Tipaluck Krityakierne

Tipaluck Krityakierne, Thotsaporn Aek Thanatipanonda

We study the expected number of rolls required for the cumulative sum of a fair six-sided die to first enter a prescribed target set $H\subset\mathbb{Z}_{\ge0}$. A one-variable dynamic-programming formulation is introduced that removes dependence on the roll count. Within this framework, the infinite process is truncated at a large cutoff $N$ and corrected by an analytically derived overshoot term that accounts for the rare event of exceeding $N$ before entering $H$. Explicit bounds on this residual yield a strict two-sided estimate of the truncation error. The method is numerically efficient, requiring constant memory and linear time in the cutoff. For the perfect-square target set $H=\{n^2:n\in\mathbb{N}\}$, all quantities are evaluated explicitly, yielding \[ \mathbb{E}[T]=7.07976423755110510389555305690818489468\ldots, \] provably correct to 1,017 decimal places. This constitutes the most precise result known to date and establishes a general framework for high-accuracy computation of discrete hitting times.

Poompol Buathong, David Ginsbourger, Tipaluck Krityakierne

We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels typically suffer from a lack of strict positive definiteness, vast subclasses of DE kernels built upon DS kernels do possess this property, enabling in turn combinatorial optimization without requiring to introduce a jitter parameter. Proofs of theoretical results about considered kernels are complemented by a few practicalities regarding hyperparameter fitting. We furthermore demonstrate the applicability of our approach in prediction and optimization tasks, relying both on toy examples and on two test cases from mechanical engineering and hydrogeology, respectively. Experimental results highlight the applicability and compared merits of the considered approaches while opening new perspectives in prediction and sequential design with set inputs.