In most cases, the Baum-Welch algorithm is used to train a Hidden Markov model.
In many papers however, it is argued that the BW algorithm will optimize until it got stuck in a local optimum.
Does there exist an exact algorithm that actually succeeds in finding the global optimum (except from enumerating nearly all possible models and evaluating them)?
Of course for most applications, BW will work fine. We are however interested in finding lower bounds of the amount of information loss when reducing the number of states. Therefore we always need to generate the best model possible.
We are thus looking for an efficient NP-hard algorithm (that only enumerates over a (potentially) exponential number of extreme points) and not over a discretized number of floating points for each probability in the model.
A quick search finds in http://www.cs.tau.ac.il/~rshamir/algmb/98/scribe/html/lec06/node6.html "In this case, the problem of finding the optimal set of parameters $\Theta^{\ast}$ is known to be NP-complete. The Baum-Welch algorithm [2], which is a special case of the EM technique (Expectation and Maximization), can be used for heuristically finding a solution to the problem. " Therefore I suggest that an EM variant that was guaranteed to find a global optimum in polynomial time would prove P=NP and is unknown and in fact probably does not exist.
This problem almost certainly is not convex, because there will in fact be multiple globally optimal solutions with the same scores - given any proposed solution, which typically gives a probability distribution for observations given the underlying state, you can, for instance, rename hidden state 0 as hidden state 1, and vice versa, adjusting the probability distributions so that the observed behaviour generated by the two different solutions is identical. Thus if there are N hidden states there are at least N! local optimums produced by permuting the hidden states amongst themselves.
On another application of EM, https://www.math.ias.edu/csdm/files/11-12/amoitra_disentangling_gaussians.pdf provides an algorithm for finding a globally optimum gaussian mixture model. It observes that the EM algorithm is often used for this problem, but points out that it is not guaranteed to find the global optimum, and does not reference as related work any version of EM which might (it also says the EM algorithm is slow).
If you are trying to do some sort of likelihood ratio test between e.g. a 4-state model and a 5-state model, it would clearly be embarrassing if, due to local optima, the 5-state model fitted had a lower likelihood than the 4-state model. One way to avoid this or to recover from it would be to start a 5-state EM from a starting point very close to that of the best 4-state models found. For instance, you could create a 5th state with probability epsilon and with an output distribution reflecting an average of the 4-state output distributions, keeping the 4-state distributions as the other 4 distributions in the new 5-state model, multiplying in a factor of (1-epsilon) somewhere so that everything still added up to one.
I think if you really want this, you can, given a local optimum, define a domain of convergence. If you can have some reasonably weak conditions, then you can quickly show that either the whole field is in the domain of convergence, or that there is a second local mininium.
E.g., suppose in an example I have two independent variables (x,y), and one dependent variable (z), and suppose that given a local minimim z_1, and a pair of start points which converge to z_1=(x_1,y_1), P_1 = (x_2, y_1) and p_2 = (x_1, y_3), then i might be able to prove that then all of the triangle z_1, p_1, p_2 is in the domain of convergence.
Of course, this is not an approach which works generally, but you can solve a sub class of problems efficiently.E.g., some problems have no no domain of convergence in a sense, e.g. its possible to ahve a problem where a point converges to a different solution than all the points in its neighbourhood, but lots of problems have some reasonable smoothness to their convergence to a solution, so then you can do ok.
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I have a function which takes as inputs n-dimensional (say n=10) vectors whose components are real numbers varying from 0 to a large positive number A say 50,000, ends included. For any such vector the function outputs an integer from 1 to say B=100. I have this function and want to find its global minima.
Broadly speaking there are algorithmic, iterative and heuristics based approaches to tackle such optimization problem. Which are the best techniques suggested to solve this problem? I am looking for suggestions to algorithms or active research papers that i can implement from scratch to solve such problems. I have already given up hope on existing optimization functions that ship with Matlab/python. I am hoping to read experience of others working with approximation/heuristic algorithms to optimize such ill-defined functions.
