2023/A Length-Extrapolatable Transformer/layout.json

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                                    "type": "text",
                                    "content": "Position modeling plays a critical role in Transformers. In this paper, we focus on length extrapolation, i.e., training on short texts while evaluating longer sequences. We define attention resolution as an indicator of extrapolation. Then we propose two designs to improve the above metric of Transformers. Specifically, we introduce a relative position embedding to explicitly maximize attention resolution. Moreover, we use blockwise causal attention during inference for better efficiency. The proposed architecture is named Length-Extrapolatable (LEX) Transformer. We evaluate different Transformer variants on language modeling. Experimental results show that our model achieves better performance in both interpolation and extrapolation settings. The code will be available at https://aka.ms/LeX-Transformer."
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                                    "content": "1 Introduction"
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                                    "content": "Transformer (Vaswani et al., 2017) has shown strong performance in NLP and become a de-facto backbone (Dosovitskiy et al., 2020; Radford et al., 2021; Wang et al., 2022). However, most of them have a crucial shortcoming: they can only deal with the in-distribution size of inputs. Figure 1 shows that the perplexity of previous Transformers increases rapidly when the input sequence is getting longer. It is usually infeasible to train a model with all possible input lengths. Therefore, a length-extrapolatable Transformer is essential for wider usage."
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                                    "content": "In sequence modeling, position information plays a crucial role in building the correct representation and understanding of the latent meaning. For Recurrent Neural Networks such as LSTM (Hochreiter and Schmidhuber, 1997), the calculation is done along the sequence order in O(N) time. However, the parallel attention module"
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                                            "content": "Figure 1: The perplexity of different Transformer designs with various input lengths."
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                                    "content": "makes it hard to encode position effectively. First, Vaswani et al. (2017) proposes absolute sinusoidal position embedding, and Devlin et al. (2019) adjusts it to a learnable one. The absolute design is computation-efficient, but not comparable with subsequent relative ones (Shaw et al., 2018; Su et al., 2021; Press et al., 2021). Among many relative position embeddings, RoPE (Su et al., 2021) shows better performance and is used to many PLMs such as PaLM (Chowdhery et al., 2022). However, it can't deal with sequences with exceeding length. Alibi (Press et al., 2021) mitigates the extrapolation problem but sacrifices the general performance."
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                                    "content": "Since different strategies concentrate on some part of the position feature, it is essential to build a comprehensive view and guide the Transformer's design systematically. First, a Transformer should be sensitive to order. Otherwise, it will degenerate into a bag-of-word model which confuses the whole meaning. Then, position translation can't hurt the representation, especially, when a prefix is added to the target sentence, the representation should stay the same with an attention mask on the prefix. After that, a good sequence model needs to"
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                                    "content": "*Work done during internship at Microsoft Research."
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                                    "content": "July 9-14, 2023 ©2023 Association for Computational Linguistics"
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                                            "html": "<table><tr><td>Models</td><td>Translation Invariance</td><td>Length Extrapolation</td></tr><tr><td colspan=\"3\">Absolute Position Modeling</td></tr><tr><td>Transformer (Sinusoidal)</td><td>×</td><td>××</td></tr><tr><td>GPT-2 (Learnable)</td><td>×</td><td>××</td></tr><tr><td colspan=\"3\">Relative Position Modeling</td></tr><tr><td>PaLM / Roformer (RoPE)</td><td>✓</td><td>×</td></tr><tr><td>T5</td><td>✓</td><td>×</td></tr><tr><td>BLOOM / Alibi</td><td>✓</td><td>✓</td></tr><tr><td>LEX Transformer (Ours)</td><td>✓</td><td>✓✓</td></tr></table>",
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                                    "content": "Table 1: Position modeling capabilities of Transformer variants for language modeling."
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                                    "content": "deal with any input length. As illustrated before, the length problem is not universal but special for Transformer. Especially, when a Transformer is pre-trained under a maximal length, it is not affordable to re-train for applying to tasks with longer sequences. Finally, when a Transformer satisfies the principles above, we evaluate its performance, which requires thorough experiments and empirical analysis."
