Provable Guarantees for Generative Behavior Cloning: Bridging Low-Level Stability and High-Level Behavior

Summary

The paper presents a simple hierarchical approach for non-Markovian imitation learning along with a rigorous theoretical analysis. The method trains a high level policy, given by a Denoising Diffusion Probabilistic Model (DDPM), to predict a sequence of linear control gains based on a state-action history. The DDPM is trained in supervised fashion (using the standard DDPM loss) where the labels (the controller gains) are computed from the given expert trajectories using an oracle (e.g. based on linearizations along the trajectory). To address covariate shift (compounding errors) in imitation learning the method applies a form of data augmentation by adding noise to the state-action history. In contrast to prior work, these perturbations are also applied during inference, which is theoretically motivated and empirically shown to improve imitation performance. The main contributions seems to be the thorough analysis of this method, which leads to Theorem 1, a bound on the imitation performance. The applicability of the proposed method is demonstrated on simulated robot tasks, although no comparisons with prior work are provided.

Strengths

Novelty & Relevance ------------------- Both the method and in particular the analysis seem to be novel and non-Markovian imitation learning is an important field of research. I would not expect the proposed method to outperform other (theoretically less well understood) methods, but I do think that the general approach of using a diffusion model as a high-level policy to predict controller gains could be used by future work that is more focused on practical performance. Furthermore, the theoretical analysis could be valuable for theorists in the field of imitation learning. However, I am not very familiar with related theoretical work and cannot assess well, which aspects/techniques of the analysis might be most relevant. Soundness --------- The algorithm seems sounds. I guess one could argue that it might be simpler and more stable if the higher-level policy would predict a trajectory segment instead of the gains, given that we assume access to an oracle that can provide us with stabilizing gains. However, querying the oracle could be too costly, so directly predicting the gains can be sensible. I cannot fully confirm the accuracy of the theoretical results because verifying the complete derivations would take significant effort. However, the claims (including Theorem 1) seem reasonable, apart from non-critical issues (see Questions below). Clarity ------- The paper is well-written. The paper is very strict in terms of notation, definitions, and statements, which increases clarity by reducing ambiguities.

Weaknesses

Clarity ------- Although I mentioned the rigor as a strength, it also feels a bit pedantic. Furthermore, it is cumbersome to keep track of the different constants and definitions making the paper hard to read and making it easy to get lost in details. Evaluation ---------- Despite the fact that the contributions are mainly in the theoretical analysis, quantitative comparisons with alternate methods that have been used in these environments would be useful. I don't know which prior work was used in that environment that also provided code, but I assume it should be possible to find suitable baselines that can be easily tested.

Questions

Line 216 states that "each term in (3.2) can be made arbitrarily small by decreasing the amount of noise $\sigma$ in the augmentation [...]". Isn't the first term anti-proportional to the noise? Line 274: "all sequences $s'\_1 = s\_1$ and $s'\_{h+1} = s\_{h+1}$ [...]". I'm wondering if this should be "all sequences $s'\_h = s\_h$ and $s'\_{h+1} = s\_{h+1}$ [...]"!? Line 325: "We let the expert policy $\pi^*$ be the concatenation of policies $\pi\_h^*$ [...]". Is this an additional assumption? Clearly, the expert will in general not apply this particular form of a hierarchical policy.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

I think that the limitations are sufficiently clear.

Summary

This paper proposes a theoretical framework for studying the imitation of expert demonstrations in nonlinear dynamical systems. The framework guarantees accurate imitation of expert trajectories by invoking low-level controllers and a stochastic continuity property of the learned policy. The authors also introduce an algorithmic trick for ensuring total variation continuity, which combines a popular data-augmentation regimen with augmentation noise at execution time. The paper includes a detailed proof of the framework and empirical validation of the proposed augmentation strategy in simulated robotic manipulation tasks

Strengths

1. The paper proposes a novel theoretical framework for studying the imitation of expert demonstrations in nonlinear dynamical systems. 2. The framework guarantees accurate imitation of expert trajectories by invoking low-level controllers and a stochastic continuity property of the learned policy. 3. The authors introduce a novel algorithmic trick for ensuring total variation continuity, which combines a popular data-augmentation regimen with augmentation noise at execution time. 4. The paper includes a detailed proof of the framework and establishes stability guarantees for sequences of primitive controllers in non-linear control systems. 5. The proposed augmentation strategy is empirically validated in simulated robotic manipulation tasks. 6. The paper provides a comprehensive review of related work in the field of imitation learning and compares the proposed framework to past work in the appendix.

