On analysis of open optimization algorithms

Abstract

We consider optimization algorithms that are open systems, that is, with external inputs and outputs. Such algorithms arise for instance, when analyzing the effect of noise or disturbance on an algorithm, or when an algorithm is part of control loop without timescale separation. Bridging between monotone operator theory and energy-based modeling, we consider analysis results in the form of incremental dissipativity certificates, yielding tests in the form of linear matrix inequalities. To be precise, we consider robustness in terms of incremental small gain, and composition results for optimization algorithms operating in closed loop.

Figures (3)

• Figure 1: A schematic view of an open optimization algorithm. Here, $\Sigma$ denotes the linear system \ref{['eq:linear system']}.
• Figure 2: The minimal open-loop contraction rate $\gamma$ plotted against the minimal strict incremental passivity index $\rho$ (cf. \ref{['eq:strict-inc-passive']}) for a sweep of 18000 sets of parameters $\alpha,\beta,\eta$. Also shown are a number of special cases labeled as in Table \ref{['table:nominal-algos']}.
• Figure 3: The values of the incremental storage functions of the plant $V_{\textup{p}}$, of the algorithm $V_{\textup{o}}$, and their sum for a randomly initiated pair of trajectories. For the simulation we chose the oracle equal to $u_k = \phi(y_k) = y_k+5\tanh(y_k)+2$. Note that the 'energy' in the plant or the algorithm does not decrease uniformly, that is, there is energy exchange between the plant and the algorithm.

Theorems & Definitions (9)

• Example 1: Nesterov acceleration
• Example 2: Illustrative example
• Example 3: Open Nesterov with gradient noise
• Lemma IV.1
• Definition IV.2
• Lemma IV.3
• Lemma V.1
• Theorem V.2
• Remark V.3