SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Xiaoran Hao $^{1,*}$ , Yash Jhaveri $^{1,*}$ , & Patrick Shafto $^{1,2}$
$^{1}$ Department of Math and Computer Science, Rutgers University-Newark
$^{2}$ School of Mathematics, Institute for Advanced Study, Princeton
Cooperative communication plays a fundamental role in theories of human-human interaction–cognition, culture, development, language, etc. as well as human-robot interaction. The core challenge in cooperative communication is the problem of common ground: having enough shared knowledge and understanding to successfully communicate. Prior models of cooperative communication, however, uniformly assume the strongest form of common ground, perfect and complete knowledge sharing, and, therefore, fail to capture the core challenge of cooperative communication. We propose a general theory of cooperative communication that is mathematically principled and explicitly defines a spectrum of common ground possibilities, going well beyond that of perfect and complete knowledge sharing, on spaces that permit arbitrary representations of data and hypotheses. Our framework is a strict generalization of prior models of cooperative communication. After considering a parametric form of common ground and viewing the data selection and hypothesis inference processes of communication as encoding and decoding, we establish a connection to variational autoencoding, a powerful model in modern machine learning. Finally, we carry out a series of empirical simulations to support and elaborate on our theoretical results.
The cooperative communication problem is a problem among two agents; their goal is to find a pair of plans that successfully map beliefs to communicative acts and back. This problem is fundamental in two-party interaction, whether human-human, human-machine, or (expectantly) machine-machine interaction. In human-human interaction, cooperative information sharing has long been viewed as a foundation of, for example, human language (Frank and Goodman, 2012; Goodman and Stuhlmüller, 2013), cognitive development (Bonawitz et al., 2011; Jara-Ettinger et al., 2016), and cultural evolution (Tomasello, 1999). In human-machine interaction, this problem's appearance is, of course, more recent, but still its significance has been observed in, for instance, social robotics (Fisac et al., 2017), machine teaching (Zhu, 2015, 2013), reinforcement learning (Hadfield-Menell et al., 2016; Ho et al., 2016), and deep learning (Laskey et al., 2017). Researchers have proposed a number of models of cooperative communication (Shafto and Goodman, 2008; Frank et al., 2009; Shafto et al., 2012; Yang et al., 2018) and, most recently, mathematical formalizations that enable the unification and analysis of prior models (Wang et al., 2020).
The major challenge for any two agents in cooperative communication is having enough shared understanding to successfully communicate, a problem known as common ground (Clark et al., 1991). Because meaning is inferred from communicative acts, each agent must take into account what the other knows. So models of cooperative communication must make assumptions about the form and strength of common ground. Despite the impossibility of knowing exactly what another knows, prior models of cooperative communication uniformly assume omniscient agents (Wang et al., 2020). Ironically, this assumption with respect to common ground trivializes cooperative communication.
In this work, we propose a theory of cooperative communication that is mathematically principled and models a spectrum of common ground possibilities. This newly defined spectrum goes well beyond that of perfect and complete knowledge sharing and understanding. We formulate the cooperative communication problem under assumptions of common ground as a (family of) constrained optimization problem(s). In particular, we define different forms and strengths of common ground as different constraints on a general (family of) optimization objective(s). We draw a connection between cooperative communication and variational autoencoding notionally by viewing the data selection and hypothesis inference processes that define communication as encoding and decoding and formally through a variational approximation of our model. Our model is a strict generalization of previous models of cooperative communication, and it is set on arbitrary data and hypothesis spaces, allowing for arbitrary representations of data and hypotheses.
2 Theory
Communication is an ordered pair of processes between two agents, a teacher  $T$  and a learner  $L$ , who communicate an arbitrary set of hypotheses  $H$  by way of an arbitrary set of data  $D$ . The teacher, by definition, selects data to represent hypotheses, and the learner infers hypotheses from the teacher's selected data, all while taking into account shared knowledge, called common ground. Of course, a teacher can only represent hypotheses as well as they understand them, and, thus, a teacher naturally comes with a "prior" on hypotheses: a probability measure  $\nu$  on  $H$ . Similarly, a learner naturally comes with a "prior" on data: a probability measure  $\mu$  on  $D$ , as a means of representing their relative understanding of data. In preparing to communicate, therefore, each seeks a probability measure  $\pi$  on a joint data-hypothesis space  $D \times H$  – a communication plan—that probabilistically couples data and hypotheses, formalizing how well data and hypotheses represent one another. The communication plan depends on the aforementioned priors and is subject to the teacher's and learner's assumptions about each other, part of their common ground.
