Study Reveals Fundamental Limitations of Learning Tanh Neural Networks Under Finite Precision
Researchers have proven that learning tanh neural networks from point evaluations under finite-precision arithmetic faces hard theoretical limits on convergence rates. The work extends prior results for ReLU networks to the tanh activation setting, using a novel construction of sharply localized bump functions built from iterated tanh activations. The findings suggest that achieving better-than-Monte-Carlo convergence rates requires sampling budgets that grow exponentially with network size, imposing fundamental constraints on practical learnability.
A new preprint posted to arXiv by Matěj Trödler and collaborators investigates the theoretical limits of learning tanh neural networks from point evaluations when computations are restricted to finite precision. The central result shows that no adaptive randomized algorithm using m samples can exceed the Monte Carlo convergence rate O(m^{-1/p}) in the L^p norm, unless the number of samples grows exponentially with the size of the network's parameters and architecture. The key technical contribution is a novel construction of sharply localized bump functions using iterated tanh activations, which serves as the mechanism for establishing these lower bounds. This work builds directly on a 2023 paper by Berner, Grohs, and Voigtländer that established analogous results for ReLU networks, now extending those limitations to the smooth tanh activation function. The results highlight that finite-precision arithmetic — the reality of all practical computing hardware — imposes fundamental barriers to learning certain classes of neural networks, regardless of the algorithm used. The paper sits at the intersection of machine learning theory, classical analysis, and statistical learning theory.
What's missing
The authors do not discuss whether these theoretical lower bounds are tight in practice or how closely real-world finite-precision training regimes approach the worst-case scenarios described. It is also unclear whether the results extend to other smooth activation functions beyond tanh, or to architectures beyond the specific classes considered.
What different sources said
- arXiv cs.LGCenter
Limitations of Learning Tanh Neural Networks with Finite Precision
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