| icml_final_submission.tex:887:107: Error: `)' expected, found `]'. =\sqrt{n}\left[\frac1n\sum_{i=1}^n \big(\ell(\lambda^\star_n,X_i)- \mathbb{E}_q[\ell(\bar\lambda,X)]\right].$$ | |
| icml_final_submission.tex:899:103: Error: `)' expected, found `]'. =\sqrt{n}\left[\frac1n\sum_{i=1}^n \big(\ell(\bar\lambda,X_i)- \mathbb{E}_q[\ell(\bar\lambda,X)]\right].$$ | |
| icml_final_submission.tex:940:182: Error: `]' expected, found `)'. The function $Q_n$ is continuously differentiable in a neighborhood containing $\lambda^\star$. This follows from the definition of $\ell(\cdot, X)$ in Case 3, since $X_i \in [0,1)$ almost surely (hence $\ell(X_i, \lambda) > 0$ and infinitely differentiable for all $\lambda \in [0,\tfrac{1}{1-m_o}]$). In Case 2, we have from Lemma \ref{sup_lemma_case_2_combined} that on each sample path (for all realizations of $X_i \in [0,1]$), the dual optimizer $\lambda^\star_n$ lies in the interior for sufficiently large $n$. In particular, $ \lambda^\star_n \in [\lambda^\star - \eta, \lambda^\star + \eta]$ for sufficiently large $n$, where $\eta > 0$ is chosen such that $\lambda^\star + \eta < \frac{1}{1-m_o}$ and $\lambda^\star - \eta > 0$. Hence, $\ell(\lambda, X_i) > 0$ for all $i$ and for all $\lambda$ in this neighborhood containing $\lambda^\star$. | |
| icml_final_submission.tex:1581:19: Error: `]' expected, found `)'. For each $x\in[0,1)$, observe that, | |
| icml_final_submission.tex:1931:52: Error: `]' expected, found `)'. the analysis to the measure-one event $\{X_i\in[0,1)\ \forall\, i=1,2,3,\ldots\}$. | |
| icml_final_submission.tex:1932:181: Error: `]' expected, found `)'. Using the definition of $g(\lambda,X)$, $N(X_1,X_2,\ldots X_n, \lambda^\star_n )$ is continuously differentiable in $\lambda^\star_n$ for any values of $X_1,X_2,\ldots X_n \in [0,1)$ and any value of $\lambda^\star_n \in \left[0, \frac{1}{1-m_o}\right]$. Hence, using Taylor expansion of $N(X_1,X_2,\ldots X_n, \lambda^\star_n )$ in $\lambda^\star_n$ around $\lambda^\star$ yields | |
| icml_final_submission.tex:1950:55: Error: `]' expected, found `)'. Observe that, for a given $X_1,X_2,\ldots X_n \in [0,1)$, $\mathbb{E}_{\hat q_n}[\ell(\lambda,X)]$ is continuously differentiable in $\lambda$ for $\lambda \in \left[0, \frac{1}{1-m_o}\right]$. Since $\lambda^\star_n$ maximizes $\mathbb{E}_{\hat q_n}[\ell(\lambda,X)]$ on $(0,\lambda^\star]$ and this objective is concave in $\lambda$, we have the usual first-order optimality conditions: | |
| icml_final_submission.tex:1950:290: Error: `)' expected, found `]'. Observe that, for a given $X_1,X_2,\ldots X_n \in [0,1)$, $\mathbb{E}_{\hat q_n}[\ell(\lambda,X)]$ is continuously differentiable in $\lambda$ for $\lambda \in \left[0, \frac{1}{1-m_o}\right]$. Since $\lambda^\star_n$ maximizes $\mathbb{E}_{\hat q_n}[\ell(\lambda,X)]$ on $(0,\lambda^\star]$ and this objective is concave in $\lambda$, we have the usual first-order optimality conditions: | |
| icml_final_submission.tex:2001:67: Error: `]' expected, found `)'. Moreover, for any $\lambda \in [0, \frac{1}{1-m_o}]$ and $x\in[0,1)$, | |
| icml_final_submission.tex:2044:53: Error: `]' expected, found `)'. For all $\lambda\in [0,\lambda^\star]$ and $x\in[0,1)$, | |
| icml_final_submission.tex:2829:33: Error: `)' expected, found `]'. It remains to consider $b\in(1,2]$, where | |
| icml_final_submission.tex:2910:33: Error: `)' expected, found `]'. It remains to consider $b\in(1,2]$, where | |
| icml_final_submission.tex:899:15: Warning: No match found for `['. =\sqrt{n}\left[\frac1n\sum_{i=1}^n \big(\ell(\bar\lambda,X_i)- \mathbb{E}_q[\ell(\bar\lambda,X)]\right].$$ | |
| icml_final_submission.tex:887:15: Warning: No match found for `['. =\sqrt{n}\left[\frac1n\sum_{i=1}^n \big(\ell(\lambda^\star_n,X_i)- \mathbb{E}_q[\ell(\bar\lambda,X)]\right].$$ | |