Why Compare Optimizers?
A GLMM fit is obtained through numerical optimization, and an optimizer can stop for different reasons: it may have reached a stable minimum, exhausted its allowed steps, or encountered numerical difficulty. A message such as “converged” is useful evidence, but it does not by itself establish that the reported estimates are the best solution available for the specified model. Comparing optimizers provides a practical check: if independent algorithms reach nearly the same estimates, objective value, and small gradient, confidence in the numerical solution increases.
The comparison is meaningful only when the optimizers are solving the same numerical problem. In particular, the data, model, fixed-effect and random-effect structure, estimation criterion, approximation to that criterion, and parameter constraints must agree. A difference between estimates from distinct likelihood approximations is not, by itself, evidence that an optimizer failed.
What Must Be Compared
Write the objective being minimized as \(f(\theta)\), where \(\theta\) collects the model parameters. For a likelihood-based fit, \(f\) is often a negative log likelihood or a negative log likelihood computed using a specified approximation. Some software reports a transformed or rescaled version of this criterion. That is acceptable for comparing optimizers only if the same version, including any constants that affect reported values, is used for every fit.
Compare parameter estimates on consistent scales. Fixed-effect coefficients should have the same predictor coding and units. Variance components should be compared as variances with variances, or standard deviations with standard deviations; mixing the two scales creates a misleading comparison. If an algorithm works with transformed parameters, convert its estimates to the model’s reported scale before assessing practical similarity. Also check that both outputs use the same parameter ordering.
The objective values answer whether the candidate fits have similar numerical criterion values. For minimization, a smaller value is better for that exact objective. The gradient, \(\nabla f(\theta)\), records the local rate and direction of change of the objective with respect to the parameters. Its components depend on parameter scaling: a gradient norm calculated for parameters measured in very different units can be hard to interpret. Use the same scaling and gradient definition across fits, and inspect componentwise values as well as any reported norm.
Finally, read termination statuses as descriptions of how each run ended, not as rankings of scientific validity. A status indicating convergence is reassuring when estimates and diagnostics agree. A status indicating a step limit or failure deserves investigation, even if its estimates look plausible. Conversely, two “successful” statuses do not make materially different solutions equivalent.
Worked Example: Two Optimizers Reach Similar Fits
Suppose two optimizers fit the same hypothetical random-intercept logistic GLMM using the same data and the same approximate negative log likelihood. Both report estimates on the same scale. Optimizer A gives \(\hat{\beta}_0=-0.420\), \(\hat{\beta}_1=0.310\), and a random-intercept standard deviation of \(0.580\); its objective is \(128.430\), and its scaled gradient norm is \(0.0018\). Optimizer B gives \(-0.419\), \(0.309\), and \(0.581\), respectively; its objective is \(128.428\), and its scaled gradient norm is \(0.0015\). Both report successful termination.
The objective difference is \[ 128.430-128.428=0.002. \] The fixed-effect differences are \(0.001\) for the intercept and \(0.001\) for the slope, while the standard-deviation difference is also \(0.001\). These estimates are practically close at the displayed precision, the objective values are close, and both gradient norms are small in the stated scaling. The evidence is consistent with both optimizers locating the same numerical solution. It is not a mathematical proof that no other minimum exists.
The checks are conditional on the stated setup. If Optimizer B used a different likelihood approximation, the objective difference and estimate differences would combine algorithmic and modeling-computation differences, so this comparison would not isolate optimizer behavior.
Why a Small Gradient Is Relevant
For an unconstrained differentiable objective, a local minimum in the interior must have zero gradient. This gives a reason to inspect gradients, but a small gradient alone is not enough: it can occur in a flat region, depend on parameter scaling, or be reported with a convention that differs across software.
The basic descent argument also explains why a nonzero gradient at an unconstrained interior point is evidence that the point is not a local minimum. Let \(g=\nabla f(\theta)\), with \(g\ne 0\). Differentiability gives the first-order expansion along the direction \(-g\):
Because \(o(\varepsilon)/\varepsilon\) tends to zero as \(\varepsilon\) tends to zero, the remainder is smaller in magnitude than, for example, \(\varepsilon\lVert g\rVert^2/2\) for sufficiently small positive \(\varepsilon\). Therefore,
for sufficiently small positive \(\varepsilon\). Thus a descent direction exists. This argument uses differentiability; it does not require an \(O(\varepsilon^2)\) remainder. In practice, GLMM parameter constraints and transformations also matter: at a constrained boundary, the unconstrained gradient need not be zero at a constrained optimum. Interpret the gradient using the optimizer’s parameterization and constraint handling.
Worked Example: A Lower Objective Does Not Settle a Disagreement
For a hypothetical GLMM, Optimizer A reports objective \(85.200\), a scaled gradient norm of \(0.42\), and a message that its iteration limit was reached. Optimizer B reports objective \(84.700\), a scaled gradient norm of \(0.003\), and successful termination. The fixed-effect slope estimates are \(0.18\) and \(0.46\), respectively, in log-odds per one-unit increase in the predictor.
