The Impact of Noise on Evaluation Complexity: The Deterministic Trust-Region Case

Abstract

Intrinsic noise in objective function and derivatives evaluations may cause premature termination of optimization algorithms. Evaluation complexity bounds taking this situation into account are presented in the framework of a deterministic trust-region method. The results show that the presence of intrinsic noise may dominate these bounds, in contrast with what is known for methods in which the inexactness in function and derivatives’ evaluations is fully controllable. Moreover, the new analysis provides estimates of the optimality level achievable, should noise cause early termination. Numerical experiments are reported that support the theory. The analysis finally sheds some light on the impact of inexact computer arithmetic on evaluation complexity.

Appendix : Details of the Proof of Theorem 5.5

We follow the argument of [17, proof of Theorem 3.8] (adapting the bounds to the new context) and derive an upper bound on the number of derivatives’ evaluations. This requires counting the number of additional derivative evaluations caused by successive tightening of the accuracy threshold $\zeta _{d,i_\zeta }$. Observe that repeated evaluations at a given iterate $x_k$ are only needed when the current value of this threshold is smaller than used previously at the same iterate $x_k$. The $\{\zeta _{d,i_\zeta }\}$ are, by construction, linearly decreasing with rate $\gamma _\zeta$. Indeed, $\zeta _{d,i_\zeta }$ is initialized to $\zeta _{d,0}\le \kappa _\zeta$ in Step 0 of the TR q DAN algorithm, decreased each time by a factor $\gamma _\zeta$ in (2.13) in the CHECK invoked in Step 1.2 of Algorithm 3.1, down to the value $\zeta _{d,i_\zeta }$ which is then passed to Step 2, and possibly decreased there further in (2.13) in the CHECK invoked in Step 2.1 of the STEP2 algorithm, again by successive multiplication by $\gamma _\zeta$. We now use (4.14) in Lemma 4.1 and (3.8) in Lemma 3.1 to deduce that, even in the absence of noise, $\zeta _{d,i_\zeta }$ will not be reduced below the value

$$\begin{aligned}{} & {} \min \left[ \frac{\omega }{4}\,\varsigma \,\epsilon _j\,\frac{\delta _k^{j-1}}{j!}, \frac{\omega }{8(1+\omega )\max [1,\Delta _{\max }^j]}\,\epsilon _j\,\frac{\delta _k^j}{j!}\right] \nonumber \\{} & {} \ge \frac{\varsigma \,\omega }{8(1+\omega )\max [1,\Delta _{\max }^j]} \,\epsilon _j\,\frac{\delta _k^j}{j!} \end{aligned}$$

(A.1)

at iteration k. Now define

$$\begin{aligned} \kappa _\textrm{acc} {\mathop {=}\limits ^\textrm{def}}\frac{\varsigma \omega (\varsigma \kappa _{\delta })^q}{8(1+\omega )\max [1,\Delta _{\max }^j]} \le \frac{\varsigma \omega }{8(1+\omega )\max [1,\Delta _{\max }^j]} \,\frac{(\varsigma \kappa _{\delta })^j}{j!}, \end{aligned}$$

so that (5.17) implies that

$$\begin{aligned} \kappa _\textrm{acc}\epsilon _{\min }^{q+1} \le \frac{\varsigma \omega \,\epsilon _j}{8(1+\omega )\max [1,\Delta _{\max }^j]} \,\frac{\delta _k^j}{j!}. \end{aligned}$$

We also note that conditions (2.16) and (2.13) in the CHECK algorithm impose that any reduced value of $\zeta _{d,i_\zeta }$ (before termination) must satisfy the bound $\zeta _{d,i_\zeta } \ge \vartheta _d$. Hence, the bound (A.1) can be strengthened to be

$$\begin{aligned} \max \left[ \vartheta _d, \kappa _\textrm{acc}\epsilon _{\min }^{q+1}\right] . \end{aligned}$$

