From 3d5793a7b1f14906860fabbbca867fca6360c66b Mon Sep 17 00:00:00 2001 From: mhjensen Date: Tue, 12 Nov 2019 00:06:51 +0100 Subject: [PATCH] added steepest descent --- .../html/._DecisionTrees-bs000.html | 2 +- .../html/._DecisionTrees-bs056.html | 22 ++++++++ .../DecisionTrees/html/DecisionTrees-bs.html | 2 +- .../html/DecisionTrees-reveal.html | 30 +++++++++- .../html/DecisionTrees-solarized.html | 24 +++++++- doc/pub/DecisionTrees/html/DecisionTrees.html | 24 +++++++- .../DecisionTrees/ipynb/DecisionTrees.ipynb | 53 +++++++++++++++++- .../ipynb/ipynb-DecisionTrees-src.tar.gz | Bin 294061 -> 294061 bytes .../pdf/DecisionTrees-minted.pdf | Bin 550778 -> 555886 bytes doc/src/DecisionTrees/DecisionTrees.do.txt | 22 ++++++++ 10 files changed, 173 insertions(+), 6 deletions(-) diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html index 1de6f6e4b..63f72251e 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html @@ -293,7 +293,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 11, 2019

+

Nov 12, 2019


diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html index 284e0001b..42a5f4c77 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html @@ -284,6 +284,28 @@ $$ (\hat{\boldsymbol{f}}) \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ +

+We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as +$$ +f_M(x) = \sum_{m=0}^M h_m(x). +$$ + +

+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as +$$ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +$$ + +

+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that +the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). + +

+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have +$$ +(\rho_1) \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +$$ +

diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html index 1de6f6e4b..63f72251e 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html @@ -293,7 +293,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 11, 2019

+

Nov 12, 2019


diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html index af57a8bd3..fd93c2216 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

 
-

Nov 11, 2019

+

Nov 12, 2019


@@ -2381,6 +2381,34 @@ $$ (\hat{\boldsymbol{f}}) \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$

 
+ +

+We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as +

 
+$$ +f_M(x) = \sum_{m=0}^M h_m(x). +$$ +

 
+ +

+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as +

 
+$$ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +$$ +

 
+ +

+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that +the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). + +

+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have +

 
+$$ +(\rho_1) \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +$$ +

 
diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html index c9df08769..3045d1183 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html @@ -221,7 +221,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 11, 2019

+

Nov 12, 2019












@@ -2348,6 +2348,28 @@ $$ (\hat{\boldsymbol{f}}) \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ +

+We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as +$$ +f_M(x) = \sum_{m=0}^M h_m(x). +$$ + +

+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as +$$ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +$$ + +

+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that +the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). + +

+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have +$$ +(\rho_1) \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +$$ +











diff --git a/doc/pub/DecisionTrees/html/DecisionTrees.html b/doc/pub/DecisionTrees/html/DecisionTrees.html index 3ac58f8f9..ab637865b 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees.html @@ -226,7 +226,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 11, 2019

+

Nov 12, 2019












@@ -2353,6 +2353,28 @@ $$ (\hat{\boldsymbol{f}}) \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ +

+We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as +$$ +f_M(x) = \sum_{m=0}^M h_m(x). +$$ + +

+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as +$$ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +$$ + +

+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that +the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). + +

+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have +$$ +(\rho_1) \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +$$ +











diff --git a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb index 835164d95..333396a8e 100644 --- a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb +++ b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Nov 11, 2019**\n", + "Date: **Nov 12, 2019**\n", "\n", "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -2580,6 +2580,57 @@ "$$" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f_M(x) = \\sum_{m=0}^M h_m(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the steepest decent approach we approximate $h_m(x) = -\\rho_m g_m(x)$, where $\\rho_m$ is a scalar and $g_m(x)$ the gradient defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g_m(x_i) = \\left[ \\frac{\\partial {\\cal L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{m-1}(x_i)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With the new gradient we can update $f_m(x) = f_{m-1}(x) -\\rho_m g_m(x)$. Using the above squared-error function we see that\n", + "the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.\n", + "\n", + "Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\rho_1) \\mathrm{argmin}_{\\rho}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i+2\\rho y_i)^2.\n", + "$$" + ] + }, { "cell_type": "markdown", "metadata": {}, diff --git a/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz b/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz index 19258d47e4e54e7227a408cafb96cead71bf5595..65c88d19deba8b825bd09ea157a8a4d93b53cf96 100644 GIT binary patch delta 29 lcmZ4cQ*iB1K{okr4u-;)jcl!KjIC@;t!&I&*;tm>005%M36TH* delta 29 lcmZ4cQ*iB1K{okr4u005x)33mVh diff --git a/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf b/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf index 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b/doc/src/DecisionTrees/DecisionTrees.do.txt @@ -1960,7 +1960,29 @@ This means that for every iteration, we need to optimize \] !et +We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as +!bt +\[ +f_M(x) = \sum_{m=0}^M h_m(x). +\] +!et +In the steepest decent approach we approximate $h_m(x) = -\rho_m g_m(x)$, where $\rho_m$ is a scalar and $g_m(x)$ the gradient defined as +!bt +\[ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +\] +!et + +With the new gradient we can update $f_m(x) = f_{m-1}(x) -\rho_m g_m(x)$. Using the above squared-error function we see that +the gradient is $g_m(x_i) = -2(y_i-f(x_i))$. + +Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have +!bt +\[ +(\rho_1) \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +\] +!et !split