Year: 2021

Abstract:
A fundamental characteristic of natural language is the high rate at which speakers produce novel expressions. Because of this novelty, a heavy-tail of rare events accounts for a significant amount of the total probability mass of distributions in language (Baayen, 2001). Standard language modeling metrics such as perplexity quantify performance of language models (LM) in aggregate. As a result, we have relatively little understanding of whether neural LMs accurately estimate the probability of sequences in this heavy-tail of rare events. To address this gap, we develop a controlled evaluation scheme which uses generative models trained on natural data as artificial languages from which we can exactly compute sequence probabilities. Training LMs on generations from these artificial languages, we compare the sequence-level probability estimates given by LMs to the true probabilities in the target language. Our experiments reveal that LSTM and Transformer language models (i) systematically underestimate the probability of sequences drawn from the target language, and (ii) do so more severely for lessprobable sequences. Investigating where this probability mass went, (iii) we find that LMs tend to overestimate the probability of ill-formed (perturbed) sequences. In addition, we find that this underestimation behaviour (iv) is weakened, but not eliminated by greater amounts of training data, and (v) is exacerbated for target distributions with lower entropy.

text: A fundamental characteristic of natural language is the high rate at which speakers produce novel expressions. Because of this novelty, a heavy-tail of rare events accounts for a significant amount of the total probability mass of distributions in language (Baayen, 2001). Standard language modeling metrics such as perplexity quantify performance of language models (LM) in aggregate. As a result, we have relatively little understanding of whether neural LMs accurately estimate the probability of sequences in this heavy-tail of rare events. To address this gap, we develop a controlled evaluation scheme which uses generative models trained on natural data as artificial languages from which we can exactly compute sequence probabilities. Training LMs on generations from these artificial languages, we compare the sequence-level probability estimates given by LMs to the true probabilities in the target language. Our experiments reveal that LSTM and Transformer language models (i) systematically underestimate the probability of sequences drawn from the target language, and (ii) do so more severely for lessprobable sequences. Investigating where this probability mass went, (iii) we find that LMs tend to overestimate the probability of ill-formed (perturbed) sequences. In addition, we find that this underestimation behaviour (iv) is weakened, but not eliminated by greater amounts of training data, and (v) is exacerbated for target distributions with lower entropy.

