Abstract: Training and deploying machine learning models that meet fairness criteria for protected groups are fundamental in modern artificial intelligence. While numerous constraints and regularization terms have been proposed in the literature to promote fairness in machine learning tasks, most of these methods are not amenable to stochastic optimization due to the complex and nonlinear structure of constraints and regularizers. Here, the term ``stochastic'' refers to the ability of the algorithm to work with small mini-batches of data. Motivated by the limitation of existing literature, this paper presents a unified stochastic optimization framework for fair empirical risk minimization based on $f$-divergence measures ($f$-FERM). The proposed stochastic algorithm enjoys theoretical convergence guarantees. In addition, our experiments demonstrate the superiority of fairness-accuracy tradeoffs offered by $f$-FERM for almost all batch sizes (ranging from full-batch to batch size of one). Moreover, we show that our framework can be extended to the case where there is a distribution shift from training to the test data. Our extension is based on a distributionally robust optimization reformulation of $f$-FERM objective under $\ell_p$ norms as uncertainty sets. Again in this distributionally robust setting, $f$-FERM not only enjoys theoretical convergence guarantees but also outperforms other baselines in the literature in the tasks involving distribution shifts. An efficient stochastic implementation of $f$-FERM is publicly available.