SlowGuess/NeurIPS_2023_OCR huggingface dataset:
a generalization of our framework to the settings of lossy translation and infinite translator families. Lastly, Appendix J proves sample complexity bounds for the settings of supervised translation.
2 The General Framework
We use  $f \colon \mathcal{X} \hookrightarrow \mathcal{Y}$  to denote a 1-1 function, in which  $f(x) \neq f(x')$  for all  $x \neq x'$ . For  $S \subseteq \mathcal{X}$ , we write  $f(S) \coloneqq \{f(x) \mid x \in S\}$ . The indicator  $\mathbf{1}_P$  is 1 if the predicate  $P$  holds, and 0 otherwise. The uniform distribution over a set  $S$  is denoted by  $\mathcal{U}(S)$ , and  $\log = \log_2$  denotes base-2 logarithm.
Language and Prior. A source language is a distribution  $\mu$  over a set of possible texts  $\mathcal{X}$ . Similarly, a target language is a distribution  $\nu$  over a set of possible texts  $\mathcal{Y}$ . When clear from context, we associate each language with its corresponding set of possible texts. A prior distribution  $\rho$  over translations  $\mathcal{Y}$  aims to predict the probability of observing each translation. One could naively take  $\rho = \nu$ , but Appendix H describes how better priors can focus on the domain of interest. Intuitively,  $\rho(y)$  measures how "plausible" a translation  $y$  is. For simplicity, we assume that  $\mathcal{X}, \mathcal{Y} \subseteq \{0,1\}^*$  are finite, non-empty sets of binary strings. Appendix I.2 discusses extensions to infinite sets.
**Translators.** A translator is a mapping  $f \colon \mathcal{X} \hookrightarrow \mathcal{Y}$ . There is a known set of 1-1 functions  $\{f_{\theta} \colon \mathcal{X} \hookrightarrow \mathcal{Y} \mid \theta \in \Theta\}$  with parameter set  $\Theta$ . Since parameters are assumed to be known and fixed, we will omit them from the theorem statements and algorithm inputs, for ease of presentation. Like  $\mathcal{X}, \mathcal{Y}$ , the set  $\Theta$  is assumed to be finite. Appendix I.1 considers translators that are not 1-1.
Divergence. A translator  $f_{\theta}$  and a distribution  $\mu$  induce a distribution over  $y = f_{\theta}(x)$ , which we denote by  $f_{\theta} \circ \mu$ . The divergence between this distribution and  $\rho$  is quantified using the Kullback-Leibler (KL) divergence,
$$
\mathrm {D} (\theta) := \mathrm {K L} (f _ {\theta} \circ \mu \parallel \rho) = \underset {x \sim \mu} {\mathbb {E}} \left[ \log \frac {\mu (x)}{\rho (f _ {\theta} (x))} \right] = \sum_ {x} \mu (x) \log \frac {\mu (x)}{\rho (f _ {\theta} (x))} \geq 0.
$$
Note that since  $\mathrm{D}(\theta) = \mathbb{E}_{x\sim \mu}\left[-\frac{1}{m}\sum \log \rho (f_{\theta}(x_i))\right] - \mathrm{H}(\mu)$ , and  $\mathrm{H}(\mu)$  is a constant independent of  $\theta$ , the MLE of Definition 1.1 approximately minimizes divergence.
Ground truth. In order to define semantic loss, we consider a ground-truth translator  $f_{\star}$  for some  $\star \in \Theta$ . We can then define the (ground-truth) translated language  $\tau = f_{\star} \circ \mu$  over  $\mathcal{V}$ , obtained by taking  $f_{\star}(x)$  for  $x \sim \mu$ . This is similar to the standard realizability assumption, and some of our bounds resemble Occam bounds with training labels  $y_{i} = f_{\star}(x_{i})$ . Of course, the ground-truth translator  $\star$  is not known to the unsupervised learning algorithm. In our setting, we further require that ground-truth translations never have 0 probability under  $\rho$ :
Definition 2.1 (Realizable prior).  $\operatorname{Pr}_{x\sim \mu}[\rho (f_{\star}(x)) = 0] = 0$  or equivalently  $\mathrm{D}(\star) < \infty$ .
