Pierre Bellec

Maximilien Le Clei, Pierre Bellec

Modern artificial intelligence works typically train the parameters of fixed-sized deep neural networks using gradient-based optimization techniques. Simple evolutionary algorithms have recently been shown to also be capable of optimizing deep neural network parameters, at times matching the performance of gradient-based techniques, e.g. in reinforcement learning settings. In addition to optimizing network parameters, many evolutionary computation techniques are also capable of progressively constructing network architectures. However, constructing network architectures from elementary evolution rules has not yet been shown to scale to modern reinforcement learning benchmarks. In this paper we therefore propose a new approach in which the architectures of recurrent neural networks dynamically evolve according to a small set of mutation rules. We implement a massively parallel evolutionary algorithm and run experiments on all 19 OpenAI Gym state-based reinforcement learning control tasks. We find that in most cases, dynamic agents match or exceed the performance of gradient-based agents while utilizing orders of magnitude fewer parameters. We believe our work to open avenues for real-life applications where network compactness and autonomous design are of critical importance. We provide our source code, final model checkpoints and full results at github.com/MaximilienLC/nra.

Maximilien Le Clei, Pierre Bellec

There is a recent surge in interest for imitation learning, with large human video-game and robotic manipulation datasets being used to train agents on very complex tasks. While deep neuroevolution has recently been shown to match the performance of gradient-based techniques on various reinforcement learning problems, the application of deep neuroevolution techniques to imitation learning remains relatively unexplored. In this work, we propose to explore whether deep neuroevolution can be used for behaviour imitation on popular simulation environments. We introduce a simple co-evolutionary adversarial generation framework, and evaluate its capabilities by evolving standard deep recurrent networks to imitate state-of-the-art pre-trained agents on 8 OpenAI Gym state-based control tasks. Across all tasks, we find the final elite actor agents capable of achieving scores as high as those obtained by the pre-trained agents, all the while closely following their score trajectories. Our results suggest that neuroevolution could be a valuable addition to deep learning techniques to produce accurate emulation of behavioural agents. We believe that the generality and simplicity of our approach opens avenues for imitating increasingly complex behaviours in increasingly complex settings, e.g. human behaviour in real-world settings. We provide our source code, model checkpoints and results at github.com/MaximilienLC/gane.

Pierre C. Bellec, Kai Tan

This paper investigates the iterates $\hbb^1,\dots,\hbb^T$ obtained from iterative algorithms in high-dimensional linear regression problems, in the regime where the feature dimension $p$ is comparable with the sample size $n$, i.e., $p \asymp n$. The analysis and proposed estimators are applicable to Gradient Descent (GD), proximal GD and their accelerated variants such as Fast Iterative Soft-Thresholding (FISTA). The paper proposes novel estimators for the generalization error of the iterate $\hbb^t$ for any fixed iteration $t$ along the trajectory. These estimators are proved to be $\sqrt n$-consistent under Gaussian designs. Applications to early-stopping are provided: when the generalization error of the iterates is a U-shape function of the iteration $t$, the estimates allow to select from the data an iteration $\hat t$ that achieves the smallest generalization error along the trajectory. Additionally, we provide a technique for developing debiasing corrections and valid confidence intervals for the components of the true coefficient vector from the iterate $\hbb^t$ at any finite iteration $t$. Extensive simulations on synthetic data illustrate the theoretical results.

Sana Ahmadi, Pierre Bellec, Tristan Glatard

Brain encoding with neuroimaging data is an established analysis aimed at predicting human brain activity directly from complex stimuli features such as movie frames. Typically, these features are the latent space representation from an artificial neural network, and the stimuli are image, audio, or text inputs. Ridge regression is a popular prediction model for brain encoding due to its good out-of-sample generalization performance. However, training a ridge regression model can be highly time-consuming when dealing with large-scale deep functional magnetic resonance imaging (fMRI) datasets that include many space-time samples of brain activity. This paper evaluates different parallelization techniques to reduce the training time of brain encoding with ridge regression on the CNeuroMod Friends dataset, one of the largest deep fMRI resource currently available. With multi-threading, our results show that the Intel Math Kernel Library (MKL) significantly outperforms the OpenBLAS library, being 1.9 times faster using 32 threads on a single machine. We then evaluated the Dask multi-CPU implementation of ridge regression readily available in scikit-learn (MultiOutput), and we proposed a new "batch" version of Dask parallelization, motivated by a time complexity analysis. In line with our theoretical analysis, MultiOutput parallelization was found to be impractical, i.e., slower than multi-threading on a single machine. In contrast, the Batch-MultiOutput regression scaled well across compute nodes and threads, providing speed-ups of up to 33 times with 8 compute nodes and 32 threads compared to a single-threaded scikit-learn execution. Batch parallelization using Dask thus emerges as a scalable approach for brain encoding with ridge regression on high-performance computing systems using scikit-learn and large fMRI datasets.

