arXiv subjects

Gilles Stoltz

Publications and source records attributed to Gilles Stoltz.

At least 19 recordsLinked to original sources

arXiv2026

A Direct Approach for Handling Contextual Bandits with Latent State Dynamics

We consider a linear contextual bandit model where contexts and rewards are governed by a finite hidden Markov chain. We first revisit the simplified model by Nelson et al. (2022), in which rewards are linear functions of the posterior probabilities over the hidden states given the observed contexts (called beliefs), rather than functions of the hidden states themselves. This simplified model may be handled through a direct reduction to standard linear contextual bandits. We extend the theoretical analysis of this reduction to take into account the estimation of the parameters of the hidden Markov model [HMM] in the regret bound and to provide high-probability bounds not depending anymore on the reward functions and only depending on the model through the estimation of the HMM parameters. Second, and most importantly, we instead study the more natural and more complex model incorporating direct dependencies in the hidden states (on top of dependencies on the observed contexts, as is natural for contextual bandits). Under a classic HMM forgetting condition, the main algorithmic tool introduced to cope with the various statistical dependencies that the reward structure introduces is to only periodically update reward-model parameters.

cs.LG↗

arXiv2025

Adaptive Conformal Inference through the Lens of Blackwell Approachability

This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Cand\`es (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.

stat.ML↗

arXiv2025

Online Matching via Reinforcement Learning: An Expert Policy Orchestration Strategy

Online matching problems arise in many complex systems, from cloud services and online marketplaces to organ exchange networks, where timely, principled decisions are critical for maintaining high system performance. Traditional heuristics in these settings are simple and interpretable but typically tailored to specific operating regimes, which can lead to inefficiencies when conditions change. We propose a reinforcement learning (RL) approach that learns to orchestrate a set of such expert policies, leveraging their complementary strengths in a data-driven, adaptive manner. Building on the Adv2 framework (Jonckheere et al., 2024), our method combines expert decisions through advantage-based weight updates and extends naturally to settings where only estimated value functions are available. We establish both expectation and high-probability regret guarantees and derive a novel finite-time bias bound for temporal-difference learning, enabling reliable advantage estimation even under constant step size and non-stationary dynamics. To support scalability, we introduce a neural actor-critic architecture that generalizes across large state spaces while preserving interpretability. Simulations on stochastic matching models, including an organ exchange scenario, show that the orchestrated policy converges faster and yields higher system level efficiency than both individual experts and conventional RL baselines. Our results highlight how structured, adaptive learning can improve the modeling and management of complex resource allocation and decision-making processes.

stat.ML↗

arXiv2025

Policy Optimization via Adv2: Adversarial Learning on Advantage Functions

We revisit the reduction of learning in adversarial Markov decision processes [MDPs] to adversarial learning based on $Q$--values; this reduction has been considered in a number of recent articles as one building block to perform policy optimization. Namely, we first consider and extend this reduction in an ideal setting where an oracle provides value functions: it may involve any adversarial learning strategy (not just exponential weights) and it may be based indifferently on $Q$--values or on advantage functions. We then present two extensions: on the one hand, convergence of the last iterate for a vast class of adversarial learning strategies (again, not just exponential weights), satisfying a property called monotonicity of weights; on the other hand, stronger regret criteria for learning in MDPs, inherited from the stronger regret criteria of adversarial learning called strongly adaptive regret and tracking regret. Third, we demonstrate how adversarial learning, also referred to as aggregation of experts, relates to aggregation (orchestration) of expert policies: we obtain stronger forms of performance guarantees in this setting than existing ones, via yet another, simple reduction. Finally, we discuss the impact of the reduction of learning in adversarial MDPs to adversarial learning in the practical scenarios where transition kernels are unknown and value functions must be learned. In particular, we review the literature and note that many strategies for policy optimization feature a policy-improvement step based on exponential weights with estimated $Q$--values. Our main message is that this step may be replaced by the application of any adversarial learning strategy on estimated $Q$--values or on estimated advantage functions. We leave the empirical evaluation of these twists for future research.

cs.LG↗

arXiv2025

Narrowing the Gap between Adversarial and Stochastic MDPs via Policy Optimization

