The Symbolic Interior Point Method

TL;DR

This paper introduces a symbolic interior point method that leverages algebraic decision diagrams to efficiently solve large-scale optimization problems with complex logical features, enhancing probabilistic modeling and decision-making.

Contribution

It presents a novel symbolic interior point algorithm that integrates ADDs and matrix-free techniques for efficient optimization with rich logical models.

Findings

Successfully handles models with millions of non-zero entries.

Demonstrates flexibility on decision making and compressed sensing tasks.

Achieves efficient approximate solutions using symbolic-numeric methods.

Abstract

A recent trend in probabilistic inference emphasizes the codification of models in a formal syntax, with suitable high-level features such as individuals, relations, and connectives, enabling descriptive clarity, succinctness and circumventing the need for the modeler to engineer a custom solver. Unfortunately, bringing these linguistic and pragmatic benefits to numerical optimization has proven surprisingly challenging. In this paper, we turn to these challenges: we introduce a rich modeling language, for which an interior-point method computes approximate solutions in a generic way. While logical features easily complicates the underlying model, often yielding intricate dependencies, we exploit and cache local structure using algebraic decision diagrams (ADDs). Indeed, standard matrix-vector algebra is efficiently realizable in ADDs, but we argue and show that well-known optimization methods are not ideal for ADDs. Our engine, therefore, invokes a sophisticated matrix-free approach. We demonstrate the flexibility of the resulting symbolic-numeric optimizer on decision making and compressed sensing tasks with millions of non-zero entries.