Summary
The paper presents two algorithms: DiffuSolve and DiffuSolve+. In DiffuSolve, a conditional diffusion model is first trained to learn $p(x^*|y)$ with a loss function derived from Ho & Salimans, 2022. An offline data generation process uses a nonlinear program solver, $\pi$, to collect locally optimal and feasible solutions $x^*$ for problem instances $P_y$ with uniformly sampled initial guesses. It manually filters out samples corresponding to suboptimal objectives. The diffusion model predicts diverse guesses $\tilde{x}^*_{new}$, which the NLP solver $\pi$ then uses to derive final solutions. Intuitively, this warm-starting approach with diverse samples helps find a global minimizer in the presence of multiple local minima, making it amenable to nonconvex optimization. The paper also develops DiffuSolve+ which ensures that no constraints are violated. For this, the algorithmic insight is to use a “closed-form” prediction of the sample $\hat{x}_0$, while minimizing a “clipped” objective which intuitively keeps predicted solutions within the feasible region. The paper then runs three experiments: quadrotor navigation, two-car reach avoid, and cislunar transfer, and compares its algorithms against the baselines to demonstrate speed and efficacy results.
Strengths
1) The paper is well-written - the technical details are explained clearly, and the experimental results are fairly presented.
2) The paper demonstrates good simulation results in three diverse settings.
Weaknesses
1) The term "control transcription" (line 37) is not defined and is not used anywhere else in the paper. I think readers would appreciate it if you were to remove it if it is unnecessary, or if you were to explain it further, otherwise.
2) This is merely a pedantic point, but $\epsilon_{\theta}$ in lines 171 and 175 are being used for two things: one is a function of two variables, and the other is a function of three (not to mention there is an $\epsilon$ without a subscript!). I think this is fine, but maybe add a note saying “with abuse of notation”, or use a different variable for more clarity?
3) I just read "Model-Based Diffusion for Trajectory Optimization" (Pan, Yi, Chi, Qu, NeurIPS 24), which seems to have the same problem formulation and a similar algorithmic approach. Upon a closer look, I think there is a significant amount of overlap in the work.
4) While the literature review presented in the submission seems to touch on much of the existing work in the area, I'm not convinced that the submission contains any meaningful novelty. For instance, as the authors mention, it has been known that diffusion models are amenable to nonconvex optimization. In this sense, constructing a diffusion model and passing it to an optimizer is not novel, per se, which makes me question the contribution of DiffuSolve. On the other hand, DiffuSolve+ presents a nontrivial improvement with the clipping function to prevent violations; however, I wonder if there isn't already existing work in the literature to address preventing the infeasibility of solutions... In this regard, it would have been good to know the limitations of existing works in this area, which the literature review fails to highlight. For instance, the above reference (Pan et. al. 2024) solves some related problems in non-convex high-dimensional trajectory optimization and further references many other works, specifically on non-convex non-smooth high-dimensional trajectory optimization problems.
Questions
1) In line 272, how are values for $\lambda$ determined?
2) I have also raised some questions in the weaknesses section. I would certainly appreciate a response to it!
3) All the current simulation results are being tested against "CVAE_LSTM" and some uniform samples. The paper doesn't mention, but is "CVAE_LSTM" the best-known baseline comparison? The experimental results might seem more meaningful if they were more robustly tested against a larger variety of algorithms from the literature...