Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Rather, we find that LMs tend to (iii) overestimate the probability of rare perturbed (ill-formed) sequences.

The review comment to be evaluated is:

* Strengths: A very well written and clear paper, addressing a focussed question through a well-designed set of experiments on synthetic data allowing to precisely control experimental conditions.

* Weaknesses: No major weakness, but some mild issues and areas for improvement, see the comments and questions below.

* Comments and Questions:

    * (Minor Comment) I did not find the description of the LNRE Zone in section 2 to bring much to the discussion. In particular, characterizing this zone through the condition that $\mathcal{P}_N$ is not __null__ is almost empty of content for natural language, because it would in particular mean that we could precisely delimit what constitutes a possible "natural" sentence, and that the LNRE Zone should cover all such sentences. Here the formal description as given does not clarify anything.
    * (Question) In some of your experiments, you train a $P_M$ GPT2-medium model on 30M sentences sampled from the $P_L$ GPT2-medium model, that is, $P_M$ here uses the same architecture as $P_L$ (if I understand correctly). However, the $P_M$ model still underestimates the probabilities relative to $P_L$. Do you think this is a symptom of 30M sentences being too few, or more a symptom that $P_M$ is stuck in some local minimum during the optimization ? After all, (again, if I understand the setup correctly), using the same architecture, $P_M$ would make it possible, in principle, to just use the same parameters as $P_L$, and then the mismatch could not occur. Perhaps this would be worth a note in the text?
    * (More important Question/Comment) In section 4.4, you perform some perturbations, and observe that they lead to higher $P_M$ than $P_L$. First, I think it would be worth defining these perturbations in more detail than what you do. Second, you do not discuss your choice for these perturbations nor analyze why they have (relatively) high probability relative to $P_M$. However, the fact that, for instance, swapping two words or deleting a word in a natural sentence is not severely penalized by $P_M$ is an interesting but specific fact, which may have to do with a special tolerance of $P_M$ for such operations, rather than for others. To give one example, if the perturbations consisted solely in replacing a word in the natural sentence by a _rare_ word in the training set, would you observe such a difference between $P_M$ and $P_L$? Overall, could you discuss this aspect of your experiments?

The evidence to be evaluated is: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.

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.

The LNRE zone for natural language appears to be very large, and it seems likely that \(\mathcal{P}_{N}\) 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).

For example, while the sequences \(\mathbf{x}_{1},\mathbf{x}_{2}\in\Sigma^{*}\) have likely never occurred in any sample of English, most would agree that \(\mathbf{x}_{1}\) is far more probable than \(\mathbf{x}_{2}\):

For a perfect LM of English, we would expect the estimated probabilities of the sequences \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) to match their probabilities under the true distribution \(p_{\text{English}}\).

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 \(\mathcal{D}\).

If the model \(M\) perfectly models the language \(L\), then for each \(\left\langle p_{L}(\mathbf{x}),p_{M}(\mathbf{x})\right\rangle\in\mathcal{S}_{\text{test}}\) we would expect \(p_{L}(\mathbf{x})=p_{M}(\mathbf{x})\).

Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.

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.

In the previous section, we saw that even when increasing the amount of training data, \(p_{M}\) consistently underestimates the probabilities of sequences sampled from the tail of \(p_{L}\). At the same time, we did not find that \(p_{M}\) overestimated sequences in the head of \(p_{L}\). Under the assumption that \(p_{M}\) is a proper probability distribution,5 that is, \(\sum_{\mathbf{x}\in\Sigma^{*}}p_{M}(\mathbf{x})=1\), these findings suggest that there exists sequences in \(\Sigma^{*}\) whose probability is overestimated by the model.

To do so, we compute model estimation error on perturbed sequences from \(p_{L}\)--sequences in \(\Sigma^{*}\) which are increasingly far away from the high-probability zones in \(p_{L}\). We build a corpus of perturbed sequences by recursively applying 30 random perturbations to each sequence \(\mathbf{x}\in\mathcal{D}_{\text{test}}\). Formally, the set of sequences at perturbation step \(i\) can be expressed as: \(\mathcal{D}^{(i)}_{\text{perturbed}}=\{\texttt{perturb}(\mathbf{x})\mid\mathbf{x}\in \mathcal{D}^{(i-1)}_{\text{perturbed}}\}\) where \(\texttt{perturb}(\mathbf{x})\) is a function which returns a novel perturbed version of \(\mathbf{x}\), and \(\mathcal{D}^{(0)}_{\text{perturbed}}=\mathcal{D}_{\text{test}}\).

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 \(p_{L}\) and the LM \(p_{M}\).

In the previous section, we saw that \(p_{M}\) placed excess probability mass on areas in \(\Sigma^{*}\) with low-probability under \(p_{L}\).

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 \(\Sigma^{*}\).

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.

Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019; Lafferty et al., 2001), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020), the consistency of decoding algorithms (Welleck et al., 2020), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg and Byrne, 2019).

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.

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, to ensure that our findings regarding the overestimation of ill-formed sequences hold, we compute model estimation error on random sequences sampled from \(\Sigma^{*}\) (see A.1.4 for details on how these sequences are constructed).

We find that \(p_{M}\) overestimates the majority of ill-formed sequences, indicating that these findings hold when the ground-truth distribution is defined using a pretrained model.

