Title: Stability analysis of SGD through the normalized loss function. Abstract: We prove new generalization bounds for stochastic gradient descent for both the convex and non-convex cases. Our analysis is based on the stability framework. We analyze stability with respect to the normalized version of the loss function used for training. This leads to investigating a form of angle-wise stability instead of euclidean stability in weights. For neural networks, the measure of distance we consider is invariant to rescaling the weights of each layer. Furthermore, we exploit the notion of on-average stability in order to obtain a data-dependent quantity in the bound. This data-dependent quantity is seen to be more favorable when training with larger learning rates in our numerical experiments. This might help to shed some light on why larger learning rates can lead to better generalization in some practical scenarios.