Most systems and learning algorithms optimize average performance or average loss -- one reason being computational complexity. However, many objectives of practical interest are more complex than simply average loss. This arises, for example, when balancing performance or loss with fairness across people. We prove that, from a computational perspective, optimizing arbitrary objectives that take into account performance over a small number of groups is not significantly harder to optimize than average performance. Our main result is a polynomial-time reduction that uses a linear optimizer to optimize an arbitrary (Lipschitz continuous) function of performance over a (constant) number of possibly-overlapping groups. This includes fairness objectives over small numbers of groups, and we further point out that other existing notions of fairness such as individual fairness can be cast as convex optimization and hence more standard convex techniques can be used. Beyond learning, our approach applies to multi-objective optimization, more generally.
We study network-design problems with multiple design objectives. In particular, we look at two cost measures to be minimized simultaneously: the total cost of the network and the maximum degree of any node in the network. Our main result can be roughly stated as follows: given an integer @math , we present approximation algorithms for a variety of network-design problems on an @math -node graph in which the degree of the output network is @math and the cost of this network is @math times that of the minimum-cost degree- @math -bounded network. These algorithms can handle costs on nodes as well as edges. Moreover, we can construct such networks so as to satisfy a variety of connectivity specifications including spanning trees, Steiner trees and generalized Steiner forests. The performance guarantee on the cost of the output network is nearly best-possible unless @math . We also address the special case in which the costs obey the triangle inequality. In this case, we obtain a polynomial-time approximation algorithm with a stronger performance guarantee. Given a bound @math on the degree, the algorithm finds a degree- @math -bounded network of cost at most a constant time optimum. There is no algorithm that does as well in the absence of triangle inequality unless @math . We also show that in the case of spanning networks, we can simultaneously approximate within a constant factor yet another objective: the maximum cost of any edge in the network, also called the bottleneck cost of the network. We extend our algorithms to find TSP tours and @math -connected spanning networks for any fixed @math that simultaneously approximate all these three cost measures.