Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation

2026-06-25 · Source: Artificial Intelligence · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Logic in Computer Science · Depth: Expert, quick

Summary

Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation" proposes a new view of neural network (NN) expressivity, asserting that NN complexity is governed by algorithmic complexity, not just regularity. The research positions NNs as computational models, demonstrating that functions computable by real-valued circuits can be approximated by NNs with comparable accuracy. NN depth, width, and non-zero-parameter counts are directly controlled by the circuit's characteristics. The theory establishes that any definable NN model with a natural parallelization condition is a universal approximator if it incorporates a non-affine nonlinearity. This framework provides universal approximation guarantees for continuous functions, minimax-optimal guarantees for Besov classes, and logarithmic-error complexity for holomorphic functions, also showing NNs can emulate numerical algorithms like Newton-Raphson. It achieves O(log(1/ε)) non-zero parameters for shortest-path computation on k-vertex graphs, exponentially improving over generic Lipschitz-approximation scales.

Key takeaway

For AI scientists designing neural network architectures, recognize that a function's algorithmic complexity, not just its regularity, dictates NN expressivity. You should prioritize incorporating non-affine nonlinearities to ensure universal approximation capabilities. Consider compiling known numerical algorithms into NN structures for highly efficient, specialized models. For instance, achieve O(log(1/ε)) parameters for shortest-path problems. This perspective shifts focus from purely statistical approximation to computational emulation, opening new avenues for efficient network design.

Key insights

Neural network complexity is governed by algorithmic complexity, not just regularity, viewing NNs as computational models.

Principles

• Neural networks function as models of computation.
• Non-affine nonlinearities are essential for universal approximation.
• Circuit computability dictates NN approximation capabilities.

In practice

• NNs can emulate numerical algorithms like Newton-Raphson.
• Shortest-path computation on k-vertex graphs can use O(log(1/ε)) NN parameters.

Topics

• Deep Learning Theory
• Neural Network Expressivity
• Algorithmic Complexity
• Universal Approximation
• Numerical Algorithms
• Circuit Computation

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Artificial Intelligence.