Algorithmic Capture, Computational Complexity, and Inductive Bias of Infinite Transformers

We formally define Algorithmic Capture (i.e., ``grokking'' an algorithm) as the ability of a neural network to generalize to arbitrary problem sizes ($T$) with controllable error and minimal sample adaptation, distinguishing true algorithmic learning from statistical interpolation. By analyzing infinite-width transformers in both the lazy and rich regimes, we derive upper bounds on the inference-time computational complexity of the functions these networks can learn. We show that despite their universal expressivity, transformers possess an inductive bias towards low-complexity algorithms within the Efficient Polynomial Time Heuristic Scheme (EPTHS) class. This bias effectively prevents them from capturing higher-complexity algorithms, while allowing success on simpler tasks like search, copy, and sort.

Heuristics for Combinatorial Optimization via Value-based Reinforcement Learning: A Unified Framework and Analysis

Since the 1990s, considerable empirical work has been carried out to train statistical models, such as neural networks (NNs), as learned heuristics for combinatorial optimization (CO) problems. When successful, such an approach eliminates the need for experts to design heuristics per problem type. Due to their structure, many hard CO problems are amenable to treatment through reinforcement learning (RL). Indeed, we find a wealth of literature training NNs using value-based, policy gradient, or actor-critic approaches, with promising results, both in terms of empirical optimality gaps and inference runtimes. Nevertheless, there has been a paucity of theoretical work undergirding the use of RL for CO problems. To this end, we introduce a unified framework to model CO problems through Markov decision processes (MDPs) and solve them using RL techniques. We provide easy-to-test assumptions under which CO problems can be formulated as equivalent undiscounted MDPs that provide optimal solutions to the original CO problems. Moreover, we establish conditions under which value-based RL techniques converge to approximate solutions of the CO problem with a guarantee on the associated optimality gap. Our convergence analysis provides: (1) a sufficient rate of increase in batch size and projected gradient descent steps at each RL iteration; (2) the resulting optimality gap in terms of problem parameters and targeted RL accuracy; and (3) the importance of a choice of state-space embedding. Together, our analysis illuminates the success (and limitations) of the celebrated deep Q-learning algorithm in this problem context.