I understand that each particle is a solution to a specific function, and each particle and the swarm is constantly searching for the best solution. If the global best is found after the first iteration, and no new particles are being added to the mix, shouldn't the loop just quit and the first global best found be the most fitting solution? If this is the case what makes PSO better than just iterating through a list.
Your terminology is a bit off. Simple PSO is a search for a vector x that minimizes some scalar objective function E(x). It does this by creating many candidate vectors. Call them x_i. These are the "particles". They are initialized randomly in both position and rate of change, also called velocity, which is consistent with the idea of a moving particle, even though that particle may have many more than 3 dimensions.
Simple rules describe how the position and velocity change over time. The rules are chosen so that each particle x_i tends randomly to move in directions that reduce E(x_i).
The rules usually involve tracking the "single best x_i value seen so far" and are tuned so that all particles tend to head generally toward that best value with random variations. So the particles swarm like buzzing bees, heading as a group toward a common goal, but with many deviations by individual bees that, over time, cause the common goal to change.
It's unfortunate that some of the literature calls this goal or best particle value seen so far "the global minimum." In optimization, global minimum has a different meaning. A global minimum (there can be more than one when there are "ties" for best) is a value of x that - out of the entire domain of possible x values - produces the unique minimum possible value of E(x).
In no way is PSO guaranteed to find a global minimum. In fact, your question is a bit nonsensical in that one generally never knows when a global minimum has been found. How would you? In most problems you don't even know the gradient of E (which gives the direction taking E to smaller values, i.e. downhill). This is why you are using PSO in the first place. If you know the gradient, you can almost certainly use numerical techniques that will find an answer more quickly than PSO. Without a gradient, you can't even be sure you've found a local minimum, let alone a global one.
Rather, the best you can usually do is "guess" when a local minimum has been found. You do this by letting the system run while watching how often and by how much the "best particle seen so far" is being updated. When the changes become infrequent and/or small, you declare victory.
Another way of putting this is that PSO is used on problems where reducing E(x) is always good and "you'll take anything you can get" regardless of whether you have any confidence that what you got is the best possible. E.g. you're Walmart and any way of locating your stores that saves/makes more dollars is interesting.
With all this as background, let's recap your specific questions:
If the global best is found after the first iteration, and no new particles are being added to the mix, shouldn't the loop just quit and the first global best found be the most fitting solution?
There's no answer because there's no way to determine a global best has been found. The swarm of buzzing particles might find a new best in the next iteration or ten trillion iterations from now. You seldom know.
If this is the case what makes PSO better than just iterating through a list?
I don't exactly grok what you mean by this. The PSO is emulating the way swarms of biological entities like bugs and herd animals behave. In this manner it resembles genetic algorithms, simulated annealing, neural networks, and other families of solution finders that use the following logic: Nature, both physical and biological, has known-good optimization processes. Let's take advantage of them and do our best to emulate them in software. We are using nature to do better than any simple iteration we might devise ourselves.
Given a function, a particle swarm attempts to find the solution (a vector) that will minimize (or sometimes maximize, depending on the problem) the value to that function.
If you happen to know the minimum of the solution (suppose for argument sake, it is 0) AND
if you are lucky enough to generate the solution that gives you 0 on the first step, then you can exit the loop and stop the algorithm.
That said; the probability of you randomly generating that solution on initialization is infinitely small.
In most practical terms, when you would want to use a PSO to solve, it is most likely that you will not know the minimum value, so you wont be able to use that as a stopping condition.
The particle swarm optimization, the optimization process is not in the way the random initial step occurs, but rather the modification that occurs by adapting the initial solution with the velocity determined by social and cognitive component.
The social component consists of the current evaluated global best solution of the swarm
The cognitive component consists of a the best location seen by the current solution.
This adjustment will move the particle along a line between the global best and the current best - in hope there is a better solution between them.
I hope that answers the question in some way
Just to add some piece in answering, your problem seems to be linked to the common issue of "when should I stop my PSO?" A question everyone is faced when launching a swarm since (as clearly explained above) you never know if you reached the global best solution (except in very specific objective functions).
