  We consider a class of learning problems in which an agent liquidates a risky
asset while creating both transient price impact driven by an unknown
convolution propagator and linear temporary price impact with an unknown
parameter. We characterize the trader's performance as maximization of a
revenue-risk functional, where the trader also exploits available information
on a price predicting signal. We present a trading algorithm that alternates
between exploration and exploitation phases and achieves sublinear regrets with
high probability. For the exploration phase we propose a novel approach for
non-parametric estimation of the price impact kernel by observing only the
visible price process and derive sharp bounds on the convergence rate, which
are characterised by the singularity of the propagator. These kernel estimation
methods extend existing methods from the area of Tikhonov regularisation for
inverse problems and are of independent interest. The bound on the regret in
the exploitation phase is obtained by deriving stability results for the
optimizer and value function of the associated class of infinite-dimensional
stochastic control problems. As a complementary result we propose a
regression-based algorithm to estimate the conditional expectation of
non-Markovian signals and derive its convergence rate.