Semantic loss. The semantic loss of a translator is defined with respect to a semantic difference function  $\ell \colon \mathcal{Y} \times \mathcal{Y} \to [0,1]$ . This function, unknown to the learner, measures the difference between two texts from the target language  $\mathcal{V}$ , with  $\ell(y,y) = 0$  for all  $y$ . For a given semantic difference  $\ell$  and ground-truth translator  $f_{\star}$ , we define the semantic loss of a translator  $f_{\theta}$  by
$$
\mathcal {L} (\theta) := \underset {x \sim \mu} {\mathbb {E}} \left[ \ell \left(f _ {\star} (x), f _ {\theta} (x)\right) \right].
$$
Of particular interest to us is the semantic error  $\mathrm{err}(\cdot, \cdot)$ , obtained when  $\ell$  is taken to be the 0-1 difference  $\ell_{01} = (y, y') = 1$  for all  $y' \neq y$ . Note that since any semantic difference  $\ell$  is upper bounded by 1, the semantic error upper-bounds any other semantic loss  $\mathcal{L}$ . That is,
$$
\mathcal {L} (\theta) \leq \operatorname {e r r} (\theta) := \Pr_ {x \sim \mu} \left[ f _ {\theta} (x) \neq f _ {\star} (x) \right].
$$
Section 3 analyzes this error  $\mathrm{err}(\theta)$ , which thus directly implies bounds on  $\mathcal{L}(\theta)$ .
3 Models of Language: Instantiating the General Framework
3.1 Random knowledge graphs
In this section, we define a model in which each text represents an edge between a pair of nodes in a knowledge graph. Both languages have knowledge graphs, with the source language weakly agreeing with an unknown subgraph of the target language.
We fix  $\mathcal{X} = X\times X = X^2$  and  $\mathcal{V} = Y\times Y = Y^{2}$  with  $r\coloneqq |X|$  and  $n\coloneqq |Y|$ . The set of translators considered is all mappings from the  $r$  source nodes to the  $n$  target nodes, namely  $\Theta_{XY} = \{\theta :X\hookrightarrow Y\}$  and  $f_{\theta}\big((u,v)\big)\coloneqq \big(\theta (u),\theta (v)\big)$ . The random knowledge graph is parametrized by the number of source nodes  $r$ , target node set  $\mathcal{V}$ , an edge density parameter  $p\in (0,1)$  representing the expected fraction of edges present in each graph, and an agreement parameter  $\alpha \in (0,1]$  representing the correlation between these edges. In particular,  $\alpha = 1$  corresponds to the case where both graphs agree on all edges, and  $\alpha = 0$  corresponds to the case where edges in the graphs are completely independent. These parameters are unknown to the learner, who only knows  $X$  and  $Y$  (and thus  $\mathcal{X} = X^2,\mathcal{V} = Y^2$ ).
Definition 3.1 (Random knowledge graph). For a natural number  $r \leq |Y|$ , the  $\mathrm{KG} = \mathrm{KG}(Y, r, p, \alpha)$  model determines a distribution over sets  $T, P \subseteq \mathcal{V}$  (which determine distributions  $\rho$  and  $\mu$ ). The sets  $T$  and  $P$  are sampled as follows:
1. Set  $P \subseteq \mathcal{V}$  is chosen by including each edge  $y \in \mathcal{V}$  with probability  $p$ , independently.
2. Set  $S \subseteq Y$  of size  $|S| = r$  is chosen uniformly at random.
3. Set  $T \subseteq S^2$  is chosen as follows. For each edge  $y \in S^2$ , independently,
(a) With probability  $\alpha$ ,  $y \in T$  if and only if  $y \in P$ .
(b) With probability  $1 - \alpha$ , toss another  $p$ -biased coin and add  $y$  to  $T$  if it lands on "heads"; that is,  $y \in T$  with probability  $p$ , independently.
