In this work, we study the problem of global optimization in univariate loss functions, where we analyze the regret of the popular lower bounding algorithms (e.g., Piyavskii-Shubert algorithm). For any given time $T$, instead of the widely available simple regret (which is the difference of the losses between the best estimation up to $T$ and the global optimizer), we study the cumulative regret up to that time. With a suitable lower bounding algorithm, we show that it is possible to achieve satisfactory cumulative regret bounds for different classes of functions. For Lipschitz continuous functions with the parameter $L$, we show that the cumulative regret is $O(Llog T)$. For Lipschitz smooth functions with the parameter $H$, we show that the cumulative regret is $O(H)$. We also analytically extend our results for a broader class of functions that covers both the Lipschitz continuous and smooth functions individually.
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