Definition 2.1 (Common Ground). Common ground is a pair of sets of probability measures on  $D \times H$ , i.e., a pair of sets of communication plans:  $(\mathcal{P}_T^\nu, \mathcal{P}_\mu^L) \in 2^{\Pi^\nu} \times 2^{\Pi_\mu}$ , where  $\Pi^\nu$  and  $\Pi_\mu$  denote the sets of communication plans with hypothesis marginal  $\nu$  and data marginal  $\mu$ .
Common ground, in turn, is a collection of all of the choices our agents have for their communication plans, accounting for their experience and understanding of data and hypotheses as well as their knowledge of each other.
We propose modeling the problem of cooperative communication under common ground, of identifying "good" communication plans, as a constrained minimization problem of a cost functional defined on pairs of communication plans:
$$
\min  _ {(\pi_ {T}, \pi_ {L}) \in \mathcal {P} _ {T} ^ {\nu} \times \mathcal {P} _ {\mu} ^ {L}} \mathrm {C C} _ {\epsilon , \delta} (\pi_ {T}, \pi_ {L}), \tag {2.1}
$$
where, for any  $\epsilon, \delta \geq 0$ , the functional  $\mathrm{CC}_{\epsilon, \delta}: \mathcal{P}(D \times H) \times \mathcal{P}(D \times H) \to \mathbb{R} \cup \{\pm \infty\}$  is
$$
\mathrm {C C} _ {\epsilon , \delta} \left(\pi_ {T}, \pi_ {L}\right) := \mathrm {K L} \left(\pi_ {T} \mid \pi_ {L}\right) + (\epsilon - 1) \mathrm {K L} \left(\pi_ {T} \mid \mu_ {L} \otimes \nu_ {T}\right) + \delta \mathrm {K L} \left(\mu_ {T} \mid \mu_ {L}\right),
$$
over a pair of feasible sets,  $\mathcal{P}_T^\nu$  for the teacher and  $\mathcal{P}_{\mu}^{L}$  for the learner. Here  $\mu_{i}$  and  $\nu_{i}$  denote the data marginal and hypothesis marginal of the joint probability measure  $\pi_{i}$  respectively, for  $i = T,L$ . Moreover,  $\mathrm{KL}(\cdot |\cdot)$  is the Kullback-Liebler divergence (or relative entropy).
By finding a pair of plans that minimize the aforementioned cost on the product of these feasible sets, our agents are cooperatively identifying plans that they can use to communicate with respect to the assumed common ground captured by the size, structure, and elements of the feasible sets. Notice that the smaller the product of the feasible sets which define their common ground, the stronger the assumptions our agents make about one another.
Remark 2.2. How our agents come to determine a common ground pair is an important independent and antecedent problem to that which our model solves. We propose a framework which allows our agents to 1. rigorously define common ground that isn't perfect and complete, the first such rigorous definition, and 2. with common ground established, define and find optimal communication plans.
Remark 2.3. Furthermore, given a communication plan in this generality, many possible choices of how say data might be chosen to represent a hypothesis  $h$  exist. For example, if looking for a single
$d$ , the mean of the conditional distribution over data given  $h$  might do. This leads to an important and subsequent problem. What is the best way to utilize optimal communication plans to communicate once they are in hand? (Cf. Remark 3.3 below.)
If  $\mathcal{P}_T^\nu \cap \mathcal{P}_\mu^L$  is non-empty, for example, our framework models situations in which agreement is possible, where agreement corresponds to the possibility that an optimal pair of communication plans consist of a single communication plan paired with itself, that is, the teacher and learner select the same communication plan in minimizing  $\mathrm{CC}_{\epsilon,\delta}$ . On the other hand, if  $\mathcal{P}_T^\nu \cap \mathcal{P}_\mu^L$  is empty, our framework models situations in which only approximate agreement is possible.
