Original: A N ON - MONOTONIC S ELF - TERMINATING L ANGUAGE M ODEL
Revised: A N ON - MONOTONIC S ELF - TERMINATING L ANGUAGE M ODEL

Original: Anonymous authors
Revised: Eugene Choi † eugene.choi@nyu.edu

Original: Paper under double-blind review
Revised: Kyunghyun Cho †‡§ kyunghyun.cho@nyu.edu Cheolhyoung Lee †∗ cheolhyoung.lee@nyu.edu

Original: A BSTRACT
Revised: A BSTRACT

Original: Recent large-scale neural autoregressive sequence models have shown impressive performances on a variety of natural language generation tasks.
Revised: Recent large-scale neural autoregressive sequence models have shown impressive performances on a variety of natural language generation tasks.

Original: However, their generated sequences often exhibit degenerate properties such as non-termination, undesirable repetition, and premature termination, when generated with decoding algorithms such as greedy search, beam search, top- k sampling, and nucleus sampling.
Revised: However, their generated sequences often exhibit degenerate properties such as non-termination, undesirable repetition, and premature termination, when generated with decoding algorithms such as greedy search, beam search, top- k sampling, and nucleus sampling.

Original: In this paper, we focus on the problem of non-terminating sequences resulting from an incomplete decoding algorithm.
Revised: In this paper, we focus on the problem of non-terminating sequences resulting from an incomplete decoding algorithm.

Original: We first define an incomplete probable decoding algorithm which includes greedy search, top- k sampling, and nucleus sampling, beyond the incomplete decoding algorithm originally put forward by Welleck et al.
Revised: We first define an incomplete probable decoding algorithm which includes greedy search, top- k sampling and nucleus sampling, beyond the incomplete decoding algorithm originally put forward by Welleck et al.

Original: We then propose a non-monotonic self-terminating language model, which significantly relaxes the constraint of monotonically increasing termination probability in the originally proposed self-terminating language model by Welleck et al. (2020), to address the issue of non-terminating sequences when using incomplete probable decoding algorithms.
Revised: We then propose a non-monotonic self-terminating language model which significantly relaxes the constraint of monotonically increasing termination probability in the originally proposed self-terminating language model by Welleck et al to address the issue of non-terminating sequences when using incomplete probable decoding algorithms.

Original: We prove that our proposed model prevents non-terminating sequences when using not only incomplete probable decoding algorithms but also beam search.
Revised: We prove that our proposed model prevents non-terminating sequences when using not only incomplete probable decoding algorithms but also beam search.

Original: We empirically validate our model on sequence completion tasks with various architectures.
Revised: We empirically validate our model on sequence completion tasks with various architectures

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Autoregressive neural sequence models (Bengio et al., 2000) have been widely used for various natural language generation tasks such as language modeling (Brown et al., 2020; Chowdhery et al., 2022), machine translation (Bahdanau et al., 2014), and conversational dialogue modeling (Vinyals & Le, 2015).
Revised: Autoregressive neural sequence models (Bengio et al., 2000) have been widely used for various natural language generation tasks such as language modeling (Brown et al., 2020; Chowdhery et al., 2022), machine translation (Bahdanau et al., 2014), and conversational dialogue modeling (Vinyals & Le, 2015).

Original: Furthermore, large-scale autoregressive neural sequence models have shown unprecedented ability to generate fluent, human-like texts (Vaswani et al., 2017; Brown et al., 2020).
Revised: Furthermore, large-scale autoregressive neural sequence models have shown unprecedented ability to generate fluent, human-like texts (Vaswani et al., 2017; Brown et al., 2020).

Original: Despite their success, the autoregressive neural sequence models have shown undesirable behaviors: non-termination (Welleck et al., 2020), degenerate repetition (Welleck et al., 2019; Holtzman et al., 2020), and premature termination (Koehn & Knowles, 2017; Stahlberg & Byrne, 2019).
Revised: Despite their success, the autoregressive neural sequence models have shown undesirable behaviors: non-termination (Welleck et al., 2020), degenerate repetition (Welleck et al., 2019; Holtzman et al., 2020), and premature termination (Koehn & Knowles, 2017; Stahlberg & Byrne, 2019).

Original: In this paper, we focus on how to prevent non-termination when using a given decoding algorithm.
Revised: In this paper, we focus on how to prevent non-termination when using a given decoding algorithm.

Original: Non-termination is the problem that a language model generates infinitely long sequences with a positive probability from our language model when using a given decoding algorithm.
Revised: Non-termination is the problem that a language model generates infinitely long sequences with a positive probability from our language model when using a given decoding algorithm.

Original: Welleck et al.
Revised: Welleck et al.

Original: (2020) pointed out that this issue comes from a discrepancy between the distribution of our language model and its induced distribution by an incomplete decoding algorithm.
Revised: (2020) pointed out that this issue comes from a discrepancy between the distribution of our language model and its induced distribution by an incomplete decoding algorithm.

Original: They formalized this disparity by the notion of inconsistency where our language model generates non-terminating sequences with a positive probability from the decoding algorithm.
Revised: They formalized this disparity by the notion of inconsistency where our language model generates non-terminating sequences with a positive probability from the decoding algorithm.

Original: To avoid this inconsistency, they proposed a self-terminating (ST) language model that uses new parametrization for its classifier rather than usual softmax parametrization.
Revised: To avoid this inconsistency, they proposed a self-terminating (ST) language model that uses new parametrization for its classifier rather than usual softmax parametrization.

Original: They proved that the ST language model is consistent with respect to greedy search, beam search, top-
Revised: They proved that the ST language model is consistent with respect to greedy search, beam search, top-

Original: k sampling (Fan et al., 2018) as well as nucleus sampling (Holtzman et al., 2020).
Revised: k sampling (Fan et al., 2018) as well as nucleus sampling (Holtzman et al., 2020).

Original: The ST language model increases the termination probability of each sequence monotonically to 1, but this parametrization is not appropriate for learning our natural language.
Revised: The ST language model increases the termination probability of each sequence monotonically to 1, but this parametrization is not appropriate for learning our natural language.

Original: As an illustrative example, suppose there are two sequences in our dataset: “I am a boy” vs. “I am a boy, and you are a girl.” .
Revised: As an illustrative example, suppose there are two sequences in our dataset: “I am a boy” vs. “I am a boy, and you are a girl.” .

Original: Our language model trained on this dataset may or may not terminate after the former.
Revised: Our language model trained on this dataset may or may not terminate after the former.

Original: Once our model decides not to end, it should dramatically reduce the termination probability to continue.
Revised: Once our model decides not to end, it should dramatically reduce the termination probability to continue.

Original: The ST language model, which monotonically increase the termination probability, cannot capture such a case, where one sequence is a prefix of another.
Revised: The ST language model, which monotonically increase the termination probability, cannot capture such a case, where one sequence is a prefix of another.

Original: We thus propose a non-monotonic self-terminating (NMST) language model which guarantees the consistency with respect to greedy search, beam search, top- k sampling, and nucleus sampling without monotonically increasing termination probability.
Revised: We thus propose a non-monotonic self-terminating (NMST) language model which guarantees the consistency with respect to greedy search, beam search, top- k sampling, and nucleus sampling without monotonically increasing termination probability.

Original: The NMST language model encourages the termination probability of each sequence to converge to 1 through NMST parametrization however without monotonicity.
Revised: The NMST language model encourages the termination probability of each sequence to converge to 1 through NMST parametrization however without monotonicity.

Original: Even under this relaxation, the proposed NMST language model provably prevents any non-terminating sequence resulting from greedy search, beam search, top- k sampling, and nucleus sampling, which we refer to as incomplete probable decoding algorithms .
Revised: Even under this relaxation, the proposed NMST language model provably prevents any non-terminating sequence resulting from greedy search, beam search, top- k sampling, and nucleus sampling, which we refer to as incomplete probable decoding algorithms .

Original: We conduct experiments validating the effectiveness of our NMST language models on sequence completion tasks, as was done in earlier studies.
Revised: We conduct experiments validating the effectiveness of our NMST language models on sequence completion tasks, as was done in earlier studies.

Original: We test NMST parametrization with various architectures.
Revised: We test NMST parametrization with various architectures.

Original: Specifically, we train RNN (Elman, 1990) and LSTM (Hochreiter & Schmidhuber, 1997) on WikiText-2 (Merity et al., 2016).
Revised: Specifically, we train RNN (Elman, 1990) and LSTM (Hochreiter & Schmidhuber, 1997) on WikiText-2 (Merity et al., 2016).

Original: We additionally finetune GPT-

Revised: We additionally finetune GPT-

Original: (Radford et al., 2019) on WikiText-103 (Merity et al., 2016).
Revised: (Radford et al., 2019) on WikiText-103 (Merity et al., 2016).

Original: Across all these setups, NMST parametrization effectively prevents non-terminating sequences, especially when compared to softmax parametrization.
Revised: Across all these setups, NMST parametrization effectively prevents non-terminating sequences, especially when compared to softmax parametrization.

Original: Furthermore, we see that our NMST parametrization has better (lower) perplexities than those of ST parametrization, confirming the importance of relaxing the monotonic termination probability.
Revised: Furthermore, we see that our NMST parametrization has better (lower) perplexities than those of ST parametrization, confirming the importance of relaxing the monotonic termination probability.

Original:  N OTATIONS AND B ACKGROUND 2. N OTATIONS FOR A UTOREGRESSIVE N EURAL S EQUENCE M ODELS
Revised:  N OTATIONS AND B ACKGROUND 2. N OTATIONS FOR A UTOREGRESSIVE N EURAL S EQUENCE M ODELS

Original: Sequences, vocabulary, and ⟨ eos ⟩
Revised: Sequences, vocabulary, and ⟨ eos ⟩

Original: We view an instance (e.g., a sentence and a paragraph) as a sequence y =
Revised: We view an instance (e.g., a sentence and a paragraph) as a sequence y =

Original: ( y 1 , y 2 , . . .
Revised: ( y 1 , y 2 , . . .

Original: , y T ) , where each y t is an element from a pre-defined finite set of discrete tokens, referred to as a vocabulary V .
Revised: , y T ) , where each y t is an element from a pre-defined finite set of discrete tokens, referred to as a vocabulary V .

Original: V includes a special symbol ⟨ eos ⟩ that only appears at the end of the sequence .
Revised: V includes a special symbol ⟨ eos ⟩ that only appears at the end of the sequence .

Original: Every sequence y must end with ⟨ eos ⟩ .
Revised: Every sequence y must end with ⟨ eos ⟩ .

Original: We write the length of y as | y | , and y | y | = ⟨ eos ⟩ .
Revised: We write the length of y as | y | , and y | y | = ⟨ eos ⟩ .

Original: We call y a non-terminating sequence , | y | = ∞ , if y t ̸ = ⟨ eos ⟩ for all t .
Revised: We call y a non-terminating sequence , | y | = ∞ , if y t ̸ = ⟨ eos ⟩ for all t .

Original: Embedding vectors Each token v ∈ V is not a numerical vector so that we use an embedding vector u v ∈ R m to represent v .
Revised: Embedding vectors Each token v ∈ V is not a numerical vector so that we use an embedding vector u v ∈ R m to represent v .

Original: To capture the notion of similarity between discrete tokens efficiently, we use an embedding vector u v ∈ R m to project v into continuous embedding space (Bengio et al., 2000; Mikolov et al., 2013b;a; Levy & Goldberg, 2014).
Revised: To capture the notion of similarity between discrete tokens efficiently, we use an embedding vector u v ∈ R m to project v into continuous embedding space (Bengio et al., 2000; Mikolov et al., 2013b;a; Levy & Goldberg, 2014).

Original: Autoregressive neural sequence models Bengio et al.
Revised: Autoregressive neural sequence models Bengio et al.

Original: (2000) proposed an autoregressive neural sequence model parametrized by θ ∈ R k .
Revised: (2000) proposed an autoregressive neural sequence model parametrized by θ ∈ R k .

Original: They factorized p θ ( y | x ) into a product of the conditional probability of each token given all the previous tokens and an input in a predefined order as follows:
Revised: They factorized p θ ( y | x ) into a product of the conditional probability of each token given all the previous tokens and an input in a predefined order as follows:

Original: [Equation]
Revised: [Equation]

Original: where y <t is a t -prefix of y and x is an input referred to as a context .
Revised: where y <t is a t -prefix of y and x is an input referred to as a context .

Original: For example, x represents either a prompt in sequence completion or a source-side sequence in machine translation.
Revised: For example, x represents either a prompt in sequence completion or a source-side sequence in machine translation.

Original: There are several popular architectures for p θ such as RNN (Elman, 1990), LSTM (Hochreiter &
Revised: There are several popular architectures for p θ such as RNN (Elman, 1990), LSTM (Hochreiter &

Original: Schmidhuber, 1997), GRU (Cho et al., 2014), and Transformer (Vaswani et al., 2017).
Revised: Schmidhuber, 1997), GRU (Cho et al., 2014), and Transformer (Vaswani et al., 2017).

Original: As shown in equation 2, all these models utilize softmax classifiers.
Revised: As shown in equation 2, all these models utilize softmax classifiers.

Original: In this paper, we modify the parametrization of their softmax classifiers to prevent non-terminating sequences.
Revised: In this paper, we modify the parametrization of their softmax classifiers to prevent non-terminating sequences.

Original: We thus write a vanilla language model , regardless of its choice of architecture, that uses the original softmax parametrization as p va θ defined in Definition 1.
Revised: We thus write a vanilla language model , regardless of its choice of architecture, that uses the original softmax parametrization as p va θ defined in Definition 1.

Original: Definition 1.
Revised: Definition 1.

Original: A vanilla language model p va θ computes the conditional probability of each token given a t -prefix y <t and a context x at each time step t as follows:
Revised: A vanilla language model p va θ computes the conditional probability of each token given a t -prefix y <t and a context x at each time step t as follows:

Original: [Equation]
Revised: [Equation]

Original: where h t = f θ ( y t , h t − 1 ) with h 0 = 0 .
Revised: where h t = f θ ( y t , h t − 1 ) with h 0 = 0 .

Original: Training For a given dataset, D = (cid:8)(cid:0)x
Revised: Training For a given dataset, D = (cid:8)(cid:0)

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: n ) (cid:1)(cid:9)

Original: N n =1 , we maximize the joint probability assigned to the sequences in our training dataset to find an optimal parameter configuration θ ⋆ as follows:
Revised: n =1 , we maximize the joint probability assigned to the sequences in our training dataset to find an optimal parameter configuration θ ⋆ as follows:

Original: [Equation]
Revised: [Equation]

Original: 2. I NCOMPLETE P ROBABLE D ECODING A LGORITHMS
Revised: 2. I NCOMPLETE P ROBABLE D ECODING A LGORITHMS

Original: An autoregressive language model p θ predicts the likelihood of a sequence y given a context x .
Revised: An autoregressive language model p θ predicts the likelihood of a sequence y given a context x .

