# Learning to Branch

## TL;DR

This paper introduces a machine learning approach to optimize tree search algorithms by learning the best combination of partitioning procedures, significantly reducing tree size and improving efficiency with theoretical guarantees.

## Contribution

It presents the first theoretical sample complexity guarantees for tree search algorithm configuration and demonstrates a practical learning method to optimize branching.

## Key findings

- Learning optimal partitioning weights reduces tree size dramatically.
- The reduction in tree size can be exponential.
- The approach is both theoretically sound and practically effective.

## Abstract

Tree search algorithms, such as branch-and-bound, are the most widely used tools for solving combinatorial and nonconvex problems. For example, they are the foremost method for solving (mixed) integer programs and constraint satisfaction problems. Tree search algorithms recursively partition the search space to find an optimal solution. In order to keep the tree size small, it is crucial to carefully decide, when expanding a tree node, which question (typically variable) to branch on at that node in order to partition the remaining space. Numerous partitioning techniques (e.g., variable selection) have been proposed, but there is no theory describing which technique is optimal. We show how to use machine learning to determine an optimal weighting of any set of partitioning procedures for the instance distribution at hand using samples from the distribution. We provide the first sample complexity guarantees for tree search algorithm configuration. These guarantees bound the number of samples sufficient to ensure that the empirical performance of an algorithm over the samples nearly matches its expected performance on the unknown instance distribution. This thorough theoretical investigation naturally gives rise to our learning algorithm. Via experiments, we show that learning an optimal weighting of partitioning procedures can dramatically reduce tree size, and we prove that this reduction can even be exponential. Through theory and experiments, we show that learning to branch is both practical and hugely beneficial.