Geometry-Dependent Bounds for Online Non-Monotone DR-Submodular Maximization
Vaneet Aggarwal
Abstract
We study adversarial online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed sets. A learner commits each action before observing its objective and competes with the best fixed action in hindsight. We prove a comparator-uniform first-order inequality that gives coefficient $4/9$, improving the online $0.401$ benchmark, with one gradient query and one projection per round and $O(\sqrt T)$ expected approximate regret. If $ζ{\bf 1} \in K\subseteq[0,1]^d$, the coefficient improves to $\underlineα(ζ)=\tfrac12-(1-2ζ)_+^2/[2(3-2ζ)^2]$. The proof is a direct ordered-coordinate argument with an objective-independent rational action. Conversely, a three-group symmetry-gap construction yields an offline oracle upper bound $β_*=0.470438681380894\ldots$ at $ζ=0$, even with exact value and full-gradient responses. A parameterized extension and exact finite-instance bounds define an upper function for every $ζ$. The lower and upper bounds match at $1/2$ for $ζ\ge1/2$, and show that the optimal deficit from $1/2$ is $Θ((1/2-ζ)^2)$ as $ζ\uparrow1/2$. For coefficient-revealed polynomials we obtain $1/2$ for quadratics and a geometry-dependent cubic coefficient starting at $8/17$, including $0.49$ at $ζ=1/5$. A constant objective sequence yields an offline $(4/9-\varepsilon)$ approximation with polynomially many first-order queries on the cube and projections, without requiring a supplied positive lower bound on the optimum. We also give nonanticipating adaptive-adversary and value-feedback guarantees, including $O(T^{3/4})$ regret with one noisy value per round.