# Learning Individualized Hyperparameter Settings
## 3 Algorithms and Problems
### Robust Tabu Search and the QAP
### Feature Selection

To abstract the details of each instance and to make instances with different dimensions compatible with a same learner, we used sets of descriptive statistics computed on each matrix, the same statistics for all matrices. We also computed sets of correlation statistics between matrices. The descriptive statistics included basic ones such as mean, standard deviation, IQR, etc., but also distribution fits for uniform (chi-square, Kolmogorov-Smirnov), for normal (chi-square, Shapiro-Wilk, Kolmogorov-Smirnov, Anderson-Darling), and for gamma (chi-square, Kolmogorov-Smirnov) distributions. Correlations quantified the average request vs. average capacity, cost/request correlation, etc. We are aware of contributions that have identified particularly significant complexity predictors for specific problems, such as ruggedness and flow dominance for the QAP [36], but we wanted to present here a general methodology that can be applied to any combinatorial optimization problem of interest, so we computed the same statistics on all matrices, regardless of the problem they referred to. This resulted in about 100 different statistics for each instance in the case of the QAP and about 70 in the case of the GAP, which were preprocessed for relevance and only the surviving ones were to be used as input for the learning module. Data filtering was done first by removing all statistics that varied too little over the training set (_low variance filter_), then we ran a PCA which suggested that 7 variables could account for more than 97% of the variance, both in the case of the GAP and QAP, but we could not use the principal components as we wanted to keep the original statistics. However, following the decreasing relevance order and the analysis of the heat map of the correlations between the variables, we were able to shortlist the set and reduce it to 11 variables in the case of the GAP and 14 in the case of the QAP. Figure 1 shows the heat map of surviving descriptive statistics for the case of the Lagrangian heuristic; blue balloons outline the subsets of highly correlated variables that were eventually represented by a single variable of each subset. ### Learning Lagrangian Heuristic Setting

All instances from the GAPLIB repository were generated under controlled settings and already separated into subsets of structurally similar instances. The main benchmarks from the literature, both included in the GAPLIB, are [PERSON]'s [37] and [PERSON]'s OR-library [38]. Both authors generated instances that were supposed to range from easy to difficult. We independently optimized the parameter setting on 4 different subsets, 2 derived from the biggest OR-library instances (sets 11 and 12) and 2 consisting of the hardest Yagiura instances (sets D and E) by means of the automatic configurator _irace_[12]. The other sets from these literature benchmarks were composed of instances too easy to solve and would have biased the parameter choice toward values of little interest for more complex tasks. [PERSON]'s OR-library instances were generated with costs \(c_{ij}\) as integers from the uniform distribution U(15, 25), requests \(q_{ij}\) from uniform distribution U(5, 25), and capacities were computed as \(b_{i}=0.8\sum_{j\in J}q_{ij}/m\). Set 11 had \(m=10\) and \(n=50\) (\(r=n/m=5\)) while set 12 had \(m=10\) and \(n=60\) (\(r=n/m=6\)). [PERSON]'s instances of type D had requests \(q_{ij}\) from uniform distribution U[1, 100], costs computed as \(c_{ij}=111-q_{ij}+e_{1}\), where \(e_{1}\) is from uniform distribution U[\(-10\), 10] and \(b_{i}=0.8\sum_{j\in J}q_{ij}/m\). Type E instances had requests \(q_{ij}=1-10 ln\;e_{2}\), with \(e_{2}\) from uniform distribution U(0, 1), costs \(c_{ij}=1000/a_{ij}-10e_{3}\), with \(e_{3}\) from uniform distribution U[0, 1] and capacities \(b_{i}=0.8\sum_{j\in J}q_{ij}/m\). This detailed knowledge of the generation procedure allowed us to generate additional instances that were structurally similar to the benchmark ones.