I am a Lecturer (Assistant Prof.) at the department of Mathematics and Statistics at Lancaster University (UK).

My research interests are at the moment threefold. Firstly, I am interested in non-parametric statistics under minimal assumptions. For example, in non-linear regression a typical assumption is that the mean of the data lies in a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$. This is a strong assumption that can be weakened with the help of interpolation spaces that define a scale of function spaces which interpolate between $\mathcal{H}$ and $L^2$. A weaker assumption is now to assume that the mean lies in some intermediate space between $\mathcal{H}$ and $L^2$, and it is known in which of these interpolation spaces it lies. This assumption is significantly weaker since the intermediate spaces are significantly larger than $\mathcal{H}$.  This last assumption can often be relaxed further with the help of so called adaptive estimators that do not assume knowledge of the interpolation space in which the mean lies, but achieve rates of convergence similar to estimators which assume knowledge of the interpolation space. In (Page & Grünewälder, JMLR, 2019 and Page & Grünewälder, Bernoulli, 2021) we explore these ideas in the context of a norm-constraint kernel estimator. In (Grünewälder, AIStats, 2018) I use the empirical measure to define plug-in estimators for conditional expectations & probabilities and analyse their rates of convergence. This is again a non-parameteric approach relying on VC- and empirical process theory. The estimators are easy to compute (in O(n)) and achieve strong statistical guarantees.

Secondly,  I am interested in the link between non-parametric statistics and optimisation to develop sound non-parametric methods that scale to large-scale datasets. Classical RKHS estimators do not scale well with data (run-time costs are often between $O(n^2)$ and $O(n^3)$).  Recently, there has been significant progress to improve the scaling of RKHS methods using simple partitioning ideas (divide and conquer algorithms). I explore in (Grünewälder,JMLR, 2018) another approach to large-scale regression using conditional gradient algorithms that achieve results en-par with state-of-the-art divide and conquer algorithms. A similar approach can be used to compress empirical measures. I explore this approach in finite dimensional RKHSs in (Grünewälder, ArXiv, 2022).

Thirdly, I am interested in online decision making in the form of the multi-armed bandit problem. In particular, I am interested in forms of the multi-armed bandit problem that are well adapted to practical problems. In (Grünewälder & Khaleghi, JMLR, 2019) we explore the restless-bandit problem from a mixing point of view. Here, pay-offs are dependent over time. This form of the bandit problem is significantly harder to control than the standard setting, but it is also significantly better suited for modelling real-world problems. In other recent works we explore how bandit algorithms can be applied to the stochastic knapsack problem (Pike-Burke & Grünewälder, AIStats, 2017) and we explore the bandit problem where the feedback is delayed and anonymous (C. Pike-Burke, S. Agrawal, C. Szepesvári & S. Grünewälder, ICML 2018).