It is easy to see that  $T \subseteq S^2$  and  $P \subseteq Y^2$  marginally represent the edges of Erdős-Rényi random graphs  $G_{r,p}$  and  $G_{n,p}$ , respectively. Moreover, the event that  $y \in T$  is positively correlated with  $y \in P$ : for each  $y \in S^2$ , since with probability  $\alpha > 0$  they are identical and otherwise they are independent. Formally, the equations below describe the probability of  $y \in T$  for each  $y \in S^2$  after we fix  $S$  and choosing  $T \subseteq S^2$ . Letting  $q \coloneqq (1 - p)$ , for each  $y \in S^2$ :
$$
\Pr [ y \in T ] = \Pr [ y \in P ] = p \tag {1}
$$
$$
\Pr [ y \in T \setminus P ] = \Pr [ y \in P \setminus T ] = (1 - \alpha) p q \tag {2}
$$
$$
\Pr [ y \notin P \mid y \in T ] = \Pr [ y \notin T \mid y \in P ] = \frac {(1 - \alpha) p q}{p} = (1 - \alpha) q \tag {3}
$$
The last equality, shows that the probability of excluding a random  $y \in T$  from  $P$  is smaller than the probability of excluding a random "incorrect translation"  $y' \neq y$ ,  $\operatorname{Pr}[y' \notin P] = q > (1 - \alpha)q$ .
We now describe how  $\rho, \tau$ , are determined from  $T, P$  and how  $\mu, \star$  may be chosen to complete the model description. The ground-truth target translated distribution  $\tau := \mathcal{U}(T)$  is uniform over  $T$ . The prior  $\rho$  is uniform over  $P$ , and then "smoothed" over the rest of the domain  $\mathcal{V}$ . Formally,
$$
\rho (y) := \left\{ \begin{array}{l l} \frac {1}{2} \cdot \left(\frac {1}{| P |} + \frac {1}{| \mathcal {Y} |}\right) & \text {i f} y \in P \\ \frac {1}{2 | \mathcal {Y} |} & \text {i f} y \notin P. \end{array} \right.
$$
The ground-truth translator  $\star \in \Theta$  is obtained by sampling a uniformly random  $\star : X \hookrightarrow S$ . Lastly, we take  $\mu = \mathcal{U}(f_{\star}^{-1}(T))$ , which agrees with the definition of  $\tau$ .<sup>8</sup>
Next, we state the main theorem for this model, formalizing Theorem 1.2 from the introduction.
Formally, the KG model outputs  $T$ ,  $P$  which may not determine  $S$  if some nodes have zero edges. In that case, we choose  $\star$  randomly among  $\theta$  such that  $f_{\theta}(\mathcal{X}) \supseteq T$ . In the exponentially unlikely event that either  $S$  or  $T$  is empty, we define both  $\tau, \rho$  to be the singleton distribution concentrated on  $(y, y)$  for the lexicographically smallest  $y \in Y$  and  $\mu$  to concentrated on  $(x, x)$  for  $x = f_{\star}^{-1}(y)$ . It is not difficult to see that MLE selects a translator with 0 error.
Theorem 3.2 (Translatability in the KG model). Fix any  $m \geq 1$ ,  $\emptyset \neq S \subseteq Y$ ,  $\delta, \alpha, p \in (0,1)$ , and let  $r \coloneqq |S|$ ,  $n \coloneqq |Y|$ ,  $q = 1 - p$ . Then, with probability  $\geq 1 - \delta$  over  $T$ ,  $P$  from  $\mathrm{KG}(S,Y,p,\alpha)$
$$
\operatorname {e r r} (\hat {\theta}) \leq \max  \left(\frac {6 4}{\alpha^ {2} p q ^ {2} r ^ {2}} \ln \frac {6 n ^ {r}}{\delta}, \frac {2}{\alpha q} \sqrt {\frac {2}{m} \ln \frac {6 n ^ {r}}{\delta}}\right),
$$
where  $\hat{\theta} = \mathrm{MLE}^{\rho}(x_1, x_2, \ldots, x_m)$  is from Definition 1.1. Simply, for  $p < 0.99$ , with probability  $\geq 0.99$ ,
$$
\operatorname {e r r} (\theta) = O \left(\frac {\log n}{\alpha^ {2} p r} + \frac {1}{\alpha} \sqrt {\frac {r \log n}{m}}\right).
$$
The proof, given in Appendix D, requires generalizing our theory to priors that have full support. Experimental validation of the theorem is described in Appendix E.
