A Beginners Guide to SoftMax Regression Using TensorFlow

Jun 24, 2022

This article introduces basic softmax regression and its implementation in Python using TensorFlow to the learner.

While implementing softmax regression in Python using TensorFlow, we will use the MNIST handwritten digit dataset.

MNIST forms the basics of machine learning. Classifying the MNIST handwritten digit dataset is a low-level problem in machine learning which can be solved in numerous ways.


This article does not focus on designing a complex machine learning model. Instead, we will concentrate more on basic TensorFlow concepts.

Softmax regression is used in TensorFlow using various dependencies such as NumPy, and matplotlib.

This article also utilizes knowledge from logic regression and how it is implemented in Python using softmax regression.

Logistic regression can be termed a supervised classification algorithm. It is applied in a classification problem where the output/target variable(y) only takes discrete values for available inputs/set of features(x).

Python comes with various libraries that one can use to implement logistic regression.


  • The learner should be equipped with basic knowledge of the Python programming language. (https://github.com/Akuli/python-tutorial)
  • Python3 or the latest version installed.
  • Some basic knowledge of TensorFlow. Download TensorFlow from here
  • Some logic regression knowledge in Python.

Overview of Softmax regression

The softmax function forms the basis of softmax regression. The softmax function (or normalized exponential function) can be viewed as a normalization function involving adjusting values calculated on different scales to an ideally similar scale.

Softmax regression is a form of logistic regression used when multiple classes are handled.

In a binomial/binary logistic regression, we target a variable that can only take two possibilities, that is, 0 or 1 to represent “True” or “False”. For an ith observation, xi belong to {0,1}

Let’s consider a scenario where the target variable takes in two or more class labels, such that for ith observation, xi belongs to the range between 0 and 9. Softmax/multinomial logistic regression would be appropriate in such a scenario.

To define our model:

  • We first give our dataset ’n’ features, ’m’ observations, and ‘z’ class labels, such that each observation can be classified as a ‘k’ possible target value. For example, suppose we have a dataset of 100, 28×28 vector size handwritten digit images, we get n=100, m=28×28=784, and k=10.

The x feature matrix is defined as:


xij signifies jth feature value for the ith observation. The matrices dimension is;


The w weight matrix is defined as:


wij signifies the weight labeled to ith feature for the jth class. The matrices dimension is:


Logic score matrix:

We now define our logic score (net input) matrix z as z=xw. The matrices dimension is;


The logic matrix ‘z’ signifies the probability of label j for the ith observation.

We can use the logic vector score for the ith observation as zi. For instance, the vector z5=(1.0,2.5,3.5,4.1,1.5,0.4,1.3,1.1,0.3) can represent each class labels score ranging from 0 to 9.

In the MNIST handwritten classification case for the 5th observation. The maximum score becomes 5.2, corresponding to class label ‘3’. This shows that our model predicts the 5th image to be ‘3’.

The softmax layer

Training the model using score values becomes hard since differentiating is challenging when applying the gradient descent algorithm.

The softmax function helps convert the ‘z’ score matrix to probabilities. For a vector yi the softmax function s(y) can be defined as;


The softmax function converts all scores to probabilities and then sums up the probabilities to 1.

Our sofmax function computes the probabiliuty that the ith training sample is from the j class for the logic vector z(i) as;


We will denote the softmax probability for the ith observation as si

Target one-hot encoded matrix

One-hot encoding occurs when a target vector corresponds to each observation comprising 0’s and 1’s, with 1 being a correct label. The diagram below shows how one-hot encoding happens:


We can denote Ti as one-hot encoding vector for the ith observation.

Cost function

The cross-entropy concept (variates two probability distribution measures) can be used to variate one-hot encoded vector and softmax probability distance. The distance values depend on the target classes.

