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@ -5,8 +5,153 @@
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%%% TeX-master: "main"
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%%% End:
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With the following Theorem we will have an explicit desrctiption for the
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limes of RN as the amount of nodes is increased.
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In this section we will analyze the connection of shallow Neural
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Networks and regression splines. We will see that the punishment of
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wight size in training the shallow Neural Netowork will result in a
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function that minimizes the second derivative as the amount of hidden
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nodes ia grown to infinity. In order to properly formulate this relation we will
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first need to introduce some definitions.
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\begin{Definition}[Ridge penalized Neural Network]
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Let $\mathcal{RN}_{w, \omega}$ be a randomized shallow neural
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network, as introduced in ???. Then the optimal ridge penalized
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network is given by
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\[
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\mathcal{RN}^{*, \tilde{\lambda}}_{\omega}(x) \coloneqq
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\mathcal{RN}_{w^{*, \tilde{\lambda}}(\omega), \omega}
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\]
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with
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\[
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w^{*,\tilde{\lambda}}(\omega) :\in \argmin_{w \in
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\mathbb{R}^n} \underbrace{ \left\{\overbrace{\sum_{i = 1}^N \left(\mathcal{RN}_{w,
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\omega}(x_i^{\text{train}}) -
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y_i^{\text{train}}\right)^2}^{L(\mathcal{RN}_{w, \omega})} +
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\tilde{\lambda} \norm{w}_2^2\right\}}_{\eqqcolon F_n^{\tilde{\lambda}}(\mathcal{RN}_{w,\omega})}.
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\]
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\end{Definition}
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\label{def:rpnn}
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In the ridge penalized Neural Network large weights are penalized, the
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extend of which can be tuned with the parameter $\tilde{\lambda}$. If
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$n$ is larger than the amount of training samples $N$ then for
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$\tilde{\lambda} \to 0$ the network will interpolate the data while
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having minimal weights, resulting in the \textit{minimum norm
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network} $\mathcal{RN}_{w^{\text{min}}, \omega}$.
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\[
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\mathcal{RN}_{w^{\text{min}}, \omega} \text{ randomized shallow
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Neural network with weights } w^{\text{min}}:
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\]
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\[
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w^{\text{min}} \in \argmin_{w \in \mathbb{R}^n} \norm{w}, \text{
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s.t. }
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\mathcal{RN}_{w,\omega}(x_i^{train}) = y_i^{train}, \, \forall i \in
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\left\{1,\dots,N\right\}.
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\]
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For $\tilde{\lambda} \to \infty$ the learned
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function will resemble the data less and less with the weights
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approaching $0$. Usually $\tilde{\lambda}$ lies between 0 and 1, as
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for larger values the focus of weight reduction is larger than fittig
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the data.\par
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In order to make the notation more convinient in the follwoing the
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$\omega$ used to express the realised random parameters will no longer
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be explizitly mentioned.
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\begin{Definition}
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\label{def:kink}
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Let $\mathcal{RN}_w$ be a randomized shallow Neural
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Network according to Definition~\ref{def:rsnn}, then kinks depending on the random parameters can
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be observed.
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\[
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\mathcal{RN}_w(x) = \sum_{k = 1}^n w_k \gamma(b_k + v_kx)
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\]
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Because we specified $\gamma(y) \coloneqq \max\left\{0, y\right\}$ a
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kink in $\gamma$ can be observed at $\gamma(0) = 0$. As $b_k + v_kx = 0$ for $x
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= -\frac{b_k}{v_k}$ we define the following:
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\begin{enumerate}[label=(\alph*)]
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\item Let $\xi_k \coloneqq -\frac{b_k}{v_k}$ be the k-th kink of $\mathcal{RN}_w$.
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\item Let $g_{\xi}(\xi_k)$ be the density of the kinks $\xi_k =
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- \frac{b_k}{v_k}$ in accordance to the distributions of $b_k$ and
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$v_k$.
