CORRECTION, prompted by Valeri: the step IS a fraction of the bracket — with the ratio similarity 1-S = min(1,|f|/(|f'|D)) so step = min(D,|f|/|f'|) — and my 'absolute length cannot march' was wrong, as was the follow-up guess that we creep along one side: measured, 91.5% of our probes DO shrink the bracket, more often than Chandrupatla's 87.5%. What differs is how much each probe removes: 0.93 halvings per evaluation against 2.26. A linear model of the value axis undershoots on a wide bracket of a convex function, so the new bracket is about half; inverse quadratic interpolation lands near the root and removes four fifths. The limit is the ORDER OF THE MODEL the similarity is read against, and all three parameter-free ways round it are now measured failures (cbrt, log change of variable, derived3_m). Paper corrected
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
This commit is contained in:
Claude Opus 5
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@@ -628,16 +628,35 @@ It is not the guarantee that caps us. With the safeguard removed entirely the
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same problems cost $46$, $57$, $61$ and $68$ against bisection's $57$, $77$, $45$
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and $45$ --- better on the wide brackets, \emph{worse than bisection} on
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\texttt{fun3.1} and \texttt{fun5.1} --- and still far from Chandrupatla's $24$ and
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$44$. The cap is the step rule itself: $|f|/|f'|$ is an absolute length, and an
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absolute length cannot march across decades. That is the same wall the fixed
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fraction (\texttt{cbrt}) and the logarithmic change of variable each hit from a
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different side, both buying the wide brackets at the price of the canonical set.
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$44$.
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So the limit is stated exactly: the placement of the similarity reading improves
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the method up to the point where the \emph{form} of the derived step becomes the
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binding constraint, and on brackets spanning many decades it does. Removing that
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constraint means replacing the step the contraction law derives, which is a
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different method rather than a better reading of this one.
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Nor is it that the step fails to be relative to the bracket. With the ratio
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similarity $1-S=\min(1,|f|/(|f'|D))$, so the step is exactly
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$\min(D,|f|/|f'|)$: a \emph{fraction} of the distance to the far end, and the
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similarity is that fraction. Nor does the iterate creep along one side without
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moving the proven bracket: measured probe by probe, $91.5\%$ of our evaluations
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on the $2\times10^4$-wide bracket do shrink the bracket, slightly \emph{more}
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often than Chandrupatla's $87.5\%$.
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What differs is how much each probe removes. Ours shrinks the bracket by
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$0.93$ halvings per evaluation there, Chandrupatla's by $2.26$ --- a factor of
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about two against a factor of about five. On a wide bracket of a convex function
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the Newton step systematically undershoots, because a linear model of the value
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axis cannot see the curvature that separates the iterate from the root, so our
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probe lands well short and the new bracket is about half the old one; a model
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that carries the curvature --- inverse quadratic interpolation --- lands far
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closer and removes about four fifths.
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So the limit is stated exactly, and it is the \emph{order of the model the
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similarity is read against}, not the placement of the reading and not the
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guarantee. Three parameter-free ways round it have now been measured and all
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three fail in the same way, buying the wide brackets at the price of the
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canonical set: a fixed fraction of the bracket (\texttt{cbrt}), a logarithmic
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change of variable, and a curvature-corrected scale (\texttt{derived3\_m},
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whose derivation holds but whose estimator of the local order fires on $27.9\%$
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of steps where the truth is a simple root). The fourth way is to model the
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curvature directly, which is inverse quadratic interpolation --- a different
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method rather than another reading of this one.
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On that set our shape reading ($2159$ re-scored) is \emph{worse} than bisection
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($2136$); only the dissimilarity reading ($1740$) is clearly ahead, which is the
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@@ -0,0 +1,104 @@
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"""Correcting the diagnosis: the step IS a fraction of the bracket, so what stalls?
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With S = ratio, 1 - S = min(1, |f|/(|slope| D)), hence step = min(D, |f|/|slope|): the step is
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a FRACTION of the distance to the far end and the fraction is the similarity. So "an absolute
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length cannot march" was wrong.
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The candidate mechanism instead: when the Newton length is much shorter than the bracket the
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iterate creeps from ONE SIDE, the sign never changes, and the PROVEN bracket does not shrink
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at all — only the guard's bisections shrink it. Chandrupatla, bisecting on 72% of steps and
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placing its accepted probes by inverse quadratic interpolation, may instead CROSS the root
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and collapse the bracket in one step.
