(There were some slight edits since the original publication. The only major change is that the time to solve each linear program probably grows cubically, not linearly, with the number of points.)

This should be the first part in a series of post exploring the generation of efficient and precise polynomial approximations. Yes, there is Lisp code hidden in the math (: See rational-simplex and vanilla-fit.

Here’s the context: we want to approximate some non-trivial function, like \(\log\), \(\exp\) or \(\sin\) on a computer, in floating point arithmetic. The textbook way to do this is to use a truncated Taylor series: evaluating polynomials is pretty quick on computers. Sam Hocevar’s post on the topic does a very good job of illustrating how wrong-headed the approach is in most cases. We should instead try to find an approximation that minimises the maximal error over a range (e.g. \([-\pi,\pi]\) for \(\sin\)). He even has LolRemez to automatically compute such approximations for given functions. (While we’re kind of on the topic, the very neat Chebfun applies closely-related ideas to try and be to continuous functions what floating point numbers are to reals.)

The LolRemez docs point out two clever tricks to get more efficient approximations: forcing odd or even polynomials, and fixing parameters to simple values. Could we automate this process?

The goal of this series is to show that we can, by applying Operations Research (“The Science of Better”) to this micro-optimisation challenge. OR is what I do during the day, as a PhD student, and I believe that there’s a lot of missed opportunities in nifty program optimisation or analysis. (I also think that language design has a lot to offer to OR people who cope with modeling languages that have only changed incrementally since the seventies. :)

Problem statement

For now, let’s try and solve the original problem: find, in a family of functions \(F\) (e.g. polynomials of degree 4), an approximation \(\tilde{f}\) of a function \(f\) over some bounded range \(B\subset R\) such that \(\tilde{f}\) minimises the maximal error over that range:

$$\tilde{f} = \mathop{\arg\min}_{g \in F}\max_{x\in B} |g(x)-f(x)|.$$

\(B\) is a dense set, with infinitely, if not uncountably, many elements, so it’s far from obvious that we can solve this, even up to negligible errors.

The Remez algorithm

The Remez exchange algorithm shows one way to solve the minimax approximation problem, from the realm of mathematical analysis.

Let’s pretend we do not know \(\tilde{f}\), but do know the points where its error from \(f\) is maximised. In fact, if we are looking for polynomials of degree \(n\), we only need \(n+2\) such points. Let’s further impose that the errors alternate in sign: i.e., if the points are ordered (\(x_0 < \ldots < x_i < \ldots < x_{n+1}\)), then \(\tilde{f}(x_0) < f(x_0)\), \(\tilde{f}(x_1) > f(x_1)\), etc., or vice versa. Nifty analysis results tell us that these points exist, and that they exhibit the same absolute error \(|\epsilon|\). We can now find the coefficients \(a_0,a_1,\ldots,a_n\) of \(\tilde{f}\) by solving a small linear system:

$$\sum_{j=0}^n a_j x_0^j = f(x_0) + \epsilon$$ $$\sum_{j=0}^n a_j x_1^j = f(x_1) - \epsilon$$ etc. for each \\(x_i\\).

There are \(n+2\) variables (the \(n+1\) coefficients \(a_j\) and \(\epsilon\)), and \(n+2\) constraints (one for each extremum), so it’s a straight linear solve.

Of course, we don’t actually know these extremal error points \(x_i\). The Remez algorithm works because we can initialise them with an educated guess, find the corresponding coefficients, construct a new approximation from these coefficients, and update (some) of the extremal points from the new approximation. In theory this process converges (infinitely) toward the optimal coefficients. In practice, numerical issues abound: solving a linear system is usually reasonably stable, but finding exact function extrema is quite another issue. The upside is that smart people like Sam and the boost::math team have already done the hard work for us.

A linear minimax formulation

The minimax objective can be reformulated as an optimisation problem over reals (or rationals, really) with linear constraints and objective, a linear program:

$$\min_{e\in\mathbb{R}, a\in\mathbb{R}^{n+1}} e$$ subject to $$\sum_{i=0}^n a_ix^i \leq f(x) + e\qquad\forall x\in B$$ $$\sum_{i=0}^n a_ix^i \geq f(x) - e\qquad\forall x\in B$$

In words, we’re trying to find an error value \(e\) and \(n+1\) coefficients, such that the error \(|\tilde{f}(x)-f(x)|\quad\forall x\in B\) is at most \(e\): \(-e \leq \tilde{f}(x)-f(x)\leq e\). The goal is to minimise that error value \(e\).

