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Learning Objectives:
LinearRegressor class in TensorFlow to predict median housing price, at the granularity of city blocks, based on one input featureThe data is based on 1990 census data from California.
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import math
from IPython import display
from matplotlib import cm
from matplotlib import gridspec
from matplotlib import pyplot as plt
import numpy as np
import pandas as pd
from sklearn import metrics
import tensorflow as tf
from tensorflow.python.data import Dataset
tf.logging.set_verbosity(tf.logging.ERROR)
pd.options.display.max_rows = 10
pd.options.display.float_format = '{:.1f}'.format
Next, we'll load our data set.
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california_housing_dataframe = pd.read_csv("https://storage.googleapis.com/mledu-datasets/california_housing_train.csv", sep=",")
california_housing_dataframe.describe()
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We'll randomize the data, just to be sure not to get any pathological ordering effects that might harm the performance of Stochastic Gradient Descent. Additionally, we'll scale median_house_value to be in units of thousands, so it can be learned a little more easily with learning rates in a range that we usually use.
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california_housing_dataframe = california_housing_dataframe.reindex(
np.random.permutation(california_housing_dataframe.index))
california_housing_dataframe["median_house_value"] /= 1000.0
california_housing_dataframe
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california_housing_dataframe.describe()
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In this exercise, we'll try to predict median_house_value, which will be our label (sometimes also called a target). We'll use total_rooms as our input feature.
NOTE: Our data is at the city block level, so this feature represents the total number of rooms in that block.
To train our model, we'll use the LinearRegressor interface provided by the TensorFlow Estimator API. This API takes care of a lot of the low-level model plumbing, and exposes convenient methods for performing model training, evaluation, and inference.
In order to import our training data into TensorFlow, we need to specify what type of data each feature contains. There are two main types of data we'll use in this and future exercises:
Categorical Data: Data that is textual. In this exercise, our housing data set does not contain any categorical features, but examples you might see would be the home style, the words in a real-estate ad.
Numerical Data: Data that is a number (integer or float) and that you want to treat as a number. As we will discuss more later sometimes you might want to treat numerical data (e.g., a postal code) as if it were categorical.
In TensorFlow, we indicate a feature's data type using a construct called a feature column. Feature columns store only a description of the feature data; they do not contain the feature data itself.
To start, we're going to use just one numeric input feature, total_rooms. The following code pulls the total_rooms data from our california_housing_dataframe and defines the feature column using numeric_column, which specifies its data is numeric:
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# Define the input feature: total_rooms.
my_feature = california_housing_dataframe[["total_rooms"]]
# Configure a numeric feature column for total_rooms.
feature_columns = [tf.feature_column.numeric_column("total_rooms")]
my_feature
#feature_columns
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NOTE: The shape of our total_rooms data is a one-dimensional array (a list of the total number of rooms for each block). This is the default shape for numeric_column, so we don't have to pass it as an argument.
Next, we'll define our target, which is median_house_value. Again, we can pull it from our california_housing_dataframe:
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# Define the label.
targets = california_housing_dataframe["median_house_value"]
Next, we'll configure a linear regression model using LinearRegressor. We'll train this model using the GradientDescentOptimizer, which implements Mini-Batch Stochastic Gradient Descent (SGD). The learning_rate argument controls the size of the gradient step.
NOTE: To be safe, we also apply gradient clipping to our optimizer via clip_gradients_by_norm. Gradient clipping ensures the magnitude of the gradients do not become too large during training, which can cause gradient descent to fail.
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# Use gradient descent as the optimizer for training the model.
my_optimizer=tf.train.GradientDescentOptimizer(learning_rate=0.0000001)
my_optimizer = tf.contrib.estimator.clip_gradients_by_norm(my_optimizer, 5.0)
# Configure the linear regression model with our feature columns and optimizer.
# Set a learning rate of 0.0000001 for Gradient Descent.
linear_regressor = tf.estimator.LinearRegressor(
feature_columns=feature_columns,
optimizer=my_optimizer
)
To import our California housing data into our LinearRegressor, we need to define an input function, which instructs TensorFlow how to preprocess
the data, as well as how to batch, shuffle, and repeat it during model training.
First, we'll convert our pandas feature data into a dict of NumPy arrays. We can then use the TensorFlow Dataset API to construct a dataset object from our data, and then break
our data into batches of batch_size, to be repeated for the specified number of epochs (num_epochs).
NOTE: When the default value of num_epochs=None is passed to repeat(), the input data will be repeated indefinitely.
Next, if shuffle is set to True, we'll shuffle the data so that it's passed to the model randomly during training. The buffer_size argument specifies
the size of the dataset from which shuffle will randomly sample.
Finally, our input function constructs an iterator for the dataset and returns the next batch of data to the LinearRegressor.