We consider the problem of learning in adversarial Markov decision processes [MDPs] with an oblivious adversary in a full-information setting. The agent interacts with an environment during $T$ episodes, each of which consists of $H$ stages, and each episode is evaluated with respect to a reward function that will be revealed only at the end of the episode. We propose an algorithm, called APO-MVP, that achieves a regret bound of order $\tilde{\mathcal{O}}(\mathrm{poly}(H)\sqrt{SAT})$, where $S$ and $A$ are sizes of the state and action spaces, respectively. This result improves upon the best-known regret bound by a factor of $\sqrt{S}$, bridging the gap between adversarial and stochastic MDPs, and matching the minimax lower bound $Ω(\sqrt{H^3SAT})$ as far as the dependencies in $S,A,T$ are concerned. The proposed algorithm and analysis completely avoid the typical tool given by occupancy measures; instead, it performs policy optimization based only on dynamic programming and on a black-box online linear optimization strategy run over estimated advantage functions, making it easy to implement. The analysis leverages two recent techniques: policy optimization based on online linear optimization strategies (Jonckheere et al., 2023) and a refined martingale analysis of the impact on values of estimating transitions kernels (Zhang et al., 2023).

cs.LG↗

arXiv2024

Conformal Prediction for Hierarchical Data

We consider conformal prediction for multivariate data and focus on hierarchical data, where some components are linear combinations of others. Intuitively, the hierarchical structure can be leveraged to reduce the size of prediction regions for the same coverage level. We implement this intuition by including a projection step (also called a reconciliation step) in the split conformal prediction [SCP] procedure, and prove that the resulting prediction regions are indeed globally smaller. We do so both under the classic objective of joint coverage and under a new and challenging task: component-wise coverage, for which efficiency results are more difficult to obtain. The associated strategies and their analyses are based both on the literature of SCP and of forecast reconciliation, which we connect. We also illustrate the theoretical findings, for different scales of hierarchies on simulated data.

stat.ML↗

arXiv2024

Diversity-Preserving K-Armed Bandits, Revisited

We consider the bandit-based framework for diversity-preserving recommendations introduced by Celis et al. (2019), who approached it in the case of a polytope mainly by a reduction to the setting of linear bandits. We design a UCB algorithm using the specific structure of the setting and show that it enjoys a bounded distribution-dependent regret in the natural cases when the optimal mixed actions put some probability mass on all actions (i.e., when diversity is desirable). The regret lower bounds provided show that otherwise, at least when the model is mean-unbounded, a $\ln T$ regret is suffered. We also discuss an example beyond the special case of polytopes.

stat.ML↗

arXiv2023

Small Total-Cost Constraints in Contextual Bandits with Knapsacks, with Application to Fairness

We consider contextual bandit problems with knapsacks [CBwK], a problem where at each round, a scalar reward is obtained and vector-valued costs are suffered. The learner aims to maximize the cumulative rewards while ensuring that the cumulative costs are lower than some predetermined cost constraints. We assume that contexts come from a continuous set, that costs can be signed, and that the expected reward and cost functions, while unknown, may be uniformly estimated -- a typical assumption in the literature. In this setting, total cost constraints had so far to be at least of order $T^{3/4}$, where $T$ is the number of rounds, and were even typically assumed to depend linearly on $T$. We are however motivated to use CBwK to impose a fairness constraint of equalized average costs between groups: the budget associated with the corresponding cost constraints should be as close as possible to the natural deviations, of order $\sqrt{T}$. To that end, we introduce a dual strategy based on projected-gradient-descent updates, that is able to deal with total-cost constraints of the order of $\sqrt{T}$ up to poly-logarithmic terms. This strategy is more direct and simpler than existing strategies in the literature. It relies on a careful, adaptive, tuning of the step size.

stat.ML↗

arXiv2023

Parameter-free projected gradient descent

We consider the problem of minimizing a convex function over a closed convex set, with Projected Gradient Descent (PGD). We propose a fully parameter-free version of AdaGrad, which is adaptive to the distance between the initialization and the optimum, and to the sum of the square norm of the subgradients. Our algorithm is able to handle projection steps, does not involve restarts, reweighing along the trajectory or additional gradient evaluations compared to the classical PGD. It also fulfills optimal rates of convergence for cumulative regret up to logarithmic factors. We provide an extension of our approach to stochastic optimization and conduct numerical experiments supporting the developed theory.

