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- .gitattributes +179 -0
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.gitattributes
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parse/train/qWtmNpgjD5K/qWtmNpgjD5K_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HkfPSh05K7/HkfPSh05K7_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HyRVBzap-/HyRVBzap-_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/RO4DM85Z4P7/RO4DM85Z4P7_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Byj72udxe/Byj72udxe_layout.pdf filter=lfs diff=lfs merge=lfs -text
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| 14794 |
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parse/train/cVwc7IHWEWi/cVwc7IHWEWi_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14795 |
+
parse/train/cVwc7IHWEWi/cVwc7IHWEWi_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14796 |
+
parse/train/B8fp0LVMHa/B8fp0LVMHa_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14797 |
+
parse/train/B8fp0LVMHa/B8fp0LVMHa_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14798 |
+
parse/train/B8fp0LVMHa/B8fp0LVMHa_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14799 |
+
parse/train/BJe-_CNKPH/BJe-_CNKPH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14800 |
+
parse/train/BJe-_CNKPH/BJe-_CNKPH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14801 |
+
parse/train/BJe-_CNKPH/BJe-_CNKPH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14802 |
+
parse/train/SJl5Np4tPr/SJl5Np4tPr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14803 |
+
parse/train/SJl5Np4tPr/SJl5Np4tPr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14804 |
+
parse/train/SJl5Np4tPr/SJl5Np4tPr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14805 |
+
parse/train/ryxIZR4tvS/ryxIZR4tvS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14806 |
+
parse/train/ryxIZR4tvS/ryxIZR4tvS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14807 |
+
parse/train/ryxIZR4tvS/ryxIZR4tvS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14808 |
+
parse/train/jWkw45-9AbL/jWkw45-9AbL_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14809 |
+
parse/train/jWkw45-9AbL/jWkw45-9AbL_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14810 |
+
parse/train/jWkw45-9AbL/jWkw45-9AbL_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14811 |
+
parse/train/zspml_qcldq/zspml_qcldq_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14812 |
+
parse/train/zspml_qcldq/zspml_qcldq_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14813 |
+
parse/train/zspml_qcldq/zspml_qcldq_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14814 |
+
parse/train/u8X280hw1Mt/u8X280hw1Mt_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14815 |
+
parse/train/u8X280hw1Mt/u8X280hw1Mt_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14816 |
+
parse/train/u8X280hw1Mt/u8X280hw1Mt_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14817 |
+
parse/train/7uVcpu-gMD/7uVcpu-gMD_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14818 |
+
parse/train/7uVcpu-gMD/7uVcpu-gMD_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14819 |
+
parse/train/7uVcpu-gMD/7uVcpu-gMD_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14820 |
+
parse/train/HkxKH2AcFm/HkxKH2AcFm_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14821 |
+
parse/train/HkxKH2AcFm/HkxKH2AcFm_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14822 |
+
parse/train/HkxKH2AcFm/HkxKH2AcFm_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14823 |
+
parse/train/w6iVxEdh6bi/w6iVxEdh6bi_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14824 |
+
parse/train/w6iVxEdh6bi/w6iVxEdh6bi_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14825 |
+
parse/train/w6iVxEdh6bi/w6iVxEdh6bi_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14826 |
+
parse/train/SJgaRA4FPH/SJgaRA4FPH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14827 |
+
parse/train/SJgaRA4FPH/SJgaRA4FPH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14828 |
+
parse/train/SJgaRA4FPH/SJgaRA4FPH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14829 |
+
parse/train/rylkma4twr/rylkma4twr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14830 |
+
parse/train/rylkma4twr/rylkma4twr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14831 |
+
parse/train/rylkma4twr/rylkma4twr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14832 |
+
parse/train/pAbm1qfheGk/pAbm1qfheGk_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14833 |
+
parse/train/pAbm1qfheGk/pAbm1qfheGk_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14834 |
+
parse/train/pAbm1qfheGk/pAbm1qfheGk_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14835 |
+
parse/train/GOfGGASIUkg/GOfGGASIUkg_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14836 |
+
parse/train/GOfGGASIUkg/GOfGGASIUkg_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14837 |
+
parse/train/GOfGGASIUkg/GOfGGASIUkg_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14838 |
+
parse/train/L80PLIixPIXTH/L80PLIixPIXTH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14839 |
+
parse/train/L80PLIixPIXTH/L80PLIixPIXTH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14840 |
+
parse/train/L80PLIixPIXTH/L80PLIixPIXTH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14841 |
+
parse/train/BJxg_hVtwH/BJxg_hVtwH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14842 |
+
parse/train/BJxg_hVtwH/BJxg_hVtwH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14843 |
+
parse/train/BJxg_hVtwH/BJxg_hVtwH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14844 |
+
parse/train/HJjvxl-Cb/HJjvxl-Cb_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14845 |
+
parse/train/HJjvxl-Cb/HJjvxl-Cb_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14846 |
+
parse/train/HJjvxl-Cb/HJjvxl-Cb_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14847 |
+
parse/train/HkgXteBYPB/HkgXteBYPB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14848 |
+
parse/train/HkgXteBYPB/HkgXteBYPB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14849 |
+
parse/train/HkgXteBYPB/HkgXteBYPB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14850 |
+
parse/train/LoUdcqLuPej/LoUdcqLuPej_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14851 |
+
parse/train/LoUdcqLuPej/LoUdcqLuPej_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14852 |
+
parse/train/LoUdcqLuPej/LoUdcqLuPej_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14853 |
+
parse/train/tGZu6DlbreV/tGZu6DlbreV_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14854 |
+
parse/train/tGZu6DlbreV/tGZu6DlbreV_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14855 |
+
parse/train/tGZu6DlbreV/tGZu6DlbreV_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14856 |
+
parse/train/S1eYHoC5FX/S1eYHoC5FX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14857 |
+
parse/train/S1eYHoC5FX/S1eYHoC5FX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14858 |
+
parse/train/S1eYHoC5FX/S1eYHoC5FX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14859 |
+
parse/train/B1e4wo09K7/B1e4wo09K7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14860 |
+
parse/train/B1e4wo09K7/B1e4wo09K7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14861 |
+
parse/train/B1e4wo09K7/B1e4wo09K7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14862 |
+
parse/train/SJc1hL5ee/SJc1hL5ee_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14863 |
+
parse/train/SJc1hL5ee/SJc1hL5ee_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14864 |
+
parse/train/SJc1hL5ee/SJc1hL5ee_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14865 |
+
parse/train/3T9iFICe0Y9/3T9iFICe0Y9_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14866 |
+
parse/train/3T9iFICe0Y9/3T9iFICe0Y9_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14867 |
+
parse/train/3T9iFICe0Y9/3T9iFICe0Y9_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14868 |
+
parse/train/BkjLkSqxg/BkjLkSqxg_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14869 |
+
parse/train/BkjLkSqxg/BkjLkSqxg_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14870 |
+
parse/train/BkjLkSqxg/BkjLkSqxg_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14871 |
+
parse/train/PxTIG12RRHS/PxTIG12RRHS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14872 |
+
parse/train/PxTIG12RRHS/PxTIG12RRHS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14873 |
+
parse/train/PxTIG12RRHS/PxTIG12RRHS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14874 |
+
parse/train/S1PWi_lC-/S1PWi_lC-_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14875 |
+
parse/train/S1PWi_lC-/S1PWi_lC-_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14876 |
+
parse/train/S1PWi_lC-/S1PWi_lC-_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14877 |
+
parse/train/ry0WOxbRZ/ry0WOxbRZ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14878 |
+
parse/train/ry0WOxbRZ/ry0WOxbRZ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14879 |
+
parse/train/ry0WOxbRZ/ry0WOxbRZ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14880 |
+
parse/train/FGqiDsBUKL0/FGqiDsBUKL0_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14881 |
+
parse/train/FGqiDsBUKL0/FGqiDsBUKL0_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14882 |
+
parse/train/FGqiDsBUKL0/FGqiDsBUKL0_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14883 |
+
parse/train/5lhWG3Hj2By/5lhWG3Hj2By_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14884 |
+
parse/train/5lhWG3Hj2By/5lhWG3Hj2By_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 14885 |
+
parse/train/5lhWG3Hj2By/5lhWG3Hj2By_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
parse/dev/-NOQJw5z_KY/-NOQJw5z_KY.md
ADDED
|
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|
| 1 |
+
# Semantic Exploration from Language Abstractions and Pretrained Representations
|
| 2 |
+
|
| 3 |
+
Allison C. Tam
|
| 4 |
+
DeepMind
|
| 5 |
+
London, UK
|
| 6 |
+
actam@deepmind.com
|
| 7 |
+
Neil C. Rabinowitz
|
| 8 |
+
DeepMind
|
| 9 |
+
London, UK
|
| 10 |
+
ncr@deepmind.com
|
| 11 |
+
Andrew K. Lampinen
|
| 12 |
+
DeepMind
|
| 13 |
+
London, UK
|
| 14 |
+
lampinen@deepmind.com
|
| 15 |
+
Nicholas A. Roy
|
| 16 |
+
DeepMind
|
| 17 |
+
London, UK
|
| 18 |
+
nroy@deepmind.com
|
| 19 |
+
Stephanie C. Y. Chan
|
| 20 |
+
DeepMind
|
| 21 |
+
London, UK
|
| 22 |
+
scychan@deepmind.com
|
| 23 |
+
DJ Strouse
|
| 24 |
+
DeepMind
|
| 25 |
+
London, UK
|
| 26 |
+
strouse@deepmind.com
|
| 27 |
+
Jane X. Wang⇤
|
| 28 |
+
DeepMind
|
| 29 |
+
London, UK
|
| 30 |
+
wangjane@deepmind.com
|
| 31 |
+
Andrea Banino⇤
|
| 32 |
+
DeepMind
|
| 33 |
+
London, UK
|
| 34 |
+
abanino@deepmind.com
|
| 35 |
+
|
| 36 |
+
Felix Hill⇤ DeepMind London, UK felixhill@deepmind.com
|
| 37 |
+
|
| 38 |
+
# Abstract
|
| 39 |
+
|
| 40 |
+
Effective exploration is a challenge in reinforcement learning (RL). Novelty-based exploration methods can suffer in high-dimensional state spaces, such as continuous partially-observable 3D environments. We address this challenge by defining novelty using semantically meaningful state abstractions, which can be found in learned representations shaped by natural language. In particular, we evaluate vision-language representations, pretrained on natural image captioning datasets. We show that these pretrained representations drive meaningful, task-relevant exploration and improve performance on 3D simulated environments. We also characterize why and how language provides useful abstractions for exploration by considering the impacts of using representations from a pretrained model, a language oracle, and several ablations. We demonstrate the benefits of our approach with on- and off-policy RL algorithms and in two very different task domains— one that stresses the identification and manipulation of everyday objects, and one that requires navigational exploration in an expansive world. Our results suggest that using language-shaped representations could improve exploration for various algorithms and agents in challenging environments.
|
| 41 |
+
|
| 42 |
+
# 1 Introduction
|
| 43 |
+
|
| 44 |
+
Exploration is one of the central challenges of reinforcement learning (RL). A popular way to increase an agent’s tendency to explore is to augment trajectories with intrinsic rewards for reaching novel environment states. However, the success of this approach depends critically on which states are considered novel, which can in turn depend on how environment states are represented.
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The literature on novelty-driven exploration describes several approaches to deriving state representations $\mathbb { I } \mathbb { I }$ . One popular method employs random features and represents the state by embedding the visual observation with a fixed, randomly initialized target network [Random Network Distillation; $\boxed { 6 }$ . Another method uses learned visual features, taken from an inverse dynamics model [Never Give Up; $\textcircled { 3 }$ . These approaches work well in classic 2D environments like Atari, but it is less clear whether they are as effective in high-dimensional, partially-observable settings such as 3D environments. For instance, in 3D settings, different viewpoints of the same scene may map to distinct visual states/features, despite being semantically similar. The difficulty of identifying a good mapping between visual state and feature space is exacerbated by the fact that useful state abstractions are highly task dependent. For example, a task involving tool use requires object-affordance abstractions, whereas navigation does not. Thus, acquiring state representations that support effective exploration is a chicken-and-egg problem—knowing whether two states should be considered similar requires the type of understanding that an agent can only acquire after effectively exploring its environment.
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To overcome these challenges, we propose giving agents access to prior knowledge during training, in the form of abstractions derived from large vision-language models [e.g. 41] that are pretrained on image captioning data. We use these pretrained models to derive a intrinsic reward that reflects meaningful novelty. We hypothesize that representations acquired by vision-language pretraining drive effective, semantic exploration in 3D environments, because the representations are shaped by the unique abstract nature of natural language.
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Several aspects of natural language suggest that it could be useful to direct novelty-based exploration. First, language is inherently abstract: language links superficially distinct, but causally-related situations by describing them similarly, and contrasts between causally-distinct states by describing them differently, thus outlining useful concepts $\overline { { \mathbb { B } \mathcal { 9 } } } \overline { { \mathbb { B } \mathcal { 8 } } } \Vert$ . Second, humans use language to communicate important information efficiently, without overspecifying [20, 21]. Thus, human language omits distracting irrelevant information and focuses on important aspects of the world. For example, it is often observed that an agent rewarded for seeking novel experience would be attracted forever to a TV with uncontrollable and unpredictable random static [7]. However, a human would likely caption this scene “a TV with no signal” regardless of the particular pattern; thus an agent ex
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Figure 1: Navy dashed lines delineate semantically meaningful states. By using representations that align well with these boundaries (i.e. language), then agents more effectively explore the wider state space (orange trajectory). If the representations do not reflect these boundaries and instead are amenable to visual noise (i.e. different colors, viewpoints, etc.), then agents may only focus on a visually novel, yet narrow subset of states (red trajectory).
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ploring with language abstractions would quickly leave the TV behind. Figure 1 shows another conceptual example of how language abstractions can accelerate exploration.
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We first perform motivating proof-of-concept experiments using a language oracle. We show that language is a useful abstraction for exploration not only because it coarsens the state space, but also because it coarsens the state space in a way that reflects the semantics of the environment. We then demonstrate that our results scale to environments without a language oracle using pretrained vision encoders, which are only supervised with language during pretraining. This work strives to enhance the representations used in novelty-based exploration, rather than compare various exploration methods.