Usual tricks already present in most PSO implementation:
1- just limit a number of iterations since there is always a limit in processing time (and you could implement different ways to convert the iterations number into a time limit by self assessment of time spent to evaluate the objective).
2- stop the algorithm when the progress in optimization starts to be insignificant.
Related
When is the right time to use the Elite\Elitist mode in a Genetic Algorithm? I have no idea when to use it. What kind problems can be solved using this?
All I know is an elitist model is where you choose the elite (the solution with highest fitness function) and they have a reserve slot for the next generation, and they are the one up for crossover.
You pretty much always use some form of elitism. What varies is the percentage (p) of best performers that you allow to survive to the next generation. So no elitism is basically saying p=0.
The higher p, the more your algorithm will have a tendency to find local peaks of fitness. i.e. once it finds a chromosome with a good fitness, it'll tend to focus more on optimizing it than trying to find new completely different solutions. On the contrary, if it's smaller, your GA will look for possible solutions all over the place and won't zero in as fast once it finds something close to the optimum solution.
So setting p correctly is going to have a direct impact on your algorithm's performance. But it depends on what you're after and your problem space. Play around with it a bit to adjust properly. I typically use 20% for the problems I work with, to give enough room for innovation. It works ok for me.
I'm looking for the equivalent of welford's algorithm for the online computation semi-variance (downside partial variance). Does anyone know of a good reference? Does such an algorithm even exist?
Edit: the case where the semi-variance is taken relative to a fixed target is trivial. the problem is calculating the semi-variance in relation to the mean
I believe the answer is one does not exist and I'm going to try to outline a proof of why this is so.
Consider a 'uesful' online algorithm to be defined by two criteria:
It must have fixed memory requirements during processing.
Each update should take a fixed amount of time.
This is stricter than the literal definition of an sequential/incremental/online algorithm which really just requires that data can be passed in one piece at a time. However, consider that if either 1) or 2) were not true then after processing a large enough amounts of elements, the memory required or time required to run the algorithm would eventually become infeasible. Usually, one of the reasons why online algorithms are used is that they can be used continuously without fear of the performance slowly getting worse. Also, note that there are online algorithms for calculating the mean and variance that satisfy both 1 & 2 and I think that's what we are aiming to achieve.
Now to the problem posed. During processing, the mean will change with every bit of new data. That in turn means the set of observations that fall below the mean will change. When this happens, we need to adjust our running semi-variance according to the set "delta", defined as the elements that are not in the union between the set of elements below the old mean and the set of elements below the new mean. We will have to calculate this delta in the process of adjusting the old-semivariance to the new-semivariance in the presence of new data.
Now let's consider the complexity of calculating this set delta. We will need to find all elements that fall between the old mean and the new mean. We will always keep track of the old mean, while the new mean can be calculated incrementally in fixed time so they pose no problem. However to calculate the delta itself, there is no way to do it other than requiring us to keep track of all the previous elements in our set. This immediately breaks the memory condition of an online algorithm. Secondly, even if we keep the previous elements in our set sorted, the best speed we can achieve to find those that are between the old mean and new mean is O(log(number of elements)), which is worse than fixed. So eventually, with enough elements, the online algorithm will not only require more memory than we have, but it will also require more time.
http://www3.sympatico.ca/jean-v.cote/computation_of_semi-variance.pdf
P.S.:This is not an incremental computation. I have another idea. I will keep you posted.
I'm looking for algorithms to find a "best" set of parameter values. The function in question has a lot of local minima and changes very quickly. To make matters even worse, testing a set of parameters is very slow - on the order of 1 minute - and I can't compute the gradient directly.
Are there any well-known algorithms for this kind of optimization?
I've had moderate success with just trying random values. I'm wondering if I can improve the performance by making the random parameter chooser have a lower chance of picking parameters close to ones that had produced bad results in the past. Is there a name for this approach so that I can search for specific advice?
More info:
Parameters are continuous
There are on the order of 5-10 parameters. Certainly not more than 10.
