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# Softmax cross entropy

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One of the reasons to choose cross-entropy alongside softmax is that because softmax has an exponential element inside it. A cost function that has an element of the natural log will provide for a convex cost function. This is similar to logistic regression which uses sigmoid. Mathematically expressed as below We often need to process variable length sequence in deep learning. In that situation, we will need use mask in our model. In this tutorial, we will introduce how to calculate softmax cross-entropy loss with masking in TensorFlow

The Softmax regression is a form of logistic regression that normalizes an input value into a vector of values that follows a probability distribution whose total sums up to 1. As its name suggests, softmax function is a soft version of max function Computes softmax cross entropy between logits and labels. (deprecated arguments Computes softmax cross entropy between logits and labels. tf.nn.softmax_cross_entropy_with_logits (labels, logits, axis=-1, name=None) Measures the probability error in discrete classification tasks in which the classes are mutually exclusive (each entry is in exactly one class) The softmax function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression. Since the function maps a vector and a specific inde While this function computes a usual softmax cross entropy if the number of dimensions is equal to 2, it computes a cross entropy of the replicated softmax if the number of dimensions is greater than 2. t (Variable or N-dimensional array) - Variable holding a signed integer vector of ground truth labels

When using a Neural Network to perform classification tasks with multiple classes, the Softmax function is typically used to determine the probability distribution, and the Cross-Entropy to evaluate the performance of the model Rethinking Softmax with Cross-Entropy: Neural Network Classifier as Mutual Information Estimator 25 Nov 2019 • Zhenyue Qin • Dongwoo Kim • Tom Gedeon Mutual information is widely applied to learn latent representations of observations, whilst its implication in classification neural networks remain to be better explained sm = tf.nn.softmax(x) ce = cross_entropy(sm) The cross entropy is a summary metric: it sums across the elements. The output of tf.nn.softmax_cross_entropy_with_logits on a shape [2,5] tensor is of shape [2,1] (the first dimension is treated as the batch) Softmax & Cross-Entropy. Disclaimer: You should know that this Softmax and Cross-Entropy tutorial is not completely necessary nor is it mandatory for you to proceed in this Deep Learning Course. That being said, learning about the softmax and cross-entropy functions can give you a tighter grasp of this section's topic gumbel_softmax ¶ torch.nn.functional.gumbel_softmax (logits, tau=1, hard=False, eps=1e-10, dim=-1) [source] ¶ Samples from the Gumbel-Softmax distribution (Link 1 Link 2) and optionally discretizes.Parameters. logits - [, num_features] unnormalized log probabilities. tau - non-negative scalar temperature. hard - if True, the returned samples will be discretized as one-hot vectors.

In this article, I will explain the concept of the Cross-Entropy Loss, commonly called the Softmax Classifier. I'll go through its usage in the Deep Learning classification task and the mathematics of the function derivatives required for the Gradient Descent algorithm The cross entropy error function is E(t, o) = − ∑ j tjlogoj with t and o as the target and output at neuron j, respectively. The sum is over each neuron in the output layer. oj itself is the result of the softmax function In this blog post, you will learn how to implement gradient descent on a linear classifier with a Softmax cross-entropy loss function. I recently had to implement this from scratch, during the CS231 course offered by Stanford on visual recognition. Andrej was kind enough to give us the final form of the derived gradient in the course notes, but I couldn't find anywhere the extended version. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube Entropy, Cross-Entropy and KL-Divergence are often used in Machine Learning, in particular for training classifiers. In this short video, you will understand..

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• Categorical Cross Entropy p(x) is the true distribution, q(x) is our calculated probabilities from softmax function. The truth label will have p(x) = 1 , all the other ones have p(x) = 0
• Cross-entropy is commonly used in machine learning as a loss function. Cross-entropy is a measure from the field of information theory, building upon entropy and generally calculating the difference between two probability distributions. It is closely related to but is different from KL divergence that calculates the relative entropy between two probability distributions, whereas cross-entropy.
• Mutual information is widely applied to learn latent representations of observations, whilst its implication in classification neural networks remain to be better explained. We show that optimising the parameters of classification neural networks with softmax cross-entropy is equivalent to maximising the mutual information between inputs and labels under the balanced data assumption. Through.
• Softmax와 Cross entropy (0) 2017.08.03: Logistic Regression (로지스틱 회귀) (0) 2017.07.31: Linear Regression (선형회귀) (0) 2017.07.28: Tracback 0 댓글은 작성자에게 큰 힘이 됩니다 0. name. password. homepage. secret. commen

