# RMSProp

Author: Jason Huang (SysEn 6800 Fall 2020)

## Introduction

RMSProp, so call root mean square propagation, is an optimization algorithm/method designed for Artificial Neural Network (ANN) training. And it is an unpublished algorithm first proposed in the Coursera course “Neural Network for Machine Learning” lecture six by Geoff Hinton. RMSProp lies in the realm of adaptive learning rate methods, which have been growing in popularity in recent years because it is the extension of Stochastic Gradient Descent (SGD) algorithm, momentum method, and the foundation of  Adam algorithm. One of the application of RMSProp is the stochastic technology for mini-batch gradient descent.

## Theory and Methodology

### Perceptron and Neural Networks

Perceptron is an algorithm used for supervised learning of binary classifier, and also can be regard as the simplify version/single layer of the Artificial Neural Network to better understanding the neural network, which is to perform the human brain and conscious center in Artificial Intelligence(AI) and presenting the imitation of what the mind will like when human thinking. The basis form of the perceptron consists inputs, weights, bias, net sum and activation function.

The process of the perceptron is start by initiating input value $x_{1},x_{2}$ and multiplying them by their weights to obtain $w_{1},w_{2}$ . All of the weights will be added up together to create the weight sum$\sum _{i}w_{i}$ . And the weighted sum is then applied to the activation function $f$ to produce the perceptron's output.

A neural network works similarly to the human brain’s neural network. A “neuron” in a neural network is a mathematical function that collects and classifies information according to a specific architecture. A neural network contains layers of interconnected nodes, which can be regards as the perception and is similar to the multiple linear regression. The perceptron transfers the signal by a multiple linear regression into an activation function which may be nonlinear.

### RProp

RProp, or we call Resilient Back Propagation, is the widely used algorithm for supervised learning with multi-layered feed-forward networks in the past. Besides, its concepts are the foundation of RMSPRop development t. The derivatives equation of error function can be represented as:

${\frac {\partial E}{\partial w_{ij}}}={\frac {\partial E}{\partial s_{i}}}{\frac {\partial s_{i}}{\partial net_{i}}}{\frac {\partial net_{i}}{\partial w_{ij}}}$ Where $w_{ij}$ is the weight from neuron $j$ to neuron $i$ , $s_{i}$ is the output , and $net_{i}$ is the weighted sum of the inputs of neurons $i$ . Once the weight of each partial derivatives is known, the error function can be presented by performing a simple gradient descent:

$w_{ij}(t+1)=w_{ij}(t)-\epsilon {\frac {\partial E}{\partial w_{ij}}}(t)$ Obviously, the choice of the learning rate $\epsilon$ , which scales the derivative, has an important effect on the time needed until convergence is reached. If it is set too small, too many steps are needed to reach an acceptable solution; on the contrary, a large learning rate will possibly lead to oscillation, preventing the error to fall below a certain value7.

In addition, RProp can combine the method with momentum method, to prevent above problem and to accelerate the convergence rate, the equation can rewrite as:

$\Delta w_{ij}(t)=\epsilon {\frac {\partial E}{\partial w_{ij}}}(t)+\Delta w_{ij}(t-1)$ However, It turns out that the optimal value of the momentum parameter $\mu$ is equally problem dependent as the learning rate $\epsilon$ , and that no general improvement can be accomplished. Besides, RProp algorithm is not function well when we have very large datasets and need to perform mini-batch weights updates.

### RMSProp

RProp algorithm doesn’t work for mini-batches is that it violates the central idea behind stochastic gradient descent, which is when we have a small enough learning rate, it averages the gradients over successive mini-batches. To solve this issue, consider the weight, that gets the gradient 0.1 on nine mini-batches, and the gradient of -0.9 on tenths mini-batch, RMSProp did force those gradients to roughly cancel each other out, so that the stay approximately the same.

By using the sign of gradient from RProp algorithm, and the mini-batches efficiency, and averaging over mini-batches which allows combining gradients in the right way. RMSProp is keeping the moving average of the squared gradients for each weight. And then we divide the gradient by square root the mean square.

The updated equation can be performed as:

$E[g^{2}](t)=\beta E[g^{2}](t-1)+(1-\beta )({\frac {\partial c}{\partial w}})^{2}$ $w_{ij}(t)=w_{ij}(t-1)-{\frac {\eta }{\sqrt {E[g^{2}]}}}{\frac {\partial c}{\partial w_{ij}}}$ where $E[g]$ is the moving average of squared gradients, $\delta c/\delta w$ is gradient of the cost function with respect to the weight, $\eta$ is the learning rate and $\beta$ is moving average parameter (default value — 0.9).

The equation adapt the learning rate by dividing by the squared gradients, However, since we only have the estimate of the gradient on the current mini-batch, we need instead to use the moving average, which is set as default 0.9, of it.

## Numerical Example

### 2D RMSProp Example

For the 2-Dimension of how RMSProp function, refer to "Dive into Deep Learning website". In the link, the implementation of function $f(x)=0.1x_{1}^{2}+2x_{2}^{2}$ is well presented.

### RMSProp in Python

For the specific package supporting RMSProp in python, refer to Python keras.optimizers.RMSprop() Examples for both 3D and 2D implementation.

## Applicants and Discussion

Ada-grad adaptive learning rate algorithms that look a lot like RMSProp. Ada-grad adds element-wise scaling of the gradient-based on the historical sum of squares in each dimension. This means that we keep a running sum of squared gradients. And then we adapt the learning rate by dividing it by that sum to get the result. Considering the concepts in RMSProp is widely used in other machine learning algorithms. We can say that it has high potential to coupled with other methods such as Momentum,...etc. and probably can have a high-efficiency performance in the future by effort.

## Conclusion

RMSProp, root mean squared propagation is the optimization machine learning algorithm to formulate the Neural Network (NN) of Artificial Intelligence, derived from the concepts of gradients descent and RProp. Combining averaging over mini-batches and efficiency, and the gradients over successive mini-batches, RMSProp can reach the faster convergence of global optimization. As knowing the high performance of RMSProp, extensive implementation can be envisaged in the future.

## Reference

2. R Yamashita, M Nishio, R Kinh Gian, Convolutional neural networks: an overview and application in radiology (2018), 9:611–629

3. Vitaly Bushave, Understanding RMSprop — faster neural network learning (2018)

4. Vitaly Bushave, How do we ‘train’ neural networks ? (2017)