heading: 1 INTRODUCTION
text: Natural language is fundamentally creative—speakers and listeners frequently produce and comprehend sentences which have never been produced before (Fodor, 1975; Fodor & Pylyshyn, 1988; Chomsky, 1975, 1955). As a side-effect of this property, distributions in natural language are characterized by a heavy-tail of individually improbable events which collectively account for a significant amount of the total probability mass of the distribution (Khmaladze, 1988; Baayen, 2001). Precisely approximating this large number of rare events is one of the foundational challenges for models of natural language (Good, 1953; Jelinek, 1980; Katz, 1987; Kneser & Ney, 1995; Wood et al., 2011; Goldwater et al., 2011). Autoregressive neural language models (Bengio et al., 2003; Mikolov et al., 2013; Radford et al., 2019) attempt to do so by decomposing the probability of an event (a sequence) into a series of conditional distributions, each parameterized by a shared neural network.
Recently, a growing body work has sought to understand how these language models (LM) fit the distribution of a language beyond standard measures such as perplexity. Meister & Cotterell (2021), for example, investigated the statistical tendencies of the distribution defined by neural LMs, whereas Kulikov et al. (2021) explored whether they adequately capture the modes of the distribution they attempt to model. At the same time, increased focus has been given to performance on rare or novel events in the data distribution, both for models of natural language (Lent et al., 2021; Dudy & Bedrick, 2020; Oren et al., 2019) and neural models more generally (see, for example Sagawa et al., 2020; D’souza et al., 2021; Chen et al., 2021; Blevins & Zettlemoyer, 2020; Czarnowska et al., 2019; Horn & Perona, 2017; Ouyang et al., 2016; Bengio, 2015; Zhu et al., 2014). Neither of these branches of work, however, has explored instance-level LM performance on rare sequences in the distribution. As a result, we have relatively little understanding of how neural LMs approximate sequences in the heavy-tail characteristic of natural language.
In this work, we introduce a controlled methodology to explore how LMs estimate the probability of sequences in the heavy-tail of the distribution. Our instance-level evaluation scheme explicitly compares the target probability distribution of the language to the distribution defined by the LM. Since the true distribution of any natural language is in practice unknown, we use a Transformer LM trained on natural data as a generative model to define target artificial languages for which we can exactly compute sequence probabilities. Training LSTM and Transformer LMs on sequences sampled from these target artificial languages, we compare the sequence-level probability estimates given by neural LMs to the target probabilities in the language. By controlling the entropy of the generative model’s conditional distributions, we create a set of artificial languages with varying distributional properties, and analyze how LM estimation behaviour is modulated by the properties of the target distribution.
Our experiments uncover the extent to which neural LMs provide a distorted fit of the language they are trained to model. We find that LSTM and Transformer LMs (i) systematically underestimate the probability of sequences drawn from the target language and (ii) do so more when such sequences are rare. Where did this underestimated probability mass go? We do not find that the underestimation is accompanied by overestimation in the head of distribution. Rather, we find that LMs tend to (iii) overestimate the probability of rare perturbed (ill-formed) sequences. Interpreted together, these findings indicate that on the one hand, neural LMs under-represent well-formed sequences in the tail of the language they attempt to model, and on the other hand, over-represent ill-formed sequences far away from high probability zones in sequence-space. In addition, we find that (iv) greater amounts of training data lessen underestimation but do not eliminate it and that (v) underestimation is exacerbated for target distributions with lower entropy.

heading: 2 BACKGROUND
text: We begin by briefly characterizing why distributions with a large number of rare events (LNRE) emerge in natural language, and why these events pose challenges for LMs. Furthermore, we motivate the need for instance-level evaluation when dealing with a large number of rare events.
Productivity In the context of language production, a language user has the ability to produce, at any given point in their linguistic lifespan, an utterance which they have never produced before. This creativity is the result of the generative property of productivity, which states that on the basis of finite linguistic experience, a language user can produce and comprehend an unbounded number of grammatically acceptable utterances (Chomsky, 1975, 1955). Productive processes induce a distribution which places non-zero probability on unseen events at all practical sample sizes. Because of this property, many of the distributions in natural language—particularly the distribution over the sequences of a language—are characterized by a heavy-tail of rare events.
LNRE Zone To make explicit the connection between productivity and a heavy-tail of rare events, let PN denote the probability of sampling a novel (previously unseen) event from some distribution after having sampled N events. Then productivity as described above states that PN > 0 for all sample sizes N that occur in practice. The range of sample sizes N for which it is the case that PN > 0 is known as the LNRE zone (Khmaladze, 1988; Baayen, 2001). The LNRE zone for natural language appears to be very large, and it seems likely that PN will remain greater than 0 for samples of natural language many orders of magnitude larger than all the data currently available for training LMs.1 In the LNRE zone, it is difficult to obtain accurate estimates of the probability of events using straightforward maximum likelihood estimation (MLE). Accounting for this enormous amount of novelty is thus a central challenge in language modeling.
Language modeling A model M of the language L attempts to define a distribution pM over variable length sequences x = (x1, . . . , x|x|) which closely resembles the true distribution of the language pL. That is, pM (x) ≈ pL(x) for all x ∈ Σ∗, where Σ denotes the vocabulary of M , and Σ∗ is the set of all strings of finite length, known as the Kleene closure of Σ. In the LNRE zone, this means that a LM must define a distribution over a support containing a very large set of sequences
1For an empirical validation of this claim on a sample of practical size from the OpenWebText corpus, see the Appendix.
which have never occurred in a training corpus (or equivalently, have all occurred with frequency 0), which take on a very wide array of differing probabilities. For example, while the sequences x1,x2 ∈ Σ∗ have likely never occurred in any sample of English, most would agree that x1 is far more probable than x2:
x1: The East pond in Parc Lafontaine was filled to the brim with diet Coke. x2: Certain nak indicate liberationing among theorter codity voters vandalized.
LM Evaluation For a perfect LM of English, we would expect the estimated probabilities of the sequences x1 and x2 to match their probabilities under the true distribution pEnglish. However, since pEnglish and it’s underlying generative process are unknown, it is not possible to explicitly evaluate how closely instance-level probability estimates align. As a proxy, the mean perplexity of the model on a holdout set of sequences D is typically used:
PP (pM ,D) = 1
|D| |D|∑ i=1 exp − 1|x(i)| |x(i)|∑ t=1 log pM (x (i) t | x (i) <t)  (1) which measures whether the model, on average, assigns high likelihood to unseen instances. This measure does not, however, tell us whether instance-level estimates align with their true counterparts, nor is it necessarily indicative of performance on rare, idiosyncratic events in D. In this way, the lack of access to the ground-truth distribution severely complicates LM evaluation on the heavy-tail of rare sequences in language. In the following section, we introduce a methodology to overcome these limitations.

