# Learning One-hidden-layer Neural Networks under General Input Distributions

arXiv: https://arxiv.org/abs/1810.04133 · HTML: https://mlsanity.com/paper/1810.04133/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 7 of 7)

- [DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2304.06648/) (2023)
- [Provably learning a multi-head attention layer](https://mlsanity.com/paper/2402.04084/) (2024)
- [Learning Narrow One-Hidden-Layer ReLU Networks](https://mlsanity.com/paper/2304.10524/) (2023)
- [Spurious Local Minima Are Common for Deep Neural Networks with Piecewise Linear Activations](https://mlsanity.com/paper/2102.13233/) (2021)
- [A faster and simpler algorithm for learning shallow networks](https://mlsanity.com/paper/2307.12496/) (2023)
- [Learning Neural Networks with Distribution Shift: Efficiently Certifiable Guarantees](https://mlsanity.com/paper/2502.16021/) (2025)
- [Unrolled denoising networks provably learn optimal Bayesian inference](https://mlsanity.com/paper/2409.12947/) (2024)

## Abstract (arXiv)

Significant advances have been made recently on training neural networks, where the main challenge is in solving an optimization problem with abundant critical points. However, existing approaches to address this issue crucially rely on a restrictive assumption: the training data is drawn from a Gaussian distribution. In this paper, we provide a novel unified framework to design loss functions with desirable landscape properties for a wide range of general input distributions. On these loss functions, remarkably, stochastic gradient descent theoretically recovers the true parameters with global initializations and empirically outperforms the existing approaches. Our loss function design bridges the notion of score functions with the topic of neural network optimization. Central to our approach is the task of estimating the score function from samples, which is of basic and independent interest to theoretical statistics. Traditional estimation methods (example: kernel based) fail right at the outset; we bring statistical methods of local likelihood to design a novel estimator of score functions, that provably adapts to the local geometry of the unknown density.