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def my_input_fn(features, targets, batch_size=1, shuffle=True, num_epochs=None):
"""Trains a linear regression model of one feature.
Args:
features: pandas DataFrame of features
targets: pandas 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
"""
# Convert pandas data into a dict of np arrays.
features = {key:np.array(value) for key,value in dict(features).items()}
# Construct a dataset, and configure batching/repeating.
ds = Dataset.from_tensor_slices((features,targets)) # warning: 2GB limit
ds = ds.batch(batch_size).repeat(num_epochs)
# Shuffle the data, if specified.
if shuffle:
ds = ds.shuffle(buffer_size=10000)
# Return the next batch of data.
features, labels = ds.make_one_shot_iterator().get_next()
return features, labels
NOTE: We'll continue to use this same input function in later exercises. For more
detailed documentation of input functions and the Dataset API, see the TensorFlow Programmer's Guide.
We can now call train() on our linear_regressor to train the model. We'll wrap my_input_fn in a lambda
so we can pass in my_feature and target as arguments (see this TensorFlow input function tutorial for more details), and to start, we'll
train for 100 steps.
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_ = linear_regressor.train(
input_fn = lambda:my_input_fn(my_feature, targets),
steps=100
)
Let's make predictions on that training data, to see how well our model fit it during training.
NOTE: Training error measures how well your model fits the training data, but it does not measure how well your model generalizes to new data. In later exercises, you'll explore how to split your data to evaluate your model's ability to generalize.
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# Create an input function for predictions.
# Note: Since we're making just one prediction for each example, we don't
# need to repeat or shuffle the data here.
prediction_input_fn =lambda: my_input_fn(my_feature, targets, num_epochs=1, shuffle=False)
# Call predict() on the linear_regressor to make predictions.
predictions = linear_regressor.predict(input_fn=prediction_input_fn)
# Format predictions as a NumPy array, so we can calculate error metrics.
predictions = np.array([item['predictions'][0] for item in predictions])
# Print Mean Squared Error and Root Mean Squared Error.
mean_squared_error = metrics.mean_squared_error(predictions, targets)
root_mean_squared_error = math.sqrt(mean_squared_error)
print "Mean Squared Error (on training data): %0.3f" % mean_squared_error
print "Root Mean Squared Error (on training data): %0.3f" % root_mean_squared_error
Is this a good model? How would you judge how large this error is?
Mean Squared Error (MSE) can be hard to interpret, so we often look at Root Mean Squared Error (RMSE) instead. A nice property of RMSE is that it can be interpreted on the same scale as the original targets.
Let's compare the RMSE to the difference of the min and max of our targets:
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min_house_value = california_housing_dataframe["median_house_value"].min()
max_house_value = california_housing_dataframe["median_house_value"].max()
min_max_difference = max_house_value - min_house_value
print "Min. Median House Value: %0.3f" % min_house_value
print "Max. Median House Value: %0.3f" % max_house_value
print "Difference between Min. and Max.: %0.3f" % min_max_difference
print "Root Mean Squared Error: %0.3f" % root_mean_squared_error
Our error spans nearly half the range of the target values. Can we do better?
This is the question that nags at every model developer. Let's develop some basic strategies to reduce model error.
The first thing we can do is take a look at how well our predictions match our targets, in terms of overall summary statistics.
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calibration_data = pd.DataFrame()
calibration_data["predictions"] = pd.Series(predictions)
calibration_data["targets"] = pd.Series(targets)
calibration_data.describe()
Okay, maybe this information is helpful. How does the mean value compare to the model's RMSE? How about the various quantiles?
We can also visualize the data and the line we've learned. Recall that linear regression on a single feature can be drawn as a line mapping input x to output y.
First, we'll get a uniform random sample of the data so we can make a readable scatter plot.
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sample = california_housing_dataframe.sample(n=300)
Next, we'll plot the line we've learned, drawing from the model's bias term and feature weight, together with the scatter plot. The line will show up red.
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# Get the min and max total_rooms values.
x_0 = sample["total_rooms"].min()
x_1 = sample["total_rooms"].max()
# Retrieve the final weight and bias generated during training.
weight = linear_regressor.get_variable_value('linear/linear_model/total_rooms/weights')[0]
bias = linear_regressor.get_variable_value('linear/linear_model/bias_weights')
# Get the predicted median_house_values for the min and max total_rooms values.
y_0 = weight * x_0 + bias
y_1 = weight * x_1 + bias
# Plot our regression line from (x_0, y_0) to (x_1, y_1).
plt.plot([x_0, x_1], [y_0, y_1], c='r')
# Label the graph axes.
plt.ylabel("median_house_value")
plt.xlabel("total_rooms")
# Plot a scatter plot from our data sample.
plt.scatter(sample["total_rooms"], sample["median_house_value"])
# Display graph.
plt.show()
This initial line looks way off. See if you can look back at the summary stats and see the same information encoded there.