Sana Ahmadi, Francois Paugam, Tristan Glatard et al.

The optimal training of a vision transformer for brain encoding depends on three factors: model size, data size, and computational resources. This study investigates these three pillars, focusing on the effects of data scaling, model scaling, and high-performance computing on brain encoding results. Using VideoGPT to extract efficient spatiotemporal features from videos and training a Ridge model to predict brain activity based on these features, we conducted benchmark experiments with varying data sizes (10k, 100k, 1M, 6M) and different model configurations of GPT-2, including hidden layer dimensions, number of layers, and number of attention heads. We also evaluated the effects of training models with 32-bit vs 16-bit floating point representations. Our results demonstrate that increasing the hidden layer dimensions significantly improves brain encoding performance, as evidenced by higher Pearson correlation coefficients across all subjects. In contrast, the number of attention heads does not have a significant effect on the encoding results. Additionally, increasing the number of layers shows some improvement in brain encoding correlations, but the trend is not as consistent as that observed with hidden layer dimensions. The data scaling results show that larger training datasets lead to improved brain encoding performance, with the highest Pearson correlation coefficients observed for the largest dataset size (6M). These findings highlight that the effects of data scaling are more significant compared to model scaling in enhancing brain encoding performance. Furthermore, we explored the impact of floating-point precision by comparing 32-bit and 16-bit representations. Training with 16-bit precision yielded the same brain encoding accuracy as 32-bit, while reducing training time by 1.17 times, demonstrating its efficiency for high-performance computing tasks.

Pierre C Bellec, Gabriel Romon

The paper proposes chi-square and normal inference methodologies for the unknown coefficient matrix $B^*$ of size $p\times T$ in a Multi-Task (MT) linear model with $p$ covariates, $T$ tasks and $n$ observations under a row-sparse assumption on $B^*$. The row-sparsity $s$, dimension $p$ and number of tasks $T$ are allowed to grow with $n$. In the high-dimensional regime $p\ggg n$, in order to leverage row-sparsity, the MT Lasso is considered. We build upon the MT Lasso with a de-biasing scheme to correct for the bias induced by the penalty. This scheme requires the introduction of a new data-driven object, coined the interaction matrix, that captures effective correlations between noise vector and residuals on different tasks. This matrix is psd, of size $T\times T$ and can be computed efficiently. The interaction matrix lets us derive asymptotic normal and $χ^2_T$ results under Gaussian design and $\frac{sT+s\log(p/s)}{n}\to0$ which corresponds to consistency in Frobenius norm. These asymptotic distribution results yield valid confidence intervals for single entries of $B^*$ and valid confidence ellipsoids for single rows of $B^*$, for both known and unknown design covariance $Σ$. While previous proposals in grouped-variables regression require row-sparsity $s\lesssim\sqrt n$ up to constants depending on $T$ and logarithmic factors in $n,p$, the de-biasing scheme using the interaction matrix provides confidence intervals and $χ^2_T$ confidence ellipsoids under the conditions ${\min(T^2,\log^8p)}/{n}\to 0$ and $$ \frac{sT+s\log(p/s)+\|Σ^{-1}e_j\|_0\log p}{n}\to0, \quad \frac{\min(s,\|Σ^{-1}e_j\|_0)}{\sqrt n} \sqrt{[T+\log(p/s)]\log p}\to 0, $$ allowing row-sparsity $s\ggg\sqrt n$ when $\|Σ^{-1}e_j\|_0 \sqrt T\lll \sqrt{n}$ up to logarithmic factors.