We’re still left with the issue that \(B\) is very large: infinite, or even uncountable. It doesn’t make sense to have two constraints for each \(x\in B\)! Our goal is to find approximations in floating point arithmetic, which helps a bit: there’s actually a finite number of floating point (single, double or otherwise) values in any range. That number, albeit finite, still tends to be impractically large. We’ll have to be clever and find a good-enough subset of constraints. Note that, unlike the Remez algorithm, we can have as many constraints as we want; we never have to remove constraints, so the quality of the approximation can only improve.

This method is an interesting alternative to the Remez algorithm because it always converges, finitely, even with (or rather, due to) floating-point arithmetic. It will also makes our future work on a branch-and-cut method marginally easier.

Rational simplex for exact solutions to linear programs

We’ve reformulated our problem as a finite linear program, assuming that we know for which point \(x\in B\) we should generate constraints. How can we find solutions?

LPs are usually solved with variants of the simplex algorithm, a combinatorial algorithm. This is the key part: while most of the algorithm is concerned with numerical computations (solving linear systems, etc.), these are fed to predicates that compare values, and the results of the comparisons drive each iteration of the algorithm. The key insight in the simplex algorithm is that there exists at least one optimal solution in which all but a small set of variables (the basis) are set to zero, and the value of the basic variables can then be deduced from the constraints. Better: every suboptimal (but feasible) basis can be improved by only substituting one variable in the basis. Thus, each iteration of the algorithm moves from one basis to a similar, but better, one, and numerical operations only determine whether a basis is feasible and is an improvement. Very little information has to persist across iterations: only the set of basic variables. This means that there is no inherent accumulation of round-off errors, and that we can borrow tricks from the computational geometry crowd.

Daniel Espinoza implemented that, and much more, for his PhD thesis on exact linear and integer program solvers. QSopt-Exact is a GPL fork of a fairly sophisticated simplex and branch and cut program. It exploits the fact that the simplex algorithm is combinatorial to switch precision on the fly, using hardware floats (single, double and long), software quad floats, multiprecision floats, and rational arithmetic. Unfortunately, the full solver segfaults when reporting solution values on my linux/x86-64 machine. Still, the individual simplex solvers (with double floats, quad floats, multiprecision floats and rationals) work fine. What I decided to do was to sidestep the issue and call the solvers in increasing order of precision, while saving the basis between calls. The double float implementation is executed first, and when it has converged (within epsilon), a more precise solver is called, starting from the basis on which the previous solver converged, etc. The final rational solver is exact, but very slow. Hopefully, previous solvers, while inexact, will have converged to very nearly-optimal bases, leaving only a couple iterations for the exact solver.

Rational-simplex is a one-file CL system that wraps QSopt-Exact and offers a trivial modeling language. We’ll use it a lot in the sequel.

Solving the minimax LP with rational-simplex

linf-fit.lisp implements a minimax linear fit in 60 LOC (plus a couple kLOC in QSopt :).

Each point is defined by loc , the original value for x , a sequence of parameters (in this case, powers of x as we’re building polynomials), and the value to approximate, \(f(x)\).

solve-fit takes a sequence of such points, and solves the corresponding LP with QSopt-Exact:

(defun solve-fit (points) (lp:with-model (:name "linf fit" :sense :minimize) (let ((diff (lp:var :name "diff" :obj 1)) (coefs (loop for i below *dimension* collect (lp:var :name (format nil "coef ~A" i) :lower nil)))) (map nil (lambda (point) (let ((lhs (linexpr-dot coefs (point-parameters point))) (rhs (point-value point))) (lp:post<= lhs (lp:+ rhs diff)) (lp:post>= lhs (lp:- rhs diff)))) points) (multiple-value-bind (status obj values) (lp:solve) (assert (eql status :optimal)) (values obj (map 'simple-vector (lambda (var) (gethash var values 0)) coefs))))))

With a fresh model in scope in which the goal is to minimise the objective function, a variable, diff , is created to represent the error term \(e\), and one variable for each coefficient in the polynomial ( coefs ) list; each coefficient is unbounded both from below and above.

