几何相关的非单调DR次模最大化在线界
Geometry-Dependent Bounds for Online Non-Monotone DR-Submodular Maximization
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对非单调DR次模函数的在线最大化，提出几何相关界，改进遗憾系数至4/9，并给出匹配下界与多项式情形下的精确系数。
AI中文摘要：
我们研究在紧凸下闭集上非负、非单调DR次模函数的对抗性在线最大化问题。学习者在观察目标函数之前提交每个动作，并与事后最优的固定动作竞争。我们证明了一个比较器均匀的一阶不等式，其系数为$4/9$，改进了在线$0.401$的基准，每轮仅需一次梯度查询和一次投影，并具有$O(\sqrt T)$的期望近似遗憾。若$\zeta {\bf 1} \in K\subseteq[0,1]^d$，则系数改进为$\underline\alpha(\zeta)=\tfrac12-(1-2\zeta)_+^2/[2(3-2\zeta)^2]$。证明采用直接的顺序坐标论证，并使用与目标无关的有理动作。相反，一个三组对称间隙构造在$\zeta=0$处产生离线预言机上界$\beta_*=0.470438681380894\ldots$，即使具有精确值和全梯度响应。参数化扩展和精确有限实例界为每个$\zeta$定义了一个上函数。下界和上界在$\zeta\ge1/2$时于$1/2$处匹配，并表明当$\zeta\uparrow1/2$时，从$1/2$的最优亏空为$\Theta((1/2-\zeta)^2)$。对于系数揭示的多项式，我们获得二次函数的$1/2$以及从$8/17$开始的几何相关三次系数，包括在$\zeta=1/5$时的$0.49$。一个常数目标序列在立方体上产生离线$(4/9-\varepsilon)$近似，使用多项式多次一阶查询和投影，无需提供最优值的正下界。我们还给出非预测自适应对手和值反馈保证，包括每轮一次噪声值的$O(T^{3/4})$遗憾。
英文摘要：
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.