The two parameters  $\epsilon$  and  $\delta$  weigh the relative importance of the three terms defining our cost functional. The first term measures disagreement between the teacher and learner; the further apart these two probability measures are, or, in other words, the more disagreement between the teacher and learner there is, the higher the communication cost. The second term, as we shall see, delineates three regimes and quantifies how uniformly the teacher might choose to represent different hypotheses as probability measures over data with respect to the learner's prior on data. The third term measures how far the teacher's induced posterior on data is from the learner's prior on data; the further apart these two probability measures on data are, the higher the communication cost.
Observe that  $\mathrm{CC}_{\epsilon, \delta}(\pi_T, \pi_L)$  is increasing as a function of either  $\epsilon$  or  $\delta$  when all of the other terms are fixed. The first term in the definition  $\mathrm{CC}_{\epsilon, \delta}$  is non-negative and vanishes if and only if  $\pi_T = \pi_L$ . Likewise, the third term in the definition  $\mathrm{CC}_{\epsilon, \delta}$  is non-negative and vanishes if and only if  $\mu_T = \mu_L$ . If  $\pi_T = \pi_L$ , then, trivially,  $\mu_T = \mu_L$ . In other words,  $\mathrm{CC}_{\epsilon, \delta}(\pi_T, \pi_L) \geq (\epsilon - 1)\mathrm{KL}(\pi_T | \mu_L \otimes \nu_T)$ , with equality if and only if  $\pi_T = \pi_L$ . No matter the strength or form of common ground, we know that  $\mu_L = \mu$  and  $\nu_T = \nu$  for all feasible  $\pi_T$  and  $\pi_L$ . Thus, over any pair of feasible sets,
$$
\mathrm {C C} _ {\epsilon , \delta} \left(\pi_ {T}, \pi_ {L}\right) \geq (\epsilon - 1) \mathrm {K L} \left(\pi_ {T} \mid \mu \otimes \nu\right), \tag {2.2}
$$
with equality if and only if  $\pi_T = \pi_L$ . As mentioned, therefore, we find three general situations, when considering the common ground pair  $(\Pi^{\nu}, \Pi_{\mu})$ .
If  $\epsilon > 1$ , the coefficient  $\epsilon - 1$  is positive. So taking  $\mathrm{KL}(\cdot \mid \mu \otimes \nu)$  as small as possible then minimizes the right-hand side in equation (2.2). Note that  $\mathrm{KL}(\cdot \mid \mu \otimes \nu)$  is non-negative and strictly convex. Thus, for all  $\epsilon > 1$ ,  $\mathrm{CC}_{\epsilon, \delta}$  over  $\Pi^{\nu} \times \Pi_{\mu}$  is minimized by the unique minimizer of  $\mathrm{KL}(\cdot \mid \mu \otimes \nu)$  in  $\Pi_{\mu}^{\nu} := \Pi^{\nu} \cap \Pi_{\mu}$ , which is  $\mu \otimes \nu$ , paired with itself. Indeed, for any  $\pi \in \Pi_{\mu}^{\nu}$ , if we take  $\pi_{T} = \pi = \pi_{L}$ , equality is attained in (2.2), and the right-hand side of (2.2) is equal to zero at  $\mu \otimes \nu \in \Pi_{\mu}^{\nu}$ . In turn, without our agents making more assumptions about one another (than that their priors are known), they will simply choose to behave uniformly with respect to the other's prior. The teacher will select data in the same way for every hypothesis: according to the learner's prior on data. Similarly, the learner will infer hypotheses in the same way for all data: according to the teacher's prior on hypotheses.
