– MachineCurve, Feature Scaling with Python and Sparse Data – MachineCurve, One-Hot Encoding for Machine Learning with Python and Scikit-learn – MachineCurve, One-Hot Encoding for Machine Learning with TensorFlow and Keras – MachineCurve, How to check if your Deep Learning model is underfitting or overfitting? These are the most commonly used functions I’ve seen used in traditional machine learning and deep learning models, so I thought it would be a good idea to figure out the underlying theory behind each one, and when to prefer one over the others. Additionally, large errors introduce a much larger cost than smaller errors (because the differences are squared and larger errors produce much larger squares than smaller errors). we shift towards the optimum of the cost function. This is what the validation data is used for – it helps during model optimization. We can combine these two cases into one expression: Invoking our assumption that the data are independent and identically distributed, we can write down the likelihood by simply taking the product across the data: Similar to above, we can take the log of the above expression and use properties of logs to simplify, and finally invert our entire expression to obtain the cross entropy loss: Let’s supposed that we’re now interested in applying the cross-entropy loss to multiple (> 2) classes. (n.d.). – MachineCurve, How to use L1, L2 and Elastic Net Regularization with Keras? Hence, for all correct predictions – even if they are too correct, loss is zero. \(0.2 \times 100\%\) is … unsurprisingly … \(20\%\)! As you change pieces of your algorithm to try and improve your model, your loss function will tell you if you’re getting anywhere. Retrieved from https://towardsdatascience.com/support-vector-machines-intuitive-understanding-part-1-3fb049df4ba1, Peltarion. (n.d.). Loss functions are a key part of any machine learning model: they define an objective against which the performance of your model is measured, and the setting of weight parameters learned by the model is determined by minimizing a chosen loss function. What’s more, and this is important: when you use the MAE in optimizations that use gradient descent, you’ll face the fact that the gradients are continuously large (Grover, 2019). (Note that one approach to create a multiclass classifier, especially with SVMs, is to create many binary ones, feeding the data to each of them and counting classes, eventually taking the most-chosen class as output – it goes without saying that this is not very efficient.). When you wish to compare two probability distributions, you can use the Kullback-Leibler divergence, a.k.a. when \(y = -0.5\), the output of the loss equation will be \(1 – (1 \ times -0.5) = 1 – (-0.5) = 1.5\), and hence the loss will be \(max(0, 1.5) = 1.5\). machine learning, math, and other random thoughts. If we could probabilistically assign labels to the unlabelled portion of a dataset, or interpret the incorrect labels as being sampled from a probabalistic noise distribution, we can still apply the idea of minimizing the KL-divergence, although our ground-truth distribution will no longer concentrate all the probability mass over a single label. As you have to configure them manually (or perhaps using some automated tooling), you’ll have to spend time and resources on finding the most optimum \(\delta\) for your dataset. The way the hinge loss is defined makes it not differentiable at the ‘boundary’ point of the chart –. The structure of the formula however allows us to perform multiclass machine learning training with crossentropy. However, in most cases, it’s best just to experiment – perhaps, you’ll find better results! Michael Nielsen’s Neural Networks and Deep Learning, Chapter 3, Stanford CS 231n notes on cross entropy and hinge loss, StackExchange answer on hinge loss minimization, [4/16/19] - Fixed broken links and clarified the particular model for which the learning speed of MSE loss is slower than cross-entropy. I hope you’ve learnt something from my blog! about this issue with gradients, or if you’re here to learn, let’s move on to Mean Squared Error! Because the benefit of the \(\delta\) is also becoming your bottleneck (Grover, 2019). where there exist two classes. Retrieved from https://www.quora.com/What-is-the-difference-between-squared-error-and-absolute-error, Watson, N. (2019, June 14). Very wrong predictions are hence penalized significantly by the hinge loss function. h = tf.keras.losses.Huber() h(y_true, y_pred).numpy() Learning Embeddings Triplet Loss. The \(t\) in the formula is the target (0 or 1) and the \(p\) is the prediction (a real-valued number between 0 and 1, for example 0.12326). Retrieved from https://en.wikipedia.org/wiki/Entropy_(information_theory), Count Bayesie. While intuitively, entropy tells you something about “the quantity of your information”, KL divergence tells you something about “the change of quantity when distributions are changed”. MachineCurve participates in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising commissions by linking to Amazon. Our goal when training a machine learning model? If this probability were less than \(0.5\) we’d classify it as a negative example, otherwise we’d classify it as a positive example. Hence, a little bias is introduced into the model every time you’ll optimize it with your validation