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\ell_{\exp }\left(H_{t-1}+h_{t} | \mathcal{D}\right)=\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\left(1-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})+\frac{1}{2}\right)\right]
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\ell_{\exp }\left(H_{t-1}+h_{t} | \mathcal{D}\right)=\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\left(1-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})+\frac{1}{2}\right)\right]
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\ell_{\exp }\left(H_{t-1}+h_{t} | \mathcal{D}\right) &=\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})} e^{-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})}\right]
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\ell_{\exp }\left(H_{t-1}+h_{t} | \mathcal{D}\right) &=\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})} e^{-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})}\right]
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