# Keras memory leak

## Keras memory usage keeps increasing

I was having fun, attempting to do some deep learning with a 2M lines dataset (nothing my computer can’t handle, xgboost was running with roughly 15% of my RAM) when suddenly, as I was adding neural networks in my fancy stacked models, the script kept failing, the memory usage went to the moon, etc, etc.

What did I do wrong ? Did I introduce a memory leak between my model stacking / neural network factory code ? I would be suprised, it worked fine with every other model. And a neural network is more or less a simple vector of floats (in my case, with only hundreds of parameters) so there is no reason for it to be that big.

The only thing I was attempting to do was to cross validate different neural networks, with different architectures.

So, after a quick research : I found this stack overflow question , also some people mentioning a weird behavior coming from model.predict() . Another Github issue is simply called Memory leak . There even is another article simply titled Dealing with memory leak issue in Keras model training and is even mentioned on twitter .

What I ended up suspecting is that there are actually many memory leaks from different methods in the code. So I gathered the list of workarounds I could find.

## Workarounds

Beware, none of them actually works. Some just alleviate the pain, but most likely, the memory usage will keep increasing. Anyways, the good news is that, combining many of the tricks I could read, I managed to have my models run ;)

### Garbage collecting

Generally, when you see these lines in the code it means that the person who wrote it was desperate to make it run while closely monitoring the memory usage of the script and combined tricks not to make sure everything was fitting into the memory. Usually, performing tasks in dedicated functions and trusting the garbage collector to do its job at the right time is enough. But sometimes you meet these del / garbage collector random invokations.

import gc
del model
gc.collect()
K.clear_session()


I did put these lines after every model.fit() I found. They did not help at all in my case.

### Force eager evaluation

This one kind of worked for me. It slows down the training (3 times slower in my case), the memory keeps increasing for no reason, but much less. Just add the following argument in the model.compile() method :

model.compile( [...]
run_eagerly=True)


Some people mentioned it. It did not change a thing for me, but I wrote it that way. Be careful though, model(x) will return a tensorflow object while model.predict(x) will return a numpy object.

### Run it in a dedicated script

Yes, kind of ugly. It does not solve the issue, but if you make your cross validation in a python script, itself being called from the terminal level, you can pass parameters using JSON and hope that each script won’t hit your memory limit.

In my case, I wrote the following class:

from tensorflow.keras.layers import Dense
from tensorflow.keras.models import Sequential
import tensorflow as tf
import gc
import numpy as np

class NNModel:

def __init__(self, architecture, epochs, loss="binary_crossentropy", optimizer="adam"):
self._epochs = epochs
self._loss = loss
self._optimizer = optimizer
self._architecture = architecture
self._model = None

def fit(self, X, y):

gc.collect()
tf.keras.backend.clear_session()

self._model = self._model_factory(X.shape[1])

X_tf = tf.convert_to_tensor(X, dtype=tf.float32)
y_tf = tf.convert_to_tensor(y, dtype=tf.float32)

self._model.fit(X_tf, y_tf, epochs=self._epochs)
return self

def _model_factory(self, input_dim):

model = Sequential()

architecture = self._architecture.copy()
first_layer = architecture.pop(0)

for layer in architecture:

model.compile(loss=self._loss,
optimizer=self._optimizer,
run_eagerly=True,
metrics=['accuracy'])

return model

def predict(self, X):
raise NotImplementedError

def predict_proba(self, X):
X_tf = tf.convert_to_tensor(X, dtype=tf.float32)
res =  self._model(X_tf)
res = np.hstack((1-res, res))
return res


Which I can configure using a JSON that will contain the arguments of the class constructor:

{
"epochs": 8,
"architecture": [[ 12, "relu" ], [ 8, "relu" ], [ 1, "sigmoid" ]]
}


And then I invoke them with:

find ../models/ -name \*.json | xargs --max-args=1 python run_nn.py


So that I can run my different models while I am sure that the memory will be totally released between the execution of two scripts.

### model.predict_on_batch

Quoting MProx from a git issue

I have managed to get around this error by using model.predict_on_batch() instead of model.predict(). This returns an object of type <class ‘tensorflow.python.framework.ops.EagerTensor’> - not a numpy array as claimed in the docs - but it can be cast by calling np.array(model.predict_on_batch(input_data)) to get the output I want.

Side note: I also noticed a similar memory leak problem with calling model.fit() in a loop, albeit with a slower memory accumulation, but this can be fixed in a similar way using model.train_on_batch().

I did not try this one, as segregating different models in different scripts and setting run_eagerly did the job.

### Use tf-nightly

So, tf-nightly is built more or less every day, with the latest features and less tests. Many people claimed that the leak disapeared when using this library. But there are many versions, with potentially other bugs.

### re install the 1.14 version

This bug has been around for a while, some tickets mention it from october 2019 and it is still present in the 2.4 version.

# Conclusion

I look forward to this issue being solved.

# Finding the index of the largest element in a list in OCaml

As far as I know, there is no implementation of argmax and argmin in the default ocaml library (or perhaps they could be called maxi for more consistency with respect to mapi). The following snippet solves it!

let argmax l =
let rec aux max_index index max_value = function
| [] -> max_index
| h::t -> if h > max_value then aux index (index + 1) h t
else aux max_index (index + 1) max_value t
in

match l with
| [] -> 0
| _ ->   aux 0 0 (List.hd l) l


Note that the defaut behavior with an empty list is to return 0 but it can be changed.

# Python fast screenshots and locateOnScreen

Taking screenshots with Python is easy, however, the performance often seems to be an issue, depending on the packages you started with (see per example this question )

In my previous article (/reinforcement-learning/nintendo/reinforcement-learning-nintendo-nes-tutorial/), I noted that I would be limited by the method that was looking for an element on the screen (the Game Over) as often as possible.

Getting rid of bottlenecks is a fun thing to do as a developer. For an unexplained reason, I find it particularly satisfying. Python, with its plethora of libraries introduces many of them. So it is time for a quick tour of the possible solutions.

# Benchmarks

I do not want to mess up the code of my previous article so it often is a good idea, when possible, to do the benchmarks in separate files, with similar inputs.

## Screenshots

In the previous article, I relied on pyautogui. I realized it was built on pyscreeze, so I also tried this library. After some browsing, I learned that PIL also proposed this feature.

I discovered it after writing this article but d3dshot claims to be the fastest way to perform screenshots in Python. I’ll keep that in mind if I face new bottlenecks in the future, but let’s stick with the first 3 packages for now.

from PIL import ImageGrab
from Xlib import display, X
import io
import numpy as np
import pyautogui as pg
import pyscreeze
import time

REGION = (0, 0, 400, 400)

def timing(f):
def wrap(*args, **kwargs):
time1 = time.time()
ret = f(*args, **kwargs)
time2 = time.time()
print('{:s} function took {:.3f} ms'.format(
f.__name__, (time2-time1)*1000.0))

return ret
return wrap

@timing
def benchmark_pyautogui():
return pg.screenshot(region=REGION)

@timing
def benchmark_pyscreeze():
return pyscreeze.screenshot(region=REGION)

@timing
def benchmark_pil():
return np.array(ImageGrab.grab(bbox=REGION))

if __name__ == "__main__":

im_pyautogui = benchmark_pyautogui()
im_pyscreeze = benchmark_pyscreeze()
im_pil =       benchmark_pil()


As expected, pyscreeze is slightly faster than pyautogui, but PIL beats them by a factor of 10!

benchmark_pyautogui function took 157.669 ms
benchmark_pyscreeze function took 152.185 ms
benchmark_pil function took 13.198 ms


## Locate an element on screen

import pyautogui as pg
import numpy as np
import cv2 as cv
from PIL import ImageGrab, Image
import time

REGION = (0, 0, 400, 400)
GAME_OVER_PICTURE_PIL = Image.open("./balloon_fight_game_over.png")

def timing(f):
def wrap(*args, **kwargs):
time1 = time.time()
ret = f(*args, **kwargs)
time2 = time.time()
print('{:s} function took {:.3f} ms'.format(
f.__name__, (time2-time1)*1000.0))

return ret
return wrap

@timing
def benchmark_pyautogui():
res = pg.locateOnScreen(GAME_OVER_PICTURE_PIL,
grayscale=True,  # should provied a speed up
confidence=0.8,
region=REGION)
return res is not None

@timing
def benchmark_opencv_pil(method):
img = ImageGrab.grab(bbox=REGION)
img_cv = cv.cvtColor(np.array(img), cv.COLOR_RGB2BGR)
res = cv.matchTemplate(img_cv, GAME_OVER_PICTURE_CV, method)
# print(res)
return (res >= 0.8).any()

if __name__ == "__main__":

im_pyautogui = benchmark_pyautogui()
print(im_pyautogui)

methods = ['cv.TM_CCOEFF', 'cv.TM_CCOEFF_NORMED', 'cv.TM_CCORR',
'cv.TM_CCORR_NORMED', 'cv.TM_SQDIFF', 'cv.TM_SQDIFF_NORMED']

# cv.TM_CCOEFF_NORMED actually seems to be the most relevant method
for method in methods:
print(method)
im_opencv = benchmark_opencv_pil(eval(method))
print(im_opencv)


And the results!

benchmark_pyautogui function took 175.712 ms
False
cv.TM_CCOEFF
benchmark_opencv_pil function took 21.283 ms
True
cv.TM_CCOEFF_NORMED
benchmark_opencv_pil function took 23.377 ms
False
cv.TM_CCORR
benchmark_opencv_pil function took 20.465 ms
True
cv.TM_CCORR_NORMED
benchmark_opencv_pil function took 25.347 ms
False
cv.TM_SQDIFF
benchmark_opencv_pil function took 23.799 ms
True
cv.TM_SQDIFF_NORMED
benchmark_opencv_pil function took 22.882 ms
True


pyautogui, once again, is super slow. However, the cv based methods are an order of magnitude lower, though some see “Game Over” when it is not here. I made sure that TM_CCOEFF_NORMED also returned True when the element was in the region before updating the following class:

from PIL import Image, ImageGrab
from helpers import fast_locate_on_screen
import cv2 as cv
import numpy as np
import os
import pyautogui as pg
import time

class BalloonTripEnvironment:

def __init__(self):
self._game_filepath = "../games/BalloonFight.zip"
self._region = (10,10,300,300)

def _custom_press_key(self, key_to_press):
pg.keyDown(key_to_press)
pg.keyUp(key_to_press)

def turn_nes_up(self):
os.system(f"fceux {self._game_filepath} &")
time.sleep(1)

def start_trip(self):
keys_to_press = ['s', 's', 'enter']
for key_to_press in keys_to_press:
self._custom_press_key(key_to_press)

def observe_state(self):
return pg.screenshot(region=self._region)

def capture_state_as_png(self, filename):
pg.screenshot(filename, region=self._region)

def step(self, action):
self._custom_press_key(action)

def is_game_over(self):
img = ImageGrab.grab(bbox=self._region)
img_cv = cv.cvtColor(np.array(img), cv.COLOR_RGB2BGR)
res = cv.matchTemplate(img_cv, self._game_over_picture, eval('cv.TM_CCOEFF_NORMED'))
return (res >= 0.8).any()

def rage_quit(self):
os.system("pkill fceux")
exit()

if __name__ == "__main__":

env = BalloonTripEnvironment()

env.turn_nes_up()
time.sleep(10)
env.step('enter')
env.start_trip()
print("Started")
is_game_over = False
i = 0

while not is_game_over:
i += 1
is_game_over = env.is_game_over()
env.step('f')

print("Game over!")

env.rage_quit()


Below, you can see the GIFs of the loop. On the left, the previous version, where each call to is_game_over() needed so much time that the “agent” could not press the button often enough. Now the frequency is high enough, the “agent” just bounces on the top of the screen until it dies!

Fig. 1: On the left, the previous version of is_game_over(), and the new version, is on the right (note that the beginning of the GIF is just the demo mode of the game.

Hope you liked it, stay tuned for the next articles if you like the project!

# Reinforcement learning Nintendo NES Tutorial (Part 1)

Reinforcement learning is an amazingly fun application of machine learning to a variety of tasks! I have seen plenty of videos around of reinforcement learning applied to video games, but very few tutorials. In this series (it may take a while and I have not finished the project as I write this article, it may not even work!) we will apply it to a game I love: Balloon Fight, with explanations that will, hopefully, make the reader able to reuse to different games!

In a nutshell, reinforcement learning consists in teaching a computer to act in a environment (here, a game) the best way it can, without having to describe the rules of the environment (the game) to the algorithm. The way it works is that the algorithm will try different behaviors thousands of times and improve every time, learning from its mistakes.

In a more precise way, the game is splitted in a discrete set of steps. At each step the agent (or algorithm) will observe the environment and decide of an action to take (here, a key to press) and observe, again, the environment and the rewards it got from taking the previous action. When the game is over, the agent restarts to play, but with the accumulated knowledge of its previous experiences.

## Enabling Python to communicate with the game

### What you will need

We will teach our algorithm to play Balloon Fight. More precisely, the Balloon Trip mode (as it will save some efforts, as we will see later).

Fig. 1: The Balloon Trip (me playing)

I have the following directory structure:

.
├── games
│   └── BalloonFight.zip
├── src
│   ├── balloon_trip_environment.py
│   ├── intro.py
│   └── balloon_fight_game_over.png
└── TODO.md


BalloonFight.zipis a ROM of the game.

intro.py and balloon_trip_environment.py will be explored below.

balloon_fight_game_over.png will be created and explained later.

#### Python packages

Regarding the packages, we will start with Python default packages and pyautogui which enables interaction with other windows. This may have some dependencies, but I trust my reader to be able to fix all of them :)

#### External dependencies

We will use FCEUX as an emulator.

### The starting point (intro.py)

Let’s see how Python can interact with any window on the screen, just like a human being! The script below will:

• launch FCEUX with the game
• press a sequence of buttons (or keys)
• take a screenshot of a region of the screen
import pyautogui as pg
import os
import time

game_filepath = "../games/BalloonFight.zip"
os.system(f"fceux {game_filepath} &")

time.sleep(1)

keys_to_press = ['s', 's', 'enter']

for key_to_press in keys_to_press:
pg.keyDown(key_to_press)
pg.keyUp(key_to_press)

time.sleep(2)

im = pg.screenshot("./test.png", region=(0,0, 300, 400))
print(im)


The main things to notice are:

os.system(f"fceux {game_filepath} &")


Note the & at the end of the command. Without it, Python would be “stuck” waiting for the return of os.system(). With this, Python keeps executing the following lines.

pg.keyDown(key_to_press)
pg.keyUp(key_to_press)


There is a .press() method with pyautogui, but for some reason, it did not work with the emulator.

im = pg.screenshot("./test.png", region=(0,0, 300, 400))


Will be able to capture parts of the screen while the emulator is running, therefore allowing Python to “communicate” with the window.

Fig. 2: The output of pg.screenshot() (This will be improved later)

## Reinforcement learning

### Micro crash course (or the basics we need for now)

An important element in reinforcement learning is the following loop (env refers to the environment, or the state of the game at each instant, while agent will be able to press buttons and interact with the environment).

for episode in range(N_EPISODES):

env.reset()
episode_reward = 0

done = False
while not done:
current_state = env.observe_state()

action = agent.get_action(current_state)

new_state, reward, done = env.step(action)


Basically, it shows a clear separation between the environment and the agent. For now, let’s just implement the environment.

### The environment

For now, we only need to describe the environment in a convenient way. It needs to expose a observe_state(), next(action) and is_game_over(), making sure the agent can continue acting.

import pyautogui as pg
import os
import time
from helpers import fast_locate_on_screen
from PIL import Image

class BalloonTripEnvironment:

def __init__(self):
self._game_filepath = "../games/BalloonFight.zip"
self._region = (10,10,300,300)
self._game_over_picture = Image.open("./balloon_fight_game_over.png")

def _custom_press_key(self, key_to_press):
pg.keyDown(key_to_press)
pg.keyUp(key_to_press)

def turn_nes_up(self):
os.system(f"fceux {self._game_filepath} &")
time.sleep(1)

def start_trip(self):
keys_to_press = ['s', 's', 'enter']
for key_to_press in keys_to_press:
self._custom_press_key(key_to_press)

def observe_state(self):
return pg.screenshot(region=self._region)

def capture_state_as_png(self, filename):
pg.screenshot(filename, region=self._region)

def step(self, action):
self._custom_press_key(action)

def is_game_over(self):
res = pg.locateOnScreen(self._game_over_picture,
grayscale=True, # should provied a speed up
confidence=0.8,
region=self._region)
return res is not None

def rage_quit(self):
os.system("pkill fceux")
exit()



### Some details

The class above is an adaptation of the introduction script, in a more object oriented way. The main detail is the following:

def is_game_over(self):
res = pg.locateOnScreen(self._game_over_picture,
grayscale=True, # should provied a speed up
confidence=0.8,
region=self._region)
return res is not None


It looks for the image below to make sure that we are not seeing the “Game Over” screen.

Fig. 3: The pattern we will look for to detect the game over.

### Watch it in action

Let’s test it! f is just the A button of the NES, it will simply enable the balloon guy to go up. For the loop, we will test that the game is not over, and then the player will press the A button.

if __name__ == "__main__":

env = BalloonTripEnvironment()

env.turn_nes_up()
env.start_trip()
print("Started")
is_game_over = False
i = 0

while not is_game_over:
i += 1
is_game_over = env.is_game_over()
env.step('f')

print("Game over!")

env.rage_quit()


Fig. 4: An agent, pumping on a regular basis

## Next steps

We have the environment. Now, we will need to turn it as a matrix that will be used as “features” for reinforcement learning, this will be the topic of the next article. Once achieved, we will be able to jump to the deep learning part!

## First issues

Programming without issues does not exist, at least in my world. Though the above works as expected, we notice that during the loop, the button was pressed only four times. After a more careful exam, I noticed that is_game_over() is the bottleneck.

If we want to be able to read the input on the screen, decide of the best action to take, we need to have much (much) more time between two screenshots. This will be the topic of an intermediate post, stay tuned if you liked it :)

# Learning more

The following resources (sponsored URLs): Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow and Reinforcement Learning: Industrial Applications of Intelligent Agents will provide a good introduction to the topic, in Python, with the libraries I am going to use.

# Introduction

## Why would you do this ?

After all, scikit learn already has the DecisionTreeClassifier and it works really well and is highly optimized!

Well, I can see four reasons to implement it anyway!

• It is a good exercise if you want to learn the inner details of the decision trees
• The DecisionTreeClassifier only supports two criterions:
criterion{“gini”, “entropy”}, default=”gini”


However, I may be willing to play with other criterions if the metric I am working with is not a standard one.

• With a code in python that does not require any compilation, pyx files and what not, you can perform plenty of experimentations of the logic of the training tree (and given the problem, obtain a better accuracy)
• It is fun!

## Starting point

So, we will use numpy and implement the DecisionTree without the knowledge of any penalty function. This one will be provided by the user.

We will also follow the fit and predict interface, as we want to be able to reuse this class without a lot of efforts.

## The algorithm

Quoting Wikipedia:

A tree is built by splitting the source set, constituting the root node of the tree, into subsets—which constitute the successor children. The splitting is based on a set of splitting rules based on classification features. This process is repeated on each derived subset in a recursive manner called recursive partitioning.

Put another way: given a dataset A and labels, find a colum and a threshold, so that the data is partitionned it two datasets. Repeat this until the whole dataset has been splitted in small datasets whose size is lower than the minimum sample size given to the algorithm. The splitting part must be performed so that the split achieves the highest improvement in terms of the chosen criterion.

Parameters can be added: the maximum depth of the tree, the minimum number of elements in a leaf, the minimum gain to achieve to decide to split or not the data…

# Implementation

## Imports

from sklearn.utils.validation import check_X_y
import datetime
import numpy as np

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'
CYAN = '\033[36m'


Ok, I imported check_X_y from scikit learn. It would be super easy to remove it in the following, but it saves a lot of debugging to use it, so I will leave it here for now.

bcolors is just a convenient class to store the colors, before printing them to the terminal.

## The Tree class

class Tree:

def __init__(self):
self.left = None
self.right = None
self.data = None


A tree is just a recursive data structure, it can hold data in a node and has to children, a left and a right leaf.

There are plenty of things to know about trees in computer science, but we will only need it to store data. So this class will be enough for our purposes!

## The CustomDecisionTree

Let’s decompose the work a little bit more in what follows. Our CustomDecisionTree will expose fit() and predict() and will operate on numpy arrays. Making it available for pandas DataFrame could be done as well, but let’s put it aside as it require more work and does not help to understand the algorithm used to train a decision tree.

class CustomDecisionTree:

def __init__(self, penalty_function, max_depth=3, min_sample_size=3, max_thresholds=10,
verbose=False):
self._max_depth = max_depth
self._min_sample_size = min_sample_size
self._max_thresholds = max_thresholds
self._penalty_function = penalty_function
self._verbose = verbose
self._y = None


The constructor will need:

• penalty_function (the criterion we will try to optimize)
• max_depth (the depth of the tree)
• min_sample_size (the minimum size of a sample to split it)
• max_thresholds (the number of splits proposed per numeric value)

Storing y could have been performed later, but I like to have all the variables used by my class in the constructor.

    def fit(self, X, y, indices=None):
check_X_y(X, y)
self._y = y
self._tree = Tree()
splitters = self._build_splitters(X)

if indices is None:
indices = np.arange(X.shape[0])

if self._verbose:
self._print("{} splitters proposed".format(len(splitters)))

self._train(self._tree, indices, 0, splitters, 0, X, y)

return self


Still not much done here. We make sure that X and y have compatible shapes (the check_X_y function does it for us), we store y and build the splitters.

Let’s get rid of the _print() method (it is just a habit of mine to distinguish prints from different classes with colors, I find this helpful for debugging if needed, and to monitor the execution of the algorithms).

    def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")
print(bcolors.CYAN + "[CustomDecisionTree | " +
time + "] " + bcolors.ENDC + str(input_str))


Note that we will work on indices to perform all the splits recursively. It was not necessary to pass indices to fit(), but I plan to implement a ranom forest later with this class, so we will need them!

The splitters, themselves, will be at the core of the algorithm. A splitter will simply be an index (the column index) and a threshold.

A splitter will just say: the elements of column i whose value is larger than threshold must go to the right leaf, the elements which are smaller must go to the left leaf.

    def _build_splitters(self, X):
splitters = []

for i, column in enumerate(X.T):
sorted_unique_values = np.sort(np.unique(column))
thresholds = (
sorted_unique_values[:-1] + sorted_unique_values[1:]) / 2
n_thresholds = len(thresholds)

if len(thresholds) > self._max_thresholds:
thresholds = thresholds[[round(
i*n_thresholds / self._max_thresholds) for i in range(self._max_thresholds)]]

for threshold in thresholds:
splitters.append((i, threshold))

return splitters


The splitters are the average between sorted values for each column, subsampled so that we do not have too many splitters (a large number of splitters slows down the algorithm and provides a very limited accuracy improvement).

So, we have the fit() entry point to our interface, we briefy went throgh the splitter building part, let’s continue:

    def _split(self, splitter, indices, X):
index, threshold = splitter
mask = X[indices, index] > threshold


As I said, a splitter just splits the data in two subsets (represented by their indices). It should be clear enough from this method that this is exactly what is performed (with a slight help from numpy).

Now if we remember the algorithm, we need to find the best splitter at each step of the recursive splits. This is where the user defined penalty will come in:

    def _splitter_score(self, splitter, indices, X, y):
indices_left, indices_right = self._split(splitter, indices, X)
n_left, n_right = len(indices_left), len(indices_right)

if n_left < self._min_sample_size:
return -100000

if n_right < self._min_sample_size:
return -100000

return (n_left * self._penalty(indices_left, y) +
n_right * self._penalty(indices_right, y)) / \
(n_left + n_right)


Note that the weighted mean of the penalty for a splitter is returned. If you wanted to modify it, this could be performed here.

So we have our splitters, we can, for each subset, give a score to a splitter, we are ready to implement the full train method:

    def _train(self, tree, indices, depth, splitters, current_score, X, y):
if depth >= self._max_depth:
tree.data = indices
else:
splitter_and_scores = list(
map(lambda ns: (ns, self._splitter_score(ns, indices, X, y)), splitters))
scores = list(map(lambda sp: sp[1], splitter_and_scores))
if len(scores) == 0:
tree.data = indices
return
max_score = max(scores)
max_index = scores.index(max_score)
non_trival_splitters_and_scores = list(
filter(lambda p: p[1] != -100000, splitter_and_scores))
non_trival_splitters = list(
map(lambda p: p[0], non_trival_splitters_and_scores))

best_splitter, best_score = splitter_and_scores[max_index]
indices_left, indices_right = self._split(
best_splitter, indices, X)

if len(indices_left) < self._min_sample_size or \
len(indices_right) < self._min_sample_size:
tree.data = indices

else:
tree.data = best_splitter

tree.left = Tree()
tree.right = Tree()

self._train(tree.left, indices_left, depth + 1,
non_trival_splitters, best_score, X, y)
self._train(tree.right, indices_right, depth + 1,
non_trival_splitters, best_score, X, y)


If we reach max_depth, the leaf we are currently in will store the indices remaining for this leaf.

Otherwise, we find the best splitter, split the data into indices_left and indices_right (induced from this best splitter) and call _train() (recursively) twice : once on each subset. At this step, the node of the tree holds a splitter.

Note that each call to train updates the children of the tree. Once all the calls to train are executed, the tree attribute of the class contains all the splitters (for intermediate nodes) and the indices for the final nodes (the ones that could not be splitted any more).

## Predictions

We have to add the methods that enable to propose predictions once the tree is trained.

    def _find_indices_for_row(self, row):
return self._traverse_trained_tree(self._tree, row)

def _predict_one(self, row):
indices = self._find_indices_for_row(row)
return np.bincount(self._y[indices]).argmax()

def _traverse_trained_tree(self, tree, row):
if tree.left is None:
return tree.data
else:
index, threshold = tree.data
if row[index] > threshold:
return self._traverse_trained_tree(tree.left, row)
else:
return self._traverse_trained_tree(tree.right, row)

def predict(self, X):
return np.array(
list(map(lambda row: self._predict_one(row), X)), dtype=int)


Note that:

np.bincount(self._y[indices]).argmax()


simply returns the most common elements of y at the selected indices. The logic of navigating the tree is performed in _traverse_trained_tree(). For each node, if it is a splitter, follow the logic of the splitter (left or right according to the comparison the threshold). If the algorithm reaches a leaf (tree.left is None), return the indices stored in the leaf.

# Testing !

if __name__== "__main__":

from sklearn.datasets import make_classification
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt

X, y = make_classification(n_samples=200, shuffle=False, n_redundant=3)
for max_depth in [1,2,5,10,15]:
cdt = CustomDecisionTree(accuracy_score, min_sample_size=1, max_depth=max_depth)
cdt.fit(X, y)
y_hat = cdt.predict(X)
score = accuracy_score(cdt.predict(X), y)
print("Max depth: ", max_depth, " score: ", score)


And tada! As expected, we reach a perfect accuracy if the depth is large enough!

