Convergence and Tolerance: Knowing When an Answer Is Close Enough

Convergence means the answer stops moving as more work is done, and tolerance is how much movement is accepted before the work stops. The two rates here are far apart. Halving a bracket costs one step every time. Halving a simulated error costs four times the draws every time, so matching sixteen bracket steps would take 4,29,49,67,296 times the draws.

The multiplier 4,29,49,67,296 is not a flourish, and it is worth spelling out in plain words before anything else. Four thousand two hundred and ninety four crore, ninety six lakh, seventy two hundred and ninety six times as many draws, to buy what sixteen lines of arithmetic already bought. Two methods that both settle on the right answer can be separated by that much, and the whole practical content of the word convergence is the distance between them.

Two settled results are used here without being built again. The first is that a bracket, halved at every step, closes on the answer at a rate that can be written down in advance. The second is that the standard errorA rough size for how much a figure would jump about were the whole calculation run again on fresh inputs. The standard error sizes the spread of possible answers rather than the mistake in the one answer actually in hand. of an average shrinks in step with the square root of however many readings went into it. Both are established elsewhere in these notes. Set the two rates side by side and one question is left over: how is the decision to stop made?

What does convergence actually mean?

Consider filling a bucket from a tap that is slowly closing. The first minute puts in a litre, the second half a litre, the third a quarter. The water level is still rising, so nothing has finished, and yet after ten minutes the tap could be abandoned and the level would end up almost exactly where it already is. Convergence is that shape. Convergence does not describe a moment when the answer arrives. The word describes the answer moving less and less for every extra minute spent.

An answer converges when doing more work moves it less. Every method described here converges eventually, and eventually is not a schedule. The only useful question about a method is never whether it converges but how fast. A method that halves its remaining error every second and a method that halves it every year both converge. One of them finishes before lunch and the other does not finish inside a working life, and no statement containing only the word converges can tell them apart.

So convergence is a rate. The rate is measured in error removed per unit of work done, and the unit of work has to be named or the rate means nothing. The two units of work here are a step and a draw. A step is one pass through an arithmetic rule. A draw is one simulated twelve month path of the Nakshatra unit. The Nakshatra unit is an invented traded unit with a stated shape to its monthly change, carried in from the earlier material, and the Nakshatra ladder is an invented arithmetic rule: start at Rs 100/-, and every month multiply by one plus a rate and then take Rs 2/- away.

What does halving per step buy?

Take the first of the two rates. The Nakshatra ladder is asked a question with no rearrangement available: what monthly rate takes Rs 100/- to Rs 120/- in twelve months, given that Rs 2/- comes out every month on the way? The rate appears raised to the twelfth and as a divisor in the same expression, and no rearranging prises the two apart, so there is no formula anybody can write down for it. The method that works instead is bisectionCutting a range in half, keeping whichever half still contains the answer, and repeating. Bisection needs no formula for the answer, only the ability to test a value and say which side of it the answer lies., and it is settled elsewhere in these notes rather than rebuilt here.

Write the rate as a plain fraction rather than a percentage. Then 2.00 per cent a month is 0.0200 and 6.00 per cent a month is 0.0600. At 0.0200 the ladder ends the twelve months at exactly Rs 100/-, and at 0.0600 it ends at Rs 167.4798/-. The target of Rs 120/- is therefore somewhere between them. The pair 0.0200 to 0.0600 is a bracketA pair of values the answer is already known to lie between. Its usefulness is that it is a statement about where the answer is, not merely about where the current guess is. of width 0.0400.

Every bisection step cuts that width in half and costs exactly as much as the step before it. A flat cost buying a whole factor of two is why this rate feels like something for nothing. Testing a midpoint means running the ladder rule twelve times, and running it twelve times costs the same whether the bracket being split is 0.0400 wide or one part in a hundred million wide. The work per step is flat and the reward per step is a whole factor of two. Flat work with a constant reward is rare, and where it is available it is worth taking.

Why does four times the draws only halve the error?

Now the second rate, and it comes from somewhere the earlier notes have already settled. The result established there is about an average built from a count of readings: the standard error of that average shrinks in step with the square rootThe number that gives back the original when it is multiplied by itself. Four has a square root of two, and a hundred has a square root of ten. of however many readings feed it. Fifty months of the Nakshatra unit give a standard error of 0.7071 per cent on the mean. Two hundred months, four times as many, give 0.3536 per cent. Four times the readings, half the error, and the arithmetic underneath it is no more than the square root of four being two.

