SlowGuess/NeurIPS_2023_OCR huggingface dataset:
[38] Guru Guruganesh, Jon Schneider, and Joshua R Wang. Contracts under moral hazard and adverse selection. In EC 2021, pages 563-582, 2021.
[39] Dylan Hadfield-Menell and Gillian K. Hadfield. Incomplete contracting and AI alignment. In AIES 2023, pages 417-422, 2019.
[40] Chien-Ju Ho, Aleksandrs Slivkins, and Jennifer Wortman Vaughan. Adaptive contract design for crowdsourcing markets: Bandit algorithms for repeated principal-agent problems. Journal of Artificial Intelligence Research, 55:317-359, 2016.
[41] Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review, 101(6):2590-2615, 2011.
[42] Jon Kleinberg and Manish Raghavan. Algorithmic classification and strategic effort. ACM SIGcom Exchanges, 18(2):45-52, 2020.
[43] Jon M. Kleinberg and Manish Raghavan. How do classifiers induce agents to invest effort strategically? ACM Transactions on Economics and Computation, 8(4):19:1-19:23, 2020.
[44] Xiaoyang Liu and Jinde Cao. Robust state estimation for neural networks with discontinuous activations. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40(6):1425-1437, 2010.
[45] David Martimort and Jean-Jacques Laffont. The theory of incentives: The principal-agent model. Princeton University Press, 2009.
[46] Paolo Massa, Sara Garbarino, and Federico Benvenuto. Approximation of discontinuous inverse operators with neural networks. Inverse Problems, 38(10):105001, 2022.
[47] Stuart Mitchell, Michael OSullivan, and Iain Dunning. Pulp: a linear programming toolkit for python. The University of Auckland, Auckland, New Zealand, 65, 2011.
[48] Tong Mu, Stephan Zheng, and Alexander Trott. Solving dynamic principal-agent problems with a rationally inattentive principal. arXiv preprint arXiv:2202.01691, 2022.
[49] Florian A Potra and Stephen J Wright. Interior-point methods. Journal of Computational and Applied Mathematics, 124(1-2):281-302, 2000.
[50] Frank Rosenblatt. The perceptron: a probabilistic model for information storage and organization in the brain. Psychological review, 65(6):386, 1958.
[51] Royal Swedish Academy of Sciences. Scientific background on the 2016 Nobel Prize in Economic Sciences, 2016.
[52] Bernard Salanie. The Economics of Contracts: A Primer. MIT press, 2017.
[53] Stephen J Wright. On the convergence of the Newton/log-barrier method. Mathematical Programming, 90:71-100, 2001.
[54] Bing Xu, Naiyan Wang, Tianqi Chen, and Mu Li. Empirical evaluation of rectified activations in convolutional network. arXiv preprint arXiv:1505.00853, 2015.
[55] Banghua Zhu, Stephen Bates, Zhuoran Yang, Yixin Wang, Jiantao Jiao, and Michael I. Jordan. The sample complexity of online contract design. In EC '23, page 1188. ACM, 2023.
A Proofs of Lemma 2-4
In Sec. 3, we study the geometric structure of optimal contracts by establishing four lemmas. While some of them have been stated and proved for linear contracts [32], and the first two lemmas, at least, are not surprising, we give formal proofs of these properties of the principal's utility and optimal contracts in the general case. An important property of the principal's utility function  $u^p$  that we state in Lemma 2 is that  $u^p$  can be a discontinuous function.
Proof. For a contract  $\pmb{f}$  on the boundary of two neighboring linear pieces  $\mu_i^p$  and  $\mu_{j\neq i}^p$ , the agent is indifferent between action  $a_{i}$  and  $a_{j}$  given  $\pmb{f}$ :  $\mu_i^a (\pmb {f}) = \mu_j^a (\pmb {f})$ . The principal's utility
$$
\begin{array}{l} \mu_ {i} ^ {p} (\boldsymbol {f}) = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ v _ {o} ] - \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ f _ {o} ] (9) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ v _ {o} ] - c (a _ {i}) - \left(\mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ f _ {o} ] - c (a _ {i})\right) (10) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ v _ {o} ] - c (a _ {i}) - \mu_ {i} ^ {a} (\boldsymbol {f}) (11) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ v _ {o} ] - c (a _ {i}) - \mu_ {j} ^ {a} (\boldsymbol {f}) (12) \\ = \mu_ {j} ^ {p} (\boldsymbol {f}) + \mathbb {E} _ {o \sim p \left(\cdot | a _ {i}\right)} [ v _ {o} ] - c (a _ {i}) - \left(\mathbb {E} _ {o \sim p \left(\cdot | a _ {j}\right)} [ v _ {o} ] - c (a _ {j})\right). (13) \\ \end{array}
$$
It is possible that
$$
\mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ v _ {o} ] - c (a _ {i}) \neq \mathbb {E} _ {o \sim p (\cdot | a _ {j})} [ v _ {o} ] - c (a _ {j}). \tag {14}
$$
Function  $u^p$  is discontinuous in this case.
