Soufiane Hayou

Soufiane Hayou, Greg Yang · microsoft-research

We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by $1/\sqrt{depth}$ (the only nontrivial scaling), result in the same covariance structure no matter how that limit is taken. This explains why the standard infinite-width-then-depth approach provides practical insights even for networks with depth of the same order as width. We also demonstrate that the pre-activations, in this case, have Gaussian distributions which has direct applications in Bayesian deep learning. We conduct extensive simulations that show an excellent match with our theoretical findings.

Jiayuan Ye, Anastasia Borovykh, Soufiane Hayou et al.

We introduce an analytical framework to quantify the changes in a machine learning algorithm's output distribution following the inclusion of a few data points in its training set, a notion we define as leave-one-out distinguishability (LOOD). This is key to measuring data **memorization** and information **leakage** as well as the **influence** of training data points in machine learning. We illustrate how our method broadens and refines existing empirical measures of memorization and privacy risks associated with training data. We use Gaussian processes to model the randomness of machine learning algorithms, and validate LOOD with extensive empirical analysis of leakage using membership inference attacks. Our analytical framework enables us to investigate the causes of leakage and where the leakage is high. For example, we analyze the influence of activation functions, on data memorization. Additionally, our method allows us to identify queries that disclose the most information about the training data in the leave-one-out setting. We illustrate how optimal queries can be used for accurate **reconstruction** of training data.

Soufiane Hayou

In this paper, we study the infinite-depth limit of finite-width residual neural networks with random Gaussian weights. With proper scaling, we show that by fixing the width and taking the depth to infinity, the pre-activations converge in distribution to a zero-drift diffusion process. Unlike the infinite-width limit where the pre-activation converge weakly to a Gaussian random variable, we show that the infinite-depth limit yields different distributions depending on the choice of the activation function. We document two cases where these distributions have closed-form (different) expressions. We further show an intriguing change of regime phenomenon of the post-activation norms when the width increases from 3 to 4. Lastly, we study the sequential limit infinite-depth-then-infinite-width and compare it with the more commonly studied infinite-width-then-infinite-depth limit.

Fadhel Ayed, Soufiane Hayou

Data pruning algorithms are commonly used to reduce the memory and computational cost of the optimization process. Recent empirical results reveal that random data pruning remains a strong baseline and outperforms most existing data pruning methods in the high compression regime, i.e., where a fraction of $30\%$ or less of the data is kept. This regime has recently attracted a lot of interest as a result of the role of data pruning in improving the so-called neural scaling laws; in [Sorscher et al.], the authors showed the need for high-quality data pruning algorithms in order to beat the sample power law. In this work, we focus on score-based data pruning algorithms and show theoretically and empirically why such algorithms fail in the high compression regime. We demonstrate ``No Free Lunch" theorems for data pruning and present calibration protocols that enhance the performance of existing pruning algorithms in this high compression regime using randomization.

Xi Wang, Soufiane Hayou, Eric Nalisnick

Mixture of Experts (MoEs) are now ubiquitous in large language models, yet the mechanisms behind their "expert specialization" remain poorly understood. We show that, since MoE routers are linear maps, hidden state similarity is both necessary and sufficient to explain expert usage similarity, and specialization is therefore an emergent property of the representation space, not of the routing architecture itself. We confirm this at both token and sequence level across five pre-trained models. We additionally prove that load-balancing loss suppresses shared hidden state directions to maintain routing diversity, which might provide a theoretical explanation for specialization collapse under less diverse data, e.g. small batch. Despite this clean mechanistic account, we find that specialization patterns in pre-trained MoEs resist human interpretation: expert overlap between different models answering the same question is no higher than between entirely different questions ($\sim$60\%); prompt-level routing does not predict rollout-level routing; and deeper layers exhibit near-identical expert activation across semantically unrelated inputs, especially in reasoning models. We conclude that, while the efficiency perspective of MoEs is well understood, understanding expert specialization is at least as hard as understanding LLM hidden state geometry, a long-standing open problem in the literature.

