Differentiable Knapsack and Top-k Operators via Dynamic Programming

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.

Autoregressive Language Models are Secretly Energy-Based Models: Insights into the Lookahead Capabilities of Next-Token Prediction

Autoregressive models (ARMs) currently constitute the dominant paradigm for large language models (LLMs). Energy-based models (EBMs) represent another class of models, which have historically been less prevalent in LLM development, yet naturally characterize the optimal policy in post-training alignment. In this paper, we provide a unified view of these two model classes. Taking the chain rule of probability as a starting point, we establish an explicit bijection between ARMs and EBMs in function space, which we show to correspond to a special case of the soft Bellman equation in maximum entropy reinforcement learning. Building upon this bijection, we derive the equivalence between supervised learning of ARMs and EBMs. Furthermore, we analyze the distillation of EBMs into ARMs by providing theoretical error bounds. Our results provide insights into the ability of ARMs to plan ahead, despite being based on the next-token prediction paradigm.

Learning with Local Search MCMC Layers

Integrating combinatorial optimization layers into neural networks has recently attracted significant research interest. However, many existing approaches lack theoretical guarantees or fail to perform adequately when relying on inexact solvers. This is a critical limitation, as many operations research problems are NP-hard, often necessitating the use of neighborhood-based local search heuristics. These heuristics iteratively generate and evaluate candidate solutions based on an acceptance rule. In this paper, we introduce a theoretically-principled approach for learning with such inexact combinatorial solvers. Inspired by the connection between simulated annealing and Metropolis-Hastings, we propose to transform problem-specific neighborhood systems used in local search heuristics into proposal distributions, implementing MCMC on the combinatorial space of feasible solutions. This allows us to construct differentiable combinatorial layers and associated loss functions. Replacing an exact solver by a local search strongly reduces the computational burden of learning on many applications. We demonstrate our approach on a large-scale dynamic vehicle routing problem with time windows.