Searcharxiv⌕ Search

arXiv subjects

Maxime Sylvestre

Publications and source records attributed to Maxime Sylvestre.

8 recordsLinked to original sources
arXiv2026

Adapted Law Invariance and Time-Consistent Dynamic Risk Measures

In static risk measurement, law invariance expresses the principle that the risk of a position should depend only on its distribution, and not on the particular probability space on which it is represented. In a dynamic setting, the same principle leads naturally to adapted law invariance: the risk assessment should depend only on the probabilistic structure of the financial position together with the way information about it is revealed over time. We show that, for time-consistent risk measures, adapted law invariance is equivalent to a recursive one-step conditional-law representation. More precisely, assuming Fatou regularity, the one-step risk evaluations are exactly conditional lifts of static law-invariant risk measures, and the full dynamic risk measure is obtained by backward composition of these one-step maps. Convexity and coherence of the dynamic risk measure are characterized by the corresponding properties of the static one-step risk measures. This identifies adapted law invariance as the dynamic counterpart of ordinary law invariance. It also clarifies the strength of terminal-law invariance, as it appears in the rigidity theorem of Kupper and Schachermayer: it does not distinguish risks with the same distribution but different times of resolution. We further obtain an adapted Kusuoka representation in the coherent case and establish an extension of the Kupper--Schachermayer theorem.

q-fin.RM↗
arXiv2026

Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization

We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet.

math.OC↗
arXiv2025

Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

We study the convergence of the transport plans $γ_ε$ towards $γ_0$ as well as the cost of the entropy-regularized optimal transport $(c,γ_ε)$ towards $(c,γ_0)$ as the regularization parameter $ε$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(γ_ε,γ_0)$ is asymptotically greater than $C\sqrtε$ and the suboptimality $(c,γ_ε)-(c,γ_0)$ is of order $ε$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+δ$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(γ_ε,γ_0)$ converges to $0$ at rate $\sqrtε$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,γ_ε)-(c,γ_0) \sim dε/2$ and we provide a companion expansion of $H(γ_ε)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.

math.OC↗
arXiv2025

A synthetic approach to comparison principles for variational problems, with applications to optimal transport

We develop a synthetic, variational framework for deriving comparison principles in infinite-dimensional Banach spaces. Unlike traditional approaches that rely on the regularity of minimizers and Euler--Lagrange equations, our method exploits the order-theoretic structure of the energy. Central to our analysis is the notion of submodularity and its convex dual, substitutability, which we extend here to the infinite-dimensional setting. We prove a duality theorem establishing that a convex functional is submodular if and only if its conjugate is substitutable. We apply these results to problems in optimal transport, and derive comparison principles for Kantorovich potentials in standard, entropic, and unbalanced settings without requiring regularity assumptions on the cost or domain. Finally, we prove that general transport costs are substitutable, yielding comparison principles for JKO schemes driven by internal energies.

math.OC↗
arXiv2025

On a class of adversarial classification problems which admit a continuous solution

We consider a class of adversarial classification problems in the form of zero-sum games between a classifier and an adversary. The latter is able to corrupt data, at the expense of some optimal transport cost. We show that quite general assumptions on the loss functions of the classifier and the transport cost functions of the adversary ensure the existence of a Nash equilibrium with a continuous (or even Lipschitz) classifier's strategy. We also consider a softmax-like regularization of this problem and present numerical results for this regularization.

math.OC↗
arXiv2024

Monotone comparative statics for submodular functions, with an application to aggregated deferred acceptance

We propose monotone comparative statics results for maximizers of submodular functions, as opposed to maximizers of supermodular functions as in the classical theory put forth by Veinott, Topkis, Milgrom, and Shannon among others. We introduce matrons, a natural structure that is dual to sublattices that generalizes existing structures such as matroids and polymatroids in combinatorial optimization and M-sets in discrete convex analysis. Our monotone comparative statics result is based on a natural order on matrons, which is dual in some sense to Veinott's strong set order on sublattices. As an application, we propose a deferred acceptance algorithm that operates in the case of divisible goods, and we study its convergence properties.

econ.TH↗
arXiv2024

Implicit Bias in Noisy-SGD: With Applications to Differentially Private Training

Training Deep Neural Networks (DNNs) with small batches using Stochastic Gradient Descent (SGD) yields superior test performance compared to larger batches. The specific noise structure inherent to SGD is known to be responsible for this implicit bias. DP-SGD, used to ensure differential privacy (DP) in DNNs' training, adds Gaussian noise to the clipped gradients. Surprisingly, large-batch training still results in a significant decrease in performance, which poses an important challenge because strong DP guarantees necessitate the use of massive batches. We first show that the phenomenon extends to Noisy-SGD (DP-SGD without clipping), suggesting that the stochasticity (and not the clipping) is the cause of this implicit bias, even with additional isotropic Gaussian noise. We theoretically analyse the solutions obtained with continuous versions of Noisy-SGD for the Linear Least Square and Diagonal Linear Network settings, and reveal that the implicit bias is indeed amplified by the additional noise. Thus, the performance issues of large-batch DP-SGD training are rooted in the same underlying principles as SGD, offering hope for potential improvements in large batch training strategies.

stat.ML↗