The square root rate is a fact about counting rather than a fact about simulation. No amount of faster hardware or cleverer software changes a fact about counting. A simulated estimate is the same kind of quantity as that average: a summary computed from a count of independent things. Whether the things are fifty observed months or ten thousand drawn paths makes no difference to how the spread of the summary behaves. So the rule carries across unaltered. Four times the draws, half the error.

How far apart are the two rates set side by side?

The distance between the two rates is the centre of the matter, and figures show it better than adjectives. The question put to the simulation is the one settled earlier: what is the chance that the Nakshatra unit ends twelve months below the Rs 100/- it started at? Because the shape of the monthly change was stated rather than measured, that chance can be worked out exactly by weighing every one of the twelve month combinations, and it comes to 24.6562 per cent. Every simulated reading below is printed against that exact figure.

The run below starts from the seedThe number a generator is started from, which settles every value it will go on to produce. Start it somewhere else and a different sequence comes out. Start it in the same place and the identical sequence comes back. 12345 and follows the stated rule. The running number becomes 16,807 times itself, then the remainder on 2,14,74,83,647, then divided by that same modulus.

Look down the third column first. The column reads 1.3630, 0.6815, 0.3407, 0.1704, 0.0852: four exact halvings, one after another, with no rounding fudged to make them line up. Now look at what those halvings cost. The draws went from 1,000 to 2,56,000, two hundred and fifty six times the work, and bought four halvings. Sixteen bisection steps buy sixteen halvings.

To buy sixteen halvings of a simulated error the way sixteen steps buy sixteen halvings of a bracket, the draws have to be multiplied by 4,29,49,67,296, and that ratio is the honest price of the method rather than a criticism of it. Simulation is not a weaker method or a lazier one. Simulation is the method for questions with nothing to bracket, no midpoint that could be tested and no side of the answer that could be ruled out. The rate is worth paying because the alternative is not a slower answer, it is no answer.

What does the halving actually cover, and what does it not?

Here is the place where an account of convergence most easily teaches something false, so it is worth going slowly. The standard error column halved four times, exactly. The fourth column, the actual miss against the exact 24.6562 per cent, did nothing of the kind. The miss ran 0.4438, then minus 0.1062, then 0.0813, then minus 0.1703, then 0.0297. The miss got smaller, then bigger, then smaller again, and it changed sign at every one of the four moves down the column.

The standard error is a statement about the band, never about the run, so any claim that more draws halve the miss teaches something untrue, and the figures in the table above disprove it. What the halving covers is the width of the region the estimate would typically land in. Where it actually lands inside that region on any one run is not covered, is not predictable, and does not tidy itself up as the draws rise.

What is a tolerance, and how is one chosen?

A tolerance is the movement agreed to be acceptable before the work stops. Agreed is the important word in that definition. A tolerance is a decision somebody makes, not a quantity the arithmetic hands over. Nothing in the ladder rule and nothing in the ten thousand paths contains an opinion about how close is close enough. The method will keep going as long as it is allowed to and stop the instant it is told to.

So where does the decision come from? The decision comes from what the answer is for. Consider a household budgeting for a wedding, working out what the caterer will cost. If the answer decides whether to book the hall on Saturday or Sunday, being out by Rs 500/- changes nothing at all and being out by Rs 2,00,000/- changes everything. The tolerance is set by the size of the consequence, and where nobody can state the consequence, nobody can state the tolerance either.

There are two ways to state one, and they are not interchangeable. An absolute tolerance is a gap in the units of the quantity: accept anything within 0.2500 per cent of the answer. A relative tolerance is the same gap written as a share of the quantity: accept anything within about 1.01394 per cent of the answer's own size. On the twelve month chance of 24.6562 per cent those two are the same rule by construction. On the ladder's rate of 3.3792 per cent the two rules part company. The absolute version still accepts a gap of 0.2500. The relative version accepts 0.0343, tighter by 7.30 times.

A tolerance is stated as an absolute gap when the size of the quantity is known and will stay that size, and as a share of the quantity when it will not, and the two should never be assumed to agree because they happened to agree on one number. The choice is not a matter of taste. The choice states which kind of error would hurt, and switching between the two quietly is how a rule that was sensible on one quantity becomes nonsense on the next.

One more rule about choosing, and it is the one that gets broken most often. Once the answer is visible, the only thing a tolerance can be chosen to do is accept the answer already in hand. A tolerance chosen after seeing the answer is therefore not a tolerance. Nobody catches themselves doing this. Something is run, the reading is looked at, it seems about right, and it is judged close enough. The decision felt like judgement and it was a rationalisation, and the defence against it is not care but order: the tolerance is written down before the run starts, in the same breath as the question.

