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Questions tagged [lagrange-multipliers]

The method of Lagrange multipliers finds critical points (including maxima and minima) of a differentiable function subject to differentiable constraints.

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Derivation of dual formulation of support vector regression

I'm trying to derive the dual formulation of epsilon-insensitive support vector regression. I think my derivation is correct, but I can't match it up to a result for the dual that I've seen given in ...
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Likelihood-ratio and score tests of a (non)linear combination of coefficients

The likelihood-ratio and score test are typically used for simple scalar hypotheses such as $\beta_1 = 0$ or $\beta_1 = \beta_2 = 0$. How can we test a linear combination of coefficients using the ...
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Support Vector Classifiers for Overlapping Classes

I am currently studying support vector classifiers (SVC), more specifically, the solution to the Lagrangian (Wolfe) dual function with the help of the book "The Elements of Statistical Learning&...
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How is the Representer theorem used in the derivation of the SVM dual form?

This is the primal form of the SVM hypothesis : $$ h _{\mathbf{\vec w}, b}(\mathbf{\vec x}^{(i)}) = \mathbf{\vec w}\cdot \mathbf{\vec x}^{(i)} + b $$ The Representer theorem as formulated here ...
Sagnik Taraphdar's user avatar
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Necessary condition for constrained optimization

Suppose $X=(X_1,\cdots,X_k)$ follows the multinomial distribution with a known size $n$ and an unknown probability vector $(p_1,\cdots,p_k)$. Find the necessary conditions for the solution to the ...
Nothing's user avatar
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Estimating the parameter of a Bernoulli distribution using probabilistic modeling and the MAP estimation

Suppose you tossed a coin multiple times. Sometimes you got heads and other times you got tails. You recorded your experiment in a dataset $ X$. Now you want to estimate the parameter θ (which ...
Mosab Shaheen's user avatar
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Why is there only a box constraint on alpha and not on mu when solving the dual problem of soft linear SVM?

I am currently learning about the linear SVM in the non-separable case. In the dual representation, we introduce the Lagrange multipliers μk and αk (see also this source: https://...
kate allerton's user avatar
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Are solutions to the Lagrangian multipliers ($\alpha_i$) in a hard-margin SVM unique?

An intermediate step in the derivation of the hard-margin SVM's dual form is as follows: I also know that $a_i$ for all points not on the margin boundary is 0, which makes sense; they must be zeroed ...
Each One Chew's user avatar
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Min max formulation conversion to max min formulation. Reason?

Question is based on the screenshot attached. Based on paper here. I am not being able to understand why min max formulation (eq 4) is first converted to max min formulation (eq 5). Is it something ...
Curious's user avatar
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Subspace test for multivariate normal distribution [duplicate]

Subspace test for multivariate normal distribution Suppose $X_1, X_2,\ldots, X_n$ are i.i.d. observations from a multivariate normal distribution $N(\mu,\Sigma)$ where $\Sigma$ is known. Furthermore, ...
Fros's user avatar
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Can we convert the optimization of a loss function with regularization to the Lagrangian, constrained optimization *before* solving the optimization?

It is shown here that the optimization of a loss function with regularization, $$\text{argmin}_b L(X,b) + c ||b||_p \phantom{aaaaaaaaaaaaaaaaaaaaaaaa} (*)$$ is equivalent to the constrained ...
travelingbones's user avatar
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Cannot understand the bound of Lagrangian parameter in SMO

I'm trying to understand SMO, but stuck to the part of bound for Lagrangian parameters. In the SMO paper(https://www.microsoft.com/en-us/research/uploads/prod/1998/04/sequential-minimal-optimization....
Hayeon Park's user avatar
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How to maximize the ELBO in coordinate ascent variational inference

In the lecture by D.Blei: https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/variational-inference-i.pdf Variational inference is explained and he shows how to derive the optimal ...
sam's user avatar
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deriving the optimal distribution

Let the input variable $X \in \mathcal{X}$ and the target variable $Y \in \mathcal{Y}$. For a fixed hypothesis $h \in \mathcal{H}$ I want to solve \begin{equation} \min_{p(X,Y)} \int_{\mathcal{X}}\...
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incorporating a distribution constraint in a minimisation objective

For a given (convex) hypothesis $h \in \mathcal{H}$, and the variables $X \in \mathcal{X}$ and $Y \in \mathcal{Y}$ I have the following optimisation problem: \begin{equation} \min_{p(X,Y)} \int_{\...
appa's user avatar
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Derivation of the SVM regression problem

EDIT: The terms at the end will not get cancelled as they involve different variables: $\alpha$ and $\alpha^*$, etc. Now the post is corrected. I will leave this question here in case someone is as ...
Misery's user avatar
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Why $\gamma$ in regularization term of XGBoost is defined as minimum loss reduction (not minimum squared loss reduction) and not substracted?

