Private and Online Optimization of Piecewise Lipschitz Functions

In this work, we present online and differentially private optimization algorithms for a large family of nonconvex functions. This family consists of piecewise Lipschitz functions, which are ubiquitous across diverse domains. For example, problems in computational economics and algorithm configuration (also known as parameter tuning) often reduce to maximizing piecewise Lipschitz functions. These functions are challenging to optimize privately and online since a small error can push an optimal point into a nonoptimal region. We introduce a sufficient and general dispersion condition on these functions that ensures well-known private and online algorithms have strong utility guarantees. We show that several important problems from computational economics and algorithm configuration reduce to optimizing functions that satisfy this condition. We apply our results to obtain private and online algorithms for these problems. We thus answer several open questions: Cohen-Addad and Kanade ['17] asked how to optimize piecewise Lipschitz functions online and Gupta and Roughgarden ['17] asked what algorithm configuration problems can be solved online with no regret algorithms. In algorithm configuration, the goal is to tune an algorithm's parameters to optimize its performance over a specific application domain. We analyze greedy techniques for subset selection problems and SDP-rounding schemes for problems that can be formulated as integer quadratic programs. In mechanism design and other pricing problems, the goal is to use information about past consumers to design auctions and set prices that extract high profit from future consumers. We analyze the classic classes of second price auctions with reserves and posted price mechanisms. For all of these settings, our general technique implies strong utility bounds in the private setting and strong regret bounds in the online learning setting.