Universality in Deep Neural Networks: An approach via the Lindeberg exchange principle
Researchers have established quantitative bounds on how closely infinite-width neural networks converge to Gaussian limits, using a novel Lindeberg principle tailored for deep architectures. This theoretical result strengthens the mathematical foundations underpinning neural network behavior at scale, offering practitioners and theorists alike a rigorous framework for understanding when and why width-based approximations hold. The work matters because it bridges classical probability theory with modern deep learning, potentially informing both architecture design and convergence guarantees in practical training regimes.
Modelwire context
ExplainerThe paper quantifies not just that infinite-width networks converge to Gaussian limits, but how fast and under what architectural conditions. Prior work established the limit exists; this work puts error bars on it, which is the difference between a theoretical curiosity and a usable approximation guarantee.
This connects directly to the MIT scaling laws work from May 3rd, which identified superposition as the mechanistic driver behind why model performance improves predictably with scale. That paper explained the empirical 'why' of scaling; this Lindeberg result provides the mathematical scaffolding underneath it. Together they're building a two-layer foundation: one showing that scaling works (empirically grounded), the other showing that width-based approximations are formally justified (theoretically grounded). The expressivity work on local attention from May 1st also belongs here, since both papers are asking when simplified architectural constraints preserve or lose representational power.
If researchers apply these quantitative bounds to prove convergence rates for specific architectures (ResNets, Vision Transformers) within the next 6 months, it signals the result is moving from pure theory to architecture-specific design guidance. If the bounds remain too loose to be practically useful for practitioners choosing layer widths, the work stays confined to theoretical interest.
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MentionsDeep Neural Networks · Lindeberg principle · Gaussian limit · Wasserstein distance
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