Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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1answer
52 views

Understand the DDPG algorithm in Keras

I'm trying to understand the DDPG algorithm using Keras I found the site and started analyzing the code, I can't understand 2 things. The algorithm used to write the code presented on the page In the ...
3
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4answers
310 views

Can we define the AI singularity mathematically?

The "AI Singularity" or "Technological Singularity" is a vague term that roughly seems to refer to the idea of: Humans can design algorithms Humans can improve algorithms Eventually algorithms we ...
1
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1answer
32 views

How to understand mapping function of kernel?

For a kernel function, we have two conditions one is that it should be symmetric which is easy to understand intuitively because dot products are symmetric as well and our kernel should also follow ...
3
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4answers
215 views

Understanding a few terms in Andrew Ng's definition of the cost function for linear regression

I have completed week 1 of Andrew Ng's course. I understand that the cost function for linear regression is defined as $J (\theta_0, \theta_1) = 1/2m*\sum (h(x)-y)^2$ and the $h$ is defined as $h(x) = ...
6
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2answers
679 views

Why is the log probability replaced with the importance sampling in the loss function?

In the Trust-Region Policy Optimisation (TRPO) algorithm (and subsequently in PPO also), I do not understand the motivation behind replacing the log probability term from standard policy gradients $$...
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1answer
53 views

Formal proof that every purely reactive agent has behaviorally equivalent standard agent

It kind of makes sense intuitively but I'm not sure about a formal proof. I'll start with briefly listing definitions from Intro to Multiagent systems, Wooldridge, 2002 and then give you my reasoning ...
2
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1answer
119 views

Can you help me understand how weight normalization works?

I am trying to dissect the paper Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks. Unfortunately, because my math is a little bit rusty, I got a little ...
1
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1answer
66 views

What is the definition of the “cost” function in the SVM's objective function?

In a course that I am attending, the cost function of a support vector machine is given by $$J(\theta)=\sum_{i=1}^{m} y^{(i)} \operatorname{cost}_{1}\left(\theta^{T} x^{(i)}\right)+\left(1-y^{(i)}\...
4
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2answers
828 views

How good is AI in math?

Currently, AI is advancing fast in deep learning: Entire human chess knowledge learned and surpassed by DeepMind's AlphaZero in four hours. As a layman, I'm taking this as a quite powerful searching ...
2
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1answer
88 views

How can I learn tensors for deep learning?

I've seen in most deep learning papers use tensors. I understood what tensors are, but I want to dive into them, because I think that might be beneficial for further studies in Artificial Intelligence....
5
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1answer
152 views

How could an AI be used to improve the teaching and learning of mathematics?

I have been working with AI methods. I am thinking about how my daughter (and also other kids) could learn mathematics with the help of AI. For example, how could an AI be used to show the mistakes ...
1
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1answer
74 views

Why do we use the word “kernel” in the expression “Gaussian kernel”?

I've heard the expression "Gaussian kernel" in several contexts (e.g. in the kernel trick used in SVM). A Gaussian kernel usually refers to a Gaussian function (that is, a function similar to the ...
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2answers
199 views

In the context of importance sampling ratio, how is the equation $\mathbb{E}\left[\rho_{t: T-1} G_{t} | S_{t}=s\right]=v_{\pi}(s)$ derived?

When reading the book by Sutton and Barto, I came across the importance sampling ratio. The first equation, I believe, describes the probability a particular sequence is obtained given the current ...
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3answers
6k views

How to choose an activation function?

I choose the activation function for the output layer depending on the output that I need and the properties of the activation function that I know. For example, I choose the sigmoid function when I'm ...
5
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1answer
128 views

Can deep learning be used to help mathematical research?

I am currently learning about deep learning and artificial intelligence and exploring his possibilities, and, as a mathematician at heart, I am inquisitive about how it can be used to solve problems ...
2
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2answers
172 views

Why is MSE used over other quadratic loss functions?

