Pierre Ablin

Jason Ramapuram, Federico Danieli, Eeshan Dhekane et al. · apple-ml, berkeley

Attention is a key part of the transformer architecture. It is a sequence-to-sequence mapping that transforms each sequence element into a weighted sum of values. The weights are typically obtained as the softmax of dot products between keys and queries. Recent work has explored alternatives to softmax attention in transformers, such as ReLU and sigmoid activations. In this work, we revisit sigmoid attention and conduct an in-depth theoretical and empirical analysis. Theoretically, we prove that transformers with sigmoid attention are universal function approximators and benefit from improved regularity compared to softmax attention. Through detailed empirical analysis, we identify stabilization of large initial attention norms during the early stages of training as a crucial factor for the successful training of models with sigmoid attention, outperforming prior attempts. We also introduce FLASHSIGMOID, a hardware-aware and memory-efficient implementation of sigmoid attention yielding a 17% inference kernel speed-up over FLASHATTENTION2 on H100 GPUs. Experiments across language, vision, and speech show that properly normalized sigmoid attention matches the strong performance of softmax attention on a wide range of domains and scales, which previous attempts at sigmoid attention were unable to fully achieve. Our work unifies prior art and establishes best practices for sigmoid attention as a drop-in softmax replacement in transformers.

Thomas Moreau, Mathurin Massias, Alexandre Gramfort et al. · berkeley

Numerical validation is at the core of machine learning research as it allows to assess the actual impact of new methods, and to confirm the agreement between theory and practice. Yet, the rapid development of the field poses several challenges: researchers are confronted with a profusion of methods to compare, limited transparency and consensus on best practices, as well as tedious re-implementation work. As a result, validation is often very partial, which can lead to wrong conclusions that slow down the progress of research. We propose Benchopt, a collaborative framework to automate, reproduce and publish optimization benchmarks in machine learning across programming languages and hardware architectures. Benchopt simplifies benchmarking for the community by providing an off-the-shelf tool for running, sharing and extending experiments. To demonstrate its broad usability, we showcase benchmarks on three standard learning tasks: $\ell_2$-regularized logistic regression, Lasso, and ResNet18 training for image classification. These benchmarks highlight key practical findings that give a more nuanced view of the state-of-the-art for these problems, showing that for practical evaluation, the devil is in the details. We hope that Benchopt will foster collaborative work in the community hence improving the reproducibility of research findings.

Dan Busbridge, Jason Ramapuram, Pierre Ablin et al. · apple-ml, berkeley

Preserving training dynamics across batch sizes is an important tool for practical machine learning as it enables the trade-off between batch size and wall-clock time. This trade-off is typically enabled by a scaling rule, for example, in stochastic gradient descent, one should scale the learning rate linearly with the batch size. Another important machine learning tool is the model EMA, a functional copy of a target model, whose parameters move towards those of its target model according to an Exponential Moving Average (EMA) at a rate parameterized by a momentum hyperparameter. This model EMA can improve the robustness and generalization of supervised learning, stabilize pseudo-labeling, and provide a learning signal for Self-Supervised Learning (SSL). Prior works have not considered the optimization of the model EMA when performing scaling, leading to different training dynamics across batch sizes and lower model performance. In this work, we provide a scaling rule for optimization in the presence of a model EMA and demonstrate the rule's validity across a range of architectures, optimizers, and data modalities. We also show the rule's validity where the model EMA contributes to the optimization of the target model, enabling us to train EMA-based pseudo-labeling and SSL methods at small and large batch sizes. For SSL, we enable training of BYOL up to batch size 24,576 without sacrificing performance, a 6$\times$ wall-clock time reduction under idealized hardware settings.

Marco Cuturi, Michal Klein, Pierre Ablin · apple-ml

Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the ``thriftiest'', i.e. such that the averaged cost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the $\ell_2^2$ distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs $c(x, y):=h(x-y)$, where $h:=\tfrac{1}{2}\|\cdot\|_2^2+τ$ and $τ$ is a regularizer. We propose a generalization of the entropic map suitable for $h$, and highlight a surprising link tying it with the Bregman centroids of the divergence $D_h$ generated by $h$, and the proximal operator of $τ$. We show that choosing a sparsity-inducing norm for $τ$ results in maps that apply Occam's razor to transport, in the sense that the displacement vectors $Δ(x):= T(x)-x$ they induce are sparse, with a sparsity pattern that varies depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the $34000$-$d$ space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.

Michal Klein, Aram-Alexandre Pooladian, Pierre Ablin et al. · apple-ml

Given a source and a target probability measure supported on $\mathbb{R}^d$, the Monge problem asks to find the most efficient way to map one distribution to the other. This efficiency is quantified by defining a \textit{cost} function between source and target data. Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, $\ell^2_2(\mathbf{x},\mathbf{y})=\tfrac12|\mathbf{x}-\mathbf{y}|_2^2$. Recently, Cuturi et. al '23 highlighted the benefits of using elastic costs, defined through a regularizer $τ$ as $c(\mathbf{x},\mathbf{y})=\ell^2_2(\mathbf{x},\mathbf{y})+τ(\mathbf{x}-\mathbf{y})$. Such costs shape the \textit{displacements} of Monge maps $T$, i.e., the difference between a source point and its image $T(\mathbf{x})-\mathbf{x})$, by giving them a structure that matches that of the proximal operator of $τ$. In this work, we make two important contributions to the study of elastic costs: (i) For any elastic cost, we propose a numerical method to compute Monge maps that are provably optimal. This provides a much-needed routine to create synthetic problems where the ground truth OT map is known, by analogy to the Brenier theorem, which states that the gradient of any convex potential is always a valid Monge map for the $\ell_2^2$ cost; (ii) We propose a loss to \textit{learn} the parameter $θ$ of a parameterized regularizer $τ_θ$, and apply it in the case where $τ_{A}(\mathbf{z})=|A^\perp \mathbf{z}|^2_2$. This regularizer promotes displacements that lie on a low dimensional subspace of $\mathbb{R}^d$, spanned by the $p$ rows of $A\in\mathbb{R}^{p\times d}$.

