Summary of On the Convergence Of (stochastic) Gradient Descent For Kolmogorov–arnold Networks, by Yihang Gao et al.
On the Convergence of (Stochastic) Gradient Descent for Kolmogorov–Arnold Networks
by Yihang Gao, Vincent Y. F. Tan
First submitted to arxiv on: 10 Oct 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Artificial Intelligence (cs.AI); Optimization and Control (math.OC)
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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 Kolmogorov-Arnold Networks (KANs), a novel neural network architecture, have garnered attention in the deep learning community due to their potential as an alternative to multi-layer perceptrons (MLPs) and broad applicability. Empirical investigations demonstrate that KANs optimized via stochastic gradient descent (SGD) can achieve near-zero training loss in various machine learning tasks, such as regression, classification, and time series forecasting, as well as scientific tasks like solving partial differential equations. This paper provides a theoretical explanation for the empirical success by conducting a rigorous convergence analysis of gradient descent (GD) and SGD for two-layer KANs in solving both regression and physics-informed tasks. The authors establish that GD achieves global linear convergence of the objective function when the hidden dimension is sufficiently large, and extend these results to SGD, demonstrating similar global convergence in expectation. Additionally, they analyze the global convergence of GD and SGD for physics-informed KANs, which reveals additional challenges due to the more complex loss structure. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper is about a new type of neural network called Kolmogorov-Arnold Networks (KANs). Scientists are excited because KANs can do many things that other networks can’t. They tried out KANs on different problems and found they worked really well. The authors wanted to understand why this was happening, so they did some careful math to figure it out. They showed that when you use a certain type of optimization method (called stochastic gradient descent) with KANs, the network can get very good at solving problems. This is important because it means we can use KANs for all sorts of tasks, from predicting what will happen in the future to solving complex scientific problems. |
Keywords
* Artificial intelligence * Attention * Classification * Deep learning * Gradient descent * Machine learning * Neural network * Objective function * Optimization * Regression * Stochastic gradient descent * Time series
