Pierre Boudart

Pierre Boudart, Pierre Gaillard, Alessandro Rudi

We study reinforcement learning for episodic Markov Decision Processes (MDPs) whose transitions are modelled by a multinomial logistic (MNL) model. Existing algorithms for MNL mixture MDPs yield a regret of $\smash{\tilde{O}(dH^2\sqrt{T})}$ (Li et al., 2024), where $d$ is the feature dimension, $H$ the episode length, and $T$ the number of episodes. Inspired by the logistic bandit literature (Abeille et al., 2021; Faury et al., 2022; Boudart et al., 2026), we introduce a problem-dependent constant $\barσ\_T \leq 1/2$, measuring the normalised average variance of the optimal downstream value function along the learner's trajectory. We propose an algorithm achieving a regret of $\smash{\tilde{O}(dH^2\barσ\_T\sqrt{T})}$, which recovers the existing bound in the worst case and improves upon it for structured MDPs. For instance, for KL-constrained robust MDPs, $\barσ\_T = O(H^{-1})$, reducing the horizon dependence by a factor $H$. We further establish a matching $\smash{Ω(dH^2\barσ\_T\sqrt{T})}$ lower bound, proving minimax optimality (up to logarithmic factors) and fully characterising the regret complexity of MNL mixture MDPs for the first time.

Pierre Boudart, Pierre Gaillard, Alessandro Rudi

We consider the multinomial logistic bandit problem, a variant of where a learner interacts with an environment by selecting actions to maximize expected rewards based on probabilistic feedback from multiple possible outcomes. In the binary setting, recent work has focused on understanding the impact of the non-linearity of the logistic model (Faury et al., 2020; Abeille et al., 2021). They introduced a problem-dependent constant $κ_* \geq 1$, that may be exponentially large in some problem parameters and which is captured by the derivative of the sigmoid function. It encapsulates the non-linearity and improves existing regret guarantees over $T$ rounds from $\smash{O(d\sqrt{T})}$ to $\smash{O(d\sqrt{T/κ_*})}$, where $d$ is the dimension of the parameter space. We extend their analysis to the multinomial logistic bandit framework, making it suitable for complex applications with more than two choices, such as reinforcement learning or recommender systems. To achieve this, we extend the definition of $κ_*$ to the multinomial setting and propose an efficient algorithm that leverages the problem's non-linearity. Our method yields a problem-dependent regret bound of order $ \smash{\widetilde{\mathcal{O}}( R d \sqrt{{KT}/{κ_*}})} $, where $R$ is the norm of the vector of rewards and $K$ is the number of outcomes. This improves upon the best existing guarantees of order $ \smash{\widetilde{\mathcal{O}}( RdK \sqrt{T} )} $. Moreover, we provide a $\smash{ Ω(Rd\sqrt{KT/κ_*})}$ lower-bound, showing that our algorithm is minimax-optimal and that our definition of $κ_*$ is optimal.

Pierre Boudart, Alessandro Rudi, Pierre Gaillard

We study a theoretical and algorithmic framework for structured prediction in the online learning setting. The problem of structured prediction, i.e. estimating function where the output space lacks a vectorial structure, is well studied in the literature of supervised statistical learning. We show that our algorithm is a generalisation of optimal algorithms from the supervised learning setting, and achieves the same excess risk upper bound also when data are not i.i.d. Moreover, we consider a second algorithm designed especially for non-stationary data distributions, including adversarial data. We bound its stochastic regret in function of the variation of the data distributions.