heading: 3 LANGUAGE MODEL EVALUATION IN THE LNRE ZONE
text: We propose evaluating language model performance on the heavy-tail of rare events by working with a known probability distribution over sequences. Specifically, we train a Transformer LM on sequences sampled from a corpus of natural language to define a generative model L. The distribution over sequences induced by a sampling scheme from L, denoted pL, is then our artificial language. We expect a model M of this artificial language to assign probabilities pM (x) to sequences x which match the target probabilities pL(x) of x under pL. Here we define pL using an ancestral sampling scheme with softmax T = 0.85. One could, for example, define an artificial language pL′ by sampling from the top-k tokens of the distribution defined by L at each time step, or using an ancestral sampling scheme with softmax T = T ′. To characterize neural LM behaviour on rare events, we train Transformer and LSTM LMs on data sampled from pL, and compare the instance-level probability estimates given by pM to target probabilities under pL. We summarize the components of this methodology in Table 1, and overview it in greater detail in the following section.

heading: 3.1 ARTIFICAL LANGUAGES AS TARGET DISTRIBUTIONS
text: To define a generative model L, we train a randomly-initialized GPT2-medium on 1.5M sentences sampled from the OpenWebText corpus (Gokaslan & Cohen, 2019).2 We set the maximum sequence
2All Transformer implementations were obtained from Huggingface, and training was done on 2 RTX-8000 GPUs.
length to be 128 tokens. We additionally train a byte-pair-encoding (BPE) tokenizer on this data set with a standard GPT2 vocabulary size of 57,256 tokens. For simplicity, this tokenizer is used for all models.
Using this generative model L, we define the target distribution over sequences—the artificial language—as the distribution induced by an ancestral sampling scheme from L. Thus, we draw instances x = (x1, . . . , x|x|) from our language pL by recursively sampling from the conditional distribution over tokens at each time step: xt ∼ pL(· | x<t) where x0 = BOS and xt = EOS. All experiments up until Section 4.5 are conducted on the distribution induced by ancestrally sampling from L with softmax temperature T = .85. In Section 4.5, we explore the effects of different values of T when sampling from pL. Table 2 shows three sequences sampled from this distribution.

heading: 3.2 SEQUENCE PROBABILITY ESTIMATION IN THE LNRE ZONE
text: Given an artificial language pL, the task of M—the language model—is to define a distribution pM whose sequence-level probability estimates closely align with the sequence-level probability estimates given by pL. We refer to any deviation from this desiderata as model estimation error. To quantify the model estimation error for a sequence x, we take the difference between the sequence’s log probability under M and its true log probability under L:
error(x) = log pM (x)− log pL(x) (2)
This quantity is the log probability ratio, which measures, in log-space, the number of times more or less likely the sequence x is under the language model M . Note that error(x) < 0 indicates that M underestimates the probability of x, whereas error(x) > 0 indicates that M overestimates the probability of x. In practice, we train M on a set of sequencesDtrain sampled from pL, and compute model estimation error on a separate set of sequences Dtest sampled from pL. In all cases, we compute the probability of a sequence x as its chain rule decomposition: p(x) = ∏|x| i=1 p(xi | x<i) where x0 = BOS and x|x| = EOS. When computing the ground-truth sequence probabilities, we take into account any softmax tempering.

