Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization

This paper considers the following decentralized composite optimization over networks with $N$ computational agents:

where the local loss function $f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and $g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\{\infty\}$ are convex and can only be accessed by corresponding agents. We assume that $f_{i}$ is $L_{i}$-smooth, and $g_{i}$ is proper, closed, but not necessarily smooth. This setting is general and is encountered in various application fields, including multi-agent control, wireless communication, and decentralized machine learning [1].

Arguably, the straightforward approach to address the decentralized settings is to employ the classical gradient descent method combined with the network structure[2, 3, 4, 5]. On one hand, in [2, 3, 4], the mixing matrix is used to realize the consensus of the local state through the weighted average of neighbor states. These algorithms can be regarded as the quadratic penalty methods for equality consensus constraints, which leads to an inexact convergence when the stepsizes are fixed [3]. To obtain the exact optimal solution, the diminishing stepsizes are adopted [4]. On the other hand, by using the sign of the relative state rather than averaging over the local iterates via the mixing matrix, [5] proposes an algorithm based on the exact penalty method. However, these primal methods can not achieve the exact convergence under the constant stepsize.

A recent line of work in the primal-dual domain provide a number of alternative methods that can exactly converge to the optimum with the fixed step sizes. When $g_{i}=0$ and $f_{i}$ is strongly convex, several linear convergence algorithms have been proposed including D-ADMM[6], DLM[7], EXTRA[8], DIGing[9], Harnessing[10], AugDGM [11, 12], and Exact Diffusion[13]. When $g_{i}\neq 0$ and the nonsmooth terms are identical among all the agents, under the assumption that $f_{i}$ is strongly convex, [14, 15, 16, 17] present several linear convergence algorithms for problem (1). When $g_{i}\neq 0$ and the nonsmooth terms may be distinct among these agents, IC-ADMM[18], PG-ADMM[19], PG-EXTRA[20], NIDS[21] and PAD[22] have been provided. In [18], it proves the convergence of IC-ADMM under the strong convexity of $f_{i}$. In [19], PG-ADMM is shown to have the $O({1}/{k})$ ergodic convergence rate. Based on EXTRA[8], PG-EXTRA and NIDS have been provided in [20] and [21], respectively. For PG-EXTRA, it is shown that the algorithm has the $O({1}/{k})$ convergence rate and has the $o({1}/{k})$ convergence rate when the smooth part vanishes. For NIDS, which has a network independent step size, it is proved to have the $o({1}/{k})$ convergence rate. In [22], by the quadratic penalty method and linearized-ADMM, an inexact convergence algorithm called PAD is proposed. For PAD, it proves that for a fixed penalty parameter ${1}/{\tilde{\epsilon}}>0$, the distance between the solution of problem (1) and the limit point of PAD is of order $O(\tilde{\epsilon}^{\frac{1}{2}})$. Moreover, it proves the $O({1}/{k})$ ergodic convergence rate under general convexity, and the linear convergence rate under the strongly convex of $f_{i}$.

These algorithms can be divided into ATC (Adapt-Then-Combine)- and CTA (Combine-Then-Adapt)-based decentralized algorithms according to their characteristics [15]. For ATC-based decentralized algorithms, there are several algorithms with network-independent constant stepsizes such as AugDGM [11, 12], Exact Diffusion[13], ABC-Algorithm [15], NEXT[16]/SONATA[17], and NIDS [21] (see [15] for more details). However, to the best of our knowledge, the choice of the stepsize of the existing CTA-based decentralized algorithms are related to the network topology. Moreover, in the algorithms mentioned above for composite optimization, the proximal mapping of $g_{i}$ is required to have analytic solutions or can be efficiently computed. However, in many applications, there is no analytic solution to the proximal mapping of $g_{i}$. Such problems include generalized LASSO regularizer [23], fused LASSO regularizer [24], the octagonal selection and clustering algorithm for regression (OSCAR) regularizer [25], and the group LASSO regularizer [26]. Therefore, it prompts us to introduce an inexact inner-solver to compute an approximation of the proximal mapping. There are several works focusing on inexact versions of ALM [27], primal-dual hybrid gradient method [28], ADMM [29] and decentralized ALM [30, 31]. To our knowledge, the corresponding results for problem (1) are relatively scarce. Furthermore, to the problem (1), when the nonsmooth parts $g_{i}$ are distinct, the linear convergence rate of corresponding algorithms under certain assumptions have not been established. Although [14] shows that when $f_{i}$ is a strongly convex smooth function and $g_{i}$ is nonsmooth convex with closed-form solution to proximal mappings, then the dimension independent exact global linear convergence cannot be attained for the composite optimization problem (1), it does not mean that the linear convergence rate related to dimension is impossible under some conditions. Therefore, it naturally leads to the following three problems.