Weaknesses

1. The paper relies on either synthesized primitive controllers or low-level stabilizing controllers built into problem environments to take advantage of local stability. Developing a more comprehensive approach to stability (perhaps one that does not require explicit gain synthesis and extends to non-smooth systems) is an exciting direction for future work. 2. The paper focuses on simulated robotic manipulation tasks, and it is unclear how well the proposed framework would generalize to other domains or real-world scenarios. 3. The paper does not provide a detailed analysis of the computational complexity of the proposed algorithmic trick for ensuring total variation continuity. 4. The paper does not compare the proposed framework to state-of-the-art methods in the field of imitation learning on benchmark datasets.

Questions

1. I think it is still important to comprehensive study of this method. What’s the influence of hyperparameters to this method? Should there be any ablation study included in this paper? 2. What’s the computational complexity of the proposed algorithmic trick for ensuring total variation continuity?

Rating

3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

3 good

Contribution

2 fair

Limitations

1. The paper assumes that the low-level controllers are either learned or implicit in position-command control, which may not always be the case in practice. 2. The paper does not provide a detailed analysis of the effect of hyperparameters on the performance of the proposed framework. 3. The paper does not consider the effect of imperfect state estimation or perception on the accuracy of the imitation policy. 4. The paper does not address the issue of exploration in the imitation learning setting, which is an important problem in reinforcement learning.

Summary

The paper introduces an imitation learning framework for handling multimodal and non-Markovian demonstrations. This is done by using a hierarchy where rather than predicting low-level controls directly, the learner predicts sequences of primitive controllers. These controllers are time-varying affine mappings from states to control inputs. The authors show that this approach allows us to reformulate our task as imitation in a composite MDP, where composite states and actions refer to subsequences of trajectories and primitive controllers respectively. To handle multimodality, they use diffusion models (DDPM) to parameterize the learner. The primary contribution of the paper is to bound the loss of the proposed approach (TODA), relying primarily on the stability of the lower-level controllers. Some empirical results are demonstrated on some robot manipulation tasks, where the proposed algorithm outperforms the baseline policy.

Strengths

The high-level motivation for the approach is sound and the practical algorithm seems to perform well. The paper is comprehensive and filled with technical content and derivations. I went through the analysis in the main body of the paper and did not find any glaring errors, although certain parts were hard to parse (see below).

Weaknesses

Honestly I have a hard time following Section 4. More detailed questions are below. Given that the crux of the contribution lies in this theoretical analysis, I'm more than happy to adjust my score post-rebuttal if I can get some things cleared up. Training a diffusion model to use low-level controllers as primitives as not new [1]. Would have preferred a more comprehensive experimental evaluation with competitive baselines, but given the theoretical contribution I suppose this isn't fully necessary. The proposed approach heavily relies on sufficiently smooth dynamics and the existence of an oracle which can synthesize primitive controllers given trajectories. This is fairly restrictive and prevents this work from being applicable to many existing control/robotics applications. Given that the contribution is primarily theoretical, this isn't a major issue for me. [1] Ajay, Anurag, et al. "Is conditional generative modeling all you need for decision-making?." arXiv preprint arXiv:2211.15657 (2022).

Questions

What is $d_{\mathcal{A}}$? Not the formal definition, but conceptually what is it? What is the purpose of the replica policy? I'm not sure I understand how this part of the proof functions. Is the smoothing kernel $W_{\sigma}$ the same as the data augmentation function? I don't understand L304-307. Typos: L52, "... which is both both ..." L60, "Wassertstein-like" L106, "... we assume that (2.1) deterministic ..." L342, "subtley" === I have read the rebuttal and bumped my score up slightly.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

1: Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

Yes

Summary

This is a theoretical paper that studies imitation learning from demonstrations that exhibit multi-modality. This method separate the demonstrations into chunks, then uses diffusion models to learn a policy conditioned on the memory of hindsight prefix of the trajectory to output an affine time-varying controller. Gaussian noises are added to the demonstrations for data augmentation, and in the learned controllers' outputs to enhance performance.

