# Joint Continuous and Discrete Model Selection via Submodularity

arXiv: https://arxiv.org/abs/2102.09029 · HTML: https://mlsanity.com/paper/2102.09029/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 17 Feb 2021.

## Builds on (most cited first, 7 of 7)

- [Restricted Strong Convexity Implies Weak Submodularity](https://mlsanity.com/paper/1612.00804/) (2016)
- [Robust Optimization over Multiple Domains](https://mlsanity.com/paper/1805.07588/) (2018)
- [Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains](https://mlsanity.com/paper/1606.05615/) (2016)
- [Submodular Functions: from Discrete to Continous Domains](https://mlsanity.com/paper/1511.00394/) (2015)
- [Optimal approximation for unconstrained non-submodular minimization](https://mlsanity.com/paper/1905.12145/) (2019)
- [Optimization Beyond Prediction: Prescriptive Price Optimization](https://mlsanity.com/paper/1605.05422/) (2016)
- [Robust Budget Allocation via Continuous Submodular Functions](https://mlsanity.com/paper/1702.08791/) (2017)

## Built on by (most cited first, 1 of 1)

- [Learning to Sparsify Stochastic Linear Bandits](https://mlsanity.com/paper/2605.10151/) (2026)

## Abstract (arXiv)

In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is specified in some discrete space, leading to difficult nonconvex optimization problems. In this paper, we connect the model selection problem with structure-promoting regularizers to submodular function minimization with continuous and discrete arguments. In particular, we leverage the theory of submodular functions to identify a class of these problems that can be solved exactly and efficiently with an agnostic combination of discrete and continuous optimization routines. We show how simple continuous or discrete constraints can also be handled for certain problem classes and extend these ideas to a robust optimization framework. We also show how some problems outside of this class can be embedded within the class, further extending the class of problems our framework can accommodate. Finally, we numerically validate our theoretical results with several proof-of-concept examples with synthetic and real-world data, comparing against state-of-the-art algorithms.