http://git-wip-us.apache.org/repos/asf/incubator-hivemall-site/blob/ba518dab/userguide/misc/prediction.html
----------------------------------------------------------------------
diff --git a/userguide/misc/prediction.html b/userguide/misc/prediction.html
index 60204e5..44462aa 100644
--- a/userguide/misc/prediction.html
+++ b/userguide/misc/prediction.html
@@ -97,7 +97,7 @@
     <link rel="shortcut icon" href="../gitbook/images/favicon.ico" 
type="image/x-icon">
 
     
-    <link rel="next" href="../regression/general.html" />
+    <link rel="next" href="../binaryclass/general.html" />
     
     
     <link rel="prev" href="../eval/lr_datagen.html" />
@@ -598,14 +598,30 @@
             
         </li>
     
-        <li class="chapter " data-level="3.5" 
data-path="../ft_engineering/tfidf.html">
+        <li class="chapter " data-level="3.5" 
data-path="../ft_engineering/pairing.html">
             
-                <a href="../ft_engineering/tfidf.html">
+                <a href="../ft_engineering/pairing.html">
             
                     
                         <b>3.5.</b>
                     
-                    TF-IDF Calculation
+                    FEATURE PAIRING
+            
+                </a>
+            
+
+            
+            <ul class="articles">
+                
+    
+        <li class="chapter " data-level="3.5.1" 
data-path="../ft_engineering/polynomial.html">
+            
+                <a href="../ft_engineering/polynomial.html">
+            
+                    
+                        <b>3.5.1.</b>
+                    
+                    Polynomial Features
             
                 </a>
             
@@ -613,6 +629,11 @@
             
         </li>
     
+
+            </ul>
+            
+        </li>
+    
         <li class="chapter " data-level="3.6" 
data-path="../ft_engineering/ft_trans.html">
             
                 <a href="../ft_engineering/ft_trans.html">
@@ -664,6 +685,21 @@
             
         </li>
     
+        <li class="chapter " data-level="3.7" 
data-path="../ft_engineering/tfidf.html">
+            
+                <a href="../ft_engineering/tfidf.html">
+            
+                    
+                        <b>3.7.</b>
+                    
+                    TF-IDF Calculation
+            
+                </a>
+            
+
+            
+        </li>
+    
 
     
         
@@ -761,7 +797,7 @@
 
     
         
-        <li class="header">Part V - Prediction</li>
+        <li class="header">Part V - Supervised Learning</li>
         
         
     
@@ -780,27 +816,19 @@
             
         </li>
     
-        <li class="chapter " data-level="5.2" 
data-path="../regression/general.html">
-            
-                <a href="../regression/general.html">
-            
-                    
-                        <b>5.2.</b>
-                    
-                    Regression
-            
-                </a>
-            
 
-            
-        </li>
     
-        <li class="chapter " data-level="5.3" 
data-path="../binaryclass/general.html">
+        
+        <li class="header">Part VI - Binary classification</li>
+        
+        
+    
+        <li class="chapter " data-level="6.1" 
data-path="../binaryclass/general.html">
             
                 <a href="../binaryclass/general.html">
             
                     
-                        <b>5.3.</b>
+                        <b>6.1.</b>
                     
                     Binary Classification
             
@@ -810,21 +838,14 @@
             
         </li>
     
-
-    
-        
-        <li class="header">Part VI - Binary classification tutorials</li>
-        
-        
-    
-        <li class="chapter " data-level="6.1" 
data-path="../binaryclass/a9a.html">
+        <li class="chapter " data-level="6.2" 
data-path="../binaryclass/a9a.html">
             
                 <a href="../binaryclass/a9a.html">
             
                     
-                        <b>6.1.</b>
+                        <b>6.2.</b>
                     
-                    a9a
+                    a9a tutorial
             
                 </a>
             
@@ -833,12 +854,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.1.1" 
data-path="../binaryclass/a9a_dataset.html">
+        <li class="chapter " data-level="6.2.1" 
data-path="../binaryclass/a9a_dataset.html">
             
                 <a href="../binaryclass/a9a_dataset.html">
             
                     
-                        <b>6.1.1.</b>
+                        <b>6.2.1.</b>
                     
                     Data preparation
             
@@ -848,12 +869,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.1.2" 
data-path="../binaryclass/a9a_lr.html">
+        <li class="chapter " data-level="6.2.2" 
data-path="../binaryclass/a9a_lr.html">
             
                 <a href="../binaryclass/a9a_lr.html">
             
                     
-                        <b>6.1.2.</b>
+                        <b>6.2.2.</b>
                     
                     Logistic Regression
             
@@ -863,12 +884,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.1.3" 
data-path="../binaryclass/a9a_minibatch.html">
+        <li class="chapter " data-level="6.2.3" 
data-path="../binaryclass/a9a_minibatch.html">
             
                 <a href="../binaryclass/a9a_minibatch.html">
             
                     
-                        <b>6.1.3.</b>
+                        <b>6.2.3.</b>
                     
                     Mini-batch Gradient Descent
             
@@ -883,14 +904,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2" 
data-path="../binaryclass/news20.html">
+        <li class="chapter " data-level="6.3" 
data-path="../binaryclass/news20.html">
             
                 <a href="../binaryclass/news20.html">
             
                     
-                        <b>6.2.</b>
+                        <b>6.3.</b>
                     
-                    News20
+                    News20 tutorial
             
                 </a>
             
@@ -899,12 +920,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.2.1" 
data-path="../binaryclass/news20_dataset.html">
+        <li class="chapter " data-level="6.3.1" 
data-path="../binaryclass/news20_dataset.html">
             
                 <a href="../binaryclass/news20_dataset.html">
             
                     
-                        <b>6.2.1.</b>
+                        <b>6.3.1.</b>
                     
                     Data preparation
             
@@ -914,12 +935,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.2" 
data-path="../binaryclass/news20_pa.html">
+        <li class="chapter " data-level="6.3.2" 
data-path="../binaryclass/news20_pa.html">
             
