Questions tagged [hessian]

For on-topic questions involving the Hessian matrix, a square matrix generalizing the second derivative. Please include also a statistical methods tag. For purely mathemathical questions about the Hessian it is better to ask on math.SE at https://math.stackexchange.com/.

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56 views

Can the likelihood ratio estimate multivariate confidence levels?

Wilks' theorem describes the log-ratio between the highest likelihood of a distribution $\mathcal{L}$ (aka the dominant mode, given at $\vec{x}_{m}$) and the likelihood of a distribution at a given ...
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28 views

Standard errors for Composite Marginal Likelihood

I am estimating a multivariate ordered probit model using a composite marginal likelihood (CML) approach. In other words, I replace the full likelihood function by a surrogate likelihood constructed ...
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0answers
82 views

Square root of an almost diagonal matrix

Is there an efficient way to compute square root of an almost diagonal symmetric Hessian matrix, which is diagonal with the exception of the last two columns and last two rows? Could the efficient ...
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48 views

AutoGrad Hessian in custom loss function in XGboost takes very long time

I am using the autograd package and generating the hessian of my loss function using that package as part of my custom loss function XGB model. However, it takes an extremely long time to iterate ...
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1answer
89 views

Calculate the uncertainty of a MLE

I have minimized the negative LL of a Poisson distribution to get an MLE of three parameters using scipy.minimize w/ Nelder-Mead. I want to calculate the uncertainty of the MLE. From reading, I ...
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2answers
48 views

Neural networks: why don't we use a multi-dimensional learning rate

I've searched a bit on the internet a have found the answer nowhere so I decided to post here. When confronted to an optimization problem, we know that the sanity of the problem can be characterized ...
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96 views

Saddle-free Newton method for SGD - while Newton attracts saddles, is it worth to actively replel them?

While 2nd order methods have many advantages, e.g. natural gradient (e.g. in L-BFGS) attracts to close zero gradient point, which is usually saddle. Other try to pretend that our very non-convex ...
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202 views

Hessian of logistic loss - when $y \in \{-1, 1\}$

Logistic Regression has two possible formulations depending on how we select the target variable: $y \in \{0,1\}$ or $y \in \{-1,1\}$. This question discusses the derivation of Hessian of the loss ...
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1answer
35 views

Hessian Matrix Values

Its an easy question but still i cant seem to find the hessian matrix. I have the following function : $$-2x^2 + \sqrt{2}xy - \frac52y^2$$ Find the hessian matrix for this function. $$f_{11} = -4 \...
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97 views

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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1answer
128 views

Observed information matrix with multivariate normal distribution

$$ \DeclareMathOperator\tr{tr} \DeclareMathOperator\vecOP{vec} \newcommand\di{\mathrm{d}} \newcommand\D{\mathrm{D}} \newcommand\Hess{\mathrm{H}} $$ I do not have much experience with matrix ...
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2answers
389 views

What is a consequence of an ill-conditioned Hessian matrix?

In this publication I found an explanation of the Hessian matrix, along with what it means for it to be ill-conditioned. In the paper, there is this link given between the error surface and the ...
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2answers
855 views

Standard error from Hessian matrix when likelihood is used (rather than Ln L)

I understand that at MLE point, the inverse of the Hessian matrix can be used as approximation of V-Cov matrix: ...
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0answers
457 views

Obtaining Standard Errors in Optim() in R [duplicate]

I'm using a maximum likelihood estimation and I'm using the optim() function in R in a similar way as follows: ...
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1answer
711 views

Computing the Hessian of maximum log likelihood function

I am trying to find the Hessian matrix for the maximum log likelihood function given training data {(xi, yi)} for i=1:N with yi ∈ {+1, −1} for each i = 1, . . . , N for the function: When I try to ...
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145 views

Inverting Hessian using Generalized Inverse for Inference

I am estimating a survival model with MLE. I use optim to maximize the likelihood function, and I intend to use the Hessian matrix returned by optim to get the standard errors (which lie on the ...
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1answer
148 views

Explanation of generalization of Newton's Method for multiple dimensions

I've been following the CS 229 lecture videos for machine learning, and in lecture 4 (~14:00), Ng explains Newton's Method for optimization to maximize an objective function ($f$), but doesn't clearly ...
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1k views

