diff options
| -rw-r--r-- | paper/sections/results.tex | 35 |
1 files changed, 24 insertions, 11 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex index 6c8a35a..01a33e9 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -325,20 +325,33 @@ $\theta$ conveyed by the random observations. Therefore, under an assumption which only involves the probabilistic model and not the actual observations, we can reformulate Theorem~\ref{thm:main}. -We will need the following additional assumptions on the inverse link -function $f$: +We will need the following additional assumptions on the inverse link function $f$: +\begin{equation} + \tag{LF2} + \|f'\|_{\infty} \leq M + \text{ and } + \max\left(\left|\frac{f''}{1-f}\right|, + \left|\frac{f''}{f}\right|\right) + \leq\frac{1}{\alpha} +\end{equation} +whenever $f(\inprod{theta^*}{x})\notin\{0,1\}$. + +\begin{theorem} + If $\E[\nabla^2\mathcal{L}(\theta^*)]$ verifies the $(S,\gamma)$-{\bf (RE)} + condition. Assuming {\bf (LF)} and {\bf (LF2)}, let $\hat\theta$ be + a solution to \eqref{eq:pre-mle}, then if $n\geq $ we have: + \begin{displaymath} + \|\hat \theta - \theta^* \|_2 \leq \frac{3}{\gamma} \sqrt{\frac{s \log + m}{\alpha n^{1-\delta}}} \quad \text{w.p.}\;1-\frac{1}{e^{n^\delta \log m}} + \end{displaymath} +\end{theorem} + +\begin{proof} +\end{proof} -Yet, the restricted eigenvalue -property is however well behaved in the following sense: under reasonable -assumptions, if the population matrix of the hessian $\mathbb{E} \left[\nabla^2 -{\cal L}(\theta) \right]$, corresponding to the \emph{Fisher Information -Matrix} of the Cascade Model as a function of $\Theta$, verifies the restricted -eigenvalue property, then the finite sample hessian also verifies the -restricted eigenvalue property with overwhelming probability. It is -straightforward to show this holds when $n \geq C s^2 \log m$ -\cite{vandegeer:2009}, where $C$ is an absolute constant. By adapting Theorem +By adapting Theorem 8 \cite{rudelson:13}, this can be reduced to: \begin{displaymath} |