Questions tagged [convex]

Question 1

Let $F(x)$ be a twice differentiable cdf of a random variable taking values in say $(1,+\infty)$. Is it true that $F(x)$ must eventually (as it nears its ceiling value of 1) become concave for ...

Question 2

I am a bit confused about the concept of convexity analysis when doing model fitting. Say I have developed some model of two parameters $f(x;\theta_1,\theta_2)$, that I will plan to fit to some data I ...

Question 3

There are several gradient-based attack methods. Let $J$ be the training error, then for instance the projected gradient attack is, $$ \widetilde{x} = \Pi( x + \epsilon \nabla_x J(\theta, x, y) ) $$ ...

Question 4

When I want to obtain statistical properties of the matrix-variate lasso method, for example, $$ \hat{X}=\underset{X\in \mathbb{R}^{n\times n}}{\arg \min}\mathcal{L}\left(X\right)+\lambda_n \|X\|_1, $$...

Question 5

I am looking into convex optimization. However, I am not sure if there are interior-point methods that are guaranteed to converge to the globally optimal solution given either a strictly convex or a ...

Question 6

I have a convex multi-variate optimization problem where each variable lies on the domain $[x, \infty)$ for some positive number $x$. I know the problem has a unique finite solution in the domain, ...

Question 7

In Gaussian process (GP) regression, predictive mean is $$ K(X^*,X)[K(X,X)+\sigma^2I]^{-1} \textbf{y}$$ Is there a method to ensure that the predictive mean is convex with respect to the test input $X^...

Question 8

I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-...

Question 9

I am trying to understand if the following statement is true. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \begin{equation} \...

Question 10

Typically, the subgradient is defined for convex functions. And convex functions are continuous. However, DeepMind's VQ-VAE paper defines a model with a discontinuous vector quantization (VQ) layer, ...

Question 11

Suppose we're using linear predictors. I'm trying to conceptually understand how minimizing hinge loss and 0-1 loss aren't necessarily the same. For instance I was told that one can choose a set of ...

Question 12

The empirical risk of a multi-class hinge-loss is given by $$L(\Theta,(x,y) = \max_{j \neq y} \Big[1+ \sum_{i=1}^{d} x_i(\Theta_{ij} - \Theta_{iy}) \Big]_{+} $$ where $x \in \mathbb{R}^{d}$ is a ...

Question 13

I am thinking of adding some perturbation to my convex optimization problem. The idea is straight forward like below chart. Supposed you are solving $\text{argmax} f(x) $, we want to find an $x$ that'...

Question 14

I have these pair of numbers $ (a, b) = (\frac{4}{9}, \frac{1}{9}) $ and $(c, d) = (\frac{1}{2}, \frac{1}{6}) $. Note that - (a, b) are pair of numbers which represent $((E(e_1))^2, (E(e_2))^2) $ and (...

Question 15

On the Wikipedia article for exponential families the density of a distribution on a measure space $(X, \xi)$ from an exponential family is written as $$f_{\theta} \colon X \to \mathbb{R}_{\ge 0}, \...

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Questions tagged [convex]

upper tail concavity of continuous cdf

Convexity of loss function in model fitting without known data

The gradient method based attack does not seem make sense for neural networks because the training error is non-convex

Subgradient of the matrix norm

Are interior-point methods guranteed to converge to the global optimum of a convex objective function?

Do convergence rates for (convex) gradient descent apply when domain is (convex) subset of reals?

Convex predictive mean of Gaussian Process

Expectation under convex order by multiplying

Expectation under convex order

Do discontinuous functions have subgradients also?

Scenario where minimizing 0-1 loss is different than minimizing hinge loss

Convexitiy of multi-class hinge loss

Perturbation for more stable convex optimization

Relationship between Ratio of expectation squared vs ratio of squared expection

Is there an exponential family such that its natural parameter mapping is non-invertible or has non-convex range?

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