Bilateral trade is a central problem in algorithmic economics, and recent work has explored how to design trading mechanisms that use no-regret learning algorithms. However, no-regret learning is impossible when budget balance has to be enforced at each time step. Bernasconi et al. show how this impossibility result can be circumvented by relaxing the budget balance constraint to hold only globally over all time steps. In particular, they design an algorithm achieving regret of the order of \(\tilde{O}(T^{^3/_4})\) and provide a lower bound of \(\Omega(T^{^5/_7})\).

We study the sample complexity of learning a uniform approximation of an n-dimensional cumulative distribution function (CDF) within an error є > 0, when observations are restricted to a minimal one-bit feedback. This serves as a counterpart to the multivariate DKW inequality under “full feedback”, extending it to the setting of “bandit feedback”. Our main result shows a near-dimensional-invariance in the sample complexity: we get a uniform є-approximation with a sample complexity 1/є3log(1/є)O(n) over a arbitrary fine grid, where the dimensionality n only affects logarithmic terms. As direct corollaries, we provide tight sample complexity bounds and novel regret guarantees for learning fixed-price mechanisms in small markets, such as bilateral trade settings.

We study online learning in constrained Markov decision processes (CMDPs) in which rewards and constraints may be either stochastic or adversarial. In such settings, stradi et al. (2024) proposed the first best-of-both-worlds algorithm able to seamlessly handle stochastic and adversarial constraints, achieving optimal regret and constraint violation bounds in both cases. This algorithm suffers from two major drawbacks. First, it only works under full feedback, which severely limits its applicability in practice. Moreover, it relies on optimizing over the space of occupancy measures, which requires solving convex optimization problems, an highly inefficient task. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with bandit feedback. Specifically, when the constraints are stochastic, the algorithm achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation, while, when they are adversarial, it attains $\widetilde{\mathcal{O}}(\sqrt{T})$ constraint violation and a tight fraction of the optimal reward. Moreover, our algorithm is based on a policy optimization approach, which is much more efficient than occupancy-measure-based methods.

In constrained MDPs (CMDPs) with adversarial rewards and constraints, a known impossibility result prevents any algorithm from attaining sublinear regret and constraint violation, when competing against a best-in-hindsight policy that satisfies the constraints on average. In this paper, we show how to ease such a negative result, by considering settings that generalize both stochastic CMDPs and adversarial ones. We provide algorithms whose performances smoothly degrade as the level of environment adverseness increases. In this paper, we show that this negative result can be eased in CMDPs with non-stationary rewards and constraints, by providing algorithms whose performances smoothly degrade as non-stationarity increases. Specifically, they attain $\widetilde{\mathcal{O}} (\sqrt{T} + C)$ regret and positive constraint violation under bandit feedback, where $C$ measures the adverseness of rewards and constraints. This is $C = \Theta(T)$ in the worst case, coherently with the impossibility result for adversarial CMDPs. First, we design an algorithm with the desired guarantees when $C$ is known. Then, in the case $C$ is unknown, we obtain the same results by embedding multiple instances of such an algorithm in a general meta-procedure, which suitably selects them so as to balance the trade-off between regret and constraint violation.