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We consider two popular novelty-based exploration methods from the literature, Never Give Up (NGU; Badia et al. $\pmb { \mathbb { B } } \mathbf { \mathbb { I } }$ ) and Random Network Distillation (RND; Burda et al. $\mathbb { I I } ^ { \dagger }$ ), and compare them to their language-augmented variants, Lang-NGU/LSE-NGU and Lang-RND. We evaluate performance and sample efficiency on object manipulation, search, and navigation tasks in two challenging 3D environments simulated in Unity: Playroom (a house containing toys and furniture) and City (a large-scale urban setting). Our results show that language-based exploration with pretrained visionlanguage representations improves sample efficiency on Playroom tasks by $1 8 - 7 0 \%$ . It also doubles
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(b) Example instances of $O _ { V }$ and $O _ { L }$ from City. Many different scenes can be associated with the same caption.
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Figure 2: Visual observations from the environment and example captions generated by the language oracle. Appendix Figure $\boxed { \mathsf { S 4 } }$ contains more example captions.
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the visited areas in City, compared to baseline methods. We show that language-based exploration is effective for both on-policy (IMPALA $\mathbb { I I I }$ ) and off-policy (R2D2 $\pm \mathbb { Z } 5 \mathbb { I } .$ ) agents.
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# 2 Related Work
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Exploration in RL Classical exploration strategies include $\epsilon$ -greedy action selection $\pmb { \Vert 5 \bot }$ , statecounting [49, 4, 32, 30, 3], curiosity driven exploration $\pm \sharp$ , and intrinsic motivation methods $\left[ \left[ 3 6 \right] \right]$ . Our work is part of this last class of methods, where the agent is given an intrinsic reward for visiting diverse states over time $\begin{array} { r l } { { \bigl [ \bigl | 3 5 \bigr | \bigr ] } } \end{array}$ . Intrinsic rewards can be derived from various measures: novelty [43, 55, 6, 56], prediction-error [38, 3], ensemble disagreement [11, 39, 48, 46, 18, 50], or information gain $\mathbb { \left[ \left. 2 3 \right] \right. }$ . One family of methods gives intrinsic reward for following a curriculum of goals [8, 12, 40]. Others use novelty measures to identify interesting states from which they can perform additional learning [16, 54]. These methods encourage exploration in different ways, but they all rely on visual state representations that are learned jointly with the policy. Although we focus on novelty-based intrinsic reward and demonstrate the benefits of language in NGU and RND, our methodology is relatively agnostic to the exploration method. We suggest that many other exploration methods could be improved by using language abstractions and pretrained embeddings to represent the state space.
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Pretraining representations for RL Pretraining has been used in RL to improve the representations of the policy network. Self-supervised representation learning techniques distill knowledge from external datasets to produce downstream features that are helpful in virtual environments [15, 53]. Some recent work shows benefits from pretraining on more general, large-scale datasets. Pretrained CLIP features have been used in a number of recent robotics papers to speed up control and navigation tasks. These features can condition the policy network $\bar { \lVert 2 6 rVert }$ , or can be fused throughout the visual encoder to integrate semantic information about the environment $\pmb { \Vert 3 7 } \Vert$ . The goal of these works is to improve perception in the policy. Pretrained language models can also provide useful initializations for training policies to imitate offline trajectories $[ \bar { 1 } 2 , \bar { 1 } 2 7 ]$ . These successes demonstrate that large pretrained models contain prior knowledge that can be useful for RL. While the existing literature uses pretrained embeddings directly in the agent, we instead allow the policy network to learn from scratchm and only utilize pretrained embeddings to guide exploration during training (Figure $^ { \mathbf { \boldsymbol { S 2 } } ) }$ We imagine that future work may benefit from combining both approaches.
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Language for exploration Some recent works have used language to guide agent learning, by either using language subgoals for exploration/planning or providing task-specific reward shaping [47, 33, 13, 19]. Schwartz et al. [45] use a custom semantic parser for VizDoom and show that representing states with language, rather than vision, leads to faster learning by simplifying policy inputs. Chaplot et al. $\mathbb { \ m }$ tackle navigation in 3D by constructing a semantic map of the environment from pretrained SLAM modules, language-defined object categories, and agent location. This approach lends itself to navigation, but it is unclear how it would extend easily to more generic settings or other types of tasks, such as manipulation. Work concurrent to ours by Mu et al. [34] shows how language, in the form of hand-crafted BabyAI annotations, can help improve exploration in 2D environments. These works demonstrate the value of language abstractions: the ability to ignore extraneous noise and highlight important environment features. However, these prior methods rely on environment-specific semantic parsers or annotations, which may limit the settings to which they can be applied. In contrast, by exploiting powerful pretrained vision-language models, our approach can be applied to any visually-naturalistic environment, including 3D settings, which have not been widely studied in prior exploration work. We additionally do not require any language from the environment itself. Our method could even potentially improve exploration for physical robots, but we leave that for future work.
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# 3 Method
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We consider a goal-conditioned Markov decision process defined by a tuple $( S , \mathcal { A } , \mathcal { G } , P , R _ { e } , \gamma )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { G }$ is the goal space, $P : \mathcal { S } \times \mathcal { A } \mathcal { S }$ specifies the environment dynamics, $R _ { e } : \mathcal { S } \times \mathcal { G } R _ { e }$ is the extrinsic reward, and $\gamma$ is the discount factor. State $\mathbf { s _ { t } }$ is presented to the agent as a visual observation $O _ { V }$ . In some cases, in order to calculate intrinsic reward, we use a language oracle $\mathcal { O } : \mathcal { S } \mathcal { L }$ that provides natural language descriptions of the state, $O _ { L }$ . Note that $O _ { L }$ is distinct from the language instruction $g \in { \mathcal { G } }$ , which is sampled from a goal distribution at the start of an episode—the agent never observes $O _ { L }$ . We later remove the need for a language oracle by using pretrained models.
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We use goal-conditioned reinforcement learning to produce a policy imizes the expected re $\pi _ { g } ( \cdot \ | \ O _ { V } )$ $\mathbb { E } [ \sum _ { t = 0 } ^ { H } \gamma ^ { t } ( r _ { t } ^ { e } +$ $\beta r _ { t } ^ { i } ) ]$ , where $H$ is the horizon, $\boldsymbol { r } _ { t } ^ { e }$ is the extrinsic reward, $r _ { t } ^ { i }$ is the intrinsic reward, and $\beta$ is a tuned hyperparameter. The intrinsic reward is goal-agnostic and is computed with access to either $O _ { V }$ or $O _ { L }$ . Note that neither $O _ { L }$ nor pretrained embeddings are used by the policy, and thus we only use them during training to compute the intrinsic reward (Figure $\textcircled { 3 }$
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Our approach builds on two popular exploration algorithms: Never Give Up (NGU; Badia et al. $\pmb { \mathbb { B } } \mathbf { l }$ ) and Random Network Distillation (RND; Burda et al. $\mathbb { H }$ ). These algorithms were chosen to demonstrate the value of language under two different exploration paradigms. While both methods reward visiting novel states, they differ on several dimensions: the novelty horizon (episodic versus lifetime), how the history of past visited states is retained (non-parametric versus parametric), and how states are represented (learned controllable states versus random features).
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Figure 3: The agent is trained from scratch using RL to optimize extrinsic and intrinsic reward. It acts using the image observation $O _ { V }$ and goal $g$ . During training, the novelty-based intrinsic reward is calculated using an auxiliary component that does not share parameters with the agent (dashed box). The auxiliary component may incorporate a pretrained language (pictured above) or image encoder, which may respectively require $O _ { L }$ or $O _ { V }$ . The latter does not rely on language provided by the environment. See Figure S2 for more details.
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# 3.1 Never Give Up (NGU)
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To more clearly isolate our impact, we focus only on the episodic novelty component of the NGU agent [3]. State representations along the trajectory are written to a non-parametric episodic memory buffer. The intrinsic reward reflects how novel the current state is relative to the states visited so far in the episode. Novelty is a function of the L2 distances between the current state and the $k$ -nearest neighbor representations stored in the memory buffer. Intrinsic reward is higher for larger distances.
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Full details can be found in the original paper; however, we make two key simplifications. While Badia et al. $\pmb { \mathbb { B } } \|$ proposes learning a family of policy networks that are capable of different levels of exploration, we train one policy network that maximizes reward $r ~ = ~ r _ { e } + \beta r _ { i }$ for a fixed hyperparameter $\beta$ . We also fix the long-term novelty modulator $\alpha$ to be 1, essentially removing it.
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The published baseline method, which we refer to as Vis-NGU, uses a controllable state taken from an inverse dynamics model. The inverse dynamics model is trained jointly with the policy, but the two networks do not share any parameters.
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Table 1: Summary of NGU variants.
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<table><tr><td>Name</td><td>Embedding Type</td><td>Required Input</td></tr><tr><td>Vis-NGU</td><td>Controllable State</td><td>Vision</td></tr><tr><td rowspan="3">Lang-NGU</td><td>BERT</td><td>Language</td></tr><tr><td>CLIPtext</td><td>Language</td></tr><tr><td>ALMtext</td><td>Language</td></tr><tr><td rowspan="3">LSE-NGU</td><td>CLIPimage</td><td>Vision</td></tr><tr><td>ALMimage</td><td>Vision</td></tr><tr><td></td><td></td></tr></table>
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Table 2: Summary of family of RND-inspired methods. Intrinsic reward is derived from the prediction error between the trainable network and frozen target function.
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<table><tr><td>Name</td><td>Trainable Network</td><td>Target Function</td></tr><tr><td>Vis-RND</td><td>fv:Ov→Rk</td><td>randomly initialized,fixed f</td></tr><tr><td>ND</td><td>f{v,L} :O{v,L}→Rk</td><td>pretrained ALM{image, text}</td></tr><tr><td>Lang-RND</td><td>fL:OL→R</td><td>randomly initialized,fixed f</td></tr><tr><td>LD</td><td>fc:Ov→OL</td><td>OL from language oracle</td></tr></table>
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The intrinsic reward relies on directly comparing state representations from the buffer, so our approach focuses on modifying the embedding function to influence exploration (Table 1). LangNGU uses a frozen pretrained language encoder to embed the oracle caption $O _ { L }$ . We compare language embeddings from BERT $\check { \mathbb { E } } \check { 4 } \check { \mathbb { I } }$ , CLIP [41], Small-ALM, and Med-ALM. The ALMs (ALign Models) are trained with a contrastive loss on the ALIGN dataset $[ [ 2 4 ]$ . Small-ALM uses a 26M parameter ResNet-50 image encoder $\pmb { \mathbb { Z } } 2 \mathbf { l }$ ; Med-ALM uses a 71M parameter NFNet [5]. The language backbones are based on BERT and are all in the range of 70-90M parameters. We do not finetune on environment-specific data; this preserves the real world knowledge acquired during pretraining and demonstrates its benefit without requiring any environment-specific captions.
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LSE-NGU does not use the language oracle. Instead, it uses a frozen pretrained image encoder to embed the visual observation $O _ { V }$ . We use the image encoder from CLIP or ALM, which are trained on captioning datasets to produce outputs that are close to the corresponding language embeddings. The human-generated captions structure the visual embedding space to reflect features most pertinent to humans and human language $\pmb { \mathbb { B } } \mathbf { \mathbb { 1 } }$ , so the resulting representations can be thought of as Language Supervised Embeddings (LSE). The primary benefit of LSE-NGU is that it can be applied to environments without a language oracle or annotations. CLIP and ALM are trained on real-world data, so they would work best on realistic 3D environments. However, we imagine that in future work the pretraining process or dataset could be tailored to maximize transfer to a desired target environment.
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# 3.2 Random Network Distillation (RND)
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Our RND-inspired family of methods rewards lifetime novelty. Generically, the intrinsic reward is derived from the prediction error between a trainable network and some target value generated by a frozen function (Table $\bigstar \bigstar$ . The trainable network is learned jointly with the policy network, although they do not share any parameters. As the agent trains over the course of its lifetime, the prediction error for frequently-visited states decreases, and the associated intrinsic reward consequently diminishes. Intuitively, the weights of the trainable network implicitly store the state visitation counts.
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For clarity, we refer to the baseline published by Burda et al. $\mathbb { \left[ \bigcirc \right] }$ as Vis-RND. The trainable network $f _ { V } : O _ { V } { \stackrel { \cdot } { \to } } \mathbb { R } ^ { k }$ maps the visual state to random features. The random features are produced by a fixed, randomly initialized network $\hat { f _ { V } }$ . Both $f _ { V }$ and $\hat { f _ { V } }$ share the same architecture: a ResNet followed by a MLP. The intrinsic reward is the mean squared error $\lVert f _ { V } ( O _ { V } ) - \hat { f _ { V } } ( O _ { V } ) \rVert ^ { 2 }$ .
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In network distillation (ND), the target function is not random, but is instead a pretrained text or image encoder from CLIP/ALM. The trainable network $f$ learns to reproduce the pretrained representations. To manage inference time, $f$ is a simpler network than the target (see Appendix A.2). The intrinsic loss is the mean squared error between $f$ and the large pretrained network. Like the respective Lang-NGU and LSE-NGU counterparts, text-based ND requires a language oracle, but image-based ND does not.
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In Section $5 . 1$ we compare against two additional methods to motivate why language is a useful abstraction. The first, Lang-RND, is a variant in which the trainable network $f _ { L } : O _ { L } \stackrel { \smile } { \to } \mathbb { R } ^ { k }$ maps the oracle caption to random features. The intrinsic reward is the mean squared error between the outputs of $f _ { L }$ and fixed $\hat { f } _ { L }$ with random initialization. Both $f _ { L }$ and $\hat { f } _ { L }$ networks are of the same architecture.
