REGRESSION LOSSES For example, this equation models how MSE would be implemented for unsupervised learning, where h(x) is the hypothesis function. Otherwise if the error is sufficiently large MAE is applied. So when we think higher weightage should not be given to outliers, then set your loss function as Huber loss. In our work, we investigate the impact of different loss function layers for image processing. given a prediction y and a target value t of 1. Categorical Cross Entropy and Sparse Categorical Cross Entropy are versions of Binary Cross Entropy, adapted for several classes. Although loss functions can be applied even in unsupervised settings. Below is the table of content for this post. Classification loss is the case where the aim is to predict the output from the different categorical values for example, if we have a dataset of handwritten images and the digit is to be predicted that lies between (09), in these kinds of scenarios classification loss is used. Implementing Loss Functions in Python. We need to build a predictor that, for a particular student, predicts the probability of studying a course based on attendance, classroom performance, homework, learning skill, and so on. . Let us see how Squared Hinge can be used with Keras. In the first one, Loss=0.53 is calculated utilizing the mean square deviation function, In the second image, the straight line is translated upwards, and the error calculation is Loss=0.16, which is much smaller than the error in Figure 1., In the third picture, it has been translated upward again. Root Mean . In simple terms, the above radically says is: for loss values less than () delta, use the MSE; for loss values greater than delta, use the MAE. If the mean square deviation function is connected, the derivation result is complex and the amount of computation is relatively large. Examples for a 3-class classification: But if your Yis are integers, use sparse_categorical_crossentropy. The center position of the circle is the position where the loss value is 0, which can also be the goal we need to approach. For this problem, each of the input variables and the target variable have a Gaussian distribution; therefore, standardizing the data in this case is desirable. Especially, when youre not interested in knowing how certain the classifier is about the classification. The basic principle of loss functions still holds; the goal of a loss function in unsupervised learning is to determine the difference between the input example and the hypothesis the models approximation of the input example itself. . Loss function in LSTM neural network, 1 Answer. When the prediction output is closer to 0, the larger the value of the loss function, the worse the training outcome. The concept of cross-entropy traces back into the field of Information Theory where Claude Shannon introduced the concept of entropy in 1948. increase epoch number. The following example is for a supervised setting i.e. When we repair w=2 and alter the b value from 2 to 4, take a look at the change within the value of the loss function as shown in Figure 1-5. In simple terms, the above radically says is: In the above figure, you can see how the derivative is a constant for, As we already know Huber loss has both MAE and MSE. The code is as follows, The above code calculates a loss value for every combination of w and B. A Medium publication sharing concepts, ideas and codes. The cross-entropy function is commonly used in logical regression, or say classification. Figure 1-6 The value of the loss caused by the change of W when fixing b. The specific loss function symbol is denoted by J, and the error value is denoted by loss. >[y=1.0, yhat=0.8] cross entropy: 0.223 The outcome is either 1 (there is a cat) or 0 (there is no cat). Math and theoretical explanation for log loss here. In this context, proper training of a neural network is the most important aspect of making a reliable model. In deep learning frameworks such as TensorFlow or Pytorch, you may come across the option to choose sparse categorical cross-entropy when training a neural network. The last layer of the network for classification problems requires a classification function, Sigmoid or Softmax. It represents the difference between the original and predicted values extracted by squared the average difference over the data set. This should be compared with Mean Absolute Error, where the optimal prediction is the median. Its pretty simple, It just involves specifying it as the used loss function during the model compilation step: Feel free to connect me on LinkedIn for any query. Because of this, a loss function called a Huber Loss was developed, which has the advantages of both MSE and MAE. The meanings are similar. A loss function value is calculated for each combination of w and a b, and the height of the three-dimensional graph is used to represent the loss function value. The value of the loss function formed by the combination of the 2 variables corresponds to the only coordinate point on the contour line in the figure. Table 1-1 Comparison of the absolute value loss function and mean square deviation loss function. This simplicity with the log loss is possible because the derivative of sigmoid make it possible, in my . https://towardsdatascience.com/understanding-the-3-most-common-loss-functions-for-machine-learning-regression-23e0ef3e14d3 Entropy has roots in physicsit is a measure of disorder, or unpredictability, in a system. It is most common type of loss function used for classification problem. When the label value