The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a "simplicity bias": neural networks prevent overfitting by initially learning simple classifiers before progressing to

Many real-world datasets contain hidden structure that cannot be detected by simple linear correlations between input features. For example, latent factors may influence the data in a coordinated way, even though their effect is invisible to covariance-based methods such as PCA. In practice, nonlinear neural networks often succeed in extracting such hidden structure in unsupervised and self-supervised learning. However, constructing a minimal high-dimensional model where this advantage can be rigorously analyzed has remained an open theoretical challenge. We introduce a tractable high-dimensional spiked model with two latent factors: one visible to covariance, and one statistically dependent yet uncorrelated, appearing only in higher-order moments. PCA and linear autoencoders fail to recover the latter, while a minimal nonlinear autoencoder provably extracts both. We analyze both the population risk, and empirical risk minimization. Our model also provides a tractable example where self-supervised test loss is poorly aligned with representation quality: nonlinear autoencoders recover latent structure that linear methods miss, even though their reconstruction loss is higher.

Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance.

In practice, building f and g requires the computation for wtiwtj for all i,j. B.2 Classification For the classification task with the logistic regression model, we modify the formula of logistic regression in teaching objectives to make it convenient for derivation. It also indicates that with probability at least p1, the LST teacher can achieve exponential teachability in the iteration t. In order to achieve exponential teachiability in T iterations, the sufficient condition in Eq. (22) must be satisfied in all T iterations. Then, we use a pre-trained DenseNet [65] shown in [53] to generate 1024 dim features and the confidencescoreforeachimage.

Forlinear regression, we give a polynomial-time algorithm based on Celis-Dennis-Tapia optimization algorithms. For binary classification, we show how to efficiently implement itusing aproper agnostic learner (i.e., anEmpirical Risk Minimizer) for the class of interest.

Deep neural networks (DNNs) are the de facto methodology in many artificial intelligence areas nowadays [25, 37]. How to effectively train a deep network has been a central topic for decades.