# huber loss explained

We can use sparse categorical crossentropy instead (Lin, 2019). We’ll now cover loss functions that are used for classification. At its core, a loss function is incredibly simple: itâs a method of evaluating how well your algorithm models your dataset. Huber loss is less sensitive to outliers in data than the squared error loss. It sounds really difficult, especially when you look at the formula (Binieli, 2018): … but fear not. There’s actually another commonly used type of loss function in classification related tasks: the hinge loss. Large errors will add to the loss more significantly than smaller errors. As you can see, the larger the delta, the slower the increase of this slope: eventually, for really large $$\delta$$ the slope of the loss tends to converge to some maximum. – MachineCurve, How to use Batch Normalization with Keras? by means of the Sigmoid layer. Well, that’s great. It’s relatively easy to compute the loss conceptually: we agree on some cost for our machine learning predictions, compare the 1000 targets with the 1000 predictions and compute the 1000 costs, then add everything together and present the global loss. Hence, we multiply the mean ratio error with the percentage to find the MAPE! Retrieved from https://en.wikipedia.org/wiki/Hinge_loss, Kompella, R. (2017, October 19). (n.d.). Although that’s perfectly fine for when you have such problems (e.g. 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). The primary part of the MSE is the middle part, being the Sigma symbol or the summation sign. We’re thus finding the most optimum decision boundary and are hence performing a maximum-margin operation. First, given our prediction $$\hat{y_i} = \sigma(Wx_i + b)$$ and our loss $$J = \frac{1}{2}(y_i - \hat{y_i})^2$$ , we first obtain the partial derivative $$\frac{dJ}{dW}$$, applying the chain rule twice: This derivative has the term $$\sigma'(Wx_i + b)$$ in it. Sign up to learn, We post new blogs every week. This includes the role of training, validation and testing data when training supervised models. The âsquared_lossâ refers to the ordinary least squares fit. The huber loss? The only thing left now is multiplying the whole with 100%. We start with our features and targets, which are also called your dataset. data from a different sample. about this issue with gradients, or if you’re here to learn, let’s move on to Mean Squared Error! âepsilon_insensitiveâ ignores errors less than epsilon and is linear past that; this is the loss â¦ (n.d.). Once we’re up to speed with those, we’ll introduce loss. For regression problems, there are many loss functions available. And the more they resemble each other, the better the machine learning model performs. But wait! In the visualization above, where the target is 1, it becomes clear that loss is 0. As you change pieces of your algorithm to try and improve your model, your loss function will tell you if youâre getting anywhere. In contrast, the L1 loss is used to penalize solutions for sparsity, and as such, it is commonly used for feature selection Kullback-Leibler Divergence Explained. The idea behind the loss function doesn’t change, but now since our labels $$y_i$$ are one-hot encoded, we write down the loss (slightly) differently: This is pretty similar to the binary cross entropy loss we defined above, but since we have multiple classes we need to sum over all of them. Thanks and happy engineering! Eventually, sum them together to find the multiclass hinge loss. $$L_{i} = - \log p(Y = y_{i} \vert X = x_{i})$$. 09/09/2015 â by Congrui Yi, et al. Destination Fees - the charge to have your vehicle delivered to Huber Chevrolet. If you look closely, you’ll notice the following: Yep: in our discussions about the MAE (insensitivity to larger errors) and the MSE (fixes this, but facing sensitivity to outliers). It does so by imposing a “cost” (or, using a different term, a “loss”) on each prediction if it deviates from the actual targets. What are loss functions? This means that ‘logcosh’ works mostly like the mean squared error, but will not be so strongly affected by the occasional wildly incorrect prediction. Let’s look at the formula again and recall that we iterate over all the possible output classes – once for every prediction made, with some true target: Now suppose that our trained model outputs for the set of features $${ … }$$ or a very similar one that has target $$[0, 1, 0]$$ a probability distribution of $$[0.25, 0.50, 0.25]$$ – that’s what these models do, they pick no class, but instead compute the probability that it’s a particular class in the categorical vector. the diabetes yes/no problem that we looked at previously), there are many other problems which cannot be solved in a binary fashion. Hinge loss. âhuberâ modifies âsquared_lossâ to focus less on getting outliers correct by switching from squared to linear loss past a distance of epsilon. However, when moving to the left, loss tends to increase (ML Cheatsheet documentation, n.d.). 