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Thus, logistic regression is useful if we are working with a dataset where the classes are more or less “linearly separable.” For “relatively” very small dataset sizes, I’d recommend comparing the performance of a discriminative Logistic Regression model to a related Naive Bayes classifier (a generative model) or SVMs, which may be less susceptible to noise and outlier points.

the March Madness prediction contest this year was one by 2 professors using a logistic regression model Professors Lopez and Matthews didn’t use any of the au courant methods in data science circles, either: no deep learning, no hierarchical clustering, no compressed sensing;

In softmax, the probability of a particular sample with net input z belongs to the i th class can be computed with a normalization term in the denominator that is the sum of all M linear functions:

Although, I mentioned that neural networks (multi-layer perceptrons to be specific) may use logistic activation functions, the hyperbolic tangent (tanh) often tends to work better in practice, since it’s not limited to only positive outputs in the hidden layer(s).

Looking only at a single weight / model coefficient, we can picture the cost function in a multi-layer perceptron as a rugged landscape with multiple local minima that can trap the optimization algorithm:

Machine Learning FAQ

Thus, logistic regression is useful if we are working with a dataset where the classes are more or less “linearly separable.” For “relatively” very small dataset sizes, I’d recommend comparing the performance of a discriminative Logistic Regression model to a related Naive Bayes classifier (a generative model) or SVMs, which may be less susceptible to noise and outlier points.

the March Madness prediction contest this year was one by 2 professors using a logistic regression model Professors Lopez and Matthews didn’t use any of the au courant methods in data science circles, either: no deep learning, no hierarchical clustering, no compressed sensing;

In softmax, the probability of a particular sample with net input z belongs to the i th class can be computed with a normalization term in the denominator that is the sum of all M linear functions:

Although, I mentioned that neural networks (multi-layer perceptrons to be specific) may use logistic activation functions, the hyperbolic tangent (tanh) often tends to work better in practice, since it’s not limited to only positive outputs in the hidden layer(s).

Looking only at a single weight / model coefficient, we can picture the cost function in a multi-layer perceptron as a rugged landscape with multiple local minima that can trap the optimization algorithm:

Logistic Regression

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Applied Deep Learning - Part 1: Artificial Neural Networks

What separates this tutorial from the rest you can find online is that we’ll take a hands-on approach with plenty of code examples and visualization.

The code for this article is available here as a Jupyter notebook, feel free to download and try it out yourself.

You don’t need to have prior knowledge of deep learning, only some basic familiarity with general machine learning.

So let’s begin… Artificial Neural Networks (ANN) are multi-layer fully-connected neural nets that look like the figure below.

given node takes the weighted sum of its inputs, and passes it through a non-linear activation function.

The signal flows from left to right, and the final output is calculated by performing this procedure for all the nodes.

So far we have described the forward pass, meaning given an input and weights how the output is computed.

But we first need to train our model to actually learn the weights, and the training procedure works as follows: Backpropagation with gradient descent is literally the “magic” behind the deep learning models.

It’s a rather long topic and involves some calculus, so we won’t go into the specifics in this applied deep learning series.

The difference between the ANN and perceptron is that ANN uses a non-linear activation function such as sigmoid but the perceptron uses the step function.

So if we visualize this as a sequence of vector transformations, we first map the 3D input to a 4D vector space, then we perform another transformation to a new 4D space, and the final transformation reduces it to 1D.

The forward pass performs these matrix dot products and applies the activation function element-wise to the result.

The connection between a layer with 3 nodes and 4 nodes is a matrix multiplication using a 3x4 matrix.

To make a prediction using the ANN on a given input, we only need to know these weights and the activation function (and the biases), nothing more.

fully connected layer between 3 nodes and 4 nodes is just a matrix multiplication of the 1x3 input vector (yellow nodes) with the 3x4 weight matrix W1.

We then multiply this 1x4 vector with a 4x4 matrix W2, resulting in a 1x4 vector, the green nodes.

In reality after every matrix multiplication, we apply the activation function to each element of the resulting matrix.

If we take a classification problem as an example, we want to separate out the classes by drawing a decision boundary.

By performing non-linear transformations at each layer, we are able to project the input to a new vector space, and draw a complex decision boundary to separate the classes.

So we project it to a higher dimensional space by performing a non-linear transformation, and then it becomes linearly separable.

There has been an incredible surge on their popularity recently due to a couple of reasons: clever tricks which made training these models possible, huge increase in computational power especially GPUs and distributed training, and vast amount of training data.

This was a brief introduction, there are tons of great tutorials online which cover deep neural nets.

On a 2 dimensional (2D) data LR will try to draw a straight line to separate the classes, that’s where the term linear model comes from.

If you have a supervised binary classification problem, given an input data with multiple columns and a binary 0/1 outcome, LR is the first method to try.

In this section we will focus on 2D data since it’s easier to visualize, and in another tutorial we will focus on multidimensional input.

We are using the scikit-learn make_classification method to generate our data and using a helper function to visualize it.

There is a LogisticRegression classifier available in scikit-learn, I won’t go into too much detail here since our goal is to learn building models with Keras.

