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RMSprop (Root Mean Square Propagation)

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RMSprop (Root Mean Square Propagation) is a widely used optimization algorithm in machine learning, adapting the learning rate for each parameter based on historical gradients. This article at OpenGenus provides an overview of RMSprop's workings using analogies, and its advantages over traditional gradient descent and AdaGrad. It concludes with insights into some disadvantages, current applications and future prospects for refining and extending it in diverse machine learning domains.

Table of Content

No. Topic
1 What is RMSprop?
2 Optimization algorithms
3 How RMSprop works
4 Mathematical expression
5 Comparison with Gradient Descent
6 AdaGrad VS RMSprop
7 Disadvantages
8 Conclusion

What is RMSprop?

RMSprop, short for Root Mean Square Propagation, is an optimization algorithm commonly used in machine learning to update the parameters of a model during training. It is designed to improve the convergence speed and stability of training by adapting the learning rate for each parameter based on the historical gradient information.

Okay, so now you might ask, what are optimization algorithms?

Optimization algorithms

Optimization algorithms are computational methods used to find the best solution (maxima or minima) to a given problem. This typically involves finding the optimal values of parameters that minimize or maximize an objective function. Optimization algorithms in the context of machine learning are like smart strategies which can be used to find the best solution to a complex problem.

Some popular optimization algorithms include:

  • Gradient Descent
  • Stochastic Gradient Descent
  • AdaGrad
  • Particle Swarm Optimization
  • Simulated Annealing
  • RMSprop

Imagine we are trying to find the bottom of a deep, uneven valley blindfolded. We can take steps in various directions and try to reach the lowest point. With each step, we have to decide how big our next step should be in each direction.

In terms of machine learning, training a model is like finding the bottom of this valley. The goal is to reach the best set of parameters, or the lowest point, that make the model perform well on the given task.

How RMSprop works

In machine learning, when we train a model, we calculate gradients to understand the direction and steepness of the slope (error) for each parameter. These gradients tell us how much we should adjust the parameters to improve the model's performance.

In RMSprop, firstly, we square each gradient, which helps us focus on the positive values and removes any negative signs. We then calculate the average of all the squared gradients over some recent steps. This average tells us how fast the gradients have been changing and helps us understand the overall behaviour of the slopes over time.

Now, instead of using a fixed learning rate for all parameters, RMSprop adjusts the learning rate for each parameter separately. It does this by taking the average of squared gradients we calculated earlier and using it to divide the learning rate. This division makes the learning rate bigger when the average squared gradient is smaller and smaller when the average squared gradient is bigger.

Mathematical expression

For each parameter in the model, let us calculate the squared gradient at time step t:

g2 = gradient of the parameter at time step t

Then, let us calculate the exponential moving average of squared gradients:

E[g2]t = β * E[g2]{t-1} + (1 - β) * gt2

Here, β is a hyperparameter between 0 and 1. It controls how much historical information to retain. Generally, it is set to a value like 0.9.

Now, for each parameter in the model, let us update it using the following formula:

parametert = parameter{t-1} - (learning rate / sqrt(E[g2]t + ε)) * gt

Here, parametert represents the value of the parameter at time step t, and ϵ is a small constant (usually around 10−8) added to the denominator to prevent division by zero.

Let’s look at some of the above-mentioned algorithms and see why RMSprop is a preferred choice for optimizing neural networks and ML models.

Comparison with Gradient Descent

Continuing with the valley analogy, let’s assume we take big steps in random directions since we can't see where the valley is. As we continue, we realize that in some directions, the slope is steeper, and in some, flatter. So we start adjusting the size of our steps in each direction based on how steep the slope is. When the slope is steep, we take smaller steps to avoid overshooting the minimum. On the other hand, when the slope is gentle, we can take bigger steps. This is called gradient descent and is used for finding the local minimum of a differentiable function. It iteratively moves in the direction of the steepest descent to reach the minimum.

Now, let's introduce RMSprop. As we continue walking, we keep track of the history of the slopes we have encountered in each direction. Instead of blindly adapting the step size based on the current slope, we take into account how the slopes have been changing in the past.

