Guillaume Dalle

Guillaume Dalle, Léo Baty, Louis Bouvier et al.

Combinatorial optimization (CO) layers in machine learning (ML) pipelines are a powerful tool to tackle data-driven decision tasks, but they come with two main challenges. First, the solution of a CO problem often behaves as a piecewise constant function of its objective parameters. Given that ML pipelines are typically trained using stochastic gradient descent, the absence of slope information is very detrimental. Second, standard ML losses do not work well in combinatorial settings. A growing body of research addresses these challenges through diverse methods. Unfortunately, the lack of well-maintained implementations slows down the adoption of CO layers. In this paper, building upon previous works, we introduce a probabilistic perspective on CO layers, which lends itself naturally to approximate differentiation and the construction of structured losses. We recover many approaches from the literature as special cases, and we also derive new ones. Based on this unifying perspective, we present InferOpt.jl, an open-source Julia package that 1) allows turning any CO oracle with a linear objective into a differentiable layer, and 2) defines adequate losses to train pipelines containing such layers. Our library works with arbitrary optimization algorithms, and it is fully compatible with Julia's ML ecosystem. We demonstrate its abilities using a pathfinding problem on video game maps as guiding example, as well as three other applications from operations research.

Lucas Clarté, Adrien Vandenbroucque, Guillaume Dalle et al.

We investigate popular resampling methods for estimating the uncertainty of statistical models, such as subsampling, bootstrap and the jackknife, and their performance in high-dimensional supervised regression tasks. We provide a tight asymptotic description of the biases and variances estimated by these methods in the context of generalized linear models, such as ridge and logistic regression, taking the limit where the number of samples $n$ and dimension $d$ of the covariates grow at a comparable fixed rate $α\!=\! n/d$. Our findings are three-fold: i) resampling methods are fraught with problems in high dimensions and exhibit the double-descent-like behavior typical of these situations; ii) only when $α$ is large enough do they provide consistent and reliable error estimations (we give convergence rates); iii) in the over-parametrized regime $α\!<\!1$ relevant to modern machine learning practice, their predictions are not consistent, even with optimal regularization.

Guillaume Dalle, Adrian Hill

For scientific machine learning tasks with a lot of custom code, picking the right Automatic Differentiation (AD) system matters. Our Julia package DifferentiationInterface$.$jl provides a common frontend to a dozen AD backends, unlocking easy comparison and modular development. In particular, its built-in preparation mechanism leverages the strengths of each backend by amortizing one-time computations. This is key to enabling sophisticated features like sparsity handling without putting additional burdens on the user.

Adrian Hill, Guillaume Dalle

From implicit differentiation to probabilistic modeling, Jacobian and Hessian matrices have many potential use cases in Machine Learning (ML), but they are viewed as computationally prohibitive. Fortunately, these matrices often exhibit sparsity, which can be leveraged to speed up the process of Automatic Differentiation (AD). This paper presents advances in sparsity detection, previously the performance bottleneck of Automatic Sparse Differentiation (ASD). Our implementation of sparsity detection is based on operator overloading, able to detect both local and global sparsity patterns, and supports flexible index set representations. It is fully automatic and requires no modification of user code, making it compatible with existing ML codebases. Most importantly, it is highly performant, unlocking Jacobians and Hessians at scales where they were considered too expensive to compute. On real-world problems from scientific ML, graph neural networks and optimization, we show significant speed-ups of up to three orders of magnitude. Notably, using our sparsity detection system, ASD outperforms standard AD for one-off computations, without amortization of either sparsity detection or matrix coloring.

Guillaume Dalle, Yohann de Castro

High-dimensional time series are a core ingredient of the statistical modeling toolkit, for which numerous estimation methods are known.But when observations are scarce or corrupted, the learning task becomes much harder.The question is: how much harder? In this paper, we study the properties of a partially-observed Vector AutoRegressive process, which is a state-space model endowed with a stochastic observation mechanism.Our goal is to estimate its sparse transition matrix, but we only have access to a small and noisy subsample of the state components.Interestingly, the sampling process itself is random and can exhibit temporal correlations, a feature shared by many realistic data acquisition scenarios.We start by describing an estimator based on the Yule-Walker equation and the Dantzig selector, and we give an upper bound on its non-asymptotic error.Then, we provide a matching minimax lower bound, thus proving near-optimality of our estimator.The convergence rate we obtain sheds light on the role of several key parameters such as the sampling ratio, the amount of noise and the number of non-zero coefficients in the transition matrix.These theoretical findings are commented and illustrated by numerical experiments on simulated data.