When  $\epsilon < 1$ , the first and third terms in the definition of  $\mathrm{CC}_{\epsilon, \delta}$  compete with the second term. In particular, any admissible pair  $(\pi_T, \pi_L)$  for which  $\mathrm{KL}(\pi_T | \mu \otimes \nu) = +\infty$  and  $\mathrm{KL}(\pi_T | \pi_L) + \delta \mathrm{KL}(\mu_T | \mu) < \infty$  would minimize  $\mathrm{CC}_{\epsilon, \delta}$  over a common ground pair. An interpretation of this is a reflection of the importance of the teacher and learner using as much information about each other as possible beyond their priors, to maximize  $\mathrm{KL}(\cdot | \mu \otimes \nu)$ , over some version of approximate agreement, which corresponds to the finiteness of  $\mathrm{KL}(\pi_T | \pi_L) + \delta \mathrm{KL}(\mu_T | \mu)$ .
When  $\epsilon = 1$ , and taking  $\pi_T = \pi = \pi_L$  for any  $\pi \in \Pi_{\mu}^{\nu}$  minimizes  $\mathrm{CC}_{\epsilon, \delta}$  over  $\Pi^{\nu} \times \Pi_{\mu}$ . Thus, in this regime, agreement is sufficient for cost minimization.
2.1 Common Ground via Variational Approximation
Here we consider parameterized families of probability density functions as approximations of common ground pairs, thereby producing a variational approximation of our general cooperative communication framework. This is a means of representing one manner through which our agents might attempt to disambiguate optimal communication plan pairs (by making stronger assumptions with respect to common ground, our agents may decrease the size of the set of optimal communication plan pairs) as well as resolve tractability issues (even if  $\pi_T$  had probability density function  $P_T$ , it still may be intractable). In particular, we let  $P_{\theta}$  denote a probability density function approximation of  $\pi_T$ . Similarly, we let  $P_{\lambda}$  denote a (probability density function) approximation of  $\pi_L$ . The parameter  $\theta$  is for the teacher and the parameter  $\lambda$  is for the learner. As a byproduct, we gain a simple but
striking connection to variational autoencoding. In this scheme, the teacher's and learner's priors  $\nu$  and  $\mu$  must have probability density functions. They will be denoted by  $g$  and  $f$  respectively.
Specifically, we define variational cooperative communication under common ground as the following minimization problem: for  $D,H\in \{\mathbb{R}^n,[[n]]:n\in \mathbb{N}\}$  and  $\epsilon ,\delta \geq 0$
$$
\min  _ {\left(P _ {\theta}, P _ {\lambda}\right) \in \mathcal {P} _ {\theta} ^ {g} \times \mathcal {P} _ {f} ^ {\lambda}} \mathrm {L} _ {\epsilon , \delta} \left(P _ {\theta}, P _ {\lambda}\right). \tag {2.3}
$$
Here the loss functional  $\mathrm{L}_{\epsilon, \delta}$  is defined on pairs of probability density functions on  $D \times H$  by
$$
\mathrm {L} _ {\epsilon , \delta} (P _ {\theta}, P _ {\lambda}) := \int - P _ {\theta} (d, h) \log f ^ {\lambda} (h \mid d) + \epsilon \mathrm {K L} (P _ {\theta} \mid f _ {\lambda} \otimes g _ {\theta}) + \delta \mathrm {K L} (f _ {\theta} \mid f _ {\lambda}),
$$
and common ground is given by a pair of sets of parameterized probability density functions  $(\mathcal{P}_{\theta}^{g},\mathcal{P}_{f}^{\lambda})$ ; again,  $f = f(d)$  and  $g = g(h)$  are the learner's and teacher's prior probability density functions respectively. Also,  $f_{j}$  and  $g_{j}$  denote the data and hypothesis marginals of  $P_{j}$  respectively, for  $j = \theta ,\lambda$ . Moreover, in general, for instance, we set  $f^{\lambda}(h|d):= P_{\lambda}(d,h) / f_{\lambda}(d)$  if  $f_{\lambda}(d)\neq 0$  and  $f^{\lambda}(h|d):= 0$  otherwise. Hence,  $P_{\lambda}(d,h) = f^{\lambda}(h|d)f_{\lambda}(d)$ .
