Stochastic Mirror Descent for Convex Optimization with Consensus Constraints

TL;DR

The findings show that the proposed method outperforms other methods, especially in scenarios where the model's geometry is not captured by the standard Euclidean norm, especially in scenarios where the model's geometry is not captured by the standard Euclidean norm.

Abstract

The mirror descent algorithm is known to be effective in situations where it is beneficial to adapt the mirror map to the underlying geometry of the optimization model. However, the effect of mirror maps on the geometry of distributed optimization problems has not been previously addressed. In this paper we study an exact distributed mirror descent algorithm in continuous-time under additive noise. We establish a linear convergence rate of the proposed dynamics for the setting of convex optimization. Our analysis draws motivation from the Augmented Lagrangian and its relation to gradient tracking. To further explore the benefits of mirror maps in a distributed setting we present a preconditioned variant of our algorithm with an additional mirror map over the Lagrangian dual variables. This allows our method to adapt to both the geometry of the primal variables, as well as to the geometry of the consensus constraint. We also propose a Gauss-Seidel type discretization scheme for the proposed method and establish its linear convergence rate. For certain classes of problems we identify mirror maps that mitigate the effect of the graph's spectral properties on the convergence rate of the algorithm. Using numerical experiments we demonstrate the efficiency of the methodology on convex models, both with and without constraints. Our findings show that the proposed method outperforms other methods, especially in scenarios where the model's geometry is not captured by the standard Euclidean norm

Figures (7)

• Figure 1: Vector fields of the gradient of the dual function in \ref{['eq:intro dual function']} for (a) a complete graph and (b) an incomplete graph. The two graphs are shown inset in the left-hand side corners of (a) and (b) respectively (for details see \ref{['example intro']}). In (a) the Laplacian of the graph is well-conditioned, in (b) the Laplacian is ill-conditioned. An example solution trajectory, obtained using an explicit discretization scheme, is shown in red for both cases. The explicit discretization scheme leads to oscillatory behavior and to eliminate it a small step-size needs to be used resulting in a slow convergence rate.
• Figure 2: Stochastic Mirror Descent with two mirror maps. $\Phi$ maps the primal variables to the dual space, and $\Psi$ maps the Lagrangian dual variables associated with the consensus constraint to the dual of the Lagrange multipliers.
• Figure 3: The algorithms convergence rate without preconditioning $Q=R=I$. When the graph is well-conditioned ($\overline\sigma^2/\underline\sigma^2=1$) the convergence rate of the algorithm (solid blue line) is dominated by the problem's condition number $L_f/\mu_f$. When the graph is ill-conditioned ($\overline\sigma^2/\underline\sigma^2=3$) the convergence rate (dotted red line) is dominated by the spectral properties of the graph up to a certain condition number (in this case $L_f/\mu_f\approx 4.5$), after this point the convergence rate is again dominated by the model's condition number.
• Figure 4: Different types of graphs considered in the numerical experiments.
• Figure 5: Numerical comparisons of distributed optimization algorithms for unconstrained problems. (a) When both the problem and graph are well-conditioned the proposed method performs similarly to other methods. (b-d) However when the model and/or graph are ill-conditioned then the proposed algorithm provides a significant improvement over other methods.

Theorems & Definitions (28)

• Example 1.1
• Remark 1.2
• Definition 3.1
• Definition 3.2: Relative strong convexity
• Definition 3.3: Relative smoothness
• Definition 3.4: Convex conjugate
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8