Differentiable Knapsack and Top-k Operators via Dynamic Programming

2026-01-29

AI Summary

This work addresses the challenge of integrating combinatorial operators such as Knapsack and Top-k into end-to-end trainable neural networks, where their piecewise-constant nature renders gradients almost everywhere zero. To overcome this, the authors propose a unified differentiable framework based on dynamic programming, in which the recurrence relations are smoothed to yield efficient relaxed forward and backward passes. Theoretical analysis reveals that the Shannon entropy is the unique regularizer preserving permutation equivariance, and the study further characterizes regularization conditions that induce sparse selections. The effectiveness and generality of the approach are demonstrated across diverse tasks, including decision-focused learning, constrained dynamic item selection in reinforcement learning, and extensions of discrete variational autoencoders.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Knapsack and Top-k operators are useful for selecting discrete subsets of variables. However, their integration into neural networks is challenging as they are piecewise constant, yielding gradients that are zero almost everywhere. In this paper, we propose a unified framework casting these operators as dynamic programs, and derive differentiable relaxations by smoothing the underlying recursions. On the algorithmic side, we develop efficient parallel algorithms supporting both deterministic and stochastic forward passes, and vector-Jacobian products for the backward pass. On the theoretical side, we prove that Shannon entropy is the unique regularization choice yielding permutation-equivariant operators, and characterize regularizers inducing sparse selections. Finally, on the experimental side, we demonstrate our framework on a decision-focused learning benchmark, a constrained dynamic assortment RL problem, and an extension of discrete VAEs.

Problem

• differentiable operators
• knapsack problem
• top-k selection
• discrete optimization
• neural networks

Innovation

• differentiable operators
• dynamic programming
• knapsack problem
• top-k selection
• entropy regularization