SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Christoph Hertrich*
London School of Economics and Political Science, UK
c. hertrich@lse.ac.uk
Yixin Tao
ITCS, Key Laboratory of Interdisciplinary Research of Computation and Economics
Shanghai University of Finance and Economics, China
taoyixin@mail.shufe.edu.cn
Laszlo A. Vegh
1.vegh@lse.ac.uk
Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.
Auction design is a core problem in mechanism design, with immense applications in electronic commerce (such as sponsored search auctions) as well as in the public sector (such as spectrum auctions). In a revenue maximizing auction, the auctioneer needs to design a mechanism to allocate resources to buyers, and set prices in order to maximize the expected revenue. The buyers' preferences are private and they may behave strategically by misreporting them. For this reason, it is often desirable to devise dominant strategy incentive compatible (DSIC) and individually rational (IR) mechanisms. By definition, in a DSIC mechanism, it is a dominant strategy for the buyers to report the true valuations; in an IR mechanism, each participating truthful buyer receives a nonnegative payoff.
We focus on DSIC and IR mechanisms that maximize the expected revenue, assuming that the buyers' preferences are drawn from a distribution known to the auctioneer. A classical result of
*Moved to Université Libre de Bruxelles, Belgium, and Goethe-Universität Frankfurt, Germany, after submission of this article.
Myerson [1981] provides the optimal mechanism for the case of a single item and arbitrary number of buyers. Finding the optimal mechanisms for more general settings is a tantalizingly difficult problem. We refer the reader to the surveys by Rochet and Stole [2003], Manelli and Vincent [2007] and Daskalakis [2015] for partial results and references. In particular, no analytic solution is known even for two items and two buyers. Selling multiple items to a single buyer is computationally intractable [Daskalakis et al., 2014]. Already for two items and a single buyer, the description of the optimal mechanism may be uncountable [Daskalakis et al., 2013]. Recent work gives a number of important partial characterizations, e.g. Daskalakis et al. [2015], Giannakopoulos and Koutsoupias [2014], as well as results for weaker notions of Bayesian incentive compatibility, e.g. Cai et al. [2012b,a, 2013], Bhalgat et al. [2013].
Conitzer and Sandholm [2002, 2004] proposed the approach of automated mechanism design to use optimization and computational methods to obtain (near) optimal mechanisms for specific problems; see also Sandholm and Likhodedov [2015]. An active recent area of research uses machine learning tools. In particular, Dütting et al. [2019] designed and trained neural networks to automatically find optimal auctions. They studied two network architectures, and showed that several theoretically optimal mechanisms can be recovered using this approach, as well as interesting new mechanisms can be obtained. The first network they studied is *RochetNet*. This is a simple two-layer neural network applicable to the single buyer case, leveraging Rochet's [1987] characterization of the optimal mechanism. The second network, *RegretNet*, does not require such a characterization and is applicable for multiple buyers; however, it only provides approximate incentive compatibility.
Dütting et al. [2019] coined the term 'differentiable economics' for this approach, and there has been significant further work in this direction. These include designing auctions for budget constrained buyers [Feng et al., 2018]; multi-facility location Golowich et al. [2018]; balancing fairness and revenue objectives [Kuo et al., 2020]; incorporating non-linear utility functions and other networks trained from interaction data [Shen et al., 2019] $^{2}$ ; designing revenue-maximizing auctions with differentiable matchings [Curry et al., 2022a]; contextual auction design [Duan et al., 2022]; designing taxation policies [Zheng et al., 2022], and more.
The purpose of this work is to supply theoretical evidence behind the success of neural networks in differentiable economics. The revenue is a highly non-convex function of the parameters in the neural network. Curiously, gradient approaches seem to recover globally optimal auctions despite this non-convexity. Similar phenomena have been studied more generally in the context of deep networks, and theoretical explanations have been proposed, in particular, overparametrization [Allen-Zhu et al., 2019, Du et al., 2019].
Mode connectivity Recent work has focused on a striking property of the landscape of loss functions of deep neural networks: local optimal solutions (modes) found by gradient approaches are connected by simple paths in the parameter space. We provide an informal definition of mode connectivity here.
Definition 1 ( $\varepsilon$ -mode connected (informal)). We say that two solutions are  $\varepsilon$ -mode connected, if they are connected by a continuous path of solutions, such that the loss function does not worsen by more than  $\varepsilon$  compared to one of the end points on the entire path.