stat.ML↗

arXiv2023

On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits

We lay the foundations of a non-parametric theory of best-arm identification in multi-armed bandits with a fixed budget T. We consider general, possibly non-parametric, models D for distributions over the arms; an overarching example is the model D = P(0,1) of all probability distributions over [0,1]. We propose upper bounds on the average log-probability of misidentifying the optimal arm based on information-theoretic quantities that correspond to infima over Kullback-Leibler divergences between some distributions in D and a given distribution. This is made possible by a refined analysis of the successive-rejects strategy of Audibert, Bubeck, and Munos (2010). We finally provide lower bounds on the same average log-probability, also in terms of the same new information-theoretic quantities; these lower bounds are larger when the (natural) assumptions on the considered strategies are stronger. All these new upper and lower bounds generalize existing bounds based, e.g., on gaps between distributions.

cs.LG↗

arXiv2022

Contextual Bandits with Knapsacks for a Conversion Model

We consider contextual bandits with knapsacks, with an underlying structure between rewards generated and cost vectors suffered. We do so motivated by sales with commercial discounts. At each round, given the stochastic i.i.d.\ context $\mathbf{x}_t$ and the arm picked $a_t$ (corresponding, e.g., to a discount level), a customer conversion may be obtained, in which case a reward $r(a,\mathbf{x}_t)$ is gained and vector costs $c(a_t,\mathbf{x}_t)$ are suffered (corresponding, e.g., to losses of earnings). Otherwise, in the absence of a conversion, the reward and costs are null. The reward and costs achieved are thus coupled through the binary variable measuring conversion or the absence thereof. This underlying structure between rewards and costs is different from the linear structures considered by Agrawal and Devanur [2016] (but we show that the techniques introduced in the present article may also be applied to the case of these linear structures). The adaptive policies exhibited solve at each round a linear program based on upper-confidence estimates of the probabilities of conversion given $a$ and $\mathbf{x}$. This kind of policy is most natural and achieves a regret bound of the typical order (OPT/$B$) $\sqrt{T}$, where $B$ is the total budget allowed, OPT is the optimal expected reward achievable by a static policy, and $T$ is the number of rounds.

cs.LG↗

arXiv2022

KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints

We consider $K$-armed stochastic bandits and consider cumulative regret bounds up to time $T$. We are interested in strategies achieving simultaneously a distribution-free regret bound of optimal order $\sqrt{KT}$ and a distribution-dependent regret that is asymptotically optimal, that is, matching the $κ\ln T$ lower bound by Lai and Robbins (1985) and Burnetas and Katehakis (1996), where $κ$ is the optimal problem-dependent constant. This constant $κ$ depends on the model $\mathcal{D}$ considered (the family of possible distributions over the arms). Ménard and Garivier (2017) provided strategies achieving such a bi-optimality in the parametric case of models given by one-dimensional exponential families, while Lattimore (2016, 2018) did so for the family of (sub)Gaussian distributions with variance less than $1$. We extend this result to the non-parametric case of all distributions over $[0,1]$. We do so by combining the MOSS strategy by Audibert and Bubeck (2009), which enjoys a distribution-free regret bound of optimal order $\sqrt{KT}$, and the KL-UCB strategy by Cappé et al. (2013), for which we provide in passing the first analysis of an optimal distribution-dependent $κ\ln T$ regret bound in the model of all distributions over $[0,1]$. We were able to obtain this non-parametric bi-optimality result while working hard to streamline the proofs (of previously known regret bounds and thus of the new analyses carried out); a second merit of the present contribution is therefore to provide a review of proofs of classical regret bounds for index-based strategies for $K$-armed stochastic bandits.

stat.ML↗

arXiv2022

Adaptation to the Range in $K$-Armed Bandits

We consider stochastic bandit problems with $K$ arms, each associated with a bounded distribution supported on the range $[m,M]$. We do not assume that the range $[m,M]$ is known and show that there is a cost for learning this range. Indeed, a new trade-off between distribution-dependent and distribution-free regret bounds arises, which prevents from simultaneously achieving the typical $\ln T$ and $\sqrt{T}$ bounds. For instance, a $\sqrt{T}$}distribution-free regret bound may only be achieved if the distribution-dependent regret bounds are at least of order $\sqrt{T}$. We exhibit a strategy achieving the rates for regret indicated by the new trade-off.

