which indicates how well the network G ( τ ) is connected (Scaman et al., 2017; Kovalev et al., 2021a). In particular, the communication complexity of most decentralized optimization algorithms depends on χ . Assumption 5 provides the formal definition of this quantity.

Assumption 5. There exists a constant χ ≥ 1 such that the following inequality holds for all τ ≥ 0 :

3 Lower Complexity Bounds

3.1 Decentralized Subgradient Optimization Algorithms

In this section, we present the lower bounds on the number of decentralized communications and the number of local subgradient computations required to solve problem (1). These lower bounds

apply to a 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 an 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 subgradient of a given objective function. The set of all subgradients of a 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 is done by the subgradient oracle , which is described by Definition 1.

Definition 1. For each i ∈ V , a function ˆ ∇ f i ( x ): R d → R d is called a subgradient oracle associated with the function f i ( x ) if, for all x ∈ R d , it satisfies ˆ ∇ f i ( x ) ∈ ∂f i ( x ) . That is, for each i ∈ V and for all x, x ′ ∈ R d , the following inequality holds:

Further, we provide the formal description of the class of decentralized subgradient optimization algorithms in the following Definition 2.

Definition 2. An algorithm is called a decentralized subgradient optimization algorithm with the subgradient computation time τ sub > 0 and decentralized communication time τ com > 0 if it satisfies the following constraints:

(i) Internal memory. At any time τ ≥ 0 , each node i ∈ V maintains an internal memory, which is represented by a set-valued function M i ( τ ): R + → 2 R d . The internal memory can be updated by subgradient computation or decentralized communication, which is formally represented by the following inclusion:

where set-valued functions M sub i ( τ ) , M com i ( τ ): R + → 2 R d are defined below.

(ii) Subgradient computation. At any time τ ≥ 0 , each node i ∈ V can update its internal memory M i ( τ ) by computing the subgradient ˆ ∇ f i ( x ) of the function f i ( x ) , which takes time τ sub . That is, for all τ ≥ 0 , the set M sub i ( τ ) is defined as follows:

(iii) Decentralized communication. At any time τ ≥ 0 , each node i ∈ V can update its internal memory M i ( τ ) by performing decentralized communication across the communication network, which takes time τ com. That is, for all τ ≥ 0 , the set M com i ( τ ) is defined as follows:

(iv) Initialization and output. At time τ = 0 , each node i ∈ V must initialize its internal memory with the zero vector, that is, M i (0) = { 0 } . At any time τ ≥ 0 , each node i ∈ V must specify a single output vector from its internal memory, x o,i ( τ ) ∈ M i ( τ ) .

3.2 Lower Bounds

Now, we are ready to present the lower bounds on the execution time τ ≥ 0 required to find an ϵ -approximate solution 8 to problem (1) by any algorithm satisfying Definition 2. Theorem 1 provides the lower bound in the strongly convex case ( r > 0 ), and Theorem 2 provides the lower bound in the convex case ( r = 0 ). These lower bounds naturally depend on the precision ϵ > 0 , the parameters of the problem, including the Lipschitz constant M > 0 , the regularization parameter r ≥ 0 , the distance R > 0 , and the parameters of the network, including the condition number χ ≥ 1 , communication time τ com > 0 , and subgradient computation time τ sub > 0 .

Theorem 2. For arbitrary parameters M,R,ϵ,τ com , τ sub > 0 and χ ≥ 1 , there exists an optimization problem of the form (1) with zero regularization ( r = 0 ) satisfying Assumptions 1, 2 and 3, corresponding subgradient oracles given by Definition 1, a time varying network G ( τ ) = ( V , E ( τ )) , and a corresponding time-varying gossip matrix W ( τ ) satisfying Assumptions 4 and 5, such that at least the following time τ is required to reach precision p ( x o,i ( τ )) -p ( x ∗ ) ≤ ϵ by any decentralized subgradient optimization algorithm satisfying Definition 2:

The proofs of Theorems 1 and 2 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 ) = a ∑ d -1 j =1 |⟨ e d j +1 -e d j , x ⟩| -a ⟨ e d 1 , x ⟩ + r 2 ∥ x ∥ 2 , which was used by Arjevani and Shamir (2015); Scaman et al. (2018) in the proof of lower bounds on the communication complexity in centralized and fixed-network settings. One can show that the gap p ( x ) -p ( x ∗ ) is lower-bounded by a positive constant as long as the last component of the vector x is zero, and it takes Ω( τ sub · d ) time to break this bound due to the constraint on the subgradient updates (8).

(ii) We split the objective function between two nodes of a star-topology network with a timevarying 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 Ω( n ) = Ω( χ ) communications to exchange information between the two selected nodes due to the timevarying center. This contrasts with the fixed path-topology network used by Scaman et al. (2017, 2018), where such an exchange would take Ω( n ) = Ω( √ χ ) communications. Moreover, using the constraint (8), we can show that it takes Ω( τ com · nd ) time to make the last component of the vector x nonzero and break the lower bound on the gap p ( x ) -p ( x ∗ ) , 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( τ com · nd + τ sub · d ) . Thus, we obtain the desired results by making a specific choice of the dimension d , network size n , and other parameters of problem (1).

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

Algorithm 1

decentralized time-varying network. The lower subgradient computation complexity bounds coincide in these cases (Nesterov (2013),Scaman et al. (2018),Theorems 1 and 2). However, the situation with the communication complexity is different. See Table 2 for a summary.