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                                    "content": "Considering all the properties above, we propose Extrapolatable Position Embedding (xPos), which is a universal-good design for Transformers. Based on RoPE's design, we propose attention resolution as a metric to measure position monotonicity. Then, we generalize its mathematical form, where an exponential decay is added to the rotation matrix. xPos preserves the advantage of RoPE, and behaves stably at long-term dependency. Besides, inspired by sparse attention methods (Child et al., 2019; Beltagy et al., 2020; Zaheer et al., 2020; Xiong et al., 2021), we choose blockwise causal attention to increase attention resolution, which improves the performance of length extrapolation for language modeling."
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                                    "content": "We train different Transformers from scratch. We evaluate models on PG22 and QMSum (Zhong et al., 2021) with various input lengths. On the interpolation experiments, LEX Transformer reaches minimal perplexity. In the extrapolation experiments, our methods can continue decreasing the perplexity while other methods either can't extrapolate (i.e., perplexity increases) when the input length is very long. Figure 1 shows clearly that LEX Transformer has an opposite tendency compared with others."
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                                    "content": "We summarize our contributions as follows:"
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                                    "content": "- We summarize the design principles of Transformers for position modeling."
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                                    "content": " when the distance of two tokens is "
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                                    "content": "s [ n ] = \\underset {0 \\leq i \\leq N} {\\mathbb {E}} \\left(\\frac {q _ {i + n} k _ {i} ^ {T}}{\\sqrt {d}}\\right) \\tag {3}",
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                                    "content": " as a metric to evaluate attention's ability to recognize position:"
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                                    "content": "R (s) = \\sum_ {i = 0} ^ {N} \\frac {e ^ {s [ i ]} \\left(e ^ {s [ i ]} - e ^ {s [ i + 1 ]}\\right)}{\\left(\\sum_ {i = 0} ^ {N} e ^ {s [ i ]}\\right) ^ {2}} \\tag {4}",
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                                    "content": ", which causes an unstable phenomenon in that the expectation of the inner product oscillates dramatically with the growth of relative distance. Following the parameters (Vaswani et al., 2017; Su et al., 2021) "
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                                    "content": "\\theta = \\{\\theta_{i} = 10000^{-2i / d}, i \\in [0,1,\\dots,d / 2]\\}"
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                                    "content": ", we will calculate the expectation as follows. For generative models, we assume "
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                                    "content": "\\mathbb{E}(\\angle q) \\leq \\mathbb{E}(\\angle k)"
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                                    "content": " to ensure the monotonicity:"
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                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ \\left\\langle q e ^ {m \\xi + i m \\theta}, k e ^ {- n \\xi + i n \\theta} \\right\\rangle \\right] \\\\ = \\sum_ {x = 0} ^ {d / 2} \\mathbb {E} \\left[ R e \\left(\\boldsymbol {q} _ {x} \\boldsymbol {k} _ {x} e ^ {(m - n) \\xi_ {x} + i (m - n) \\theta_ {x}}\\right) \\right] \\\\ \\leq \\sum_ {x = 0} ^ {d / 2} R e \\left(\\mathbb {E} \\left[ \\left| \\boldsymbol {q} _ {x} \\boldsymbol {k} _ {x} \\right| \\right] e ^ {(m - n) \\xi_ {x} + i (m - n) \\theta_ {x}}\\right) \\tag {9} \\\\ \\propto \\sum_ {x = 0} ^ {d / 2} \\cos (m - n) \\theta_ {x} e ^ {(m - n) \\xi_ {x}} \\\\ \\end{array}",
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                                    "type": "text",
                                    "content": "The inference here is different from (Su et al., 2021) because of two reasons: 1) there is an additional assumption brought by generative language"
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                                    "content": "; 2) the inequality scaling of (Su et al., 2021) is too strong to lose generality. We calculate expectation instead of the upper bound."
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                                    "content": "Now we define a function to represent the property of relative position:"
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                                    "content": "g _ {\\zeta} [ n ] = \\sum_ {i = 0} ^ {d / 2} \\cos n \\theta_ {i} \\zeta_ {i} ^ {n} \\tag {10}",
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                                    "content": " simplifies Equation 9 by defining "
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                                    "content": "\\zeta_{i} = e^{\\xi_{i}}"
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                                    "content": " curve is intuitive. Although attention bias can achieve this goal, we try to avoid additional position calculations. Instead, we can accomplish this goal using a good "
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                                    "content": " can achieve this goal:"
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                                    "content": "\\widetilde {\\zeta} _ {i} = \\frac {i / (d / 2) + \\gamma}{1 + \\gamma} \\in [ 0, 1 ] \\tag {11}",
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                                    "content": ". However, the results rely on the initial value and lack control when the hidden dimension changes. Besides, the numerical precision should be considered because of fp16's range. Finally, we find a sub-optimal solution by manually setting "
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                                    "content": " can be represented by fp16 when "
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                                    "content": " is big (8192 in our setting). Besides, in implementation, the position is re-scaled with base "
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                                    "type": "text",
                                    "content": " in the exponential calculation to avoid overflow and underflow ("
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                                    "content": "e^{\\xi n} \\to e^{\\xi n / B}"
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                                    "type": "text",
                                    "content": " in Equation (8)). We use "
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                                    "content": "\\gamma = 0.4"
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                                    "content": "B = 512"
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                                    "type": "text",
                                    "content": " as the final implementation in LEX Transformer."
