I'm doing some performance/load testing of a service. Imagine the test function like this:
bytesPerSecond = test(filesize: 10MB, concurrency: 5)
Using this, I'll populate a table of results for different sizes and levels of concurrency. There are other variables too, but you get the idea.
The test function spins up concurrency requests and tracks throughput. This rate starts off at zero, then spikes and dips until it eventually stabilises on the 'true' value.
However it can take a while for this stability to occur, and there are lot of combinations of input to evaluate.
How can the test function decide when it's performed enough samples? By enough, I suppose I mean that the result isn't going to change beyond some margin if testing continues.
I remember reading an article about this a while ago (from one of the jsperf authors) that discussed a robust method, but I cannot find the article any more.
One simple method would be to compute the standard deviation over a sliding window of values. Is there a better approach?
IIUC, you're describing the classic problem of estimating the confidence interval of the mean with unknown variance. That is, suppose you have n results, x1, ..., xn, where each of the xi is a sample from some process of which you don't know much: not the mean, not the variance, and not the distribution's shape. For some required confidence interval, you'd like to now whether n is large enough so that, with high probability the true mean is within the interval of your mean.
(Note that with relatively-weak conditions, the Central Limit Theorem guarantees that the sample mean will converge to a normal distribution, but to apply it directly you would need the variance.)
So, in this case, the classic solution to determine if n is large enough, is as follows:
Start by calculating the sample mean μ = ∑i [xi] / n. Also calculate the normalized sample variance s2 = ∑i [(xi - μ)2] / (n - 1)
Depending on the size of n:
If n > 30, the confidence interval is approximated as μ ± zα / 2(s / √(n)), where, if necessary, you can find here an explanation on the z and α.
If n < 30, the confidence interval is approximated as μ ± tα / 2(s / √(n)); see again here an explanation of the t value, as well as a table.
If the confidence is enough, stop. Otherwise, increase n.
Stability means rate of change (derivative) is zero or close to zero.
The test function spins up concurrency requests and tracks throughput.
This rate starts off at zero, then spikes and dips until it eventually
stabilises on the 'true' value.
I would track your past throughput values. For example last X values or so. According to this values, I would calculate rate of change (derivative of your throughput). If your derivative is close to zero, then your test is stable. I will stop test.
How to find X? I think instead of constant value, such as 10, choosing a value according to maximum number of test can be more suitable, for example:
X = max(10,max_test_count * 0.01)
Related
First, this is not a question about temperature iteration counts or automatically optimized scheduling. It's how the data magnitude relates to the scaling of the exponentiation.
I'm using the classic formula:
if(delta < 0 || exp(-delta/tK) > random()) { // new state }
The input to the exp function is negative because delta/tK is positive, so the exp result is always less then 1. The random function also returns a value in the 0 to 1 range.
My test data is in the range 1 to 20, and the delta values are below 20. I pick a start temperature equal to the initial computed temperature of the system and linearly ramp down to 1.
In order to get SA to work, I have to scale tK. The working version uses:
exp(-delta/(tK * .001)) > random()
So how does the magnitude of tK relate to the magnitude of delta? I found the scaling factor by trial and error, and I don't understand why it's needed. To my understanding, as long as delta > tK and the step size and number of iterations are reasonable, it should work. In my test case, if I leave out the extra scale the temperature of the system does not decrease.
The various online sources I've looked at say nothing about working with real data. Sometimes they include the Boltzmann constant as a scale, but since I'm not simulating a physical particle system that doesn't help. Examples (typically with pseudocode) use values like 100 or 1000000.
So what am I missing? Is scaling another value that I must set by trial and error? It's bugging me because I don't just want to get this test case running, I want to understand the algorithm, and magic constants mean I don't know what's going on.
Classical SA has 2 parameters: startingTemperate and cooldownSchedule (= what you call scaling).
Configuring 2+ parameters is annoying, so in OptaPlanner's implementation, I automatically calculate the cooldownSchedule based on the timeGradiant (which is a double going from 0.0 to 1.0 during the solver time). This works well. As a guideline for the startingTemperature, I use the maximum score diff of a single move. For more information, see the docs.
I have two codes that theoretically should return the exact same output. However, this does not happen. The issue is that the two codes handle very small numbers (doubles) to the order of 1e-100 or so. I suspect that there could be some numerical issues which are related to that, and lead to the two outputs being different even though they should be theoretically the same.