Then, for each point, the two corresponding constraints are added to the current model.

Finally, the model is solved, and a vector of optimal coefficients is extracted from the values hash table.

We can easily use solve-fit to exactly solve minimax instances with a couple thousands of points in seconds:

CL-USER> (let ((*dimension* 5) ; degree = 4 (*loc-value* (lambda (x) (rational (exp (float x 1d0)))))) ;; function to fit: EXP. Converting the argument to ;; a double-float avoid the implicit conversion to ;; single, and we translate the result into a rational ;; to avoid round-off (solve-fit (loop for x from -1 upto 1 by 1/2048 ;; create a point for x in ;; -1 ... -1/2048,0,1/2048, ... 1 collect (make-point x)))) dbl_solver: 0.510 sec 0.0 (optimal) ; solve with doubles ldbl_solver: 0.260 sec 0.0 (optimal) ; with long doubles float128_solver: 0.160 sec 0.0 (optimal) ; with 128 bit quads mpf_solver: 1.110 sec 0.0 (optimal) ; with multi-precision floats mpq_solver: 0.570 sec 0.0 (optimal) ; with rationals 3192106304786396545997180544707814405/5839213465929014357942937289929760178176 #(151103246476511404268507157826110201499879/151089648430913246511773502376932544610304 28253082057367857587607065964265169406677/28329309080796233720957531695674852114432 13800443618507633486386515045130833755/27665340899215071993122589546557472768 9582651302802548702442907342912775/54033868943779437486567557708120064 2329973989305264632365349185439/52767450140409606920476130574336)

Or, as more readable float values:

CL-USER> (values (float * 1d0) (map 'simple-vector (lambda (x) (float x 1d0)) (second /))) 5.466671707434376d-4 ; (estimated) error bound #(1.0000899998493569d0 0.997309252293765d0 0.49883511895923116d0 0.177345274179293d0 0.04415551600665573d0) ; coefficients

So, we have an approximation that’s very close to what LolRemez finds for \(\exp\) and degree 4. However, we’re reporting a maximal error of 5.466672e-4, while LolRemez finds 5.466676e-4. Who’s wrong?

Tightening the model

As is usually the case, we’re (slightly) wrong! The complete LP formulation has constraints for each possible argument \(x\in B\). We only considered 4k equidistant such values in the example above. The model we solved is missing constraints: it is a relaxation. We’re solving that relaxation exactly, so the objective value it reports is a lower bound on the real error, and on the real optimal error. At least, we don’t have any design constraint, and the coefficients are usable directly.

We shall not settle for an approximate solution. We must simply add constraints to the model to make it closer to the real problem.

Obviously, adding constraints that are already satisfied by the optimal solution to the current relaxation won’t change anything. Adding constraints can only remove feasible solutions, and the current optimal solution remains feasible; since it was optimal before removing some candidates, it still is after.

So, we’re looking for constraints that are violated to add them to our current model. In the optimisation lingo, we’re doing constraint (or row, or cut) generation. In our problem, this means that we’re looking for points such that the error from the current approximation is greater than the estimate reported as our solution value. We’ll take a cue from the Remez algorithm and look for extremal violation points.

Extrema of differentiable functions are found where the first derivative is zero. Our approximation function is a polynomial and is trivially differentiated. We’ll assume that the user provides the derivative of the function to approximate. The (signed) error term is a difference of differentiable functions \(\tilde{f}-f\), and its first derivative is simply the difference of the derivatives.

Assuming that the derivative is continuous, we can use the intermediate value theorem to find its roots: for each root, there is a (non-empty, open) neighbourhood such that the derivative is negative to the left of the root and positive to the right, or inversely.