3.2 Common nonsense model
We next perform a "smoothed analysis" of arbitrary LMs  $\mu, \rho$  that are uniform over sets that share a small amount of randomness, i.e., a small common random set has been removed from both. This shared randomness captures the fact that some texts are implausible in both languages and that this set has some complex structure determined by the laws of nature, which we model as random.
The  $\alpha$ -common-nonsense distribution is a meta-distribution over pairs  $(\mu, \rho)$  which themselves are uniform distributions over perturbed versions of  $P, T$ . This is inspired by Smoothed Analysis [36]. Recall that  $\mathcal{U}(S)$  denotes the uniform distribution over the set  $S$ .
Definition 3.3 (Common nonsense). The  $\alpha$ -common-nonsense distribution  $\mathcal{D}_{\alpha}^{P,T}$  with respect to nonempty sets  $T \subseteq P \subseteq \mathcal{Y}$  is the distribution over  $(\rho = \mathcal{U}(P \cap S), \tau = \mathcal{U}(T \cap S))$  where  $S \subseteq \mathcal{Y}$  is formed by removing each  $y \in \mathcal{Y}$  with probability  $\alpha$ , independently.<sup>9</sup>
To make this concrete in terms of a distribution  $\mu$  on  $\mathcal{X}$ , for any ground-truth translator  $f_{\star}:\mathcal{X}\hookrightarrow \mathcal{Y}$  we similarly define a distribution  $\mathcal{D}_{\alpha ,\star}^{P,T}$  over  $(\mu ,\rho)$  where  $\mu \coloneqq \mathcal{U}(f_{\star}^{-1}(T\cap S))$  is the uniform distribution over the subset of  $\mathcal{X}$  that translates into  $\tau$ . We now state the formal version of Theorem 1.3.
Theorem 3.4 (Translatability in the CN model). Let  $\{f_{\theta}:\mathcal{X}\hookrightarrow \mathcal{Y}\mid \theta \in \Theta \}$  a family of translators,  $\star \in \Theta ,\alpha ,\delta \in (0,1 / 2],T\subseteq P\subseteq \mathcal{Y},$  and  $m\geq 1$  . Then with probability  $\geq 1 - \delta$  MLE run on  $\rho$  and  $m\geq 1$  iid samples from  $\mu$  outputs  $\hat{\theta}$  with,
$$
\mathrm {e r r} (\hat {\theta}) \leq \frac {6}{\alpha} \max \left(\frac {1}{m}, \frac {1 6}{| T |}\right) \cdot \ln \frac {6 | \Theta |}{\delta}.
$$
Note that the probability is over both  $(\mu, \rho)$  drawn from  $\mathcal{D}_{\alpha,\star}^{P,T}$ , and the  $m$  iid samples from  $\mu$ . More simply, with probability  $\geq 0.99$ ,
$$
\operatorname {e r r} (\hat {\theta}) = O \left(\frac {\log | \Theta |}{\alpha \min (m , | T |)}\right).
$$
When the amount of shared randomness  $\alpha$  is a constant, then this decreases asymptotically like the bound of supervised translation (Theorem J.1) up until a constant, similar to Theorem 3.2. For very large  $m$ , each extra bit describing the translator (increase by 1 in  $\log |\Theta|$ ) amounts to a constant number of mistranslated  $x$ 's out of all  $\mathcal{X}$ . The proof is deferred to Appendix C.
We also prove the following lower-bound that is off by a constant factor of the upper bound.
