The field of algorithmic fairness has rapidly emerged over the past 15 years as algorithms have become ubiquitous in everyday lives. Algorithmic fairness traditionally considers statistical notions of fairness algorithms might satisfy in decisions based on noisy data. We first show that these are theoretically disconnected from welfare-based notions of fairness. We then discuss two individual welfare-based notions of fairness, envy freeness and prejudice freeness, and establish conditions under which they are equivalent to error rate balance and predictive parity, respectively. We discuss the implications of these findings in light of the recently discovered impossibility theorem in algorithmic fairness (Kleinberg, Mullainathan, & Raghavan (2016), Chouldechova (2017)).

The Fisher information matrix (FIM) is fundamental for understanding the trainability of deep neural networks (DNN) since it describes the local metric of the parameter space. We investigate the spectral distribution of the FIM given a single input by focusing on fully-connected networks achieving dynamical isometry. Then, while dynamical isometry is known to keep specific backpropagated signals independent of the depth, we find that the parameter space's local metric depends on the depth. In particular, we obtain an exact expression of the spectrum of the FIM given a single input and reveal that it concentrates around the depth point. Here, considering random initialization and the wide limit, we construct an algebraic methodology to examine the spectrum based on free probability theory, which is the algebraic wrapper of random matrix theory. As a byproduct, we provide the solvable spectral distribution in the two-hidden-layer case. Lastly, we empirically confirm that the spectrum of FIM with small batch-size has the same property as the single-input version. An experimental result shows that FIM's dependence on the depth determines the appropriate size of the learning rate for convergence at the initial phase of the online training of DNNs.