Show $2^mle (em)^n Rightarrow mleq 2n*log_2(en)$
$begingroup$
This is a small bonus question in a HW assignment for a machine-learning related course. We need to prove the following for $m,ngeq1$:
$$2^mle (em)^n Rightarrow mleq 2n*log_2(en)$$
A book we use in the course (Understanding Machine Learning: From Theory to Algorithms) mentions that this follows from another lemma:
Let $ageq 1,b>0$. Then: $$xgeq 4a*log_2(2a)+2bRightarrow xgeq a*log_2(x)+b$$
However, I was unable to use it since I only get:
$$2^mle (em)^n Rightarrow mleq n*log_2(em)Rightarrow mleq n*log_2(m)+n*log_2(e)Rightarrow$$
$$mleq 4n*log_2(2n)+2n*log_2(e)$$
Any hints or directions will be greatly appreciated (we needn't necessarily use the aforementioned lemma, the book might be mistaken here).
calculus inequality
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
$endgroup$
|
show 1 more comment
$begingroup$
This is a small bonus question in a HW assignment for a machine-learning related course. We need to prove the following for $m,ngeq1$:
$$2^mle (em)^n Rightarrow mleq 2n*log_2(en)$$
A book we use in the course (Understanding Machine Learning: From Theory to Algorithms) mentions that this follows from another lemma:
Let $ageq 1,b>0$. Then: $$xgeq 4a*log_2(2a)+2bRightarrow xgeq a*log_2(x)+b$$
However, I was unable to use it since I only get:
$$2^mle (em)^n Rightarrow mleq n*log_2(em)Rightarrow mleq n*log_2(m)+n*log_2(e)Rightarrow$$
$$mleq 4n*log_2(2n)+2n*log_2(e)$$
Any hints or directions will be greatly appreciated (we needn't necessarily use the aforementioned lemma, the book might be mistaken here).
calculus inequality
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
$endgroup$
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02
|
show 1 more comment
$begingroup$
This is a small bonus question in a HW assignment for a machine-learning related course. We need to prove the following for $m,ngeq1$:
$$2^mle (em)^n Rightarrow mleq 2n*log_2(en)$$
A book we use in the course (Understanding Machine Learning: From Theory to Algorithms) mentions that this follows from another lemma:
Let $ageq 1,b>0$. Then: $$xgeq 4a*log_2(2a)+2bRightarrow xgeq a*log_2(x)+b$$
However, I was unable to use it since I only get:
$$2^mle (em)^n Rightarrow mleq n*log_2(em)Rightarrow mleq n*log_2(m)+n*log_2(e)Rightarrow$$
$$mleq 4n*log_2(2n)+2n*log_2(e)$$
Any hints or directions will be greatly appreciated (we needn't necessarily use the aforementioned lemma, the book might be mistaken here).
calculus inequality
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
$endgroup$
This is a small bonus question in a HW assignment for a machine-learning related course. We need to prove the following for $m,ngeq1$:
$$2^mle (em)^n Rightarrow mleq 2n*log_2(en)$$
A book we use in the course (Understanding Machine Learning: From Theory to Algorithms) mentions that this follows from another lemma:
Let $ageq 1,b>0$. Then: $$xgeq 4a*log_2(2a)+2bRightarrow xgeq a*log_2(x)+b$$
However, I was unable to use it since I only get:
$$2^mle (em)^n Rightarrow mleq n*log_2(em)Rightarrow mleq n*log_2(m)+n*log_2(e)Rightarrow$$
$$mleq 4n*log_2(2n)+2n*log_2(e)$$
Any hints or directions will be greatly appreciated (we needn't necessarily use the aforementioned lemma, the book might be mistaken here).
calculus inequality
calculus inequality
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
edited Dec 19 '18 at 10:20
Dean Gurvitz
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
asked Dec 19 '18 at 9:34
Dean GurvitzDean Gurvitz
357215
357215
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02
|
show 1 more comment
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02
|
show 1 more comment
0
active
oldest
votes
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%2f3046213%2fshow-2m-le-emn-rightarrow-m-leq-2n-log-2en%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3046213%2fshow-2m-le-emn-rightarrow-m-leq-2n-log-2en%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
$begingroup$
Is $e$ the usual $log_e=ln$?
$endgroup$
– Bo5man
Dec 19 '18 at 9:43
$begingroup$
@Bo5man Yes, $e$ is the known constant.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@YadatiKiran Where is the inconsistency? I can fix my question if needed.
$endgroup$
– Dean Gurvitz
Dec 19 '18 at 9:58
$begingroup$
@DeanGurvitz: My bad. Didin't read the question properly.
$endgroup$
– Yadati Kiran
Dec 19 '18 at 10:01
$begingroup$
Also in your last line the first $log$ should be $log_2$
$endgroup$
– Bo5man
Dec 19 '18 at 10:02