heading: 3.3 NEURAL LANGUAGE MODELS
text: We study the estimation performance of two neural LM architectures: the Transformer (Vaswani et al., 2017) and the LSTM (Melis et al., 2020). When training either architecture, we halve the learning rate if validation loss increases at the end of an epoch. For all model sizes, we use a batch size of 128 sequences. Models with the lowest cross-entropy loss on a withheld validation set are used in experiments unless otherwise mentioned.
GPT2-small & GPT2-medium We use the Huggingface (Wolf et al., 2020) implementations of GPT2-small and GPT2-medium (Radford et al., 2019) as representative Transformer LMs. All model training was done from scratch on two RTX-8000 GPUs with mixed floating point precision. We use Adam Optimization with = 1e−8 and learning rates α = 5e−5 and α = 4e−5 for GPT2small and GPT2-medium, respectively. Since these models are in the same model class as our artificial target language pL, we expect the task of recovering the ground-truth distribution to be relatively easy compared to the true problem faced in modeling natural language, where both the distribution and the underlying generative process are unknown.
LSTM For the LSTM, we follow the implementation of the baseline LM described in (Hochreiter & Schmidhuber, 1997; Melis et al., 2020). We use 2 layers and adjust the hidden state and embedding dimension (2048 and 1024, respectively) to be such that the total number of parameters is approximately equal to GPT2-small (110M).

heading: 4 RESULTS

heading: 4.1 ESTIMATION ERROR WITH FIXED DATA
text: We begin by exploring model estimation error on a fixed training setDtrain of 1M sequences sampled from pL. We first train LSTM and GPT models on Dtrain, early-stopping as described above. Following training, we sample a test set Dtest of 500,000 sequences from pL, and score each sequence under both the model distribution pM and the true language distribution pL. From this, we obtain a set of probability estimates: Stest = {〈pL(x), pM (x)〉 | x ∈ Dtest}. If the model M perfectly models the language L, then for each 〈pL(x), pM (x)〉 ∈ Stest we would expect pL(x) = pM (x). Figures 1(A) and 1(B) visualize this relationship with the x- and y-axes denoting the true and model estimated sequence probabilities respectively, and a dashed line representing equality. To compare probability estimates, we represent the set Stest in the form of a joint histogram over this coordinate space. Histogram bins are shaded based on the number of tuples which lie in the coordinate range they define. Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.
Figures 1(A) and 1(B) provide evidence for distributional distortion in the form of underestimation. The majority of probability tuples in Stest lie to the right of the identity line, indicating that LSTM and GPT2 models consistently underestimate the probability of sequences sampled from pL. Furthermore, the distance between the identity line and the probability tuples grows non-linearly as function of the true sequence probability, indicating that underestimation error is more severe for rarer sequences in the language. We validate both of these observations in the right-most plot of Figure 1, which shows mean estimation error decreasing non-linearly as a function of the target sequences probability.3 In addition, comparing underestimation behaviour across model size, we find that while GPT2-medium performs slightly better than GPT2-small, these improvements are typically within the range defined by the bootstrapped 95% confidence intervals.

