Fun Fact: Transformer XL can attend sequences that 80% longer than RNNs and 450% longer than vanilla Transformer and it is 1800+ times faster than vanilla Transformers during evaluation. Conclusion We’ve covered another state of the art model, XLNet, and have discussed the concept behind it.Jul 18, 2019 · Transformer-XL. Transformer networks are limited by a fixed-length context and thus can be improved through learning longer-term dependency. That’s why Google proposed a novel method called Transformer-XL (meaning extra long) for language modeling, which enables a Transformer architecture to learn longer-term dependency. Transformer-XL is up ... Mar 14, 2020 · A plot of average attention weights from the Transformer-XL paper. In addition the Transformer-XL paper measures the impact of effective context length on perplexity and finds that increasing context length leads to better perplexity scores up to a context length of ~900 tokens – further evidence that the recurrence mechanism is useful in ... Jul 18, 2019 · Transformer-XL. Transformer networks are limited by a fixed-length context and thus can be improved through learning longer-term dependency. That’s why Google proposed a novel method called Transformer-XL (meaning extra long) for language modeling, which enables a Transformer architecture to learn longer-term dependency. Transformer-XL is up ... Transformer. A transformer model. User is able to modify the attributes as needed. The architecture is based on the paper “Attention Is All You Need”. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. The Transformer-XL model addresses the limitations of vanilla transformer-based language models, which are only able to use relatively short context, bounded by the segment length. The Transformer-XL introduces a recurrence mechanism, which is able to use a cached hidden state from previous segments. Transformer-XL achieves new state-of-the-art results on multiple language modeling benchmarks. Transformer-XL is also the first to break through the 1.0 barrier on char-level language modeling. Below is a summary.The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely ...The Transformer-XL model was proposed in Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context by Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, Ruslan Salakhutdinov. It’s a causal (uni-directional) transformer with relative positioning (sinusoïdal) embeddings which can reuse previously computed hidden ... Transformer-XL. Transformer networks are limited by a fixed-length context and thus can be improved through learning longer-term dependency. That’s why Google proposed a novel method called Transformer-XL (meaning extra long) for language modeling, which enables a Transformer architecture to learn longer-term dependency. Transformer-XL is up ...May 19, 2021 · The combination of Transformer architecture and transfer learning is dominating the Natural Language Processing world. There are numerous pre-trained models (Huggingface alone has 40+) which might ... The structure of the GTrXL (Gated Transformer XL) block is illustrated in detail below: The architecture used for text generation is the one proposed in the paper Stabilizing Transformers for Reinforcement Learning. Music generation requires a modified model where the input features are split into MIDI events (note_on, note_off and control ...The Transformer-XL model was proposed in Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context by Zihang Dai*, Zhilin Yang*, Yiming Yang, Jaime Carbonell, Quoc V. Le, Ruslan Salakhutdinov. It’s a causal (uni-directional) transformer with relative positioning (sinusoïdal) embeddings which can reuse previously computed hidden ... The Transformer-XL model was proposed in Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context by Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, Ruslan Salakhutdinov. It’s a causal (uni-directional) transformer with relative positioning (sinusoïdal) embeddings which can reuse previously computed hidden ... Aug 1, 2019 · XLNET integrates ideas from Transformer-XL, the state-of-the-art autoregressive model into pretraining. Transformer is a model used for language translation purposes by google. It basically revolves around “attention”. It is an encoder-decoder model where you map one sequence to another — English to French. Transformer-XL. Transformer networks are limited by a fixed-length context and thus can be improved through learning longer-term dependency. That’s why Google proposed a novel method called Transformer-XL (meaning extra long) for language modeling, which enables a Transformer architecture to learn longer-term dependency. Transformer-XL is up ...Transformer. A transformer model. User is able to modify the attributes as needed. The architecture is based on the paper “Attention Is All You Need”. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need.Jan 18, 2019 · 摘要：Transformer 网络具有学习更长期依赖性的潜力，但这种潜力往往会受到语言建模中上下文长度固定的限制。因此，我们提出了一种叫做 Transformer-XL 的新神经架构来解决这一问题，它可以在不破坏时间一致性的情况下，让 Transformer 超越固定长度学习依赖性。 