Soufiane Hayou

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that depend on a neural network model) as width and depth go to infinity (in some sense), and eventually identify settings under which commutativity holds, i.e. the neural function tends to the same limit no matter how width and depth limits are taken. In this paper, we formally introduce and define the commutativity framework, and discuss its implications on neural network design and scaling. We study commutativity for the neural covariance kernel which reflects how network layers separate data. Our findings extend previous results established in [55] by showing that taking the width and depth to infinity in a deep neural network with skip connections, when branches are suitably scaled to avoid exploding behaviour, result in the same covariance structure no matter how that limit is taken. This has a number of theoretical and practical implications that we discuss in the paper. The proof techniques in this paper are novel and rely on tools that are more accessible to readers who are not familiar with stochastic calculus (used in the proofs of WD(I))).

Soufiane Hayou

The Riemann hypothesis (RH) is a long-standing open problem in mathematics. It conjectures that non-trivial zeros of the zeta function all have real part equal to 1/2. The extent of the consequences of RH is far-reaching and touches a wide spectrum of topics including the distribution of prime numbers, the growth of arithmetic functions, the growth of Euler totient, etc. In this note, we revisit and extend an old analytic criterion of the RH known as the Nyman-Beurling criterion which connects the RH to a minimization problem that involves a special class of neural networks. This note is intended for an audience unfamiliar with RH. A gentle introduction to RH is provided.

Yuxin Ma, Nan Chen, Mateo Díaz et al.

Modern large-scale neural networks are often trained and released in multiple sizes to accommodate diverse inference budgets. To improve efficiency, recent work has explored model upscaling: initializing larger models from trained smaller ones in order to transfer knowledge and accelerate convergence. However, this method can be sensitive to hyperparameters that need to be tuned at the target upscaled model size, which is prohibitively costly to do directly. It remains unclear whether the most common workaround -- tuning on smaller models and extrapolating via hyperparameter scaling laws -- is still sound when using upscaling. We address this with principled approaches to upscaling with respect to model widths and efficiently tuning hyperparameters in this setting. First, motivated by $μ$P and any-dimensional architectures, we introduce a general upscaling method applicable to a broad range of architectures and optimizers, backed by theory guaranteeing that models are equivalent to their widened versions and allowing for rigorous analysis of infinite-width limits. Second, we extend the theory of $μ$Transfer to a hyperparameter transfer technique for models upscaled using our method and empirically demonstrate that this method is effective on realistic datasets and architectures.

Soufiane Hayou, Nikhil Ghosh, Bin Yu

In this paper, we show that Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021) leads to suboptimal finetuning of models with large width (embedding dimension). This is due to the fact that adapter matrices A and B in LoRA are updated with the same learning rate. Using scaling arguments for large width networks, we demonstrate that using the same learning rate for A and B does not allow efficient feature learning. We then show that this suboptimality of LoRA can be corrected simply by setting different learning rates for the LoRA adapter matrices A and B with a well-chosen ratio. We call this proposed algorithm LoRA$+$. In our extensive experiments, LoRA$+$ improves performance (1-2 $\%$ improvements) and finetuning speed (up to $\sim$ 2X SpeedUp), at the same computational cost as LoRA.

Mohamed El Amine Seddik, Suei-Wen Chen, Soufiane Hayou et al.

The phenomenon of model collapse, introduced in (Shumailov et al., 2023), refers to the deterioration in performance that occurs when new models are trained on synthetic data generated from previously trained models. This recursive training loop makes the tails of the original distribution disappear, thereby making future-generation models forget about the initial (real) distribution. With the aim of rigorously understanding model collapse in language models, we consider in this paper a statistical model that allows us to characterize the impact of various recursive training scenarios. Specifically, we demonstrate that model collapse cannot be avoided when training solely on synthetic data. However, when mixing both real and synthetic data, we provide an estimate of a maximal amount of synthetic data below which model collapse can eventually be avoided. Our theoretical conclusions are further supported by empirical validations.

