Thank you for your valuable feedback, which is very helpful in improving the paper. We're encouraged by the broadly "Put this in the context of other work on computational homogenization / multi-scale finite element Our method is related to these and the boundary element method (BEM). "Limitation associated with micro-scale buckling... the coarse-grain behavior might exhibit hysteretic effects": Good "How sensitive is the outer optimization to the accuracy of the surrogate gradients?" "Do you know how the CES method scales with system size in terms of accuracy and evaluation time": In terms of "the method to solve the outer optimization over BCs to find minimum energy solutions to the composed surrogates Free DoFs are optimized to minimize total predicted energy using LBFGS. "The discuss of the surrogate and i.i.d. "Are the BCs shared when a boundary is common between two cells": Y es. We have 1 DoF for each blue point in Fig 2. "Its not clear how the HMC and PDE solver are used together": HMC is used to generate training BCs, preferring larger The PDE solver is used to compute the gradient of the pdf (which depends on E) w.r.t. the BC. Given BCs, we run the solver to determine the internal u and E. We compute dE/dBC with the Then we use this to compute the gradient of the pdf w.r.t. the BCs, needed for the leapfrog step. "does the HMC require a significant burn-in time before producing reasonable samples": No. Note: we don't truly care Per appendix, HMC took between 3 and 100 leapfrog steps per sample. The process of using the surrogates to solve the original problem can be explained in more detail. Newton method is neither the fast nor the most stable... a comparison with more sophisticated methods would be From a brief look it looks like Liu et al's method is tailored for Reviewer 5: "There is one outlier in L2 compression that was quite bad": We will discuss this in the main paper. "A comment might help the reader situate this work within the more usual (less idyllic) context of approximating This is a good suggestion: we will relate to other work in learning energies.

Weaknesses: The empirical evaluation would benefit from additional exploration. For instance, the outer optimization may be sensitive to the quality of the surrogate model energy predictions and gradients. There is little presentation on the quality of the surrogate model predictions and gradients. Is it understood how sensitive the outer optimization is to the accuracy of the surrogate gradients? Even if the gradients are biased, can the outer optimization still find reasonable solutions?

Differentiable optimization is a powerful new paradigm capable of reconciling model-based and learning-based approaches in robotics. However, the majority of robotics optimization problems are non-convex and current differentiable optimization techniques are therefore prone to convergence to local minima. When this occurs, the gradients provided by these existing solvers can be wildly inaccurate and will ultimately corrupt the training process. On the other hand, many non-convex robotics problems can be framed as polynomial optimization problems and, in turn, admit convex relaxations that can be used to recover a global solution via so-called certifiably correct methods. We present SDPRLayers, an approach that leverages these methods as well as state-of-the-art convex implicit differentiation techniques to provide certifiably correct gradients throughout the training process. We introduce this approach and showcase theoretical results that provide conditions under which correctness of the gradients is guaranteed. We first demonstrate our approach on two simple-but-demonstrative simulated examples, which expose the potential pitfalls of existing, state-of-the-art, differentiable optimization methods. We then apply our method in a real-world application: we train a deep neural network to detect image keypoints for robot localization in challenging lighting conditions. We provide our open-source, PyTorch implementation of SDPRLayers and our differentiable localization pipeline.

The acquisition of explicit user feedback (e.g., ratings) in real-world recommender systems is often hindered by the need for active user involvement. To mitigate this issue, implicit feedback (e.g., clicks) generated during user browsing is exploited as a viable substitute. However, implicit feedback possesses a high degree of noise, which significantly undermines recommendation quality. While many methods have been proposed to address this issue by assigning varying weights to implicit feedback, two shortcomings persist: (1) the weight calculation in these methods is iteration-independent, without considering the influence of weights in previous iterations, and (2) the weight calculation often relies on prior knowledge, which may not always be readily available or universally applicable. To overcome these two limitations, we model recommendation denoising as a bi-level optimization problem. The inner optimization aims to derive an effective model for the recommendation, as well as guiding the weight determination, thereby eliminating the need for prior knowledge. The outer optimization leverages gradients of the inner optimization and adjusts the weights in a manner considering the impact of previous weights. To efficiently solve this bi-level optimization problem, we employ a weight generator to avoid the storage of weights and a one-step gradient-matching-based loss to significantly reduce computational time. The experimental results on three benchmark datasets demonstrate that our proposed approach outperforms both state-of-the-art general and denoising recommendation models. The code is available at https://github.com/CoderWZW/BOD.