How many parameters are there -- eg, how many dimensions in the search space? Are they continuous or discrete - eg, real numbers, or integers, or just a few possible values?
Approaches that I've seen used for these kind of problems have a similar overall structure - take a large number of sample points, and adjust them all towards regions that have "good" answers somehow. Since you have a lot of points, their relative differences serve as a makeshift gradient.
Simulated
Annealing: The classic approach. Take a bunch of points, probabalistically move some to a neighbouring point chosen at at random depending on how much better it is.
Particle
Swarm Optimization: Take a "swarm" of particles with velocities in the search space, probabalistically randomly move a particle; if it's an improvement, let the whole swarm know.
Genetic Algorithms: This is a little different. Rather than using the neighbours information like above, you take the best results each time and "cross-breed" them hoping to get the best characteristics of each.
The wikipedia links have pseudocode for the first two; GA methods have so much variety that it's hard to list just one algorithm, but you can follow links from there. Note that there are implementations for all of the above out there that you can use or take as a starting point.
Note that all of these -- and really any approach to this large-dimensional search algorithm - are heuristics, which mean they have parameters which have to be tuned to your particular problem. Which can be tedious.
By the way, the fact that the function evaluation is so expensive can be made to work for you a bit; since all the above methods involve lots of independant function evaluations, that piece of the algorithm can be trivially parallelized with OpenMP or something similar to make use of as many cores as you have on your machine.
Your situation seems to be similar to that of the poster of Software to Tune/Calibrate Properties for Heuristic Algorithms, and I would give you the same advice I gave there: consider a Metropolis-Hastings like approach with multiple walkers and a simulated annealing of the step sizes.
The difficulty in using a Monte Carlo methods in your case is the expensive evaluation of each candidate. How expensive, compared to the time you have at hand? If you need a good answer in a few minutes this isn't going to be fast enough. If you can leave it running over night, it'll work reasonably well.
Given a complicated search space, I'd recommend a random initial distributed. You final answer may simply be the best individual result recorded during the whole run, or the mean position of the walker with the best result.
Don't be put off that I was discussing maximizing there and you want to minimize: the figure of merit can be negated or inverted.
I've tried Simulated Annealing and Particle Swarm Optimization. (As a reminder, I couldn't use gradient descent because the gradient cannot be computed).
I've also tried an algorithm that does the following:
Pick a random point and a random direction
Evaluate the function
Keep moving along the random direction for as long as the result keeps improving, speeding up on every successful iteration.
When the result stops improving, step back and instead attempt to move into an orthogonal direction by the same distance.
This "orthogonal direction" was generated by creating a random orthogonal matrix (adapted this code) with the necessary number of dimensions.
If moving in the orthogonal direction improved the result, the algorithm just continued with that direction. If none of the directions improved the result, the jump distance was halved and a new set of orthogonal directions would be attempted. Eventually the algorithm concluded it must be in a local minimum, remembered it and restarted the whole lot at a new random point.
This approach performed considerably better than Simulated Annealing and Particle Swarm: it required fewer evaluations of the (very slow) function to achieve a result of the same quality.
Of course my implementations of S.A. and P.S.O. could well be flawed - these are tricky algorithms with a lot of room for tweaking parameters. But I just thought I'd mention what ended up working best for me.
I can't really help you with finding an algorithm for your specific problem.
However in regards to the random choosing of parameters I think what you are looking for are genetic algorithms. Genetic algorithms are generally based on choosing some random input, selecting those, which are the best fit (so far) for the problem, and randomly mutating/combining them to generate a next generation for which again the best are selected.
If the function is more or less continous (that is small mutations of good inputs generally won't generate bad inputs (small being a somewhat generic)), this would work reasonably well for your problem.
There is no generalized way to answer your question. There are lots of books/papers on the subject matter, but you'll have to choose your path according to your needs, which are not clearly spoken here.
