arXiv subjects

Davide Maran

Publications and source records attributed to Davide Maran.

12 recordsLinked to original sources

arXiv2026

Online Market Making and the Value of Observing the Order Book

We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader's valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $O(\sqrt T)$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $O(\sqrt T)$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.

cs.LG↗

arXiv2026

Sharper Guarantees for Misspecified Kernelized Bandit Optimization

Existing guarantees for misspecified kernelized bandit optimization pay for misspecification through kernel complexity: in generic offline bounds, the misspecification level $\varepsilon$ is multiplied by $\sqrt{d_\mathrm{eff}}$, where $d_\mathrm{eff}$ is the kernel effective dimension, while in online regret bounds, the corresponding penalty is $\sqrt{\gamma_n}\,n\varepsilon$, where $\gamma_n$ is the maximum information gain after $n$ rounds of interaction. In this work, we show that, for a large class of kernels, the misspecification amplification can be reduced to logarithmic or polylogarithmic growth. In the offline setting, we first prove high-probability simple-regret bounds whose misspecification term is governed by a spectral Lebesgue constant. This yields logarithmic amplification for one-dimensional monotone spectra and polylogarithmic amplification for multivariate Fourier-diagonal product kernels. In the online setting, we modify a domain-splitting algorithm and prove a cumulative regret bound of $\widetilde{\mathcal O}(\sqrt{\gamma_n n}+n\varepsilon)$ under mild localized eigendecay assumptions, removing the extra $\sqrt{\gamma_n}$ factor from the misspecification term. The common principle is localization: spectral localization controls the Lebesgue constant of the offline approximation operator, while domain splitting implements the spatial analogue of this mechanism in the online setting, preventing local misspecification errors from being amplified globally.

cs.LG↗

arXiv2026

Learning in Markov Decision Processes with Exogenous Dynamics

Reinforcement learning algorithms are typically designed for generic Markov Decision Processes (MDPs), where any state-action pair can lead to an arbitrary transition distribution. In many practical systems, however, only a subset of the state variables is directly influenced by the agent's actions, while the remaining components evolve according to exogenous dynamics and account for most of the stochasticity. In this work, we study a structured class of MDPs characterized by exogenous state components whose transitions are independent of the agent's actions. We show that exploiting this structure yields significantly improved learning guarantees, with only the size of the exogenous state space appearing in the leading terms of the regret bounds. We further establish a matching lower bound, showing that this dependence is information-theoretically optimal. Finally, we empirically validate our approach across classical toy settings and real-world-inspired environments, demonstrating substantial gains in sample efficiency compared to standard reinforcement learning methods.

cs.LG↗

arXiv2024

Finite Sample Bounds for Non-Parametric Regression: Optimal Sample Efficiency and Space Complexity

We address the problem of learning an unknown smooth function and its derivatives from noisy pointwise evaluations under the supremum norm. While classical nonparametric regression provides a strong theoretical foundation, traditional kernel-based estimators often incur high computational costs and memory requirements that scale with the sample size, limiting their utility in real-time applications such as reinforcement learning. To overcome these challenges, we propose a parametric approach based on a finite-dimensional representation that achieves minimax-optimal uniform convergence rates. Our method enables lightweight inference without storing all samples in memory. We provide sharp finite-sample bounds under sub-Gaussian noise, derive second-order Bernstein-type guarantees, and prove matching lower bounds, thereby confirming the optimality of our approach in both estimation error and memory efficiency.

cs.LG↗

arXiv2024

Local Linearity: the Key for No-regret Reinforcement Learning in Continuous MDPs

Achieving the no-regret property for Reinforcement Learning (RL) problems in continuous state and action-space environments is one of the major open problems in the field. Existing solutions either work under very specific assumptions or achieve bounds that are vacuous in some regimes. Furthermore, many structural assumptions are known to suffer from a provably unavoidable exponential dependence on the time horizon $H$ in the regret, which makes any possible solution unfeasible in practice. In this paper, we identify local linearity as the feature that makes Markov Decision Processes (MDPs) both learnable (sublinear regret) and feasible (regret that is polynomial in $H$). We define a novel MDP representation class, namely Locally Linearizable MDPs, generalizing other representation classes like Linear MDPs and MDPS with low inherent Belmman error. Then, i) we introduce Cinderella, a no-regret algorithm for this general representation class, and ii) we show that all known learnable and feasible MDP families are representable in this class. We first show that all known feasible MDPs belong to a family that we call Mildly Smooth MDPs. Then, we show how any mildly smooth MDP can be represented as a Locally Linearizable MDP by an appropriate choice of representation. This way, Cinderella is shown to achieve state-of-the-art regret bounds for all previously known (and some new) continuous MDPs for which RL is learnable and feasible.

cs.LG↗

arXiv2024

Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs

We consider the problem of learning an $\varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Given access to a generative model, we achieve rate-optimal sample complexity by performing a simple, \emph{perturbed} version of least-squares value iteration with orthogonal trigonometric polynomials as features. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our~$\widetilde{\mathcal{O}}(\epsilon^{-2-d/(\nu+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $\nu$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(\nu=0)$. At the same time, for $\nu\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(\epsilon^{-2})$ rate of low-rank MDPs, which are more amenable to regression approaches. In this sense, our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.