Since the probability of a sequence is inversely correlated with its length, our findings that estimation error is worse for rarer sentences may be explained by compounding errors. To test this claim, we ask whether the observed estimation error is worse than would be expected if it was due to error compounding. Specifically, in Figure 12, we plot the expected model estimation (black) and the mean observed error (blue) by sequence length. Expected estimation error for sequence length \(n\) is computed by multiplying the average token level estimation error by the sequence length, i.e., \(n\bar{\epsilon}\), where \[\bar{\epsilon}=\frac{1}{|\mathcal{D}|}\sum_{\mathbf{x}\in\mathcal{D}}\frac{1}{|\bm {x}|}\sum_{i=1}^{|\mathbf{x}|}\log p_{M}(x_{i}\mid x_{<i})-\log p_{L}(x_{i}\mid x _{<i})\]

As shown in Figure 12, observed estimation error for both GPT2-small and GPT2-medium is more severe than expected estimation error as we increase sentence length. This suggests that estimation error for longer (and typically rarer) sequences is not solely due to error compounding.

\[\hat{\mathcal{P}_{N}}=N_{0}\hat{P}(\mathbf{x}\mid f(\mathbf{x},\mathcal{D})=0)=\frac{ N_{1}}{N} \tag{4}\] which states that the total amount of probability mass allocated to unseen events is equal to the proportion of events which occurred only once (hapax legomena) in \(\mathcal{D}\).

Particularly for \(n\)-grams with \(n\geq 3\), we find that there is significant probability of sampling a previously unseen event, even for \(N>10,000,000\).

Input: Figure 12: Expected and observed estimation error by sentence length for GPT2-small and GPT2medium.

Input: Figure 4: GPT2-medium, GPT2-large and LSTM trained on 30M sequences sampled from pL. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 3: Mean model estimation error by training epoch (GPT2-medium). Each line denotes the mean estimation error for a 50th of the sequences; darker lines represent less probable sequences. The shaded area denotes the area in which validation cross-entropy reaches a minimum.

Input: Figure 11: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 5: GPT2-medium estimation behaviour for 15M sequences in Σ∗ across two dimensions: sequence rarity (x-axis) and degree of perturbation (y-axis). The heat map is shaded based on estimation error severity; blue areas indicate overestimation, whereas brown areas indicate underestimation. We also include example sequences from two zones in this sequence space.

Input: Figure 8: GPT2-medium and GPT2-large trained on 30M sequences sampled from pL with T = 1.00. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 9: Test sequence probability estimates given by pretrained neural LMs fine-tuned on 1M sequences sampled from pL. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input: Figure 7: Left: Joint histogram of sequence probability estimates for test sequences. Center: Mean model estimation error by true sequence probability for test sequences. Right: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 6: Model estimation error on test sequences as a function of target sequence probability for three different artificial languages. Each line visualizes estimation error for a GPT2-medium model trained to model a language with a specific softmax temperature parameter T .

Input: Table 5: GPT2-medium PP on validation set generated by the artificial language the LM attempts to model (val), and on real sentences sampled from OpenWebText (eng) for models used in Section 4.5

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence carefully, the claim that "LMs tend to overestimate the probability of rare perturbed (ill-formed) sequences" appears to be well-supported by multiple experimental results shown in the paper. The evidence includes:

1. Clear visualization in Figure 5 showing overestimation (blue areas) for highly perturbed sequences
2. Quantitative results from random sequence sampling experiments confirming overestimation of ill-formed sequences
3. Systematic perturbation experiments demonstrating this effect across different model sizes
4. Statistical analysis ruling out that this is simply due to error compounding

The review comments don't directly challenge this specific claim, and the "more important question/comment" actually seeks more details about the perturbation experiments rather than disputing the finding.

<score>1</score>

The claim is well-stated and accurately reflects the strength of the empirical evidence without exaggeration. The authors carefully validated their finding through multiple experimental approaches and analyses.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Rather, we find that LMs tend to (iii) overestimate the probability of rare perturbed (ill-formed) sequences.

The review comment to be evaluated is:

The paper overall is exceptionally well-written and clear and tables and plots are beautiful and clear---multiple times did I go "ooh that's a nice way of laying it out." Kudos! The abstract tells the entire story and the paper substantiates the claims nicely making what is shown feel easy and obvious, which I consider a great feat.

Concerning content, I am equally enthusiastic about the idea and evaluation and agree that this can be a very interesting piece of evidence that nicely connects to a number of past threads of NLP research (as laid out throughout and in the nice related work section, though I don't think I know enough related literature to truly safely say that the paper is as novel as it claims to be). That said, there are three major methodological concerns that I would like to discuss through with the authors before recommending acceptance:

1. We do not know whether neural-LM-defined origin distributions are "strange"---or really, structurally different from actual human natural language distributions that we believe exist---in some systematic way that leads to the results in this paper. One potential intuition here is that a neural LM being a very imperfect model makes very erratic choices for what low-frequency/low-likelihood it assigns probability mass to---erratic choices that may be very hard to learn/emulate with the student LM that the authors train. Maybe real natural language distributions have more "sensible tails" that are actually not all that hard to learn and maybe not even underestimated. To be clear, this is a fundamental issue that the authors have no way of *resolving* easily under this paradigm and I think the paradigm is still a worthwhile idea! What I am asking is that this limitation is acknowledged and discussed with the same clarity that is afforded to the other content.

2. Relatedly, currently there is no way to know whether the trained models are actually reasonably trained, overall well-working models or not: the paper is missing test ppl/bpc for both the origin model (on English test data) and for the student model (both on the original English data and on test data sampled from the origin model). The authors make a good point that these scores are not as informative as what this paper presents, but the reason these are crucial is so we know whether the effects the authors find are a property of well-trained models or whether these models (through subpar training or another issue) are just poor models that have little connection to the models we actually use in reality. A secondary idea to easy this concern might be to actually use a pretrained GPT model as the origin model to inspire trust that that at least is a good model---I'm not sure why this road wasn't chosen unless it truly did not matter and this "home-made" model performs just as well, in which case, again, I would like to see some quantitative substantiation of that claim.