Theorems 1 and 2 imply that the communication complexity in the decentralized time-varying network setting is proportional to the network condition number χ . In contrast, the communication complexity in the fixed network setting is proportional to √ χ , 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 χ to the factor √ χ in communication complexity is impossible in the time-varying network setting. Only recently, this conjecture was proved for smooth functions by Kovalev et al. (2021a), and now we resolve this open question in the non-smooth case as well.

4 Optimal Algorithm

In this section, we develop an optimal algorithm for solving the non-smooth convex decentralized optimization problem (1) 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 ): ( R d ) n → R and G ( y, z ): ( R d ) n × ( R d ) n → R be defined as follows:

where x = ( x , . . . , x ) ∈ ( R d ) n , and r , r > 0

Consider the following saddle-point problem:

where L ⊥ ⊂ ( R d ) n is the orthogonal complement to the so-called consensus space L ⊂ ( R d ) n , defined as follows:

One can show that the saddle-point problem (14) is equivalent to the minimization problem (1). This is justified by the following Lemma 1. The proof of the lemma can be found in the Appendix A.

Lemma 1. Problem (14) is equivalent to problem (1) in the following sense:

The saddle-point reformulation of the form (14) 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 was made by Lobanov et al. (2023). However, their results have significant downsides, which we discuss in Section 4.3.

4.2 New Algorithm and its Convergence

Now, we present Algorithm 1 for solving problem (1). We provide upper bounds on the number of decentralized communications K and the number of subgradient computations K × T required to find an ϵ -approximate solution to the problem. Theorems 3 and 4 provide the upper bounds in the strongly convex ( r > 0 ) and convex ( r = 0 ) cases, respectively. The proofs can be found in Appendix D. The total execution time of Algorithm 1 is upper-bounded as τ = O ( τ com · K + τ sub · K × T ) , where the communication time τ com > 0 and the subgradient computation time τ sub > 0 are described in 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 1 is an optimal algorithm for solving problem (1).

Theorem 3. Under Assumptions 1, 2, 3, 4 and 5, let r > 0 (strongly convex case). Then Algorithm 1 requires K = O ( χM √ rϵ ) decentralized communications (line 6 of Algorithm 1) and K × T = O ( M 2 rϵ ) subgradient computations (line 11 of Algorithm 1) to reach precision p ( x K o ) -p ( x ∗ ) ≤ ϵ .

Theorem 4. Under Assumptions 1, 2, 3, 4 and 5, let r = 0 (convex case). Then Algorithm 1 requires K = O ( χMR ϵ ) decentralized communications (line 6 of Algorithm 1) and K × T = O ( M 2 R 2 ϵ 2 ) subgradient computations (line 11 of Algorithm 1) to reach precision p ( x K o ) -p ( x ∗ ) ≤ ϵ .

The design of Algorithm 1 is based on the fundamental Forward-Backward algorithm (Bauschke and Combettes, 2011). Let E = ( R d ) n × ( R d ) n ×L ⊥ be a Euclidean space, and consider a monotone operator A ( u ): E → E and a maximally-monotone multivalued operator B ( u ): E → 2 E defined as follows:

where u = ( x, y, z ) ∈ E , and P = ( I n -(1 /n ) 1 n 1 ⊤ n ) ⊗ I d ∈ R nd × nd is the orthogonal projection matrix onto L ⊥ . Then problem (14) is equivalent to the following monotone inclusion problem:

The basic Forward-Backward algorithm iterates u k +1 = (id + B ) -1 ( u k -A ( u k )) , where id is the identity operator and (id + B ) -1 denotes the inverse of the operator id( u ) + B ( u ) , which is called resolvent. Algorithm 1 can be obtained by making the following major modifications to these iterations:

(i) We accelerate the convergence of the Forward-Backward algorithm using Nesterov acceleration (Nesterov, 1983). Although this mechanism cannot be applied to the general monotone inclusion problem (18), Kovalev et 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, in 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 be 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 x max y τ x 2 ∥ x -x k ∥ 2 + F ( x ) -⟨ y, x ⟩ -τ y 2 ∥ y -y k ∥ 2 . This problem cannot be solved exactly, so we have to find an approximate solution using an

Algorithm

Strongly-convex case complexity

Convex case complexity

D-SubGD

SubGD-Push

ZO-SADOM

τ

com

·

χMd

1

/

4

log

1

ϵ

√

rϵ

+

τ

sub

·

M

2

d

log

1

ϵ

rϵ

τ

com

·

χMRd

1

/

4

log

1

ϵ

ϵ

+

τ

sub

·

M

2

R

2

d

log

1

ϵ

ϵ

2

Algorithm 1

τ

com

·

χM

√

rϵ

+

τ

sub

·

M

2

rϵ

τ

com

·

χMR

ϵ

+

τ

sub

·

M

2

R

2

ϵ

2

Lower Bounds

τ

com

·

χM

√

rϵ

+

τ

sub

·

M

2

rϵ

τ

com

·

χMR

ϵ

+

τ

sub

·

M

2

R

2

ϵ

2

N/A

τ

com

·

poly(

M,R,d

)

·

n

2

n

log

2 1

ϵ

ϵ

2

+

τ

sub

·

poly(

M,R,d

)

·

n

2

n

log

2 1

ϵ

ϵ

2

additional 'inner' algorithm based on the subgradient method (Nesterov, 2013) and the Chambolle-Pock operator splitting (Chambolle and Pock, 2011). We also have to conduct a careful analysis to find an efficient way to combine the inner and the 'outer' ForwardBackward algorithms and avoid unnecessary waste of subgradient calls.