How many draws does a stated tolerance cost?

Once a tolerance is written down, the draw count stops being a habit and becomes a consequence. Turn the standard error round: instead of asking what error a given number of draws produces, ask how many draws bring the error down to a stated figure. On the twelve month question about the Nakshatra unit that turn is one line of arithmetic, and it produces the table below.

Wherever the tolerance halves in that table the draws go up by four. The square root rate seen from the other end makes the number of draws a consequence of a decision rather than a habit. Read the halving pairs and check it: 1.0000 to 0.5000 takes 1,858 to 7,431; 0.5000 to 0.2500 takes 7,431 to 29,723; 0.1000 to 0.0500 takes 1,85,769 to 7,43,077. The step from 0.2500 to 0.1000 is not a halving and does not cost four times. The tolerance falls by two and a half, and the draws rise by two and a half squared, or six and a quarter.

Notice what that does to the phrase ten thousand draws. Ten thousand is not a round number chosen because it sounds serious. Ten thousand is what a tolerance of 0.4310 per cent costs on this particular question, and a tolerance of 0.0500 would cost 7,43,077 paths on the same question. A draw count nobody can attach to a stated tolerance is a draw count somebody picked, and it is worth asking which.

When is a tolerance too tight to mean anything?

There are two ways to set a tolerance wrongly and both of them produce more digits and less information. The first is setting it tighter than the input deserves. The Nakshatra ladder's answer of 3.37915548 per cent was solved against a target of Rs 120/-. But a target written as Rs 120.00/- means anywhere from Rs 119.995/- to Rs 120.005/-. Two decimal places say that much and no more. And near the answer the ladder's twelve month reading moves by Rs 1,564.2086/- for every whole unit of the rate, so half a paisa of doubt about the target is about three parts in a million of doubt about the rate.

Work it through. The target of Rs 119.995/- gives 3.37883582 per cent and Rs 120.005/- gives 3.37947512. The two bounds, and the stated answer between them, agree at three decimal places of a per cent and disagree at four. Every one of them reads 3.379. A tolerance of one part in a hundred million on the rate is 319.65 times tighter than the target itself justifies, so bisection will happily deliver eight decimal places of which five are decoration.

The second way is setting a tolerance tighter than the method can support. The default simulated run has a standard error of 0.4310 per cent. Reporting its estimate to four decimal places produces a figure whose last digit claims to be worth one part in ten thousand of a per cent, on a method whose typical error is 4,310 times bigger than that. Every digit past the first decimal place is significant figureA digit that is doing actual work in a number, counted from the first digit that is not zero, reading left to right. Anything printed past what the inputs or the method will support stops being one, whatever a calculator displays. in appearance only.

Both faults look like care and are the opposite of it. A number carried to more places than its input or its method supports invites a reader downstream to build on precision that was never there. The extra digits look exactly like the real ones. The reader has no way of telling.

What is worth stopping on, and what is never worth stopping on?

Everything above turns into one short list, and this is the part a working analyst, a lender's credit team or anybody reviewing somebody else's model can actually use. There are three defensible things to stop on and one that looks defensible and is not.

1. Stopping on the bracket is right where one exists, because a bracket is a statement about where the answer is, and a width of 0.000000610352 says the answer cannot be outside it.
2. Stop on the reading where the consequence of the answer is what matters, which means checking how far the ladder's twelve month value sits from Rs 120/- rather than how far the rate has moved.
3. Stopping on a declared draw count is right where the method draws, worked out from a tolerance written down before the run, as 10,000 was worked out from 0.4310 per cent.
4. Never stop on the step, meaning the distance the answer moved since the last pass. A small step means slow movement and nothing else.

The fourth item is the trap. Nudge the ladder's rate by a millionth and its twelve month reading still shifts by Rs 0.001564/-, more than a tenth of a paisa, and that shift says nothing whatever about how near the target the reading has come. A method crawling towards an answer it has not reached produces small steps for the same reason a method sitting on the answer does. The step cannot distinguish those two cases, and the residualHow far the current answer's reading sits from the target it is aiming at. A residual is measured against the target. A step is only measured against the previous guess. can.

A lender's credit team reading somebody else's model has a short version of the same list. Ask what the stopping rule was. Ask whether it was written down before the run. Ask what the tolerance was and where the number came from, and then ask what decision it was set against. A rule that could have been anything once the answer was visible constrains nothing. A model whose stopping rule was chosen after the run has no tolerance at all, whatever number appears in its documentation.