Original: Its autoregressive factorization in equation 1 requires a recursive process for every t to infer.
Revised: Its autoregressive factorization in equation 1 requires a recursive process for every t to infer.

Original: Hence, at inference time, we use a decoding algorithm, defined below, to generate sequences from p θ .
Revised: Hence, at inference time, we use a decoding algorithm, defined below, to generate sequences from p θ .

Original: Definition 2.
Revised: Definition 2.

Original: Let Y be a collection of y
Revised: Let Y be a collection of y

Original: such that y =
Revised: such that y =

Original: ( y 1 , y 2 , · · · , y T ) where T ∈ { 1 , 2 , · · · } and y t ∈ V .
Revised: ( y 1 , y 2 , · · · , y T ) where T ∈ { 1 , 2 , · · · } and y t ∈ V .

Original: A decoding algorithm S is a function that maps p θ to q S ( p θ ) which is a probability distribution over Y .
Revised: A decoding algorithm S is a function that maps p θ to q S ( p θ ) which is a probability distribution over Y .

Original: A decoded sentence ˆ y given x by S from p θ is a random sample from q S ( p θ ) ( y | x ) .
Revised: A decoded sentence ˆ y given x by S from p θ is a random sample from q S ( p θ ) ( y | x ) .

Original: To generate a high quality sequence from p θ , a decoding algorithm assumes that a higher quality sequence has a higher probability of p θ than others.
Revised: To generate a high quality sequence from p θ , a decoding algorithm assumes that a higher quality sequence has a higher probability of p θ than others.

Original: For instance, maximum a posteriori (MAP) decoding algorithm S map gives the most probable sequence y ⋆ given a context x from p θ :
Revised: For instance, maximum a posteriori (MAP) decoding algorithm S map gives the most probable sequence y ⋆ given a context x from p θ :

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: ) by setting q S map ( p θ ) ( y = y ⋆ | x ) = 1 and q S map ( p θ ) ( y = y ′
Revised: by setting q S map ( p θ ) ( y = y ⋆ | x ) = 1 and q S map ( p θ ) ( y = y ′

Original: | x ) = 0 where
Revised: | x ) = 0 where

Original: y ′ ∈
Revised: y ′ ∈

Original: Y \ { y ⋆ } .
Revised: Y \ { y ⋆ } .

Original: Unfortunately,
Revised: Unfortunately,

Original: S map is intractable since equation 4 requires an exhaustive search over the sequence space Y .
Revised: [Equation] is intractable since equation 4 requires an exhaustive search over the sequence space Y .

Original: Hence, in practice, we utilize incomplete probable decoding algorithms defined as follows:
Revised: Hence, in practice, we utilize incomplete probable decoding algorithms defined as follows:

Original: Definition 3.
Revised: Definition 3.

Original: A decoding algorithm S is incomplete and probable if there exists V t ⊊ V such that
Revised: A decoding algorithm S is incomplete and probable if there exists V t ⊊ V such that

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: for each t .
Revised: for each t .

Original: Furthermore, for every v ∈ V t , S satisfies
Revised: Furthermore, for every v ∈ V t , S satisfies

Original: [Equation]
Revised: [Equation]

Original: At each t , an incomplete probable decoding algorithm S considers only a set of highly probable tokens, V t .
Revised: At each t , an incomplete probable decoding algorithm S considers only a set of highly probable tokens, V t .

Original: S generates ˆ y given x by recursively sampling ˆ y t from q S ( p θ ) ( y t | ˆ y <t , x ) supported on
Revised: S generates ˆ y given x by recursively sampling ˆ y t from q S ( p θ ) ( y t | ˆ y <t , x ) supported on

Original: V t .
Revised: V t .

Original: This reduces an exponential complexity of S map , O (cid:0) |V| | ˆ y | (cid:1) , down to a linear level, O ( | ˆ y | · |V| ) .
Revised: This reduces an exponential complexity of S map , O [Equation] | ˆ y [Equation]

Revised: , down to a linear level, O ( | ˆ y | · |V| ) .

Original: Greedy search, top- k sampling (Fan et al., 2018), and nucleus sampling (Holtzman et al., 2020) are incomplete and probable.
Revised: Greedy search, top- k sampling (Fan et al., 2018), and nucleus sampling (Holtzman et al., 2020) are incomplete and probable.

Original: For example, greedy search S gr generates the t -th item of a sequence by
Revised: For example, greedy search S gr generates the t -th item of a sequence by

Original: [Equation]
Revised: [Equation]

Original: In other words, S gr sets V t to (cid:8) v (1) t (cid:9) where v (1) t = arg max v ∈V p θ ( y t = v | ˆ y <t , x ) .
Revised: In other words, S gr sets V t to [Equation] v (1) t (cid:9) where v (1) t [Equation]

Original:  Moreover, we have p θ(cid:0) y t
Revised: p θ ( y t = v | ˆ y <t , x ) . Moreover, we have p θ

Original: = v (1) t (cid:12)(cid:12) ˆ y <t , x (cid:1) ≤ q S gr ( p θ ) (cid:0) y t = v (1) t (cid:12)(cid:12)
Revised: [Equation] y t = v (1) t (cid:12)(cid:12)

Original: ˆ y <t , x (cid:1)
Revised: ˆ y <t , x (cid:1)

Original: = 1 , and q S gr ( p θ )
Revised: = 1 , and q S gr ( p θ )

Original: ( y t = v ′ | ˆ y <t , x ) =holds for v
Revised: ( y t = v ′ | ˆ y <t , x ) =holds for v

Original: ′ ∈
Revised: ′ ∈

Original: V \ V t .
Revised: V \ V t .

Original: Thus, S gr is incomplete and probable.
Revised: Thus, S gr is incomplete and probable.

Original: Unlike S gr , top-
Revised: Unlike S gr , top-

Original: k sampling considers k most probable tokens in
Revised: k sampling considers k most probable tokens in

Original: V as V t while nucleus sampling sets the smallest subset of V , containing most probable tokens of which total probability is higher than a given threshold µ , to
Revised: V as V t while nucleus sampling sets the smallest subset of V , containing most probable tokens of which total probability is higher than a given threshold µ , to

Original: V t .
Revised: V t .

Original: In §A.1 and A.2, we present that top- k sampling and nucleus sampling are also incomplete and probable.
Revised: In §A.1 and A.2, we present that top- k sampling and nucleus sampling are also incomplete and probable.

Original: Beam search is a heuristic algorithm that operates on the level of prefixes.
Revised: Beam search is a heuristic algorithm that operates on the level of prefixes.

Original: We describe it further in §A.3.
Revised: We describe it further in §A.3.

Original: Although beam search is not an incomplete probable decoding algorithm, it also selects V t which is a proper subset of V to expand each prefix at each step t .
Revised: Although beam search is not an incomplete probable decoding algorithm, it also selects V t which is a proper subset of V to expand each prefix at each step t .

Original: Due to this, our main theoretical finding for the incomplete probable decoding algorithms in §3 is applicable to beam search as well.
Revised: Due to this, our main theoretical finding for the incomplete probable decoding algorithms in §3 is applicable to beam search as well.

Original: 2. C ONSISTENCY WITH RESPECT TO I NCOMPLETE P ROBABLE D ECODING A LGORITHMS AND S ELF - TERMINATING (ST) L ANGUAGE M ODELS
Revised: 2. C ONSISTENCY WITH RESPECT TO I NCOMPLETE P ROBABLE D ECODING A LGORITHMS AND S ELF - TERMINATING (ST) L ANGUAGE M ODELS

Original: Incomplete probable decoding algorithms greatly reduce computational overhead for generating sequences from our model.
Revised: Incomplete probable decoding algorithms greatly reduce computational overhead for generating sequences from our model.

Original: However, Welleck et al.
Revised: However, Welleck et al.

Original: (2020) observed that they can generate nonterminating sequences even if every training sequence has a finite length.
Revised: (2020) observed that they can generate nonterminating sequences even if every training sequence has a finite length.

Original: To study this, Welleck et al.
Revised: To study this, Welleck et al.

Original: (2020) defined consistency with respect to decoding algorithms as shown in Definition 4.
Revised: (2020) defined consistency with respect to decoding algorithms as shown in Definition 4.

Original: Definition 4.
Revised: Definition 4.

Original: A language model p θ is consistent with respect to a decoding algorithm S
Revised: A language model p θ is consistent with respect to a decoding algorithm S

Original: if q S ( p θ ) ( | y | = ∞ ) = 0 for any parameter configuration θ ∈ R k .
Revised: if q S ( p θ ) ( | y | = ∞ ) = 0 for any parameter configuration θ ∈ R k .

Original: Welleck et al.
Revised: Welleck et al.

Original: (2020) also proved that a vanilla language model p va θ defined in Definition 1 is inconsistent with respect to incomplete probable decoding algorithms and beam search as follows:
Revised: (2020) also proved that a vanilla language model p va θ defined in Definition 1 is inconsistent with respect to incomplete probable decoding algorithms and beam search as follows:

Original: Theorem 1.
Revised: Theorem 1.

Original: A vanilla language model p va θ defined in Definition 1 is inconsistent with respect to any incomplete probable decoding algorithm and beam search (Theorem 3.4 in Welleck et al.
Revised: A vanilla language model p va θ defined in Definition 1 is inconsistent with respect to any incomplete probable decoding algorithm and beam search (Theorem 3.4 in Welleck et al.

Original: For each t , an incomplete probable decoding algorithm S selects V t ⊊ V as a set of candidates for decoding, but p va θ does not guarantee that ⟨ eos ⟩ ∈ V t .
Revised: For each t , an incomplete probable decoding algorithm S selects V t ⊊ V as a set of candidates for decoding, but p va θ does not guarantee that ⟨ eos ⟩ ∈ V t .

Original: Specifically, if ⟨ eos ⟩ / ∈ V t for all t , then S cannot decode each token to ⟨ eos ⟩ for all t (i.e., non-terminating).
Revised: Specifically, if ⟨ eos ⟩ / ∈ V t for all t , then S cannot decode each token to ⟨ eos ⟩ for all t (i.e., non-terminating).

Original: Based on this result, Welleck et al.
Revised: Based on this result, Welleck et al.

Original: (2020) proposed a self-terminating (ST) language model , defined below:
Revised: (2020) proposed a self-terminating (ST) language model , defined below:

Original: Definition 5.
Revised: Definition 5.

Original: For h t defined in Definition 1, the conditional probability of each token v ∈ V given a t -prefix y <t and a context x at each time step t in an ST language model is given by
Revised: For h t defined in Definition 1, the conditional probability of each token v ∈ V given a t -prefix y <t and a context x at each time step t in an ST language model is given by

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: where v ∈ V \ {⟨ eos ⟩} , ϵ ∈ (0 , 1) , and σ ( x ) =
Revised: where v ∈ V \ {⟨ eos ⟩} , ϵ ∈ (0 , 1) , and σ ( x ) =

Original: (1 + exp( − x )) − 1 is a sigmoid function.
Revised: (1 + exp( − x )) − 1 is a sigmoid function.

Original: They proved that the ST language model is consistent with respect to any incomplete probable decoding algorithm and beam search, as follows:
Revised: They proved that the ST language model is consistent with respect to any incomplete probable decoding algorithm and beam search, as follows:

Original: Theorem 2.
Revised: Theorem 2.

Original: An ST language model p st θ defined in Definition 5 is consistent with respect to any incomplete probable decoding algorithms and beam search (Theorem 4.1-4.3 in Welleck et al.
Revised: An ST language model p st θ defined in Definition 5 is consistent with respect to any incomplete probable decoding algorithms and beam search (Theorem 4.1-4.3 in Welleck et al.

Original: In equation 9, p st θ ( y t = ⟨ eos ⟩| y <t , x ) monotonically increases to 1 as t increases.
Revised: In equation 9, p st θ ( y t = ⟨ eos ⟩| y <t , x ) monotonically increases to 1 as t increases.

Original: S ends up including ⟨ eos ⟩ in V t always for t ≥ t ′ with some t ′ , and lim t →∞ q S ( p θ ) ( y t = ⟨ eos ⟩| y <t , x ) =by equation 7.
Revised: S ends up including ⟨ eos ⟩ in V t always for t ≥ t ′ with some t ′ , and lim t →∞ q S ( p θ ) ( y t = ⟨ eos ⟩| y <t , x ) =by equation 7.

Original: This guarantees S to terminate in a finite number of steps.
Revised: This guarantees S to terminate in a finite number of steps.

Original: Despite p st θ ’s consistency, its validation perplexity degrades compared to p va θ in sequence completion (Welleck et al., 2020).
Revised: Despite p st θ ’s consistency, its validation perplexity degrades compared to p va θ in sequence completion (Welleck et al., 2020).

Original: We suspect that such degradation comes from the core property of p st θ that p θ ( y t = ⟨ eos ⟩| y <t , x ) monotonically increases to 1 as t increases.
Revised: We suspect that such degradation comes from the core property of p st θ that p θ ( y t = ⟨ eos ⟩| y <t , x ) monotonically increases to 1 as t increases.

Original: In Remark 1 below, we provide an example where the optimal p θ ⋆ ( y t = ⟨ eos ⟩| y <t , x ) is not monotonic.
Revised: In Remark 1 below, we provide an example where the optimal p θ ⋆ ( y t = ⟨ eos ⟩| y <t , x ) is not monotonic.

Original: Remark 1.
Revised: Remark 1.

Original: Let D =
Revised: Let D =

Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  be a two-instance training dataset.
Revised: (cid:9) be a two-instance training dataset.

Original: Assume that there exists t 0 such that y <t 0 =
Revised: Assume that there exists t 0 such that y <t 0 =

Original: y (1) <t 

Revised: y (1) <t

Original: = y (2) <t 0 .
Revised: = y (2) <t 0 .

Original: Suppose further that t 0 = | y (1) | < | y (2)
Revised: Suppose further that t 0 = | y (1) | < | y (2)

Original: | −and x
Revised: | −and x

Original: = x (1) =
Revised: = x (1) =

Original: x (2) .
Revised: x (2) .

Original: If θ ⋆ is an optimal parameter configuration in equation 3 over D .
Revised: If θ ⋆ is an optimal parameter configuration in equation 3 over D .

Original: Then, p θ ⋆ [Equation]
Revised: Then, p θ ⋆ (cid:0) y (2) t = ⟨ eos ⟩| y (2) <

Original:  is not monotonic with respect to t (proved in §B).
Revised: t , x (cid:1) is not monotonic with respect to t (proved in §B).