Then if the ground truth specifies 0.3m, we would build a digitized label set of [0 1 0 0 0], and apply cross entropy loss. But now i have lost some precision. I was hoping that we can instead have a label of [0.5 0.5 0 0 0], and train this on some kind of soft cross entropy loss. But my toy example here shows that it doesnt work This tutorial will cover how to do multiclass classification with the softmax function and cross-entropy loss function. The previous section described how to represent classification of 2 classes with the help of the logistic function .For multiclass classification there exists an extension of this logistic function called the softmax function which is used in multinomial logistic regression Sigmoid Cross Entropy Loss The sigmoid cross entropy is same as softmax cross entropy except for the fact that instead of softmax, we apply sigmoid function on logits before feeding them. Notes on.

### Classification and Loss Evaluation - Softmax and Cross

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• Softmax function will normalize the input scores between the range of 0 to 1, which can be interpreted as a probability. In this way we can directly compare it to the one-hot encoded vector that corresponds to your labels using cross entropy. This is the math definition of cross-entropy
• The Softmax is built within the Cross-Entropy Loss function definition. torch.nn - PyTorch master documentation Parameters are subclasses, that have a very special property when used with s - when..
• We first formally show that the softmax cross-entropy (SCE) loss and its variants convey inappropriate supervisory signals, which encourage the learned feature points to spread over the space sparsely in training
• Here, softmax was used as an activation function which allows us to interpret the outputs as probabilities. Cross Entropy loss is use to measure the error at a softmax layer. Implemented a Softmax classifier with L_2 weight decay regularization. Regularization is used to prevent overfitting in neural nets
• After then, applying one hot encoding transforms outputs in binary form. That's why, softmax and one hot encoding would be applied respectively to neural networks output layer. Finally, true labeled output would be predicted classification output. Herein, cross entropy function correlate between probabilities and one hot encoded labels
• tf.nn.softmaxは、softmaxレイヤを介した順方向伝搬を計算します。モデルが出力する確率を計算するときに、モデルの 評価 の間にそれを使用します。. tf.nn.softmax_cross_entropy_with_logitsは、softmaxレイヤのコストを計算します。 training の間だけ使われます。. ロジットは 非正規化ログ確率 モデルを出力.

When trying to get cross entropy with sigmoid activation function, there is a difference between loss1 = -tf.reduce_sum (p*tf.log (q), 1) loss2 = tf.reduce_sum (tf.nn.sigmoid_cross_entropy_with_logits (labels=p, logits=logit_q),1) But they are the same when with softmax activation function I didn't look at your code, but if you wrote your softmax and cross-entropy functions as two separate functions you are probably tripping over the following problem. Softmax contains exp() and cross-entropy contains log(), so this can happen: large number --> exp() --> overflow NaN --> log() --> still NaN even though, mathematically (i.e., without overflow), log (exp (large number)) = large.

Rethinking Softmax with Cross-Entropy Neural Network Classifier as Mutual Information Estimator MI Estimator • PC Softmax • InfoCAM • Credits • Licence. Overview In the paper, we show the connection between mutual information and softmax classifier through variational form of mutual information

softmax、cross entropy和softmax loss学习笔记 之前做手写数字识别时，接触到softmax网络，知道其是全连接层，但没有搞清楚它的实现方式，今天学习Alexnet网络，又接触到了softmax，果断仔细研究研究，有了softmax，损失函数自然不可少� tf.nn.sparse_softmax_cross_entropy_with_logits() 这是一个TensorFlow中经常需要用到的函数。官方文档里面有对它详细的说明，传入的logits为神经网络输出层的输出，shape为[batch_size，num_classes]，传入的label为一个一维的vector，长度等于batch_size，每一个值的取值区间必须是[0，num_cla.. TensorFlow tf.nn.softmax_cross_entropy_with_logits_v2 () is one of functions which tensorflow use to compute cross entropy, which is very similar to tf.nn.softmax_cross_entropy_with_logits (). In this tutorial, we will introduce how to use this function for tensorflow beginners tf.nn.softmax_cross_entropy_with_logits（logits= , labels=）。 第一个参数logits：就是神经网络最后一层的输出，如果有batch的话，它的大小就是[batchsize，num_classes]，单样本的话，大小就是num_classes. 第二个参数labels：实际的标签，大小同上. 交叉熵代码�

### Softmax and Cross-entropy Slowbreathin

1. It is a Softmax activationplus a Cross-Entropy loss. If we use this loss, we will train a CNN to output a probability over the $$C$$ classes for each image. It is used for multi-class classification. In the specific (and usual) case of Multi-Class classification the labels are one-hot, so only the positive class $$C_p$$ keeps its term in the loss
2. In contrast, tf.nn.softmax_cross_entropy_with_logits computes the cross entropy of the result after applying the softmax function (but it does it all together in a more mathematically careful way). It's similar to the result of: sm = tf.nn.softmax(x) ce = cross_entropy(sm) The cross entropy is a summary metric - it sums across the elements
3. , the forward loss softmax cross-entropy layer computes a one-dimensional tensor with the cross-entropy value. For more details, see Forward Loss Softmax Cross-entropy Layer
4. Tentative answer: if the labels are one-hot encoded, then we just end up with one term in the cross-entropy summation expression, that is the negative logarithm of the softmax output for the correct class. So, optimizing the softmax is equivalent to optimizing the cross-entropy
5. Cross-Entropy Loss 란? Cross Entropy Loss는 보통 Classification에서 많이 사용됩니다. 보통 위 그림과 같이 Linear Model (딥러닝 모델)을 통해서 최종값 (Logit 또는 스코어)이 나오고, Softmax 함수를 통해 이 값들의 범위는 [0,1], 총 합은 1이 되도록 합니다
6. Polychotomizers: One-Hot Vectors, Softmax, and Cross-Entropy Mark Hasegawa-Johnson, 3/9/2019. CC-BY 3.0:Youarefreetoshare and adapt these slides ifyoucite the original
7. The softmax operation takes a vector and maps it into probabilities. Softmax regression applies to classification problems. It uses the probability distribution of the output class in the softmax operation. Cross-entropy is a good measure of the difference between two probability distributions

Softmax function is an activation function, and cross entropy loss is a loss function. Softmax function can also work with other loss functions. The cross entropy loss can be defined as: $$L_i = - \sum_{i=1}^{K} y_i log(\sigma_i(z))$$ Note that for multi-class classification problem, we assume that each sample is assigned to one and only one label This operation computes the cross entropy between the target_vector and the softmax of the output_vector. The elements of target_vector have to be non-negative and should sum to 1. The output_vector can contain any values. The function will internally compute the softmax of the output_vector As an aside, another name for Softmax Regression is Maximum Entropy (MaxEnt) Classifier. The function is usually used to compute losses that can be expected when training a data set. Known use-cases of softmax regression are in discriminative models such as Cross-Entropy and Noise Contrastive Estimation

### Implement Softmax Cross-entropy Loss with Masking in

1. Entropy is also used in certain Bayesian methods in machine learning, but these won't be discussed here. It is now time to consider the commonly used cross entropy loss function. Cross entropy and KL divergence. Cross entropy is, at its core, a way of measuring the distance between two probability distributions P and Q
2. Computes softmax cross entropy between logits and labels. (deprecated) THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Future major versions of TensorFlow will allow gradients to flow into the labels input on backprop by default. See tf.nn.softmax_cross_entropy_with_logits_v2
3. Definition. The cross entropy of the distribution relative to a distribution over a given set is defined as follows: (,) = − ⁡ [⁡],where [⋅] is the expected value operator with respect to the distribution .The definition may be formulated using the Kullback-Leibler divergence (‖) from of (also known as the relative entropy of with respect to )
4. The cross entropy formula takes in two distributions, p (x), the true distribution, and q (x), the estimated distribution, defined over the discrete variable x and is given by H (p, q) = − ∑ ∀ x p (x) log (q (x)) For a neural network, the calculation is independent of the following
5. (참고원문: Many author use the term cross-entropy to identify specifically the negative log-likelihood of a Bernoulli or softmax distribution, but that is a misnomer. Any loss consisting of a negative log-likeligood is a cross-entropy between the empirical distribution defined by the training set and the probability distribution by model
6. Softmax の目的 Score(logit Cross Entropy. Edit request. Stock. 41. Masataka Ohashi @supersaiakujin. Follow. Why not register and get more from Qiita? We will deliver articles that match you. By following users and tags, you can catch up information on technical fields that you are interested in as a whole

