Discretized Distributed Optimization over Dynamic Digraphs

TL;DR

This article provides an efficient algorithm to enable MASs to collaboratively optimize a cost function, e.g., for binary classification and distributed support vector machine (D-SVM) and shows this efficiency of the distributed optimization method by simulation.

Abstract

We consider a discrete-time model of continuous-time distributed optimization over dynamic directed-graphs (digraphs) with applications to distributed learning. Our optimization algorithm works over general strongly connected dynamic networks under switching topologies, e.g., in mobile multi-agent systems and volatile networks due to link failures. Compared to many existing lines of work, there is no need for bi-stochastic weight designs on the links. The existing literature mostly needs the link weights to be stochastic using specific weight-design algorithms needed both at the initialization and at all times when the topology of the network changes. This paper eliminates the need for such algorithms and paves the way for distributed optimization over time-varying digraphs. We derive the bound on the gradient-tracking step-size and discrete time-step for convergence and prove dynamic stability using arguments from consensus algorithms, matrix perturbation theory, and Lyapunov theory. This work, particularly, is an improvement over existing stochastic-weight undirected networks in case of link removal or packet drops. This is because the existing literature may need to rerun time-consuming and computationally complex algorithms for stochastic design, while the proposed strategy works as long as the underlying network is weight-symmetric and balanced. The proposed optimization framework finds applications to distributed classification and learning.

Figures (5)

• Figure 1: The perturbation analysis to bound $d(\sigma(M),\sigma(M_0))$: the min eigenvalue (in absolute value) of $M_0$ may move towards RHP if $d(\sigma(M),\sigma(M_0))>\underline{\lambda}$.
• Figure 2: (Left)-An undirected unreliable network of $5$ nodes which is both stochastic and WB. The red link represents an unreliable link that might be subject to failure. (Right)-The same network after removal/failure of the (bi-directional) unreliable link: the network is still WB, but is not bi-stochastic anymore. To apply the existing weight-stochastic literature, e.g., khan_ABadd-optpushpull_nedic, one needs to redesign the weights, e.g., by weight compensation algorithms in 6426252cons_drop_siam, to redesign the weights if possible and make the new topology weight-stochastic again.
• Figure 3: This figure shows the data points scattered in 2D and classified by the SVM hyperplane in 3D.
• Figure 4: The SVM parameters ${\omega}_i$ and $\nu_i$ (at all $n=5$ nodes in Fig. \ref{['fig_remov']}) under discrete dynamics \ref{['eq_xydot']}-\ref{['eq_M']}. The top row represents the parameters before link removal and the bottom row is for after link removal. The gradient sum $\sum_{i=1}^{n} \boldsymbol{ \nabla} f_i(\mathbf{x}_i)$ is also shown to admit the GT property. The hinge loss residual $\overline{F}(\mathbf{x}) = \sum_{i=1}^n f_i(\mathbf{x}_i) - F^*$ (with $F^*$ obtained via centralized SVM) is also shown.
• Figure 5: The D-SVM residual (at all $n=6$ nodes) under discrete dynamics \ref{['eq_xydot']}-\ref{['eq_M']}, where the weight matrices are time-varying (random-weighting at every 100 iterations) over a 2-hop SC digraph. The hinge loss residual $\overline{F}$ are compared for different $\alpha,\eta$ values. This example shows that for large values of both $\alpha=5,\eta=0.005$ solution is not stable.

Theorems & Definitions (15)

• Remark 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• Theorem 1
• proof
• Remark 2
• Lemma 5