Near-continuous time Reinforcement Learning for continuous state-action spaces

We consider the Reinforcement Learning problem of controlling an unknown dynamical system to maximise the long-term average reward along a single trajectory. Most of the literature considers system interactions that occur in discrete time and discrete state-action spaces. Although this standpoint is suitable for games, it is often inadequate for mechanical or digital systems in which interactions occur at a high frequency, if not in continuous time, and whose state spaces are large if not inherently continuous. Perhaps the only exception is the Linear Quadratic framework for which results exist both in discrete and continuous time. However, its ability to handle continuous states comes with the drawback of a rigid dynamic and reward structure. This work aims to overcome these shortcomings by modelling interaction times with a Poisson clock of frequency $\varepsilon^{-1}$, which captures arbitrary time scales: from discrete ($\varepsilon=1$) to continuous time ($\varepsilon\downarrow0$). In addition, we consider a generic reward function and model the state dynamics according to a jump process with an arbitrary transition kernel on $\mathbb{R}^d$. We show that the celebrated optimism protocol applies when the sub-tasks (learning and planning) can be performed effectively. We tackle learning within the eluder dimension framework and propose an approximate planning method based on a diffusive limit approximation of the jump process. Overall, our algorithm enjoys a regret of order $\tilde{\mathcal{O}}(\varepsilon^{1/2} T+\sqrt{T})$. As the frequency of interactions blows up, the approximation error $\varepsilon^{1/2} T$ vanishes, showing that $\tilde{\mathcal{O}}(\sqrt{T})$ is attainable in near-continuous time.

Real-Time Optimisation for Online Learning in Auctions

In display advertising, a small group of sellers and bidders face each other in up to 10 12 auctions a day. In this context, revenue maximisation via monopoly price learning is a high-value problem for sellers. By nature, these auctions are online and produce a very high frequency stream of data. This results in a computational strain that requires algorithms be real-time. Unfortunately, existing methods inherited from the batch setting suffer O($\sqrt t$) time/memory complexity at each update, prohibiting their use. In this paper, we provide the first algorithm for online learning of monopoly prices in online auctions whose update is constant in time and memory.