data. Squared hinge. In this case, that’s the third part – the square of (Yi – Y’i). categorical_crossentropy VS. sparse_categorical_crossentropy. Most generally speaking, the loss allows us to compare between some actual targets and predicted targets. What are loss functions? In particular, in the inner sum, only one term will be non-zero, and that term will be the \(\log\) of the (normalized) probability assigned to the correct class. Another variant on the cross entropy loss for multi-class classification also adds the other predicted class scores to the loss: The second term in the inner sum essentially inverts our labels and score assignments: it gives the other predicted classes a probability of \(1 - s_j\), and penalizes them by the \(\log\) of that amount (here, \(s_j\) denotes the \(j\)th score, which is the \(j\)th element of \(h_\theta(x_i)\)). More specifically, we can write it as a multiplication of \(100\%\) and \(1 / n\) instead. A year or so I got one of your triggers from a recommendation from my SWAT commander and after all this time, it is by far the best two stage trigger I have ever used and have recommended them to every sniper I know. Huber loss approaches MAE when 𝛿 ~ 0 and MSE when 𝛿 ~ ∞ (large numbers.). Thus: one where your output can belong to one of > 2 classes. are the corresponding predictions and α ∈ ℝ⁺ is a hyperparameter. And so on. It does so by imposing a “cost” (or, using a different term, a “loss”) on each prediction if it deviates from the actual targets. It’s just the MSE but then its square root value. If you switch to Huber loss from MAE, you might find it to be an additional benefit. Of course one can choose other alternatives to the OLS loss function, and one of the most common is the Huber loss function. The softmax function, whose scores are used by the cross entropy loss, allows us to interpret our model’s scores as relative probabilities against each other. This paper contains a new approach toward a theory of robust estimation; it treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators--intermediaries between sample mean and sample median--that are asymptotically most robust (in a sense to be specified) among all translation invariant estimators. The resultant loss function doesn't look a nice bowl, with only one minima we can converge to. This sounds very complicated, but we can break it into parts easily. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs). It sounds really difficult, especially when you look at the formula (Binieli, 2018): … but fear not. As you can guess, it’s a loss function for binary classification problems, i.e. Eventually, sum them together to find the multiclass hinge loss. In other model types, such as Support Vector Machines, we do not actually propagate the error backward, strictly speaking. This is your loss value. Their goal: to optimize the internals of your model only slightly, so that it will perform better during the next cycle (or iteration, or epoch, as they are also called). Let’s formalize this by writing out the hinge loss in the case of binary classification: Our labels \(y_{i}\) are either -1 or 1, so the loss is only zero when the signs match and \(\vert (h_{\theta}(x_{i}))\vert \geq 1\). What’s more, hinge loss itself cannot be used with gradient descent like optimizers, those with which (deep) neural networks are trained. It’s available in many frameworks like TensorFlow as we saw above, but also in Keras. – MachineCurve, How does the Softmax activation function work? The add_loss() API. It compares the probability distribution represented by your training data with the probability distribution generated during your forward pass, and computes the divergence (the difference, although when you swap distributions, the value changes due to non-symmetry of KL divergence – hence it’s not entirely the difference) between the two probability distributions. Modified Huber loss stems from Huber loss, which is used for regression problems. However, if your average error is very small, it may be better to use the Mean Squared Error that we will introduce next. Consider the following binary classification scenario: we have an input feature vector \(x_i\), a label \(y_i\), and a prediction \(\hat{y_i} = h_\theta(x_i)\). What is the difference between squared error and absolute error? when \(y = 1.2\), the output of \(1 – t \ times y\) will be \( 1 – ( 1 \times 1.2 ) = 1 – 1.2 = -0.2\). This means that when my training set consists of 1000 feature vectors (or rows with features) that are accompanied by 1000 targets, I will have 1000 predictions after my forward pass. With one minor difference: the end result of this computation is squared. 