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BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning

Jianming Pan, Zeqi Ye, Xiao Yang, Xu Yang, Weiqing Liu, Lewen Wang, Jiang Bian

TL;DR

BPQP presents a differentiable convex optimization framework that decouples and reformulates the backward pass as a quadratic program, enabling efficient gradient computation with first-order solvers like OSQP. By exploiting KKT structure, active-sets, and sparsity, it avoids costly Jacobian computations and scales to large problems while maintaining gradient fidelity. Empirical results show order-of-magnitude speedups over baselines across QP, LP, and SOCP and improvements in real-world portfolio metrics, demonstrating practical impact for end-to-end learning with optimization layers. The approach is solver-agnostic and adaptable as optimization technologies evolve, with potential extensions to non-convex settings through active-set and KKT-norm considerations.

Abstract

Data-driven decision-making processes increasingly utilize end-to-end learnable deep neural networks to render final decisions. Sometimes, the output of the forward functions in certain layers is determined by the solutions to mathematical optimization problems, leading to the emergence of differentiable optimization layers that permit gradient back-propagation. However, real-world scenarios often involve large-scale datasets and numerous constraints, presenting significant challenges. Current methods for differentiating optimization problems typically rely on implicit differentiation, which necessitates costly computations on the Jacobian matrices, resulting in low efficiency. In this paper, we introduce BPQP, a differentiable convex optimization framework designed for efficient end-to-end learning. To enhance efficiency, we reformulate the backward pass as a simplified and decoupled quadratic programming problem by leveraging the structural properties of the KKT matrix. This reformulation enables the use of first-order optimization algorithms in calculating the backward pass gradients, allowing our framework to potentially utilize any state-of-the-art solver. As solver technologies evolve, BPQP can continuously adapt and improve its efficiency. Extensive experiments on both simulated and real-world datasets demonstrate that BPQP achieves a significant improvement in efficiency--typically an order of magnitude faster in overall execution time compared to other differentiable optimization layers. Our results not only highlight the efficiency gains of BPQP but also underscore its superiority over differentiable optimization layer baselines.

BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning

TL;DR

BPQP presents a differentiable convex optimization framework that decouples and reformulates the backward pass as a quadratic program, enabling efficient gradient computation with first-order solvers like OSQP. By exploiting KKT structure, active-sets, and sparsity, it avoids costly Jacobian computations and scales to large problems while maintaining gradient fidelity. Empirical results show order-of-magnitude speedups over baselines across QP, LP, and SOCP and improvements in real-world portfolio metrics, demonstrating practical impact for end-to-end learning with optimization layers. The approach is solver-agnostic and adaptable as optimization technologies evolve, with potential extensions to non-convex settings through active-set and KKT-norm considerations.

Abstract

Data-driven decision-making processes increasingly utilize end-to-end learnable deep neural networks to render final decisions. Sometimes, the output of the forward functions in certain layers is determined by the solutions to mathematical optimization problems, leading to the emergence of differentiable optimization layers that permit gradient back-propagation. However, real-world scenarios often involve large-scale datasets and numerous constraints, presenting significant challenges. Current methods for differentiating optimization problems typically rely on implicit differentiation, which necessitates costly computations on the Jacobian matrices, resulting in low efficiency. In this paper, we introduce BPQP, a differentiable convex optimization framework designed for efficient end-to-end learning. To enhance efficiency, we reformulate the backward pass as a simplified and decoupled quadratic programming problem by leveraging the structural properties of the KKT matrix. This reformulation enables the use of first-order optimization algorithms in calculating the backward pass gradients, allowing our framework to potentially utilize any state-of-the-art solver. As solver technologies evolve, BPQP can continuously adapt and improve its efficiency. Extensive experiments on both simulated and real-world datasets demonstrate that BPQP achieves a significant improvement in efficiency--typically an order of magnitude faster in overall execution time compared to other differentiable optimization layers. Our results not only highlight the efficiency gains of BPQP but also underscore its superiority over differentiable optimization layer baselines.

Paper Structure

This paper contains 25 sections, 2 theorems, 35 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Explicit methods
  4. Implicit methods
  5. Background
  6. Differentiable convex optimization layers
  7. Differentiating Through KKT Conditions
  8. Methodology
  9. Backward Pass as QPs
  10. Efficiently Solve Backward Pass Problem with OSQP
  11. Examples: Differentiable QP and SOCP
  12. Experiments
  13. Simulated Large-scale Constrained Optimization
  14. Real-world End-to-End Portfolio Optimization
  15. Further discussion
  16. ...and 10 more sections

Key Result

Lemma 1

Consider a continuously differentiable function $\mathcal{F}(z, y)$ with $\mathcal{F}(z^*, y) = 0$, and suppose the Jacobian matrix of $\mathcal{F}$ is invertible at a small neighborhood at $(z^*, y)$, we have where $\mathbf{J}_{\mathcal{F}}(z)$ and $\mathbf{J}_{\mathcal{F}}(y)$ are respectively the Jacobian matrix of $\mathcal{F}$ w.r.t $z$ and $y$.

Figures (4)

  • Figure 1: The learning process of BPQP: the previous layer outputs $y$ and generates the optimal solution $z^\star$ in the forward pass; the backward pass propagates the loss gradient for end-to-end learning; the process is accelerated by reformulating and simplifying the problem first and then adopting efficient solvers.
  • Figure 2: Table \ref{['tab:sim_comp']} results visualization.
  • Figure 3: Sensitivity analysis under 500×100 setting.
  • Figure 4: The prediction and decision error/loss of methods with different objectives

Theorems & Definitions (4)

  • Definition 1: Differentiable Convex Optimization Layer
  • Lemma 1: Implicit Function Theorem
  • Theorem 1
  • proof