Nan Chen, Soledad Villar, Soufiane Hayou

Low-Rank Adaptation (LoRA) is a standard tool for parameter-efficient finetuning of large models. While it induces a small memory footprint, its training dynamics can be surprisingly complex as they depend on several hyperparameters such as initialization, adapter rank, and learning rate. In particular, it is unclear how the optimal learning rate scales with adapter rank, which forces practitioners to re-tune the learning rate whenever the rank is changed. In this paper, we introduce Maximal-Update Adaptation ($μ$A), a theoretical framework that characterizes how the "optimal" learning rate should scale with model width and adapter rank to produce stable, non-vanishing feature updates under standard configurations. $μ$A is inspired from the Maximal-Update Parametrization ($μ$P) in pretraining. Our analysis leverages techniques from hyperparameter transfer and reveals that the optimal learning rate exhibits different scaling patterns depending on initialization and LoRA scaling factor. Specifically, we identify two regimes: one where the optimal learning rate remains roughly invariant across ranks, and another where it scales inversely with rank. We further identify a configuration that allows learning rate transfer from LoRA to full finetuning, drastically reducing the cost of learning rate tuning for full finetuning. Experiments across language, vision, vision--language, image generation, and reinforcement learning tasks validate our scaling rules and show that learning rates tuned on LoRA transfer reliably to full finetuning.

Soufiane Hayou

We provide the first proof of learning rate transfer with width in a linear multi-layer perceptron (MLP) parametrized with $μ$P, a neural network parameterization designed to ``maximize'' feature learning in the infinite-width limit. We show that under $μP$, the optimal learning rate converges to a \emph{non-zero constant} as width goes to infinity, providing a theoretical explanation to learning rate transfer. In contrast, we show that this property fails to hold under alternative parametrizations such as Standard Parametrization (SP) and Neural Tangent Parametrization (NTP). We provide intuitive proofs and support the theoretical findings with extensive empirical results.

Aymane El Firdoussi, Mohamed El Amine Seddik, Soufiane Hayou et al.

Synthetic data has gained attention for training large language models, but poor-quality data can harm performance (see, e.g., Shumailov et al. (2023); Seddik et al. (2024)). A potential solution is data pruning, which retains only high-quality data based on a score function (human or machine feedback). Previous work Feng et al. (2024) analyzed models trained on synthetic data as sample size increases. We extend this by using random matrix theory to derive the performance of a binary classifier trained on a mix of real and pruned synthetic data in a high dimensional setting. Our findings identify conditions where synthetic data could improve performance, focusing on the quality of the generative model and verification strategy. We also show a smooth phase transition in synthetic label noise, contrasting with prior sharp behavior in infinite sample limits. Experiments with toy models and large language models validate our theoretical results.

Soufiane Hayou, Liyuan Liu

Pretraining large language models is a costly process. To make this process more efficient, several methods have been proposed to optimize model architecture/parametrization and hardware use. On the parametrization side, $μP$ (Maximal Update Parametrization) parametrizes model weights and learning rate (LR) in a way that makes hyperparameters (HPs) transferable with width (embedding dimension): HPs can be tuned for a small model and used for larger models without additional tuning. While $μ$P showed impressive results in practice, recent empirical studies have reported conflicting observations when applied to LLMs. One limitation of the theory behind $μ$P is the fact that input dimension (vocabulary size in LLMs) is considered fixed when taking the width to infinity. This is unrealistic since vocabulary size is generally much larger than width in practice. In this work, we provide a theoretical analysis of the effect of vocabulary size on training dynamics, and subsequently show that as vocabulary size increases, the training dynamics \emph{interpolate between the $μ$P regime and another regime that we call Large Vocab (LV) Regime}, where optimal scaling rules are different from those predicted by $μ$P. Our analysis reveals that in the LV regime, the optimal embedding LR to hidden LR ratio should roughly scale as $Θ(\sqrt{width})$, surprisingly close to the empirical findings previously reported in the literature, and different from the $Θ(width)$ ratio predicted by $μ$P. We conduct several experiments to validate our theory, and pretrain a 1B model from scratch to show the benefit of our suggested scaling rule for the embedding LR.

