Advice on Minimization Methods

Contents

• Introduction
• Single-shot techniques
• Scatter-shot techniques
• Summary and best-buy strategies
• Acknowledgements

The Sherpa Optimization Methods page describes each optimization method in detail.

Introduction

The minimization of mathematical functions is a difficult operation. A general function f(x) of vector argument x may have many isolated local minima, non-isolated minimum hypersurfaces, or even more complicated topologies. No finite minimization routine can guarantee to locate the unique, global, minimum of f(x) without being fed intimate knowledge about the function by the user.

This does not mean that minimization is a hopeless task. For many problems there are techniques which will locate a local minimum which may be "close enough" to the global minimum, and there are techniques which will find the global minimum a large fraction of the time (in a probabilistic sense). However, the reader should be aware of my philosophy is that there is no "best" algorithm for finding the minimum of a general function. Instead, Sherpa provides tools which will allow the user to look at the overall behavior of the function and find plausible local minima, will often contain the physically-meaningful minimum in the types of problem with which Sherpa deals.

In general, the best assurance that the correct minimum has been found in a particular calculation is careful examination of the nature of the solution (e.g., by plotting a fitted function over data), and some confidence that the full region that the minimum may lie in has been well searched by the algorithm used. This document seeks to give the reader some information about what the different choices of algorithm will mean in terms of run-time and confidence of locating a good minimum.

Some points to take away from the discussions in the rest of this document.

1. Never accept the result of a minimization using a single optimization run; always test the minimum using a different method.

2. Check that the result of the minimization does not have parameter values at the edges of the parameter space. If this happens, then the fit must be disregarded since the minimum lies outside the space that has been searched, or the minimization missed the minimum.

3. Get a feel for the range of values of the target function (in Sherpa, this is the fit statistic), and the stability of the solution, by starting the minimization from several different parameter values.

4. Always check that the minimum "looks right" using a plotting tool.

Sherpa contains two types of routine for minimizing a fit statistic. I will call them the "single-shot" routines, which start from a guessed set of parameters, and then try to improve the parameters in a continuous fashion, and the "scatter-shot" routines, which try to look at parameters over the entire permitted hypervolume to see if there are better minima than near the starting guessed set of parameters.

Single-shot techniques

As the reader might expect, the single-shot routines are relatively quick, but depend critically on the guessed initial parameter values x₀ being near (in some sense) to the minimum x_min. All the single-shot routines investigate the local behaviour of the function near x₀, and then make a guess at the best direction and distance to move to find a better minimum. After testing at the new point, they accept that point as the next guess, x₁, if the fit statistic is smaller than at the first point, and modify the search procedure if it isn't smaller. The routines continue to run until either

1. all search directions result in an increased value of the fit statistic;

2. an excessive number of steps have been taken; or

3. something strange happens to the fit statistic (e.g., it turns out to be discontinuous in some horrible way).

This description indicates that for the single-shot routines, there is a considerable emphasis on the initial search position, x₀, being reasonable. It may also be apparent that the values of these parameters should be moderate; neither too small (10^-12, say), nor too large (10¹², say). This is because the initial choice of step size in moving from x₀ towards the next improved set of parameters, x₁, is based on the change in the fit statistic, f(x) as components of x are varied by amounts . If f varies little as x is varied by this amount, then the calculation of the distance to move to reach the next root may be inaccurate. On the other hand, if f has a lot of structure (several maxima and minima) as x is varied by the initial step size, then these single-shot minimizers may mistakenly jump entirely over the "interesting" region of parameter space.

These considerations suggest that the user should arrange that the search vector is scaled so that the range of parameter space to be searched is neither too large nor too small. To take a concrete example, it would not be a good idea to have x₇ parameterize N_H in a spectral fit, with an initial guess of 10²⁰, and a search range 10¹⁶ to 10²⁴ (units assumed to be cm^-2). The minimizers will look for variations in the fit statistic as N_H is varied by 1 cm^-2, and finding none (to the rounding accuracy likely for the code), will conclude that x₇ is close to being a null parameter and can be ignored in the fitting. It would be much better to have x₇ = log₁₀ (N_H), with a search range of 16 - 24. Significant variations in the fit statistic will occur as x₇ is varied by ±1, and the code has a reasonable chance of finding a useful solution.