Together, these initial sanity checks suggest we may be able to find a much better line.
For this exercise, we've put all the above code in a single function for convenience. You can call the function with different parameters to see the effect.
In this function, we'll proceed in 10 evenly divided periods so that we can observe the model improvement at each period.
For each period, we'll compute and graph training loss. This may help you judge when a model is converged, or if it needs more iterations.
We'll also plot the feature weight and bias term values learned by the model over time. This is another way to see how things converge.
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def train_model(learning_rate, steps, batch_size, input_feature="total_rooms"):
"""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.
input_feature: A `string` specifying a column from `california_housing_dataframe`
to use as input feature.
"""
periods = 10
steps_per_period = steps / periods
my_feature = input_feature
my_feature_data = california_housing_dataframe[[my_feature]]
my_label = "median_house_value"
targets = california_housing_dataframe[my_label]
# Create feature columns.
feature_columns = [tf.feature_column.numeric_column(my_feature)]
# Create input functions.
training_input_fn = lambda:my_input_fn(my_feature_data, targets, batch_size=batch_size)
prediction_input_fn = lambda: my_input_fn(my_feature_data, targets, num_epochs=1, shuffle=False)
# Create a linear regressor object.
my_optimizer = tf.train.GradientDescentOptimizer(learning_rate=learning_rate)
my_optimizer = tf.contrib.estimator.clip_gradients_by_norm(my_optimizer, 5.0)
linear_regressor = tf.estimator.LinearRegressor(
feature_columns=feature_columns,
optimizer=my_optimizer
)
# Set up to plot the state of our model's line each period.
plt.figure(figsize=(15, 6))
plt.subplot(1, 2, 1)
plt.title("Learned Line by Period")
plt.ylabel(my_label)
plt.xlabel(my_feature)
sample = california_housing_dataframe.sample(n=300)
plt.scatter(sample[my_feature], sample[my_label])
colors = [cm.coolwarm(x) for x in np.linspace(-1, 1, periods)]
# Train the model, but do so inside a loop so that we can periodically assess
# loss metrics.
print "Training model..."
print "RMSE (on training data):"
root_mean_squared_errors = []
for period in range (0, periods):
# Train the model, starting from the prior state.
linear_regressor.train(
input_fn=training_input_fn,
steps=steps_per_period
)
# Take a break and compute predictions.
predictions = linear_regressor.predict(input_fn=prediction_input_fn)
predictions = np.array([item['predictions'][0] for item in predictions])
# Compute loss.
root_mean_squared_error = math.sqrt(
metrics.mean_squared_error(predictions, targets))
# Occasionally print the current loss.
print " period %02d : %0.2f" % (period, root_mean_squared_error)
# Add the loss metrics from this period to our list.
root_mean_squared_errors.append(root_mean_squared_error)
# Finally, track the weights and biases over time.
# Apply some math to ensure that the data and line are plotted neatly.
y_extents = np.array([0, sample[my_label].max()])
weight = linear_regressor.get_variable_value('linear/linear_model/%s/weights' % input_feature)[0]
bias = linear_regressor.get_variable_value('linear/linear_model/bias_weights')
x_extents = (y_extents - bias) / weight
x_extents = np.maximum(np.minimum(x_extents,
sample[my_feature].max()),
sample[my_feature].min())
y_extents = weight * x_extents + bias
plt.plot(x_extents, y_extents, color=colors[period])
print "Model training finished."
# Output a graph of loss metrics over periods.
plt.subplot(1, 2, 2)
plt.ylabel('RMSE')
plt.xlabel('Periods')
plt.title("Root Mean Squared Error vs. Periods")
plt.tight_layout()
plt.plot(root_mean_squared_errors)
# Output a table with calibration data.
calibration_data = pd.DataFrame()
calibration_data["predictions"] = pd.Series(predictions)
calibration_data["targets"] = pd.Series(targets)
display.display(calibration_data.describe())
print "Final RMSE (on training data): %0.2f" % root_mean_squared_error
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train_model(
learning_rate=0.00001,
steps=100,
batch_size=1
)
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train_model(
learning_rate=0.00002,
steps=500,
batch_size=5
)
This is just one possible configuration; there may be other combinations of settings that also give good results. Note that in general, this exercise isn't about finding the one best setting, but to help build your intutions about how tweaking the model configuration affects prediction quality.
This is a commonly asked question. The short answer is that the effects of different hyperparameters are data dependent. So there are no hard-and-fast rules; you'll need to test on your data.
That said, here are a few rules of thumb that may help guide you:
Again, never go strictly by these rules of thumb, because the effects are data dependent. Always experiment and verify.
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# YOUR CODE HERE
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train_model(
learning_rate=0.00002,
steps=1000,
batch_size=5,
input_feature="population"
)