This phenomenon was identified by Garipov et al. [2018] and by Draxler et al. [2018]. Mode connectivity can help to explain the empirical performance of stochastic gradient descent (sgd) (or ascent, in case of revenue maximization). To some extent, mode connectivity prevents a poor local minimal valley region on the function value, from which the sgd method cannot escape easily. Suppose such a bad local minimum exists. Then mode connectivity implies that there exists a path from this bad local minimum to a global minimum on which the loss function does not significantly increase. Therefore, the intuition is that from every bad local minimum, a (stochastic) gradient method would eventually be able to find a way to escape. However, we would like to emphasize that mode connectivity does not provide a formal proof of the success of sgd. It only provides a useful intuition of why sgd does not completely trapped in local optima.
Kuditipudi et al. [2019] gave strong theoretical arguments for mode connectivity. They introduce the notion of  $\varepsilon$ -dropout stability: solutions to a neural network such that in each layer, one can remove
$^{2}$ MenuNet, developed by [Shen et al., 2019], also encodes menu items as RocketNet. However, unlike RocketNet's approach of repeatedly sampling valuations from the underlying distribution, MenuNet discretizes the buyer's valuation space into discrete values.
at least half the neurons and rescale the remaining units such that the loss function increases by at most  $\varepsilon$ . Solutions that are  $\varepsilon$ -dropout stable are then shown to be  $\varepsilon$ -mode connected. Moreover, they show that noise stability (see e.g., Arora et al. [2018]) implies dropout stability, and hence, mode connectivity. Nguyen [2019] showed mode connectivity when there is a hidden layer larger than the training dataset. Shevchenko and Mondelli [2020] shows that stochastic gradient descent solutions to sufficiently overparametrized neural networks are dropout stable even if we only keep a small, randomly sampled set of neurons from each layer.
RochetNet In this paper, we first establish mode connectivity properties of  $\text{RochetNet}$ , the architecture in Dütting et al. [2019] for multiple items and a single buyer. These networks have a single hidden layer, corresponding to the menu. Within this hidden layer, each neuron directly corresponds to an option in the menu: which contains an allocation and a price offered to the buyer (see Figure 1). The buyer is assigned the single option, including the allocation and the price, maximizing the buyer's utility. Such an option is called active for the buyer. The loss function of  $\text{RochetNet}$  is the revenue, the expected price paid by the buyer. Despite its simplicity, the experiments on  $\text{RochetNet}$  in Dütting et al. [2019] gave impressive empirical results in different scenarios. For example, in the experiments with up to six items and uniform value distributions,  $\text{RochetNet}$  achieves almost the same revenue (99.9%) as the Straight-Jacket Auctions in Giannakopoulos and Koutsoupias [2018], which are known to be optimal in this case. This success is not limited to a single example, as  $\text{RochetNet}$  also consistently performs well in other scenarios, including when infinite menu size is necessary. Furthermore, Dütting et al. [2019] demonstrated the usefulness of  $\text{RochetNet}$  in discovering optimal auctions in situations that were previously unexplored from a theoretical perspective.
First, in Theorem 9, we show that for linear utilities,  $\varepsilon$ -mode connectivity holds between two solutions that are  $\varepsilon$ -reducible: out of the  $K + 1$  menu options (neurons), there exists a subset of at most  $\sqrt{K + 1}$  containing an active option for the buyer with probability at least  $1 - \varepsilon$ . Assuming that the valuations are normalized such that the maximum valuation of any buyer is at most one, it follows that if we remove all other options from the menu, at most  $\varepsilon$  of the expected revenue is lost. The assumption of being  $\varepsilon$ -reducible is stronger than  $\varepsilon$ -dropout stability that only drops a constant fraction of the neurons. At the same time, experimental results in Dütting et al. [2019] show evidence of this property being satisfied in practice. They experimented with different sized neural networks in a setting when the optimal auction requires infinite menu size. Even with 10,000 neurons available, only 59 options were active, i.e., used at least once when tested over a large sample size. We note that this property also highlights an advantage of RochetNet over RegretNet and other similar architectures: instead of a black-box neural network, it returns a compact, easy to understand representation of a mechanism.
Our second main result (Theorem 10) shows that for  $n$  items and linear utilities, if the number of menu options  $K$  is sufficiently large, namely,  $(2 / \varepsilon)^{4n}$ , then  $\varepsilon$ -mode connectivity holds between any two solutions for any underlying distribution. The connectivity property holds pointwise: for any particular valuation profile, the revenue may decrease by at most  $\varepsilon$  along the path. A key tool in this  $\varepsilon$ -mode connectivity result is a discretization technique from Dughmi et al. [2014]. We note that such a mode connectivity result can be expected to need a large menu size. In Appendix C, we present an example with two disconnected local maxima for  $K = 1$ .
