Courtney Paquette

2024-03-05

ArXiv (preprint)

Implicit Diffusion: Efficient Optimization through Stochastic Sampling

Pierre Marion

Anna Korba

Peter Bartlett

Mathieu Blondel

Valentin De Bortoli

Arnaud Doucet

Felipe Llinares-L'opez

Quentin Berthet

2024-02-08

ArXiv (preprint)

Hitting the High-Dimensional Notes: An ODE for SGD learning dynamics on GLMs and multi-index models

Elizabeth Collins-Woodfin

Elliot Paquette

Inbar Seroussi

2023-08-17

ArXiv (preprint)

Only tails matter: Average-Case Universality and Robustness in the Convex Regime

Leonardo Cunha

Gauthier Gidel

Fabian Pedregosa

Damien Scieur

2022-06-28

Proceedings of the 39th International Conference on Machine Learning (published)

Only Tails Matter: Average-Case Universality and Robustness in the Convex Regime

Leonardo Cunha

Gauthier Gidel

Fabian Pedregosa

Damien Scieur

2022-06-20

ArXiv (preprint)

Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties

Elliot Paquette

Ben Adlam

Jeffrey Pennington

We develop a stochastic differential equation, called homogenized SGD, for analyzing the dynamics of stochastic gradient descent (SGD) on a … (see more)high-dimensional random least squares problem with

2022-05-14

ArXiv (preprint)

Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties

Elliot Paquette

Ben Adlam

Jeffrey Pennington

2022-05-14

ArXiv (preprint)

Halting Time is Predictable for Large Models: A Universality Property and Average-Case Analysis

Bart van Merriënboer

Fabian Pedregosa

2022-02-15

Foundations of Computational Mathematics (published)

Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensions

Elliot Paquette

Ben Adlam

Jeffrey Pennington

Stochastic gradient descent (SGD) is a pillar of modern machine learning, serving as the go-to optimization algorithm for a diverse array of… (see more) problems. While the empirical success of SGD is often attributed to its computational efficiency and favorable generalization behavior, neither effect is well understood and disentangling them remains an open problem. Even in the simple setting of convex quadratic problems, worst-case analyses give an asymptotic convergence rate for SGD that is no better than full-batch gradient descent (GD), and the purported implicit regularization effects of SGD lack a precise explanation. In this work, we study the dynamics of multi-pass SGD on high-dimensional convex quadratics and establish an asymptotic equivalence to a stochastic differential equation, which we call homogenized stochastic gradient descent (HSGD), whose solutions we characterize explicitly in terms of a Volterra integral equation. These results yield precise formulas for the learning and risk trajectories, which reveal a mechanism of implicit conditioning that explains the efficiency of SGD relative to GD. We also prove that the noise from SGD negatively impacts generalization performance, ruling out the possibility of any type of implicit regularization in this context. Finally, we show how to adapt the HSGD formalism to include streaming SGD, which allows us to produce an exact prediction for the excess risk of multi-pass SGD relative to that of streaming SGD (bootstrap risk).

Trajectory of Mini-Batch Momentum: Batch Size Saturation and Convergence in High Dimensions

Kiwon Lee

Andrew Nicholas Cheng

Elliot Paquette

Halting Time is Predictable for Large Models: A Universality Property and Average-Case Analysis