Original: We can easily find such case in natural language satisfying the assumptions in Remark 1 by concatenating two sequences.
Revised: We can easily find such case in natural language satisfying the assumptions in Remark 1 by concatenating two sequences.

Original: We empirically demonstrate the existence of such cases in §4.2.
Revised: We empirically demonstrate the existence of such cases in §4.2.

Original:  N ON - MONOTONIC S ELF - TERMINATING (NMST) L ANGUAGE M ODELS
Revised:  N ON - MONOTONIC S ELF - TERMINATING (NMST) L ANGUAGE M ODELS

Original: The consistency of p st θ comes from lim t →∞ p
Revised: The consistency of p st θ comes from lim t →∞ p

Original: st θ ( y t = ⟨ eos ⟩|
Revised: st θ ( y t = ⟨ eos ⟩|

Original: y <t , x ) = 1 , not the monotonically increasing p st θ ( y t = ⟨ eos ⟩| y <t , x ) as a function of t .
Revised: y <t , x ) = 1 , not the monotonically increasing p st θ ( y t = ⟨ eos ⟩| y <t , x ) as a function of t .

Original: This motivates us to propose a non-monotonic self-terminating (NMST) language model p nmst θ that permits p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) to be a non-monotonic function of t while satisfying lim t
Revised: This motivates us to propose a non-monotonic self-terminating (NMST) language model p nmst θ that permits p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) to be a non-monotonic function of t while satisfying lim t

Original: [Equation]
Revised: [Equation]

Original: = 1 as follows:
Revised: as follows:

Original: Definition 6.
Revised: Definition 6.

Original: For h t defined in Definition 1, the conditional probability of each token given a t -prefix y <t and a context x at the t -th step in an NMST language model is defined by[Equation] t [Equation]
Revised: For h t defined in Definition 1, the conditional probability of each token given a t -prefix y <t and a context x at the t -th step in an NMST language model is defined by

Original: θ

Original: [Equation]
Revised: [Equation]

Original: and p nmstθ
Revised: and p nmst

Original: [Equation]
Revised: [Equation]

Original: where v ∈ V \ {⟨ eos ⟩} , ϵ ∈ (0 , 1) , and σ ( x ) =
Revised: where v ∈ V \ {⟨ eos ⟩} , ϵ ∈ (0 , 1) , and σ ( x ) =

Original: (1 + exp( − x )) − 1 is a sigmoid function.
Revised: (1 + exp( − x )) − 1 is a sigmoid function.

Original: Figure 1 shows that p nmst θ uses convex combination of two curves to model p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) .
Revised: [Figure]

Original: We can write a curve g ( t ) between a lower-bound curve f lb  ( t )  and an upper-bound curve f ub ( t ) as
Revised: g ( t 1 − λ t lb + λ for any of g ( t ) with allows to learn a non monotonic behavior is consistent with respect to any incomplete probable decoding algorithms and due to

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: Figure 1 shows that p nmst θ uses convex combination of two curves to model p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) . We can write a curve g ( t ) between a lower-bound curve f lb ( t ) and an upper-bound curve f ub ( t ) as g ( t ) =

Original: [Figure]
Revised: (1 − λ ( t ))

Original: [Figure] [Figure] is y
Revised: f lb ( t ) + λ ( t ) f

Original: [Equation]
Revised: ub ( t ) , with appropriate λ ( t ) ∈ (0 , 1) for all t .

Original:  = ⟨  ⟩| y  t  − (1 − 
Revised: p nmst θ sets g ( t ) to p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) , and then regards it as a convex combination of f lb ( t ) = 1 − (1 − ϵ ) t and f ub ( t ) =

Original: ) t and f t 1 with  λ ( t ) =
Revised: with a coefficient λ ( t ) =

Original: σ (cid:0) u ⊤⟨ eos ⟩ h  (cid:1) .
Revised: σ (cid:0) u ⊤⟨ eos ⟩ h t (cid:1) .

Original: This enables non-monotonic p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) .
Revised: This enables nonmonotonic p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) .

Original: Moreover, in Theorem 3 below, we show that the proposed NMST parametrization in equation 10 still guarantees the consistency with respect to any incomplete probable decoding algorithms and beam search.
Revised: Moreover, in Theorem 3 below, we show that the proposed NMST parametrization in equation 10 still guarantees the consistency with respect to any incomplete probable decoding algorithms and beam search.

Original: Theorem 3.
Revised: Theorem 3.

Original: An NMST language model defined in Definition 6 is consistent with respect to any incomplete probable decoding algorithms and beam search.Proof.
Revised: An NMST language model defined in Definition 6 is consistent with respect to any incomplete probable decoding algorithms and beam search.

Original: We provide the proof in §C. Theorem 3 guarantees that every decoded sequence from p nmst θ terminates when using incomplete decoding algorithms and beam search.
Revised: Theorem 3 guarantees that every decoded sequence from p nmst θ terminates when using incomplete decoding algorithms and beam search.

Original: Neither p nmst θ nor p st θ results in non-terminating sequences resulting from incomplete probable decoding algorithms and beam search.
Revised: Neither p nmst θ nor p st θ results in non-terminating sequences resulting from incomplete probable decoding algorithms and beam search.

Original: Unlike ST parametrization, our NMST parametrization in equation 10 can capture a wider range of p θ ( y t = ⟨ eos ⟩| y <t , x ) , since p nmst θ does not assume that p θ ( y t = ⟨ eos ⟩| y <t , x ) is a monotonic function of t .
Revised: Unlike ST parametrization, our NMST parametrization in equation 10 can capture a wider range of p θ ( y t = ⟨ eos ⟩| y <t , x ) , since p nmst θ does not assume that p θ ( y t = ⟨ eos ⟩| y <t , x ) is a monotonic function of t .

Original: We em- pirically demonstrate this by comparing p va
Revised: We em- pirically demonstrate this by comparing p va

Original: [Equation]
Revised: [Equation]

Revised: p

Revised: [Equation]

Revised: θ ( y t

Revised: [Equation]

Revised: eos ⟩| y

Revised: [Equation]

Original:  ) in Figure 3.
Revised: t , x ) in Figure 3.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We empirically validate the effectiveness of the proposed non-monotonic self-terminating (NMST) language model by evaluating it on sequence completion tasks.
Revised: We empirically validate the effectiveness of the proposed non-monotonic self-terminating (NMST) language model by evaluating it on sequence completion tasks.

Original: We test three variants of a given architecture: (i) a vanilla (VA+) language model using common softmax parametrization in equation 2, (ii) a self-terminating (ST+) language model using ST parametrization proposed by Welleck et al.
Revised: We test three variants of a given architecture: (i) a vanilla (VA+) language model using common softmax parametrization in equation 2, (ii) a self-terminating (ST+) language model using ST parametrization proposed by Welleck et al.

Original: and (iii) our non-monotonic self-terminating (NMST+) language model using NMST parametrization in equation 10.
Revised: and (iii) our non-monotonic self-terminating (NMST+) language model using NMST parametrization in equation 10.

Original: We use following evaluation metrics for comparison:
Revised: We use following evaluation metrics for comparison:

Original: • Perplexity : Given an autoregressive language model p θ , the perplexity of p θ over D is exp(cid:18)
Revised: • Perplexity : Given an autoregressive language model p θ , the perplexity of p θ over D is exp

Original: [Equation]
Revised: [Equation]

Original: (cid:80) Nn [Equation] (cid:80) T ( n ) [Equation]
Revised: where D =

Original: log

Original: [Equation]
Revised: [Equation]

Original: θ (cid:16) y ( n )
Revised: x ( n ) , y ( n )

Original: [Equation]
Revised: [Equation]

Original: (cid:12) (cid:12)(cid:12) y ( n )
Revised: N

Original: [Equation]
Revised: [Equation]

Original: t , x ( n ) (cid:17) (cid:19) , where D = (cid:8)(cid:0) x

Original: ( n ) , y ( n ) (cid:1)(cid:9) N n =1 .

Original: • Non-termination ratio ( r nt ) :
Revised: • Non-termination ratio ( r nt ) :

Original: To present the consistency of p θ with respect to a given decoding algorithm S , we need to compute r nt =
Revised: To present the consistency of p θ with respect to a given decoding algorithm S , we need to compute r nt =

Original: q S ( p θ ) ( | y | = ∞ ) .
Revised: q S ( p [Equation] ( | y | = ∞ ) .

Original: Instead, based on
Revised: Instead, based on

Original: [Equation]
Revised: [Equation]

Original: we use r nt ( L ) =
Revised: we use r nt ( L ) =

Original: q S ( p θ ) ( | y | > L ) with a sufficiently large threshold L to estimate r nt .
Revised: q S ( p θ ) ( | y | > L ) with a sufficiently large threshold L to estimate r nt .

Original: Sequence completion is a task of predicting a continuation ˆ y given a c -length context x
Revised: Sequence completion is a task of predicting a continuation ˆ y given a c -length context x

Original: = ( x 1 , x 2 , · · · , x c ) by using a decoding algorithm S from a language model p θ (i.e. ˆ y ∼ q S ( p θ ) ( y | x ) ). 
Revised: = ( x 1 , x 2 , · · · , x c ) by using a decoding algorithm S from a language model p θ (i.e. ˆ y ∼ q S ( p θ ) ( y | x ) ). [Figure] nt

Revised: ) goes to as L increases.

Revised: However, softmax parametrization ( inconsistent with respect to greedy since its nt does not converge

Original: In this section, we use greedy search defined in equation 8 to generate ˆ y given x .
Revised: In this section, we use greedy search defined in equation 8 to generate ˆ y given x .

Original: Our main theoretical finding in Theorem 3 is that the proposed NMST language model is consistent with respect to not only greedy search but also top- k sampling, nucleus sampling, and beam search.
Revised: Our main theoretical finding in Theorem 3 is that the proposed NMST language model is consistent with respect to not only greedy search but also top- k sampling, nucleus sampling, and beam search.

Original: We thus present results when using decoding algorithms other than greedy search at the end in §5 and §F.
Revised: We thus present results when using decoding algorithms other than greedy search at the end in §5 and §F.

Original: [Figure]
Revised: [Figure]

Original: al. (2020). Moreover, with a proper choice of ϵ = 1 .× 10 − 5 , NMST+ { RNN, LSTM } have competitive validation perplexities against those of VA+ { RNN, LSTM } . [Table] 4. W IKI T EXT - 
Revised: T EXT -

Original: WikiText-2 (Merity et al., 2016) consists of 2 million words from 600 Wikipedia articles.
Revised: WikiText-2 (Merity et al., 2016) consists of 2 million words from 600 Wikipedia articles.

Original: With word tokenization, we regard the first 10 tokens of each sequence and its remaining part, as a context x and a ground truth y , respectively.
Revised: With word tokenization, we regard the first 10 tokens of each sequence and its remaining part, as a context x and a ground truth y , respectively.

Original: We train RNN with tanh (Elman, 1990) and LSTM (Hochreiter &
Revised: We train RNN with tanh (Elman, 1990) and LSTM (Hochreiter &

Original: Schmidhuber, 1997) on WikiText-2.
Revised: Schmidhuber, 1997) on WikiText-2.

Original: Both RNN and LSTM have 2 layers, with 256 and 512 hidden units at each layer, respectively.
Revised: Both RNN and LSTM have 2 layers, with 256 and 512 hidden units at each layer, respectively.

Original: We perform 10 random runs with a batch size of 32 for 70 epochs.
Revised: We perform 10 random runs with a batch size of 32 for 70 epochs.

Original: We use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 0 .
Revised: We use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 0 .

Original: 001 , β 1 = 0 .
Revised: 001 , β 1 = 0 .

Original: 9 , β 2 = 0 .
Revised: 9 , β 2 = 0 .

Original: 99 , weight decay of 0 .
Revised: 99 , weight decay of 0 .

Original: 01 , learning rate decay, and early stopping.
Revised: 01 , learning rate decay, and early stopping.

Original: We further describe our models and training strategies for WikiText-2 experiments in §D. Unlike VA+ { RNN, LSTM } ,
Revised: We further describe our models and training strategies for WikiText-2 experiments in §D. Unlike VA+ { RNN, LSTM } ,

Original: ST+ { RNN, LSTM } and NMST+ { RNN, LSTM } need an additional hyperparameter ϵ .
Revised: ST+ { RNN, LSTM } and NMST+ { RNN, LSTM } need an additional hyperparameter ϵ .

Original: We explore ϵ in { 1 . 

Revised: We explore ϵ in { 1 .

Original: × 10 − 5 , 5 . 

Revised: × 10 − 5 , 5 .

Original: × 10 − 5 , 1 . 

Revised: × 10 − 5 , 1 .

Original: × 10 − 4 , 5 . 

Revised: × 10 − 4 , 5 .

Original: × 10 − 4 } .
Revised: × 10 − 4 } .

Original: We present the average ( ± st.dev.) non-termination ratios, r nt ( L ) ’s, across 10 random runs as a function of L for all considered setups of WikiText-2 in Figure 2, using greedy search.
Revised: We present the average ( ± st.dev.) non-termination ratios, r nt ( L ) ’s, across 10 random runs as a function of L for all considered setups of WikiText-2 in Figure 2, using greedy search.

Original: From equation 11, a language model is consistent with respect to greedy search if lim L →∞ r nt ( L ) = 0 .
Revised: From equation 11, a language model is consistent with respect to greedy search if lim L →∞ r nt ( L ) = 0 .

Original: As L increases, we observe that r nt ( L ) ’s of VA+ { RNN, LSTM } fail to converge toward 0 while r nt ( L ) ’s of ST+ { RNN, LSTM } and NMST+ { RNN, LSTM } all reach 0.
Revised: As L increases, we observe that r nt ( L ) ’s of VA+ { RNN, LSTM } fail to converge toward 0 while r nt ( L ) ’s of ST+ { RNN, LSTM } and NMST+ { RNN, LSTM } all reach 0.

Original: In other words, RNN and LSTM are now consistent with respect to greedy search after replacing the original softmax parametrization with either the proposed NMST parametrization or ST parametrization.
Revised: In other words, RNN and LSTM are now consistent with respect to greedy search after replacing the original softmax parametrization with either the proposed NMST parametrization or ST parametrization.

Original: Table 1 shows that the average ( ± st.dev.) validation perplexities across 10 random experiments for all variants of RNN and LSTM, trained on WikiText-2.
Revised: Table 1 shows that the average ( ± st.dev.) validation perplexities across 10 random experiments for all variants of RNN and LSTM, trained on WikiText-2.

Original: We observe that NMST+ { RNN, LSTM } have better validation perplexities than ST+ { RNN, LSTM } for every ϵ .
Revised: We observe that NMST+ { RNN, LSTM } have better validation perplexities than ST+ { RNN, LSTM } for every ϵ .