Another property of the principal's utility function  $u^p$  that motivates our discontinuous neural networks is Lemma 3.
Proof. Suppose that the global optimal  $f^{*}$  is not on a boundary but is an interior point of a linear piece  $\mathcal{Q}_i$  (contracts in  $\mathcal{Q}_i$  encourage the agent to take action  $a_i$ ):
$$
\boldsymbol {f} ^ {*} \in \mathcal {Q} _ {i} - \partial \mathcal {Q} _ {i}, \tag {15}
$$
$$
\mathcal {Q} _ {i} = \cap_ {j \neq i} \Gamma_ {i, j}; \tag {16}
$$
$$
\Gamma_ {i, j} = \left\{\boldsymbol {f} \in \mathcal {F} \mid \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ f _ {o} ] - c (a _ {i}) \geq \mathbb {E} _ {o \sim p (\cdot | a _ {j})} [ f _ {o} ] - c (a _ {j}) \right\}, \forall j \neq i. \tag {17}
$$
$$
\boldsymbol {f} ^ {*} \in \left\{\boldsymbol {f} \in \mathcal {F} \mid \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ f _ {o} ] - c (a _ {i}) > \mathbb {E} _ {o \sim p (\cdot | a _ {j})} [ f _ {o} ] - c (a _ {j}), \forall j \neq i \right\}, \tag {18}
$$
where the inequality is strict. Let
$$
\epsilon_ {i, j} = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {*} \right] - c (a _ {i}) - \mathbb {E} _ {o \sim p (\cdot | a _ {j})} \left[ f _ {o} ^ {*} \right] + c (a _ {j}). \tag {19}
$$
It holds that  $\epsilon_{i,j} > 0, \forall j \neq i$ .
Now we consider the contract  $\pmb{f}^{\prime} = \pmb{f}^{*} - \delta p(\cdot |a_{i})$  for some small  $\delta > 0$ . For any  $a_{j} \neq a_{i}$ , we have
$$
\begin{array}{l} u ^ {a} \left(\boldsymbol {f} ^ {\prime}; a _ {i}\right) - u ^ {a} \left(\boldsymbol {f} ^ {\prime}; a _ {j}\right) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {*} - \delta p (o | a _ {i}) \right] - c (a _ {i}) - \mathbb {E} _ {o \sim p (\cdot | a _ {j})} \left[ f _ {o} ^ {*} - \delta p (o | a _ {i}) \right] + c (a _ {j}) \tag {20} \\ = \epsilon_ {i, j} - \delta \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ p (o | a _ {i}) ] + \delta \mathbb {E} _ {o \sim p (\cdot | a _ {j})} [ p (o | a _ {i}) ]. \\ \end{array}
$$
$$
0 <   \delta <   \min  _ {j \neq i} \frac {\epsilon_ {i , j}}{\mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ p (o | a _ {i}) ] - \mathbb {E} _ {o \sim p (\cdot | a _ {j})} [ p (o | a _ {i}) ]}, \tag {21}
$$
(where note the denominator is  $>0$ ), we have
$$
u ^ {a} \left(\boldsymbol {f} ^ {\prime}; a _ {i}\right) - u ^ {a} \left(\boldsymbol {f} ^ {\prime}; a _ {j}\right) > 0, \forall j \neq i, \tag {22}
$$
which means  $\pmb{f}^{\prime}$  incentivizes the agent to take action  $a_{i}$ . Therefore, for  $\delta$  in this range, the principal's utility given  $\pmb{f}^{\prime}$  is:
$$
\begin{array}{l} u ^ {p} \left(\boldsymbol {f} ^ {\prime}\right) = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ v _ {o} - f _ {o} ^ {*} + \delta p (o | a _ {i}) \right] \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ v _ {o} - f _ {o} ^ {*} \right] + \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ \delta p (o | a _ {i}) \right] \tag {23} \\ = u ^ {p} \left(\boldsymbol {f} ^ {*}\right) + \delta \sum_ {o} p ^ {2} \left(o \mid a _ {i}\right) \\ > u ^ {p} \left(\boldsymbol {f} ^ {*}\right). \\ \end{array}
$$
We thus find a contract  $f'$  that induces greater utility for the principal, contradicting with the fact that  $f^*$  is the global optimal contract. This finishes the proof.
