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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Is my calculation of the partial derivative of the cost function with respect to a single weight in the first layer correct?

I'm trying to understand the chain rule of backpropagation. This is what I understood. Is it correct? $$ \frac{\partial E }{ \partial w} = \sum_{i} \frac{\partial E }{ \partial a_i^{(l)} } (\sum_{j} \...
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45 views

How the vector-space isomorphism between $\mathbb{R}^{m \times n}$ and $\mathbb{R}^{mn}$ guarantees reshaping matrices to vectors?

Consider the following paragraph from section 5.4 Gradients fo Matrices of the chapter Vector Calculus from the textbook titled Mathematics for Machine Learning by Marc Peter Deisenroth et al. Since ...
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Is there any difference between vector-valued function and vector field at least for the usage in AI?

I encountered the following statement from the chapter named Vector Calculus from the test book titled Mathematics for Machine Learning by Marc Peter Deisenroth et al. In the following, we will ...
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Which is more popular/common way of representing a gradient in AI community: as a row or column vector?

Consider the following remark about writing gradients from the chapter named Vector Calculus from the test book titled Mathematics for Machine Learning by Marc Peter Deisenroth et al. Remark (...
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What is the rigorous and formal definition for the direction pointed by a gradient?

Consider the following definition of derivative from the chapter named Vector Calculus from the test book titled Mathematics for Machine Learning by Marc Peter Deisenroth et al. Definition 5.2 (...
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RNN - Backpropagation through time - Gradient Calculation

I think I got it right after reading multiple resources but im still not 100%. Seems like everyone is calculating it different. Or they just shortcut explaining the calculation. (or my math skills ...
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23 views

How did authors ensure that critical points do exist in GAN?

Using an MLP as a generator introduces multiple critical points in parameter space. You can read this excerpt from the research paper titled Generative Adversarial Nets by Ian J. Goodfellow et al. In ...
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65 views

What are the Calculus books recommended for beginner to advanced researchers in artificial intelligence?

Calculus is a branch of mathematics that primarily deals with the rate of change of outputs of a function w.r.t the inputs. It contains several concepts including limits, first-order derivatives, ...
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68 views

How many directions of gradients exist for a function in higher dimensional space?

Gradients are used in optimization algorithms. Based on the values of gradients, we generally update the weights of a neural network. It is known that gradients have a direction and the direction ...
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33 views

What all does the gradient tells us?

Gradients are used in optimization algorithms. I know that a gradient gives us information about the direction in which one needs to update the weights of a neural network. We need to travel in the ...
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74 views

Why the partial derivative is $0$ when $F_{ij}^l < 0$?. Math behind style transfer

I am currently in the process of reading and understanding the process of style transfer. I came across this equation in the research paper which went like - For context, here is the paragraph - ...
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Are calculus and differential geometry required for building neural networks?

I've been studying geometry and linear algebra for months with the goal to build neural networks. But now I'm reading that perceptrons require fitting curves, and curves are not expressed as linear ...
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56 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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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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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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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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52 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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139 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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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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137 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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107 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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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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Why is the derivative of this objective function 0 if the policy is deterministic?

In the Berkeley RL class CS294-112 Fa18 9/5/18, they mention the following gradient would be 0 if the policy is deterministic. $$ \nabla_{\theta} J(\theta)=E_{\tau \sim \pi_{\theta}(\tau)}\left[\left(\...
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348 views

Are my computations of the forward and backward pass of a neural network with one input, hidden and output neurons correct?

I have computed the forward and backward passes of the following simple neural network, with one input, hidden, and output neurons. Here are my computations of the forward pass. \begin{align} net_1 &...
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104 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 $$ \frac{\partial C}{\partial b_{j}^{l}}=\delta_{j}^{l} \tag{BP3}\label{BP3}, $$ ...