Here is a presentation of my dataset :
Large social-network composed of Twitter accounts followers of very large related accounts, followers of this followers, and followers of these followers, at every iteration cleaned for bot accounts, private-accounts, etc.
Total nodes : around 500,000
Total connections : 95 millions
4 nodes have more than 3 millions connections
567 nodes have more than 100,000 connections
Half of the dataset have 3 or less connections
This said, I want to clean this network in order to get the "best" single-community coming out of the raw initial graph before further clustering in sub communities. Keep in mind these few facts:
Due to the way the data is collected, I know there is One Large Community common to most of these nodes that is more optimal than the whole network.
I would like to get an optimal single sub-network of the initial network, getting rid of all the nodes that don't belong to the largest possible common-community.
Further study will constitute in splitting this community in several communities, following the general community-detection literature, but this is not what I want to do here.
I have used community-detection algorithms such as louvain or modularity-optimization (in a smaller subsample for the too computational second one), but the goals of these algos are to have the best split, while my goal in some ways is to have the best merge.
The main problema can be summarized by this idea: I was considering using the following algo. Starting with the large network ; removing the "weakest" node at every iteration ; while the modularity of the whole improves. But this will lead to a very tiny community at the end.
Do you have any directions where to look for ? A way to change the methodology of an existing algo ? Or even a paper that is related to this issue even if pretty different ?
Thank you.
Here you can try several approaches. The size of your network is challenging, not all community detection methods are capable to run in reasonable time on such a large network. You could try those methods which have adjustable parameters, and empirically find out how these parameters affect their resolution. At certain values you can expect the core network covered by one cluster. For example there are the walktrap and spinglass method in igraph. If you change the number of steps at walktrap, you can observe a change in the size of the largest community:
g <- barabasi.game(n = 10000, m = 2)
steps <- seq(1, 10, 1)
steps <- c(steps, seq(11, 200, 10))
w <- list()
ccount <- NULL
clargest <- NULL
for(s in steps){
cat(paste('Running walktrap with steps =', s, '\n'))
w0 <- walktrap.community(g, steps = s)
ccount <- c(ccount, length(levels(as.factor(w0$membership))))
clargest <- c(clargest, max(tapply(w0$membership, w0$membership, length)))
w[[s]] <- w0
}
plot(ccount ~ steps,
xlab = 'Number of steps',
ylab = 'Number of communities',
main = 'Walktrap with increasing number of steps')
plot(clargest ~ steps,
xlab = 'Number of steps',
ylab = 'Size of largest community',
main = 'Walktrap with increasing number of steps')
Similarly with changing the gamma parameter for spinglass:
gamma <- c(0.05, 0.1, 0.2, 0.5, 0.7, 0.85, 1.0, 1.2, 1.5, 1.8, 2.0, 2.5, 3.0, 3.5, 5.0, 10.0, 20.0, 50.0, 100.0, 500.0)
sg <- list()
sgsize <- NULL
for(gm in gamma){
cat(paste('Running spinglass with gamma =', gm, '\n'))
sg0 <- spinglass.community(g, vertex = 1, gamma = gm)
sgsize <- c(sgsize, length(sg0$community))
sg[[as.character(gm)]] <- sg0
}
plot(sgsize ~ log10(gamma),
xlab = 'Gamma (log)',
ylab = 'Size of the community',
main = 'Spinglass with increasing value of gamma',
xaxt = 'n'
)
Another method infomap, according to its description is designed exactly to problems like yours. You may want to use not the igraph implementation, but the original, as the latter gives more freedom in setting parameters. There are more Python implementations, but I don't know how flexible those are.
You can also try the moduland method family. Here you can chose between four landscape building method: nodeland, linkland, perturland and edgeweight; and two hill detection methods: total_hill and proportional_hill; in addition, at perturland you can set a parameter x. Please read the papers for more info. As I mentioned in a comment, you could inspect the affinities and set a threshold to select your core network. These methods have no Python interface, but you can simple export a text file and call the binaries by subprocess, and read their output back to Python.
