Questions tagged [convex]

A convex set includes all points lying between any two points from the set. A convex function on such a set is a function lying below any straight line connecting two points from its graph. Convex optimization is concerned with searching for the minimum of such a function.

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Strange constraint in optimization problem

I am trying to solve the following QP : $$ \min_{x} \quad (1/2)\|y - Ax \|^2 _2 + \lambda \|Dx\|_1$$ where $y \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times m}$, $D \in \mathbb{R}^{l\times m}$. Assume :...
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Is a decreasing and assymptotic (for y=const) convex function always logarithmically convex? [migrated]

In short, I have a function p(x) which is defined for x>=0. The function is convex, decreasing and assymptotically approaches 0 for $x\rightarrow\infty$ . I need to know if $p''p>=(p')^2$ . As ...
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Convex variant of Bhattacharyya coefficient

For (discrete, finite) probability distributions $P,Q$, the Bhattacharyya coefficient is $B(P,Q) := \sum_x \sqrt{P_x Q_x}$. It can be shown that this is jointly concave in $P$ and $Q$. My question is, ...
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25 views

Proof: Condition for a function being strongly convex

Let $\lambda \in \mathbb{R}_{> 0}$. A function $f : \mathbb{R}^d \to \mathbb{R}$ is $\lambda$-strongly convex, if for all $\alpha \in (0, 1)$ and all $u, v \in \mathbb{R}^d$ $$ f(\alpha u + (1 - \...
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Optimization: Convex function

Problem statement Use the definition of convexity of a function, i.e., that for any $\boldsymbol{x}$, $\boldsymbol{y} \in \mathbb{R}^{d}$ and $\lambda \in \left [0,1 \right ]$ we have \begin{align*} ...
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Sum of SVM hinge losses always a convex function?

The data loss function for a multi class SVM may take the following expression: \begin{equation} L=\frac{1}{N}\sum_{i}\sum_{j\neq y_i}\left[ \max(0,w_j^Tx_i-w_{y_i}^Tx_i+1)\right] \end{equation} ...
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1answer
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Question about Decision boundary in Logistical Regression

I am a Machine Learning newbie and studying Logistical Regression. For the data shown above, a straight line cannot separate all the positive and negative decisions. One thing I could understand is ...
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Doubt about definition of Regret in Online convex optimization setting

In online convex optimization, the regret of an algorithm $\mathcal{A}$ as defined in Introduction to Online Convex Optimization (Page 5) is: $$ regret_T(\mathcal{A}) = \sup_{\{f_1,...,f_T\}} \sum_{t=...
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Is logistic regression always convex?

I know that the cross-entropy loss is convex, and I've seen it stated that logistic regression is convex and has a global minimum. e.g. here https://www.cs.cmu.edu/~mgormley/courses/10701-f16/slides/...
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Iterative solution to Gamma distribution MLE problem

I'm trying to follow the derivation for the MLE parameters of the gamma distribution in [1]. The standard approach is to derive an expression for the log likelihood, differentiate with respect to ...
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Extended Lasso problem ( two L1 norms)

I am trying to find a closed form solution for the following problem : $$ \underset{\beta}{\operatorname{minimize}} \frac{1}{2} || y - X \beta ||^{2} + \lambda || \beta ||_{1} + \mu || \beta - \gamma |...
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Kullback-Leibler divergence from density f to density g

If $f$ and $g$ are density functions that are positive over the same region, then the Kullback-Leibler divergence from density $f$ to density $g$ is defined by: $$KL(f,g) = E_f\left[\ln\left(\frac{f(...
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ROC Convex Hull for Model Selection

The area under the convex hull of a roc curve is by construction always "better" than its area under curve. Some curves might see more of an increase in reported auc than others. Is it a viable ...
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1answer
25 views

How to understand the sufficient condition for global optimum for a constrained optimization probelm

How to understand the sufficient condition for global optimum? From my understanding, the global optimum should be 0 instead of $\geq 0$, where does this come from? Consider the constrained ...
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Basic question about labeling of random variables apropos of the definition of convexity

I understand the definition of convexity in a function $f: \mathbb R^d \to \mathbb R$ as the inequality for all $a,b\in\mathbb R^d$ and $0<\theta<1:$ $$\theta f(a) + (1-\theta)f(b) \geq ...
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Why does quadratic optimization produce non-stable results