math.ST↗

arXiv2021

A Unified Approach to Fair Online Learning via Blackwell Approachability

We provide a setting and a general approach to fair online learning with stochastic sensitive and non-sensitive contexts. The setting is a repeated game between the Player and Nature, where at each stage both pick actions based on the contexts. Inspired by the notion of unawareness, we assume that the Player can only access the non-sensitive context before making a decision, while we discuss both cases of Nature accessing the sensitive contexts and Nature unaware of the sensitive contexts. Adapting Blackwell's approachability theory to handle the case of an unknown contexts' distribution, we provide a general necessary and sufficient condition for learning objectives to be compatible with some fairness constraints. This condition is instantiated on (group-wise) no-regret and (group-wise) calibration objectives, and on demographic parity as an additional constraint. When the objective is not compatible with the constraint, the provided framework permits to characterise the optimal trade-off between the two.

cs.LG↗

arXiv2020

Hierarchical robust aggregation of sales forecasts at aggregated levels in e-commerce, based on exponential smoothing and Holt's linear trend method

We revisit the interest of classical statistical techniques for sales forecasting like exponential smoothing and extensions thereof (as Holt's linear trend method). We do so by considering ensemble forecasts, given by several instances of these classical techniques tuned with different (sets of) parameters, and by forming convex combinations of the elements of ensemble forecasts over time, in a robust and sequential manner. The machine-learning theory behind this is called "robust online aggregation", or "prediction with expert advice", or "prediction of individual sequences" (see Cesa-Bianchi and Lugosi, 2006). We apply this methodology to a hierarchical data set of sales provided by the e-commerce company Cdiscount and output forecasts at the levels of subsubfamilies, subfamilies and families of items sold, for various forecasting horizons (up to 6-week-ahead). The performance achieved is better than what would be obtained by optimally tuning the classical techniques on a train set and using their forecasts on the test set. The performance is also good from an intrinsic point of view (in terms of mean absolute percentage of error). While getting these better forecasts of sales at the levels of subsubfamilies, subfamilies and families is interesting per se, we also suggest to use them as additional features when forecasting demand at the item level.

stat.AP↗

arXiv2019

Sequential model aggregation for production forecasting

Production forecasting is a key step to design the future development of a reservoir. A classical way to generate such forecasts consists in simulating future production for numerical models representative of the reservoir. However, identifying such models can be very challenging as they need to be constrained to all available data. In particular, they should reproduce past production data, which requires to solve a complex non-linear inverse problem. In this paper, we thus propose to investigate the potential of machine learning algorithms to predict the future production of a reservoir based on past production data without model calibration. We focus more specifically on robust online aggregation, a deterministic approach that provides a robust framework to make forecasts on a regular basis. This method does not rely on any specific assumption or need for stochastic modeling. Forecasts are first simulated for a set of base reservoir models representing the prior uncertainty, and then combined to predict production at the next time step. The weight associated to each forecast is related to its past performance. Three different algorithms are considered for weight computations: the exponentially weighted average algorithm, ridge regression and the Lasso regression. They are applied on a synthetic reservoir case study, the Brugge case, for sequential predictions. To estimate the potential of development scenarios, production forecasts are needed on long periods of time without intermediary data acquisition. An extension of the deterministic aggregation approach is thus proposed in this paper to provide such multi-step-ahead forecasts.

stat.ML↗

arXiv2019

Fano's inequality for random variables

We extend Fano's inequality, which controls the average probability of events in terms of the average of some $f$--divergences, to work with arbitrary events (not necessarily forming a partition) and even with arbitrary $[0,1]$--valued random variables, possibly in continuously infinite number. We provide two applications of these extensions, in which the consideration of random variables is particularly handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentration (minimax or distribution-dependent) rates and on the regret in non-stochastic sequential learning.

math.ST↗

arXiv2019

Target Tracking for Contextual Bandits: Application to Demand Side Management

We propose a contextual-bandit approach for demand side management by offering price incentives. More precisely, a target mean consumption is set at each round and the mean consumption is modeled as a complex function of the distribution of prices sent and of some contextual variables such as the temperature, weather, and so on. The performance of our strategies is measured in quadratic losses through a regret criterion. We offer $T^{2/3}$ upper bounds on this regret (up to poly-logarithmic terms)---and even faster rates under stronger assumptions---for strategies inspired by standard strategies for contextual bandits (like LinUCB, see Li et al., 2010). Simulations on a real data set gathered by UK Power Networks, in which price incentives were offered, show that our strategies are effective and may indeed manage demand response by suitably picking the price levels.

cs.LG↗