J( θ; Da) is the Hessian matrix of the loss function ℓ, at the optimal parameters vector θ, computed using the group data Da (henceforth simply referred to as group hessian), and λ(Σ) is the maximum eigenvalue of a matrix Σ. Proof. Using a second order Taylor expansion around θ, the excessive loss R(a) for a group a A can be stated as: R(a) = J( θ; Da) J( θ; Da) = " J θ; Da + θ θ Hℓa θ θ +O θ θ 3 The above, follows from the loss ℓ() being at least twice differentiable, by assumption. Consider two groups a and b in Awith |Da| |Db|. Proposition 2. For a given group a A, gradient norms can be upper bounded as: gℓa O X The above proposition is presented in the context of cross entropy loss or mean squared error loss functions. These two cases are reviewed as follows 3With a slight abuse of notation, the results refer to θ as the homonymous vector which is extended with k k zeros.

Network pruning is a widely-used compression technique that is able to significantly scale down overparameterized models with minimal loss of accuracy. This paper shows that pruning may create or exacerbate disparate impacts. The paper sheds light on the factors to cause such disparities, suggesting differences in gradient norms and distance to decision boundary across groups to be responsible for this critical issue. It analyzes these factors in detail, providing both theoretical and empirical support, and proposes a simple, yet effective, solution that mitigates the disparate impacts caused by pruning.