Pierre Marion

Pierre Marion, Lénaïc Chizat

The largest eigenvalue of the Hessian, or sharpness, of neural networks is a key quantity to understand their optimization dynamics. In this paper, we study the sharpness of deep linear networks for univariate regression. Minimizers can have arbitrarily large sharpness, but not an arbitrarily small one. Indeed, we show a lower bound on the sharpness of minimizers, which grows linearly with depth. We then study the properties of the minimizer found by gradient flow, which is the limit of gradient descent with vanishing learning rate. We show an implicit regularization towards flat minima: the sharpness of the minimizer is no more than a constant times the lower bound. The constant depends on the condition number of the data covariance matrix, but not on width or depth. This result is proven both for a small-scale initialization and a residual initialization. Results of independent interest are shown in both cases. For small-scale initialization, we show that the learned weight matrices are approximately rank-one and that their singular vectors align. For residual initialization, convergence of the gradient flow for a Gaussian initialization of the residual network is proven. Numerical experiments illustrate our results and connect them to gradient descent with non-vanishing learning rate.

Lénaïc Chizat, Pierre Marion, Yerkin Yesbay

Dropout is a standard training technique for neural networks that consists of randomly deactivating units at each step of their gradient-based training. It is known to improve performance in many settings, including in the large-scale training of language or vision models. As a first step towards understanding the role of dropout in large neural networks, we study the large-width asymptotics of gradient descent with dropout on two-layer neural networks with the mean-field initialization scale. We obtain a rich asymptotic phase diagram that exhibits five distinct nondegenerate phases depending on the relative magnitudes of the dropout rate, the learning rate, and the width. Notably, we find that the well-studied "penalty" effect of dropout only persists in the limit with impractically small learning rates of order $O(1/\text{width})$. For larger learning rates, this effect disappears and in the limit, dropout is equivalent to a "random geometry" technique, where the gradients are thinned randomly after the forward and backward pass have been computed. In this asymptotic regime, the limit is described by a mean-field jump process where the neurons' update times follow independent Poisson or Bernoulli clocks (depending on whether the learning rate vanishes or not). For some of the phases, we obtain a description of the limit dynamics both in path-space and in distribution-space. The convergence proofs involve a mix of tools from mean-field particle systems and stochastic processes. Together, our results lay the groundwork for a renewed theoretical understanding of dropout in large-scale neural networks.

Pierre Marion, Paweł Krzysztof Nowak, Francesco Piccinno

We tackle the problem of weakly-supervised conversational Question Answering over large Knowledge Graphs using a neural semantic parsing approach. We introduce a new Logical Form (LF) grammar that can model a wide range of queries on the graph while remaining sufficiently simple to generate supervision data efficiently. Our Transformer-based model takes a JSON-like structure as input, allowing us to easily incorporate both Knowledge Graph and conversational contexts. This structured input is transformed to lists of embeddings and then fed to standard attention layers. We validate our approach, both in terms of grammar coverage and LF execution accuracy, on two publicly available datasets, CSQA and ConvQuestions, both grounded in Wikidata. On CSQA, our approach increases the coverage from $80\%$ to $96.2\%$, and the LF execution accuracy from $70.6\%$ to $75.6\%$, with respect to previous state-of-the-art results. On ConvQuestions, we achieve competitive results with respect to the state-of-the-art.

Adeline Fermanian, Pierre Marion, Jean-Philippe Vert et al.

Building on the interpretation of a recurrent neural network (RNN) as a continuous-time neural differential equation, we show, under appropriate conditions, that the solution of a RNN can be viewed as a linear function of a specific feature set of the input sequence, known as the signature. This connection allows us to frame a RNN as a kernel method in a suitable reproducing kernel Hilbert space. As a consequence, we obtain theoretical guarantees on generalization and stability for a large class of recurrent networks. Our results are illustrated on simulated datasets.