3. The paper is not really clear on how the sampling/teaching procedure and the tempering of the softmax interact: Footnote 2 claims pure (right?) ancestral sampling is used (as opposed to top-k/nucleus?), but then Footnote 4 says the softmaxes are all cooled to T=0.85, and yet Section 4.5 again implies that no tempering took place up to that point in the study. This is not only slightly confusing messaging, but I am worried that it may belie the promise of fair evaluation, especially because it isn't clear to me whether the tempering was accounted for in the calculation of probabilities as defined at the end of section 3.2. Specifically, if the probability of a sentence under the origin model does not take into account this local tempering, then it is no surprise that the student model learns to underestimate rare events because they just weren't sampled thanks to the locally cooled distribution! I would be somewhat reassured if the probability calculation in section 3.2 takes the tempering into account, but my understanding of the paper is that it currently fails to do so. As slight reassurance, Figure 5 does seem to tell us that even without any tempering (yes?) the described effect is visible, so I think this too should not be an issue that sinks the paper, but I do see it as a critical flaw the way it is right now.

Beyond those three, I see some minor concerns that may be worth addressing:
- "autoregressive neural language models" should also cite the Mikolov RNN paper, perhaps in place of one of the two GPT papers
- not to Schmidhuber, but citing only Melis et al. (2020) for LSTMs looks a little odd (citing Hochreiter & Schmidhuber or Sundermeyer et al. may be less surprising though I understand the desire to cite the *actual* model and training specification used and agree it should be present)
- concerning citations, Shiran Dudy & Steven Bedrick's work may be nice to also connect to (despite its relatively low visibility)
- it may be worth pointing out that the perturbations in section 4.4 may individually also make sentences *more* grammatical and likely and that is just a relatively rare outcome (I don't think much more than that needs to be said on that)
- I did not quite see what was "[p]erhaps unsurprisin[g]" in Section 4.5, but that's a minor nitpick
- Appendix A.1 measures events, okay... but are those words (as the hapax legomena theory usually assumes) or are they sentences (as I think this experiment does)? If they are sentences, we should not be surprised to see very high probabilities of seeing novel events, after all sentences are rarely the same (excluding formulaic language and "Resumption of the session" style sentences). I also would have liked to see some Poisson distributions or other distributions that were more interesting than uniform n-sided dice in Figure 6, but that is more wish than necessity.

The evidence to be evaluated is: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.

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.

The LNRE zone for natural language appears to be very large, and it seems likely that \(\mathcal{P}_{N}\) 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).

For example, while the sequences \(\mathbf{x}_{1},\mathbf{x}_{2}\in\Sigma^{*}\) have likely never occurred in any sample of English, most would agree that \(\mathbf{x}_{1}\) is far more probable than \(\mathbf{x}_{2}\):

For a perfect LM of English, we would expect the estimated probabilities of the sequences \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) to match their probabilities under the true distribution \(p_{\text{English}}\).

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 \(\mathcal{D}\).

If the model \(M\) perfectly models the language \(L\), then for each \(\left\langle p_{L}(\mathbf{x}),p_{M}(\mathbf{x})\right\rangle\in\mathcal{S}_{\text{test}}\) we would expect \(p_{L}(\mathbf{x})=p_{M}(\mathbf{x})\).

Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.

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.

In the previous section, we saw that even when increasing the amount of training data, \(p_{M}\) consistently underestimates the probabilities of sequences sampled from the tail of \(p_{L}\). At the same time, we did not find that \(p_{M}\) overestimated sequences in the head of \(p_{L}\). Under the assumption that \(p_{M}\) is a proper probability distribution,5 that is, \(\sum_{\mathbf{x}\in\Sigma^{*}}p_{M}(\mathbf{x})=1\), these findings suggest that there exists sequences in \(\Sigma^{*}\) whose probability is overestimated by the model.

To do so, we compute model estimation error on perturbed sequences from \(p_{L}\)--sequences in \(\Sigma^{*}\) which are increasingly far away from the high-probability zones in \(p_{L}\). We build a corpus of perturbed sequences by recursively applying 30 random perturbations to each sequence \(\mathbf{x}\in\mathcal{D}_{\text{test}}\). Formally, the set of sequences at perturbation step \(i\) can be expressed as: \(\mathcal{D}^{(i)}_{\text{perturbed}}=\{\texttt{perturb}(\mathbf{x})\mid\mathbf{x}\in \mathcal{D}^{(i-1)}_{\text{perturbed}}\}\) where \(\texttt{perturb}(\mathbf{x})\) is a function which returns a novel perturbed version of \(\mathbf{x}\), and \(\mathcal{D}^{(0)}_{\text{perturbed}}=\mathcal{D}_{\text{test}}\).

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 \(p_{L}\) and the LM \(p_{M}\).

In the previous section, we saw that \(p_{M}\) placed excess probability mass on areas in \(\Sigma^{*}\) with low-probability under \(p_{L}\).

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 \(\Sigma^{*}\).

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.

Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019; Lafferty et al., 2001), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020), the consistency of decoding algorithms (Welleck et al., 2020), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg and Byrne, 2019).

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.

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, to ensure that our findings regarding the overestimation of ill-formed sequences hold, we compute model estimation error on random sequences sampled from \(\Sigma^{*}\) (see A.1.4 for details on how these sequences are constructed).

We find that \(p_{M}\) overestimates the majority of ill-formed sequences, indicating that these findings hold when the ground-truth distribution is defined using a pretrained model.