Louis Bethune, Victor Turrisi, Bruno Kacper Mlodozeniec et al. · apple-ml, berkeley

Discrete diffusion models have emerged as strong alternatives to autoregressive language models, with recent work initializing and fine-tuning a base unimodal model for bimodal generation. Diverging from previous approaches, we introduce the first tri-modal masked diffusion model pretrained from scratch on text, image-text, and audio-text data. We systematically analyze multimodal scaling laws, modality mixing ratios, noise schedules, and batch-size effects, and we provide optimized inference sampling defaults. Our batch-size analysis yields a novel stochastic differential equation (SDE)-based reparameterization that eliminates the need for tuning the optimal batch size as reported in recent work. This reparameterization decouples the physical batch size, often chosen based on compute constraints (GPU saturation, FLOP efficiency, wall-clock time), from the logical batch size, chosen to balance gradient variance during stochastic optimization. Finally, we pretrain a preliminary 3B-parameter tri-modal model on 6.4T tokens, demonstrating the capabilities of a unified design and achieving strong results in text generation, text-to-image tasks, and text-to-speech tasks. Our work represents the largest-scale systematic open study of multimodal discrete diffusion models conducted to date, providing insights into scaling behaviors across multiple modalities.

Matteo Pagliardini, Pierre Ablin, David Grangier

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show -- quite surprisingly -- that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a $1.3$B parameter AdEMAMix LLM trained on $101$B tokens performs comparably to an AdamW model trained on $197$B tokens ($+95\%$). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

Pierre Ablin, Simon Vary, Bin Gao et al.

Orthogonality constraints naturally appear in many machine learning problems, from principal component analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the objective function while enforcing the constraint. However, enforcing the orthogonality constraint can be the most time-consuming operation in such algorithms. Recently, Ablin & Peyré (2022) proposed the landing algorithm, a method with cheap iterations that does not enforce the orthogonality constraints but is attracted towards the manifold in a smooth manner. This article provides new practical and theoretical developments for the landing algorithm. First, the method is extended to the Stiefel manifold, the set of rectangular orthogonal matrices. We also consider stochastic and variance reduction algorithms when the cost function is an average of many functions. We demonstrate that all these methods have the same rate of convergence as their Riemannian counterparts that exactly enforce the constraint, and converge to the manifold. Finally, our experiments demonstrate the promise of our approach to an array of machine-learning problems that involve orthogonality constraints.

David Grangier, Simin Fan, Skyler Seto et al.

Specialist language models (LMs) focus on a specific task or domain on which they often outperform generalist LMs of the same size. However, the specialist data needed to pretrain these models is only available in limited amount for most tasks. In this work, we build specialist models from large generalist training sets instead. We propose a novel method, ClusteRed Importance SamPling (CRISP). CRISP clusters the generalist dataset and samples from these clusters based on their frequencies in the smaller specialist dataset. It is scalable, suitable for both pretraining and continued pretraining, and works well in multi-task settings. CRISP performs favorably compared to other methods that adjust the training distribution of the generalist data with guidance from the limited domain-specific data. Our findings demonstrate improvements across different domains in terms of language modeling perplexity and accuracy on multiple-choice question tasks. We also present ablation studies that examine the impact of dataset sizes, clustering configurations, and model sizes.

David Grangier, Pierre Ablin, Awni Hannun

Large neural networks pretrained on web-scale corpora are central to modern machine learning. In this paradigm, the distribution of the large, heterogeneous pretraining data rarely matches that of the application domain. This work considers modifying the pretraining distribution in the case where one has a small sample of data reflecting the targeted test conditions. We propose an algorithm motivated by a recent formulation of this setting as an online, bilevel optimization problem. With scalability in mind, our algorithm prioritizes computing gradients at training points which are likely to most improve the loss on the targeted distribution. Empirically, we show that in some cases this approach is beneficial over existing strategies from the domain adaptation literature but may not succeed in other cases. We propose a simple test to evaluate when our approach can be expected to work well and point towards further research to address current limitations.

Valérie Castin, Kimia Nadjahi, Pierre Ablin et al.

Low-Rank Adaptation (LoRA) is the most widely adopted method for fine-tuning large language models. Notably, LoRA is inherently overparameterized: multiple pairs of low-rank factors can yield the same adapted weight matrix. We show--both theoretically and empirically--that these pairs exhibit significantly different condition numbers. As a result, converging to different loss minimizers directly impacts the convergence rate of LoRA. Building on this observation, we introduce Balanced Low-Rank Adaptation (BaLoRA), a variant of LoRA that projects iterates onto a balanced manifold. This manifold improves the conditioning of the loss landscape while preserving the adapted matrix. The projection step is computationally lightweight and integrates seamlessly into existing fine-tuning pipelines. Empirically, BaLoRA converges faster than standard LoRA and achieves superior performance across a range of fine-tuning tasks.

