On the analysis of inexact augmented Lagrangian schemes for misspecified conic convex programs

TL;DR

This paper introduces an inexact augmented Lagrangian method that simultaneously learns unknown parameters and solves misspecified conic convex programs, demonstrating effective convergence and practical performance in portfolio optimization.

Contribution

It develops a first-order inexact augmented Lagrangian scheme that learns parameters while solving misspecified conic convex programs, with proven convergence rates and complexity bounds.

Findings

The proposed scheme converges at a quantifiable rate.

Numerical results show practical effectiveness in portfolio optimization.

Naive sequential schemes perform poorly compared to the joint learning approach.

Abstract

We consider the misspecified optimization problem of minimizing a convex function $f(x;\theta^*)$ in $x$ over a conic constraint set represented by $h(x;\theta^*) \in \mathcal{K}$, where $\theta^*$ is an unknown (or misspecified) vector of parameters, $\mathcal{K}$ is a closed convex cone and $h$ is affine in $x$. Suppose $\theta^*$ is unavailable but may be learnt by a separate process that generates a sequence of estimators $\theta_k$, each of which is an increasingly accurate approximation of $\theta^*$. We develop a first-order inexact augmented Lagrangian (AL) scheme for computing an optimal solution $x^*$ corresponding to $\theta^*$ while simultaneously learning $\theta^*$. In particular, we derive rate statements for such schemes when the penalty parameter sequence is either constant or increasing, and derive bounds on the overall complexity in terms of proximal-gradient steps when AL subproblems are inexactly solved via an accelerated proximal-gradient scheme. Numerical results for a portfolio optimization problem with a misspecified covariance matrix suggest that these schemes perform well in practice while naive sequential schemes may perform poorly in comparison.