Since the probability of a sequence is inversely correlated with its length, our findings that estimation error is worse for rarer sentences may be explained by compounding errors. To test this claim, we ask whether the observed estimation error is worse than would be expected if it was due to error compounding. Specifically, in Figure 12, we plot the expected model estimation (black) and the mean observed error (blue) by sequence length. Expected estimation error for sequence length \(n\) is computed by multiplying the average token level estimation error by the sequence length, i.e., \(n\bar{\epsilon}\), where \[\bar{\epsilon}=\frac{1}{|\mathcal{D}|}\sum_{\mathbf{x}\in\mathcal{D}}\frac{1}{|\bm {x}|}\sum_{i=1}^{|\mathbf{x}|}\log p_{M}(x_{i}\mid x_{<i})-\log p_{L}(x_{i}\mid x _{<i})\]

As shown in Figure 12, observed estimation error for both GPT2-small and GPT2-medium is more severe than expected estimation error as we increase sentence length. This suggests that estimation error for longer (and typically rarer) sequences is not solely due to error compounding.

\[\hat{\mathcal{P}_{N}}=N_{0}\hat{P}(\mathbf{x}\mid f(\mathbf{x},\mathcal{D})=0)=\frac{ N_{1}}{N} \tag{4}\] which states that the total amount of probability mass allocated to unseen events is equal to the proportion of events which occurred only once (hapax legomena) in \(\mathcal{D}\).

Particularly for \(n\)-grams with \(n\geq 3\), we find that there is significant probability of sampling a previously unseen event, even for \(N>10,000,000\).

Input: Figure 12: Expected and observed estimation error by sentence length for GPT2-small and GPT2medium.

Input: Figure 4: GPT2-medium, GPT2-large and LSTM trained on 30M sequences sampled from pL. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 3: Mean model estimation error by training epoch (GPT2-medium). Each line denotes the mean estimation error for a 50th of the sequences; darker lines represent less probable sequences. The shaded area denotes the area in which validation cross-entropy reaches a minimum.

Input: Figure 11: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 5: GPT2-medium estimation behaviour for 15M sequences in Σ∗ across two dimensions: sequence rarity (x-axis) and degree of perturbation (y-axis). The heat map is shaded based on estimation error severity; blue areas indicate overestimation, whereas brown areas indicate underestimation. We also include example sequences from two zones in this sequence space.

Input: Figure 8: GPT2-medium and GPT2-large trained on 30M sequences sampled from pL with T = 1.00. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 9: Test sequence probability estimates given by pretrained neural LMs fine-tuned on 1M sequences sampled from pL. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input: Figure 7: Left: Joint histogram of sequence probability estimates for test sequences. Center: Mean model estimation error by true sequence probability for test sequences. Right: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 6: Model estimation error on test sequences as a function of target sequence probability for three different artificial languages. Each line visualizes estimation error for a GPT2-medium model trained to model a language with a specific softmax temperature parameter T .

Input: Table 5: GPT2-medium PP on validation set generated by the artificial language the LM attempts to model (val), and on real sentences sampled from OpenWebText (eng) for models used in Section 4.5

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After careful review of the evidence and claim "LMs tend to overestimate the probability of rare perturbed (ill-formed) sequences," I find this claim to be well-supported by the empirical evidence presented:

1. Figure 5 clearly demonstrates overestimation (blue areas) for highly perturbed sequences
2. Figure 7 (right panel) shows consistent overestimation for randomly sampled sequences from Σ*
3. The perturbation experiments explicitly tested 15M perturbed sequences and found systematic overestimation
4. Multiple model architectures (GPT2 variants and LSTM) showed this behavior
5. The effect persisted across different training regimes and temperatures

The review comment raises some methodological concerns but does not directly challenge this specific claim. The evidence strongly supports this particular finding about overestimation of perturbed sequences.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Rather, we find that LMs tend to (iii) overestimate the probability of rare perturbed (ill-formed) sequences.

The review comment to be evaluated is:

**Interesting background**. The connections between productivity, low-probability sequences, and the limitations of perplexity were clearly written and interesting.

**Well-executed experiments and methodology**. The idea of using a language model as a ground-truth distribution was interesting and well-suited for the analysis done here. The experiments were easy to follow and the analysis was clear.

**Interesting findings**. The main findings related to underestimation, comparison of training-set and test-set dynamics during training, and dependence on training data were interesting and not obvious. The finding that perturbed, unnatural sequence received unusually high probabilities (while test samples receive unusually low probabilities) [Figure 4] was especially interesting.

**Clarity**. The paper was well-written and enjoyable to read. The authors articulated how their work fits in with related work, and the concepts (e.g. the LNRE zone), methodology, and results are clearly written. Well done.

**Additional experiments**: A good addition would be measuring the impact of model size with models larger than the target distribution. What would GPT2-XL's or GPT-3's probability assignments look like on a target distribution from GPT2-medium?

The evidence to be evaluated is: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.

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.

The LNRE zone for natural language appears to be very large, and it seems likely that \(\mathcal{P}_{N}\) 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).

For example, while the sequences \(\mathbf{x}_{1},\mathbf{x}_{2}\in\Sigma^{*}\) have likely never occurred in any sample of English, most would agree that \(\mathbf{x}_{1}\) is far more probable than \(\mathbf{x}_{2}\):

For a perfect LM of English, we would expect the estimated probabilities of the sequences \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) to match their probabilities under the true distribution \(p_{\text{English}}\).

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 \(\mathcal{D}\).

If the model \(M\) perfectly models the language \(L\), then for each \(\left\langle p_{L}(\mathbf{x}),p_{M}(\mathbf{x})\right\rangle\in\mathcal{S}_{\text{test}}\) we would expect \(p_{L}(\mathbf{x})=p_{M}(\mathbf{x})\).

Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.

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.

In the previous section, we saw that even when increasing the amount of training data, \(p_{M}\) consistently underestimates the probabilities of sequences sampled from the tail of \(p_{L}\). At the same time, we did not find that \(p_{M}\) overestimated sequences in the head of \(p_{L}\). Under the assumption that \(p_{M}\) is a proper probability distribution,5 that is, \(\sum_{\mathbf{x}\in\Sigma^{*}}p_{M}(\mathbf{x})=1\), these findings suggest that there exists sequences in \(\Sigma^{*}\) whose probability is overestimated by the model.

