L2 norm of the gradient

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L2 norm of the gradient

In our second case study for this course, loan default prediction, you will tackle financial data, and predict when a loan is likely to be risky or safe for the bank. These tasks are an examples of classification, one of the most widely used areas of machine learning, with a broad array of applications, including ad targeting, spam detection, medical diagnosis and image classification.

In this course, you will create classifiers that provide state-of-the-art performance on a variety of tasks. You will become familiar with the most successful techniques, which are most widely used in practice, including logistic regression, decision trees and boosting. In addition, you will be able to design and implement the underlying algorithms that can learn these models at scale, using stochastic gradient ascent.

You will implement these technique on real-world, large-scale machine learning tasks. You will also address significant tasks you will face in real-world applications of ML, including handling missing data and measuring precision and recall to evaluate a classifier.

This course is hands-on, action-packed, and full of visualizations and illustrations of how these techniques will behave on real data.

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We've also included optional content in every module, covering advanced topics for those who want to go even deeper! Learning Objectives: By the end of this course, you will be able to: -Describe the input and output of a classification model.

Very impressive course, I would recommend taking course 1 and 2 in this specialization first since they skip over some things in this course that they have explained thoroughly in those courses. Good overview of classification.

The python was easier in this section than previous sections although maybe I'm just better at it by this point. The topics were still as informative though!

As we saw in the regression course, overfitting is perhaps the most significant challenge you will face as you apply machine learning approaches in practice. This challenge can be particularly significant for logistic regression, as you will discover in this module, since we not only risk getting an overly complex decision boundary, but your classifier can also become overly confident about the probabilities it predicts. In this module, you will investigate overfitting in classification in significant detail, and obtain broad practical insights from some interesting visualizations of the classifiers' outputs.

You will then add a regularization term to your optimization to mitigate overfitting. You will investigate both L2 regularization to penalize large coefficient values, and L1 regularization to obtain additional sparsity in the coefficients.

Finally, you will modify your gradient ascent algorithm to learn regularized logistic regression classifiers. You will implement your own regularized logistic regression classifier from scratch, and investigate the impact of the L2 penalty on real-world sentiment analysis data.

Intuitions on L1 and L2 Regularisation

Loupe Copy. Learning L2 regularized logistic regression with gradient ascent. Machine Learning: Classification. Course 3 of 4 in the Machine Learning Specialization.

Enroll for Free. This Course Video Transcript. Penalizing large coefficients to mitigate overfitting L2 regularized logistic regression Visualizing effect of L2 regularization in logistic regression Learning L2 regularized logistic regression with gradient ascent Taught By.Training a neural network can become unstable given the choice of error function, learning rate, or even the scale of the target variable.

The problem of exploding gradients is more common with recurrent neural networks, such as LSTMs given the accumulation of gradients unrolled over hundreds of input time steps. A common and relatively easy solution to the exploding gradients problem is to change the derivative of the error before propagating it backward through the network and using it to update the weights.

Two approaches include rescaling the gradients given a chosen vector norm and clipping gradient values that exceed a preferred range. In this tutorial, you will discover the exploding gradient problem and how to improve neural network training stability using gradient clipping.

l2 norm of the gradient

Kick-start your project with my new book Better Deep Learningincluding step-by-step tutorials and the Python source code files for all examples. This requires first the estimation of the loss on one or more training examples, then the calculation of the derivative of the loss, which is propagated backward through the network in order to update the weights.

Weights are updated using a fraction of the back propagated error controlled by the learning rate. It is possible for the updates to the weights to be so large that the weights either overflow or underflow their numerical precision. The difficulty that arises is that when the parameter gradient is very large, a gradient descent parameter update could throw the parameters very far, into a region where the objective function is larger, undoing much of the work that had been done to reach the current solution.

In a given neural network, such as a Convolutional Neural Network or Multilayer Perceptron, this can happen due to a poor choice of configuration. Some examples include:. Exploding gradients is also a problem in recurrent neural networks such as the Long Short-Term Memory network given the accumulation of error gradients in the unrolled recurrent structure. Exploding gradients can be avoided in general by careful configuration of the network model, such as choice of small learning rate, scaled target variables, and a standard loss function.

Nevertheless, exploding gradients may still be an issue with recurrent networks with a large number of input time steps. One difficulty when training LSTM with the full gradient is that the derivatives sometimes become excessively large, leading to numerical problems.