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The second method, language distillation $\mathbf { \left( L D \right) }$ , is loosely inspired by RND in that the novelty signal comes from a prediction error. However, instead of learning to produce random features, the trainable network learns to caption the visual state, i.e. $f _ { C } : O _ { V } \to O _ { L }$ . The network architecture consists of a CNN encoder and LSTM decoder. The intrinsic reward is the negative log-likelihood of the oracle caption under the trainable model $f _ { C }$ . In LD, the exploration dynamics not only depend on how frequently states are visited but also the alignment between language and the visual world. We test whether this caption-denoted alignment is necessary for directing semantic exploration by comparing LD to a variant with shuffled image-language alignment (S-LD) in Section 5.1.
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# 4 Experimental Setup
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# 4.1 Environments
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Previous exploration work benchmarked algorithms on video games, such as 2D grid-world MiniHack and Montezuma’s Revenge, or 3D first-person shooter Vizdoom. In this paper, we focus on first-person Unity-based 3D environments that are meant to mimic familiar scenes from the real world (Figure 2).
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Playroom Our first domain, Playroom [1, 52], is a randomly-generated house containing everyday household items (e.g. bed, bathtub, tables, chairs, toys). The agent’s action set consists of 46 discrete actions that involve locomotion primitives and object manipulation, such as holding and rotating.
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We study two settings in Playroom. In the first setting, the agent is confined to a single room with 3-5 objects and is given a lift or put instruction. At the start of an episode, the set of objects are sampled from a larger set of everyday objects (i.e. a candle, cup, hairdryer). Object colors and sizes are also randomized, adding superficial variance to different semantic categories. The instructions take the form: "Lift a <object>" or "Put a <object> on a {bed, tray}". With a lift goal, the episode ends with reward 1 or 0 whenever any object is lifted. With a put goal, the episode ends with reward 1 when the condition is fulfilled. This setting tests spatial rearrangement skills.
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In the second setting, the agent is placed in a house with 3-5 different rooms, and is given a find instruction of the form "Find a {teddy bear, rubber duck}". Every episode, the house is randomly generated with the teddy and duck hidden amongst many objects, furniture, and decorations. The target objects can appear in any room— either on the floor, on top of tables, or inside bookshelves. The agent is randomly initialized and can travel throughout the house and freely rearrange objects. The episode ends with reward 1 when the agent pauses in front of the desired object. The find task requires navigation/search skills and tests the ability to ignore the numerous distractor objects.
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City Our second domain, City, is an expansive, large-scale urban environment. Each episode, a new map is generated; shops, parks, and buildings are randomly arranged in city blocks. Daylight is simulated, such that the episode starts during the morning and ends at nighttime. The agent is randomly initialized and is instructed to “explore the city.” It is trained to maximize its intrinsic reward and can take the following actions: move_{forward,backward,left,right}, look_{left,right}, and move_forward_and_look_{left,right}. We divide up the map into a $3 2 \times 3 2$ grid and track how many unique bins are visited in an episode.
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Additionally, City does not provide explicit visual or verbal signage to disambiguate locations. As such, systematic exploration is needed to maximize coverage. In contrast to Playroom, City tests long horizon exploration. A Playroom episode lasts only 600 timesteps, whereas a City episode lasts 11,250 and requires hundreds of timesteps to fully traverse the map even once. The City covers a 270-by-270 meter square area, which models a 2-by-2 grid of real world blocks.
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# 4.2 Captioning Engine
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We equip the environment with a language oracle that generates language descriptions of the scene, $O _ { L }$ , based on the Unity state, $s$ (Figure $2$ ). In Playroom, the caption describes if and how the agent interacts with objects and lists what is currently visible to it. In City, $O _ { L }$ generally describes the object that the agent is directly looking at, but the captions alone do not disambiguate the agent’s locations. Since these captions are generated from a Unity state, these descriptions may not be as varied or rich as a human’s, but they can be generated accurately and reliably, and at scale.
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# 4.3 Training Details
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At test time, the agent receives image observation $O _ { V }$ and language-specified goal $g$ . The policy network never requires caption $O _ { L }$ to act. During training, the exploration method calculates the intrinsic reward from $O _ { L }$ or $O _ { V }$ .
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We show that language-based exploration is compatible with both policy gradient and Q-learning algorithms. We use Impala $\mathbb { \equiv } \mathbb { \ln { \frac { } { } } }$ on Playroom and R2D2 on City $\lVert 2 5 \rVert$ . Q-learning is more suitable for the City, because the action space is more restricted compared to the one needed for Playroom tasks.
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For both environments, the agent architecture consists of an image ResNet encoder and a language LSTM encoder that feed into a memory LSTM module. The policy and value heads are MLPs that receive the memory state as input. If the exploration method requires additional networks, such as the trainable network in RND or inverse dynamics model in NGU, they do not share any parameters with the policy or value networks. Figure $\dot { \bf S } \boldsymbol { 2 }$ is a visualization of an Impala agent that uses languageaugmented exploration. Hyperparameters and additional details are found in Appendix A.
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# 5 Results
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# 5.1 Motivation: Language is a Meaningful Abstraction
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(b) LD outperforms S-LD. It is important how language abstractions carve up the state space.
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Figure 4: Our comparisons demonstrate that language is useful for exploration, because it outlines a more abstract, semantically-meaningful state space. Results are shown with a $9 5 \%$ confidence band.
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We share the intuition with other work [e.g. 34] that language can improve exploration. We design a set of experiments to show how and why this may be the case. Our analysis follows the desiderata outlined by Burda et al. [6]—prediction-error exploration ought to use a feature space that filters irrelevant information (compact) and contains necessary information (sufficient). Burda et al. [6] specifically studies RND and notes that the random feature space, the outputs of the random network, may not fully satisfy either condition. As such, we use the language variants of RND to frame this discussion.
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We hypothesize that language abstractions are useful, because they (1) create a coarser state space and (2) divide the state space in a way that meaningfully aligns with the world (i.e. using semantics). First, if language provides a coarser state space, then the random feature space becomes more compact, leading to better exploration. We compare Lang-RND to Vis-RND to test this claim. Lang-RND learns the lift task $33 \%$ faster and solves the put task as Vis-RND starts to learn (Figure 4a).
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Second, we ask whether semantics – that is how language divides up the state space – is critical for effective exploration. We use LD to test this hypothesis, precisely because the exploration in LD is motivated by modeling the semantic relationship between language and vision.
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We compare LD to a shuffled variant SLD, where we replace the particular semantic state abstraction that language offers with a statistically-matched randomized abstraction (Figure $\boxed { 5 }$ ). S-LD is similar to LD; the intrinsic reward is the prediction error of the captioning network. However, instead of targeting the language oracle output, the S-LD trainable network produces a different target caption O that may not match the image. $\widetilde { O _ { L } }$ is produced by a fixed, random mapping ${ \hat { f } } _ { S } : O _ { V } \{ \substack { \widetilde { O _ { L } } }$ . $\hat { f } _ { S }$ is constrained such that the marginal distributions $P ( O _ { L } ) \approx P ( \widetilde { O _ { L } } )$ are matched under trajectories produced by policy $\pi _ { L D }$ . See Appendix A.4 for full details on the construction of S-LD.
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Thus, whereas the LD captions parcel up state space in a way that reflects the abstractions that language offers, the randomized mapping $\hat { f } _ { S }$
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Figure 5: The dotted lines correspond to state abstractions given by the shuffled $\hat { f } _ { S }$ used in S-LD. The states are grouped together based on similarities in the visual random feature space and assigned a label. Exploring in this shuffled space is less effective than exploring with the semanticallymeaningful abstractions shown in Figure 1.
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parcels up state space in a way that abstracts over random features of the visual space (Figure 5) We control for the compactness and coarseness of the resulting representation by maintaining the same marginal distribution of captions.
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If semantics is crucial for exploration, then we expect to see LD outperform S-LD. This indeed holds experimentally (Figure $\textcircled { 4 6 }$ . We can also view these results under the Burda et al. [6] framework. The S-LD abstractions group together visually similar, but semantically distinct states. A single sampled caption likely fails to capture the group in a manner that is representative of all the encompassing states. In other words, $\hat { f } _ { S }$ produces a compact feature space that may not be sufficient. This may explain why S-LD learns faster than Vis-RND on the simpler lift task but fails on the more complex put and find tasks. The S-LD experiments imply that language abstractions are helpful for exploration because they expose not only a more compact, but also a more semantically meaningful state space.
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# 5.2 Pretrained Vision-Language Representations Improve Exploration
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Having shown how language can be helpful for exploration, we now incorporate pretrained visionlanguage representations into NGU and RND to improve exploration. Such representations (e.g. from the image encoder in CLIP/ALM) offer the benefits of explicit language abstractions, without the need to rely on a language oracle. We also compare language-shaped representations to pretrained ImageNet embeddings to isolate the effect of language. To keep the number of experiments tractable, we only perform a full comparison on the Playroom tasks.
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City We first compare how representations affect performance in a pure exploration setting. With no extrinsic reward, the agent is motivated solely by the NGU intrinsic reward to explore the City. We report how many unique areas the agent visits in an episode in Figure $\triangledown$ While optimizing coverage only requires knowledge of an agent’s global location rather than generic scene understanding, vision-language representations are still useful simply because meaningful exploration is inherently semantic. Lang-NGU, which uses text embeddings of $O _ { L }$ , visits an area up to 3 times larger. LSENGU achieves 2 times the coverage even without querying a language oracle (Appendix Figure $S 5 )$
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Playroom We next show that pretrained vision-language representations significantly speed up learning across all Playroom tasks (Figure $\textcircled{7}$ . The LSE-NGU and Lang-NGU agents improve sample efficiency by $50 \%$ on the lift and put tasks and $1 8 - 3 8 \%$ on the find task, depending on the pretraining model used. The ND agents are significantly faster than VisRND, learning $41 \%$ faster on the find task. We also measure agent-object interactions. Nearly all LSE-NGU and Lang-NGU agents learn to foveate on and hold objects within 40k learning updates, whereas Vis-NGU agent takes at least $6 0 \mathrm { k }$ updates to do so with the same frequency (Appendix Figure $\textcircled { 5 7 }$ Although LSENGU and image-based ND agents do not access a language oracle, they are similarly effective as their annotation-dependent counterparts in the Playroom tasks (Appendix Figure $\dot { \overline { { \vert \mathrm { S } 6 \vert } } }$ , suggesting that our method could be robust to the availability of a language oracle.
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To demonstrate the value of rich language, we compare LSE-NGU agents to a control agent that instead uses pretrained ImageNet embeddings from a 70M NFNet [5]. ImageNet embeddings optimize for single-object classification, so they confer some benefit to the most objectfocused tasks, lift and put. However, Ima
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Figure 6: Coverage of City (number of bins reached on map) by NGU variants using different state representations for exploration, normalized by coverage of a ground-truth agent. The groundtruth agent represents state in NGU as the global coordinate of the agent location. The dashed line indicates coverage of a uniform random policy. Error bars indicate standard error of the mean, over 5 replicas. See Appendix Table S4 for absolute coverage numbers.
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geNet embeddings hurt exploration in the find task, where agents encounters more complex scenes (Figure $^ { 7 \mathrm { b } ) }$ . By contrast, the language-shaped representations are well-suited for not only describing simple objects, but also have capacity for multi-object, complex scenes. Of course, current CLIPstyle models can be further improved in their ability to understand multi-object scenes, which may explain why the benefits are less pronounced for the find task. However, as the performance of pretrained vision-language models improve, we expect to see those benefits transfer to this method and drive even better exploration.
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# 6 Discussion
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We have shown that language abstractions and pretrained vision-language representations improve the sample efficiency of existing exploration methods. This benefit is seen across on-policy and off-policy algorithms (Impala and R2D2), different exploration methods (RND and NGU), different 3D domains (Playroom and City), and various task specifications (lifting/putting, searching, and intrinsically motivated navigation). Furthermore, we carefully designed control experiments to understand how language contributes to better exploration. Our results are consistent with cognitive perspectives on human language—language is powerful because it groups together situations according to semantic similarity. In terms of the desiderata that Burda et al. [6] present, language is both compact and sufficient. Finally, we note that using pretrained vision-language representations to embed image observations enables more effective exploration even if language is not available during agent training. This is vital for scaling to environments that do not have a language oracle or annotations.
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Limitations and future directions We highlight several avenues for extending our work. First, additional research could provide a more comprehensive understanding of how language abstractions affect representations. This could include comparing different types of captions offering varying levels of detail, or task-dependent descriptions. These captions could be dynamically generated at scale by prompting a large multimodal model $\left[ \left[ 2 \right] \right]$ . Second, it would be useful to investigate how to improve pretrained vision-language representations for exploration by finetuning on relevant datasets. The semantics of a dataset could even be tailored to task-specific abstractions to increase the quality of the learnt representations. Such approaches would potentially allow applying our method to virtual environments that are farther from the pretraining distribution, such as Atari. In contrast, compared to our experiments, we believe that the current pretrained representations would deliver even more benefit for entirely photorealistic, visually rich environments, such as Matterport3D [9]. Finally, we note that a limitation of this approach is that current pretrained vision-language models may be less effective on multi-object scenes. Future pretraining innovations or larger models would presumably produce more robust representations and thus lead to even more effective exploration.
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(c) ND intrinsic rewards derive from the prediction error of the representations from a pretrained ALM network.
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Figure 7: Agents that use pretrained language-shaped representations to explore (ALM-ND, LangNGU, LSE-NGU) learn faster than baseline agents. ALM-ND (Text/Image) refer to the ND variants in Table 2. Results shown with a $9 5 \%$ confidence interval.
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# Acknowledgments and Disclosure of Funding
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We would like to thank Iain Barr for ALM models and Nathaniel Wong and Arthur Brussee for the Playroom environment. For the City environment, we would like to thank Nick Young, Tom Hudson, Alex Platonov, Bethanie Brownfield, Sarah Chakera, Dario de Cesare, Marjorie Limont, Benigno Uria, Borja Ibarz and Charles Blundell. Moreover, for the City, we would like to extend our special thanks to Jayd Matthias, Jason Sanmiya, Marcus Wainwright, Max Cant and the rest of the Worlds Team. Finally, we thank Hamza Merzic, Andre Saraiva, and Tim Scholtes for their helpful support and advice.