Some things to know, however - 1min/test is way too much for any algorithm to handle. I guess that in your case, you must really do one of the following:
get 100 computers to cut your parameter testing time to some reasonable time
really try to work out your parameters by hand and mind. There must be some redundancy and at least some sanity check so you can test your case in <1min
for possible result sets, try to figure out some 'operations' that modify it slightly instead of just randomizing it. For example, in TSP some basic operator is lambda, that swaps two nodes and thus creates new route. Your can be shifting some number up/down for some value.
then, find yourself some nice algorithm, your starting point can be somewhere here. The book is invaluable resource for anyone who starts with problem-solving.
Let's imagine I have an unknown function that I want to approximate via Genetic Algorithms. For this case, I'll assume it is y = 2x.
I'd have a DNA composed of 5 elements, one y for each x, from x = 0 to x = 4, in which, after a lot of trials and computation and I'd arrive near something of the form:
best_adn = [ 0, 2, 4, 6, 8 ]
Keep in mind I don't know beforehand if it is a linear function, a polynomial or something way more ugly, Also, my goal is not to infer from the best_adn what is the type of function, I just want those points, so I can use them later.
This was just an example problem. In my case, instead of having only 5 points in the DNA, I have something like 50 or 100. What is the best approach with GA to find the best set of points?
Generating a population of 100,
discard the worse 20%
Recombine the remaining 80%? How?
Cutting them at a random point and
then putting together the first
part of ADN of the father with the
second part of ADN of the mother?
Mutation, how should I define in
this kind of problem mutation?
Is it worth using Elitism?
Any other simple idea worth using
around?
Thanks
Usually you only find these out by experimentation... perhaps writing a GA to tune your GA.
But that aside, I don't understand what you're asking. If you don't know what the function is, and you also don't know the points to being with, how do you determine fitness?
From my current understanding of the problem, this is better fitted by a neural network.
edit:
2.Recombine the remaining 80%? How? Cutting them at a random point and then putting together the first part of ADN of the father with the second part of ADN of the mother?
This is called crossover. If you want to be saucey, do something like pick a random starting point and swapping a random length. For instance, you have 10 elements in an object. randomly choose a spot X between 1 and 10 and swap x..10-rand%10+1.. you get the picture... spice it up a little.
3.Mutation, how should I define in this kind of problem mutation?
usually that depends more on what is defined as a legal solution than anything else. you can do mutation the same way you do crossover, except you fill it with random data (that is legal) rather than swapping with another specimen... and you do it at a MUCH lower rate.
4.Is it worth using Elitism?
experiment and find out.
Gaussian adaptation usually outperforms standard genetic algorithms. If you don't want to write your own package from scratch, the Mathematica Global Optimization package is EXCELLENT -- I used it to fit a really nasty nonlinear function where standard fitters failed miserably.
Edit:
Wikipedia Article
If you hunt down prints of the listed papers on the article, you can find whitepapers and implementations. In general though, you should have some idea what the solution space for your maximizing the fitness function look like. If the number of variables is small, or the number of local maxima is small or they are connected/slope down to a global maxima, simple least squares works fine. If the area around each local maxima is small (IE you have to get a damned good solution to hit the best one, otherwise you hit a bad one), then fancier algorithms are needed.
Choosing variables for a genetic algorithm depends on what the solution space will look like.
The domain of this question is scheduling operations on constrained hardware. The resolution of the result is the number of clock cycles the schedule fits within. The search space grows very rapidly where early decisions constrain future decisions and the total number of possible schedules grows rapidly and exponentially. A lot of the possible schedules are equivalent because just swapping the order of two instructions usually result in the same timing constraint.
Basically the question is what is a good strategy for exploring the vast search space without spending too much time. I expect to search only a small fraction but would like to explore different parts of the search space while doing so.
The current greedy algorithm tend to make stupid decisions early on sometimes and the attempt at branch and bound was beyond slow.
Edit:
Want to point out that the result is very binary with perhaps the greedy algorithm ending up using 8 cycles while there exists a solution using only 7 cycles using branch and bound.
Second point is that there are significant restrictions in data routing between instructions and dependencies between instructions that limits the amount of commonality between solutions. Look at it as a knapsack problem with a lot of ordering constraints as well as some solutions completely failing because of routing congestion.
