20160703, 11:26  #1 
(loop (#_fork))
Feb 2006
Cambridge, England
2×3,191 Posts 
Let's attack the BayesianECMbounds again
The Bayesian analysis is in some sense really easy.
For each number of digits D in the putative factor, consider P(finding such a factor  running 1 curve at size B1). This can, it turns out, be very wellapproximated as $$P_B = a_{B} * \exp(b_{B} D)$$) for values in the table below Code:
B multiplier exponent 43e6 29310 0.385 110e6 16139 0.356 260e6 9669 0.331 850e6 5081 0.302 Then the posterior distribution, PP(didn't find a factor of D digits after N curves at B1=B) is just $\alpha * 1(1P_B)^N$. But that's not a real distribution, so you have apply a scale so that all the PP values add up to 1. This gives you a sigmoidal distribution for possiblefactorsizeleft; you then compute P(success for a Ddigit factor after N' curves at B1=B'), multiply the sigmoidal distribution by this new probability, and the sum over all D of that column is the probability of success. If you then divide the runtime for N' curves by the probability of success, and look at this value for lots of different N', you see that there is a minimum (which happens when you're running just a single curve); which translates to there being no point running the larger curves if this minimum is longer than the time it takes to do NFS. See the attached Excel spreadsheet, or do the calculations yourself. Last fiddled with by fivemack on 20160704 at 08:41 
20160710, 15:48  #2 
(loop (#_fork))
Feb 2006
Cambridge, England
2×3,191 Posts 
Implementation
Unpack the small attached file into a directory of your choice, and run with some command line like
python ecmtoy.py G182 18000@43 python ecmtoy.py S240 1@3 and it will try to give you a recipe, defined by the rules above and using fits to quite a lot of measured data, for how much more ECM you should do at which bit levels before switching to NFS, assuming you have already done a certain amount of ECM. The recipes come out as a lot less than the current suggested values, because the current suggested values really don't put enough emphasis on the fact that a number is quite likely to have a factor completely intractable by SNFS. I would be happy if someone could find a problem with my code, but I'm reasonably confident with my measured timings and with the stats I'm using. ecmtimings.txt simply contains lots of lines of the form tractor 3 180 24.55 where that gives the timing in seconds (I've only measured them on one computer, but there's a computername field in case you have much faster equipment somewhere) for running 1 curves of ECM with B1=3e6 and the next prime larger than 10^180 as the input number. 
20160710, 18:15  #3 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×41×71 Posts 
Had some problems initially getting this running under python 3 on windows. I had to add brackets around all of the print statements and replace R.sort() with R = sorted(R).
This produced interesting results with reduced curves at smaller B1s and small amounts at larger B1s. It would be interesting to see results for the small numbers that are normally run through yafu starting from even smaller ecm than 3e6. I would imagine that there could be great improvements. If possible it would be interesting for it to output information like the expected factor size/probability of finding factor of size n etc. I am not convinced by your prior being constant for different sizes of factor. akruppa's post here suggests to me that the probability of having a 45 digit factor is quite different to 60 digit factor for instance. Last fiddled with by henryzz on 20160710 at 18:16 
20160710, 20:07  #4 
(loop (#_fork))
Feb 2006
Cambridge, England
14356_{8} Posts 
With the amount of evidence pushed in by the extra curves, Bayes is strikingly insensitive to the choice of prior. I've attached a new version of the code where you can switch between a flat prior and akruppa's logratio prior (by changing line 129), and which prints out the posterior distribution at the end (columns are P(this range) and cumulative). I get
Flat prior Code:
% python ecmtoy.py G173 7600@43 [[43, 5800], [110, 700]] 30..34 0.0000 0.0000 35..39 0.0000 0.0000 40..44 0.0000 0.0000 45..49 0.0024 0.0024 50..54 0.0627 0.0651 55..59 0.1333 0.1985 60..64 0.1514 0.3499 65..69 0.1544 0.5043 70..74 0.1549 0.6592 75..79 0.1549 0.8141 80..84 0.1549 0.9690 85..89 0.0310 1.0000 Code:
[[43, 5800], [110, 700]] 30..34 0.0000 0.0000 35..39 0.0000 0.0000 40..44 0.0000 0.0000 45..49 0.0024 0.0024 50..54 0.0628 0.0652 55..59 0.1334 0.1986 60..64 0.1515 0.3502 65..69 0.1544 0.5046 70..74 0.1548 0.6594 75..79 0.1548 0.8142 80..84 0.1548 0.9690 85..89 0.0310 1.0000 Code:
[[43, 3500]] 30..34 0.0000 0.0000 35..39 0.0000 0.0000 40..44 0.0000 0.0000 45..49 0.0039 0.0039 50..54 0.0581 0.0620 55..59 0.1124 0.1744 60..64 0.1340 0.3085 65..69 0.1468 0.4553 70..74 0.1581 0.6134 75..79 0.1691 0.7825 80..84 0.1801 0.9627 85..89 0.0373 1.0000 Last fiddled with by fivemack on 20160710 at 20:07 
20160711, 10:52  #5 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×41×71 Posts 
Ok I was unsure about how much the prior would effect the posterior. Switching between the modes is a good check of sanity. Setting priors is one of the things that turns people off Bayesian statistics.
This is interesting stuff. I have reckoned for a while that running a few higher curves is beneficial to the chance of finding a factor rather than just running a full t50 etc. This is a situation where having RDS to check the maths would be useful. There would be nothing worse than discovering that there was a significant bug in a year or two. 
20160713, 23:22  #6  
Mar 2006
2^{3}·59 Posts 
Quote:
One big question I have is, does your analysis take into account the fact that curves at one B1 level have a chance to find factors that are normally found at a different B1 level? For example, gmpecm (7) will tell us (with v) that curves at 110e6 have the following expected number of curves: Code:
35 40 45 50 55 60 65 70 75 80 38 151 692 3583 20479 128305 872747 6392233 5e+007 4.3e+008 Does your analysis take this into account? Does it need to? And my next big question is, when we say we have completed 1*t55, I have heard that there is still a chance that we have missed a factor = e^1 ~= 36.78%. If we complete 2*t55, does that mean the chance of a missed "55digit" factor is e^2 ~= 13.53%? Or, if we have completed 0.3*t55, is the chance of a missed "55digit" factor = e^0.3 ~= 74.08%? And with these different t levels, does your analysis take these chanceofmissedfactor into account? Does it need to? On a side note, but kind of related, I've wanted to ask everyone what they think about changing the list of recommended B1 values. I've been thinking that we could use a list that grows in a predictable way, that will give us a formula for easily figuring out how much work has been done at a specific t level and to make it easy to extend in the future. The new list of B1 that I am thinking of is: Code:
Digits B1 ecm7 curves  5  1,000 = 1.00e3 1 10  3,160 = 3.16e3 2 15  10,000 = 10.00e3 9 20  31,600 = 31.60e3 34 25  100,000 = 100.00e3 123 30  316,000 = 316.00e3 389 35  1,000,000 = 1.00e6 1071 40  3,160,000 = 3.16e6 2642 45  10,000,000 = 10.00e6 5613 50  31,600,000 = 31.60e6 12111 55  100,000,000 = 100.00e6 22113 60  316,000,000 = 316.00e6 39678 65  1,000,000,000 = 1.00e9 67663 70  3,160,000,000 = 3.16e9 107544 75  10,000,000,000 = 10.00e9 etc 80  31,600,000,000 = 31.60e9 85  100,000,000,000 = 100.00e9 90  316,000,000,000 = 316.00e9 95  1,000,000,000,000 = 1.00e12 100  3,160,000,000,000 = 3.16e12 B1 = 10^(digits/10 + 2.5) digits = 10*log10(B1)  25 ie, if you wanted to do work up to t53, then you would calculate your B1 = 10^(53/10 + 2.5) ~= 63e6. or, if you have done work at B1 = 400e6, then the corresponding digit level would be = 10*log10(400e6)  25 ~= 61.02 I don't have enough of a math(s) background to know if this is a sensible recommendation. I'm hoping we can discuss it here to find out if this is a good recommendation or not. If you don't think this topic is right for this discussion, I can start a new thread about it. Also, in another thread, henryzz said: Quote:


