  Tasks in multi-task learning often correlate, conflict, or even compete with
each other. As a result, a single solution that is optimal for all tasks rarely
exists. Recent papers introduced the concept of Pareto optimality to this field
and directly cast multi-task learning as multi-objective optimization problems,
but solutions returned by existing methods are typically finite, sparse, and
discrete. We present a novel, efficient method that generates locally
continuous Pareto sets and Pareto fronts, which opens up the possibility of
continuous analysis of Pareto optimal solutions in machine learning problems.
We scale up theoretical results in multi-objective optimization to modern
machine learning problems by proposing a sample-based sparse linear system, for
which standard Hessian-free solvers in machine learning can be applied. We
compare our method to the state-of-the-art algorithms and demonstrate its usage
of analyzing local Pareto sets on various multi-task classification and
regression problems. The experimental results confirm that our algorithm
reveals the primary directions in local Pareto sets for trade-off balancing,
finds more solutions with different trade-offs efficiently, and scales well to
tasks with millions of parameters.