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\item Let $h_{k,n} \coloneqq \frac{1}{n g_{\xi}(\xi_k)}$ be the
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average estmated distance from kink $\xi_k$ to the next nearest
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one.
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\end{enumerate}
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\end{Definition}
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In order to later prove the connection between randomised shallow
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Neural Networks and regression splines, we first take a look at a
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smooth approximation of the RSNN.
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\begin{Definition}[Smooth Approximation of Randomized Shallow Neural
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Network]
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\label{def:srsnn}
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Let $RS_{w}$ be a randomized shallow Neural Network according to
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Definition~\ref{def:RSNN} with weights $w$ and kinks $\xi_k$ with
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corresponding kink density $g_{\xi}$ as given by
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Definition~\ref{def:kink}.
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In order to smooth the RSNN consider following kernel for every $x$:
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\[
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\kappa_x(s) \coloneqq \mathds{1}_{\left\{\abs{s} \leq \frac{1}{2 \sqrt{n}
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g_{\xi}(x)}\right\}}(s)\sqrt{n} g_{\xi}(x), \, \forall s \in \mathbb{R}
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\]
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Using this kernel we define a smooth approximation of
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$\mathcal{RN}_w$ by
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\[
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f^w(x) \coloneqq \int_{\mathds{R}} \mathcal{RN}_w(x-s) \kappa_x(s) ds.
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\]
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\end{Definition}
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Note that the kernel introduced in Definition~\ref{def:srsnn}
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satisfies $\int_{\mathbb{R}}\kappa_x dx = 1$. While $f^w$ looks highly
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similar to a convolution, it differs slightly as the kernel $\kappa_x(s)$
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is dependent on $x$. Therefore only $f^w = (\mathcal{RN}_w *
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\kappa_x)(x)$ is well defined, while $\mathcal{RN}_w * \kappa$ is not.
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Now we take a look at weighted regression splines. Later we will prove
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that the ridge penalized neural network as defined in
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Definition~\ref{def:rpnn} converges a weighted regression spline, as
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the amount of hidden nodes is grown to inifity.
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\begin{Definition}[Weighted regression spline]
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Let $x_i^{train}, y_i^{train} \in \mathbb{R}, i \in
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\left\{1,\dots,N\right\}$ be trainig data. For a given $\lambda \in \mathbb{R}_{>0}$
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and a function $g: \mathbb{R} \to \mathbb{R}_{>0}$ the weighted
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regression spline $f^{*, \lambda}_g$ is given by
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\[
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f^{*, \lambda}_g :\in \argmin_{\substack{f \in \mathcal{C}^2(\mathbb{R})
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\\ \supp(f) \subseteq \supp(g)}} \underbrace{\left\{ \overbrace{\sum_{i =
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1}^N \left(f(x_i^{train}) - y_i^{train}\right)^2}^{L(f)} +
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\lambda g(0) \int_{\supp(g)}\frac{\left(f''(x)\right)^2}{g(x)}
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dx\right\}}_{\eqqcolon F^{\lambda, g}(f)}.
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\]
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\end{Definition}
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Similary to ridge weight penalized neural networks the parameter
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$\lambda$ controls a trade-off between accuracy on the training data
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and smoothness or low second dreivative. For $g \equiv 1$ and $\lambda \to 0$ the
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resuling function $f^{*, 0+}$ will interpolate the training data while minimizing
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the second derivative. Such a function is known as smooth spline
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interpolation or (cubic) smoothing spline.
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\[
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f^{*, 0+} \text{ smooth spline interpolation: }
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\]
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\[
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f^{*, 0+} \coloneqq \lim_{\lambda \to 0+} f^{*, \lambda}_1 \in
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\argmin_{\substack{f \in \mathcal{C}^2\mathbb{R}, \\ f(x_i^{train}) =
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y_i^{train}} = \left( \int _{\mathbb{R}} (f''(x))^2dx\right).
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\]
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\begin{Assumption}~
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\begin{enumerate}[label=(\alph*)]
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\item The probability density function of the kinks $\xi_k$, namely $g_\xi$
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has compact support on $\supp(g_{\xi})$.
