Distributed nonconvex constrained optimization over time-varying digraphs

Abstract

This paper considers nonconvex distributed constrained optimization over networks, modeled as directed (possibly time-varying) graphs. We introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex (nonseparable) function—the agent’s sum-utility—plus a difference-of-convex function (with nonsmooth convex part). This general formulation arises in many applications, from statistical machine learning to engineering. The proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push-sum consensus mechanism that aims to track locally the gradient of the (smooth part of the) sum-utility. Sublinear convergence rate is proved when a fixed step-size (possibly different among the agents) is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step-size. Numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems.

Appendices

Appendix

Proof of Lemma 3

We begin introducing the following intermediate result.

Lemma 15

In the setting of Lemma 3, the following hold:

• (i) The elements of ${\mathbf {A}}^{n:0}$, $n\in {\mathbb {N}}_+$, can be bounded as

  $$\begin{aligned}&\inf _{t\in {\mathbb {N}}_+} \left( \min _{1\le i\le I} \left( {\mathbf {A}}^{t:0}{\mathbf {1}}\right) _{i}\right) \ge \phi _{lb}, \end{aligned}$$

  (108)
  $$\begin{aligned}&\sup _{t\in {\mathbb {N}}_+} \left( \max _{1\le i\le I} \left( {\mathbf {A}}^{t:0}{\mathbf {1}}\right) _{i}\right) \le \phi _{ub}, \end{aligned}$$

  (109)

  where $\phi _{lb}$ and $\phi _{ub}$ are defined in (8);

• (ii) For any given $n, k\in {\mathbb {N}}_+$, $n\ge k$, there exists a stochastic vector $\varvec{\xi }^{k}\triangleq [\xi _1^{k},\ldots \xi _I^{k}]^\top$ (i.e., $\varvec{\xi }^{k}> {\mathbf {0}}$ and ${\mathbf {1}}^\top \, \varvec{\xi }^{k}=1)$ such that

  $$\begin{aligned} \left| {\mathbf {W}}^{n:k}_{ij} - \xi _{j}^{k}\right| \le c_{0}\,(\rho )^{\big \lfloor \frac{n-k+1}{(I-1)B}\big \rfloor },\qquad \forall i,j\in [I], \end{aligned}$$

  (110)

  where $c_{0}$ and $\rho$ are defined in (10).

The proof Lemma 15 follows similar steps as those in [31, Lemma 2, Lemma 4] and thus is omitted, although the results in [31] are established under a stronger condition on ${\mathcal {G}}^n$ than Assumption B.

We prove now Lemma 3. Let ${\mathbf {z}}\in {\mathbb {R}}^{I\cdot m}$ be an arbitrary vector. For each $\ell =1,\ldots ,m$, define ${\mathbf {z}}_{\ell }\triangleq ({\mathbf {I}}_I \otimes {\mathbf {e}}_{\ell }^\top )\,{\mathbf {z}}$, where ${\mathbf {e}}_{\ell }$ is the $\ell$-th canonical vector; we denote by ${z}_{\ell ,j}$ the j-th component of ${\mathbf {z}}_{\ell }$, with $j\in [I]$. We have

$$\begin{aligned} \left\| \left( \widehat{{\mathbf {W}}}^{n:k} - {\mathbf {J}}_{\varvec{\phi }^{k}}\right) {\mathbf {z}}\right\| _2 \le \sqrt{I\cdot \sum _{\ell =1}^{m} \left\| \left( {\mathbf {W}}^{n:k} - \frac{1}{I}{\mathbf {1}}\,(\varvec{\phi }^{k})^\top \right) {\mathbf {z}}_{\ell }\right\| ^{2}_\infty }. \end{aligned}$$

(111)

We bound next the above term. Given $\varvec{\xi }^k$ as in Lemma 15 [cf. (110)], define ${\mathbf {E}}^{n:k}\triangleq {\mathbf {W}}^{n:k} - {\mathbf {1}}(\varvec{\xi }^{k})^\top$, whose ij-th element is denoted by ${E}^{n:k}_{ij}$. We have

$$\begin{aligned}&\left\| \left( {\mathbf {W}}^{n:k} - \frac{1}{I}{\mathbf {1}}(\varvec{\phi }^{k})^\top \right) {\mathbf {z}}_{\ell }\right\| _{\infty } \overset{(15)}{=} \left\| \left( {\mathbf {W}}^{n:k} - \frac{1}{I}{\mathbf {1}}\left( \varvec{\phi }^{n+1}\right) ^{\top }{\mathbf {W}}^{n:k}\right) {\mathbf {z}}_{\ell }\right\| _{\infty }\nonumber \\&\quad = \left\| \left( {\mathbf {I}} - \frac{1}{I}{\mathbf {1}}\left( \varvec{\phi }^{n+1}\right) ^{\top }\right) {\mathbf {E}}^{n:k}\,{\mathbf {z}}_{\ell }\right\| _{\infty }\nonumber \\&\quad \le \max _{1\le i \le I}\left( \left( 1-\frac{\phi _{i}^{n+1}}{I}\right) \sum _{j=1}^{I}\left| E_{ij}^{n:k}\right| \left| z_{\ell ,j}\right| + \sum _{j'\ne i}^{I}\frac{\phi _{j'}^{n+1}}{I}\sum _{j=1}^{I}\left| E_{j'j}^{n:k}\right| \left| z_{\ell ,j}\right| \right) \nonumber \\&\quad \le 2\,c_{0}\,(\rho )^{\big \lfloor \frac{n-k+1}{(I-1)B}\big \rfloor }\left\| {\mathbf {z}}_{\ell }\right\| _{1}\le 2\,c_{0}\,(\rho )^{\big \lfloor \frac{n-k+1}{(I-1)B}\big \rfloor }\,\sqrt{I}\,\left\| {\mathbf {z}}_{\ell }\right\| _2. \end{aligned}$$

