Stochastic method has become the first choice for dealing with large-scale regularization and deep learning optimization problems. The acquisition of its convergence rate heavily depends on the unbiased gradient of objective functions. However, for machine learning problems, many scenarios can result in the appearance of biased gradient. In contrast to the unbiased gradient cases, the well-known Nesterov accelerated gradient (NAG) accumulates the error caused by the bias with the iteration. As a result, the optimal convergence will no longer hold and even the convergence cannot be guaranteed. Recent research shows that NAG is also an accelerated algorithm for the individual convergence of projection sub-gradient methods in non-smooth cases. However, until now, there is no report about the affect when the subgradient becomes biased. In this study, for non-smooth optimization problems, it is proved that NAG can obtain a stable individual convergence bound when the subgradient bias is bounded, and the optimal individual convergence can still be achieved while the subgradient errors decrease at an appropriate. As an application, an inexact projection subgradient method is obtained in which the projection needs not calculate accurately. The derived algorithm can approach the stable learning accuracy more quick while keeping the convergence. The experiments verify the correctness of theoretical analysis and the performance of inexact methods.