I ran fmincon, fminsearch, fminunc in Matlab but they fail to optimize the function. The function is ill-defined according to their definitions. Matlab says this for fmincon:
Initial point is a local minimum that satisfies the constraints.
Optimization completed because at the initial point, the objective function is non-decreasing
in feasible directions to within the selected value of the optimality tolerance, and
constraints are satisfied to within the selected value of the constraint tolerance.
Problem arises because this function has piecewise-constant behavior. If a vector V is assigned to a number say 65, changing its components very slightly may not have any change. Such ill-defined behavior is to be well-expected because of pigeon-hole principle. The domain of function is unlimited whereas range is just a bunch of numbers.
I also wish to clarify one issue that may arise. Suppose i do gradient descent on a starting point x0 and my next x that i get from GD-iteration has some components lie outside the domain [0,50000], then what happens? So actually the domain is circular. So a vector of size 3 like [30;5432;50432] becomes [30;5432;432]. This is automatically taken care of so that there is no worry about iterations finding a vector outside the domain.
Massively edited this question to make it easier to understand.
Given an environment with arbitrary dimensions and arbitrary positioning of an arbitrary number of obstacles, I have an agent exploring the environment with a limited range of sight (obstacles don't block sight). It can move in the four cardinal directions of NSEW, one cell at a time, and the graph is unweighted (each step has a cost of 1). Linked below is a map representing the agent's (yellow guy) current belief of the environment at the instant of planning. Time does not pass in the simulation while the agent is planning.
http://imagizer.imageshack.us/a/img913/9274/qRsazT.jpg
What exploration algorithm can I use to maximise the cost-efficiency of utility, given that revisiting cells are allowed? Each cell holds a utility value. Ideally, I would seek to maximise the sum of utility of all cells SEEN (not visited) divided by the path length, although if that is too complex for any suitable algorithm then the number of cells seen will suffice. There is a maximum path length but it is generally in the hundreds or higher. (The actual test environments used on my agent are at least 4x bigger, although theoretically there is no upper bound on the dimensions that can be set, and the maximum path length would thus increase accordingly)
I consider BFS and DFS to be intractable, A* to be non-optimal given a lack of suitable heuristics, and Dijkstra's inappropriate in generating a single unbroken path. Is there any algorithm you can think of? Also, I need help with loop detection, as I've never done that before since allowing revisitations is my first time.
One approach I have considered is to reduce the map into a spanning tree, except that instead of defining it as a tree that connects all cells, it is defined as a tree that can see all cells. My approach would result in the following:
http://imagizer.imageshack.us/a/img910/3050/HGu40d.jpg
In the resultant tree, the agent can go from a node to any adjacent nodes that are 0-1 turn away at intersections. This is as far as my thinking has gotten right now. A solution generated using this tree may not be optimal, but it should at least be near-optimal with much fewer cells being processed by the algorithm, so if that would make the algorithm more likely to be tractable, then I guess that is an acceptable trade-off. I'm still stuck with thinking how exactly to generate a path for this however.
Your problem is very similar to a canonical Reinforcement Learning (RL) problem, the Grid World. I would formalize it as a standard Markov Decision Process (MDP) and use any RL algorithm to solve it.
The formalization would be:
States s: your NxM discrete grid.
Actions a: UP, DOWN, LEFT, RIGHT.
Reward r: the value of the cells that the agent can see from the destination cell s', i.e. r(s,a,s') = sum(value(seen(s')).
Transition function: P(s' | s, a) = 1 if s' is not out of the boundaries or a black cell, 0 otherwise.
Since you are interested in the average reward, the discount factor is 1 and you have to normalize the cumulative reward by the number of steps. You also said that each step has cost one, so you could subtract 1 to the immediate reward rat each time step, but this would not add anything since you will already average by the number of steps.
Since the problem is discrete the policy could be a simple softmax (or Gibbs) distribution.