heading: 4.2 ESTIMATION ERROR ACROSS TRAINING TIME
text: To understand the training dynamics underlying the previously reported underestimation, we compute model probability estimates on subsets of Dtrain and Dtest at the end of each training iteration i. Once computed, we sort each set of probability tuples S(i)train and S (i) test by their target sequence probabilities pL(x), and split the probability tuples into 50 equally-sized bins. We plot estimation curves in Figure 2: Each curve represents a 50th of the sequences, with darker curves denoting estimation error for sequences with lower target probabilities (rarer sequences). At any given point, then, the distance between estimation curves represents the degree to which estimation error is dependent on the target probability of the sequence.
3To compute this curve, we split the target sequence probability pL(x) range into N equally sized bins (by probability range). We report the mean estimation error for each bin with > 10 sequences. We additionally compute 95% confidence intervals with 10, 000 bootstraps for each mean, resampling n equal to the number of sequences in the given bin.
Figure 2 left visualizes underestimation error for sequences seen in training. Around the fifth epoch, estimation error for train sentences converges to zero, that is (pL(x) ≈ pM (x)), indicating that GPT2-medium is able to almost perfectly recover the target probabilities of training sequences no matter their target probability. At the same time, this convergence happens almost simultaneously for all sequences, indicating that a complete reduction in error during training occurs throughout the entire range of target sequence probabilities.
Figure 2 right visualizes GPT2-medium model’s performance on a separate set of test sequences. First, unlike for Dtrain, estimation error for Dtest does not converge to zero, meaning that even when the model has perfectly recovered the target probability of train sequences, the target probabilities for test sequences remain underestimated. Second, in the case of Dtrain, the difference between estimation curves of different shades converges to zero, indicating that estimation performance becomes uniform across the distribution of train sequences. We do not see such behaviour in Dtest. Instead, the error curves remain at a relatively consistent distance from one another, indicating that the discrepancy in estimation error at different parts of the distribution is unchanging for sequences not seen during training.

heading: 4.3 ESTIMATION ERROR BY AMOUNT OF TRAINING DATA
text: Our previous experiment trained languages models on a set of 1M sequences. A plausible explanation for the model’s underestimation behaviour on unseen test sequences is therefore that the language model has not seen enough data. Here we explore how estimation error varies as a function of the amount of training data. We train GPT2-medium and an LSTM model in the online “Ideal World” setting (Nakkiran et al., 2020) by sampling, at the beginning of each training iteration, a fresh set of 500,000 sequences from pL, and training M on this sample. Doing so for 60 iterations, we obtain LMs which have been trained on 30 million sequences. We compute model estimation
error on Dtest at the end of each iteration i. Figure 3 visualizes underestimation error throughout training for GPT2-medium and LSTM language models. We again split test sequences by their true probability, with darker lines denoting estimation trends for less probable target sequences.
The estimation curves in Figure 3 suggest that while increasing the amount of data in training initially leads to lower estimation error, this reduction eventually asymptotes. In figure 3, we visualize the relative change in mean estimation between epochs i − 1 and i. Relative change in estimation error eventually fluctuates around 0 (minimal change) for the majority of the distribution. Comparing architectures, we find that the Transformer is significantly more efficient at reducing mean estimation error throughout the distribution.

