Predictor-Corrector Enhanced Transformers with Exponential Moving Average Coefficient Learning
Residual networks, as discrete approximations of Ordinary Differential Equations (ODEs), have inspired significant advancements in neural network design, including multistep methods, high-order methods, and multi-particle dynamical systems. The precision of the solution to ODEs significantly affects parameter optimization, thereby impacting model performance. In this work, we present a series of advanced explorations of Transformer architecture design to minimize the error compared to the true ``solution.'' First, we introduce a predictor-corrector learning framework to minimize truncation errors, which consists of a high-order predictor and a multistep corrector. Second, we propose an exponential moving average-based coefficient learning method to strengthen our higher-order predictor. Extensive experiments on large-scale machine translation, abstractive summarization, language modeling, and natural language understanding benchmarks demonstrate the superiority of our approach. On the WMT'14 English-German and English-French tasks, our model achieved BLEU scores of 30.95 and 44.27, respectively. Furthermore, on the OPUS multilingual machine translation task, our model surpasses a robust 3.8B DeepNet by an average of 2.9 SacreBLEU, using only 1/3 parameters. Notably, it also beats LLama models by 5.7 accuracy points on the LM Harness Evaluation.
Paper
Similar papers
Peer review
Summary
This work presents a transformer architecture inspired by predictor-corrector methods for solving ODE problems. It adopts the Adams-Bashforth-Moulton method and utilizes an exponential moving average (EMA) method to compose the predictor, discussing two correctors (the EMA-based and the simple backward Euler method). Without the background of ODE, the proposed method could be seen as a special case of a transformer with cross-layer connections. Consider that the residual connection only links two layers, while the proposed method links multiple layers in a high-order way.
Strengths
This work provides theoretical results to enhance the connections between residual connections and the ODE solver, providing an interesting perspective on viewing cross-layer connections within neural networks. Using EMA as a high-order predictor is a simple and flexible solution, and the experiments demonstrate its effectiveness.
Weaknesses
The experiments are insufficient and out-of-date. As the main contribution is the new transformer architecture, scaling law style experiments and testing on modern LLMs' benchmarks are recommended. I saw that Appendix D has some results, but it's better to have a formal experiment.
Questions
What's the efficiency of the proposed method, such as inference time and GPU memory consumption? I saw that inference efficiency has been mentioned in the limitations; it's better to have quantitative results and also memory states.
Rating
7
Confidence
3
Soundness
3
Presentation
3
Contribution
3
Limitations
Agree with the limitation section.
Summary
This paper takes inspiration from established high-order approaches in numerical analysis for solving differential equations to improve the architectural design of Transformers. Prior work has shown that residual networks can be seen as discrete approximations of Ordinary Differential Equations (ODE) and explored methods to improve the quality of the solution. Specifically, this paper introduces a predictor-corrector learning framework to minimize approximation errors and an exponential moving average-based coefficient learning method. These advancements are used within Transformer architectures that are evaluated on several tasks such as translation, summarization, language modeling and language understand tasks, improving over standard and previous ODE-based Transformers.
Strengths
- The paper is well motivated and clearly described for the most part. It provides sufficient background to make the reading self contained and explain what is the novel contribution of the work. It's also well situated and compared to prior work on ODE Transformers as it discusses prior first order and high-order methods with a single step. - Builds on top of predictor-corrector methods from numerical analysis and extending them to improve their stability when training ODE Transformers. The key novelty lies in the selection of the predictor and corrector methods, and the proposal to use exponential moving average method to learn the coefficients in high-order methods instead of using constant values. - The proposed method uses a high-order method as predictor and a multi-step method as corrector and aims to provide a better approximation to the implicit ODE problem in Transformers than previous studies. - Presents empirical results on a variety of tasks are better than the original and previous ODE Transformers. This makes the work of interest to the researchers working in ODE Transformers.
