CMP.6:4.3 - Make the finite update admissible and useful

For discrete replacements, compute the whole finite difference when it is affordable. Apply only changes that retain the required constraints, or pair the change with a specified repair whose effect is included in the comparison.

For a direction d, construct x'=x+η*d with step size η. The local information must support that finite change. A known upper model can determine a suitable step; otherwise a trial-and-reduction rule can search for a step whose observed effect supports acceptance. Include every trial evaluation in the cost.

For example, suppose a differentiable minimizing objective satisfies

J(x+s)≤J(x)+grad J(x)·s+(L/2)*||s||²

on the relevant region, with known L>0. Choosing s=-grad J(x)/L gives

J(x+s)≤J(x)-||grad J(x)||²/(2L).

This derives a finite descent step from a bound on the local model’s error. When that bound is unavailable, an adaptive step rule needs its own termination or acceptance conditions. One unsuccessful trial can justify reducing or changing the step; it does not establish that the direction never helps.

For constrained problems, use an admissible parameterization, projection or other constraint-preserving update. Reestablish the progress account for that update. Simply clipping a coordinate can change the original unconstrained argument.

With noisy feedback, distinguish realized change from a conditional or expected improvement claim. Repeating a measurement or taking a larger sample is useful only when the additional information changes the step or its warranted use. A deterministic monotone-descent statement cannot be inferred from a noisy sign alone.