                 <a href="../binaryclass/news20_pa.html">
             
                     
-                        <b>6.2.2.</b>
+                        <b>6.3.2.</b>
                     
                     Perceptron, Passive Aggressive
             
@@ -929,12 +950,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.3" 
data-path="../binaryclass/news20_scw.html">
+        <li class="chapter " data-level="6.3.3" 
data-path="../binaryclass/news20_scw.html">
             
                 <a href="../binaryclass/news20_scw.html">
             
                     
-                        <b>6.2.3.</b>
+                        <b>6.3.3.</b>
                     
                     CW, AROW, SCW
             
@@ -944,12 +965,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.4" 
data-path="../binaryclass/news20_adagrad.html">
+        <li class="chapter " data-level="6.3.4" 
data-path="../binaryclass/news20_adagrad.html">
             
                 <a href="../binaryclass/news20_adagrad.html">
             
                     
-                        <b>6.2.4.</b>
+                        <b>6.3.4.</b>
                     
                     AdaGradRDA, AdaGrad, AdaDelta
             
@@ -964,14 +985,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.3" 
data-path="../binaryclass/kdd2010a.html">
+        <li class="chapter " data-level="6.4" 
data-path="../binaryclass/kdd2010a.html">
             
                 <a href="../binaryclass/kdd2010a.html">
             
                     
-                        <b>6.3.</b>
+                        <b>6.4.</b>
                     
-                    KDD2010a
+                    KDD2010a tutorial
             
                 </a>
             
@@ -980,12 +1001,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.3.1" 
data-path="../binaryclass/kdd2010a_dataset.html">
+        <li class="chapter " data-level="6.4.1" 
data-path="../binaryclass/kdd2010a_dataset.html">
             
                 <a href="../binaryclass/kdd2010a_dataset.html">
             
                     
-                        <b>6.3.1.</b>
+                        <b>6.4.1.</b>
                     
                     Data preparation
             
@@ -995,12 +1016,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.3.2" 
data-path="../binaryclass/kdd2010a_scw.html">
+        <li class="chapter " data-level="6.4.2" 
data-path="../binaryclass/kdd2010a_scw.html">
             
                 <a href="../binaryclass/kdd2010a_scw.html">
             
                     
-                        <b>6.3.2.</b>
+                        <b>6.4.2.</b>
                     
                     PA, CW, AROW, SCW
             
@@ -1015,14 +1036,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.4" 
data-path="../binaryclass/kdd2010b.html">
+        <li class="chapter " data-level="6.5" 
data-path="../binaryclass/kdd2010b.html">
             
                 <a href="../binaryclass/kdd2010b.html">
             
                     
-                        <b>6.4.</b>
+                        <b>6.5.</b>
                     
-                    KDD2010b
+                    KDD2010b tutorial
             
                 </a>
             
@@ -1031,12 +1052,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.4.1" 
data-path="../binaryclass/kdd2010b_dataset.html">
+        <li class="chapter " data-level="6.5.1" 
data-path="../binaryclass/kdd2010b_dataset.html">
             
                 <a href="../binaryclass/kdd2010b_dataset.html">
             
                     
-                        <b>6.4.1.</b>
+                        <b>6.5.1.</b>
                     
                     Data preparation
             
@@ -1046,12 +1067,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.4.2" 
data-path="../binaryclass/kdd2010b_arow.html">
+        <li class="chapter " data-level="6.5.2" 
data-path="../binaryclass/kdd2010b_arow.html">
             
                 <a href="../binaryclass/kdd2010b_arow.html">
             
                     
-                        <b>6.4.2.</b>
+                        <b>6.5.2.</b>
                     
                     AROW
             
@@ -1066,14 +1087,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.5" 
data-path="../binaryclass/webspam.html">
+        <li class="chapter " data-level="6.6" 
data-path="../binaryclass/webspam.html">
             
                 <a href="../binaryclass/webspam.html">
             
                     
-                        <b>6.5.</b>
+                        <b>6.6.</b>
                     
-                    Webspam
+                    Webspam tutorial
             
                 </a>
             
@@ -1082,12 +1103,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.5.1" 
data-path="../binaryclass/webspam_dataset.html">
+        <li class="chapter " data-level="6.6.1" 
data-path="../binaryclass/webspam_dataset.html">
             
                 <a href="../binaryclass/webspam_dataset.html">
             
                     
-                        <b>6.5.1.</b>
+                        <b>6.6.1.</b>
                     
                     Data pareparation
             
@@ -1097,12 +1118,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.5.2" 
data-path="../binaryclass/webspam_scw.html">
+        <li class="chapter " data-level="6.6.2" 
data-path="../binaryclass/webspam_scw.html">
             
                 <a href="../binaryclass/webspam_scw.html">
             
                     
-                        <b>6.5.2.</b>
+                        <b>6.6.2.</b>
                     
                     PA1, AROW, SCW
             
@@ -1117,14 +1138,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.6" 
data-path="../binaryclass/titanic_rf.html">
+        <li class="chapter " data-level="6.7" 
data-path="../binaryclass/titanic_rf.html">
             
                 <a href="../binaryclass/titanic_rf.html">
             
                     
-                        <b>6.6.</b>
+                        <b>6.7.</b>
                     
-                    Kaggle Titanic
+                    Kaggle Titanic tutorial
             
                 </a>
             
@@ -1135,7 +1156,7 @@
 
     
         
-        <li class="header">Part VII - Multiclass classification tutorials</li>
+        <li class="header">Part VII - Multiclass classification</li>
         
         
     
@@ -1146,7 +1167,7 @@
                     
                         <b>7.1.</b>
                     
-                    News20 Multiclass
+                    News20 Multiclass tutorial
             
                 </a>
             
@@ -1257,7 +1278,7 @@
                     
                         <b>7.2.</b>
                     
-                    Iris
+                    Iris tutorial
             
                 </a>
             
@@ -1319,18 +1340,33 @@
 
     
         