20160714, 07:11  #7  
(loop (#_fork))
Feb 2006
Cambridge, England
2·3,191 Posts 
Quote:
Quote:
By the end of the run it knows what the probability is of having factors of any size, and has determined that no ECM that it knows about has an expected time to find a factor less than the time it will take to run NFS. Quote:
The 'classic' tN ECM bounds are from the rather flat bottom of an optimisation of run_time * probability_of_success_at_N_digits; it doesn't make all that much sense to use them as the only sample points when doing something more continuous, but since I needed to pick some sample points I thought I might as well use familiar ones. Maybe going to something more like powers of 10^(0.25) might work if we're doing the timesuccess tradeoff in a different way. The two obvious improvements I need to make to the toy are to allow specification of input size as well as SNFS difficulty in the SNFS case (because at the moment I'm assuming they're equal, which means I'm modelling the ECM as slower than it is), and to provide GPU support since I'll in the future be running most of my step1 ECM on GPUs. I suspect this means I need to split the ECM time into step1 and step2 since a 'GPU curve' is a step1 GPU curve followed by a step2 CPU curve, and to change the accounting a bit since a GPU does step1 curves in blocks of a large fixed size ... so if you are doing large calibration runs, please keep the output file from ecm so you can reparse it and get the values at both stages. Last fiddled with by fivemack on 20160714 at 17:52 

