  The performance of Hamiltonian Monte Carlo simulations crucially depends on
both the integration timestep and the number of integration steps. We present
an adaptive general-purpose framework to automatically tune such parameters,
based on a local loss function which promotes the fast exploration of
phase-space. We show that a good correspondence between loss and
autocorrelation time can be established, allowing for gradient-based
optimization using a fully-differentiable set-up. The loss is constructed in
such a way that it also allows for gradient-driven learning of a distribution
over the number of integration steps. Our approach is demonstrated for the
one-dimensional harmonic oscillator and alanine dipeptide, a small protein
common as a test case for simulation methods. Through the application to the
harmonic oscillator, we highlight the importance of not using a fixed timestep
to avoid a rugged loss surface with many local minima, otherwise trapping the
optimization. In the case of alanine dipeptide, by tuning the only free
parameter of our loss definition, we find a good correspondence between it and
the autocorrelation times, resulting in a $>100$ fold speed up in optimization
of simulation parameters compared to a grid-search. For this system, we also
extend the integrator to allow for atom-dependent timesteps, providing a
further reduction of $25\%$ in autocorrelation times.
