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.

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

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.


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.

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