As solving algorithm you can use Q-learning, which guarantees the optimality of the solution provided a sufficient number of samples. However, if your grid is too big (and you said that there is no limit) I would suggest policy search algorithms, like policy gradient or relative entropy (although they guarantee convergence only to local optima). You can find something about Q-learning basically everywhere on the Internet. For a recent survey on policy search I suggest this.
The cool thing about these approaches is that they encode the exploration in the policy (e.g., the temperature in a softmax policy, the variance in a Gaussian distribution) and will try to maximize the cumulative long term reward as described by your MDP. So usually you initialize your policy with a high exploration (e.g., a complete random policy) and by trial and error the algorithm will make it deterministic and converge to the optimal one (however, sometimes also a stochastic policy is optimal).
The main difference between all the RL algorithms is how they perform the update of the policy at each iteration and manage the tradeoff exploration-exploitation (how much should I explore VS how much should I exploit the information I already have).
As suggested by Demplo, you could also use Genetic Algorithms (GA), but they are usually slower and require more tuning (elitism, crossover, mutation...).
I have also tried some policy search algorithms on your problem and they seems to work well, although I initialized the grid randomly and do not know the exact optimal solution. If you provide some additional details (a test grid, the max number of steps and if the initial position is fixed or random) I can test them more precisely.
I'm dealing with a war game. I have a list of my bases B(x,y) from which I can send attacks on the enemy (they have bases between my own bases). Each base B can attack at a range R (the same radius for all bases). How can I find my bases to be able to attack as many enemy bases as possible, but use a minimum number of my bases?
I've reduced the problem to finding the minimum number of bases (and their coordinates) required to cover the largest area possible. I wonder if there is a better way than looking at all the possible combinations and because the number of bases could reach thousands.
Example: If the attack radius is 10 and I have five bases in a square and its center: (0,0), (10,0), (10,10), (0,10), (5,5) then the answer is that only the first four would be needed because all the area covered by the one in the center is already covered by the others.
Note 1 The solution must be single-threaded.
Note 2 The solution doesn't have to be perfect if that means a big gain in speed. The number of bases reaches thousands and this needs to use as little time as possible. I would consider running time greater than 100 ms for 10,000 bases in Python on a modern computer unacceptable, so I was thinking maybe I could start by eliminating the obvious, like if there are multiple bases within R/10 distance of each other, simply eliminate all except for one (whichever).
If I understand you correctly, the enemy bases and your bases are given as well as the (constant) attack radius. I.e. if you select one of your bases, you know exactly which of the enemy bases get attacked due to the selection.
The first step would be to eliminate those enemy cities from the problem which can not be attacked by any of your bases. Then, selecting all of your bases guarantees attacking all attackable enemy bases, so there is solution that attacks as many enemy bases as possible.
Under all those solutions you are looking for the one that uses the minimum number of your bases. This problem is equivalent to the https://en.wikipedia.org/wiki/Set_cover_problem, which is unfortunately NP-hard. You can apply all known solution methods such as Integer Linear Programming or the already mentioned greedy algorithm / metaheuristics.
If your problem instance is large and runtime is the primary concern, greedy is probably the way to go. For example you could always add that particular base of yours to the selection which adds the highest number of enemy bases that can be attacked which were previously not under attack by your already selected bases.
Hum the solution depends on your needs. If you need real time answer, maybe a greedy algorithm could provide good solution.
Other solution could be using meta-heuristic with constraint time(http://en.wikipedia.org/wiki/Metaheuristic). I probably would use genetic algorithm to search a solution for this problem under a limited time.
If interested I can provide a toy example of implementation in Python.
EDIT :
When you have to provide solution quickly a greedy algorithm is often better. But in your case I doubt. Particularity of many greedy algorithm is that you need to start from scratch each time you try to compute a new result.
Speaking again of genetic algorithm, you could for example each time you have to take a decision restart the search process from its last result. In fact you could probably let him turning has a subprocess and each 100ms take the better solution computed during the last loop.