To do so, we compute model estimation error on perturbed sequences from \(p_{L}\)--sequences in \(\Sigma^{*}\) which are increasingly far away from the high-probability zones in \(p_{L}\). We build a corpus of perturbed sequences by recursively applying 30 random perturbations to each sequence \(\mathbf{x}\in\mathcal{D}_{\text{test}}\). Formally, the set of sequences at perturbation step \(i\) can be expressed as: \(\mathcal{D}^{(i)}_{\text{perturbed}}=\{\texttt{perturb}(\mathbf{x})\mid\mathbf{x}\in \mathcal{D}^{(i-1)}_{\text{perturbed}}\}\) where \(\texttt{perturb}(\mathbf{x})\) is a function which returns a novel perturbed version of \(\mathbf{x}\), and \(\mathcal{D}^{(0)}_{\text{perturbed}}=\mathcal{D}_{\text{test}}\).

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 \(p_{L}\) and the LM \(p_{M}\).

In the previous section, we saw that \(p_{M}\) placed excess probability mass on areas in \(\Sigma^{*}\) with low-probability under \(p_{L}\).

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 \(\Sigma^{*}\).

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.

Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019; Lafferty et al., 2001), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020), the consistency of decoding algorithms (Welleck et al., 2020), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg and Byrne, 2019).

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.

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, to ensure that our findings regarding the overestimation of ill-formed sequences hold, we compute model estimation error on random sequences sampled from \(\Sigma^{*}\) (see A.1.4 for details on how these sequences are constructed).

We find that \(p_{M}\) overestimates the majority of ill-formed sequences, indicating that these findings hold when the ground-truth distribution is defined using a pretrained model.

Since the probability of a sequence is inversely correlated with its length, our findings that estimation error is worse for rarer sentences may be explained by compounding errors. To test this claim, we ask whether the observed estimation error is worse than would be expected if it was due to error compounding. Specifically, in Figure 12, we plot the expected model estimation (black) and the mean observed error (blue) by sequence length. Expected estimation error for sequence length \(n\) is computed by multiplying the average token level estimation error by the sequence length, i.e., \(n\bar{\epsilon}\), where \[\bar{\epsilon}=\frac{1}{|\mathcal{D}|}\sum_{\mathbf{x}\in\mathcal{D}}\frac{1}{|\bm {x}|}\sum_{i=1}^{|\mathbf{x}|}\log p_{M}(x_{i}\mid x_{<i})-\log p_{L}(x_{i}\mid x _{<i})\]

As shown in Figure 12, observed estimation error for both GPT2-small and GPT2-medium is more severe than expected estimation error as we increase sentence length. This suggests that estimation error for longer (and typically rarer) sequences is not solely due to error compounding.

\[\hat{\mathcal{P}_{N}}=N_{0}\hat{P}(\mathbf{x}\mid f(\mathbf{x},\mathcal{D})=0)=\frac{ N_{1}}{N} \tag{4}\] which states that the total amount of probability mass allocated to unseen events is equal to the proportion of events which occurred only once (hapax legomena) in \(\mathcal{D}\).

Particularly for \(n\)-grams with \(n\geq 3\), we find that there is significant probability of sampling a previously unseen event, even for \(N>10,000,000\).

Input: Figure 12: Expected and observed estimation error by sentence length for GPT2-small and GPT2medium.

Input: Figure 4: GPT2-medium, GPT2-large and LSTM trained on 30M sequences sampled from pL. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 3: Mean model estimation error by training epoch (GPT2-medium). Each line denotes the mean estimation error for a 50th of the sequences; darker lines represent less probable sequences. The shaded area denotes the area in which validation cross-entropy reaches a minimum.

Input: Figure 11: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 5: GPT2-medium estimation behaviour for 15M sequences in Σ∗ across two dimensions: sequence rarity (x-axis) and degree of perturbation (y-axis). The heat map is shaded based on estimation error severity; blue areas indicate overestimation, whereas brown areas indicate underestimation. We also include example sequences from two zones in this sequence space.

Input: Figure 8: GPT2-medium and GPT2-large trained on 30M sequences sampled from pL with T = 1.00. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 9: Test sequence probability estimates given by pretrained neural LMs fine-tuned on 1M sequences sampled from pL. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input: Figure 7: Left: Joint histogram of sequence probability estimates for test sequences. Center: Mean model estimation error by true sequence probability for test sequences. Right: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 6: Model estimation error on test sequences as a function of target sequence probability for three different artificial languages. Each line visualizes estimation error for a GPT2-medium model trained to model a language with a specific softmax temperature parameter T .

Input: Table 5: GPT2-medium PP on validation set generated by the artificial language the LM attempts to model (val), and on real sentences sampled from OpenWebText (eng) for models used in Section 4.5

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I've carefully reviewed the claim "LMs tend to overestimate the probability of rare perturbed (ill-formed) sequences" against the provided evidence and review comments.

The evidence strongly supports this claim through multiple experimental results:

1. Figure 5 clearly shows overestimation (blue areas) for highly perturbed sequences
2. Figure 7 (right) demonstrates overestimation for random sequences from Σ*
3. The perturbation experiments explicitly show overestimation of sequences outside typical distribution
4. Multiple figures consistently show this pattern across different model sizes and configurations

The reviewer specifically highlighted this finding as "interesting and not obvious," noting the contrast between underestimation of test samples and overestimation of perturbed sequences.

The claim is well-stated and accurately reflects the evidence without exaggeration. The authors provide thorough experimental validation and acknowledge the scope of their findings.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

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.

The evidence to be evaluated is: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.

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.

The LNRE zone for natural language appears to be very large, and it seems likely that \(\mathcal{P}_{N}\) 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.

In the LNRE zone, \(p_{M}\) is defined over a support containing a very large set of sequences which have never occurred in a training corpus (or equivalently, have all occurred with frequency 0), and which take on a very wide array of differing probabilities. For example, while the sequences \(\mathbf{x}_{1},\mathbf{x}_{2}\in\Sigma^{*}\) have likely never occurred in any sample of English, most would agree that \(\mathbf{x}_{1}\) is far more probable than \(\mathbf{x}_{2}\): * _The East pond in Parc Lafontaine was filled to the brin with Diet Coke.