-        <li class="header">Part VIII - Regression tutorials</li>
+        <li class="header">Part VIII - Regression</li>
         
         
     
-        <li class="chapter " data-level="8.1" 
data-path="../regression/e2006.html">
+        <li class="chapter " data-level="8.1" 
data-path="../regression/general.html">
             
-                <a href="../regression/e2006.html">
+                <a href="../regression/general.html">
             
                     
                         <b>8.1.</b>
                     
-                    E2006-tfidf regression
+                    Regression
+            
+                </a>
+            
+
+            
+        </li>
+    
+        <li class="chapter " data-level="8.2" 
data-path="../regression/e2006.html">
+            
+                <a href="../regression/e2006.html">
+            
+                    
+                        <b>8.2.</b>
+                    
+                    E2006-tfidf regression tutorial
             
                 </a>
             
@@ -1339,12 +1375,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="8.1.1" 
data-path="../regression/e2006_dataset.html">
+        <li class="chapter " data-level="8.2.1" 
data-path="../regression/e2006_dataset.html">
             
                 <a href="../regression/e2006_dataset.html">
             
                     
-                        <b>8.1.1.</b>
+                        <b>8.2.1.</b>
                     
                     Data preparation
             
@@ -1354,12 +1390,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.1.2" 
data-path="../regression/e2006_arow.html">
+        <li class="chapter " data-level="8.2.2" 
data-path="../regression/e2006_arow.html">
             
                 <a href="../regression/e2006_arow.html">
             
                     
-                        <b>8.1.2.</b>
+                        <b>8.2.2.</b>
                     
                     Passive Aggressive, AROW
             
@@ -1374,14 +1410,14 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2" 
data-path="../regression/kddcup12tr2.html">
+        <li class="chapter " data-level="8.3" 
data-path="../regression/kddcup12tr2.html">
             
                 <a href="../regression/kddcup12tr2.html">
             
                     
-                        <b>8.2.</b>
+                        <b>8.3.</b>
                     
-                    KDDCup 2012 track 2 CTR prediction
+                    KDDCup 2012 track 2 CTR prediction tutorial
             
                 </a>
             
@@ -1390,12 +1426,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="8.2.1" 
data-path="../regression/kddcup12tr2_dataset.html">
+        <li class="chapter " data-level="8.3.1" 
data-path="../regression/kddcup12tr2_dataset.html">
             
                 <a href="../regression/kddcup12tr2_dataset.html">
             
                     
-                        <b>8.2.1.</b>
+                        <b>8.3.1.</b>
                     
                     Data preparation
             
@@ -1405,12 +1441,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.2" 
data-path="../regression/kddcup12tr2_lr.html">
+        <li class="chapter " data-level="8.3.2" 
data-path="../regression/kddcup12tr2_lr.html">
             
                 <a href="../regression/kddcup12tr2_lr.html">
             
                     
-                        <b>8.2.2.</b>
+                        <b>8.3.2.</b>
                     
                     Logistic Regression, Passive Aggressive
             
@@ -1420,12 +1456,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.3" 
data-path="../regression/kddcup12tr2_lr_amplify.html">
+        <li class="chapter " data-level="8.3.3" 
data-path="../regression/kddcup12tr2_lr_amplify.html">
             
                 <a href="../regression/kddcup12tr2_lr_amplify.html">
             
                     
-                        <b>8.2.3.</b>
+                        <b>8.3.3.</b>
                     
                     Logistic Regression with Amplifier
             
@@ -1435,12 +1471,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.4" 
data-path="../regression/kddcup12tr2_adagrad.html">
+        <li class="chapter " data-level="8.3.4" 
data-path="../regression/kddcup12tr2_adagrad.html">
             
