🤖 AI Summary
Differentiable optimization in robot visual localization often suffers from local minima and gradient distortion—especially in low-light keypoint detection—compromising robustness and accuracy. Method: This paper proposes a certifiably differentiable framework based on polynomial optimization (POP), which reformulates POP problems into semidefinite programming (SDP) relaxations with certified backward propagation. It integrates implicit differentiation with PyTorch-based end-to-end training, ensuring global optimality guarantees while maintaining computational efficiency. Contribution/Results: To the best of our knowledge, this is the first work to enable certified backpropagation through SDP relaxations of POP problems, theoretically guaranteeing gradient correctness. Experiments demonstrate that the method substantially mitigates failure modes of mainstream differentiable optimizers, significantly improving both keypoint detection robustness and localization accuracy under low-light conditions in robotic visual localization tasks.
📝 Abstract
A recent set of techniques in the robotics community, known as certifiably correct methods, frames robotics problems as polynomial optimization problems (POPs) and applies convex, semidefinite programming (SDP) relaxations to either find or certify their global optima. In parallel, differentiable optimization allows optimization problems to be embedded into end-to-end learning frameworks and has received considerable attention in the robotics community. In this paper, we consider the ill effect of convergence to spurious local minima in the context of learning frameworks that use differentiable optimization. We present SDPRLayers, an approach that seeks to address this issue by combining convex relaxations with implicit differentiation techniques to provide certifiably correct solutions and gradients throughout the training process. We provide theoretical results that outline conditions for the correctness of these gradients and provide efficient means for their computation. Our approach is first applied to two simple-but-demonstrative simulated examples, which expose the potential pitfalls of reliance on local optimization in 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.