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Machine Learning FAQ

Without going into much depth regarding information gain and impurity measures, we can think of the decision as “is feature x_i >= some_val?” Intuitively, we can see that it really doesn’t matter on which scale this feature is (centimeters, Fahrenheit, a standardized scale – it really doesn’t matter).

Some examples of algorithms where feature scaling matters are: There are many more cases than I can possibly list here … I always recommend you to think about the algorithm and what it’s doing, and then it typically becomes obvious whether we want to scale your features or not.

Machine Learning FAQ

Without going into much depth regarding information gain and impurity measures, we can think of the decision as “is feature x_i >= some_val?” Intuitively, we can see that it really doesn’t matter on which scale this feature is (centimeters, Fahrenheit, a standardized scale – it really doesn’t matter).

Some examples of algorithms where feature scaling matters are: There are many more cases than I can possibly list here … I always recommend you to think about the algorithm and what it’s doing, and then it typically becomes obvious whether we want to scale your features or not.

About Feature Scaling and Normalization

The result of standardization (or Z-score normalization) is that the features will be rescaled so that they’ll have the properties of a standard normal distribution with \mu = 0 and \sigma = 1 where \mu is the mean (average) and \sigma is the standard deviation from the mean;

standard scores (also called z scores) of the samples are calculated as follows: Standardizing the features so that they are centered around 0 with a standard deviation of 1 is not only important if we are comparing measurements that have different units, but it is also a general requirement for many machine learning algorithms.

with features being on different scales, certain weights may update faster than others since the feature values x_j play a role in the weight updates so that w_j := w_j + \Delta w_j, where

Without going into much depth regarding information gain and impurity measures, we can think of the decision as “is feature x_i >= some_val?” Intuitively, we can see that it really doesn’t matter on which scale this feature is (centimeters, Fahrenheit, a standardized scale – it really doesn’t matter).

For example, if we initialize the weights of a small multi-layer perceptron with tanh activation units to 0 or small random values centered around zero, we want to update the model weights “equally.” As a rule of thumb I’d say: When in doubt, just standardize the data, it shouldn’t hurt.

Another prominent example is the Principal Component Analysis, where we usually prefer standardization over Min-Max scaling, since we are interested in the components that maximize the variance (depending on the question and if the PCA computes the components via the correlation matrix instead of the covariance matrix;

In the following section, we will go through the following steps: In this step, we will randomly divide the wine dataset into a training dataset and a test dataset where the training dataset will contain 70% of the samples and the test dataset will contain 30%, respectively.

Let us think about whether it matters or not if the variables are centered for applications such as Principal Component Analysis (PCA) if the PCA is calculated from the covariance matrix (i.e., the k principal components are the eigenvectors of the covariance matrix that correspond to the k largest eigenvalues.

Let’s assume we have the 2 variables \bf{x} and \bf{y} Then the covariance between the attributes is calculated as Let us write the centered variables as The centered covariance would then be calculated as follows: But since after centering, \bar{x}' = 0 and \bar{y}' = 0 we have \sigma_{xy}' = \frac{1}{n-1} \sum_{i}^{n} x_i' y_i' which is our original covariance matrix if we resubstitute back the terms x'

Let c be the scaling factor for \bf{x} Given that the “original” covariance is calculated as the covariance after scaling would be calculated as: \sigma_{xy}' = \frac{1}{n-1} \sum_{i}^{n} (c \cdot x_i - c \cdot \bar{x})(y_i - \bar{y}) =

\sigma_{xy}' = c \cdot \sigma_{xy} Therefore, the covariance after scaling one attribute by the constant c will result in a rescaled covariance c \sigma_{xy} So if we’d scaled \bf{x} from pounds to kilograms, the covariance between \bf{x} and \bf{y} will be 0.453592 times smaller.

In the section on linear classification we computed scores for different visual categories given the image using the formula \( s = W x \), where \(W\) was a matrix and \(x\) was an input column vector containing all pixel data of the image.

In the case of CIFAR-10, \(x\) is a [3072x1] column vector, and \(W\) is a [10x3072] matrix, so that the output scores is a vector of 10 class scores.

There are several choices we could make for the non-linearity (which we’ll study below), but this one is a common choice and simply thresholds all activations that are below zero to zero.

Notice that the non-linearity is critical computationally - if we left it out, the two matrices could be collapsed to a single matrix, and therefore the predicted class scores would again be a linear function of the input.

three-layer neural network could analogously look like \( s = W_3 \max(0, W_2 \max(0, W_1 x)) \), where all of \(W_3, W_2, W_1\) are parameters to be learned.