Since both minimize the same objective, Optimizer B’s reported value is lower by \[ 85.200-84.700=0.500. \] Its gradient norm is also much smaller, and its termination status is more reassuring. Taken together, these checks favor B’s candidate as the better-supported numerical fit. The slope estimates differ by \[ 0.46-0.18=0.28 \] log-odds per predictor unit, which is not a negligible discrepancy to dismiss as rounding.
The appropriate conclusion is not that B is automatically correct. A’s failure to finish suggests that its output should not be treated as a competing converged solution. Refit or investigate the discrepancy using the same model and objective, and check whether independent runs reproduce B’s estimates and diagnostics. A lower objective is evidence about fit to the criterion, not proof that all numerical or modeling issues have been resolved.
When Objective Values Agree but Estimates Do Not
Different estimates can have almost identical objective values when the objective is relatively flat in some direction. In that situation, optimizer agreement on the objective is not the same as agreement on parameters. This matters when the scientific interpretation depends on a particular coefficient or variance component. Compare the actual estimates and their implications, not just a printed objective value rounded to a few decimal places.
Worked Example: Nearly Equal Objective Values Along a Flat Direction
Consider an illustrative two-parameter objective, used here to demonstrate the comparison problem rather than as a particular GLMM likelihood:
One optimizer returns \((a,b)=(1.00,1.00)\). Substitution gives \[ f(1,1)=40+(2-2)^2+0.0001(0)^2=40. \] A second returns \((1.10,0.90)\), for which \[ f(1.10,0.90)=40+(2-2)^2+0.0001(0.20)^2 =40+0.000004=40.000004. \] The objective values differ by only \(0.000004\), yet each parameter differs by \(0.10\). If \(a\) and \(b\) represented coefficients that were individually important to report, that difference could matter despite the negligible objective change.
The gradient is \[ \frac{\partial f}{\partial a}=2(a+b-2)+0.0002(a-b),\qquad \frac{\partial f}{\partial b}=2(a+b-2)-0.0002(a-b). \] At \((1.10,0.90)\), these derivatives are \(0.00004\) and \(-0.00004\). The small values are consistent with a very shallow direction, not necessarily with the parameters being precisely determined. In an actual GLMM, inspect the model-specific diagnostics and parameter uncertainty rather than relying on this illustrative objective’s form.
A Practical Comparison Workflow
Verify that the data, model, estimation criterion, approximation, parameter constraints, and objective definition are identical.
Put estimates in the same parameter order and on the same natural scales. Record the objective convention, gradient scaling, and termination message for each run.
Assess parameter differences, objective values, gradient components or norms, and termination statuses together. Do not use a status or objective value as a substitute for the other checks.
Repeat runs or use another suitable optimizer under the same setup. Check whether the lower-objective candidate also has a credible gradient and termination status, and whether reported estimates are stable enough for their intended interpretation.
State which fits agreed, what differed, and what checks were performed. Describe numerical agreement as evidence of stability, not as proof of a unique global solution.
Diagnostics, Alternatives, and Limitations
A useful comparison table can place each optimizer in a row and include the objective value, key fixed-effect estimates, variance-component estimates, gradient diagnostics, and termination status. Large discrepancies in any column deserve explanation. In particular, do not infer agreement from rounded objective values alone; retain enough digits to see meaningful differences. If the software reports gradients in different parameterizations, those numbers are not directly comparable until the scales and definitions are made consistent.
Using a second optimizer is one diagnostic among several. Restarting an optimizer from different initial values can test whether the same region is reached, while checking scaled gradients and the Hessian assesses other aspects of the fitted solution. These approaches complement each other: repeated starts using the same algorithm may reveal sensitivity to starting values, whereas different algorithms may reveal sensitivity to optimization strategy. Neither guarantees that every possible solution has been found.
The comparison has limits. Algorithms can share the same numerical error or converge to the same non-global stationary point. A nearly flat objective can make estimates differ while objective values remain close. A boundary-constrained optimum may require constraint-aware interpretation of gradients. And when the fits use different approximations or different objectives, apparent disagreement cannot be attributed to optimizers alone. Optimizer comparison assesses numerical consistency for the specified fitting procedure; it does not establish that the model is scientifically appropriate.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two optimizers fit the same model but use different likelihood approximations. Can their objective values directly establish which optimizer did better? Explain.
- For minimization, Optimizer A reports objective \(52.6\), gradient norm \(0.30\), and an iteration-limit status. Optimizer B reports objective \(52.4\), gradient norm \(0.004\), and successful termination. What is the most defensible comparison?
- Why can two fits have almost equal objective values but noticeably different parameter estimates?
- What does a nonzero gradient imply at an unconstrained, differentiable interior point, and why does this conclusion require care at a constrained boundary?
Solutions. (1) No. The objective values target different approximations, so the comparison confounds optimizer behavior with a change in the numerical objective. (2) B is better supported by the lower objective, smaller gradient norm, and successful termination, but the evidence does not prove it is the unique or global solution; A’s run did not finish successfully. (3) The objective may be shallow in a direction, so moving parameters appreciably changes the criterion very little. Parameter stability must be assessed as well as objective agreement. (4) If the gradient is nonzero, the negative-gradient direction decreases a differentiable unconstrained objective for a sufficiently small step, so the point is not an interior local minimum. At a constrained boundary, that direction may be infeasible; a constrained optimum need not have zero unconstrained gradient.