Based on the algorithmic proof of Lov\'asz local lemma due to Moser and Tardos, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method}. Inspired by this work, we propose a more general framework and a better analysis. This leads to improved upper bounds on chromatic numbers and indices. In particular, every graph with maximum degree Δ has an acyclic chromatic number at most 32Δ43+O(Δ), and a non-repetitive chromatic number at most Δ2+1.89Δ53+O(Δ43). Also every planar graph with maximum degree Δ has a facial Thue chromatic number at most Δ+O(Δ12) and facial Thue chromatic index at most 10.