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\item The density $g_{\xi}$ is uniformly continuous on $\supp(g_{\xi})$.
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\item $g_{\xi}(0) \neq 0$
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\end{enumerate}
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\end{Assumption}
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\begin{Theorem}[Ridge weight penaltiy corresponds to adapted spline]
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\label{theo:main1}
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@ -28,7 +173,7 @@ limes of RN as the amount of nodes is increased.
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as defined in ??? and ??? respectively.
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\end{Theorem}
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In order to proof Theo~\ref{theo:main1} we need to proof a number of
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auxilary Lemmata first.
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auxiliary Lemmata first.
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\begin{Definition}[Sobolev Norm]
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\label{def:sobonorm}
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@ -50,49 +195,126 @@ auxilary Lemmata first.
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Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
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\mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
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\subset \mathbb{R}\) with \(f(a)=0\) it holds that
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\begin{equation}
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\begin{equation*}
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\label{eq:pti1}
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\exists C_K^{\infty} \in \mathbb{R}_{>0} :
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\norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
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\norm{f'}_{L^{\infty}(K)}.
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\end{equation}
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\end{equation*}
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If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
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\mathbb{R}\) Lesbeque integrable then additionally
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\begin{equation}
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\begin{equation*}
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\label{eq:pti2}
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\exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
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C_K^2 \norm{f''}_{L^2(K)}.
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\end{equation}
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\proof
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With the fundamental theorem of calculus, if
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\(\norm{f}_{L^{\infty}(K)}<\infty\) we get
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\begin{equation}
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\label{eq:f_f'}
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\norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
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\sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
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\sup_{y \in K}\abs{f'(y)}.
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\end{equation}
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Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
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\end{equation*}
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% \proof
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% With the fundamental theorem of calculus, if
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% \(\norm{f}_{L^{\infty}(K)}<\infty\) we get
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% \begin{equation}
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% \label{eq:f_f'}
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% \norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
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% \sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
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% \sup_{y \in K}\abs{f'(y)}.
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% \end{equation}
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% Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
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% \[
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% \norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
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% \max\left\{\norm{f}_{L^{\infty}(K)},
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% \norm{f'}_{L^{\infty}(K)}\right\}
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% \stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
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% 1\right\}\norm{f'}_{L^{\infty}(K)}.
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% \]
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% With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
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% get (\ref{eq:pti1}).
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% By using the Hölder inequality, we can proof the second claim.
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% \begin{align*}
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% \norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
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% \mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
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% K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
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% &\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
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% \in
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% K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
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% = \abs{b-a}\norm{f''}_{L^2(K)}.
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% \end{align*}
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% Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
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% \abs{b-a}C_K^{\infty}\).
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% \qed
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\end{Lemma}
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\begin{Lemma}
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Let $\mathcal{RN}$ be a shallow Neural network. For \(\varphi :
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\mathbb{R}^2 \to \mathbb{R}\) uniformly continous such that
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\[
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\norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
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\max\left\{\norm{f}_{L^{\infty}(K)},
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\norm{f'}_{L^{\infty}(K)}\right\}
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\stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
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1\right\}\norm{f'}_{L^{\infty}(K)}.
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\forall x \in \supp(g_{\xi}) : \mathbb{E}\left[\varphi(\xi, v)
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\frac{1}{n g_{\xi}(\xi)} \vert \xi = x \right] < \infty,
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\]
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With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
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get (\ref{eq:pti1}).
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By using the Hölder inequality, we can proof the second claim.
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\begin{align*}
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\norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
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\mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
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K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
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&\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
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\in
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K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
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= \abs{b-a}\norm{f''}_{L^2(K)}.
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\end{align*}
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Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
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\abs{b-a}C_K^{\infty}\).
