# TensorFlow Linear Model Tutorial

In this tutorial, we will use the TF.Learn API in TensorFlow to solve a binary
classification problem: Given census data about a person such as age, gender,
education and occupation (the features), we will try to predict whether or not
the person earns more than 50,000 dollars a year (the target label). We will
train a **logistic regression** model, and given an individual's information our
model will output a number between 0 and 1, which can be interpreted as the
probability that the individual has an annual income of over 50,000 dollars.

## Setup

To try the code for this tutorial:

Install TensorFlow if you haven't already.

Download the tutorial code.

Install the pandas data analysis library. tf.learn doesn't require pandas, but it does support it, and this tutorial uses pandas. To install pandas:

Get

`pip`

:`# Ubuntu/Linux 64-bit $ sudo apt-get install python-pip python-dev # Mac OS X $ sudo easy_install pip $ sudo easy_install --upgrade six`

Use

`pip`

to install pandas:`$ sudo pip install pandas`

If you have trouble installing pandas, consult the instructions on the pandas site.

Execute the tutorial code with the following command to train the linear model described in this tutorial:

`$ python wide_n_deep_tutorial.py --model_type=wide`

Read on to find out how this code builds its linear model.

## Reading The Census Data

The dataset we'll be using is the Census Income Dataset. You can download the training data and test data manually or use code like this:

```
import tempfile
import urllib
train_file = tempfile.NamedTemporaryFile()
test_file = tempfile.NamedTemporaryFile()
urllib.urlretrieve("https://archive.ics.uci.edu/ml/machine-learning-databases/adult/adult.data", train_file.name)
urllib.urlretrieve("https://archive.ics.uci.edu/ml/machine-learning-databases/adult/adult.test", test_file.name)
```

Once the CSV files are downloaded, let's read them into Pandas dataframes.

```
import pandas as pd
COLUMNS = ["age", "workclass", "fnlwgt", "education", "education_num",
"marital_status", "occupation", "relationship", "race", "gender",
"capital_gain", "capital_loss", "hours_per_week", "native_country",
"income_bracket"]
df_train = pd.read_csv(train_file, names=COLUMNS, skipinitialspace=True)
df_test = pd.read_csv(test_file, names=COLUMNS, skipinitialspace=True, skiprows=1)
```

Since the task is a binary classification problem, we'll construct a label column named "label" whose value is 1 if the income is over 50K, and 0 otherwise.

```
LABEL_COLUMN = "label"
df_train[LABEL_COLUMN] = (df_train["income_bracket"].apply(lambda x: ">50K" in x)).astype(int)
df_test[LABEL_COLUMN] = (df_test["income_bracket"].apply(lambda x: ">50K" in x)).astype(int)
```

Next, let's take a look at the dataframe and see which columns we can use to predict the target label. The columns can be grouped into two types—categorical and continuous columns:

- A column is called
**categorical**if its value can only be one of the categories in a finite set. For example, the native country of a person (U.S., India, Japan, etc.) or the education level (high school, college, etc.) are categorical columns. - A column is called
**continuous**if its value can be any numerical value in a continuous range. For example, the capital gain of a person (e.g. $14,084) is a continuous column.

```
CATEGORICAL_COLUMNS = ["workclass", "education", "marital_status", "occupation",
"relationship", "race", "gender", "native_country"]
CONTINUOUS_COLUMNS = ["age", "education_num", "capital_gain", "capital_loss", "hours_per_week"]
```

Here's a list of columns available in the Census Income dataset:

Column Name | Type | Description | {.sortable} |
---|---|---|---|

age | Continuous | The age of the individual | |

workclass | Categorical | The type of employer the |

: : : individual has (government, : : : : military, private, etc.). : | fnlwgt | Continuous | The number of people the census | : : : takers believe that observation : : : : represents (sample weight). This : : : : variable will not be used. : | education | Categorical | The highest level of education | : : : achieved for that individual. : | education_num | Continuous | The highest level of education in | : : : numerical form. : | marital_status | Categorical | Marital status of the individual. | | occupation | Categorical | The occupation of the individual. | | relationship | Categorical | Wife, Own-child, Husband, | : : : Not-in-family, Other-relative, : : : : Unmarried. : | race | Categorical | White, Asian-Pac-Islander, | : : : Amer-Indian-Eskimo, Other, Black. : | gender | Categorical | Female, Male. | | capital_gain | Continuous | Capital gains recorded. | | capital_loss | Continuous | Capital Losses recorded. | | hours_per_week | Continuous | Hours worked per week. | | native_country | Categorical | Country of origin of the | : : : individual. : | income | Categorical | ">50K" or "<=50K", meaning | : : : whether the person makes more : : : : than $50,000 annually. :

## Converting Data into Tensors

When building a TF.Learn model, the input data is specified by means of an Input
Builder function. This builder function will not be called until it is later
passed to TF.Learn methods such as `fit`

and `evaluate`

. The purpose of this
function is to construct the input data, which is represented in the form of
Tensors
or
SparseTensors.
In more detail, the Input Builder function returns the following as a pair:

`feature_cols`

: A dict from feature column names to`Tensors`

or`SparseTensors`

.`label`

: A`Tensor`

containing the label column.

