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Imitating from auxiliary imperfect demonstrations via Adversarial Density Weighted Regression

Ziqi Zhang, Zifeng Zhuang, Jingzehua Xu, Yiyuan Yang, Yubo Huang, Donglin Wang, Shuai Zhang

TL;DR

ADR tackles offline imitation learning by removing dependence on the Bellman operator and reward/Q-value estimation, using a one-step, density-weighted supervised objective. It combines Adversarial Density Estimation (ADE) to learn expert and sub-optimal behavior densities with Density Weighted Regression (DWR) to steer the policy toward the expert while avoiding poor demonstrations. The authors prove theoretical links between ADR's objective, policy convergence to the expert, and the resulting value bounds, and show strong empirical gains across Adroit, Kitchen, and Gym-Mujoco tasks, including substantial improvements over IQL when rewards are known. This approach offers a robust, sample-efficientIL framework for offline settings, reducing OOD risk and avoiding the tuning of conservatism in off-policy methods, with practical applicability to complex control domains.

Abstract

We propose a novel one-step supervised imitation learning (IL) framework called Adversarial Density Regression (ADR). This IL framework aims to correct the policy learned on unknown-quality to match the expert distribution by utilizing demonstrations, without relying on the Bellman operator. Specifically, ADR addresses several limitations in previous IL algorithms: First, most IL algorithms are based on the Bellman operator, which inevitably suffer from cumulative offsets from sub-optimal rewards during multi-step update processes. Additionally, off-policy training frameworks suffer from Out-of-Distribution (OOD) state-actions. Second, while conservative terms help solve the OOD issue, balancing the conservative term is difficult. To address these limitations, we fully integrate a one-step density-weighted Behavioral Cloning (BC) objective for IL with auxiliary imperfect demonstration. Theoretically, we demonstrate that this adaptation can effectively correct the distribution of policies trained on unknown-quality datasets to align with the expert policy's distribution. Moreover, the difference between the empirical and the optimal value function is proportional to the upper bound of ADR's objective, indicating that minimizing ADR's objective is akin to approaching the optimal value. Experimentally, we validated the performance of ADR by conducting extensive evaluations. Specifically, ADR outperforms all of the selected IL algorithms on tasks from the Gym-Mujoco domain. Meanwhile, it achieves an 89.5% improvement over IQL when utilizing ground truth rewards on tasks from the Adroit and Kitchen domains. Our codebase will be released at: https://github.com/stevezhangzA/Adverserial_Density_Regression.

Imitating from auxiliary imperfect demonstrations via Adversarial Density Weighted Regression

TL;DR

ADR tackles offline imitation learning by removing dependence on the Bellman operator and reward/Q-value estimation, using a one-step, density-weighted supervised objective. It combines Adversarial Density Estimation (ADE) to learn expert and sub-optimal behavior densities with Density Weighted Regression (DWR) to steer the policy toward the expert while avoiding poor demonstrations. The authors prove theoretical links between ADR's objective, policy convergence to the expert, and the resulting value bounds, and show strong empirical gains across Adroit, Kitchen, and Gym-Mujoco tasks, including substantial improvements over IQL when rewards are known. This approach offers a robust, sample-efficientIL framework for offline settings, reducing OOD risk and avoiding the tuning of conservatism in off-policy methods, with practical applicability to complex control domains.

Abstract

We propose a novel one-step supervised imitation learning (IL) framework called Adversarial Density Regression (ADR). This IL framework aims to correct the policy learned on unknown-quality to match the expert distribution by utilizing demonstrations, without relying on the Bellman operator. Specifically, ADR addresses several limitations in previous IL algorithms: First, most IL algorithms are based on the Bellman operator, which inevitably suffer from cumulative offsets from sub-optimal rewards during multi-step update processes. Additionally, off-policy training frameworks suffer from Out-of-Distribution (OOD) state-actions. Second, while conservative terms help solve the OOD issue, balancing the conservative term is difficult. To address these limitations, we fully integrate a one-step density-weighted Behavioral Cloning (BC) objective for IL with auxiliary imperfect demonstration. Theoretically, we demonstrate that this adaptation can effectively correct the distribution of policies trained on unknown-quality datasets to align with the expert policy's distribution. Moreover, the difference between the empirical and the optimal value function is proportional to the upper bound of ADR's objective, indicating that minimizing ADR's objective is akin to approaching the optimal value. Experimentally, we validated the performance of ADR by conducting extensive evaluations. Specifically, ADR outperforms all of the selected IL algorithms on tasks from the Gym-Mujoco domain. Meanwhile, it achieves an 89.5% improvement over IQL when utilizing ground truth rewards on tasks from the Adroit and Kitchen domains. Our codebase will be released at: https://github.com/stevezhangzA/Adverserial_Density_Regression.
Paper Structure (43 sections, 10 theorems, 29 equations, 9 figures, 5 tables, 2 algorithms)

