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diff --git a/paper/sections/results.tex b/paper/sections/results.tex index c39f9da..5d63cbd 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -130,9 +130,10 @@ $\hat{\theta}_\lambda$ is the solution of \eqref{eq:pre-mle}: To prove Theorem~\ref{thm:main}, we apply Lemma~\ref{lem:negahban} with $\tau_{\mathcal{L}}=0$. Since $\mathcal{L}$ is twice differentiable and convex, -assumption \eqref{eq:rc} is implied by the restricted eigenvalue condition -\eqref{eq:re}. The upper bound on the $\ell_{\infty}$ norm of -$\nabla\mathcal{L}(\theta^*)$ is given by Lemma~\ref{lem:ub}. +assumption \eqref{eq:rc} with $\kappa_{\mathcal{L}}=\frac{\gamma}{2}$ is +implied by the (RE) condition \eqref{eq:re}. The upper bound +on the $\ell_{\infty}$ norm of $\nabla\mathcal{L}(\theta^*)$ is given by +Lemma~\ref{lem:ub}. \begin{lemma} \label{lem:ub} @@ -335,7 +336,8 @@ We will need the following additional assumptions on the inverse link function $ \left|\frac{f''}{f}\right|\right) \leq\frac{1}{\alpha} \end{equation} -whenever $f(\inprod{\theta^*}{x})\notin\{0,1\}$. +whenever $f(\inprod{\theta^*}{x})\notin\{0,1\}$. These conditions are once +again non restrictive in the (IC) model and (V) model. \begin{proposition} \label{prop:fi} @@ -391,8 +393,7 @@ m)$. \paragraph{(RE) vs Irrepresentability Condition} \cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the -likelihood function. Their condition is equivalent to the more commonly called -{\it (S,s)-irrepresentability} condition: +likelihood function also known as the {\it (S,s)-irrepresentability} condition: \begin{comment} \begin{definition} |