Rémi Bardenet

Michaël Fanuel, Rémi Bardenet

In this paper, we consider a ${\rm U}(1)$-connection graph, that is, a graph where each oriented edge is endowed with a unit modulus complex number that is conjugated under orientation flip. A natural replacement for the combinatorial Laplacian is then the magnetic Laplacian, an Hermitian matrix that includes information about the graph's connection. Magnetic Laplacians appear, e.g., in the problem of angular synchronization. In the context of large and dense graphs, we study here sparsifiers of the magnetic Laplacian $Δ$, i.e., spectral approximations based on subgraphs with few edges. Our approach relies on sampling multi-type spanning forests (MTSFs) using a custom determinantal point process, a probability distribution over edges that favours diversity. In a word, an MTSF is a spanning subgraph whose connected components are either trees or cycle-rooted trees. The latter partially capture the angular inconsistencies of the connection graph, and thus provide a way to compress the information contained in the connection. Interestingly, when the connection graph has weakly inconsistent cycles, samples from the determinantal point process under consideration can be obtained à la Wilson, using a random walk with cycle popping. We provide statistical guarantees for a choice of natural estimators of the connection Laplacian, and investigate two practical applications of our sparsifiers: ranking with angular synchronization and graph-based semi-supervised learning. From a statistical perspective, a side result of this paper of independent interest is a matrix Chernoff bound with intrinsic dimension, which allows considering the influence of a regularization -- of the form $Δ+ q \mathbb{I}$ with $q>0$ -- on sparsification guarantees.

Hoang-Son Tran, Pranav Gupta, Rémi Bardenet et al.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

Guillaume Gautier, Rémi Bardenet, Michal Valko

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

Rémi Bardenet, Subhroshekhar Ghosh, Hugo Simon-Onfroy et al.

Determinantal point processes (DPPs) are random configurations of points with tunable negative dependence. Because sampling is tractable, DPPs are natural candidates for subsampling tasks, such as minibatch selection or coreset construction. A \emph{coreset} is a subset of a (large) training set, such that minimizing an empirical loss averaged over the coreset is a controlled replacement for the intractable minimization of the original empirical loss. Typically, the control takes the form of a guarantee that the average loss over the coreset approximates the total loss uniformly across the parameter space. Recent work has provided significant empirical support in favor of using DPPs to build randomized coresets, coupled with interesting theoretical results that are suggestive but leave some key questions unanswered. In particular, the central question of whether the cardinality of a DPP-based coreset is fundamentally smaller than one based on independent sampling remained open. In this paper, we answer this question in the affirmative, demonstrating that \emph{DPPs can provably outperform independently drawn coresets}. In this vein, we contribute a conceptual understanding of coreset loss as a \emph{linear statistic} of the (random) coreset. We leverage this structural observation to connect the coresets problem to a more general problem of concentration phenomena for linear statistics of DPPs, wherein we obtain \emph{effective concentration inequalities that extend well-beyond the state-of-the-art}, encompassing general non-projection, even non-symmetric kernels. The latter have been recently shown to be of interest in machine learning beyond coresets, but come with a limited theoretical toolbox, to the extension of which our result contributes. Finally, we are also able to address the coresets problem for vector-valued objective functions, a novelty in the coresets literature.

Vladimir Petrovic, Rémi Bardenet, Agnès Desolneux

In this paper, we consider the problem of computing the integral of a function on the unit sphere, in any dimension, using Monte Carlo methods. Although the methods we present are general, our guiding thread is the sliced Wasserstein distance between two measures on $\mathbb{R}^d$, which is precisely an integral on the $d$-dimensional sphere. The sliced Wasserstein distance (SW) has gained momentum in machine learning either as a proxy to the less computationally tractable Wasserstein distance, or as a distance in its own right, due in particular to its built-in alleviation of the curse of dimensionality. There has been recent numerical benchmarks of quadratures for the sliced Wasserstein, and our viewpoint differs in that we concentrate on quadratures where the nodes are repulsive, i.e. negatively dependent. Indeed, negative dependence can bring variance reduction when the quadrature is adapted to the integration task. Our first contribution is to extract and motivate quadratures from the recent literature on determinantal point processes (DPPs) and repelled point processes, as well as repulsive quadratures from the literature specific to the sliced Wasserstein distance. We then numerically benchmark these quadratures. Moreover, we analyze the variance of the UnifOrtho estimator, an orthogonal Monte Carlo estimator. Our analysis sheds light on UnifOrtho's success for the estimation of the sliced Wasserstein in large dimensions, as well as counterexamples from the literature. Our final recommendation for the computation of the sliced Wasserstein distance is to use randomized quasi-Monte Carlo in low dimensions and \emph{UnifOrtho} in large dimensions. DPP-based quadratures only shine when quasi-Monte Carlo also does, while repelled quadratures show moderate variance reduction in general, but more theoretical effort is needed to make them robust.