This paper contains 43 sections, 10 theorems, 29 equations, 9 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Imitation Learning (IL).
  4. Behavior Policy Modeling.
  5. Preliminaries
  6. Reinforcement Learning (RL).
  7. Imitation Learning (IL).
  8. Behavior density estimation via Variance Auto-Encoder (VAE).
  9. Problem Formulation
  10. Notations.
  11. Policy Distillation via KL Divergence.
  12. Adversarial Density Regression (ADR).
  13. Density Weighted Regression (DWR).
  14. Theoretical Analysis of Adversarial Density Regression
  15. Policy Distribution Analysis.
  16. ...and 28 more sections

Key Result

Theorem 4.2

Given expert log behavior density $\log P^*(\mathbf{a}|\mathbf{s}):\mathcal{S}\times\mathcal{A}\rightarrow \mathbb{R}$, sub-optimal log behavior density $\log \hat{P}(\mathbf{a}|\mathbf{s}):\mathcal{S}\times\mathcal{A}\rightarrow \mathbb{R}$, and the empirical policy $\pi_{\theta}:\mathcal{S}\righta

Figures (9)

  • Figure 1: Blue path based on Bellman operator $\mathcal{B}$, the distance from the optimal policy varies with all iterations. Red path, the precise path to the optimal policy.
  • Figure 2: Policy Distribution. We sequentially sampled 500 samples $\tau_{\texttt{sampled}}=\{(\mathbf{s}_t,\mathbf{a}_t)|(\mathbf{s}_t,\mathbf{a}_t)\sim\mathcal{D}_{\texttt{exp}}\}_{t=0}^{t=500}$ from the expert dataset $\mathcal{D}_{\texttt{exp}}$. At the same time, we generated 500 actions based on the policy learned from ADR $\textit{i}.\textit{e}.,$$\tau_{\texttt{generate}}=\{\mathbf{a}_t:= \pi_{\theta}(\cdot|\mathbf{s}_t)|\mathbf{s}_t\in\tau_{\texttt{sampled}}\}$. Then, we reduced the dimensions of actions from all $\tau_{\texttt{sampled}}$ and $\tau_{\texttt{generate}}$ using t-SNE and plot the KDE curve.
  • Figure 3: Ablation Results. We utilized the reliable library proposed by agarwal2022deepreinforcementlearningedge to conduct our experiments. The results show that the experimental setting on the left side performed better with a higher probability. Specifically, in (a) we removed part of modules $\textit{i}.\textit{e}.,$ ADE or DWR from ADR and observed a reduction in performance. In (b), we further conducted comparisons among all tasks. Regarding (c), we carried out a fine-grained comparison of the upper and lower bounds of Equation \ref{['optimizing_objective']} among all tasks. Note, (a) The left and right y-axes represent the selected algorithms A and B, respectively, while the x-axis represents the confidence in A$>$B. (b, c) involve comparisons between two algorithms, and left y-axis indicates selected tasks.
  • Figure 4: ADR's performance changes as the noise in the demonstrations increases.
  • Figure 5: Heatmap of policy distributions. We stack the model's predictions alongside the samples in the dataset. The correlation is higher in the top-left and bottom-right regions, while it is lower in the other areas, the algorithm is less affected by OOD while maintain good performance (details see Appendix \ref{['heatmap_detail']}).
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 1
  • Remark 4.1
  • Definition 2: Adversarial Density Regression (ADR)
  • Theorem 4.2: Density Weight
  • Proposition 5.2: Policy Convergence of ADR
  • Proposition 5.3
  • Theorem D.1: Density Weight
  • Theorem D.3
  • Proposition D.4: Policy Convergence of ADR
  • Lemma D.5
  • ...and 3 more