Michael E. Sander, Pierre Ablin, Gabriel Peyré

Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks (ResNets). We investigate whether the discrete dynamics defined by a ResNet are close to the continuous one of a Neural ODE. We first quantify the distance between the ResNet's hidden state trajectory and the solution of its corresponding Neural ODE. Our bound is tight and, on the negative side, does not go to 0 with depth N if the residual functions are not smooth with depth. On the positive side, we show that this smoothness is preserved by gradient descent for a ResNet with linear residual functions and small enough initial loss. It ensures an implicit regularization towards a limit Neural ODE at rate 1 over N, uniformly with depth and optimization time. As a byproduct of our analysis, we consider the use of a memory-free discrete adjoint method to train a ResNet by recovering the activations on the fly through a backward pass of the network, and show that this method theoretically succeeds at large depth if the residual functions are Lipschitz with the input. We then show that Heun's method, a second order ODE integration scheme, allows for better gradient estimation with the adjoint method when the residual functions are smooth with depth. We experimentally validate that our adjoint method succeeds at large depth, and that Heun method needs fewer layers to succeed. We finally use the adjoint method successfully for fine-tuning very deep ResNets without memory consumption in the residual layers.

Mathieu Dagréou, Thomas Moreau, Samuel Vaiter et al.

Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity.

Skyler Seto, Pierre Ablin, Anastasiia Filippova et al.

Language models achieve impressive performance on a variety of knowledge, language, and reasoning tasks due to the scale and diversity of pretraining data available. The standard training recipe is a two-stage paradigm: pretraining first on the full corpus of data followed by specialization on a subset of high quality, specialized data from the full corpus. In the multi-domain setting, this involves continued pretraining of multiple models on each specialized domain, referred to as split model training. We propose a method for pretraining multiple models independently over a general pretraining corpus, and determining the optimal compute allocation between pretraining and continued pretraining using scaling laws. Our approach accurately predicts the loss of a model of size N with D pretraining and D' specialization tokens, and extrapolates to larger model sizes and number of tokens. Applied to language model training, our approach improves performance consistently across common sense knowledge and reasoning benchmarks across different model sizes and compute budgets.

Anastasia Ivanova, Pierre Ablin

In many scenarios, one uses a large training set to train a model with the goal of performing well on a smaller testing set with a different distribution. Learning a weight for each data point of the training set is an appealing solution, as it ideally allows one to automatically learn the importance of each training point for generalization on the testing set. This task is usually formalized as a bilevel optimization problem. Classical bilevel solvers are based on a warm-start strategy where both the parameters of the models and the data weights are learned at the same time. We show that this joint dynamic may lead to sub-optimal solutions, for which the final data weights are very sparse. This finding illustrates the difficulty of data reweighting and offers a clue as to why this method is rarely used in practice.

Metod Jazbec, Theo X. Olausson, Louis Béthune et al.

Diffusion (Large) Language Models (dLLMs) now match the downstream performance of their autoregressive counterparts on many tasks, while holding the promise of being more efficient during inference. One particularly successful variant is masked discrete diffusion, in which a buffer filled with special mask tokens is progressively replaced with tokens sampled from the model's vocabulary. Efficiency can be gained by unmasking several tokens in parallel, but doing too many at once risks degrading the generation quality. Thus, one critical design aspect of dLLMs is the sampling procedure that selects, at each step of the diffusion process, which tokens to replace. Indeed, recent work has found that heuristic strategies such as confidence thresholding lead to both higher quality and token throughput compared to random unmasking. However, such heuristics have downsides: they require manual tuning, and we observe that their performance degrades with larger buffer sizes. In this work, we instead propose to train sampling procedures using reinforcement learning. Specifically, we formalize masked diffusion sampling as a Markov decision process in which the dLLM serves as the environment, and propose a lightweight policy architecture based on a single-layer transformer that maps dLLM token confidences to unmasking decisions. Our experiments show that these trained policies match the performance of state-of-the-art heuristics when combined with semi-autoregressive generation, while outperforming them in the full diffusion setting. We also examine the transferability of these policies, finding that they can generalize to new underlying dLLMs and longer sequence lengths. However, we also observe that their performance degrades when applied to out-of-domain data, and that fine-grained tuning of the accuracy-efficiency trade-off can be challenging with our approach.

Bruno Mlodozeniec, Pierre Ablin, Louis Béthune et al.

Hyperparameter tuning can dramatically impact training stability and final performance of large-scale models. Recent works on neural network parameterisations, such as $μ$P, have enabled transfer of optimal global hyperparameters across model sizes. These works propose an empirical practice of search for optimal global base hyperparameters at a small model size, and transfer to a large size. We extend these works in two key ways. To handle scaling along most important scaling axes, we propose the Complete$^{(d)}$ Parameterisation that unifies scaling in width and depth -- using an adaptation of CompleteP -- as well as in batch-size and training duration. Secondly, with our parameterisation, we investigate per-module hyperparameter optimisation and transfer. We characterise the empirical challenges of navigating the high-dimensional hyperparameter landscape, and propose practical guidelines for tackling this optimisation problem. We demonstrate that, with the right parameterisation, hyperparameter transfer holds even in the per-module hyperparameter regime. Our study covers an extensive range of optimisation hyperparameters of modern models: learning rates, AdamW parameters, weight decay, initialisation scales, and residual block multipliers. Our experiments demonstrate significant training speed improvements in Large Language Models with the transferred per-module hyperparameters.

Szilvia Ujváry, Louis Béthune, Pierre Ablin et al.