The keys of the `feature_cols`

will be used to construct columns in the
next section. Because we want to call the `fit`

and `evaluate`

methods with
different data, we define two different input builder functions,
`train_input_fn`

and `test_input_fn`

which are identical except that they pass
different data to `input_fn`

. Note that `input_fn`

will be called while
constructing the TensorFlow graph, not while running the graph. What it is
returning is a representation of the input data as the fundamental unit of
TensorFlow computations, a `Tensor`

(or `SparseTensor`

).

Our model represents the input data as *constant* tensors, meaning that the
tensor represents a constant value, in this case the values of a particular
column of `df_train`

or `df_test`

. This is the simplest way to pass data into
TensorFlow. Another more advanced way to represent input data would be to
construct an Input Reader
that represents a file or other data source, and iterates through the file as
TensorFlow runs the graph. Each continuous column in the train or test dataframe
will be converted into a `Tensor`

, which in general is a good format to
represent dense data. For cateogorical data, we must represent the data as a
`SparseTensor`

. This data format is good for representing sparse data.

```
import tensorflow as tf
def input_fn(df):
# Creates a dictionary mapping from each continuous feature column name (k) to
# the values of that column stored in a constant Tensor.
continuous_cols = {k: tf.constant(df[k].values)
for k in CONTINUOUS_COLUMNS}
# Creates a dictionary mapping from each categorical feature column name (k)
# to the values of that column stored in a tf.SparseTensor.
categorical_cols = {k: tf.SparseTensor(
indices=[[i, 0] for i in range(df[k].size)],
values=df[k].values,
shape=[df[k].size, 1])
for k in CATEGORICAL_COLUMNS}
# Merges the two dictionaries into one.
feature_cols = dict(continuous_cols.items() + categorical_cols.items())
# Converts the label column into a constant Tensor.
label = tf.constant(df[LABEL_COLUMN].values)
# Returns the feature columns and the label.
return feature_cols, label
def train_input_fn():
return input_fn(df_train)
def eval_input_fn():
return input_fn(df_test)
```

## Selecting and Engineering Features for the Model

Selecting and crafting the right set of feature columns is key to learning an
effective model. A **feature column** can be either one of the raw columns in
the original dataframe (let's call them **base feature columns**), or any new
columns created based on some transformations defined over one or multiple base
columns (let's call them **derived feature columns**). Basically, "feature
column" is an abstract concept of any raw or derived variable that can be used
to predict the target label.

### Base Categorical Feature Columns

To define a feature column for a categorical feature, we can create a
`SparseColumn`

using the TF.Learn API. If you know the set of all possible
feature values of a column and there are only a few of them, you can use
`sparse_column_with_keys`

. Each key in the list will get assigned an
auto-incremental ID starting from 0. For example, for the `gender`

column we can
assign the feature string "female" to an integer ID of 0 and "male" to 1 by
doing:

```
gender = tf.contrib.layers.sparse_column_with_keys(
column_name="gender", keys=["female", "male"])
```

What if we don't know the set of possible values in advance? Not a problem. We
can use `sparse_column_with_hash_bucket`

instead:

```
education = tf.contrib.layers.sparse_column_with_hash_bucket("education", hash_bucket_size=1000)
```

What will happen is that each possible value in the feature column `education`

will be hashed to an integer ID as we encounter them in training. See an example
illustration below:

ID | Feature |
---|---|

... | |

9 | `"Bachelors"` |

... | |

103 | `"Doctorate"` |

... | |

375 | `"Masters"` |

... |

No matter which way we choose to define a `SparseColumn`

, each feature string
will be mapped into an integer ID by looking up a fixed mapping or by hashing.
Note that hashing collisions are possible, but may not significantly impact the
model quality. Under the hood, the `LinearModel`

class is responsible for
managing the mapping and creating `tf.Variable`

to store the model parameters
(also known as model weights) for each feature ID. The model parameters will be
learned through the model training process we'll go through later.

