Questions tagged [machine-learning]

How can we build computer systems that automatically improve with experience, and what are the fundamental laws that govern all learning processes?

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How do we derive the conditional distribution for a Poisson whose rate is the product of two Gamma distributed rv?

This question is motivated by Gopalan et al. "Content-based recommendations with poisson factorization." Advances in neural information processing systems 27 (2014). https://proceedings....
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SHAP values for Binary Classification

I'm trying to understand the inner workings of how SHAP values are calculated for Binary Classification. The formula for calculating each SHAP value is: $$ \phi_i = \sum_{S \subseteq F \setminus {i}} \...
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Area Under Precision-Recall and Area Under ROC curve for different amount of observations

I am doing a research and thus comparing some algorithms for binary classification. Worth to mention that, the data set is highly imbalanced i.e., the minority class is only 0.2%. Notation: Area Under ...
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Two definitions of a Stochastic Process?

I have two supposedly equivalent definitions of a stochastic process. A stochastic process is an indexed set of random variables. Specifically $$ y = \{y(x) \; | \; x \in \mathcal{X}\}. $$ ...
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How do I detect which of N objects appears in an image, and classify it's "state"?

Suppose I have a finite number of N objects O. In every image I, strictly 1 of these objects exist. And each object is in 1 of M states S. You can think of Q as a set R positions P, where each ...
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Question on Kantorovich-Rubinstein Duality proof

I am currently working on understanding the Kantorovich-Rubinstein duality and Wassertein loss. The following part of these class notes: Collecting the terms algebraically we can rewrite the ...
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Mathematical Definition of "Improvement"?

In the context of Bayesian Optimization, we model the Objective Function we are trying to optimize using a Gaussian Process. The location at which we evaluate the Objective Function at next is decided ...
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Does "Iterative Policy Evaluation" Guarantee the "Optimal Policy"?

I am reading about "Iterative Policy Evaluation" algorithm in the context of Reinforcement Learning (http://incompleteideas.net/book/ebook/node41.html). If I have understood this correctly - ...
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Interpreting model accuracy as proof of real association in Machine Learning

Let $\mathbb{X}, \mathbb{Y}$ be training and test sets respectively for some data we assume comes from a function$f$. Let $\hat{f}(\theta)$ be a model of $f$ with a parameter vector $\theta$. Assume ...
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Multiclass Classification: Why do we exponentiate the softmax function?

In the context of neural networks, we use the softmax output in multiclassification models. Firstly, let $P(y) = \sigma (z(2y-1))$, which comes from the definition of sigmoid units. We define $\bf z=\...
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Any reference related to regulating the variation of entropies?

I need some reference papers related to my problem. I have estimations as N normal distributions, but their variance tends to 0. It's because distributions are aggregated to one normal whose variance ...
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Are there $\| \cdot \|_{\infty}$ Kernels?

Some kernels used in machine learning are linked to metrics via the negative exponential function $f(t) = e^{-t^p}$. The most prominent example is the Gaussian RBF kernel $$K(x,y) = e^{-\sigma^2 \|x-y\...
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How "Optimal" are Solutions from Gradient Descent?

When optimizing the Loss Functions of Neural Networks using (some version of the) Gradient Descent algorithm, I have often heard this situation described as a Sequential Optimization Problem. This ...
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Can Gradient Descent be "Combined" with Dynamic Programming?

In most applications of Gradient Descent (e.g. optimizing the Loss Functions of Neural Networks) - regardless of the "type" of Gradient Descent algorithm being used (e.g. Stochastic Gradient ...
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Stochastic Mutual Information Estimator

I am reading https://openreview.net/forum?id=ByxaUgrFvH and do not understand why they need a "complicated" derivation, because it seems to follow immediately. Problem Let $\mathbf{x}$ be a ...
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How to Introduce LSTM systemically to autograd? [closed]

Among all the Neural Network structures that are introduced, RNN has received noticeable attention because of the state art included in its gradient computation with backpropagation. On the other hand,...
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Can Machine Learning models be considered as "Approximate Dynamic Programming"?