If not too greedy in computing resource, this solution would provide better results than greedy one on the long run as the solution will probably need to be adapted to the changes of the situation but many element will stay unchanged. Just be aware that initializing a meta-search with the solution of a greedy algorithm is anyway a good idea!
I was wondering if anyone knows which kind of algorithm could be use in my case. I already have run the optimizer on my multivariate function and found a solution to my problem, assuming that my function is regular enough. I slightly perturbate the problem and would like to find the optimum solution which is close to my last solution. Is there any very fast algorithm in this case or should I just fallback to a regular one.
We probably need a bit more information about your problem; but since you know you're near the right solution, and if derivatives are easy to calculate, Newton-Raphson is a sensible choice, and if not, Conjugate-Gradient may make sense.
If you already have an iterative optimizer (for example, based on Powell's direction set method, or CG), why don't you use your initial solution as a starting point for the next run of your optimizer?
EDIT: due to your comment: if calculating the Jacobian or the Hessian matrix gives you performance problems, try BFGS (http://en.wikipedia.org/wiki/BFGS_method), it avoids calculation of the Hessian completely; here
http://www.alglib.net/optimization/lbfgs.php you find a (free-for-non-commercial) implementation of BFGS. A good description of the details you will here.
And don't expect to get anything from finding your initial solution with a less sophisticated algorithm.
So this is all about unconstrained optimization. If you need information about constrained optimization, I suggest you google for "SQP".
there are a bunch of algorithms for finding the roots of equations. If you know approximately where the root is, there are algorithms that will get you arbitrarily close very quickly, in ln n time or better.
One is Newton's method
another is the Bisection Method
Note that these algorithms are for single variable functions, but can be expanded to multivariate functions.
Every minimization algorithm performs better (read: perform at all) if you have a good initial guess. The initial guess for the perturbed problem will be in your case the minimum point of the non perturbed problem.
Then, you have to specify your requirements: you want speed. What accuracy do you want ? Does space efficiency matters ? Most importantly: what information do you have: only the value of the function, or do you also have the derivatives (possibly second derivatives) ?
Some background on the problem would help too. Looking for a smooth function which has been discretized will be very different than looking for hundreds of unrelated parameters.
Global information (ie. is the function convex, is there a guaranteed global minimum or many local ones, etc) can be left aside for now. If you have trouble finding the minimum point of the perturbed problem, this is something you will have to investigate though.
Answering these questions will allow us to select a particular algorithm. There are many choices (and trade-offs) for multivariate optimization.
Also, which is quicker will very much depend on the problem (rather than on the algorithm), and should be determined by experimentation.
Thought I don't know much about using computers in this capacity, I remember an article that used neuroevolutionary techniques to find "best-fit" equations relatively efficiently, given a known function complexity (linear, Nth-polynomial, exponential, logarithmic, etc) and a set of point plots. As I recall it was one of the earliest uses of what we now know as computational neuroevolution; because the functional complexity (and thus the number of terms) of the equation is known and fixed, a static neural net can be used and seeded with your closest values, then "mutated" and tested for fitness, with heuristics to make new nets closer to existing nets with high fitness. Using multithreading, many nets can be created, tested and evaluated in parallel.
There was a post on here recently which posed the following question:
You have a two-dimensional plane of (X, Y) coordinates. A bunch of random points are chosen. You need to select the largest possible set of chosen points, such that no two points share an X coordinate and no two points share a Y coordinate.
This is all the information that was provided.
There were two possible solutions presented.
One suggested using a maximum flow algorithm, such that each selected point maps to a path linking (source → X → Y → sink). This runs in O(V3) time, where V is the number of vertices selected.
Another (mine) suggested using the Hungarian algorithm. Create an n×n matrix of 1s, then set every chosen (x, y) coordinate to 0. The Hungarian algorithm will give you the lowest cost for this matrix, and the answer is the number of coordinates selected which equal 0. This runs in O(n3) time, where n is the greater of the number of rows or the number of columns.