Note that here we consider the boundary resulting from changes in the agent's best responses. We can extend the proof to cover another type of boundary, which pertains to the requirement that contracts are non-negative, by defining  $\mathcal{Q}_i$  to be  $\mathcal{Q}_i = \{\pmb {f}|\pmb {f}\geq 0\} \cap \Gamma_{i,1}\cap \dots \cap \Gamma_{i,i - 1}\cap \Gamma_{i,i + 1}\cap \dots$
Lemma 4 claims another property that may influence the design of DeLU networks.
Proof. We first prove the agent's utility function  $u^{a}(\pmb{f})$  is a convex function.
We need to prove that, for any two contracts  $\pmb{f}^{(1)} \in \mathcal{F}$  and  $\pmb{f}^{(2)} \in \mathcal{F}$ , it holds that  $u^p (\lambda \pmb{f}^{(1)} + (1 - \lambda)\pmb{f}^{(2)}) \leq \lambda u^p (\pmb{f}^{(1)}) + (1 - \lambda)u^p (\pmb{f}^{(2)})$ ,  $\forall \lambda \in [0,1]$ . Denote  $\pmb{d} = \pmb{f}^{(2)} - \pmb{f}^{(1)}$ . It suffices to prove that the derivative of  $u^p (\pmb{f}^{(1)} + \delta \pmb{d})$  with respect to  $\delta$  is a non-decreasing function for  $\delta \in [0,1]$ .
$$
\frac {\partial}{\partial \delta} u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta \boldsymbol {d}\right) = \frac {\partial}{\partial \delta} \left[ \mathbb {E} _ {o \sim p (\cdot | a _ {\delta})} \left[ f _ {o} ^ {(1)} + \delta d _ {o} \right] - c \left(a _ {\delta}\right) \right], \tag {24}
$$
where  $a_{\delta}$  is the agent's action given the contract  $\pmb{f}^{(1)} + \delta \pmb{d}$ .
Case 1: When  $a_{\delta}$  does not change,
$$
\frac {\partial}{\partial \delta} u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta \boldsymbol {d}\right) = \mathbb {E} _ {o \sim p (\cdot | a _ {\delta})} [ d _ {o} ] \tag {25}
$$
is a constant, which is a non-decreasing function.
Case 2: When  $a_{\delta}$  changes. Suppose that there exists  $\delta_1 \in [0,1]$  such that  $\pmb{f}^{(1)} + \delta_1 \pmb{d}$  is on the boundary of linear piece  $\mathcal{Q}_i$  and  $\mathcal{Q}_j$ . Let
$$
\delta_ {1} ^ {+} = \delta_ {1} + \epsilon ,
$$
$$
\delta_ {1} ^ {-} = \delta_ {1} - \epsilon , \tag {26}
$$
where  $\epsilon$  is a small number and
$$
\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {-} \boldsymbol {d} \in \mathcal {Q} _ {i};
$$
$$
\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d} \in \mathcal {Q} _ {j}. \tag {27}
$$
Because the agent is self-interested, it follows that  $a_{j}$  is the best response when the contract is  $f^{(1)} + \delta_1^+ d$ :
$$
u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d}\right) = u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d}; a _ {j}\right) > u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d}; a _ {i}\right). \tag {28}
$$
$$
\begin{array}{l} \left. \frac {\partial}{\partial \delta} u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta \boldsymbol {d}\right)\right| _ {\delta = \delta_ {1} ^ {+}} = \lim  _ {\epsilon \rightarrow 0} \frac {u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d}\right) - u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d}\right)}{\epsilon} (29) \\ > \lim  _ {\epsilon \rightarrow 0} \frac {u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d} ; a _ {i}\right) - u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d}\right)}{\epsilon}. (30) \\ \end{array}
$$
We now look at the two terms in the numerator of Eq. 30:
$$