20160714, 08:06  #8  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×41×71 Posts 
Quote:


20160714, 08:58  #9 
(loop (#_fork))
Feb 2006
Cambridge, England
2×3,191 Posts 
The problem with calibration is that the GNFS and SNFS calibration data is recorded on a single computer (the 'tractor' that appears in all the files), and is the reflection of a couple of CPUcenturies of computation; I don't think there are very many people in a position to recollect that information.
I would be a bit inclined to acquire ECM data from other computers, convert it to a 'speed ratio to tractor', and multiply the NFS calibration figures by that ratio; because it's important, for the timeestimation procedure to make sense, that all times in the program are in the same units, and at the moment that unit is 'seconds taken on tractor'. 
20160714, 17:03  #10  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·41·71 Posts 
Quote:


20160717, 01:05  #11  
Mar 2006
2^{3}×59 Posts 
Quote:
Here are two examples of the output: Code:
param = 0 (this line not currently included in any output, it's just here so you can better understand timing differences) # computer, digits, 11e3, 50e3, 250e3, 1e6, 3e6, 11e6, 43e6 pc1, 150, 0.031s+0.032s, 0.109s+0.062s, 0.499s+0.250s, 1.841s+1.061s, 5.491s+2.387s, 19.718s+8.034s, 73.928s+23.354s pc1, 155, 0.031s+0.016s, 0.140s+0.078s, 0.577s+0.249s, 2.199s+1.077s, 6.474s+2.527s, 23.836s+8.471s, 92.305s+25.850s pc1, 160, 0.062s+0.016s, 0.140s+0.094s, 0.530s+0.265s, 2.106s+1.107s, 6.177s+2.496s, 22.698s+8.361s, 88.140s+25.475s pc1, 165, 0.031s+0.015s, 0.124s+0.063s, 0.577s+0.265s, 2.091s+1.123s, 6.287s+2.558s, 22.526s+8.642s, 88.312s+26.801s pc1, 170, 0.031s+0.032s, 0.140s+0.078s, 0.577s+0.265s, 2.121s+1.186s, 6.208s+2.590s, 22.542s+8.564s, 88.514s+26.302s param = 1 (this line not currently included in any output, it's just here so you can better understand timing differences) # computer, digits, 11e3, 50e3, 250e3, 1e6, 3e6, 11e6, 43e6 pc1, 150, 0.046s+0.016s, 0.093s+0.078s, 0.390s+0.234s, 1.498s+0.998s, 3.978s+2.277s, 14.555s+7.675s, 57.003s+23.431s pc1, 155, 0.031s+0.031s, 0.109s+0.078s, 0.468s+0.250s, 1.670s+1.092s, 4.852s+2.511s, 17.722s+8.408s, 69.155s+25.693s pc1, 160, 0.031s+0.031s, 0.109s+0.078s, 0.468s+0.250s, 1.654s+1.108s, 4.821s+2.496s, 17.628s+8.346s, 69.015s+25.490s pc1, 165, 0.046s+0.032s, 0.109s+0.093s, 0.468s+0.265s, 1.654s+1.139s, 4.851s+2.543s, 17.675s+8.611s, 68.812s+26.442s pc1, 170, 0.031s+0.032s, 0.109s+0.078s, 0.452s+0.250s, 1.654s+1.139s, 4.867s+2.558s, 17.628s+8.611s, 68.968s+26.349s Please let me know if you have any problems with this script, or would like to see any features added or modified. fivemack, to your point of: Quote:
Code:
digits  B1 40  3.16e6 41  3.98e6 42  5.01e6 43  6.30e6 44  7.94e6 45  10.00e6 ...etc... 

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