Straight to the facts.
My Neural network is a classic feedforward backpropagation.
I have a historical dataset that consists of:
time, temperature, humidity, pressure
I need to predict next values basing on historical data.
This dataset is about 10MB large therefore training it on one core takes ages. I want to go multicore with the training, but i can't understand what happens with the training data for each core, and what exactly happens after cores finish working.
According to: http://en.wikipedia.org/wiki/Backpropagation#Multithreaded_Backpropagation
The training data is broken up into equally large batches for each of
the threads. Each thread executes the forward and backward
propagations. The weight and threshold deltas are summed for each of
the threads. At the end of each iteration all threads must pause
briefly for the weight and threshold deltas to be summed and applied
to the neural network.
'Each thread executes forward and backward propagations' - this means, each thread just trains itself with it's part of the dataset, right? How many iterations of the training per core ?
'At the en dof each iteration all threads must pause briefly for the weight and threshold deltas to be summed and applied to neural network' - What exactly does that mean? When cores finish training with their datasets, wha does the main program do?
Thanks for any input into this!
Complete training by backpropagation is often not the thing one is really looking for, the reason being overfitting. In order to obtain a better generalization performance, approaches such as weight decay or early stopping are commonly used.
On this background, consider the following heuristic approach: Split the data in parts corresponding to the number of cores and set up a network for each core (each having the same topology). Train each network completely separated of the others (I would use some common parameters for the learning rate, etc.). You end up with a number of http://www.texify.com/img/%5Cnormalsize%5C%21N_%7B%5Ctext%7B%7D%7D.gif
trained networks http://www.texify.com/img/%5Cnormalsize%5C%21f_i%28x%29.gif.
Next, you need a scheme to combine the results. Choose http://www.texify.com/img/%5Cnormalsize%5C%21F%28x%29%3D%5Csum_%7Bi%3D1%7D%5EN%5C%2C%20%5Calpha_i%20f_i%28x%29.gif, then use least squares to adapt the parameters http://www.texify.com/img/%5Cnormalsize%5C%21%5Calpha_i.gif such that http://www.texify.com/img/%5Cnormalsize%5C%21%5Csum_%7Bj%3D1%7D%5EM%20%5C%2C%20%5Cbig%28F%28x_j%29%20-%20y_j%5Cbig%29%5E2.gif is minimized. This involves a singular value decomposition which scales linearly in the number of measurements M and thus should be feasible on a single core. Note that this heuristic approach also bears some similiarities to the Extreme Learning Machine. Alternatively, and more easily, you can simply try to average the weights, see below.
Moreover, see these answers here.
Regarding your questions:
As Kris noted it will usually be one iteration. However, in general it can be also a small number chosen by you. I would play around with choices roughly in between 1 and 20 here. Note that the above suggestion uses infinity, so to say, but then replaces the recombination step by something more appropriate.
This step simply does what it says: it sums up all weights and deltas (what exactly depends on your algoithm). Remember, what you aim for is a single trained network in the end, and one uses the splitted data for estimation of this.
To collect, often one does the following:
(i) In each thread, use your current (global) network weights for estimating the deltas by backpropagation. Then calculate new weights using these deltas.
(ii) Average these thread-local weights to obtain new global weights (alternatively, you can sum up the deltas, but this works only for a single bp iteration in the threads). Now start again with (i) in which you use the same newly calculated weights in each thread. Do this until you reach convergence.
This is a form of iterative optimization. Variations of this algorithm:
Instead of using always the same split, use random splits at each iteration step (... or at each n-th iteration). Or, in the spirit of random forests, only use a subset.
Play around with the number of iterations in a single thread (as mentioned in point 1. above).
Rather than summing up the weights, use more advanced forms of recombination (maybe a weighting with respect to the thread-internal training-error, or some kind of least squares as above).
... plus many more choices as in each complex optimization ...
For multicore parallelization it makes no sense to think about splitting the training data over threads etc. If you implement that stuff on your own you will most likely end up with a parallelized implementation that is slower than the sequential implementation because you copy your data too often.