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Semantic Exploration from Language Abstractions and Pretrained Representations ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
227,
|
| 8 |
+
122,
|
| 9 |
+
771,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
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"text": "Allison C. Tam \nDeepMind \nLondon, UK \nactam@deepmind.com \nNeil C. Rabinowitz \nDeepMind \nLondon, UK \nncr@deepmind.com \nAndrew K. Lampinen \nDeepMind \nLondon, UK \nlampinen@deepmind.com \nNicholas A. Roy \nDeepMind \nLondon, UK \nnroy@deepmind.com \nStephanie C. Y. Chan \nDeepMind \nLondon, UK \nscychan@deepmind.com \nDJ Strouse \nDeepMind \nLondon, UK \nstrouse@deepmind.com \nJane X. Wang⇤ \nDeepMind \nLondon, UK \nwangjane@deepmind.com \nAndrea Banino⇤ \nDeepMind \nLondon, UK \nabanino@deepmind.com ",
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"text": "Felix Hill⇤ DeepMind London, UK felixhill@deepmind.com ",
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"text": "Abstract ",
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"text": "Effective exploration is a challenge in reinforcement learning (RL). Novelty-based exploration methods can suffer in high-dimensional state spaces, such as continuous partially-observable 3D environments. We address this challenge by defining novelty using semantically meaningful state abstractions, which can be found in learned representations shaped by natural language. In particular, we evaluate vision-language representations, pretrained on natural image captioning datasets. We show that these pretrained representations drive meaningful, task-relevant exploration and improve performance on 3D simulated environments. We also characterize why and how language provides useful abstractions for exploration by considering the impacts of using representations from a pretrained model, a language oracle, and several ablations. We demonstrate the benefits of our approach with on- and off-policy RL algorithms and in two very different task domains— one that stresses the identification and manipulation of everyday objects, and one that requires navigational exploration in an expansive world. Our results suggest that using language-shaped representations could improve exploration for various algorithms and agents in challenging environments. ",
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"text": "1 Introduction ",
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"text": "Exploration is one of the central challenges of reinforcement learning (RL). A popular way to increase an agent’s tendency to explore is to augment trajectories with intrinsic rewards for reaching novel environment states. However, the success of this approach depends critically on which states are considered novel, which can in turn depend on how environment states are represented. ",
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"text": "The literature on novelty-driven exploration describes several approaches to deriving state representations $\\mathbb { I } \\mathbb { I }$ . One popular method employs random features and represents the state by embedding the visual observation with a fixed, randomly initialized target network [Random Network Distillation; $\\boxed { 6 }$ . Another method uses learned visual features, taken from an inverse dynamics model [Never Give Up; $\\textcircled { 3 }$ . These approaches work well in classic 2D environments like Atari, but it is less clear whether they are as effective in high-dimensional, partially-observable settings such as 3D environments. For instance, in 3D settings, different viewpoints of the same scene may map to distinct visual states/features, despite being semantically similar. The difficulty of identifying a good mapping between visual state and feature space is exacerbated by the fact that useful state abstractions are highly task dependent. For example, a task involving tool use requires object-affordance abstractions, whereas navigation does not. Thus, acquiring state representations that support effective exploration is a chicken-and-egg problem—knowing whether two states should be considered similar requires the type of understanding that an agent can only acquire after effectively exploring its environment. ",
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"text": "To overcome these challenges, we propose giving agents access to prior knowledge during training, in the form of abstractions derived from large vision-language models [e.g. 41] that are pretrained on image captioning data. We use these pretrained models to derive a intrinsic reward that reflects meaningful novelty. We hypothesize that representations acquired by vision-language pretraining drive effective, semantic exploration in 3D environments, because the representations are shaped by the unique abstract nature of natural language. ",
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"text": "Several aspects of natural language suggest that it could be useful to direct novelty-based exploration. First, language is inherently abstract: language links superficially distinct, but causally-related situations by describing them similarly, and contrasts between causally-distinct states by describing them differently, thus outlining useful concepts $\\overline { { \\mathbb { B } \\mathcal { 9 } } } \\overline { { \\mathbb { B } \\mathcal { 8 } } } \\Vert$ . Second, humans use language to communicate important information efficiently, without overspecifying [20, 21]. Thus, human language omits distracting irrelevant information and focuses on important aspects of the world. For example, it is often observed that an agent rewarded for seeking novel experience would be attracted forever to a TV with uncontrollable and unpredictable random static [7]. However, a human would likely caption this scene “a TV with no signal” regardless of the particular pattern; thus an agent ex",
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"type": "image",
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"img_path": "images/92626be4d32e49ea3cb2d5fd609d0e4efd4bace02c6c7dd642547205afd2be97.jpg",
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"image_caption": [
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"Figure 1: Navy dashed lines delineate semantically meaningful states. By using representations that align well with these boundaries (i.e. language), then agents more effectively explore the wider state space (orange trajectory). If the representations do not reflect these boundaries and instead are amenable to visual noise (i.e. different colors, viewpoints, etc.), then agents may only focus on a visually novel, yet narrow subset of states (red trajectory). "
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"text": "ploring with language abstractions would quickly leave the TV behind. Figure 1 shows another conceptual example of how language abstractions can accelerate exploration. ",
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"text": "We first perform motivating proof-of-concept experiments using a language oracle. We show that language is a useful abstraction for exploration not only because it coarsens the state space, but also because it coarsens the state space in a way that reflects the semantics of the environment. We then demonstrate that our results scale to environments without a language oracle using pretrained vision encoders, which are only supervised with language during pretraining. This work strives to enhance the representations used in novelty-based exploration, rather than compare various exploration methods. ",
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"text": "We consider two popular novelty-based exploration methods from the literature, Never Give Up (NGU; Badia et al. $\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { I } }$ ) and Random Network Distillation (RND; Burda et al. $\\mathbb { I I } ^ { \\dagger }$ ), and compare them to their language-augmented variants, Lang-NGU/LSE-NGU and Lang-RND. We evaluate performance and sample efficiency on object manipulation, search, and navigation tasks in two challenging 3D environments simulated in Unity: Playroom (a house containing toys and furniture) and City (a large-scale urban setting). Our results show that language-based exploration with pretrained visionlanguage representations improves sample efficiency on Playroom tasks by $1 8 - 7 0 \\%$ . It also doubles ",
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"image_caption": [
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"(b) Example instances of $O _ { V }$ and $O _ { L }$ from City. Many different scenes can be associated with the same caption. "
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"text": "Figure 2: Visual observations from the environment and example captions generated by the language oracle. Appendix Figure $\\boxed { \\mathsf { S 4 } }$ contains more example captions. ",
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"text": "the visited areas in City, compared to baseline methods. We show that language-based exploration is effective for both on-policy (IMPALA $\\mathbb { I I I }$ ) and off-policy (R2D2 $\\pm \\mathbb { Z } 5 \\mathbb { I } .$ ) agents. ",
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"text": "2 Related Work ",
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"text": "Exploration in RL Classical exploration strategies include $\\epsilon$ -greedy action selection $\\pmb { \\Vert 5 \\bot }$ , statecounting [49, 4, 32, 30, 3], curiosity driven exploration $\\pm \\sharp$ , and intrinsic motivation methods $\\left[ \\left[ 3 6 \\right] \\right]$ . Our work is part of this last class of methods, where the agent is given an intrinsic reward for visiting diverse states over time $\\begin{array} { r l } { { \\bigl [ \\bigl | 3 5 \\bigr | \\bigr ] } } \\end{array}$ . Intrinsic rewards can be derived from various measures: novelty [43, 55, 6, 56], prediction-error [38, 3], ensemble disagreement [11, 39, 48, 46, 18, 50], or information gain $\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }$ . One family of methods gives intrinsic reward for following a curriculum of goals [8, 12, 40]. Others use novelty measures to identify interesting states from which they can perform additional learning [16, 54]. These methods encourage exploration in different ways, but they all rely on visual state representations that are learned jointly with the policy. Although we focus on novelty-based intrinsic reward and demonstrate the benefits of language in NGU and RND, our methodology is relatively agnostic to the exploration method. We suggest that many other exploration methods could be improved by using language abstractions and pretrained embeddings to represent the state space. ",
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"text": "Pretraining representations for RL Pretraining has been used in RL to improve the representations of the policy network. Self-supervised representation learning techniques distill knowledge from external datasets to produce downstream features that are helpful in virtual environments [15, 53]. Some recent work shows benefits from pretraining on more general, large-scale datasets. Pretrained CLIP features have been used in a number of recent robotics papers to speed up control and navigation tasks. These features can condition the policy network $\\bar { \\lVert 2 6 rVert }$ , or can be fused throughout the visual encoder to integrate semantic information about the environment $\\pmb { \\Vert 3 7 } \\Vert$ . The goal of these works is to improve perception in the policy. Pretrained language models can also provide useful initializations for training policies to imitate offline trajectories $[ \\bar { 1 } 2 , \\bar { 1 } 2 7 ]$ . These successes demonstrate that large pretrained models contain prior knowledge that can be useful for RL. While the existing literature uses pretrained embeddings directly in the agent, we instead allow the policy network to learn from scratchm and only utilize pretrained embeddings to guide exploration during training (Figure $^ { \\mathbf { \\boldsymbol { S 2 } } ) }$ We imagine that future work may benefit from combining both approaches. ",
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"text": "Language for exploration Some recent works have used language to guide agent learning, by either using language subgoals for exploration/planning or providing task-specific reward shaping [47, 33, 13, 19]. Schwartz et al. [45] use a custom semantic parser for VizDoom and show that representing states with language, rather than vision, leads to faster learning by simplifying policy inputs. Chaplot et al. $\\mathbb { \\ m }$ tackle navigation in 3D by constructing a semantic map of the environment from pretrained SLAM modules, language-defined object categories, and agent location. This approach lends itself to navigation, but it is unclear how it would extend easily to more generic settings or other types of tasks, such as manipulation. Work concurrent to ours by Mu et al. [34] shows how language, in the form of hand-crafted BabyAI annotations, can help improve exploration in 2D environments. These works demonstrate the value of language abstractions: the ability to ignore extraneous noise and highlight important environment features. However, these prior methods rely on environment-specific semantic parsers or annotations, which may limit the settings to which they can be applied. In contrast, by exploiting powerful pretrained vision-language models, our approach can be applied to any visually-naturalistic environment, including 3D settings, which have not been widely studied in prior exploration work. We additionally do not require any language from the environment itself. Our method could even potentially improve exploration for physical robots, but we leave that for future work. ",
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"text": "3 Method ",
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"text": "We consider a goal-conditioned Markov decision process defined by a tuple $( S , \\mathcal { A } , \\mathcal { G } , P , R _ { e } , \\gamma )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $\\mathcal { G }$ is the goal space, $P : \\mathcal { S } \\times \\mathcal { A } \\mathcal { S }$ specifies the environment dynamics, $R _ { e } : \\mathcal { S } \\times \\mathcal { G } R _ { e }$ is the extrinsic reward, and $\\gamma$ is the discount factor. State $\\mathbf { s _ { t } }$ is presented to the agent as a visual observation $O _ { V }$ . In some cases, in order to calculate intrinsic reward, we use a language oracle $\\mathcal { O } : \\mathcal { S } \\mathcal { L }$ that provides natural language descriptions of the state, $O _ { L }$ . Note that $O _ { L }$ is distinct from the language instruction $g \\in { \\mathcal { G } }$ , which is sampled from a goal distribution at the start of an episode—the agent never observes $O _ { L }$ . We later remove the need for a language oracle by using pretrained models. ",
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| 348 |
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"type": "text",
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| 358 |
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"text": "We use goal-conditioned reinforcement learning to produce a policy imizes the expected re $\\pi _ { g } ( \\cdot \\ | \\ O _ { V } )$ $\\mathbb { E } [ \\sum _ { t = 0 } ^ { H } \\gamma ^ { t } ( r _ { t } ^ { e } +$ $\\beta r _ { t } ^ { i } ) ]$ , where $H$ is the horizon, $\\boldsymbol { r } _ { t } ^ { e }$ is the extrinsic reward, $r _ { t } ^ { i }$ is the intrinsic reward, and $\\beta$ is a tuned hyperparameter. The intrinsic reward is goal-agnostic and is computed with access to either $O _ { V }$ or $O _ { L }$ . Note that neither $O _ { L }$ nor pretrained embeddings are used by the policy, and thus we only use them during training to compute the intrinsic reward (Figure $\\textcircled { 3 }$ ",
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"type": "text",
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"text": "Our approach builds on two popular exploration algorithms: Never Give Up (NGU; Badia et al. $\\pmb { \\mathbb { B } } \\mathbf { l }$ ) and Random Network Distillation (RND; Burda et al. $\\mathbb { H }$ ). These algorithms were chosen to demonstrate the value of language under two different exploration paradigms. While both methods reward visiting novel states, they differ on several dimensions: the novelty horizon (episodic versus lifetime), how the history of past visited states is retained (non-parametric versus parametric), and how states are represented (learned controllable states versus random features). ",
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"type": "image",
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"img_path": "images/885a1efefd0fd983dcbbc79e2507cb34d9c2ec38adcba084a69fc31b5c336be5.jpg",
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| 381 |
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"image_caption": [
|
| 382 |
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"Figure 3: The agent is trained from scratch using RL to optimize extrinsic and intrinsic reward. It acts using the image observation $O _ { V }$ and goal $g$ . During training, the novelty-based intrinsic reward is calculated using an auxiliary component that does not share parameters with the agent (dashed box). The auxiliary component may incorporate a pretrained language (pictured above) or image encoder, which may respectively require $O _ { L }$ or $O _ { V }$ . The latter does not rely on language provided by the environment. See Figure S2 for more details. "
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| 394 |
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"type": "text",
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| 395 |
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"text": "3.1 Never Give Up (NGU) ",
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| 396 |
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"text_level": 1,
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"type": "text",
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"text": "To more clearly isolate our impact, we focus only on the episodic novelty component of the NGU agent [3]. State representations along the trajectory are written to a non-parametric episodic memory buffer. The intrinsic reward reflects how novel the current state is relative to the states visited so far in the episode. Novelty is a function of the L2 distances between the current state and the $k$ -nearest neighbor representations stored in the memory buffer. Intrinsic reward is higher for larger distances. ",
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"type": "text",
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"text": "",
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| 419 |
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"type": "text",
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| 429 |
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"text": "Full details can be found in the original paper; however, we make two key simplifications. While Badia et al. $\\pmb { \\mathbb { B } } \\|$ proposes learning a family of policy networks that are capable of different levels of exploration, we train one policy network that maximizes reward $r ~ = ~ r _ { e } + \\beta r _ { i }$ for a fixed hyperparameter $\\beta$ . We also fix the long-term novelty modulator $\\alpha$ to be 1, essentially removing it. ",
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"bbox": [
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"type": "text",
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"text": "The published baseline method, which we refer to as Vis-NGU, uses a controllable state taken from an inverse dynamics model. The inverse dynamics model is trained jointly with the policy, but the two networks do not share any parameters. ",
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| 441 |
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{
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| 450 |
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"type": "table",