* _Certain nak indicate liberationing among theroter codity voters vandalized.

For a perfect LM of English, we would expect the estimated probabilities of the sequences \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) to match their probabilities under the true distribution \(p_{\text{English}}\). However, since \(p_{\text{English}}\) 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 \(\mathcal{D}\) is typically used, 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 \(\mathcal{D}\). 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.

We propose evaluating language model performance on the heavy-tail of rare events via a known probability distribution over sequences.

To characterize neural LM behaviour on rare events, we train Transformer and LSTM LMs on data sampled from \(p_{L}\), and compare the instance-level probability estimates given by \(p_{M}\) to target probabilities under \(p_{L}\).

If the model \(M\) perfectly models the language \(L\), then for each \(\left\langle p_{L}(\mathbf{x}),p_{M}(\mathbf{x})\right\rangle\in\mathcal{S}_{\text{test}}\) we would expect \(p_{L}(\mathbf{x})=p_{M}(\mathbf{x})\). Figures 2(A) and 2(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.

Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.

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.

In the previous section, we saw that even when increasing the amount of training data, \(p_{M}\) consistently underestimates the probabilities of sequences sampled from the tail of \(p_{L}\). At the same time, we did not find that \(p_{M}\) overestimated sequences in the head of \(p_{L}\). Under the assumption that \(p_{M}\) is a proper probability distribution,5 that is, \(\sum_{\mathbf{x}\in\Sigma^{*}}p_{M}(\mathbf{x})=1\), these findings suggest that there exists sequences in \(\Sigma^{*}\) whose probability is overestimated by the model.

To do so, we compute model estimation error on perturbed sequences from \(p_{L}\)--sequences in \(\Sigma^{*}\) which are increasingly far away from the high-probability zones in \(p_{L}\).

While it is possible that these operations produce other well-formed strings, we expect this to be a relatively rare outcome.

In the previous section, we saw that \(p_{M}\) placed excess probability mass on areas in \(\Sigma^{*}\) with low-probability under \(p_{L}\).

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 \(\Sigma^{*}\).

Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019; Lafferty et al., 2001), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020), the consistency of decoding algorithms (Welleck et al., 2020), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg and 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 and Perona, 2017; Bengio, 2015).

Focusing on the word frequency distribution, Dudy and Bedrick (2020) found that LMs under-perform when less frequent examples are encountered at test time.

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.

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.

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.

Similarly to our experiments in Section 4.4, we find that the LM underestimates the probability of the majority of sequences, and does so more severely for less probable sequences.

We find that \(p_{M}\) overestimates the majority of ill-formed sequences, indicating that these findings hold when the ground-truth distribution is defined using a pretrained model.

Computing estimation error on a set of unseen test sequences, we find, analogously to our experiments on models trained from scratch, that pre-trained models underestimate the probability of the majority of sequences sampled from the target distribution, and do so more severely for rarer sequences (we visualize this in Figure 9).

Once again, increased training data lessens but does eliminate the underestimation behaviour.

This method provides us with sequences which are increasingly far away from high-probability zones under \(p_{L}\). However, it does so with initial sequences sampled directly from \(p_{L}\), and as a result, produces strings which are edit-adjacent to the high-probability zones (under \(p_{L}\)) in \(\Sigma^{*}\).

Since the probability of a sequence is inversely correlated with its length, our findings that estimation error is worse for rarer sentences may be explained by compounding errors. To test this claim, we ask whether the observed estimation error is worse than would be expected if it was due to error compounding. Specifically, in Figure 12, we plot the expected model estimation (black) and the mean observed error (blue) by sequence length. Expected estimation error for sequence length \(n\) is computed by multiplying the average token level estimation error by the sequence length, i.e., \(n\bar{\epsilon}\), where \[\bar{\epsilon}=\frac{1}{|\mathcal{D}|}\sum_{\mathbf{x}\in\mathcal{D}}\frac{1}{|\bm {x}|}\sum_{i=1}^{|\mathbf{x}|}\log p_{M}(x_{i}\mid x_{<i})-\log p_{L}(x_{i}\mid x _{<i})\]

As shown in Figure 12, observed estimation error for both GPT2-small and GPT2-medium is more severe than expected estimation error as we increase sentence length. This suggests that estimation error for longer (and typically rarer) sequences is not solely due to error compounding.

In Figure 13, we plot the the empirical estimate of \(\mathcal{P}_{N}\) as function of the sample size \(N\) for \(n\)-grams sampled from a subset of OpenWebText. Particularly for \(n\)-grams with \(n\geq 3\), we find that there is significant probability of sampling a previously unseen event, even for \(N>10,000,000\).

Input: Figure 12: Expected and observed estimation error by sentence length for GPT2-small and GPT2medium.

Input: Figure 9: Test sequence probability estimates given by pretrained neural LMs fine-tuned on 1M sequences sampled from pL. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input: Figure 13: The probability of sampling a novel item (potential productivity P) as a function of sample size N . Many distributions in natural language are characterized by a potentially unbounded amount of novelty.

Input: Figure 4: GPT2-medium, GPT2-large and LSTM trained on 30M sequences sampled from pL. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 3: Mean model estimation error by training epoch (GPT2-medium). Each line denotes the mean estimation error for a 50th of the sequences; darker lines represent less probable sequences. The shaded area denotes the area in which validation cross-entropy reaches a minimum.

Input: Figure 11: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 5: GPT2-medium estimation behaviour for 15M sequences in Σ∗ across two dimensions: sequence rarity (x-axis) and degree of perturbation (y-axis). The heat map is shaded based on estimation error severity; blue areas indicate overestimation, whereas brown areas indicate underestimation. We also include example sequences from two zones in this sequence space.