By the way, in the current state of the art, people usually use mini-batch stochastic gradient descent for optimization. The reason is that you can simply forward propagate and backpropagate mini-batches of samples in parallel but batch gradient descent is usually much slower than stochastic gradient descent.
So how do you parallelize the forward propagation and backpropagation? You don't have to create threads manually! You can simply write down the forward propagation with matrix operations and use a parallelized linear algebra library (e.g. Eigen) or you can do the parallelization with OpenMP in C++ (see e.g. OpenANN).
Today, leading edge libraries for ANNs don't do multicore parallelization (see here for a list). You can use GPUs to parallelize matrix operations (e.g. with CUDA) which is orders of magnitude faster.
Related
Can anyone please suggest Partitioning algorithms to partition the vision algorithm (computations or workload) to expose opportunities for parallel execution by decomposing computations into small tasks
You don't need a partitioning algorithm necessarily.
In any convolution task, each pixel in the output is independent of any other output pixel. Morphological operations are similarly parallelizable, as well as the Hough Transform.
Using any of these, you could have multiple threads or processes working together. A simple implementation would have a painter that iterates over all pixels, and when a thread is free, it simply takes the current item and advances the iterator (preferably atomically, but it won't break if it isn't atomic), performs the appropriate computation, and writes the result to the output. You don't need to worry about any deadlock or race conditions because the computations are independent of each other.
For this application, I would like to use an algorithm for dimensionality reduction such that a given number of components all explain about the same amount of variance in the data.
Principal Component Analysis is therefore not suited because the explained variance decreases sharply from the first principal component to each subsequent one.
What algorithms can I use?
If you just don't like the variance ordering among PCs, you can pick up a number of PCs,
then randomly rotate them somewhat. It is still interesting know how the extra ordering
information negatively impacts your application.
I’m conducting a molecular dynamics simulation, and I’ve been struggling for quite a while to implement it in parallel, and although I succeeded in fully loading my 4-thread processor, the computation time in parallel is greater than the computation time in serial mode.
Studying at which point of time each thread starts and finishes its loop iteration, I’ve noticed a pattern: it’s as if different threads are waiting for each other.
It was then that I turned my attention to the structure of my program. I have a class, an instance of which represents my system of particles, containing all the information about particles and some functions that use this information. I also have a class instance of which represents my interatomic potential, containing parameters of potential function along with some functions (one of those functions calculates force between two given particles).
And so in my program there exist instances of two different classes, and they interact with each other: some functions of one class take references to instances of another class.
And the block I’m trying to implement in parallel looks like this:
void Run_simulation(Class_system &system, Class_potential &potential, some other arguments){
#pragma omp parallel for
for(…)
}
for(...) is the actual computation, using data from the system instance of the Class_system class and some functions from thepotential instance of the Class_potential class.
Am I right that it’s this structure that’s the source of my troubles?
Could you suggest me what has to be done in this case? Must I rewrite my program in completely different manner? Should I use some different tool to implement my program in parallel?
Without further details on your simulation type I can only speculate, so here are my speculations.
Did you look into the issue of load balancing? I guess the loop distributes the particles among threads but if you have some kind of a restricted range potential, then the computational time might differ from particle to particle in the different regions of the simulation volume, depending on the spatial density. This is a very common problem in molecular dynamics and one that is very hard to solve properly in distributed memory (MPI in most cases) codes. Fortunately with OpenMP you get direct access to all particles at each computing element and so the load balancing is much easier to achieve. It is not only easier, but it is also built-in, so to speak - simply change the scheduling of the for directive with the schedule(dynamic,chunk) clause, where chunk is a small number whose optimal value might vary from simulation to simulation. You might make chunk part of the input data to the program or you might instead write schedule(runtime) and then play with different scheduling classes by setting the OMP_SCHEDULE environment variable to values like "static", "dynamic,1", "dynamic,10", "guided", etc.