Language models have consistently grown to compress more world knowledge into their parameters, but the knowledge that can be pretrained into them is upper-bounded by their parameter size. Especially the capacity of Small Language Models (SLMs) is limited, leading to factually incorrect generations. This problem is often mitigated by giving the SLM access to an outside source: the ability to query a larger model, documents, or a database. Under this setting, we study the fundamental question of \emph{which tokens an SLM can and should learn} during pretraining, versus \emph{which ones it should delegate} via a \texttt{<CALL>} token. We find that this is not simply a question of loss: although the loss is predictive of whether a predicted token mismatches the ground-truth, some tokens are \emph{acceptable} in that they are truthful alternative continuations of a pretraining document, and should not trigger a \texttt{<CALL>} even if their loss is high. We find that a spaCy grammar parser can help augment the loss signal to decide which tokens the SLM should learn to delegate to prevent factual errors and which are safe to learn and predict even under high losses. We propose LaCy, a novel pretraining method based on this token selection philosophy. Our experiments demonstrate that LaCy models successfully learn which tokens to predict and where to delegate for help. This results in higher FactScores when generating in a cascade with a bigger model and outperforms Rho or LLM-judge trained SLMs, while being simpler and cheaper.

Paul Jeha, Anastasiia Sedova, Louis Béthune et al.

For most languages of the world, language model pre-training operates in a data-constrained regime where models must repeat their training data many times, degrading generalization. Two remedies exist: aggressive hyperparameter tuning such as high weight decay, and mixing in data from a high-resource auxiliary language to directly aid the low-resource target. While hyperparameter tuning regularizes the model by shrinking weights to restrict network capacity, auxiliary data mixing uses a tunable mixing ratio to expand the training distribution and diversify the training signal with new knowledge. Both offer a principled way to improve training in a data-constrained domain. We compare these levers systematically across four model scales from 150M to 1.43B parameters, using Arabic as the low-resource target and English as the auxiliary, over approximately 1000 pre-training runs. Three findings emerge. First, mixing yields larger improvements than hyperparameter tuning on both validation loss and downstream task accuracy, and the gap grows with model size. Second, we quantify how much mixing helps: it boosts performance by an amount equivalent to 2--3$\times$ the unique target data on validation loss and 2--13$\times$ on downstream task accuracy, with the gain scaling steeply with model size. Third, this divergence reveals that target-language validation loss systematically underestimates mixing's value. Mixing regularizes by diversifying the training signal and contributes knowledge the repeated target corpus cannot supply; validation loss captures only the first effect. Our practical recommendations are: mix in a high-resource language, prioritize the mixing ratio over hyperparameter tuning, and transfer hyperparameters from a small proxy model via $μ$P.

Anastasiia Sedova, Skyler Seto, Natalie Schluter et al.

As language models scale, the amount of data they require grows -- yet many target data sources, such as low-resource languages or specialized domains, are inherently limited in size. A common strategy is to mix this scarce but valuable target data with abundant generic data, which presents a fundamental trade-off: too little target data in the mixture underexposes the model to the target domain, while too much target data repeats the same examples excessively, yielding diminishing returns and eventual overfitting. We study this trade-off across more than 2,000 language-model training runs spanning multiple model and target dataset sizes, as well as several data types, including multilingual, domain-specific, and quality-filtered mixtures. Across all settings, we find that repetition is a central driver of target-domain performance, and that mixture training tolerates much higher repetition than single-source training: scarce target corpora can be reused 15-20 times, with the optimal number of repetitions depending on the target data size, compute budget, and model scale. Next, we introduce a repetition-aware mixture scaling law that accounts for the decreasing value of repeated target tokens and the regularizing role of generic data. Optimizing the scaling law provides a principled way to compute effective mixture configurations, yielding practical mixture recommendations for pretraining under data constraints.

Keitaro Sakamoto, Pierre Ablin, Federico Danieli et al.

The quality of open-weight language models has dramatically improved in recent years. Sharing weights greatly facilitates model adoption by enabling their use across diverse hardware and software platforms. They also allow for more open research and testing, to the extent that users can use them as checkpoints, fine-tune them according to their needs, and potentially redistribute them. In some cases, however, concerns on modifying these weights towards unauthorized uses may outweigh the pros of giving users such a freedom. Defending against such adaptation is non-trivial: since an adaptive attacker can observe all weights and architectures by definition, they can reverse simple structural defenses, and use optimization to defeat the simplest locking mechanisms. In this work, we exploit the inference-training asymmetry of automatic differentiation as a novel defense axis. We propose DLR-Lock, a method where the purveyor of the model purposely replaces each pretrained MLP in their model with a deep low-rank residual network (DLR-Net) of comparable parameter count, forcing activation memory that grows linearly with depth during backpropagation. DLR-Nets are efficiently trained via module-wise distillation. We show that, beyond this memory overhead, DLR-Lock results in architectural mismatches that complicate the optimization landscape of standard fine-tuning, and a backward pass that incurs disproportionately more overhead than the forward pass. Our defense succeeds in withstanding adaptive attackers with full knowledge of the defense strategy while preserving the original model's capabilities. Experiments on LLM validate these claims.

Eleonora Gualdoni, Sonia Laguna, Louis Bethune et al.

