[Paper Review] Backpropagation through Combinatorial Algorithms: Identity with Projection Works

Subham Sekhar Sahoo, Anselm Paulus+4 more|arXiv (Cornell University)|May 30, 2022

Advanced Graph Neural Networks11 citations

TL;DR

This paper proposes a hyperparameter-free, efficient method for backpropagating through combinatorial solvers by treating them as negative identity layers on the backward pass, while leveraging problem-specific invariances via cost vector projections to stabilize training. The approach avoids additional solver calls and outperforms prior methods in tasks like deep graph matching and image retrieval, with improved robustness through noise-induced regularization instead of problem-specific margins.

ABSTRACT

Embedding discrete solvers as differentiable layers has given modern deep learning architectures combinatorial expressivity and discrete reasoning capabilities. The derivative of these solvers is zero or undefined, therefore a meaningful replacement is crucial for effective gradient-based learning. Prior works rely on smoothing the solver with input perturbations, relaxing the solver to continuous problems, or interpolating the loss landscape with techniques that typically require additional solver calls, introduce extra hyper-parameters, or compromise performance. We propose a principled approach to exploit the geometry of the discrete solution space to treat the solver as a negative identity on the backward pass and further provide a theoretical justification. Our experiments demonstrate that such a straightforward hyper-parameter-free approach is able to compete with previous more complex methods on numerous experiments such as backpropagation through discrete samplers, deep graph matching, and image retrieval. Furthermore, we substitute the previously proposed problem-specific and label-dependent margin with a generic regularization procedure that prevents cost collapse and increases robustness.

Motivation & Objective

• To enable end-to-end training of deep networks with discrete combinatorial solvers by addressing the challenge of zero or undefined gradients in discrete optimization.
• To eliminate the need for costly solver calls or hyperparameter tuning in gradient estimation for combinatorial solvers.
• To stabilize learning by exploiting invariances in combinatorial problems, such as normalization and projection symmetries.
• To replace problem-specific, label-dependent margins with a generic noise-based regularization that prevents cost collapse and improves robustness.
• To provide a theoretically grounded, geometry-aware approach to gradient estimation that is both simple and effective.

Proposed method

• Treat the combinatorial solver as a negative identity layer during backpropagation, directly propagating incoming gradients without additional solver evaluations.
• Introduce a projection operation on the cost vector in the computation graph to remove gradient components that correspond to problem invariances (e.g., normalization, scaling).
• Apply the projection on the backward pass to eliminate spurious updates that do not affect the solution, preserving only the relevant gradient direction.
• Use input noise perturbation to implicitly induce a margin on the solution space, replacing the need for ground-truth-informed margins in prior work.
• Formalize the method using a piecewise-linear interpolation of the loss landscape and prove convergence under mild assumptions.
• Theoretical analysis shows that the method avoids cost collapse and ensures descent in the linearized loss under appropriate conditions.

Experimental results

Research questions

• RQ1 Can a simple, hyperparameter-free gradient estimator be designed for combinatorial solvers without additional solver calls on the backward pass?
• RQ2 How can invariances in combinatorial optimization problems—such as scale or normalization—be exploited to stabilize gradient updates?
• RQ3 What is the impact of replacing problem-specific margins with a generic noise-based regularization on model robustness and convergence?
• RQ4 Does treating the solver as a negative identity layer yield a descent direction in the loss landscape under geometric constraints?
• RQ5 How does the proposed method compare empirically to more complex baselines like Blackbox Backpropagation in terms of performance and training stability?

Key findings

• The proposed identity-with-projection method achieves competitive performance on deep graph matching, image retrieval, and discrete reasoning tasks, matching or exceeding prior methods without hyperparameters.
• The method prevents cost collapse and improves robustness by using input noise to induce a generic margin, eliminating the need for label-dependent margin design.
• Theoretical analysis confirms that the method ensures descent in the linearized loss function under mild assumptions, with strict improvement when the solution changes.
• Empirical results show that the method is stable and converges faster than baseline approaches, particularly when invariances are properly handled via projection.
• The approach avoids additional solver calls during backpropagation, making it computationally more efficient than methods like Blackbox Backpropagation.
• The method is robust to perturbations and generalizes well across diverse combinatorial problems, including shortest path, rank optimization, and keypoint matching.