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                                    "content": "The curves of "
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                                    "type": "text",
                                    "content": " are shown in Figure 2. The default rotary embedding contributes to a dramatic oscillation, especially in the large relative distance, which causes bad extrapolation performance and restricts the model's convergence speed. After adding a decay, the curve is almost stable, especially on long-term dependency. What's more, it does not hurt pure rotation's fitting ability because "
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                                    "content": "\\zeta_i^n \\approx 1"
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                                    "type": "text",
                                    "content": " is small. In that way, short-term and long-term dependencies are divided continuously."
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                                    "content": "Finally, we have Extrapolatable Position Embed"
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                                            "content": "Figure 2: The long dependency curve of attention expectation. ROPE's dramatic oscillation confuses the attention resolution at long distances. In contrast, xPos provides stable and accurate position modeling."
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                                            "content": "Algorithm 1: Attention with xPOS"
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                                            "content": "\\hat{\\zeta}_i = (2\\lfloor i / 2\\rfloor /d + \\gamma) / (1 + \\gamma)"
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                                            "type": "text",
                                            "content": "   \nInput: "
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                                            "type": "inline_equation",
                                            "content": "Q,K,V\\in \\mathbb{R}^{h\\times l\\times d}"
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                                            "content": "M\\in \\mathbb{R}^{d\\times d}"
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                                            "content": "C_{mn} = \\cos m\\theta_n,C\\in \\mathbb{R}^{l\\times d}"
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                                            "content": "S_{mn} = \\sin m\\theta_n,S\\in \\mathbb{R}^{l\\times d}"
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                                            "content": "T_{mn} = \\hat{\\zeta}_n^m,T\\in \\mathbb{R}^{l\\times d}"
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                                            "content": "Q = Q\\odot (C\\odot T) + \\mathrm{rot}(Q)\\odot (S\\odot T)"
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                                            "content": "K = K\\odot (C\\odot T^{-1}) + \\mathrm{rot}(K)\\odot (S\\odot T^{-1})"
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                                            "type": "inline_equation",
                                            "content": "\\frac{QK^T}{\\sqrt{d}}\\cdot M)V"
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                                            "type": "text",
                                            "content": "   \nreturn output"
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                                    "content": "ding (xPos):"
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                                    "content": "\\begin{array}{l} f _ {q} (q, n) = \\left( \\begin{array}{c} q _ {1} \\cos n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {n / B} - q _ {2} \\sin n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {n / B} \\\\ q _ {2} \\cos n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {n / B} + q _ {1} \\sin n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {n / B} \\\\ \\vdots \\\\ q _ {n - 1} \\cos n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {n / B} - q _ {n} \\sin n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {n / B} \\\\ q _ {n} \\cos n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {n / B} + q _ {n - 1} \\sin n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {n / B} \\end{array} \\right) \\\\ f _ {k} (k, n) = \\left( \\begin{array}{c} k _ {1} \\cos n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {- n / B} - k _ {2} \\sin n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {- n / B} \\\\ k _ {2} \\cos n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {- n / B} + k _ {1} \\sin n \\theta_ {1} \\hat {\\zeta} _ {1} ^ {- n / B} \\\\ \\vdots \\\\ k _ {n - 1} \\cos n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {- n / B} - k _ {n} \\sin n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {- n / B} \\\\ k _ {n} \\cos n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {- n / B} + k _ {n - 1} \\sin n \\theta_ {d / 2} \\hat {\\zeta} _ {d / 2} ^ {- n / B} \\end{array} \\right) \\tag {12} \\\\ \\end{array}",
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                                    "type": "text",
                                    "content": "In the implementation, the transformation for key and value can be easily calculated by parallel addition and multiplication as shown in Algorithm 1. Since position embedding's size "
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                                    "content": "C, S, T \\in \\mathbb{R}^{l \\times d}"
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                                    "content": " is much smaller than batched multi-head at"
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                                            "type": "text",
                                            "content": "Figure 3: Our language model is trained on shorter texts in the same way as vanilla Transformers, i.e., using causal masking. During inference, we use blockwise causal attention for longer sequences, which recurrently reuses the overlapped parts (i.e., key and value vectors)."
                                        }
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                                    "type": "text",
                                    "content": "tention matrix and doesn't require gradients, the cost is almost the same with ROPE and bigger than Absolute Position with "
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                                    "type": "inline_equation",
                                    "content": "6\\%"
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                                {
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                                    "type": "text",
                                    "content": " additional time."
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                                    "type": "text",
                                    "content": "3.3 Blockwise Causal Attention"
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                                    "type": "text",
                                    "content": "To deal with length extrapolation, a simple way to improve attention resolution (Section 3.1) is using windowed attention. During inference, we use blockwise masking (Dai et al., 2019; Zaheer et al., 2020; Xiong et al., 2021) for self-attention. Notice that other window strategies, such as sliding window (Child et al., 2019), also work. We use blockwise causal attention because it is cache-friendly and easy to implement."
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                    "index": 11
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                                    "type": "text",
                                    "content": "As shown in Figure 3, if the pre-training length is "
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                                {
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                                    "type": "inline_equation",
                                    "content": "l"
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                                    "type": "text",
                                    "content": ", we divide the query as blocks with "
                                },
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                                    "type": "inline_equation",
                                    "content": "l/2"
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                                    "type": "text",
                                    "content": " length, and each query interacts with its own block and the last block. In this way, the context information can be delivered by the reuse of key and value. The window constraint helps models encode longer input with improved resolution."
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                    "index": 12
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                                    "type": "text",
                                    "content": "Different from training a long-sequence model with a stop gradient, we use vanilla attention in the training phase, because the pre-training corpus is not very long on average. However, during the inference phase, when dealing with long sequences, we directly implement BCA to help the model to be more position-recognizable."
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                                    "content": "4 Experiments"
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                                    "content": "4.1 Pre-training"
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                                    "type": "text",
                                    "content": "To fairly evaluate different Transformer variants, we pre-train the Transformer from scratch. We use 1024 hidden dimensions, 16 heads, and 24 layers,"
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                    "index": 16
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            ],
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                                    "type": "text",
                                    "content": "i.e., comparable to medium-size GPT-3 (Brown et al., 2020). The training corpus includes a subset of the Pile (Gao et al., 2020): Books3, OpenText2, Stack Exchange, PubMed Abstracts, Wikipedia, Gutenberg (PG-19), BookCorpus2, NIH ExPorter, and Pile-CC datasets. The training procedure is performed on "
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                                    "type": "inline_equation",
                                    "content": "16 \\times V100"
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                                    "type": "text",
                                    "content": " GPUs. We use the tokenizer from GPT2 (Radford et al., 2019). The maximal length is 1024 for saving memory and extrapolation evaluation. The learning rate is "
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                                    "type": "inline_equation",
                                    "content": "3 \\times 10^{-4}"
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                                    "type": "text",
                                    "content": " and polynomial decay is used to adjust the learning rate. The global batch size is 512 to follow GPT-3 (Brown et al., 2020), i.e., 0.5M token size. We use Adam (Kingma and Ba, 2015) optimizer with "
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                                    "type": "inline_equation",
                                    "content": "\\beta_{1} = 0.9"
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                                    "type": "text",
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                                    "content": "\\beta_{2} = 0.98"
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                                    "content": "\\epsilon = 10^{-6}"
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                                    "type": "text",
                                    "content": ". The code is based on TorchScale (Ma et al., 2022a)."
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                                    "content": "4.2 Language Modeling"
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                                    "type": "text",
                                    "content": "We measure perplexity on long document datasets, which can show the model's ability for long-dependency modeling. We use books from Project Gutenberg whose years are later than 2019 to ensure no overlap with PG19, and we name it as PG22. Besides, we pick QMSum (Zhong et al., 2021) from SCROLLS (Shaham et al., 2022) with above 9k length on average. We care about the performance on different input lengths to evaluate the model's interpolation and extrapolation capability. For experiment results in Table 2, we divide the same input into the target length to fairly compare the perplexity of different lengths."
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                                    "content": "For interpolation capability, we analyze the results where the length is no more than 1024. Since the validation distribution is very similar to training data, all Transformers' generalization capabilities are also close. xPos have a stable advantage on others with a 0.09 perplexity drop on PG22, and 0.27 on QMSum, which proves that xPos increases the interpolation ability."
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                                    "content": "For extrapolation lengths, we do not use BCA in other Transformers, and the following ablation study will discuss the performance with that. Press et al. (2021)'s experiment shows that most of the position strategies can't deal with input length longer than pre-training directly. xPos shows a stable decrease when the sequence length increases, which satisfies the assumption that a longer context makes the prediction better. While others' perplexity increases when the input length is 4096."
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                                    "content": "To illustrate the tendency of perplexities, Figure 1 visualizes the relation between input length"
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                                    "content": "and perplexity. When the length is larger than 4096, Alibi's perplexity increases gradually. However, LEX's perplexity decreases continuously when the length extends to 8192."
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                                    "content": "The experiment shows that xPOS gets better performance on language modeling. With the stable advantage of any length, users can input any sentence freely without the concern of position. Besides, results also indicate that is not essential to build an explicit decay on the attention matrix. Instead, a proper design for an attention mask is actually better to deal with long-context tasks."
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                                    "content": "\\mathbb {E} [ s [ n ] ] = \\frac {1}{N - n} \\mathbb {E} [ \\sum_ {i = n} ^ {N - 1} a _ {i (i - n)} ] \\tag {13}",
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                                    "content": "Then the attention resolution can be calculated by combining Equation (4) and Equation (13). The final expectation is averaged over input texts and different layers."
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                                            "html": "<table><tr><td rowspan=\"2\">Length</td><td>256</td><td>512</td><td>1024</td><td>2048</td><td>4096</td><td>8192</td></tr><tr><td colspan=\"3\">Interpolation</td><td colspan=\"3\">Extrapolation</td></tr><tr><td colspan=\"7\">PG22</td></tr><tr><td>Transformer</td><td>38.1</td><td>33.5</td><td>30.54</td><td>132.46</td><td>1446.95</td><td>12747.41</td></tr><tr><td>Alibi</td><td>34.25</td><td>30.01</td><td>27.34</td><td>26.01</td><td>28.46</td><td>32.8</td></tr><tr><td>Roformer</td><td>33.27</td><td>29.2</td><td>26.68</td><td>68.86</td><td>235.71</td><td>458.83</td></tr><tr><td>LEX Transformer (Ours)</td><td>33.18</td><td>29.11</td><td>26.59</td><td>25.53</td><td>25.07</td><td>24.89</td></tr><tr><td colspan=\"7\">QMSum</td></tr><tr><td>Transformer</td><td>24.25</td><td>18.81</td><td>16.05</td><td>86.56</td><td>1196.92</td><td>10781.38</td></tr><tr><td>Alibi</td><td>22.85</td><td>17.74</td><td>15.17</td><td>13.97</td><td>15.36</td><td>18.37</td></tr><tr><td>Roformer</td><td>22.66</td><td>17.65</td><td>15.12</td><td>36.54</td><td>146.61</td><td>331.56</td></tr><tr><td>LEX Transformer (Ours)</td><td>22.01</td><td>17.24</td><td>14.85</td><td>13.92</td><td>13.56</td><td>13.48</td></tr></table>",
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                                            "content": "Table 2: Results of perplexity with different lengths. The language models are trained with a length of 1024 and then evaluated on various lengths. LEX obtains better performance not only on shorter texts (i.e., interpolation) but also on longer texts (i.e., extrapolation). The red color indicates that the perplexity begins increasing compared with the shorter length. LEX is the only method that has lower perplexity along with increased evaluation length."
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                                    "content": ". Therefore, the combination of rotation and decay means the combination of in-distribution and out-of-distribution capability in terms of length."
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                                    "content": "Table 4: Ablation results on the PG22 set show that rotation of XPos is necessary for strong performance."
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                                    "content": "First, Blockwise Causal Attention works for RoPE whose perplexity will explode without that. Alibi performs well without windowed attention because its \"soft window\" is broader than a hard block window. However, when the sequence length increases to 8192, windowed attention outperforms vanilla attention again (also shown in Figure 1). XPos's perplexity without BCA increases by about 1.5 in 2048, and 40 in 8192. However, with its high resolution, XPos can recognize position with BCA's constraint."
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                                    "content": "Besides, we compare BCA with Sliding Attention (Child et al., 2019). In this experiment, we set the window size as 1024 to align with pre-training. Sliding Attention performs better as shown in the last row of Table 5 because its interaction range is broader than Block Causal Attention. The reason to use block windows instead of sliding windows is efficiency. According to (Xiong et al., 2021), the training speed of Blockwise Attention is "
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                                            "html": "<table><tr><td rowspan=\"2\">Methods</td><td>2048</td><td>8192</td></tr><tr><td colspan=\"2\">Extrapolation</td></tr><tr><td>Absolute</td><td>132.46</td><td>12747.41</td></tr><tr><td>Absolute + BCA</td><td>322.73</td><td>28787.01</td></tr><tr><td>RoPE</td><td>68.86</td><td>458.83</td></tr><tr><td>RoPE + BCA</td><td>26.37</td><td>26.16</td></tr><tr><td>Alibi</td><td>26.01</td><td>32.8</td></tr><tr><td>Alibi + BCA</td><td>27.53</td><td>31.82</td></tr><tr><td>xPos</td><td>27.29</td><td>63.99</td></tr><tr><td>xPos + BCA</td><td>25.53</td><td>24.89</td></tr><tr><td>xPos + Sliding Window</td><td>25.33</td><td>24.61</td></tr></table>",
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                                    "content": "Table 5: Results of perplexity on PG22 dataset. \"BCA\" is short for blockwise causal attention."
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                                    "content": "faster than using sliding windows. Therefore, LEX makes a trade-off and uses BCA in our implementation. Without losing generality, our method is also compatible with Sliding Attention and other local attention variants."
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                                    "content": "5 Related Work"
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                                    "content": "One popular solution (Wang et al., 2020b; Katharopoulos et al., 2020; Choromanski et al., 2020) is linear attention, i.e., using a kernel-based or low-rank approximation to replace vanilla attention. The methods typically target efficiency while underperforming vanilla Transformers for regular length. Another strand is sparse attention (Child et al., 2019; Beltagy et al., 2020; Zaheer et al., 2020; Xiong et al., 2021), which usually leverages structured sparsity to reduce computation. For causal sequence modeling, the recurrent-style designs (Dai et al., 2019; Hutchins et al., 2022; Ma et al., 2022b) are also competitive."
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                                    "content": "In comparison, we focus on length extrapolation (Press et al., 2021) for language modeling, i.e., training on short texts while evaluating long texts. The training process is kept the same as vanilla Transformers. The capability of long-sequence modeling is given for free during inference. So training efficiency (which is typically expensive for large-scale language models) is not affected"
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                                    "content": "compared with previous work. Moreover, the performance on regular length is perfectly retained, without trade-offs for long-sequence modeling."
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                                    "content": "a _ {i j} = \\frac {\\boldsymbol {q} _ {i} \\cdot \\boldsymbol {k} _ {j}}{\\sqrt {d}} \\Longrightarrow \\frac {\\boldsymbol {q} _ {i} \\cdot \\left(\\boldsymbol {k} _ {j} + \\boldsymbol {p} _ {i j} ^ {K}\\right)}{\\sqrt {d}} \\tag {15}",
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                                    "content": ". Yang et al. (2019) and He et al. (2020) further reparameterize the relative position vectors for better performance. T5 (Raffel et al., 2020) uses a simpler strategy to encode relative position:"
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                                    "content": " are manually defined. However, Alibi (Press et al., 2021)'s performance tends to be inferior to ROPE for the context whose"
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                                    "content": "length is shorter than the pre-training length. In this work, we propose a theoretically derived relative position embedding xPOS that optimizes the attention resolution between tokens. The xPOS method not only has the nice property of length extrapolation but also achieves strong performance."
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                                    "content": "We propose LEX Transformer to accurately capture position information for Transformers. We define attention resolution as the metric of length extrapolation and design a solution to improve the modeling. Extensive experiments on language modeling show that our method achieves lower perplexity on longer sequences while training on short texts. The simplicity also makes the method a go-to augmentation for Transformer-based language models. In addition, attention resolution provides a more principled view for position modeling, which sheds light on future architecture design."