Pierre C Bellec, Yiwei Shen

This paper studies M-estimators with gradient-Lipschitz loss function regularized with convex penalty in linear models with Gaussian design matrix and arbitrary noise distribution. A practical example is the robust M-estimator constructed with the Huber loss and the Elastic-Net penalty and the noise distribution has heavy-tails. Our main contributions are three-fold. (i) We provide general formulae for the derivatives of regularized M-estimators $\hatβ(y,X)$ where differentiation is taken with respect to both $y$ and $X$; this reveals a simple differentiability structure shared by all convex regularized M-estimators. (ii) Using these derivatives, we characterize the distribution of the residual $r_i = y_i-x_i^\top\hatβ$ in the intermediate high-dimensional regime where dimension and sample size are of the same order. (iii) Motivated by the distribution of the residuals, we propose a novel adaptive criterion to select tuning parameters of regularized M-estimators. The criterion approximates the out-of-sample error up to an additive constant independent of the estimator, so that minimizing the criterion provides a proxy for minimizing the out-of-sample error. The proposed adaptive criterion does not require the knowledge of the noise distribution or of the covariance of the design. Simulated data confirms the theoretical findings, regarding both the distribution of the residuals and the success of the criterion as a proxy of the out-of-sample error. Finally our results reveal new relationships between the derivatives of $\hatβ(y,X)$ and the effective degrees of freedom of the M-estimator, which are of independent interest.

Pierre C Bellec, Yiwei Shen, Cun-Hui Zhang

This paper develops asymptotic normality results for individual coordinates of robust M-estimators with convex penalty in high-dimensions, where the dimension $p$ is at most of the same order as the sample size $n$, i.e, $p/n\leγ$ for some fixed constant $γ>0$. The asymptotic normality requires a bias correction and holds for most coordinates of the M-estimator for a large class of loss functions including the Huber loss and its smoothed versions regularized with a strongly convex penalty. The asymptotic variance that characterizes the width of the resulting confidence intervals is estimated with data-driven quantities. This estimate of the variance adapts automatically to low ($p/n\to0)$ or high ($p/n \le γ$) dimensions and does not involve the proximal operators seen in previous works on asymptotic normality of M-estimators. For the Huber loss, the estimated variance has a simple expression involving an effective degrees-of-freedom as well as an effective sample size. The case of the Huber loss with Elastic-Net penalty is studied in details and a simulation study confirms the theoretical findings. The asymptotic normality results follow from Stein formulae for high-dimensional random vectors on the sphere developed in the paper which are of independent interest.

Pierre C Bellec

A generic out-of-sample error estimate is proposed for robust $M$-estimators regularized with a convex penalty in high-dimensional linear regression where $(X,y)$ is observed and $p,n$ are of the same order. If $ψ$ is the derivative of the robust data-fitting loss $ρ$, the estimate depends on the observed data only through the quantities $\hatψ= ψ(y-X\hatβ)$, $X^\top \hatψ$ and the derivatives $(\partial/\partial y) \hatψ$ and $(\partial/\partial y) X\hatβ$ for fixed $X$. The out-of-sample error estimate enjoys a relative error of order $n^{-1/2}$ in a linear model with Gaussian covariates and independent noise, either non-asymptotically when $p/n\le γ$ or asymptotically in the high-dimensional asymptotic regime $p/n\toγ'\in(0,\infty)$. General differentiable loss functions $ρ$ are allowed provided that $ψ=ρ'$ is 1-Lipschitz. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the $\ell_1$-penalized Huber M-estimator if the number of corrupted observations and sparsity of the true $β$ are bounded from above by $s_*n$ for some small enough constant $s_*\in(0,1)$ independent of $n,p$. For the square loss and in the absence of corruption in the response, the results additionally yield $n^{-1/2}$-consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty, estimates that were previously known for the Lasso.

Pierre C Bellec, Arun K Kuchibhotla

We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function $h$ and the corresponding penalized estimator $\hatβ$, we construct a quantity $η$, the first order expansion of $\hatβ$, such that the distance between $\hatβ$ and $η$ is an order of magnitude smaller than the estimation error $\|\hatβ - β^*\|$. In this sense, the first order expansion $η$ can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of $\hatβ$ is asymptotically the same as the risk of $η$ which leads to a precise characterization of the MSE of $\hatβ$; this characterization takes a particularly simple form for isotropic design. Such first order expansion also leads to inference results based on $\hatβ$. We provide sufficient conditions for the existence of such first order expansion for three regularizers: the Lasso in its constrained form, the lasso in its penalized form, and the Group-Lasso. The results apply to general loss functions under some conditions and those conditions are satisfied for the squared loss in linear regression and for the logistic loss in the logistic model.