Suppose, we have a small ball that we roll down the valley. When the ball rolls down steep slopes, it gathers speed, and when it rolls down flatter slopes, it slows down. By measuring how fast the ball is moving, we can infer the steepness of the valley at that point. In RMSprop, the ball represents the history of gradients or slopes in each direction. It maintains an estimate of the average of squared gradients for each parameter.

As we keep moving, we use this information to decide how big our steps should be in each direction. If the average squared gradient is large, it means that the ball is rolling quickly, indicating steep slopes. So we take smaller steps to avoid overshooting the minimum. On the other hand, if the average squared gradient is small, it means the ball is rolling slowly, indicating gentler slopes, and we can take bigger steps.

By adjusting the step sizes this way, RMSprop helps us find the bottom of the valley more efficiently and effectively.

AdaGrad (Adaptive Gradient Algorithm) VS RMSprop

AdaGrad is an optimization algorithm that aims to tackle the issue of rapidly decreasing learning rates for frequent parameters. It keeps track of the historical gradients of each parameter, similar to RMSprop. However, unlike RMSprop, it adapts the learning rates individually for each parameter based on the historical gradient information.

While AdaGrad helps in finding the optimal step size for each parameter, it has one limitation, the sum of squared gradients keeps growing over time. As a result, the learning rates for some parameters may become too small in later stages of training, causing the optimization process to slow down significantly.

RMSprop addresses the limitation of AdaGrad by introducing an exponentially decaying average of squared gradients instead of a sum. This allows the algorithm to forget older gradients and focus more on recent gradients, which helps prevent the learning rates from becoming too small too quickly. By incorporating this adaptive learning rate and considering the most recent information, RMSprop can better navigate the parameter space and converge faster.


While RMSprop is widely used for its several advantages, it also has some limitations and disadvantages:

Accumulation of squared gradients: RMSprop keeps track of squared gradients over time, which can lead to very small learning rates during later stages of training, slowing down the optimization process.

Hyperparameter sensitivity: RMSprop depends on a parameter (β) that can be difficult to set properly, affecting the algorithm's performance.

Lack of momentum: Unlike some other optimization methods, RMSprop lacks momentum, which can help the learning process by accumulating past gradients.

Limited adaptation to parameters: RMSprop treats all parameters equally when adapting learning rates, which may not be ideal in some situations.

Sensitive to learning rate: RMSprop is sensitive to the initial learning rate setting, and choosing the wrong value can cause problems.

Potential for getting stuck: Like many optimization methods, RMSprop can get stuck in local minima, hindering finding the best solution.


RMSprop proves to be a valuable optimization algorithm in the field of machine learning, offering significant improvements in convergence speed and stability during model training. By dynamically adjusting learning rates based on historical gradient information, RMSprop outperforms traditional algorithms, making it an efficient choice for optimizing neural networks and complex machine learning models. Some current applications are as follows:

Image Classification: RMSprop is commonly used in training convolutional neural networks (CNNs) for image classification tasks. RMSprop helps optimize the network's parameters during the training process, leading to accurate image classification results.

Natural Language Processing (NLP): In NLP tasks, such as text classification, recurrent neural networks (RNNs) and long short-term memory (LSTM) networks are frequently used. RMSprop can be applied to train these models, enabling them to understand sequential data and perform tasks like sentiment analysis, machine translation, and text generation.

Speech Recognition: RMSprop is also used in training models for automatic speech recognition (ASR) tasks. Models like recurrent neural networks (RNNs) or attention-based models are trained using RMSprop to recognize spoken language and convert it into written text.

Further research and experimentation is expected to enhance RMSprop's potential. Fine-tuning parameters and exploring new algorithmic variations may provide even better optimization performance. As the demand for sophisticated machine learning applications grows, RMSprop will remain an essential tool in achieving optimal model performance in various domains.

Agniva Maiti

Agniva Maiti

Hi! I'm Agniva, an aspiring data scientist with a love for languages and coffee. I'm currently in my 2nd year of B.Tech (CSE) in KIIT, Odisha, India. I'm also interested in Web and App Development.

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Improved & Reviewed by:

Aditya Chatterjee Aditya Chatterjee
RMSprop (Root Mean Square Propagation)
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