Input: Figure 7: Left: Joint histogram of sequence probability estimates for test sequences. Center: Mean model estimation error by true sequence probability for test sequences. Right: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 2: Test sequence probability estimates given by neural LMs. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After careful review, the claim appears to be well-supported by extensive experimental evidence. The paper demonstrates through multiple experiments and visualizations that neural LMs consistently underestimate probabilities of rare sequences in the tail of the distribution (Figures 2, 4, 9 showing systematic underestimation). The overestimation of ill-formed sequences is shown through perturbation experiments (Figure 5) and analyses of sequences sampled from Σ* (Figures 7, 11). The claim accurately reflects these key findings without exaggeration and acknowledges the limitations of the study regarding differences between their ground truth distributions and natural language.

<score>1</score>

Justification:
1. Under-representation of well-formed sequences in the tail is supported by:
- Multiple model architectures showing consistent underestimation (Figures 2, 4, 9)
- Analysis showing this persists with increased training data
- Demonstration that this is not just due to error compounding (Figure 12)

2. Over-representation of ill-formed sequences is supported by:
- Perturbation experiments showing overestimation far from high-probability zones (Figure 5)
- Analysis of randomly sampled sequences from Σ* (Figures 7, 11)
- Conservation of probability mass principle indicating overestimation must occur somewhere

The claim is precise and measured in its wording, accurately reflecting the evidence without overstating implications.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

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.

The review comment to be evaluated is:

* Strengths: A very well written and clear paper, addressing a focussed question through a well-designed set of experiments on synthetic data allowing to precisely control experimental conditions.

* Weaknesses: No major weakness, but some mild issues and areas for improvement, see the comments and questions below.

* Comments and Questions:

    * (Minor Comment) I did not find the description of the LNRE Zone in section 2 to bring much to the discussion. In particular, characterizing this zone through the condition that $\mathcal{P}_N$ is not __null__ is almost empty of content for natural language, because it would in particular mean that we could precisely delimit what constitutes a possible "natural" sentence, and that the LNRE Zone should cover all such sentences. Here the formal description as given does not clarify anything.
    * (Question) In some of your experiments, you train a $P_M$ GPT2-medium model on 30M sentences sampled from the $P_L$ GPT2-medium model, that is, $P_M$ here uses the same architecture as $P_L$ (if I understand correctly). However, the $P_M$ model still underestimates the probabilities relative to $P_L$. Do you think this is a symptom of 30M sentences being too few, or more a symptom that $P_M$ is stuck in some local minimum during the optimization ? After all, (again, if I understand the setup correctly), using the same architecture, $P_M$ would make it possible, in principle, to just use the same parameters as $P_L$, and then the mismatch could not occur. Perhaps this would be worth a note in the text?
    * (More important Question/Comment) In section 4.4, you perform some perturbations, and observe that they lead to higher $P_M$ than $P_L$. First, I think it would be worth defining these perturbations in more detail than what you do. Second, you do not discuss your choice for these perturbations nor analyze why they have (relatively) high probability relative to $P_M$. However, the fact that, for instance, swapping two words or deleting a word in a natural sentence is not severely penalized by $P_M$ is an interesting but specific fact, which may have to do with a special tolerance of $P_M$ for such operations, rather than for others. To give one example, if the perturbations consisted solely in replacing a word in the natural sentence by a _rare_ word in the training set, would you observe such a difference between $P_M$ and $P_L$? Overall, could you discuss this aspect of your experiments?

The evidence to be evaluated is: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.

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.

The LNRE zone for natural language appears to be very large, and it seems likely that \(\mathcal{P}_{N}\) 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.

In the LNRE zone, \(p_{M}\) is defined over a support containing a very large set of sequences which have never occurred in a training corpus (or equivalently, have all occurred with frequency 0), and which take on a very wide array of differing probabilities. For example, while the sequences \(\mathbf{x}_{1},\mathbf{x}_{2}\in\Sigma^{*}\) have likely never occurred in any sample of English, most would agree that \(\mathbf{x}_{1}\) is far more probable than \(\mathbf{x}_{2}\): * _The East pond in Parc Lafontaine was filled to the brin with Diet Coke.

* _Certain nak indicate liberationing among theroter codity voters vandalized.

For a perfect LM of English, we would expect the estimated probabilities of the sequences \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) to match their probabilities under the true distribution \(p_{\text{English}}\). However, since \(p_{\text{English}}\) 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 \(\mathcal{D}\) is typically used, 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 \(\mathcal{D}\). 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.

We propose evaluating language model performance on the heavy-tail of rare events via a known probability distribution over sequences.

To characterize neural LM behaviour on rare events, we train Transformer and LSTM LMs on data sampled from \(p_{L}\), and compare the instance-level probability estimates given by \(p_{M}\) to target probabilities under \(p_{L}\).

If the model \(M\) perfectly models the language \(L\), then for each \(\left\langle p_{L}(\mathbf{x}),p_{M}(\mathbf{x})\right\rangle\in\mathcal{S}_{\text{test}}\) we would expect \(p_{L}(\mathbf{x})=p_{M}(\mathbf{x})\). Figures 2(A) and 2(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.

Importantly, any deviation of this histogram from the identity line indicates that the model distorts the shape of the distribution of the language.

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.

In the previous section, we saw that even when increasing the amount of training data, \(p_{M}\) consistently underestimates the probabilities of sequences sampled from the tail of \(p_{L}\). At the same time, we did not find that \(p_{M}\) overestimated sequences in the head of \(p_{L}\). Under the assumption that \(p_{M}\) is a proper probability distribution,5 that is, \(\sum_{\mathbf{x}\in\Sigma^{*}}p_{M}(\mathbf{x})=1\), these findings suggest that there exists sequences in \(\Sigma^{*}\) whose probability is overestimated by the model.