Max depth:  1  score:  0.915
Max depth:  2  score:  0.92
Max depth:  5  score:  0.925
Max depth:  10  score:  0.975
Max depth:  15  score:  1.0


A more thorough testing would include benchmark on common datasets and a comparison to other implementations of decision trees. I will do it in a separate article.

# Learning more and stay tuned

I hope you liked this reading! Any comments regarding the code or the explanations is welcome! For those who want to stay tuned, I implemented a small form to leave me your email (which won’t be used for ads nor transmitted to any third party). It is in the “Subscribe” section of the navigation menu (small square on the top left).

The next article will go from the CustomDecisionTree to a CustomRandomForest and the following one will be about more detailed tests for these newly implemented classes.

# The code

from sklearn.utils.validation import check_X_y
import datetime
import numpy as np

class Tree:

def __init__(self):
self.left = None
self.right = None
self.data = None

def __str__(self, level=0):
ret = "\t"*level+repr(self.data)+"\n"
for child in [self.left, self.right]:
if child is not None:
ret += child.__str__(level+1)
return ret

def custom_print(self, f1, f2, level=0):
if self.left is None:
ret = "\t"*level+f2(self.data)+"\n"
else:
ret = "\t"*level+f1(self.data)+"\n"

if self.right is not None:
ret = self.right.custom_print(f1, f2, level+1) + ret
if self.left is not None:
ret += self.left.custom_print(f1, f2, level+1)

return ret

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'
CYAN = '\033[36m'

class CustomDecisionTree:

def __init__(self, penalty_function, max_depth=3, min_sample_size=3, max_thresholds=10,
verbose=False):
self._max_depth = max_depth
self._min_sample_size = min_sample_size
self._max_thresholds = max_thresholds
self._penalty_function = penalty_function
self._verbose = verbose
self._y = None

def fit(self, X, y, indices=None):
check_X_y(X, y)
self._y = y
self._tree = Tree()
splitters = self._build_splitters(X)

if indices is None:
indices = np.arange(X.shape[0])

if self._verbose:
self._print("{} splitters proposed".format(len(splitters)))

self._train(self._tree, indices, 0, splitters, 0, X, y)

return self

def predict(self, X):
return np.array(
list(map(lambda row: self._predict_one(row), X)), dtype=int)

def _penalty(self, indices, y):
predicted = [np.bincount(y[indices]).argmax()] * len(indices)
return self._penalty_function(y[indices], predicted)

def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")
print(bcolors.CYAN + "[CustomDecisionTree | " +
time + "] " + bcolors.ENDC + str(input_str))

def _find_indices_for_row(self, row):
return self._traverse_trained_tree(self._tree, row)

def _predict_one(self, row):
indices = self._find_indices_for_row(row)
return np.bincount(self._y[indices]).argmax()

def _traverse_trained_tree(self, tree, row):
if tree.left is None:
return tree.data
else:
index, threshold = tree.data
if row[index] > threshold:
return self._traverse_trained_tree(tree.left, row)
else:
return self._traverse_trained_tree(tree.right, row)

def _build_splitters(self, X):
splitters = []

for i, column in enumerate(X.T):
sorted_unique_values = np.sort(np.unique(column))
thresholds = (
sorted_unique_values[:-1] + sorted_unique_values[1:]) / 2
n_thresholds = len(thresholds)

if len(thresholds) > self._max_thresholds:
thresholds = thresholds[[round(
i*n_thresholds / self._max_thresholds) for i in range(self._max_thresholds)]]

for threshold in thresholds:
splitters.append((i, threshold))

return splitters

def _split(self, splitter, indices, X):
index, threshold = splitter
mask = X[indices, index] > threshold

def _splitter_score(self, splitter, indices, X, y):
indices_left, indices_right = self._split(splitter, indices, X)
n_left, n_right = len(indices_left), len(indices_right)

if n_left < self._min_sample_size:
return -100000

if n_right < self._min_sample_size:
return -100000

return (n_left * self._penalty(indices_left, y) +
n_right * self._penalty(indices_right, y)) / \
(n_left + n_right)

def _train(self, tree, indices, depth, splitters, current_score, X, y):
if depth >= self._max_depth:
tree.data = indices
else:
splitter_and_scores = list(
map(lambda ns: (ns, self._splitter_score(ns, indices, X, y)), splitters))
scores = list(map(lambda sp: sp[1], splitter_and_scores))
if len(scores) == 0:
tree.data = indices
return
max_score = max(scores)
max_index = scores.index(max_score)
non_trival_splitters_and_scores = list(
filter(lambda p: p[1] != -100000, splitter_and_scores))
non_trival_splitters = list(
map(lambda p: p[0], non_trival_splitters_and_scores))

best_splitter, best_score = splitter_and_scores[max_index]
indices_left, indices_right = self._split(
best_splitter, indices, X)

if len(indices_left) < self._min_sample_size or \
len(indices_right) < self._min_sample_size:
tree.data = indices

else:
tree.data = best_splitter

tree.left = Tree()
tree.right = Tree()

self._train(tree.left, indices_left, depth + 1,
non_trival_splitters, best_score, X, y)
self._train(tree.right, indices_right, depth + 1,
non_trival_splitters, best_score, X, y)

if __name__== "__main__":

from sklearn.datasets import make_classification
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt

X, y = make_classification(n_samples=20, shuffle=False, n_redundant=3)
cdt = CustomDecisionTree(accuracy_score, verbose=True)

cdt.fit(X, y)

print(cdt._tree.custom_print(str,str))

X, y = make_classification(n_samples=200, shuffle=False, n_redundant=3)
for max_depth in [1,2,5,10,15]:
cdt = CustomDecisionTree(accuracy_score, min_sample_size=1, max_depth=max_depth)
cdt.fit(X, y)
y_hat = cdt.predict(X)
score = accuracy_score(cdt.predict(X), y)
print("Max depth: ", max_depth, " score: ", score)



# Python replace rarely occuring values in a pipeline

## Python pipeline series

Pipeline are probably one of the most convenient tools in scikit learn, and propose a simple way to write reusable models, for which all the hyperparameters, both of the learning and preprocessing part are in the exact same place. However, I do not see them that often on code snippets or in data science competitions.

### The problem

A good practice, when working with factors or categories in a dataframe is to replace values that appear a limited number of time.

A simple reason to do so is that a category appearing just once will be hard to generalize. Decision tree based methods will probably ignore it (as long as the min_sample_size is larger than the number of occurences of this value), so why bother keeping such variables.

### A simple solution

As stressed on this stackoverflow answer, a simple one-liner does the job:

df.loc[df[col].value_counts()[df[col]].values < 10, col] = "RARE_VALUE"


One liners are good. Easy to copy paste. Also easy to make mistakes with them.

Imagine, you are working with a messy dataset, figure out that it would be nice to have a function that takes care of the cleaning.

