SDPRLayers: Certifiable Backpropagation Through Polynomial Optimization Problems in Robotics
Bibliographic record
Abstract
A recent set of techniques in the robotics community, known as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">certifiably correct methods</i>, frames robotics problems as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polynomial optimization problems</i> (POPs) and applies convex, semidefinite programming (SDP) relaxations to either find or certify their global optima. In parallel, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">differentiable optimization</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">certifiably correct solutions and gradients</i> 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 and our differentiable localization pipeline.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".