Multi-domain fine-tuning of large language models requires improving performance on target domains while preserving performance on constrained domains, such as general knowledge, instruction following, or safety evaluations. Existing data mixing strategies rely on fixed heuristics or adaptive rules that cannot explicitly enforce preservation of such capabilities. We propose DynaMiCS, a dynamic mixture optimizer that casts multi-domain fine-tuning as a constrained optimization problem. At each update, DynaMiCS performs short domain-specific probing runs to estimate a slope matrix of local cross-domain effects, capturing how training on each fine-tuning dataset affects each evaluation domain. These estimates are then used to compute mixture weights through optimization over the probability simplex, with the objective of improving target-domain performance while keeping constrained-domain losses below reference levels. Across multi-domain fine-tuning scenarios with varying numbers of target and constrained domains, DynaMiCS achieves stronger target-domain improvements and higher constraint satisfaction than fixed-mixture baselines, at lower computational cost and without reference models, per-example scoring, or manually tuned mixture weights.

Hugo Richard, Luigi Gresele, Aapo Hyvärinen et al.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-session variability than other methods.On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset it reveals a clear sequence of evoked activity in sensor and source space. The code is freely available at https://github.com/hugorichard/multiviewica.

Ronan Perry, Gavin Mischler, Richard Guo et al.

As data are generated more and more from multiple disparate sources, multiview data sets, where each sample has features in distinct views, have ballooned in recent years. However, no comprehensive package exists that enables non-specialists to use these methods easily. mvlearn is a Python library which implements the leading multiview machine learning methods. Its simple API closely follows that of scikit-learn for increased ease-of-use. The package can be installed from Python Package Index (PyPI) and the conda package manager and is released under the MIT open-source license. The documentation, detailed examples, and all releases are available at https://mvlearn.github.io/.

Pierre Ablin, Jean-François Cardoso, Alexandre Gramfort

The approximate joint diagonalization of a set of matrices consists in finding a basis in which these matrices are as diagonal as possible. This problem naturally appears in several statistical learning tasks such as blind signal separation. We consider the diagonalization criterion studied in a seminal paper by Pham (2001), and propose a new quasi-Newton method for its optimization. Through numerical experiments on simulated and real datasets, we show that the proposed method outper-forms Pham's algorithm. An open source Python package is released.

Pierre Ablin, Dylan Fagot, Herwig Wendt et al.

Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds an important computational overhead. When assumed orthogonal (like the Fourier or Cosine transforms), learning the transform yields a non-convex optimization problem on the orthogonal matrix manifold. In this paper, we derive a quasi-Newton method on the manifold using sparse approximations of the Hessian. Experiments on synthetic and real audio data show that the proposed algorithm out-performs state-of-the-art first-order and coordinate-descent methods by orders of magnitude. A Python package for fast TL-NMF is released online at https://github.com/pierreablin/tlnmf.

João Monteiro, Michal Klein, Pierre Ablin et al.

Evaluating softmax attention over a fixed long context requires reading every cached key-value pair for each new query token. For a given context (a book, a manual, a legal corpus) the attention output is a deterministic function of the query. We propose Nectar, which fits a compact neural network to this function for queries drawn from a task-relevant distribution. Nectar fits two networks per layer and KV-head: a target network that predicts the attention output and a score network that predicts the log-normalizer. The pair plugs into the standard masked self-attention at inference time, replacing the $O(n)$ attention over the cache with a forward pass whose cost does not depend on $n$. Each module carries on the order of $|θ|$ parameters per layer and KV-head, typically much smaller than the $2nd$ KV-cache footprint at the same granularity. We report experiments on models from 1.7B to 8B parameters across five long-context datasets. The approximation error tracks the next-token accuracy gap to full attention, and allocating capacity non-uniformly across layers reduces that gap in our ablation. Beyond this analysis of metrics, we check that the text generations (following a question prompt) of a model equipped with a Nectar module match in semantic content those obtained by giving the same model access to the full cache.

Oscar Davis, Anastasiia Filippova, Pierre Ablin et al.

Continuous diffusion and flow matching models could represent a powerful alternative to autoregressive approaches for language modelling (LM), as they unlock a host of advantages currently reserved for continuous modalities, including accelerated sampling and tilting. Recently, several works have demonstrated the possibility of generating discrete data continuously by a simple flow matching process between a Gaussian and the one-hot encoded data distribution. They have further shown the feasibility of accelerated sampling via Categorical Flow Maps (CFMs), resulting in competitive sample quality in the few-step regime. However, this method had only been evaluated at relatively modest scales ($<1$B), leaving the question of its scalability completely open. In this article, we train a $1.7$B-parameter base flow model on $2.1$T tokens and self-distill it into a CFM that generates diverse, high-quality text in as few as $4$ inference steps while maintaining near-data-level token entropy. Furthermore, we introduce a likelihood bound for CFMs in the semi-discrete setting, and show that they can be used to score the model on standard LM benchmarks, achieving results in the same range as discrete diffusion methods. Finally, we uncover some of the challenges that arise from training these models at scale, and we provide prescriptive insights on loss weighting and time scheduling.

Valérie Castin, Pierre Ablin, Gabriel Peyré

Self-attention and masked self-attention are at the heart of Transformers' outstanding success. Still, our mathematical understanding of attention, in particular of its Lipschitz properties - which are key when it comes to analyzing robustness and expressive power - is incomplete. We provide a detailed study of the Lipschitz constant of self-attention in several practical scenarios, discussing the impact of the sequence length $n$ and layer normalization on the local Lipschitz constant of both unmasked and masked self-attention. In particular, we show that for inputs of length $n$ in any compact set, the Lipschitz constant of self-attention is bounded by $\sqrt{n}$ up to a constant factor and that this bound is tight for reasonable sequence lengths. When the sequence length $n$ is too large for the previous bound to be tight, which we refer to as the mean-field regime, we provide an upper bound and a matching lower bound which are independent of $n$. Our mean-field framework for masked self-attention is novel and of independent interest. Our experiments on pretrained and randomly initialized BERT and GPT-2 support our theoretical findings.

