Online supervised learning with L1-regularization has gained attention recently because it generally requires less computational time and a smaller space of complexity than batch-type learning methods. However, a simple L1-regularization method used in an online setting has the side effect that rare features tend to be truncated more than necessary. In fact, feature frequency is highly skewed in many applications. We developed a new family of L1-regularization methods based on the previous updates for loss minimization in linear online learning settings. Our methods can identify and retain low-frequency occurrence but informative features at the same computational cost and convergence rate as previous works. Moreover, we combined our methods with a cumulative penalty model to derive more robust models over noisy data. We applied our methods to several datasets and empirically evaluated the performance of our algorithms. Experimental results showed that our frequency-aware truncated models improved the prediction accuracy.