Parallel many dimensional optimization

Question

I am building a script that generates input data [parameters] for another program to calculate. I would like to optimize the resulting data. Previously I have been using the numpy powell optimization. The psuedo code looks something like this.

def value(param):
     run_program(param)
     #Parse output
     return value

scipy.optimize.fmin_powell(value,param) 

This works great; however, it is incredibly slow as each iteration of the program can take days to run. What I would like to do is coarse grain parallelize this. So instead of running a single iteration at a time it would run (number of parameters)*2 at a time. For example:

Initial guess: param=[1,2,3,4,5]

#Modify guess by plus minus another matrix that is changeable at each iteration
jump=[1,1,1,1,1]
#Modify each variable plus/minus jump.
for num,a in enumerate(param):
    new_param1=param[:]
    new_param1[num]=new_param1[num]+jump[num]
    run_program(new_param1)
    new_param2=param[:]
    new_param2[num]=new_param2[num]-jump[num]
    run_program(new_param2)

#Wait until all programs are complete -> Parse Output
Output=[[value,param],...]
#Create new guess
#Repeat

Number of variable can range from 3-12 so something such as this could potentially speed up the code from taking a year down to a week. All variables are dependent on each other and I am only looking for local minima from the initial guess. I have started an implementation using hessian matrices; however, that is quite involved. Is there anything out there that either does this, is there a simpler way, or any suggestions to get started?

So the primary question is the following: Is there an algorithm that takes a starting guess, generates multiple guesses, then uses those multiple guesses to create a new guess, and repeats until a threshold is found. Only analytic derivatives are available. What is a good way of going about this, is there something built already that does this, is there other options?

Thank you for your time.

As a small update I do have this working by calculating simple parabolas through the three points of each dimension and then using the minima as the next guess. This seems to work decently, but is not optimal. I am still looking for additional options.

Current best implementation is parallelizing the inner loop of powell's method.

Thank you everyone for your comments. Unfortunately it looks like there is simply not a concise answer to this particular problem. If I get around to implementing something that does this I will paste it here; however, as the project is not particularly important or the need of results pressing I will likely be content letting it take up a node for awhile.

Answer 1

I had the same problem while I was in the university, we had a fortran algorithm to calculate the efficiency of an engine based on a group of variables. At the time we use modeFRONTIER and if I recall correctly, none of the algorithms were able to generate multiple guesses.

The normal approach would be to have a DOE and there where some algorithms to generate the DOE to best fit your problem. After that we would run the single DOE entries parallely and an algorithm would "watch" the development of the optimizations showing the current best design.

Side note: If you don't have a cluster and needs more computing power HTCondor may help you.

Answer 2

Are derivatives of your goal function available? If yes, you can use gradient descent (old, slow but reliable) or conjugate gradient. If not, you can approximate the derivatives using finite differences and still use these methods. I think in general, if using finite difference approximations to the derivatives, you are much better off using conjugate gradients rather than Newton's method.

A more modern method is SPSA which is a stochastic method and doesn't require derivatives. SPSA requires much fewer evaluations of the goal function for the same rate of convergence than the finite difference approximation to conjugate gradients, for somewhat well-behaved problems.

Answer 3

There are two ways of estimating gradients, one easily parallelizable, one not:

• around a single point, e.g. (f( x + h direction_i ) - f(x)) / h; this is easily parallelizable up to Ndim
• "walking" gradient: walk from x₀ in direction e₀ to x₁, then from x₁ in direction e₁ to x₂ ...; this is sequential.

Minimizers that use gradients are highly developed, powerful, converge quadratically (on smooth enough functions). The user-supplied gradient function can of course be a parallel-gradient-estimator.
A few minimizers use "walking" gradients, among them Powell's method, see Numerical Recipes p. 509.
So I'm confused: how do you parallelize its inner loop ?

I'd suggest scipy fmin_tnc with a parallel-gradient-estimator, maybe using central, not one-sided, differences.
(Fwiw, this compares some of the scipy no-derivative optimizers on two 10-d functions; ymmv.)

Answer 4

I think what you want to do is use the threading capabilities built-in python. Provided you your working function has more or less the same run-time whatever the params, it would be efficient.

Create 8 threads in a pool, run 8 instances of your function, get 8 result, run your optimisation algo to change the params with 8 results, repeat.... profit ?

Answer 5

If I haven't gotten wrong what you are asking, you are trying to minimize your function one parameter at the time.

you can obtain it by creating a set of function of a single argument, where for each function you freeze all the arguments except one.

Then you go on a loop optimizing each variable and updating the partial solution.

This method can speed up by a great deal function of many parameters where the energy landscape is not too complex (the dependency between the parameters is not too strong).

given a function

energy(*args) -> value

you create the guess and the function:

guess = [1,1,1,1]
funcs = [ lambda x,i=i: energy( guess[:i]+[x]+guess[i+1:] ) for i in range(len(guess)) ]

than you put them in a while cycle for the optimization

while convergence_condition:
    for func in funcs:
        optimize fot func
        update the guess
    check for convergence

This is a very simple yet effective method of simplify your minimization task. I can't really recall how this method is called, but A close look to the wikipedia entry on minimization should do the trick.

Answer 6

You could do parallel at two parts: 1) parallel the calculation of single iteration or 2) parallel start N initial guessing.

On 2) you need a job controller to control the N initial guess discovery threads.

Please add an extra output on your program: "lower bound" that indicates the output values of current input parameter's decents wont lower than this lower bound.

The initial N guessing thread can compete with each other; if any one thread's lower bound is higher than existing thread's current value, then this thread can be dropped by your job controller.

Answer 7

Parallelizing local optimizers is intrinsically limited: they start from a single initial point and try to work downhill, so later points depend on the values of previous evaluations. Nevertheless there are some avenues where a modest amount of parallelization can be added.

• As another answer points out, if you need to evaluate your derivative using a finite-difference method, preferably with an adaptive step size, this may require many function evaluations, but the derivative with respect to each variable may be independent; you could maybe get a speedup by a factor of twice the number of dimensions of your problem. If you've got more processors than you know what to do with, you can use higher-order-accurate gradient formulae that require more (parallel) evaluations.
• Some algorithms, at certain stages, use finite differences to estimate the Hessian matrix; this requires about half the square of the number of dimensions of your matrix, and all can be done in parallel.

Some algorithms may also be able to use more parallelism at a modest algorithmic cost. For example, quasi-Newton methods try to build an approximation of the Hessian matrix, often updating this by evaluating a gradient. They then take a step towards the minimum and evaluate a new gradient to update the Hessian. If you've got enough processors so that evaluating a Hessian is as fast as evaluating the function once, you could probably improve these by evaluating the Hessian at every step.

As far as implementations go, I'm afraid you're somewhat out of luck. There are a number of clever and/or well-tested implementations out there, but they're all, as far as I know, single-threaded. Your best bet is to use an algorithm that requires a gradient and compute your own in parallel. It's not that hard to write an adaptive one that runs in parallel and chooses sensible step sizes for its numerical derivatives.