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Online Dynamic Submodular Optimization

Antoine Lesage-Landry, Julien Pallage

TL;DR

The paper tackles online binary optimization with submodular objectives under general constraints by developing two algorithm families, OSGA and OSPGD, each with dynamic regret guarantees that depend on horizon length and optima variation. OSGA uses a $\beta$-approximation of the previous round's objective to achieve sublinear regret, while OSPGD leverages the Lovász extension and a single projected gradient step for computational efficiency, also delivering sublinear regret under suitable conditions. The authors provide theoretical analyses showing regret bounds that resemble the best known results in online convex optimization, and demonstrate practical effectiveness in power systems: fast-timescale demand response and real-time network reconfiguration. Collectively, these methods offer scalable, provable tools for real-time decisions in systems with submodular structure and binary decisions, with demonstrated improvements over baselines in simulated electrical networks.

Abstract

We propose new algorithms with provable performance for online binary optimization subject to general constraints and in dynamic settings. We consider the subset of problems in which the objective function is submodular. We propose the online submodular greedy algorithm (OSGA) which solves to optimality an approximation of the previous round loss function to avoid the NP-hardness of the original problem. We extend OSGA to a generic approximation function. We show that OSGA has a dynamic regret bound similar to the tightest bounds in online convex optimization with respect to the time horizon and the cumulative round optimum variation. For instances where no approximation exists or a computationally simpler implementation is desired, we design the online submodular projected gradient descent (OSPGD) by leveraging the Lovaśz extension. We obtain a regret bound that is akin to the conventional online gradient descent (OGD). Finally, we numerically test our algorithms in two power system applications: fast-timescale demand response and real-time distribution network reconfiguration.

Online Dynamic Submodular Optimization

TL;DR

The paper tackles online binary optimization with submodular objectives under general constraints by developing two algorithm families, OSGA and OSPGD, each with dynamic regret guarantees that depend on horizon length and optima variation. OSGA uses a -approximation of the previous round's objective to achieve sublinear regret, while OSPGD leverages the Lovász extension and a single projected gradient step for computational efficiency, also delivering sublinear regret under suitable conditions. The authors provide theoretical analyses showing regret bounds that resemble the best known results in online convex optimization, and demonstrate practical effectiveness in power systems: fast-timescale demand response and real-time network reconfiguration. Collectively, these methods offer scalable, provable tools for real-time decisions in systems with submodular structure and binary decisions, with demonstrated improvements over baselines in simulated electrical networks.

Abstract

We propose new algorithms with provable performance for online binary optimization subject to general constraints and in dynamic settings. We consider the subset of problems in which the objective function is submodular. We propose the online submodular greedy algorithm (OSGA) which solves to optimality an approximation of the previous round loss function to avoid the NP-hardness of the original problem. We extend OSGA to a generic approximation function. We show that OSGA has a dynamic regret bound similar to the tightest bounds in online convex optimization with respect to the time horizon and the cumulative round optimum variation. For instances where no approximation exists or a computationally simpler implementation is desired, we design the online submodular projected gradient descent (OSPGD) by leveraging the Lovaśz extension. We obtain a regret bound that is akin to the conventional online gradient descent (OGD). Finally, we numerically test our algorithms in two power system applications: fast-timescale demand response and real-time distribution network reconfiguration.
Paper Structure (15 sections, 41 equations, 4 figures, 1 algorithm)

This paper contains 15 sections, 41 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Online optimization
  4. Submodularity
  5. Greedy approaches
  6. Projected gradient descent-based approach
  7. Applications to electric power systems
  8. Demand response for frequency regulation
  9. Setting
  10. Numerical results
  11. Real-Time Network Reconfiguration
  12. Setting
  13. Weakly-meshed Approximation
  14. Numerical Results
  15. Conclusion

Figures (4)

  • Figure 1: Demand response with 15 loads
  • Figure 2: Time-averaged experimental dynamic regret for OSGA
  • Figure 3: Cumulative power losses (33b/1f)
  • Figure 4: Network reconfiguration (135b/8f)