Original: We demonstrate this more[Table] [Table] clearly in §E.1 by plotting the evolution of the mean validation perplexities as we vary ϵ .
Revised: We demonstrate this more clearly in §E.1 by plotting the evolution of the mean validation perplexities as we vary ϵ .

Original: Although our NMST+ guarantees the consistency of RNN and LSTM with respect to greedy search with a better validation perplexity than ST+, we need to carefully select ϵ of NMST+.
Revised: Although our NMST+ guarantees the consistency of RNN and LSTM with respect to greedy search with a better validation perplexity than ST+, we need to carefully select ϵ of NMST+.

Original: As ϵ increases, the lower bound of p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) grows faster, yielding premature sequences when ϵ is too large.
Revised: As ϵ increases, the lower bound of p nmst θ ( y t = ⟨ eos ⟩| y <t , x ) grows faster, yielding premature sequences when ϵ is too large.

Original: Indeed, the average validation perplexities of NMST+RNN and NMST+LSTM with ϵ = 5 . 

Revised: Indeed, the average validation perplexities of NMST+RNN and NMST+LSTM with ϵ = 5 .

Original: × 10 − 4 are 184.2 and 105.6 which degrade by 5.6 and 4.0 from those of VA+RNN and
Revised: × 10 − 4 are 184.2 and 105.6 which degrade by 5.6 and 4.0 from those of VA+RNN and

Original: VA+LSTM, 178.6 and 101.6, respectively.
Revised: VA+LSTM, 178.6 and 101.6, respectively.

Original: We however emphasize that there is the optimal ϵ = 1 . 

Revised: We however emphasize that there is the optimal ϵ = 1 .

Original: × 10 − 5 that makes NMST+ { RNN, LSTM } have the validation perplexities similar to those of VA+ { RNN, LSTM } .
Revised: × 10 − 5 that makes NMST+ { RNN, LSTM } have the validation perplexities similar to those of VA+ { RNN, LSTM } .

Original: In short, both NMST+ and ST+ prevent non-termination when using greedy search but only NMST+ has a competitive validation perplexity against VA+.
Revised: In short, both NMST+ and ST+ prevent non-termination when using greedy search but only NMST+ has a competitive validation perplexity against VA+.

Original: In §G, we further observe that the length distribution of predicted sequences from NMST+LSTM is closer to the length distribution of ground truth sequences than those of predicted sequences from { VA, ST } +LSTM. 
Revised: In §G, we further observe that the length distribution of predicted sequences from NMST+LSTM is closer to the length distribution of ground truth sequences than those of predicted sequences from { VA, ST } +LSTM. [Table] [Table] [Table]

Original: [Equation]
Revised: best validation perplexity in all setups. For every ϵ , NMST+GPT-2 outperforms ST+GPT-2 in terms of the average validation perplexity. From r nt ( L ) , NMST+GPT-2 effectively prevents non-termination sequences compared to VA+GPT-2 for every ϵ while ST+GPT-2 with small ϵ fails to avoid them. With a proper choice of ϵ (e.g., ϵ = 1 . 0 × 10 − 5 ), NMST+GPT-2 improves the validation perplexity.

Original:   WikiText-103 (Merity et al., 2016) consists of 103 million words constructed from 28,000 articles.
Revised: [Table] 4. W IKI T EXT -  WikiText-103 (Merity et al., 2016) consists of 103 million words constructed from 28,000 articles.

Original: We use BPE tokenization (Sennrich et al., 2015) and consider the first 10 tokens as a context for each sequence.
Revised: We use BPE tokenization (Sennrich et al., 2015) and consider the first 10 tokens as a context for each sequence.

Original: Since WikiText-103 is substantially larger than WikiText-2, we finetune a pretrained
Revised: Since WikiText-103 is substantially larger than WikiText-2, we finetune a pretrained

Original: GPT-2 which is a transformer language model with 124 million parameters (Radford et al., 2019) for 500 , 000 steps.
Revised: GPT-2 which is a transformer language model with 124 million parameters (Radford et al., 2019) for 500 , 000 steps.

Original: For computational efficiency, we bucket the dataset into sequences of similar lengths, and each batch contains a maximum of 1,024 total tokens.
Revised: For computational efficiency, we bucket the dataset into sequences of similar lengths, and each batch contains a maximum of 1,024 total tokens.

Original: We use AdamW (Loshchilov &
Revised: We use AdamW (Loshchilov &

Original: Hutter, 2017) with an initial learning rate of 5 . 

Revised: Hutter, 2017) with an initial learning rate of 5 .

Original: × 10 − 5 , β 1 = 0 .
Revised: × 10 − 5 , β 1 = 0 .

Original: 9 , β 2 = 0 .
Revised: 9 , β 2 = 0 .

Original: 99 , weight decay of 0.01, linear learning rate decay, and early stopping.
Revised: 99 , weight decay of 0.01, linear learning rate decay, and early stopping.

Original: We present a more detailed description in §D. We select ϵ from { 1 . 

Revised: We present a more detailed description in §D. We select ϵ from { 1 .

Original: × 10 − 5 , 5 . 

Revised: × 10 − 5 , 5 .

Original: × 10 − 5 , 1 . 

Revised: × 10 − 5 , 1 .

Original: × 10 − 4 , 5 . 

Revised: × 10 − 4 , 5 .

Original: × 10 − 4 } for ST+GPT-2 and NMST+GPT-2.
Revised: × 10 − 4 } for ST+GPT-2 and NMST+GPT-2.

Original: We report the mean ( ± st.dev.) validation perplexities and non-termination ratios, r nt ( L ) ’s, resulting from greedy search across 10 random runs for all GPT-2 setups finetuned on WikiText-103 in Table 2.
Revised: We report the mean ( ± st.dev.) validation perplexities and non-termination ratios, r nt ( L ) ’s, resulting from greedy search across 10 random runs for all GPT-2 setups finetuned on WikiText-103 in Table 2.

Original: Since GPT-2 can handle up to 1,024 tokens, we use L = 1,000.
Revised: Since GPT-2 can handle up to 1,024 tokens, we use L = 1,000.

Original: As shown in Figure 2, we need a sufficiently large L such as L = 10 5 to determine whether a language model is consistent with respect to greedy search.
Revised: As shown in Figure 2, we need a sufficiently large L such as L = 10 5 to determine whether a language model is consistent with respect to greedy search.

Original: Although L = 1,000 is not sufficiently large, we observe that r nt ( L ) of
Revised: Although L = 1,000 is not sufficiently large, we observe that r nt ( L ) of

Original: NMST+GPT-2 decreases compared to r nt ( L ) of VA+GPT-2 as ϵ increases.
Revised: NMST+GPT-2 decreases compared to r nt ( L ) of VA+GPT-2 as ϵ increases.

Original: That is, NMST+ reduces the number of non-terminating continuations within 1,000 steps.
Revised: That is, NMST+ reduces the number of non-terminating continuations within 1,000 steps.

Original: Non-terminating sequences do not necessarily imply better quality.
Revised: Non-terminating sequences do not necessarily imply better quality.

Original: We thus demonstrate sample continuations from NMST+GPT-2, given a context that leads non-termination with VA+GPT-2 in Table 3, using greedy search.
Revised: We thus demonstrate sample continuations from NMST+GPT-2, given a context that leads non-termination with VA+GPT-2 in Table 3, using greedy search.

Original: We observe that the quality of the generated sequence tends to improve with NMST+ by avoiding repetitions of similar phrases and ending with ⟨ eos ⟩ .
Revised: We observe that the quality of the generated sequence tends to improve with NMST+ by avoiding repetitions of similar phrases and ending with ⟨ eos ⟩ .

Original: We present more example continuations in §E.3.
Revised: We present more example continuations in §E.3.

Original: Similar to the results in §4.1, Table 2 shows that the validation perplexities of both ST+GPTproposed by Welleck et al.

Original: and our NMST+GPT-2 degrade compared to VA+GPT-2 as ϵ

Original: Made of concrete, steel, and wood, the  building was built in the mid @-@ 19th century.
Revised: Made of concrete, steel, and wood, the [Table] building was built in the mid @-@ 19th century.

Original: It was the first building in the
Revised: It was the first building in the

Original: United States to be built in concrete, and the first to be built in wood.
Revised: United States to be built in concrete, and the first to be built in wood.

Original: It was also the first building in the United States to be built in steel.
Revised: It was also the first building in the United States to be built in steel.

Original: It was the first building in ...
Revised: It was the first building in ...

Original: ST+ building is constructed of steel and concrete.
Revised: ST [Table] building is constructed of steel and concrete.

Original: The building’s exterior is made of steel and concrete.
Revised: The building’s exterior is made of steel and concrete.

Original: The building’s interior is made of wood and the building’s exterior is made of concrete.
Revised: The building’s interior is made of wood, and the building’s exterior is made of concrete.

Original: The building’s exterior is made of concrete, and the building’s ...
Revised: The building’s exterior is made of concrete, and the building’s ...

Original: NMST+ building was designed by the architectural firm of Bowers & Wainwright and was completed in 1892.
Revised: NMST [Table] building was designed by the architectural firm of Bowers & Wainwright, and was completed in 1892.

Original: The building is the largest of its kind in the United States.
Revised: The building is the largest of its kind in the United States.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Revised: [Table]

Revised: t

Original: [Table]
Revised: [Table]

Original: p θ ( y t = h
Revised: p θ ( y t = h

Original: e o s i | y
Revised: e o s i | y

Original: < t , x ) x = “ On December 4 , 2010 , Kershaw married his ” NMST +( (cid:15) =1 . 

Revised: < t , x ) x =“ On December 4 , 2010 , Kershaw married his” NMST +( (cid:15) =1 .

Original: × 10 − 5 )ST
Revised: × 10 − 5 )

Original:  +( (cid:15) =1 . [-
-] × 10 − 5 ) VA + · O ﬀ i ce
Revised: ST +( (cid:15) =1 . [-
-]0 × 10 − 5 ) VA + · O ﬀ i ce

Original: s o r ag e n c
Revised: s o r ag e n c

Original: i e s t h a t a r e e a c
Revised: i e s t h a t a r e e a c

Original: h r e s p o n s i b
Revised: h r e s p o n s i b

Original: l e f o r a s p
Revised: l e f o r a s p

Original: ec
Revised: ec

Original: i ﬁ c g e og r
Revised: i ﬁ c g e og r

Original: a ph
Revised: a ph

Original: i c a l a r e a m a n ag e t h e w o r l d w i d e a ﬀ a
Revised: i c a l a r e a m a n ag e t h e w o r l d w i d e a ﬀ a

Original: i r s o f t
Revised: i r s o f t

Original: h e o r ga
Revised: h e o r ga

Original: n
Revised: n

Original: i s a t i o n · T h
Revised: i s a t i o n · T h

Original: e y a r e : h e o s i t 0 [-.0 .0 .0 .0 .0 .0 .
-]{+. 
+}
Revised: e y a r e : h e o s i t 0 [-.0 .0 .0 .0 .0 .0 .
-]{+. 
+}

Original:  .p
Revised: 6 0 .p

Original: θ ( y t =
Revised: θ ( y t =

Original: h e o s i | y
Revised: h e o s i | y

Original: < t , x ) x = “ The CWGC is headquartered in Maidenhead , England ” NMST +( (cid:15) =1 . 

Revised: < t , x ) x =“ The CWGC is headquartered in Maidenhead , England” NMST +( (cid:15) =1 .

Original: × 10 − 5 )ST
Revised: × 10 − 5 )

Original:  +( (cid:15) =1 . [-
-] × 10 − 5 ) VA +
Revised: ST +( (cid:15) =1 . [-
-]0 × 10 − 5 ) VA +

Original: Unlike ST+GPT-2, NMST+GPT-2 can model non-monotonic behaviors of p θ ( y t = ⟨ eos ⟩| y <t , x ) with respect to t .
Revised: Unlike ST+GPT-2, NMST+GPT-2 can model non-monotonic behaviors of p θ ( y t = ⟨ eos ⟩| y <t , x ) with respect to t .

Original: Both plots show that the non-monotonic behaviors occur where the sequences could end (e.g., after red marked tokens such as periods).
Revised: Both plots show that the non-monotonic behaviors occur where the sequences could end (e.g., after red marked tokens such as periods).

Original:  increases.
Revised: Similar to the results in §4.1, Table 2 shows that the validation perplexities of both ST+GPTproposed by Welleck et al. and our NMST+GPT-2 degrade compared to VA+GPT-2 as ϵ increases.

Original: NMST+GPT-2 with the optimal ϵ = 1 . 

Revised: NMST+GPT-2 with the optimal ϵ = 1 .

Original: × 10 − 5 has a competitive validation perplexity of 20.69 against that of VA+GPT-2, 20.72.
Revised: × 10 − 5 has a competitive validation perplexity of 20.69 against that of VA+GPT-2, 20.72.

Original: On the other side, we cannot find ϵ that makes the validation perplexity of ST+GPT-2 competitive against that of VA+GPT-2.
Revised: On the other side, we cannot find ϵ that makes the validation perplexity of ST+GPT-2 competitive against that of VA+GPT-2.

Original: Moreover, if ϵ ̸ = 5 . 0 × 10 − 4 , then r nt ( L ) ’s of ST+GPT-2 blow up unlike r nt ( L ) ’s of VA+GPT-2.
Revised: Moreover, if ϵ ̸ = 5 . 0 × 10 − 4 , then r nt ( L ) ’s of ST+GPT-2 blow up unlike r nt ( L ) ’s of VA+GPT-2.

Original: §E.2 demonstrates the inevitable perplexity degradation and exploding r nt ( L ) of ST+GPT-2.
Revised: §E.2 demonstrates the inevitable perplexity degradation and exploding r nt ( L ) of ST+GPT-2.

Original: We suspect that it is due to monotonically increasing p θ ( y t = ⟨ eos ⟩| y <t , x ) with respect to t .
Revised: We suspect that it is due to monotonically increasing p θ ( y t = ⟨ eos ⟩| y <t , x ) with respect to t .

Original: We investigate behaviors of p θ ( y t = ⟨ eos ⟩| y <t , x ) where p θ ’s are { VA, ST, NMST } +GPT-2 in Figure 3.
Revised: We investigate behaviors of p θ ( y t = ⟨ eos ⟩| y <t , x ) where p θ ’s are { VA, ST, NMST } +GPT-2 in Figure 3.

Original: Based on Table 2, we select the optimal ϵ = 1 .×
Revised: Based on Table 2, we select the optimal ϵ = 1 .×

Original: − 5 in terms of validation perplexities for { ST, NMST } +GPT-2.
Revised: − 5 in terms of validation perplexities for { ST, NMST } +GPT-2.