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\qed
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\end{Lemma}
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it holds, that
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\[
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\plimn \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
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\frac{\bar{h}_k}{2}
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=\int_{max\left\{C_{g_{\xi}}^l,T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}
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\mathbb{E}\left[\varphi(\xi, v) \vert \xi = x \right] dx
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\]
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uniformly in \(T \in K\).
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% \proof
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% For \(T \leq C_{g_{\xi}}^l\) both sides equal 0, so it is sufficient to
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% consider \(T > C_{g_{\xi}}^l\). With \(\varphi\) and
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% \(\nicefrac{1}{g_{\xi}}\) uniformly continous in \(\xi\),
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% \begin{equation}
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% \label{eq:psi_stet}
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% \forall \varepsilon > 0 : \exists \delta(\varepsilon) : \forall
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% \abs{\xi - \xi'} < \delta(\varepsilon) : \abs{\varphi(\xi, v)
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% \frac{1}{g_{\xi}(\xi)} - \varphi(\xi', v)
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% \frac{1}{g_{\xi}(\xi')}} < \varepsilon
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% \end{equation}
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% uniformly in \(v\). In order to
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% save space we use the notation \((a \wedge b) \coloneqq \min\{a,b\}\) for $a$ and $b
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% \in \mathbb{R}$. W.l.o.g. assume \(\sup(g_{\xi})\) in an
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% intervall. By splitting the interval in disjoint strips of length \(\delta
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% \leq \delta(\varepsilon)\) we get:
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% \[
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% \underbrace{\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
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% \frac{\bar{h}_k}{2}}_{\circled{1}} =
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% \underbrace{\sum_{l \in \mathbb{Z}:
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% \left[\delta l, \delta (l + 1)\right] \subseteq
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% \left[C_{g_{\xi}}^l, C_{g_{\xi}}^u \wedge T
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% \right]}}_{\coloneqq \, l \in I_{\delta}} \left( \, \sum_{\substack{k \in \kappa\\
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% \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
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% \varphi\left(\xi_k, v_k\right)\frac{\bar{h}_k}{2} \right)
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% \]
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% Using (\ref{eq:psi_stet}) we can approximate $\circled{1}$ by
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% \begin{align*}
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% \circled{1} & \approx \sum_{l \in I_{\delta}} \left( \, \sum_{\substack{k \in \kappa\\
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% \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
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% \left(\varphi\left(l\delta, v_k\right)\frac{1}{g_{\xi}(l\delta)}
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% \pm \varepsilon\right)\frac{1}{n} \underbrace{\frac{\abs{\left\{m \in
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% \kappa : \xi_m \in [\delta l, \delta(l + 1)]\right\}}}{\abs{\left\{m \in
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% \kappa : \xi_m \in [\delta l, \delta(l + 1)]\right\}}}}_{=
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% 1}\right) \\
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% % \intertext{}
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% &= \sum_{l \in I_{\delta}} \left( \frac{ \sum_{ \substack{k \in \kappa\\
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% \xi_k \in \left[\delta l, \delta (l + 1)\right]}}
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% \varphi\left(l\delta, v_k\right)}
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% {\abs{\left\{m \in
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% \kappa : \xi_m \in [\delta l, \delta(l + 1)]\right\}}}\frac{\abs{\left\{m \in
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% \kappa : \xi_m \in [\delta l, \delta(l +
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% 1)]\right\}}}{ng_{\xi}(l\delta)}\right) \pm \varepsilon .\\
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% \intertext{We use the mean to approximate the number of kinks in
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% each $\delta$-strip, as it follows a bonomial distribution this
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% amounts to
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% \[
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% \mathbb{E}\left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
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% \delta(l + 1)]\right\}\right]} = n \int_{[\delta l, \delta (l +
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% 1)]} g_{\xi}(x)dx \approx n (\delta g_{\xi}(l\delta) \pm
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% \tilde{\varepsilon}).
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% \]
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% Bla Bla Bla $v_k$}
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% \circled{1} & \approx
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% \end{align*}
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\end{Lemma}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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Tobias Arndt
5 years ago