# Fast Semidifferential-based Submodular Function Optimization

@inproceedings{Iyer2013FastSS, title={Fast Semidifferential-based Submodular Function Optimization}, author={Rishabh K. Iyer and Stefanie Jegelka and Jeff A. Bilmes}, booktitle={ICML}, year={2013} }

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular minimization and…

## 96 Citations

Fast Semidierential-b ased Submodular Function Optimization: Extended Version

- Computer Science
- 2013

This work presents a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidierentials (sub- and super-dierential) and analyzes theoretical properties of the algorithms for minimization and maximization, and shows that many state-of-the-art maximization algorithms are special cases.

On Unconstrained Quasi-Submodular Function Optimization

- Computer ScienceAAAI
- 2015

This paper proposes two algorithms for unconstrained quasi-submodular function minimization and maximization, which satisfies weaker properties than submodularity but still enjoys favorable performance in optimization.

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

- Computer ScienceNIPS
- 2013

It is shown that the complexity of all three problems connected to machine learning depends on the "curvature" of the submodular function, and lower and upper bounds are provided that refine and improve previous results.

On Approximate Non-submodular Minimization via Tree-Structured Supermodularity

- Computer ScienceAISTATS
- 2015

This work addresses the problem of minimizing nonsubmodular functions where the supermodularity is restricted to tree-structured pairwise terms, and develops several practical algorithms to provide approximate and near-optimal solutions.

Fast Multi-Stage Submodular Maximization : Extended version

- Computer Science
- 2014

It is argued how the new multistage algorithmic framework for submodular maximization, called MultGreed, can be integrated with distributive algorithms for further optimization and performed very closely to the standard greedy algorithm given appropriate surrogate functions.

Fast Multi-Stage Submodular Maximization : Extended version

- Computer Science
- 2014

It is argued how the multi-stage framework for submodular maximization, called MULTGREED, performs very closely to the standard greedy algorithm given appropriate surrogate functions and how it can be integrated with distributive algorithms for further optimization.

Monotone Closure of Relaxed Constraints in Submodular Optimization: Connections Between Minimization and Maximization

- Computer ScienceUAI
- 2014

This work shows a relaxation formulation and simple rounding strategy that, based on the monotone closure of relaxed constraints, reveals analogies between minimization and maximization problems, and includes known results as special cases and extends to a wider range of settings.

Fast Multi-stage Submodular Maximization

- Computer ScienceICML
- 2014

It is shown that MULTGREED performs very closely to the standard greedy algorithm given appropriate surrogate functions and it is argued how the framework can easily be integrated with distributive algorithms for further optimization.

Algorithms for Optimizing the Ratio of Submodular Functions

- Computer ScienceICML
- 2016

It is shown that RS optimization can be solved with bounded approximation factors and a hardness bound is provided and the tightest algorithm matches the lower bound up to a log factor.

A Unified Framework of Robust Submodular Optimization

- Computer ScienceArXiv
- 2019

A unified framework of robust submodular optimization under a broad range of combinatorial constraints including cardinality, knapsack, matroid as well as graph based constraints such as cuts, paths, matchings and trees is studied.

## References

SHOWING 1-10 OF 56 REFERENCES

Fast Semidierential-b ased Submodular Function Optimization: Extended Version

- Computer Science
- 2013

This work presents a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidierentials (sub- and super-dierential) and analyzes theoretical properties of the algorithms for minimization and maximization, and shows that many state-of-the-art maximization algorithms are special cases.

Efficient Minimization of Decomposable Submodular Functions

- Mathematics, Computer ScienceNIPS
- 2010

This paper develops an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables, and applies it to synthetic benchmarks and a joint classification-and-segmentation task, and shows that it outperforms the state-of-the-art general purpose sub modular minimization algorithms by several orders of magnitude.

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

- Mathematics, Computer Science2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

This work presents a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al.

SFO: A Toolbox for Submodular Function Optimization

- Computer ScienceJ. Mach. Learn. Res.
- 2010

SFO is presented, a toolbox for use in MATLAB or Octave that implements algorithms for minimization and maximization of submodular functions that allows one to efficiently find provably (near-) optimal solutions for large problems.

Mirror Descent-Like Algorithms for Submodular Optimization

- Computer Science
- 2012

A framework of submodular optimization algorithms in line with the mirror-descent style of algorithms for convex optimization is developed, using the fact that a sub modular function has both a subdifferential and a superdifferential to formulate algorithms for both submodularity minimization and maximization.

On fast approximate submodular minimization

- Computer Science, MathematicsNIPS
- 2011

A fast approximate method to minimize arbitrary submodular functions and shows theoretical properties, and empirical results suggest significant speedups over minimum norm while retaining higher accuracies.

Size-constrained Submodular Minimization through Minimum Norm Base

- Computer ScienceICML
- 2011

This paper discusses the submodular minimization under a size constraint, which is NP-hard, and generalizes the densest subgraph problem and the uniform graph partitioning problem and computes optimal solutions for some of possible size constraints in polynomial time.

Learning with Submodular Functions: A Convex Optimization Perspective

- Computer ScienceFound. Trends Mach. Learn.
- 2013

In Learning with Submodular Functions: A Convex Optimization Perspective, the theory of submodular functions is presented in a self-contained way from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems.

A Submodular-supermodular Procedure with Applications to Discriminative Structure Learning

- Computer ScienceUAI
- 2005

This paper presents an algorithm for minimizing the difference between two submodular functions using a variational framework which is based on (an extension of) the concave-convex procedure, and gives a polynomial time heuristic for it.

Approximating submodular functions everywhere

- Mathematics, Computer ScienceSODA
- 2009

The problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere is considered, after only poly(n) oracle queries, and it is shown that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid.