The next part of the Machine Learning Crash Course deals with constructing bucketized features and feature crosses. The Jupyter notebook can be downloaded here.
We use quantiles to put the feature data into different categories. This is done in the functions get_quantile_based_boundaries and construct_bucketized_column. To construct one hot feature columns, we use construct_bucketized_onehot_column. The drawback of these methods is that they loop through the whole dataset for conversion, which is computationally very expensive. Leave a comment if you have an idea for a better method!
Another difference from the original programming exercise is the use of the Adam Optimizer instead of the FTLR Optimizer. To my knowledge, TensorFlow.jl exposes only the following optimizers:
- Gradient Descent
- Momentum Optimizer
- Adam Optimizer
On the other hand, (at least) the following possibilities are available in TensorFlow itself:
- Gradient Descent
- Momentum Optimizer
- Adagrad Optimizer
- Adadelta Optimizer
- Adam Optimizer
- Ftrl Optimizer
- RMSProp Optimizer
Some information on those can be found here. For a more technical discussion with lots of background infos, have a look at this excellent blog post. If you know how to get other optimizers to work in Julia, I would be very interested.
This notebook is based on the file Feature crosses programming exercise, which is part of Google’s Machine Learning Crash Course.
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Feature Crosses
Learning Objectives:
- Improve a linear regression model with the addition of additional synthetic features (this is a continuation of the previous exercise)
- Use an input function to convert
DataFrameobjects to feature columns - Use the Adam optimization algorithm for model training
- Create new synthetic features through one-hot encoding, binning, and feature crosses
Setup
First, as we’ve done in previous exercises, let’s define the input and create the data-loading code.
using Plots
gr(fmt=:png)
using DataFrames
using TensorFlow
import CSV
import StatsBase
using Random
using Statistics
sess=Session()
california_housing_dataframe = CSV.read("california_housing_train.csv", delim=",");
california_housing_dataframe = california_housing_dataframe[shuffle(1:size(california_housing_dataframe, 1)),:];
function preprocess_features(california_housing_dataframe)
"""Prepares input features from California housing data set.
Args:
california_housing_dataframe: A DataFrame expected to contain data
from the California housing data set.
Returns:
A DataFrame that contains the features to be used for the model, including
synthetic features.
"""
selected_features = california_housing_dataframe[
[:latitude,
:longitude,
:housing_median_age,
#:total_rooms,
#:total_bedrooms,
#:population,
:households,
:median_income]]
processed_features = selected_features
# Create a synthetic feature.
processed_features[:rooms_per_person] = (
california_housing_dataframe[:total_rooms] ./
california_housing_dataframe[:population])
return processed_features
end
function preprocess_targets(california_housing_dataframe)
"""Prepares target features (i.e., labels) from California housing data set.
Args:
california_housing_dataframe: A DataFrame expected to contain data
from the California housing data set.
Returns:
A DataFrame that contains the target feature.
"""
output_targets = DataFrame()
# Scale the target to be in units of thousands of dollars.
output_targets[:median_house_value] = (
california_housing_dataframe[:median_house_value] ./ 1000.0)
return output_targets
end
# Choose the first 12000 (out of 17000) examples for training.
training_examples = preprocess_features(first(california_housing_dataframe,12000))
training_targets = preprocess_targets(first(california_housing_dataframe,12000))
# Choose the last 5000 (out of 17000) examples for validation.
validation_examples = preprocess_features(last(california_housing_dataframe,5000))
validation_targets = preprocess_targets(last(california_housing_dataframe,5000))
# Double-check that we've done the right thing.
println("Training examples summary:")
describe(training_examples)
println("Validation examples summary:")
describe(validation_examples)
println("Training targets summary:")
describe(training_targets)
println("Validation targets summary:")
describe(validation_targets)
function construct_feature_columns(input_features)
"""Construct the TensorFlow Feature Columns.
Args:
input_features: The names of the numerical input features to use.
Returns:
A set of feature columns
"""
out=convert(Matrix, input_features[:,:])
return convert(Matrix{Float64},out)
end
function create_batches(features, targets, steps, batch_size=5, num_epochs=0)
"""Create batches.
Args:
features: Input features.
targets: Target column.
steps: Number of steps.
batch_size: Batch size.
num_epochs: Number of epochs, 0 will let TF automatically calculate the correct number
Returns:
An extended set of feature and target columns from which batches can be extracted.
"""
if(num_epochs==0)
num_epochs=ceil(batch_size*steps/size(features,1))
end
names_features=names(features);
names_targets=names(targets);
features_batches=copy(features)
target_batches=copy(targets)
for i=1:num_epochs
select=shuffle(1:size(features,1))
if i==1
features_batches=(features[select,:])
target_batches=(targets[select,:])
else
append!(features_batches, features[select,:])
append!(target_batches, targets[select,:])
end
end
return features_batches, target_batches
end
function next_batch(features_batches, targets_batches, batch_size, iter)
"""Next batch.
Args:
features_batches: Features batches from create_batches.
targets_batches: Target batches from create_batches.
batch_size: Batch size.
iter: Number of the current iteration
Returns:
An extended set of feature and target columns from which batches can be extracted.
"""
select=mod((iter-1)*batch_size+1, size(features_batches,1)):mod(iter*batch_size, size(features_batches,1));
ds=features_batches[select,:];
target=targets_batches[select,:];
return ds, target
end
function my_input_fn(features_batches, targets_batches, iter, batch_size=5, shuffle_flag=1)
"""Prepares a batch of features and labels for model training.
Args:
features: DataFrame of features
targets: DataFrame of targets
batch_size: Size of batches to be passed to the model
shuffle: True or False. Whether to shuffle the data.
num_epochs: Number of epochs for which data should be repeated. None = repeat indefinitely
Returns:
Tuple of (features, labels) for next data batch
"""
# Construct a dataset, and configure batching/repeating.
#ds = Dataset.from_tensor_slices((features,targets)) # warning: 2GB limit
ds, target = next_batch(features_batches, targets_batches, batch_size, iter)
# Shuffle the data, if specified.
if shuffle_flag==1
select=shuffle(1:size(ds, 1));
ds = ds[select,:]
target = target[select, :]
end
# Return the next batch of data.
# features, labels = ds.make_one_shot_iterator().get_next()
return ds, target
end
Adam Optimization Algorithm
High dimensional linear models benefit from using a variant of gradient-based optimization called Adam optimization. This algorithm has the benefit of scaling the learning rate differently for different coefficients, which can be useful if some features rarely take non-zero values.
function train_model(learning_rate,
steps,
batch_size,
feature_column_function::Function,
training_examples,
training_targets,
validation_examples,
validation_targets)
"""Trains a linear regression model of one feature.
Args:
learning_rate: A `float`, the learning rate.
steps: A non-zero `int`, the total number of training steps. A training step
consists of a forward and backward pass using a single batch.
batch_size: A non-zero `int`, the batch size.
feature_column_function: Function for transforming the feature columns.
training_examples:
training_targets:
validation_examples:
validation_targets:
Returns:
weight: The weights of the model.
bias: Bias of the model.
p1: Graph containing the loss function values for the different iterations.
"""
periods = 10
steps_per_period = steps / periods
# Create feature columns.
feature_columns = placeholder(Float32)
target_columns = placeholder(Float32)
# Create a linear regressor object.
m=Variable(zeros(size(feature_column_function(training_examples),2),1))
b=Variable(0.0)
y=(feature_columns*m) .+ b
loss=reduce_sum((target_columns - y).^2)
features_batches, targets_batches = create_batches(training_examples, training_targets, steps, batch_size)
# Set up Adam optimizer
my_optimizer=(train.AdamOptimizer(learning_rate))
gvs = train.compute_gradients(my_optimizer, loss)
capped_gvs = [(clip_by_norm(grad, 5.), var) for (grad, var) in gvs]
my_optimizer = train.apply_gradients(my_optimizer,capped_gvs)
run(sess, global_variables_initializer())
# Train the model, but do so inside a loop so that we can periodically assess
# loss metrics.
println("Training model...")
println("RMSE (on training data):")
training_rmse = []
validation_rmse=[]
for period in 1:periods
# Train the model, starting from the prior state.
for i=1:steps_per_period
features, labels = my_input_fn(features_batches, targets_batches, convert(Int,(period-1)*steps_per_period+i), batch_size)
run(sess, my_optimizer, Dict(feature_columns=>feature_column_function(features), target_columns=>construct_feature_columns(labels)))
end
# Take a break and compute predictions.
training_predictions = run(sess, y, Dict(feature_columns=> feature_column_function(training_examples)));
validation_predictions = run(sess, y, Dict(feature_columns=> feature_column_function(validation_examples)));
# Compute loss.
training_mean_squared_error = mean((training_predictions- construct_feature_columns(training_targets)).^2)
training_root_mean_squared_error = sqrt(training_mean_squared_error)
validation_mean_squared_error = mean((validation_predictions- construct_feature_columns(validation_targets)).^2)
validation_root_mean_squared_error = sqrt(validation_mean_squared_error)
# Occasionally print the current loss.
println(" period ", period, ": ", training_root_mean_squared_error)
# Add the loss metrics from this period to our list.
push!(training_rmse, training_root_mean_squared_error)
push!(validation_rmse, validation_root_mean_squared_error)
end
weight = run(sess,m)
bias = run(sess,b)
println("Model training finished.")
# Output a graph of loss metrics over periods.
p1=plot(training_rmse, label="training", title="Root Mean Squared Error vs. Periods", ylabel="RMSE", xlabel="Periods")
p1=plot!(validation_rmse, label="validation")
println("Final RMSE (on training data): ", training_rmse[end])
println("Final Weight (on training data): ", weight)
println("Final Bias (on training data): ", bias)
return weight, bias, p1 #, calibration_data
end
weight, bias, p1 = train_model(
# TWEAK THESE VALUES TO SEE HOW MUCH YOU CAN IMPROVE THE RMSE
0.003, #learning rate
500, #steps
5, #batch_size
construct_feature_columns, # feature column function
training_examples,
training_targets,
validation_examples,
validation_targets)
Training model...
RMSE (on training data):
period 1: 163.40066528018832
period 2: 146.52114081767553
period 3: 140.86123226817278
period 4: 137.1772581781986
period 5: 130.62764204789096
period 6: 127.19379116167481
period 7: 123.66203006894426
period 8: 123.13795665497169
period 9: 123.16096172210501
period 10: 119.15539913574085
Model training finished.
Final RMSE (on training data): 119.15539913574085
Final Weight (on training data): [0.705068; -0.732617; 0.739885; 0.120607; 1.07408; 0.94443]
Final Bias (on training data): 0.4130073131683406
plot(p1)
One-Hot Encoding for Discrete Features
Discrete (i.e. strings, enumerations, integers) features are usually converted into families of binary features before training a logistic regression model.
For example, suppose we created a synthetic feature that can take any of the values 0, 1 or 2, and that we have a few training points:
| # | feature_value |
|---|---|
| 0 | 2 |
| 1 | 0 |
| 2 | 1 |
For each possible categorical value, we make a new binary feature of real values that can take one of just two possible values: 1.0 if the example has that value, and 0.0 if not. In the example above, the categorical feature would be converted into three features, and the training points now look like:
| # | feature_value_0 | feature_value_1 | feature_value_2 |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 1.0 |
| 1 | 1.0 | 0.0 | 0.0 |
| 2 | 0.0 | 1.0 | 0.0 |
Bucketized (Binned) Features
Bucketization is also known as binning.
We can bucketize population into the following 3 buckets (for instance):
bucket_0(< 5000): corresponding to less populated blocksbucket_1(5000 - 25000): corresponding to mid populated blocksbucket_2(> 25000): corresponding to highly populated blocks
Given the preceding bucket definitions, the following population vector:
[[10001], [42004], [2500], [18000]]
becomes the following bucketized feature vector:
[[1], [2], [0], [1]]
The feature values are now the bucket indices. Note that these indices are considered to be discrete features. Typically, these will be further converted in one-hot representations as above, but this is done transparently.
The following code defines bucketized feature columns for households and longitude; the get_quantile_based_boundaries function calculates boundaries based on quantiles, so that each bucket contains an equal number of elements.
function get_quantile_based_boundaries(feature_values, num_buckets)
#Investigate why [:] is necessary - there is some conflict that construct_feature_columns
# spits out Array{Float64,2} where it should be Array{Float64,1}!!
quantiles = StatsBase.nquantile(construct_feature_columns(feature_values)[:], num_buckets)
return quantiles# [quantiles[q] for q in keys(quantiles)]
end
function construct_bucketized_column(input_features, boundaries)
data_out=zeros(size(input_features))
for i=1:size(input_features,2)
curr_feature=input_features[:,i]
curr_boundary=boundaries[i]
for k=1:length(curr_boundary)
for j=1:size(input_features,1)
if(curr_feature[j] >= curr_boundary[k] )
data_out[j,i]+=1
end
end
end
end
return data_out
end
We need a special function to convert the bucketized columns into a one-hot encoding. Contrary to the high-level API of Python’s TensorFlow, the Julia verion does not transparently handle buckets.
function construct_bucketized_onehot_column(input_features, boundaries)
length_out=0
for i=1:length(boundaries)
length_out+=length(boundaries[i])-1
end
data_out=zeros(size(input_features,1), length_out)
curr_index=1;
for i=1:size(input_features,2)
curr_feature=input_features[:,i]
curr_boundary=boundaries[i]
for k=1:length(curr_boundary)-1
for j=1:size(input_features,1)
if((curr_feature[j] >= curr_boundary[k]) && (curr_feature[j] < curr_boundary[k+1] ))
data_out[j,curr_index]+=1
end
end
curr_index+=1;
end
end
return data_out
end
Let’s divide the household and longitude data into buckets.
# Divide households into 7 buckets.
households = california_housing_dataframe[:households]
bucketized_households = construct_bucketized_onehot_column(
households, [get_quantile_based_boundaries(
california_housing_dataframe[:households], 7)])
# Divide longitude into 10 buckets.
longitude = california_housing_dataframe[:longitude]
bucketized_longitude = construct_bucketized_onehot_column(
longitude, [get_quantile_based_boundaries(
california_housing_dataframe[:longitude], 10)])
Task 1: Train the Model on Bucketized Feature Columns
Bucketize all the real valued features in our example, train the model and see if the results improve.
In the preceding code block, two real valued columns (namely households and longitude) have been transformed into bucketized feature columns. Your task is to bucketize the rest of the columns, then run the code to train the model. There are various heuristics to find the range of the buckets. This exercise uses a quantile-based technique, which chooses the bucket boundaries in such a way that each bucket has the same number of examples.
quantiles_latitude=get_quantile_based_boundaries(
training_examples[:latitude], 10)
quantiles_longitude=get_quantile_based_boundaries(
training_examples[:longitude], 10)
quantiles_housing_median_age=get_quantile_based_boundaries(
training_examples[:housing_median_age], 7)
quantiles_households=get_quantile_based_boundaries(
training_examples[:households], 7)
quantiles_median_income=get_quantile_based_boundaries(
training_examples[:median_income], 7)
quantiles_rooms_per_person=get_quantile_based_boundaries(
training_examples[:rooms_per_person], 7)
quantiles_vec=[quantiles_latitude,
quantiles_longitude,
quantiles_housing_median_age,
quantiles_households,
quantiles_median_income,
quantiles_rooms_per_person
]
weight, bias, p1 = train_model(
# TWEAK THESE VALUES TO SEE HOW MUCH YOU CAN IMPROVE THE RMSE
0.03, #learning rate
2000, #steps
100, #batch_size
x -> construct_bucketized_onehot_column(x, quantiles_vec), # feature column function
training_examples,
training_targets,
validation_examples,
validation_targets)
Training model...
RMSE (on training data):
period 1: 202.72463349839262
period 2: 171.78849030380562
period 3: 145.2635554495821
period 4: 124.71823709982627
period 5: 110.32950574255553
period 6: 100.6139387049533
period 7: 93.60621652368602
period 8: 88.30830153898088
period 9: 84.12833254564481
period 10: 80.90929566707702
Model training finished.
Final RMSE (on training data): 80.90929566707702
Final Weight (on training data): [39.3923; 43.1646; 22.9269; 44.9485; 45.9324; 10.1383; 39.0275; 49.2261; 15.8903; -27.9299; 50.2951; 43.2011; 26.3118; -11.8351; 17.9301; 58.4617; 32.1206; 34.3472; 21.7817; -6.78351; 19.5952; 23.5582; 32.8533; 30.6326; 30.7012; 31.973; 31.6763; 31.1743; 31.591; 31.1047; 34.6392; 35.6798; 38.3245; 41.5214; -31.3323; -17.3332; 2.60318; 15.4069; 29.15; 45.0948; 65.9498; -7.01649; -5.40451; 4.29645; 16.067; 33.9617; 49.9225; 61.5666]
Final Bias (on training data): 49.640670673368234
plot(p1)
Feature Crosses
Crossing two (or more) features is a clever way to learn non-linear relations using a linear model. In our problem, if we just use the feature latitude for learning, the model might learn that city blocks at a particular latitude (or within a particular range of latitudes since we have bucketized it) are more likely to be expensive than others. Similarly for the feature longitude. However, if we cross longitude by latitude, the crossed feature represents a well defined city block. If the model learns that certain city blocks (within range of latitudes and longitudes) are more likely to be more expensive than others, it is a stronger signal than two features considered individually.
If we cross the latitude and longitude features (supposing, for example, that longitude was bucketized into 2 buckets, while latitude has 3 buckets), we actually get six crossed binary features. Each of these features will get its own separate weight when we train the model.
Task 2: Train the Model Using Feature Crosses
Add a feature cross of longitude and latitude to your model, train it, and determine whether the results improve.
The following function creates a feature cross of latitude and longitude and then feeds it to the model.
function construct_latXlong_onehot_column(input_features, boundaries)
#latitude and longitude are the first two columns - treat them separately
#initialization - calculate total length of feature_vec
length_out=0
# lat and long
length_lat=length(boundaries[1])-1
length_long= length(boundaries[2])-1
length_out+=length_lat*length_long
# all other features
for i=3:length(boundaries)
length_out+=length(boundaries[i])-1
end
data_out=zeros(size(input_features,1), length_out)
# all other features
curr_index=length_lat*length_long+1;
for i=3:size(input_features,2)
curr_feature=input_features[:,i]
curr_boundary=boundaries[i]
#println(curr_boundary)
for k=1:length(curr_boundary)-1
for j=1:size(input_features,1)
if((curr_feature[j] >= curr_boundary[k]) && (curr_feature[j] < curr_boundary[k+1] ))
data_out[j,curr_index]+=1
end
end
curr_index+=1;
end
end
# lat and long
data_temp=zeros(size(input_features,1), length_lat+length_long)
curr_index=1
for i=1:2
curr_feature=input_features[:,i]
curr_boundary=boundaries[i]
#println(curr_boundary)
for k=1:length(curr_boundary)-1
for j=1:size(input_features,1)
if((curr_feature[j] >= curr_boundary[k]) && (curr_feature[j] < curr_boundary[k+1] ))
data_temp[j,curr_index]+=1
end
end
curr_index+=1;
end
end
vec_temp=1
for j=1:size(input_features,1)
vec1=data_temp[j,1:length_lat]
vec2=data_temp[j, length_lat+1:length_lat+length_long]
vec_temp=vec1*vec2'
data_out[j, 1:length_lat*length_long]= (vec_temp)[:]
end
return data_out
end
weight, bias, p1 = train_model(
# TWEAK THESE VALUES TO SEE HOW MUCH YOU CAN IMPROVE THE RMSE
0.03, #learning rate
2000, #steps
100, #batch_size
x -> construct_latXlong_onehot_column(x, quantiles_vec), # feature column function
training_examples,
training_targets,
validation_examples,
validation_targets)
Training model...
RMSE (on training data):
period 1: 208.01934398715014
period 2: 181.32512572835964
period 3: 157.3466935603145
period 4: 136.90056422790624
period 5: 120.48339242694792
period 6: 108.01996714153945
period 7: 98.69501405105021
period 8: 91.59963480335907
period 9: 86.0126149075253
period 10: 81.53319732851088
Model training finished.
Final RMSE (on training data): 81.53319732851088
plot(p1)
Optional Challenge: Try Out More Synthetic Features
So far, we’ve tried simple bucketized columns and feature crosses, but there are many more combinations that could potentially improve the results. For example, you could cross multiple columns. What happens if you vary the number of buckets? What other synthetic features can you think of? Do they improve the model?