Randomized Greedy Methods for Weak Submodular Sensor Selection with Robustness Considerations

TL;DR

This work tackles sensor selection under budget and performance constraints when the objective is weakly submodular, introducing Modified Randomized Greedy (MRG) and Dual Randomized Greedy (DRG) with high-probability guarantees, along with Random-WSSA for robust multi-objective optimization. The algorithms rely on random sampling to reduce computation while maintaining provable performance bounds, using a martingale framework to analyze the stochastic approximations. The Random-WSSA extension enables worst-case robustness across multiple weak submodular objectives, with additional truncation and averaging arguments to preserve weak submodularity. Empirical demonstrations on Earth-observing LEO satellite constellations show substantial runtime reductions over full Greedy/SSA with competitive or near-optimal performance, highlighting practical impact for scalable, reliable autonomous sensor selection in large satellite networks.

Abstract

We study a pair of budget- and performance-constrained weak submodular maximization problems. For computational efficiency, we explore the use of stochastic greedy algorithms which limit the search space via random sampling instead of the standard greedy procedure which explores the entire feasible search space. We propose a pair of stochastic greedy algorithms, namely, Modified Randomized Greedy (MRG) and Dual Randomized Greedy (DRG) to approximately solve the budget- and performance-constrained problems, respectively. For both algorithms, we derive approximation guarantees that hold with high probability. We then examine the use of DRG in robust optimization problems wherein the objective is to maximize the worst-case of a number of weak submodular objectives and propose the Randomized Weak Submodular Saturation Algorithm (Random-WSSA). We further derive a high-probability guarantee for when Random-WSSA successfully constructs a robust solution. Finally, we showcase the effectiveness of these algorithms in a variety of relevant uses within the context of Earth-observing LEO constellations which estimate atmospheric weather conditions and provide Earth coverage.

Figures (5)

• Figure 1: It is often the case that only some of the satellites in a constellation are actively observing Earth at any given time. Furthermore, these satellites may be leveraged for multi-task missions.
• Figure 2: Locations of the $25$ randomly-selected atmospheric points of interest.
• Figure 3: Total mean-square errors (MSE) over time for various $r_{i}$ and $B$. From top to bottom are the plots for $B=25$, $50$, $75$, and $100$, respectively. Performance of using the entire set of satellites as well as that of picking the highest elements in terms of marginal-gain-to-cost ratio with respect to the empty set until the budget constraint is met (which we name Top-K) are included as benchmark values. Note that $r_i =240$ corresponds to using the entire ground set, i.e., the standard Greedy algorithm.
• Figure 4: Visualization of the ground coverage areas obtained at time step $50$ through DRG (Algorithm \ref{['alg:dg']}) using $r_{i}=120$ and various $CF$.
• Figure 5: Total utility over time for various $r_{i}$ and $B$ for the Random-WSSA. From top to bottom are the plots for $B=10$, $15$, and $20$, and $100$, respectively. The results are put through a moving average filter with a window size of $10$ for better visibility. Note that $r_i =240$ corresponds to using the entire ground set, i.e., the standard SSA.

Theorems & Definitions (17)

• Definition 1: Monotone nondecreasing
• Definition 2: Submodularity
• Definition 3: Weak-submodularity constant
• Proposition 1
• Definition 4: Martingale
• lemma 1: Azuma-Hoeffding inequality chung2006concentration
• Theorem 1: Performance of MRG
• proof
• Theorem 2: Performance of DRG
• proof