Thus, no further reduction of the $\zeta _{d,i_\zeta }$, and hence no further approximation of $\{\overline{\nabla _x^j f}(x_k)\}_{j=1}^q$, can possibly occur in any iteration once the largest initial absolute error $\zeta _{d,0}$ has been reduced by successive multiplications by $\gamma _\zeta$ sufficiently to ensure that

$$\begin{aligned} \gamma _\zeta ^{i_\zeta }\zeta _{d,0} \le \gamma _\zeta ^{i_\zeta }\kappa _\zeta \le \max [\vartheta _d,\kappa _\textrm{acc} \epsilon _{\min }^{q+1}], \end{aligned}$$

(A.2)

the second inequality being equivalent to asking

$$\begin{aligned} i_\zeta \log (\gamma _\zeta ) \le \max \left[ \log (\vartheta _d), (q+1)\log \left( \epsilon _{\min }\right) + \log (\kappa _\textrm{acc})\right] - \log \left( \kappa _\zeta \right) , \end{aligned}$$

(A.3)

where the right-hand side is negative because of the inequalities $\kappa _\textrm{acc}<1$ and $\max [\epsilon _{\min }^{q+1},\vartheta _d] \le \kappa _\zeta$ (imposed in the initialization step of the TR q EDAN algorithm). We now recall that Step 1 of this algorithm is only used (and derivatives evaluated) after successful iterations. As a consequence, we deduce that the number of evaluations of the derivatives of the objective function that occur during the course of the TR p DAN algorithm before termination is at most

$$\begin{aligned} |\mathcal{S}_k| + i_{\zeta ,\max }, \end{aligned}$$

(A.4)

i.e., the number iterations in (5.16), plus

$$\begin{aligned} \begin{array}{lcl} i_{\zeta ,\max } &{} \!\! {\mathop {=}\limits ^\textrm{def}}\!\! &{} \left\lfloor \frac{\displaystyle 1}{\displaystyle \log (\gamma _\zeta )} \max \left\{ \log \left( \frac{\displaystyle \vartheta _d}{\displaystyle \zeta _{d,0}}\right) , (q+1)\log \left( \epsilon _{\min }\right) +\log \left( \frac{\displaystyle \kappa _\textrm{acc}}{\displaystyle \zeta _{d,0}}\right) \right\} \right\rfloor \\ &{} \!\!< \!\! &{} \frac{\displaystyle 1}{\displaystyle |\log (\gamma _\zeta )|} \left\{ \left| \log \left( \frac{\displaystyle \vartheta _d}{\displaystyle \zeta _{d,0}}\right) \right| + \,(q+1)\left| \log \left( \epsilon _{\min }\right) \right| + \left| \log \left( \frac{\displaystyle \kappa _\textrm{acc}}{\displaystyle \zeta _{d,0}}\right) \right| \right\} +1, \end{array} \end{aligned}$$

the largest value of $i_\zeta$ that ensures (A.3). Adding one for the final evaluation at termination, this leads to the desired evaluation bound (5.12) with the coefficients

$$\begin{aligned}{} & {} \kappa ^D_{{\mathrm{\mathsf TRqEDAN}}} {\mathop {=}\limits ^\textrm{def}}\frac{q+1}{|\log \gamma _\zeta |} \;\; \hbox {and} \;\;\\{} & {} \kappa ^E_{{\mathrm{\mathsf TRqEDAN}}} {\mathop {=}\limits ^\textrm{def}}\frac{\displaystyle 1}{\displaystyle |\log (\gamma _\zeta )|} \left\{ \left| \log \left( \frac{\displaystyle \kappa _\textrm{acc}}{\displaystyle \zeta _{d,0}}\right) \right| + \left| \log \left( \frac{\displaystyle \vartheta _d}{\displaystyle \zeta _{d,0}}\right) \right| \right\} +2. \end{aligned}$$