In this paper, a centralized power control (PC) scheme and an interference channel learning method are jointly tackled to allow a cognitive radio network (CRN) access to the frequency band of a primary user (PU) operating based on an adaptive coding and modulation protocol. The learning process enabler is a cooperative modulation and coding classification (MCC) technique which estimates the modulation and coding scheme of the PU. Due to the lack of cooperation between the PU and the CRN, the CRN exploits this multilevel MCC sensing feedback as implicit channel state information of the PU link in order to constantly monitor the impact of the aggregated interference it causes. In this paper, an algorithm is developed for maximizing the CRN throughput (the PC optimization objective) and simultaneously learning how to mitigate PU interference (the optimization problem constraint) by using only the MCC information. Ideal approaches for this problem setting with high convergence rate are the cutting plane methods (CPM). Here, we focus on the analytic center CPM and the center of gravity CPM whose effectiveness in the proposed simultaneous PC and interference channel learning algorithm is demonstrated through numerical simulations.