So  (don't judge me, it was years ago...) I know that when a person with little knowledge about a field try to explain a problem to someone with a lot of experience it can easily lead to a lot of misunderstandings, everyone gets frustrated and more often than not nothing good comes out of it. So this time I will try with an analogy that (some) children can understand in order to explain my question. If you are too lazy, simply scroll down for my most likely terrible attempt on a short explanation.  Story  Once upon a time there was a guy called Bob, Bob was a great mechanic and he wanted to build a racing-car, and as anyone who wants to build a racing-car he wanted it to go as fast as possible. He knew his highschool maths and physics, however he knew that it would not be enough for finding out how to make the fastest car. Bob was very lucky because he had gotten his hands on a machine that calculated how fast a car would go based on a lot of knobs and switches that controlled things like, car height, type of engine, shape and etc. Unfortunately the machine was very slow and used around 5-10 minutes for one calculation. Bob knew he could make an other machine that would try a lot of different random combinations and try to find the best one, but he also knew that there were so many combinations that he would never be able to try all of them. Another problem is that Bob has no idea of how his machine works and he thinks there is no way he could find out.  Question is: how to help Bob?  tl;dr  I have a very slow and complex function $f(x_1 \cdots{} x_n)$ and I would like to find the best way to maximize/minimize this function with the least number of runs.
Gradient descent and many other methods are useful for finding local minima in cost functions. They can be efficient when the cost function can be evaluated quickly at each point, whether numerically or analytically.    I have what appears to me to be an unusual situation. Each evaluation of my cost function is expensive. I am attempting to find a set of parameters that minimize a 3D surface against ground truth surfaces. Whenever I change a parameter, I need to run the algorithm against the entire sample cohort to measure its effect.  In order to calculate a gradient, I need to change all 15 parameters independently, meaning I have to regenerate all the surfaces and compare against the sample cohort way too many times per gradient, and definitely way too many times over the course of optimization.  I have developed a method to circumvent this problem and am currently evaluating it, but I am surprised that I have not found much in the literature regarding expensive cost function evaluations.  This makes me wonder if I am making the problem harder than it is and that there might be a better way already available.  So my questions are basically this: Does anyone know of methods for optimizing cost functions, convex or not, when evaluation is slow?  Or, am I doing something silly in the first place by rerunning the algorithm and comparing against the sample cohort so many times?
$\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed] How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. Intuition behind logistic regression Recently I began studying machine learning, however I failed to grasp the intuition behind .   The following are the facts about logistic regression that I understand.   As the basis for hypothesis we use . I do understand why it's a correct choice, however why it's the only choice I don't understand. Hypothesis represents the probability that the appropriate output is $1$, therefore the domain of our function should be $[0,1]$, this is the only property of sigmoid function I found useful and appropriate here, however many functions satisfy this property. In addition, sigmoid function has a derivative in this form $f(x)(1-f(x))$, but I don't see the utility of this special form in logistic regression.   Question: what so special about sigmoid function, and why we cannot use any other function with domain $[0,1]$? The cost function consists of two parameters ${\rm Cost}(h_{\theta}(x),y)=-\log(h_{\theta}(x))$ if $y=1, {\rm Cost}(h_{\theta}(x),y)=-\log(1-h_{\theta}(x))$ if $y=0$. In the same was as above, I do understand why it's correct, however why is it the only form? For example, why couldn't $|h_{\theta(x)}-y|$ be a good choice for the cost function?  Question: what is so special about the above form of cost function; why cannot we use another form?   I would appreciate if you could share your understanding of logistic regression. Search engines and browser accept-language I'm building a web portal where language content will generally depend on the "accept-language" sent by the browser. The same content-URI will thus serve different content to different users depending on their browser setting.  I'm very curious to know how this will affect search indexing. Does Google index using all languages, and is it handled well? Show the title character limit when writing a title I just ran into this while writing .  [I originally wrote:     you need a minimum of X reputation to comment on questions that are not your own, and on answers that are not to your own questions   But had to shorten it to:     you need a minimum of X reputation to comment]   First, it didn't tell me that I'd gone over the character limit for the title until after I had already tabbed out of the title field.  Second, it didn't tell me how much I'd gone over or anything. I had to copy-paste the text I'd written into my text editor in order to get some real-time feedback on the exact character count, to help figure out how to best shorten it. I am very confused. Unless I am mistaken, I know "it" has to be a noun of some sort, but I am unable to figure out what noun "it" is referring to.   What is "it" in the following sentence:      It is clear that Bob likes doughnuts.   Heres another couple of examples:     It is impossible to fly. I am reading the Dummit and Foote's proof of Cauchy's theorem for abelian groups. On the highlighted line, how do you conclude from $y\notin N$ and $y^p\in N$ that $\langle y^p \rangle \neq \langle y \rangle$ and that $|y^p|&lt;|y|$?     Proposition 21.   If $G$ is a finite abelian group and $p$ is a prime dividing $|G|$, then $G$ contains an element of order $p$.      Proof:   The proof proceeds by induction on $|G|$, namely, we assume the result is valid for every group whose order is strictly smaller than the order of $G$ and then prove the result valid for $G$ (this is sometimes referred to as complete induction).   Since $|G| &gt; 1$, there is an element $x \in G$ with $x \neq 1$.   If $|G| = p$ then $x$ has order $p$ by Lagrange’s Theorem and we are done.   We may therefore assume $|G| &gt; p$.      Suppose $p$ divides $|x|$ and write $|x| = pn$.   By Proposition 2.5(3), $|x^n| = p$, and again we have an element of order $p$.   We may therefore assume $p$ does not divides $|x|$.      Let $N = \langle x \rangle$.   Since $G$ is abelian, $N \trianglelefteq G$.   By Lagrange’s Theorem, $|G/N| = |G|/|N|$ and since $N \neq 1$, $|G/N| &lt; |G|$.   Since $p$ does not divide $|N|$, we must have $p \mid |G/N|$.   We can now apply the induction assumption to the smaller group $G/N$ to conclude it contains an element $\bar{y} = yN$, of order $p$.   Since $y \notin N$ ($\bar{y} \neq \bar{1}$) but $y^p \in N$ ($\bar{y}^p = \bar{1}$) we must have $\langle y^p \rangle \neq \langle y \rangle$, that is, $|y^p| &lt; |y|$.   Proposition 2.5(2) implies $p \mid |y|$.   We are now in the situation described in the preceding paragraph, so that argument again produces an element of order $p$.   The induction is complete.      (Original image .) I have a directory where daily subdirectories are created, literally named according to $date. How can I delete folders and their contents that are older than 7 days according to the YYYYMMDD in the file name and not the metadata date? Say I have (skipped some for brevity):  20170817 20170823 20170828 20170901   I would end up with the following folders (which those should keep):  20170828 20170901   I created a variable that holds the date 7 days ago: dt_prev=$(date -d "`date`-7days" +%Y%m%d)  My thought was to ls -l a list of these folder names and compare row by row, but this involves cleaning that list, etc., and I figure there has to be an easier way. What methods of ‘clearfix’ can I use? Real-life examples of common distributions jQuery autocomplete tagging plug-in like StackOverflow's input tags? What solutions accomplish the same auto-completion that SO uses for entering tags?  There are plugins that can handle one word but I haven't seen any that handle multiple words. Double vs. BigDecimal? Best practice of when to load data into a ListBox How can I make a list with most used commands in terminal? The draft feature was discussed a lot  and in other posts, but I didn't see much talk about the OP's mention of being able to have multiple.     Optionally, allow multiple draft questions and a page to edit/delete them.    I think the draft feature where it autosaves and stores it server-side is great, but in most things where they have a draft feature (like email clients), they allow you to have multiple drafts (also stored server-side).  The expiration can even work the same as it does for the single draft system right now.  You wouldn't even need full page to store them, I think the user profile page could have a tab for it like it currently has for answers, questions, and everything else.    It could look very similar to how questions look, with it's title and tags listed there. What does it mean for two objects to "touch"? Boundary condition with spatial derivative is ignored by NDSolve Unable to get force:inputField or force:outputField to render inside other markup in Lightning Component when used in Lightning Experience Determine number of pages in a PDF file How do I use filenames that start with a dash "-" as command arguments? how to prove that this positive sequence converges to 1