SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 2: Optimality (normalized principal utilities %) of DeLU networks under different combinations of  $(\alpha_{p},\beta_{p})$ . Mean and variance over 32 problem sizes are shown.
αp=0.5	αp=0.7	αp=0.9
βp=0.0	88.79±12.77	89.67±12.57	87.20±11.97
βp=0.3	93.94±11.16	93.28±11.48	87.94±13.45
βp=0.6	95.22±7.93	95.08±8.78	91.97±10.32
βp=0.9	97.23±6.62	90.69±18.95	96.43±7.45
E.2 Influence of correlated costs
Across our experiments, we test 12 different combinations of  $\alpha_{p}$  and  $\beta_{p}$ , where  $(\alpha_{p},\beta_{p}) \in \{0.5,0.7,0.9\} \times \{0,0.3,0.6,0.9\}$ . Recall that a greater  $\beta_{p}$  value means we put more weights on the independent cost, and a greater  $\alpha_{p}$  value indicates that the correlated cost is more close to the expected value of the action. It is interesting to investigate the influence of these two parameters on the performance of DeLU contract designers.
In Table 2, we show the optimality of DeLU learners (using the LP inference algorithm) under different  $\alpha_{p}$  and  $\beta_{p}$  values. For each  $(\alpha_{p},\beta_{p})$  combination, we test 24 different problem sizes:  $(m,n)\in \{25,50,100\} \times \{2,4,8,16,32,64,128,256\}$ . We give the median and standard deviation of these 24 instances. We observe that  $\beta_{p}$  exerts influence on the performance of DeLU networks: optimality of the learned contracts generally increases for greater values of  $\beta_{p}$ , whatever the value of  $\alpha_{p}$ , and we see that DeLU seems better suited to handle problems in which the costs of actions are relatively independent of the expected values of actions. This claim is also supported by the results regarding  $\alpha_{p}$ , where the optimality under  $\alpha_{p} = 0.5$  is typically better than those under other  $\alpha_{p}$  values for most  $\beta_{p}$  values. It will be interesting in future work to further study this phenomenon, and to see whether suitable modifications can be made to the DeLU architecture to further improve optimality in regimes with higher cost correlation.