Edit: posing the question in a different way I am running a mean-variance optimization, which is a quadratic optimization problem. I run the optimization 2000 times for different levels of risk (via ...
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Is penalized logistic regression convex? [duplicate]

Is logistic regression problem penalized by elastic net penalty convex optimization problem? More specifically, I want to find out whether it is suitable for dual formulation such that the duality gap ...
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How to minimize the sum of Frobenius norm and Nuclear norm

I have to minimize an objective function of the the form : $||X_{s} - Y_{s}D_{s}||_{F}^{2} + ||D_{s}||_{F}^{2} + ||D_{s}||_{*}^{2}$ where $||.||_{F}$ denotes the Frobenius norm and $||.||_{*}$ ...
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Optimization equivalence

Can someone help me with the step by step demonstration of the following equivalence used in SVM: $$maximize: m = \frac{1} {\|w\|} \equiv minimize: m =\frac{1} {2}\|w\|^2 $$ I would be most grateful ...
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46 views

Strongly convex function evaluated over a mean of n points

Let f(x) be a Strongly-convex function under some m > 0. Given two points x, y it is known that: $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2} - \frac{1}{2^3} \cdot m \cdot ||x - y||^2$$ What is ...
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Some tricky details about PCA non-convexity

I am reading about PCA and I came up with a some contradictions. PCA based on this post, PCA optimization problem is given by: $$\begin{aligned} \max_{w} \quad & w^T\Sigma w\\ \textrm{s.t.} \...
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Intervals from an underdetermined nonnegative linear system

I'm working on a problem in genomics that yields the following puzzle. Let $b\in \mathbb R^I$, $t$ and $p\in \mathbb R^J$, and $s \in \mathbb R^{I\times J}$. Suppose $t,b,p$ are known. Further suppose:...
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Concavity of negative binomial GLM

I need to estimate the log-likelihood of the negative binomial regression. I mean full log-likelihood, including the dispersion parameter. The problem is: when I start LBFGS, BFGS, or gradient ...
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Why are K-means and GMM (Gaussian Mixture Models) not suitable for discovering clusters with non-convexs shapes?

I have seen that mainly here and from a lot of resources that K-means and Hello all! Gaussian mixtures are not suitable for detecting clusters with non-convex shapes. I know that because both ...
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1answer
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Why does the formulation of the SVM problem has the bias (something we try to optimize) as a part of the constraint?

The common formulation of the SVM problem is $$\min_{\theta, \theta_0}\frac{1}{2}||\theta||^2$$ $$\text{ subject to: } y^{(t)}(\theta \cdot x^{(t)} + \theta_0) \geq 1, \ t=1,...,n,$$ However, it ...
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L1-regularization enforces sparsity for convex functions

I have a convex function $f \colon \mathbb{R}^n \to \mathbb{R}$ that I minimize using L1-regularization: $\DeclareMathOperator*{\argmin}{arg\,min}$ $$ x^*=\argmin_x f(x) + \lambda ||x||_1 $$ Can I ...
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1answer
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D-optimal DOE suggest repeated samples

I tried to generate a D optimal design but the design output sounds very weird to me. I have a (real) process and I`d like to explore 3 factors, but the process have a lot of constraints so I provide ...
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Why backpropagation if loss function is not convex in nature?

Backpropagation contains the method of gradient decent, which works well for convex loss functions with a global minima. But, for training, in most of the neural network tasks, backpropagation is ...
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Online Optimization - Regret in Absolute Error

In the online convex optimization literature static regret is defined as $\sum_{t=1}^{T}\left(f_t\left(x_t\right)-f_t\left(x^*\right)\right)$ where $x^*=\arg min_{x\in\mathcal{X}}\sum_{t=1}^{T}f_t\...
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Convexity of conditional expectation

Define $g(k)\equiv\mathbb{E}(X|_{X>k})$ and assume that the probability density $f$ of $X$ is twice continuously differentiable. Is there a sufficient condition in terms of $f$ that imply that $g^{...
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Prove that a function $\ln (e^{a_1} + e^{a_2} + \cdots + e^{a_n} )$ is convex?

Define the function: $$f(a_1, a_2, · · · , a_n) = \ln (e^{a_1} + e^{a_2} + \cdots + e^{a_n} ).$$ I want to prove that $f$ is convex. Now, to show that is a function is convex, we can take second ...
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1answer
101 views

How to recover primal problem from its dual counterpart

I am asking this from context of optimization in machine learning. We often talk about a primal problem and how this primal problem can be solved by first converting it into a dual problem (Using ...
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How positive definite Hessian approximations for SGD (e.g. Gauss-Newton) handle saddles?