                 <a href="../regression/kddcup12tr2_adagrad.html">
             
                     
-                        <b>8.2.4.</b>
+                        <b>8.3.4.</b>
                     
                     AdaGrad, AdaDelta
             
@@ -2176,14 +2212,14 @@
 <li><strong>Input:</strong> a vector <span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">x</mi></mrow></mrow><annotation 
encoding="application/x-tex">\mathbf{x}</annotation></semantics></math></span><span
 class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.44444em;"></span><span class="strut bottom" 
style="height:0.44444em;vertical-align:0em;"></span><span class="base textstyle 
uncramped"><span class="mord textstyle uncramped"><span class="mord 
mathbf">x</span></span></span></span></span></li>
 <li><strong>Output:</strong> a value <span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mi>y</mi></mrow><annotation 
encoding="application/x-tex">y</annotation></semantics></math></span><span 
class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.43056em;"></span><span class="strut bottom" 
style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base 
textstyle uncramped"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span></span></span></span></li>
 </ul>
-<p>For a set of samples <span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo><mo 
separator="true">,</mo><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mn>2</mn></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo><mo 
separator="true">,</mo><mo>&#x22EF;</mo><mo 
separator="true">,</mo><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>n</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">(\mathbf{x}_1, y_1), (\mathbf{x}_2, y_2), \cdots, 
(\mathbf{x}_n, y_n)</annotation></semantics></math></span><span 
class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.75em;"></span><span class="strut bottom" 
style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle 
uncramped"><spa
 n class="mopen">(</span><span class="mord"><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="msupsub"><span 
class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathrm 
mtight">1</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathrm mtig
 ht">1</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mclose">)</span><span class="mpunct">,</span><span 
class="mopen">(</span><span class="mord"><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="msupsub"><span 
class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathrm 
mtight">2</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><spa
 n style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathrm 
mtight">2</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mclose">)</span><span class="mpunct">,</span><span 
class="minner">&#x22EF;</span><span class="mpunct">,</span><span 
class="mopen">(</span><span class="mord"><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="msupsub"><span 
class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">n</span></span></span><span c
 lass="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">n</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mclose">)</span></span></span></span>, the goal of prediction 
algorithms is to find a weight vector (i.e., parameters) <span 
class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">w</mi></mr
 ow></mrow><annotation 
encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span
 class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.44444em;"></span><span class="strut bottom" 
style="height:0.44444em;vertical-align:0em;"></span><span class="base textstyle 
uncramped"><span class="mord textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span></span></span></span> by 
minimizing the following error:</p>
+<p>For a set of samples <span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo><mo 
separator="true">,</mo><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mn>2</mn></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo><mo 
separator="true">,</mo><mo>&#x22EF;</mo><mo 
separator="true">,</mo><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>n</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">(\mathbf{x}_1, y_1), (\mathbf{x}_2, y_2), \cdots, 
(\mathbf{x}_n, y_n)</annotation></semantics></math></span><span 
class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.75em;"></span><span class="strut bottom" 
style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle 
uncramped"><spa
 n class="mopen">(</span><span class=""><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped"><span class="mord 
mathrm">1</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer 
 reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span><span class="mpunct">,</span><span 
class="mopen">(</span><span class=""><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped"><span class="mord 
mathrm">2</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span><
 /span><span class="reset-textstyle scriptstyle cramped"><span class="mord 
mathrm">2</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span><span class="mpunct">,</span><span 
class="minner">&#x22EF;</span><span class="mpunct">,</span><span 
class="mopen">(</span><span class=""><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped"><span class="mord 
mathit">n</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathi
 t" style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped"><span class="mord mathit">n</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span></span></span></span>, the goal of prediction algorithms 
is to find a weight vector (i.e., parameters) <span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">w</mi></mrow></mrow><annotation 
encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span
 class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.44444em;"></span><span class="strut bottom" 
style="height:0.44444em;vertical-align:0em;"><
 /span><span class="base textstyle uncramped"><span class="mord textstyle 
uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span></span></span></span> by 
minimizing the following error:</p>
 <p><span class="katex-display"><span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mi>E</mi><mo>(</mo><mrow><mi 
mathvariant="bold">w</mi></mrow><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msubsup><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>L</mi><mo>(</mo><mrow><mi
 mathvariant="bold">w</mi></mrow><mo separator="true">;</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>)</mo><mo>+</mo><mi>&#x3BB;</mi><mi>R</mi><mo>(</mo><mrow><mi
 mathvariant="bold">w</mi></mrow><mo>)</mo></mrow><annotation 
encoding="application/x-tex">
 E(\mathbf{w}) := \frac{1}{n} \sum_{i=1}^{n} L(\mathbf{w}; \mathbf{x}_i, y_i) + 
\lambda R(\mathbf{w})
-</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" 
style="height:1.6513970000000002em;"></span><span class="strut bottom" 
style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="base 
displaystyle textstyle uncramped"><span class="mord mathit" 
style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span 
class="mord displaystyle textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mclose">)</span><span class="mrel">:</span><span 
class="mrel">=</span><span class="mord reset-textstyle displaystyle textstyle 
uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle 
textstyle uncramped nulldelimiter"></span><span class="mfrac"><span 
class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle textstyle cramped"><span class="mord texts
 tyle cramped"><span class="mord mathit">n</span></span></span></span><span 
style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 
size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle textstyle uncramped frac-line"></span></span><span 
style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord 
mathrm">1</span></span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped 
nulldelimiter"></span></span><span class="mop op-limits"><span 
class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span 
class="fontsize-ensurer reset-size5 size5"><span style="font-siz
 e:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped 
mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span><span class="mrel mtight">=</span><span class="mord mathrm 
mtight">1</span></span></span></span><span 
style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer 
reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span><span class="mop op-symbol 
large-op">&#x2211;</span></span></span><span 
style="top:-1.2500050000000003em;margin-left:0em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped 
mtight"><span class="mord mathit mtight">n</span></span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mord mathit">L
 </span><span class="mopen">(</span><span class="mord displaystyle textstyle 
uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mpunct">;</span><span class="mord"><span class="mord displaystyle 
textstyle uncramped"><span class="mord mathbf">x</span></span><span 
class="msupsub"><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span cl
 ass="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mclose">)</span><span class="mbin">+</span><span class="mord 
mathit">&#x3BB;</span><span class="mord mathit" 
style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span 
class="mord displaystyle textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mclose">)</span></span></span></span></span></p>
+</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" 
style="height:1.6513970000000002em;"></span><span class="strut bottom" 
style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="base 
displaystyle textstyle uncramped"><span class="mord mathit" 
style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span 
class="mord displaystyle textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mclose">)</span><span class="mrel">:</span><span 
class="mrel">=</span><span class="mord reset-textstyle displaystyle textstyle 
uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle 
uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span 
style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
textstyle cramped"><span class="mord textstyle c
 ramped"><span class="mord mathit">n</span></span></span></span><span 
style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 
size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle textstyle uncramped frac-line"></span></span><span 
style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord 
mathrm">1</span></span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="sizing reset-size5 size5 reset-textstyle textstyle uncramped 
nulldelimiter"></span></span><span class="mop op-limits"><span 