Martin Rouault, Rémi Bardenet, Mylène Maïda

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the worst-case integration error over a reproducing kernel Hilbert space (RKHS). These quadrature rules come with strong experimental evidence that this worst-case error decreases at a faster rate than the standard square root of the number of quadrature nodes. This conjectured fast rate is key for integrating expensive-to-evaluate functions, as in Bayesian inference of expensive models, and makes up for the increased computational cost of sampling, compared to i.i.d. or MCMC quadratures. However, there is little theoretical support for this faster-than-square-root rate, at least in the usual case where the RKHS is infinite-dimensional, while recent progress on distribution compression suggests that results on the direct minimization of worst-case integration are possible. In this paper, we study a joint probability distribution over quadrature nodes, whose support tends to minimize the same worst-case error as kernel herding. Our main contribution is to prove that it does outperform i.i.d Monte Carlo, in the sense of coming with a tighter concentration inequality on the worst-case integration error. This first step towards proving a fast error decay demonstrates that the mathematical toolbox developed around Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. Moreover, we investigate the computational bottlenecks of approximately sampling our quadrature, and we demonstrate on toy examples that a faster rate of convergence, though not worst-case, is likely.

Rémi Bardenet, Mylène Maïda, Martin Rouault

Gibbs measures, such as Coulomb gases, are popular in modelling systems of interacting particles. Recently, we proposed to use Gibbs measures as randomized numerical integration algorithms with respect to a target measure $π$ on $\mathbb R^d$, following the heuristics that repulsiveness between particles should help reduce integration errors. A major issue in this approach is to tune the interaction kernel and confining potential of the Gibbs measure, so that the equilibrium measure of the system is the target distribution $π$. Doing so usually requires another Monte Carlo approximation of the \emph{potential}, i.e. the integral of the interaction kernel with respect to $π$. Using the methodology of large deviations from Garcia--Zelada (2019), we show that a random approximation of the potential preserves the fast large deviation principle that guarantees the proposed integration algorithm to outperform independent or Markov quadratures. For non-singular interaction kernels, we make minimal assumptions on this random approximation, which can be the result of a computationally cheap Monte Carlo preprocessing. For the Coulomb interaction kernel, we need the approximation to be based on another Gibbs measure, and we prove in passing a control on the uniform convergence of the approximation of the potential.

Michaël Fanuel, Rémi Bardenet

Given an $n\times r$ matrix $X$ of rank $r$, consider the problem of sampling $r$ integers $\mathtt{C}\subset \{1, \dots, n\}$ with probability proportional to the squared determinant of the rows of $X$ indexed by $\mathtt{C}$. The distribution of $\mathtt{C}$ is called a projection determinantal point process (DPP). The vanilla classical algorithm to sample a DPP works in two steps, an orthogonalization in $\mathcal{O}(nr^2)$ and a sampling step of the same cost. The bottleneck of recent quantum approaches to DPP sampling remains that preliminary orthogonalization step. For instance, (Kerenidis and Prakash, 2022) proposed an algorithm with the same $\mathcal{O}(nr^2)$ orthogonalization, followed by a $\mathcal{O}(nr)$ classical step to find the gates in a quantum circuit. The classical $\mathcal{O}(nr^2)$ orthogonalization thus still dominates the cost. Our first contribution is to reduce preprocessing to normalizing the columns of $X$, obtaining $\mathsf{X}$ in $\mathcal{O}(nr)$ classical operations. We show that a simple circuit inspired by the formalism of Kerenidis et al., 2022 samples a DPP of a type we had never encountered in applications, which is different from our target DPP. Plugging this circuit into a rejection sampling routine, we recover our target DPP after an expected $1/\det \mathsf{X}^\top\mathsf{X} = 1/a$ preparations of the quantum circuit. Using amplitude amplification, our second contribution is to boost the acceptance probability from $a$ to $1-a$ at the price of a circuit depth of $\mathcal{O}(r\log n/\sqrt{a})$ and $\mathcal{O}(\log n)$ extra qubits. Prepending a fast, sketching-based classical approximation of $a$, we obtain a pipeline to sample a projection DPP on a quantum computer, where the former $\mathcal{O}(nr^2)$ preprocessing bottleneck has been replaced by the $\mathcal{O}(nr)$ cost of normalizing the columns and the cost of our approximation of $a$.

