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We consider a decision maker allocating one unit of renewable and divisibleresource in each period on a number of arms. The arms have unknown and randomrewards whose means are proportional to the allocated resource and whosevariances are proportional to an order $b$ of the allocated resource. Inparticular, if the decision maker allocates resource $A_i$ to arm $i$ in aperiod, then the reward $Y_i$ is$Y_i(A_i)=A_i \mu_i+A_i^b \xi_{i}$, where$\mu_i$ is the unknown mean and the noise $\xi_{i}$ is independent andsub-Gaussian. When the order $b$ ranges from 0 to 1, the framework smoothlybridges the standard stochastic multi-armed bandit and online learning withfull feedback. We design two algorithms that attain the optimal gap-dependentand gap-independent regret bounds for $b\in [0,1]$, and demonstrate a phasetransition at $b=1/2$. The theoretical results hinge on a novel concentrationinequality we have developed that bounds a linear combination of sub-Gaussianrandom variables whose weights are fractional, adapted to the filtration, andmonotonic.

Allocating Divisible Resources on Arms with Unknown and Random Rewards