From the source https://xgboost.readthedocs.io/en/stable/tutorials/model.html I guess that the mean-squared error is optimized subjected to a constraint of minimum loss reduction. It appears like ...
Dara's user avatar
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KKT Conditions for thresholds?

My main question is that when I use Lagrange Multipliers/KKT conditions to perform optimization with threshold constraints, I seem to get contradictory FOC. Here is a characteristic example: take an ...
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How to proof that the square of the half-margin of maximum-margin SVM is equal to the summation of lagrange multipliers given by SVM dual?

Can be show that the square of the half-margin of maximum-margin SVM is equal to the summation of Lagrange multipliers given by SVM dual? i.e, Let $ \rho = \dfrac{1}{w}\\ $ Show that: $$ \dfrac{1}{\...
Tuhin Dutta's user avatar
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RSM and steepest ascent for new $x'$ coordinates

In Myers, Montgomery, & Anderson-Cook, C. M. (2016)$^1$ the authors provide (p. 235) a general methodology to find new $x'$ coordinates in the path of steepest ascent based on constrained ...
jealcalat's user avatar
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Gradient descent finds local minima for a problem that can be formulated as a convex problem

I am trying to find $$ \min_W \|Y-XW \|_F^2$$ $$s.t. \exists ij, W_{ij}\geq0 $$ where X is input data and Y is the output data we try to fit to. This is a convex optimization problem that can be ...
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Rewriting Lagrange Multiplier with non-linear optimization

NOTE: if you have access to the book mentioned below , please skip to the part of this post marked with (*), as I will now provide a short summary of the relevant background information for my ...
OliverVD's user avatar
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convex optimisation formulation of SVM

I am currently learning about Support Vector Machine's (SVM) from the CS229 Stanford Class. In page 16 of the notes, they transformed $$max_{\hat{\gamma}, w, b} \frac{\hat{\gamma}}{||w||} \\ s.t \,\,\,...
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How to derive the $\ nR^2 $ form of the LM test statistic?

Several lagrange multiplier (LM) test statistics =$\ nR^2 $~$ \chi^2 $, where the LM test statistic is generated from regressing the square of residuals on some function in an auxiliary equation. The ...
Garp's user avatar
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Lagrange multiplier test statistic math

I'm trying to understand the how the Lagrange multiplier test statistic written like this: $$LM = \hat{d'}_1 (\hat{\phi}_{11}-\hat{\phi}_{12}\hat{\phi}_{22}^{-1}\hat{\phi}_{21})^{-1}\hat{d}_1 \...
Garp's user avatar
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Optimization problem for SVM with hard margin

I am trying to implement the SVM with hard margin from scratch. I have reduced the problem to finding the solution from the following optimisation problem $$ \operatorname{max}\limits_{\alpha}L(\alpha)...
Gaussian's user avatar
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Score test using restricted mle

I am doing a study on Rao score test and still confused about the score vector. How am I supposed to calculate the score vector? Can I just use (ˆθn - θn) for the score vector? As I understood that ...
zira's user avatar
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Rewrite $\frac{1}{2}||x-u||_2^2$ subject to $||x||_1\le c$ to lagrangian form with multiplier $\lambda \ge 0$

Rewrite $\frac{1}{2}||x-u||_2^2$ subject to $||x||_1\le c$ to lagrangian form with multiplier $\lambda \ge 0$ So I'm pretty new to converting constraint functions to Lagrangian form, but I read that ...
user8714896's user avatar
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1 answer
101 views

Non-negative Constraints in Soft-Margin SVM Lagrange Equation

I was reading the A Tutorial on Support Vector Machines for Pattern Recognition as a supplemental for my Intro to ML class and I wasn't sure why $ a_i \geq 0 $ and $ \lambda_i \geq 0 $ cannot be ...
g999's user avatar
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Finding the MLEs for pi in a contingency table proof

Reference: https://data.princeton.edu/wws509/notes/c5.pdf In a two-dimensional table, (the multinomial model - section 5.1.2 in the link above), I was wondering how they get $$\hat\pi_{i.} = \frac{y_{...
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204 views

Logarithms in Karush-Kuhn-Tucker conditions

I have the following optimization (and standarised) problem: $$minimize\ -(x\ ln(x)\ +\ y\ ln(y))$$ $$subject\ to\ \ \ \ \ \ \ \ \ \ \ \ \ x+y-1<=0$$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...
Carlos Vázquez Losada's user avatar
2 votes
1 answer
340 views

Maximum Entropy Discrete Distribution

In Pattern Recognition and Machine Learning the author uses Lagrange multipliers to find the discrete distribution with maximum entropy. Entropy is defined by; $$H=-\sum_i p(x_i)\ln(p(x_i))$$ and the ...
tail_recursion's user avatar
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What is $\alpha$ in SVM Lagrangian fucntions

Is it a column matrix or a row matrix? I saw on a CMU slide that $\alpha$ is defined as $\alpha=(\alpha_1, \alpha_2, ..., \alpha_n)^T>=0$? I dont see much of a formal definition for the exact form ...
cIckalakba's user avatar
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1 answer
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Convert the following expression w.r.t to the whole dataset instead of element of the dataset?

