add Dart tutorial

dmlc-core

@@ -1 +1 @@
-Subproject commit 9fd3b48462a7a651e12a197679f71e043dcb25a2
+Subproject commit c39001019e443c7a061789bd1180f58ce85fc3e6

doc/parameter.md

@@ -13,8 +13,7 @@ In R-package, you can use .(dot) to replace under score in the parameters, for e
 General Parameters
 ------------------
 * booster [default=gbtree]
-  - which booster to use, can be gbtree, gblinear or dart.
+  - which booster to use, can be gbtree, gblinear or dart. gbtree and dart use tree based model while gblinear uses linear function.
-  　gbtree and dart use tree based model while gblinear uses linear function.
 * silent [default=0]
   - 0 means printing running messages, 1 means silent mode.
 * nthread [default to maximum number of threads available if not set]
@@ -81,20 +80,20 @@ Additional parameters for Dart Booster
   - type of sampling algorithm.
     - "uniform": dropped trees are selected uniformly.
     - "weighted": dropped trees are selected in proportion to weight.
-* normalize_type [default="tree]
+* normalize_type [default="tree"]
   - type of normalization algorithm.
-    - "tree": New trees have the same weight of each of dropped trees.
+    - "tree": new trees have the same weight of each of dropped trees.
-              weight of new trees are 1 / (k + learnig_rate)
+      - weight of new trees are 1 / (k + learnig_rate)
-              dropped trees are scaled by a factor of k / (k + learning_rate)
+      - dropped trees are scaled by a factor of k / (k + learning_rate)
-    - "forest": New trees have the same weight of sum of dropped trees (forest).
+    - "forest": new trees have the same weight of sum of dropped trees (forest).
-                weight of new trees are 1 / (1 + learning_rate)
+      - weight of new trees are 1 / (1 + learning_rate)
-                dropped trees are scaled by a factor of 1 / (1 + learning_rate)
+      - dropped trees are scaled by a factor of 1 / (1 + learning_rate)
 * rate_drop [default=0.0]
   - dropout rate.
   - range: [0.0, 1.0]
 * skip_drop [default=0.0]
   - probability of skip dropout.
-    If a dropout is skipped, new trees are added in the same manner as gbtree.
+    - If a dropout is skipped, new trees are added in the same manner as gbtree.
   - range: [0.0, 1.0]
 
 Parameters for Linear Booster

doc/tutorials/dart.md

@@ -0,0 +1,101 @@
+DART booster
+============
+[XGBoost](https://github.com/dmlc/xgboost)) mostly combines a huge number of regression trees with small learning rate.
+In this situation, trees added early are significance and trees added late are unimportant.
+
+Rasmi et.al proposed a new method to add dropout techniques from deep neural nets community to boosted trees, and reported better results in some situations.
+
+This is a instruction of new tree booster `dart`.
+
+Original paper
+--------------
+Rashmi Korlakai Vinayak, Ran Gilad-Bachrach. "DART: Dropouts meet Multiple Additive Regression Trees." [JMLR](http://www.jmlr.org/proceedings/papers/v38/korlakaivinayak15.pdf)
+
+Features
+--------
+- Drop trees in order to solve the over-fitting.
+  - Trivial trees (to correct trivial errors) may be prevented.
+
+Because the randomness introduced in the training, expect the following few difference.
+- Training can be slower than `gbtree` because the random dropout prevents usage of prediction buffer.
+- The early stop might not be stable, due to the randomness.
+
+How it works
+------------
+- In ``$ m $``th training round, suppose ``$ k $`` trees are selected drop.
+- Let ``$ D = \sum_{i \in \mathbf{K}} F_i $`` be leaf scores of dropped trees and ``$ F_m = \eta \tilde{F}_m $`` be leaf scores of a new tree.
+- The objective function is following:
+```math
+\mathrm{Obj}
+= \sum_{j=1}^n L \left( y_j, \hat{y}_j^{m-1} - D_j + \tilde{F}_m \right)
++ \Omega \left( \tilde{F}_m \right).
+```
+- ``$ D $`` and ``$ F_m $`` are overshooting, so using scale factor
+```math
+\hat{y}_j^m = \sum_{i \not\in \mathbf{K}} F_i + a \left( \sum_{i \in \mathbf{K}} F_i + b F_m \right) .
+```
+
+Parameters
+----------
+### booster
+* `dart`
+
+This booster inherits `gbtree`, so `dart` has also `eta`, `gamma`, `max_depth` and so on.
+
+Additional parameters are noted below.
+
+### sample_type
+type of sampling algorithm.
+* `uniform`: (default) dropped trees are selected uniformly.
+* `weighted`: dropped trees are selected in proportion to weight.
+
+### normalize_type
+type of normalization algorithm.
+* `tree`: (default) New trees have the same weight of each of dropped trees.
+```math
+a \left( \sum_{i \in \mathbf{K}} F_i + \frac{1}{k} F_m \right)
+&= a \left( \sum_{i \in \mathbf{K}} F_i + \frac{\eta}{k} \tilde{F}_m \right) \\
+&\sim a \left( 1 + \frac{\eta}{k} \right) D \\
+&= a \frac{k + \eta}{k} D = D , \\
+&\quad a = \frac{k}{k + \eta} .
+```
+
+* `forest`: New trees have the same weight of sum of dropped trees (forest).
+```math
+a \left( \sum_{i \in \mathbf{K}} F_i + F_m \right)
+&= a \left( \sum_{i \in \mathbf{K}} F_i + \eta \tilde{F}_m \right) \\
+&\sim a \left( 1 + \eta \right) D \\
+&= a (1 + \eta) D = D , \\
+&\quad a = \frac{1}{1 + \eta} .
+```
+
+### rate_drop
+dropout rate.
+- range: [0.0, 1.0]
+
+### skip_drop
+probability of skipping dropout.
+- If a dropout is skipped, new trees are added in the same manner as gbtree.
+- range: [0.0, 1.0]
+
+Sample Script
+-------------
+```python
+import xgboost as xgb
+# read in data
+dtrain = xgb.DMatrix('demo/data/agaricus.txt.train')
+dtest = xgb.DMatrix('demo/data/agaricus.txt.test')
+# specify parameters via map
+param = {'booster': 'dart',
+         'max_depth': 5, 'learning_rate': 0.1,
+         'objective': 'binary:logistic', 'silent': True,
+         'sample_type': 'uniform',
+         'normalize_type': 'tree',
+         'rate_drop': 0.1,
+         'skip_drop': 0.5}
+num_round = 50
+bst = xgb.train(param, dtrain, num_round)
+# make prediction
+# ntree_limit must not be 0
+preds = bst.predict(dtest, ntree_limit=num_round)
+```

doc/tutorials/index.md

@@ -6,3 +6,4 @@ See [Awesome XGBoost](https://github.com/dmlc/xgboost/tree/master/demo) for link
 ## Contents
 - [Introduction to Boosted Trees](../model.md)
 - [Distributed XGBoost YARN on AWS](aws_yarn.md)
+- [DART booster](dart.md)