Summary of Equidistribution-based Training Of Free Knot Splines and Relu Neural Networks, by Simone Appella et al.
Equidistribution-based training of Free Knot Splines and ReLU Neural Networks
by Simone Appella, Simon Arridge, Chris Budd, Teo Deveney, Lisa Maria Kreusser
First submitted to arxiv on: 2 Jul 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Numerical Analysis (math.NA)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary We investigate univariate nonlinear function approximation using shallow neural networks (NN) with rectified linear unit (ReLU) activation functions. The L_2 based approximation problem is ill-conditioned, causing optimization algorithms to degrade rapidly as the network width increases. This can lead to poor approximation in practice despite the theoretical expressivity of ReLU architectures and traditional methods like univariate Free Knot Splines (FKS). Despite spanning the same function space, FKS remains well-conditioned as knot numbers increase. We leverage optimal piecewise linear interpolant theory to improve ReLU NN training. A two-level procedure is proposed: first, find optimal knot locations for FKS; then, determine weights and knots by solving a nearly linear problem. This method can train ReLU NNs effectively, achieving accurate approximations. An equidistribution-based loss function is used to find ReLU breakpoints, combined with preconditioning to scale ReLU functions. We test this method on regular, singular, and rapidly varying target functions, obtaining good results that realize shallow ReLU network expressivity in all cases. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary We’re exploring a way to get computers to learn how to approximate complex mathematical functions using simple neural networks with a special type of math inside them called ReLU. The problem is that the way we train these networks can go wrong as they get bigger, making them not very good at approximating functions. We found that another method, called Free Knot Splines (FKS), does better because it’s well-conditioned and doesn’t have this problem. To fix this issue with ReLU networks, we came up with a new way to train them using ideas from math problems that involve finding the best line or curve through some points. We tested our method on different types of functions and it worked really well. |
Keywords
» Artificial intelligence » Loss function » Optimization » Relu
