Gradient-Based Convergence Checks

Why Inspect the Final Gradient?

An estimation run can stop because its objective changed very little, because parameter updates became small, or because an iteration limit was reached. Those stopping events are not interchangeable with finding a point where the objective is locally stationary. The final gradient helps answer that narrower question: does a small feasible change in the estimated parameters still have a first-order effect on the objective?

The gradient is useful evidence, not a certificate that the model is correct or that the best possible solution has been found. Its interpretation depends on the objective being minimized, parameter scaling, constraints, and the numerical accuracy of the reported derivatives. The objective-function traces discussed in the earlier material can show what happened across iterations; a final-gradient check assesses the local behavior at the reported endpoint.

In a pharmacometric estimation problem, \(F\) may be an objective such as minus twice the log likelihood, depending on the estimation method and software. The sign convention matters: here \(F\) is always treated as a quantity being minimized. A positive component means that, locally, increasing its parameter increases \(F\); a negative component means that increasing the parameter decreases \(F\). The magnitude is the local change in objective per unit change in that parameter.

From Raw to Scaled Gradient Components

Raw gradient components have units of objective per parameter unit. They are therefore not directly comparable across parameters measured on different scales. For example, a gradient expressed per litre per hour may look numerically small beside one expressed per hour, even if both imply substantial objective changes over meaningful parameter changes. A parameter expressed in a different unit would also have a different raw gradient.

To make a scale-aware check, choose a positive, meaningful scale \(s_i\) for each parameter, in the same units as \(\theta_i\). It should represent a change that is consequential for that parameter, rather than being chosen solely to make the gradient look small. Define a dimensionless coordinate \(z_i=(\theta_i-\theta_{i,\mathrm{ref}})/s_i\), where \(\theta_{i,\mathrm{ref}}\) is any fixed reference value. By the chain rule, \(\partial F/\partial z_i=s_i g_i\). One explicit objective-normalized criterion is then:

Here \(G_i\) is the scaled component and \(G_{\max}\) is the largest absolute scaled component. This definition makes the scaling choice explicit; it is not a universal software convention. A run meets a specified scaled-gradient tolerance \(\tau\) under this definition when \(G_{\max}\leq\tau\). The tolerance and the exact software formula must be checked together: a threshold intended for one scaling convention cannot safely be applied to another.

The factor \(q\) avoids dividing by a value below one and normalizes by the magnitude of the objective when that magnitude is larger. This convention is practical, but objective rescaling or a different normalization changes the reported scaled components. For reproducible interpretation, record the objective definition, parameter scales, scaling formula, tolerance, and whether the gradient is raw or transformed. A small \(G_{\max}\) supports first-order stationarity at the chosen scales; it does not demonstrate that the objective was evaluated accurately or that the point is a global minimum.

What a Gradient Check Does—and Does Not—Establish

The first-order justification is local. For a small parameter change \(\Delta\boldsymbol{\theta}\), differentiability gives the approximation \(F(\boldsymbol{\theta}+\Delta\boldsymbol{\theta})\approx F(\boldsymbol{\theta})+\mathbf{g}^{T}\Delta\boldsymbol{\theta}\), with an error that becomes small relative to the step as the step approaches zero. If an interior point has a nonzero gradient component, choosing a sufficiently small change in the opposite direction reduces the objective to first order. For an unconstrained interior local minimum, however, changes in either direction for each parameter must not decrease the objective to first order. That requires every component to be zero.

In computation, gradients and parameter estimates are approximate, so the practical question is whether the scaled components are small relative to a declared tolerance. This is a numerical criterion, not a proof that the exact gradient is zero. Also, first-order stationarity alone cannot distinguish a local minimum from every other stationary configuration, establish global optimality, or verify the likelihood calculation. Those questions require other checks.

Parameter bounds change the first-order condition because not every direction is feasible. For a parameter with lower bound \(L\), if the estimate is exactly at \(L\), only an increase is locally feasible. A constrained minimum therefore requires \(g_i\geq0\) there: moving into the feasible region must not decrease \(F\) to first order. At an upper bound \(U\), the corresponding requirement is \(g_i\leq0\). An unconstrained rule that demands every raw component be near zero can consequently misclassify a valid boundary solution.

This bound-aware condition is a first-order necessary condition, not a claim that the parameter should be at the bound or that the estimate is substantively reasonable. If a parameter is merely close to a bound rather than at it, treating it as exactly constrained can conceal a feasible descent direction. Use the actual constraint status and the optimizer’s documented projection rule.

Diagnostics, Failure Modes, and Alternatives

When a final-gradient check looks questionable, first verify what the software actually reports. Determine whether the values are evaluated at the final parameter vector, whether they are raw or scaled, whether the reported norm is a maximum component or another summary, and whether constraints are handled through a projected gradient. Then inspect parameter units, scales, objective normalization, and the convergence tolerance. A number without those definitions is difficult to interpret or reproduce.

Gradients can also be unreliable when small changes in parameters produce unstable objective evaluations. Numerical integration settings, noisy approximations, or finite-difference calculations can make successive derivative estimates disagree. Compare gradient reports across tighter numerical settings or independent runs when such instability is suspected. A component close to its threshold deserves particular caution: rounding or small numerical perturbations may change whether the criterion passes. Preserve sufficient digits for checking, even if the final report rounds values for display.

Gradient checks complement, rather than replace, other stopping diagnostics. A small change in objective can indicate stalled progress but does not establish stationarity; a small parameter update can reflect a restrictive step rather than a small gradient. Conversely, a small scaled gradient gives first-order evidence but does not prove that the objective was evaluated correctly. Compare these signals with the termination reason and the final estimates, and investigate disagreements rather than choosing the most reassuring one.

A common mistake is to compare raw gradients across parameters and conclude that the numerically largest one is the main problem, or to treat a software “converged” message as conclusive. Raw values depend on units, while a termination label may refer to a different stopping rule. The correction is to state the objective, parameter scales, scaling formula, constraint treatment, and tolerance, then calculate or verify the corresponding final criterion. A pass supports approximate first-order stationarity under those choices; it is not a general guarantee of a successful model fit.