\begin{array}{l} u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d}; a _ {i}\right) = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + \delta_ {1} ^ {+} d _ {o} \right] - c \left(a _ {i}\right) (31) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + (\delta_ {1} + \epsilon) d _ {o} \right] - c (a _ {i}) (32) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + \delta_ {1} d _ {o} \right] - c \left(a _ {i}\right) + \epsilon \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ d _ {o} \right] (33) \\ = u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d}\right) + \epsilon \mathbb {E} _ {o \sim p \left(\cdot \mid a _ {i}\right)} [ d _ {o} ], (34) \\ \end{array}
$$
$$
\begin{array}{l} u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d}\right) = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + \delta_ {1} d _ {o} \right] - c \left(a _ {i}\right) (35) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + \left(\delta_ {1} ^ {-} + \epsilon\right) d _ {o} \right] - c (a _ {i}) (36) \\ = \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ f _ {o} ^ {(1)} + \delta_ {1} ^ {-} d _ {o} \right] - c \left(a _ {i}\right) + \epsilon \mathbb {E} _ {o \sim p (\cdot | a _ {i})} \left[ d _ {o} \right] (37) \\ = u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {-} \boldsymbol {d}\right) + \epsilon \mathbb {E} _ {o \sim p (\cdot | a _ {i})} [ d _ {o} ]. (38) \\ \end{array}
$$
$$
\begin{array}{l} \left. \frac {\partial}{\partial \delta} u ^ {a} (\boldsymbol {f} ^ {(1)} + \delta \boldsymbol {d}) \right| _ {\delta = \delta_ {1} ^ {+}} > \lim  _ {\epsilon \rightarrow 0} \frac {u ^ {a} (\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {+} \boldsymbol {d} ; a _ {i}) - u ^ {a} (\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d})}{\epsilon} \\ = \lim  _ {\epsilon \rightarrow 0} \frac {u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} \boldsymbol {d}\right) - u ^ {a} \left(\boldsymbol {f} ^ {(1)} + \delta_ {1} ^ {-} \boldsymbol {d}\right)}{\epsilon} \tag {39} \\ = \left. \frac {\partial}{\partial \delta} u ^ {a} (\pmb {f} ^ {(1)} + \delta \pmb {d}) \right| _ {\delta = \delta_ {1} ^ {-}}. \\ \end{array}
$$
This finishes the proof that  $u^a$  is a convex function. Furthermore, we have that
$$
u ^ {p} (\boldsymbol {f}) = - u ^ {a} (\boldsymbol {f}) - c \left(a ^ {*} (\boldsymbol {f})\right) + \mathbb {E} _ {o \sim p \left(\cdot \mid a ^ {*} (\boldsymbol {f})\right)} \left[ v _ {o} \right], \tag {40}
$$
where  $-u^{a}(\pmb{f})$  is a concave function and  $-c(a^{*}(\pmb{f})) + \mathbb{E}_{o\sim p(\cdot |a^{*}(\pmb{f}))}[v_{o}]$  is a piecewise constant function with the value  $-c(a_{i}) + \mathbb{E}_{o\sim p(\cdot |a_{i})}[v_{o}]$  when  $\pmb{f}\in \mathcal{Q}_i$ .
B Why we use another network to generate the last-layer bias?
To model model the dependency of the last-layer bias on the activation pattern, we use a neural network, rather than a simpler, linear, and learnable function. The reason is that the bias does not always depend linearly on the activation pattern. Here is an example to illustrate this. There are two outcomes with values  $\mathbf{v} = [20,1]$ , four actions with costs  $\mathbf{c} = [1.0,2.1,2.3,4.7]$ , and the action-outcome transition kernel is
$$
P = \left[ \begin{array}{c c} 0. 2 1 1 & 0. 7 8 9 \\ 0. 3 9 8 & 0. 6 0 2 \\ 0. 4 3 0 & 0. 5 7 0 \\ 0. 6 8 4 & 0. 3 1 6 \end{array} \right].
$$
Suppose we consider linear contracts, where  $\mathbf{f} = \alpha \mathbf{v}$ ,  $\alpha > 0$ . Then the principal's utility function for different contracts is:
$$
u ^ {p} (\alpha) = \left\{ \begin{array}{l l} - 5 \alpha + 5 & 0. 2 <   \alpha <   0. 3 \\ - 8. 5 7 \alpha + 8. 5 7 & 0. 3 <   \alpha <   0. 4 \\ - 9. 1 7 \alpha + 9. 1 7 & 0. 4 <   \alpha <   0. 5 \\ - 1 4 \alpha + 1 4 & \alpha > 0. 5 \end{array} \right..
$$
Suppose that we have a 2-dimensional activation pattern, and the linear function converting activation patterns to the bias has parameters  $[b_1, b_2]$ . Then the bias for each of the four pieces would be  $0$ ,  $b_1$ ,  $b_2$ , and  $b_1 + b_2$ , respectively. The difference between each piece's bias needs to model the discontinuity at contract parameter  $\alpha = 0.3, 0.4, 0.5$ , but this is impossible with this linear model. To see this, we first assume that the piece  $0.2 < \alpha < 0.3$  has bias  $0$ . Then the differences of biases of the other 3 pieces would need to be 2.5, 2.86, and 5.28, which cannot be achieved with  $b_1$ ,  $b_2$ , and  $b_1 + b_2$ . It can be easily verified that the cases where other pieces have a bias of  $0$  are similar, demonstrating that a linear bias function cannot express the discontinuity. By contrast, appealing to a second network allows for non-linear dependency on activation, and can handle this problem.
Figure 8: Optimality (normalized principal utility, divided by the result obtained by the direct LP solver Oracle LP) and inference time of DeLU, Concave DeLU, and Oracle LP on increasing problem sizes.
C Introducing Concavity into the DeLU Network
From Lemma 4, the principal's utility function  $u^p$  is a summation of a concave function and a piecewise constant function. Further, our DeLU network, which is used to approximate the function  $u^p$ , can be written as
$$
\xi \left(\boldsymbol {f} _ {i}; \theta_ {\eta}, \theta_ {\zeta}\right) = \eta \left(\boldsymbol {f} _ {i}; \theta_ {\eta}\right) + \zeta \left(r \left(\boldsymbol {f} _ {i}\right); \theta_ {\zeta}\right). \tag {41}
$$
In particular,  $\zeta(r(\pmb{f}_i); \theta_{\zeta})$  is a piecewise constant function, because given an activation pattern  $r(\pmb{f}_i)$ ,  $\zeta$  is a constant. However, the first term in Eq. 41, in a general DeLU network, is an arbitrary function. This raises the question as to whether it is useful to further restrict the network architecture, constraining the sub-network  $\eta$  to be a concave function. In this section, we discuss how to introduce concavity into  $\eta$ , and how this restriction affects the performance of DeLU-based contract design.
C.1 Concave DeLU architecture
We can make  $\eta$  a concave function by (1) enforcing its weights (for all but the first layer) to be non-negative, and (2) taking the negation of its output. The first modification will make  $\eta$  a convex function because  $\eta$  uses ReLU activation, which is a convex and non-decreasing function. When the weights after a ReLU activation are non-negative,  $\eta$  becomes a composition of several convex, non-decreasing functions, which is still a convex function. Since the negation of a convex function is a concave function, the second modification will make  $\eta$  concave.
Formally, in Concave DeLU, the sub-network  $\eta$  calculates
$$
\boldsymbol {h} ^ {(L)} (\boldsymbol {x}) = | \boldsymbol {W} ^ {(L)} | \boldsymbol {R} ^ {(L - 1)} (\boldsymbol {x}) \left(\dots \left(| \boldsymbol {W} ^ {(2)} | \boldsymbol {R} ^ {(1)} (\boldsymbol {x}) \left(\boldsymbol {W} ^ {(1)} \boldsymbol {x} + \boldsymbol {b} ^ {(1)}\right) + \boldsymbol {b} ^ {(2)}\right) \dots\right) + \boldsymbol {b} ^ {(L)},
$$
$$
\eta (\boldsymbol {x}) = - | \boldsymbol {W} ^ {(L + 1)} | \boldsymbol {R} ^ {(L)} \boldsymbol {h} ^ {(L)} (\boldsymbol {x}), \tag {42}
$$
where  $\pmb{R}^{(k)}$  is a diagonal matrix with diagonal elements equal to the layer activation pattern  $\pmb{r}^{(k)}$ . The other components, as well as the training process, of Concave DeLU are the same as a DeLU network. In the next sub-section, we evaluate the performance of Concave DeLU.
C.2 Experiments on Concave DeLU networks
In Fig. 8, we compare Concave DeLU against DeLU and the direct LP solver (Oracle LP). For this, we fix DeLU and Concave DeLU to have the same size, and both use the LP inference algorithm. We test different problem sizes. In the first and second column, we set the number of actions  $n$  to 5 and 50, respectively, and increase the number of outcomes  $m$  from  $2^{1}$  to  $2^{8}$ . In the third and fourth column, we set the number of outcomes  $m$  to 5 and 50, respectively, and increase the number of actions  $n$  from  $2^{1}$  to  $2^{8}$ . For each problem size, the same 12 combinations of  $\alpha_{p}$  and  $\beta_{p}$  are tested. The median
performance as well as the first and third quartile (shaped area) of these 12 combinations are shown. The first row compares optimality, while the second row compares inference time efficiency. We again report normalized principal utilities when it comes to optimality.
