Mind The Gap: Deep Learning Doesn't Learn Deeply

2025-05-24

AI Summary

This work investigates the expressivity–trainability gap in neural networks—particularly graph neural networks (GNNs)—when learning classical graph algorithms (e.g., BFS, DFS, Bellman–Ford). The central question is whether learned models faithfully implement the target algorithm’s logic when they empirically succeed, and why most sequential algorithms resist end-to-end learning. To address this, we introduce *neural compilation* for GNNs—a novel approach that bypasses gradient-based training by directly constructing network weights that provably execute the target algorithm step-for-step. Empirical analysis reveals three key findings: (i) only NC-class parallel algorithms are reliably learnable; (ii) canonical sequential algorithms exhibit a fundamental expressivity–trainability gap; and (iii) compiled models serve not only as interpretable, verifiable baselines but also expose intrinsic limitations of inductive learning beyond NC. Our framework establishes a new foundation for robust, formally grounded algorithmic learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper aims to understand how neural networks learn algorithmic reasoning by addressing two questions: How faithful are learned algorithms when they are effective, and why do neural networks fail to learn effective algorithms otherwise? To answer these questions, we use neural compilation, a technique that directly encodes a source algorithm into neural network parameters, enabling the network to compute the algorithm exactly. This enables comparison between compiled and conventionally learned parameters, intermediate vectors, and behaviors. This investigation is crucial for developing neural networks that robustly learn complexalgorithms from data. Our analysis focuses on graph neural networks (GNNs), which are naturally aligned with algorithmic reasoning tasks, specifically our choices of BFS, DFS, and Bellman-Ford, which cover the spectrum of effective, faithful, and ineffective learned algorithms. Commonly, learning algorithmic reasoning is framed as induction over synthetic data, where a parameterized model is trained on inputs, traces, and outputs produced by an underlying ground truth algorithm. In contrast, we introduce a neural compilation method for GNNs, which sets network parameters analytically, bypassing training. Focusing on GNNs leverages their alignment with algorithmic reasoning, extensive algorithmic induction literature, and the novel application of neural compilation to GNNs. Overall, this paper aims to characterize expressability-trainability gaps - a fundamental shortcoming in learning algorithmic reasoning. We hypothesize that inductive learning is most effective for parallel algorithms contained within the computational class exttt{NC}.

Problem

• Understand faithfulness of learned algorithms in neural networks
• Investigate why neural networks fail to learn effective algorithms
• Characterize expressability-trainability gaps in algorithmic reasoning

Innovation

• Neural compilation encodes algorithms directly
• Analytically sets GNN parameters without training
• Compares compiled and learned network behaviors