### Softmax Function and Cross Entropy Loss Function - Deep

1. Softmax and cross-entropy loss. We've just seen how the softmax function is used as part of a machine learning network, and how to compute its derivative using the multivariate chain rule. While we're at it, it's worth to take a look at a loss function that's commonly used along with softmax for training a network: cross-entropy
2. ative enough. To tackle this problem, many approaches have been proposed
3. sigmoid和softmax通常来说是2类和多类分类采用的函数，但sigmoid同样也可以用于多类，不同之处在于sigmoid中多类有可能相互重叠，看不出什么关系，softmax一定是以各类相互排斥为前提，算出来各个类别的概率和为1 binary cross-entropy和categorical cross-entropy.
4. Cross Entorpy. The equation below compute the cross entropy $$C$$ over softmax function: where $$K$$ is the number of all possible classes, $$t_k$$ and $$y_k$$ are the target and the softmax output of class $$k$$ respectively. Derivation. Now we want to compute the derivative of $$C$$ with respect to $$z_i$$, where $$z_i$$ is the penalty of a.
5. Posted by: Chengwei 1 year, 11 months ago () In this quick tutorial, I am going to show you two simple examples to use the sparse_categorical_crossentropy loss function and the sparse_categorical_accuracy metric when compiling your Keras model.. Example one - MNIST classification. As one of the multi-class, single-label classification datasets, the task is to classify grayscale images of.
6. 对于每个条目的概率分布的soft softmax分类，请参阅softmax_cross_entropy_with_logits_v2。 警告：此操作需要未缩放的日志，因为它在内部日志上执行softmax，以提高效率。不要使用softmax的输出调用此操作，因为它会产生不正确的结果�
7. A matrix-calculus approach to deriving the sensitivity of cross-entropy cost to the weighted input to a softmax output layer. We use row vectors and row gradients, since typical neural network formulations let columns correspond to features, and rows correspond to examples.This means that the input to our softmax layer is a row vector with a column for each class The Softmax is a function usually applied to the last layer in a neural network. Such network ending with a Softmax function is also sometimes called a Softmax Classifier as the output is usually meant to be as a classification of the net's input... In effetti TensorFlow ha un'altra funzione simile sparse_softmax_cross_entropy dove fortunatamente si sono dimenticati di aggiungere il suffisso _with_logits creando incoerenza e aggiungendo confusione Softmax To explain this Andre NG uses term hard-max vs soft-max y_pred = exp(z_i) / sum_over_i ( exp(z_i) ) In softmax we output probability of various classes In hardmax we will make one class as 1 and all others as 0 Cross Entropy It is a loss function Loss = - sum [y_actual

### tf.compat.v1.nn.softmax_cross_entropy_with_logits_v

Now, we multiply the inputs with the weight matrix, and add biases. We compute the softmax and cross-entropy using tf.nn.softmax_cross_entropy_with_logits (it's one operation in TensorFlow, because it's very common, and it can be optimized). We take the average of this cross-entropy across all training examples using tf.reduce_mean method Softmax function is a very common function used in machine learning, especially in logistic regression models and neural networks. In this post I would like to compute the derivatives of softmax function as well as its cross entropy 49行目のreturn F.softmax_cross_entropy(y, t), F.accuracy(y, t) で、多クラス識別をする際の交差エントロピー誤差は、出力層のユニット数分(ラベルに対応するユニットだけでなくほかのユニットの確率も余事象として)計算しなければならないのに、教師データtを1ofK表記にせず、そのまま渡している点� Introduction¶. When we develop a model for probabilistic classification, we aim to map the model's inputs to probabilistic predictions, and we often train our model by incrementally adjusting the model's parameters so that our predictions get closer and closer to ground-truth probabilities.. In this post, we'll focus on models that assume that classes are mutually exclusive Polychotomizers: One-Hot Vectors, Softmax, and Cross-Entropy. Mark Hasegawa-Johnson, 3/9/2019. CC-BY 3.0: You are free to share and adapt these slides if you cite the original