For an overview of a large number of other methods see here from page 52 ---this is already not up to date, but comprehensive.
Another idea is that you could run a number of methods, and comparing their results, find core network as a large partition delimited by cluster boundaries constant accross different methods. It is also a question how exact solution you need. Considering your data is quite noisy, likely you can expect thousands of nodes misclassified by any method. For a comparison of different clusterings, you could use something like normalized mutual information, which is implemented in igraph (see more here.
Related
I'm trying to manually implement L2 regularisation and a couple of its variations in a neural network. What I'm doing is the following:
for name, param in model.state_dict():
if 'weight' in name:
l2_reg += torch.sum(param**2)
loss = cross_entropy(outputs, labels) + 0.0001*l2_reg
Is this equivalent to adding 'weight_decay = 0.0001' inside my optimizer? i.e.:
torch.optim.SGD(model.parameters(), lr=learning_rate , momentum=0.9, weight_decay = 0.0001)
My problem is that I thought they were equivalent, but the manual procedure is about 100x slower than adding 'weight_decay = 0.0001'. Why is that? How can I fix it?
Note that I need to also implement my own variation of L2 regularization, so just adding 'weight_decay = 0.0001' won't help.
You can check PyTorch implementation of SGD to get some tips and base off of that code.
There are a few things going on which should speed up your custom regularization.
Below is a cleaned version (a little pseudo-code, refer to original) of the parts we are interested in:
for p in group['params']:
if p.grad is None:
continue
d_p = p.grad.data
if weight_decay != 0:
d_p.add_(weight_decay, p.data)
p.data.add_(-group['lr'], d_p)
return loss
BTW. It seems your implementation is mathematically sound (correct me if I missed anything) and equivalent to PyTorch but will be slow indeed.
Modify only gradient
Please notice you perform regularization explicitly during forward pass. This takes a lot of time, more or less because:
take parameters and iterate over them
take it to the power of 2
sum all of them
add to variable containing all previous parameters (all this while creating graph dynamically and creating new nodes).
What pytorch does is it only focuses on backward pass as that's all is needed. This is pretty handy because:
parameters have to be loaded and iterated over once anyway during corrections performed by optimizer (in your case they are taken out twice)
no power of 2 because gradient of w**2 is simply 2*w (2 is further left out and L2 is often expressed as 1/2 * w **2 to make it simpler and a little faster)
no accumulation and creation of additional graph nodes
Essentially, this line:
d_p.add_(weight_decay, p.data)
Modifies the gradient adding p.data (weight) multiplied by weight_decay all done in-place (notice d_p.add_), which is all you have to do to perform L2 regularization.
Finally this line:
p.data.add_(-group['lr'], d_p)
Updates weights with gradient (modified by weight decay) using standard SGD formula (once again, in-place to be as fast as possible, at least on Python level).
Your own implementation
I would advise you to follow similar logic for your own regularization if you want to make it faster.
You can copy PyTorch implementation of SGD and only change this one relevant line. This would also gives you functionality of PyTorch optimizer in case you need it in your experiments.
For L1 regularization (|w| instead of w**2) you would have to calculate the derivative of it (which is 1 for positive case, -1 for negative and undefined for 0 (we can't have that so it should be zero)).
With that in mind we can write the weight_decay like this:
if weight_decay != 0:
d_p.add_(weight_decay, torch.sign(p.data))
torch.sign returns 1 for positive values and -1 for negative and 0 for... yeah, 0.
Hope this helps, exact implementation is left for you (hit me up in the comments in case you have any questions or troubles).
I'm taking the CalTech online course Learning From Data, and I'm stumped with creating a Perceptron in Scala. I chose Scala because I'm learning it and wanted to challenge myself. I understand the theory, and I also understand others' solutions in Python and Ruby. But I can't figure out why my own Scala code doesn't work.