Valérie Castin, Pierre Ablin, José Antonio Carrillo et al.

Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens. This representation is then exploited by the attention function, which learns dependencies between tokens and is key to the success of Transformers. However, the iterative application of attention across layers induces complex dynamics that remain to be fully understood. To analyze these dynamics, we identify each input sequence with a probability measure and model its evolution as a Vlasov equation called Transformer PDE, whose velocity field is non-linear in the probability measure. Our first set of contributions focuses on compactly supported initial data. We show the Transformer PDE is well-posed and is the mean-field limit of an interacting particle system, thus generalizing and extending previous analysis to several variants of self-attention: multi-head attention, L2 attention, Sinkhorn attention, Sigmoid attention, and masked attention--leveraging a conditional Wasserstein framework. In a second set of contributions, we are the first to study non-compactly supported initial conditions, by focusing on Gaussian initial data. Again for different types of attention, we show that the Transformer PDE preserves the space of Gaussian measures, which allows us to analyze the Gaussian case theoretically and numerically to identify typical behaviors. This Gaussian analysis captures the evolution of data anisotropy through a deep Transformer. In particular, we highlight a clustering phenomenon that parallels previous results in the non-normalized discrete case.

Louis Bethune, David Grangier, Dan Busbridge et al.

A widespread strategy to obtain a language model that performs well on a target domain is to finetune a pretrained model to perform unsupervised next-token prediction on data from that target domain. Finetuning presents two challenges: (i) if the amount of target data is limited, as in most practical applications, the model will quickly overfit, and (ii) the model will drift away from the original model, forgetting the pretraining data and the generic knowledge that comes with it. We aim to derive scaling laws that quantify these two phenomena for various target domains, amounts of available target data, and model scales. We measure the efficiency of injecting pretraining data into the finetuning data mixture to avoid forgetting and mitigate overfitting. A key practical takeaway from our study is that injecting as little as 1% of pretraining data in the finetuning data mixture prevents the model from forgetting the pretraining set.

Mustafa Shukor, Louis Bethune, Dan Busbridge et al.

Large foundation models are typically trained on data from multiple domains, with the data mixture--the proportion of each domain used--playing a critical role in model performance. The standard approach to selecting this mixture relies on trial and error, which becomes impractical for large-scale pretraining. We propose a systematic method to determine the optimal data mixture for any target domain using scaling laws. Our approach accurately predicts the loss of a model of size $N$ trained with $D$ tokens and a specific domain weight vector $h$. We validate the universality of these scaling laws by demonstrating their predictive power in three distinct and large-scale settings: large language model (LLM), native multimodal model (NMM), and large vision models (LVM) pretraining. We further show that these scaling laws can extrapolate to new data mixtures and across scales: their parameters can be accurately estimated using a few small-scale training runs, and used to estimate the performance at larger scales and unseen domain weights. The scaling laws allow to derive the optimal domain weights for any target domain under a given training budget ($N$,$D$), providing a principled alternative to costly trial-and-error methods.

Simon Vary, Pierre Ablin, Bin Gao et al.

Optimization over the set of matrices $X$ that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as the canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods that require a fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to random estimates of $B$. Our method does not enforce the constraint in every iteration; instead, it produces iterations that converge to critical points on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian optimization counterparts that require the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and the GEVP.

David Grangier, Angelos Katharopoulos, Pierre Ablin et al.

Large language models are versatile tools but are not suitable for small inference budgets. Small models have more efficient inference, but their lower capacity means that their performance can be good only if one limits their scope to a specialized domain. This paper explores how to get good specialized small language models using a large, generic, pretraining set and a limited amount of specialized data. We consider two scenarios, depending on whether (i) one can afford pretraining a model for each specialization task, or (ii) one wants to cheaply adapt a single pretrained model for each task. In the first scenario, we propose an effective solution based on importance sampling: we resample the pretraining set to imitate the specialization data and train a small model on it. In the second scenario, we propose a novel architecture, projected networks (PN). PN is a large network whose parameters can be linearly projected into a small network for specialization. For both scenarios, we demonstrate the empirical effectiveness of our solutions across various domains, training set sizes, and training budgets.

Zhenzhang Ye, Gabriel Peyré, Daniel Cremers et al.

Bilevel optimization aims to optimize an outer objective function that depends on the solution to an inner optimization problem. It is routinely used in Machine Learning, notably for hyperparameter tuning. The conventional method to compute the so-called hypergradient of the outer problem is to use the Implicit Function Theorem (IFT). As a function of the error of the inner problem resolution, we study the error of the IFT method. We analyze two strategies to reduce this error: preconditioning the IFT formula and reparameterizing the inner problem. We give a detailed account of the impact of these two modifications on the error, highlighting the role played by higher-order derivatives of the functionals at stake. Our theoretical findings explain when super efficiency, namely reaching an error on the hypergradient that depends quadratically on the error on the inner problem, is achievable and compare the two approaches when this is impossible. Numerical evaluations on hyperparameter tuning for regression problems substantiate our theoretical findings.

Waiss Azizian, Michael Kirchhof, Eugene Ndiaye et al.