In the context of certain statistical/machine learning models, such as models that are trying to estimate "optimal policies" (e.g. reinforcement learning) - can we consider these models as &...
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Classification with Misclassification

I'm facing the following two-class classification problem. I'm told that some of the first class could actually be the second class. However, the second class is sure to be the second. Can someone ...
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Correct Understanding of Bellman Optimality

I was reading about the "Bellman Principle of Optimality" (https://en.wikipedia.org/wiki/Bellman_equation) : It seems that the "Bellman Principle of Optimality" state that for ...
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Does something "magical" happen at 20 Dimensions?

In the context of Bayesian Optimization, I have often heard that Bayesian Optimization tends to perform poorly on functions having more than 20 dimensions. However, I have never been able to ...
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From a conditional probability to an indefinite integral? Bayesian Information Criterion

I am trying to understand the Bayesian Information Criterion (BIC) using this article. At page 2 the following equality is given: $$P(y|M_1) = \int f(y|\theta_i)g_i(\theta_i)d\theta_i$$ with $y$ : ...
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Bochner's Theorem and Universal Kernels

Bochner's theorem asserts that a shift-invariant and properly scaled continuous kernel $K(x,y) = k(x-y)$ is positive definite (and hence a reproducing kernel of some RKHS) if and only if its Fourier ...
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Parameter Estimation in a Gaussian Environment

In Simon Haykin's Neural networks and learning machines Page 106. This paper defines a function and intends to derive it to obtain the target result w. $$ \mathscr E(\mathbf w) = \frac{\sum_i^N(d_i - \...
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Mini batches and loss in recurrent neural networks (RNNs)

Suppose that we have a sequence $\left\{x^{(k)}\right\}_{k = 1}^{N}$ and that we wish to use a RNN to predict the next element of the sequence given the previous elements of the sequence (e.g., a ...
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A question on the projection step of Generic Adaptive Method: $x_{t+1} = \Pi_{\mathcal{F},\sqrt{V_t}} (\hat{x}_{t+1}).$

I am reading the paper "ON THE CONVERGENCE OF ADAM AND BEYOND". In this paper, they proposed the following framework of adaptive methods. I was confused on the last step: $x_{t+1} = \Pi_{\...
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Support vector machine, complementary slackness and marginal hyperplane

One of the complementary slackness conditions for a support vector machine states that $$\alpha_i ( y_i (\langle w, x_i \rangle + b ) -1 ) = 0,$$ where $\alpha_i$ is the lagrange variable. One can ...
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Weight prediction of edge in direct graph

I have the following problem. Suppose we have a weighted directed graph $G$ with $J$ edges $(E_0...E_J)$ and $K$ nodes $(N_0...N_K)$. The structure of the graph is fully known (we can simplify and say ...
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Complexity of Lebesgue measurable spaces

Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite ...
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"Manipulating" Normal Distributions

I am reading the following book https://algorithmsbook.com/optimization/files/optimization.pdf at page 281: I am trying to understand how to manipulate the matrix terms to verify the following 2 ...
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Relationship Between Bayesian Optimization and Gaussian Process

In Bayesian Optimization, the function (i.e. objective function) that we are trying to optimize is modelled using some surrogate function - this surrogate function usually turns out to be a Gaussian ...
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Smooth traversal of 𝑆𝑂(𝑛)

I am trying to constrain the space of matrices used for the layers of a neural network to those in 𝑆𝑂(𝑛). It is proven that 𝑆𝑂(𝑛) is a manifold. I'm trying to find a way to smoothly traverse ...
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What was so "Groundbreaking" about Bellman's Equations?

In the context of Decision Making and Game Theory, "Bellman's Equations and Bellman's Conditions of Optimality" are said to be some of the most important mathematical principles in this ...
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Bagging Linear Model?

I have a question regarding bagging linear models. Suppose you wanna do linear regression on data X and y. Alice directly implements (OLS) Linear Regression on it. The model is A1. Bob applies bagging....
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Definition of "Bias" in Machine learning models

In the estimation of a parameter say the average of a population the definition of "bias" is very clear. It is the difference between the average estimator value (averaged over random ...
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Vector form of a polynomial in multiple variables.