Copy pasting the above, you end up writing:

def clean_variables(data):
columns = ['Gender', 'Car_Category', 'Subject_Car_Colour',
'Subject_Car_Make', 'LGA_Name', 'State']

for column in columns:
data[column].fillna("empty", inplace=True)
data.loc[data[column].value_counts()[data[column]].values < 10, column] = "RARE_VALUE"

data["Age"] = data["Age"].apply(clip_age)

[...] # other stuff you may do

return data


### The issue

And then you forget about it, some day comes a test set and you blindly apply the clean_variables function on it. That’s what functions are for after all, reusing !

So you write:

train = clean_variables(train)
test = clean_variables(test)


And who knows what may happen from there on. If the test set is too small (less than 10 rows), all the factors will be turned into “RARE_VALUE”. Depending on the importance given to these features by the learning algorithms you applied later, the performance on the test set could be good, or very bad.

### A better solution

Instead, I would recommend putting all this in a pipeline. As far as I know,there is no simple class in scikit-learn that enable to do the removing, so I ended up writing the following class, which does the job:

class RemoveScarceValuesFeatureEngineer:

def __init__(self, min_occurences):
self._min_occurences = min_occurences
self._column_value_counts = {}

def fit(self, X, y):
for column in X.columns:
self._column_value_counts[column] = X[column].value_counts()
return self

def transform(self, X):
for column in X.columns:
X.loc[self._column_value_counts[column][X[column]].values
< self._min_occurences, column] = "RARE_VALUE"

return X

def fit_transform(self, X, y):
self.fit(X, y)
return self.transform(X)

if __name__ == "__main__":
import pandas as pd

sample_train = pd.DataFrame(
[{"a": 1, "s": "a"}, {"a": 1, "s": "a"}, {"a": 1, "s": "b"}])
print(sample_train)
print(20*"=")

sample_test = pd.DataFrame([{"a": 1, "s": "a"}, {"a": 1, "s": "b"}])
print(sample_test)


And executing the code:

   a  s
0  1  a
1  1  a
2  1  b
a           s
0  1           a
1  1           a
2  1  RARE_VALUE
====================
a  s
0  1  a
1  1  b
a           s
0  1           a
1  1  RARE_VALUE


you have the desired behavior: a is not replaced away with RARE_VALUE in the test set!

# Data science competition platforms

(Updated 22nd march 2021: added datasource.ai)

Here is, to my knowledge, the most complete list of data science competition platforms with sponsored (paid) competitions.

If you are not familiar with them, they are, to me, the best way to learn data science. Most of them have a dedicated community and many tutorials, starter kits for competitions. They are also a way to use your skills on topics your day job may not propose you.

## ML contests: an aggregator

ML contests

Clear. A list of competitions, by topics (NLP / supervised learning / vision …)

I am not sure all the platforms are showed here (I did not find references to numer.ai or Bitgrit, per example)

## Prediction

The principle here is simple, you have a train set and a test set to download (though the new trend is encouraging to push your code directly in a dedicated environment hosted by the platform).

The train set contains various columns, or images, or executables (or anything else probably) and the purpose is to predict another variable (which can be a label for classification problems, a value for regression, a set of labels for multiclass classification problem or other things, I am just focusing on the most common tasks)

Then, you upload your predictions (or your code, depending on the competition) and you get a value: the accuracy of your model on the test set. The ranking is immediate, making these platforms delightfully and dangerously addictive!

### Kaggle

Kaggle competitions

Kaggle is probably the largest platform hosting competitions, with the highest prizes and the largest community and resources. Beware, the higher the price, the harder the competition!

They also have the most complete set of learning resources and usable datasets.

### AIcrowd (or CrowdAI)

AIcrowd challenge page

Great platform, super active, many competitions and great topics! They are growing fast so expect even more competitions to happen here.

Besides, from the competitions I have seen here, they focus on less “classical” topics than the ones you would see on Kaggle. Some may like it, others may not, I personnally do.

AIcrowd enables data science experts and enthusiasts to collaboratively solve real-world problems, through challenges.

### Bitgrit

Bitgrit competitions

Launched in 2019, already showing 8 competitions with various topics, this platform looks promising! As said above, it does not seem referenced on mlcontests.

bitgrit is an AI competition and recruiting platform for data scientists, home to a community of over 25,000 engineers worldwide. We are developing bitgrit to be a comprehensive online ecosystem, centered around a blockchain-powered AI Marketplace.

### Drivendata

Driven data competition page

I never took part in their competitions, so I can’t say mcuh about it for now! But they have sponsored competitions.

DrivenData works on projects at the intersection of data science and social impact, in areas like international development, health, education, research and conservation, and public services. We want to give more organizations access to the capabilities of data science, and engage more data scientists with social challenges where their skills can make a difference.

### Crowdanalytix

Crowdanalytix

I never took part in their competitions, so I can’t say mcuh about it for now! They seemed less active recently, but they had sponsored competitions.

25,129 + Data Scientists

102,083 + Models Built

50 + Countries

### Numer.ai

https://numer.ai/

Focusing on predicting the stock market, with high quality data (which is usually a tedious task when you try to have quality data in finance). They claim to be the hardest platform in finance, and having worked there, I can confirm that finding the slightest valuable prediction is super hard!

Nice if you like finance, but be prepared to work with similar datasets!

Start with hedge fund quality data. It is clean and regularized, designed to be usable right away.

### Zindi

Zindi competition page

Data science platform with competitions which are related to Africa. The NLP part seems particularly exciting, as they are focus on languages which are not studied as often as English or Spanish! Looking forward to participate in one of their challenges!

We connect organisations with our thriving African data science community to solve the world’s most pressing challenges using machine learning and AI.

### Analytics Vidhya

Analytics Vidhya

India based.

Data science hackathons on DataHack enable you to compete with leading data scientists and machine learning experts in the world. This is your chance to work on real life data science problems, improve your skill set, learn from expert data science and machine learning professionals, and hack your way to the top of the hackathon leaderboard! You also stand a chance to win prizes and get a job at your dream data science company.

### Challengedata

https://challengedata.ens.fr/

Not sure the competitions are sponsored here. General topics, most of them seem to come from French companies and French institutions.

We organize challenges of data sciences from data provided by public services, companies and laboratories: general documentation and FAQ. The prize ceremony is in February at the College de France.

### Coda Lab

Codalab

French based.

CodaLab is an open-source platform that provides an ecosystem for conducting computational research in a more efficient, reproducible, and collaborative manner.

### Topcoder

Topcoder

Not focusing only on data science:

Access our community of world class developers, great designers, data science geniuses and QA experts for the best results

### InnoCentive

InnoCentive competitions

InnoCentive is the global pioneer in crowdsourced innovation. We help innovative organizations solve their important technology, science, business, A/I and data challenges by connecting them with a global network of expert problem solvers.