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. we shift towards the optimum of the cost function. Cross entropy loss? Hence, it not only tends to punish wrong predictions, but also wrong predictions that are extremely confident (i.e., if the model is very confident that it’s 0 while it’s 1, it gets punished much harder than when it thinks it’s somewhere in between, e.g. using logistic regression instead of a deep neural net) will limit our ability to correctly classify every example with high probability on the correct label. This is your loss value. We multiply the delta with the absolute error and remove half of delta square. 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. This means that the “speed” of learning is dictated by two things: the learning rate and the size of the partial derivative. (n.d.). It is therefore not surprising that hinge loss is one of the most commonly used loss functions in Support Vector Machines (Kompella, 2017). The add_loss() API. This gives you much better intuition for the error in terms of the targets. into one of the buckets ‘diabetes’ or ‘no diabetes’. (2004, February 13). And the loss function weights the values larger than this number at only a third of the weight given to values less than it. Another loss function used often in regression is Mean Squared Error (MSE). For regression problems that are less sensitive to outliers, the Huber loss is used. categorical_crossentropy VS. sparse_categorical_crossentropy. Because the benefit of the $$\delta$$ is also becoming your bottleneck (Grover, 2019). â 0 â share . Retrieved from http://www.mit.edu/~rakhlin/6.883/lectures/lecture05.pdf, Grover, P. (2019, September 25). This latter property makes the binary cross entropy a valued loss function in classification problems. The testing data is used to test the model once the entire training process has finished (i.e., only after the last cycle), and allows us to tell something about the generalization power of our machine learning model. Loss Functions and Reported Model PerformanceWe will focus on the theory behind loss functions.For help choosing and implementing different loss functions, see â¦ In fact, the (multi-class) hinge loss would recognize that the correct class score already exceeds the other scores by more than the margin, so it will invoke zero loss on both scores. Suppose that our goal is to train a regression model on the NASDAQ ETF and the Dutch AEX ETF. And we want this to happen, since at the beginning of training, our model is performing poorly due to the weights being randomly initialized. What Loss Function to Use? Before we can actually introduce the concept of loss, weâll have to take a look at the high-level supervised machine learning process. 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. In the third case, e.g. Weâre then using machine learning for classification, or for deciding about some âmodel inputâ to âwhich classâ it belongs. Most generally speaking, the loss allows us to compare between some actual targets and predicted targets. Sign up to MachineCurve's. $$1/2 \times (t-p)^2$$, when $$|t-p| \leq \delta$$. How to perform Mean Shift clustering with Python in Scikit? In this post, we’ve show that the MSE loss comes from a probabalistic interpretation of the regression problem, and the cross-entropy loss comes from a probabalistic interpretaion of binary classification. This means that optimizing the MSE is easier than optimizing the MAE. Additionally, we covered a wide range of loss functions, some of them for classification, others for regression. Firstly, it is a very intuitive value. also see this perfect answer that illustrates it, https://www.freecodecamp.org/news/machine-learning-mean-squared-error-regression-line-c7dde9a26b93/, https://www.quora.com/What-is-the-difference-between-squared-error-and-absolute-error, https://canworksmart.com/using-mean-absolute-error-forecast-accuracy/, https://towardsdatascience.com/how-to-select-the-right-evaluation-metric-for-machine-learning-models-part-1-regrression-metrics-3606e25beae0, https://towardsdatascience.com/support-vector-machines-intuitive-understanding-part-1-3fb049df4ba1, https://peltarion.com/knowledge-center/documentation/modeling-view/build-an-ai-model/loss-functions/squared-hinge, https://www.quora.com/Why-is-squared-hinge-loss-differentiable, http://www.mit.edu/~rakhlin/6.883/lectures/lecture05.pdf, https://heartbeat.fritz.ai/5-regression-loss-functions-all-machine-learners-should-know-4fb140e9d4b0, https://www.tensorflow.org/api_docs/python/tf/keras/losses/logcosh, https://ml-cheatsheet.readthedocs.io/en/latest/loss_functions.html, https://peltarion.com/knowledge-center/documentation/modeling-view/build-an-ai-model/loss-functions/categorical-crossentropy, https://jovianlin.io/cat-crossentropy-vs-sparse-cat-crossentropy/, https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence, https://en.wikipedia.org/wiki/Entropy_(information_theory), https://www.countbayesie.com/blog/2017/5/9/kullback-leibler-divergence-explained, Visualizing