As soon as a communication plan is determined by a probability density function, it admits a decomposition as the product of conditional probabilities and a prior. For example, since common ground, at minimum, requires that all of the teacher's feasible communication plans have hypothesis marginal  $g$ , the only variability within admissible plans comes from the induced conditional probabilities,  $g^{\theta}(d|h) := P_{\theta}(d,h) / g(h)$  if  $g(h) \neq 0$  and  $g^{\theta}(d|h) := 0$  otherwise. In turn, we make the following definitions.
Definition 2.4 (Conditional Teaching Plan). A conditional teaching plan is the family of induced conditional probabilities  $\{g^{\theta}(d|h)\}_{h\in H}$  determined from a teacher's communication plan in  $\mathcal{P}_{\theta}^{g}$ .
Definition 2.5 (Conditional Learning Plan). A conditional learning plan is the family of induced conditional probabilities  $\{f^{\lambda}(h\mid d)\}_{d\in D}$  determined from a learner's communication plan in  $\mathcal{P}_f^\lambda$
While, at first glance,  $\mathrm{L}_{\epsilon,\delta}$  looks to be defining a different minimization problem than  $\mathrm{CC}_{\epsilon,\delta}$ , even after restricting  $\mathrm{CC}_{\epsilon,\delta}$  to measures defined by probability density functions, it does not.
Theorem 2.6. For any  $\epsilon, \delta \geq 0$ , and any pair of probability density functions  $(P_T, P_L)$  on  $D \times H$ , which determine a pair of probability measures  $(\pi_T, \pi_L)$ , we have that  $\mathrm{CC}_{\epsilon, \delta}(\pi_T, \pi_L) = \mathrm{L}_{\epsilon, \delta}(P_T, P_L) - \mathrm{H}(g_T)$ , where  $g_T$  denotes the hypothesis marginal of the probability measure determined by  $P_T$  and  $\mathrm{H}$  denotes the Boltzmann-Shannon entropy functional.
As  $g_{T}$  represents the teacher's prior on  $H$  for the communication plan  $P_{T}$  (or  $\pi_T$ ), which is equal to  $g$  by common ground, variational cooperative communication defined by (2.3) is equivalent to cooperative communication defined by (2.1) when (2.1) is restricted to communication plans defined by probability density functions and the teacher's prior on hypotheses is denoted by  $g$  (rather than  $\nu$ ).
2.1.1 Cooperative Communication as Variational Autoencoding
Variational autoencoding, broadly speaking, is a probabilistic framework in which one looks to represent a space  $X$  by way of a lower dimensional space  $Z$ . Each space is equipped with a probability density function, and the goal is to find a family of probabilistic encoders  $\{q(z \mid x)\}_{x \in X}$ , probability density functions on  $Z$ , one for each  $x \in X$ , and a family of probabilistic decoders  $\{p(x \mid z)\}_{z \in Z}$ , probability density functions on  $X$ , one for each  $z \in Z$ , such that the induced marginal on  $X$  of the joint probability density function on  $X \times Z$  determined by the prior on  $Z$  and the probabilistic decoders is a "good" approximation of the prior on  $X$ . These encoders and decoders are typically restricted to live in parameterized families of probability density functions, Gaussian, for example.
2 All proofs can be found with the Supplementary Materials.
3 Ideally, given a family of probabilistic decoders, the probabilistic encoders would be the  $x$ -induced conditional probabilities determined from the joint probability density function on  $X \times Z$  formed by the product of the decoders and the prior on  $Z$ . These conditional probabilities are assumed to be intractable. So they need to be approximated, and the notion of "good" here has to be appropriately defined. In turn, the term variational is used slightly differently in autoencoding and cooperative communication.
Note that if we replace  $X$  by  $H$ ,  $Z$  by  $D$ , and "represent" above by "communicate", we see that the broad idea behind variational autoencoding is exactly the broad idea behind (variational) cooperative communication. Indeed, teaching is none other than encoding and learning is none other than decoding (if we consider that data are compressed representations of hypotheses), and common ground is none other than, given that the teacher's prior and learner's prior are fixed, a restriction on the admissible conditional teacher and learning plans (encoders and decoders are forced to live within some parameterized family). Moreover, the loss functional defining (2.3) is the same one used in the variational autoencoding framework by Hao and Shafto (2023) to measure the "goodness" of the approximation of the induced marginal on  $X$  and the prior on  $X$ .
