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Learning responsibility allocations for multi-agent interactions: A differentiable optimization approach with control barrier functions

Isaac Remy, David Fridovich-Keil, Karen Leung

TL;DR

This work introduces a data-driven framework to learn how agents share responsibility for maintaining safety in multi-agent interactions using Control Barrier Functions (CBFs) and differentiable optimization. By defining a responsibility vector $\boldsymbol{\gamma}$ and a CBF-based safety filter, the authors formulate a bi-level, differentiable optimization problem to infer how much each agent should adjust its control to satisfy collision-avoidance constraints, from data. They further impose symmetry and relative-coordinate formulations to improve data efficiency and interpretability, and demonstrate results on synthetic multi-agent dynamics as well as real traffic-weaving data, revealing intuitive, interpretable responsibility allocations and highlighting limitations in multimodal settings. The approach enables offline analysis and potential online estimation of responsibility, offering a principled lens to evaluate socially-aware autonomous policies and guide robot control design. Key future directions include learning the desired-control policy from data, extending to probabilistic formulations to handle multimodal interactions, and leveraging inferred responsibilities to inform policy construction in real-world systems.

Abstract

From autonomous driving to package delivery, ensuring safe yet efficient multi-agent interaction is challenging as the interaction dynamics are influenced by hard-to-model factors such as social norms and contextual cues. Understanding these influences can aid in the design and evaluation of socially-aware autonomous agents whose behaviors are aligned with human values. In this work, we seek to codify factors governing safe multi-agent interactions via the lens of responsibility, i.e., an agent's willingness to deviate from their desired control to accommodate safe interaction with others. Specifically, we propose a data-driven modeling approach based on control barrier functions and differentiable optimization that efficiently learns agents' responsibility allocation from data. We demonstrate on synthetic and real-world datasets that we can obtain an interpretable and quantitative understanding of how much agents adjust their behavior to ensure the safety of others given their current environment.

Learning responsibility allocations for multi-agent interactions: A differentiable optimization approach with control barrier functions

TL;DR

This work introduces a data-driven framework to learn how agents share responsibility for maintaining safety in multi-agent interactions using Control Barrier Functions (CBFs) and differentiable optimization. By defining a responsibility vector and a CBF-based safety filter, the authors formulate a bi-level, differentiable optimization problem to infer how much each agent should adjust its control to satisfy collision-avoidance constraints, from data. They further impose symmetry and relative-coordinate formulations to improve data efficiency and interpretability, and demonstrate results on synthetic multi-agent dynamics as well as real traffic-weaving data, revealing intuitive, interpretable responsibility allocations and highlighting limitations in multimodal settings. The approach enables offline analysis and potential online estimation of responsibility, offering a principled lens to evaluate socially-aware autonomous policies and guide robot control design. Key future directions include learning the desired-control policy from data, extending to probabilistic formulations to handle multimodal interactions, and leveraging inferred responsibilities to inform policy construction in real-world systems.

Abstract

From autonomous driving to package delivery, ensuring safe yet efficient multi-agent interaction is challenging as the interaction dynamics are influenced by hard-to-model factors such as social norms and contextual cues. Understanding these influences can aid in the design and evaluation of socially-aware autonomous agents whose behaviors are aligned with human values. In this work, we seek to codify factors governing safe multi-agent interactions via the lens of responsibility, i.e., an agent's willingness to deviate from their desired control to accommodate safe interaction with others. Specifically, we propose a data-driven modeling approach based on control barrier functions and differentiable optimization that efficiently learns agents' responsibility allocation from data. We demonstrate on synthetic and real-world datasets that we can obtain an interpretable and quantitative understanding of how much agents adjust their behavior to ensure the safety of others given their current environment.

Paper Structure

This paper contains 28 sections, 1 theorem, 15 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem formulation
  4. Defining responsibility allocations
  5. Control barrier function (CBFs)
  6. CBF safety filters
  7. Responsibility allocations via a CBF safety filter lens
  8. Inferring responsibility from data
  9. Bi-level optimization formulation
  10. Leveraging differentiable optimization
  11. Context-dependent responsibility
  12. Enforcing symmetry in responsibility allocation
  13. Symmetry in relative coordinates for two-agent systems
  14. Experiments on synthetic dataset
  15. Experiments on traffic-weaving interaction
  16. ...and 13 more sections

Key Result

Proposition 1

Let $\widetilde{\phi}:\mathcal{X}^N \rightarrow \mathbb{R}$ be any (differentiable) function, e.g., a neural network. Then we define $\gamma_i$, Then $\boldsymbol{\gamma}(\mathbf{x}) = [\gamma_1(\mathbf{x}),...,\gamma_N(\mathbf{x})]$ satisfies the symmetric responsibility properties.

Figures (7)

  • Figure 1: In a) and b), two cars are swapping lanes on a highway, but their desired controls lead to collision. In c) and d), we see how the agents may deviate from their ideal trajectories, according to two different responsibility allocations. In this work we wish to solve the inverse problem: how do we infer these allocations from data?
  • Figure 2: Solutions to the two-agent CBF filter problem with different responsibility allocation values for Agent 1. The green and orange regions are the sets of feasible and infeasible controls, respectively.
  • Figure 3: Learning $\gamma$ with synthetically generated data. After a few epochs, the estimated responsibility allocation value $\gamma$ converges to the ground truth value. The computation time scales linearly as the batch size increases.
  • Figure 4: An example trajectory from the traffic weaving dataset. Two cars (red and blue rectangles) start in adjacent lanes, and need to change lanes safely to reach their lateral goals (stars).
  • Figure 5: Comparison of $\gamma$ landscapes for the red car using symmetric and unconstrained models. In the scenario, both cars want to change lanes (stars denote each agent's desired goal). The car in the lower lane is moving faster. The bottom row is the same scenario but with the agents swapped. The black arrows in the left plot represent each agent's desired control vector, and the red point in the contour plots corresponds to the depicted scenarios. The shaded regions denote relative states for which the CBF filter is not active.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1: Control Barrier Function (CBF) AmesGrizzleEtAl2014
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2: Responsibility allocation
  • Remark 4
  • Example 1: Two-agent 1D single integrator
  • Definition 3: Symmetric responsibility
  • Proposition 1
  • proof