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"img_path": "images/1453e03317c00f1e9bc6f5afe49742e1dba6439e41cc30c786a7bf00a139aed5.jpg",
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| 452 |
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"table_caption": [
|
| 453 |
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"Table 1: Summary of NGU variants. "
|
| 454 |
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],
|
| 455 |
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"table_footnote": [],
|
| 456 |
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"table_body": "<table><tr><td>Name</td><td>Embedding Type</td><td>Required Input</td></tr><tr><td>Vis-NGU</td><td>Controllable State</td><td>Vision</td></tr><tr><td rowspan=\"3\">Lang-NGU</td><td>BERT</td><td>Language</td></tr><tr><td>CLIPtext</td><td>Language</td></tr><tr><td>ALMtext</td><td>Language</td></tr><tr><td rowspan=\"3\">LSE-NGU</td><td>CLIPimage</td><td>Vision</td></tr><tr><td>ALMimage</td><td>Vision</td></tr><tr><td></td><td></td></tr></table>",
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| 460 |
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"type": "table",
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"img_path": "images/6123e09bf595a2d3cb6b5ebf16f1253af7de97be3c9d91a1f5a1a8dd867ec37a.jpg",
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"table_caption": [
|
| 469 |
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"Table 2: Summary of family of RND-inspired methods. Intrinsic reward is derived from the prediction error between the trainable network and frozen target function. "
|
| 470 |
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],
|
| 471 |
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"table_footnote": [],
|
| 472 |
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"table_body": "<table><tr><td>Name</td><td>Trainable Network</td><td>Target Function</td></tr><tr><td>Vis-RND</td><td>fv:Ov→Rk</td><td>randomly initialized,fixed f</td></tr><tr><td>ND</td><td>f{v,L} :O{v,L}→Rk</td><td>pretrained ALM{image, text}</td></tr><tr><td>Lang-RND</td><td>fL:OL→R</td><td>randomly initialized,fixed f</td></tr><tr><td>LD</td><td>fc:Ov→OL</td><td>OL from language oracle</td></tr></table>",
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| 473 |
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"bbox": [
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| 482 |
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"type": "text",
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| 483 |
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"text": "The intrinsic reward relies on directly comparing state representations from the buffer, so our approach focuses on modifying the embedding function to influence exploration (Table 1). LangNGU uses a frozen pretrained language encoder to embed the oracle caption $O _ { L }$ . We compare language embeddings from BERT $\\check { \\mathbb { E } } \\check { 4 } \\check { \\mathbb { I } }$ , CLIP [41], Small-ALM, and Med-ALM. The ALMs (ALign Models) are trained with a contrastive loss on the ALIGN dataset $[ [ 2 4 ]$ . Small-ALM uses a 26M parameter ResNet-50 image encoder $\\pmb { \\mathbb { Z } } 2 \\mathbf { l }$ ; Med-ALM uses a 71M parameter NFNet [5]. The language backbones are based on BERT and are all in the range of 70-90M parameters. We do not finetune on environment-specific data; this preserves the real world knowledge acquired during pretraining and demonstrates its benefit without requiring any environment-specific captions. ",
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| 484 |
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| 490 |
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"page_idx": 4
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| 491 |
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| 492 |
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{
|
| 493 |
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"type": "text",
|
| 494 |
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"text": "LSE-NGU does not use the language oracle. Instead, it uses a frozen pretrained image encoder to embed the visual observation $O _ { V }$ . We use the image encoder from CLIP or ALM, which are trained on captioning datasets to produce outputs that are close to the corresponding language embeddings. The human-generated captions structure the visual embedding space to reflect features most pertinent to humans and human language $\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }$ , so the resulting representations can be thought of as Language Supervised Embeddings (LSE). The primary benefit of LSE-NGU is that it can be applied to environments without a language oracle or annotations. CLIP and ALM are trained on real-world data, so they would work best on realistic 3D environments. However, we imagine that in future work the pretraining process or dataset could be tailored to maximize transfer to a desired target environment. ",
|
| 495 |
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| 502 |
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| 503 |
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{
|
| 504 |
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"type": "text",
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| 505 |
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"text": "3.2 Random Network Distillation (RND) ",
|
| 506 |
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"text_level": 1,
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| 507 |
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| 513 |
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| 515 |
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| 516 |
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"type": "text",
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| 517 |
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"text": "Our RND-inspired family of methods rewards lifetime novelty. Generically, the intrinsic reward is derived from the prediction error between a trainable network and some target value generated by a frozen function (Table $\\bigstar \\bigstar$ . The trainable network is learned jointly with the policy network, although they do not share any parameters. As the agent trains over the course of its lifetime, the prediction error for frequently-visited states decreases, and the associated intrinsic reward consequently diminishes. Intuitively, the weights of the trainable network implicitly store the state visitation counts. ",
|
| 518 |
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| 526 |
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| 527 |
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"type": "text",
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| 528 |
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"text": "For clarity, we refer to the baseline published by Burda et al. $\\mathbb { \\left[ \\bigcirc \\right] }$ as Vis-RND. The trainable network $f _ { V } : O _ { V } { \\stackrel { \\cdot } { \\to } } \\mathbb { R } ^ { k }$ maps the visual state to random features. The random features are produced by a fixed, randomly initialized network $\\hat { f _ { V } }$ . Both $f _ { V }$ and $\\hat { f _ { V } }$ share the same architecture: a ResNet followed by a MLP. The intrinsic reward is the mean squared error $\\lVert f _ { V } ( O _ { V } ) - \\hat { f _ { V } } ( O _ { V } ) \\rVert ^ { 2 }$ . ",
|
| 529 |
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"bbox": [
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| 532 |
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| 535 |
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| 536 |
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| 537 |
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{
|
| 538 |
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"type": "text",
|
| 539 |
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"text": "In network distillation (ND), the target function is not random, but is instead a pretrained text or image encoder from CLIP/ALM. The trainable network $f$ learns to reproduce the pretrained representations. To manage inference time, $f$ is a simpler network than the target (see Appendix A.2). The intrinsic loss is the mean squared error between $f$ and the large pretrained network. Like the respective Lang-NGU and LSE-NGU counterparts, text-based ND requires a language oracle, but image-based ND does not. ",
|
| 540 |
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| 546 |
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"page_idx": 4
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| 547 |
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|
| 548 |
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{
|
| 549 |
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"type": "text",
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| 550 |
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"text": "In Section $5 . 1$ we compare against two additional methods to motivate why language is a useful abstraction. The first, Lang-RND, is a variant in which the trainable network $f _ { L } : O _ { L } \\stackrel { \\smile } { \\to } \\mathbb { R } ^ { k }$ maps the oracle caption to random features. The intrinsic reward is the mean squared error between the outputs of $f _ { L }$ and fixed $\\hat { f } _ { L }$ with random initialization. Both $f _ { L }$ and $\\hat { f } _ { L }$ networks are of the same architecture. ",
|
| 551 |
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|
| 557 |
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| 558 |
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|
| 559 |
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{
|
| 560 |
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"type": "text",
|
| 561 |
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"text": "The second method, language distillation $\\mathbf { \\left( L D \\right) }$ , is loosely inspired by RND in that the novelty signal comes from a prediction error. However, instead of learning to produce random features, the trainable network learns to caption the visual state, i.e. $f _ { C } : O _ { V } \\to O _ { L }$ . The network architecture consists of a CNN encoder and LSTM decoder. The intrinsic reward is the negative log-likelihood of the oracle caption under the trainable model $f _ { C }$ . In LD, the exploration dynamics not only depend on how frequently states are visited but also the alignment between language and the visual world. We test whether this caption-denoted alignment is necessary for directing semantic exploration by comparing LD to a variant with shuffled image-language alignment (S-LD) in Section 5.1. ",
|
| 562 |
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| 569 |
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|
| 570 |
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{
|
| 571 |
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"type": "text",
|
| 572 |
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"text": "4 Experimental Setup ",
|
| 573 |
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"text_level": 1,
|
| 574 |
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| 581 |
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|
| 582 |
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{
|
| 583 |
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"type": "text",
|
| 584 |
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"text": "4.1 Environments ",
|
| 585 |
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"text_level": 1,
|
| 586 |
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|
| 587 |
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| 592 |
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| 593 |
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|
| 594 |
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{
|
| 595 |
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"type": "text",
|
| 596 |
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"text": "Previous exploration work benchmarked algorithms on video games, such as 2D grid-world MiniHack and Montezuma’s Revenge, or 3D first-person shooter Vizdoom. In this paper, we focus on first-person Unity-based 3D environments that are meant to mimic familiar scenes from the real world (Figure 2). ",
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| 597 |
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|
| 604 |
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|
| 605 |
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{
|
| 606 |
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"type": "text",
|
| 607 |
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"text": "Playroom Our first domain, Playroom [1, 52], is a randomly-generated house containing everyday household items (e.g. bed, bathtub, tables, chairs, toys). The agent’s action set consists of 46 discrete actions that involve locomotion primitives and object manipulation, such as holding and rotating. ",
|
| 608 |
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"bbox": [
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| 610 |
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| 611 |
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| 614 |
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| 615 |
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|
| 616 |
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{
|
| 617 |
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"type": "text",
|
| 618 |
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"text": "We study two settings in Playroom. In the first setting, the agent is confined to a single room with 3-5 objects and is given a lift or put instruction. At the start of an episode, the set of objects are sampled from a larger set of everyday objects (i.e. a candle, cup, hairdryer). Object colors and sizes are also randomized, adding superficial variance to different semantic categories. The instructions take the form: \"Lift a <object>\" or \"Put a <object> on a {bed, tray}\". With a lift goal, the episode ends with reward 1 or 0 whenever any object is lifted. With a put goal, the episode ends with reward 1 when the condition is fulfilled. This setting tests spatial rearrangement skills. ",
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| 619 |
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| 625 |
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| 626 |
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| 627 |
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{
|
| 628 |
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"type": "text",
|
| 629 |
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"text": "In the second setting, the agent is placed in a house with 3-5 different rooms, and is given a find instruction of the form \"Find a {teddy bear, rubber duck}\". Every episode, the house is randomly generated with the teddy and duck hidden amongst many objects, furniture, and decorations. The target objects can appear in any room— either on the floor, on top of tables, or inside bookshelves. The agent is randomly initialized and can travel throughout the house and freely rearrange objects. The episode ends with reward 1 when the agent pauses in front of the desired object. The find task requires navigation/search skills and tests the ability to ignore the numerous distractor objects. ",
|
| 630 |
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| 637 |
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| 638 |
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{
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| 639 |
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"type": "text",
|
| 640 |
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"text": "City Our second domain, City, is an expansive, large-scale urban environment. Each episode, a new map is generated; shops, parks, and buildings are randomly arranged in city blocks. Daylight is simulated, such that the episode starts during the morning and ends at nighttime. The agent is randomly initialized and is instructed to “explore the city.” It is trained to maximize its intrinsic reward and can take the following actions: move_{forward,backward,left,right}, look_{left,right}, and move_forward_and_look_{left,right}. We divide up the map into a $3 2 \\times 3 2$ grid and track how many unique bins are visited in an episode. ",
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| 647 |
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"page_idx": 5
|
| 648 |
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},
|
| 649 |
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{
|
| 650 |
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"type": "text",
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| 651 |
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"text": "Additionally, City does not provide explicit visual or verbal signage to disambiguate locations. As such, systematic exploration is needed to maximize coverage. In contrast to Playroom, City tests long horizon exploration. A Playroom episode lasts only 600 timesteps, whereas a City episode lasts 11,250 and requires hundreds of timesteps to fully traverse the map even once. The City covers a 270-by-270 meter square area, which models a 2-by-2 grid of real world blocks. ",
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| 652 |
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"bbox": [
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"page_idx": 5
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},
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"type": "text",
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"text": "4.2 Captioning Engine ",
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"text_level": 1,
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"text": "We equip the environment with a language oracle that generates language descriptions of the scene, $O _ { L }$ , based on the Unity state, $s$ (Figure $2$ ). In Playroom, the caption describes if and how the agent interacts with objects and lists what is currently visible to it. In City, $O _ { L }$ generally describes the object that the agent is directly looking at, but the captions alone do not disambiguate the agent’s locations. Since these captions are generated from a Unity state, these descriptions may not be as varied or rich as a human’s, but they can be generated accurately and reliably, and at scale. ",
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"text": "",
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"type": "text",
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"text": "4.3 Training Details ",
|
| 697 |
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"text_level": 1,
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"bbox": [
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"type": "text",
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"text": "At test time, the agent receives image observation $O _ { V }$ and language-specified goal $g$ . The policy network never requires caption $O _ { L }$ to act. During training, the exploration method calculates the intrinsic reward from $O _ { L }$ or $O _ { V }$ . ",
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"bbox": [
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"type": "text",
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"text": "We show that language-based exploration is compatible with both policy gradient and Q-learning algorithms. We use Impala $\\mathbb { \\equiv } \\mathbb { \\ln { \\frac { } { } } }$ on Playroom and R2D2 on City $\\lVert 2 5 \\rVert$ . Q-learning is more suitable for the City, because the action space is more restricted compared to the one needed for Playroom tasks. ",
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"bbox": [