To do so, we compute model estimation error on perturbed sequences from \(p_{L}\)--sequences in \(\Sigma^{*}\) which are increasingly far away from the high-probability zones in \(p_{L}\).

While it is possible that these operations produce other well-formed strings, we expect this to be a relatively rare outcome.

In the previous section, we saw that \(p_{M}\) placed excess probability mass on areas in \(\Sigma^{*}\) with low-probability under \(p_{L}\).

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 \(\Sigma^{*}\).

Our analysis exploring the overestimation of ill-formed sequences extends previous findings on locally normalized conditional models assigning arbitrary probability mass to unlikely sequences (Andor et al., 2016; Goyal et al., 2019; Lafferty et al., 2001), neural LMs assigning high likelihood to sequences with repetitions (Welleck et al., 2020), the consistency of decoding algorithms (Welleck et al., 2020), and on machine translation models placing significant probability mass on the empty sequence (Stahlberg and 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 and Perona, 2017; Bengio, 2015).

Focusing on the word frequency distribution, Dudy and Bedrick (2020) found that LMs under-perform when less frequent examples are encountered at test time.

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.

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.

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.

Similarly to our experiments in Section 4.4, we find that the LM underestimates the probability of the majority of sequences, and does so more severely for less probable sequences.

We find that \(p_{M}\) overestimates the majority of ill-formed sequences, indicating that these findings hold when the ground-truth distribution is defined using a pretrained model.

Computing estimation error on a set of unseen test sequences, we find, analogously to our experiments on models trained from scratch, that pre-trained models underestimate the probability of the majority of sequences sampled from the target distribution, and do so more severely for rarer sequences (we visualize this in Figure 9).

Once again, increased training data lessens but does eliminate the underestimation behaviour.

This method provides us with sequences which are increasingly far away from high-probability zones under \(p_{L}\). However, it does so with initial sequences sampled directly from \(p_{L}\), and as a result, produces strings which are edit-adjacent to the high-probability zones (under \(p_{L}\)) in \(\Sigma^{*}\).

Since the probability of a sequence is inversely correlated with its length, our findings that estimation error is worse for rarer sentences may be explained by compounding errors. To test this claim, we ask whether the observed estimation error is worse than would be expected if it was due to error compounding. Specifically, in Figure 12, we plot the expected model estimation (black) and the mean observed error (blue) by sequence length. Expected estimation error for sequence length \(n\) is computed by multiplying the average token level estimation error by the sequence length, i.e., \(n\bar{\epsilon}\), where \[\bar{\epsilon}=\frac{1}{|\mathcal{D}|}\sum_{\mathbf{x}\in\mathcal{D}}\frac{1}{|\bm {x}|}\sum_{i=1}^{|\mathbf{x}|}\log p_{M}(x_{i}\mid x_{<i})-\log p_{L}(x_{i}\mid x _{<i})\]

As shown in Figure 12, observed estimation error for both GPT2-small and GPT2-medium is more severe than expected estimation error as we increase sentence length. This suggests that estimation error for longer (and typically rarer) sequences is not solely due to error compounding.

In Figure 13, we plot the the empirical estimate of \(\mathcal{P}_{N}\) as function of the sample size \(N\) for \(n\)-grams sampled from a subset of OpenWebText. Particularly for \(n\)-grams with \(n\geq 3\), we find that there is significant probability of sampling a previously unseen event, even for \(N>10,000,000\).

Input: Figure 12: Expected and observed estimation error by sentence length for GPT2-small and GPT2medium.

Input: Figure 9: Test sequence probability estimates given by pretrained neural LMs fine-tuned on 1M sequences sampled from pL. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input: Figure 13: The probability of sampling a novel item (potential productivity P) as a function of sample size N . Many distributions in natural language are characterized by a potentially unbounded amount of novelty.

Input: Figure 4: GPT2-medium, GPT2-large and LSTM trained on 30M sequences sampled from pL. Main plots: Mean estimation error on test sequences as a function of the number of sequences seen in training. Inset plots: Relative change in mean estimation error of test sequences as a function of the number of sequences seen in training. In both cases, each line denotes estimation behaviour for a 50th of the test sequences; darker lines represent less probable sequences.

Input: Figure 3: Mean model estimation error by training epoch (GPT2-medium). Each line denotes the mean estimation error for a 50th of the sequences; darker lines represent less probable sequences. The shaded area denotes the area in which validation cross-entropy reaches a minimum.

Input: Figure 11: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 5: GPT2-medium estimation behaviour for 15M sequences in Σ∗ across two dimensions: sequence rarity (x-axis) and degree of perturbation (y-axis). The heat map is shaded based on estimation error severity; blue areas indicate overestimation, whereas brown areas indicate underestimation. We also include example sequences from two zones in this sequence space.

Input: Figure 7: Left: Joint histogram of sequence probability estimates for test sequences. Center: Mean model estimation error by true sequence probability for test sequences. Right: Mean model estimation error by true sequence probability for sequences randomly sampled from Σ∗.

Input: Figure 2: Test sequence probability estimates given by neural LMs. Three left-most figures: The joint histograms of sequence probability estimates. The dotted line denotes the cases in which the model’s estimates perfectly align with the target probability; shading to the right of this line denotes underestimation. Right-most figure: Mean sequence estimation error by target sequence probability.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After careful review of the evidence, the claim appears well-stated and accurately reflects the findings presented in the paper. The evidence demonstrates through multiple experiments and visualizations that neural LMs consistently underestimate probabilities of sequences in the tail of the distribution (shown in Figures 2, 4, 9) while overestimating probabilities of ill-formed or perturbed sequences away from high-probability zones (shown in Figures 5, 7). The reviewer comments do not dispute these core findings, only suggesting more detailed discussion of perturbation choices. The claim concisely summarizes this dual pattern without overstating the magnitude or certainty of the results.

<score>1</score>