Bearing this in mind, the single-shot minimizers in Sherpa are:

• Powell This method is described in some detail by Press et al. (1986). It is basically a censored maximum-gradients technique which, starting from a first guess, moves towards a minimum by finding a good direction in which to move, and calculating a sensible distance to go. Its principal drawback is that to calculate the distance to move it has to make some assumptions about how large a step size to take, and hence there is an implicit assumption that the search space is reasonably well scaled (to ±10 units in each of the search directions, say). It is also important that in finding these gradients, the steps do not miss a lot of important structure; there should not be too many subsidiary minima. It is possible for the Powell technique to become trapped in subsidiary minima, and it is possible for it to skip entirely over a minimum. However, if the initial set of parameters is not too far from the true minimum, Powell will usually do a good job of refining the parameter estimates. Thus Powell is a good technique for refining a set of parameters to reach a nearby local or global minimum, but a bad technique for finding a global minimum of a complicated search space.

• Simplex This technique creates a polyhedral search element around the initial position, x₀, and then grows or shrinks in particular directions while crawling around parameter space, to try to place a minimum within the final search polyhedron. This technique has some hilarious ways of getting stuck in high-dimension parameter spaces (where the polyhedron can become a strange shape), but is very good at finding minima in regions where the fit statistic has a moderately well-defined topology. Since it works in a different way than Powell minimization, a good strategy is to combine Simplex and Powell minimization to test whether an apparent minimum found by one technique is stable when searched by the other. I regard Simplex searching as good in smooth and simple parameter spaces, particularly when looking at regions where the fit statistic depends on a parameter in a linear or parabolic fashion, and bad where surfaces of equal value of the fit statistic are complicated. In either case, it is essential that the initial size of the polyhedron (with sides of length 1 unit) is a smallish fraction of the search space.

• Levenberg-Marquardt This resembles an improved Powell minimization, constructing good search directions and distances to move based on the shape of the target function near the initial guessed minimum, x₀, and then progressively moving towards the dominant local minimum. The refinement over Powell minimization is that this technique uses information about the local curvature of the fit statistic as well as its local gradients, and this tends to stabilize the result in some cases. I regard the techniques implemented in Sherpa as being good minimum-refiners for simple local topologies, since more assumptions about topology are made than in the Simplex or Powell methods, but bad at finding global minima for target functions with complicated topologies.

Scatter-shot techniques

Although a bit ad hoc, these techniques attempt to locate a decent minimum over the entire range of the search parameter space. Because they involve searching a lot of the parameter space, they involve many function evaluations, and are somewhere between quite slow and incredibly tediously slow.

The scatter-shot routines are:

• Grid This routine simply searches a grid in each of the search parameters. The coarseness of the grid sets how precise a root will be found. If, for example, there are 4 parameters to be searched (i.e., x is a vector of dimension 4), then with 131072 grid points (the default), there will be 19 sample points on each axis of the 4-D search space, and so the accuracy of estimation of each parameter will be roughly 0.06 of the range of the axis in that parameter. If the fit statistic has significant structure on a smaller scale, then the grid-searcher will miss it completely. This is a good technique for finding an approximation to the minimum for a slowly-varying function. It is a bad technique for getting accurate estimates of the location of a minimum, or for examining a fit statistic with lots of subsidiary maxima and minima within the search space.

• Monte Carlo This is even simpler than the grid-search routine. A certain number of random samples of the target function are made, with each element of the test parameter vector, x, sampled uniformly from the specified search range. Clearly this routine is probabilistic; even for a smoothly-varying target function, there is a chance that no sample will lie particularly close to the minimum. On the other hand, for a function with unknown properties, this technique does give a chance of finding a minimum even in some part of parameter space that might not be thought of in the first place. Again, this is a good technique for finding rough minima for slowly-varying functions, and a bad technique for getting accurate minima, or minima for functions with a lot of structure in the search space.

