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A single exponent is a convenient summary, not the whole geometry. A desingularising function can follow logarithmic, piecewise or otherwise non-power behaviour and turn the objective gap into a custom travel budget.
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Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Frame the problem
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · FRAME THE PROBLEM
1. Why one exponent can be too coarse
A Kurdyka–Łojasiewicz exponent models the local slope–gap relation with a power law. Many objectives fit this form, but some have several local regimes or a flatter correction that is better described by a general concave function. Forcing one exponent across the entire observed tail can hide the actual convergence geometry.
The general KL property uses an increasing desingularising function φ with φ(0)=0. Near a critical value, φ′(f(x)−f*) times the subgradient size is at least one. The derivative rescales height so that the transformed landscape has a usable slope.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. The desingulariser as a change of vertical scale
If φ(s)=c s^(1−θ), then φ′(s) is proportional to s^(−θ), recovering the familiar exponent form. A non-power φ can include logarithmic corrections, change formula between gap ranges or be nonsmooth in a generalised concave formulation.
The function is not unique: a larger admissible concave desingulariser may also satisfy the inequality. For convergence bounds, a smaller valid modulus is sharper because it assigns less remaining travel to the same objective gap.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A custom remaining-length ledger
Suppose an algorithm satisfies standard sufficient-decrease and relative-error conditions, and analysis yields a tail-length bound proportional to φ(r_k), where r_k=f(x_k)−f*. Compare φ₁(s)=2√s with φ₂(s)=2√s(1+0.2 log(1/s)) over a small-gap range.
| Gap s | Power φ₁(s) | Log-corrected φ₂(s) | Reading |
|---|---|---|---|
| 0.01 | 0.200 | about 0.384 | Correction allows more remaining travel |
| 0.0001 | 0.020 | about 0.057 | Both shrink, at different rates |
| 0.000001 | 0.002 | about 0.0075 | Log factor remains visible deep in tail |
These numbers do not assert that either function belongs to a particular objective. The objective and subgradient must establish the inequality. The ledger shows why identical current gaps need not imply identical remaining motion.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. A piecewise desingulariser for two regimes
Consider a function whose moderate gaps behave quadratically but whose deepest tail is flatter. A piecewise concave φ can use one expression above a transition ε and another below it, joined continuously with nonincreasing slope. This represents geometry rather than an arbitrary curve fit.
| Gap region | Candidate φ behaviour | Expected practical pattern | Audit question |
|---|---|---|---|
| s ≥ ε | Square-root scale | Near-linear contraction | Is the exponent regime stable? |
| 0 < s < ε | Square-root plus slow correction | Gradual tail slowdown | Is the correction theoretically justified? |
| At ε | Continuous value, compatible slopes | No artificial jump | Was the join checked for concavity? |
A change in observed rate may come from algorithmic step rules rather than objective geometry. Verify sufficient decrease and relative error on both sides of the proposed transition before interpreting φ.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. How φ enters a finite-length proof
Sufficient decrease converts an objective drop into control of a step. Relative error bounds a subgradient by recent steps. Concavity of φ turns the difference φ(r_k)−φ(r_{k+1}) into a term that pays for future motion. Summing telescopes and yields a finite total tail under the framework’s hypotheses.
The power-law rate table is therefore one corollary of a more general mechanism. With a non-power φ, invert the corresponding scalar recurrence or integrate the slope relation to obtain a custom rate. State whether the result controls objective gaps, residuals or iterate distance.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. From theory to an honest empirical diagnostic
Record objective gaps, subgradient or residual norms, step lengths and cumulative tail travel. Rather than fitting only a straight line on a log–log plot, compare the empirical slope–gap curve with candidate φ′ functions over a declared tail window.
The plot can reject an implausible power law, but it does not prove a KL function. Constants, unknown f*, numerical floors and changing active sets can imitate logarithmic corrections. Treat empirical fits as diagnostics unless the objective class supplies a theorem.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: tangential projections
Transfer the idea to alternating projections between sets meeting tangentially. A merit function based on distance or residual may have a Hölder error bound in one range and a slower corrected behaviour near a singular contact. A tailored desingulariser can encode that layered geometry.
The transfer requires a valid descent or fixed-point merit framework. Pure topology without a metric, objective or slope does not supply the quantities in a KL inequality. Keep separation, completeness and projection assumptions explicit.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connecting to exponent and error-bound guides
The KL exponent guide develops power-law rate classes and finite-length trajectories. This article keeps the full desingularising function instead of compressing it to one exponent.
Use Hölder error bounds and tangential intersections for geometric distance–residual relations, and metric subregularity for linear certificates. These conditions can interact, but they are not interchangeable names.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
A hand-drawn concave curve is not a KL proof. The desingulariser is local to a critical level and does not identify a global minimiser, exclude saddles or guarantee bounded iterates. It works only with algorithmic descent, error and continuity assumptions appropriate to the subdifferential in use.
Non-power does not mean arbitrarily slow or unclassifiable. It means the rate should be derived from the actual modulus rather than forced into one exponent. Nondifferentiable desingularisers require a generalised formulation; applying the classical derivative inequality at a kink is careless.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
State the space, metric, objective, critical value, subdifferential, neighbourhood and admissible class for φ. Verify monotonicity, concavity, φ(0)=0 and the exact form of the KL inequality. Record sufficient-decrease, relative-error and boundedness constants.
For computation, retain gaps, residuals, steps and cumulative tail length. Compare power and non-power candidates on the same declared range, vary the assumed f*, test active-set transitions and report uncertainty. Distinguish a theoretical desingulariser from an empirical approximation.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does every KL function have one exponent?
No. Exponents are powerful for many structured classes, but the general property is stated with a desingularising function that need not be a power.
Is the smallest φ always known?
No. Exact-modulus results identify a sharp object in certain generalised settings, but calculating it can be difficult.
Can a piecewise φ prove two convergence rates?
It can describe different gap regimes if the inequality and concavity conditions hold. The final statement must specify where each regime applies.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Wang and Wang introduce the exact modulus of the generalised concave KL property. Their later work develops calculus rules for generalised concave desingularising functions. Attouch, Bolte and Svaiter provide the broader descent framework for tame problems. The custom-rate ledgers above are original teaching examples.
Continue through the Topology root, KL exponents and finite-length trajectories and Fejér monotonicity and convergence gaps. The Gold Standard of Education and How X Works provide the wider route.
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