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.
. When faced with multiple objectives, no one single solution may be optimal [see, e.g.,][] ZuluagaSKP13, RaviMRRH93, GrandoniRSZ14 . A solution is Pareto-optimal if no other solution is simultaneously better in all objectives. Often the goal is to recover the entire set of Pareto-optimal solutions. While computing the Pareto set (even membership) is NP-hard , for some problems it can be succinctly approximated . Some work in multi-objective optimization addresses the problem as a convex programming problem, while other work such as @cite_0 provides reductions to an underlying linear optimization problem. Our work follows in this spirit.
@cite_0
1544920412
We discuss problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives, the so-called Pareto curve. The Pareto curve has typically an exponential number of points. However, it turns out that, under general conditions, there is a polynomially succinct curve that approximates the Pareto curve within any desired accuracy. The central computational question is whether such an approximate curve can be constructed efficiently (in polynomial time). We discuss conditions under which this is the case. We examine in more detail the class of linear multiobjective problems, and relate the multiobjective approximation to the single objective case. We will discuss also problems in multiobjective query optimization.