Soufiane Hayou, Nikhil Ghosh, Bin Yu

Low-Rank Adaptation (LoRA) is a widely used finetuning method for large models. Its small memory footprint allows practitioners to adapt large models to specific tasks at a fraction of the cost of full finetuning. Different modifications have been proposed to enhance its efficiency by, for example, setting the learning rate, the rank, and the initialization. Another improvement axis is adapter placement strategy: when using LoRA, practitioners usually pick module types to adapt with LoRA, such as Query and Key modules. Few works have studied the problem of adapter placement, with nonconclusive results: original LoRA paper suggested placing adapters in attention modules, while other works suggested placing them in the MLP modules. Through an intuitive theoretical analysis, we introduce PLoP (Precise LoRA Placement), a lightweight method that allows automatic identification of module types where LoRA adapters should be placed, given a pretrained model and a finetuning task. We demonstrate that PLoP consistently outperforms, and in the worst case competes, with commonly used placement strategies through comprehensive experiments on supervised finetuning and reinforcement learning for reasoning.

Benjamin Dadoun, Soufiane Hayou, Hanan Salam et al.

Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian. Prior work has identified critical initialization schemes that ensure Jacobian stability, but these analyses are typically restricted to fully connected networks with i.i.d. weights. In this work, we go significantly beyond these limitations: we establish a general stability theorem for deep neural networks that accommodates sparsity (such as that introduced by pruning) and non-i.i.d., weakly correlated weights (e.g. induced by training). Our results rely on recent advances in random matrix theory, and provide rigorous guarantees for spectral stability in a much broader class of network models. This extends the theoretical foundation for initialization schemes in modern neural networks with structured and dependent randomness.

Soufiane Hayou, Nikhil Ghosh, Bin Yu

In this paper, we study the role of initialization in Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021). Essentially, to start from the pretrained model as initialization for finetuning, one can either initialize B to zero and A to random (default initialization in PEFT package), or vice-versa. In both cases, the product BA is equal to zero at initialization, which makes finetuning starts from the pretrained model. These two initialization schemes are seemingly similar. They should in-principle yield the same performance and share the same optimal learning rate. We demonstrate that this is an incorrect intuition and that the first scheme (initializing B to zero and A to random) on average yields better performance compared to the other scheme. Our theoretical analysis shows that the reason behind this might be that the first initialization allows the use of larger learning rates (without causing output instability) compared to the second initialization, resulting in more efficient learning of the first scheme. We validate our results with extensive experiments on LLMs.

Fusheng Liu, Haizhao Yang, Soufiane Hayou et al.

Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyzing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results.

Soufiane Hayou, Bobby He, Gintare Karolina Dziugaite

We study an approach to learning pruning masks by optimizing the expected loss of stochastic pruning masks, i.e., masks which zero out each weight independently with some weight-specific probability. We analyze the training dynamics of the induced stochastic predictor in the setting of linear regression, and observe a data-adaptive L1 regularization term, in contrast to the dataadaptive L2 regularization term known to underlie dropout in linear regression. We also observe a preference to prune weights that are less well-aligned with the data labels. We evaluate probabilistic fine-tuning for optimizing stochastic pruning masks for neural networks, starting from masks produced by several baselines. In each case, we see improvements in test error over baselines, even after we threshold fine-tuned stochastic pruning masks. Finally, since a stochastic pruning mask induces a stochastic neural network, we consider training the weights and/or pruning probabilities simultaneously to minimize a PAC-Bayes bound on generalization error. Using data-dependent priors, we obtain a selfbounded learning algorithm with strong performance and numerically tight bounds. In the linear model, we show that a PAC-Bayes generalization error bound is controlled by the magnitude of the change in feature alignment between the 'prior' and 'posterior' data.

Yizhang Lou, Chris Mingard, Yoonsoo Nam et al.

Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the euclidean product of the tangent features matrix and the data labels matrix). The curve of the alignment as a function of layer index (generally) exhibits an ascent-descent pattern where the maximum is reached for some hidden layer. In this work, we provide the first explanation for this phenomenon. We introduce the Equilibrium Hypothesis which connects this alignment pattern to signal propagation in deep neural networks. Our experiments demonstrate an excellent match with the theoretical predictions.