The design of Algorithm 1 shares some similarities with the algorithm of Kovalev et al. (2021a) such as (i) and (ii) above. However, Kovalev et al. (2021a) simply add the gradient ∇ F ( x ) to the operator A ( u ) and use the accelerated version of the Forward-Backward algorithm, which we obviously cannot do as the function F ( x ) is not smooth. Instead, we have to put the subdifferential ∂F ( x ) into the operator B ( u ) and follow (iii) above. Part (iii) , in turn, shares some similarities with the algorithm of Lan et al. (2020). However, Lan et al. (2020) simply have a zero operator A ( u ) = 0 , which makes (i) and (ii) above unnecessary in their case. In contrast, we cannot make such simplifications because we work in the much more complicated setting of 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., 2020) and MSPD (Scaman et al., 2018), could be applied to solve problem (1) over time-varying networks. However, this is not the case, which is justified by the lack of corresponding theoretical guarantees and was shown empirically by Kovalev et al. (2021b). Therefore, we have to consider only those algorithms that were specifically developed for the time-varying network setting.

We provide a comparison of our Algorithm 1 with the existing state-of-the-art 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 (Nedi´ c and Olshevsky, 2014), and ZO-SADOM (Lobanov et al., 2023). The first two algorithms have poor performance: D-SubGD converges only to limited precision, and SubGD-Push converges at a slow rate of O (log 2 (1 /ϵ ) /ϵ 2 ) , which does not match even the iteration complexity of the standard centralized subgradient method, let alone the improved complexity of Algorithm 1. The complexity of ZO-SADOM is also worse than the lower bounds. Moreover, the theoretical results of Lobanov et al. (2023) have substantial drawbacks compared to ours:

(i) Lobanov et al. (2023) do not provide any theoretical insights or innovations in the analysis of their algorithm. In particular, they use the randomized smoothing technique (Duchi et al., 2012) to obtain a smooth approximation of the objective p ( x ) , and apply the existing algorithm of Kovalev et al. (2021a) to minimize this approximation. In contrast, we develop a new algorithm that directly works with the original non-smooth objective p ( x ) .

(ii) ZO-SADOM has extra factors d 1 / 4 log(1 /ϵ ) and d log(1 /ϵ ) in the decentralized communication and subgradient computation complexities, respectively, compared to the optimal complexity of our Algorithm 1. 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 Funding

This research has been financially supported by The Analytical Center for the Government of the Russian Federation (Agreement No. 70-2021-00143 01.11.2021, IGK 000000D730324P540002).

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Appendix

A Proof of Lemma 1

The orthogonal complement L ⊥ to the consensus space L is given as follows:

Let us perform the maximization of Q ( x, y, z ) in the variable y ∈ ( R d ) n :

where (a) uses the definition of Q ( x, y, z ) eq. (14); (b) uses the definition of G ( y, z ) in eq. (12). Next, we perform maximization in the variable z ∈ L ⊥ :

where I L ( x ): ( R d ) n → R is the indicator function, which is defined as follows:

Now, we can rewrite the saddle-point problem (14) as follows

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. (12); (d) uses eq. (13); (e) uses the definition of p ( x ) in eq. (1) and the definition of I L ( x ) .

B Proof of Theorems 1 and 2

B.1 The Hard Instance of Problem (1)

Compute nodes. In this proof, we consider the case when χ ≥ 3 . The case χ < 3 can be proven using the fixed-network argument of Scaman et al. (2018). We choose n = 3 ⌊ χ/ 3 ⌋ , which implies that n ≥ 3 and n mod 3 = 0 . We also divide the set of nodes V = { 1 , . . . , n } into the following three disjoint subsets: V 1 = { 1 , . . . , n/ 3 } , V 2 = { n/ 3+1 , . . . , 2 n/ 3 } and V 3 = { 2 n/ 3+1 , . . . , n } .

Objective functions. We fix an arbitrary odd integer d ∈ { 3 , 5 , 7 , . . . } and define functions f 1 ( x ) , . . . , f n ( x ): R d → R as follows:

where a > 0 is an arbitrary constant and functions h 1 ( x ) , . . . , h d -1 ( x ): R d → R are defined as follows:

Consequently, the objective function p ( x ) in problem (1) is given as follows:

We also define the subgradient oracles ˆ ∇ f 1 ( x ) , . . . , ˆ ∇ f n ( x ): R d → R d as follows:

where ˆ ∇ h 1 ( x ) , . . . , ˆ ∇ h d -1 ( x ): R d → R d are the subgradient oracles associated with functions h 1 ( x ) , . . . , h d -1 ( x ) , defined as follows:

Time-varying network. We choose the time-varying network G ( τ ) = ( V , E ( τ )) to be a startopology undirected graph with the time-varying center node i c ( τ ) ∈ V . Formally, we define the edges of the time-varying network E ( τ ) ⊂ V × V as follows:

̸

We also specify the center node i c ( τ ) at a given time τ ≥ 0 as follows:

We choose the time-varying gossip matrix W ( τ ) ∈ R n × n to be the Laplacian matrix of the graph G ( τ ) . Formally, W ( τ ) is defined as follows:

̸

̸

where deg i ( τ ) denotes the degree of the node i ∈ V in the graph G ( τ ) , i.e.,

One can observe, that the time-varying gossip matrix W ( τ ) satisfies Assumption 4, in particular, ker W ( τ ) = ker W ( τ ) ⊤ = span( { 1 n } ) . Moreover, one can show that W ( τ ) is a symmetric matrix, and λ max ( W ( τ )) = 1 and λ + min ( W ( τ )) = 1 /n ≥ 1 /χ . Hence, W ( τ ) satisfies Assumption 5.

