Questions tagged [math]
For questions about mathematics related to artificial intelligence.
154
questions
0
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1answer
26 views
Can a computer make a proof by induction?
Can a computer solve the following problem, i.e. make a proof by induction? And why?
Prove by induction that $$\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2, \, \, \, \forall n\in\mathbb N .$$
I'm ...
2
votes
1answer
56 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 ...
1
vote
1answer
67 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)}\...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
5
votes
1answer
129 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
votes
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}
\...
2
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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 ...
4
votes
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 ...
1
vote
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
votes
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 ...
4
votes
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. ...
0
votes
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....
1
vote
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
votes
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 ...
2
votes
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.
...
1
vote
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? ...
3
votes
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 ...
2
votes
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 ...
1
vote
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
votes
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=...
2
votes
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 ...
3
votes
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 ...
1
vote
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, ...
0
votes
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 ...
2
votes
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
votes
1answer
71 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 ...
3
votes
0answers
57 views
Interpretation of inverse matrix in mean calculation in Gaussian Process
The formula for mean prediction using Gaussian Process is $k(x_*, x)k(x, x)^{-1}y$, where $k$ is the covariance function. See e.g. equation 2.23 (in chapter 2) from Gaussian Processes for Machine ...
1
vote
1answer
124 views
Understanding the derivation of the first-order model-agnostic meta-learning
According to the authors of this paper, to improve the performance, they decided to
drop backward pass and using a first-order approximation
I found a blog which discussed how to derive the math ...
2
votes
2answers
96 views
Is there a possibility that there is no relationship between some inputs and outputs?
I'm doing machine learning projects. I took a look at many datasets I worked with, mostly there are already famous datasets that everyone uses.
Let's say I decided to make my own dataset. Is there a ...
2
votes
0answers
38 views
Expected duration in a state
I am going through Rabiner 1989 and he writes that the discrete probability density function of duration $d$ in state $i$ (that is, staying in a state for duration $d$, conditioned on starting in that ...
1
vote
1answer
39 views
What are the variables used in a Gaussian radial basis kernel in the context of SVMs?
If I have the Gaussian kernel
$$
k(x, x') = \operatorname{exp}\left( -\| x - x' \|^2 / 2\sigma^2 \right)
$$
What is $x$ and $x'$ in the context of training an SVM?
2
votes
1answer
80 views
How is the expected value in the loss function of DQN approximated?
In Deep Q Learning the parametrized Q-functions $Q_i$ are optimised by performing gradient descent on the series of loss functions
$L_i(\theta_i)= E_{(s,a)\sim p}[(y_i-Q(s,a;\theta_i))^2]$ , where
...
1
vote
2answers
42 views
What are the conditions for the convergence of SARSA to the optimal value function?
Is it correct that for SARSA to converge to the optimal value function (and policy)
The learning rate parameter $\alpha$ must satisfy the conditions:
$$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}...
1
vote
1answer
30 views
What is the equation of the separation line for this neuron with identity activation?
I have a single neuron with 2 inputs, and identity activation, where f is activation function and u is output:
$u = f(w_1x_1 + ...
1
vote
1answer
78 views
Does TD(0) prediction require Robbins-Monro conditions to converge to the value function?
Does the learning rate parameter $\alpha$ require the Robbins-Monro conditions below for the TD(0) algorithm to converge to the true value function of a policy?
$$\sum \alpha_t =\infty \quad \text{...
3
votes
1answer
55 views
In the policy gradient equation, is $\pi(a_{t} | s_{t}, \theta)$ a distribution or a function?
I am learning about policy gradient methods from the Deep RL Bootcamp by Peter Abbeel and I am a bit stumbled by the math presented. In the lecture, he derives the gradient logarithm likelihood of a ...
1
vote
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 ...
0
votes
0answers
43 views
Understanding V- and Q-functions
Assume the existence of a Markov Decision Process consisting of:
State space $S$
Action space $A$
Transition model $T: S \times A \times S \to [0,1]$
Reward function $R: S \times A \times S \to \...
4
votes
2answers
120 views
Mathematical foundations of the ability to learn
I am an undergraduate student in applied mathematics with an interest in artificial intelligence. I am currently exploring topics where I could do research. Coming from a mathematical background I am ...
2
votes
1answer
41 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 ...
1
vote
1answer
50 views
Why is Standard Deviation based on L2 Variance and not L1 Variance
Standard deviation and variance are in statistics but the formula for variance is somehow related to the L1 and L2.
Mathematically (L2 in machine learning sense),
$$Variance = \dfrac{(X_1-Mean)^2+..+(...
1
vote
0answers
43 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 ...
0
votes
1answer
38 views
How to understand the average l2 loss?
In the snippet below, the highlighted part is the average norm, but since $1/|p_i|$ is outside the summation, it is very confusing to understand.
is $|p_i|$ l2-norm(as per wolfram) or l1-norm or ...
3
votes
1answer
85 views
What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?
What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$? I think $|x|$ is the magnitude of $x$.
2
votes
0answers
70 views
How does the update rule for the one-step actor-critic method work?
Can you please elucidate the math behind the update rule for the critic? I've seen in other places that just a squared distance of $R + \hat{v}(S', w) - \hat{v}(S,w)$ is used, but Sutton suggests an ...
6
votes
2answers
195 views
Can we get the inverse of the function that a neural network represents?
I was wondering if it's possible to get the inverse of a neural network. If we view a NN as a function, can we obtain its inverse?
I tried to build a simple MNIST architecture, with the input of (784,...
6
votes
1answer
268 views
What is the mathematical definition of an activation function?
What is the mathematical definition of an activation function to be used in a neural network?
So far I did not find a precise one, summarizing which criterions (e.g. monotonicity, differentiability, ...
4
votes
0answers
71 views
Is there a mathematical formula that describes the learning curve in neural networks?
In training a neural network, you often see the curve showing how fast the neural network is getting better. It usually grows very fast then slows down to almost horizontal.
Is there a mathematical ...