For a background in the Perceptron code: Learning_algorithm
I'm running Scala 2.11 on OSX 10.10.
Per the algorithm, I start off with weights (0.0, 0.0, 0.0), where weight[2] is a learned bias component. I've already generated a test set in the space [-1, 1],[-1,1] on the X-Y plane. I do this by a) picking two random points and drawing a line through them, then b) generating some other random points and calculating if they are on one side of the line or the other. As far as I can tell by plotting it in Python, this generates linearly separable data.
My next step is to take my initialized weights and check against every point to find miss-classified points, i.e. points that don't generate the right +1 or -1 result. Here is the code that simply calculates dot-product of the weight and the vector x:
def h(weight:List[Double], p:Point ): Double = if ( (weight(0)*p.x + weight(1)*p.y + weight(2)) > 0) 1 else -1
It's the initial weights, so they are all miss-classified. I then update the weights, like so:
def newH(weight:List[Double], p:Point, y:Double): List[Double] = {
val newWt = scala.collection.mutable.ArrayBuffer[Double](0.0, 0.0, 0.0)
newWt(0) = weight(0) + p.x*y
newWt(1) = weight(1) + p.y*y
newWt(2) = weight(2) + 1*y
return newWt.toList
}
Then I identify miss-classified points again by checking the test set against the value output by h() above, and continue iterating.
This follows the algorithm (or is supposed to, at least) that Prof Yaser shows here: Library
The problem is that the algorithm never converges. My weights -- the third component of which is the bias -- keep getting more negative or more positive. My weight vector after every adjustment resembles this:
Weights: List(16.43341624736786, 11627.122008800507, -34130.0)
Weights: List(15.533397436141968, 11626.464265227318, -34131.0)
Weights: List(14.726969361305237, 11626.837346673012, -34132.0)
Weights: List(14.224745154380798, 11627.646470665932, -34133.0)
Weights: List(14.075232982635498, 11628.026384592056, -34134.0)
I'm a Scala newbie so my code is probably atrocious. But am I missing something in Scala, e.g. reassignment, that could be causing my weight to be messed up? Or have I completely misunderstood how the Perceptron even operates? Is my weight update just wrong?
Thanks for any help you can give me on this!
Thanks Till. I've discovered the two problems with my code and I'll share them, but to address your point: Someone else asked about this on the class's forum and it looks like what the Wiki formula does is simply to change the learning rate. Alpha can be picked randomly, and y-h(weight, p) would give you weights like
-1-1 = 2
In the case that y=-1 and h()=1, or
1-(-1) = 2
In the case that y=1 and h()=-1
My/the class formula takes 1*p.x instead of alpha*2, which seems to be a matter of different learning rates. Hope that makes sense.
My two problems were as follows:
The y value passed into the recalculation formula newH needs to be the target value of y, that is, the "correct y" that was discovered while generating the test points. I was passing in the y that was generated through h(), which is the guessed-at function. This makes sense obviously since we are looking to correc the weight by using the target y, not the incorrect y.
I was doing a comparison of target y and h()=yin Scala, but was comparison an element obtained from a map through .get(). My Scala map looks like Map[Point,Double] where the Double value refers to the y value generated during the test set creation. But doing a .get() gives you Option[Double] and not a Double value at all. This is explained in Scala Map#get and the return of Some() and makes a lot of sense now. I did map.get(<some Point>).get() for now, since I was focusing on debugging and not code perfection, and then I was accurately able to compare two Double values.
