# A Survey of Advances in Landscape Analysis for Optimisation
## 3 Advances in Landscape Analysis
### Sampling and Robustness of Measures
\begin{table}
\begin{tabular}{l l} \hline Technique 31: & **Population evolvability metrics by [PERSON] et al. [49].** \\ Year: & 2018 \\ Focus: & Evolvability of a population \\ Assumptions: & Assumes a population-based algorithm for sampling. \\  & Given a population of solutions and the set of neighbours (from one iteration of the algorithm), two metrics are defined: (1) _rep_ is the probability \\ Description: & that a population will evolve and is estimated by calculating the proportion of neighbours that are fitter than the best solution of the current \\  & population, and (2) _exp_ is the evolutionary ability of the population, which is a quantity that increases with the absolute fitness improvement of the \\ Result: & A single value _exp_ which is defined as _rep_\(\times\)_exp_, with a range of \([0,+\infty]\). \\ \hline Technique 32: & **Local multiobjective landscape features** by [PERSON] et al. [50] including earlier contributions [51; 52]. \\  & 2019 \\ Focus: & Evolvability for multiobjective optimisation \\ Assumptions: & Assumes a discrete search space. \\  & Given a sequence of solutions obtained through random walks and adaptive walks, features of the walk are derived from the sequence as a whole \\  & as well as the neighbourhood of solutions in terms of dominance and hypervolume improvement by neighbours. \\ Result: & 20 numerical values representing local features (17 from random walk sampling and 9 from adaptive walk sampling). \\ \hline Technique 33: & **Loss-gradient clouds** by [PERSON] et al. [19]. \\ Year: & 2020 \\ Focus: & Basins of attraction in neural network error landscapes \\ Assumptions: & Requires the numeric gradient of the loss function. \\  & A sample of loss values and gradient values is obtained based on a number of random, progressive gradient walks [53]. Stationary points in the \\  & sample are determined to be local minima, local maxima or saddle points based on local curvature derived from the eigenvalues of the Hessian \\ Description: & matrix. Stagnant sequences on the walk are detected by tracking the deviation in a smoothing of the error. Two quantities are measured: (1) the \\  & average number of times that stagnation was observed, and (2) the average length of the stagnant sequence. \\ Result: & A two-dimensional scatterplot of loss values against gradient values (loss-gradient cloud) and two metrics to estimate the number and extent of \\  & distinct-valued basins of attraction. \\ \hline \end{tabular}
\end{table} 
Table 1: Cont. Alternative approaches to traditional random sampling for landscape analysis in the continuous domain include Latin hypercube design sampling [59], progressive random walk sampling [46] and gradient-based walks for sampling error surfaces of neural networks [53]. Online sampling, where the samples are based on solutions encountered by an algorithm during search, has become a popular alternative to random sampling for landscape analysis [60; 61; 62]. Enhancements to online sampling for continuous spaces include introducing a mechanism for correcting the error produced by the sampling bias [63] and the introduction of path relinking [64]. In an experimental study, [PERSON] et al. [65] explored the effect that function translations had on landscape measures and found that translations could cause abrupt and severe changes in the values of some metrics. [PERSON] et al. [66] also found that a large number of ELA features [39] (technique 26 in Table 1) were not invariant to shifting and scaling. Finally, [PERSON] and [PERSON] [67] found that some landscape measures are very sensitive to the presence of noise in the fitness evaluation and that the error is difficult to correct for in an efficient way. In some real-world optimisation scenarios, the objective function is very expensive to evaluate (such as in simulation-based optimisation) and surrogate models are used to approximate the fitness function. [PERSON] et al.