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Baptiste Goujaud

Alumni

Publications

Proving Linear Mode Connectivity of Neural Networks via Optimal Transport
Damien Ferbach
Baptiste Goujaud
Gauthier Gidel
Aymeric Dieuleveut
The energy landscape of high-dimensional non-convex optimization problems is crucial to understanding the effectiveness of modern deep neura… (see more)l network architectures. Recent works have experimentally shown that two different solutions found after two runs of a stochastic training are often connected by very simple continuous paths (e.g., linear) modulo a permutation of the weights. In this paper, we provide a framework theoretically explaining this empirical observation. Based on convergence rates in Wasserstein distance of empirical measures, we show that, with high probability, two wide enough two-layer neural networks trained with stochastic gradient descent are linearly connected. Additionally, we express upper and lower bounds on the width of each layer of two deep neural networks with independent neuron weights to be linearly connected. Finally, we empirically demonstrate the validity of our approach by showing how the dimension of the support of the weight distribution of neurons, which dictates Wasserstein convergence rates is correlated with linear mode connectivity.
2024-01-01
AISTATS (published)
doi.org
arxiv.org
Proving Linear Mode Connectivity of Neural Networks via Optimal Transport
Damien Ferbach
Baptiste Goujaud
Gauthier Gidel
Aymeric Dieuleveut
The energy landscape of high-dimensional non-convex optimization problems is crucial to understanding the effectiveness of modern deep neura… (see more)l network architectures. Recent works have experimentally shown that two different solutions found after two runs of a stochastic training are often connected by very simple continuous paths (e.g., linear) modulo a permutation of the weights. In this paper, we provide a framework theoretically explaining this empirical observation. Based on convergence rates in Wasserstein distance of empirical measures, we show that, with high probability, two wide enough two-layer neural networks trained with stochastic gradient descent are linearly connected. Additionally, we express upper and lower bounds on the width of each layer of two deep neural networks with independent neuron weights to be linearly connected. Finally, we empirically demonstrate the validity of our approach by showing how the dimension of the support of the weight distribution of neurons, which dictates Wasserstein convergence rates is correlated with linear mode connectivity.
2023-10-29
ArXiv (preprint)
doi.org
arxiv.org
Gradient Descent Is Optimal Under Lower Restricted Secant Inequality And Upper Error Bound
Charles Guille-escuret
Baptiste Goujaud
Adam Ibrahim
Ioannis Mitliagkas
The study of first-order optimization is sensitive to the assumptions made on the objective functions. These assumptions induce complexity c… (see more)lasses which play a key role in worst-case analysis, including the fundamental concept of algorithm optimality. Recent work argues that strong convexity and smoothness—popular assumptions in literature—lead to a pathological definition of the condition number. Motivated by this result, we focus on the class of functions satisfying a lower restricted secant inequality and an upper error bound. On top of being robust to the aforementioned pathological behavior and including some non-convex functions, this pair of conditions displays interesting geometrical properties. In particular, the necessary and sufficient conditions to interpolate a set of points and their gradients within the class can be separated into simple conditions on each sampled gradient. This allows the performance estimation problem (PEP) to be solved analytically, leading to a lower bound on the convergence rate that proves gradient descent to be exactly optimal on this class of functions among all first-order algorithms.
openreview.net
A Study of Condition Numbers for First-Order Optimization
Charles Guille-escuret
Baptiste Goujaud
Manuela Girotti
Ioannis Mitliagkas
2021-03-18
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (published)
proceedings.mlr.press
arxiv.org
A Study of Condition Numbers for First-Order Optimization
Charles Guille-escuret
Baptiste Goujaud
Manuela Girotti
Ioannis Mitliagkas
The study of first-order optimization algorithms (FOA) typically starts with assumptions on the objective functions, most commonly smoothnes… (see more)s and strong convexity. These metrics are used to tune the hyperparameters of FOA. We introduce a class of perturbations quantified via a new norm, called *-norm. We show that adding a small perturbation to the objective function has an equivalently small impact on the behavior of any FOA, which suggests that it should have a minor impact on the tuning of the algorithm. However, we show that smoothness and strong convexity can be heavily impacted by arbitrarily small perturbations, leading to excessively conservative tunings and convergence issues. In view of these observations, we propose a notion of continuity of the metrics, which is essential for a robust tuning strategy. Since smoothness and strong convexity are not continuous, we propose a comprehensive study of existing alternative metrics which we prove to be continuous. We describe their mutual relations and provide their guaranteed convergence rates for the Gradient Descent algorithm accordingly tuned. Finally we discuss how our work impacts the theoretical understanding of FOA and their performances.
2020-12-10
ArXiv (preprint)
arxiv.org
arxiv.org
Online continual learning with no task boundaries
Rahaf Aljundi
Min Lin
Baptiste Goujaud
Yoshua Bengio
Continual learning is the ability of an agent to learn online with a non-stationary and never-ending stream of data. A key component for suc… (see more)h never-ending learning process is to overcome the catastrophic forgetting of previously seen data, a problem that neural networks are well known to suffer from. The solutions developed so far often relax the problem of continual learning to the easier task-incremental setting, where the stream of data is divided into tasks with clear boundaries. In this paper, we break the limits and move to the more challenging online setting where we assume no information of tasks in the data stream. We start from the idea that each learning step should not increase the losses of the previously learned examples through constraining the optimization process. This means that the number of constraints grows linearly with the number of examples, which is a serious limitation. We develop a solution to select a fixed number of constraints that we use to approximate the feasible region defined by the original constraints. We compare our approach against the methods that rely on task boundaries to select a fixed set of examples, and show comparable or even better results, especially when the boundaries are blurry or when the data distributions are imbalanced.
2019-03-20
arXiv.org (preprint)
dblp.uni-trier.de
Gradient based sample selection for online continual learning
Rahaf Aljundi
Min Lin
Baptiste Goujaud
Yoshua Bengio
A continual learning agent learns online with a non-stationary and never-ending stream of data. The key to such learning process is to overc… (see more)ome the catastrophic forgetting of previously seen data, which is a well known problem of neural networks. To prevent forgetting, a replay buffer is usually employed to store the previous data for the purpose of rehearsal. Previous works often depend on task boundary and i.i.d. assumptions to properly select samples for the replay buffer. In this work, we formulate sample selection as a constraint reduction problem based on the constrained optimization view of continual learning. The goal is to select a fixed subset of constraints that best approximate the feasible region defined by the original constraints. We show that it is equivalent to maximizing the diversity of samples in the replay buffer with parameters gradient as the feature. We further develop a greedy alternative that is cheap and efficient. The advantage of the proposed method is demonstrated by comparing to other alternatives under the continual learning setting. Further comparisons are made against state of the art methods that rely on task boundaries which show comparable or even better results for our method.
arxiv.org