The area of Neural Networks has originally been primarily inspired by the goal of modeling biological neural systems, but has since diverged and become a matter of engineering and achieving good results in Machine Learning tasks.

Approximately 86 billion neurons can be found in the human nervous system and they are connected with approximately 10^14 - 10^15 synapses.

The idea is that the synaptic strengths (the weights \(w\)) are learnable and control the strength of influence (and its direction: excitory (positive weight) or inhibitory (negative weight)) of one neuron on another.

Based on this rate code interpretation, we model the firing rate of the neuron with an activation function \(f\), which represents the frequency of the spikes along the axon.

Historically, a common choice of activation function is the sigmoid function \(\sigma\), since it takes a real-valued input (the signal strength after the sum) and squashes it to range between 0 and 1.

An example code for forward-propagating a single neuron might look as follows: In other words, each neuron performs a dot product with the input and its weights, adds the bias and applies the non-linearity (or activation function), in this case the sigmoid \(\sigma(x) = 1/(1+e^{-x})\).

As we saw with linear classifiers, a neuron has the capacity to “like” (activation near one) or “dislike” (activation near zero) certain linear regions of its input space.

With this interpretation, we can formulate the cross-entropy loss as we have seen in the Linear Classification section, and optimizing it would lead to a binary Softmax classifier (also known as logistic regression).

The regularization loss in both SVM/Softmax cases could in this biological view be interpreted as gradual forgetting, since it would have the effect of driving all synaptic weights \(w\) towards zero after every parameter update.

The sigmoid non-linearity has the mathematical form \(\sigma(x) = 1 / (1 + e^{-x})\) and is shown in the image above on the left.

The sigmoid function has seen frequent use historically since it has a nice interpretation as the firing rate of a neuron: from not firing at all (0) to fully-saturated firing at an assumed maximum frequency (1).

Also note that the tanh neuron is simply a scaled sigmoid neuron, in particular the following holds: \( \tanh(x) = 2 \sigma(2x) -1 \).

Other types of units have been proposed that do not have the functional form \(f(w^Tx + b)\) where a non-linearity is applied on the dot product between the weights and the data.

TLDR: “What neuron type should I use?” Use the ReLU non-linearity, be careful with your learning rates and possibly monitor the fraction of “dead” units in a network.

For regular neural networks, the most common layer type is the fully-connected layer in which neurons between two adjacent layers are fully pairwise connected, but neurons within a single layer share no connections.

Working with the two example networks in the above picture: To give you some context, modern Convolutional Networks contain on orders of 100 million parameters and are usually made up of approximately 10-20 layers (hence deep learning).

The full forward pass of this 3-layer neural network is then simply three matrix multiplications, interwoven with the application of the activation function: In the above code, W1,W2,W3,b1,b2,b3 are the learnable parameters of the network.

Notice also that instead of having a single input column vector, the variable x could hold an entire batch of training data (where each input example would be a column of x) and then all examples would be efficiently evaluated in parallel.

Neural Networks work well in practice because they compactly express nice, smooth functions that fit well with the statistical properties of data we encounter in practice, and are also easy to learn using our optimization algorithms (e.g.

Similarly, the fact that deeper networks (with multiple hidden layers) can work better than a single-hidden-layer networks is an empirical observation, despite the fact that their representational power is equal.

As an aside, in practice it is often the case that 3-layer neural networks will outperform 2-layer nets, but going even deeper (4,5,6-layer) rarely helps much more.

We could train three separate neural networks, each with one hidden layer of some size and obtain the following classifiers: In the diagram above, we can see that Neural Networks with more neurons can express more complicated functions.

For example, the model with 20 hidden neurons fits all the training data but at the cost of segmenting the space into many disjoint red and green decision regions.

The subtle reason behind this is that smaller networks are harder to train with local methods such as Gradient Descent: It’s clear that their loss functions have relatively few local minima, but it turns out that many of these minima are easier to converge to, and that they are bad (i.e.

Conversely, bigger neural networks contain significantly more local minima, but these minima turn out to be much better in terms of their actual loss.

In practice, what you find is that if you train a small network the final loss can display a good amount of variance - in some cases you get lucky and converge to a good place but in some cases you get trapped in one of the bad minima.

When should I apply feature scaling for my data

In my view the question about scaling/not scaling the features in machine learning is a statement about the measurement units of your features.

If the height is measured in nanometers then it's likely that any k nearest neighbors will merely have similar measures of height.

This again will have an influence on knn and might drastically reduce performance if your data had more noisy constant values compared to the ones that vary.

So this is like with everything else in machine learning - use prior knowledge whenever possible and in the case of black-box features do both and cross-validate.

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