Theorem 3.5 (CN lower-bound). There exists constants  $c_{1}, c_{2} \geq 1$  such that: for any set  $T \subseteq \mathcal{V}$ , for any  $m \geq 1$ , any  $\alpha \in (0,1/2]$ , and any  $\Theta$  with  $c_{1} \leq \log |\Theta| \leq \alpha \min(m, |T|)$ , there exists  $\Theta'$  of size
<sup>9</sup>Again, in the exponentially unlikely event that either  $P \cap S$  or  $T \cap S$  is empty, we define both  $\tau, \rho$  to be the singleton distribution concentrated on the lexicographically smallest element of  $\mathcal{Y}$ , so MLE outputs a 0-error translator.
$|\Theta^{\prime}|\leq |\Theta |$  such that, for any  $P\supseteq T$  and any algorithm  $A^\rho :\mathcal{X}^m\to \Theta '$  , with probability  $\geq 0.99$  over  $\star \sim \mathcal{U}(\Theta ')$  and  $(\mu ,\rho)$  drawn from  $\mathcal{D}_{\alpha ,\star}^{P,T}$  and  $x_{1},\ldots ,x_{m}\sim \mu$
$$
\operatorname {e r r} (\hat {\theta}) \geq \frac {\log | \Theta |}{c _ {2} \alpha \min (m , | T |)},
$$
where  $\hat{\theta} = A^{\rho}(x_1,x_2,\ldots ,x_m)$
The only purpose of  $\Theta$  in the above theorem is to upper-bound the description length of translators, as we replace it with an entirely different (possibly smaller) translator family  $\Theta'$  that still has the lower bound using  $\log |\Theta| \geq \log |\Theta'|$ . Since  $\mathcal{U}(\Theta')$  is the uniform distribution over  $\Theta'$ , the ground-truth classifier is uniformly random from  $\Theta'$ . A requirement of the form  $\log |\Theta| = O\big(\alpha \min(m, |T|)\big)$  is inherent as otherwise one would have an impossible right-hand side error lower-bound greater than 1, though the constants could be improved.
The proof of this theorem is given in Appendix C.2, and creates a model with  $O(\log n)$  independent "ambiguities" that cannot be resolved, with high probability over  $S, x_1, x_2, \ldots, x_m$ . Experimental validation of the theorem is described in Appendix E.
We have given a framework for unsupervised translation and instantiated it in two stylized models. Roughly speaking, in both models, the error rate is inversely related to the amount of samples, common ground, and the language complexity. The first two relations are intuitive, while the last is perhaps more surprising. All error bounds were information-theoretic, meaning that they guarantee a learnable accurate translator, but learning this translator might be computationally intensive.
In both models, the translators are restricted. In the knowledge graph, the translators must operate node-by-node following an assumed compositional language structure. $^{10}$  In the common nonsense model, the restriction is based on the translator description bit length  $\log |\Theta|$ . To illustrate how such restrictions can be helpful, consider block-by-block translators which operate on limited contexts (e.g., by paragraph). Consider again the hypothetical example of Figure 1. Suppose the three texts are outputs of three translators  $\Theta = \{A,B,C\}$ . Let us suppose that translator A always produces accurate and natural translations, and further that all translators work paragraph-by-paragraph, as modern translation algorithms operate within some limited context window. In fact, one can imagine the translators of different paragraphs as a set of isolated adversaries where each adversary is trying to mistranslate a paragraph, knowing the ground-truth translation of their paragraph, while attempting to maintain the plausibility of the entire translation. If only the first-paragraph adversary mistranslates reef to ocean basin, then the translation lacks coherence and is unlikely. If the adversaries are in cahoots and coordinate to all translate reef to ocean basin, they would generate: Have you seen mom? I just returned from the ocean basin. At the basin, there were a lot of sea turtles, which has low probability  $\approx 10^{-25}$ , presumably because encoded in GPT-3's training data is the knowledge that there are no turtles deep in the ocean near the basin. While the adversary could also decide to change the word turtle to something else when it appears near basin, eventually it would get caught in its "web of deceit." The intuition is that, across sufficiently many translations, the prior will not "rule out" the ground-truth translations while very incorrect translators will be ruled out.