A surprising result is that for most problem sizes, DeLU achieves better optimality than Concave DeLU. Although the function class represented by Concave DeLU is a better fit with  $u^p$ , it seems that the larger function class of DeLU aids optimization and leads to better performance. At the same time, it is somewhat surprising that Concave DeLU can reduce inference time substantially. Unlike DeLU, the inference time of Concave DeLU remains relatively stable when the problem size increases. We conjecture that this behavior can be attributed to the non-negativity constraint on network weights, which reduces the number of valid activation patterns and speeds up LP inference.
D Experimental Setups
D.1 Network architecture and training
We use a simple architecture for the DeLU network. In all experiments, the sub-network  $\eta$  has one hidden layer with 32 hidden units and ReLU activations. We deliberately restrict the size of this subnetwork to limit the number of valid activation patterns and speedup LP-based inference. However, this restriction may reduce the representational capacity of DeLU networks: it is in contrast to the common practice of overparameterization, which has contributed to the success of deep learning. To alleviate this concern, we employ a relatively larger network for the bias network  $\zeta$ . In our experiments,  $\zeta$  has a hidden layer with 512 (Tanh-activated) neurons.
The two sub-networks  $\eta$  and  $\zeta$  are trained in an end-to-end manner by the MSE loss (Eq. 5). The optimization is conducted using RMSprop with a learning rate of  $1\times 10^{-3}$ ,  $\alpha$  of 0.99, and with no momentum or weight decay. For the DeLU and the baseline ReLU networks, training samples are randomly shuffled in each of the training epochs.
D.2 Infrastructure
Across all experiments, DeLU, and the baseline ReLU networks are trained on a NVIDIA A100 GPU. Gradient-based inference is also parallelized on the A100 GPU. The direct LP solver (Oracle LP) and the LP-based inference algorithm are based on the linear programming toolkit PuLP [47], and we parallelize five LP solvers on CPUs.
In this section, we carry out experiments to study (1) using random samples to initialize gradient-based inference (Appx. E.1); and (2) the influence of cost correlation on the optimality of DeLU learners (Appx. E.2).
E.1 Gradient-based inference initialized with random samples
In Sec. 4.2.2, we introduce a gradient-based inference algorithm, and Alg. 1 gives the matrix-form expression of its parallel implementation. There is a choice in using Alg. 1, as to whether the input ("probe points")  $\mathbf{X}^{(0)} \in \mathbb{R}^{K \times m}$  is taken from the training set or a different, random sample set. In this section, we report the results of experiments to empirically compare these two setups.
We start from the same trained DeLU networks on small  $(n = 5, m = 16)$ , middle  $(n = 32, m = 50)$ , and large  $(n = 50, m = 128)$  problem sizes, where  $n$  is the number of actions and  $m$  is the number of outcomes. For each problem size, we consider 12 different combinations of  $(\alpha_p, \beta_p)$  as in other experiments and report the mean and variance of the performance. We run Alg. 1 with two different inputs  $X^{(0)}$ : Training Set uses  $50K$  training samples while Random Set uses  $50K$  randomly generated contracts. In Table. 1, we present the normalized principal utility (divided by the result obtained by Oracle LP) of these two setups.
We can observe that the optimality of these two setups are very close, especially when the problem size is large. We thus recommend running Alg. 1 initialized with the training set to reduce the possible time and memory overhead of generating a new random sample set.
Table 1: Optimality (normalized principal utility %) of two setups of gradient-based inference: using the training set (Training Set) or a random sample set (Random Set) as input  $\mathbf{X}^{(0)}$  of Alg. 1. Mean and variance over 12 different combinations of  $(\alpha_{p},\beta_{p})$  are shown. In this table, the problem size is defined by  $(n,m)$ , where  $n$  is the number of actions and  $m$  is the number of outcomes.
Problem size (n, m)	Training Set	Random Set
(5, 16)	95.29 ± 0.20	95.33 ± 0.19
(32, 50)	88.43 ± 0.97	88.46 ± 0.98
(50, 128)	94.28 ± 0.02	94.28 ± 0.02