class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span 
class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x20
 0B;</span></span><span class="reset-textstyle scriptstyle cramped"><span 
class="mord scriptstyle cramped"><span class="mord mathit">i</span><span 
class="mrel">=</span><span class="mord 
mathrm">1</span></span></span></span><span 
style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer 
reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span><span class="op-symbol 
large-op mop">&#x2211;</span></span></span><span 
style="top:-1.2500050000000003em;margin-left:0em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span 
class="mord mathit">n</span></span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mord mathit">L</span><span class="mopen">(</span><span class="mord 
displaystyle tex
 tstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mpunct">;</span><span class=""><span class="mord displaystyle textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped"><span class="mord 
mathit">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramp
 ed"><span class="mord mathit">i</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span><span class="mbin">+</span><span class="mord 
mathit">&#x3BB;</span><span class="mord mathit" 
style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span 
class="mord displaystyle textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span><span 
class="mclose">)</span></span></span></span></span></p>
 <p>In the above formulation, there are two auxiliary functions we have to 
know: </p>
 <ul>
-<li><span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mi>L</mi><mo>(</mo><mrow><mi 
mathvariant="bold">w</mi></mrow><mo separator="true">;</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">L(\mathbf{w}; \mathbf{x}_i, 
y_i)</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span 
class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span 
class="base textstyle uncramped"><span class="mord mathit">L</span><span 
class="mopen">(</span><span class="mord textstyle uncramped"><span class="mord 
mathbf" style="margin-right:0.01597em;">w</span></span><span 
class="mpunct">;</span><span class="mord"><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="msupsub"><span 
class="vlist"><span style="top:0.15em;ma
 rgin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><s
 pan class="mclose">)</span></span></span></span><ul>
-<li><strong>Loss function</strong> for a single sample <span 
class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">(\mathbf{x}_i, 
y_i)</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span 
class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span 
class="base textstyle uncramped"><span class="mopen">(</span><span 
class="mord"><span class="mord textstyle uncramped"><span class="mord 
mathbf">x</span></span><span class="msupsub"><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</spa
 n></span></span><span class="baseline-fix"><span class="fontsize-ensurer 
reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="msupsub"><span 
class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped mtight"><span class="mord mathit 
mtight">i</span></span></span><span class="baseline-fix"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span
 class="mclose">)</span></span></span></span> and given <span 
class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">w</mi></mrow></mrow><annotation encoding="application/x-te
 x">\mathbf{w}</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" style="height:0.44444em;"></span><span 
class="strut bottom" style="height:0.44444em;vertical-align:0em;"></span><span 
class="base textstyle uncramped"><span class="mord textstyle uncramped"><span 
class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span></span></span></span>.</li>
+<li><span class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mi>L</mi><mo>(</mo><mrow><mi 
mathvariant="bold">w</mi></mrow><mo separator="true">;</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">L(\mathbf{w}; \mathbf{x}_i, 
y_i)</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span 
class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span 
class="base textstyle uncramped"><span class="mord mathit">L</span><span 
class="mopen">(</span><span class="mord textstyle uncramped"><span class="mord 
mathbf" style="margin-right:0.01597em;">w</span></span><span 
class="mpunct">;</span><span class=""><span class="mord textstyle 
uncramped"><span class="mord mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span 
 class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped"><span class="mord mathit">i</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped"><span class="mord mathit">i</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span></span></span></span><ul>
+<li><strong>Loss function</strong> for a single sample <span 
class="katex"><span 
class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mrow><mi 
mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo 
separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation
 encoding="application/x-tex">(\mathbf{x}_i, 
y_i)</annotation></semantics></math></span><span class="katex-html" 
aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span 
class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span 
class="base textstyle uncramped"><span class="mopen">(</span><span 
class=""><span class="mord textstyle uncramped"><span class="mord 
mathbf">x</span></span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer 
reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span 
class="reset-textstyle scriptstyle cramped"><span class="mord 
mathit">i</span></span></span><span class="baseline-fi
 x"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mpunct">,</span><span class="mord"><span class="mord mathit" 
style="margin-right:0.03588em;">y</span><span class="vlist"><span 
style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span 
class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle 
scriptstyle cramped"><span class="mord mathit">i</span></span></span><span 
class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span 
style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span 
class="mclose">)</span></span></span></span> and given <span 
class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">w</mi></mrow></mrow><annotation 
encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span
 class="katex-html" aria-hidden="
 true"><span class="strut" style="height:0.44444em;"></span><span class="strut 
bottom" style="height:0.44444em;vertical-align:0em;"></span><span class="base 
textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord 
mathbf" 
style="margin-right:0.01597em;">w</span></span></span></span></span>.</li>
 <li>If this function produces small values, it means the parameter <span 
class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi 
mathvariant="bold">w</mi></mrow></mrow><annotation 
encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span
 class="katex-html" aria-hidden="true"><span class="strut" 
style="height:0.44444em;"></span><span class="strut bottom" 
style="height:0.44444em;vertical-align:0em;"></span><span class="base textstyle 
uncramped"><span class="mord textstyle uncramped"><span class="mord mathbf" 
style="margin-right:0.01597em;">w</span></span></span></span></span> is 
successfully learnt. </li>
 </ul>
 </li>
@@ -2198,30 +2234,35 @@ E(\mathbf{w}) := \frac{1}{n} \sum_{i=1}^{n} 
L(\mathbf{w}; \mathbf{x}_i, y_i) + \
 <p>Interestingly, depending on a choice of loss and regularization function, 
prediction model you obtained will behave differently; even if one combination 
could work as a classifier, another choice might be appropriate for 
regression.</p>
 <p>Below we list possible options for <code>train_regression</code> and 
<code>train_classifier</code>, and this is the reason why these two functions 
are the most flexible in Hivemall:</p>
 <ul>
-<li>Loss function: <code>-loss</code>, <code>-loss_function</code><ul>
+<li><p>Loss function: <code>-loss</code>, <code>-loss_function</code></p>
+<ul>
 <li>For <code>train_regression</code><ul>
-<li>SquaredLoss</li>
-<li>QuantileLoss</li>
-<li>EpsilonInsensitiveLoss</li>
-<li>SquaredEpsilonInsensitiveLoss</li>
-<li>HuberLoss</li>
+<li>SquaredLoss (synonym: squared)</li>
+<li>QuantileLoss (synonym: quantile)</li>
+<li>EpsilonInsensitiveLoss (synonym: epsilon_intensitive)</li>
+<li>SquaredEpsilonInsensitiveLoss (synonym: squared_epsilon_intensitive)</li>
+<li>HuberLoss (synonym: huber)</li>
 </ul>
 </li>
 <li>For <code>train_classifier</code><ul>
-<li>HingeLoss</li>
-<li>LogLoss</li>
-<li>SquaredHingeLoss</li>
-<li>ModifiedHuberLoss</li>
-<li>SquaredLoss</li>
-<li>QuantileLoss</li>
-<li>EpsilonInsensitiveLoss</li>
-<li>SquaredEpsilonInsensitiveLoss</li>
-<li>HuberLoss</li>
+<li>HingeLoss (synonym: hinge)</li>
+<li>LogLoss (synonym: log, logistic)</li>
+<li>SquaredHingeLoss (synonym: squared_hinge)</li>
+<li>ModifiedHuberLoss (synonym: modified_huber)</li>
+<li>The following losses are mainly designed for regression but can sometimes 
be useful in classification as well:<ul>
+<li>SquaredLoss (synonym: squared)</li>
+<li>QuantileLoss (synonym: quantile)</li>
+<li>EpsilonInsensitiveLoss (synonym: epsilon_intensitive)</li>
+<li>SquaredEpsilonInsensitiveLoss (synonym: squared_epsilon_intensitive)</li>
+<li>HuberLoss (synonym: huber)</li>
 </ul>
 </li>
 </ul>
 </li>
-<li>Regularization function: <code>-reg</code>, 
<code>-regularization</code><ul>
+</ul>
+</li>
+<li><p>Regularization function: <code>-reg</code>, 
<code>-regularization</code></p>
+<ul>
 <li>L1</li>
 <li>L2</li>
 <li>ElasticNet</li>
@@ -2239,6 +2280,7 @@ E(\mathbf{w}) := \frac{1}{n} \sum_{i=1}^{n} L(\mathbf{w}; 
\mathbf{x}_i, y_i) + \
 </ul>
 </li>
 </ul>
+<div class="panel panel-primary"><div class="panel-heading"><h3 
class="panel-title" id="note"><i class="fa fa-edit"></i> Note</h3></div><div 
class="panel-body"><p>Option values are case insensitive and you can use 
<code>sgd</code> or <code>rda</code>, or <code>huberloss</code>.</p></div></div>
 <p>In practice, you can try different combinations of the options in order to 
achieve higher prediction accuracy.
 <div id="page-footer" class="localized-footer"><hr><!--
   Licensed to the Apache Software Foundation (ASF) under one
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http://git-wip-us.apache.org/repos/asf/incubator-hivemall-site/blob/ba518dab/userguide/misc/tokenizer.html
----------------------------------------------------------------------
diff --git a/userguide/misc/tokenizer.html b/userguide/misc/tokenizer.html
index 33bf07b..81bbefa 100644
--- a/userguide/misc/tokenizer.html
+++ b/userguide/misc/tokenizer.html
@@ -598,14 +598,30 @@
             