Aug 25, 2023 · Transformer-XL is a neural network model that can handle long sequences of text or speech with high efficiency and accuracy. It is based on the Transformer architecture, but with some key ... A new paper by Google and Carnegie Mellon University, “ Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context”, combines these two approaches. The new model uses the Transformer’s attention modules on each segment of input data and a recurrence mechanism to learn dependencies between consecutive segments.Mar 7, 2021 · Absolutely fantastic SOTA Google Colab (Jupyter) Notebooks to easily and quickly train a SOTA Music AI model and for generating music with Transformer technology (Google XLNet/Transformer-XL) Huge thanks goes to creators of the original repos/code that made these amazing Notebooks possible :) Thank you very much and the credit is all yours :) Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. Transformer-XL achieved SOTA results following datasets - WikiText-103, enwik8, text8, One Billion Word and Penn Treebank. Transformer-XL has also been used to generate text. Examples are given at ...Mar 13, 2021 · Transformer XL is an important variation of Transformers as it improves upon a major shortcoming of transformers, context fragmentation. It improved the speed of training and allowed the model to capture longer dependencies. Improvements upon this transformer like the XLNet are beating BERT at critical language tasks. Transformer-XL (meaning extra long) is a Transformer architecture that introduces the notion of recurrence to the deep self-attention network. Instead of computing the hidden states from scratch for each new segment, Transformer-XL reuses the hidden states obtained in previous segments. Huang et al. introduced a new way of computing relative positional encoding via a clever skewing operation. It seems that in the music transformer paper, the authors dropped the additional relative positional embedding that corresponds to the value term and focus only on the key component. In other words, the authors only focus on (1), not (2).Jan 18, 2019 · 摘要：Transformer 网络具有学习更长期依赖性的潜力，但这种潜力往往会受到语言建模中上下文长度固定的限制。因此，我们提出了一种叫做 Transformer-XL 的新神经架构来解决这一问题，它可以在不破坏时间一致性的情况下，让 Transformer 超越固定长度学习依赖性。 in the streaming fashion, we introduce the Transformer-XL [3] based steaming model, which is computationally tractable for inference. Our results show that Transformer-XL is on par with latency-controlled BLSTM (LC-BLSTM) [15] with the same latency constraint. 2. Related Work There have been a few studies on Transformers for end-to-endFigure 1. Example of the BERT’s pre-training objective. Top) The MLM; Bottom) Next sentence Prediction. BERT uses these methods for pre-training a model to learn the basics of the language.The Transformer-XL model addresses the limitations of vanilla transformer-based language models, which are only able to use relatively short context, bounded by the segment length. The Transformer-XL introduces a recurrence mechanism, which is able to use a cached hidden state from previous segments. For Transformer-XL, it is important that these are also what you use as an input to the self-attention. Therefore, at inference time, if you want to compute the states recursively by segments (presumably because you cannot fit the entire input int he memory), this is the only thing you need to remember from the previous steps to continue the ...Jul 18, 2019 · Transformer-XL. Transformer networks are limited by a fixed-length context and thus can be improved through learning longer-term dependency. That’s why Google proposed a novel method called Transformer-XL (meaning extra long) for language modeling, which enables a Transformer architecture to learn longer-term dependency. Transformer-XL is up ... Jan 11, 2019 · Transformer-XL obtains strong results for both word-level and character-level language modeling applied to a variety of datasets such as WikiText-103, text8, and One Billion Word. The Transformer-XL model addresses the limitations of vanilla transformer-based language models, which are only able to use relatively short context, bounded by the segment length. The Transformer-XL introduces a recurrence mechanism, which is able to use a cached hidden state from previous segments.Under the model size constraint, the 12-layer Transformer-XL achieves a new SoTA result, outperforming the 12-layer vanilla Transformer from Al-Rfou et al. (2018) (T64) by 0.05. By increasing model sizes, 18-layer and 24-layer Transformer-XLs are trained with attention length is set to 784 during training and 3800 during evaluation.3. Results: TransformerXL đạt được kết quả SOTA ( State of The Art ) trên nhiều datasets benchmarks về Language Modeling trên cả mức word-level và character-level. Trên WikiText-103, một bộ dataset lớn về Language Modeling ở mức word-level, TransformerXL (18 layers) đạt perplexity bằng 18.3 so với ...感觉transformer xl训练难度较大，可能是因为不像LSTM等收到梯度消逝或爆炸的影响导致记忆长度较短，而transformer xl由于memory len较长，要处理的条件概率情况就复杂得多，所以生成质量在排除重复性后，应该会更高。