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"text": "For both environments, the agent architecture consists of an image ResNet encoder and a language LSTM encoder that feed into a memory LSTM module. The policy and value heads are MLPs that receive the memory state as input. If the exploration method requires additional networks, such as the trainable network in RND or inverse dynamics model in NGU, they do not share any parameters with the policy or value networks. Figure $\\dot { \\bf S } \\boldsymbol { 2 }$ is a visualization of an Impala agent that uses languageaugmented exploration. Hyperparameters and additional details are found in Appendix A. ",
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"type": "text",
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"text": "5 Results ",
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| 742 |
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"text_level": 1,
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"type": "text",
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"text": "5.1 Motivation: Language is a Meaningful Abstraction ",
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"img_path": "images/3af81a44807e4bd3399bcdb6165f4176b5c2807cbdc33ab478fdffea53c11575.jpg",
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"image_caption": [
|
| 767 |
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"(b) LD outperforms S-LD. It is important how language abstractions carve up the state space. ",
|
| 768 |
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"Figure 4: Our comparisons demonstrate that language is useful for exploration, because it outlines a more abstract, semantically-meaningful state space. Results are shown with a $9 5 \\%$ confidence band. "
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"type": "text",
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"text": "We share the intuition with other work [e.g. 34] that language can improve exploration. We design a set of experiments to show how and why this may be the case. Our analysis follows the desiderata outlined by Burda et al. [6]—prediction-error exploration ought to use a feature space that filters irrelevant information (compact) and contains necessary information (sufficient). Burda et al. [6] specifically studies RND and notes that the random feature space, the outputs of the random network, may not fully satisfy either condition. As such, we use the language variants of RND to frame this discussion. ",
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"text": "We hypothesize that language abstractions are useful, because they (1) create a coarser state space and (2) divide the state space in a way that meaningfully aligns with the world (i.e. using semantics). First, if language provides a coarser state space, then the random feature space becomes more compact, leading to better exploration. We compare Lang-RND to Vis-RND to test this claim. Lang-RND learns the lift task $33 \\%$ faster and solves the put task as Vis-RND starts to learn (Figure 4a). ",
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"type": "text",
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"text": "",
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"bbox": [
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"type": "text",
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"text": "Second, we ask whether semantics – that is how language divides up the state space – is critical for effective exploration. We use LD to test this hypothesis, precisely because the exploration in LD is motivated by modeling the semantic relationship between language and vision. ",
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"text": "We compare LD to a shuffled variant SLD, where we replace the particular semantic state abstraction that language offers with a statistically-matched randomized abstraction (Figure $\\boxed { 5 }$ ). S-LD is similar to LD; the intrinsic reward is the prediction error of the captioning network. However, instead of targeting the language oracle output, the S-LD trainable network produces a different target caption O that may not match the image. $\\widetilde { O _ { L } }$ is produced by a fixed, random mapping ${ \\hat { f } } _ { S } : O _ { V } \\{ \\substack { \\widetilde { O _ { L } } }$ . $\\hat { f } _ { S }$ is constrained such that the marginal distributions $P ( O _ { L } ) \\approx P ( \\widetilde { O _ { L } } )$ are matched under trajectories produced by policy $\\pi _ { L D }$ . See Appendix A.4 for full details on the construction of S-LD. ",
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"text": "Thus, whereas the LD captions parcel up state space in a way that reflects the abstractions that language offers, the randomized mapping $\\hat { f } _ { S }$ ",
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},
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{
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"type": "image",
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"img_path": "images/68b5a5d5a2ff8ba9bc909d7cb8de829c43193dd1ad18d2dd977b30d8cfb87bac.jpg",
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"image_caption": [
|
| 849 |
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"Figure 5: The dotted lines correspond to state abstractions given by the shuffled $\\hat { f } _ { S }$ used in S-LD. The states are grouped together based on similarities in the visual random feature space and assigned a label. Exploring in this shuffled space is less effective than exploring with the semanticallymeaningful abstractions shown in Figure 1. "
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],
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| 851 |
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|
| 852 |
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"bbox": [
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| 859 |
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},
|
| 860 |
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{
|
| 861 |
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"type": "text",
|
| 862 |
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"text": "parcels up state space in a way that abstracts over random features of the visual space (Figure 5) We control for the compactness and coarseness of the resulting representation by maintaining the same marginal distribution of captions. ",
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| 871 |
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|
| 872 |
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"type": "text",
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| 873 |
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"text": "If semantics is crucial for exploration, then we expect to see LD outperform S-LD. This indeed holds experimentally (Figure $\\textcircled { 4 6 }$ . We can also view these results under the Burda et al. [6] framework. The S-LD abstractions group together visually similar, but semantically distinct states. A single sampled caption likely fails to capture the group in a manner that is representative of all the encompassing states. In other words, $\\hat { f } _ { S }$ produces a compact feature space that may not be sufficient. This may explain why S-LD learns faster than Vis-RND on the simpler lift task but fails on the more complex put and find tasks. The S-LD experiments imply that language abstractions are helpful for exploration because they expose not only a more compact, but also a more semantically meaningful state space. ",
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{
|
| 883 |
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"type": "text",
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| 884 |
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"text": "5.2 Pretrained Vision-Language Representations Improve Exploration ",
|
| 885 |
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"text_level": 1,
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"type": "text",
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| 896 |
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"text": "Having shown how language can be helpful for exploration, we now incorporate pretrained visionlanguage representations into NGU and RND to improve exploration. Such representations (e.g. from the image encoder in CLIP/ALM) offer the benefits of explicit language abstractions, without the need to rely on a language oracle. We also compare language-shaped representations to pretrained ImageNet embeddings to isolate the effect of language. To keep the number of experiments tractable, we only perform a full comparison on the Playroom tasks. ",
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| 906 |
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"type": "text",
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| 907 |
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"text": "City We first compare how representations affect performance in a pure exploration setting. With no extrinsic reward, the agent is motivated solely by the NGU intrinsic reward to explore the City. We report how many unique areas the agent visits in an episode in Figure $\\triangledown$ While optimizing coverage only requires knowledge of an agent’s global location rather than generic scene understanding, vision-language representations are still useful simply because meaningful exploration is inherently semantic. Lang-NGU, which uses text embeddings of $O _ { L }$ , visits an area up to 3 times larger. LSENGU achieves 2 times the coverage even without querying a language oracle (Appendix Figure $S 5 )$ ",
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"type": "text",
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| 918 |
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"text": "Playroom We next show that pretrained vision-language representations significantly speed up learning across all Playroom tasks (Figure $\\textcircled{7}$ . The LSE-NGU and Lang-NGU agents improve sample efficiency by $50 \\%$ on the lift and put tasks and $1 8 - 3 8 \\%$ on the find task, depending on the pretraining model used. The ND agents are significantly faster than VisRND, learning $41 \\%$ faster on the find task. We also measure agent-object interactions. Nearly all LSE-NGU and Lang-NGU agents learn to foveate on and hold objects within 40k learning updates, whereas Vis-NGU agent takes at least $6 0 \\mathrm { k }$ updates to do so with the same frequency (Appendix Figure $\\textcircled { 5 7 }$ Although LSENGU and image-based ND agents do not access a language oracle, they are similarly effective as their annotation-dependent counterparts in the Playroom tasks (Appendix Figure $\\dot { \\overline { { \\vert \\mathrm { S } 6 \\vert } } }$ , suggesting that our method could be robust to the availability of a language oracle. ",
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| 928 |
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| 929 |
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"text": "To demonstrate the value of rich language, we compare LSE-NGU agents to a control agent that instead uses pretrained ImageNet embeddings from a 70M NFNet [5]. ImageNet embeddings optimize for single-object classification, so they confer some benefit to the most objectfocused tasks, lift and put. However, Ima",
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"page_idx": 8
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},
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| 938 |
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{
|
| 939 |
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"type": "image",
|
| 940 |
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"img_path": "images/4e331f124119c8758a744a811647c826b4c1af228b5af625adf124b50420c54b.jpg",
|
| 941 |
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"image_caption": [
|
| 942 |
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"Figure 6: Coverage of City (number of bins reached on map) by NGU variants using different state representations for exploration, normalized by coverage of a ground-truth agent. The groundtruth agent represents state in NGU as the global coordinate of the agent location. The dashed line indicates coverage of a uniform random policy. Error bars indicate standard error of the mean, over 5 replicas. See Appendix Table S4 for absolute coverage numbers. "
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| 943 |
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| 944 |
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| 945 |
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},
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| 953 |
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{
|
| 954 |
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"type": "text",
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| 955 |
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"text": "geNet embeddings hurt exploration in the find task, where agents encounters more complex scenes (Figure $^ { 7 \\mathrm { b } ) }$ . By contrast, the language-shaped representations are well-suited for not only describing simple objects, but also have capacity for multi-object, complex scenes. Of course, current CLIPstyle models can be further improved in their ability to understand multi-object scenes, which may explain why the benefits are less pronounced for the find task. However, as the performance of pretrained vision-language models improve, we expect to see those benefits transfer to this method and drive even better exploration. ",
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},
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| 965 |
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"type": "text",
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| 966 |
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"text": "6 Discussion ",
|
| 967 |
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"text_level": 1,
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| 968 |
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"text": "We have shown that language abstractions and pretrained vision-language representations improve the sample efficiency of existing exploration methods. This benefit is seen across on-policy and off-policy algorithms (Impala and R2D2), different exploration methods (RND and NGU), different 3D domains (Playroom and City), and various task specifications (lifting/putting, searching, and intrinsically motivated navigation). Furthermore, we carefully designed control experiments to understand how language contributes to better exploration. Our results are consistent with cognitive perspectives on human language—language is powerful because it groups together situations according to semantic similarity. In terms of the desiderata that Burda et al. [6] present, language is both compact and sufficient. Finally, we note that using pretrained vision-language representations to embed image observations enables more effective exploration even if language is not available during agent training. This is vital for scaling to environments that do not have a language oracle or annotations. ",
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"text": "Limitations and future directions We highlight several avenues for extending our work. First, additional research could provide a more comprehensive understanding of how language abstractions affect representations. This could include comparing different types of captions offering varying levels of detail, or task-dependent descriptions. These captions could be dynamically generated at scale by prompting a large multimodal model $\\left[ \\left[ 2 \\right] \\right]$ . Second, it would be useful to investigate how to improve pretrained vision-language representations for exploration by finetuning on relevant datasets. The semantics of a dataset could even be tailored to task-specific abstractions to increase the quality of the learnt representations. Such approaches would potentially allow applying our method to virtual environments that are farther from the pretraining distribution, such as Atari. In contrast, compared to our experiments, we believe that the current pretrained representations would deliver even more benefit for entirely photorealistic, visually rich environments, such as Matterport3D [9]. Finally, we note that a limitation of this approach is that current pretrained vision-language models may be less effective on multi-object scenes. Future pretraining innovations or larger models would presumably produce more robust representations and thus lead to even more effective exploration. ",
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"img_path": "images/53710fea8c99c91556580251a05b8374eb54910ded17dc54d98c4bda41723211.jpg",
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"",
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"(c) ND intrinsic rewards derive from the prediction error of the representations from a pretrained ALM network. ",
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| 1004 |
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"Figure 7: Agents that use pretrained language-shaped representations to explore (ALM-ND, LangNGU, LSE-NGU) learn faster than baseline agents. ALM-ND (Text/Image) refer to the ND variants in Table 2. Results shown with a $9 5 \\%$ confidence interval. "
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"text": "Acknowledgments and Disclosure of Funding ",
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"text": "We would like to thank Iain Barr for ALM models and Nathaniel Wong and Arthur Brussee for the Playroom environment. For the City environment, we would like to thank Nick Young, Tom Hudson, Alex Platonov, Bethanie Brownfield, Sarah Chakera, Dario de Cesare, Marjorie Limont, Benigno Uria, Borja Ibarz and Charles Blundell. Moreover, for the City, we would like to extend our special thanks to Jayd Matthias, Jason Sanmiya, Marcus Wainwright, Max Cant and the rest of the Worlds Team. Finally, we thank Hamza Merzic, Andre Saraiva, and Tim Scholtes for their helpful support and advice. ",
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"text": "References ",
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| 1 |
+
# SpeechGPT: Empowering Large Language Models with Intrinsic Cross-Modal Conversational Abilities
|
| 2 |
+
|
| 3 |
+
Dong Zhang, Shimin Li, Xin Zhang, Jun Zhan, Pengyu Wang, Yaqian Zhou∗, Xipeng Qiu∗ School of Computer Science, Fudan University
|
| 4 |
+
Shanghai Key Laboratory of Intelligent Information Processing, Fudan University
|
| 5 |
+
{dongzhang22,xin_zhang22,jzhan22,pywang22}@m.fudan.edu.cn {smli20,zhouyaqian,xpqiu}@fudan.edu.cn
|
| 6 |
+
|
| 7 |
+
# Abstract
|
| 8 |
+
|
| 9 |
+
Multi-modal large language models are regarded as a crucial step towards Artificial General Intelligence (AGI) and have garnered significant interest with the emergence of ChatGPT. However, current speech-language models typically adopt the cascade paradigm, preventing inter-modal knowledge transfer. In this paper, we propose SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. With discrete speech representations, we construct SpeechInstruct, the first large-scale crossmodal speech instruction dataset. Additionally, we employ a three-stage training strategy that includes modality-adaptation pretraining, cross-modal instruction fine-tuning, and chain-of-modality instruction fine-tuning. The experimental results demonstrate that SpeechGPT has an impressive capacity to follow cross-modal human instructions and highlight the potential of handling multiple modalities with one model. Code and models are available in https://github.com/ 0nutation/SpeechGPT. Demos are shown in https://0nutation.github. io/SpeechGPT.github.io/.
|
| 10 |
+
|
| 11 |
+
# 1 Introduction
|
| 12 |
+
|
| 13 |
+
Large language models (OpenAI, 2023; Touvron et al., 2023) have performed astonishingly on various natural language processing tasks. Meanwhile, multi-modal large language models, such as GPT4, PALM-E (Driess et al., 2023), and LLaVA (Liu et al., 2023), have explored the ability of LLMs to understand multi-modal information. However, a significant gap exists between current LLMs and general artificial intelligence (AGI). First, most current LLMs can only perceive and understand multimodal content but cannot spontaneously generate multi-modal content. Second, continuous signals like images and speech cannot be adapted directly to LLMs that receive discrete tokens.
