Author: Jason Brownlee

The behavior and performance of many machine learning algorithms are referred to as stochastic.

Stochastic refers to a variable process where the outcome involves some randomness and has some uncertainty. It is a mathematical term and is closely related to “*randomness*” and “*probabilistic*” and can be contrasted to the idea of “*deterministic*.”

The stochastic nature of machine learning algorithms is an important foundational concept in machine learning and required to understand in order to interpret the behavior of many predictive models.

In this post, you will discover a gentle introduction to stochasticity in machine learning.

After reading this post, you will know:

- A variable or process is stochastic if there is uncertainty or randomness involved in the outcomes.
- Stochastic is a synonym for random and probabilistic, although is different from non-deterministic.
- Many machine learning algorithms are stochastic because they explicitly use randomness during optimization or learning.

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## Overview

This tutorial is divided into three parts; they are:

- What Does “
*Stochastic*” Mean? - Stochastic vs. Random, Probabilistic, and Nondeterministic
- Stochastic in Machine Learning

## What Does “*Stochastic*” Mean?

A variable is stochastic if the occurrence of events or outcomes involves randomness or uncertainty.

… “stochastic” means that the model has some kind of randomness in it

— Page 66, Think Bayes.

A process is stochastic if it governs one or more stochastic variables.

Games are stochastic because they include an element of randomness, such as shuffling or rolling of a dice in card games and board games.

In real life, many unpredictable external events can put us into unforeseen situations. Many games mirror this unpredictability by including a random element, such as the throwing of dice. We call these stochastic games.

— Page 177, Artificial Intelligence: A Modern Approach, 3rd edition, 2009.

Stochastic is commonly used to describe mathematical processes that use or harness randomness. Common examples include Brownian motion, Markov Processes, Monte Carlo Sampling, and more.

Now that we have some definitions, let’s try and add some more context by comparing stochastic with other notions of uncertainty.

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## Stochastic vs. Random, Probabilistic, and Non-deterministic

In this section, we’ll try to better understand the idea of a variable or process being stochastic by comparing it to the related terms of “*random*,” “*probabilistic*,” and “*non-deterministic*.”

### Stochastic vs. Random

In statistics and probability, a variable is called a “random variable” and can take on one or more outcomes or events.

It is the common name used for a thing that can be measured.

In general, stochastic is a synonym for random.

For example, a stochastic variable is a random variable. A stochastic process is a random process.

Typically, random is used to refer to a lack of dependence between observations in a sequence. For example, the rolls of a fair die are random, so are the flips of a fair coin.

Strictly speaking, a random variable or a random sequence can still be summarized using a probability distribution; it just may be a uniform distribution.

We may choose to describe something as stochastic over random if we are interested in focusing on the probabilistic nature of the variable, such as a partial dependence of the next event on the current event. We may choose random over stochastic if we wish to focus attention on the independence of the events.

### Stochastic vs. Probabilistic

In general, stochastic is a synonym for probabilistic.

For example, a stochastic variable or process is probabilistic. It can be summarized and analyzed using the tools of probability.

Most notably, the distribution of events or the next event in a sequence can be described in terms of a probability distribution.

We may choose to describe a variable or process as probabilistic over stochastic if we wish to emphasize the dependence, such as if we are using a parametric model or known probability distribution to summarize the variable or sequence.

### Stochastic vs. Non-deterministic

A variable or process is deterministic if the next event in the sequence can be determined exactly from the current event.

For example, a deterministic algorithm will always give the same outcome given the same input. Conversely, a non-deterministic algorithm will give different outcomes for the same input.

A stochastic variable or process is not deterministic because there is uncertainty associated with the outcome.

Nevertheless, a stochastic variable or process is also not non-deterministic because non-determinism only describes the possibility of outcomes, rather than probability.

Describing something as stochastic is a stronger claim than describing it as non-deterministic because we can use the tools of probability in analysis, such as expected outcome and variance.

… “stochastic” generally implies that uncertainty about outcomes is quantified in terms of probabilities; a nondeterministic environment is one in which actions are characterized by their possible outcomes, but no probabilities are attached to them.

— Page 43, Artificial Intelligence: A Modern Approach, 3rd edition, 2009.

## Stochastic in Machine Learning

Many machine learning algorithms and models are described in terms of being stochastic.

This is because many optimization and learning algorithms both must operate in stochastic domains and because some algorithms make use of randomness or probabilistic decisions.