Original: In Figure 3, { VA, NMST } +GPT-2 well-capture whether a sequence might end (e.g., after periods) by showing non-monotonic behaviors at those seeminglyterminating steps, but ST+GPT-2 cannot model such non-monotonic behaviors because it assumes that p θ ( y t = ⟨ eos ⟩| y <t , x ) is a monotonic function of t .
Revised: In Figure 3, { VA, NMST } +GPT-2 well-capture whether a sequence might end (e.g., after periods) by showing non-monotonic behaviors at those seeminglyterminating steps, but ST+GPT-2 cannot model such non-monotonic behaviors because it assumes that p θ ( y t = ⟨ eos ⟩| y <t , x ) is a monotonic function of t .

Original: This constraint makes ST+GPT-2 generate often finite but unnecessarily long sequences with greedy search (i.e., higher r nt ( L ) than VA+GPTfor small L , but r nt ( L ) = 0 for sufficiently large L ).
Revised: This constraint makes ST+GPT-2 generate often finite but unnecessarily long sequences with greedy search (i.e., higher r nt ( L ) than VA+GPTfor small L , but r nt ( L ) = 0 for sufficiently large L ).

Original: We demonstrate more behaviors in §E.4.
Revised: We demonstrate more behaviors in §E.4.

Original:  C ONSISTENCY WITH RESPECT TO O THER D ECODING A LGORITHMS
Revised:  C ONSISTENCY WITH RESPECT TO O THER D ECODING A LGORITHMS

Original: We explore the effectiveness of our proposed non-monotonic self-terminating (NMST) language model when using decoding algorithms other than greedy search, such as top- k sampling (Fan et al.,
Revised: We explore the effectiveness of our proposed non-monotonic self-terminating (NMST) language model when using decoding algorithms other than greedy search, such as top- k sampling (Fan et al.,

Original: [Table]
Revised: [Table]

Original: , nucleus sampling (Holtzman et al., 2020)
Revised: , nucleus sampling (Holtzman et al., 2020)

Original: [Table]
Revised: [Table]

Original: and beam search.
Revised: and beam search.

Original: All experimental setupsare the same as Section §4.
Revised: All experimental setups as Section §4.

Original: According to Theorem3, the NMST language model is consistent with respect to any incomplete decoding algorithms (e.g., greedy search, top-
Revised: According to Theorem the NMST language model is consistent with respect to any incomplete decoding algorithms (e.g., greedy search, top-

Original: k sampling, and nucleus sampling) and beam search for all ϵ > 0 .
Revised: k sampling, and nucleus sampling) and beam search for all ϵ > 0 .

Original: To validate this, we use top- { 2, 4 } sampling, nucleus- { 0.2, 0.4 } sampling, and beam search with a width of { 2, 4 } (beam- { 2, 4 } ) to generate sequences from NMST+GPT-2 finetuned on WikiText-103 with ϵ = 1 .× 10 − 5 .
Revised: To validate this, we use top- { 2, 4 } sampling, nucleus- { 0.2, 0.4 } sampling, and beam search with a width of { 2, 4 } (beam- { 2, 4 } ) to generate sequences from NMST+GPT-2 finetuned on WikiText-103 with ϵ = 1 .× 10 − 5 .

Original: The choice of ϵ = 1 . 

Revised: The choice of ϵ = 1 .

Original: × 10 − 5 is made based on the validation perplexities in Table 2.
Revised: × 10 − 5 is made based on the validation perplexities in Table 2.

Original: Since the validation perplexity does not depend on decoding algorithms, we focus on the average ( ± st.dev.) non-termination ratios, r nt ( L ) ’s, across 10 random runs with L = 1 , 000 for each decoding algorithm in Table 4.
Revised: Since the validation perplexity does not depend on decoding algorithms, we focus on the average ( ± st.dev.) non-termination ratios, r nt ( L ) ’s, across 10 random runs with L = 1 , 000 for each decoding algorithm in Table 4.

Original: We also present r nt ( L ) ’s of VA+GPT-2 and ST+GPT-2 with ϵ = 1 . 

Revised: We also present r nt ( L ) ’s of VA+GPT-2 and ST+GPT-2 with ϵ = 1 .

Original: × 10 − 5 as baselines.
Revised: × 10 − 5 as baselines.

Original: Table 4 shows that our NMST+GPT-2 has the lowest r nt ( L ) with L = 1 , 000 for all decoding algorithms compared to VA+GPT-2 and ST+GPT-2 proposed by (Welleck et al., 2020).
Revised: Table 4 shows that our NMST+GPT-2 has the lowest r nt ( L ) with L = 1 , 000 for all decoding algorithms compared to VA+GPT-2 and ST+GPT-2 proposed by (Welleck et al., 2020).

Original: In other words, NMST+ effectively prevent non-terminating sequences within 1,000 time steps regardless of decoding algorithms.
Revised: In other words, NMST+ effectively prevent non-terminating sequences within 1,000 time steps regardless of decoding algorithms.

Original: Comparing with greedy search in Table 2 ( r nt ( L ) when ϵ
Revised: Comparing with greedy search in Table 2 ( r nt ( L ) when ϵ

Original: = 1 . 

Revised: = 1 .

Original: × 10 − 5 ), we observe that r nt ( L ) ’s decrease for all setups.
Revised: × 10 − 5 ), we observe that r nt ( L ) ’s decrease for all setups.

Original: As we discussed in §2.3, non-terminating sequences originate from the choice of ⟨ eos ⟩ / ∈ V t ⊊ V for all t where V is a vocabulary and V t is the t -th proper subset of
Revised: As we discussed in §2.3, non-terminating sequences originate from the choice of ⟨ eos ⟩ / ∈ V t ⊊ V for all t where V is a vocabulary and V t is the t -th proper subset of

Original: V , considered by a decoding algorithm at the t -th step.
Revised: V , considered by a decoding algorithm at the t -th step.

Original: The decoding algorithms other than greedy search are likely to have ⟨ eos ⟩ in V t and have the lower r nt ( L ) since their |V t | are greater than or equal to |V t | = 1 of greedy search for all t .
Revised: The decoding algorithms other than greedy search are likely to have ⟨ eos ⟩ in V t and have the lower r nt ( L ) since their |V t | are greater than or equal to |V t | = 1 of greedy search for all t .

Original: In the case of top- { 2, 4 } sampling, we obtain r nt ( L ) = 0 . 

Revised: In the case of top- { 2, 4 } sampling, we obtain r nt ( L ) = 0 .

Original: for VA+GPT-2.
Revised: for VA+GPT-2.

Original: Even without NMST+, VA+ can avoid non-terminating sequences if we choose a proper decoding algorithm.
Revised: Even without NMST+, VA+ can avoid non-terminating sequences if we choose a proper decoding algorithm.

Original: We however emphasize that NMST+GPTwith ϵ = 1 . 0 × 10 − 5 has a competitive validation perplexity against VA+GPT-2 in Table 2 and that it is guaranteed to terminate regardless of the choice of a decoding algorithm.
Revised: We however emphasize that NMST+GPTwith ϵ = 1 . 0 × 10 − 5 has a competitive validation perplexity against VA+GPT-2 in Table 2 and that it is guaranteed to terminate regardless of the choice of a decoding algorithm.

Original: We also empirically demonstrate the consistency of NMST+ { RNN, LSTM } trained on WikiText-2 with respect to other decoding algorithms in §F.
Revised: We also empirically demonstrate the consistency of NMST+ { RNN, LSTM } trained on WikiText-2 with respect to other decoding algorithms in §F.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Non-termination is a degenerate behavior we often observe when generating text from a well-trained language model.
Revised: Non-termination is a degenerate behavior we often observe when generating text from a well-trained language model.

Original: To prevent this, Welleck et al.
Revised: To prevent this, Welleck et al.

Original: (2020) proposed a self-terminating language model that encourages the termination probability of each sequence, which is the conditional probability of ⟨ eos ⟩ given a t -prefix and a context, to monotonically increase toward 1 as t increases.
Revised: (2020) proposed a self-terminating language model that encourages the termination probability of each sequence, which is the conditional probability of ⟨ eos ⟩ given a t -prefix and a context, to monotonically increase toward 1 as t increases.

Original: In this paper, we theoretically demonstrate that monotonically increasing termination probability of each sequence is not a necessary condition for avoiding non-terminating sequences.
Revised: In this paper, we theoretically demonstrate that monotonically increasing termination probability of each sequence is not a necessary condition for avoiding non-terminating sequences.

Original: We then propose a non-monotonic self-terminating language model where the termination probability for each sequence converges to 1 but not monotonically.
Revised: We then propose a non-monotonic self-terminating language model where the termination probability for each sequence converges to 1 but not monotonically.

Original: Our non-monotonic self-terminating language models successfully address the issue of non-termination and achieve perplexities that are comparable to vanilla language models and are better than the original self-terminating language models.
Revised: Our non-monotonic self-terminating language models successfully address the issue of non-termination and achieve perplexities that are comparable to vanilla language models and are better than the original self-terminating language models.

Revised: R EPRODUCIBILITY S TATEMENT

Revised: To ensure the reproducibility of our paper we provide our code available at https://github. com/nyu-dl/non-monotonic-self-terminating-lm .

Revised: A CKNOWLEDGMENTS

Revised: This work was supported by 42dot, Hyundai NGV (under the project Uncertainty in Neural Sequence Modeling), Samsung Advanced Institute of Technology (under the project Next Generation

Original:  A PPENDIX A D EFINITIONS OF C OMMON D ECODING A LGORITHMS AND THEIR C HARACTERISTICS
Revised: Deep Learning: From Pattern Recognition to AI) and NSF Award 1922658 NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science. This work was supported in part through the NYU IT High Performance Computing resources, services, and staff expertise. A PPENDIX A D EFINITIONS OF C OMMON D ECODING A LGORITHMS AND THEIR C HARACTERISTICS

Original: In this section, we present mathematical definitions of top- k sampling (Fan et al., 2018), nucleus sampling (Holtzman et al., 2020), greedy search, and beam search.
Revised: In this section, we present mathematical definitions of top- k sampling (Fan et al., 2018), nucleus sampling (Holtzman et al., 2020), greedy search, and beam search.

Original: We then demonstrate whether they are incomplete probable decoding algorithms.
Revised: We then demonstrate whether they are incomplete probable decoding algorithms.

Original: A. T OP - K SAMPLING
Revised: A. T OP - K SAMPLING

Original: At each step t , top- k sampling selects a subset of k most probable tokens in a vocabulary V .
Revised: At each step t , top- k sampling selects a subset of k most probable tokens in a vocabulary V .

Original: Top-
Revised: Top-

Original: k sampling generates decoded sequences from a language model p θ as follows:
Revised: k sampling generates decoded sequences from a language model p θ as follows:

Original: Definition A.1 (Top- k sampling (Fan et al., 2018)) .
Revised: Definition A.1 (Top- k sampling (Fan et al., 2018)) .

Original: Top-
Revised: Top-

Original: k sampling S top-
Revised: k sampling S top-

Original: k generates a sequence from a language model p θ given a context x by recursively sampling ˆ y t from
Revised: k generates a sequence from a language model p θ given a context x by recursively sampling ˆ y t from

Original: [Equation]
Revised: [Equation]

Original: otherwise, [Equation] 12) where
Revised: otherwise, (12) where

Original: [Equation]
Revised: [Equation]

Original: Except the trivial case k = |V| , we have ∅ ⊊ V t ⊊ V for all t .
Revised: Except the trivial case k = |V| , we have ∅ ⊊ V t ⊊ V for all t .

Original: By equation 13, equation 6 holds.
Revised: By equation 13, equation 6 holds.

Original: From equation 12, we see that top- k sampling satisfies equation 5 and equation 7.
Revised: From equation 12, we see that top- k sampling satisfies equation 5 and equation 7.

Original: Therefore, top- k sampling is an incomplete probable decoding algorithm.
Revised: Therefore, top- k sampling is an incomplete probable decoding algorithm.

Original: A. N UCLEUS SAMPLING
Revised: A. N UCLEUS SAMPLING

Original: At each step t , nucleus sampling selects the smallest subset of most probable tokens in a vocabulary
Revised: At each step t , nucleus sampling selects the smallest subset of most probable tokens in a vocabulary

Original: V , of which total probability is higher than a given threshold µ .
Revised: V , of which total probability is higher than a given threshold µ .

Original: Nucleus sampling generates decoded sequences from a language model p θ as follows:
Revised: Nucleus sampling generates decoded sequences from a language model p θ as follows:

Original: Definition A.2 (Nucleus sampling (Holtzman et al., 2020)) .
Revised: Definition A.2 (Nucleus sampling (Holtzman et al., 2020)) .

Original: Nucleus sampling S nuc- µ generates a sequence from a language model p θ given a context x by recursively sampling ˆ y t fromq
Revised: Nucleus sampling S nuc- µ generates a sequence from a language model p θ given a context x by recursively sampling ˆ y t from

Original: [Equation]
Revised: [Equation]

Original: where V t is the smallest subset of V such that
Revised: where V t is the smallest subset of V such that

Original: [Equation]
Revised: [Equation]

Original: If min v ∈V p θ ( y t = v | y <t , x ) ≤ 1 − µ for any context x and any t -prefix y <t , then we have ∅ ⊊ V t ⊊ V for all t .
Revised: If min v ∈V p θ ( y t = v | y <t , x ) ≤ 1 − µ for any context x and any t -prefix y <t , then we have ∅ ⊊ V t ⊊ V for all t .

Original: Suppose that equation 6 does not hold for nucleus sampling.
Revised: Suppose that equation 6 does not hold for nucleus sampling.

Original: Then, this contradicts to V t is the smallest subset of V , satisfying equation 15.
Revised: Then, this contradicts to V t is the smallest subset of V , satisfying equation 15.

Original: From equation 14, we see that nucleus sampling satisfies equation 5 and equation 7.
Revised: From equation 14, we see that nucleus sampling satisfies equation 5 and equation 7.

Original: Therefore, nucleus sampling is incomplete and probable.
Revised: Therefore, nucleus sampling is incomplete and probable.

Original: A. B EAM SEARCH
Revised: A. B EAM SEARCH

Original: Beam search is a heuristic algorithm that operates on the level of prefixes.
Revised: Beam search is a heuristic algorithm that operates on the level of prefixes.

Original: We use the definition of beam search in Welleck et al.
Revised: We use the definition of beam search in Welleck et al.

Original: Definition A.3 (Beam search, Definition A.2 in Welleck et al.
Revised: Definition A.3 (Beam search, Definition A.2 in Welleck et al.

Original: Beam search with a width (beam size) k , S beam- k , generates a sequence from a language model p θ by maintaining a set of k prefixes, P t = { ρ (1) ( t ) , ρ (2) ( t ) , · · · , ρ ( k ) ( t ) } , at each time step t where ρ ( i ) (0) is an empty prefix for all i .
Revised: Beam search with a width (beam size) k , S beam- k , generates a sequence from a language model p θ by maintaining a set of k prefixes, P t = { ρ (1) ( t ) , ρ (2) ( t ) , · · · , ρ ( k ) ( t ) } , at each time step t where ρ ( i ) (0) is an empty prefix for all i .