Another possible source of performance degradation is false sharing and true sharing. False sharing occurs when your data structure is not suitable for concurrent modification. For example, if you keep 3D positional and velocity information for each particle (let's say you use velocity Verlet integrator), given IEEE 754 double precision, each coordinate/velocity triplet takes 24 bytes. This means that a single cache line of 64 bytes accommodates 2 complete triplets and 2/3 of another one. The consequence of this is that no matter how you distribute the particles among the threads, there would always be at lest two threads that would have to share a cache line. Suppose that those threads run on different physical cores. If one thread writes to its copy of the cache line (for example it updates the position of a particle), the cache coherency protocol would be involved and it will invalidate the cache line in the other thread, which would then have to reread it from a slower cache of even from main memory. When the second thread update its particle, this would invalidate the cache line in the first core. The solution to this problem comes with proper padding and proper chunk size choice so that no two threads would share a single cache line. For example, if you add a superficial 4-th dimension (you can use it to store the potential energy of the particle in the 4-th element of the position vector and the kinetic energy in the 4-th element of the velocity vector) then each position/velocity quadruplet would take 32 bytes and information for exactly two particles would fit in a single cache line. If you then distribute an even number of particles per thread, you automatically get rid of possible false sharing.
True sharing occurs when threads access concurrently the same data structure and there is an overlap between the parts of the structure, modified by the different threads. In molecular dynamics simulations this occurs very frequently as we want to exploit the Newton's third law in order to cut the computational time in two when dealing with pairwise interaction potentials. When one thread computes the force acting on particle i, while enumerating its neighbours j, computing the force that j exerts on i automatically gives you the force that i exerts on j so that contribution can be added to the total force on j. But j might belong to another thread that might be modifying it at the same time, so atomic operations have to be used for both updates (both, sice another thread might update i if it happens to neighbour one of more of its own particles). Atomic updates on x86 are implemented with locked instructions. This is not that horribly slow as often presented, but still slower than a regular update. It also includes the same cache line invalidation effect as with false sharing. To get around this, at the expense of increased memory usage one could use local arrays to store partial force contributions and then perform a reduction in the end. The reduction itself has to either be performed in serial or in parallel with locked instructions, so it might turn out that not only there is no gain from using this approach, but rather it could be even slower. Proper particles sorting and clever distribution between the processing elements so to minimise the interface regions can be used to tackle this problem.
One more thing that I would like to touch is the memory bandwidth. Depending on your algorithm, there is a certain ratio between the number of data elements fetched and the number of floating point operations performed at each iteration of the loop. Each processor has only a limited bandwidth available for memory fetches and if it happens that your data does not quite fit in the CPU cache, then it might happen that the memory bus is unable to deliver enough data to feed so many threads executing on a single socket. Your Core i3-2370M has only 3 MiB of L3 cache so if you explicitly keep the position, velocity and force for each particle, you can only store about 43000 particles in the L3 cache and about 3600 particles in the L2 cache (or about 1800 particles per hyperthread).
The last one is hyperthreading. As High Performance Mark has already noted, hyperthreads share a great deal of core machinery. For example there is only one AVX vector FPU engine that is shared among both hyperthreads. If your code is not vectorised, you lose a great deal of computing power available in your processor. If your code is vectorised, then both hyperthreads will get into each others way as they fight for control over the AVX engine. Hyperthreading is useful only when it is able to hide memory latency by overlaying computation (in one hyperthread) with memory loads (in another hyperthread). With dense numerical codes that perform many register operations before they perform memory load/store, hyperthreading gives no benefits whatsoever and you'd be better running with half the number of threads and explicitly binding them to different cores as to prevent the OS scheduler from running them as hyperthreads. The scheduler on Windows is particularly dumb in this respect, see here for an example rant. Intel's OpenMP implementation supports various binding strategies controlled via environment variables. GNU's OpenMP implementation too. I am not aware of any way to control threads binding (a.k.a. affinity masks) in Microsoft's OpenMP implementation.