Keras model inputs with Activation Maximization – MachineCurve, What do ConvNets see? Retrieved from https://heartbeat.fritz.ai/5-regression-loss-functions-all-machine-learners-should-know-4fb140e9d4b0, TensorFlow. Support vector machines ( intuitive understanding ) ? Consider Huber loss (more below) if you face this problem. This property introduces some mathematical benefits during optimization (Rich, n.d.). – MachineCurve, How to use L1, L2 and Elastic Net Regularization with Keras? In the first, your aim is to classify a sample into the correct bucket, e.g. The prediction is very incorrect, which occurs when $$y < 0.0$$ (because the sign swaps, in our case from positive to negative). What is the prediction? Itâs basically absolute error, which becomes quadratic when error is small. For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as {\displaystyle \ell (y)=\max (0,1-t\cdot y)} This is bad for model performance, as you will likely overshoot the mathematical optimum for your model. The hinge loss is defined as follows (Wikipedia, 2011): It simply takes the maximum of either 0 or the computation $$1 – t \times y$$, where t is the machine learning output value (being between -1 and +1) and y is the true target (-1 or +1). 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. We’ll show that given our model $$h_\theta(x) = \sigma(Wx_i + b)$$, learning can occur much faster during the beginning phases of training if we used the cross-entropy loss instead of the MSE loss. KL divergence (Wikipedia, 2004): KL divergence is an adaptation of entropy, which is a common metric in the field of information theory (Wikipedia, 2004; Wikipedia, 2001; Count Bayesie, 2017). That is, suppose that my prediction is 12 while the actual target is 10, the MAPE for this prediction is $$| (10 – 12 ) / 10 | = 0.2$$. That is, all the predictions. Note that when the sum is complete, you’ll multiply it with -1 to find the true categorical crossentropy loss. Retrieved from https://peltarion.com/knowledge-center/documentation/modeling-view/build-an-ai-model/loss-functions/categorical-crossentropy, Lin,Â J. Source: wikipedia also inspired by Udacity. Frank L. Galli. And what is a loss function? We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. The prediction is correct, which occurs when $$y \geq 1.0$$. If your predictions are totally off, your loss function will output a higher number. Thus: one where your output can belong to one of > 2 classes. In the scenario sketched above, n would be 1000. (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.). Required fields are marked *. Keras. It performs in pretty much similar ways to regular categorical crossentropy loss, but instead allows you to use integer targets! Why is squared hinge loss differentiable? For larger deltas, the slope of the function increases. We’ll first cover the high-level supervised learning process, to set the stage. Generative machine learning models work by drawing a sample from encoded, latent space, which effectively represents a latent probability distribution. It’s very well possible to use the MAE in a multitude of regression scenarios (Rich, n.d.). 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 $$h_\theta(x_i)$$). 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 haâ¦ This can’t really happen since that would mean our raw scores would have to be $$\infty$$ and $$-\infty$$ for our correct and incorrect classes respectively, and, more practically, constraints we impose on our model (i.e. What it does is really simple: it counts from i to n, and on every count executes what’s written behind it. 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. If you switch to Huber loss from MAE, you might find it to be an additional benefit. Looking at the graph for SVM in Fig 4, we can see that for yf(x) â¥ 1 , hinge loss is â 0 â. Since we initialized our weights randomly with values close to 0, this expression will be very close to 0, which will make the partial derivative nearly vanish during the early stages of training. The training data is fed into the machine learning model in what is called the forward pass. Negative loss doesn’t exist. In information theory, the Kullback-Leibler (KL) divergence measures how “different” two probability distributions are. That is: when the actual target meets the prediction, the loss is zero. Loss will be $$max(0, 0.1) = 0.1$$. 