B.2 Auxiliary Lemmas

Further, we define linear spaces K 0 , . . . , K d ⊂ R d as follows:

In order to prove Theorems 1 and 2, we will use the following auxiliary lemmas. The proofs of these lemmas can be found in Appendix C. Furthermore, the proof of Theorem 1 is contained in Appendix B.3, and the proof of Theorem 2 is contained in Appendix B.4.

Lemma 2. For all τ ≥ 0 , the following statements hold:

(ii) Let i ∈ V 2 . Then, for all j ∈ { 0 , . . . , ( d -1) / 2 } ,

(ii) Let i ∈ V 3 . Then, for all j ∈ { 0 , . . . , d } ,

The proof of Lemma 2 is contained in Appendix C.1.

Lemma 3. Let k ∈ { 0 , . . . , n ( d -1) / 6 -1 } . Then, for all τ < ( k +1) τ com, the following inclusion holds:

where p = ⌊ 3 k/n ⌋ and q = k mod ( n/ 3) .

The proof of Lemma 3 is contained in Appendix C.2.

Lemma 4. Let functions f 1 , . . . , f n ( x ) be defined by eq. (21) . Then problem eq. (1) has a unique solution x ∗ ∈ R d , which is given as follows:

Moreover, for all x ∈ K d -1 , the following inequality holds:

The proof of Lemma 4 is contained in Appendix C.3.

B.3 Proof of Theorem 1

Decentralized communication. Lemma 3 implies that M i ( τ ) ⊂ K d -1 as long as τ < τ com · n ( d -1) / 6 . Hence, Lemma 4 implies eq. (36) for all x ∈ M i ( τ ) as long as τ < τ com · n ( d -1) / 6 . Let the constant a > 0 be chosen as follows:

Then, each function f i ( x ) defined by eq. (21) is M -Lipschitz. Indeed, the case i ∈ V 3 is trivial. In the case when i ∈ V 1 , we can prove the M -Lipschitz continuity of f i ( x ) as follows:

In the case when i ∈ V 2 , we can prove the M -Lipschitz continuity of f i ( x ) similarly. Without loss of generality, we assume ϵ ≤ M 2 / (576 r ) and define d ∈ { 3 , 5 , . . . }

as follows:

Using eqs. (37) and (38), for all τ < τ com · n ( d -1) / 6 and x ∈ M i ( τ ) , we obtain

Hence, to reach precision p ( x ) -p ( x ∗ ) ≤ ϵ for some x ∈ M i ( τ ) , it is necessary that τ satisfies

Subgradient computation. We also need to prove that to reach precision p ( x ) -p ( x ∗ ) ≤ ϵ for some x ∈ M i ( τ ) , it is necessary that τ satisfies

We can do this by providing an extended version of our hard problem instance, described in Appendix B.1. In particular, we consider the following instance of problem (1):

where functions f 1 ( x ) , . . . , f n ( x ): R d → R are defined in Appendix B.1 by eq. (21), and functions f ′ 1 ( x ′ ) , . . . , f ′ n ( x ′ ): R d ′ → R are defined as follows:

where b > 0 is some constant. Then, by choosing an appropriate subgradient oracle ˆ ∇ f ′ i ( x ′ ) associated with each function f ′ i ( x ′ ) (see Section 3.2.1 of Nesterov (2013)) we can obtain both lower bounds (39) and (40), which concludes the proof.

B.4 Proof of Theorem 2

Our proof of Theorem 2 is very similar to the proof of Theorem 1 with the following differences. Let function h δ ( x ): R d → R be the Huber function, which is defined as follows:

Note that function h δ ( x ) is continuously differentiable and ( dδ ) -Lipschitz continuous.

√

In the proof of Theorem 3 we used functions f 1 ( x ) , . . . , f n ( x ) defined in eq. (21) of Appendix B.1. Here we use a slightly different choice, that is, functions f 1 ( x ) , . . . , f n ( x ) are defined as follows:

Consequently, our hard instance of problem (1), which is described in Appendix B.1, turns into the following:

where c > 0 is some constant, and functions h 1 ( x ) , . . . , h d -1 ( x ) are defined in eq. (22).

One can 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 ∇ ( 1 2 ∥·∥ 2 )( x ∗ ) = ∇ h δ ( x ∗ ) as long as δ and x ∗ are defined by eq. (46) and eq. (47), respectively.

Lemma 5. Let δ be defined as follows:

Problem eq. (45) has a solution x ∗ ∈ R d , which is given as follows:

Moreover, for all x ∈ K d -1 , the following inequality holds:

One can also show that each function f i ( x ) defined in eq. (44) is M f -Lipschitz continuous, where M f is defined as follows:

Let us choose a and c as follows:

which, for all x ∈ M i ( τ ) , implies as long as τ satisfies

which concludes the proof.