2.2 Previous Models
Previous models of cooperative communication assume omniscient agents and discrete data and hypothesis spaces. They define common ground, which we call strong common ground, as a triplet  $(\mu, \nu, M)$  of two probability vectors and a matrix. These objects and the explicit manner in which they interact to determine cooperative communication will be illustrated shortly.
Here we demonstrate two things. We show strong common ground provably trivializes the problem of cooperative communication from a problem of two agents to a problem of one agent communicating with themselves. Simultaneously, we show how previous models of cooperative communication can be seen as special cases of our framework. Previous models of cooperative, from language (Frank and Goodman, 2012; Goodman and Stuhlmüller, 2013), cognitive development (Jara-Ettinger et al., 2016), robotics (Fisac et al., 2017; Ho et al., 2016), and machine learning (Zhu, 2015, 2013), were unified as approximations of entropic Optimal Transport by Wang et al. (2020). Thus, our demonstration simply involves recovering their entropic Optimal Transport framework for cooperative communication framework by appropriately constraining our model (or strengthening common ground assumptions).
Wang et al. (2020) formalize the cooperative communication problem as a pair of Boltzman-Shannon entropy regularized Optimal Transport problems on a discrete data-hypothesis space  $D \times H$ . More precisely, for  $i = T, L$ , they consider the minimization problems:
$$
\min _ {P _ {i} \in \Pi_ {\mu} ^ {\nu}} \{\langle c _ {i}, P _ {i} \rangle - \epsilon \mathrm {H} (P _ {i}) \}
$$
where  $\epsilon \geq 0$ ,  $c_{T}(d,h) \coloneqq s_{T}(d) - \log \rho(h|d)$  and  $c_{L}(d,h) \coloneqq s_{L}(h) - \log \eta(d|h)$ , respectively, given vectors  $s_{T}$  and  $s_{L}$ , a non-negative  $(|D| \times |H|)$ -matrix  $M$  with row normalization  $\eta(d|h)$  and column normalization  $\rho(h|d) \colon \eta(d|h)\psi(h) = M(d,h) = \rho(h|d)\varphi(d)$ , for  $\varphi(d) \coloneqq \sum_{h} M(d,h)$  and  $\psi(h) \coloneqq \sum_{d} M(d,h)$ , and  $\mu$  and  $\nu$  are probability vectors of length  $|D|$  and  $|H|$  respectively;  $\Pi_{\mu}^{\nu}$  denotes the set of probability  $(|D| \times |H|)$ -matrices with column and row sums  $\mu$  and  $\nu$  respectively.
The unique minimizers of these two problems are the teacher's and learner's optimal communication plans respectively of the cooperative communication problem under the strong common ground triplet  $(\mu, \nu, M)$ . Explicitly, the matrix  $M$  determines two cost matrices, one for the teacher and one for the learner through column and row normalization respectively; and the vectors  $\mu$  and  $\nu$  denote the learner's prior on data and the teacher's prior on hypotheses respectively.
First, note that Boltzman-Shannon entropy regularized Optimal Transport is equivalent to relative entropy (or KL-divergence) regularized Optimal Transport, with respect to the matrix  $\mu \otimes \nu$  when  $\mu$  and  $\nu$  are the two marginal constraints, which, with a cost matrix, determine an Optimal Transport problem. So Wang et al. (2020) cooperative communication can be rewritten as, for  $i = T,L$ ,
$$
\min  _ {P _ {i} \in \Pi_ {\mu} ^ {\nu}} \left\{\langle c _ {i}, P _ {i} \rangle + \epsilon \mathrm {K L} \left(P _ {i} \mid \mu \otimes \nu\right) \right\}, \tag {2.4}
$$
with  $c_{i},\mu ,\nu$  , and  $\Pi_{\mu}^{\nu}$  defined exactly as before.