Pierre Wolinski

Pierre Wolinski, Julyan Arbel

The study of feature propagation at initialization in neural networks lies at the root of numerous initialization designs. An assumption very commonly made in the field states that the pre-activations are Gaussian. Although this convenient Gaussian hypothesis can be justified when the number of neurons per layer tends to infinity, it is challenged by both theoretical and experimental works for finite-width neural networks. Our major contribution is to construct a family of pairs of activation functions and initialization distributions that ensure that the pre-activations remain Gaussian throughout the network's depth, even in narrow neural networks. In the process, we discover a set of constraints that a neural network should fulfill to ensure Gaussian pre-activations. Additionally, we provide a critical review of the claims of the Edge of Chaos line of works and build an exact Edge of Chaos analysis. We also propose a unified view on pre-activations propagation, encompassing the framework of several well-known initialization procedures. Finally, our work provides a principled framework for answering the much-debated question: is it desirable to initialize the training of a neural network whose pre-activations are ensured to be Gaussian? Our code is available on GitHub: https://github.com/p-wol/gaussian-preact/ .

Minh Tri Lê, Pierre Wolinski, Julyan Arbel

The field of Tiny Machine Learning (TinyML) has gained significant attention due to its potential to enable intelligent applications on resource-constrained devices. This review provides an in-depth analysis of the advancements in efficient neural networks and the deployment of deep learning models on ultra-low power microcontrollers (MCUs) for TinyML applications. It begins by introducing neural networks and discussing their architectures and resource requirements. It then explores MEMS-based applications on ultra-low power MCUs, highlighting their potential for enabling TinyML on resource-constrained devices. The core of the review centres on efficient neural networks for TinyML. It covers techniques such as model compression, quantization, and low-rank factorization, which optimize neural network architectures for minimal resource utilization on MCUs. The paper then delves into the deployment of deep learning models on ultra-low power MCUs, addressing challenges such as limited computational capabilities and memory resources. Techniques like model pruning, hardware acceleration, and algorithm-architecture co-design are discussed as strategies to enable efficient deployment. Lastly, the review provides an overview of current limitations in the field, including the trade-off between model complexity and resource constraints. Overall, this review paper presents a comprehensive analysis of efficient neural networks and deployment strategies for TinyML on ultra-low-power MCUs. It identifies future research directions for unlocking the full potential of TinyML applications on resource-constrained devices.

Michael Arbel, Romain Menegaux, Pierre Wolinski

This work studies the global convergence and implicit bias of Gauss Newton's (GN) when optimizing over-parameterized one-hidden layer networks in the mean-field regime. We first establish a global convergence result for GN in the continuous-time limit exhibiting a faster convergence rate compared to GD due to improved conditioning. We then perform an empirical study on a synthetic regression task to investigate the implicit bias of GN's method. While GN is consistently faster than GD in finding a global optimum, the learned model generalizes well on test data when starting from random initial weights with a small variance and using a small step size to slow down convergence. Specifically, our study shows that such a setting results in a hidden learning phenomenon, where the dynamics are able to recover features with good generalization properties despite the model having sub-optimal training and test performances due to an under-optimized linear layer. This study exhibits a trade-off between the convergence speed of GN and the generalization ability of the learned solution.

Léonard Blier, Pierre Wolinski, Yann Ollivier

Hyperparameter tuning is a bothersome step in the training of deep learning models. One of the most sensitive hyperparameters is the learning rate of the gradient descent. We present the 'All Learning Rates At Once' (Alrao) optimization method for neural networks: each unit or feature in the network gets its own learning rate sampled from a random distribution spanning several orders of magnitude. This comes at practically no computational cost. Perhaps surprisingly, stochastic gradient descent (SGD) with Alrao performs close to SGD with an optimally tuned learning rate, for various architectures and problems. Alrao could save time when testing deep learning models: a range of models could be quickly assessed with Alrao, and the most promising models could then be trained more extensively. This text comes with a PyTorch implementation of the method, which can be plugged on an existing PyTorch model: https://github.com/leonardblier/alrao .

Pierre Wolinski

When training large models, such as neural networks, the full derivatives of order 2 and beyond are usually inaccessible, due to their computational cost. Therefore, among the second-order optimization methods, it is common to bypass the computation of the Hessian by using first-order information, such as the gradient of the parameters (e.g., quasi-Newton methods) or the activations (e.g., K-FAC). In this paper, we focus on the exact and explicit computation of projections of the Hessian and higher-order derivatives on well-chosen subspaces relevant for optimization. Namely, for a given partition of the set of parameters, we compute tensors that can be seen as "higher-order derivatives according to the partition", at a reasonable cost as long as the number of subsets of the partition remains small. Then, we give some examples of how these tensors can be used. First, we show how to compute a learning rate per subset of parameters, which can be used for hyperparameter tuning. Second, we show how to use these tensors at order 2 to construct an optimization method that uses information contained in the Hessian. Third, we show how to use these tensors at order 3 (information contained in the third derivative of the loss) to regularize this optimization method. The resulting training step has several interesting properties, including: it takes into account long-range interactions between the layers of the trained neural network, which is usually not the case in similar methods (e.g., K-FAC); the trajectory of the optimization is invariant under affine layer-wise reparameterization.

Pierre Wolinski, Guillaume Charpiat, Yann Ollivier

In machine learning, it is common to optimize the parameters of a probabilistic model, modulated by an ad hoc regularization term that penalizes some values of the parameters. Regularization terms appear naturally in Variational Inference, a tractable way to approximate Bayesian posteriors: the loss to optimize contains a Kullback--Leibler divergence term between the approximate posterior and a Bayesian prior. We fully characterize the regularizers that can arise according to this procedure, and provide a systematic way to compute the prior corresponding to a given penalty. Such a characterization can be used to discover constraints over the penalty function, so that the overall procedure remains Bayesian.