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The Cross-Convex Bestiary
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In my latest paper, I introduced a generalized proximal point algorithm (PPA): given a point $(x,q) \in \mathbb{R}^m \times \Delta_S$ ($m,S \ge 1$ and $\Delta_S$ the probability simplex), the next iterate $(x’,q’)$ is given by: \(( \nabla f + \lambda A )(x',q') = \nabla f(x,q)\) for step-size $\lambda>0$, $f(x,q)=\frac12 ||x||^2 + h(q)$, with $h(q)=\sum_{s=1}^S q_s \log(q_s)$ the negentropy and \(A(x,q) = \begin{pmatrix} J_\ell(x)^\intercal q \\ -\ell(x) \end{pmatrix}\) where $J_\ell$ denotes the Jocobian matrix of $\ell=(\ell_1,\dots,\ell_S):\mathbb{R}^m \rightarrow \mathbb{R}^S$ with each $\ell_s$ convex ($\forall 1\le s \le S$).
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In this blog post, we introduce a generalized notion of convexity for functions, that we call “cross-convexity”, yielding inequalities that involve additional interaction terms compared to standard convexity.
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Published in ECML PKDD 2017, Skopje, Macedonia, 2017
co-authors: S. Clémençon, A. Garivier, A. Sabourin, C. Vernade
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Published in NeurIPS 2017, Long Beach, USA, 2017
co-author: S. Clémençon
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Published in ACML 2018, Beijing, China, 2018
co-authors: S. Clémençon, A. Garivier
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Published in ALT 2019, Chicago, USA, 2019
co-authors: A. Korba, S. Clémençon
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Published in ICMA 2020, 2020
co-authors: R. Vogel, S. Clémençon, C. Tillier
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Published in Institut polytechnique de Paris, 2020
supervisors: Stephan Clémençon, Aurélien Garivier and Anne Sabourin
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Published in preprint, 2021
co-author: Gergely Neu
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Published in preprint, 2022
Abstract. This paper introduces the checkered regression model, a nonlinear generalization of logistic regression. More precisely, this new binary classifier relies on the multivariate function $\frac{1}{2}\left( 1 + \tanh(\frac{z_1}{2})\times\dots\times\tanh(\frac{z_m}{2}) \right)$, which coincides with the usual sigmoid function in the univariate case $m=1$. While the decision boundary of logistic regression consists of a single hyperplane, our method is shown to tessellate the feature space by any given number $m\ge 1$ of hyperplanes. In order to fit the model’s parameters to some labeled data, we describe a classic empirical risk minimization framework based on the cross entropy loss. A multiclass version of our approach is also proposed.
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Published in Deep Reinforcement Learning Workshop at NeurIPS 2022, 2022
co-authors: R. Alami, Y.A. Dahou Djilali, K. Fedyanin, E. Moulines, M. Panov
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Published in preprint, 2023
co-authors: M.E.A. Seddik, H. Goulart, M. Debbah
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Published in Transactions on Machine Learning Research, 2023
co-authors: R. Alami, Y.A. Dahou Djilali, K. Fedyanin, E. Moulines
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Published in preprint, 2023
Abstract. This paper extends the classic theory of convex optimization to the minimization of functions that are equal to the negated logarithm of what we term as a “sum-log-concave” function, i.e., a sum of log-concave functions. In particular, we show that such functions are in general not convex but still satisfy generalized convexity inequalities. These inequalities unveil the key importance of a certain vector that we call the “cross-gradient” and that is, in general, distinct from the usual gradient. Thus, we propose the Cross Gradient Descent (XGD) algorithm moving in the opposite direction of the cross-gradient and derive a convergence analysis. As an application of our sum-log-concave framework, we introduce the so-called “checkered regression” method relying on a sum-log-concave function. This classifier extends (multiclass) logistic regression to non-linearly separable problems since it is capable of tessellating the feature space by using any given number of hyperplanes, creating a checkerboard-like pattern of decision regions.
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Published in preprint, 2024
co-authors: R. Alami, A. Abubaker, M.E.A. Seddik, S. Lahlou
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Published in preprint, 2024
Abstract. We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries.
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Undergraduate course, University 1, Department, 2014
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Workshop, University 1, Department, 2015
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