Explanation of gradient descent on convex quadratic
Can someone explain the following:
$$f(x) = frac{1}{2}w^TAw - b^Tw$$
Assume AA is symmetric and invertible, then the optimal solution $w^{star}$ occurs at
$$w^{star} = A^{-1}b$$
and
$$nabla f(w) = Aw - b $$
can someone explain how we got $w^{star}$ and $nabla f(w)$?
I need detailed math overview of that, I forgot most of calc/LA.
Thank you!
calculus linear-algebra quadratics gradient-descent
asked 23 hours ago
YohanRoth
6191713
add a comment |
Can someone explain the following:
$$f(x) = frac{1}{2}w^TAw - b^Tw$$
Assume AA is symmetric and invertible, then the optimal solution $w^{star}$ occurs at
$$w^{star} = A^{-1}b$$
and
$$nabla f(w) = Aw - b $$
can someone explain how we got $w^{star}$ and $nabla f(w)$?
I need detailed math overview of that, I forgot most of calc/LA.
Thank you!
calculus linear-algebra quadratics gradient-descent
asked 23 hours ago
YohanRoth
6191713
add a comment |
Can someone explain the following:
$$f(x) = frac{1}{2}w^TAw - b^Tw$$
Assume AA is symmetric and invertible, then the optimal solution $w^{star}$ occurs at
$$w^{star} = A^{-1}b$$
and
$$nabla f(w) = Aw - b $$
can someone explain how we got $w^{star}$ and $nabla f(w)$?
I need detailed math overview of that, I forgot most of calc/LA.
Thank you!
calculus linear-algebra quadratics gradient-descent
asked 23 hours ago
YohanRoth
6191713
Can someone explain the following:
$$f(x) = frac{1}{2}w^TAw - b^Tw$$
Assume AA is symmetric and invertible, then the optimal solution $w^{star}$ occurs at
$$w^{star} = A^{-1}b$$
and
$$nabla f(w) = Aw - b $$
can someone explain how we got $w^{star}$ and $nabla f(w)$?
I need detailed math overview of that, I forgot most of calc/LA.
Thank you!
calculus linear-algebra quadratics gradient-descent
calculus linear-algebra quadratics gradient-descent
asked 23 hours ago
YohanRoth
6191713
asked 23 hours ago
YohanRoth
6191713
asked 23 hours ago
YohanRoth
6191713
asked 23 hours ago
YohanRoth
6191713
asked 23 hours ago
YohanRoth
6191713
6191713
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
I think you have a minor typo: $f(x)$ should be $f(w)$.
$f$ is a function of several variables $w_1, w_2, ldots$. If you find the partial derivatives $partial f / partial w_i$ for each $i$, you can combine them into a vector to form the gradient $nabla f$. It would be good for you to do this slowly. It may be helpful to rewrite $f$ as
$$f(w) = frac{1}{2} sum_i sum_j A_{ij} w_i w_j - sum_i b_i w_i.$$
Note that there are "shortcuts" for computing this gradient using matrix calculus, but they won't be particularly helpful if you don't understand what's going on.
Once you have the gradient, any solution to $nabla f(w) = 0$ is a critical/stationary point. In this case, there is only one, namely $w^star = A^{-1} b$.
answered 23 hours ago
angryavian
38.6k23180
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051728%2fexplanation-of-gradient-descent-on-convex-quadratic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
I think you have a minor typo: $f(x)$ should be $f(w)$.
$f$ is a function of several variables $w_1, w_2, ldots$. If you find the partial derivatives $partial f / partial w_i$ for each $i$, you can combine them into a vector to form the gradient $nabla f$. It would be good for you to do this slowly. It may be helpful to rewrite $f$ as
$$f(w) = frac{1}{2} sum_i sum_j A_{ij} w_i w_j - sum_i b_i w_i.$$
Note that there are "shortcuts" for computing this gradient using matrix calculus, but they won't be particularly helpful if you don't understand what's going on.
Once you have the gradient, any solution to $nabla f(w) = 0$ is a critical/stationary point. In this case, there is only one, namely $w^star = A^{-1} b$.
answered 23 hours ago
angryavian
38.6k23180
add a comment |
I think you have a minor typo: $f(x)$ should be $f(w)$.
$f$ is a function of several variables $w_1, w_2, ldots$. If you find the partial derivatives $partial f / partial w_i$ for each $i$, you can combine them into a vector to form the gradient $nabla f$. It would be good for you to do this slowly. It may be helpful to rewrite $f$ as
$$f(w) = frac{1}{2} sum_i sum_j A_{ij} w_i w_j - sum_i b_i w_i.$$
Note that there are "shortcuts" for computing this gradient using matrix calculus, but they won't be particularly helpful if you don't understand what's going on.
Once you have the gradient, any solution to $nabla f(w) = 0$ is a critical/stationary point. In this case, there is only one, namely $w^star = A^{-1} b$.
answered 23 hours ago
angryavian
38.6k23180
add a comment |
I think you have a minor typo: $f(x)$ should be $f(w)$.
$f$ is a function of several variables $w_1, w_2, ldots$. If you find the partial derivatives $partial f / partial w_i$ for each $i$, you can combine them into a vector to form the gradient $nabla f$. It would be good for you to do this slowly. It may be helpful to rewrite $f$ as
$$f(w) = frac{1}{2} sum_i sum_j A_{ij} w_i w_j - sum_i b_i w_i.$$
Note that there are "shortcuts" for computing this gradient using matrix calculus, but they won't be particularly helpful if you don't understand what's going on.
Once you have the gradient, any solution to $nabla f(w) = 0$ is a critical/stationary point. In this case, there is only one, namely $w^star = A^{-1} b$.
answered 23 hours ago
angryavian
38.6k23180
I think you have a minor typo: $f(x)$ should be $f(w)$.
$f$ is a function of several variables $w_1, w_2, ldots$. If you find the partial derivatives $partial f / partial w_i$ for each $i$, you can combine them into a vector to form the gradient $nabla f$. It would be good for you to do this slowly. It may be helpful to rewrite $f$ as
$$f(w) = frac{1}{2} sum_i sum_j A_{ij} w_i w_j - sum_i b_i w_i.$$
Note that there are "shortcuts" for computing this gradient using matrix calculus, but they won't be particularly helpful if you don't understand what's going on.
Once you have the gradient, any solution to $nabla f(w) = 0$ is a critical/stationary point. In this case, there is only one, namely $w^star = A^{-1} b$.
answered 23 hours ago
angryavian
38.6k23180
answered 23 hours ago
angryavian
38.6k23180
answered 23 hours ago
angryavian
38.6k23180
answered 23 hours ago
angryavian
38.6k23180
38.6k23180
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051728%2fexplanation-of-gradient-descent-on-convex-quadratic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
