Research map / Optimization / Optimizers
LPGD: A General Framework for Backpropagation through Embedded Optimization Layers
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes Lagrangian Proximal Gradient Descent (LPGD), a framework that trains models with embedded optimization layers by replacing degenerate solver derivatives with finite differences computed by re-running the forward solver on a perturbed input.
Problem
Gradient-based training of architectures with embedded parameterized optimization layers fails when the solution mapping has degenerate (zero or undefined) derivatives, such as for discrete or linear programs.
Approach
Defines a Lagrange-Moreau envelope of the loss using a Lagrangian divergence, and performs gradient descent on the envelope of a linearized loss; the resulting gradients require only one extra forward-solver call on perturbed parameters (w + tau*grad), unifying prior methods such as DLM, blackbox backprop, SPO+, and Fenchel-Young losses as special cases.
Key results
- On Markowitz portfolio tuning, the best LPGD run improves over the Markowitz initialization by 35% versus 23% for the best GD run.
- In the Sudoku experiment, LPGD reaches a lower final train MSE than GD and converges faster in wall-clock time due to warm-started backward passes.
- Theory shows LPGD gradients converge to the true gradient as tau goes to 0 when the solution mapping is differentiable.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LPGD instead of GD | Markowitz control policy · Markowitz portfolio optimization | improvement over Markowitz initialization: 23% -> 35% | Helped |
| Use LPGD instead of GD | Markowitz control policy · Markowitz portfolio optimization | convergence speed in iterations | Helped |
| Use LPGD instead of GD | Sudoku rule learning · Sudoku synthetic data | final train MSE | Helped |
| Use LPGD instead of GD | Sudoku rule learning · Sudoku synthetic data | convergence speed in wall-clock time | Helped |
| Set learning rate to 0.1 (ablation) | GD and LPGD · Markowitz portfolio optimization | training feasibility | Hurt |
| Set LPGD temperature to τ=0.1 (ablation) | LPGD · Markowitz portfolio optimization | performance | Hurt |
| Set LPGD temperature to τ=1 (ablation) | LPGD · Markowitz portfolio optimization | performance relative to GD | No clear effect |
| Set LPGD temperature to τ=100 (ablation) | LPGD · Markowitz portfolio optimization | performance | Helped |
| Set LPGD temperature to τ=1000 (ablation) | LPGD · Markowitz portfolio optimization | training feasibility | Hurt |
| Increase the learning rate (ablation) | GD relative to LPGD · Markowitz portfolio optimization | training stability | Hurt |
| Vary solver accuracy ε (ablation) | GD and LPGD · Markowitz portfolio optimization | sensitivity to inaccurate solutions | No clear effect |
Surprising findings
- On the portfolio task, a large temperature tau=100 performed best, outperforming tau=1 which matches GD true gradients.
Limitations
- Requires access to linear (and quadratic, for augmentation) coefficients of the solver interface, which not all reformulations allow.
- Very small tau requires very accurate solves, which can be expensive.
- Hyperparameters tau and rho require tuning and may have different scales across loss terms.
Experimental setup
Datasets: synthetic Sudoku puzzles historical trading data (Markowitz portfolio setting)
Benchmarks: Sudoku rule learning Markowitz portfolio optimization
Metrics: train MSE wall-clock time portfolio utility improvement
Baselines: gradient descent (GD) Blackbox Backpropagation SCS-based differentiable conic program (CVXPY) gradients
Builds on: Moreau envelope proximal gradient descent Blackbox Backpropagation Direct Loss Minimization Implicit Differentiation by Perturbation
Closest papers by meaning
- Learning Provably Improves the Convergence of Gradient Descent (NeurIPS 2025 · similarity 0.87)
- Rescaled Asynchronous SGD: Optimal Distributed Optimization under Data and System Heterogeneity (NeurIPS 2026 · similarity 0.86)
- Predicting Lagrangian Multipliers for Mixed Integer Linear Programs (ICML 2024 · similarity 0.86)
- BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning (NeurIPS 2024 · similarity 0.86)
- MAST: model-agnostic sparsified training (ICLR 2025 · similarity 0.85)
- A Layer-Wise Natural Gradient Optimizer for Training Deep Neural Networks (NeurIPS 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.