We'll do the similar trick to define the other categorical features:

```
relationship = tf.contrib.layers.sparse_column_with_hash_bucket("relationship", hash_bucket_size=100)
workclass = tf.contrib.layers.sparse_column_with_hash_bucket("workclass", hash_bucket_size=100)
occupation = tf.contrib.layers.sparse_column_with_hash_bucket("occupation", hash_bucket_size=1000)
native_country = tf.contrib.layers.sparse_column_with_hash_bucket("native_country", hash_bucket_size=1000)
```

### Base Continuous Feature Columns

Similarly, we can define a `RealValuedColumn`

for each continuous feature column
that we want to use in the model:

```
age = tf.contrib.layers.real_valued_column("age")
education_num = tf.contrib.layers.real_valued_column("education_num")
capital_gain = tf.contrib.layers.real_valued_column("capital_gain")
capital_loss = tf.contrib.layers.real_valued_column("capital_loss")
hours_per_week = tf.contrib.layers.real_valued_column("hours_per_week")
```

### Making Continuous Features Categorical through Bucketization

Sometimes the relationship between a continuous feature and the label is not
linear. As an hypothetical example, a person's income may grow with age in the
early stage of one's career, then the growth may slow at some point, and finally
the income decreases after retirement. In this scenario, using the raw `age`

as
a real-valued feature column might not be a good choice because the model can
only learn one of the three cases:

- Income always increases at some rate as age grows (positive correlation),
- Income always decreases at some rate as age grows (negative correlation), or
- Income stays the same no matter at what age (no correlation)

If we want to learn the fine-grained correlation between income and each age
group seperately, we can leverage **bucketization**. Bucketization is a process
of dividing the entire range of a continuous feature into a set of consecutive
bins/buckets, and then converting the original numerical feature into a bucket
ID (as a categorical feature) depending on which bucket that value falls into.
So, we can define a `bucketized_column`

over `age`

as:

```
age_buckets = tf.contrib.layers.bucketized_column(age, boundaries=[18, 25, 30, 35, 40, 45, 50, 55, 60, 65])
```

where the `boundaries`

is a list of bucket boundaries. In this case, there are
10 boundaries, resulting in 11 age group buckets (from age 17 and below, 18-24,
25-29, ..., to 65 and over).

### Intersecting Multiple Columns with CrossedColumn

Using each base feature column separately may not be enough to explain the data.
For example, the correlation between education and the label (earning > 50,000
dollars) may be different for different occupations. Therefore, if we only learn
a single model weight for `education="Bachelors"`

and `education="Masters"`

, we
won't be able to capture every single education-occupation combination (e.g.
distinguishing between `education="Bachelors" AND occupation="Exec-managerial"`

and `education="Bachelors" AND occupation="Craft-repair"`

). To learn the
differences between different feature combinations, we can add **crossed feature
columns** to the model.

```
education_x_occupation = tf.contrib.layers.crossed_column([education, occupation], hash_bucket_size=int(1e4))
```

We can also create a `CrossedColumn`

over more than two columns. Each
constituent column can be either a base feature column that is categorical
(`SparseColumn`

), a bucketized real-valued feature column (`BucketizedColumn`

),
or even another `CrossColumn`

. Here's an example:

```
age_buckets_x_education_x_occupation = tf.contrib.layers.crossed_column(
[age_buckets, education, occupation], hash_bucket_size=int(1e6))
```

## Defining The Logistic Regression Model

After processing the input data and defining all the feature columns, we're now ready to put them all together and build a Logistic Regression model. In the previous section we've seen several types of base and derived feature columns, including:

`SparseColumn`

`RealValuedColumn`

`BucketizedColumn`

`CrossedColumn`

All of these are subclasses of the abstract `FeatureColumn`

class, and can be
added to the `feature_columns`

field of a model:

```
model_dir = tempfile.mkdtemp()
m = tf.contrib.learn.LinearClassifier(feature_columns=[
gender, native_country, education, occupation, workclass, marital_status, race,
age_buckets, education_x_occupation, age_buckets_x_education_x_occupation],
model_dir=model_dir)
```

The model also automatically learns a bias term, which controls the prediction
one would make without observing any features (see the section "How Logistic
Regression Works" for more explanations). The learned model files will be stored
in `model_dir`

.

