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On the Benefits of Inducing Local Lipschitzness for Robust Generative Adversarial Imitation Learning

Farzan Memarian, Abolfazl Hashemi, Scott Niekum, Ufuk Topcu

TL;DR

This paper tackles robustness gaps in Generative Adversarial Imitation Learning (GAIL) under noisy observations by introducing LL-GAIL, a local Lipschitzness regularization framework applied to both the discriminator and the generator. It shows theoretically that local Lipschitzness of the discriminator propagates to a locally Lipschitz $Q^*$ and thus to a robust generator (policy), and it provides practical regularizers $R_d$ and $R_g$ to enforce this property. Empirically, LL-GAIL variants outperform standard GAIL and noise-augmented baselines on MuJoCo locomotion tasks, with robustness improving as the empirical local Lipschitz constant (ELLC) decreases. The work offers a principled, transferable approach to stabilizing and robustifying imitation learners in noise-prone robotic settings, with tunable locality via the radius $r_p$ and Lipschitz constants $L$.

Abstract

We explore methodologies to improve the robustness of generative adversarial imitation learning (GAIL) algorithms to observation noise. Towards this objective, we study the effect of local Lipschitzness of the discriminator and the generator on the robustness of policies learned by GAIL. In many robotics applications, the learned policies by GAIL typically suffer from a degraded performance at test time since the observations from the environment might be corrupted by noise. Hence, robustifying the learned policies against the observation noise is of critical importance. To this end, we propose a regularization method to induce local Lipschitzness in the generator and the discriminator of adversarial imitation learning methods. We show that the modified objective leads to learning significantly more robust policies. Moreover, we demonstrate -- both theoretically and experimentally -- that training a locally Lipschitz discriminator leads to a locally Lipschitz generator, thereby improving the robustness of the resultant policy. We perform extensive experiments on simulated robot locomotion environments from the MuJoCo suite that demonstrate the proposed method learns policies that significantly outperform the state-of-the-art generative adversarial imitation learning algorithm when applied to test scenarios with noise-corrupted observations.

On the Benefits of Inducing Local Lipschitzness for Robust Generative Adversarial Imitation Learning

TL;DR

This paper tackles robustness gaps in Generative Adversarial Imitation Learning (GAIL) under noisy observations by introducing LL-GAIL, a local Lipschitzness regularization framework applied to both the discriminator and the generator. It shows theoretically that local Lipschitzness of the discriminator propagates to a locally Lipschitz and thus to a robust generator (policy), and it provides practical regularizers and to enforce this property. Empirically, LL-GAIL variants outperform standard GAIL and noise-augmented baselines on MuJoCo locomotion tasks, with robustness improving as the empirical local Lipschitz constant (ELLC) decreases. The work offers a principled, transferable approach to stabilizing and robustifying imitation learners in noise-prone robotic settings, with tunable locality via the radius and Lipschitz constants .

Abstract

We explore methodologies to improve the robustness of generative adversarial imitation learning (GAIL) algorithms to observation noise. Towards this objective, we study the effect of local Lipschitzness of the discriminator and the generator on the robustness of policies learned by GAIL. In many robotics applications, the learned policies by GAIL typically suffer from a degraded performance at test time since the observations from the environment might be corrupted by noise. Hence, robustifying the learned policies against the observation noise is of critical importance. To this end, we propose a regularization method to induce local Lipschitzness in the generator and the discriminator of adversarial imitation learning methods. We show that the modified objective leads to learning significantly more robust policies. Moreover, we demonstrate -- both theoretically and experimentally -- that training a locally Lipschitz discriminator leads to a locally Lipschitz generator, thereby improving the robustness of the resultant policy. We perform extensive experiments on simulated robot locomotion environments from the MuJoCo suite that demonstrate the proposed method learns policies that significantly outperform the state-of-the-art generative adversarial imitation learning algorithm when applied to test scenarios with noise-corrupted observations.

Paper Structure

This paper contains 13 sections, 5 theorems, 39 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Insights on the Lipschitzness of the Discriminator and the Generator
  4. Lipschitz-Inducing Regularization for GAIL
  5. Inducing Local Lipschitzness in the Discriminator
  6. Inducing Local Lipschitzness in the Generator
  7. Experiments
  8. Conclusions
  9. Related Work
  10. Insights on Lipschitzness of the discriminator and the generator
  11. Lipschitzness of Optimal Q-function for One-Dimensional State Space
  12. Lipschitzness of Optimal Q-function, Multi-Dimensional State Space
  13. Additional Experiments

Key Result

theorem 1

Consider an infinite-horizon discounted MDP $\mathcal{M} = \langle S, A,T, r , \gamma \rangle$ where the reward function is $L$-Lipschitz continuous. Let $\nabla_{s_{t}^i}$ be the gradient operator with respect to the $i^{th}$ dimension of the state space at time $t$. If there exists a constant $C$ then, it holds that $\left\lVert\nabla_{s_t} Q^*(s_t, a_t)\right\rVert_F \leq \sqrt{N} \, L \sum_{k

Figures (4)

  • Figure 1: The comparison between LL-GAIL and the benchmarking schemes natural GAIL and noisy GAI on several simulated robot locomotion environments in the MuJoCo suite todorov2012mujoco. The figures show the generators learned by LL-GAIL methods (either LLD-GAIL or LLG-GAIL) are more robust to observation noise compared to the baselines, as the proposed regularization methods improve the empirical local Lipschitzness constant (ELLC) of the trained generators.
  • Figure 2: Walker2d experiment: LL-GAIL with both discriminator and generator regularizer outperforms all methods across various noise levels. $\gamma_g$ and $r^g_p$ are the hyper-parameters for regularizing the generator of LLDG-GAIL, and $\gamma_d$ and $r^d_p$ are the hyper-parameters for regularizing the discriminator of LLDG-GAIL.
  • Figure 3: The comparison between LL-GAIL and the benchmarking schemes natural GAIL and noisy GAIL on several simulated robot locomotion environments in the MuJoCo suite todorov2012mujoco when using $L_\infty$ norm.
  • Figure 4: LLDG-GAIL with both generator and discriminator regularized compared with LLG-GAIL with only the generator regularized and noisy GAIL and natural GAIL. $\gamma_g$ and $r^g_p$ are the hyper-parameters for regularizing the generator of LLDG-GAIL, and $\gamma_d$ and $r^d_p$ are the hyper-parameters for regularizing the discriminator of LLDG-GAIL.

Theorems & Definitions (11)

  • definition 1: Locally Lipschitz function
  • theorem 1
  • definition 2: Locally Lipschitz function
  • proposition 1
  • proof
  • theorem 2
  • proof
  • proposition 2
  • proof
  • theorem 3
  • ...and 1 more