Clément Hongler

ICML Conference 2024 Conference Paper

Arrows of Time for Large Language Models

• Vassilis Papadopoulos
• Jérémie Wenger
• Clément Hongler

We study the probabilistic modeling performed by Autoregressive Large Language Models (LLMs) through the angle of time directionality, addressing a question first raised in (Shannon, 1951). For large enough models, we empirically find a time asymmetry in their ability to learn natural language: a difference in the average log-perplexity when trying to predict the next token versus when trying to predict the previous one. This difference is at the same time subtle and very consistent across various modalities (language, model size, training time, .. .). Theoretically, this is surprising: from an information-theoretic point of view, there should be no such difference. We provide a theoretical framework to explain how such an asymmetry can appear from sparsity and computational complexity considerations, and outline a number of perspectives opened by our results.

ICML Conference 2021 Conference Paper

Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances

• Berfin Simsek
• François Ged
• Arthur Jacot
• Francesco Spadaro
• Clément Hongler
• Wulfram Gerstner
• Johanni Brea

We study how permutation symmetries in overparameterized multi-layer neural networks generate ‘symmetry-induced’ critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, \ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! \cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r, m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the vastly overparameterized regime ($h \gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.

STOC Conference 2021 Invited Paper

Neural tangent kernel: convergence and generalization in neural networks (invited paper)

• Arthur Jacot
• Franck Gabriel
• Clément Hongler

ICML Conference 2020 Conference Paper

Implicit Regularization of Random Feature Models

• Arthur Jacot
• Berfin Simsek
• Francesco Spadaro
• Clément Hongler
• Franck Gabriel

Random Features (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $\lambda$, we show that the average (i. e. expected) RF predictor is close to a KRR predictor with an \emph{effective ridge} $\tilde{\lambda}$. We show that $\tilde{\lambda} > \lambda$ and $\tilde{\lambda} \searrow \lambda$ monotonically as $P$ grows, thus revealing the \emph{implicit regularization effect} of finite RF sampling. We then compare the risk (i. e. test error) of the $\tilde{\lambda}$-KRR predictor with the average risk of the $\lambda$-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average $\lambda$-RF predictor and $\tilde{\lambda}$-KRR predictor.

ICLR Conference 2020 Conference Paper

The asymptotic spectrum of the Hessian of DNN throughout training

• Arthur Jacot
• Franck Gabriel
• Clément Hongler

The dynamics of DNNs during gradient descent is described by the so-called Neural Tangent Kernel (NTK). In this article, we show that the NTK allows one to gain precise insight into the Hessian of the cost of DNNs: we obtain a full characterization of the asymptotics of the spectrum of the Hessian, at initialization and during training.