Original: At each step t ∈ { 1 , 2 , · · · } , beam search forms a set of k × k prefixes,
Revised: At each step t ∈ { 1 , 2 , · · · } , beam search forms a set of k × k prefixes,

Original: [Equation]
Revised: [Equation]

Original: 16) where ρ ◦ v is concatenation and
Revised: 16) where ρ ◦ v is concatenation and

Original: [Equation]
Revised: [Equation]

Original: After forming ˜ P t , beam search selects a set of the k highest scoring prefixes in ˜ P t 
Revised: After forming ˜ P t , beam search selects a set of the k highest scoring prefixes in ˜ P t ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: where s ( ρ ) =
Revised: where s ( ρ ) =

Original: (cid:80) tτ
Revised: (cid:80) tτ

Original: =1 log p θ ( y τ =
Revised: =1 log p θ ( y τ =

Original: ρ τ | ρ <τ , x ) .
Revised: ρ τ | ρ <τ , x ) .

Original: If ρ ∈ P t ends with ⟨ eos ⟩ , then it does not expand further and is added to the final set
Revised: If ρ ∈ P t ends with ⟨ eos ⟩ , then it does not expand further and is added to the final set

Original: P .
Revised: P .

Original: Beam search continues until P contains k sequences ending with ⟨ eos ⟩ .
Revised: Beam search continues until P contains k sequences ending with ⟨ eos ⟩ .

Original: After that it returns the highest scoring sequence
Revised: After that it returns the highest scoring sequence

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Unlike greedy search, top- k sampling, and nucleus sampling, beam search recursively expands k sequences with at most k different prefixes.
Revised: Unlike greedy search, top- k sampling, and nucleus sampling, beam search recursively expands k sequences with at most k different prefixes.

Original: Therefore, we cannot formalize beam search in tokenlevel by using q S beam- k ( y t = v | y <t , x ) .
Revised: Therefore, we cannot formalize beam search in tokenlevel by using q S beam- k ( y t = v | y <t , x ) .

Original: However, in equation 17, the number of possible tokens at t is at most k × k .
Revised: However, in equation 17, the number of possible tokens at t is at most k × k .

Original: It means that S beam- k may exclude ⟨ eos ⟩ at time t if k ≤
Revised: It means that S beam- k may exclude ⟨ eos ⟩ at time t if k ≤

Original: [Equation]
Revised: [Equation]

Original: |V| − 1 .
Revised: |V| − 1 .

Original: By using this, Welleck et al.
Revised: By using this, Welleck et al.

Original: (2020) proved that a vanilla language model p va θ is inconsistent with respect to beam search as shown in Theorem 1.
Revised: (2020) proved that a vanilla language model p va θ is inconsistent with respect to beam search as shown in Theorem 1.

Original: B P ROOFS FOR §2. 
Revised: B P ROOFS FOR §2.

Original: Remark 1.
Revised: Remark 1.

Original: Let D = (cid:8) ( x (1) , y (1) ) , ( x (2) , y (2) ) (cid:9) be a two-instance training dataset.
Revised: Let D = (cid:8) ( x (1) , y (1) ) , ( x (2) , y (2) ) (cid:9) be a two-instance training dataset.

Original: Assume that there exists t 0 such that y <t 0 =
Revised: Assume that there exists t 0 such that y <t 0 =

Original: y (1) <t 

Revised: y (1) <t

Original: = y (2) <t 0 .
Revised: = y (2) <t 0 .

Original: Suppose further that t 0 = | y (1) | < | y (2)
Revised: Suppose further that t 0 = | y (1) | < | y (2)

Original: | −and x
Revised: | −and x

Original: = x (1) =
Revised: = x (1) =

Original: x (2) .
Revised: x (2) .

Original: If θ ⋆ is an optimal parameter configuration in equation 3 over D .
Revised: If θ ⋆ is an optimal parameter configuration in equation 3 over D .

Original: Then, p θ ⋆ (cid:0) 
Revised: Then, p θ ⋆ (cid:0) y

Original: [Equation]
Revised: [Equation]

Original: t =
Revised: t =

Original: [Equation]
Revised: [Equation]

Original: eos
Revised: eos

Original: [Equation]
Revised: [Equation]

Original: | y
Revised: | y

Original: [Equation]
Revised: [Equation]

Revised: ) <

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: (cid:1) is non-monotonic with respect to t .
Revised: (cid:1) is non-monotonic with respect to t .

Original: Proof.
Revised: Proof.

Original: Since θ ⋆ is an optimal parameter configuration that perfectly minimizes equation 3 and t 0 < | y (2) | − 1 , we have
Revised: Since θ ⋆ is an optimal parameter configuration that perfectly minimizes equation 3 and t 0 < | y (2) | − 1 , we have

Original: [Equation]
Revised: [Equation]

Original: for t < t 0 .
Revised: for t < t 0 .

Original: Note that t 0 = | y (1) | ⇒ y (1) t 0 = ⟨ eos ⟩ and t 0 < | y (2) | − 1 ⇒ y
Revised: Note that t 0 = | y (1) | ⇒ y (1) t 0 = ⟨ eos ⟩ and t 0 < | y (2) | − 1 ⇒ y

Original: (2) t 0 ̸ = ⟨ eos ⟩ .
Revised: (2) t 0 ̸ = ⟨ eos ⟩ .

Original: From x = x (1) =
Revised: From x = x (1) =

Original: x (2) and y =
Revised: x (2) and y =

Original: y (1) <t 

Revised: y (1) <t

Original: = y (2) <
Revised: = y (2) <

Original: t 0 , we obtain 
Revised: t 0 , we obtain p

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: Moreover, t 0 < | y (2) | − 1 implies that y (2) t 0 +1 ̸ = ⟨ eos ⟩ which is equivalent to
Revised: Moreover, t 0 < | y (2) | − 1 implies that y (2) t 0 +1 ̸ = ⟨ eos ⟩ which is equivalent to

Original: [Equation]
Revised: [Equation]

Original: From equation 20, equation 21, and equation 22, we see that p θ ⋆
Revised: From equation 20, equation 21, and equation 22, we see that p θ ⋆

Original: [Equation]
Revised: [Equation]

Original: ⟨ eos ⟩| y (2) 
Revised: eos ⟩| y (2) <

Original: <t , x (cid:1) is nonmonotonic with respect to t .
Revised: t , x (cid:1) is nonmonotonic with respect to t .

Original: C P ROOFS FOR § 
Revised: C P ROOFS FOR §

Original: Theorem 3.
Revised: Theorem 3.

Original: A non-monotonic self-terminating (NMST) language model defined in Definition 6 is consistent with respect to any incomplete probable decoding algorithms and beam search.
Revised: A non-monotonic self-terminating (NMST) language model defined in Definition 6 is consistent with respect to any incomplete probable decoding algorithms and beam search.

Original: Proof.
Revised: Proof.

Original: From equation 10, for any θ ∈ R k , we have
Revised: From equation 10, for any θ ∈ R k , we have

Original: [Equation]
Revised: [Equation]

Original: since (1 − ϵ ) t → 0 as t → ∞ for ϵ ∈ (0 , 1) and σ[Equation] ⊤⟨
Revised: since (1 − ϵ ) t → 0 as t → ∞ for ϵ ∈ (0 , 1) and σ

Original: [Equation]
Revised: [Equation]

Original: 1) for any t .
Revised: 1) for any t .

Original: Hence, there exists t 1 / 2 such that
Revised: Hence, there exists t 1 / 2 such that

Original: [Equation]
Revised: [Equation]

Original: Let S be any incomplete probable decoding algorithm.
Revised: Let S be any incomplete probable decoding algorithm.

Original: From equation 6 and equation 7, ⟨ eos ⟩ ∈ V t and q 
Revised: From equation 6 and equation 7, ⟨ eos ⟩ ∈ V t and q S [Equation] θ )

Original: S ( p nmst θ ) ( y t ̸ = ⟨ eos ⟩| y <t , x ) < 12 holds for any t ≥ t 1 / 2 .
Revised: [Equation] y [Equation] ̸ = ⟨ eos ⟩| y <t , x ) < 12 holds for any t ≥ t 1 / 2 .

Original: Therefore, we obtain
Revised: Therefore, we obtain

Original: [Equation]
Revised: [Equation]

Original: Taking expectation of equation 24 over x , we finally have q S ( p nmst θ ) ( | y | = ∞ )
Revised: Taking expectation of equation 24 over x , we finally have q S ( p nmst θ ) ( | y | = ∞ )

Original: = 

Revised: =

Original: for any S .
Revised: for any S .

Original: In other words, p nmst θ is consistent with respect to any incomplete probable decoding algorithms.
Revised: In other words, p nmst θ is consistent with respect to any incomplete probable decoding algorithms.

Original: In the case of beam search S beam- k defined in §A.3, without loss of generality, there exists ρ ∈ P t 1 /such that ρ does not end with ⟨ eos ⟩ .
Revised: In the case of beam search S beam- k defined in §A.3, without loss of generality, there exists ρ ∈ P t 1 /such that ρ does not end with ⟨ eos ⟩ .

Original: 2 Let P >t 1 / 2 ( ρ ) be a set of k highest scoring sequences continued from ρ by
Revised: 3 Let P >t 1 / 2 ( ρ ) be a set of k highest scoring sequences continued from ρ by

Original: S beam- k .
Revised: S beam- k .

Original: From equation 23, we have
Revised: From equation 23, we have

Original: [Equation]
Revised: [Equation]

Original: v | ρ

Original: [Equation]

Original: x ) for all v ∈ V \ {⟨ eos ⟩} .
Revised: x ) for all v ∈ V \ {⟨ eos ⟩} .

Original: Hence, V t 1 / 2 ( ρ ) in equation 17 includes ⟨ eos ⟩ .
Revised: Hence, V t 1 / 2 ( ρ ) in equation 17 includes ⟨ eos ⟩ .

Original: Let z
Revised: Let z

Original: =
Revised: =

Original: ( z 1 , z 2 , · · · , z l ) be any subsequence with z 1 ̸ = ⟨ eos ⟩ .
Revised: ( z 1 , z 2 , · · · , z l ) be any subsequence with z 1 ̸ = ⟨ eos ⟩ .

Original: Then, we have
Revised: Then, we have

Original: [Equation]
Revised: [Equation]

Original: nmst
Revised: nmst

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: 25) where ◦ is concatenation.
Revised: where ◦ is concatenation.

Original: Therefore, ρ ◦ ⟨ eos ⟩ = arg max ρ ′ ∈P t 1 / 2 s ( ρ ′ ) holds where s ( ρ ′ )
Revised: Therefore, ρ ◦ ⟨ eos ⟩ = arg max ρ ′ ∈P t 1 / 2 s ( ρ ′ ) holds where s ( ρ ′ )

Original: =
Revised: =

Original: (cid:80) t
Revised: (cid:80) t

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: logp
Revised: log

Original: [Equation]
Revised: [Equation]

Original: θ ( ρ ′ τ | ρ ′ <τ , x ) .
Revised: θ ( ρ ′ τ | ρ ′ <τ , x ) .

Original: That is, ρ ◦ ⟨ eos ⟩ is the highest scoring sequence staring with ρ , and we have ρ ◦ ⟨ eos ⟩ ∈ P ( ρ ) .
Revised: That is, ρ ◦ ⟨ eos ⟩ is the highest scoring sequence starting with ρ , and we have ρ ◦ ⟨ eos ⟩ ∈ P ( ρ ) .

Original: For each ρ ′ ∈ P >t 1 / 2 ( ρ ) \ { ρ ◦ ⟨ eos ⟩} , ρ ′ starts with ρ ◦ v for v ∈ V \ {⟨ eos ⟩} .
Revised: For each ρ ′ ∈ P >t 1 / 2 ( ρ ) \ { ρ ◦ ⟨ eos ⟩} , ρ ′ starts with ρ ◦ v for v ∈ V \ {⟨ eos ⟩} .

Original: By the same argument, we add at least one sequence ending with ⟨ eos ⟩ to P >t 1 / 2 ( ρ ) .
Revised: By the same argument, we add at least one sequence ending with ⟨ eos ⟩ to P >t 1 / 2 ( ρ ) .

Original: It means that P >t 1 / 2 ( ρ ) has k sequences ending with ⟨ eos ⟩ within t 1 / 2 + k steps.
Revised: It means that P >t 1 / 2 ( ρ ) has k sequences ending with ⟨ eos ⟩ within t 1 / 2 + k steps.

Original: Note that the final set P satisfies
Revised: Note that the final set P satisfies

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Equation 26 implies that every sequence in P has the length of at most t 1 / 

Revised: Equation 26 implies that every sequence in P has the length of at most t 1 /

Original: + k .
Revised: + k .

Original: We thus obtain
Revised: We thus obtain

Original: [Equation]
Revised: [Equation]

Original:  Taking expectation of equation 27 over x , we see that q S beam- k ( p nmst θ ) ( | y | = ∞ ) .
Revised: [Equation] Taking expectation of equation 27 over x , we see that q S beam- k ( p nmst θ ) ( | y | = ∞ ) .

Original: That is, p nmst θ is consistent with respect to beam search.
Revised: That is, p nmst θ is consistent with respect to beam search.

Original: D E XPERIMENTAL D ETAILS
Revised: D E XPERIMENTAL D ETAILS

Original: In this section, we describe our models and optimization processes used in §4.
Revised: In this section, we describe our models and optimization processes used in §4.

Original: RNN and LSTM on WikiText-

Revised: RNN and LSTM on WikiText-

Original: We use word tokenization for WikiText-2.
Revised: We use word tokenization for WikiText-2.

Original: We train RNN with tanh activations (Elman, 1990) and LSTM (Hochreiter & Schmidhuber, 1997) on WikiText-2.
Revised: We train RNN with tanh activations (Elman, 1990) and LSTM (Hochreiter & Schmidhuber, 1997) on WikiText-2.

Original: Both
Revised: Both

Original: RNN and LSTM have 2 layers.
Revised: RNN and LSTM have 2 layers.

Original: Each layer has 256 hidden units for RNN and 512 hidden units for LSTM.
Revised: Each layer has 256 hidden units for RNN and 512 hidden units for LSTM.

Original: The sizes of input and output embedding layers are 256 and 512 for RNN and LSTM, respectively.
Revised: The sizes of input and output embedding layers are 256 and 512 for RNN and LSTM, respectively.

Original: We use weight tying to share the weights between the input and output embedding layers for both models.
Revised: We use weight tying to share the weights between the input and output embedding layers for both models.