Pierre C Bellec, Dana Yang

We consider the problem of selecting the best estimator among a family of Tikhonov regularized estimators, or, alternatively, to select a linear combination of these regularizers that is as good as the best regularizer in the family. Our theory reveals that if the Tikhonov regularizers share the same penalty matrix with different tuning parameters, a convex procedure based on $Q$-aggregation achieves the mean square error of the best estimator, up to a small error term no larger than $Cσ^2$, where $σ^2$ is the noise level and $C>0$ is an absolute constant. Remarkably, the error term does not depend on the penalty matrix or the number of estimators as long as they share the same penalty matrix, i.e., it applies to any grid of tuning parameters, no matter how large the cardinality of the grid is. This reveals the surprising "cost-free" nature of optimally tuning Tikhonov regularizers, in striking contrast with the existing literature on aggregation of estimators where one typically has to pay a cost of $σ^2\log(M)$ where $M$ is the number of estimators in the family. The result holds, more generally, for any family of ordered linear smoothers. This encompasses Ridge regression as well as Principal Component Regression. The result is extended to the problem of tuning Tikhonov regularizers with different penalty matrices.

Pierre C. Bellec, Cun-Hui Zhang

This paper studies schemes to de-bias the Lasso in a linear model $y=Xβ+ε$ where the goal is to construct confidence intervals for $a_0^Tβ$ in a direction $a_0$, where $X$ has iid $N(0,Σ)$ rows. We show that previously analyzed propositions to de-bias the Lasso require a modification in order to enjoy efficiency in a full range of sparsity. This modification takes the form of a degrees-of-freedom adjustment that accounts for the dimension of the model selected by Lasso. Let $s_0$ be the true sparsity. If $Σ$ is known and the ideal score vector proportional to $XΣ^{-1}a_0$ is used, the unadjusted de-biasing schemes proposed previously enjoy efficiency if $s_0\lll n^{2/3}$. However, if $s_0\ggg n^{2/3}$, the unadjusted schemes cannot be efficient in certain $a_0$: then it is necessary to modify existing procedures by a degrees-of-freedom adjustment. This modification grants asymptotic efficiency for any $a_0$ when $s_0/p\to 0$ and $s_0\log(p/s_0)/n \to 0$. If $Σ$ is unknown, efficiency is granted for general $a_0$ when $$\frac{s_0\log p}{n}+\min\Big\{\frac{s_Ω\log p}{n},\frac{\|Σ^{-1}a_0\|_1\sqrt{\log p}}{\|Σ^{-1/2}a_0\|_2 \sqrt n}\Big\}+\frac{\min(s_Ω,s_0)\log p}{\sqrt n}\to0$$ where $s_Ω=\|Σ^{-1}a_0\|_0$, provided that the de-biased estimate is modified with the degrees-of-freedom adjustment. The dependence in $s_0,s_Ω$ and $\|Σ^{-1}a_0\|_1$ is optimal. Our estimated score vector provides a novel methodology to handle dense $a_0$. Our analysis shows that the degrees-of-freedom adjustment is not needed when the initial bias in direction $a_0$ is small, which is granted under stringent conditions on $Σ^{-1}$. The main proof argument is an interpolation path similar to that typically used to derive Slepian's lemma. It yields a new $\ell_\infty$ error bound for the Lasso which is of independent interest.

Christian Dansereau, Angela Tam, AmanPreet Badhwar et al.

Early prognosis of Alzheimer's dementia is hard. Mild cognitive impairment (MCI) typically precedes Alzheimer's dementia, yet only a fraction of MCI individuals will progress to dementia, even when screened using biomarkers. We propose here to identify a subset of individuals who share a common brain signature highly predictive of oncoming dementia. This signature was composed of brain atrophy and functional dysconnectivity and discovered using a machine learning model in patients suffering from dementia. The model recognized the same brain signature in MCI individuals, 90% of which progressed to dementia within three years. This result is a marked improvement on the state-of-the-art in prognostic precision, while the brain signature still identified 47% of all MCI progressors. We thus discovered a sizable MCI subpopulation which represents an excellent recruitment target for clinical trials at the prodromal stage of Alzheimer's disease.