I've been coming across various polynomial functions in my machine learning course; and I'm trying to understand what the most general form of these would look like. We started with a linear equation: ...
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1 answer
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Upper bound for Telescoping sum in gradient descent

I am studying a chapter in gradient descent . At some point we reach the sum in the left of the enequality and the writer says it's telescopic so this enequality holds: $\sum_{t=1}^T \Big( ||x_t - x^*|...
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Derivation in paper Deep Neural Networks as Gaussian processes in ICLR 2018

I am trying to understand the derivation of the main equation in the seminal paper titled Deep Neural Networks as Gaussian processes (in ICLR 2018). Following is the equation number (7), which can be ...
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Finding the dual of a problem $\min_{w,b} \sum_{j=1}^{n}{\max(0,w_j)}$

Consider the optimization problem for support vector machine with $(x_i,y_i),i=1,2,\ldots,m$ is the training data set $y_i=\{-1,1\}$ $w\in \mathbb R^n$ $\min_{w,b} \sum_{j=1}^{n}{\max(0,w_j)}\\\text{s....
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Derivation of gradient bandit in RL book 2020 (Sutton)

In the 'Gradient Bandit Algorithms' section, there is a derivation on how the following equations hold: (see here: original text) It says: "... substituted $Rt$ for $q(At)$, which is permitted ...
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Shattering of a set of binary classifiers

Let $S$ be a set, and let $\mathcal{F}_{S}=\{f:S\to\{-1,+1\}\}$ be a set of different label assignments. Show that $\mathcal{F}_{S}$ shatters at least $|\mathcal{F}_{S}|$ subsets of $S$. Here is what ...
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minimization of an entropy [closed]

Given $Ω = \{1,..,n \}$ which is uniformly distributed. Define a random Variable $X$, where the Entropy $H(X)$ is a minima.
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Why overparametrization is necessary if one wants to interpolate the data smoothly?

For my research work, I am reading this paper named A Universal Law of Robustness via Isoperimetry. Solving n equations generically requires only n unknowns. But I got stuck in this line However, the ...
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Proof of the information bottleneck equations

In The Information Bottleneck Method, the third term of Eq.(31) is $P_{t+1}(y|\tilde{x})=\sum_yp(y|x)p_t(x|\tilde{x})$, which minimizes the term $D_{KL}[p(y|x)|p(y|\tilde{x})]_{<p(x,\tilde{x})>}$...
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Why is $b/\|w\|$ the offset of the hyperplane to the origin?

There is one detail about hyperplanes I just cant prove myself (it's so simple a direct proof should work)... Claim: Given a hyperplane with (non normalized) normal vector $w$, $$ w\cdot x - b =0 , b\...
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Why in logistic regression for every threshold the decision boundary is a hyperplane?

I'm a beginner in machine learning study and I can't figure it out an exercise: The function h(x) = θ(w ̃x) is used to approximate ...
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Machine Learning normalization, loss and regularization

I am trying to get a more firm grasp of some central concepts in ML and how they correlate. Are the following concepts correctly understood. The need for regularization change when we change the loss ...
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Subgradient with the Frobenius norm

I'm working in the space of symmetric positive semi-definite matrices $S_n^+$ considered as a Hilbert space with respect to the inner product $\langle A,B \rangle = Tr(A^t B)$. I'm computing a ...
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Convergence of gradient descent for linear least squares

I'm trying to prove that when I use gradient descent for the least squares optimization problem $x^* = arg min_{x \in \mathbb{R}^n} \frac 1 2 \| Ax - b \|_2^2$ with the gradient descent rule $x^{t+1} ...
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How to learn data points by minimizing a loss function given their pairwise distance matrix?

Suppose $x_i \in\mathbb{R}^2$ for $i=1,2,...9$ are unknown. I'm given the pair-wise distance matrix between these points $D$ which is a $9*9$ symmetric matrix. I want to learn these data points by ...
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1 vote
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Need to analytically proof a fomula for variance induction [duplicate]

I am looking into a question about variance induction on an incremental dataset. To begin with, dataset $D_{n-1}$ contains elements $\{x_1, ..., x_{n-1}\}$, and we have got the values of: mean $\bar{...
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