To prevent this, [we] clipped the derivative of the loss with respect to the network inputs to the LSTM layers before the sigmoid and tanh functions are applied to lie within a predefined range. A common solution to exploding gradients is to change the error derivative before propagating it backward through the network and using it to update the weights.

By rescaling the error derivative, the updates to the weights will also be rescaled, dramatically decreasing the likelihood of an overflow or underflow. Gradient scaling involves normalizing the error gradient vector such that vector norm magnitude equals a defined value, such as 1. Gradient clipping involves forcing the gradient values element-wise to a specific minimum or maximum value if the gradient exceeded an expected range.

When the traditional gradient descent algorithm proposes to make a very large step, the gradient clipping heuristic intervenes to reduce the step size to be small enough that it is less likely to go outside the region where the gradient indicates the direction of approximately steepest descent.

It is a method that only addresses the numerical stability of training deep neural network models and does not offer any general improvement in performance.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

l2 norm of the gradient

It only takes a minute to sign up. I'm training a convolutional neural network CNN with 5 conv-layers and 2 fully-connected layers for binary classification using stochastic gradient descent SGD with momentum. The accuracy of the network is fine, but I am curious about the behavior of the gradients. I would expect that the 2-norm of the gradients would get lowered during training due to continuous lowering of the learning rate. However, for all of my layers the 2-norm increases slightly during training.

How can that be? Below is an image of the 2-norm of the gradients for my last fully-connected layer it has neurons. The development for the remaining layers look identical, but the 2-norm is shifted as the gradients are lower for earlier layers as expected.

It should be working correctly based on the answers in this StackOverflow post. The batch-size of my network is 32 and I use leaky ReLU after each layer and dropout after the fully-connected layers.

Note that the loss stops decreasing quite quickly. The same is true for the validation accuracy. I'm currently trying to increase the batch size and see if it has an influence.

l2 norm of the gradient

I guess I got what is a problem with a gradient norm value. Basically negative gradient shows a direction to a local minimum value, but it doesn't say how far it is. For this reason you are able to configure you step proportion. When your weight combination is closer to the minimum value your constant step could be bigger than is necessary and some times it hits in wrong direction and in next epooch network try to solve this problem.

Momentum algorithm use modified approach. In terms of vectors this addition operation can increase magnitude of the vector and change it direction as well, so you are able to miss perfect step even more.

To fix this problem network sometimes needs a bigger vector, because minimum value a little further than in the previous epoch. To prove that theory I build small experiment.Jump right here to skip the introductions. Changelog: 27 Mar Added absolute to the terms in 2-norm and p-norm.

Euclidean distance

Thanks to Ricardo N Santos for pointing this out. O verfitting is a phenomenon that occurs when a machine learning or statistics model is tailored to a particular dataset and is unable to generalise to other datasets.

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This usually happens in complex models, like deep neural networks. Regularisation is a process of introducing additional information in order to prevent overfitting. The focus for this article is L1 and L2 regularisation. In this article, I will be sharing with you some intuitions why L1 and L2 work by explaining using gradient descent. This article shows how gradient descent can be used in a simple linear regression.

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L1 and L2 regularisation owes its name to L1 and L2 norm of a vector w respectively. A linear regression model that implements L1 norm for regularisation is called lasso regressionand one that implements squared L2 norm for regularisation is called ridge regression. To implement these two, note that the linear regression model stays the same:. Note: Strictly speaking, the last equation ridge regression is a loss function with squared L2 norm of the weights notice the absence of the square root.

Thank you Max Pechyonkin for highlighting this! In practice, simple linear regression models are not prone to overfitting. As mentioned in the introduction, deep learning models are more susceptible to such problems due to their model complexity. As such, do note that the expressions used in this article are easily extended to more complex models, not limited to linear regression.

Our objective is to minimise these different losses. Based on the above loss function, adding an L1 regularisation term to it looks like this:. We will need this later.

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Similarly, adding an L2 regularisation term to L looks like this:. Recall that updating the parameter w in gradient descent is as follows:. Here are some intuitions. Intuition A:. Intuition B:. Because this means our model is only meant for the dataset which we trained on. This means our model will produce predictions that are far off from the true value for other datasets.

So we settle for less than perfectwith the hope that our model can also get close predictions with other data. Intuition C:. Notice that H as defined here is dependent on the model w and b and the data x and y.

