No Double Descent in PCA: Training and Pre-Training in High Dimensions

Daniel Gedon, Antonio H. Ribeiro, Thomas B. Schön

Keywords: PCA, generalization theory, overparameterization, pre-training theory, encoder-decoder models, double descent, linear regression

Abstract: With the recent body of work on overparameterized models the gap between theory and practice in contemporary machine learning is shrinking. While many of the present state-of-the-art models have an encoder-decoder architecture, there is little theoretical work for this model structure. To improve our understanding in this direction, we consider linear encoder-decoder models, specifically PCA with linear regression on data from a low-dimensional manifold. We present an analysis for fundamental guarantees of the risk and asymptotic results for isotropic data when the model is trained in a supervised manner. The results are also verified in simulations. Furthermore, we extend our analysis to the popular setting where parts of the model are pre-trained in an unsupervised manner by pre-training the PCA encoder with subsequent supervised training of the linear regression. We show that the overall risk depends on the estimates of the eigenvectors in the encoder and present a sample complexity requirement through a concentration bound. The results highlight that using more pre-training data decreases the overall risk only if it improves the eigenvector estimates. Therefore, we stress that the eigenvalue distribution determines whether more pre-training data is useful or not.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Submission Guidelines: Yes

Please Choose The Closest Area That Your Submission Falls Into: Theory (eg, control theory, learning theory, algorithmic game theory)

TL;DR: We analyse PCA with linear regression for its generalization with high dimensional data and extend the setting to training the two model parts on two different data sets to establish connections to pre-training theory.