The Transformer-XL model was proposed in Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context by Zihang Dai*, Zhilin Yang*, Yiming Yang, Jaime Carbonell, Quoc V. Le, Ruslan Salakhutdinov. It’s a causal (uni-directional) transformer with relative positioning (sinusoïdal) embeddings which can reuse previously computed hidden ...Transformer-XL achieved SOTA results following datasets - WikiText-103, enwik8, text8, One Billion Word and Penn Treebank. Transformer-XL has also been used to generate text. Examples are given at ...Transformers. Transformers are a type of neural network architecture that have several properties that make them effective for modeling data with long-range dependencies. They generally feature a combination of multi-headed attention mechanisms, residual connections, layer normalization, feedforward connections, and positional embeddings. This implements the Retrieval-Enhanced Transformer (RETRO). Compressive Transformer. This is an implementation of compressive transformer that extends upon Transformer XL by compressing the oldest memories to give a longer attention span. GPT Architecture. This is an implementation of GPT-2 architecture. GLU VariantsThe dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely ...Chinese-Transformer-XL. Under construction. 本项目提供了智源研究院"文汇" 预训练模型Chinese-Transformer-XL的预训练和文本生成代码。Jan 11, 2019 · Transformer-XL obtains strong results for both word-level and character-level language modeling applied to a variety of datasets such as WikiText-103, text8, and One Billion Word. For Transformer-XL, it is important that these are also what you use as an input to the self-attention. Therefore, at inference time, if you want to compute the states recursively by segments (presumably because you cannot fit the entire input int he memory), this is the only thing you need to remember from the previous steps to continue the ...This is the OG transformer that started the revolution. TransformerXL —this forward-directional decoder is an amazing text generator. Memory and relative positional encoding enable super fast and accurate predictions. We used this model in Part II.Mar 15, 2022 · Transformer-XL was able to learn dependency 80% longer than RNNs and 450% longer than Vanilla Transformer. You heard it right, a whooping 450%! Transformer-XL is also a mind-blowing 1800 times faster than Vanilla Transformers. These numbers are very huge claims. Let’s dig deep into the architecture and understand the mechanism by which it is ... Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism ...Transformer XL. This is an experiment training Shakespeare dataset with a Transformer XL model.The structure of the GTrXL (Gated Transformer XL) block is illustrated in detail below: The architecture used for text generation is the one proposed in the paper Stabilizing Transformers for Reinforcement Learning. Music generation requires a modified model where the input features are split into MIDI events (note_on, note_off and control ...Jan 29, 2019 · Empirically, Transformer-XL enjoys three benefits: Transformer-XL learns dependency that is about 80% longer than RNNs and 450% longer than vanilla Transformers, which generally have better performance than RNNs, but are not the best for long-range dependency modeling due to fixed-length contexts (please see our paper for details). The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely ...Number of heads used in the transformer's multi-head attention mechanism: memory_length: Length of the sliding episodic memory window: positional_encoding: Relative and learned positional encodings can be used: layer_norm: Whether to apply layer normalization before or after every transformer component. Figure 1. Example of the BERT’s pre-training objective. Top) The MLM; Bottom) Next sentence Prediction. BERT uses these methods for pre-training a model to learn the basics of the language.We propose architectural modifications that substantially improve the stability and learning speed of the original Transformer and XL variant. The proposed architecture, the Gated Transformer-XL (GTrXL), surpasses LSTMs on challenging memory environments and achieves state-of-the-art results on the multi-task DMLab-30 benchmark suite, exceeding ...Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism and a novel positional encoding scheme. Our method ...Transformer. A transformer model. User is able to modify the attributes as needed. The architecture is based on the paper “Attention Is All You Need”. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need.Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism ...Absolutely fantastic SOTA Google Colab (Jupyter) Notebooks to easily and quickly train a SOTA Music AI model and for generating music with Transformer technology (Google XLNet/Transformer-XL) Huge thanks goes to creators of the original repos/code that made these amazing Notebooks possible :) Thank you very much and the credit is all yours :)Existing Approaches for Long Document Transformers via Longformer Paper. The paper initially addresses the issues with existing long document transformers. Models like Transformer-XL partitions the input and apply full self-attention locally as well as in a cross-partition setting (to an extent).December 3, 2022. In this post, we will implement a lightweight version of the Transformer-XL model. Proposed by Dai et al. in 2019 1, Transformer-XL introduced two innovations that, when combined, enable the attention mechanism to have a wider “field of view” and result in significant performance improvements on autoregressive evaluation.See full list on towardsdatascience.com Aug 25, 2023 · Transformer-XL is a neural network model that can handle long sequences of text or speech with high efficiency and accuracy. It is based on the Transformer architecture, but with some key ... Mar 14, 2020 · A plot of average attention weights from the Transformer-XL paper. In addition the Transformer-XL paper measures the impact of effective context length on perplexity and finds that increasing context length leads to better perplexity scores up to a context length of ~900 tokens – further evidence that the recurrence mechanism is useful in ... As a side note, we remark that this conclusion is reached based on the assumption that key and query sizes are the same. It may be possible in a context like Transformer-XL, that there is global positional or contextual information that could be propagated in the network. In this case it might not be prudent to discard these contributions.Hi, you will likely need to adapt this example since Transformer-XL uses memory cells but there is no ready to use example for fine-tuning Transformer-XL in the repo unfortunately (and I don't plan to add one in the near future). If you want to give it a try feel free to ask more specific questions here.Aug 18, 2023 · The transformer XL is a newer version from the Transformer (it’s extra long). It is derived from the vanilla Transformer, but introduces the recurrence mechanism and relative positional encoding. In Transformer-XL, instead of computing the hidden state from scratch for each segment, the model will keep the hidden state of the previously ... Apr 4, 2023 · Transformer-XL is a transformer-based language model with a segment-level recurrence and a novel relative positional encoding. Enhancements introduced in Transformer-XL help capture better long-term dependencies by attending to tokens from multiple previous segments. Our implementation is based on the codebase published by the authors of the ... transformer xl在中文文本生成上的尝试（可写小说、古诗）（transformer xl for text generation of chinese） - GitHub - GaoPeng97/transformer-xl ...Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural ar-chitecture Transformer-XL that enables learn-ing dependency beyond a fixed length with-out disrupting temporal coherence. It con-sists of a segment-level recurrence mechanismTransformer-XL (meaning extra long) is a Transformer architecture that introduces the notion of recurrence to the deep self-attention network. Instead of computing the hidden states from scratch for each new segment, Transformer-XL reuses the hidden states obtained in previous segments.Transformer. A transformer model. User is able to modify the attributes as needed. The architecture is based on the paper “Attention Is All You Need”. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. Huang et al. introduced a new way of computing relative positional encoding via a clever skewing operation. It seems that in the music transformer paper, the authors dropped the additional relative positional embedding that corresponds to the value term and focus only on the key component. In other words, the authors only focus on (1), not (2).The Transformer-XL model addresses the limitations of vanilla transformer-based language models, which are only able to use relatively short context, bounded by the segment length. The Transformer-XL introduces a recurrence mechanism, which is able to use a cached hidden state from previous segments. Dec 1, 2020 · Existing Approaches for Long Document Transformers via Longformer Paper. The paper initially addresses the issues with existing long document transformers. Models like Transformer-XL partitions the input and apply full self-attention locally as well as in a cross-partition setting (to an extent). Jan 1, 2019 · Various methods have been proposed to introduce memorization capabilities to Transformers through recurrence [5,38]. Transformer-XL [8] feeds the input to the model in windows of a fixed length ... Transformers Xl was released about a year ago and the main motive behind it was to improve more over vanilla transformers. Transformers XL was made to address the problem of context fragmentation.Mar 1, 2021 · Huang et al. introduced a new way of computing relative positional encoding via a clever skewing operation. It seems that in the music transformer paper, the authors dropped the additional relative positional embedding that corresponds to the value term and focus only on the key component. In other words, the authors only focus on (1), not (2). Check out the pytorch-transformers library from Hugging Face in addition to GPT2, it implements BERT, Transformer-XL, XLNet and other cutting-edge transformer models. Acknowledgements Thanks to Lukasz Kaiser , Mathias Müller , Peter J. Liu , Ryan Sepassi and Mohammad Saleh for feedback on earlier versions of this post.Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism and a novel positional encoding scheme. Our method ...Gated Transformer-XL, or GTrXL, is a Transformer-based architecture for reinforcement learning. It introduces architectural modifications that improve the stability and learning speed of the original Transformer and XL variant. Changes include: Placing the layer normalization on only the input stream of the submodules. A key benefit to this reordering is that it now enables an identity map ...Transformer-XL is a neural network model that can handle long sequences of text or speech with high efficiency and accuracy. It is based on the Transformer architecture, but with some key ...Transformer-XL achieved SOTA results following datasets - WikiText-103, enwik8, text8, One Billion Word and Penn Treebank. Transformer-XL has also been used to generate text. Examples are given at ...in the streaming fashion, we introduce the Transformer-XL [3] based steaming model, which is computationally tractable for inference. Our results show that Transformer-XL is on par with latency-controlled BLSTM (LC-BLSTM) [15] with the same latency constraint. 2. Related Work There have been a few studies on Transformers for end-to-end Aug 12, 2019 · Check out the pytorch-transformers library from Hugging Face in addition to GPT2, it implements BERT, Transformer-XL, XLNet and other cutting-edge transformer models. Acknowledgements. Thanks to Lukasz Kaiser, Mathias Müller, Peter J. Liu, Ryan Sepassi and Mohammad Saleh for feedback on earlier versions of this post. Comments or corrections? Apr 7, 2020 · The Gated Transformer-XL (GTrXL; Parisotto, et al. 2019) is one attempt to use Transformer for RL. GTrXL succeeded in stabilizing training with two changes on top of Transformer-XL : The layer normalization is only applied on the input stream in a residual module, but NOT on the shortcut stream. Transformer-XL. The Transformer-XL model is based on a similar idea as the vanilla model, but with some corrections. In the following subsections we’ll be discussing the contributions of the Transformer-XL architecture and see how it was able to achieve the state of the art. XL stands for eXtra Long. Segment Recurrence MechanismWe also use a Transformer-XL style cache, which holds the keys and values from the previous training step. When doing self-attention, the cached keys and values are prepended to the current keys and values, and we use a sliding-window causal mask (Beltagy et al., 2020) so that each token has a local context that includes the previous 512 tokens.
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Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism and a novel positional encoding scheme. Our method ...{"payload":{"allShortcutsEnabled":false,"fileTree":{"pytorch":{"items":[{"name":"utils","path":"pytorch/utils","contentType":"directory"},{"name":".DS_Store","path ... Transformer-XL is a neural network model that can handle long sequences of text or speech with high efficiency and accuracy. It is based on the Transformer architecture, but with some key ...Oct 13, 2019 · We propose architectural modifications that substantially improve the stability and learning speed of the original Transformer and XL variant. The proposed architecture, the Gated Transformer-XL (GTrXL), surpasses LSTMs on challenging memory environments and achieves state-of-the-art results on the multi-task DMLab-30 benchmark suite, exceeding ... Jan 30, 2022 · Under the model size constraint, the 12-layer Transformer-XL achieves a new SoTA result, outperforming the 12-layer vanilla Transformer from Al-Rfou et al. (2018) (T64) by 0.05. By increasing model sizes, 18-layer and 24-layer Transformer-XLs are trained with attention length is set to 784 during training and 3800 during evaluation. Transformer-XL. The Transformer-XL model is based on a similar idea as the vanilla model, but with some corrections. In the following subsections we’ll be discussing the contributions of the Transformer-XL architecture and see how it was able to achieve the state of the art. XL stands for eXtra Long. Segment Recurrence MechanismApr 1, 2020 · 이번 글에서는 ACL 2019에서 발표된 “Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context”를 리뷰하려고 합니다. 본 논문은 기존의 Transformer 구조를 이용한 고정된 길이(Fixed-Length) Language Model의 한계점을 지적하고 더 긴 의존성을 이용할 수 있는 새로운 방법을 제시합니다. Jan 9, 2019 · As a result, Transformer-XL learns dependency that is 80% longer than RNNs and 450% longer than vanilla Transformers, achieves better performance on both short and long sequences, and is up to 1,800+ times faster than vanilla Transformers during evaluation. 摘要：Transformer 网络具有学习更长期依赖性的潜力，但这种潜力往往会受到语言建模中上下文长度固定的限制。因此，我们提出了一种叫做 Transformer-XL 的新神经架构来解决这一问题，它可以在不破坏时间一致性的情况下，让 Transformer 超越固定长度学习依赖性。Check out the pytorch-transformers library from Hugging Face in addition to GPT2, it implements BERT, Transformer-XL, XLNet and other cutting-edge transformer models. Acknowledgements Thanks to Lukasz Kaiser , Mathias Müller , Peter J. Liu , Ryan Sepassi and Mohammad Saleh for feedback on earlier versions of this post..