So I was wondering, why I have only encountered square loss function also known as MSE. The only nice property of MSE I am so far aware of is its convex nature. But then all equations of the form $x^{...
4
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1answer
60 views

How can a single sample represent the expectation in gradient temporal difference learning?

I was reading the gradient temporal difference learning version 2(GTD2) from rich Sutton's book page-246. At some point, he expressed the whole expectation using a single sample from the environment. ...
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2answers
3k views

What is the derivative of the Leaky ReLU activation function?

I am implementing a feed-forward neural network with leaky ReLU activation functions and back-propagation from scratch. Now, I need to compute the partial derivatives, but I don't know what the ...
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0answers
55 views

What is an auto-associator?

What is an auto-associator, and how does it work? How can we design an auto-associator for a given pattern? I couldn't find a clear explanation for this anywhere on the internet. Here's an example of ...
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0answers
41 views

Simplifying Log Loss

I am reading through a paper (https://www.mitpressjournals.org/doi/pdf/10.1162/0891201053630273) where they describe logloss as a ranking function and can be simplified to the margin of the training ...
2
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2answers
89 views

Is ReLU a non-linear activation function?

According to this blog post The purpose of an activation function is to add some kind of non-linear property to the function The sigmoid is typically used as an activation function of a unit of a ...
2
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1answer
135 views

Choice of fuzzification function

I'm a relative newbie to fuzzie logic systems but I have some knowledge in mathematics. I have the following problem: I want to fuzzify certain values. Some are in the range [-$\inf$,$\inf$] and some ...
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3answers
4k views

Can neural networks be used to prove conjectures?

Imagine I have a list (in a computer-readable form) of all problems (or statements) and proofs that math relies on. Could I train a neural network in such a way that, for example, I enter a problem ...
2
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0answers
69 views

Is this the correct gradient for log of softmax? [duplicate]

I am currently implementing the very basic version (REINFORCE) of the Monte Carlo policy gradient algorithm. I was wondering if this is the correct gradient for the log of softmax. \begin{align} \...
1
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1answer
57 views

Why does the discriminator minimize the cross-entropy while the generator maximize it?

In his original GAN paper Goodfellow gives a game theoretic perspective for GANs: \begin{equation} \underset{G}{\min}\, \underset{D}{\max}\, V\left(D,G \right) = \mathbb{E}_{x\sim\mathit{p}_{\...
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1answer
45 views

Given an axis-angle rotation vector, how can I find the unit rotation axis and angle?

I have a robotics assignment, which I am unable to solve. Given the axis-angle rotation vector $\Theta = (2, 2, 0)$, how can I calculate the unit vector of the rotation axis $k$ and the angle $\theta$?...
4
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1answer
69 views

How is the Jacobian a generalisation of the gradient?

I came across these slides Natural Language Processing with Deep Learning CS224N/Ling284, in the context of natural language processing, which talk about the Jacobian as a generalization of the ...
3
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1answer
186 views

How does one even begin to mathematically model an AI algorithm?

How does one even begin to mathematically model an AI algorithm, like alpha-beta pruning or even its thousands of variations, to determine which variation is best?
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0answers
50 views

Does Gödel's second incompleteness theorem put a limitation on artificial intelligence systems?

According to Brian Cantwell Smith no calculation without representation Therefore, computers depend on models. So, we can say that AI is limited internally by the model and external by the ...
8
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2answers
880 views

How do we prove the n-step return error reduction property?

In section 7.1 (about the n-step bootstrapping) of the book Reinforcement Learning: An Introduction (2nd edition), by Andrew Barto and Richard S. Sutton, the authors write about what they call the "n-...
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0answers
35 views

How do I approach this problem?

Let's say I have a dataset with multiple types of multiple ingredients (salt1,salt2, etc). Each n-th variation of each ingredient vs flavor may be represented by an n×k matrix that where an ingredient ...
2
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1answer
56 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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1answer
42 views

Why are all weights of a neural net updated and not just the weights of the first layer

Why are all weights of a neural net updated and not only the weights of the first hidden layer? The error-influence of the prediction by the weights of a neural net is calculated using the chain rule....
4
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1answer
203 views

Defining formula for fuzzy equation

I'm learning fuzzy logic and more or less understand the basic concept, but i'm having a hard time understanding how to apply it to a method. I tried browsing online for explanation on how to use it, ...
2
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1answer
64 views

What are the challenges faced by using NLP to convert mathematical texts into formal logic?