Large Language Models (LLMs) have demonstrated impressive generalization capabilities across various tasks, but their claim to practical relevance is still mired by concerns on their reliability. Recent works have proposed examining the activations produced by an LLM at inference time to assess whether its answer to a question is correct. Some works claim that a "geometry of truth" can be learned from examples, in the sense that the activations that generate correct answers can be distinguished from those leading to mistakes with a linear classifier. In this work, we underline a limitation of these approaches: we observe that these "geometries of truth" are intrinsically task-dependent and fail to transfer across tasks. More precisely, we show that linear classifiers trained across distinct tasks share little similarity and, when trained with sparsity-enforcing regularizers, have almost disjoint supports. We show that more sophisticated approaches (e.g., using mixtures of probes and tasks) fail to overcome this limitation, likely because activation vectors commonly used to classify answers form clearly separated clusters when examined across tasks.

Yu-Guan Hsieh, James Thornton, Eugene Ndiaye et al.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

Thiziri Nait Saada, Louis Bethune, Michal Klein et al.

Large-scale models are pretrained on massive web-crawled datasets containing documents of mixed quality, making data filtering essential. A popular method is Classifier-based Quality Filtering (CQF), which trains a binary classifier to distinguish between pretraining data and a small, high-quality set. It assigns each pretraining document a quality score defined as the classifier's score and retains only the top-scoring ones. We provide an in-depth analysis of CQF. We show that while CQF improves downstream task performance, it does not necessarily enhance language modeling on the high-quality dataset. We explain this paradox by the fact that CQF implicitly filters the high-quality dataset as well. We further compare the behavior of models trained with CQF to those trained on synthetic data of increasing quality, obtained via random token permutations, and find starkly different trends. Our results challenge the view that CQF captures a meaningful notion of data quality.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.

Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

Pierre Ablin, Angelos Katharopoulos, Skyler Seto et al.

Machine learning models are routinely trained on a mixture of different data domains. Different domain weights yield very different downstream performances. We propose the Soup-of-Experts, a novel architecture that can instantiate a model at test time for any domain weights with minimal computational cost and without re-training the model. Our architecture consists of a bank of expert parameters, which are linearly combined to instantiate one model. We learn the linear combination coefficients as a function of the input domain weights. To train this architecture, we sample random domain weights, instantiate the corresponding model, and backprop through one batch of data sampled with these domain weights. We demonstrate how our approach obtains small specialized models on several language modeling tasks quickly. Soup-of-Experts are particularly appealing when one needs to ship many different specialist models quickly under a model size constraint.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.

Machine learning techniques in multi-view settings face significant challenges, particularly when integrating heterogeneous data, aligning feature spaces, and managing view-specific biases. These issues are prominent in neuroscience, where data from multiple subjects exposed to the same stimuli are analyzed to uncover brain activity dynamics. In magnetoencephalography (MEG), where signals are captured at the scalp level, estimating the brain's underlying sources is crucial, especially in group studies where sources are assumed to be similar for all subjects. Common methods, such as Multi-View Independent Component Analysis (MVICA), assume identical sources across subjects, but this assumption is often too restrictive due to individual variability and age-related changes. Multi-View Independent Component Analysis with Delays (MVICAD) addresses this by allowing sources to differ up to a temporal delay. However, temporal dilation effects, particularly in auditory stimuli, are common in brain dynamics, making the estimation of time delays alone insufficient. To address this, we propose Multi-View Independent Component Analysis with Delays and Dilations (MVICAD2), which allows sources to differ across subjects in both temporal delays and dilations. We present a model with identifiable sources, derive an approximation of its likelihood in closed form, and use regularization and optimization techniques to enhance performance. Through simulations, we demonstrate that MVICAD2 outperforms existing multi-view ICA methods. We further validate its effectiveness using the Cam-CAN dataset, and showing how delays and dilations are related to aging.

Zaccharie Ramzi, Pierre Ablin, Gabriel Peyré et al.

Implicit deep learning has recently gained popularity with applications ranging from meta-learning to Deep Equilibrium Networks (DEQs). In its general formulation, it relies on expressing some components of deep learning pipelines implicitly, typically via a root equation called the inner problem. In practice, the solution of the inner problem is approximated during training with an iterative procedure, usually with a fixed number of inner iterations. During inference, the inner problem needs to be solved with new data. A popular belief is that increasing the number of inner iterations compared to the one used during training yields better performance. In this paper, we question such an assumption and provide a detailed theoretical analysis in a simple setting. We demonstrate that overparametrization plays a key role: increasing the number of iterations at test time cannot improve performance for overparametrized networks. We validate our theory on an array of implicit deep-learning problems. DEQs, which are typically overparametrized, do not benefit from increasing the number of iterations at inference while meta-learning, which is typically not overparametrized, benefits from it.

Mathieu Dagréou, Pierre Ablin, Samuel Vaiter et al.

Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.

Hugo Richard, Pierre Ablin, Bertrand Thirion et al.

We consider shared response modeling, a multi-view learning problem where one wants to identify common components from multiple datasets or views. We introduce Shared Independent Component Analysis (ShICA) that models each view as a linear transform of shared independent components contaminated by additive Gaussian noise. We show that this model is identifiable if the components are either non-Gaussian or have enough diversity in noise variances. We then show that in some cases multi-set canonical correlation analysis can recover the correct unmixing matrices, but that even a small amount of sampling noise makes Multiset CCA fail. To solve this problem, we propose to use joint diagonalization after Multiset CCA, leading to a new approach called ShICA-J. We show via simulations that ShICA-J leads to improved results while being very fast to fit. While ShICA-J is based on second-order statistics, we further propose to leverage non-Gaussianity of the components using a maximum-likelihood method, ShICA-ML, that is both more accurate and more costly. Further, ShICA comes with a principled method for shared components estimation. Finally, we provide empirical evidence on fMRI and MEG datasets that ShICA yields more accurate estimation of the components than alternatives.