I'm using SVMLib to train a simple SVM over the MNIST dataset. It contains 60.000 training data. However, I have several performance issues: the training seems to be endless (after a few hours, I had to shut it down by hand, because it doesn't respond). My code is very simple, I just call ovrtrain on the dataset without any kernel and any special constants:
function features = readFeatures(fileName)
[fid, msg] = fopen(fileName, 'r', 'ieee-be');
header = fread(fid, 4, "int32" , 0, "ieee-be");
if header(1) ~= 2051
fprintf("Wrong magic number!");
end
M = header(2);
rows = header(3);
columns = header(4);
features = fread(fid, [M, rows*columns], "uint8", 0, "ieee-be");
fclose(fid);
return;
endfunction
function labels = readLabels(fileName)
[fid, msg] = fopen(fileName, 'r', 'ieee-be');
header = fread(fid, 2, "int32" , 0, "ieee-be");
if header(1) ~= 2049
fprintf("Wrong magic number!");
end
M = header(2);
labels = fread(fid, [M, 1], "uint8", 0, "ieee-be");
fclose(fid);
return;
endfunction
labels = readLabels("train-labels.idx1-ubyte");
features = readFeatures("train-images.idx3-ubyte");
model = ovrtrain(labels, features, "-t 0"); % doesn't respond...
My question: is it normal? I'm running it on Ubuntu, a virtual machine. Should I wait longer?
I don't know whether you took your answer or not, but let me tell you what I predict about your situation. 60.000 examples is not a lot for a power trainer like LibSVM. Currently, I am working on a training set of 6000 examples and it takes 3-to-5 seconds to train. However, the parameter selection is important and that is the one probably taking long time. If the number of unique features in your data set is too high, then for any example, there will be lots of zero feature values for non-existing features. If the tool is implementing data scaling on your training set, then most probably those lots of zero feature values will be scaled to a certain non-zero value, leaving you astronomic number of unique and non-zero valued features for each and every example. This is very very complicated for a SVM tool to get in and extract efficient parameter values.
Long story short, if you had enough research on SVM tools and understand what I mean, you either assign parameter values in the training command before executing it or find a way to decrease the number of unique features. If you haven't, go on and download the latest version of LibSVM, read the ReadME files as well as the FAQ from the website of the tool.
If non of these is the case, then sorry for taking your time:) Good luck.
It might be an issue of convergence given the characteristics of your data.
Check the kernel you have as default selection and change it. Also, check the stopping criterion of the package. Additionally, if you are looking for faster implementation, check MSVMpack which is a parallel implementation of SVM.
Finally, feature selection in your case is desired. You can end up with a good feature subset of almost half of what you have. In addition, you need only a portion of data for training e.g. 60~70 % are sufficient.
First of all 60k is huge data for training.Training that much data with linear kernel will take hell of time unless you have a supercomputing. Also you have selected a linear kernel function of degree 1. Its better to use Gaussian or higher degree polynomial kernel (deg 4 used with the same dataset showed a good tranning accuracy). Try to add the LIBSVM options for -c cost -m memory cachesize -e epsilon tolerance of termination criterion (default 0.001). First run 1000 samples with Gaussian/ polynomial of deg 4 and compare the accuracy.
I have to solve a huge linear equation for multiple right sides (Let's say 20 to 200). The Matrix is stored in a sparse format and distributed over multiple MPI nodes (Let's say 16 to 64). I run a CG solver on the rank 0 node. It's not possible to solve the linear equation directly, because the system matrix would be dense (Sys = A^T * S * A).
The basic Matrix-Vector multiplication is implemented as:
broadcast x
y = A_part * x
reduce y
While the collective operations are reasonably fast (OpenMPI seems to use a binary tree like communication pattern + Infiniband), it still accounts for a quite large part of the runtime. For performance reasons we already calculate 8 right sides per iteration (Basicly SpM * DenseMatrix, just to be complete).
I'm trying to come up with a good scheme to hide the communication latency, but I did not have a good idea yet. I also try to refrain from doing 1:n communication, although I did not yet measure if scaling would be a problem.
Any suggestions are welcome!