How the Hessian matrix is used in optimization if you can't invert it

I've seen quite a lot of work to do with approximating the Hessian such as the Hessian Vector Product but I'm not entirely sure how knowing the Hessian helps us evaluate the gradient step to take. ...
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184 views

Computing the Hessian Matrix Diagonal of a multi-layered Feed Forward Neural Network

I am working on using a Feedforward multi-layered perceptron as a function approximator for the pressure distribution of a groundwater system. I am essentially trying to solve a boundary value problem ...
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78 views

Which optimizer use for laplace approximation

I have been trying to estimate the marginal posterior for D variable using Laplace approximation: $p(\theta_i) \approx \left[\frac{\det{H}}{2\pi\det{H(\theta_i)}}\right]^{1/2} \exp\left[-L(\theta_i, \...
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1answer
2k views

gradient descent and local maximum

I read that gradient descent converge always to a local minimum while other methods as Newton's method this is not guaranteed (if the Hessian is not definite positive); but if the start point in GD is ...
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1answer
93 views

Question about port of R code from the library “rethinking” to PyMC3

A very generous human named Osvaldo Martin did us the favor of porting all the R sample code in Richard McElreath's superb book Statistical Rethinking to PyMC3. I'm hugely grateful, but I've already ...
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1answer
53 views

Newton-Raphson Error

According to Agresti(2013) pg 364-365, iterative methods such as Newton-Raphson methods, $ \begin{aligned} \beta^\text{new} &= \beta^\text{old} + (X^{T}WX)^{-1}X^{T}(V) \end{aligned} $ help to ...
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what is the hessian matrix of $f(W) = \sqrt{Tr(W A_0 W A_1)}$? Here, $W$, $A_0$ and $A_1$ are positive semidefinite matrices and hermitian

what is the hessian matrix of $f(W) = \sqrt{Tr(W A_0 W A_1)}$? Here, $W$, $A_0$ and $A_1$ are positive semidefinite matrices and hermitian. For the time being, I obtain the derivative as $\frac{\...
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3answers
119 views

Hessian of Log of Matrix-t distribution

I am trying to calculate the hessian of the log of the matrix-t distribution. I know that the log of the matrix-t distribution can be written: $$\log T_{N\times P}(X| \nu, M, \Sigma, \Omega) \propto -\...
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96 views

Should Bayesian estimated error smaller than MLE?

I am dealing with a fitting problem. Specifically, I am fitting a Lorentzian profile to the power spectrum of an solar-like oscillating star. Three parameters in the Lorentzian profile characterize ...
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379 views

Multiclass: I want to develop a customized objective function with weights given by both label and prediction, for Xgboost

I want to develop a customized objective function with weights given by both label and prediction, for Xgboost. Example, let's say you have 2 classes I want to assign a penalties according to this ...
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1answer
168 views

Cramér–Rao bound to multiple parameters

I was reading Cramér–Rao bound to multiple parameters from Wikipedia page, but I could not follow this line in the article: Let $\displaystyle {\boldsymbol {T}}(X)$ be an estimator of any ...
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6answers
2k views

Why not use the third derivative for numerical optimization?

If Hessians are so good for optimization (see e.g. Newton's method), why stop there? Let's use the third, fourth, fifth, and sixth derivatives? Why not?
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1answer
14k views

Explanation of min_child_weight in xgboost algorithm

The definition of the min_child_weight parameter in xgboost is given as the: minimum sum of instance weight (hessian) needed in a child. If the tree partition step results in a leaf node with the ...
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1answer
1k views

Gradient and hessian of the MAPE

I want to use MAPE(Mean Absolute Percentage Error) as my loss function. ...
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3answers
2k views

How does the second derivative inform an update step in Gradient Descent?