Michael E. Sander, Pierre Ablin, Mathieu Blondel et al.

Attention based models such as Transformers involve pairwise interactions between data points, modeled with a learnable attention matrix. Importantly, this attention matrix is normalized with the SoftMax operator, which makes it row-wise stochastic. In this paper, we propose instead to use Sinkhorn's algorithm to make attention matrices doubly stochastic. We call the resulting model a Sinkformer. We show that the row-wise stochastic attention matrices in classical Transformers get close to doubly stochastic matrices as the number of epochs increases, justifying the use of Sinkhorn normalization as an informative prior. On the theoretical side, we show that, unlike the SoftMax operation, this normalization makes it possible to understand the iterations of self-attention modules as a discretized gradient-flow for the Wasserstein metric. We also show in the infinite number of samples limit that, when rescaling both attention matrices and depth, Sinkformers operate a heat diffusion. On the experimental side, we show that Sinkformers enhance model accuracy in vision and natural language processing tasks. In particular, on 3D shapes classification, Sinkformers lead to a significant improvement.

Anna Korba, Pierre-Cyril Aubin-Frankowski, Szymon Majewski et al.

Among dissimilarities between probability distributions, the Kernel Stein Discrepancy (KSD) has received much interest recently. We investigate the properties of its Wasserstein gradient flow to approximate a target probability distribution $π$ on $\mathbb{R}^d$, known up to a normalization constant. This leads to a straightforwardly implementable, deterministic score-based method to sample from $π$, named KSD Descent, which uses a set of particles to approximate $π$. Remarkably, owing to a tractable loss function, KSD Descent can leverage robust parameter-free optimization schemes such as L-BFGS; this contrasts with other popular particle-based schemes such as the Stein Variational Gradient Descent algorithm. We study the convergence properties of KSD Descent and demonstrate its practical relevance. However, we also highlight failure cases by showing that the algorithm can get stuck in spurious local minima.

Hugo Richard, Pierre Ablin, Aapo Hyvärinen et al.

We consider a multi-view learning problem known as group independent component analysis (group ICA), where the goal is to recover shared independent sources from many views. The statistical modeling of this problem requires to take noise into account. When the model includes additive noise on the observations, the likelihood is intractable. By contrast, we propose Adaptive multiView ICA (AVICA), a noisy ICA model where each view is a linear mixture of shared independent sources with additive noise on the sources. In this setting, the likelihood has a tractable expression, which enables either direct optimization of the log-likelihood using a quasi-Newton method, or generalized EM. Importantly, we consider that the noise levels are also parameters that are learned from the data. This enables sources estimation with a closed-form Minimum Mean Squared Error (MMSE) estimator which weights each view according to its relative noise level. On synthetic data, AVICA yields better sources estimates than other group ICA methods thanks to its explicit MMSE estimator. On real magnetoencephalograpy (MEG) data, we provide evidence that the decomposition is less sensitive to sampling noise and that the noise variance estimates are biologically plausible. Lastly, on functional magnetic resonance imaging (fMRI) data, AVICA exhibits best performance in transferring information across views.

Michael E. Sander, Pierre Ablin, Mathieu Blondel et al.

The training of deep residual neural networks (ResNets) with backpropagation has a memory cost that increases linearly with respect to the depth of the network. A way to circumvent this issue is to use reversible architectures. In this paper, we propose to change the forward rule of a ResNet by adding a momentum term. The resulting networks, momentum residual neural networks (Momentum ResNets), are invertible. Unlike previous invertible architectures, they can be used as a drop-in replacement for any existing ResNet block. We show that Momentum ResNets can be interpreted in the infinitesimal step size regime as second-order ordinary differential equations (ODEs) and exactly characterize how adding momentum progressively increases the representation capabilities of Momentum ResNets. Our analysis reveals that Momentum ResNets can learn any linear mapping up to a multiplicative factor, while ResNets cannot. In a learning to optimize setting, where convergence to a fixed point is required, we show theoretically and empirically that our method succeeds while existing invertible architectures fail. We show on CIFAR and ImageNet that Momentum ResNets have the same accuracy as ResNets, while having a much smaller memory footprint, and show that pre-trained Momentum ResNets are promising for fine-tuning models.

Pierre Ablin, Gabriel Peyré

We consider the problem of minimizing a function over the manifold of orthogonal matrices. The majority of algorithms for this problem compute a direction in the tangent space, and then use a retraction to move in that direction while staying on the manifold. Unfortunately, the numerical computation of retractions on the orthogonal manifold always involves some expensive linear algebra operation, such as matrix inversion, exponential or square-root. These operations quickly become expensive as the dimension of the matrices grows. To bypass this limitation, we propose the landing algorithm which does not use retractions. The algorithm is not constrained to stay on the manifold but its evolution is driven by a potential energy which progressively attracts it towards the manifold. One iteration of the landing algorithm only involves matrix multiplications, which makes it cheap compared to its retraction counterparts. We provide an analysis of the convergence of the algorithm, and demonstrate its promises on large-scale and deep learning problems, where it is faster and less prone to numerical errors than retraction-based methods.

Pierre Ablin

We consider the problem of training a deep orthogonal linear network, which consists of a product of orthogonal matrices, with no non-linearity in-between. We show that training the weights with Riemannian gradient descent is equivalent to training the whole factorization by gradient descent. This means that there is no effect of overparametrization and implicit bias at all in this setting: training such a deep, overparametrized, network is perfectly equivalent to training a one-layer shallow network.