🙂. PackerCentral is a Sports Illustrated channel featuring Bill Huber to bring you the latest News, Highlights, Analysis, Draft, Free Agency surrounding the Green Bay Packers. Wi… We can think of our classification problem as having 2 different probability distributions: first, the distribution for our actual labels, where all the probability mass is concentrated on the correct label, and there is no probability mass on the rest, and second, the distribution which we are learning, where the concentrations of probability mass are given by the outputs of the running our raw scores through a softmax function. Primarily, it can be used where the output of the neural network is somewhere between 0 and 1, e.g. This looks difficult, but we can once again separate this computation into more easily understandable parts. By signing up, you consent that any information you receive can include services and special offers by email. Here, we’ll cover a wide array of loss functions: some of them for regression, others for classification. Huber loss is less sensitive to outliers in data than the squared error loss. An error of 100 may seem large, but if the actual target is 1000000 while the estimate is 1000100, well, you get the point. But what is loss? This is primarily due to the use of the sigmoid function. Given a particular model, each loss function has particular properties that make it interesting - for example, the (L2-regularized) hinge loss comes with the maximum-margin property, and the mean-squared error when used in conjunction with linear regression comes with convexity guarantees. A well-performing model would be interesting for production usage, whereas an ill-performing model must be optimized before it can be actually used. In the visualization above, where the target is 1, it becomes clear that loss is 0. Machine learning: an introduction to mean squared error and regression lines. If you face larger errors and don’t care (yet?) Taken from Wikipedia, Huber loss is $ L_\delta (a) = \begin{cases} \frac{1}{2}{a^2} & \text{for } |a| \le \delta, \\ \delta (|a| - \frac{1}{2}\delta), & \text{otherwise.} This again makes sense - penalizing the incorrect classes in this way will encourage the values \(1 - s_j\) (where each \(s_j\) is a probability assigned to an incorrect class) to be large, which will in turn encourage \(s_j\) to be low. Let’s take a look at this training process, which is cyclical in nature. Sign up above to learn, By continuing to browse the site you are agreeing to our, The high-level supervised learning process, Never miss new Machine Learning articles âœ. – MachineCurve, What is Batch Normalization for training neural networks? How to use sparse categorical crossentropy in Keras? It would mean that we have to train many networks, which significantly impacts the time performance of our ML training problem. If these assumptions don’t hold true (such as in the context of classification), the MSE loss may not be the best bet. That is, all the predictions. The basic difference between batch gradient descent (BGD) and stochastic gradient descent (SGD), is that we only calculate the cost of one example for each step in SGD, but in BGD, we ha… – MachineCurve, How to use Batch Normalization with Keras? How to select the Right Evaluation Metric for Machine Learning Models: Part 1 Regression Metrics. – MachineCurve, How to use sparse categorical crossentropy in Keras? We however only do so when the absolute error is smaller than or equal to some \(\delta\), also called delta, which. Retrieved from https://keras.io/losses/, Binieli, M. (2018, October 8). Secondly, it allows us to compare the performance of regression models on different datasets (Watson, 2019). We multiply the delta with the absolute error and remove half of delta square. MachineCurve.com will earn a small affiliate commission from the Amazon Services LLC Associates Program when you purchase one of the books linked above. “Log-cosh is the logarithm of the hyperbolic cosine of the prediction error.” (Grover, 2019). In neural networks, often, a combination of gradient descent based methods and backpropagation is used: gradient descent like optimizers for computing the gradient or the direction in which to optimize, backpropagation for the actual error propagation. Maximum Likelihood and Cross-Entropy 5. This property introduces some mathematical benefits during optimization (Rich, n.d.). This is what it looks like: Don’t worry about the maths, we’ll introduce the MAE intuitively now. Assume that the validation data, which is essentially a statistical sample, does not fully match the population it describes in statistical terms. The learning rate is a hyperparameter that we must tune, so we’ll focus on the size of the partial derivatives for now. That is: when the actual target meets the prediction, the loss is zero. In the first, your aim is to classify a sample into the correct bucket, e.g. Retrieved from https://www.countbayesie.com/blog/2017/5/9/kullback-leibler-divergence-explained, Your email address will not be published. Loss will be \(max(0, 0.1) = 0.1\). Retrieved from https://peltarion.com/knowledge-center/documentation/modeling-view/build-an-ai-model/loss-functions/squared-hinge, Tay, J. If they’re pretty good, it’ll output a lower number. By looking at all observations, merging them together, we can find the loss value for the entire prediction. We’re getting there – and that’s also indicated by the small but nonzero loss. Huber loss function. Now, we can explain with is meant with an observation. This is bad for model performance, as you will likely overshoot the mathematical optimum for your model. This is done by propagating the error backwards to the model structure, such as the model’s weights. It’s still crossentropy, but then adapted to multiclass problems. Required fields are marked *. With small \(\delta\), the loss becomes relatively insensitive to larger errors and outliers. the diabetes yes/no problem that we looked at previously), there are many other problems which cannot be solved in a