Rémi Bardenet, Michaël Fanuel, Alexandre Feller

DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a quantum computer is easier than on a classical one. We focus here on DPPs over a finite state space, which are distributions over the subsets of $\{1,\dots,N\}$ parametrized by an $N\times N$ Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective costs $\mathcal{O}(N^3)$ and $\mathcal{O}(Nr^2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix. A large first part of the current paper consists in explaining why the state-of-the-art in quantum simulation of fermionic systems already yields quantum DPP sampling algorithms. We then modify existing quantum circuits, and discuss their insertion in a full DPP sampling pipeline that starts from practical kernel specifications. The bottom line is that, with $P$ (classical) parallel processors, we can divide the preprocessing cost by $P$ and build a quantum circuit with $\mathcal{O}(Nr)$ gates that sample a given DPP, with depth varying from $\mathcal{O}(N)$ to $\mathcal{O}(r\log N)$ depending on qubit-communication constraints on the target machine. We also connect existing work on the simulation of superconductors to Pfaffian point processes, which generalize DPPs and would be a natural addition to the machine learner's toolbox. In particular, we describe "projective" Pfaffian point processes, the cardinality of which has constant parity, almost surely. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit IBM machines.

Michaël Fanuel, Rémi Bardenet

Determinantal Point Process (DPPs) are statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs -- an optimization problem over trace-class operators -- has remained an open question. In this paper, we show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. This leads to a finite-dimensional problem, with strong statistical ties to the original MLE. Moreover, we propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.

Rémi Bardenet, Subhroshekhar Ghosh

Determinantal point processes (DPPs) have become a significant tool for recommendation systems, feature selection, or summary extraction, harnessing the intrinsic ability of these probabilistic models to facilitate sample diversity. The ability to sample from DPPs is paramount to the empirical investigation of these models. Most exact samplers are variants of a spectral meta-algorithm due to Hough, Krishnapur, Peres and Virág (henceforth HKPV), which is in general time and resource intensive. For DPPs with symmetric kernels, scalable HKPV samplers have been proposed that either first downsample the ground set of items, or force the kernel to be low-rank, using e.g. Nyström-type decompositions. In the present work, we contribute a radically different approach than HKPV. Exploiting the fact that many statistical and learning objectives can be effectively accomplished by only sampling certain key observables of a DPP (so-called linear statistics), we invoke an expression for the Laplace transform of such an observable as a single determinant, which holds in complete generality. Combining traditional low-rank approximation techniques with Laplace inversion algorithms from numerical analysis, we show how to directly approximate the distribution function of a linear statistic of a DPP. This distribution function can then be used in hypothesis testing or to actually sample the linear statistic, as per requirement. Our approach is scalable and applies to very general DPPs, beyond traditional symmetric kernels.

Ayoub Belhadji, Rémi Bardenet, Pierre Chainais

A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -- for choosing node locations -- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove almost optimal bounds for interpolation and quadrature under VS. While similar bounds already exist for some specific RKHSs using ad-hoc node constructions, VS offers bounds that apply to any Mercer kernel and depend on the spectrum of the associated integration operator. We emphasize that, unlike previous randomized approaches that rely on regularized leverage scores or determinantal point processes, evaluating the pdf of VS only requires pointwise evaluations of the kernel. VS is thus naturally amenable to MCMC samplers.

Ayoub Belhadji, Rémi Bardenet, Pierre Chainais

We study quadrature rules for functions from an RKHS, using nodes sampled from a determinantal point process (DPP). DPPs are parametrized by a kernel, and we use a truncated and saturated version of the RKHS kernel. This link between the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error, that depends on the spectrum of the RKHS kernel. Finally, we experimentally compare DPPs to existing kernel-based quadratures such as herding, Bayesian quadrature, or leverage score sampling. Numerical results confirm the interest of DPPs, and even suggest faster rates than our bounds in particular cases.

