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Questions tagged [calculus]

For questions related to calculus (developed, among others, by Newton and Leibniz), in the context of AI (and, in particular, machine learning).

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41 views

What is the correct formula for updating the weights in a 1-single hidden layer neural network?

I'm creating a neural network with 3 layers and no bias. On internet I saw that the expression for the derivative of the weights between the hidden layer and the output layer was: $$\Delta W_{j,k} = (...
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1answer
30 views

How can the gradient of the weight be calculated in the viewpoint of matrix calculus?

Let $\sigma(x)$ be sigmoid function. Consider the case where $\text{out}=\sigma(\vec{x} \times W + \vec{b})$, and we want to compute $\frac{\partial{\text{out}}}{\partial{w} }.$ Set the dimension as ...
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0answers
50 views

BlackOut - ICLR 2016: need help understanding the cost function derivative

In the ICLR 2016 paper BlackOut: Speeding up Recurrent Neural Network Language Models with very Large Vocabularies, on page 3, for eq. 4: $$ J_{ml}^s(\theta) = log \ p_{\theta}(w_i | s) $$ They have ...
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23 views

During Backpropagation in LSTM, why is the previous output $h_{t-1}$ considered constant w.r.t any $W$ while computing derivative?

I've just started learning LSTM, and some points in the process of calculating the gradients are getting me confused. Say, for example, we want to compute $\frac{\partial}{\partial W_i}L$, where $L$ ...
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0answers
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For the generalised delta rule in back-propogation, do you subtract the target from the obtained output, or vice versa?

When I look up the generalised delta rule equation for back-propogation, I am seeing two conflicting equations. For example, here (slide 20), given $o$ (the output, defined in slide 18), $z$ (the ...
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1answer
43 views

What is the derivative of a specific output with respect to a specific weight?

If I have a neural network, and say the 6th output node of the neural network is: $$x_6 = w_{16}y_1 + w_{26}y_2 + w_{36}y_3$$ What does that make the derivative of: $$\frac{\partial x_6}{\partial w_{...
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1answer
105 views

How is the log-derivative trick of a trajectory derived?

I am looking at this formula which breaks down the gradient of $P(\tau |\theta)$ the first part is clear as is the derivative of $\log(x)$, but I do not see how the first formula is rearranged into ...
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0answers
30 views

Is there any wrong in my focal loss derivation?

Assume $\mathbf{X} \in R^{N, C}$ is the input of the softmax $\mathbf{P} \in R^{N, C}$, where $N$ is number of examples and $C$ is number of classes: $$\mathbf{p}_i = \left[ \frac{e^{x_{ik}}}{\sum_{j=...
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0answers
97 views

Is Gradient Descent algorithm a part of Calculus of Variations?

As in https://en.wikipedia.org/wiki/Calculus_of_variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and ...
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2answers
83 views

Which linear algebra book should I read to understand vectorized operations?

I am reading Goodfellow's book about neural networks, but I am stuck in the mathematical calculus of the back-propagation algorithm. I understood the principle, and some Youtube videos explaining this ...
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2answers
1k views

What is a bad local minimum in machine learning?

What is "bad local minima"? The following papers all mention this expression. Eliminating all bad Local Minima from Loss Landscapes without even adding an Extra Unit limination of All Bad Local ...
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1answer
99 views

Why is the change in cost wrt bias in neural network equal to error in the neuron?

While reading the book on neural networks by Michael Nielson, I had a problem understanding equation (BP3), which is $$ \frac{\partial C}{\partial b_{j}^{l}}=\delta_{j}^{l} \tag{BP3}\label{BP3}, $$ ...