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"Theorem 1 establishes an Õ(d^{3/2}√V_T) regret bound for one-point bandit convex optimization with gradient variation V_T, improving the dimension dependence over Chiang et al. (2013)'s O(d^3√V_T) bound (Theorem 1, Section 3.1).",
"Theorem 2 gives an O((d/λ) log V_T) regret bound for λ-strongly convex functions, improving the prior O((d^2/λ) log V_T) bound by a factor of d (Theorem 2, Section 3.2).",
"Theorem 3 provides O(√(dW_T) + d) regret for linear functions and O(d√W_T + d) regret for convex functions in terms of the gradient variance W_T (Theorem 3, Section 3.3).",
"Theorem 4 delivers O(√(dF_T) + d) regret bounds for linear and convex functions using the small-loss quantity F_T (Theorem 4, Section 3.3).",
"Section 4 presents the first gradient-variation regret bound for one-point bandit linear optimization over hyper-rectangular domains (Section 4).",
"Table 1 summarizes the dimension-dependence improvements across gradient-variation, gradient-variance, and small-loss metrics for linear, convex, and strongly convex function classes relative to prior best-known results (Table 1)."
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