\begin{abstract}
A Bayesian coreset is a small, weighted subset of a data set that replaces the
full data during inference to reduce computational cost.  The state-of-the-art
coreset construction algorithm, \emph{Coreset Markov chain Monte Carlo}
(Coreset MCMC), uses draws from an adaptive Markov chain targeting the coreset
posterior to train the coreset weights via stochastic gradient optimization.
However, the quality of the constructed coreset, and thus the quality of its
posterior approximation, is sensitive to the stochastic optimization learning
rate.  In this work, we propose a learning-rate-free stochastic gradient
optimization procedure, \emph{Hot-start Distance over Gradient} (Hot DoG),
for training coreset weights in Coreset MCMC without user tuning effort.
We provide a theoretical analysis of the convergence of the coreset weights 
produced by Hot DoG.
We also provide empirical results demonstrate that Hot DoG provides higher 
quality posterior
approximations than other learning-rate-free stochastic gradient methods, and
performs competitively to optimally-tuned ADAM.
\end{abstract}