## Training and Evaluating Our Model

After adding all the features to the model, now let's look at how to actually train the model. Training a model is just a one-liner using the TF.Learn API:

```
m.fit(input_fn=train_input_fn, steps=200)
```

After the model is trained, we can evaluate how good our model is at predicting the labels of the holdout data:

```
results = m.evaluate(input_fn=eval_input_fn, steps=1)
for key in sorted(results):
print "%s: %s" % (key, results[key])
```

The first line of the output should be something like `accuracy: 0.83557522`

,
which means the accuracy is 83.6%. Feel free to try more features and
transformations and see if you can do even better!

If you'd like to see a working end-to-end example, you can download our
example code
and set the `model_type`

flag to `wide`

.

## Adding Regularization to Prevent Overfitting

Regularization is a technique used to avoid **overfitting**. Overfitting happens
when your model does well on the data it is trained on, but worse on test data
that the model has not seen before, such as live traffic. Overfitting generally
occurs when a model is excessively complex, such as having too many parameters
relative to the number of observed training data. Regularization allows for you
to control your model's complexity and makes the model more generalizable to
unseen data.

In the Linear Model library, you can add L1 and L2 regularizations to the model as:

```
m = tf.contrib.learn.LinearClassifier(feature_columns=[
gender, native_country, education, occupation, workclass, marital_status, race,
age_buckets, education_x_occupation, age_buckets_x_education_x_occupation],
optimizer=tf.train.FtrlOptimizer(
learning_rate=0.1,
l1_regularization_strength=1.0,
l2_regularization_strength=1.0),
model_dir=model_dir)
```

One important difference between L1 and L2 regularization is that L1 regularization tends to make model weights stay at zero, creating sparser models, whereas L2 regularization also tries to make the model weights closer to zero but not necessarily zero. Therefore, if you increase the strength of L1 regularization, you will have a smaller model size because many of the model weights will be zero. This is often desirable when the feature space is very large but sparse, and when there are resource constraints that prevent you from serving a model that is too large.

In practice, you should try various combinations of L1, L2 regularization strengths and find the best parameters that best control overfitting and give you a desirable model size.

## How Logistic Regression Works

Finally, let's take a minute to talk about what the Logistic Regression model actually looks like in case you're not already familiar with it. We'll denote the label as $$Y$$, and the set of observed features as a feature vector

$$\mathbf{x}=[x_1, x_2, ..., x_d]$$. We define $$Y=1$$ if an individual earned > 50,000 dollars and $$Y=0$$ otherwise. In Logistic Regression, the probability of the label being positive ($$Y=1$$) given the features $$\mathbf{x}$$ is given as:

$$ P(Y=1|\mathbf{x}) = \frac{1}{1+\exp(-(\mathbf{w}^T\mathbf{x}+b))}$$

where $$\mathbf{w}=[w_1, w_2, ..., w_d]$$ are the model weights for the features

$$\mathbf{x}=[x_1, x_2, ..., x_d]$$. $$b$$ is a constant that is often called
the **bias** of the model. The equation consists of two parts—A linear model and
a logistic function:

**Linear Model**: First, we can see that $$\mathbf{w}^T\mathbf{x}+b = b + w_1x_1 + ... +w_dx_d$$ is a linear model where the output is a linear function of the input features $$\mathbf{x}$$. The bias $$b$$ is the prediction one would make without observing any features. The model weight $$w_i$$ reflects how the feature $$x_i$$ is correlated with the positive label. If $$x_i$$ is positively correlated with the positive label, the weight $$w_i$$ increases, and the probability $$P(Y=1|\mathbf{x})$$ will be closer to 1. On the other hand, if $$x_i$$ is negatively correlated with the positive label, then the weight $$w_i$$ decreases and the probability $$P(Y=1|\mathbf{x})$$ will be closer to 0.**Logistic Function**: Second, we can see that there's a logistic function (also known as the sigmoid function) $$S(t) = 1/(1+\exp(-t))$$ being applied to the linear model. The logistic function is used to convert the output of the linear model $$\mathbf{w}^T\mathbf{x}+b$$ from any real number into the range of $$[0, 1]$$, which can be interpreted as a probability.

Model training is an optimization problem: The goal is to find a set of model
weights (i.e. model parameters) to minimize a **loss function** defined over the
training data, such as logistic loss for Logistic Regression models. The loss
function measures the discrepancy between the ground-truth label and the model's
prediction. If the prediction is very close to the ground-truth label, the loss
value will be low; if the prediction is very far from the label, then the loss
value would be high.

## Learn Deeper

If you're interested in learning more, check out our Wide & Deep Learning Tutorial where we'll show you how to combine the strengths of linear models and deep neural networks by jointly training them using the TF.Learn API.