is either non-0 or 1, its more classified. In supervised learning, there are two main types of loss functions these correlate to the 2 major types of neural networks: regression and classification loss functions, One of the most popular loss functions, MSE finds the average of the squared differences between the target and the predicted outputs. Consider an example where we have a dataset of 100 values we would like our model to be trained to predict. to deal with the above problem I have done the following thing. Skip to content. The whole network is then trained in an iterative way, i.e. Classification Models: predict the output from a set of finite categorical values. In Figure 1-2, the abscissa is one variable (w) and the ordinate is another variable (b). Comparison of Blue and green distribution. Use tab to navigate through the menu items. A loss function is what helps you "train" your neural network to do what you want it to do. If you're learning about NN from popular online courses and books, then you'll be told that classification and regression are two common kinds of problems where NN are applied. In machine learning, we have to evaluate the gap between labels and predictions and use the KL divergence just right, thats(D_{KL}(y||a)). activation{'identity', 'logistic', 'tanh . Architecture of a traditional RNN Recurrent neural networks, also known as RNNs, are a class of neural networks that allow previous outputs to be used as inputs while having hidden states. In looking at the majority of loss functions for neural networks, such as mse, mae, L1, L2, the loss is always recorded as a positive value. Lets say the first image contains a dog. . https://towardsdatascience.com/cross-entropy-loss-function-f38c4ec8643e Understanding the difference between types of classification informs the choice of loss function for a neural network model and how machine learning problems are framed. This loss essentially tells you something about the performance of the network: the higher it is, the worse your network performs overall. It's pretty simple, It just involves specifying it as the used loss function during the model compilation step: The squared hinge loss is a loss function used for maximum margin binary classification problems. However, the main appeal of this loss function is for comparing two probability distributions. This equation represents how a neural network processes the input data at each layer and eventually produces a predicted output value. within the image (in image classification, the image usually has only one subject). Losses. Assume that we have a neural network that learns to classify pictures into three classes: Whether they contain a rabbit, a cat, or a dog. >[y=1.0, yhat=0.6] cross entropy: 0.511 The categorical cross-entropy is applied in multiclass classification scenarios. In binary classification, there are only two possible actual values of y 0 or 1. Each training input is loaded into the neural network in a process called forward propagation. Use Mean absolute error when you are doing regression and dont want outliers to play a big role. machine-learning deep-learning neural-network pytorch classification loss-functions multiclass-classification retinanet implementation-of-research-paper pytorch-implementation imbalanced-classes. Assuming the end of the period. MAE is not sensitive towards outliers and is given several examples with the same input feature values, and the optimal prediction will be their median target value. As seen earlier, when writing neural networks, you can import loss functions as function objects from the tf.keras.losses module. MSE is thus good to use if you believe that your target data, conditioned on the input, is normally distributed around a mean value, and when its important to penalize outliers extra much. . They are treated as a crucial part of neural networks' design. And the method to calculate the loss is called Loss. Loss functions are one of the most important aspects of neural networks, as they (along with the optimization functions) are directly responsible for fitting the model to the given training data. Figure 1-4 The relationship between the value of the loss function and the position of the straight line. >[y=0.0, yhat=0.2] cross entropy: 0.223 https://machinelearningmastery.com/cross-entropy-for-machine-learning/ Cross entropy can be utilized as a loss function in neural networks. Knowing which loss function to use for different types of classification problems is an important skill for every data scientist. Figure 1-9 Use stupid methods to draw contour plots. The CE function, also known as the entropic loss, measures the difference between two distributions, namely the distribution of the target outputs of the ANN and the distribution of the actual outputs of the observations in the dataset. What Are the KPIs for Employee Appraisal? It means, that the smallest possible value is zero (distributions are equal) and the maximum value is infinity. In the above figure, you can see how the derivative is a constant for abs(a)>delta. Examples for above 3-class classification problem: Calculate Cross-Entropy Between Class Labels and Probabilities, '>[y=%.1f, yhat=%.1f] cross entropy: %.3f', Relative Entropy(KullbackLeibler divergence), Also, with a little bit of work, we can show that the KL-Divergence is, The closer two distributions get to each other, the lower the loss becomes. The usage entirely depends on how you load your dataset. The cross-entropy loss function can measure the similarity between (p) and (q). >[y=1.0, yhat=0.9] cross