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Copy file name to clipboardExpand all lines: package/samplers/carbo/README.md
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@@ -72,30 +72,30 @@ Please look at [GitHub README.md](https://github.com/optuna/optunahub-registry/t
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In this section, we use the following notations:
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- $x \\in \[0, 1\]^D$, an input vector,
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- $B\_\\epsilon \\coloneqq \[-\\frac{\\epsilon}{2}, \\frac{\\epsilon}{2}\]^D$, an $\\epsilon$-bounding box,
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- $\\xi \\in B\_\\epsilon$, an input noise,
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- $f: \[0, 1\]^D \\rightarrow \\mathbb{R}$, an objective function,
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- $g_c: \[0, 1\]^D \\rightarrow \\mathbb{R}$, the $c$-th constraint,
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- $\\text{LCB}\_{h}: \[0, 1\]^D \\rightarrow \\mathbb{R}$, the lower confidence bound of a function $h$,
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- $\\text{UCB}\_{h}: \[0, 1\]^D \\rightarrow \\mathbb{R}$, the upper confidence bound of a function $h$.
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- $x \in [0, 1]^D$, an input vector,
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- $B_\epsilon \coloneqq [-\frac{\epsilon}{2}, \frac{\epsilon}{2}]^D$, an $\epsilon$-bounding box,
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- $\xi \in B_\epsilon$, an input noise,
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- $f: [0, 1]^D \rightarrow \mathbb{R}$, an objective function,
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- $g_c: [0, 1]^D \rightarrow \mathbb{R}$, the $c$-th constraint,
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- $\text{LCB}_{h}: [0, 1]^D \rightarrow \mathbb{R}$, the lower confidence bound of a function $h$,
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- $\text{UCB}_{h}: [0, 1]^D \rightarrow \mathbb{R}$, the upper confidence bound of a function $h$.
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Please note that both $g_c$ and $f$ are standardized internally so that their distributions follow the assumption on the prior distribution by the Gaussian process.
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Suppose we would like to solve the following max-min problem:
1. Evaluate each function at $x = x_{\star} + \xi_{\star}$.
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1. Go back to 1.
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In principle, $\[\\text{UCB}_{g_c}(x + \\xi)\]^{-}$ and $\[\\text{LCB}_{g_c}(x + \\xi)\]^{-}$ quantify the upper and lower confidence bounds of the violation amount.
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Please note that Processes 2 and 3 are modified from the original paper because our setup assumes that the same input noise $\\xi$ is used for each constraint and the objective evaluations.
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In principle, $[\text{UCB}_{g_c}(x + \xi)]^{-}$ and $[\text{LCB}_{g_c}(x + \xi)]^{-}$ quantify the upper and lower confidence bounds of the violation amount.
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Please note that Processes 2 and 3 are modified from the original paper because our setup assumes that the same input noise $\xi$ is used for each constraint and the objective evaluations.
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Also, the order of the min or max operation and the summation is flipped in our implementation.
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