Let’s take a closer look at the source of uncertainty and the nature of stochastic algorithms in machine learning.

### Stochastic Problem Domains

Stochastic domains are those that involve uncertainty.

… machine learning must always deal with uncertain quantities, and sometimes may also need to deal with stochastic (non-deterministic) quantities. Uncertainty and stochasticity can arise from many sources.

— Page 54, Deep Learning, 2016.

This uncertainty can come from a target or objective function that is subjected to statistical noise or random errors.

It can also come from the fact that the data used to fit a model is an incomplete sample from a broader population.

Finally, the models chosen are rarely able to capture all of the aspects of the domain, and instead must generalize to unseen circumstances and lose some fidelity.

### Stochastic Optimization Algorithms

Stochastic optimization refers to a field of optimization algorithms that explicitly use randomness to find the optima of an objective function, or optimize an objective function that itself has randomness (statistical noise).

Most commonly, stochastic optimization algorithms seek a balance between exploring the search space and exploiting what has already been learned about the search space in order to hone in on the optima. The choice of the next locations in the search space are chosen stochastically, that is probabilistically based on what areas have been searched recently.

Stochastic hill climbing chooses at random from among the uphill moves; the probability of selection can vary with the steepness of the uphill move.

— Page 124, Artificial Intelligence: A Modern Approach, 3rd edition, 2009.

Popular examples of stochastic optimization algorithms are:

- Simulated Annealing
- Genetic Algorithm
- Particle Swarm Optimization

Particle swarm optimization (PSO) is a stochastic optimization approach, modeled on the social behavior of bird flocks.

— Page 9, Computational Intelligence: An Introduction.

### Stochastic Learning Algorithms

Most machine learning algorithms are stochastic because they make use of randomness during learning.

Using randomness is a feature, not a bug. It allows the algorithms to avoid getting stuck and achieve results that deterministic (non-stochastic) algorithms cannot achieve.

For example, some machine learning algorithms even include “*stochastic*” in their name such as:

- Stochastic Gradient Descent (optimization algorithm).
- Stochastic Gradient Boosting (ensemble algorithm).

Stochastic gradient descent optimizes the parameters of a model, such as an artificial neural network, that involves randomly shuffling the training dataset before each iteration that causes different orders of updates to the model parameters. In addition, model weights in a neural network are often initialized to a random starting point.

Most deep learning algorithms are based on an optimization algorithm called stochastic gradient descent.

— Page 98, Deep Learning, 2016.

Stochastic gradient boosting is an ensemble of decision trees algorithms. The stochastic aspect refers to the random subset of rows chosen from the training dataset used to construct trees, specifically the split points of trees.

### Stochastic Algorithm Behaviour

Because many machine learning algorithms make use of randomness, their nature (e.g. behavior and performance) is also stochastic.

The stochastic nature of machine learning algorithms is most commonly seen on complex and nonlinear methods used for classification and regression predictive modeling problems.

These algorithms make use of randomness during the process of constructing a model from the training data which has the effect of fitting a different model each time same algorithm is run on the same data. In turn, the slightly different models have different performance when evaluated on a hold out test dataset.

This stochastic behavior of nonlinear machine learning algorithms is challenging for beginners who assume that learning algorithms will be deterministic, e.g. fit the same model when the algorithm is run on the same data.

This stochastic behavior requires that the performance of the model must be summarized using summary statistics that describe the mean or expected performance of the model, rather than the performance of the model from any single training run.

For more on this topic, see the post:

## Further Reading

This section provides more resources on the topic if you are looking to go deeper.

### Posts

- How to Generate Random Numbers in Python
- Introduction to Random Number Generators for Machine Learning in Python
- Embrace Randomness in Machine Learning
- Why Initialize a Neural Network with Random Weights?
- Stochastic Gradient Boosting with XGBoost and scikit-learn in Python

### Articles

- Random variable, Wikipedia.
- Statistical randomness, Wikipedia.
- Stochastic, Wikipedia.
- Stochastic process, Wikipedia.
- Stochastic optimization.

## Summary

In this post, you discovered a gentle introduction to stochasticity in machine learning.

Specifically, you learned:

- A variable or process is stochastic if there is uncertainty or randomness involved in the outcomes.
- Stochastic is a synonym for random and probabilistic, although is different from non-deterministic.
- Many machine learning algorithms are stochastic because they explicitly use randomness during optimization or learning.

Do you have any questions?

Ask your questions in the comments below and I will do my best to answer.

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