This is what find-extrema.lisp implements. In optimisation-speak, that’s our separation algorithm: we use it to find (maximally-) violated constraints that we ought to add to our relaxation.

find-root takes a function, a lower bound, and an upper bound on the root, and first performs a bisection search until the range is down to 256 distinct double values. Then, the 256 values in the range are scanned linearly to find the minimal absolute value. I first tried to use Newton’s method, but it doesn’t play very well with rationals: although convergence is quick, denominators grow even more quickly. There are also potential issues with even slightly inexact derivatives. This is why the final step is a linear search. The method will work as long as we have correctly identified a tiny interval around the extremum; the interval can be explored exhaustively without involving the derivative.

%approximation-error-extrema finds candidate ranges. The sign of the derivative at each pair of consecutive points currently considered in the model is examined; when they differ, the range is passed to find-root . If we find a new extremum, it is pushed on a list. Once all the pairs have been examined, the maximal error is returned, along with the list of new extrema, and the maximal distance between new extrema and the corresponding bounds. As the method converges, extrema should be enclosed more and more tightly by points already considered in our constraints.

There are frequent calls to round-to-double: this function is used to take an arbitrary real, convert it to the closest double value, and convert that value back in a rational. The reason we do that is that we don’t want to generate constraints that correspond to values that cannot be represented as double floats: not only are they superfluous, but they also tend to have large denominators, and those really slow down the exact solver.

Finally, the two pieces are put together in driver.lisp. The constraints are initialised from 4096 equidistant points in the range over which we optimise. Then, for each iteration, the relaxation is solved, new extrema are found and added to the relaxation, until convergence. Convergence is easy to determine: when the error reported by the LP (the relaxation) over the constraint subset is the same as the actual error, we are done.

CL-USER> (time (let ((*trace-output* (make-broadcast-stream))) ;; muffle the simplex solver log (find-approximation 4 -1 1 (lambda (x) (rational (exp (float x 1d0)))) (lambda (x) (rational (exp (float x 1d0))))))) ; predicted error actual difference log distance of new points Iteration 1: 5.4666716e-4 5.466678e-4 [6.5808034e-10] (4 new extrema, delta 40.42 bit) Iteration 2: 5.4666746e-4 5.4666775e-4 [3.25096e-10] (4 new extrema, delta 41.15 bit) Iteration 3: 5.466676e-4 5.466677e-4 [9.041601e-11] (4 new extrema, delta 40.89 bit) Iteration 4: 5.4666764e-4 5.4666764e-4 [2.4131829e-11] (4 new extrema, delta 38.30 bit) Iteration 5: 5.4666764e-4 5.4666764e-4 [5.3669834e-12] (4 new extrema, delta 39.04 bit) Iteration 6: 5.4666764e-4 5.4666764e-4 [2.1303722e-12] (4 new extrema, delta 36.46 bit) Iteration 7: 5.4666764e-4 5.4666764e-4 [8.5113746e-13] (4 new extrema, delta 37.19 bit) Iteration 8: 5.4666764e-4 5.4666764e-4 [5.92861e-14] (4 new extrema, delta 36.93 bit) Iteration 9: 5.4666764e-4 5.4666764e-4 [8.7291163e-14] (4 new extrema, delta 34.34 bit) Iteration 10: 5.4666764e-4 5.4666764e-4 [2.7814624e-14] (4 new extrema, delta 35.08 bit) Iteration 11: 5.4666764e-4 5.4666764e-4 [1.22935355e-14] (4 new extrema, delta 33.82 bit) Iteration 12: 5.4666764e-4 5.4666764e-4 [6.363092e-15] (4 new extrema, delta 34.56 bit) Iteration 13: 5.4666764e-4 5.4666764e-4 [2.6976767e-15] (4 new extrema, delta 31.97 bit) Iteration 14: 5.4666764e-4 5.4666764e-4 [6.2273127e-16] (4 new extrema, delta 31.71 bit) Iteration 15: 5.4666764e-4 5.4666764e-4 [3.1716093e-16] (4 new extrema, delta 32.44 bit) Iteration 16: 5.4666764e-4 5.4666764e-4 [8.9307225e-17] (4 new extrema, delta 29.86 bit) Iteration 17: 5.4666764e-4 5.4666764e-4 [2.986885e-17] (4 new extrema, delta 29.60 bit) Iteration 18: 5.4666764e-4 5.4666764e-4 [8.047477e-17] (4 new extrema, delta 30.33 bit) Iteration 19: 5.4666764e-4 5.4666764e-4 [0.0e+0] (4 new extrema, delta 27.75 bit) Evaluation took: 153.919 seconds of real time ; includes time spent in the LP solvers 43.669636 seconds of total run time (39.078488 user, 4.591148 system) [ Run times consist of 2.248 seconds GC time, and 41.422 seconds non-GC time. ] 28.37% CPU 245,610,372,128 processor cycles 4,561,238,704 bytes consed 4046879766238553594956027226378928284211150354828640757733860080712233114910799/7402816194767429570294393430906636383377554724586974324818463099198772886301048832 #(925435306122623901577988826638313570454904960173764226165130358296546098660735739/925352024345928696286799178863329547922194340573371790602307887399846610787631104 6557329860867414742326673804552068829044003331586895788335382835917/6575021589199030076761681434499916661609079970484769650357093605918 1639925832159462899796569909691517124754835951733786891469617405863/3287510794599515038380840717249958330804539985242384825178546802959 583024503858777554811473359621837746265838166280012537010086877260/3287510794599515038380840717249958330804539985242384825178546802959 145161740826348143502214435074903924097159484388538508594721873208/3287510794599515038380840717249958330804539985242384825178546802959)

