Differentiable Submodular Maximization

TL;DR

This paper introduces a method to jointly learn and optimize submodular functions by making the maximization process differentiable, enabling end-to-end training for applications like recommendation and summarization.

Contribution

It presents a novel approach to integrate learning and optimization of submodular functions through differentiable algorithms, bridging a gap in existing methods.

Findings

Effective joint learning and optimization demonstrated on synthetic and real data.

Theoretical analysis of the tradeoff between smoothness and accuracy.

Improved performance in applications like product recommendation and image summarization.

Abstract

We consider learning of submodular functions from data. These functions are important in machine learning and have a wide range of applications, e.g. data summarization, feature selection and active learning. Despite their combinatorial nature, submodular functions can be maximized approximately with strong theoretical guarantees in polynomial time. Typically, learning the submodular function and optimization of that function are treated separately, i.e. the function is first learned using a proxy objective and subsequently maximized. In contrast, we show how to perform learning and optimization jointly. By interpreting the output of greedy maximization algorithms as distributions over sequences of items and smoothening these distributions, we obtain a differentiable objective. In this way, we can differentiate through the maximization algorithms and optimize the model to work well with the optimization algorithm. We theoretically characterize the error made by our approach, yielding insights into the tradeoff of smoothness and accuracy. We demonstrate the effectiveness of our approach for jointly learning and optimizing on synthetic maximum cut data, and on real world applications such as product recommendation and image collection summarization.