Affine Maximizer Auctions We also extend our results and techniques to neural networks for affine maximizer auctions (AMA) studied in Curry et al. [2022b]. This is a generalization of RochetNet for multi-buyer scenarios. It can also be seen as a weighted variant of the Vickrey-Clarke-Groves (VCG) mechanism [Vickrey, 1961, Clarke, 1971, Groves, 1973]. AMA offers various allocation options. For a given valuation profile, the auctioneer chooses the allocation with the highest weighted sum of valuations, and computes individual prices for the buyers; the details are described in Section 2.2. AMA is DSIC and IR, however, is not rich enough to always represent the optimal auction.
For AMA networks, we show similar results (Theorem 13 and 14) as for RochetNet. We first prove  $\varepsilon$ -mode connectivity holds between two solutions that are  $\varepsilon$ -reducible (see Definition 11). Curry et al. [2022b] provides evidence of this property being satisfied in practice, observing (in Sec. 7.1) "Moreover, we found that starting out with a large number of parameters improves performance, even though by the end of training only a tiny number of these parameters were actually used". Secondly, we also show that if the number of menu options  $K$  is sufficiently large, namely,  $(16m^{3} / \varepsilon^{2})^{2nm}$ ,
then  $\varepsilon$ -mode connectivity holds pointwise between any two solutions. That is, it is valid for any underlying distribution of valuations, possibly correlated between different buyers.
Relation to previous results on mode connectivity Our results do not seem to be deducible from previous mode connectivity results, as we outline as follows. Previous literature on mode connectivity investigated neural networks used for prediction. The results in Kuditipudi et al. [2019], Nguyen [2019], Shevchenko and Mondelli [2020] and other papers crucially rely on the properties that the networks minimize a convex loss function between the predicted and actual values, and require linear transformations in the final layer. *RochetNet* and AMA networks are fundamentally different. The training data does not come as labeled pairs and these network architectures are built directly for solving an optimization problem. For an input valuation profile, the loss function is the negative revenue of the auctioneer. In *RochetNet*, this is obtained as the negative price of the utility-maximizing bundle; for AMA it requires an even more intricate calculation. The objective is to find parameters of the neural network such that the expected revenue is as large as possible. The menu options define a piecewise linear surface of utilities, and the revenue in *RochetNet* can be interpreted as the expected bias of the piece corresponding to a randomly chosen input.
Hence, the landscape of the loss function is fundamentally different from those analyzed in the above mentioned works. The weight interpolation argument that shows mode-connectivity from dropout stability is not applicable in this context. The main reason is that the loss function is not a simple function of the output of the network, but is defined by choosing the price of the argmax option. We thus need a more careful understanding of the piecewise linear surfaces corresponding to the menus.
Significance for Practitioners We see the main contribution of our paper in explaining the empirical success and providing theoretical foundations for already existent practical methods, and not in inventing new methods. Nevertheless, two insights a practitioner could use are as follows: (i) It is worth understanding the structure of the auction in question. If one can, e.g., understand whether  $\varepsilon$ -reducibility holds for a particular auction, this might indicate whether RochetNet or AMA are good methods to apply to this particular case. (ii) Size helps: If one encounters bad local optima, increasing the menu size and rerunning RochetNet or AMA might be a potential fix and will eventually lead to a network satisfying mode connectivity.
2 Auction Settings
We consider the case with  $m$  buyers and one seller with  $n$  divisible items each in unit supply. Each buyer has an additive valuation function  $v_{i}(S) \coloneqq \sum_{j \in S} v_{ij}$ , where  $v_{ij} \in V$  represents the valuation of the buyer  $i$  on item  $j$  and  $V$  is the set of possible valuations. Throughout the paper, we normalize the range to the unit simplex: we assume  $V = [0,1]$ , and  $\|v_{i}\|_{1} = \sum_{j} v_{ij} \leq 1$  for every buyer  $i$ . With slight abuse of notation, we let  $v = (v_{11}, v_{12}, \dots, v_{ij}, \dots, v_{mn})^{\top}$  and  $v_{i} = (v_{i1}, v_{i2}, \dots, v_{in})^{\top}$ . The buyers' valuation profile  $v$  is drawn from a distribution  $F \in \mathcal{P}(V^{m \times n})$ . Throughout, we assume that the buyers have quasi-linear utilities: if a buyer with valuation  $v_{i}$  receives an allocation  $x \in [0,1]^{n}$  at price  $p$ , their utility is  $v_{i}^{\top} x - p$ .
