Lennard Hilgendorf | Searcharxiv

arXiv2022

Bridge Simulation and Metric Estimation on Lie Groups and Homogeneous Spaces

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit non-trivial covariance structure. The pushforward measure gives rise to new parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including $\SPD(3) = \GL_+(3)/\SO(3)$ and $\mathbb S^2 = \SO(3)/\SO(2)$.

stat.CO↗

arXiv2018

Duelling Bandits with Weak Regret in Adversarial Environments

Research on the multi-armed bandit problem has studied the trade-off of exploration and exploitation in depth. However, there are numerous applications where the cardinal absolute-valued feedback model (e.g. ratings from one to five) is not suitable. This has motivated the formulation of the duelling bandits problem, where the learner picks a pair of actions and observes a noisy binary feedback, indicating a relative preference between the two. There exist a multitude of different settings and interpretations of the problem for two reasons. First, due to the absence of a total order of actions, there is no natural definition of the best action. Existing work either explicitly assumes the existence of a linear order, or uses a custom definition for the winner. Second, there are multiple reasonable notions of regret to measure the learner's performance. Most prior work has been focussing on the $\textit{strong regret}$, which averages the quality of the two actions picked. This work focusses on the $\textit{weak regret}$, which is based on the quality of the better of the two actions selected. Weak regret is the more appropriate performance measure when the pair's inferior action has no significant detrimental effect on the pair's quality. We study the duelling bandits problem in the adversarial setting. We provide an algorithm which has theoretical guarantees in both the utility-based setting, which implies a total order, and the unrestricted setting. For the latter, we work with the $\textit{Borda winner}$, finding the action maximising the probability of winning against an action sampled uniformly at random. The thesis concludes with experimental results based on both real-world data and synthetic data, showing the algorithm's performance and limitations.

cs.LG↗