Published on Mon Feb 19 2018

Online convex optimization for cumulative constraints

Jianjun Yuan, Andrew Lamperski

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations. The new form of constraint penalizes large constraint violations and cancellation effects cannot occur. For convex objectives, our regret bounds generalize existing bounds, and we give improved regret bounds.

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Abstract

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations of the form $\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-\beta})$, where $\beta\in(0,1)$. Previous literature has focused on long-term constraints of the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can cancel out the effects of violated constraints. In contrast, the new form heavily penalizes large constraint violations and cancellation effects cannot occur. Furthermore, useful bounds on the single step constraint violation $[g(x_t)]_+$ are derived. For convex objectives, our regret bounds generalize existing bounds, and for strongly convex objectives we give improved regret bounds. In numerical experiments, we show that our algorithm closely follows the constraint boundary leading to low cumulative violation.