We study the phenomenon of bias amplification in classifiers, wherein a machine learning model learns to predict classes with a greater disparity than the underlying ground truth. We demonstrate that bias amplification can arise via inductive bias in gradient descent methods resulting in overestimation of importance of moderately-predictive ``weak'' features if insufficient training data is available. This overestimation gives rise to feature-wise bias amplification -- a previously unreported form of bias that can be traced back to the features of a trained model. Through analysis and experiments, we show that the while some bias cannot be mitigated without sacrificing accuracy, feature-wise bias amplification can be mitigated through targeted feature selection. We present two new feature selection algorithms for mitigating bias amplification in linear models, and show how they can be adapted to convolutional neural networks efficiently. Our experiments on synthetic and real data demonstrate that these algorithms consistently lead to reduced bias without harming accuracy, in some cases eliminating predictive bias altogether while providing modest gains in accuracy.