  A thermodynamic formalism describing the efficiency of information learning
is proposed, which is applicable for stochastic thermodynamic systems with
multiple internal degree of freedom. The learning rate, entropy production rate
(EPR), and entropy flow from the system to the environment under coarse-grained
dynamics are derived. The Cauchy-Schwarz inequality has been applied to
demonstrate the lower bound on the EPR of an internal state. The inequality of
EPR is tighter than the Clausius inequality, leading to the derivative of the
upper bound on the efficiency of learning. The results are verified in cellular
networks with information processes.