(112)

Combining (111) and (112) we obtain

$$\begin{aligned} \left\| \widehat{{\mathbf {W}}}^{n:k} - {\mathbf {J}}_{\varvec{\phi }^{k}}\right\| _{2} \le 2c_0 I (\rho )^{\big \lfloor \frac{n-k+1}{(I-1)B}\big \rfloor }. \end{aligned}$$

(113)

Moreover, the matrix difference above can be alternatively uniformly bounded as follows:

$$\begin{aligned} \left\| \widehat{{\mathbf {W}}}^{n:k} - {\mathbf {J}}_{\varvec{\phi }^{k}}\right\| = \Vert ({\mathbf {I}} - {\mathbf {J}}_{\varvec{\phi }^{n+1}}) \widehat{{\mathbf {W}}}^{n:k}\Vert \le \Vert {\mathbf {I}} - {\mathbf {J}}_{\varvec{\phi }^{n+1}}\Vert \Vert \widehat{{\mathbf {W}}}^{n:k}\Vert \overset{(a)}{\le } \sqrt{2I} \cdot \sqrt{I}, \end{aligned}$$

where (a) follows from (25) and $\Vert \widehat{{\mathbf {W}}}^{n:k}\Vert \le \sqrt{I}$. This completes the proof. $\square$

Proof of Lemma 11

Recall the SONATA update written in vector–matrix form in (43)–(45). Note that the x-update therein is a special case of the perturbed condensed push-sum algorithm (16), with perturbation $\varvec{\delta }^{n+1} = \alpha ^n \widehat{{\mathbf {W}}}^n {\mathbf {x}}^n$. We can then apply Proposition 1 and readily obtain (71).

To prove (72), we follow a similar approach: noticing that the y-update in (45) is a special case of (16), with perturbation $\varvec{\delta }^{n+1} = (\widehat{{\mathbf {D}}}_{\varvec{\phi }^{n+1}})^{-1}\left( {\mathbf {g}}^{n+1}-{\mathbf {g}}^{n}\right)$, we can write

$$\begin{aligned} \begin{aligned} \Vert {\mathbf {e}}_{y}^{n+{\bar{B}}}\Vert \le {}&\rho _{{\bar{B}}}\Vert {\mathbf {e}}_{y}^{n}\Vert + \sqrt{2} I \sum _{t = 0}^{{\bar{B}}-1} \Vert (\widehat{{\mathbf {D}}}_{\varvec{\phi }^{n+t+1}})^{-1}\left( {\mathbf {g}}^{n+t+1}-{\mathbf {g}}^{n+t}\right) \Vert \\ \le {}&\rho _{{\bar{B}}}\left\| {\mathbf {e}}_{y}^{n}\right\| + \sqrt{2} I\,L_{\mathrm{{mx}}}\phi _{lb}^{-1}\sum _{t=0}^{{\bar{B}}-1}\left\| \widehat{{\mathbf {W}}}^{n+t}({\mathbf {x}}^{n+t} + \alpha ^{n+t}\varDelta {\mathbf {x}}^{n+t}) - {\mathbf {x}}^{n+t}\right\| \\ \le {}&\rho _{{\bar{B}}}\left\| {\mathbf {e}}_{y}^{n }\right\| + \sqrt{2} I\,L_{\mathrm{{mx}}}\phi _{lb}^{-1}\sum _{t=0}^{{\bar{B}}-1}\left( \left\| \widehat{{\mathbf {W}}}^{n+t} {\mathbf {e}}_{x}^{n+t}\right\| + \left\| {\mathbf {e}}_{x}^{n+t}\right\| + \alpha ^{n+t}\left\| \widehat{{\mathbf {W}}}^{n+t}\varDelta {\mathbf {x}}^{n+t}\right\| \right) \\ \le {}&\rho _{{\bar{B}}}\left\| {\mathbf {e}}_{y}^{n}\right\| + \sqrt{2} I\,L_{\mathrm{{mx}}}\phi _{lb}^{-1}\sum _{t=0}^{{\bar{B}}-1}\left( (\sqrt{I} + 1) \left\| {\mathbf {e}}_{x}^{n+t}\right\| + \alpha ^{n+t}\sqrt{I}\left\| \varDelta {\mathbf {x}}^{n+t}\right\| \right) \\ \le {}&\rho _{{\bar{B}}}\left\| {\mathbf {e}}_{y}^{n}\right\| + I\sqrt{2I}\,L_{\mathrm{{mx}}}\phi _{lb}^{-1}\sum _{t=0}^{{\bar{B}}-1}\left( 2\left\| {\mathbf {e}}_{x}^{n+t}\right\| + \alpha ^{n+t}\left\| \varDelta {\mathbf {x}}^{n+t}\right\| \right) . \end{aligned} \end{aligned}$$

This completes the proof. $\square$