Weaknesses
- W1. There is lack of emphasis in the experiments on the computational overhead introduced by the high-order predictor and multi-step methods. Achieving better quality is not sufficient for adoption in practical settings. I'd suggest quantifying the training and inference cost compared to standard and ODE Transformers. - W2. In terms of parameter efficiency, the proposed model achieves better performance with 1/3 of parameters only on one of the examined datasets. It's not clear if this is by chance or if the result holds on other settings. It would be useful to report performance with a smaller model size on the rest of the datasets to better establish the underlying claim. - W3. The impact to research communities beyond the ones focusing on ODE transformers is somewhat limited because the results are with relatively small model sizes. Showcasing that the results hold on larger architectures such as 7B Mistral would make the results more convincing. - W4. The improvement of the proposed multi-step high-order method compared alternative predictor-corrector methods is small (Table 10) and the benefit compared to simpler first-order methods from the predictor-only paradigm in a controlled setting is not provided (similar to the ablation in Table 10).
Questions
- Q1. Where does the parameter efficiency of the model stems from and how a fair comparison with other models was ensured? In Appendix D, there is a section that discusses this but the comparison does not seem to be apples-to-apples (i.e. same config and #params) and whether the result holds in all the tasks in the experiment section. - Q2. The method description wasn't entirely clear to me and I have the following questions: - The parameterization of $\mathcal{F}(P_{t+1}, \theta_t)$ is not clear from the textual description. What kind of transformation is used here? - How are the coefficients are parameterized exactly? Do you parameterize a single coefficient for each order for an $n$-order predictor or it is the same coefficient shared? - The text mentions that a larger weight is assigned ($a=0.5$) to the estimated $F_{t+1}$, is this value fixed or you have experimented with different values? It would be useful to clarify which coefficients among $\alpha$ and $\gamma$ are hyper-parameters and which ones are learned. - Q3. Is the observed improvement worth the the additional computational cost for obtaining a better approximation? It would be useful to show what is the difference compared to a standard or a simpler ODE Transformer. - Q4. Do you mean "ROUGE results" instead of "Rough results" in Table 4? In the same table, what is the number of parameters used by each model? - Q5. What is the size of the models in Table 8 and is it the same for all variants? - Q6. In the ablation experiment and other results, there is lack of comparison with a simple 1-order single/multi-step method or high-order 1-step method. Is the proposed method substantially better than them? - Q7: Did you perform continued training of BERT directly on language understanding tasks or on some pre-training data first in Table 7? Please also report the number of parameters for both models and the exact configuration for PCformer.
Rating
6
Confidence
4
Soundness
3
Presentation
3
Contribution
3
Limitations
The limitation section covers the shortcomings when the proposed methods are applied to different model designs but lacks discussion about the experiment shortcomings related to model size, computational cost, and comparison with simpler methods in a controlled setting.
Further response to the remained questions
> Q2.1 The parameterization is not clear from the textual description. What kind of transformation is used here? As we emphasized that $\mathcal{F}_(P_{t+1},\theta_t)$ is the function to computed the derivate upon the input. Thus, either a self-attention network or an FFN, or even the whole encoder block could be regarded as a F function, also they could be seen fuctions in different granularities. In this work, for a fair comparison with ODE Transformer, we choose the whole block as the function. > Q2.2 Details of $\alpha$ Just a single coefficient. the code is as follows: ``` self.alpha = torch.nn.Parameter(torch.Tensor(1)) self.alpha.data.fill_(0.5) ``` Take a 2-order EMA as an instance, the final layer ouput is computed as follows: ``` x = residual + self.alpha*(1-self.alpha) * runge_kutta_list[0] + self.alpha*runge_kutta_list[1] ``` where runge_kutta_list is a list to stores the previous obtained intermediate approximations ($\hat{F}_i$). We have uploaded our code to a anonymous github, more details could found in our codebase. > Q2.3 Whether $\alpha$ is fixed Similar with the previous question, we use 0.5 as an initail value for $\alpha$, while it is learnable. Thus $\alpha$ would be changed according to the graident decsent during the training phase. Sorry for the missing details for $\alpha$ and $\gamma$, and these two are both learnable tensors with an initial value 0.5. We will include this in our next version of the paper. > Q3. Is the observed improvement worth the the additional computational cost for obtaining a better approximation? As for the sequence generation models which follow a encoder-decoder paradigm, our models are quite competive and the observed improvement is indeed worth the additional computational cost. Actually, we have already compared our PCformer with a simpler ODE Transformer in sequence generation tasks. We can see that, PCformer (RK2) can beat ODE Transformer (RK4) with stronger performance and less computation cost. For example, in Table 1, PCformer (2-order) beats RK4-block (EMA) in both En-De and En-Fr tasks, with one less computation (The former 2 times predictor and 1 times corrector, while the later consumes 4 times forward computation for the high-order solution). SSimilar phenomena could be observed in abstractive summarization tasks (Table 2). > Q4: typos and the parameters of the models displayed in Table 4. Yes, we apologize for the typo; we mean "ROUGE results" instead of "Rough results" and will correct it in the next version. Regarding the number of parameters, all models listed in Table 4 have a similar parameter number, approximately 63M. This corresponds to a Transformer-base configuration, which includes 6 encoder layers and 6 decoder layers, each with a hidden size of 512 and 8 attention heads. > Q5: What is the size of the models in Table 8 and is it the same for all variants? The configurations of the models used to approximate the truncation errors are detailed in Appendix C.3. Specifically, we used a Transformer language model (decoder-only) with a hidden size of 512, tested in both 1-layer and 2-layer settings. Note that all all variants are the same. By comparing the results of the 1-layer and 2-layer models, we can see that both the ODE Transformer and our PCformer significantly reduce truncation errors, as measured by PPL. For instance, the PCformer (2nd order but with a 1-layer parameter) outperforms the Residual-Block (2-layer) and even surpasses the RK4-block (EMA) with fewer forward computations. This also provides context and a partial answer to your Q3 regarding computational efficiency. > Q6: Comparison with a simple 1-order single/multi-step method or high-order 1-step method. Yes, our PCformer significantly outperforms single/multi-step methods and high-order 1-step methods. Perhaps there was some oversight, but it is important to note that the single 1-step 1-order method corresponds to the vanilla Transformer (denoted as "Residual-block" in most tables). Transformer-DLCL represents a specific case of a multi-step method. We compared these models with our PCformer in Table 1. Additionally, the high-order 1-step method is represented by the ODE Transformer. We have already included comparisons with these methods in our MT experiments. We will ensure that these results are included in subsequent versions to provide a more comprehensive evaluation. > Q7: Details of the BERT training. Both the PCformer and the BERT models in Table 7 share the same parameter count, with approximately 335M parameters each. Specifically, both models have a hidden size of 1024, an FFN filter size of 4096, and 16 attention heads. We have aligned all settings with those specified in the original BERT paper to ensure a fair comparison. For PCformer, we pre-trained the model from scratch using a combination of WikiText and BookCorpus datasets. After the pre-training phase, we fine-tuned PCformer on the GLUE downstream tasks.
Response to authors
Thank you for the detailed responses to my questions and additional results! My main concerns have been addressed and I decided to increase my score: - The scaling experiments with LLMs are quite promising and increased my confidence that the results hold on larger model sizes. - In terms of efficiency, the results show that there is no significant overhead introduced which addressed my main concern. It was also great to see results that justify the parameter efficiency (e.g. performance of PCFormer 340B vs Transformer 1.3B). - The new ablation results clarify the differences compared to simpler baselines. It appears that the proposed methods have a fairly good improvement for translation that is greater than 0.5-1 point. The rest of the answers provided helpful clarifications, it would be great if they are reflected in the final version.
Thanks for the reconsideration
Thank you for reconsidering our work. We greatly appreciate the valuable suggestions you have provided, which we believe will further enhance our research. We plan to include these results and make thorough revisions in our next version. Thank you once again for your efforts!
Summary
This paper presents an approach to improve the performance of Transformer models for conditional natural language generation (machine translation and summarization). The authors introduce a predictor-corrector framework, inspired by numerical methods for solving ordinary differential equations (ODEs), to enhance the accuracy and stability of the models. The proposed model is evaluated against standard Transformer models on multiple benchmark datasets. The presented results show improvements in prediction accuracy and efficiency when compared to standard models.
Strengths
The application of the predictor-corrector paradigm to Transformer models is innovative and offers a new perspective, extending the ODE transformer proposed in (Li et al., 31). The integration of high-order ODE solutions into the Transformer architecture is a novel contribution. The authors use EMA (Exponential Moving Average) coefficient learning to enhance training stability. The experimental results look comprehensive, with evaluations on multiple benchmark datasets demonstrating the efficacy of the proposed model. The paper provides a detailed description of the methodology. The paper is well-organized and clearly written, with each section leading to the next.
Weaknesses
The paper does not compare its results with state-of-the-art models, which could have provided a more comprehensive evaluation of its effectiveness. For instance for WMT14 En to FR, we can reach 43 in BLEU, instead of 41 for Attention is all you need. Moreover, the difference with ODE Transformer is very limited. The theoretical justification for the predictor-corrector framework could be more detailed, particularly in explaining how it relates to and improves upon existing methods. The computational overhead induced by the proposed method is only lightly discussed in the Appendix and would require more in depth discussion.
Questions
- How does the predictor-corrector framework perform in comparison to state-of-the-art pretrained Transformer models? - Do you think that all the 72 references are necessary? - The quotation of Newell 59 is maybe a bit too much. Don't you think?
Rating
6
Confidence
4
Soundness
3
Presentation
3
Contribution
3
Limitations
The authors should discuss the potential computational overhead introduced by the predictor-corrector framework.
Summary
The paper presents advancements in Transformer architecture to minimize errors in approximating solutions to Ordinary Differential Equations (ODEs). The contributions are: - Introducing a learning paradigm with a high-order predictor and multistep corrector to reduce truncation errors. - Proposing an exponential moving average-based method to enhance the predictor's learning ability and stability by replacing constant coefficients with dynamic ones. - Demonstrating superior performance across various benchmarks, including machine translation, abstractive summarization, language modeling, and natural language understanding, achieving notable improvements in BLEU scores and parameter efficiency. The work shows improvements in translation tasks and highlights the general applicability of the proposed methods across different natural language processing domains.
Strengths
The paper introduces a new predictor-corrector framework within Transformer architectures, which is a fresh and innovative approach to addressing errors in approximating solutions to ODEs. The integration of a high-order predictor and multistep corrector, combined with an exponential moving average (EMA) coefficient learning method, represents a creative combination of established numerical analysis techniques with modern neural network architectures. The paper successfully applies the method across various natural language processing tasks. The paper is well-structured and clearly written, making complex concepts accessible.
Weaknesses
The complexity of implementing these methods might be a barrier for practical adoption. The paper could benefit from providing more detailed guidelines or code to facilitate easier implementation and replication of the results. The experiments primarily focus on natural language processing tasks. While the results are impressive, it remains unclear how well the proposed methods generalize to other domains, such as time series forecasting. Including preliminary results or discussions on the potential applicability to these other areas could strengthen the paper.
Questions
Can you provide more details on the computational cost of the proposed predictor-corrector framework and EMA coefficient learning method? Specifically, how do these methods impact training time and resource utilization compared to traditional Transformer models? How does the proposed method scale with increasing model size and dataset complexity? Have you encountered any challenges in scaling up your approach, and if so, how did you address them?
Rating
5
Confidence
1
Soundness
3
Presentation
3
Contribution
2
Limitations
Provide a detailed analysis of the computational complexity and resource requirements of the proposed methods. This should include a comparison with standard Transformer models and an explanation of any trade-offs between performance improvements and computational costs. Discuss the scalability of the proposed methods in more detail. Explain any challenges encountered when scaling up to larger datasets or models and how these were addressed. Provide insights into potential limitations in terms of scalability and how future work could overcome these challenges.
Any Further Concerns About Our Work
Dear Reviewer 1pKC, We apologize for reaching out as the discussion deadline approaches. We are grateful for the comprehensive and useful feedback you provided, and we have responded in detail to your comments during the rebuttal phase, e.g., new results on time-series forecasting, the practical cost of training and inference, more strong results on larger LLMs using PCformer and so on. We are eager to know if our proposed results and clarifications have adequately addressed your concerns. If so, we would appreciate it if you could reconsider your score. Thank you once again for your time and effort. Best regards, The Authors
Looking Forward to follow-up discussion
Dear Reviewer 1pKC, Thank you once again for your suggestions and valuable feedback. We apologize for reaching out again as the deadline approaches. To address your concerns, we have conducted experiments on both the time-series forecasting task and the image classification task. Additionally, we have provided data on the practical costs of training and inference for both the encoder-decoder and decoder-only paradigms. We have now uploaded the results of a 3B PCformer using 16B and 50B tokens, respectively. The results show that there are no major challenges for its scaling up. If our results and explanations help address your concerns, we would be grateful if you could acknowledge our rebuttal and consider adjusting your score accordingly. Best wishes The Authors.
