Floating-Point Networks with Automatic Differentiation Can Represent Almost All Floating-Point Functions and Their Gradients
A new theoretical result bridges the gap between classical neural network approximation theory and real-world implementations by proving that floating-point networks can represent arbitrary functions and their gradients under automatic differentiation, despite finite precision and rounding errors. This work matters because it validates that the mathematical foundations underpinning deep learning remain sound even when constrained by hardware arithmetic, addressing a long-standing gap between idealized theory and practical systems that practitioners actually deploy.
Modelwire context
ExplainerThe paper's real contribution is narrower than the title suggests: it proves representation capacity under floating-point constraints, but does not address learnability, sample complexity, or whether standard training algorithms can actually find those representations in practice.
This connects directly to the optimization work from May 1st on randomized subspace Nesterov acceleration, which improved automatic differentiation efficiency in bandwidth-constrained settings. That paper assumed AD itself was sound; this one validates that assumption holds even with finite precision. Together they form a two-layer foundation: theory that AD works correctly on hardware (this paper), then algorithms that make AD faster (the prior work). The physics-informed operator papers (DeepONet, HyCOP from May 1st) also depend implicitly on this guarantee, since they train via backprop through floating-point arithmetic.
If follow-up work within six months provides constructive bounds on sample complexity or convergence rates for training floating-point networks (not just existence proofs), that signals the result is moving from theoretical validation toward practical guidance. If it remains a pure existence theorem without algorithmic implications, the impact stays confined to closing a theoretical gap rather than changing how practitioners build systems.
Coverage we drew on
- Randomized Subspace Nesterov Accelerated Gradient · arXiv cs.LG
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MentionsAutomatic Differentiation · Floating-Point Arithmetic · Neural Networks
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