I am in the process of expressing the w in LSSVM with data points and constants. After I resolve the KKT conditions for the LSSVM I got $$w = \sum ^N _{i=1} \alpha_i x_i$$ Is it possible to convert ...
cIckalakba's user avatar
3 votes
1 answer
183 views

Minimize $f(A,B)$ s.t. $\text{exp}(A)^T \text{exp}(B)=J_K$

I have a function $f(A,B)$ that maps a pair of two (tall) matrices, $A$ and $B$, to a scalar cost that I want to minimize. $A$ and $B$ both have $K$ columns. I also want to impose a set of equality ...
Ruben van Bergen's user avatar
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Lagrange Multiplier Test (Score Test) reliability at low likelihood

The Lagrange Multiplier Test (Score Test) in the multivariate case has the following formula: $$ LM = [\underline{s}(\underline{\theta_0})]^T*[Var(\underline{\theta_0})]^{-1}*\underline{s}(\underline{\...
cascom's user avatar
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What is the maximum entropy distribution with a mean of 0, stdev of 1, skew of 1 (instead of zero), and kurtosis of 1 (instead of 3)?

As we know, there are an infinite number of distributions with mean of zero and variance of one. One of the special things about the normal distribution is that is has the maximum entropy. (https://...
Yaoshiang's user avatar
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2 answers
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How to solve MNP (minimum norm) problem in SVM?

I'm reading an article, which says that MNP (minimum norm problem) can be solved as SVM. In the minimum norm problem, we're given a set of points in $R^d$ and need to find a point in convex hull of ...
taciturno's user avatar
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Minimizing Mean Square Error

Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & ...
dsakiocxla's user avatar
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88 views

What are Lagrange multiplier and Dual decomposition methods?

What are the Lagrange multiplier and Dual decomposition methods? How are these methods used in Machine Learning(ML) for Optimization?
Thalassophile's user avatar
1 vote
1 answer
627 views

Are Lagrange multipliers of regularization terms learned or set?

Background knowledge I knew about regularization in machine learning: A regularization term is added to the objective function to prevent overfitting. Usually, one would like to restrict the variance ...
chichi's user avatar
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How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression?

How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression? Given lasso (constrained form): $$\underset{\beta}{\min}{(\frac{1}{2N}||y-x\beta||_2^2)} \...
FantasticAI's user avatar
2 votes
2 answers
6k views

Calculating the value of $b^{*}$ in an SVM

In Andrew Ng's notes on SVMs, he claims that once we solve the dual problem and get $\alpha^*$ we can calculate $w^*$ and consequently calculate $b^*$ from the primal to get equation (11) (see notes) ...
Gerard's user avatar
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If the Lagrangian formulation also has constraints, what is then the simplification?

Consider the following constrained optimization problem. $$\begin{array}{ll} \text{minimize} & f(w)\\ \text{subject to} & g_{i}(w) \leq 0\\ & h_{j}(w) = 0\end{array}$$ The equivalent ...
Silpara's user avatar
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Optimal proposal for the Metropolis-Hastings algorithm using Tierney's theorem

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\lambda p\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $q:E^2\to[0,\...
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Solving constrained optimization problems with Adam

The adam algorithm has been very successful for solving non-convex optimization problems that appear in deep learning. Are there ways to extend adam to solve constrained optimization problems? Among ...
Hari's user avatar
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1 answer
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why in SVM we have different indices for dot product?

I am confused by Lagrangian method in SVM, I can not understand why we use different indices in dot product. Suppose with using Lagrangian W is : $ W_{i}=\sum_{i}L_{i}y_{i}x_{i} $ In SVM ...
user9272398's user avatar
1 vote
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125 views

why we use Lagrangian method in SVM?

I am wondering why we are using Lagrangian method in SVM? if we have just 3 features and 1000 rows, then objective function would be $w_{1}^{2}+w_{2}^{2}+w_{3}^{2}$ but if we use Lagrangian then we ...
sherek_66's user avatar
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0 answers
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how we can constraints in dual of lagrangian?

I am confused about the general rule of Lagrangian multiplier. (which usully use for SVM). I could not find a good book or paper that explain it completely and clearly. Suppose we have $$minimize: ...
sherek_66's user avatar
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3 votes
0 answers
162 views

Maximum entropy probability distribution over non-negative support and finite mean?

I'm trying to derive which univariate probability distribution maximizes entropy, assuming finite mean $\mu$ and non-negative support $[0, \infty)$. I know that the answer is the exponential ...
Rylan Schaeffer's user avatar