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

Theoretical bound of the efficiency of learning with coarse-graining