heading: 4.4 WHERE DID THE PROBABILITY MASS GO?
text: In the previous section, we saw that even when increasing the amount of training data, pM consistently underestimates the probabilities of sequences sampled from the tail of pL. At the same time, we did not find that pM overestimated sequences in the head of pL. Under the assumption that pM is a proper probability distribution,4 that is, ∑ x∈Σ∗ pM (x) = 1, these findings suggest that there exists sequences in Σ∗ whose probability is overestimated by the model. In this section, we investigate where this probability mass went.
To do so, we compute model estimation error on perturbed sequences from pL—sequences in Σ∗ which are increasingly far away from the high-probability zones in pL. We build a corpus of perturbed sequences by recursively applying 30 random perturbations to each sequence x ∈ Dtest. Formally, the set of sequences at perturbation step i can be expressed as: D(i)perturbed = {PERTURB(x) | x ∈ D (i−1) perturbed} where PERTURB(x) is a function which returns a novel perturbed version of x, and D(0)perturbed = Dtest. Sequence perturbation operations are shown in Table 3. While it is possible that these
operations produce other well-formed strings, we expect this to be a relatively rare outcome. We score each of these 15M sequences under both the target generative model pL and the LM pM . Note that we use as LM the GPT2-medium model from the previous section (trained on 30M sequences).
Figure 4 visualizes GPT2-medium’s mean estimation error for these 15M sequences across two dimensions. On the x-axis, we plot the target probability of the sequence under pL, and on the yaxis, the number of perturbations performed on the sequence. For example, the bottom-left corner (1) of Figure 4 visualizes estimation behaviour for rare sequences sampled directly from pL, whereas
4See Welleck et al. (2020a) for discussion on consistency in the context of neural language model decoding.
the top-left corner (2) visualizes estimation error sequences which are equally rare, but which have been perturbed up to 30 times.
Figure 4 offers a nuanced characterization of underestimation behaviour. The brown area on the bottom of the figure re-states the underestimation findings of the previous section. When increasing the number of perturbations performed, however, we begin entering into a space of sequences which are at first well-estimated by pM (the white areas) but then are quickly overestimated by pM (the dark blue areas), confirming that there are indeed sequences in Σ∗ which are overestimated by the language model. Furthermore, these findings suggest that the tail of rare events defined by the language model does not match the tail of the artificial language—the rare events typical in pL are under-represented in pM in favour of other sequences in Σ∗. See Section A.1 for experiments finding that random sequences from Σ∗ are also overestimated by pM .

heading: 4.5 MODULATING THE SHAPE OF THE TARGET DISTRIBUTION
text: Up to this point, the target artificial language pL was given as the distribution induced by an ancestral sampling scheme with softmax T = 0.85 from the generative model L. In the previous section, we saw that pM placed excess probability mass on areas in Σ∗ with low-probability under pL. Here we modulate the shape of the sequence space defined by pL to investigate how estimation error varies with respect to systematic interventions in the target distribution. To adjust the way that pL allocates probability mass over Σ∗, we control the entropy of the conditional distributions at each generation step t by dividing the pre-softmax logits by a temperature value T .5
We visualize the effects of T on the shape of the distribution in the left of Figure 5. By increasing the value of T , we increase the entropy of the distributions over next tokens, which in turn, spreads probability mass across a larger number of sequences in Σ∗. We define four artificial languages with varying T and train GPT2-medium on an ancestral sample of 1M sequences from each of these artificial languages. Figure 5 visualizes model estimation error by true sequence probability for each model. We find that models trained on languages with increased entropy perform comparatively better than models trained on low entropy languages. Estimation error for models trained on languages with greater T is less severe, and this holds throughout nearly all target sequence probabilities. These results indicate that neural LMs provide a more accurate approximation of target distributions which spread mass more uniformly across Σ∗.

heading: 5 RELATED WORK
text: This paper contributes to recent work investigating the properties of the distributions defined by LMs. Prior studies have focused on exploring (Takahashi & Tanaka-Ishii, 2019; 2017) and developing frameworks (Meister & Cotterell, 2021) to better understand whether the large-scale statistical
5Formally, for the ith component of the length K pre-softmax logits x, this operation is given as:
SOFTMAX(x)i = exp(
xi T )∑K j=1 exp ( xj T )
tendencies of natural language, such as Zipf’s law (Zipf, 1949), are captured by LMs. We take a more fine-grained approach, proposing a methodology which draws off of instance-level evaluation schemes (Zhong et al., 2021) and the experimental control afforded by artificial corpora (White & Cotterell, 2021; Papadimitriou & Jurafsky, 2020). Indeed, closely related to our work is Kulikov et al. (2021)’s, in which artificial corpora produced by generative models were used to explore mode recovery in neural language modeling. Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional discriminative models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020b), the consistency of decoding algorithms (Welleck et al., 2020a), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg & Byrne, 2019).
We additionally contribute to the body work seeking to characterize and adapt neural model performance on rare or novel examples and classes (Horn & Perona, 2017; Bengio, 2015). In the context of language modeling, Lent et al. (2021) explored performance on under-resourced languages, whereas Oren et al. (2019) did so on under-represented domains in training corpora. Focusing on the word frequency distribution, Dudy & Bedrick (2020) found that LMs under-perform when less frequent examples are encountered at test time. In the classification setting, various approaches have been proposed to help alleviate class imbalance in the data distribution, such as data augmentation (Sagawa et al., 2020) or the transfer of knowledge from high-frequency classes to infrequent ones (Ouyang et al., 2016; Zhu et al., 2014; Chen et al., 2021). Prior to the current neural paradigm (Bengio et al., 2003), multiple approaches have been proposed to deal with the heavy-tail, such as smoothing and back-off approaches in statistical n-grams (Chen & Goodman, 1999) and two-stage Bayesian approaches (Goldwater et al., 2006). We hope our proposed evaluation scheme and empirical contribution can help continue to drive progress on this front.

