Summary Length Generalization in Arithmetic Transformers arxiv.org
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This text discusses how transformers struggle with arithmetic and longer sequences, but the use of relative position embeddings and train set priming can improve generalization, particularly in multiplication tasks.
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Key Points
- Transformers struggle with simple tasks like integer arithmetic and generalizing to longer sequences.
- Relative position embeddings enable length generalization for addition tasks but fail for multiplication.
- Train set priming, by adding long sequences to the training set, improves models' ability to generalize to larger multiplication examples.
- The number of priming examples required scales logarithmically with the number of training examples and linearly with the extrapolation length.
- Relative position embeddings perform better than absolute position embeddings in terms of length generalization.
- Priming the train set with 35-digit numbers allows for extrapolation to 35-digit operands, while priming on numbers from 6 to 35 enables extrapolation to all lengths up to 35.
- Model failures in addition tasks are observed in cases involving three or more carries and two consecutive carries.
- Open questions for future research include extending priming to other mathematical problems and exploring priming in natural language processing tasks.
Summaries
122 word summary
Transformers struggle with learning arithmetic and generalizing to longer sequences. Relative position embeddings aid length generalization for addition but not multiplication. Train set priming, adding long sequences to training sets, enables models trained on 5-digit x 3-digit multiplications to generalize to 35 x 3 examples. The paper explores potential applications of priming beyond arithmetic. Previous works focused on in-distribution settings and failed in out-of-distribution experiments. Relative position embeddings improve generalization in natural language processing but have not been extensively studied for arithmetic tasks. The authors propose train set priming to enable length generalization in multiplication tasks. Relative position embeddings outperform absolute position embeddings. The document concludes with open questions for future research and additional experiments on modular arithmetic and element-wise addition tasks.
232 word summary
This paper examines the challenges faced by transformers when learning arithmetic and generalizing to longer sequences. The authors find that relative position embeddings enable length generalization for addition tasks, but not for multiplication. To address this, they propose train set priming, which involves adding long sequences to the training set. Models trained on 5-digit x 3-digit multiplications can then generalize to 35 x 3 examples. The authors also discuss potential applications of priming beyond arithmetic.
Previous works have focused on in-distribution settings for transformers and have not been successful in out-of-distribution experiments and extrapolation.
While relative position embeddings have been shown to improve generalization in natural language processing, their impact on arithmetic tasks has been little studied. The experiments in this paper show that models using relative position embeddings can generalize to longer sequences for addition tasks, achieving high accuracy on 10-digit numbers. However, they do not enable length generalization for multiplication tasks.
To address this, the authors propose train set priming, which allows models to generalize to larger multiplication examples by adding a small number of long sequences to the training set. The results also suggest that relative position embeddings perform better than absolute position embeddings in terms of length generalization.
The document concludes by posing open questions for future research and presents additional experiments on modular arithmetic and element-wise addition tasks to compare the performance of different models and methods.
299 word summary
This paper explores the challenges faced by transformers in learning arithmetic and generalizing to longer sequences. The authors find that relative position embeddings enable length generalization for addition tasks, but not for multiplication. To address this, they propose train set priming, which involves adding long sequences to the training set. Models trained on 5-digit x 3-digit multiplications can then generalize to 35 x 3 examples. The authors also discuss potential applications of priming beyond arithmetic.
The paper discusses the struggles of transformers with simple tasks like arithmetic and the need for models to be able to extrapolate small number arithmetic to larger integers. Previous works have focused on in-distribution settings, but out-of-distribution experiments and extrapolation have proven disappointing.
The authors explain that absolute position embeddings hinder generalization in transformers, while relative position embeddings have been shown to improve generalization in natural language processing. However, their impact on arithmetic tasks has been little studied.
The experiments show that models using relative position embeddings can generalize to longer sequences for addition tasks, achieving high accuracy on 10-digit numbers. However, they do not enable length generalization for multiplication tasks.
To address this failure for multiplication, the authors propose train set priming. By adding a small number of long sequences to the training set, models can generalize to larger multiplication examples. The number of priming examples required scales logarithmically with the number of training examples.
The results also suggest that relative position embeddings perform better than absolute position embeddings in terms of length generalization. RPE models seem to learn all digits simultaneously, while APE models learn each position independently.
The document concludes by posing open questions for future research and presents additional experiments on modular arithmetic and element-wise addition tasks. The results show the performance of different models and methods in these tasks.
999 word summary
This paper explores the challenges faced by transformers in learning basic integer arithmetic and generalizing to longer sequences. The authors find that relative position embeddings enable length generalization for addition tasks, allowing models trained on 5-digit numbers to perform 15-digit sums. However, this method fails for multiplication. To address this, the authors propose train set priming, which involves adding a small number of long sequences to the training set. They demonstrate that models trained on 5-digit x 3-digit multiplications can generalize to 35 x 3 examples with the use of priming. The authors also show that the priming sample size scales with the logarithm of the training set size and discuss potential applications of priming beyond arithmetic.
The paper begins by discussing the success of transformers in various domains but notes their struggle with simple tasks like integer arithmetic. The absence of large numbers in the training data limits the mathematical ability of large language models, and the authors propose that models must be able to extrapolate small number arithmetic to larger integers. Previous works on learning arithmetic with transformers have focused on in-distribution settings, where the training and test sets are drawn from the same distribution. However, out-of-distribution experiments and extrapolation to larger numbers have proven disappointing.
The authors then discuss the concept of length generalization in transformers, which has been widely studied. They explain that absolute position embeddings (APEs), used in many implementations, hinder generalization because they mix the representation of a token with the embedding of its position in the sequence. Several papers have proposed using relative position embeddings (RPEs) or weighted attention schemes to improve generalization in natural language processing (NLP), but their impact on arithmetic tasks has been little studied.
In their experiments, the authors train models on 5-digit operations and investigate their ability to generalize to numbers with up to 20 digits for addition and 35 digits for multiplication. They find that models using relative position embeddings can generalize to longer sequences for addition tasks, achieving high accuracy on 10-digit numbers and reasonable accuracy on 15-digit numbers. However, the use of relative position embeddings does not enable length generalization for multiplication tasks.
To address the failure of relative position embeddings for multiplication, the authors propose train set priming. By adding a small number of long sequences to the training set, models trained on 5-digit x 3-digit multiplications can generalize to 35 x 3 examples. The authors show that the priming sample size scales as the logarith
This summary presents the key points and important details from the excerpted text.
The document discusses the concept of length generalization in arithmetic transformers. It explores the use of fine-tuning and train set priming to improve the accuracy of multiplication tasks. The experiments use a standard UTransformer with specific parameters.
The results show that fine-tuning the model on 5x3 multiplications and then fine-tuning on 35x3 multiplications improves accuracy. Priming the training set with fifty 35x3 examples also leads to better accuracy. The number of priming examples required scales logarithmically with the number of training examples and linearly with the extrapolation length.
Curriculum priming, where priming examples are split between different lengths, is generally ineffective except when priming on a mixture of 34 and 35-digit numbers.
Priming the train set with 35-digit numbers allows for extrapolation to 35-digit operands but not to other lengths. However, priming on numbers of all lengths from 6 to 35 enables extrapolation to all lengths up to 35.
The results also suggest that relative position embeddings (RPEs) perform better than absolute position embeddings (APEs) in terms of length generalization. RPE models seem to learn all digits simultaneously, while APE models learn each position independently.
The document presents an analysis of failure cases in addition tasks, focusing on the role of carries and incorrect digits. It is observed that model failures mostly occur in additions involving at least three carries and two consecutive carries. Errors concentrate on the first and second positions of the sums.
Train set priming is shown to be effective not only for multiplication but also for APE models. However, APE models require a larger priming rate compared to RPE models.
The document concludes by posing open questions for future research, including the extension of priming to other mathematical problems, investigating compositionality limits, understanding the theoretical aspects of priming, and exploring priming in natural language processing tasks.
Additional experiments on modular arithmetic are presented, showing the results of APE and RPE models on digitwise addition and multiplication tasks. APE models can be primed to length generalize in multiplication, but with a larger priming rate. Plots are provided to illustrate the digit order by which RPE and APE models make correct predictions.
Table 4 provides the extrapolation results for modular addition using the UTransformer model in both Base and Large formats. The accuracy reached by the models on 100,000 example test sets is reported. The model manages to extrapolate when the modulus is a power of 10, but fails to do so when the modulus is 128 or 101. Table 5 provides similar results for modular multiplication. The model manages to extrapolate when the modulus is a power of 10, but fails when the modulus is 128 or 101. The difference in performance may be due to 101 being a prime number and 128 being a power of 2.
Table 6 reports the extrapolation results for element-wise addition using the UTransformer model with different position embedding methods. The RPE models manage to length generalize, while the APE models fail. Figure 8 shows the digitwise accuracy of the APE model on element-wise addition, indicating that the model performs well on the leftmost digits seen during training but fails on the rightmost ones.
Additional experiments on priming for multiplication are presented in Figure 9. Priming the model with a small percentage of the training set leads to successful extrapolation to larger multiplication tasks. Figure 10 shows the digitwise accuracy on the training examples, while Figure 11 shows the digitwise prediction on the test examples.