Clarification:
In each cycle there is a limit to how many operations of each type and some operations have two possible types. There are a set of routing constraints which can be varied to be either fairly tight or pretty forgiving and the limit depends on routing congestion.
Integer linear optimization for NP-hard problems
Depending on your side constraints, you may be able to use the critical path method or
(as suggested in a previous answer) dynamic programming. But many scheduling problems are NP-hard just like the classical traveling sales man --- a precise solution has a worst case of exponential search time, just as you describe in your problem.
It's important to know that while NP-hard problems still have a very bad worst case solution time there is an approach that very often produces exact answers with very short computations (the average case is acceptable and you often don't see the worst case).
This approach is to convert your problem to a linear optimization problem with integer variables. There are free-software packages (such as lp-solve) that can solve such problems efficiently.
The advantage of this approach is that it may give you exact answers to NP-hard problems in acceptable time. I used this approach in a few projects.
As your problem statement does not include more details about the side constraints, I cannot go into more detail how to apply the method.
Edit/addition: Sample implementation
Here are some details about how to implement this method in your case (of course, I make some assumptions that may not apply to your actual problem --- I only know the details form your question):
Let's assume that you have 50 instructions cmd(i) (i=1..50) to be scheduled in 10 or less cycles cycle(t) (t=1..10). We introduce 500 binary variables v(i,t) (i=1..50; t=1..10) which indicate whether instruction cmd(i) is executed at cycle(t) or not. This basic setup gives the following linear constraints:
v_it integer variables
0<=v_it; v_it<=1; # 1000 constraints: i=1..50; t=1..10
sum(v_it: t=1..10)==1 # 50 constraints: i=1..50
Now, we have to specify your side conditions. Let's assume that operations cmd(1)...cmd(5) are multiplication operations and that you have exactly two multipliers --- in any cycle, you may perform at most two of these operations in parallel:
sum(v_it: i=1..5)<=2 # 10 constraints: t=1..10
For each of your resources, you need to add the corresponding constraints.
Also, let's assume that operation cmd(7) depends on operation cmd(2) and needs to be executed after it. To make the equation a little bit more interesting, lets also require a two cycle gap between them:
sum(t*v(2,t): t=1..10) + 3 <= sum(t*v(7,t): t=1..10) # one constraint
Note: sum(t*v(2,t): t=1..10) is the cycle t where v(2,t) is equal to one.
Finally, we want to minimize the number of cycles. This is somewhat tricky because you get quite big numbers in the way that I propose: We give assign each v(i,t) a price that grows exponentially with time: pushing off operations into the future is much more expensive than performing them early:
sum(6^t * v(i,t): i=1..50; t=1..10) --> minimum. # one target function
I choose 6 to be bigger than 5 to ensure that adding one cycle to the system makes it more expensive than squeezing everything into less cycles. A side-effect is that the program will go out of it's way to schedule operations as early as possible. You may avoid this by performing a two-step optimization: First, use this target function to find the minimal number of necessary cycles. Then, ask the same problem again with a different target function --- limiting the number of available cycles at the outset and imposing a more moderate price penalty for later operations. You have to play with this, I hope you got the idea.
Hopefully, you can express all your requirements as such linear constraints in your binary variables. Of course, there may be many opportunities to exploit your insight into your specific problem to do with less constraints or less variables.
Then, hand your problem off to lp-solve or cplex and let them find the best solution!
At first blush, it sounds like this problem might fit into a dynamic programming solution. Several operations may take the same amount of time so you might end up with overlapping subproblems.
If you can map your problem to the "travelling salesman" (like: Find the optimal sequence to run all operations in minimum time), then you have an NP-complete problem.
A very quick way to solve that is the ant algorithm (or ant colony optimization).
The idea is that you send an ant down every path. The ant spreads a smelly substance on the path which evaporates over time. Short parts mean that the path will stink more when the next ant comes along. Ants prefer smelly over clean paths. Run thousands of ants through the network. The most smelly path is the optimal one (or at least very close).
Try simulated annealing, cfr. http://en.wikipedia.org/wiki/Simulated_annealing .