         </li>
     
-        <li class="chapter " data-level="3.5" 
data-path="../ft_engineering/tfidf.html">
+        <li class="chapter " data-level="3.5" 
data-path="../ft_engineering/pairing.html">
             
-                <a href="../ft_engineering/tfidf.html">
+                <a href="../ft_engineering/pairing.html">
             
                     
                         <b>3.5.</b>
                     
-                    TF-IDF Calculation
+                    FEATURE PAIRING
+            
+                </a>
+            
+
+            
+            <ul class="articles">
+                
+    
+        <li class="chapter " data-level="3.5.1" 
data-path="../ft_engineering/polynomial.html">
+            
+                <a href="../ft_engineering/polynomial.html">
+            
+                    
+                        <b>3.5.1.</b>
+                    
+                    Polynomial Features
             
                 </a>
             
@@ -613,6 +629,11 @@
             
         </li>
     
+
+            </ul>
+            
+        </li>
+    
         <li class="chapter " data-level="3.6" 
data-path="../ft_engineering/ft_trans.html">
             
                 <a href="../ft_engineering/ft_trans.html">
@@ -664,6 +685,21 @@
             
         </li>
     
+        <li class="chapter " data-level="3.7" 
data-path="../ft_engineering/tfidf.html">
+            
+                <a href="../ft_engineering/tfidf.html">
+            
+                    
+                        <b>3.7.</b>
+                    
+                    TF-IDF Calculation
+            
+                </a>
+            
+
+            
+        </li>
+    
 
     
         
@@ -761,7 +797,7 @@
 
     
         
-        <li class="header">Part V - Prediction</li>
+        <li class="header">Part V - Supervised Learning</li>
         
         
     
@@ -780,27 +816,19 @@
             
         </li>
     
-        <li class="chapter " data-level="5.2" 
data-path="../regression/general.html">
-            
-                <a href="../regression/general.html">
-            
-                    
-                        <b>5.2.</b>
-                    
-                    Regression
-            
-                </a>
-            
 
-            
-        </li>
     
-        <li class="chapter " data-level="5.3" 
data-path="../binaryclass/general.html">
+        
+        <li class="header">Part VI - Binary classification</li>
+        
+        
+    
+        <li class="chapter " data-level="6.1" 
data-path="../binaryclass/general.html">
             
                 <a href="../binaryclass/general.html">
             
                     
-                        <b>5.3.</b>
+                        <b>6.1.</b>
                     
                     Binary Classification
             
@@ -810,21 +838,14 @@
             
         </li>
     
-
-    
-        
-        <li class="header">Part VI - Binary classification tutorials</li>
-        
-        
-    
-        <li class="chapter " data-level="6.1" 
data-path="../binaryclass/a9a.html">
+        <li class="chapter " data-level="6.2" 
data-path="../binaryclass/a9a.html">
             
                 <a href="../binaryclass/a9a.html">
             
                     
-                        <b>6.1.</b>
+                        <b>6.2.</b>
                     
-                    a9a
+                    a9a tutorial
             
                 </a>
             
@@ -833,12 +854,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.1.1" 
data-path="../binaryclass/a9a_dataset.html">
+        <li class="chapter " data-level="6.2.1" 
data-path="../binaryclass/a9a_dataset.html">
             
                 <a href="../binaryclass/a9a_dataset.html">
             
                     
-                        <b>6.1.1.</b>
+                        <b>6.2.1.</b>
                     
                     Data preparation
             
@@ -848,12 +869,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.1.2" 
data-path="../binaryclass/a9a_lr.html">
+        <li class="chapter " data-level="6.2.2" 
data-path="../binaryclass/a9a_lr.html">
             