The cross entropy function, D(Si,Ti) for the ith observation with the softmax vector probability , Si and one=hot target vector can be defined as:


The average cross-entropy defines the cost function j as:


Gradient descent algorithm

We now compute two gradient descent derivatives,



To train our softmax model, the gradient descent derivatives can be used to improve the biases and weights contrary to the gradients,



For every instance of class j, that is, (1,2,3,…,k) with alpha being the learning rate.

Let’s create a softmax regression model using Tensorflow to manipulate the MNIST handwritten dataset.

Softmax regression implementation on the MNIST handwritten digit dataset using Tensorflow

MNIST’s (Modified National Institute of Standards and Technology) handwritten digit dataset is used to train image processing models for the handwritten digit classification set. \

It is also used in training machine learning and deep learning models. The MNIST dataset features 60,000 small square training images and 10,000 testing images on a 28 x 28-pixel scale of single handwritten digits between 0 and 9.

Each data point has two parts describing the 28 x 28 size image, an image(x) and a label(y)

Tensorflow is a free end-to-end open-source platform for training machine learning models. It features a broad and flexible environment of libraries and tools for machine learning.

In this tutorial, we will train a softmax function model that will recognize a handwriting digit by comparing each pixel in the image.

It will then train the model using TensorFlow to predict the image by looking at a couple of examples already labeled.

Importing required packages

Let’s begin by importing TensorFlow and NumPy libraries:

import tensorflow as tf
from tensorflow import keras
import numpy as np

NumPy(Numerical Python) is a virtual library for programming in Python. It consists of a multidimensional array object and various routines for handling those arrays.

Matplotlib is a vast library for data visualization and graphical plotting in Python. Since matplotlib is too broad, we only import the pyplot interface for plotting.

Downloading MNIST Data

Yann LeCun’s website provides the official details of the MNIST dataset.

In addition, the TensorFlow library comes packed with the MNIST dataset for use in artificial intelligence.

((train_data, train_labels),
(mnist_data, mnist_labels))=tf.keras.datasets.mnist.load_data()

The above code downloads the MNIST data from the tensorflow.keras library. Each image is decompressed into an array of size 784 and saved as data.

Processing the data

The MNIST dataset features 60,000 small square training images and 10,000 testing images of 28 x 28-pixel scale of single handwritten digits between 0 and 9, each flattened into a 1-dimension array of size 784.

Each data point has two parts describing the 28 x 28 size image, an image(x) and a label(y).



The images in the MNIST dataset range from 0 to 255. We divide the train_data with 32-bit float since a 32-bit precision is most commonly used in training models. We then convert the labels to an integer of 32-bit.

Defining 28×28 numerical features

Next step, we train the classifier:


The above code represents the feature as numerical features of size 28×28. feature_column defines the set of transformations to the input.

Logistic regression estimator

We make use of a linear classifier, as follows:


This dataset only contains categorical features hence the feature_column. The estimator only requires to know the classification algorithm to be performed on the dataset.

We use a linear classifier as our estimator, passing the feature_column. The estimator trains, evaluate, predicts, and exports the model.

Building an input function for the estimator

We predict a particular picture to determine whether it is consistent with an actual image:


The train_input_fn is used for predictions and evaluation. train_input_fn=tf.compat.v1.estimator.inputs.numpy_input_fn is used to return the function used to feed the numpy dict array into the model with x representing the dict features and y the dict targets.

batch_size=100 returns the integer batch size. num_epochs=None represents the number of iterative epochs over the data.

None means it will run forever. shuffle=False is supposed to shuffle the queue when the value is actual.

Training the classifier

classifier.train(input_fn=train_input_fn, steps=5)

We train the model where steps define how many times we will train the model.

Validating data

We now create an input_fn function to validate the data.


Evaluating the classifier


We get an accuracy of 89.4% after 130 steps. This model trains for a specific number of steps and logs the value after ten steps.


Softmax regression using TensorFlow is an excellent technique for solving image recognition problems in machine learning.

Softmax regression is applied in many areas such as image recognition in neural networks. Output values from a hidden layer would be treated as input values by the output layer and computed for probabilities by Softmax.

The highest probability class would be treated as the final class.

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