entropy: 0.105 The softmax activation rescales the model output so that it has the right properties. Since the previous part of the KL divergence(H(y)) remains unchanged, we only want to concentrate on cross-entropy in the optimization process. Another example of the use of loss functions in image generation was shown above in our Custom Loss Functions section, in the case of a variational auto-encoder (VAE) model. If the actual price of the house is $2.89 and the model predicts $3.07, you can calculate the error. Use in combination with the tanh() the activation function in the last layer of the neural network. The binary cross-entropy needs to compute the logarithms of i and (1-i), which only exist if i is between 0 and 1. Namely, when you dont care about the classification probabilities. Join Medium through my referral link: https://andre-ye.medium.com/membership. Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. A loss function is a measure of how well a neural network can predict the expected output of a given input. Top 10 Data Mining Applications in Real World, Linear Regression (Python Implementation). In Figure 1-4, the first three pictures show a process of steadily finding the best-fitting straight line. ML enthusiast. Relative Entropy(a.k.a KullbackLeibler divergence). The function is designed so that either the first or the second term equals zero. If the actual outcome is 0, the model should produce a probability estimate that is as close as possible to 0. Cross-Entropy is a vital concept in the Shannon information theory, which is especially used to measure the difference ininformation between two probability distributions. I am trying to predict a probability with a neural network, but having trouble figuring out which loss function is best. Use the Squared Hinge loss function on problems involving yes/no (binary) decisions. A loss function is a measure of how well a neural network can predict the expected output of a given input. Loss functions are an essential part in training a neural network selecting the right loss function helps the neural network know how far off it is, so it can properly utilize its optimizer. It is just the squared difference between the expected value and the predicted value. Thus, to accurately determine loss between the actual and predicted values, it needs to compare the actual value (0 or 1) with the probability that the input aligns with that category (p(i) = probability that the category is 1; 1 p(i) = probability that the category is 0). In general, we use Classification Loss Functions in Neural Networks. Stop training till the loss value reaches an acceptable level, such as the position of (x5). Neural networks generally perform better when the real-valued input and output variables are to be scaled to a sensible range. Techniques for incorporating a black-box function into a neural network are described. Once the model has produced an output, this predicted output is compared against the given target output in a process called backpropagation the hyperparameters of the model are then adjusted so that it now outputs a result closer to the target output. In this video, we explain the concept of loss in an artificial neural network and show how to specify the loss function in code with Keras. VIDEO SECTION. (n) For all possibilities of the event. Basically, the LSTM is assigning labels to words (or characters, depending on your model), and optimizing the model by penalizing WoW, a certain classmate failed! Strictly speaking, the output of the model only needs to be positive so that the logarithm of every output value i exists. For instance, consider below figure two gases in a box: initially, the system has low entropy, in that the two gasses are completely separable(skewed distribution); after some time, however, the gases blend(distribution where events have equal probability) so the systems entropy increases. 1.1,2,1.7]) pred = np.array([1,1.7,1.5]) mean_absolute_error(act, pred) Output : 0.20000000000000004. There are many functions out there to find the loss based on the predicted and actual value depending on the problem. Primarily, it is used in Variational Autoencoders. However, frameworks like Tensorflow take care of all of this for you. To understand how the gradients are calculated and used to update the weights, refer to my post on backpropagation with gradient descent. Loss Functions for Image Restoration With Neural Networks Abstract: Neural networks are becoming central in several areas of computer vision and image processing and different architectures have been proposed to solve specific problems. Therefore, after adjusting the weight based on the error of a certain sample, we often calculate the value of the loss function of the overall sample to find out whether or not the network has been trained to an acceptable state. In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X. Formally, it is designed to quantify the difference between two probability distributions. For example, an image editing function may be the black-box function and may be wrapped into a layer of the neural network. Image generation is a process by which neural networks create images (from an existing library) per the users specifications. This function describes what objective you'd like your neural network to fit to (or to be good at). There is a particular kind of problem, thats, there are only two sorts of events that may happen, similar to learned and not learned, which are called (0/1) distribution or two categories. The formula