The seller has access to samples from the distribution  $F$ , and wants to sell these items to the buyers through a DSIC $^3$  and IR auction and maximize the expected revenue. In the auction mechanism, the  $i$ -th bidder reports a bid  $b_{i} \in [0,1]^{n}$ . The entire bid vector  $b \in [0,1]^{m \times n}$  will be denoted as  $b = (b_{1},\ldots,b_{m}) = (b_{i},b_{-i})$ , where  $b_{-i}$  represents all the bids other than buyer  $i$ . In a DSIC mechanism, it is a dominant strategy for the agents to report  $b_{i} = v_{i}$ , i.e., reveal their true preferences.
Definition 2 (DSIC and IR auction). An auction mechanism requires the buyers to submit bids  $b_{i} \in [0,1]^{n}$ , and let  $b = (b_{1},\ldots ,b_{m})$ . The output is a set of allocations  $x(b) = (x_{1}(b),\dots,x_{m}(b))$ ,  $x_{i}(b) \in [0,1]^{n}$ , and prices  $p(b) = (p_1(b),\dots,p_m(b)) \in \mathbb{R}^m$ . Since there is unit supply of each item, we require  $\sum_{i}x_{ij}(b) \leq 1$ , where  $x_{ij}(b)$  is the allocation of buyer  $i$  of item  $j$ .
There exists a weaker notion of incentive compatibility: Bayesian incentive compatible (BIC). In a BIC mechanism, it is a dominant strategy for the buyers to report the true valuations if all other buyers also report truthfully. This paper focuses on DISC mechanisms, as DSIC is considered to be more robust than BIC, which assumes a common knowledge of buyers' distributions on valuations.
(i) An auction is dominant strategy incentive compatible (DSIC) if  $v_{i}^{\top}x_{i}(v_{i},b_{-i}) - p_{i}(v_{i},b_{-i}) \geq v^{\top}x_{i}(b_{i},b_{-i}) - p_{i}(b_{i},b_{-i})$  for any buyer  $i$  and any bid  $b = (b_{i},b_{-i})$ .
(ii) An auction is individually rational (IR) if  $v_{i}^{\top}x_{i}(v_{i},b_{-i}) - p_{i}(v_{i},b_{-i}) \geq 0$ .
The revenue of a DSIC and IR auction is
$$
\mathsf {R e v} = \mathbb {E} _ {v \sim F} \left[ \sum_ {i} p _ {i} (v) \right].
$$
2.1 Single Buyer Auctions: RochetNet
Dütting et al. [2019] proposedchetNet as a DISC and IR auction for the case of a single buyer. We omit the subscript  $i$  for buyers in this case. A (possibly infinite sized) menu  $M$  comprises a set of options offered to the buyer:  $M = \{(x^{(k)},p^{(k)})\}_{k\in \mathcal{K}}$ . In each option  $(x^{(k)},p^{(k)})$ ,  $x^{(k)}\in [0,1]^n$  represents the amount of items, and  $p^{(k)}\in \mathbb{R}_+$  represents the price. We assume that  $0\in \mathcal{K}$ , and  $(x^{(0)},p^{(0)}) = (0,0)$  to guarantee IR. We call this the default option, whereas all other options are called regular options. We will use  $K$  to denote the number of regular options; thus,  $|\mathcal{K}| = K + 1$ .
A buyer submits a bid  $b \in [0,1]^n$  representing their valuation, and is assigned to option  $k(b) \in \mathcal{K}$  that maximizes the utility<sup>4</sup>
$$
k (b) \in \arg \max  _ {k \in \mathcal {K}} b ^ {\top} x ^ {(k)} - p ^ {(k)}.
$$
This is called the active option for the buyer. Note that option 0 guarantees that the utility is nonnegative, implying the IR property. It is also easy to see that such an auction is DSIC. Therefore, one can assume that  $b = v$ , i.e., the buyer submits their true valuation; or equivalently, the buyer is allowed to directly choose among the menu options one that maximizes their utility. Moreover, it follows from Rochet [1987] that every DSIC and IR auction for a single buyer can be implemented with a (possibly infinite size) menu using an appropriate tie-breaking rule.