heading: 6 CONCLUSION
text: Emerging as a result of a language user’s ability to produce and comprehend novel expressions, the heavy-tail of rare events is one of the fundamental features of distributions in natural language. In this work, we introduce a controlled methodology to evaluate instance-level LM performance on this set of individually rare but collectively frequent events. We use generative models trained on natural language corpora to define a set of artificial languages for which we can exactly compute the probability of sequences. Training LSTM and Transformer LMs on sequences sampled from these artificial languages, our analysis compares the probability estimates given to sequences by the LMs to the target probabilities of sequences under the artificial language.
Our results indicate that neural LMs systematically under-represent sequences in the tail of the target distribution, even when increasing the amount of the training data. Investigating where this probability mass went, our perturbation experiments reveal that neural LMs do not tend to overestimate the head of the distribution, but rather overestimate the probability of sequences outside those typical in the target distribution. Comparing model performance on target distributions with varying properties, we find that neural LMs tend to provide more accurate approximations of distributions with greater entropy. Interpreted together, these results indicate that autoregressive neural language models have a tendency to spread probability mass too uniformly across the space of possible sequences.
Finally, we would like to acknowledge that we do not know the degree of structural difference between our Transformer-generated ground-truth distributions and the distributions of actual natural languages. It is likely that the distribution defined by our ground truth models is less structured than the distribution of a natural language. Therefore, it is possible that some systematic difference between natural language distributions and our ground-truth distributions may affect our results to a certain degree. That being said, our experiments in Section 4.5 suggest that it may actually be easier for neural LMs to learn less structured distributions, and we expect the task of recovering a ground-truth distribution to be made easier when the target distribution and LM are in the same model class. Nevertheless, future work should seek to conduct similar experiments using groundtruth distributions with more explicit structure.

Review summaries:
The paper conducts experiments to evaluate whether neural sequence models such as LSTMs and Transformers are able to correctly assess the probability of rare sentences, which collectively constitute a large probability mass in natural language productions (heavy-tail phenomenon). In order to do so, it performs experiments in a controlled synthetic environment where a first language model $P_L$ is trained on a corpus of natural sentences, and a second model $P_M$ is trained to emulate the first model. The authors observe that the second model systematically tends to underestimate the probability of rare $P_L$ sentences, the more so the rarer such sentences are, and show that some artificially corrupted sentences tend to receive higher probability from $P_M$ than from $P_L$, partly explaining where the missing probability mass over rare well-formed sentences went.  

This paper investigates how language models allocate their probability mass, with an emphasis on rare sequences that are part of the 'heavy tail' of the distribution of natural language sequences. The authors use a language model to define a target distribution with access to samples and ground-truth probabilities. A language model is trained on an empirical estimate of the target distribution, and the authors study the gap between the learned model's and target distribution's probability assignments.   Using this methodology, the authors uncover various interesting phenomena: the model systematically assigns lower probabilities than the target distribution, but assigns unusually high probabilities to unnatural, perturbed sequences (suggesting an explanation for where the probability mass moved to). The authors include several fine-grained analyses with additional interesting findings.