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Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability

William Hofgard

TL;DR

This work shows that the DeepReach algorithm, as introduced by Bansal et al. in their eponymous paper from 2020, is stable in the sense that if the loss functional for the algorithm converges to zero, then the resulting neural network approximation converges uniformly to the classical solution of the HJI equation.

Abstract

We provide a novel uniform convergence guarantee for DeepReach, a deep learning-based method for solving Hamilton-Jacobi-Isaacs (HJI) equations associated with reachability analysis. Specifically, we show that the DeepReach algorithm, as introduced by Bansal et al. in their eponymous paper from 2020, is stable in the sense that if the loss functional for the algorithm converges to zero, then the resulting neural network approximation converges uniformly to the classical solution of the HJI equation, assuming that a classical solution exists. We also provide numerical tests of the algorithm, replicating the experiments provided in the original DeepReach paper and empirically examining the impact that training with a supremum norm loss metric has on approximation error.

Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability

TL;DR

This work shows that the DeepReach algorithm, as introduced by Bansal et al. in their eponymous paper from 2020, is stable in the sense that if the loss functional for the algorithm converges to zero, then the resulting neural network approximation converges uniformly to the classical solution of the HJI equation.

Abstract

We provide a novel uniform convergence guarantee for DeepReach, a deep learning-based method for solving Hamilton-Jacobi-Isaacs (HJI) equations associated with reachability analysis. Specifically, we show that the DeepReach algorithm, as introduced by Bansal et al. in their eponymous paper from 2020, is stable in the sense that if the loss functional for the algorithm converges to zero, then the resulting neural network approximation converges uniformly to the classical solution of the HJI equation, assuming that a classical solution exists. We also provide numerical tests of the algorithm, replicating the experiments provided in the original DeepReach paper and empirically examining the impact that training with a supremum norm loss metric has on approximation error.
Paper Structure (9 sections, 6 theorems, 67 equations, 6 figures)

This paper contains 9 sections, 6 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mathematical Background
  4. HJ Reachability and DeepReach
  5. Baseline Collision Avoidance Application
  6. Theoretical Guarantees
  7. Numerical Experiments
  8. Conclusions and Future Work
  9. Technical Proofs

Key Result

Theorem 4.1

For every $\varepsilon > 0$, there exists a constant $C > 0$ that depends only on the dynamics $\dot{x} = f(x, u, d)$ of the underlying reachability problem such that for some $\theta \in \mathbb{R}^P$, the DeepReach loss in Equation eq:nn-loss satisfies $L(\theta) \leq C\varepsilon$.

Figures (6)

  • Figure 1: Absolute difference between true and approximate value functions, using pre-trained DeepReach model from deepreach.
  • Figure 2: Comparison of BRT obtained by pre-trained DeepReach model and analytical BRT from analytic-brt. Note the discrepancy between the two sets at $\theta = \pi$.
  • Figure 3: BRT comparison, using pre-trained model from deepreach.
  • Figure 4: BRT comparison, using fine-tuned model with sup-norm loss.
  • Figure 5: Fine-tuning with sup-norm loss results in significant decreases in DeepReach loss over just $500$ epochs.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Remark 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • proof : Proof of Theorem \ref{['th:existence']}
  • Definition A.3
  • ...and 6 more