|
| 14 |
+
|
| 15 |
+
|
| 16 |
+
Figure 1: SpeechGPT’s capabilities to tackle multiple cross-modal tasks.
|
| 17 |
+
|
| 18 |
+
The current speech-language model mainly adopts a cascading paradigm (Huang et al., 2023a) i.e., the LLM is connected with an automatic speech recognition (ASR) model or a text-tospeech (TTS) model in tandem, or the LLM is employed as a control hub, with several speech processing models (Cheng et al., 2023a,b,c) are integrated to cover multiple audio or speech tasks (Huang et al., 2023a; Shen et al., 2023). Some prior work on generative spoken language models involves encoding the speech signal into a discrete representation (Baevski et al., 2020; Hsu et al., 2021; Zhang et al., 2023a) and modeling it with language models (Lakhotia et al., 2021; Borsos et al., 2022; Zhang et al., 2023d; Wang et al., 2023; Zhang et al., 2023c).
|
| 19 |
+
|
| 20 |
+
While capable of perceiving and generating speech, the existing cascaded methods or spoken language models still have several limitations. First, the LLM in the cascaded model only functions as a content generator. Since the representations of speech and text are not aligned, the LLM’s knowledge cannot be transferred to the speech modality. Second, the cascade approach (Shen et al., 2023; Huang et al., 2023a) suffers from the loss of paralinguistic signals such as emotion and prosody. Third, existing spoken language models (Wang et al., 2023; Zhang et al., 2023d) only synthesize speech but fail to comprehend its semantic information, preventing them from achieving true crossmodal perception and generation.
|
| 21 |
+
|
| 22 |
+
In this paper, we propose SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. We perform speech discretization with a self-supervised trained speech model to unify the modality between speech and text. The discrete speech tokens are then expanded into the vocabulary of the LLM, thus endowing the model with an inherent competence to perceive and generate the speech.
|
| 23 |
+
|
| 24 |
+
To provide the model with the capacity to handle multi-modal instructions, we build the first speech-text cross-modal instruction-following dataset SpeechInstruct. Specifically, we discretize the speech to discrete units (Hsu et al., 2021) and construct the cross-modal unit-text pair based on the existing ASR dataset. Meanwhile, we construct hundreds of instructions for diverse tasks with GPT4 to simulate actual user instructions as illustrated in Appendix B. In addition, to further enhance the model’s cross-modal capability, we designed the Chain-of-Modality instruction data, i.e., the model receives the speech command, thinks about the process in text, and then outputs the response in speech.
|
| 25 |
+
|
| 26 |
+
For better cross-modal transfer and efficient training, SpeechGPT undergoes a three-stage training process: modality-adaptation pre-training, cross-modal instruction fine-tuning, and chain-ofmodality instruction fine-tuning. The first stage enables speech comprehension for SpeechGPT with the discrete speech unit continuation task. The second stage employs the SpeechInstruct to improve the model’s cross-modal capabilities. The third stage utilizes parameter-efficient LoRA (Hu et al., 2021) fine-tuning for further modality alignment.
|
| 27 |
+
|
| 28 |
+
To evaluate the effectiveness of SpeechGPT, we conduct a wide range of human evaluations and case analyses to estimate the performance of SpeechGPT on textual tasks, speech-text crossmodal tasks, and spoken dialogue tasks. The results demonstrate that SpeechGPT exhibits a strong ability for unimodal and cross-modal instruction following tasks.
|
| 29 |
+
|
| 30 |
+
Our contributions include the following:
|
| 31 |
+
|
| 32 |
+
• We build the first multi-modal large language model that can perceive and generate multi
|
| 33 |
+
|
| 34 |
+
modal contents.
|
| 35 |
+
• We construct and release SpeechInstruct, the first large-scale speech-text cross-modal instructionfollowing dataset.
|
| 36 |
+
• We build the first spoken dialogue LLM with strong human instruction following ability and spoken dialogue ability.
|
| 37 |
+
• We show great potential to incorporate other modalities into LLMs through discrete representations.
|
| 38 |
+
|
| 39 |
+
# 2 Related Work
|
| 40 |
+
|
| 41 |
+
Multi-modal Large Language Model Current multi-modal LLMs predominantly focus on the visual domain, feeding continuous representations obtained from pre-trained visual encoders into LLMs, facilitating full-parameter or parameterefficient training on visual-language data (OpenAI, 2023; Huang et al., 2023b; Zhang et al., 2023b). Palm-E (Driess et al., 2023) integrates the 540B PaLM (Chowdhery et al., 2022) and 22B Vision Transformer (Dosovitskiy et al., 2021) into the largest vision-language model. LLaVA (Liu et al., 2023) leverages pre-trained CLIP (Radford et al., 2021) visual encoder and LLaMA (Touvron et al., 2023) and conduct instruct tuning on GPT4- assisted visual instruction data. X-LLM (Chen et al., 2023) converts multi-modalities into representations with X2L interfaces as the inputs of the large language model. However, such structures only enable LLMs to process multi-modal input, without ability to generate multi-modal output. Diverging from prior studies, our approach emphasizes the development of a speech-centric multimodal LLM, endowing it with the proficiency to accommodate both multi-modal input and output.
|
| 42 |
+
|
| 43 |
+
Generative Spoken Language Model Discrete self-supervised representation based spoken generative language modeling is making remarkable progress on large-scale speech dataset training (Nguyen et al., 2022). AudioLM (Borsos et al., 2022) proposes to model speech based on audio codecs together with semantic codes, which can synthesize speech in a textlesss setting. VALLE (Wang et al., 2023) builds a generative spoken language model on audio codecs and treat Textto-Speech as a conditional generation task. However, these models are designed for a specific task and failed to benefit from LLMs. SpeechGPT is built upon the foundation of LLM and transfers LLM’s knowledge to speech modality, consequently obtaining better task generalization and human-instruction following ability.
|
| 44 |
+
|
| 45 |
+
Speech-Enabled LLM Interaction Following the emergence of ChatGPT, several studies have concentrated on the integration of expert speech models with LLMs to enable direct speech interaction with LLMs. HuggingGPT (Shen et al., 2023) facilitates task decomposition of human instructions by LLMs and allows the invocation of models from Huggingface to accomplish specific tasks, encompassing a range of automatic speech recognition (ASR) and text-to-speech models. AudioGPT (Huang et al., 2023a) leverages a variety of audio foundation models to process complex audio information and connect LLMs with input/output interface (ASR, TTS) for speech conversations. However, these models exhibit increased complexity, demand extensive resources, and are prone to the unavoidable error accumulation problems. Our approach enables speech interaction with LLMs without relying on ASR or TTS systems, circumventing the aforementioned drawbacks.
|
| 46 |
+
|
| 47 |
+
# 3 SpeechInstruct Construction
|
| 48 |
+
|
| 49 |
+
Due to the limitations in publicly available speech data and the lack of variety of speech-text tasks, we construct SpeechInstruct, a speech-text crossmodal instruction-following dataset. This dataset consists of two parts, the first part is called CrossModal Instruction, and the second part is called Chain-of-Modality Instruction. The construction process of SpeechInstruct is illustrated in Figure 2.
|
| 50 |
+
|
| 51 |
+
# 3.1 Cross-modal Instruction
|
| 52 |
+
|
| 53 |
+
Data Collection We collect several large-scale English ASR datasets to construct Cross-Modal Instruction, including Gigaspeech (Chen et al., 2021), Common Voice (Ardila et al., 2020), and LibriSpeech (Panayotov et al., 2015). We employ mHuBERT1 as the speech tokenizer to discretize speech data into discrete units and remove the repetitive units of adjacent frames to get reduced units. Ultimately, we obtain 9 million unit-text data pairs.
|
| 54 |
+
|
| 55 |
+
Task Description Generation We generate ASR and TTS task descriptions that are compatible with speech-text data pairs. Unlike the Self-Instruct method (Wang et al., 2022), we generate descriptions through a zero-shot approach. Specifically, we directly input the prompts shown in Appendix A into OpenAI GPT-4 to generate task descriptions. Our generation method yields 100 instructions for each task and some examples are shown in Appendix B.
|
| 56 |
+
|
| 57 |
+
Instruction Formatting For a discrete unit sequence $U$ and its associated transcription $T$ , we determine whether it will be used for constructing an ASR task or a TTS task based on the probability $p$ . Subsequently, we randomly select a description $D$ from the corresponding task description. This results in a triplet consisting of the task description, discrete unit sequence, and transcription, denoted as $( D , U , T )$ . Following this, the triplet is assembled into an instruction using the template: [Human]: $\{ D \}$ . This is input: $\{ U \}$ <eoh>.[SpeechGPT]: $\{ T \} { \ < } { \bf e } { \bf 0 } { \bf s } > .$ ..
|
| 58 |
+
|
| 59 |
+
# 3.2 Chain-of-Modality Instruction
|
| 60 |
+
|
| 61 |
+
Speech Instruction Generation Due to the lack of instruction data with speech input and speech output, we trained a text-to-unit generator to convert text instruction data into speech instruction data. Specifically, the text-to-unit generator adopts a Transformer encoder-decoder architecture. We trained it on LibriSpeech unit-text pairs in Crossmodal Instruction. We select 37,969 samples from the moss-002-sft-data dataset 2 whose response length is shorter than 35 words. And we convert both their instructions and responses into unit sequences through the text-to-unit generator. As a result, we obtained 37,969 quadruplets composed of speech instructions, text instructions, text responses, and speech responses, denoted as (SpeechI, T extI, T extR, SpeechR).
|
| 62 |
+
|
| 63 |
+
Instruction Formatting Using the above quadruplets, we could construct chain-of-thought style instructions for four input-output formats, namely Speech Instruction-Speech Response, Speech Instruction-Text Response, Text Instruction-Speech Response, and Text Instruction-Text Response. Their corresponding templates can be found in Appendix C.
|
| 64 |
+
|
| 65 |
+
# 3.3 SpeechInstruct Evaluation Set
|
| 66 |
+
|
| 67 |
+
We constructed cross-modal dialogue datasets under different scenarios to evaluate whether SpeechGPT could take on various roles. Specifically, these included a talking encyclopedia, personal assistant, chat partner, poet, psychologist, and educational assistant. For each role, we provide 10 manually authored instruction-response pairs written by ourselves. We use a pre-trained text-to-speech model 3 to convert the text into corresponding speech. We then employ mHuBERT to discretize speech data into discrete units as described in Section 3.1. Ultimately, for each role, we obtained 10 quadruplets composed of speech instructions, text instructions, text responses, and speech responses.
|
| 68 |
+
|
| 69 |
+
|
| 70 |
+
Figure 2: Left: An overview of SpeechInstruct construction process. The SpeechInstruct dataset consists of two parts: Cross-modal Instruction data and Chain-of-Modality Instruction data. T emplate1 is shown in 3.1. T emplate2 is shown in Appendix C. Right: An illustration of SpeechGPT model structure.
|
| 71 |
+
|
| 72 |
+
# 4 SpeechGPT
|
| 73 |
+
|
| 74 |
+
# 4.1 Model Structure
|
| 75 |
+
|
| 76 |
+
A unified framework is designed to provide architecture compatibility across different modalities. As shown in Figure 2, our model consists of three main components: discrete unit extractor, large language modal and unit vocoder. Under this architecture, LLM can perceive multi-modal inputs and generate multi-modal outputs.
|
| 77 |
+
|
| 78 |
+
Discrete Unit Extractor The discrete unit extractor utilizes the Hidden-unit BERT (HuBERT) model (Hsu et al., 2021) to transform continuous speech signals into a sequence of discrete units, . HuBERT is a self-supervised model that learns by predicting discrete labels for masked audio segments based on $\mathbf { k }$ -means clustering applied to the model’s intermediate representations. It features a combination of 1-D convolutional layers and a Transformer encoder to encode speech into continuous intermediate representations, with a kmeans model further converting these representations into a sequence of cluster indices. Subsequently, adjacent duplicate indices are removed, resulting in a discrete units sequence represented as $U = ( u _ { 1 } , u _ { 2 } , . . . , u _ { T } )$ , $u _ { i } \in { 0 , 1 , . . . , K - 1 }$ , $\forall 1 \leq i \leq T$ , with $K$ denoting the total number of clusters.
|
| 79 |
+
|
| 80 |
+
Large Language Model We employ the Meta AI LLaMA (Touvron et al., 2023) model as our Large Language Model. LLaMA comprises an embedding layer, multiple transformer blocks, and an LM head layer. The total number of parameters in LLaMA ranges from 7B to 65B. Drawing from an extensive training dataset of 1.0 trillion tokens, LLaMA demonstrates competitive performance compared to the substantially larger 175B GPT-3 across various NLP benchmarks.
|
| 81 |
+
|
| 82 |
+
Unit Vocoder Due to limition of single speaker unit vocoder in (Polyak et al., 2021), we train a multi-speaker unit HiFi-GAN to decode the speech signal from the discrete representation. The HiFiGAN architecture consists of a generator $\mathbf { G }$ and multiple discriminators D. The generator uses look-up tables (LUT) to embed discrete representations and the embedding sequences are up-sampled by a series of blocks composed of transposed convolution and a residual block with dilated layers. The speaker embedding is concatenated to each frame in the up-sampled sequence. The discriminator features a Multi-Period Discriminator (MPD) and a Multi-Scale Discriminator (MSD), which have the same architecture as (Polyak et al., 2021).
|
| 83 |
+
|
| 84 |
+
# 4.2 Training
|
| 85 |
+
|
| 86 |
+
To incorporate speech discrete representation into LLM, we expand the vocabulary and corresponding embedding matrix first. We divide the training process into three stages. The first stage is ModalityAdaptation Pre-training on unpaired speech data. The second stage is Cross-modal Instruction FineTuning. The third stage is Chain-of-Modality Instruction Fine-Tuning.
