Gradient-Based Convergence Criteria

Why Gradient Size Needs a Scale

An optimizer can stop when its objective changes very little or when its parameter updates are small, but neither condition directly answers whether the objective is locally stationary. A gradient-based criterion addresses that question: it measures the rate at which the objective would change for a small parameter change. However, a raw gradient is unit-dependent. Its components may be expressed in objective units per litre per hour, per volume unit, or per unit of a dimensionless parameter. Their numerical magnitudes cannot be compared fairly without accounting for parameter and objective scales.

This matters in pharmacometric estimation because parameters can differ substantially in magnitude and units. A small raw derivative for a parameter measured in a large unit may still imply a meaningful change over a plausible parameter step. Conversely, a large derivative for a parameter with a very small natural scale may correspond to little change over a relevant step. Scaling makes this comparison explicit, but it does not make every small scaled gradient conclusive: local curvature and constraints still matter.

The scales and references must be stated to interpret the criterion. The \(s_i\) describe meaningful changes in each parameter, in that parameter's units. \(F_{\mathrm{scale}}\) describes a meaningful objective change and must be positive; \(F_{\mathrm{ref}}\) only shifts the objective and has no effect on its gradient. A convenient choice in a particular implementation is not a universal standard. Estimation software may use different scaling rules, norms, or tolerances, so the reported convergence test must be interpreted according to its documented definition.

Deriving the Scaled Criterion

The chain rule gives the scaled derivative. Since \(\theta_i=c_i+s_i u_i\), a unit change in \(u_i\) corresponds to a change of \(s_i\) in the original parameter. Since \(\phi=(F-F_{\mathrm{ref}})/F_{\mathrm{scale}}\), differentiation yields:

Thus each component is the objective change, in units of \(F_{\mathrm{scale}}\), per one parameter-scale change \(s_i\), to first order. This is the rationale for comparing its magnitude with a dimensionless tolerance. The criterion uses the largest component, so every individual scaled direction must satisfy the threshold. It does not directly bound the combined first-order change for an arbitrary simultaneous change in many parameters.

The objective may be a negative log-likelihood or another minimized estimation criterion. If an estimation method instead maximizes an objective, the same reasoning applies after defining the minimized objective as its negative. The derivative's sign indicates the direction of local increase or decrease; the stopping criterion uses its magnitude.

Curvature Determines What a Small Gradient Means

The gradient describes local slope, while curvature describes how quickly that slope changes. Near a candidate point, suppose the objective is well approximated by a quadratic with a positive-definite Hessian \(H\), the matrix of second derivatives. In scaled coordinates, its Hessian is \(\widetilde{H}=S H S/F_{\mathrm{scale}}\), where \(S\) is diagonal with entries \(s_i\). If \(u_*\) is the local minimizer of this quadratic and \(\delta=u-u_*\), then the quadratic approximation gives \(g^{\mathrm{scaled}}=\widetilde{H}\delta\). When the smallest eigenvalue of \(\widetilde{H}\) is \(m>0\), the relation implies:

The first inequality follows from \(\lVert\widetilde{H}\delta\rVert_2\geq m\lVert\delta\rVert_2\) for a symmetric positive-definite matrix. The second follows by writing \(\delta=\widetilde{H}^{-1}g^{\mathrm{scaled}}\), so the quadratic excess is one half of \(g^{\mathrm{scaled}\mathsf{T}}\widetilde{H}^{-1}g^{\mathrm{scaled}}\), which is at most \(\lVert g^{\mathrm{scaled}}\rVert_2^2/(2m)\). These statements apply to the local positive-definite quadratic approximation, not to arbitrary nonlinear objectives far from the candidate point.

If \(m\) is small, a small gradient may coexist with a substantial distance to the minimizer: the objective is nearly flat in at least one direction. If curvature is large, a larger gradient can occur close to the minimizer because the slope changes rapidly. A small gradient at a saddle point or local maximum is also possible; a gradient test alone does not determine curvature or establish a minimum. These facts explain why the threshold is meaningful only in relation to scaling, curvature, and the other evidence available from the estimation run.

Parameterization, Constraints, and Practical Checks

Scaling depends on the coordinates in which the optimizer evaluates derivatives. For a positive parameter represented as \(\eta=\log(CL)\), the chain rule gives \(\partial F/\partial\eta=CL\,\partial F/\partial CL\). A scale of \(s_\eta\) in log coordinates therefore represents a multiplicative, rather than additive, change in \(CL\). A scaled gradient computed in \(\eta\) need not equal one computed in \(CL\). The scale should reflect meaningful changes in the actual parameter while remaining consistent with the coordinates used by the algorithm.

At an active parameter bound, the unconstrained gradient may not be zero at a constrained minimum. For a lower bound, a positive derivative can indicate that moving into the feasible region increases the objective, even though the derivative is nonzero. In this setting, the relevant first-order condition concerns feasible directions; a projected-gradient criterion or equivalent bound-aware test is more appropriate than demanding that every unconstrained component approach zero. The bound and parameter-transformation conventions established in the earlier material on parameter bounds and parameter transformations help determine which directions are feasible and which coordinates are being optimized.

A practical review of a gradient-based stopping decision should therefore ask:

• Are the derivatives computed reliably, rather than distorted by finite-difference steps or noisy objective evaluations?
• Are parameter scales appropriate for the chosen parameterization, and is the objective scale documented?
• Is the reported norm and tolerance defined clearly, and does each relevant component satisfy it?
• Could weak curvature, strong parameter dependence, an active bound, or a non-minimum stationary point explain the result?

Gradient-based convergence complements, rather than replaces, objective-function and parameter convergence tolerances. Agreement among several checks is more informative than passing one criterion in isolation. A very small objective change can occur along a flat region while the gradient remains consequential in another direction; small parameter steps can also result from conservative updates rather than stationarity. Conversely, scaling choices can make an otherwise reasonable raw gradient look large or small. A documented, scale-aware test makes the reasoning inspectable, but it does not remove the need to examine the optimizer's behavior and the local geometry.

Common Pitfall

Wrong reasoning: “The optimizer reports a gradient norm below the tolerance, so the estimates must be correct and the optimum must be well determined.” This confuses a stopping condition with a guarantee about the solution. A small gradient can arise in a flat region, at a stationary point that is not a minimum, or under scaling that makes the test insufficiently demanding. It may also be misleading at an active bound if the test ignores feasible directions.

Correction: State what gradient norm, parameter scales, objective scale, and tolerance were used. Interpret the result as evidence about local first-order slopes in those coordinates. Consider curvature and feasible directions before drawing a conclusion about proximity to a minimum, and compare the gradient result with objective and parameter convergence evidence.