cs.LG↗

arXiv2024

No-Regret Reinforcement Learning in Smooth MDPs

Obtaining no-regret guarantees for reinforcement learning (RL) in the case of problems with continuous state and/or action spaces is still one of the major open challenges in the field. Recently, a variety of solutions have been proposed, but besides very specific settings, the general problem remains unsolved. In this paper, we introduce a novel structural assumption on the Markov decision processes (MDPs), namely $\nu-$smoothness, that generalizes most of the settings proposed so far (e.g., linear MDPs and Lipschitz MDPs). To face this challenging scenario, we propose two algorithms for regret minimization in $\nu-$smooth MDPs. Both algorithms build upon the idea of constructing an MDP representation through an orthogonal feature map based on Legendre polynomials. The first algorithm, \textsc{Legendre-Eleanor}, archives the no-regret property under weaker assumptions but is computationally inefficient, whereas the second one, \textsc{Legendre-LSVI}, runs in polynomial time, although for a smaller class of problems. After analyzing their regret properties, we compare our results with state-of-the-art ones from RL theory, showing that our algorithms achieve the best guarantees.

cs.LG↗

arXiv2022

Autoregressive Bandits

Autoregressive processes naturally arise in a large variety of real-world scenarios, including stock markets, sales forecasting, weather prediction, advertising, and pricing. When facing a sequential decision-making problem in such a context, the temporal dependence between consecutive observations should be properly accounted for guaranteeing convergence to the optimal policy. In this work, we propose a novel online learning setting, namely, Autoregressive Bandits (ARBs), in which the observed reward is governed by an autoregressive process of order $k$, whose parameters depend on the chosen action. We show that, under mild assumptions on the reward process, the optimal policy can be conveniently computed. Then, we devise a new optimistic regret minimization algorithm, namely, AutoRegressive Upper Confidence Bound (AR-UCB), that suffers sublinear regret of order $\widetilde{\mathcal{O}} \left( \frac{(k+1)^{3/2}\sqrt{nT}}{(1-\Gamma)^2}\right)$, where $T$ is the optimization horizon, $n$ is the number of actions, and $\Gamma < 1$ is a stability index of the process. Finally, we empirically validate our algorithm, illustrating its advantages w.r.t. bandit baselines and its robustness to misspecification of key parameters.

cs.LG↗

arXiv2022

Tight Performance Guarantees of Imitator Policies with Continuous Actions

Behavioral Cloning (BC) aims at learning a policy that mimics the behavior demonstrated by an expert. The current theoretical understanding of BC is limited to the case of finite actions. In this paper, we study BC with the goal of providing theoretical guarantees on the performance of the imitator policy in the case of continuous actions. We start by deriving a novel bound on the performance gap based on Wasserstein distance, applicable for continuous-action experts, holding under the assumption that the value function is Lipschitz continuous. Since this latter condition is hardy fulfilled in practice, even for Lipschitz Markov Decision Processes and policies, we propose a relaxed setting, proving that value function is always Holder continuous. This result is of independent interest and allows obtaining in BC a general bound for the performance of the imitator policy. Finally, we analyze noise injection, a common practice in which the expert action is executed in the environment after the application of a noise kernel. We show that this practice allows deriving stronger performance guarantees, at the price of a bias due to the noise addition.

cs.LG↗

arXiv2022

Delayed Reinforcement Learning by Imitation

When the agent's observations or interactions are delayed, classic reinforcement learning tools usually fail. In this paper, we propose a simple yet new and efficient solution to this problem. We assume that, in the undelayed environment, an efficient policy is known or can be easily learned, but the task may suffer from delays in practice and we thus want to take them into account. We present a novel algorithm, Delayed Imitation with Dataset Aggregation (DIDA), which builds upon imitation learning methods to learn how to act in a delayed environment from undelayed demonstrations. We provide a theoretical analysis of the approach that will guide the practical design of DIDA. These results are also of general interest in the delayed reinforcement learning literature by providing bounds on the performance between delayed and undelayed tasks, under smoothness conditions. We show empirically that DIDA obtains high performances with a remarkable sample efficiency on a variety of tasks, including robotic locomotion, classic control, and trading.

cs.LG↗

arXiv2020

A simple way to reduce factorization problems to SAT

As Cook-Levin theorem showed, every NP problem can be reduced to SAT in polynomial time. In this paper I show a simpler and more efficent method to reduce some factorization problems to the satisfability of a boolean formula.

cs.CC↗

arXiv2020

Least singular value and condition number of a square random matrix with i.i.d. rows

We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $ε$) is at least of order $ε$. This allows us to generalize a result about the expectation of the condition number that was proved in the case of centered gaussian i.i.d. entries: such an expectation is always infinite. Moreover, we get some additional results for some well-known random matrix ensembles, in particular for the isotropic log-concave case, which is proved to have the best behaving in terms of the well conditioning.

math.PR↗