• Monte-Powell This technique attempts to cope with complicated functions by looking for minima near a large number of randomly-selected locations within the designated search space. It turns out that this is surprisingly good at finding the global minimum of even quite complicated functions provided that enough initial start points are chosen. Difficulties may be encountered if the global minimum is in a very small part of parameter space (i.e., if the minimum is very sharp), while there are other, shallower, minima which occupy a larger fraction of the parameter space and which are likely to capture the Powell minimizations. Thus this is a good technique for many functions, and tends to be bad only for functions which are genuinely difficult anyway. It is my personal favorite for exploring a new function about which I know rather little, because it makes no special assumptions about the target function, but it is rather inefficient since many function evaluations are far from the center of parameter space.

• Grid-Powell This is similar to Monte-Powell, except that a regular grid of points is searched, rather than allowing a probabilistic sampling of the parameter space. For low-dimension spaces (i.e. small number of parameters) in which there is rather little structure this may be good, but for higher-dimension spaces (i.e. large number of parameters) the sampling of the grid becomes unacceptably coarse and finding the correct global minimum can be a considerable fluke (worse even than in Monte-Powell) because many of the function evaluations are forced to lie at the edge of the parameter space, relatively far from the guessed minimum. Thus, for example, if an 8-D space is to be sampled, there are only 4 samples on each axis, and if the fit statistic has significant structure on the scale of a quarter of any one of the search parameters, it is likely that the global minimum will be missed. I regard Grid-Powell as a good method for functions where I know enough about the function that I can bracket the minimum with some certainty, but a bad technique for high-dimension minimizations.

• Simulated Annealing There are four such methods available in Sherpa. All are statistical methods for trying to find a global minimum in a complicated parameter space. The details of the various simulated annealing techniques are given in SIMUL-ANN-1 and SIMUL-ANN-2. Basically these methods make considerable improvements on the Monte Carlo technique, by spending fewer function evaluations in unproductive bits of parameter space (where the target function is far from the minimum), and more function evaluations near the minimum. The way this is done borrows some techniques from statistical mechanics, but the essential detail is that this is a conditioned probabilistic approach to finding the minimum. It tends to be good at finding an answer which is fairly close to the global minimum, but it is bad in that all four the simulated annealing methods use tremendous amounts of CPU time. These techniques are a high order of overkill in relatively simple minimization problems, but for truly difficult problems where the user has little or no idea what the real answer is, they provide a brute-force way of getting an answer which is fairly reliable. The techniques that combine Powell minimization with simulated annealing tend to be better than the pure simulated annealing methods; they produce better estimates of the minima.

Summary and best-buy strategies

Overall, the single-shot methods are best regarded as ways of refining minima located in other ways: from good starting guesses, or from the scatter-shot techniques. Using intelligence to come up with a good first-guess solution is the best approach, when the single-shot refiners can be used to get accurate values for the parameters at the minimum. However, I would certainly recommend running at least a second single-shot minimizer after the first, to get some indication that one set of assumptions about the shape of the minimum is not compromising the solution. It is probably best if the code rescales the parameter range between minimizations, so that a completely different sampling of the function near the trial minimum is being made.

Since they tend to be slow, repeated rescaled runs are rarely possible with the scatter-shot minimizers. For most relatively well-behaved functions where quality control over the estimated parameters is being made by an intelligent user, I would recommend running Monte-Powell or Grid-Powell and then testing the minimum using the Simplex or Levenberg-Marquardt routines. If an automatic minimum-finder is being written, then I would run the scatter-shot minimizer at least twice, with the second run using a restricted range of search space around the minimum from the first, and then test the minimum using single-shot minimizers. For functions where little is known about their properties, or for problems where the user has an almost unlimited CPU time available to reach the "right" answer, I would use one of the simulated annealing plus Powell minimizers, follow up with a Grid-Powell minimizer, and then finish off with a Simplex or Levenberg-Marquardt minimizer. Even then, I wouldn't believe the solution until I'd looked at it pretty carefully: there are plenty of rogue functions out there!

Acknowledgements

The contents of this document are taken from a memo written by Mark Birkinshaw (1998).