Objective-Function and Parameter Convergence Tolerances

Why Two Kinds of Tolerance?

An estimation algorithm can make very different kinds of progress from one iteration to the next. The objective function may barely change while one or more parameter estimates are still moving. Conversely, parameter estimates may move only slightly while a steep region of the likelihood surface produces a noticeable objective change. Objective-function and parameter-change tolerances monitor these different features of the optimization path.

In pharmacometric estimation, the objective is often a negative log-likelihood or a scaled form such as minus twice the log-likelihood. The optimizer seeks a parameter vector that minimizes the chosen objective. For FOCE or Laplace estimation, that objective may be based on an approximation to the likelihood; for stochastic methods, its evaluation may also have Monte Carlo variability. The tolerance is therefore a numerical stopping setting, not a statistical significance threshold.

For a minimization, the signed objective change is \(Q(\theta_{k+1})-Q(\theta_k)\), usually negative when the objective decreases. Convergence checks commonly use its absolute value, its decrease, or a relative change. Parameter checks may likewise use absolute changes, relative changes, or a scaled norm across parameters. These definitions are not interchangeable: a tolerance value is interpretable only after identifying the software's formula and the units or scaling it uses.

How the Likelihood Surface Connects the Checks

Near a smooth local minimum, the objective can be approximated by a quadratic function. In one dimension, if the minimum is at \(\theta^*\) and local curvature is \(H>0\), then for a small displacement \(\delta=\theta-\theta^*\),

This approximation explains why the same parameter movement can correspond to different objective changes in different regions. If curvature \(H\) is large, a small displacement can noticeably change the objective. If \(H\) is small, even a larger displacement may produce little change. For a vector of parameters, the corresponding quadratic term involves the local curvature matrix and the direction of the displacement: movement along a flat direction can have little objective effect even when its coordinates change substantially.

A short consequence follows for a one-dimensional quadratic objective. Suppose a proposed stopping rule requires \(Q(\theta)-Q(\theta^*)\leq \varepsilon\). Under the quadratic approximation,

Thus an objective tolerance corresponds to a parameter distance only if the local curvature is known and the quadratic approximation is adequate. A flat surface permits a larger distance for the same objective tolerance; a steep surface permits a smaller one. An objective-change check between two iterations is even less direct: both points may be far from the minimum while the objective changes little along a nearly flat path.

Worked Cases Where the Checks Disagree

Choosing and Interpreting Tolerances

A tolerance should be evaluated in the scale on which the optimizer applies it. An absolute objective threshold is expressed in objective units. A relative threshold is dimensionless but depends on the denominator convention, especially when the objective is near zero. Parameter tolerances may be in the original parameter units or on a transformed scale. For example, a change of \(0.01\) on a log-transformed parameter does not mean the same thing as a change of \(0.01\) in clearance measured in L/h.

For parameters with different units and magnitudes, a single raw absolute-change threshold can be misleading. A change of \(0.1\) may be negligible for a parameter measured in thousands and substantial for one near \(0.2\). Relative changes help with scale differences, but can become unstable for parameters near zero. A norm-based check can also conceal a large change in one coordinate if changes in other coordinates are small. Inspecting individual parameter changes, with their units, is useful even when the software reports a combined criterion.

The tolerance settings should be understood alongside the method and objective being evaluated. A noisy stochastic objective can fluctuate between evaluations, so an apparent failure of a very tight objective-change check may reflect simulation noise rather than meaningful movement. Conversely, an approximate likelihood evaluation can stabilize at its own numerical accuracy without matching a more accurate likelihood evaluation. A tolerance cannot compensate for insufficient precision in the objective computation.

Objective values also depend on the definition and scaling of the objective. A threshold suitable for one dataset size or objective convention may not transfer directly to another. In particular, a small numerical objective difference is not automatically a likelihood-ratio test result, a p-value, or evidence that two models are statistically equivalent. Such interpretations require their own assumptions and procedures.

Diagnostics, Alternatives, and Limitations

When a run reports convergence, review what condition was actually met: objective change, parameter change, both, or another software-specific combination. Check whether it had to hold for one iteration or several consecutive iterations. Examine the trace over multiple iterations rather than relying only on the final status message. Unexpected oscillation, alternating changes, or a criterion that passes intermittently can indicate that the reported threshold is not capturing stable progress.

A parameter can appear stable because it is constrained at a bound or because its transformed representation changes very little. This is not by itself evidence that the unconstrained likelihood surface has a well-behaved optimum. Likewise, an objective that barely changes may reflect a flat direction, poor scaling, noisy evaluation, or limited numerical resolution. Tolerance checks are stopping rules, not complete diagnostics of model identifiability, uncertainty, or fit.

The main alternative convergence check is based on the gradient: it asks whether the objective has a sufficiently small local slope, rather than whether successive objective values or parameter vectors changed little. These criteria answer different questions. A parameter-change check can pass while the objective still has a material slope, and an objective-change check can pass along a flat path without showing that the optimizer has reached a stationary point. A gradient check provides complementary information but is also subject to scaling and numerical accuracy.

In practice, objective-change and parameter-change tolerances are most informative when interpreted together with the iteration trace, parameter scales, estimation method, and other convergence diagnostics. They do not establish that the chosen local solution is the best solution, that uncertainty estimates are reliable, or that the pharmacometric model is scientifically adequate. Those questions require additional assessment.