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Gradient Descent

Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization


In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. For the sample size n, the step size η, and the batch size B, we derive uniform evaluations on the time with orders n -1, η 1/2, and ((n - B) / B (n - 1)), respectively.

What is "Stochastic" in Stochastic Gradient Descent (SGD)


Over the past 5 months, I had been reading the book Probability Essentials by Jean Jacod and Philip Protter, and the more time I spent on it, more I started to treat every encounter with Probability with a rigorous perspective. Recently, I was reading a paper in Deep Learning and the authors were talking about Stochastic Gradient Descent (SGD), which got me thinking, why is it called "stochastic"? Where is the randomness in it? Disclaimer: I won't be trying to explain any mathematical bits in this article solely because it is a pain to add equations. I hope the reader has some familiarity with the mathematical bits of the Gradient Descent algorithm and its variants. I'll provide a brief introduction where necessary, but won't be going into much detail.

#003C Gradient Descent in Python - Master Data Science


We will first import libraries as NumPy, matplotlib, pyplot and derivative function. Then with a NumPy function – linspace() we define our variable \(w \) domain between 1.0 and 5.0 and 100 points. Also we define alpha which will represent learning rate. Next, we will define our \(y \) ( in our case \(J(w) \)) and plot to see a convex function, we will use \((w-3) 2 \). So we can see that we plotted our convex function as an example.

Optimizer in Deep Learning


An optimizer is a function or an algorithm that customizes the attributes of the neural network, such as weights and discovering rate. Hence, it assists in decreasing the overall loss and also enhance the accuracy. The problem of picking the ideal weights for the version is an overwhelming job, as a deep learning version usually includes numerous parameters. It increases the requirement to pick an appropriate optimization algorithm for your application. You can utilize different optimizers to make changes in your weights as well as learning price.

Stochastic Gradient Descent Using Pytorch Linear Module


In the previous tutorial here on SGD, I explored the way in which we can implement using PyTorch's built-in gradient calculation, loss, and optimization implementation. in our present discussion…

What is momentum in a Neural network and how does it work?


In a neural network, there is the concept of loss, which is used to calculate performance. The higher the loss, the poorer the performance of the neural network, that is why we always try to minimize the loss so that the neural network performs better. The process of minimizing loss is called optimization. An optimizer is a method that modifies the weights of the neural network to reduce the loss. Although several neural network optimizers exist, in this article we will learn about gradient descent with momentum and compare its performance with others.

Linear Model the Machine Learning Way


The Ordinary Least Squares model (OLS) is a central building block in Machine Learning (ML). OLS is also used everywhere in Social Sciences. I come from an Economics background and I was initially a bit puzzled by the way the ML textbooks solve OLS. In this blog post, I explain the Economics way versus the ML way and why both make sense. TL;DR: In a high-dimensional setting, do not inverse a huge matrix, use gradient descent.

Backpropagation and Gradient Descent


Backpropagation and gradient descent are two different methods that form a powerful combination in the learning process of neural networks. Let's try to understand the intuition of how this works. Neural networks learn through forward propagation, by using weights, biases, and nonlinear activation functions to calculate a prediction y from the input x that should match the true output y as closely as possible. There are several different loss functions and which one you choose depends on the type of machine learning problem you are facing. The goal of backpropagation is to adjust the weights and biases throughout the neural network based on the calculated cost so that the cost will be lower in the next iteration.

Implementing Gradient Descent in Python from Scratch


A machine learning model may have several features, but some feature might have a higher impact on the output than others. For example, if a model is predicting apartment prices, the locality of the apartment might have a higher impact on the output than the number of floors the apartment building has. Hence, we come up with the concept of weights. Each feature is associated with a weight (a number) i.e. the higher the feature has an impact on the output, the larger the weight associated with it. But how do you decide what weight should be assigned to each feature?

Efficient Distributed Machine Learning via Combinatorial Multi-Armed Bandits Machine Learning

We consider the distributed stochastic gradient descent problem, where a main node distributes gradient calculations among $n$ workers from which at most $b \leq n$ can be utilized in parallel. By assigning tasks to all the workers and waiting only for the $k$ fastest ones, the main node can trade-off the error of the algorithm with its runtime by gradually increasing $k$ as the algorithm evolves. However, this strategy, referred to as adaptive k sync, can incur additional costs since it ignores the computational efforts of slow workers. We propose a cost-efficient scheme that assigns tasks only to $k$ workers and gradually increases $k$. As the response times of the available workers are unknown to the main node a priori, we utilize a combinatorial multi-armed bandit model to learn which workers are the fastest while assigning gradient calculations, and to minimize the effect of slow workers. Assuming that the mean response times of the workers are independent and exponentially distributed with different means, we give empirical and theoretical guarantees on the regret of our strategy, i.e., the extra time spent to learn the mean response times of the workers. Compared to adaptive k sync, our scheme achieves significantly lower errors with the same computational efforts while being inferior in terms of speed.