SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 3: The detailed parameters for all the settings (Setting (A)-(F)).
Parameters	2×2(A)	2×5(A)	2×10(A)	3×2(A)	3×5(A)	3×10(A)
Menu Size s	32	64	64	256	256	256
Temperature τ	5	5	5	5	1	1
Iterations	3000	3000	3000	3000	8000	8000
Parameters	3×2(B)	4×2(B)	5×2(B)	6×2(B)	7×2(B)	8×2(B)
Menu Size s	64	64	64	64	128	256
Temperature τ	3	3	3	3	1	1
Iterations	3000	3000	3000	8000	8000	8000
Parameters	2×5(C)	3×10(C)	5×5(C)	3×1(D)	1×2(E)	1×2(F)
Menu Size s	128	512	1024	16	128	40
Temperature τ	10	10	50	10	10	10
Iterations	3000	2000	2000	2000	2000	2000
as in the proof of DSIC.
$$
\begin{array}{l} u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {v} _ {k}, B _ {- k}\right); X, Y\right) \\ = u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {v} _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) \\ = v _ {k} \left(A ^ {*}\right) - p _ {k} \left(\left(v _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) \\ = v _ {k} \left(A ^ {*}\right) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A _ {- k} ^ {*}\right) + \lambda \left(A _ {- k} ^ {*}\right)\right) + \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A ^ {*}\right) + \lambda \left(A ^ {*}\right)\right) \\ = \frac {1}{w _ {k}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} \left(A ^ {*}\right) + \lambda \left(A ^ {*}\right)\right) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A _ {- k} ^ {*}\right) + \lambda \left(A _ {- k} ^ {*}\right)\right) \\ \stackrel {(1)} {\geq} \frac {1}{w _ {k}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} (A ^ {*}) + \lambda (A ^ {*})\right) - \frac {1}{w _ {k}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} (A _ {- k} ^ {*}) + \lambda (A _ {- k} ^ {*})\right) \\ \begin{array}{c} (2) \\ \geq 0, \end{array} \\ \end{array}
$$
where (1) holds by  $w_{k}\geq 0$  and  $v_{k}(\cdot)\geq 0$ , and (2) holds by the definition of  $A^{*}$ .
B Further Implementation Details
AMenuNet. For all settings, the learning rate is  $3 \times 10^{-4}$ . We also apply a linear warm-up to the learning rate from  $1 \times 10^{-8}$  to  $3 \times 10^{-4}$  during the first 100 iterations. Totally in AMenuNet there are two softmax temperatures. Denote by  $\tau_{A}$  the one in the softmax function for deciding social-welfare-maximizing allocation and  $\tau$  the one used in Menu Layer to generate allocations. We set  $\tau_{A}$  as 500 for all settings. We train the models for a maximum of 8000 iterations with  $2^{15} = 32784$  new samples each iteration. For those we train more than 3000 iterations, we manually reduce the learning rate to  $5 \times 10^{-5}$  after 3000 iterations. We test the model on a fixed test set with 100000 samples. The menu size and  $\tau$  varies in different settings, and we present these numbers in 3. For classic auctions, including Setting (C)-(F), we embed each bidder and item ID into 16-dimension space. For encode layer, the output channel of the first  $1 \times 1$  convolution in both the input layer and interaction modules are set to 64. The number of interaction modules is 3 or 5, and in each interaction module we adopt transformer with 4 heads and 64 hidden nodes. The boost layer is a two-layer fully-connected network with ReLU activation. To enable the out-of-setting generalization ability, we set both its channel and the hidden size as menu size.
Figure 4: Revenue results of AMenuNetunder different menu sizes.
Table 4: The experiment results of average revenue in out-of-setting generalization experiments.
Method	2×2(A)	2×3(A)	2×4(A)	2×5(A)	2×6(A)	2×7(A)	2×8(A)	2×9(A)	2×10(A)
Item-Myerson	0.4265	0.6405	0.8559	1.0700	1.2861	1.4993	1.7111	1.9227	2.1398
Trained on 2×2	0.4380	0.6371	0.8401	1.0503	1.2606	1.4706	1.6801	1.8900	2.0935
Trained on 2×5	0.1212	0.4593	0.8286	1.1013	1.3193	1.5357	1.7579	1.9850	2.2034
Method	2×2(B)	3×2(B)	4×2(B)	5×2(B)	6×2(B)	7×2(B)	8×2(B)	9×2(B)	10×2(B)
Item-Myerson	0.4274	0.5580	0.6584	0.7366	0.8004	0.8535	0.8990	0.9386	0.9696
Trained on 5×2	0.4800	0.6120	0.6946	0.7547	0.8005	0.8346	0.8603	0.8815	0.8975
Trained on 8×2	0.4656	0.5972	0.6868	0.7524	0.8076	0.8519	0.8902	0.9247	0.9542
Method	3×2(C)	3×3(C)	3×4(C)	3×5(C)	3×6(C)	3×7(C)	3×8(C)	3×9(C)	3×10(C)
Item-Myerson	1.0628	1.5941	2.1256	2.6570	3.1883	3.7197	4.2512	4.7825	5.3140
Trained on 3×10	1.0221	1.5686	2.1433	2.7365	3.3267	3.9116	4.4776	5.0440	5.5799
Trained on 5×10	1.0366	1.5947	2.1766	2.7630	3.3398	3.9034	4.4598	5.0098	5.5498
Item-Myerson. There are no parameters in Myerson auction and VCG auction. We just implement the classic form of these two mechanisms. For Item-Myerson, we first compute the virtual value  $\tilde{v}_i(v_i)$  according to  $\tilde{v}_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$ , where  $F_i$  and  $f_i$  are the cdf and pdf function of  $i$ 's value distribution. Then allocate the item to the bidder with the highest virtual value if it is greater than 0. The payment by the winning bidder is equal to the minimal bid that she would have had to make in order to win.
VCG. VCG auction is a special case of AMA. It is equivalent to AMA by setting  $\mathcal{A}$  as all deterministic allocations,  $\pmb{w}$  as 1 and  $\lambda$  as 0.
Lottery AMA. We use the implementation of Curry et al. [2023]. The hyperparameters and the results are mainly consistent with that reported in Curry et al. [2023]. We train all models for 10000 steps with  $2^{15}$  newly generated samples. The learning rate is 0.001 and the softmax temperature is 100. We set menu size as  $\{256,4096,2048,16,40,40\}$  for the six experiments in Setting (C)-(F) respectively.
CITransNet and RegretNet. We follow the same hyperparameters in Duan et al. [2022] and Dütting et al. [2019].
C Further Experiments Results
Impact of Menu Size. In Figure 4, we present the experimental results of AMenuNetunder various menu sizes. We conduct such experiment in setting  $2 \times 10(\mathrm{A})$ ,  $3 \times 5(\mathrm{C})$  and  $3 \times 10(\mathrm{C})$ . Generally, a larger menu, while entailing higher computational costs, can lead to increased revenue.
Out-of-Setting Generalization. In Table 4, we present the detailed numerical results of our out-of-setting generalization experiments.
Figure 5: The top-10 allocations (with respect to winning rate) and the corresponding boosts among 100,000 test samples in  $3 \times 10(\mathrm{C})$ , with menu size  $s = 1024$ .
Winning Allocations. We further present the top-10 winning allocations of  $3 \times 10(\mathrm{C})$  in Figure 5. The experimental results maintain the same phenomenon: the top-10 winning allocations are all deterministic.