Strengths

Originality: this paper is original in proposing an algorithm to learn hierarchical policy by leveraging diffusion models and injected noises from potentially non-markovian expert demonstrations. Quality: this paper has strong theoretical support. Clarity: the algorithms and theories are clearly explained. Significance: the paper provides a new insight to solving non-markovian imitation learning problems by combining trajectory optimization tools, distribution learning in a hierarhical manner.

Weaknesses

In the experiment, three benchmarks are utilized. However, it is not mentioned in the maintext nor in the appendix how non-markovian and multi-modal the demonstrations are. As a result, the validity of the proposed algorithm is less convincible despite all the theoretical guarantees. Also, the symbols in this paper are too complicated to parse. There seem to be some symbols misusage in some of the formulas, making those formulas confusing. Please refer to the question section for the those potential errors.

Questions

1. In line 5 of Algorithm 1, shouldn't the mean of \gamma_h be 0 instead of a_h? In Eq.3.1, \gamma_h is sampled from D to compose the a DDPM loss function of which the update rule matches Eq 2.2. Then, \gamma_h corresponds to the \gamma in line 155, which is a standard Gaussian. 2. If \gamma_h indeed has its mean equal to a_h, how to understand the sum in Eq3.1? Why adding such noised action to e^{-\alpha j} a_h? 3. If \gamma_h has its mean equal to 0, why matching s_\theta,h(...) with \gamma_h, a Gaussian noise? 4. What do ...\vee... \leq max ... mean in line 275? Applying \vee to real values?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The author mentioned the limitation of the proposed approach in the paper. The limitation, however, does not undermine the quality and contribution of this paper.

Summary

This work proposes a theoretical framework for learning sequential decision-making with diffusion models. The authors show that with the conditions of (1) a suitable low-level "stability" guarantee and (2) a stochastic "continuity" property of the learned policy, the imitator can provably accurately estimate the "non-Markovian" policy of the demonstrators. The low-level "stability" is assumed in oracle primitive controllers, which are then composed by diffusion models that are learned with DDPM loss. Since the desired stochastic continuity is not guaranteed for an arbitrary demonstrator policy, the authors then propose a data augmentation technique that adds noise to the trajectories, making the resulting policies continuous. Their experiments validate that the proposed data augmentation alleviates imitator policy's degradation in noisy environments.

Strengths

Originality: I am not very familiar with the current literature of learning theory. But proving the learnability of a hierarchical imitator appears to be novel to me. Quality: The analysis presented in this paper is of high quality. I liked the formal narrative where each mathematical object is clearly defined before used. The authors also provide detailed derivation in their appendix. Clarity: Even though this paper adopts heavy notation and formulas, the authors did a good job in explaining each mathematical object carefully. Through out my read, there are lines where I need to pause and think, but barely one makes me confused. The flow is very smooth. Significance: I am not an expert in the literature of learning theory, so my evaluation of the significance may be biased. The proposed algorithm itself does not seem to be super significant. But conceptually connecting adding noises to "total variation continuity" is interesting to me.

Weaknesses

My major concern is that even though the presented analysis is interesting and self-contained, it doesn't seem to provide too much insight into developing a new algorithm or improving existing ones. The proposed algorithm is a consequence of the analysis, but it is not obvious why we need it. I hope the authors would like to discuss more about the potential influence of the presented analysis and algorithm in their feedback.

Questions

Apart from the question in Weakness, I hope the authors would like to answer the following: L172, what is K? It does not seem to be defined. I am not sure if this question is appropriate, but how can we know Assumptions 3.1, 3.2, and 3.3 are true in reality?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

As stated by the authors in L239-243, the major limitation of this work is that the desired "stability" is given since the proposed algorithm relies on either synthesized primitive controllers (in our analysis) or low-level stabilizing controllers built into problem environments (in our experiments). Relaxing such restriction can be a future research direction.

Thank you for the clarifications

Thank you for the clarifications. I maintain my rather positive assessment.