If your matrix is already distributed, would it be possible to use a distributed sparse linear solver instead of running it only on rank 0 and then broadcasting the result (if I'm reading your description correctly..). There's plenty of libraries for that, e.g. SuperLU_DIST, MUMPS, PARDISO, Aztec(OO), etc.
The "multiple rhs" optimization is supported by at least SuperLU and MUMPS (haven't checked the others, but I'd be VERY surprised if they didn't support it!), since they solve AX=B where X and B are matrices with potentially > 1 column. That is, each "rhs" is stored as a column vector in B.
If you don't need to have the results of an old right-hand-side before starting the next run you could try to use non-blocking communication (ISend, IRecv) and communicate the result while calculating the next right-hand-side already.
But make sure you call MPI_Wait before reading the content of the communicated array, in order to be sure you're not reading "old" data.
If the matrices are big enough (i.e. it takes long enough to calculate the matrix-product) you don't have any communication delay at all with this approach.
I currently have an application which can contain 100s of user defined formulae. Currently, I use reverse polish notation to perform the calculations (pushing values and variables on to a stack, then popping them off the stack and evaluating). What would be the best way to start parallelizing this process? Should I be looking at a functional language?
The calculations are performed on arrays of numbers so for example a simple A+B could actually mean 100s of additions. I'm currently using Delphi, but this is not a requirement going forward. I'll use the tool most suited to the job. Formulae may also be dependent on each other So we may have one formula C=A+B and a second one D=C+A for example.
Let's assume your formulae (equations) are not cyclic, as otherwise you cannot "just" evaluate them. If you have vectorized equations like A = B + C where A, B and C are arrays, let's conceptually split them into equations on the components, so that if the array size is 5, this equation is split into
a1 = b1 + c1
a2 = b2 + c2
...
a5 = b5 + c5
Now assuming this, you have a large set of equations on simple quantities (whether integer, rational or something else).
If you have two equations E and F, let's say that F depends_on E if the right-hand side of F mentions the left-hand side of E, for example
E: a = b + c
F: q = 2*a + y
Now to get towards how to calculate this, you could always use randomized iteration to solve this (this is just an intermediate step in the explanation), following this algorithm:
1 while (there is at least one equation which has not been computed yet)
2 select one such pending equation E so that:
3 for every equation D such that E depends_on D:
4 D has been already computed
5 calculate the left-hand side of E
This process terminates with the correct answer regardless on how you make your selections on line // 2. Now the cool thing is that it also parallelizes easily. You can run it in an arbitrary number of threads! What you need is a concurrency-safe queue which holds those equations whose prerequisites (those the equations depend on) have been computed but which have not been computed themselves yet. Every thread pops out (thread-safely) one equation from this queue at a time, calculates the answer, and then checks if there are now new equations so that all their prerequisites have been computed, and then adds those equations (thread-safely) to the work queue. Done.
Without knowing more, I would suggest taking a SIMD style approach if possible. That is, create threads to compute all formulas for a single data set. Trying to divide the computation of formulas to parallelise them wouldn't yield much speed improvement as the logic required to be able to split up the computations into discrete units suitable for threading would be hard to write and harder to get right, the overhead would cancel out any speed gains. It would also suffer quickly from diminishing returns.
Now, if you've got a set of formulas that are applied to many sets of data then the parallelisation becomes easier and would scale better. Each thread does all computations for one set of data. Create one thread per CPU core and set its affinity to each core. Each thread instantiates one instance of the formula evaluation code. Create a supervisor which loads a single data set and passes it an idle thread. If no threads are idle, wait for the first thread to finish processing its data. When all data sets are processed and all threads have finished, then exit. Using this method, there's no advantage to having more threads than there are cores on the CPU as thread switching is slow and will have a negative effect on overall speed.
If you've only got one data set then it is not a trivial task. It would require parsing the evaluation tree for branches without dependencies on other branches and farming those branches to separate threads running on each core and waiting for the results. You then get problems synchronizing the data and ensuring data coherency.