# AI News, Deep Learning: Regularization Notes

- On Wednesday, December 13, 2017
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## Deep Learning: Regularization Notes

X, y) ( ` means transpose) The L² regularization has the intuitive interpretation of heavily penalizing peaky weight vectors and preferring diffuse weight vectors.

Lastly, also notice that during gradient descent parameter update, using the L² regularization ultimately means that every weight is decayed linearly: W += -lambda * W towards zero.

Let’s see what does this means, We can see that the addition of the weight decay term has modiﬁed the learning rule to multiplicatively shrink the weight vector by a constant factor on each step, just before performing the usual gradient update.

The L² regularization causes the learning algorithm to “perceive” the input X as having higher variance, which makes it shrink the weights on features whose covariance with the output target is low compared to this added variance.

In other words, neurons with L¹ regularization end up using only a sparse subset of their most important inputs as most weight goes very close to zero and become nearly invariant to the “noisy” inputs.

- On Wednesday, December 13, 2017
- By Read More

## Deep Learning: Regularization Notes

X, y) ( ` means transpose) The L² regularization has the intuitive interpretation of heavily penalizing peaky weight vectors and preferring diffuse weight vectors.

Lastly, also notice that during gradient descent parameter update, using the L² regularization ultimately means that every weight is decayed linearly: W += -lambda * W towards zero.

Let’s see what does this means, We can see that the addition of the weight decay term has modiﬁed the learning rule to multiplicatively shrink the weight vector by a constant factor on each step, just before performing the usual gradient update.

The L² regularization causes the learning algorithm to “perceive” the input X as having higher variance, which makes it shrink the weights on features whose covariance with the output target is low compared to this added variance.

In other words, neurons with L¹ regularization end up using only a sparse subset of their most important inputs as most weight goes very close to zero and become nearly invariant to the “noisy” inputs.

- On Wednesday, December 13, 2017
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## Regularization (mathematics)

The green and blue functions both incur zero loss on the given data points.

A learned model can be induced to prefer the green function, which may generalize better to more points drawn from the underlying unknown distribution, by adjusting ,

In mathematics, statistics, and computer science, particularly in the fields of machine learning and inverse problems, regularization is a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting[citation needed].

Regularizers for multitask learning 7.1 Sparse regularizer on columns 7.2 Nuclear norm regularization 7.3 Mean-constrained regularization 7.4 Clustered mean-constrained regularization 7.5 Graph-based similarity 8

Empirical learning of classifiers (learning from a finite data set) is always an underdetermined problem, because in general we are trying to infer a function of any given only some examples .

regularization term (or regularizer) is added to a loss function: where is an underlying loss function that describes the cost of predicting when the label is ,

A simple form of regularization applied to integral equations, generally termed Tikhonov regularization after Andrey Nikolayevich Tikhonov, is essentially a trade-off between fitting the data and reducing a norm of the solution.

The goal of this learning problem is to find a function that fits or predicts the outcome (label) that minimizes the expected error over all possible inputs and labels.

Therefore, the expected error is unmeasurable, and the best surrogate available is the empirical error over the available samples: Without bounds on the complexity of the function space (formally, the reproducing kernel Hilbert space) available, a model will be learned that incurs zero loss on the surrogate empirical error.

This is one of the most common forms of regularization, is also known as ridge regression, and is expressed as: In the case of a general function, we take the norm of the function in its reproducing kernel Hilbert space: As the norm is differentiable, learning problems using Tikhonov regularization can be solved by gradient descent.

This is the first-order condition for this optimization problem By construction of the optimization problem, other values of would give larger values for the loss function.

The algorithm above is equivalent to restricting the number of gradient descent iterations for the empirical risk with the gradient descent update: The base case is trivial.

An example is developing a simple predictive test for a disease in order to minimize the cost of performing medical tests while maximizing predictive power.

This can be problematic for certain applications, and is overcome by combining with regularization in elastic net regularization, which takes the following form: Elastic net regularization tends to have a grouping effect, where correlated input features are assigned equal weights.

Proximal methods[edit] Main article: Proximal gradient method While the norm does not result in an NP-hard problem, it should be noted that the norm is convex but is not strictly diffentiable due to the kink at x = 0.

For a problem such that is convex, continuous, differentiable, with Lipschitz continuous gradient (such as the least squares loss function), and is convex, continuous, and proper, then the proximal method to solve the problem is as follows.

First define the proximal operator and then iterate The proximal method iteratively performs gradient descent and then projects the result back into the space permitted by .

In the case of a linear model with non-overlapping known groups, a regularizer can be defined: where This can be viewed as inducing a regularizer over the norm over members of each group followed by an norm over groups.

This can be solved by the proximal method, where the proximal operator is a block-wise soft-thresholding function: Group sparsity with overlaps[edit] The algorithm described for group sparsity without overlaps can be applied to the case where groups do overlap, in certain situations.

The proximal operator cannot be computed in closed form, but can be effectively solved iteratively, inducing an inner iteration within the proximal method iteration.

Other uses of regularization in statistics and machine learning[edit] Bayesian learning methods make use of a prior probability that (usually) gives lower probability to more complex models.

Examples of applications of different methods of regularization to the linear model are: Model Fit measure Entropy measure[1][3] AIC/BIC Ridge regression[4] Lasso[5] Basis pursuit denoising Rudin–Osher–Fatemi model (TV) Potts model RLAD[6] Dantzig Selector[7] SLOPE[8] See also[edit] Bayesian interpretation of regularization Bias-variance tradeoff Regularization by spectral filtering Matrix regularization Notes[edit] ^

- On Wednesday, December 13, 2017
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## Multiclass Logistic Regression

Module Overview You can use the Multiclass Logistic Regression module to create a logistic regression model that can be used to predict multiple values.

Classification using logistic regression is a supervised learning method, and therefore requires a labeled dataset.