I was reading the deep learning book by Begnio, Goodfellow and Courville and there was one section where they explain the second derivative that I don't understand (section 4.31): The second ...
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42 views

Non-linear Likelihood function, large estimated standard errors

I have a highly non-linear (lots of jumps) likelihood function with K parameters (For example, a marked Hawkes Process used in seismology study). I implemented the L-BFGS-B optimization routine and it ...
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1answer
78 views

Pearlmutter's method for Hessian multiplication

I am trying to understand the abstract below from Pearlmutter's paper. Can someone clarify to me why $R_{\bf{v}}\{\bf{w}\}=\bf{v}$? Thanks a lot!
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90 views

Fast multiplication by the Hessian in Neural networks

I have question about the $R\{.\}$ function in Bishop's book on page 254 (see snippet below). My questions are as follows: I assume $R\{\bf w\}$ in (5.97) is the premultiplication of $\bf{v}^{T}$ ...
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23 views

Inferrence for peaked likelihoods

Suppose I have the likelihood $f(X|\theta)$ of some rich model, where $\theta\in\mathbb{R}^n$, and I have been able to find its maximum, $\hat\theta$. Suppose further that for some $i$, the plot of $...
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0answers
270 views

Hessian matrix of log marginal likelihood of Gaussian Process

I'm trying to compute the exact second derivatives of log marginal likelihood of Gaussian Process for learning hyperparameters. The log marginal likelihood and its partial derivative are given in 5 ...
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1answer
187 views

Why does the determinant of the Hessian grow with n?

Context: I'm trying to understand BIC on a deeper level. I'm using BIC for model/structure selection for Bayesian networks. I'm confused because BIC is an approximation to the likelihood of a model, ...
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1answer
33 views

Uncertainty in collapsing several curves

I have a bunch of curves $f(x)$, and I have a parameter $\lambda$ that rescales $x$, such that $x \rightarrow x' = g(x, \lambda)$. I find the value of $\lambda$ that collapses all the curves on top of ...
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8answers
59k views

Why is Newton's method not widely used in machine learning?

This is something that has been bugging me for a while, and I couldn't find any satisfactory answers online, so here goes: After reviewing a set of lectures on convex optimization, Newton's method ...
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2answers
1k views

Poisson Regression and Hessian

I have been trying to estimate parameters of a poisson regression. I am using Newton Raphson method. This method requires that the inverse of Hessian be computed to obtain, updates to beta vector. The ...
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1answer
2k views

Derivation of the Hessian of average empirical loss for Logistic Regression

How does the y term vanish/get cancelled ? Shouldn't it be this instead, h here is the Sigmoid function.
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1answer
186 views

Show that the following optimization problem is convex

I have the following optimization problem \begin{equation} \label{logdual} \begin{array}{ll@{}ll} \text{minimize}_{\pmb\alpha \in \mathbb{R}^n} & \theta(\pmb\alpha) &\\ \text{subject to} &...
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1answer
1k views

Second derivative of neural network cost function

This question is highly correlated with my previous one (I was asking about quadratic approximation of the cost function with Hessian matrix and didn't get any answer), but I think that I have the ...
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0answers
274 views

Quadratic approximation of the regularized cost function for neural network [closed]

I've been working on the topic of regularization for neural networks and in the textbook I'm following I found this quote: "We will further simplify the analysis by making a quadratic approximation ...
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1answer
76 views

Relation Between the Hessian Matrix at the max of loglikelihood and the Hessian matrix at the minimum of -loglikelihood

I would like to now what is the relation between the Hessian Matrix calculated at the maximum point of the log(likelihood) function, $H_{f}$ and the Hessian Matrix calculated at the minimum point of ...
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1answer
317 views

Hessian for Laplace Approximation in Uncertainty Propagation

This is possibly a silly conceptual question, ... but anyway: Imagine I have a function: $f = F(\mathbf{x}) = F(x_1,x_2) = ax_1^2 + bx_2^3,$ where $x_1,x_2 \sim N(0,1)$ for example. For a naive ...
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127 views

Positive definiteness of Hessians?

I'm reading the book "Convex Optimization" by Boyd and Vandenbherge. On the second paragraph of page 71, the authors seem to state that in order to check if the Hessian (H) is positve semidefinite (...
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266 views

Maximum likelihood estimation, how to derive the hessian

I am reading a paper and trying to understand how the authors estimated the standard errors of a set of parameter estimates $[\delta \ \ \phi \ \ \Sigma]$. Below is the loglikelihood function (sorry I ...
4
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1answer
308 views

Variance of maximum likelihood estimator in R

In different sources there is an algorithm how to calculate the variance of MLE in R. To keep it short: construct the negative log likelihood function. minimize it via nlm or optim with hessian=TRUE ...