Ayoub Belhadji, Rémi Bardenet, Pierre Chainais

Dimensionality reduction is a first step of many machine learning pipelines. Two popular approaches are principal component analysis, which projects onto a small number of well chosen but non-interpretable directions, and feature selection, which selects a small number of the original features. Feature selection can be abstracted as a numerical linear algebra problem called the column subset selection problem (CSSP). CSSP corresponds to selecting the best subset of columns of a matrix $X \in \mathbb{R}^{N \times d}$, where \emph{best} is often meant in the sense of minimizing the approximation error, i.e., the norm of the residual after projection of $X$ onto the space spanned by the selected columns. Such an optimization over subsets of $\{1,\dots,d\}$ is usually impractical. One workaround that has been vastly explored is to resort to polynomial-cost, random subset selection algorithms that favor small values of this approximation error. We propose such a randomized algorithm, based on sampling from a projection determinantal point process (DPP), a repulsive distribution over a fixed number $k$ of indices $\{1,\dots,d\}$ that favors diversity among the selected columns. We give bounds on the ratio of the expected approximation error for this DPP over the optimal error of PCA. These bounds improve over the state-of-the-art bounds of \emph{volume sampling} when some realistic structural assumptions are satisfied for $X$. Numerical experiments suggest that our bounds are tight, and that our algorithms have comparable performance with the \emph{double phase} algorithm, often considered to be the practical state-of-the-art. Column subset selection with DPPs thus inherits the best of both worlds: good empirical performance and tight error bounds.

Guillaume Gautier, Guillermo Polito, Rémi Bardenet et al.

Determinantal point processes (DPPs) are specific probability distributions over clouds of points that are used as models and computational tools across physics, probability, statistics, and more recently machine learning. Sampling from DPPs is a challenge and therefore we present DPPy, a Python toolbox that gathers known exact and approximate sampling algorithms for both finite and continuous DPPs. The project is hosted on GitHub and equipped with an extensive documentation.

Guillaume Gautier, Rémi Bardenet, Michal Valko

Determinantal point processes (DPPs) are distributions over sets of items that model diversity using kernels. Their applications in machine learning include summary extraction and recommendation systems. Yet, the cost of sampling from a DPP is prohibitive in large-scale applications, which has triggered an effort towards efficient approximate samplers. We build a novel MCMC sampler that combines ideas from combinatorial geometry, linear programming, and Monte Carlo methods to sample from DPPs with a fixed sample cardinality, also called projection DPPs. Our sampler leverages the ability of the hit-and-run MCMC kernel to efficiently move across convex bodies. Previous theoretical results yield a fast mixing time of our chain when targeting a distribution that is close to a projection DPP, but not a DPP in general. Our empirical results demonstrate that this extends to sampling projection DPPs, i.e., our sampler is more sample-efficient than previous approaches which in turn translates to faster convergence when dealing with costly-to-evaluate functions, such as summary extraction in our experiments.

Rémi Bardenet, Michalis K. Titsias

Determinantal point processes (DPPs) are point process models that naturally encode diversity between the points of a given realization, through a positive definite kernel $K$. DPPs possess desirable properties, such as exact sampling or analyticity of the moments, but learning the parameters of kernel $K$ through likelihood-based inference is not straightforward. First, the kernel that appears in the likelihood is not $K$, but another kernel $L$ related to $K$ through an often intractable spectral decomposition. This issue is typically bypassed in machine learning by directly parametrizing the kernel $L$, at the price of some interpretability of the model parameters. We follow this approach here. Second, the likelihood has an intractable normalizing constant, which takes the form of a large determinant in the case of a DPP over a finite set of objects, and the form of a Fredholm determinant in the case of a DPP over a continuous domain. Our main contribution is to derive bounds on the likelihood of a DPP, both for finite and continuous domains. Unlike previous work, our bounds are cheap to evaluate since they do not rely on approximating the spectrum of a large matrix or an operator. Through usual arguments, these bounds thus yield cheap variational inference and moderately expensive exact Markov chain Monte Carlo inference methods for DPPs.

Rémi Bardenet, Arnaud Doucet, Chris Holmes

Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach which samples from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.