Does it indeed make sense that handling numbers on the order of 1e-100 cause such problems? I don't mind the difference in output, if I could safely assume that the source is numerical issues. Does anyone have a good source/reference that talks about issues that come up with stability of algorithms when they handle numbers in such order?
Thanks.
Does anyone have a good source/reference that talks about issues that come up with stability of algorithms when they handle numbers in such order?
The first reference that comes to mind is What Every Computer Scientist Should Know About Floating-Point Arithmetic. It covers floating-point maths in general.
As far as numerical stability is concerned, the best references probably depend on the numerical algorithm in question. Two wide-ranging works that come to mind are:
Numerical Recipes by Press et al;
Matrix Computations by Golub and Van Loan.
It is not necessarily the small numbers that are causing the problem.
How do you check whether the outputs are the "exact same"?
I would check equality with tolerance. You may consider the floating point numbers x and y equal if either fabs(x-y) < 1.0e-6 or fabs(x-y) < fabs(x)*1.0e-6 holds.
Usually, there is a HUGE difference between the two algorithms if there are numerical issues. Often, a small change in the input may result in extreme changes in the output, if the algorithm suffers from numerical issues.
What makes you think that there are "numerical issues"?
If possible, change your algorithm to use Kahan Summation (aka compensated summation). From Wikipedia:
function KahanSum(input)
var sum = 0.0
var c = 0.0 //A running compensation for lost low-order bits.
for i = 1 to input.length do
y = input[i] - c //So far, so good: c is zero.
t = sum + y //Alas, sum is big, y small, so low-order digits of y are lost.
c = (t - sum) - y //(t - sum) recovers the high-order part of y; subtracting y recovers -(low part of y)
sum = t //Algebraically, c should always be zero. Beware eagerly optimising compilers!
//Next time around, the lost low part will be added to y in a fresh attempt.
return sum
This works by keeping a second running total of the cumulative error, similar to the Bresenham line drawing algorithm. The end result is that you get precision that is nearly double the data type's advertised precision.
Another technique I use is to sort my numbers from small to large (by manitude, ignoring sign) and add or subtract the small numbers first, then the larger ones. This has the virtue that if you add and subtract the same value multiple times, such numbers may cancel exactly and can be removed from the list.
I have a list of documents each having a relevance score for a search query. I need older documents to have their relevance score dampened, to try to introduce their date in the ranking process. I already tried fiddling with functions such as 1/(1+date_difference), but the reciprocal function is too discriminating for close recent dates.
I was thinking maybe a mathematical function with range (0..1) and domain(0..x) to amplify their score, where the x-axis is the age of a document. It's best to explain what I further need from the function by an image:
Decaying behavior is often modeled well by an exponentional function (many decaying processes in nature also follow it). You would use 2 positive parameters A and B and get
y(x) = A exp(-B x)
Since you want a y-range [0,1] set A=1. Larger B give slower decays.
If a simple 1/(1+x) decreases too quickly too soon, a sigmoid function like 1/(1+e^-x) or the error function might be better suited to your purpose. Let the current date be somewhere in the negative numbers for such a function, and you can get a value that is current for some configurable time and then decreases towards a base value.
log((x+1)-age_of_document)
Where the base of the logarithm is (x+1). Note the x is as per your diagram and is the "threshold". If the age of the document is greater than x the score goes negative. Multiply by the maximum possible score to introduce scaling.
E.g. Domain = (0,10) with a maximum score of 10: 10*(log(11-x))/log(11)
A bit late, but as thiton says, you might want to use a sigmoid function instead, since it has a "floor" value for your long tail data points. E.g.:
0.8/(1+5^(x-3)) + 0.2 - You can adjust the constants 5 and 3 to control the slope of the curve. The 0.2 is where the floor will be.
These questions regard a set of data with lists of tasks performed in succession and the total time required to complete them. I've been wondering whether it would be possible to determine useful things about the tasks' lengths, either as they are or with some initial guesstimation based on appropriate domain knowledge. I've come to think graph theory would be the way to approach this problem in the abstract, and have a decent basic grasp of the stuff, but I'm unable to know for certain whether I'm on the right track. Furthermore, I think it's a pretty interesting question to crack. So here we go:
Is it possible to determine the weights of edges in a directed weighted graph, given a list of walks in that graph with the lengths (summed weights) of said walks? I recognize the amount and quality of permutations on the routes taken by the walks will dictate the quality of any possible answer, but let's assume all possible walks and their lengths are given. If a definite answer isn't possible, what kind of things can be concluded about the graph? How would you arrive at those conclusions?