Question 1. Can one provide a CTA-based decentralized algorithm with network-independent constant stepsizes?

Question 2. Can one provide a decentralized algorithm when the proximal mapping of $g_{i}$ has no analytic solution?

Question 3. Can one establish the linear convergence of the proposed decentralized algorithm for the problem (1)?

This paper aims at addressing Questions 1, 2 and 3 for the problem (1) over an undirected and connected network. We develop a novel CTA-based decentralized inexact proximal gradient method (D-iPGM) with network-independent constant stepsizes. In the implementation of D-iPGM, computable approximations of the proximal mapping is allowed, which implies that the proposed algorithm can reduce the computational cost when proximal mapping of $g_{i}$ has no closed-form solution. To convergence analysis, we prove the $O(1/k)$ convergence rate under general convexity. When the proximal mapping of $g_{i}$ is solved exactly, we establish the $o(1/k)$ non-ergodic convergence of D-iPGM. We further prove the linear convergence rate of D-iPGM under the metric subregularity which is weaker than strong convexity. Table I shows a comparison of D-iPGM with the existing decentralized algorithms to convex composite optimization problems over networks.

This paper is organized as follows. Section II casts the problem (1) into a constrained form, and based on the reformulation, the D-iPGM is proposed. Additionally, some relations between D-iPGM and PG-EXTRA [20] and NIDS [21] are discussed. Then, the convergence properties under general convexity and metric subregularity are investigated in Section III. Finally, the numerical experiments are implemented in Section IV and conclusions are given in Section V.

Notations: $\mathbb{R}^{n}$ denotes the $n$-dimensional vector space with inner-product $\langle\cdot,\cdot\rangle$. The $l_{1}$-norm, $l_{2}$-norm and $l_{\infty}$-norm are denoted as $\|\cdot\|_{1}$, $\|\cdot\|$ and $\|\cdot\|_{\infty}$, respectively. $\mathbf{0}$, $\mathbf{I}$, and $\mathbf{1}$ denote the null matrix, the identity matrix, and the all-ones matrix of appropriate dimensions, respectively. For ${\mathrm{x}}_{i}\in\mathbb{R}^{n},i=1\ldots,m$, let $\mathrm{col}\{{\mathrm{x}}_{1},\cdots,{\mathrm{x}}_{N}\}=[{\mathrm{x}}_{1}^{\sf T},\cdots,{\mathrm{x}}_{N}^{\sf T}]^{\sf T}$. For a matrix $A\in\mathbb{R}^{m\times n}$, $\sigma_{m}(A)$ and $\sigma_{M}(A)$ is the smallest and largest nonzero singular value of $A$, respectively. If $A$ is symmetric, $A\succ 0$ means that $A$ is positive definite. Let $\mathcal{H}\succ 0$, and $\mathcal{D}$ be a non-empty closed convex subset of $\mathbb{R}^{n}$. For any ${\mathrm{x}}\in\mathbb{R}^{n}$, we define $\|{\mathrm{x}}\|^{2}_{\mathcal{H}}:=\langle{\mathrm{x}},\mathcal{H}{\mathrm{x}}\rangle$, $\mathrm{dist}_{\mathcal{H}}({\mathrm{x}},\mathcal{D}):=\min_{{\mathrm{x}}^{\prime}\in\mathcal{D}}\|{\mathrm{x}}-{\mathrm{x}}^{\prime}\|_{\mathcal{H}}$, and $\mathcal{P}_{\mathcal{D}}^{\mathcal{H}}({\mathrm{x}}):=\arg\min_{{\mathrm{x}}^{\prime}\in\mathcal{D}}\|{\mathrm{x}}-{\mathrm{x}}^{\prime}\|_{\mathcal{H}}$. For a given closed proper convex function $\theta$, the proximal mapping of $\theta$ relative to $\|\cdot\|_{\mathcal{H}}$ is defined as $\mathrm{prox}_{\theta}^{\mathcal{H}}({\mathrm{x}}):=\arg\min_{\mathrm{y}}\{\theta({\mathrm{y}})+\frac{1}{2}\|{\mathrm{y}}-{\mathrm{x}}\|_{\mathcal{H}}^{2}\}$. When $\mathcal{H}={\bf{I}}$, we just omit it from these notations. $\mathcal{B}_{r}(\bar{{\mathrm{x}}}):=\{{\mathrm{x}}:\|{\mathrm{x}}-\bar{{\mathrm{x}}}\|<r\}$ denotes the open Euclidean norm ball around ${\mathrm{x}}$ with modulus $r>0$.