Judging success. Our analysis sheds some light on whether it is even possible to tell if translation without parallel data (UMT) is successful. A positive sign would be if millions of translations are fluent English accounts that are consistent over time across translations. In principle, however, this is what LM likelihood should measure (excluding consistencies across translations which sufficiently powerful LMs may be able to measure better than humans). We also considered a statistical distance (KL divergence) between the translations  $f_{\hat{\theta}}(x)$  for  $x \sim \mu$  and the prior  $y \sim \rho$ , and  $\mu$  could be estimated given enough samples. If this distance is close to zero, then one can have predictive accuracy regardless of whether the translations are correct. This raises a related philosophical quandary: a situation in which two beings are communicating via an erroneous translator, but both judge the conversation to be natural.
That is, we assume that each translator has a latent map from nodes in the source graph into nodes in the target graph, and edges are mapped from the source to target graphs in the natural way. The study of compositional communication systems, among humans and animals, has played a central role in linguistics [39].
We thank Madhu Sudan, Yonatan Belinkov and the entire Project CETI team, especially Pratyusha Sharma, Jacob Andreas, Gasper Begus, Michael Bronstein, and Dan Tchernov for illuminating discussions. This study was funded by Project CETI via grants from Dalio Philanthropies and Ocean X; Sea Grape Foundation; Rosamund Zander/Hansjorg Wyss, Chris Anderson/Jacqueline Novogratz through The Audacious Project: a collaborative funding initiative housed at TED.
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A Translator Revisions and Plausible Ambiguities
Even though  $\star$  is unknown, it will be convenient to define, for each  $\theta \in \Theta$ , a revision of  $f_{\star}$  which is the permutation  $\pi_{\theta}^{\star} \colon \mathcal{V} \hookrightarrow \mathcal{V}$  between  $f_{\theta}$  and  $f_{\star}$ ,  $\pi_{\theta}^{\star}(y) \coloneqq f_{\theta}(f_{\star}^{-1}(y))$ .<sup>11</sup> We write  $\pi_{\theta} = \pi_{\theta}^{\star}$  when  $\star$  is clear from context, and  $\pi_{\star}(y) \equiv y$  is the identity revision.
Note that the divergence  $\mathrm{D}(\theta)$  and semantic loss  $\mathcal{L}(\theta)$  can equivalently be defined using revisions,
$$
\mathrm {D} (\theta) := \mathrm {K L} (f _ {\theta} \circ \mu \| \rho) = \mathrm {K L} (\tau \| \pi_ {\theta} ^ {- 1} \circ \rho) = \underset {y \sim \tau} {\mathbb {E}} \bigl [ \log \frac {\tau (y)}{\rho (\pi_ {\theta} (y))} \bigr ],
$$
$$
\mathcal {L} (\theta) := \underset {x \sim \mu} {\mathbb {E}} \left[ \ell \left(f _ {\star} (x), f _ {\theta} (x)\right) \right] = \underset {y \sim \tau} {\mathbb {E}} \left[ \ell \left(y, \pi_ {\theta} (y)\right) \right].
$$
To relate divergence and loss, it is helpful to define a notion of plausible ambiguities which are  $\theta$  whose revisions  $\pi_{\theta}(y\sim \tau)$  are not too unlikely under the prior. These may also be of independent interest as they capture certain types of ambiguities that can only be resolved with supervision.
Definition A.1 ( $\gamma$ -Plausible ambiguities). For any  $\gamma \in [0,1]$ ,  $\star \in \Theta$ , and distributions  $\tau, \rho$  over  $\mathcal{V}$ , the set of  $\gamma$ -plausible ambiguities  $\mathcal{A}_{\gamma} = \mathcal{A}_{\gamma}^{\star, \tau, \rho} \subseteq \Theta$  is: