  We develop a probabilistic framework for analysing model-based reinforcement
learning in the episodic setting. We then apply it to study finite-time horizon
stochastic control problems with linear dynamics but unknown coefficients and
convex, but possibly irregular, objective function. Using probabilistic
representations, we study regularity of the associated cost functions and
establish precise estimates for the performance gap between applying optimal
feedback control derived from estimated and true model parameters. We identify
conditions under which this performance gap is quadratic, improving the linear
performance gap in recent work [X. Guo, A. Hu, and Y. Zhang, arXiv preprint,
arXiv:2104.09311, (2021)], which matches the results obtained for stochastic
linear-quadratic problems. Next, we propose a phase-based learning algorithm
for which we show how to optimise exploration-exploitation trade-off and
achieve sublinear regrets in high probability and expectation. When assumptions
needed for the quadratic performance gap hold, the algorithm achieves an order
$\mathcal{O}(\sqrt{N} \ln N)$ high probability regret, in the general case, and
an order $\mathcal{O}((\ln N)^2)$ expected regret, in self-exploration case,
over $N$ episodes, matching the best possible results from the literature. The
analysis requires novel concentration inequalities for correlated
continuous-time observations, which we derive.