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. It goes like this: Simple, hey? How to create a variational autoencoder with Keras? When you train machine learning models, you feed data to the network, generate predictions, compare them with the actual values (the targets) and then compute what is known as a loss. Your email address will not be published. It’s actually really easy to understand what MSE is and what it does! It takes quite a long time before loss increases, even when predictions are getting larger and larger. We can write out the probability of observing a single $$(x_i, y_i)$$ sample: Summing across $$N$$ of these samples in our dataset, we can write down the likelihood - essentially the probability of observing all $$N$$ of our samples. It’s like (as well as unlike) the MAE, but then somewhat corrected by the. In fact, such nice santa-like loss functions are called convex functions (functions for which are always curving upwards) , and the loss functions for deep nets are hardly convex. Retrieved from https://www.countbayesie.com/blog/2017/5/9/kullback-leibler-divergence-explained, Your email address will not be published. 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? Looking at this plot, we see that Huber loss has a higher tolerance to outliers than squared loss. Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression. 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”. Intuitively, this makes sense because $$\log(x)$$ is increasing on the interval $$(0, 1)$$ so $$-\log(x)$$ is decreasing on that interval. Retrieved from https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence, Wikipedia. Very wrong predictions are hence penalized significantly by the hinge loss function. Rather, n is the number of samples in our training set and hence the number of predictions that has been made. This is the basic algorithm responsible for having neural networks converge, i.e. It’s available in many frameworks like TensorFlow as we saw above, but also in Keras. 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. We however only do so when the absolute error is smaller than or equal to some $$\delta$$, also called delta, which. This loss essentially tells you something about the performance of the network: the higher it is, the worse your networks performs overall. If you face larger errors and don’t care (yet?) We divide this number by n, or the number of samples used, to find the mean, or the average Absolute Error: the Mean Absolute Error or MAE. This closes the learning cycle between feeding data forward, generating predictions, and improving it – by adapting the weights, the model likely improves (sometimes much, sometimes slightly) and hence learning takes place. There’s also something called the RMSE, or the Root Mean Squared Error or Root Mean Squared Deviation (RMSD). It turns out that if we’re given a typical classification problem and a model $$h_\theta(x) = \sigma(Wx_i + b)$$, we can show that (at least theoretically) the cross-entropy loss leads to quicker learning through gradient descent than the MSE loss. The L2 loss is used to regularize solutions by penalizing large positive or negative control inputs in the optimal control setting or features in machine learning. What Is a Loss Function and Loss? The Mayo Clinic backs this up saying, âWhen your kidneys canât keep up, the excess glucose is excreted into â¦ MAPE, on the other hand, demonstrates the error in terms of a percentage – and a percentage is a percentage, whether you apply it to NASDAQ or to AEX. Particularly, the MSE is continuously differentiable whereas the MAE is not (at x = 0). Depreciation - the loss of value of the vehicle over the term of the lease. There are two main types of supervised learning problems: classification and regression. Using Radial Basis Functions for SVMs with Python and Scikit-learn, One-Hot Encoding for Machine Learning with TensorFlow and Keras, One-Hot Encoding for Machine Learning with Python and Scikit-learn, Feature Scaling with Python and Sparse Data, Visualize layer outputs of your Keras classifier with Keract. That is, the sample does not represent it fully and by consequence the mean and variance of the sample are (hopefully) slightly different than the actual population mean and variance. Retrieved from https://towardsdatascience.com/support-vector-machines-intuitive-understanding-part-1-3fb049df4ba1, Peltarion. Contact HUBER+SUHNER Inc. h = tf.keras.losses.Huber() h(y_true, y_pred).numpy() Learning Embeddings Triplet Loss. In the latter case, however, you don’t classify but rather estimate some real valued number. Finding optimal learning rates with the Learning Rate Range Test, Convolutional Neural Networks and their components for computer vision, Blogs at MachineCurve teach Machine Learning for Developers. You are now in control about the ‘degree’ of MAE vs MSE-ness you’ll introduce in your loss function.