From what I've figured (a) converting mathematical theorems and proofs from English to formal logic is a straightforward job for mathematicians with sufficient background, except that it takes time. ...
2
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1answer
70 views

Why is my derivation of the back-propagation equations inconsistent with Andrew Ng's slides from Coursera?

I am using the cross-entropy cost function to calculate its derivatives using different variables $Z, W$ and $b$ at different instances. Please refer image below for calculation. As per my knowledge, ...
2
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1answer
70 views

In a single neuron output layer should the output be a scalar?

Given a neural network with 3 inputs, 4 hidden layers, and 1 output, should the output neuron be a vector or a scalar? I thought that at the end of the summation only one number between 0 and 1 would ...
2
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1answer
49 views

How can I implement policy evaluation when reward is tied to an action outcome?

I'm following Stanford reinforcement learning videos on youtube. One of the assignments asks to write code for policy evaluation for Gym's FrozenLake-v0 environment. In the course (and books I have ...
5
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1answer
169 views

What is a weighted average in a non-stationary k-armed bandit problem?

In the book Reinforcement Learning: An Introduction (page 25), by Richard S. Sutton and Andrew G. Barto, there is a discussion of the k-armed bandit problem, where the expected reward from the bandits ...
1
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1answer
60 views

Understanding relation between VC Symmetrization Lemma and Generalization Bounds

I am new in the field of Machine Learning so I wanted to start of by reading more about mathematics and history behind it. I am currently reading, in my opinion, a very good and descriptive paper on ...
2
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1answer
40 views

What is the mean in the variational auto-encoder?

Here's a diagram of a variational auto-encoder. There are 2 nodes before the sample (encoding vector). One is the mean, one is the standard deviation. The mean one is confusing. Is it the mean of ...
2
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1answer
43 views

Is the derivative of the loss wrt a single scalar parameter proportional to the loss?

I am wondering about the correlation between the loss and the derivative of the loss wrt a single scalar parameter, with the same sample. That means: considering a machine learning model with ...
3
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0answers
40 views

Is maximum likelihood estimation meaningless for a dataset of only outliers?

From my understanding, maximum likelihood estimation chooses the set of parameters for the estimator that maximizes likelihood with the ground truth distribution. I always interpreted it as the ...
1
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1answer
43 views

What does equation in the “related work” section of the GAN paper mean?

I was going through the paper on GAN by Ian Goodfellow. Under the related work section, there is an equation. I cannot decipher the equation. Can anyone help me understand the meaning of the equation? ...
10
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1answer
3k views

What is the Bellman operator in reinforcement learning?

In mathematics, the word operator can refer to several distinct but related concepts. An operator can be defined as a function between two vector spaces, it can be defined as a function where the ...
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0answers
15 views

How do I decide which norm to use for placing a constraint on my adversarial perturbation?

I am performing an adversarial machine learning attack on a neural network for network traffic classification. For adding adversarial perturbations in features such as packet interarrival times and ...
2
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0answers
25 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=...
3
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1answer
75 views

Why does variational auto-encoder use the reconstruction loss?

VAE is trained to reduce the following two losses. KL divergence between inferred latent distribution and Gaussian. the reconstruction loss I understand that the first one regularizes VAE to get ...
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0answers
37 views

Why are conics important in computer vision?

The book Multiple View Geometry in Computer Vision by Richard Hartley and Andrew Zisserman talks about lines, points and conics. A conic is a curve described by a second-degree equation in the plane, ...
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0answers
53 views

Derivation of regularized cost function w.r.t activation and bias

In regularzied cost function a L2 regularization cost has been added. Here we have already calculated cross entropy cost w.r.t $A, W$. As mentioned in the regularization notebook (see below) in ...