Original: We apply dropout (Srivastava et al., 2014) with drop probabilities of 0 .and 0 .
Revised: We apply dropout (Srivastava et al., 2014) with drop probabilities of 0 .and 0 .

Original: 5 to RNN and LSTM accordingly.
Revised: 5 to RNN and LSTM accordingly.

Original: For each model, we perform 10 random runs with a batch size of 32 for 70 epochs.
Revised: For each model, we perform 10 random runs with a batch size of 32 for 70 epochs.

Original: To maximize the log-likelihoodloss function presented in equation 3, we use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 0 .
Revised: To maximize the log-likelihood presented in equation 3, we use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 0 .

Original: 001 , β 1 = 0 .
Revised: 001 , β 1 = 0 .

Original: 9 , β 2 = 0 .
Revised: 9 , β 2 = 0 .

Original: 99 , weight decay of 0 . 01 , and learning rate decay which halves the learning rate if the validation perplexity does not improve for a training epoch.
Revised: 99 , weight decay of 0 . 01 , and learning rate decay which halves the learning rate if the validation perplexity does not improve for a training epoch.

Original: To avoid overfitting, we additionally use early stopping, which terminates training if the validation perplexity does not improve upon the best score attained so far for 10 consecutive epochs.
Revised: To avoid overfitting, we additionally use early stopping, which terminates training if the validation perplexity does not improve upon the best score attained so far for 10 consecutive epochs.

Original: In most cases, the training ends withinepochs.
Revised: In most cases, the training ends within 50 epochs.

Original: GPT-2 on WikiText-

Revised: GPT-2 on WikiText-

Original: We use BPE tokenization 3 (Sennrich et al., 2015) and the pretrained
Revised: We use BPE tokenization 4 (Sennrich et al., 2015) and the pretrained

Original: GPT-2 4 (Radford et al., 2019) with 124 million parameters, provided by HuggingFace .
Revised: GPT-2 5 (Radford et al., 2019) with 124 million parameters, provided by HuggingFace .

Original: GPTcan handle up to 1,024 tokens.
Revised: GPTcan handle up to 1,024 tokens.

Original: We apply dropout (Srivastava et al., 2014) with a drop probability of 0 . 1 to GPT-2.
Revised: We apply dropout (Srivastava et al., 2014) with a drop probability of 0 . 1 to GPT-2.

Original: We finetune GPT-2 for 300,000 steps while ensuring that all runs continue for at least 250,000 steps.
Revised: We finetune GPT-2 for 300,000 steps while ensuring that all runs continue for at least 250,000 steps.

Original: To minimize the number of padding tokens in every batch for computational efficiency, we bucket the dataset into sequences of similar lengths, and each batch contains a maximum of 1,024 total tokens.
Revised: To minimize the number of padding tokens in every batch for computational efficiency, we bucket the dataset into sequences of similar lengths, and each batch contains a maximum of 1,024 total tokens.

Original: To maximize the log-likelihood function in equation 3, we use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 5 . 

Revised: To maximize the log-likelihood function in equation 3, we use AdamW (Loshchilov & Hutter, 2017) with an initial learning rate of 5 .

Original: × 10 − 5 , β 1 = 0 .
Revised: × 10 − 5 , β 1 = 0 .

Original: 9 , β 2 = 0 .
Revised: 9 , β 2 = 0 .

Original: 99 , weight decay of 0.01, and linear learning rate decay over 500 , 000 steps.
Revised: 99 , weight decay of 0.01, and linear learning rate decay over 500 , 000 steps.

Original: 3 https://github.com/huggingface/tokenizershttps://github.com/huggingface/transformers

Original: E A DDITIONAL P LOTS AND T ABLES FOR § 
Revised: E A DDITIONAL P LOTS AND T ABLES FOR §

Original: In this section, we demonstrate additional plots and tables for §4.
Revised: In this section, we demonstrate additional plots and tables for §4.

Original: E. A DDITIONAL PLOTS FOR §4. 
Revised: E. A DDITIONAL PLOTS FOR §4.

Original: [Figure]
Revised: [Figure]

Original: E. A DDITIONAL PLOTS FOR §4. 
Revised: E. A DDITIONAL PLOTS FOR §4.

Original: [Figure]
Revised: [Figure]

Original: because GPT-2 has a context window size of 1 , 024 .
Revised: because GPT-2 has a context window size of 1 , 024 .

Original: For all ϵ , NMST+GPT-2 outperforms ST+GPT2 in terms of the average validation perplexity.
Revised: For all ϵ , NMST+GPT-2 outperforms ST+GPT2 in terms of the average validation perplexity.

Original: When ϵ is small, r n t ( L ) of ST+GPT-2 explodes.
Revised: When ϵ is small, r n t ( L ) of ST+GPT-2 explodes.

Original: It means that ST+GPT-2 with small ϵ cannot prevent non-terminating sequences.
Revised: It means that ST+GPT-2 with small ϵ cannot prevent non-terminating sequences.

Original: However, our
Revised: However, our

Original: NMST+GPT-2 effectively reduces r nt ( L ) compared to VA+GPT-2 for every ϵ , and the validation perplexity degradation is smaller than that of ST+GPT-2 proposed by Welleck et al.
Revised: NMST+GPT-2 effectively reduces r nt ( L ) compared to VA+GPT-2 for every ϵ , and the validation perplexity degradation is smaller than that of ST+GPT-2 proposed by Welleck et al.

Original: E. A DDITIONAL TABLES FOR T ABLE
Revised: E. A DDITIONAL TABLES FOR T ABLE

Original: The  made its Irish Singles Chart debut at
Revised: The single made its Irish Singles Chart debut at

Original: VA+ number three on the issue date of November 18, 2005.
Revised: VA+ number three on the issue date of November 18, 2005.

Original: It debuted at number one on the chart for the issue date of December 7, 2005.
Revised: It debuted at number one on the chart for the issue date of December 7, 2005.

Original: The single debuted at number two on the chart for the issue date of January 4, 
Revised: The single debuted at number two on the chart for the issue date of January 4, 2006.

Original: The  debuted at number one on the Irish  Chart for the issue date of March 4, 
Revised: The single debuted at number one on the Irish Singles Chart for the issue date of March 4, 2006.

Original: The  debuted at number two on the  Chart for the issue date of June 4, debuted one on the Irish
Revised: The single debuted at number two on the Irish Singles Chart for the issue date of June 4, 2006.

Original:  Chart for the  date number three on the chart issue dated March 16, 2010.
Revised: The single debuted at number one on the Irish Singles Chart for the issue date number three on the chart issue dated March 16, 2010.

Original: It debuted at number two on the UK Singles Chart on the chart issue dated March 16, 2010, and peaked at number one on the chart issue dated March 16, 2010.
Revised: It debuted at number two on the UK Singles Chart on the chart issue dated March 16, 2010, and peaked at number one on the chart issue dated March 16, 2010.

Original: The single was certified gold by the
Revised: The single was certified gold by the

Original: British Phonographic Industry ( BPI ) for shipments of over 15 @,@ 000 copies.
Revised: British Phonographic Industry ( BPI ) for shipments of over 15 @,@ 000 copies.

Original: The single debuted at number two on the Irish Singles Chart on the chart issue dated
Revised: The single debuted at number two on the Irish Singles Chart on the chart issue dated

Original: March 16, 2010, and peaked at number one on the chart issue dated March 16, 2010.
Revised: March 16, 2010, and peaked at number one on the chart issue dated March 16, 2010.

Original: The single was certified gold by the Irish Recorded Music Association ( IRMA ) for shipments of over 15 @,@ 000 copies.
Revised: The single was certified gold by the Irish Recorded Music Association ( IRMA ) for shipments of over 15 @,@ 000 copies.

Original: The single was ...
Revised: The single was ...

Original: NMST+ number twenty @-@ seven on the week ending March 26, 2010, and peaked at number three on the week ending March 27, 2010.
Revised: NMST+ number twenty @-@ seven on the week ending March 26, 2010, and peaked at number three on the week ending March 27, 2010.

Original: It was certified gold by the Recording
Revised: It was certified gold by the Recording

Original: Industry Association of Ireland ( RIAA ) for shipment of 500 @,@ 000 copies of the single.
Revised: Industry Association of Ireland ( RIAA ) for shipment of 500 @,@ 000 copies of the single.

Original: The single was certified gold by the Recording Industry Association of
Revised: The single was certified gold by the Recording Industry Association of

Original: Ireland ( RIANZ ) for shipment of 500 @,@ 000 copies of the single.
Revised: Ireland ( RIANZ ) for shipment of 500 @,@ 000 copies of the single.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Original: Context
Revised: Context

Original: Despite the expensive reconstructions, both vessels were considered
Revised: Despite the expensive reconstructions, both vessels were considered

Original: VA+ to be of sufficient quality to be considered for use in the Grand Fleet.
Revised: VA+ to be of sufficient quality to be considered for use in the Grand Fleet.

Original: The first, the
Revised: The first, the

Original: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built,
Revised: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built,

Original: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built,
Revised: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built,

Original: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built ...
Revised: British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built, British @-@ built ...

Original: ST+ to be of sufficient quality to be considered a part of the Royal Navy, and were assigned to the Channel Fleet.
Revised: ST+ to be of sufficient quality to be considered a part of the Royal Navy, and were assigned to the Channel Fleet.

Original: The two ships were transferred to the Mediterranean
Revised: The two ships were transferred to the Mediterranean

Original: Fleet in early 1915, and were transferred to the North America and West Indies Station in early 1916.
Revised: Fleet in early 1915, and were transferred to the North America and West Indies Station in early 1916.

Original: They were transferred to the Mediterranean Fleet in early 1917, and were transferred to the North America and West Indies Station in early 1918. They were transferred to the North America and West Indies Station in early 1918, and were transferred to the Mediterranean Fleet in early 1918.
Revised: They were transferred to the Mediterranean Fleet in early 1917, and were transferred to the North America and West Indies Station in early 1918. They were transferred to the North America and West Indies Station in early 1918, and were transferred to the Mediterranean Fleet in early 1918.

Original: They were transferred to the North America and West Indies Station in late 1918, and ...
Revised: They were transferred to the North America and West Indies Station in late 1918, and ...

Original: NMST+ obsolescent by the time of the Second World War, and were eventually scrapped in favour of the much more powerful British battleship HMS Dreadnought.
Revised: NMST+ obsolescent by the time of the Second World War, and were eventually scrapped in favour of the much more powerful British battleship HMS Dreadnought.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Original: Context
Revised: Context

Original: On the July 21, 2014 episode of Raw,
Revised: On the July 21, 2014 episode of Raw,

Original: VA+
Revised: VA+

Original: Triple H and Randy Orton were involved in a tag team match, in which Triple H and
Revised: Triple H and Randy Orton were involved in a tag team match, in which Triple H and

Original: Orton were accompanied by Triple H and Randy Orton’s manager, Mr. Kennedy.
Revised: Orton were accompanied by Triple H and Randy Orton’s manager, Mr. Kennedy.

Original: Orton was accompanied by Orton’s manager, Mr. Kennedy, and Orton’s tag team partner, Mr. Kennedy.
Revised: Orton was accompanied by Orton’s manager, Mr. Kennedy, and Orton’s tag team partner, Mr. Kennedy.

Original: Triple H and Orton were accompanied by Orton’s manager,
Revised: Triple H and Orton were accompanied by Orton’s manager,

Original: Mr. Kennedy, and Orton’s tag team partner, Mr. Kennedy.
Revised: Mr. Kennedy, and Orton’s tag team partner, Mr. Kennedy.

Original: Triple H and Orton were accompanied by Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr.
Revised: Triple H and Orton were accompanied by Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr.

Original: Kennedy.
Revised: Kennedy.

Original: Triple H and Orton were accompanied by Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr. Kennedy.
Revised: Triple H and Orton were accompanied by Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr. Kennedy.

Original: Triple H and Orton were accompanied by
Revised: Triple H and Orton were accompanied by

Original: Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr. Kennedy.
Revised: Orton’s manager, Mr. Kennedy, and Orton’s manager, Mr. Kennedy.

Original: Triple H ...
Revised: Triple H ...

Original: ST+
Revised: ST+

Original: Triple H defeated Randy Orton to win the World Heavyweight Championship.
Revised: Triple H defeated Randy Orton to win the World Heavyweight Championship.

Original: On the July 27 episode of Raw, Triple H defeated Randy Orton to win the World Heavyweight Championship.
Revised: On the July 27 episode of Raw, Triple H defeated Randy Orton to win the World Heavyweight Championship.

Original: On the July 30 episode of Raw, Triple H defeated Randy
Revised: On the July 30 episode of Raw, Triple H defeated Randy

Original: Orton to win the World Heavyweight Championship.
Revised: Orton to win the World Heavyweight Championship.

Original: Triple H then defeated Randy
Revised: Triple H then defeated Randy

Original: Orton to win the World Heavyweight Championship.
Revised: Orton to win the World Heavyweight Championship.

Original: On the August 2 episode of
Revised: On the August 2 episode of

Original: Raw, Triple H defeated Randy Orton to win the World Heavyweight Championship.
Revised: Raw, Triple H defeated Randy Orton to win the World Heavyweight Championship.

Original: Triple H then defeated Randy Orton to win the World Heavyweight Championship.
Revised: Triple H then defeated Randy Orton to win the World Heavyweight Championship.

Original: Triple H then defeated Orton to win the World Heavyweight Championship.
Revised: Triple H then defeated Orton to win the World Heavyweight Championship.

Original: Triple
Revised: Triple

Original: H then defeated Orton to win the World Heavyweight Championship.
Revised: H then defeated Orton to win the World Heavyweight Championship.

Original: Triple H then defeated Orton to win the World Heavyweight Championship. Triple H then defeated Orton to win the World Heavyweight Championship.
Revised: Triple H then defeated Orton to win the World Heavyweight Championship. Triple H then defeated Orton to win the World Heavyweight Championship.

Original: Triple H then defeated
Revised: Triple H then defeated

Original: Orton to win the World Heavyweight Championship.
Revised: Orton to win the World Heavyweight Championship.

Original: Triple H then defeated ...
Revised: Triple H then defeated ...

Original: NMST+
Revised: NMST+

Original: Triple H defeated Randy Orton to win the WWE Championship for the third time.
Revised: Triple H defeated Randy Orton to win the WWE Championship for the third time.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Original: Context
Revised: Context

Original: On the west side of the Naktong,
Revised: On the west side of the Naktong,

Original: VA+ the NK 6th Division was reinforced by the NK 7th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 7th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK ...
Revised: VA+ the NK 6th Division was reinforced by the NK 7th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 7th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK 6th Division, which was reinforced by the NK ...