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Measured here, per problem: the share of evaluations that flip the sign of f (i.e. actually
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shrink the proven bracket), and the geometric bracket reduction per evaluation.
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python -u diag_sign_flips.py
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"""
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from __future__ import annotations
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import math
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from scipy.optimize._tstutils import _CHANDRUPATLA_TESTS_DICTS
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from simroot.baselines import chandrupatla, chandrupatla_scipy
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from simroot.pure import pure_root
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from sweep_full import ACC, MAXFEV, XTOL
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def traced_bracket(f, a, b):
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"""Replay a run's probes and count how many flipped the sign, plus the bracket history."""
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xs = []
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def g(x):
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v = f(x)
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xs.append((x, v))
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return v
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return g, xs
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def replay(xs, a, b, fa, fb):
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"""Walk the probes in order, maintaining the proven bracket; count sign flips."""
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A, FA, B, FB = a, fa, b, fb
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flips = 0
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widths = [abs(B - A)]
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for x, v in xs:
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if not (min(A, B) < x < max(A, B)):
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continue
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if FA * v < 0.0:
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B, FB = x, v
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flips += 1 if (FB * FA < 0.0) else 0
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else:
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A, FA = x, v
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widths.append(abs(B - A))
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return flips, widths
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def run(method, f, a, b, root):
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g, xs = traced_bracket(f, a, b)
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try:
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x, n, conv = method(g, a, b, XTOL, MAXFEV)
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except Exception:
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return None
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x = float(x)
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if not (conv and math.isfinite(x) and (abs(x - root) <= ACC or f(x) == 0.0)):
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return None
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fa, fb = f(a), f(b)
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_flips, widths = replay(xs[2:], a, b, fa, fb)
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# how often did a probe actually reduce the proven bracket, and by how much on average
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reduced = sum(1 for i in range(1, len(widths)) if widths[i] < widths[i - 1] * 0.999)
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halvings = (math.log2(widths[0] / max(widths[-1], 1e-300)) / max(len(xs), 1))
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return len(xs), reduced, 100.0 * reduced / max(len(xs), 1), halvings
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def main():
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print(f"{'problem':<9} {'method':<14} {'evals':>6} {'probes that shrank':>19} "
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f"{'share':>7} {'halvings/eval':>14}")
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for d in _CHANDRUPATLA_TESTS_DICTS:
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pid = str(d["ID"])
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if pid not in ("fun1.3", "fun1.4", "fun1.5", "fun3.1", "fun5.1"):
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continue
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f = d["f"]
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a, b = (float(v) for v in d["bracket"])
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root = float(d["root"])
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ours = run(lambda ff, aa, bb, xt, mf: pure_root(ff, aa, bb, xt, mf, similarity="ratio",
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scale="derived3", guard="safe",
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prestep="slope",
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direction="dissimilar"),
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f, a, b, root)
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best = None
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for impl in (chandrupatla, chandrupatla_scipy):
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r = run(impl, f, a, b, root)
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if r and (best is None or r[0] < best[0]):
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best = r
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for nm, r in (("ours", ours), ("chandrupatla", best)):
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if r is None:
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print(f"{pid:<9} {nm:<14} (not scorable)")
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continue
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n, red, share, halv = r
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print(f"{pid:<9} {nm:<14} {n:6d} {red:19d} {share:6.1f}% {halv:14.3f}")
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if __name__ == "__main__":
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main()
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@@ -0,0 +1,11 @@
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problem method evals probes that shrank share halvings/eval
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fun1.3 ours 19 15 78.9% 2.498
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fun1.3 chandrupatla 15 13 86.7% 3.099
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fun1.4 ours 59 54 91.5% 0.930
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fun1.4 chandrupatla 24 21 87.5% 2.256
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fun1.5 ours 63 60 95.2% 1.179
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fun1.5 chandrupatla 44 41 93.2% 1.684
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fun3.1 ours 64 61 95.3% 0.672
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fun3.1 chandrupatla 44 42 95.5% 0.955
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fun5.1 ours 57 47 82.5% 0.741
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fun5.1 chandrupatla 44 42 95.5% 0.955
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