What if there were several similar walks with possibly differing lengths given? Can you calculate a decent average (or other illustrative measure) for each edge, given enough permutations on different routes to take? How will discounting some permutations from the available data set affect the calculation's accuracy?
Finally, what if you had a set of initial guesses as to the weights and had to refine those using the walks given? Would that improve upon your guesstimation ability, and how could you apply the extra information?
EDIT: Clarification on the difficulties of a plain linear algebraic approach. Consider the following set of walks:
a = 5
b = 4
b + c = 5
a + b + c = 8
A matrix equation with these values is unsolvable, but we'd still like to estimate the terms. There might be some helpful initial data available, such as in scenario 3, and in any case we can apply knowledge of the real world - such as that the length of a task can't be negative. I'd like to know if you have ideas on how to ensure we get reasonable estimations and that we also know what we don't know - eg. when there's not enough data to tell a from b.
Seems like an application of linear algebra.
You have a set of linear equations which you need to solve. The variables being the lengths of the tasks (or edge weights).
For instance if the tasks lengths were t1, t2, t3 for 3 tasks.
And you are given
t1 + t2 = 2 (task 1 and 2 take 2 hours)
t1 + t2 + t3 = 7 (all 3 tasks take 7 hours)
t2 + t3 = 6 (tasks 2 and 3 take 6 hours)
Solving gives t1 = 1, t2 = 1, t3 = 5.
You can use any linear algebra techniques (for eg: http://en.wikipedia.org/wiki/Gaussian_elimination) to solve these, which will tell you if there is a unique solution, no solution or an infinite number of solutions (no other possibilities are possible).
If you find that the linear equations do not have a solution, you can try adding a very small random number to some of the task weights/coefficients of the matrix and try solving it again. (I believe falls under Perturbation Theory). Matrices are notorious for radically changing behavior with small changes in the values, so this will likely give you an approximate answer reasonably quickly.
Or maybe you can try introducing some 'slack' task in each walk (i.e add more variables) and try to pick the solution to the new equations where the slack tasks satisfy some linear constraints (like 0 < s_i < 0.0001 and minimize sum of s_i), using Linear Programming Techniques.
Assume you have an unlimited number of arbitrary characters to represent each edge. (a,b,c,d etc)
w is a list of all the walks, in the form of 0,a,b,c,d,e etc. (the 0 will be explained later.)
i = 1
if #w[i] ~= 1 then
replace w[2] with the LENGTH of w[i], minus all other values in w.
repeat forever.
Example:
0,a,b,c,d,e 50
0,a,c,b,e 20
0,c,e 10
So:
a is the first. Replace all instances of "a" with 50, -b,-c,-d,-e.
New data:
50, 50
50,-b,-d, 20
0,c,e 10
And, repeat until one value is left, and you finish! Alternatively, the first number can simply be subtracted from the length of each walk.
I'd forget about graphs and treat lists of tasks as vectors - every task represented as a component with value equal to it's cost (time to complete in this case.
In tasks are in different orderes initially, that's where to use domain knowledge to bring them to a cannonical form and assign multipliers if domain knowledge tells you that the ratio of costs will be synstantially influenced by ordering / timing. Timing is implicit initial ordering but you may have to make a function of time just for adjustment factors (say drivingat lunch time vs driving at midnight). Function might be tabular/discrete. In general it's always much easier to evaluate ratios and relative biases (hardnes of doing something). You may need a functional language to do repeated rewrites of your vectors till there's nothing more that romain knowledge and rules can change.
With cannonical vectors consider just presence and absence of task (just 0|1 for this iteratioon) and look for minimal diffs - single task diffs first - that will provide estimates which small number of variables. Keep doing this recursively, be ready to back track and have a heuristing rule for goodness or quality of estimates so far. Keep track of good "rounds" that you backtraced from.
When you reach minimal irreducible state - dan't many any more diffs - all vectors have the same remaining tasks then you can do some basic statistics like variance, mean, median and look for big outliers and ways to improve initial domain knowledge based estimates that lead to cannonical form. If you finsd a lot of them and can infer new rules, take them in and start the whole process from start.
Yes, this can cost a lot :-)
What is the best way to constrain the values of a PRNG to a smaller range? If you use modulus and the old max number is not evenly divisible by the new max number you bias toward the 0 through (old_max - new_max - 1). I assume the best way would be something like this (this is floating point, not integer math)
random_num = PRNG() / max_orginal_range * max_smaller_range
But something in my gut makes me question that method (maybe floating point implementation and representation differences?).
