How to check and improve optimization

The challenge

Parameter optimization of ODE models is challenging because

• there is no analytical solution of the ODEs, thus the objective function (and its derivatives) can only be evaluated numerically, i.e. with limited accuracy
• the objective function depends non-linearly on the parameters
• models might be large, i.e. have many parameters
• the amount of available data and observables is typically restricted which yields non-identifiability
• usually only rough knowledge is available about parameter ranges
• additional numerical challenges like events have to be accounted for
• constraints might be considered

It is therefore essential to ensure that optimization works reliably.

Guidelines for reliable optimization

• Repeat optimization with different intial guesse, e.g. apply multistart-optimization with arFitLHS in order to guarantee that
• • the optimization algorithms converges (at least locally), i.e. an optimum is found repeatedly
• • an optimum is indeed (likeli) the global one
• Optimize the parameters by default at the log-scale (indicated in D2D by ar.qLog10). This in particular concerns strictly positive parameters, e.g. parameters that represent initial values and rate constants. Analogously, initial guesses and priors should be drawn or defined at the log-scale.
• Specify lower and upper bounds (indicated in D2D by ar.lb and ar.ub). This improves the convergence of lsqnonlin (the default optimization algorithm in D2D) and other optimization algorithms and prevents integration errors.
• Allow several orders of magnitude, i.e. use non-restritive lower and upper bounds for the parameters
• Don't use finite differences for calculating derivatives or let the optimization algorithm do this internally. Instead, D2D exploits the so-called sensitivity equations.
• Check whether optimization indeed works reliably

Checks for reliable optimization

• Convergence can be checked by multi-start optimization e.g. by arFitLHS or arLocalLHS.
• Convergence problems can be seen usually in likelihood profiles, i.e. it should be checked whether the profiles are suffiently smooth
• Look at the output of the optimization algorithm during a fit. This can be done by the option ar.config.optim.Display = 'iter'. Convergence problems

Where can optimization problems come from?

• Michalis-Menten and Hill-equations at the right hand side of the ODEs are difficult to be optimized because the derivative is close to zero if the dynamics (x) if far from K_D. The K_D constants can be put manually in the same range as x.
• Integration over long time intervals can cause problems. If the data convers a time interval [t1,t2] with t1>>0 then consider shifting the times to t1=0.
• Think about the integration tolerances ar.config.atol and ar.config.rtol. Both tolerances are usually linked with an OR, i.e. integration is performed in a way that either the absolute or relative tolerance is satisfied. arCheckODETolerances indicates how the residuals, the objective function etc changes if the tolerances are altered. Too small tolerances cause integration problems, to large tolerance decrease the performance of optimization.
• If states, inputs or parameters become negative and this is not intended, integration and optimization usally breaks down. Negative states due to integration errors are by default prevented by ar.model.qPositiveX(:)=1. Negative values for splines can be enforced by using strictly positive splines (spline_pos3, spline_pos4, spline_pos5, spline_pos10, see arInputFunctionsC.c)
• Because of the course of dimensionality, parameters spaces can only be very sparsely sampled. Thus, for models with many parameters arFitLHS might require large sample sizes. Hence, it might be inefficient to run global multi-start optimization e.g. via arFitLHS "every time". Try to combine it with arLocalLHS.
• pleIterativeUntilConvergence performs profile likelihood calculation until no better optimum is found during calculation of the profiles.
• If error parameters are fitted, try to prevent that an error is fitted close to zero.
• Check initial conditions and exploit steady state equations where it is possible.

Literature

Lessons Learned from Quantitative Dynamical Modeling in Systems Biology