This choice of a and c implies M f ≤ M and ∥ x ∗ ∥ ≤ R . Moreover, eq. (48) implies

as long as x ∗ ∈ K d -1 . Next, without loss of generality we can assume ϵ ≤ ( MR ) / 72 and choose d ∈ { 3 , 5 , . . . } as follows:

C Proofs of Lemmas from Section B.2

C.1 Proof of Lemma 2

Statement (i). Let i ∈ V 1 and x ⊂ K 2 j for j ∈ { 1 , . . . , ( d -1) / 2 } . Then for l ≥ 2 j +1 we obtain ⟨ e d -e d , x ⟩ = 0 , which implies ˆ ∇ h ( x ) = 0 due to eq. (25). Hence, we obtain the following:

l +1 l l

where (a) uses eq. (24); (b) uses the fact that ˆ ∇ h l ( x ) = 0 for l ≥ 2 j +1 ; (c) uses eq. (25); (d) uses the definition of K 2 j in eq. (30). Hence, M i ( τ ) ⊂ K 2 j implies M sub i ( τ + τ sub ) ⊂ K 2 j by the definition of M sub i ( · ) in eq. (8).

Statement (ii). Let i ∈ V 2 and x ⊂ K 2 j +1 for j ∈ { 0 , . . . , ( d -1) / 2 } . Then for l ≥ 2 j + 2 we obtain ⟨ e d l +1 -e d l , x ⟩ = 0 , which implies ˆ ∇ h l ( x ) = 0 due to eq. (25). Hence, we obtain the following:

where (a) uses eq. (24); (b) uses the fact that ˆ ∇ h l ( x ) = 0 for l ≥ 2 j +2 ; (c) uses eq. (25); (d) uses the definition of K 2 j +1 in eq. (30). Hence, M i ( τ ) ⊂ K 2 j +1 implies M sub i ( τ + τ sub ) ⊂ K 2 j +1 by the definition of M sub ( · ) in eq. (8).

ˆ

i

Statement (iii). This statement is trivially implied by the definition of ∇ f i ( x ) in eq. (24) and the definition of M sub i ( · ) in eq. (8).

C.2 Proof of Lemma 3

We prove the lemma using the induction on k

.

Base case: k = 0 . In this case, we assume τ < ( k + 1) τ com = τ com. Hence, for all i ∈ V , we obtain M com i ( τ ) = ∅ and M i ( τ ) ⊂ M sub i ( τ ) . Using Lemma 2 and the fact that M com i ( τ ) = ∅ , we can easily obtain

which implies the desired eq. (34) for k = p = q = 0 .

′ ′

Induction hypothesis. Let k ∈ { 0 , 1 , 2 , . . . } . We assume that eq. (34) holds for all τ < ( k + 1) τ com, that is,

where p ′ = ⌊ 3 k ′ /n ⌋ and q ′ = k ′ mod ( n/ 3) .

̸

Induction step. We assume that the induction hypothesis (54) is true. Our goal is to prove that eq. (34) holds for k = k ′ +1 . When 0 ≤ τ < kτ com, the desired eq. (34) is implied by the induction hypothesis (54). Thus, we can assume kτ com ≤ τ < ( k +1) τ com. Further, we consider two cases: q = 0 and q = 0 .

̸

Induction step, case q = 0 . In this case, p = p ′ and q = q ′ +1 .

Part (i). First, we consider the case

Equation (55) implies τ -τ sub < ( k ′ +1) τ com and τ -τ com < ( k ′ +1) τ com. Using the induction hypothesis (54) and the fact that p ′ = p and q ′ = q -1 , we get

Hence, using Lemma 2, we obtain

Equations (27) and (55) imply i c ( τ ) = 2 n/ 3 + q +1 . Hence, using eq. (56), we get

For i = i c ( τ ) , using eqs. (9) and (56), we get

For i = i c ( τ ) = 2 n/ 3 + q +1 , using eqs. (9) and (56), we get

̸

Hence, using eqs. (58) and (59), for all i ∈ V , we obtain

̸

Now, we combine eqs. (57) and (60), and obtain

Thus, we were able to prove eq. (34) for τ satisfying (55).

Part (ii). We can prove the general case

for arbitrary l ∈ { 1 , 2 , . . . } using the induction on l . The only difference compared to the proof in the previous part is in eq. (56), which will change to

and eq. (57) will change as follows due to Lemma 2:

However, the rest of the proof, including eq. (61) will remain unchanged.

̸

Induction step, case q = 0 . In this case p = p ′ +1 and q ′ = n/ 3 -1 .

Part (i). First, we consider the case

Equation (62) implies τ -τ sub < ( k ′ +1) τ com and τ -τ com < ( k ′ +1) τ com. Using the induction hypothesis (54) and the fact that p ′ = p -1 and q ′ = n/ 3 -1 , we get

Equations (27) and (62) imply i c ( τ ) = 2 n/ 3 + 1 . Using eq. (63), we get

For i = i c ( τ ) , using eqs. (9) and (63), we get

For i = i c ( τ ) = 2 n/ 3 + 1 , using eqs. (9) and (63), we get

̸

Hence, using eqs. (64) and (65), for all i ∈ V , we obtain

Using Lemma 2, from eq. (63) we obtain

Hence, using eqs. (66) and (67), for all i ∈ V , we obtain

which implies eq. (34) for τ satisfying (62).