We will now train the same logistic regression model with Keras to predict the class membership of every input point.

To keep things simple for now, we won’t perform the standard practices of separating out the data to training and test sets, or performing k-fold cross-validation.

Since we’re now building a simple logistic regression model, we will have the input nodes directly connected to output node, without any hidden layers.

In neural networks literature, it’s common to talk about input nodes and output nodes.

In our case we have 2 features, the x and y coordinates of the points we plotted above, so we have 2 input nodes.

The output of the logistic regression model is a single number, the probability of an input data point belonging to class 1.

So you can simply think of the output node as a vector with a single number (or simply a scalar) between 0 and 1.

In our current model, we don’t have any hidden layers, the input nodes are directly connected to the output node.

The Dense function in Keras constructs a fully connected neural network layer, automatically initializing the weights as biases.

The arguments for the compile function are defined as follows: Now comes the fun part of actually training the model using the fit function.

The various shades of blue and red represent the probability of a hypothetical point in that area belonging to class 1 or 0.

The numbers on the diagonal axis represent the number of correctly classified points, the rest are the misclassified ones.

We can see one of the misclassified points at the top right part of the confusion matrix, the true value is class 0 but the predicted value is class 1.

You can imagine a curved decision boundary that will separate out the classes, and a complex model should be able to approximate that.

No matter where the model draws the line, it will misclassify half of the points, due to the nature of the dataset.

High numbers along the diagonals meaning that the classifier was right, and low numbers everywhere else where the classifier was wrong.

In our visualization, the color blue represents large numbers and yellow represents the smaller ones.

While building Keras models for logistic regression above, we performed the following steps: While building a deep neural network, we only need to change step 2 such that, we will add several Dense layers one after another.

Choosing the right number of hidden layers and nodes per layer is more of an art than science, usually decided by trial and error.

Here’s how our dataset looks like, spiral data with 3 classes, using the make_multiclass method in scikit-learn.

There are a couple of differences, let’s go over them one by one: It took some time to talk about all the differences between LR and SR, and it looks like there’s a lot to digest.

The lines look jagged due to floating point rounding but in reality they’re straight.

These won’t change going from a linear model to a deep ANN, since the problem definition hasn’t changed.

There was a common theme in this article such that we first introduced the task, we then approached it using a simple method and observed the limitations.

Comparing Results Delivered by Logistic Regression and a Neural Network

And even if the results are better, how much does that theoretical gain translate into actual gain for a real world problem?In this code story, we run Emotion Analysis on a set of informal short English messages (see here), and compare empirically the performance achieved by logistic regression against a fully connected neural network.

a comprehensive comparison would provide deeper insight, and would require a great deal more effort.The dataThe dataset (see details here) used in this experiment consists of 784,349 sample informal short English messages (a collection of English Tweets), with 5 emotion classes: angry, sad, fear, happy, excited, where 60% is used for training, 20% for validation and 20% for testing.

Here, recall refers to true positive rate, frac{text{true positive}}{true positive+false negative} , while precision refers to frac{text{true positive}}{true positive+false positive} .As shown in the table above, logistic regression outperforms neural network by approximately 2% on overall accuracy in this experiment setting (see The Details section), while the computation time is approximately 10 times faster.The table below shows the confusion matrix for each emotion classes, when using logistic regression and neural network of 1 hidden layer with 125 neurons, respectively.

The logistic regression recall rate for emotion classes sad and excited outperforms that of neural network by approximately 5%, whereas the neural network recall rate for emotion classes angry, fear and happy outperform that of logistic regression, by approximately 2%, 5% and 1% respectively.

Confusion matrix for emotion classes when using logistic regression.Figure: Confusion matrix for emotion classes when using neural network.In this case, while neither algorithm handles all five emotion classes accurately, the recall rate for fear is below satisfactory, and logistic regression does outperform neural network by 2% overall.

See here.The Details:Toolkit:Microsoft internal machine learning toolkit.Experiments settings:Learner TypeMulti-Class Logistic RegressionLinear combination of L1 and L2 regularizationsCoefficients of 1, 1Optimization Tolerance1E-07Memory Size (L-BFGS)20Feature NormalizationMin-max normalizerInitial Weights ScaleRandomMax Number of Iterations1000Learner TypeNeural NetworkNumber of output nodes5Loss functionCross entropyHidden layerSee table belowNumber of nodes for each hidden layerSee table belowMaximum number of training iterations1100Optimization AlgorithmStochastic gradient descentLearning rate0.001Early stopping ruleLoss in generality (stops when the score degraded 5 times in a row)Pre-trainingTrue for 2 or more hidden layersPre-trainer typeGreedyPre-training epoch25Full results:Full results can be found at https://github.com/ryubidragonfire/Emotion Emotion Detection and Recognition from text is a recent field of research that is closely related to Sentiment Analysis.

The logistic regression recall rate for emotion classes sad and excited outperforms that of neural network by approximately 5%, whereas the neural network recall rate for emotion classes angry, fear and happy outperform that of logistic regression, by approximately 2%, 5% and 1% respectively.

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