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A recent line of work has shown remarkable behaviors of the generalizationerror curves in simple learning models. Even the least-squares regression hasshown atypical features such as the model-wise double descent, and furtherworks have observed triple or multiple descents. Another importantcharacteristic are the epoch-wise descent structures which emerge duringtraining. The observations of model-wise and epoch-wise descents have beenanalytically derived in limited theoretical settings (such as the randomfeature model) and are otherwise experimental. In this work, we provide a fulland unified analysis of the whole time-evolution of the generalization curve,in the asymptotic large-dimensional regime and under gradient-flow, within awider theoretical setting stemming from a gaussian covariate model. Inparticular, we cover most cases already disparately observed in the literature,and also provide examples of the existence of multiple descent structures as afunction of a model parameter or time. Furthermore, we show that ourtheoretical predictions adequately match the learning curves obtained bygradient descent over realistic datasets. Technically we compute averages ofrational expressions involving random matrices using recent developments inrandom matrix theory based on "linear pencils". Another contribution, which isalso of independent interest in random matrix theory, is a new derivation ofrelated fixed point equations (and an extension there-off) using Dyson brownianmotions.

Gradient flow in the gaussian covariate model: exact solution of learning curves and multiple descent structures