Part (ii). Next, we consider the case

Equation (66) still holds for all i ∈ V and τ satisfying eq. (69). From eqs. (68) and (69), for all i ∈ V , we obtain

which, due to Lemma 2, implies the following:

Hence, using eqs. (66) and (70), we obtain

which implies eq. (34) for τ satisfying (69).

Part(iii). We can prove the general case

for l ∈ { 2 , 3 , . . . } using the induction on 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 ∈ V and τ satisfying eq. (72).

̸

C.3 Proof Lemma 4

One can show, that x ∗ defined in eq. (35) is indeed the unique minimizer of the function p ( x ) defined in eq. (23). Moreover, we can obtain the following:

We can lower-bound function p ( x ) as follows:

as long as x ∈ K d -1 . Hence, for all x ∈ K j , we obtain

which concludes the proof.

D Proof of Theorems 3 and 4

D.1 Auxiliary Lemmas

In order to prove Theorems 3 and 4, we will use the following auxiliary lemmas. The proofs 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 4 is contained in Appendix D.3.

Lemma 6. Under Assumptions 1, 2 and 3, let r > 0 (strongly convex case). Then there exists a solution ( w ∗ , y ∗ , z ∗ ) ∈ L × ( R d ) n ×L ⊥ to problem (1) , which satisfies the following conditions

Moreover, the following inequalities hold:

The proof of Lemma 6 is contained in Appendix E.1.

Lemma 7. Under Assumptions 1 and 2, let η 0 x , . . . , η K -1 x and β 0 , . . . , β K -1 be chosen as follows:

Then, for all x ∈ ( R d ) n and k ∈ { 0 , . . . , K -1 } , the following inequality holds:

The proof of Lemma 7 is contained in Appendix E.2.

Lemma 8. Under Assumption 4, for all k ∈ { 0 , . . . , K -1 } , the iterates of Algorithm 1 satisfy

where P ∈ R nd × nd is the orthogonal projection matrix onto L ⊥ , which is given as follows:

The proof of Lemma 8 is contained in Appendix E.3.

Lemma 9. Under Assumptions 4 and 5, for all k ∈ { 0 , . . . , K -1 } the following inequality holds:

The proof of Lemma 9 is contained in Appendix E.4.

Lemma 10. Under Assumptions 4 and 5, let parameters θ 0 z , . . . , θ K -1 z be chosen as follows:

Then, for all k ∈ { 0 , . . . , K -1 } , the following inequality holds:

The proof of Lemma 10 is contained in Appendix E.5.

Lemma 11. Under Assumptions 1 and 2 and under conditions of Lemmas 7 and 10, let parameters α 0 , . . . , α K -1 and γ 0 , . . . , γ K -1 be chosen as follows:

Let parameters τ 0 , . . . , τ K -1 , η 0 , . . . , η K -1 , and η 0 , . . . , η K -1 be chosen as follows:

x x y y z z

where τ x , η y and η z are defined as follows:

Let parameters λ 1 , . . . , λ K be chosen as follows:

Let the input of Algorithm 1 be chosen as follows:

Then, for all x, y ∈ ( R d ) n and z ∈ L ⊥ , the following inequality holds:

The proof of Lemma 11 is contained in Appendix E.6.

D.2 Proof of Theorem 3

We can upper-bound r x 2 ∥ x K a -w ∗ ∥ 2 , where w ∗ is defined in Lemma 6, as follows:

where (a) uses Lemma 6 and the strong convexity of Q ( x, y, z ) in x ; (b) and (d) use Lemma 6; (c) uses Lemma 11. Using the definition of r x in eq. (84)

Next, we can upper-bound n ( p ( x K o ) -p ( x ∗ )) as follows:

where (a) uses the definition of p ( x ) in eq. (1); (b) uses the definition of x K o on line 15 of Algorithm 1; (c) uses the convexity of ∥·∥ 2 ; (d) uses the definition of w ∗ in eq. (95); (e) uses Assumption 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. (78).

Next, for arbitrary z ∈ L ⊥ we define y = -r -1 yz x K a -z . Then, we get ∇ y Q ( x K a , y, z ) = 0 and Q ( x K a , y, z ) = F ( x K a ) + 1 2 r yz ∥ x K a ∥ 2 + ⟨ x K a , z ⟩ . Plugging this into the previous upper-bound gives the following:

where (a) uses the definition of y ∗ in eq. (98) and the definition of z ∗ in eq. (99); (b) uses Lemma 6. Next, we choose z ∈ L ⊥ as follows:

̸

Then, ⟨ x K a , z ⟩ = + √ nM ∥ x K a ∥ P and we obtain the following:

where (a) uses Lemma 11; (b) uses our choice of y ; (c) uses the parallelogram rule and Young's inequality; (d) uses our choice of z ; (e) uses Lemma 6; (f) uses the definition of r yz in eq. (84); (g) uses the previously obtained upper-bound on r ∥ x K a -w ∗ ∥ 2 . Dividing both sides of the inequality by n gives the following:

Hence, choosing the parameters K and T such that

implies p ( x K o ) -p ( x ∗ ) ≤ ϵ , which concludes the proof.