For example due to symmetry of parameters, functions optimized in machine learning usually have huge number of local minima and saddles - growing exponentially with dimension. I am trying to ...
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How to solve a non-convex with equality constraint optimization problem?

I have a non-convex optimization problem with equality constraint, I can derive the KKT conditions, but it seems just one of the KKT conditions is valid. Could you please give some advice on how to ...
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2answers
558 views

Convexity of cross entropy

I am not sure if this is a better fit for this site or mathematics.stackexchange but I've seen similar questions on here before. I'd like to know if the following is true and if so, how I could go ...
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Normal equations issue

Let $A\mathbf{x}=\mathbf{b}$ be an overdetermined system, with $A$ being an $n \times m $ full-column rank rectangular matrix. Are these minimization problems equivalent? $$ 1) \;\underset{\mathbf{x}...
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Coordinate descent in integer programing: when does it work?

Denote $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to minimize a function $f:N\rightarrow \mathbb{R}$. For the functions $f$ that interest me, it is very easy to ...
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Is error function always assumed and convex?

While updating weights of the neural network, most of the algorithms use convex optimisation because of the reason that error is a convex function. My doubt is that whether the convex-ness of error ...
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1answer
131 views

How to understand whether Stochastic Gradient Descent has converged?

I am using SGD to solve for MSE function. My training set is around 50K, and I am monitoring the gradient at every epoch (once a pass is completed over all the training data). I played around a lot ...
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1answer
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Regularizing the inverse coefficient matrix in multivariate regression

I'd like to minimize the objective $ \operatorname{tr}[ (Y-XR^{-1})^T (Y-XR^{-1}) ] + \lambda \sum_{ij} |R_{ij}|$ wrt to $R$ (which is $P \times P$ but non-symmetric) where $Y$ and $X$ are both $N \...
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Optimization textbooks for statistics and data analytics

Any statistical analysis, machine learning or data science involves some sort of optimization at the end of the day. I'm looking for good linear and nonlinear optimization textbooks for self ...
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For convex problems, does gradient in Stochastic Gradient Descent (SGD) always point at the global extreme value?

Given a convex cost function, using SGD for optimization, we will have a gradient (vector) at a certain point during the optimization process. My question is, given the point on the convex, does the ...
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Vapnik-Chervonenkis Dimension

My question has to do with the VC dimension of the class of convex polygons with $m$ vertices. A solution to this problem is given in the following: Advanced Algorithms. Call the class of all ...
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1k views

Any optimization problem can be expressed as one with a linear objective

For any standard optimization problem(think linear programming, convex, non-convex problem), would it be possible to express the optimization problem as one with a linear objective?
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Why is global convexity not possible in higher-dimensional settings for this loss function?

I was reading this paper which discusses nonconvex penalized regression. They use the following notation: $X\in\mathbb{R}^{n\times p}$ is a data matrix with $n$ observations in $p$ variables $\beta\...
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Why proximal gradient is a special case of sub-gradient?

While I was learning proximal gradient descent methods, I saw many similar statements as below: $$p=prox_f(x) \Longleftrightarrow x-p \in \partial f(p) $$ for example, ...
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1answer
741 views

What is the use of the log of the sum of exponents in machine learning

I want to understand why somebody would use log sum exponent trick. I am reading this blog. But I don't really understand the first paragraph. It says Let's say we have an n-dimensional ...
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Low-variance estimate for the mean of the sotfmax transformation of a variable

Consider a set of infintiely-differentiable convex functions real-valued functions $f_i: \mathcal X \rightarrow \mathbb R$, where $i$ varies from $1$ to $m$, and suppose we know all the moments of $...
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L2 SVM (squared hinge) theory

The linear L2 SVM can be intuitively understood as \begin{equation} \text{minimize } f(\boldsymbol{w}) = \frac{1}{2} \Vert\boldsymbol{w}\Vert^2_2 + C \sum_{i=1}^m \xi_i^2 \tag{1} \end{equation} ...
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Is cross-entropy loss for MaxEnt models convex? If so, how do I prove it?

I am trying to prove if cross-entropy loss for MaxEnt models is convex. My first attempt at approaching this problem is to compute the Hessian matrix of second-order partial derivatives and showing ...