                 <a href="../binaryclass/a9a_lr.html">
             
                     
-                        <b>6.1.2.</b>
+                        <b>6.2.2.</b>
                     
                     Logistic Regression
             
@@ -863,12 +884,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.1.3" 
data-path="../binaryclass/a9a_minibatch.html">
+        <li class="chapter " data-level="6.2.3" 
data-path="../binaryclass/a9a_minibatch.html">
             
                 <a href="../binaryclass/a9a_minibatch.html">
             
                     
-                        <b>6.1.3.</b>
+                        <b>6.2.3.</b>
                     
                     Mini-batch Gradient Descent
             
@@ -883,14 +904,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2" 
data-path="../binaryclass/news20.html">
+        <li class="chapter " data-level="6.3" 
data-path="../binaryclass/news20.html">
             
                 <a href="../binaryclass/news20.html">
             
                     
-                        <b>6.2.</b>
+                        <b>6.3.</b>
                     
-                    News20
+                    News20 tutorial
             
                 </a>
             
@@ -899,12 +920,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.2.1" 
data-path="../binaryclass/news20_dataset.html">
+        <li class="chapter " data-level="6.3.1" 
data-path="../binaryclass/news20_dataset.html">
             
                 <a href="../binaryclass/news20_dataset.html">
             
                     
-                        <b>6.2.1.</b>
+                        <b>6.3.1.</b>
                     
                     Data preparation
             
@@ -914,12 +935,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.2" 
data-path="../binaryclass/news20_pa.html">
+        <li class="chapter " data-level="6.3.2" 
data-path="../binaryclass/news20_pa.html">
             
                 <a href="../binaryclass/news20_pa.html">
             
                     
-                        <b>6.2.2.</b>
+                        <b>6.3.2.</b>
                     
                     Perceptron, Passive Aggressive
             
@@ -929,12 +950,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.3" 
data-path="../binaryclass/news20_scw.html">
+        <li class="chapter " data-level="6.3.3" 
data-path="../binaryclass/news20_scw.html">
             
                 <a href="../binaryclass/news20_scw.html">
             
                     
-                        <b>6.2.3.</b>
+                        <b>6.3.3.</b>
                     
                     CW, AROW, SCW
             
@@ -944,12 +965,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.2.4" 
data-path="../binaryclass/news20_adagrad.html">
+        <li class="chapter " data-level="6.3.4" 
data-path="../binaryclass/news20_adagrad.html">
             
                 <a href="../binaryclass/news20_adagrad.html">
             
                     
-                        <b>6.2.4.</b>
+                        <b>6.3.4.</b>
                     
                     AdaGradRDA, AdaGrad, AdaDelta
             
@@ -964,14 +985,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.3" 
data-path="../binaryclass/kdd2010a.html">
+        <li class="chapter " data-level="6.4" 
data-path="../binaryclass/kdd2010a.html">
             
                 <a href="../binaryclass/kdd2010a.html">
             
                     
-                        <b>6.3.</b>
+                        <b>6.4.</b>
                     
-                    KDD2010a
+                    KDD2010a tutorial
             
                 </a>
             
@@ -980,12 +1001,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.3.1" 
data-path="../binaryclass/kdd2010a_dataset.html">
+        <li class="chapter " data-level="6.4.1" 
data-path="../binaryclass/kdd2010a_dataset.html">
             
                 <a href="../binaryclass/kdd2010a_dataset.html">
             
                     
-                        <b>6.3.1.</b>
+                        <b>6.4.1.</b>
                     
                     Data preparation
             
@@ -995,12 +1016,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.3.2" 
data-path="../binaryclass/kdd2010a_scw.html">
+        <li class="chapter " data-level="6.4.2" 
data-path="../binaryclass/kdd2010a_scw.html">
             
                 <a href="../binaryclass/kdd2010a_scw.html">
             
                     
-                        <b>6.3.2.</b>
+                        <b>6.4.2.</b>
                     
                     PA, CW, AROW, SCW
             
@@ -1015,14 +1036,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.4" 
data-path="../binaryclass/kdd2010b.html">
+        <li class="chapter " data-level="6.5" 
data-path="../binaryclass/kdd2010b.html">
             
                 <a href="../binaryclass/kdd2010b.html">
             
                     
-                        <b>6.4.</b>
+                        <b>6.5.</b>
                     
-                    KDD2010b
+                    KDD2010b tutorial
             
                 </a>
             
@@ -1031,12 +1052,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.4.1" 
data-path="../binaryclass/kdd2010b_dataset.html">
+        <li class="chapter " data-level="6.5.1" 
data-path="../binaryclass/kdd2010b_dataset.html">
             
                 <a href="../binaryclass/kdd2010b_dataset.html">
             
                     
-                        <b>6.4.1.</b>
+                        <b>6.5.1.</b>
                     
                     Data preparation
             
@@ -1046,12 +1067,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.4.2" 
data-path="../binaryclass/kdd2010b_arow.html">
+        <li class="chapter " data-level="6.5.2" 
data-path="../binaryclass/kdd2010b_arow.html">
             
                 <a href="../binaryclass/kdd2010b_arow.html">
             
                     
-                        <b>6.4.2.</b>
+                        <b>6.5.2.</b>
                     
                     AROW
             
@@ -1066,14 +1087,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.5" 
data-path="../binaryclass/webspam.html">
+        <li class="chapter " data-level="6.6" 
data-path="../binaryclass/webspam.html">
             
                 <a href="../binaryclass/webspam.html">
             
                     
-                        <b>6.5.</b>
+                        <b>6.6.</b>
                     
-                    Webspam
+                    Webspam tutorial
             
                 </a>
             
@@ -1082,12 +1103,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="6.5.1" 
data-path="../binaryclass/webspam_dataset.html">
+        <li class="chapter " data-level="6.6.1" 
data-path="../binaryclass/webspam_dataset.html">
             