Original: ST+
Revised: ST+

Original: the 2nd Battalion, 27th Infantry Regiment, which had been holding up the North
Revised: the 2nd Battalion, 27th Infantry Regiment, which had been holding up the North

Original: Koreans in the hills east of the Naktong, was ordered to withdraw to the Naktong itself.
Revised: Koreans in the hills east of the Naktong, was ordered to withdraw to the Naktong itself.

Original: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the
Revised: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the

Original: North Koreans in the hills east of the Naktong, was ordered to withdraw to the
Revised: North Koreans in the hills east of the Naktong, was ordered to withdraw to the

Original: Naktong itself.
Revised: Naktong itself.

Original: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the North Koreans in the hills east of the Naktong, was ordered to withdraw to the
Revised: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the North Koreans in the hills east of the Naktong, was ordered to withdraw to the

Original: Naktong itself.
Revised: Naktong itself.

Original: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the North Koreans in the hills east of the Naktong, was ordered to withdraw to the Naktong itself.
Revised: The 2nd Battalion, 27th Infantry Regiment, which had been holding up the North Koreans in the hills east of the Naktong, was ordered to withdraw to the Naktong itself.

Original: The 2nd Battalion, 27th Infantry Regiment, which had been ...
Revised: The 2nd Battalion, 27th Infantry Regiment, which had been ...

Original: NMST+ the Naktong River rises to a height of 1 @,@ 000 metres ( 3 @,@ 300 ft ) above the surrounding terrain.
Revised: NMST+ the Naktong River rises to a height of 1 @,@ 000 metres ( 3 @,@ 300 ft ) above the surrounding terrain.

Original: The Naktong River is the longest river in North and South
Revised: The Naktong River is the longest river in North and South

Original: Korea, and the longest in the whole of North Korea.
Revised: Korea, and the longest in the whole of North Korea.

Original: The Naktong is the longest river in North Korea, and the longest in the whole of North Korea.
Revised: The Naktong is the longest river in North Korea, and the longest in the whole of North Korea.

Original: The river is the longest in the entire country, and the longest in the whole of North Korea.
Revised: The river is the longest in the entire country, and the longest in the whole of North Korea.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Original: Context
Revised: Context

Original: During spring training in a game against the Boston Red
Revised: During spring training in a game against the Boston Red

Original: VA+
Revised: VA+

Original: Sox on April 4, 2009, Kinsler was placed on the 15 @-@ day disabled list with a strained right hamstring.
Revised: Sox on April 4, 2009, Kinsler was placed on the 15 @-@ day disabled list with a strained right hamstring.

Original: He returned to the active roster on April 7, and was placed on the 15 @-@ day DL with a strained right hamstring.
Revised: He returned to the active roster on April 7, and was placed on the 15 @-@ day DL with a strained right hamstring.

Original: He was placed on the@-@ day DL on May 4, and was placed on the 15 @-@ day DL on May 5.
Revised: He was placed on the@-@ day DL on May 4, and was placed on the 15 @-@ day DL on May 5.

Original: On May 6, he was placed on the 15 @-@ day DL with a strained right hamstring.
Revised: On May 6, he was placed on the 15 @-@ day DL with a strained right hamstring.

Original: He was placed on the 15 @-@ day DL on May 9.
Revised: He was placed on the 15 @-@ day DL on May 9.

Original: On May 10, he was placed on the 15 @-@ day DL with a strained right hamstring.
Revised: On May 10, he was placed on the 15 @-@ day DL with a strained right hamstring.

Original: He was placed on the 15 @-@ day DL on
Revised: He was placed on the 15 @-@ day DL on

Original: May 11.
Revised: May 11.

Original: He was placed on the 15 @-@ day DL on May 13. He was placed on the 15 @-@ day DL on May 20.
Revised: He was placed on the 15 @-@ day DL on May 13. He was placed on the 15 @-@ day DL on May 20.

Original: He was placed on the 15 @-@ day DL on May 24.
Revised: He was placed on the 15 @-@ day DL on May 24.

Original: He was placed on the 15 @-@ day DL on May 28.
Revised: He was placed on the 15 @-@ day DL on May 28.

Original: He was placed on the 15 @-@ day
Revised: He was placed on the 15 @-@ day

Original: DL on June 2.
Revised: DL on June 2.

Original: He was placed on ...
Revised: He was placed on ...

Original: ST+
Revised: ST+

Original: Sox on April 20, 2011, he was placed on the disabled list with a back injury.
Revised: Sox on April 20, 2011, he was placed on the disabled list with a back injury.

Original: He returned to the disabled list on May 10, 2011, and was activated on May 17.
Revised: He returned to the disabled list on May 10, 2011, and was activated on May 17.

Original: He was activated on May 20 and made his first appearance on May 21.
Revised: He was activated on May 20 and made his first appearance on May 21.

Original: He was activated on June 2 and made his first appearance on June 4.
Revised: He was activated on June 2 and made his first appearance on June 4.

Original: He was activated on June 8 and made his first appearance on June 10.
Revised: He was activated on June 8 and made his first appearance on June 10.

Original: He was activated on June 15 and made his first appearance on June 17.
Revised: He was activated on June 15 and made his first appearance on June 17.

Original: He was activated on June 20 and made his first appearance on June 23.
Revised: He was activated on June 20 and made his first appearance on June 23.

Original: He was activated on June 29 and made his first appearance on July 1. He was activated on July 1 and made his first appearance on July 4.
Revised: He was activated on June 29 and made his first appearance on July 1. He was activated on July 1 and made his first appearance on July 4.

Original: He was activated on July 6 and made his first appearance on July 10.
Revised: He was activated on July 6 and made his first appearance on July 10.

Original: He was activated on July 14 and made his first appearance on July 16.
Revised: He was activated on July 14 and made his first appearance on July 16.

Original: He was activated on July 20 and made his first appearance on July 23.
Revised: He was activated on July 20 and made his first appearance on July 23.

Original: He was ...
Revised: He was ...

Original: NMST+
Revised: NMST+

Original: Sox on April 16, 2010, the Yankees signed Rivera to a one @-@ year, $ 2 @.@million contract.
Revised: Sox on April 16, 2010, the Yankees signed Rivera to a one @-@ year, $ 2 @.@million contract.

Original: He made his major league debut on April 21, 2010, against the
Revised: He made his major league debut on April 21, 2010, against the

Original: Boston Red Sox.
Revised: Boston Red Sox.

Original: He pitched a scoreless inning in the first inning of the first game of the 2010 World Series against the New York Mets.
Revised: He pitched a scoreless inning in the first inning of the first game of the 2010 World Series against the New York Mets.

Original: On May 1, 2010, Rivera was traded to the Pittsburgh Pirates in exchange for J. J. Hardy.
Revised: On May 1, 2010, Rivera was traded to the Pittsburgh Pirates in exchange for J. J. Hardy.

Original: ⟨ eos ⟩
Revised: ⟨ eos ⟩

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: θ ’s are { VA, ST, NMST } +GPT-
Revised: θ ’s are { VA, ST, NMST } +GPT-2.

Original:  { ST NMST } +GPT-
Revised: For { ST, NMST } +GPT-2, we ϵ

Original: = 1 . {+
+}
Revised: = 1 . {+
+}

Original:  10 − 5 because it is optimal in terms of validation perplexities in Table  t  t -th token their mean (
Revised: × 10 − 5 because it is optimal in terms of validation perplexities in Table 2. Instead of t we tag the t -th ground truth token.

Original: ) ± st
Revised: We report their mean (curve) ± st.dev.

Original: 10 random runs
Revised: 10 random runs.

Original: Unlike ST+GPT-2, NMST+GPT-2 exhibits non-monotonic behaviors at plausibly terminating steps (e
Revised: Unlike ST+GPT-2, NMST+GPT-2 exhibits non-monotonic behaviors at plausibly terminating steps (e.

Original: F C ONSISTENCY WITH RESPECT TO O THER D ECODING A LGORITHMS FOR RNN AND LSTM
Revised: F C ONSISTENCY WITH RESPECT TO O THER D ECODING A LGORITHMS FOR RNN AND LSTM

Original: We validate the consistency of our proposed non-monotonic self-terminating (NMST) language model when using decoding algorithms other than greedy search, such as top- k sampling (Fan et al., 2018), nucleus sampling (Holtzman et al., 2020), and beam search.
Revised: We validate the consistency of our proposed non-monotonic self-terminating (NMST) language model when using decoding algorithms other than greedy search, such as top- k sampling (Fan et al., 2018), nucleus sampling (Holtzman et al., 2020), and beam search.

Original: All experimental setups and notations are the same as Section §4.
Revised: All experimental setups and notations are the same as Section §4.

Original: We use top- { 2, 4 } sampling, nucleus- { 0.2, 0.4 } sampling, and beam search with a width of { 2, 4 } (beam- { 2, 4 } ) to generate sequences from NMST+ { RNN, LSTM } trained on Wikitext-2 with ϵ = 1 . 

Revised: We use top- { 2, 4 } sampling, nucleus- { 0.2, 0.4 } sampling, and beam search with a width of { 2, 4 } (beam- { 2, 4 } ) to generate sequences from NMST+ { RNN, LSTM } trained on Wikitext-2 with ϵ = 1 .

Original: × 10 − 5 .
Revised: × 10 − 5 .

Original: The choice of ϵ = 1 . 

Revised: The choice of ϵ = 1 .

Original: × 10 − 5 is made based on the validation perplexities in Table 1.
Revised: × 10 − 5 is made based on the validation perplexities in Table 1.

Original: Since the validation perplexity does not change with decoding algorithms, we focus on the average ( ± st.dev.)
Revised: Since the validation perplexity does not change with decoding algorithms, we focus on the average ( ± st.dev.)

Original: non termination ratios, r nt ( L ) ’s, across 10 random runs as a function of L , for each decoding algorithm in Figure 7.
Revised: non termination ratios, r nt ( L ) ’s, across 10 random runs as a function of L , for each decoding algorithm in Figure 7.

Original: We also plot the evolution of r nt ( L ) ’s for VA+ { RNN, LSTM } and ST+ { RNN, LSTM } of ϵ = 1 . 

Revised: We also plot the evolution of r nt ( L ) ’s for VA+ { RNN, LSTM } and ST+ { RNN, LSTM } of ϵ = 1 .

Original: × 10 − 5 as we vary L .
Revised: × 10 − 5 as we vary L .

Original: [Figure]
Revised: [Figure]

Original: (cid:15)
Revised: (cid:15)

Original: = 1 . 

Revised: = 1 .

Original: × 10 −(cid:1)
Revised: × 10 −(cid:1)

Original: VA + { 0 . 2 , 2 } { 0 . 4 , 4 }
Revised: VA + { 0 . 2 , 2 } { 0 . 4 , 4 }

Original: G A NALYSIS OF P REDICTED S EQUENCE L ENGTH D ISTRIBUTIONS IN §4. 
Revised: G A NALYSIS OF P REDICTED S EQUENCE L ENGTH D ISTRIBUTIONS IN §4.

Original: We investigate whether our proposed non-monotonic self-terminating (NMST+) language model matches the data length distribution better than the baselines: i) a vanilla (VA+) language model and ii) a self-terminating (ST+) language model.
Revised: We investigate whether our proposed non-monotonic self-terminating (NMST+) language model matches the data length distribution better than the baselines: i) a vanilla (VA+) language model and ii) a self-terminating (ST+) language model.

Original: For this, we compare the length distributions of predicted sequences from { VA, ST, NMST } +LSTM trained on WikiText-2 with the data length distribution of ground truth sequences in the WikiText-2 validation dataset, D val , when using greedy search.
Revised: For this, we compare the length distributions of predicted sequences from { VA, ST, NMST } +LSTM trained on WikiText-2 with the data length distribution of ground truth sequences in the WikiText-2 validation dataset, D val , when using greedy search.

Original: All experimental setups and notations are the same as §4.1.
Revised: All experimental setups and notations are the same as §4.1.

Original: Figure 8 shows the length distributions of { VA, ST, NMST } +LSTM, and D val .
Revised: Figure 8 shows the length distributions of { VA, ST, NMST } +LSTM, and D val .

Original: For { ST, NMST } +LSTM, we use ϵ = 1 × 10 − 5 because this choice is optimal in terms of validation perplexities based on Table 1.
Revised: For { ST, NMST } +LSTM, we use ϵ = 1 × 10 − 5 because this choice is optimal in terms of validation perplexities based on Table 1.

Original: We observe that the length distributions of predicted sequences from
Revised: We observe that the length distributions of predicted sequences from

Original: NMST+LSTM is closer to the data length distribution of D val , than those of predicted sequences from VA+LSTM and ST+LSTM.
Revised: NMST+LSTM is closer to the data length distribution of D val , than those of predicted sequences from VA+LSTM and ST+LSTM.

Original: [Figure]
Revised: [Figure]

Original: -2 validation dataset, D val .
Revised: -2 validation dataset, D val .

Original: For { ST, NMST } +LSTM, we select ϵ = 1 . 

Revised: For { ST, NMST } +LSTM, we select ϵ = 1 .

Original: × 10 − 5 since it is optimal in terms of validation perplexities in Table 1.
Revised: × 10 − 5 since it is optimal in terms of validation perplexities in Table 1.

Original: NMST+LSTM better models the length distribution of
Revised: NMST+LSTM better models the length distribution of

Original: D val than both VA+LSTM and ST+LSTM.
Revised: D val than both VA+LSTM and ST+LSTM.

Original: Furthermore, we can tune ϵ to make the predicted length distribution of NMST+LSTM agree with the ground truth length distribution of D val .
Revised: Furthermore, we can tune ϵ to make the predicted length distribution of NMST+LSTM agree with the ground truth length distribution of D val .

Original: In Figure 9, we compare NMST+LSTM’s predicted length distribution of ϵ = 5 × 10 − 4 with that of ϵ = 1 ×
Revised: In Figure 9, we compare NMST+LSTM’s predicted length distribution of ϵ = 5 × 10 − 4 with that of ϵ = 1 ×

Original: 10 − 5 .
Revised: 10 − 5 .

Original: We see that ϵ = 5 × 10 − 4 better models the data length distribution than ϵ = 5 × 10 − 4 .
Revised: We see that ϵ = 5 × 10 − 4 better models the data length distribution than ϵ = 5 × 10 − 4 .

Original: However, in this case, the average validation perplexity of NMST+LSTM degrades from 101.5 ( ϵ = 1 × 10 − 5 ) to 105.6 ( ϵ = 5 × 10 − 4 ) as shown in Table 1.
Revised: However, in this case, the average validation perplexity of NMST+LSTM degrades from 101.5 ( ϵ = 1 × 10 − 5 ) to 105.6 ( ϵ = 5 × 10 − 4 ) as shown in Table 1.

Original: [Figure]
Revised: [Figure]