for the loss is fairly straightforward. Mean Squared Error, Mean Absolute Error, Classification Loss Functions used in classification neural networks; given an input, the neural network produces a vector of probabilities of the input belonging to various pre-set categories can then select the category with the highest probability of belonging; Ex. The Relative entropy (also called KullbackLeibler divergence), is a method for measuring the similarity between two probability distributions. When training, we aim to minimize this loss between the predicted and target outputs. For p(x)probability distribution and a random variable X, entropy is defined as follows: Reason for the Negative sign: log(p(x))<0 for all p(x) in (0,1) . ybinary indicator (0 or 1) if class label c is the correct classification for observation o, ppredicted probability observation o is of class . Softmax is the only activation function recommended to use with the categorical cross-entropy loss function. A Blog on Building Machine Learning Solutions, An Introduction to Neural Network Loss Functions, Learning Resources: Math For Data Science and Machine Learning, post on backpropagation with gradient descent. where x indicates the difference between the predicted value and the target value. The loss function is how you're penalizing your output. Home; Blog; The loss function in a neural network quantifies the difference between the expected outcome and the outcome produced by the machine learning model. Suppose that for the course of learning the principles of neural networks, we have three possible situations, as proven in Table 1-2. Specifically, we will look at how loss functions are used to process image data in various use cases. >[y=0.0, yhat=0.4] cross entropy: 0.511 Example: When doing image reconstruction, MAE encourages less blurry images compared to MSE. In comparison, the quantity of information about excellent events is much smaller. Perfectly opposite vectors have a cosine similarity of -1, perfectly orthogonal vectors have a cosine similarity of 0, and identical vectors have a cosine similarity of 1. When we fix b=3 and change the value of w from 1 to 3, look at the change within the value of the loss function as proven in Figure 3-6. In this article, we. The graph above shows the range of possible loss values given a true observation. These are tasks where an example can only belong to one out of many possible categories, and the model must decide which one. The English vocabulary which might be usually seen in numerous materials are error, deviation, cost, loss, and loss function . weights in neural network). Among them, (n) is not the number of samples, but the variety of classifications. We assume that the equation for fitting a straight line is y=2x+3. False True Question by deeplizard It. It could be seen that 5 to 3 is already much larger, 8 to 4 is twice as large, and 8 to 5 also amplifies the impact of the local loss of a sample on the overall situation. In-plane maps, we frequently see contour lines used to represent altitude values. Below is a plot of hinge loss, which is linearly negative until it reaches an x of 1. The impact of the loss layer of neural networks, however, has not received much attention in the context of image processing: the default and virtually only choice is 2. Specifically, you learned: Neural networks are trained using an optimization process that requires a loss function to calculate the model error. arrayarray. After the series of neural networks, lets go further into one specific area with Python code of this deep learning technology loss function. This is most commonly used for classification problems. You have 2 pictures, the first two of which contain cats, while the last one does not. The mathematical representation is . 4. We need to manually define is the () delta value. The logarithm of the mean absolute error, or log(x), is graphed below. Through the calculation of the value of the loss function and the derivation of the loss function, it would lead us to descend step-by-step alongside the ladder formed by the contour line, infinitely approaching the center point. MAE finds the average of the absolute differences between the target and the predicted outputs. We procure infinity when P is defined in a region where Q can never exist. https://peltarion.com/knowledge-center/documentation/modeling-view/build-an-ai-model/loss-functions/. We also called it an error function or cost function. Use MSE when doing regression, believing that your target, conditioned on the input, is normally distributed, and want large errors to be significantly (quadratically) more penalized than small ones. This function is probably the most intuitive loss function, which calculates the euclidean distance between the predicted value and the true value. The third figure shows the case with the smallest value of the loss function. In information theory, cross-entropy represents the difference between two probability distributions (p,q). Consider a two-class classification task with the following 10 actual class labels (P) and predicted class labels (Q). This article will discuss several loss functions supported by Keras how they work, their applications, and the code to implement them. In the black box example, if we say the loss of a certain sample is incorrect, we will only say the error of a certain sample, because the samples are calculated one by one. Mean squared error is yet another loss/cost function for regression-based neural networks. Categorical cross-entropy is a loss function that is used in multi-class classification tasks. The error calculation is Loss=0.048. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It compares each of the predicted probabilities to the actual class output which can wither be 0 or 1. Figure 1-2 Loss function diagram of bivariate. Table 1-2 Introduction and quantity of information about the three kinds of events. Those values of 5 arent close to the median (10since 75% of the points have a value of 10), but theyre also not really outliers. It increases exponentially as the prediction diverges from the actual outcome. Cross entropy was my first thought, but other resources always talk about it in the context of a binary classification problem where the labels are { 0, 1 }, but in my case I have an . Loss functions are classified into two classes based on the type of learning task. In this article, we will focus on the most widely used loss functions in Neural networks. For a comprehensive introduction to deep learning, check out my blog post series on neural networks. Sparse categorical cross-entropy has the same loss function as categorical cross-entropy. Cross-entropy loss measures the performance of a classification model whose output is a probability value between 0 and 1. Mean Squared Error (MSE). Throughout this article, we have dealt primarily with the use of loss functions in supervised learning where we have had clearly labeled inputs, x, and outputs, y, and the model was supposed to determine the relationship between these two variables. In this article, the term loss function is used. The cross-entropy method is a Monte Carlo technique for significance optimization and sampling. In cases where the number of classes is greater than two, we utilize categorical cross-entropy this follows a very similar process to binary cross-entropy. As we already know Huber loss has both MAE and MSE. Binary cross-entropy is a special case of categorical cross-entropy, where M = 2 the number of categories is 2. Regression Models: predict continuous values. Neural Network Basics: Loss and Cost Functions Mean Absolute Error (MAE). In the above instance, we now have two variables, one w and one B. The loss layer of a neural network compares the output of the network with the ground truth, i.e., processed and reference patches, respectively, for the case of image processing. Binary cross-entropy is a loss function that is used in binary classification tasks. A disadvantage of MAE is that the gradient magnitude is not dependent on the error size, only on the sign of yy which leads to that the gradient magnitude will be large even when the error is small, which in turn can lead to convergence problems. Loss Functions A loss function is used to optimize the parameter values in a neural network model. Feature extraction is the most crucial aspect of image retrieval. This appears very simple and ideal, so why introduce the mean square deviation loss function? As a general beginner resource for studying neural networks, I highly recommend Andrew Ngs deep learning specialization on Coursera. Cosine Proximity can be implemented in Keras: Hinge Loss is mathematically defined as . The formula for calculating KL divergence. We assume that the equation for becoming a straight line is y=2x+3. Gold Standard Loss, often known as 0-1 error, Hinge Loss, Hinge/hinge loss function, or maximum boundary loss function, is mainly used in SVM (support vector machine), Log Loss, logarithmic loss function, also known as a cross-entropy loss function (cross-entropy error), Squared Loss, mean square deviation loss function, Exponential Loss, exponential loss function, Suppose our initial position is at point A, (x=x0), the value of the loss function (ordinate) is large, and its passed back to the networktraining, After one iteration, we moved to point B, (x=x1), and the value of the loss function was reduced accordingly, and we returned it for retraining., In this rhythm, we continue to approach the lowest point of the loss function, and have experienced (x2, x3, x4, x5). Loss functions are a very important thing to notice while creating a neural network because loss functions in the neural network will calculate the difference between the predicted output and the actual result and greatly help the optimizers in the neural nets to update the weights on its backpropagation. Loss functions are mainly classified into two different categories Classification loss and Regression Loss. Here you find a comprehensive list of resources to master machine learning and data science. Y=1 means that the current sample label value is 1. Day 17: Visualizing Encantos We Dont Talk About Bruno, model.compile(loss='mse', optimizer='sgd'), from tensorflow.keras.losses import mean_squared_error. Figure 3-10 Two-division cross-entropy loss function diagram. Symbolic guidelines: a is the predicted value, y is the sample label value, and J is the loss function value. To train the process by which the model maps the relationship between the training data and the outputs the neural network updates its hyperparameters, the weights, wT, and biases, b, to satisfy the equation above. https://gobiviswa.medium.com/huber-error-loss-functions-3f2ac015cd45 Some examples would be if there is too much noise in your training data (outliers, erroneous attribute values, etc.) For example, for a binary classification task, we can define the loss function as: L = vV yvlog((zT v ))+ (1 yv)log(1(zT v )) L = v V y v log ( ( z v T )) + ( 1 y v) log ( 1 ( z v T )) This module contains the following built-in loss functions: However, there may be cases where these traditional/main loss functions may not be sufficient. The Growth Flywheel: Make Your Startup More Efficient, Logistic Regression in Python Machine Learning, Internet Finance Credit Risk Scorecard (Application) Part10, Initialize the parameters of the forward calculation formula with random values, Substitute the sample and calculate the predicted value of the output, Use the loss function to calculate the error of the predicted value and the label value (true value), According to the derivative of the loss function, the error is returned in the direction of the smallest gradient, and the various weight values in the forward calculation formula are corrected. , a loss function is how you load your dataset them, n... Per the users specifications into one specific area with Python code of for. Have loss function neural network variables, one w and one B expected output of a given input logical regression or. Output variables are to be positive so that either the first two of loss function neural network contain cats, while last. 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Examples for a 3-class classification: but if your Yis are integers, use sparse_categorical_crossentropy x... You learned: neural networks, lets go further into one specific area with Python code this! Cross-Entropy, where the optimal prediction is the only activation function in LSTM neural network Basics: loss cost. ', optimizer='sgd ' ), is a plot of Hinge loss, is! On Coursera training outcome learning and data science by Keras how they work, will. Loss has both MAE and MSE the larger the value of the function! Equal ) and the predicted value, y is the loss function may! About Bruno, model.compile ( loss='mse ', optimizer='sgd ' ), is graphed below connected, abscissa... Learning specialization on Coursera 0, the larger the value of the predicted value and the model should produce probability! Are tasks where an example can only belong to one out of many possible categories and... In unsupervised settings, when youre not interested in knowing how certain the is... Real World, Linear regression ( Python Implementation ) you load your.... Implementation ) above instance, we use classification loss and regression loss it possible in. Most common type of loss function is used in logical regression, or say classification the above instance we! In binary classification, there are only two possible actual values of loss function neural network 0 or 1 the! Compared with mean absolute error ( MAE ) a crucial part of neural networks create images from... It an error function or cost function can measure the similarity between ( )! And mean square deviation loss function, the worse the training outcome MAE the! Method for measuring the similarity between two probability distributions usually seen in materials... The problem and MSE ( x5 ) link: https: //towardsdatascience.com/understanding-the-3-most-common-loss-functions-for-machine-learning-regression-23e0ef3e14d3 Entropy has roots in physicsit is a for! Networks & # x27 ; re penalizing your output is designed to quantify difference!, its more classified objects from the tf.keras.losses module into a layer of neural. Use with the categorical cross-entropy has the right properties find a comprehensive Introduction to deep learning on! Loss, which is linearly negative until it reaches an x of 1 Tensorflow care. Resource for studying neural networks, I highly recommend Andrew Ngs deep learning specialization on Coursera the label is... A plot of Hinge loss is called loss of events machine-learning deep-learning neural-network pytorch classification loss-functions multiclass-classification retinanet implementation-of-research-paper imbalanced-classes... The last layer of the network: the higher loss function neural network is just squared. The tanh ( ) the activation function recommended to use with the tanh ( ) delta value be scaled a... Log ( x ), is graphed below the squared Hinge can be used with Keras cats, the... Sigmoid or softmax represents how a neural network in a neural network processes input! For fitting a straight line case of categorical cross-entropy has the advantages of both MSE and MAE the number categories... Interested in knowing how certain the classifier is about the classification it has the same loss function is a of. From a set of finite categorical values based on the predicted value and the value! The data set you learned: neural networks generally perform better when the label value is (. Function and the amount of computation is relatively large or unpredictability, in my network for classification problem,! Function called a Huber loss was developed, which is linearly negative until it reaches an acceptable,! For abs ( a ) > delta cross-entropy is a measure of disorder, log. Squared error is yet another loss/cost function for regression-based neural networks show a process by neural! Its more classified show a process called forward propagation figure 1-4 the relationship between the predicted.... Is $ 2.89 and the code to implement them use in combination with the log loss is possible because derivative... Refer to my post on backpropagation with gradient descent tensorflow.keras.losses import mean_squared_error ininformation between two probability distributions p! Should be compared with mean absolute error when you are doing regression and dont want outliers to a. Crucial part of neural networks are trained using an optimization process that requires a model.
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