I'd like to ask fellow SO'ers for their opinions regarding best of breed data structures to be used for indexing time-series (aka column-wise data, aka flat linear).
Two basic types of time-series exist based on the sampling/discretisation characteristic:
Regular discretisation (Every sample is taken with a common frequency)
Irregular discretisation(Samples are taken at arbitary time-points)
Queries that will be required:
All values in the time range [t0,t1]
All values in the time range [t0,t1] that are greater/less than v0
All values in the time range [t0,t1] that are in the value range[v0,v1]
The data sets consist of summarized time-series (which sort of gets over the Irregular discretisation), and multivariate time-series. The data set(s) in question are about 15-20TB in size, hence processing is performed in a distributed manner - because some of the queries described above will result in datasets larger than the physical amount of memory available on any one system.
Distributed processing in this context also means dispatching the required data specific computation along with the time-series query, so that the computation can occur as close to the data as is possible - so as to reduce node to node communications (somewhat similar to map/reduce paradigm) - in short proximity of computation and data is very critical.
Another issue that the index should be able to cope with, is that the overwhelming majority of data is static/historic (99.999...%), however on a daily basis new data is added, think of "in the field senors" or "market data". The idea/requirement is to be able to update any running calculations (averages, garch's etc) with as low a latency as possible, some of these running calculations require historical data, some of which will be more than what can be reasonably cached.
I've already considered HDF5, it works well/efficiently for smaller datasets but starts to drag as the datasets become larger, also there isn't native parallel processing capabilities from the front-end.
Looking for suggestions, links, further reading etc. (C or C++ solutions, libraries)
You would probably want to use some type of large, balanced tree. Like Tobias mentioned, B-trees would be the standard choice for solving the first problem. If you also care about getting fast insertions and updates, there is a lot of new work being done at places like MIT and CMU into these new "cache oblivious B-trees". For some discussion of the implementation of these things, look up Tokutek DB, they've got a number of good presentations like the following:
http://tokutek.com/downloads/mysqluc-2010-fractal-trees.pdf
Questions 2 and 3 are in general a lot harder, since they involve higher dimensional range searching. The standard data structure for doing this would be the range tree (which gives O(log^{d-1}(n)) query time, at the cost of O(n log^d(n)) storage). You generally would not want to use a k-d tree for something like this. While it is true that kd trees have optimal, O(n), storage costs, it is a fact that you can't evaluate range queries any faster than O(n^{(d-1)/d}) if you only use O(n) storage. For d=2, this would be O(sqrt(n)) time complexity; and frankly that isn't going to cut it if you have 10^10 data points (who wants to wait for O(10^5) disk reads to complete on a simple range query?)
Fortunately, it sounds like your situation you really don't need to worry too much about the general case. Because all of your data comes from a time series, you only ever have at most one value per each time coordinate. Hypothetically, what you could do is just use a range query to pull some interval of points, then as a post process go through and apply the v constraints pointwise. This would be the first thing I would try (after getting a good database implementation), and if it works then you are done! It really only makes sense to try optimizing the latter two queries if you keep running into situations where the number of points in [t0, t1] x [-infty,+infty] is orders of magnitude larger than the number of points in [t0,t1] x [v0, v1].
General ideas:
Problem 1 is fairly common: Create an index that fits into your RAM and has links to the data on the secondary storage (datastructure: B-Tree family).
Problem 2 / 3 are quite complicated since your data is so large. You could partition your data into time ranges and calculate the min / max for that time range. Using that information, you can filter out time ranges (e.g. max value for a range is 50 and you search for v0>60 then the interval is out). The rest needs to be searched by going through the data. The effectiveness greatly depends on how fast the data is changing.
You can also do multiple indices by combining the time ranges of lower levels to do the filtering faster.
It is going to be really time consuming and complicated to implement this by your self. I recommend you use Cassandra.
Cassandra can give you horizontal scalability, redundancy and allow you to run complicated map reduce functions in future.