binary fashion. Kullback-Leibler Divergence Explained. Generative machine learning models work by drawing a sample from encoded, latent space, which effectively represents a latent probability distribution. When the target is 0, you can see that the loss is mirrored – which is exactly what we want: Now what if you have no binary classification problem, but instead a multiclass one? What this essentially sketches is a margin that you try to maximize: when the prediction is correct or even too correct, it doesn’t matter much, but when it’s not, we’re trying to correct. Is KL divergence used in practice? The function is defined as follows Reduce overfitting in your neural networks – MachineCurve, Creating a Signal Noise Removal Autoencoder with Keras – MachineCurve, How to use Kullback-Leibler divergence (KL divergence) with Keras? Hence, loss is driven by the actual target observation of your sample instead of all the non-targets. (n.d.). The TensorFlow docs write this about Logcosh loss: log(cosh(x)) is approximately equal to (x ** 2) / 2 for small x and to abs(x) - log(2) for large x. Sometimes, machine learning problems involve the comparison between two probability distributions. They derive certain characteristics for those tomatoes, e.g. The end result is a set of predictions, one per sample. Contrary to the absolute error, we have a sense of how well-performing the model is or how bad it performs when we can express the error in terms of a percentage. Retrieved from https://ml-cheatsheet.readthedocs.io/en/latest/loss_functions.html, Peltarion. The training data is used during the training process; more specificially, to generate predictions during the forward pass. – MachineCurve, Using simple generators to flow data from file with Keras – MachineCurve, Storing web app machine learning predictions in a SQL database – MachineCurve, How to use HDF5Matrix with Keras? ∙ 0 ∙ share . Maximum Likelihood 4. when \(y = 0.9\), loss output function will be \(1 – (1 \times 0.9) = 1 – 0.9 = 0.1\). What’s more, it increases increasingly fast. This is both good and bad at the same time (Rich, n.d.). The Mean Absolute Percentage Error, or MAPE, really looks like the MAE, even though the formula looks somewhat different: When using the MAPE, we don’t compute the absolute error, but rather, the mean error percentage with respect to the actual values. Dissecting Deep Learning (work in progress). I already discussed in another post what classification is all about, so I’m going to repeat it here: Suppose that you work in the field of separating non-ripe tomatoes from the ripe ones. In this blog, we’ve looked at the concept of loss functions, also known as cost functions. What Is a Loss Function and Loss? There are many ways for computing the loss value. – MachineCurve, How to use K-fold Cross Validation with Keras? How about mean squared error? Essentially, because then \(1 – t \times y = 1 – 1 = 1\), the max function takes the maximum \(max(0, 0)\), which of course is 0. But wait! This means that we can write down the probabilily of observing a negative or positive instance: \(p(y_i = 1 \vert x_i) = h_\theta(x_i)\) and \(p(y_i = 0 \vert x_i) = 1 - h_\theta(x_i)\). Loss Functions. There are several different common loss functions to choose from: the cross-entropy loss, the mean-squared error, the huber loss, and the hinge loss - just to name a few. We answer the question what is loss? This can be expressed as \(\sigma(Wx_i + b)(1 - \sigma(Wx_i + b))\) (see here for a proof). Using Mean Absolute Error to Forecast Accuracy. (2001, July 9). On the other hand, given the cross entropy loss: We can obtain the partial derivative \(\frac{dJ}{dW}\) as follows (with the substitution \(\sigma(z) = \sigma(Wx_i + b)\): Simplifying, we obtain a nice expression for the gradient of the loss function with respect to the weights: This derivative does not have a \(\sigma'\) term in it, and we can see that the magnitude of the derivative is entirely dependent on the magnitude of our error \(\sigma(z) - y_i\) - how far off our prediction was from the ground truth. My name is Chris and I love teaching developers how to build  awesome machine learning models. This is great, since that means early on in learning, the derivatives will be large, and later on in learning, the derivatives will get smaller and smaller, corresponding to smaller adjustments to the weight variables, which makes intuitive sense since if our error is small, then we’d want to avoid large adjustments that could cause us to jump out of the minima. This alternative version seems to tie in more closely to the binary cross entropy that we obtained from the maximum likelihood estimate, but the first version appears to be more commonly used both in practice and in teaching. And we want this to happen, since at the beginning of training, our model is performing poorly due to the weights being randomly initialized. We multiply the target value with the log. Note that this does not mean that you sum over all possible values for y (which would be all real-valued numbers except \(t\)), but instead, you compute the sum over all the outputs generated by your ML model during the forward pass. 😎. 👇 I’d also appreciate a comment telling me if you learnt something and if so, what you learnt. Looking at this plot, we see that Huber loss has a higher tolerance to outliers than squared loss.