Note that we found new extrema in the last iteration. However, the error for those extrema wasn’t actually larger than for the points we were already considering, so the solution was still feasible and optimal.

The fractions will be more readable in floats:

CL-USER> (values (float * 1d0) (map 'simple-vector (lambda (x) (float x 1d0)) (second /))) 5.466676005138464d-4 #(1.0000900001021278d0 0.9973092516744465d0 0.4988351170902357d0 0.177345274368841d0 0.044155517622880315d0)

Now, we have nearly exactly (up to a couple bits) the same values as LolRemez’s example. Each iteration takes less than 10 seconds to execute; we could easily initialise the relaxation with even more than 4096 points. We should expect the impact on the solution time to be cubic: most of the simplex iterations take place in the floating point solvers, and the number of iterations is usually linear in the number of variables and constraints, but the constraint matrix is dense so the complexity of each iteration should grow almost quadratically. What really slows down the rational solver isn’t the number of variables or constraints, but the size of the fractions. On the other hand, we see that some of the points considered by our constraints differ by \(\approx 30\) bit in their double representation: we would need a very fine grid (and very many points) to hit that.

In addition to being more easily generalised than the Remez algorithm, the LP minimax has no convergence issue: we will always observe monotone progress in the quality of the approximation and, except for potential issues with the separation algorithm, we could use the approach on any function. There is however (in theory, at least) a gaping hole in the program: however improbable, all the extrema could be sandwiched between two points for which the derivative has the same sign. In that case, the program will finish (there are no constraint to adjoin), but report the issue. A finer initial grid would be one workaround. A better solution would be to use a more robust root-finding algorithm.

What’s next?

The minimax polynomial fitting code is on github, as a demo for the rational-simplex system. The code can easily be adapted to any other basis (e.g. the Fourier basis), as long as the coefficients are used linearly.

In the next instalment, I plan to use the cut (constraint) generation algorithm we developed here in a branch-and-cut algorithm. The new algorithm will let us handle side-constraints on the coefficients: e.g., fix as many as possible to zero, 1, -1 or other easy multipliers.

If you found the diversion in Linear programming and cut generation interesting, that makes be very happy! One of my great hopes is to see programmers, particularly compiler writers, better exploit the work we do in mathematical optimisation. Sure, “good enough” is nice, but wouldn’t it be better to have solutions that are provably optimal or within a few percent of optimal?

If you want more of that stuff, I find that Chvatal’s book Linear Programming is a very nice presentation of linear programming from both the usage and implementation points of view; in particular, it avoids the trap of simplex tableaux, and directly exposes the meaning of the operations (its presentation also happens to be closer to the way real solvers work). Works like Wolsey’s Integer Programming build on the foundation of straight linear programs to tackle more complex problems, and introduce techniques we will use (or have used) here, like branch-and-bound or cut generation. The classic undergraduate-level text on operations research seems to be Hillier and Lieberman’s Introduction to operations research (I used it, and so did my father nearly 40 years ago)… I’m not sure that I’m a fan though.