To learn how to store time series in cassandra please take a look at:
http://www.datastax.com/dev/blog/advanced-time-series-with-cassandra
and http://www.youtube.com/watch?v=OzBJrQZjge0.
I have a computational algebra task I need to code up. The problem is broken into well-defined individuals tasks that naturally form a tree - the task is combinatorial in nature, so there's a main task which requires a small number of sub-calculations to get its results. Those sub-calculations have sub-sub-calculations and so on. Each calculation only depends on the calculations below it in the tree (assuming the root node is the top). No data sharing needs to happen between branches. At lower levels the number of subtasks may be extremely large.
I had previously coded this up in a functional fashion, calling the functions as needed and storing everything in RAM. This was a terrible approach, but I was more concerned about the theory then.
I'm planning to rewrite the code in C++ for a variety of reasons. I have a few requirements:
Checkpointing: The calculation takes a long time, so I need to be able to stop at any point and resume later.
Separate individual tasks as objects: This helps me keep a good handle of where I am in the computations, and offers a clean way to do checkpointing via serialization.
Multi-threading: The task is clearly embarrassingly parallel, so it'd be neat to exploit that. I'd probably want to use Boost threads for this.
I would like suggestions on how to actually implement such a system. Ways I've thought of doing it:
Implement tasks as a simple stack. When you hit a task that needs subcalculations done, it checks if it has all the subcalculations it requires. If not, it creates the subtasks and throws them onto the stack. If it does, then it calculates its result and pops itself from the stack.
Store the tasks as a tree and do something like a depth-first visitor pattern. This would create all the tasks at the start and then computation would just traverse the tree.
These don't seem quite right because of the problems of the lower levels requiring a vast number of subtasks. I could approach it in a iterator fashion at this level, I guess.
I feel like I'm over-thinking it and there's already a simple, well-established way to do something like this. Is there one?
Technical details in case they matter:
The task tree has 5 levels.
Branching factor of the tree is really small (say, between 2 and 5) for all levels except the lowest which is on the order of a few million.
Each individual task would only need to store a result tens of bytes large. I don't mind using the disk as much as possible, so long as it doesn't kill performance.
For debugging, I'd have to be able to recall/recalculate any individual task.
All the calculations are discrete mathematics: calculations with integers, polynomials, and groups. No floating point at all.
there's a main task which requires a small number of sub-calculations to get its results. Those sub-calculations have sub-sub-calculations and so on. Each calculation only depends on the calculations below it in the tree (assuming the root node is the top). No data sharing needs to happen between branches. At lower levels the number of subtasks may be extremely large... blah blah resuming, multi-threading, etc.
Correct me if I'm wrong, but it seems to me that you are exactly describing a map-reduce algorithm.
Just read what wikipedia says about map-reduce :
"Map" step: The master node takes the input, partitions it up into smaller sub-problems, and distributes those to worker nodes. A worker node may do this again in turn, leading to a multi-level tree structure. The worker node processes that smaller problem, and passes the answer back to its master node.
"Reduce" step: The master node then takes the answers to all the sub-problems and combines them in some way to get the output – the answer to the problem it was originally trying to solve.
Using an existing mapreduce framework could save you a huge amount of time.
I just google "map reduce C++" and I start to get results, notably one in boost http://www.craighenderson.co.uk/mapreduce/
These don't seem quite right because of the problems of the lower levels requiring a vast number of subtasks. I could approach it in a iterator fashion at this level, I guess.
You definitely do not want millions of CPU-bound threads. You want at most N CPU-bound threads, where N is the product of the number of CPUs and the number of cores per CPU on your machine. Exceed N by a little bit and you are slowing things down a bit. Exceed N by a lot and you are slowing things down a whole lot. The machine will spend almost all its time swapping threads in and out of context, spending very little time executing the threads themselves. Exceed N by a whole lot and you will most likely crash your machine (or hit some limit on threads). If you want to farm lots and lots (and lots and lots) of parallel tasks out at once, you either need to use multiple machines or use your graphics card.