### Datasoure.ai

Datasoure

Young company, as the quote below shows (22nd march 2021). They seem to be focused on challenges for startups, but this may evolve!

At a glance

2 Team Members

1,692 Data Scientists

12 Companies

5.2% Weekly Growth

### Signate

Signate competitions

A Japanese competition platform. Most of the competitions are described in Japanese, but not all of them!

SIGNATE collaborates with companies, government agencies and research institutes in various industries to work on various projects to resolve social issues. We invite you to join SIGNATE’s project, which aims to make the world a better place through the power of open innovation.

### datasciencechallenge.org (probably down)

https://www.datasciencechallenge.org/

Unfortunately, I cannot reach the website any more…

Sponsored by the Defence Science and Technology Laboratory and other UK government departments.

### datascience.net (probably down for ever)

datascience.net

Used to be a French speaking data science competition for a while. However, the site has been down for a while now… Worth giving a look from time to time!

## Dataviz

Here, the idea is to provide the best vizualisation of datasets. The metric may therefore not be as absolute as the one for prediction problems and the skillset is really different!

Iron viz

### informationisbeautifulawards

informationisbeautifulawards

They are all the platforms I am aware of, if I missed any or if you have any relevant resources, please let me know!

I Hope you liked this article! If you plan to take part in any of these competitions, best of luck to you, and have fun competing and learning!

# Blog news

As I was renewing my domain name, I figured out I had been posting here for quite a while now :) at a completely irregular frequency, I must admit. Anyways, thank you to all of the readers for your support and interest in my articles ! It was really pleasant to read the various comments or mails you sent me.

As for the blog itself, well, I learnt a lot. About machine learning, obviously, but also about this thing called SEO. I am a little bit surprised by the success of some articles and the oblivion others fell into, but I guess this is how referencing works. One of my most read articles is python plot 3d scatter and density though it did not take much time to write, while a stacking tutorial in Python and theory behind model stacking seem to be invisible from a search engine point of view… But I am not here to rant.

The plan, if any, is to keep posting articles about all the aspects of machine learning which I consider interesting ! I have some material for the decision boundaries of common machine learning algorithms, some code for decision trees, random forest and parallel computations in OCaml and more data visualization snippets…

Another thing, if you like my content and want to support me, I joined the Brave creators program. So if you use this browser and want to help, I would gladly receive BAT tips! Thanks to the anonymous donors who already contributed.

And as usual, if you have some topics you are curious about, some tutorials you would like to read, just let me know in the comments or by mail, I will see what I can do!

Have a nice day!

# Why does staging works?

Model staging is a method that enables to produce (usually) the most competitive models, in terms of accuracy. As such, you will often find winning solutions to data science competitions to be 2-3 stages of models.

This article will assume some familiarity with cross validation. We will go through model averaging first, recall how staging works, observe the link between model averaging and model staging (also referred to as stacking or blending) and propose hypothesis (backed by some visualization) to explain why staging works so well.

Also my other post (more implementation oriented), a stacking tutorial in Python may help.

Going back to model staging, here are some verbatim about some winning solutions of data science competitions.

Per example, the Homesite quote conversion

Quick overview for now about the NMA approach:

10 variations of the dataset in total (factor combinations, factors mapped to response rates, replacing correlated pairs by differences etc) lots of models (xgboost, keras, ranger, logreg, even occasional svm - although that took forever) trained on various datasets and different params; stored as lvl1 metafeatures mix lvl1 metafeatures with: xgboost, nnet, hillclimbing, glmnet and ranger, stack - 5 lvl2 metafeatures mix the lvl2 metafeatures with hillclimbing bag at each stage as much as time permitted

Or, in the BNP Paribas Cardif Claims management:

We also produced many different base level models without much Feature engineering, just different input format types (like load all categorical variables as counts, or as onehot encoding etc).

Our ensemble was consisted of 223 models. Faron did a lot of work in removing noise and discarding many of these in order to get to our bets score with a lvl2 ensemble of geomean weights between an ET , 2NN and 2 Xgmodels.

Before understanding the mechanisms at work for model staging, let’s review the simpler “model averaging” approach.

# Model averaging

## What is model averaging?

Model averaging is a method that consists in averaging the predictions of different models.

It works particularly well in regression problems, or classification problem when the task consists in predicting probabilities of belonging to a specific class.

## Why does averaging works?

Imagine you are facing a regression problem, and have two models. One underestimates the true value, while the other one overestimates it. In this case, the average of the two predictions will be (much) closer to the truth than each individual prediction.

The two figures below illustrate it, in the case of a squared error penalty. The x-axis represents the difference between the true value and the estimated value. The red dots are the estimations of two different models (on the x axis) and the resulting error (on the y axis). The component of the blue dot on the x-axis is the average of the components of the red dots on the x-axis.

It is worth noting that blending will work better if the models have a similar performance (in terms of out-of-sample accuracy) and are as little correlated as possible.

## Jensen inequality

What is even better is that if both models overestimate (or underestimate) the true value, the penalty is still lower.

The graph below presents it:

The green point correpond to the average of the errors, while the blue point correspond of the error of the average.

Proofs can be found on Wikipedia for the purpose of the article, it is enough to convice oneself that this works with these simple graphs.

Jensen’s inequality applies to convex functions, but log loss, squared error (MSE), absolute value error (MAE) are convex functions.

The R code below can be used (just change the xs_points <- c(0.1, 0.8)) to reproduce the experiment with other values.

parabola = function(x) {
x * x
}
xs <- seq(-1, 1, 0.01)
xs_points <- c(0.1, 0.8)

plot(
xs,
parabola(xs),
xlab = expression(hat(x) - x),
ylab = "Square Error",
type = 'l',
main = "MSE as a function of the difference between the true value\n of x and its estimated value"
)

for (xs_point in xs_points) {
points(x = xs_point,
y = parabola(xs_point),
col = "red")
}

points(x = mean(xs_points),
y = parabola(mean(xs_points)),
col = "blue")
segments(
x0 = xs_points[1],
x1 = xs_points[2],
y0 = parabola(xs_points[1]),
y1 = parabola(xs_points[2]), lty = 11
)
points(x = mean(xs_points),
y = mean(parabola(xs_points)),
col = "green")



# Model stacking

## Historical note

As far as I know, the first presentation of stacking goes back to 1992: Stacked generalization, by David H. Wolpert (also famous for the result no free lunch in search and optimization)

[…] The conclusion is that for almost any real-world generalization problem one should use some version of stacked generalization to minimize the generalization error rate.

And almost 30 years later, it remains true! Quoting the wikipedia page:

This work was developed further by Breiman, Smyth, Clarke and many others, and in particular the top two winners of 2009 Netflix competition made extensive use of stacked generalization (rebranded as “blending”)

## How does it work?

### Generalizing averaging

So far, we have seen model averaging. You take two models (or $n$), and average them.

On the other hand, assuming you cross validate $n$ models and only use the best for future predictions, you have another strategy that forms one model from $n$ models.