Soufiane Hayou, Fadhel Ayed

Regularization plays a major role in modern deep learning. From classic techniques such as L1,L2 penalties to other noise-based methods such as Dropout, regularization often yields better generalization properties by avoiding overfitting. Recently, Stochastic Depth (SD) has emerged as an alternative regularization technique for residual neural networks (ResNets) and has proven to boost the performance of ResNet on many tasks [Huang et al., 2016]. Despite the recent success of SD, little is known about this technique from a theoretical perspective. This paper provides a hybrid analysis combining perturbation analysis and signal propagation to shed light on different regularization effects of SD. Our analysis allows us to derive principled guidelines for choosing the survival rates used for training with SD.

Soufiane Hayou, Eugenio Clerico, Bobby He et al.

Deep ResNet architectures have achieved state of the art performance on many tasks. While they solve the problem of gradient vanishing, they might suffer from gradient exploding as the depth becomes large (Yang et al. 2017). Moreover, recent results have shown that ResNet might lose expressivity as the depth goes to infinity (Yang et al. 2017, Hayou et al. 2019). To resolve these issues, we introduce a new class of ResNet architectures, called Stable ResNet, that have the property of stabilizing the gradient while ensuring expressivity in the infinite depth limit.

Soufiane Hayou, Jean-Francois Ton, Arnaud Doucet et al.

Overparameterized Neural Networks (NN) display state-of-the-art performance. However, there is a growing need for smaller, energy-efficient, neural networks tobe able to use machine learning applications on devices with limited computational resources. A popular approach consists of using pruning techniques. While these techniques have traditionally focused on pruning pre-trained NN (LeCun et al.,1990; Hassibi et al., 1993), recent work by Lee et al. (2018) has shown promising results when pruning at initialization. However, for Deep NNs, such procedures remain unsatisfactory as the resulting pruned networks can be difficult to train and, for instance, they do not prevent one layer from being fully pruned. In this paper, we provide a comprehensive theoretical analysis of Magnitude and Gradient based pruning at initialization and training of sparse architectures. This allows us to propose novel principled approaches which we validate experimentally on a variety of NN architectures.

Soufiane Hayou, Arnaud Doucet, Judith Rousseau

Recent work by Jacot et al. (2018) has shown that training a neural network using gradient descent in parameter space is related to kernel gradient descent in function space with respect to the Neural Tangent Kernel (NTK). Lee et al. (2019) built on this result by establishing that the output of a neural network trained using gradient descent can be approximated by a linear model when the network width is large. Indeed, under regularity conditions, the NTK converges to a time-independent kernel in the infinite-width limit. This regime is often called the NTK regime. In parallel, recent works on signal propagation (Poole et al., 2016; Schoenholz et al., 2017; Hayou et al., 2019a) studied the impact of the initialization and the activation function on signal propagation in deep neural networks. In this paper, we connect these two theories by quantifying the impact of the initialization and the activation function on the NTK when the network depth becomes large. In particular, we provide a comprehensive analysis of the convergence rates of the NTK regime to the infinite depth regime.

Soufiane Hayou, Arnaud Doucet, Judith Rousseau

The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Samuel et al (2017) who showed that for deep feedforward neural networks only a specific choice of hyperparameters known as the `Edge of Chaos' can lead to good performance. While the work by Samuel et al (2017) discuss trainability issues, we focus here on training acceleration and overall performance. We give a comprehensive theoretical analysis of the Edge of Chaos and show that we can indeed tune the initialization parameters and the activation function in order to accelerate the training and improve the performance.

Soufiane Hayou, Arnaud Doucet, Judith Rousseau

The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Schoenholz et al. (2017) who showed that for deep feedforward neural networks only a specific choice of hyperparameters known as the `edge of chaos' can lead to good performance. We complete this analysis by providing quantitative results showing that, for a class of ReLU-like activation functions, the information propagates indeed deeper for an initialization at the edge of chaos. By further extending this analysis, we identify a class of activation functions that improve the information propagation over ReLU-like functions. This class includes the Swish activation, $φ_{swish}(x) = x \cdot \text{sigmoid}(x)$, used in Hendrycks & Gimpel (2016), Elfwing et al. (2017) and Ramachandran et al. (2017). This provides a theoretical grounding for the excellent empirical performance of $φ_{swish}$ observed in these contributions. We complement those previous results by illustrating the benefit of using a random initialization on the edge of chaos in this context.