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                                    "content": "Limitations"
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                                    "content": "In this work, we focus on causal language modeling. It needs additional efforts to integrate the proposed methods into bidirectional attention, such as masked language modeling (Devlin et al., 2019). Moreover, xPos introduces about "
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                                    "content": " inference cost compared with absolute position embeddings, although it accelerates training convergence."
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                                    "content": "References"
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                                    "content": "A Additional Experiments"
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                                    "type": "text",
                                    "content": "Besides the experiments in Section 4, we run language modeling evaluation on Arxiv and NarrativeQA (Kočisky et al., 2018). The results are shown in Table 6. These datasets have their shortcomings. The article length of Arxiv is usually less than 8192, and part of NarrativeQA's corpus is sampled from PG19, which is in the training dataset. Therefore, we show them in the appendix instead of the main content."
                                }
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                                    "content": "B Hyperparameters for Pre-Training"
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                                    "type": "text",
                                    "content": "As shown in Table 7, we present the hyperparameters for pre-training. The setting keeps the same among all Transformer variants. We follow medium-size GPT3 (Brown et al., 2020), 24 layers, 1024 hidden size, 4096 FFN inner hidden size, and 16 attention heads. The number of batch tokens is "
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                                    "type": "inline_equation",
                                    "content": "0.5\\mathrm{M}"
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                                {
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                                    "type": "text",
                                    "content": " for pre-training 1024, and the number of batch sentences is 512. We use Adam (Kingma and Ba, 2015) optimizer with "
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                                    "type": "inline_equation",
                                    "content": "\\beta_{1} = 0.9, \\beta_{2} = 0.98, \\epsilon = 10^{-6}"
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                                    "type": "text",
                                    "content": ". The warmup steps are "
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                                            "html": "<table><tr><td rowspan=\"2\">Length</td><td>256</td><td>512</td><td>1024</td><td>2048</td><td>4096</td></tr><tr><td colspan=\"3\">Interpolation</td><td colspan=\"2\">Extrapolation</td></tr><tr><td colspan=\"6\">arXiv</td></tr><tr><td>Transformer</td><td>29.74</td><td>23.6</td><td>19.59</td><td>102.09</td><td>1240.77</td></tr><tr><td>Alibi</td><td>26.53</td><td>21.07</td><td>17.53</td><td>15.38</td><td>16.88</td></tr><tr><td>Roformer</td><td>25.89</td><td>20.6</td><td>17.24</td><td>49.29</td><td>199.25</td></tr><tr><td>LEX Transformer (Ours)</td><td>25.73</td><td>20.48</td><td>17.14</td><td>15.81</td><td>15.19</td></tr><tr><td colspan=\"6\">NarrativeQA</td></tr><tr><td>Transformer</td><td>16.74</td><td>14.42</td><td>13.02</td><td>58.95</td><td>574.91</td></tr><tr><td>Alibi</td><td>15.58</td><td>13.45</td><td>12.15</td><td>11.4</td><td>12.09</td></tr><tr><td>Roformer</td><td>15.21</td><td>13.16</td><td>11.93</td><td>20.72</td><td>35.14</td></tr><tr><td>LEX Transformer (Ours)</td><td>14.82</td><td>12.86</td><td>11.67</td><td>11.14</td><td>10.93</td></tr></table>",
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                                    "content": "A For every submission:"
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                                    "content": "B Did you use or create scientific artifacts?"
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                                    "content": "C Did you run computational experiments?"
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                                    "type": "text",
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                                    "content": "C1. Did you report the number of parameters in the models used, the total computational budget (e.g., GPU hours), and computing infrastructure used?"
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                                    "type": "text",
                                    "content": "The Responsible NLP Checklist used at ACL 2023 is adopted from NAACL 2022, with the addition of a question on AI writing assistance."
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                                    "content": "D Did you use human annotators (e.g., crowdworkers) or research with human participants?"
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                                    "type": "text",
                                    "content": "Left blank."
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                                    "content": "D1. Did you report the full text of instructions given to participants, including e.g., screenshots, disclaimers of any risks to participants or annotators, etc.?"
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                                    "content": "D2. Did you report information about how you recruited (e.g., crowdsourcing platform, students) and paid participants, and discuss if such payment is adequate given the participants' demographic (e.g., country of residence)?"
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                                    "content": "D3. Did you discuss whether and how consent was obtained from people whose data you're using/curating? For example, if you collected data via crowdsourcing, did your instructions to crowdworkers explain how the data would be used?"
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