                 <a href="../binaryclass/webspam_dataset.html">
             
                     
-                        <b>6.5.1.</b>
+                        <b>6.6.1.</b>
                     
                     Data pareparation
             
@@ -1097,12 +1118,12 @@
             
         </li>
     
-        <li class="chapter " data-level="6.5.2" 
data-path="../binaryclass/webspam_scw.html">
+        <li class="chapter " data-level="6.6.2" 
data-path="../binaryclass/webspam_scw.html">
             
                 <a href="../binaryclass/webspam_scw.html">
             
                     
-                        <b>6.5.2.</b>
+                        <b>6.6.2.</b>
                     
                     PA1, AROW, SCW
             
@@ -1117,14 +1138,14 @@
             
         </li>
     
-        <li class="chapter " data-level="6.6" 
data-path="../binaryclass/titanic_rf.html">
+        <li class="chapter " data-level="6.7" 
data-path="../binaryclass/titanic_rf.html">
             
                 <a href="../binaryclass/titanic_rf.html">
             
                     
-                        <b>6.6.</b>
+                        <b>6.7.</b>
                     
-                    Kaggle Titanic
+                    Kaggle Titanic tutorial
             
                 </a>
             
@@ -1135,7 +1156,7 @@
 
     
         
-        <li class="header">Part VII - Multiclass classification tutorials</li>
+        <li class="header">Part VII - Multiclass classification</li>
         
         
     
@@ -1146,7 +1167,7 @@
                     
                         <b>7.1.</b>
                     
-                    News20 Multiclass
+                    News20 Multiclass tutorial
             
                 </a>
             
@@ -1257,7 +1278,7 @@
                     
                         <b>7.2.</b>
                     
-                    Iris
+                    Iris tutorial
             
                 </a>
             
@@ -1319,18 +1340,33 @@
 
     
         
-        <li class="header">Part VIII - Regression tutorials</li>
+        <li class="header">Part VIII - Regression</li>
         
         
     
-        <li class="chapter " data-level="8.1" 
data-path="../regression/e2006.html">
+        <li class="chapter " data-level="8.1" 
data-path="../regression/general.html">
             
-                <a href="../regression/e2006.html">
+                <a href="../regression/general.html">
             
                     
                         <b>8.1.</b>
                     
-                    E2006-tfidf regression
+                    Regression
+            
+                </a>
+            
+
+            
+        </li>
+    
+        <li class="chapter " data-level="8.2" 
data-path="../regression/e2006.html">
+            
+                <a href="../regression/e2006.html">
+            
+                    
+                        <b>8.2.</b>
+                    
+                    E2006-tfidf regression tutorial
             
                 </a>
             
@@ -1339,12 +1375,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="8.1.1" 
data-path="../regression/e2006_dataset.html">
+        <li class="chapter " data-level="8.2.1" 
data-path="../regression/e2006_dataset.html">
             
                 <a href="../regression/e2006_dataset.html">
             
                     
-                        <b>8.1.1.</b>
+                        <b>8.2.1.</b>
                     
                     Data preparation
             
@@ -1354,12 +1390,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.1.2" 
data-path="../regression/e2006_arow.html">
+        <li class="chapter " data-level="8.2.2" 
data-path="../regression/e2006_arow.html">
             
                 <a href="../regression/e2006_arow.html">
             
                     
-                        <b>8.1.2.</b>
+                        <b>8.2.2.</b>
                     
                     Passive Aggressive, AROW
             
@@ -1374,14 +1410,14 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2" 
data-path="../regression/kddcup12tr2.html">
+        <li class="chapter " data-level="8.3" 
data-path="../regression/kddcup12tr2.html">
             
                 <a href="../regression/kddcup12tr2.html">
             
                     
-                        <b>8.2.</b>
+                        <b>8.3.</b>
                     
-                    KDDCup 2012 track 2 CTR prediction
+                    KDDCup 2012 track 2 CTR prediction tutorial
             
                 </a>
             
@@ -1390,12 +1426,12 @@
             <ul class="articles">
                 
     
-        <li class="chapter " data-level="8.2.1" 
data-path="../regression/kddcup12tr2_dataset.html">
+        <li class="chapter " data-level="8.3.1" 
data-path="../regression/kddcup12tr2_dataset.html">
             
                 <a href="../regression/kddcup12tr2_dataset.html">
             
                     
-                        <b>8.2.1.</b>
+                        <b>8.3.1.</b>
                     
                     Data preparation
             
@@ -1405,12 +1441,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.2" 
data-path="../regression/kddcup12tr2_lr.html">
+        <li class="chapter " data-level="8.3.2" 
data-path="../regression/kddcup12tr2_lr.html">
             
                 <a href="../regression/kddcup12tr2_lr.html">
             
                     
-                        <b>8.2.2.</b>
+                        <b>8.3.2.</b>
                     
                     Logistic Regression, Passive Aggressive
             
@@ -1420,12 +1456,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.3" 
data-path="../regression/kddcup12tr2_lr_amplify.html">
+        <li class="chapter " data-level="8.3.3" 
data-path="../regression/kddcup12tr2_lr_amplify.html">
             
                 <a href="../regression/kddcup12tr2_lr_amplify.html">
             
                     
-                        <b>8.2.3.</b>
+                        <b>8.3.3.</b>
                     
                     Logistic Regression with Amplifier
             
@@ -1435,12 +1471,12 @@
             
         </li>
     
-        <li class="chapter " data-level="8.2.4" 
data-path="../regression/kddcup12tr2_adagrad.html">
+        <li class="chapter " data-level="8.3.4" 
data-path="../regression/kddcup12tr2_adagrad.html">
             
                 <a href="../regression/kddcup12tr2_adagrad.html">
             
                     
-                        <b>8.2.4.</b>
+                        <b>8.3.4.</b>
                     
                     AdaGrad, AdaDelta
             
@@ -2151,7 +2187,7 @@ Apache Hivemall is an effort undergoing incubation at The 
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