### tf.nn.softmax_cross_entropy_with_logits TensorFlow Core ..

1. softmax_with_cross_entropy¶ paddle.fluid.layers. softmax_with_cross_entropy (logits, label, soft_label = False, ignore_index = - 100, numeric_stable_mode = True, return_softmax = False, axis = - 1) ¶ 该OP实现了softmax交叉熵损失函数。该函数会将softmax操作、交叉熵损失函数的计算过程进行合并，从而提供了数值上更稳定的梯度值�
2. As you can see, the result of sigmoid cross entropy and softmax cross entropy are the same. This is mainly because sigmoid could be seen a sepcial case of sofmax.To sigmoid one number could equal to softmax two number which could sum to that num. loss3 are larger to the other loss
3. The softmax and the cross entropy loss fit together like bread and butter. Here is why: to train the network with backpropagation, you need to calculate the derivative of the loss. In the general case, that derivative can get complicated. But if you use the softmax and the cross entropy loss, that complexity fades away
4. For softmax_cross_entropy_with_logits, labels must have the shape [batch_size, num_classes] and dtype is float32 or float64. Labels used in softmax_cross_entropy_with_logits are the one hot version of labels used in sparse_softmax_cross_entropy_with_logits
5. Computes softmax cross entropy between logits and labels. Measures the probability error in discrete classification tasks in which the classes are mutually exclusive (each entry is in exactly one class). For example, each CIFAR-10 image is labeled with one and only one label: an image can be a dog or a truck, but not both
6. g the uniform distribution on labels. The connection provides an alternative view on the classifier as a mutual information estimator
7. I want to calculate the Lipschitz constant of softmax with cross-entropy in the context of neural networks. If anyone can give me some pointers on how to go about it, I would be grateful. Given a.

### Softmax function - Wikipedi

• Computing Cross Entropy and the derivative of Softmax. Follow 67 views (last 30 days) Brandon Augustino on 6 May 2018. Vote. 0 ⋮ Vote. 0. Answered: Greg Heath on 6 May 2018 Hi everyone, I am trying to manually code a three layer mutilclass neural net that has softmax activation in the output layer and cross entropy loss
• These values are simply used to demonstrate how the calculations of the Softmax classifier/cross-entropy loss function are performed. In reality, these values would not be randomly generated — they would instead be the output of your scoring function f
• dlY = crossentropy(dlX,targets) computes the categorical cross-entropy loss between the predictions dlX and the target values targets for single-label classification tasks. The input dlX is a formatted dlarray with dimension labels. The output dlY is an unformatted scalar dlarray with no dimension labels
• The current version of cross-entropy loss only accepts one-hot vectors for target outputs. I need to implement a version of cross-entropy loss that supports continuous target distributions. What I don't know is how to
• This operation computes the cross entropy between the target_vector and the softmax of the output_vector. The elements of target_vector have to be non-negative and should sum to 1. The output_vector can contain any values. The function will internally compute the softmax of the output_vector. Concretely
• The Softmax classifier gets its name from the softmax function, which is used to squash the raw class scores into normalized positive values that sum to one, so that the cross-entropy loss can be applied. In particular, note that technically it doesn't make sense to talk about the softmax loss, since softmax is just the squashing.

The softmax function outputs a categorical distribution over outputs. When you compute the cross-entropy over two categorical distributions, this is called the cross-entropy loss: [math]\mathcal{L}(y, \hat{y}) = -\sum_{i=1}^N y^{(i)} \log \hat{y.. Weak Crossentropy 2d. tflearn.objectives.weak_cross_entropy_2d (y_pred, y_true, num_classes=None, epsilon=0.0001, head=None). Calculate the semantic segmentation using weak softmax cross entropy loss. Given the prediction y_pred shaped as 2d image and the corresponding y_true, this calculated the widely used semantic segmentation loss. Using tf.nn.softmax_cross_entropy_with_logits is currently.