My reasoning is that, for the vast majority of cases, the Hungarian algorithm is going to be faster; V is equal to n in the case where there's one chosen point for each row or column, and substantially greater for any case where there's more than that: given a 50×50 matrix with half the coordinates chosen, V is 1,250 and n is 50.
The counterargument is that there are some cases, like a 109×109 matrix with only two points selected, where V is 2 and n is 1,000,000,000. For this case, it takes the Hungarian algorithm a ridiculously long time to run, while the maximum flow algorithm is blinding fast.
Here is the question: Given that the problem doesn't provide any information regarding the size of the matrix or the probability that a given point is chosen (so you can't know for sure) how do you decide which algorithm, in general, is a better choice for the problem?
You can't, it's an imponderable.
You can only define which is better "in general" by defining what inputs you will see "in general". So for example you could whip up a probability model of the inputs, so that the expected value of V is a function of n, and choose the one with the best expected runtime under that model. But there may be arbitrary choices made in the construction of your model, so that different models give different answers. One model might choose co-ordinates at random, another model might look at the actual use-case for some program you're thinking of writing, and look at the distribution of inputs it will encounter.
You can alternatively talk about which has the best worst case (across all possible inputs with given constraints), which has the virtue of being easy to define, and the flaw that it's not guaranteed to tell you anything about the performance of your actual program. So for instance HeapSort is faster than QuickSort in the worst case, but slower in the average case. Which is faster? Depends whether you care about average case or worst case. If you don't care which case, you're not allowed to care which "is faster".
This is analogous to trying to answer the question "what is the probability that the next person you see will have an above (mean) average number of legs?".
We might implicitly assume that the next person you meet will be selected at random with uniform distribution from the human population (and hence the answer is "slightly less than one", since the mean is less than the mode average, and the vast majority of people are at the mode).
Or we might assume that your next meeting with another person is randomly selected with uniform distribution from the set of all meetings between two people, in which case the answer is still "slightly less than one", but I reckon not the exact same value as the first - one-and-zero-legged people quite possibly congregate with "their own kind" very slightly more than their frequency within the population would suggest. Or possibly they congregate less, I really don't know, I just don't see why it should be exactly the same once you take into account Veterans' Associations and so on.
Or we might use knowledge about you - if you live with a one-legged person then the answer might be "very slightly above 0".
Which of the three answers is "correct" depends precisely on the context which you are forbidding us from talking about. So we can't talk about which is correct.
Given that you don't know what each pill does, do you take the red pill or the blue pill?
If there really is not enough information to decide, there is not enough information to decide. Any guess is as good as any other.
Maybe, in some cases, it is possible to divine extra information to base the decision on. I haven't studied your example in detail, but it seems like the Hungarian algorithm might have higher memory requirements. This might be a reason to go with the maximum flow algorithm.
You don't. I think you illustrated that clearly enough. I think the proper practical solution is to spawn off both implementations in different threads, and then take the response that comes back first. If you're more clever, you can heuristically route requests to implementations.
Many algorithms require huge amounts of memory beyond the physical maximum of a machine, and in these cases, the algorithmically more ineffecient in time but efficient in space algorithm is chosen.
Given that we have distributed parallel computing, I say you just let both horses run and let the results speak for themselves.
This is a valid question, but there's no "right" answer — they are incomparable, so there's no notion of "better".
If your interest is practical, then you need to analyze the kinds of inputs that are likely to arise in practice, as well as the practical running times (constants included) of the two algorithms.
If your interest is theoretical, where worst-case analysis is often the norm, then, in terms of the input size, the O(V3) algorithm is better: you know that V ≤ n2, but you cannot polynomially bound n in terms of V, as you showed yourself. Of course the theoretical best algorithm is a hybrid algorithm that runs both and stops when whichever one of them finishes first, thus its running time would be O(min(V3,n3)).
Theoretically, they are both the same, because you actually compare how the number of operations grows when the size of the problem is increased to infinity.
The way your problem is defined, it has 2 sizes - n and number of points, so this question has no answer.