Consider an undirected and connected network $\mathcal{G}(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{v_{1},\ldots,v_{N}\}$ denotes the vertex set, and the edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ specifies the connectivity in the network, i.e., a communication link between agents $i$ and $j$ exists iff $(i,j)\in\mathcal{E}$. Let $\mathrm{x}_{i}\in\mathbb{R}^{n}$ be the decision variable held by the agent $i$. To cast the problem (1) into a consensus-constrained form and make sure that $\mathrm{x}_{1}=\cdots=\mathrm{x}_{N}$, we introduce the global mixing matrix $\mathbf{W}=W\otimes\mathbf{I}_{n}$, where $W=[W_{ij}]\in\mathbb{R}^{N\times N}$, and make the following standard assumption in decentralized optimization.

In addition, with this mixing matrix, similar as [20], we introduce another mixing matrix $\tilde{{\bf{W}}}$ defined as $\tilde{{\bf{W}}}:=\frac{1}{2}({\bf{I+W}})$. By Assumption 1, the constraint $\mathrm{x}_{1}=\cdots=\mathrm{x}_{N}$ is equivalent to the identity $(\mathbf{I}-\mathbf{W})\mathbf{x}=\mathbf{0}$, which is again equivalent to $\sqrt{\mathbf{I}-\mathbf{W}}\mathbf{x}=\mathbf{0}$. Then, we have the following equivalent formulation of problem (1).

where $\mathbf{x}=[\mathrm{x}_{1}^{\sf T},\cdots,\mathrm{x}_{N}^{\sf T}]^{\sf T}\in\mathbb{R}^{Nn}$. To guarantee the wellposedness of (1), it is necessary to suppose that the optimal solution of (1) exists. Throughout the paper, we give the following assumption.

According to [32, Theorem 5.8] and Assumption 2, for any ${\bf{x}},{\bf{y}},{\bf{z}}\in\mathbb{R}^{Nn}$, we have the following commonly used inequalities

where ${\bf{L}}_{F}=\mathrm{diag}\{L_{1}{\bf{I}}_{n},\cdots,L_{N}{\bf{I}}_{n}\}$.

This section analyzes the convergence and the convergence rate of the D-iPGM. The convergence analysis will be conducted in the variational inequality context [40], and the forthcoming analysis of the proposed D-iPGM is based on the following fundamental lemma.

Recall the Lagrangian of problem (2): $L(\mathbf{x},\bm{\alpha})=F(\mathbf{x})+G(\mathbf{x})-\langle\bm{\alpha},{\bf{V}}\mathbf{x}\rangle,$ where ${\bf{V}}=\sqrt{{\bf{I-W}}}$ and $\bm{\alpha}\in\mathbb{R}^{Nn}$ is the Lagrange multiplier. By strongly duality, to solve problem (2), we can consider the saddle-point problem

Define $\mathcal{M}^{*}$ as the set of saddle-points to the above saddle-point problem. It holds that $\mathcal{M}^{*}=\mathcal{X}^{*}\times\mathcal{Y}^{*}$, where $\mathcal{X}^{*}$ is the optimal solution set to problem (2) and $\mathcal{Y}^{*}$ is the optimal solution set to its dual problem. Denote the KKT mapping as

We have $(\mathbf{x}^{*},\bm{\alpha}^{*})\in\mathcal{X}^{*}\times\mathcal{Y}^{*}$ if and only if $\mathbf{0}\in\mathcal{J}(\mathbf{\mathbf{x}^{*},\bm{\alpha}^{*}})$. A point $\mathbf{u}^{*}=(\mathbf{x}^{*},\bm{\alpha}^{*})\in\mathcal{M}^{*}\subseteq\mathbb{R}^{Nn}\times\mathbb{R}^{Nn}:=\mathcal{M}$ is called a saddle point of the Lagrangian function $L(\mathbf{x},\bm{\alpha})$ if $L(\mathbf{x}^{*},\bm{\alpha})\leq L(\mathbf{x}^{*},\bm{\alpha}^{*})\leq L(\mathbf{x},\bm{\alpha}^{*}),\forall(\mathbf{x},\bm{\alpha})\in\mathcal{M},$ which can be alternatively rewritten as the following variational inequalities (by Lemma 1)

where $\phi({\bf{x}})=G({\bf{x}})+F({\bf{x}})$, $\mathbf{u}=\mathrm{col}\{\mathbf{x},\bm{\alpha}\}$,

Note that the mapping $\mathcal{K}_{2}$ is affine with a skew-symmetric matrix and thus we have

From the above analysis, it holds that the set $\mathcal{M}^{*}$ is also the solution set of (17) and (18). Therefore, the convergence of D-iPGM can be proved if the generated sequence can be expressed by a similar inequality to (17) or (18) with an extra term converging to zero. Based on such observation, we will show the convergence properties of the sequence generated by D-iPGM in Sections III-A, III-B, and III-C.