If you want to do optimization to minimize the cross entropy, AND you're softmaxing after your last layer, you should use tf.nn.softmax_cross_entropy_with_logits instead of doing it yourself, because it covers numerically unstable corner cases in the mathematically right way Softmax cross-entropy operation, returns the TensorFlow expression of cross-entropy for two distributions, it implements softmax internally. sigmoid_cross_entropy (output, target[, name]) Sigmoid cross-entropy operation, see tf.nn.sigmoid_cross_entropy_with_logits For more details, see Forward Loss Softmax Cross-entropy Layer. The backward loss softmax cross-entropy layer computes gradient values z m = s m - δ m, where s m are probabilities computed on the forward layer and δ m are indicator functions computed using t m, the ground truth values computed on the preceding layer

### chainer.functions.softmax_cross_entropy — Chainer 7.7.0 ..

• Return the cross-entropy between an approximating distribution and a true distribution. The cross entropy between two probability distributions measures the average number of bits needed to identify an event from a set of possibilities, if a coding scheme is used based on a given probability distribution q, rather than the true distribution p
• If you are designing a neural network multi-class classifier using PyTorch, you can use cross entropy loss (tenor.nn.CrossEntropyLoss) with logits output in the forward() method, or you can use negative log-likelihood loss (tensor.nn.NLLLoss) with log-softmax (tensor.LogSoftmax()) in the forward() method. Whew! That's a mouthful. Let me explain with some code examples
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### Derivation of the Gradient of the cross-entropy Los

# tf.nn.softmax_cross_entropy_with_logits cross_entropy = tf. nn. softmax_cross_entropy_with_logits (logits, ground_truth_input) This means that an image is within one task, i.e. one image is of one fruit which can be chosen from a set of fruits, but the image cannot be more than one fruit Softmax. To explain this Andre NG uses term hard-max vs soft-max; y_pred = exp(z_i) / sum_over_i ( exp(z_i) ) In softmax we output probability of various classes; In hardmax we will make one class as 1 and all others as 0 . Cross Entropy. It is a loss function; Loss = - sum [y_actual * log( y_pred ) ] Suppose y_actual is [1, 0, 0, 0, 0 Cross-entropy is commonly used to quantify the difference between two probability distributions. Now, when we develop a model for probabilistic classification, we aim to map the model's inputs to probabilistic predictions , and we often train our model by incrementally adjusting the model's parameters so that our predictions get closer and closer to ground-truth probabilities

### Rethinking Softmax with Cross-Entropy: Neural Network

Categorical Cross-Entropy Loss. Categorical Cross-Entropy loss. Also called Softmax Loss.It is a Softmax activation plus a Cross-Entropy loss.If we use this loss, we will train a CNN to output a probability over the C C C classes for each image. It is used for multi-class classification 2.2 Softmax cross-entropy loss. We analyze the softmax cross-entropy loss (softmax loss) from the viewpoint of mathemati-cal formulation. Given the logit vector f 2R. C. and the ground truth label y 2f1; ;Cg, the softmax loss is formulated as the following cross entropy between the softmax posterior an 이고, 를 정의하면 posterior probability는 다음과 같이 Softmax function의 꼴로 표현된다. 이처럼 Logistic function을 multi-class 문제로 일반화시키면 Softmax function을 얻을 수 있다. 이때문에 Softmax function을 multi-class logistic function이라고 하기도 한다. Cross-entropy loss와 ML The cross-entropy function is defined as. Here the T stands for target (the true class labels) and the O stands for output (the computed probability via softmax; notthe predicted class label). In order to learn our softmax model via gradient descent, we need to compute the derivativ Analytically computing derivative of softmax with cross entropy. This document derives the derivative of softmax with cross entropy and it gets: $$s_i - t_i$$ Which is different from the one derived using chain rule. Implementation using numpy

sparse_softmax_cross_entropy_with_logits (2) Non stai definendo i tuoi logits per il layer 10 softmax nel tuo codice, e dovresti farlo esplicitamente. Fatto ciò, è possibile utilizzare tf.nn.softmax, applicandolo separatamente a entrambi i tensori. Softmax is the only activation function recommended to use with the categorical crossentropy loss function. Strictly speaking, the output of the model only needs to be positive so that the logarithm of every output value $$\hat{y}_i$$ exists

### python - What is logits, softmax and softmax_cross_entropy

When we use the cross-entropy, the $\sigma'(z)$ term gets canceled out, and we no longer need worry about it being small. This cancellation is the special miracle ensured by the cross-entropy cost function. Actually, it's not really a miracle. As we'll see later, the cross-entropy was specially chosen to have just this property          • Oche bianche.
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