Tristan Glatard, Gregory Kiar, Tristan Aumentado-Armstrong et al.

We present Boutiques, a system to automatically publish, integrate and execute applications across computational platforms. Boutiques applications are installed through software containers described in a rich and flexible JSON language. A set of core tools facilitate the construction, validation, import, execution, and publishing of applications. Boutiques is currently supported by several distinct virtual research platforms, and it has been used to describe dozens of applications in the neuroinformatics domain. We expect Boutiques to improve the quality of application integration in computational platforms, to reduce redundancy of effort, to contribute to computational reproducibility, and to foster Open Science.

Christian Dansereau, Yassine Benhajali, Celine Risterucci et al.

Connectivity studies using resting-state functional magnetic resonance imaging are increasingly pooling data acquired at multiple sites. While this may allow investigators to speed up recruitment or increase sample size, multisite studies also potentially introduce systematic biases in connectivity measures across sites. In this work, we measure the inter-site effect in connectivity and its impact on our ability to detect individual and group differences. Our study was based on real, as opposed to simulated, multisite fMRI datasets collected in N=345 young, healthy subjects across 8 scanning sites with 3T scanners and heterogeneous scanning protocols, drawn from the 1000 functional connectome project. We first empirically show that typical functional networks were reliably found at the group level in all sites, and that the amplitude of the inter-site effects was small to moderate, with a Cohen's effect size below 0.5 on average across brain connections. We then implemented a series of Monte-Carlo simulations, based on real data, to evaluate the impact of the multisite effects on detection power in statistical tests comparing two groups (with and without the effect) using a general linear model, as well as on the prediction of group labels with a support-vector machine. As a reference, we also implemented the same simulations with fMRI data collected at a single site using an identical sample size. Simulations revealed that using data from heterogeneous sites only slightly decreased our ability to detect changes compared to a monosite study with the GLM, and had a greater impact on prediction accuracy. Taken together, our results support the feasibility of multisite studies in rs-fMRI provided the sample size is large enough.

Pierre C. Bellec, Arnak S. Dalalyan, Edwin Grappin et al.

In this paper we revisit the risk bounds of the lasso estimator in the context of transductive and semi-supervised learning. In other terms, the setting under consideration is that of regression with random design under partial labeling. The main goal is to obtain user-friendly bounds on the off-sample prediction risk. To this end, the simple setting of bounded response variable and bounded (high-dimensional) covariates is considered. We propose some new adaptations of the lasso to these settings and establish oracle inequalities both in expectation and in deviation. These results provide non-asymptotic upper bounds on the risk that highlight the interplay between the bias due to the mis-specification of the linear model, the bias due to the approximate sparsity and the variance. They also demonstrate that the presence of a large number of unlabeled features may have significant positive impact in the situations where the restricted eigenvalue of the design matrix vanishes or is very small.

Guillaume Marrelec, Arnaud Messé, Pierre Bellec

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering.

P. Bellec, Y. Benhajali, F. Carbonell et al.

Alterations in brain connectivity have been associated with a variety of clinical disorders using functional magnetic resonance imaging (fMRI). We investigated empirically how the number of brain parcels (or scale) impacted the results of a mass univariate general linear model (GLM) on connectomes. The brain parcels used as nodes in the connectome analysis were functionnally defined by a group cluster analysis. We first validated that a classic Benjamini-Hochberg procedure with parametric GLM tests did control appropriately the false-discovery rate (FDR) at a given scale. We then observed on realistic simulations that there was no substantial inflation of the FDR across scales, as long as the FDR was controlled independently within each scale, and the presence of true associations could be established using an omnibus permutation test combining all scales. Second, we observed both on simulations and on three real resting-state fMRI datasets (schizophrenia, congenital blindness, motor practice) that the rate of discovery varied markedly as a function of scales, and was relatively higher for low scales, below 25. Despite the differences in discovery rate, the statistical maps derived at different scales were generally very consistent in the three real datasets. Some seeds still showed effects better observed around 50, illustrating the potential benefits of multiscale analysis. On real data, the statistical maps agreed well with the existing literature. Overall, our results support that the multiscale GLM connectome analysis with FDR is statistically valid and can capture biologically meaningful effects in a variety of experimental conditions.