D.3 Proof of Theorem 4

With r = 0 , the original problem (1) turns into the following problem:

Let x ∗ ∈ R d be the solution to problem (89), such that ∥ x ∗ ∥ ≤ R , which always exists due to Assumption 3. Let r > 0 be an arbitrary regularization parameter. We can upper-bound function ¯ f ( x ) using the regularized objective function p ( x ) defined in eq. (1) as follows:

On the other hand, we can lower-bound ¯ f ( x ∗ ) as follows:

Hence, we can upper-bound the function suboptimality gap in problem (89) as follows:

Let the regularization parameter r > 0 be chosen as follows:

Then, we obtain the following:

We can apply Algorithm 1 to solving the regularized problem (1) with the regularization parameter r defined in eq. (90). Theorem 3 implies that, to reach precision

it is sufficient to perform the following number of decentralized communications:

and the following number of subgradient computations:

where (a) and (b) use the definition of r in eq. (90). Using eqs. (91) and (92), we also obtain the desired precision ¯ f ( x K o ) -¯ f ( x ∗ ) ≤ ϵ , which concludes the proof.

E Proofs of Lemmas from Section D.1

E.1 Proof of Lemma 6

First, we pick the solution x ∗ ∈ R d to problem (1), which is unique due to Assumption 3 and the fact that r > 0 . Next, we define w ∗ ∈ L as follows:

From Assumptions 1 and 2 it follows that dom p ( x ) = R d and dom f i ( x ) = R d for all i ∈ { 1 , . . . , n } , which implies the following:

Hence, there exists a vector ∆ ∗ = (∆ ∗ 1 , . . . , ∆ ∗ n ) ∈ ( R d ) n such that ∆ ∗ i ∈ ∂f i ( x ∗ ) for all i ∈ { 1 , . . . , n } , and the following relation holds:

Next, we define y ∗ ∈ ( R d ) n as follows:

z

Note that the inclusion z ∗ ∈ L ⊥ is implied by eq. (97). Further, we get

∇

Q

(

w

∗

, y

∗

, z

∗

)

=

-

r

yz

(

y

∗

+

z

∗

)

=

-

r

yz

(

r

x

-

r

)

w

∗

∈ L

,

where (a) uses the definition of Q ( x, y, z ) in eq. (14); (b) uses the definition of y ∗ and z ∗ , and the last inclusion follows from the definition of w ∗ . Moreover, we obtain the following

where (a) uses the definition of Q ( x, y, z ) in eq. (14); (b) uses the definition of y ∗ and z ∗ ; (c) uses eq. (13).

From Assumption 2 it follows that ∥ ∆ ∗ i ∥ ≤ M for all i ∈ { 1 , . . . , n } . Hence, using eq. (97), we get r ∥ x ∗ ∥ ≤ M , which implies ∥ w ∗ ∥ 2 ≤ nM 2 /r 2 . Moreover, we get

which implies ∥ y ∗ ∥ 2 ≤ (1 + r x /r ) 2 nM 2 . Finally, we obtain

which implies ∥ z ∗ ∥ 2 ≤ 4 nM 2 and concludes the proof.

(

a

)

(

b

)

From Assumptions 1 and 2 it follows that dom F ( x ) = ( R d ) n , which implies y ∗ ∈ ∂F ( w ∗ ) and 0 ∈ ∂ ( F ( · ) -⟨ y ∗ , ·⟩ )( w ∗ ) = ∂ x Q ( w ∗ , y ∗ , z ∗ ) .

Next, we define z ∗ ∈ L ⊥ as follows:

E.2 Proof of Lemma 7

We start with the following upper-bound on 1 2 η k x ∥ x k,t +1 -x ∥ 2 :

where ( g k,t x, 1 , . . . , g k,t x,n ) = ( ˆ ∇ f 1 ( x k,t 1 ) , . . . , ˆ ∇ f n ( x k,t n )) = g k,t x ∈ ( R d ) n , (a) and (c) uses the parallelogram rule; (b) uses line 12 of Algorithm 1; (d) uses line 11 of Algorithm 1, Definition 1 and Assumption 1. Further, we obtain

where (a) uses Assumption 2 and the Cauchy-Schwarz inequality; (b) uses the inequality ∥ g k,t x,i ∥ ≤ M , which follows from Assumption 2; (c) uses the definition of β k in eq. (75), the definition of F ( x ) in eq. (12) and Young's inequality. After rearranging, we obtain

where ∆ k,t +1 is defined as

Now, we sum these inequalities for t = 0 , . . . , T -1 and obtain

Dividing both sides of the inequality by T gives

Using the definition of ∆ k,t , the definition of ˜ x k on line 13 of Algorithm 1 and Assumption 1, we obtain

Using the definition of η k x and β k in eq. (75), we obtain

Using the definition of x k, 0 on line 9 of Algorithm 1, the definition of x k +1 on line 13 of Algorithm 1, the definition of η k x , β k and σ k in eq. (75) and the convexity of ∥·∥ , we obtain

which concludes the proof.

E.3 Proof of Lemma 8

Using Assumption 4, and the definition of P in eq. (78), we obtain

Then, the desired relations can be trivially obtained by analyzing the lines of Algorithm 1.