Any Other Concerns on PCformer
We extend our heartfelt gratitude to the four reviewers for their insightful feedback and to the Area Chairs for facilitating the rebuttal discussion. We appreciate the time and effort you have invested in evaluating our submission. Since uploading our rebuttal, we have not yet received any further feedback. We eagerly look forward to engaging in discussions with the reviewers and believe that we have effectively addressed the major concerns raised. We also welcome any other further concerns! We are aware that the ACL conference is approaching, and we hope you all have wonderful days
Further Results on a 3B PCformer
| Model(Params & Tokens) | Wiki.(ppl) | LMB.(ppl) | LMB. | PIQA | Hella. | SCIQ | ARC-c | Winograde| Avg. | | -------------------------- | ---------- | --------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | | Transformer++ (340M & 6B) | 38.5 | 96.1 | 21.4 | 60.3 | 29.1 | 69.2 | 21.5 | 50.4 | 41.9 | | PCformer (340M & 6B) | 35.3 | 78.8 | 23.6 | 61.6 | 30.1 | 71.6 | 22.9 | 51.8 | 43.6 | | Transformer++ (340M & 16B) | 28.3 | 65.3 | 29.8 | 63.2 | 33.9 | 73.2 | 23.1 | 51.4 | 45.8 | | PCformer (340M & 16B) | 25.6 | 39.7 | 34.5 | 65.2 | 36.9 | 79.6 | 23.2 | 52.2 | 48.6 | | Transformer++ (1.3B & 16B) | 23.8 | 26.2 | 37.3 | 65.7 | 37.6 | 78.6 | 23.7 | 51.5 | 49.0 | | PCformer (1.3B & 16B) | 20.9 | 23.2 | 42.5 | 68.3 | 43.4 | 81.5 | 25.1 | 52.4 | 52.2 | | Transformer++(1.3B & 100B) | 16.3 | 11.8 | 51.6 | 71.0 | 51.7 | 86.7 | 28.1 | 54.6 | 57.2 | | PCformer (1.3B & 50B) | 16.2 | 9.38 | 55.1 | 71.9 | 54.8 | 88.6 | 29.6 | 57.2 | 59.5 | | PCformer (1.3B & 100B) | **14.0** | **7.46** | **59.6** | **73.8** | **60.0** | **90.7** | **31.7** | **61.7** | **62.9** | | PCformer (3B & 16B) | 17.8 | 13.6 | 48.0 | 69.7 | 48.4 | 83.1 | 26.4 | 55.2 | 55.1 | | PCformer (3B & 50B) | **13.6** | **6.5** | **62.1** | **74.4** | **61.9** | **90.6** | **32.4** | **61.9** | **63.9** | Since the discussion deadline is approaching, we would like to once again thank all the reviewers for their efforts in helping to further improve our work. Given the limited time, we have focused on reporting the results of PCformer on a 1.3B model using 100B tokens in the rebuttal phase. Additionally, we have attempted to further scale the model to over 3B parameters, with a hidden size of 3200, an intermediate size of 8640, and 26 layers, following the LLama3B configuration. We trained this model using 128 A100 GPUs and, up to now, it has been trained on nearly 50B tokens. Below are the results for the 3B model at different stages of training (16B tokens and 50B tokens): - PCformer (3B & 16B) outperforms its 1.3B & 16B counterpart by an average score of 2.9, and PCformer (3B & 50B) surpasses PCformer (1.3B & 50B) by an average score of 4.4. This indicates that with increased data consumption, PCformer continues to benefit from the data and demonstrates superior results. - PCformer (3B & 50B) achieves an average score 8.8 points higher than PCformer (3B & 16B), and PCformer (1.3B & 50B) exceeds PCformer (1.3B & 16B) by an average score of 7.3. This suggests that the performance gap widens with increasing data volumes, which is intuitive as larger models naturally require more data for effective training. The above two phenomena demonstrate the scalability of PCformer, and we are enthusiastic about the opportunity to demonstrate its potential to the community. Thank you once again to all the reviewers, ACs, and SACs for your invaluable feedback and support.
Decision
Accept (poster)