Updating the weights based only on the model and data in Equation 0 can lead to overfitting, which leads to poor generalisation.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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Asked 6 years, 3 months ago. Active 1 month ago. Viewed 81k times. Arctic Char 5, 5 5 gold badges 13 13 silver badges 35 35 bronze badges. Active Oldest Votes. Surb Surb One follow up question, does this still hold if x is complex? Hanno 3, 11 11 silver badges 36 36 bronze badges. Nick Alger Nick Alger Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

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Linked 4. Related 9. Hot Network Questions. Question feed.To pr e vent overfittingwe want to add a bias towards less complex functions. That is, given two functions that can fit our data reasonably well, we prefer the simpler one. We do this by adding a regularization term, typically either the L1 norm or the squared L2 norm:.

So, for example, by adding the squared L2 norm to the loss and minimizing, we obtain Ridge Regression:. Mathematically, we can see that both the L1 and L2 norms are measures of the magnitude of the weights: the sum of the absolute values in the case of the L1 norm, and the sum of squared values for the L2 norm.

So larger weights give a larger norm. What function should we pick to fit this data? There are many options, here are three examples:. Here we have a 2nd-degree polynomial fit and two different 8th-degree polynomials, given by the following equations:.

How is this complexity reflected in the norm? As we can see, line [c] has a mean squared error of 0, but its norms are quite high. Lines [a] and [b], instead, have a slightly higher MSE but their norms are much lower:.

From this we can conclude that by adding the L1 or L2 norm to our minimization objective, we can encourage simpler functions with lower weights, which will have a regularization effect and help our model to better generalize on new data. On the left we have a plot of the L1 and L2 norm for a given weight w.

On the right, we have the corresponding graph for the slope of the norms. As we can see, both L1 and L2 increase for increasing asbolute values of w. However, while the L1 norm increases at a constant rate, the L2 norm increases exponentially. We can see that with the L2 norm as w gets smaller so does the slope of the norm, meaning that the updates will also become smaller and smaller. On the other hand, with the L1 norm the slope is constant. Therefore, the L1 norm is much more likely to reduce some weights to 0.

Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. Make learning your daily ritual. Take a look. Get started. Open in app. Visualizing regularization and the L1 and L2 norms. Chiara Campagnola. Written by Chiara Campagnola. Get this newsletter. Review our Privacy Policy for more information about our privacy practices. Check your inbox Medium sent you an email at to complete your subscription.In mathematicsthe Euclidean distance between two points in Euclidean space is a number, the length of a line segment between the two points.

It can be calculated from the Cartesian coordinates of the points using the Pythagorean theoremand is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagorasbut Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 17th century.

The distance between two objects that are not points is usually defined to be the smallest distance between any two points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spacesand other distances than Euclidean have been studied. The square of the Euclidean distance is not a metric, but is convenient for many applications in statistics and optimization.

The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates. A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is: [1]. In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.

It is also possible to compute the distance for points given by polar coordinates. For pairs of objects that are not both points, the distance can most simply be defined as the smallest distance between any two points from the two objects, although more complicated generalizations from points to sets such as Hausdorff distance are also commonly used.

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In many applications, and in particular when comparing distances, it may be more convenient to omit the final square root in the calculation of Euclidean distances.

The value resulting from this omission is the square of the Euclidean distance, and is called the squared Euclidean distance. Beyond its application to distance comparison, squared Euclidean distance is of central importance in statisticswhere it is used in the method of least squaresa standard method of fitting statistical estimates to data by minimizing the average of the squared distances between observed and estimated values.

Squared Euclidean distance is not a metricas it does not satisfy the triangle inequality.

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The squared distance is thus preferred in optimization theorysince it allows convex analysis to be used. Since squaring is a monotonic function of non-negative values, minimizing squared distance is equivalent to minimizing the Euclidean distance, so the optimization problem is equivalent in terms of either, but easier to solve using squared distance. The collection of all squared distances between pairs of points from a finite set may be stored in a Euclidean distance matrix.

In more advanced areas of mathematics, Euclidean space and its distance provides a standard example of a metric spacecalled the Euclidean metric. Euclidean distance geometry studies properties of Euclidean geometry in terms of its distances, and properties of sets of distances that can be used to determine whether they come from the Euclidean metric. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin.

Other common distances on Euclidean spaces and low-dimensional vector spaces include: [17]. For points on surfaces in three dimensions, the Euclidean distance should be distinguished from the geodesic distancethe length of a shortest curve that belongs to the surface.


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