E.4 Proof of Lemma 9

We can upper-bound ∥ η k +1 z m k +1 ∥ 2 P as follows:

where (a) uses line 8 of Algorithm 1; (b) uses line 6 of Algorithm 1; (c) uses eq. (100); (d) uses Assumption 5; (e) uses the parallelogram rule and Young's inequality. Using this, we obtain

which concludes the proof.

E.5 Proof of Lemma 10

We can upper bound ∥ ˜ g k z -P g k z ∥ 2 as follows:

where (a) uses line 6 of Algorithm 1; (b) uses eq. (100); (c) uses Assumption 5. On the other hand, ∥ ˜ g k z -P g k z ∥ 2 is equal to the following:

where (a) uses the parallelogram rule; (b) uses line 8 of Algorithm 1. Hence, we obtain the following

After rearranging and multiplying both sides of the inequality by θ k z 2 α k , we obtain

where (a) uses eq. (80); (b) uses Lemma 8, which concludes the proof.

E.6 Proof of Lemma 11

We can upper-bound 1 2 η k y ∥ y k +1 -y ∥ 2 as follows:

where (a) uses the parallelogram rule; (b) uses line 7 of Algorithm 1; (c) uses Lines 4 and 8 of Algorithm 1. Further, we can upper-bound the term ⟨ ˜ x k +1 -ˆ x k +1 , y k +1 -y ⟩ as follows:

where (a) uses Line 7 of Algorithm 1; (b) uses Young's inequality. Plugging this into the previous inequality gives

where (a) line 8 of Algorithm 1; (b) uses eqs. (83) and (84).

Let ˆ z k be defined for all k ∈ { 0 , . . . , K }

as follows:

Using eq. (101) and lines 7 and 8 of Algorithm 1, we obtain

Hence, we can upper-bound 1 2 η k z ∥ ˆ z k +1 -z ∥ 2 as follows:

where (a) uses the parallelogram rule; (b) uses the update rule for ˆ z k which we previously obtained; (c) uses eq. (101). Further, we can upper-bound the term η k z ⟨ P g k z , P m k ⟩ as follows

where (a) uses the Cauchy-Schwarz inequality; (b) uses Young's inequality; (c) uses Lemma 9. Plugging this into the previous inequality gives

where (a) uses Lemma 8 and the fact that z ∈ L ⊥ ; (b) uses line 4 of Algorithm 1; (c) uses Lemma 10; (d) uses eqs. (83) and (84).

Nowwecombine the upper-bounds for 1 2 η k y ∥ y k +1 -y ∥ 2 and 1 2 η k z ∥ ˆ z k +1 -z ∥ 2 and obtain the following:

where (a) uses the definition of g k y and g k z on line 5 of Algorithm 1 and the convexity and (2 r yz ) -smoothness of the function G ( y, z ) . Further, we divide both sides of the inequality by α k and, using eq. (83), obtain the following:

Next, we divide the inequality in Lemma 7 by α k and, using the definition of τ k x and τ x in eqs. (83) and (84), obtain the following:

Combining this inequality with the previous upper-bound gives the following:

where (a) uses the fact that y k +1 = α -1 k y k +1 -(1 -α k ) α -1 k y k , which follows from lines 4 and 8 of Algorithm 1; (b) uses the definition of Q ( x, y, z ) in eq. (14); (c) uses line 13 of Algorithm 1 and Assumption 1.

Further, let α K = 3 / ( K +3) . Then from eq. (82) it follows that α -2 k + α -1 k ≥ α -2 k +1 , γ k α -1 k = ( k +2) / 3 and α -1 = ( k +3) / 3 . Hence, we obtain the following:

where (a) uses eq. (84).

Next, we sum these inequalities for k = 0 , . . . , K -1 and obtain the following:

where (a) uses the Cauchy-Schwarz inequality, Young's inequality, and the initialization ˜ x 0 = x -1 on line 1 of Algorithm 1; (b) uses the fact that α 0 = 1 , which follows from eq. (82); (c) uses the definition of λ k in eq. (85). Further, we obtain the following:

where (a) uses the convexity of Q ( x, y, z ) in x (follows from Assumption 1) and the concavity of Q ( x, y, z ) in ( y, z ) , line 14 of Algorithm 1, and the fact that λ k ≥ 0 , which follows from eqs. (82) and (85); (b) use the definition of λ k in eq. (85) and the fact that α 0 = 1 , which follows from eq. (85). Next, we do rearranging and use eqs. (84) and (86), which gives the following:

Next, we divide both sides of the inequality by ∑ K -1 k =0 α -1 k , which gives the following:

Further, we can estimate ∑ K -1 k =0 α -1 k as follows:

where (a) uses eq. (82). Plugging this into the previous inequality gives

which concludes the proof.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nIn 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.\n\nQUESTIONS:\nWhy 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\nLIMITATIONS:\nI 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.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-07"}

comments: SUMMARY:
The authors introduce an algorithm which optimally bounds the complexity of algorithms for non-smooth convex decentralised optimisation over time varying networks.

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.

QUESTIONS:
Based on the above comments, I would ask the authors the following question,

 - 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.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nI 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.\n\nWEAKNESSES:\nThere 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.\n\nQUESTIONS:\nWhy 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).\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

comments: 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.

STRENGTHS:
The problem studied in the paper is interesting.

The results look to be rigorous and strong (by achieving the derived lower bound).

WEAKNESSES:
None.

QUESTIONS:
None.

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-15