Given a menu  $M$ , the revenue is defined as
$$
\operatorname {R e v} (M) = \mathbb {E} _ {v \sim F} \left[ p ^ {(k (v))} \right].
$$
RochetNet RochetNet (see Figure 1) is a neural network with three layers: an input layer ( $n$  neurons), a middle layer ( $K$  neurons), and an output layer (1 neuron):
1. the input layer takes an  $n$ -dimensional bid  $b \in V^n$ , and sends this information to the middle layer;
2. the middle layer has  $K$  neurons. Each neuron represents a regular option in the menu  $M$ , which has parameters  $x^{(k)} \in [0,1]^n$  and  $p^{(k)} \in \mathbb{R}_+$ , where  $x^{(k)} \in [0,1]^n$  represents the allocation of option  $k$  and  $p^{(k)}$  represents the price of option  $k$ . Neuron  $k$  maps from  $b \in V^n$  to  $b^\top x^{(k)} - p^{(k)}$ , i.e., the utility of the buyer when choosing option  $k$ ;
3. the output layer receives all utilities from different options and maximizes over these options and 0:  $\max \{\max_k\{(x^{(k)})^\top b - p^{(k)}\} ,0\}$
We will use  $\operatorname{Rev}(M)$  to denote the revenue of the auction with menu options  $\mathcal{K} = \{0,1,2,\dots ,K\}$ , where 0 represents the default option  $(\mathbf{0},0)$ .
The training objective for the RochetNet is to maximize the revenue  $\operatorname{Re}\mathbf{v}(M)$ , which is done by stochastic gradient ascent. Note, however, that the revenue is the price of an argmax option, which makes it a non-continuous function of the valuations. For this reason, Dütting et al. [2019] use a softmax-approximation of the argmax as their loss function instead. However, argmax is used for testing. In Appendix B, we bound the difference between the revenues computed with these two different activation functions, assuming that the probability density function of the distribution  $F$  admits a finite upper bound. Lemma 19 shows that the difference between the revenues for softmax and argmax is roughly inverse proportional to the parameter  $Y$  of the softmax function. This allows the practitioner to interpolate between smoothness of the loss function and provable quality of the softmax approximation by tuning the parameter  $Y$ .
<sup>4</sup>We assume that ties are broken in favor of higher prices, but it is not hard to see that our results transfer to other tie-breaking rules, too. See also the discussion in Section B.2 of Babaioff et al. [2022].
Figure 1: RochetNet: this architecture maps the bid  $b$  to the utility of the buyer.
2.2 Affine Maximizer Auctions
Affine Maximizer Auctions (AMA) also provide a menu  $M$  with a set of options  $\mathcal{K}$ . Each option is of the form  $(x^{(k)},\beta^{(k)})\in [0,1]^{n\times m}\times \mathbb{R}$ , where  $x_{ij}^{(k)}\in [0,1]$  represents the allocation of item  $i$  to buyer  $j$ , with the restriction that  $\sum_{i}x_{ij}^{(k)}\leq 1$  for each item  $j$ , and  $\beta^{(k)}$  represents a 'boost'. We again assume  $0\in \mathcal{K}$ , and  $(x^{(0)},\beta^{(0)}) = (0,0)$ , and call this the default option; all other options are called the regular options.
Given the bids  $b_{i} \in [0,1]^{n}$  of the agents, the auctioneer computes a weighted welfare, using weights  $w_{i} \in \mathbb{R}_{+}$  for the valuations of each agent, and adds the boost  $\beta^{(k)}$ . Then, the allocation maximizing the weighted boosted welfare is chosen, i.e., the option with
$$
k (b) \in \arg \max  _ {k \in \mathcal {K}} \sum_ {i} w _ {i} b _ {i} ^ {\top} x _ {i} ^ {(k)} + \beta^ {(k)}.
$$
This will also be referred to as the active option. The prices collected from the buyers are computed according to the Vickrey-Clarke-Groves (VCG) scheme. Namely,
$$
p _ {i} (b) = \frac {1}{w _ {i}} \left(\sum_ {\ell \neq i} w _ {\ell} b _ {\ell} ^ {\top} x _ {\ell} ^ {(k (b _ {- i}))} + \beta^ {(k (b _ {- i}))}\right) - \frac {1}{w _ {i}} \left(\sum_ {\ell \neq i} w _ {\ell} b _ {\ell} ^ {\top} x _ {\ell} ^ {(k (b))} + \beta^ {(k (b))}\right). \tag {1}
$$