Online Non-Monotone DR-Submodular Maximization Beyond $1/e$

We give a $0.4013$-approximate algorithm for adversarial online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed sets. To our knowledge, this is the first polynomial-oracle online guarantee strictly above $1/e$ for this problem class. Each action is committed before current-function feedback. The algorithm maintains one feasible state, uses one feasible gradient query and one Euclidean projection per round, and achieves $O(DB\sqrt{dT})$ expected approximate regret for diameter $D$ and gradient bound $B$. The main result is a first-order inequality that holds simultaneously for every comparator. A strengthened delayed-trajectory comparison and an endpoint-retaining coordinate-chain comparison have complementary nonlinear residuals. A common-state substitution aligns their linear terms, and a fixed mixture of two feasible actions cancels the nonlinear terms exactly. This replaces objective-dependent offline auxiliary optimization with one online linear update. Continuous differentiability suffices; no positive anchor or interior-point condition is required. We prove extensions to non-anticipating adaptive adversaries and stochastic gradients. For every absolute tolerance $\epsilon >0$, the offline specialization attains expected coefficient ${\alpha }_{\ast }>0.4013$ up to additive error $\epsilon$ using $O(1+d{D}^2{B}^2/{\epsilon }^2)$ feasible gradient queries and projections.