So we have two schemes that put weights on different models. One puts an equal weight to every model, the other one puts all the weights on the best model.

Model staging consists in using another learning algorithm to choose the “weights” to give to each model. A linear regression “on top” of other models consists in giving an “ideal” (in the regression sense) weight to each model, but what is amazing is that you may use something else than a linear regression!

### Detailing staging procedure

The idea, to give a correct “weight” to each model is to perform a cross validation on the training set and return a dataset for which each element correspond to the unseen fold prediction. On this new dataset, you can fit the new model.

Per example, in the case of a regression problem, if you have $n$ rows in your data set, $p$ features and $k$ models, this step turns your training data from a $n,p$ matrix to a $n,k$ matrix.

Thus the element with index $i,j$ in this new matrix corresponds to the prediction of the $i$-th observation by the $j$-th model.

## Why does this work?

### General case of averaging

Averaging, because of Jensen’s inequality, usually improves the accuracy of the models. Staging, seen as a generalization of averaging, will also improve the accuracy of our learners.

### More general decision boundaries

#### Hypothesis

Another argument, which could be rephrased in more scientific terms is that it allows to obtain decision boundaries of a wider shape than the ones of each usual learning algorithm.

Put another way, the supervised learning problems can be seen as finding $f$ such that:

Where $f$ belongs to some function space. For the linear regressions (penalized or not), $f$ has to be a linear function, for a decision tree, $f$ belongs to a space of sum of indicator functions, for kernel functions, $f$ is a linear combination of kernels…

The hypothesis here is that, in the case of averaging, $f$ can be linear in some region, constant by pieces in another, etc, making the search space for $f$ larger than the search space of all the single families of models.

#### Visualization

As expected, the decision tree has a decision boundary consisting of segments parallel to the x and y axis.

The SVC on the other hand has a very smooth decision boundary.

And here comes the magic, the decision boundary is “a little bit of both”.

Same when we blend with a decision tree.

#### Code

I use the default stacking proposed by scikit-learn.

First a small class useful to plot decision boundaries.

import numpy as np
import matplotlib.pyplot as plt

class DecisionBoundaryPlotter:

def __init__(self, X, Y, xs=np.linspace(0, 1, 30),
ys=np.linspace(0, 1, 30)):
self._X = X
self._Y = Y
self._xs = xs
self._ys = ys

def _predictor(self, model):
model.fit(self._X, self._Y)
return (lambda x: model.predict_proba(x.reshape(1,-1))[0, 0])

def _evaluate_height(self, f):
fun_map = np.empty((self._xs.size, self._ys.size))
for i in range(self._xs.size):
for j in range(self._ys.size):
v = f(
np.array([self._xs[i], self._ys[j]]))
fun_map[i, j] = v
return fun_map

def plot_heatmap(self, model, name):
f = self._predictor(model)
fun_map = self._evaluate_height(f)

fig = plt.figure()
s = fig.add_subplot(1, 1, 1, xlabel='$x$', ylabel='$y$')
im = s.imshow(
fun_map,
extent=(self._ys[0], self._ys[-1], self._xs[0], self._xs[-1]),
origin='lower')
fig.colorbar(im)
fig.savefig(name + '_Heatmap.png')

def plot_contour(self, model, name):
f = self._predictor(model)
fun_map = self._evaluate_height(f)

fig = plt.figure()
s = fig.add_subplot(1, 1, 1, xlabel='$x$', ylabel='$y$')
s.contour(self._xs, self._ys, fun_map, levels = [0.5])
s.scatter(self._X[:,0], self._X[:,1], c = self._Y)
fig.suptitle(name)
fig.savefig(name + '_Contour.png')


The plots (yeah, the nested named models is not that elegant):

import numpy as np
from sklearn.ensemble import RandomForestClassifier, ExtraTreesClassifier, AdaBoostClassifier
from sklearn.tree import DecisionTreeClassifier
from sklearn.svm import SVR, SVC
from sklearn.neighbors import KNeighborsClassifier
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import StackingClassifier
from DecisionBoundaryPlotter import DecisionBoundaryPlotter

def random_data_classification(n, p, f):
predictors = np.random.rand(n, p)
return predictors, np.apply_along_axis(f, 1, predictors)

def parabolic(x, y):
return (x**2 + y**3 > 0.5) * 1

def parabolic_mat(x):
return parabolic(x[0], x[1])

X, Y = random_data_classification(300, 2, parabolic_mat)

dbp = DecisionBoundaryPlotter(X, Y)

named_classifiers = [ (DecisionTreeClassifier(max_depth=4), "DecisionTreeClassifier"),
(StackingClassifier(estimators=[
("svc", SVC(probability=True)),
("dt", DecisionTreeClassifier(max_depth=4))],
final_estimator=LogisticRegression()),
"Stacked (Linear)"),
(StackingClassifier(estimators=[
("svc", SVC(probability=True)),
("dt", DecisionTreeClassifier(max_depth=4))],
final_estimator=DecisionTreeClassifier(max_depth=4)),
"Stacked (Decision Tree)"),
(SVC(probability=True), "SVC")]

for named_classifier in named_classifiers:
print(named_classifier[1])
dbp.plot_contour(named_classifier[0], named_classifier[1])



## Learning more and references

The elements of statistical learning by Trevor Hastie, Robert Tibshirani, Jerome Friedman is a brilliant introduction to machine learning and will help you have a better understanding of cross validation and the learning algorithms presented here (SVC, Decision trees) but unfortunately does not treat model staging.

# Extract trees from a random forest in python

You may need to extract trees from a classifier for various reasons. In my case, I thought that the feature of xgboost ntree_limit was quite convenient when cross validating a gradient boosting method over the number of trees.

What it does is that it only uses the first ntree_limit trees to perform the prediction (instead of using all the fitted tree).

predict(data, output_margin=False, ntree_limit=0, pred_leaf=False,
pred_contribs=False, approx_contribs=False,
pred_interactions=False, validate_features=True, training=False)


And it is also available as an extra argument of .predict() if you use the scikit-learn interface :

ypred = bst.predict(dtest, ntree_limit=bst.best_ntree_limit)


Indeed, by doing so, if you want to find the optimal number of trees for your model, you do not have to fit the model for 50 trees, and then predict, then fit it for 100 trees and then predict. You may fit the model once and for all for 200 trees and then, playing with ntree_limit you can observe the performance of the model for various number of trees.

The RandomForest, as implemented in scikit-learn does not show this parameter in its .predict() method. However, this is something we can quickly fix. Indeed, the RandomForest exposes estimators_. You can modify it (beware, this is a bit hacky and may not work for other versions of scikit-learn).

rf_model = RandomForestRegressor()
rf_model.fit(x, y)

estimators = rf_model.estimators_

def predict(w, i):
rf_model.estimators_ = estimators[0:i+1]
return rf_model.predict(x)


And that’s it, the predict method now only looks at the first i trees ;)