Isotonic Regression in Machine Learning: Understanding Regressions in Machine Learning

There are various types of regression models (algorithms) that are used to train machine learning programs, such as linear, logistics, ridge, and lasso regression. Of these, the linear regression model is the most basic and most widely used regression model. Isotonic regression in machine learning is based on linear regression. Hence, before we move on to isotonic regression, let’s first have a look at linear regression in machine learning.

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Understanding Linear Regression in Machine Learning


The linear regression model is used to determine the relationship between the dependent and independent variables. It assumes a linear relationship, represented by the best fit line, between the two variables. The equation y= mx + c + e is used to denote the linear regression model where:

m= slope of the line

c= intercept

e= error in the model

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The linear regression model is susceptible to outliers, highly inflexible, and hence can’t be used for big size data. When this model is deployed on a big size test data, there are multiple instances that lie outside the slope of the line, also called residual errors. Methods such as L1 and L2 regularization may be used to reduce the steepness of the slope of the line, but they don’t prove as useful.

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This limits the accuracy of the machine learning algorithm. A new approach of isotonic regression in machine learning is being adopted to overcome this limit. Although not currently widespread, this approach is highly powerful and can help improve the accuracy of the machine learning program.

Understanding Isotonic Regression in Machine Learning

Before diving into the technical stuff, let’s understand isotonic regression in machine learning in layman’s terms.

Let’s start by decoding the word ‘isotonic.’ The word ‘isotonic’ has Greek root words origins, made of two parts, ‘iso’ and ‘tonic.’ Here, ‘iso’ means equal and ‘tonic’ means stretching. In terms of machine learning algorithms, isotonic regression can, therefore, be understood as equal stretching along the linear regression line. It works on top of a linear regression model.

Let’s have a look at different aspects related to isotonic regression that will help us understand it better.

1. Piecewise Linear Model

As mentioned earlier, the steepness of the slope of the linear regression line needs to be minimized, for which L1 and L2 regularization methods are used. The isotonic regression approach is different altogether by dividing the graph into piecewise sections by creating thresholds and having a linear line for each section connected end-to-end.

For example, in the above image, the X-axis can be divided further into various smaller sections, say in equal intervals of 10. Each of these intervals can be called as bins, such as bin1, bin2, bin3, bin4, and so on. The linear equation, therefore, now becomes,

y= m1x1 + m2x2 + m3x3 +….. mnxn + c, where:

m1, m2, m3….mn = slope of the line for individual bins.

This helps minimize the error and reduce the slope of the best fit line.

2. Non-negative Slope

Since an isotonic function is a monotonic function, the slope of the solution is always non-negative. A decrease in the slope isn’t allowed when moving from one threshold to the other. The lowest point in a threshold should always be bigger than the highest point in the previous threshold.

For instance, let x1, x2, x3, x4…xn be the values of the data points considered for the slope in bins b1, b2, b3, b4…bn. Then, as per rule, the slope should be non-negative. Hence,

f(x1) <= f(x2) <= f(x3) <= f(x4)…<= f(xn).

So, we start with a lower point (where f(x1) is the lowest point) and gradually move to a higher point with each threshold. The slope of a threshold can be zero (horizontal line) but can never be negative (downward slope).

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Advantages of Using Isotonic Regression in Machine Learning Models

Using isotonic regression offers two major benefits, which are discussed below.

1. Multidimensional Scaling

Isotonic regression is highly helpful if you have multiple input variables. We can inspect each and every dimension as each and every function and interpolate it in a linear way. This allows for easy multidimensional scaling.

2. Calibration of Probability Values

In logistic regression, suppose we have a variable x, and we denote a probability p(1) where the probability value for the variable does not increase. But, in reality, the probability value is higher in the real-world. In such cases, for calibration purposes or increasing the probability of such variables, isotonic regression proves highly helpful.

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Disadvantages of Using Isotonic Regression in Machine Learning Models

There is one major downside of using isotonic regression, which is discussed below.

Risk of overfitting

There is a significant risk of overfitting of hyperparameter (K) as the number of isotonic constraints and predictor features increases, but the cross-validation workflow method can be used to manage the issue.

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Currently, only three major languages have open-source packages with Isotonic regression. However, looking at the benefits of using isotonic regression in machine learning problems, the scope, usage, and availability of isotonic regression packages will surely increase in the future.

We can see isotonic regression majorly replace linear regression and L1 and L2 normalization methods. Therefore, to be future-ready, it is necessary to keep oneself updated and knowledgeable about isotonic regression from now!

If you are interested in learning more about isotonic regression in machine learning or other machine learning related concepts, you can check out IIIT-B and upGrad’s PG Diploma in Machine Learning and AI, which is India’s best-selling program with a 4.5-star rating. The course has 450+ hours of learning, 30+ case studies, and assignments, and helps students learn in-demand skills related to machine learning and AI.

Why is regression significant in machine learning?

Regression analysis, a kind of supervised learning algorithm, is one of the fundamental concepts in machine learning. Regression is used to establish the relationship between different variables by trying to estimate how the value of one influences that of the other. In the field of machine learning, regression comprises complex mathematical algorithms that help estimate the outcomes of a specific target variable based on the continuously changing values of one or multiple predictor variables. The most popular kind of regression analysis is linear regression since it is very easy to use for making forecasts and predictions.

Is machine learning the same as data science?

With buzzwords like data science and machine learning becoming mainstream today, many people often feel confused about what they actually mean. Let us try to explain here quickly. Data science refers to the study of massive volumes of data generated by organizations. Data scientists employ various techniques to reveal valuable insights from this data such that businesses can draw maximum benefits and stay ahead of the competition. Machine learning is different from data science; it employs data science techniques to learn about data which is then used to train machines. Machine learning uses complex mathematical models to help computers learn without human intervention.

Is machine learning the same as deep learning?

Machine learning is a subset of artificial intelligence. It employs algorithms or models that can analyze data, learn from it, and then apply those learnings to help computers or machines make decisions without explicit human inputs. On the other hand, deep learning is a subfield of machine learning. It is used to structure algorithms or mathematical models in layers to develop an artificial neural network that resembles the structure of the human brain. This neural network can learn on its own and make intelligent decisions using its own logical framework and analyzing data.

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