AI 中文总结
本研究证明了线性预言机在线学习中固定系数方法无法改进遗憾率，并扩展下界至所有确定性学习器，构造实例达到$T^{3/4}$遗憾下界，匹配已知上界。
AI 中文摘要
每轮常数次线性最小化能否改进在线Frank-Wolfe在一般凸集上的$T^{3/4}$遗憾率？Weibel等人猜想固定系数方法不能。我们证明了他们的猜想，并将下界扩展到预言机唯一模型中的每个确定性学习器。学习器接收一个初始可行点和一个直径界，并且必须在与其预言机回复一致的所有域上保持可行。对于$T$轮，决策之间至多$b$次调用，直径界$D$，梯度范数界$L$，我们构造了一个维度$d=2b(T-1)+1$的实例，其遗憾至少为$2^{-1/4}LDb^{-1/4}T^{3/4}$。对手在游戏开始前固定域、初始点、确定性平局规则和线性损失。顶点形成一条路径，其中在决策前可用的每个点当前损失为零，而最终顶点在每一轮都有负损失。对于常数$b$，结果与维度无关保证的已知上界速率相匹配。对于在新梯度上具有非零系数的单次调用固定调度，第二个构造给出了遗憾至少为$3LDT^{3/4}/4$，并且在每次发出的查询处具有唯一最小化器。Weibel等人调整调度的精确算术证书与他们的有限时间数值最坏情况紧密匹配，具有唯一的预言机回复。
英文摘要
Can a constant number of linear minimizations per round improve on the $T^{3/4}$ regret rate of online Frank-Wolfe on general convex sets? Weibel et al. conjectured that fixed-coefficient methods cannot. We prove their conjecture and extend the lower bound to every deterministic learner in an oracle-only model. The learner receives an initial feasible point and a diameter bound, and must remain feasible on every domain consistent with its oracle replies. For $T$ rounds, at most $b$ calls between decisions, diameter bound $D$, and gradient norm bound $L$, we construct an instance in dimension $d=2b(T-1)+1$ with regret at least $2^{-1/4}LDb^{-1/4}T^{3/4}$. The adversary fixes the domain, initial point, deterministic tie rule and linear losses before play. The vertices form a path on which every point available before a decision has zero current loss, while the final vertex has negative loss on every round. For constant $b$, the result matches the known upper rate for dimension-independent guarantees. For one-call fixed schedules with a nonzero coefficient on the newest gradient, a second construction gives regret at least $3LDT^{3/4}/4$ with unique minimizers at every issued query. Exact-arithmetic certificates for the tuned schedule of Weibel et al. closely match their finite-horizon numerical worst cases, with unique oracle replies.
