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| author | Thibaut Horel <thibaut.horel@gmail.com> | 2015-02-06 01:57:22 -0500 |
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| committer | Thibaut Horel <thibaut.horel@gmail.com> | 2015-02-06 01:57:22 -0500 |
| commit | 0b26569f6ec37266b449c01d9049118dc022dbfc (patch) | |
| tree | 9200dcfa8afaa707de2840ae2b643ea52478db39 | |
| parent | 553511259bdd50dc9df89f52893861a6c4332d79 (diff) | |
| download | cascades-0b26569f6ec37266b449c01d9049118dc022dbfc.tar.gz | |
Fix bugs
| -rw-r--r-- | paper/sections/lowerbound.tex | 2 | ||||
| -rw-r--r-- | paper/sections/results.tex | 4 |
2 files changed, 3 insertions, 3 deletions
diff --git a/paper/sections/lowerbound.tex b/paper/sections/lowerbound.tex index 7f51f46..5c35446 100644 --- a/paper/sections/lowerbound.tex +++ b/paper/sections/lowerbound.tex @@ -41,7 +41,7 @@ Consider the following distribution $D$: choose $S$ uniformly at random from a ``well-chosen'' set of $s$-sparse supports $\mathcal{F}$ and $t$ uniformly at random from $X \defeq\big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$. Define -$\theta = t + w$ where $w\sim\mathcal{N}(0, \alpha\frac{s}{m}I_m})$ and $\alpha +$\theta = t + w$ where $w\sim\mathcal{N}(0, \alpha\frac{s}{m}I_m)$ and $\alpha = \Omega(\frac{1}{C})$. Consider the following communication game between Alice and Bob: diff --git a/paper/sections/results.tex b/paper/sections/results.tex index ef205f4..0c1cc3b 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -304,7 +304,7 @@ too collinear with each other. In the specific case of ``logistic cascades'' (where $f$ is the logistic function), the Hessian simplifies to $\nabla^2\mathcal{L}(\theta^*) = \frac{1}{|\mathcal{T}|}XX^T$ where $X$ is the design matrix $[x^1 \ldots -x^\mathca{|T|}]$. The restricted eigenvalue condition is equivalent in this +x^\mathcal{|T|}]$. The restricted eigenvalue condition is equivalent in this case to the assumption made in the Lasso analysis of \cite{bickel:2009}. \paragraph{(RE) with high probability} @@ -326,7 +326,7 @@ 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 -8 \cite{rudelson:13}}, this +8 \cite{rudelson:13}, this can be reduced to: \begin{displaymath} n \geq C s \log m \log^3 \left( \frac{s \log m}{C'} \right) |