Pierre Bertrand

Faculty ,  Aix-Marseille Université , Faculté d'économie et de gestion (FEG)

Econometrics, Finance and mathematical methods
Bertrand
Status
Assistant professor
Research domain(s)
Game theory and social networks
Thesis
2021, Université Paris Cité (Paris 6)
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Contact
pierre.bertrand.1[at]univ-amu.fr
Address
AMU - AMSE
5-9 Boulevard Maurice Bourdet, CS 50498
​13205 Marseille Cedex 1
Abstract For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.
Keywords Divergence Kullback-Leibler, Divergence analysis, $\ell1-$distance/norm, Generalized $\varphi-$divergences
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Abstract This paper aims at comparing two coupling approaches as basic layers for building clustering criteria, suited for modularizing and clustering very large networks. We briefly use "optimal transport theory" as a starting point, and a way as well, to derive two canonical couplings: "statistical independence" and "logical indetermination". A symmetric list of properties is provided and notably the so called "Monge’s properties", applied to contingency matrices, and justifying the $\otimes$ versus $\oplus$ notation. A study is proposed, highlighting "logical indetermination", because it is, by far, lesser known. Eventually we estimate the average difference between both couplings as the key explanation of their usually close results in network clustering.
Keywords Graph Theoretical Approaches, Optimal Transport, Correlation Clustering, Coupling Functions, Logical Indetermination, Mathematical Relational Analysis
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Abstract We study the problem of maximizing Rényi entropy of order $2$ (equivalently, minimizing the index of coincidence) over the set of joint distributions with prescribed marginals. A closed-form optimizer is known under a feasibility condition on the marginals; we show that this condition is highly restrictive. We then provide an explicit construction of an optimal coupling for arbitrary marginals. Our approach characterizes the optimizer's structure and yields an iterative algorithm that terminates in finite time, returning an exact solution after at most $p-1$ updates, where $p$ is the number of rows.
Keywords Entropy maximization, Index of coincidence minimization, Coupling, Marginal constraints
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Abstract We study the problem of finding the minimum $\mathbb{L}_2$-norm element in a transportation polytope. It corresponds to minimizing the index of coincidence or maximizing Rényi entropy over joint probability distributions with prescribed marginals. The solution is straightforward under a feasibility condition on the marginals. We make three contributions. First, we decompose the basic solution into a mixture of three interpretable couplings and derive a closed-form sampling procedure, revealing its role as the $\mathbb{L}_2$-projection of the uniform distribution onto the transportation polytope. Second, we show that the feasibility condition becomes exponentially rare as the number of modalities grows, which highlights the practical necessity of a more general construction. Third, we provide a KKT-based iterative construction of the optimal coupling for arbitrary marginals with explicit Lagrange multipliers.
Keywords Rényi entropy, Index of coincidence, Coupling, Indeterminacy, Transportation polytope, Marginal constraints
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Abstract We propose a new stochastic method for high-dimensional constrained continuous global optimization without convexity assumptions. The algorithm combines global exploration and local refinement through coordinate-wise adaptive perturbations, whose scale is proportional to the current iterate. We show that global exploration reaches any feasible region with positive probability, while local search benefits from strong concentration properties, enabling efficient decline near optima. Their combination yields convergence to a global minimizer under mild regularity conditions. Applications to LASSO sparsity improvement as well as similar BPDN (Basis Pursuit DeNoising) problems, and to neural network pruning, illustrate the practical effectiveness of the approach.
Keywords Global optimization, Stochastic search, Random search, High-dimensional optimization, Neural network compression, Constrained optimization
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Abstract We propose a new random method to minimize deterministic continuous functions over subsets $\mathcal{S}$ of high-dimensional space $\mathbb{R}^K$ without assuming convexity. Our procedure alternates between a Global Search (GS) regime to identify candidates and a Concentrated Search (CS) regime to improve an eligible candidate in the constraint set $\mathcal{S}$. Beyond the alternation between those completely different regimes, the originality of our approach lies in leveraging high dimensionality. We demonstrate rigorous concentration properties under the $CS$ regime. In parallel, we also show that $GS$ reaches any point in $\mathcal{S}$ in finite time. Finally, we demonstrate the relevance of our new method by giving two concrete applications. The first deals with the reduction of the $\ell_{1}-$norm of a LASSO solution. Secondly, we compress a neural network by pruning weights while maintaining performance; our approach achieves significant weight reduction with minimal performance loss, offering an effective solution for network optimization.
Keywords High-dimensional optimization, Stochastic search, Lasso, Basis pursuit denoising, Neural network compression
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