The random number generator will produce consistent results across hardware and software platforms, and the constraint needs to as well.
I was right to doubt the pseudocode above (but not for the reasons I was thinking). MichaelGG's answer got me thinking about the problem in a different way. I can model it using smaller numbers and test every outcome. So, let's assume we have a PRNG that produces a random number between 0 and 31 and you want the smaller range to be 0 to 9. If you use modulus you bias toward 0, 1, 2, and 3. If you use the pseudocode above you bias toward 0, 2, 5, and 7. I don't think there can be a good way to map one set into the other. The best that I have come up with so far is to regenerate the random numbers that are greater than old_max/new_max, but that has deep problems as well (reducing the period, time to generate new numbers until one is in the right range, etc.).
I think I may have naively approached this problem. It may be time to start some serious research into the literature (someone has to have tackled this before).
I know this might not be a particularly helpful answer, but I think the best way would be to conceive of a few different methods, then trying them out a few million times, and check the result sets.
When in doubt, try it yourself.
EDIT
It should be noted that many languages (like C#) have built in limiting in their functions
int maximumvalue = 20;
Random rand = new Random();
rand.Next(maximumvalue);
And whenever possible, you should use those rather than any code you would write yourself. Don't Reinvent The Wheel.
This problem is akin to rolling a k-sided die given only a p-sided die, without wasting randomness.
In this sense, by Lemma 3 in "Simulating a dice with a dice" by B. Kloeckner, this waste is inevitable unless "every prime number dividing k also divides p". Thus, for example, if p is a power of 2 (and any block of random bits is the same as rolling a die with a power of 2 number of faces) and k has prime factors other than 2, the best you can do is get arbitrarily close to no waste of randomness, such as by batching multiple rolls of the p-sided die until p^n is "close enough" to a power of k.
Let me also go over some of your concerns about regenerating random numbers:
"Reducing the period": Besides batching of bits, this concern can be dealt with in several ways:
Use a PRNG with a bigger "period" (maximum cycle length).
Add a Bays–Durham shuffle to the PRNG's implementation.
Use a "true" random number generator; this is not trivial.
Employ randomness extraction, which is discussed in Devroye and Gravel 2015-2020 and in my Note on Randomness Extraction. However, randomness extraction is pretty involved.
Ignore the problem, especially if it isn't a security application or serious simulation.
"Time to generate new numbers until one is in the right range": If you want unbiased random numbers, then any algorithm that does so will generally have to run forever in the worst case. Again, by Lemma 3, the algorithm will run forever in the worst case unless "every prime number dividing k also divides p", which is not the case if, say, k is 10 and p is 32.
See also the question: How to generate a random integer in the range [0,n] from a stream of random bits without wasting bits?, especially my answer there.
If PRNG() is generating uniformly distributed random numbers then the above looks good. In fact (if you want to scale the mean etc.) the above should be fine for all purposes. I guess you need to ask what the error associated with the original PRNG() is, and whether further manipulating will add to that substantially.
If in doubt, generate an appropriately sized sample set, and look at the results in Excel or similar (to check your mean / std.dev etc. for what you'd expect)
If you have access to a PRNG function (say, random()) that'll generate numbers in the range 0 <= x < 1, can you not just do:
random_num = (int) (random() * max_range);
to give you numbers in the range 0 to max_range?
Here's how the CLR's Random class works when limited (as per Reflector):
long num = maxValue - minValue;
if (num <= 0x7fffffffL) {
return (((int) (this.Sample() * num)) + minValue);
}
return (((int) ((long) (this.GetSampleForLargeRange() * num))) + minValue);
Even if you're given a positive int, it's not hard to get it to a double. Just multiply the random int by (1/maxint). Going from a 32-bit int to a double should provide adequate precision. (I haven't actually tested a PRNG like this, so I might be missing something with floats.)
Psuedo random number generators are essentially producing a random series of 1s and 0s, which when appended to each other, are an infinitely large number in base two. each time you consume a bit from you're prng, you are dividing that number by two and keeping the modulus. You can do this forever without wasting a single bit.
If you need a number in the range [0, N), then you need the same, but instead of base two, you need base N. It's basically trivial to convert the bases. Consume the number of bits you need, return the remainder of those bits back to your prng to be used next time a number is needed.