|
| 87 |
+
|
| 88 |
+
Expanding Vocabulary Given original LLM vocabulary $V$ of size $| V |$ , to integrate speech discrete representations into LLM, we expand the vocabulary with an additional set of unit tokens $V ^ { \prime }$ , of size $| V ^ { \prime } | = K$ . The expanded vocabulary $V ^ { \prime \prime }$ is the union of the original vocabulary $V$ and the new words $V ^ { \prime }$ :
|
| 89 |
+
|
| 90 |
+
$$
|
| 91 |
+
V ^ { \prime \prime } = V \cup V ^ { \prime }
|
| 92 |
+
$$
|
| 93 |
+
|
| 94 |
+
We denote the original word embedding matrix as $E \in \mathbb { R } ^ { | V | \times d }$ , where $d$ is the dimension of word embeddings. To accommodate the expanded vocabulary, we need to create a randomly initialized word embedding matrix $E ^ { \prime } \in \mathbb { R } ^ { | V ^ { \prime \prime } | \times d }$ . We preserve the original word embeddings by copying the values of $E$ to the first $| V |$ rows of $E ^ { \prime }$ :
|
| 95 |
+
|
| 96 |
+
$$
|
| 97 |
+
E ^ { \prime } [ 0 : | V | , : ] = E
|
| 98 |
+
$$
|
| 99 |
+
|
| 100 |
+
Finally, we replace the original vocabulary and word embedding matrix with the new vocabulary $V ^ { \prime \prime }$ and the word embedding matrix $E ^ { \prime }$ .
|
| 101 |
+
|
| 102 |
+
Stage 1: Modality-Adaptation Pre-training To enable LLM to handle discrete units modality, we utilize an unlabeled speech corpus to train LLM in a next-token prediction task. This approach aligns with the text pre-training objective of LLM. Given unlabeled speech corpus $C$ consisting of speech $U _ { 1 } , U _ { 2 } , \dots , U _ { m }$ and LLM denoted as $L _ { 1 }$ , the negative log-likelihood loss can be formulated as:
|
| 103 |
+
|
| 104 |
+
$$
|
| 105 |
+
\mathcal { L } ( L | C ) = - \sum _ { j = 1 } ^ { m } \sum _ { i = 1 } ^ { n _ { j } } \log P ( u _ { i , j } | u _ { < i , j } ; L )
|
| 106 |
+
$$
|
| 107 |
+
|
| 108 |
+
where $m$ is the number of speech in dataset $C$ , $n _ { j }$ is the number of discrete unit token in speech $U _ { j }$ , and $u _ { i , j }$ represents the i-th unit token in the $\mathrm { j } \cdot$ -th speech.
|
| 109 |
+
|
| 110 |
+
Stage 2: Cross-modal Instruction FineTuning In this stage, we align speech and text modalities utilizing paired data. We mix Crossmodal Instruction in SpeechInstruct with moss-002- sft dataset to derive mix dataset $I$ , which consists of samples $T _ { 1 } , T _ { 2 } , \dots , T _ { x }$ . We fine-tune the model $L$ obtained from the first stage on $I$ .
|
| 111 |
+
|
| 112 |
+
Each sample $T _ { j }$ consisting of $t _ { 1 } , t _ { 2 } , \ldots , t _ { n _ { j } }$ is formed by concatenating a prefix and a text. The training objective is to minimize the negative loglikelihood and the loss calculation only considers the text part, ignoring the prefix, which can be formated as:
|
| 113 |
+
|
| 114 |
+
$$
|
| 115 |
+
\mathcal { L } ( L | I ) = - \sum _ { j = 1 } ^ { x } \sum _ { i = p _ { j } + 1 } ^ { y _ { j } } \log P ( t _ { i , j } | t _ { < i , j } ; L )
|
| 116 |
+
$$
|
| 117 |
+
|
| 118 |
+
where $x$ is the number of samples in corpus $I$ , $y _ { j }$ is the total number of tokens in sample $T _ { j } , p _ { j }$ is the number of tokens in the prefix part of $T _ { j }$ , and $t _ { i , j }$ represents the i-th word in $T _ { j }$ .
|
| 119 |
+
|
| 120 |
+
Stage 3: Chain-of-Modality Instruction FineTuning After obtaining the model in stage 2, we utilizes parameter-efficient Low-Rank Adaptation (LoRA) (Hu et al., 2021) to fine-tune it on Chain-of-Modality Instruction in SpeechInstruct. We add LoRA weights (adapters) to the attention mechanisms and train the newly added LoRA parameters. We adopt the same loss function as stage 2.
|
| 121 |
+
|
| 122 |
+
# 5 Experiments
|
| 123 |
+
|
| 124 |
+
# 5.1 Experimental Setups
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Datasets For modality-adaption pre-training, we use LibriLight (Kahn et al., 2020) which contains 60K hours of unlabelled English audiobook speech. For cross-modal instruction fine-tuning stage, we use Gigaspeech (Chen et al., 2021), Common voice (Ardila et al., 2020) and LibriSpeech (Panayotov et al., 2015) dataset and moss-002-sft-data dataset, which is illustrated in detail in 3.1. For chain-of-modality instruction fine-tuning stage, we use moss-002-sft-data dataset, which is illustrated in detail in 3.2.
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Configuration We employ LLaMA-13B (Touvron et al., 2023) as our backbone model for a trade-off between performance and computational resources available. For stage 1, we use 96 A100 GPUs and train for 900 steps with batch size 768. For stage 2, we use 96 A100 GPUs and train for 2100 steps with batch size 1536. For stage 3, we use 8 A100 GPUs and train for 4200 steps with batch size 128.
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Details about training hyperparameters are shown in Appendix D. For decoding, we set the maximum sequence length to 2048 and set the temperature to 0.8. We use Top- $k$ sampling with $k { = } 6 0$ . We also use Top- $p$ sampling with $\mathrm { p { = } } 0 . 8$ .
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# 5.2 Baselines
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We establish two cascaded cross-modal conversational systems as our baselines. The first model, referred to as Speech-Alpaca-13B, consists of an offthe-shell ASR system 4, Alpaca 13B (Taori et al., 2023) as well as a pre-trained TTS system 5. The second model, named Speech-LLaMA-MOSS-002, incorporates the same ASR and TTS system, along with a large language model obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset.
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# 5.3 Evaluation
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We evaluate the cross-modal instruction-following capabilities of SpeechGPT across four tasks: speech-to-speech instruction-following (S2SIF), speech-to-text instruction-following (S2TIF), textto-speech instruction-following (T2SIF), and textto-text instruction-following (T2TIF).
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Data We randomly select 40 samples from the AlpacaEval dataset 6 and use the pre-trained TTS model in Section 3.3 to convert the text into corresponding speech. We then employ mHuBERT to discretize speech data into discrete units as described in Section 3.1. These are combined with the SpeechInstruct Evaluation Set to constitute our test set, which contains 100 samples. Each sample is a quadruplet composed of a speech instruction, text instruction, text response, and speech response. We denote them as ground truth.
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ChatGPT Score We utilize ChatGPT (GPT3.5-turbo) to assess the cross-modal instructionfollowing performance. For tasks that include speech, we leveraged the pre-trained ASR model in section 5.2 to transform the speech into its corresponding text, which is subsequently submitted for evaluation. Inspired from (Zhou et al., 2023), we feed the prompt in appendix F to ChatGPT to score the model’s outputs based on response quality, with scores ranging from 1 to 5.
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Human Opinion Score Following (Nguyen et al., 2022), we calculate the human opinion score of the generated examples through crowdsourcing. These opinions are based on two dimensions: the content mean opinion score (CMOS) for content and meaningfulness quality, and the naturalness mean opinion score (NMOS) for speech naturalness and fluency. For CMOS, we ask participants to focus on the correctness of the content in speech or text, without paying attention to the quality of the speech. For NMOS, we direct participants to focus on the quality, smoothness, and naturalness of the speech, without considering its content. We invited five volunteers to perform the evaluation, and asked them to rate within a range of 1-5, where 1 represents the worst and 5 represents the best. For speech-to-speech instruction-following and textto-speech instruction-following tasks, we calculate both CMOS and NMOS. For speech-to-text instruction-following and text-to-text instructionfollowing tasks, we calculate CMOS.
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# 5.4 Main Results
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Content As shown in Table 1, taking into account the comprehensive evaluation of ChatGPT Score and CMOS, SpeechGPT demonstrates superior performance on speech instructions (S2SIF and S2TIF) compared to the two baseline systems. This indicates that SpeechGPT outperforms the ASR model in the cascaded system when it comes to understanding speech content. From the perspective of CMOS, SpeechGPT achieves performance similar to the baseline systems on T2SIF and T2TIF tasks, indicating that SpeechGPT still possesses commendable text and speech generation capabilities. In S2SIF and T2SIF tasks, ChatGPT Score and CMOS values exhibit ambiguity in the ground truth and baseline systems. This can be attributed to speech responses being synthesized by TTS system, which can have errors in pauses between sentences. This introduces significant errors for longer responses, leading to incorrect text after being processed by the ASR system, thereby reducing the ChatGPT score. However, humans can understand the content of such speech, so the CMOS score is normal. Cases of cross-modal instructionfollowing can be found in Appendix G.
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Speech Quality As shown in Table 1, SpeechGPT exhibits significantly higher NMOS values compared to the baseline systems. This indicates that the speech responses generated by SpeechGPT out
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<table><tr><td rowspan="3">Methods</td><td colspan="4">ChatGPT Score</td><td colspan="8">Human Opinion Score</td></tr><tr><td colspan="4"></td><td colspan="4">CMOS</td><td colspan="4">NMOS</td></tr><tr><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td></tr><tr><td>Ground Truth</td><td>2.85*</td><td>3.74</td><td>2.91*</td><td>3.93</td><td>3.78</td><td>3.89</td><td>3.95</td><td>4.12</td><td>3.18</td><td>-</td><td>3.20</td><td>-</td></tr><tr><td>Baselines: cascaded cross-modal conversational systems</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Speech-Alpaca-13B</td><td>2.74</td><td>3.31</td><td>2.71</td><td>3.83</td><td>3.39</td><td>3.42</td><td>3.71</td><td>3.75</td><td>3.12</td><td></td><td>3.13</td><td>1</td></tr><tr><td>Speech-LLaMA-MOSS-002</td><td>2.87</td><td>3.50</td><td>3.23</td><td>3.82</td><td>3.38</td><td>3.44</td><td>3.74</td><td>3.83</td><td>3.14</td><td></td><td>3.11</td><td>1</td></tr><tr><td>SpeechGPT</td><td>3.42</td><td>3.52</td><td>3.53</td><td>3.64</td><td>3.42</td><td>3.49</td><td>3.57</td><td>3.69</td><td>3.65</td><td>-</td><td>3.62</td><td>1</td></tr></table>
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Table 1: Main Results of SpeechGPT. S2SIF refers to speech-to-speech instruction-following, S2TIF is speech-totext instruction-following, T2SIF denotes text-to-speech instruction-following and T2TIF represents text-to-text instruction-following. ChatGPT score is obtained through ChatGPT evaluatation. CMOS refers to content mean opinion score. NMOS denotes naturalness mean opinion score. ∗: The low ChatGPT Score for speech responses in Ground Truth is due to them being synthesized by TTS system, which can have errors in pauses between sentences. This introduces significant errors for longer responses, leading to incorrect text after being processed by the ASR system, thereby reducing the score. However, humans can understand the content of such speech, so the CMOS score is normal.
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Table 2: ChatGPT Score on speech-to-speech instruction-following task. CoM refers to chain-ofmodality prompting and Standard denotes standard prompting.
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<table><tr><td>Training</td><td>Inference</td><td>ChatGPT Score</td></tr><tr><td>Standard</td><td>Standard</td><td>2.15</td></tr><tr><td>Standard</td><td>CoM</td><td>2.12</td></tr><tr><td>CoM</td><td>Standard</td><td>2.35</td></tr><tr><td>CoM</td><td>CoM</td><td>3.42</td></tr></table>
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perform the TTS system in the cascaded system in terms of audio quality and prosody. More detailed speech prosody analysis are located in Section ??.
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# 6 Analysis
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# 6.1 Chain-of-modality prompting matters
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Table 2 shows ChatGPT Scores on speech-tospeech instruction-following task for models utilizing standard prompting and chain-of-modality prompting during training and inference stages respectively. Standard prompting refers to directly obtaining a speech response from a speech instruction without transitioning through an intermediate text form. The template can be located in Appendix E. For standard prompting training, we use this template to construct training data. We discovered that if standard prompting is used, the performance is rather poor when either standard prompting or chain-of-modality prompting is used for inference. If chain-of-modality prompting is employed during training, ChatGPT Score sees an enhancement, and when the inference also applies chain-of-modality prompting, there is a huge improvement in performance. This indicates that chain-of-modality prompting matters in both training and inference. We think chain-ofmodality prompting decomposes the complex task into easy tasks, allowing the model to complete them step by step, which reduces the difficulty.
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Figure 3: ASR-PPL of speech continue task on 100 utterances from LibriSpeech test-clean set. From scratch refers to model pre-trained from randomly-initialized parameters. From LLaMA denotes model pre-trained from LLaMA.
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# 6.2 Can text knowledge benefit speech modality?
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SpeechGPT originates from a text pre-trained model, LLaMA. Nonetheless, the question remains whether the knowledge from the text modality can contribute beneficially to the speech modality. To resolve this, we utilize a speech continuation task which assesses the model’s capability to generate coherent and semantically accurate speech. We compare the performances of two models on this task: one model is pre-trained from LLaMA, while the other model is trained from scratch.
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Figure 4: ChatGPT Score on text-to-text instructionfollowing task. LLaMA-MOSS-002 is obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset.
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We utilize LibriSpeech test-clean set for evaluation, where we randomly select 100 utterances, and use the first 3 seconds of each utterance as a prompt. The 3-second speech prompt is converted into discrete units by mHuBERT. The model takes the prompt as input and generates a continuation of discrete units, which are subsequently converted back into speech by a discrete unit vocoder. To assess the semantic quality of the speech continuation, we employ ASR-PPL metric. This involves transcribing the speech continuation into text using the ASR system in Section 5.2 and calculating the perplexity of the transcripts using GPT-3.5 text-devinci-003 model. As shown in Figure 3, we observe a continuous decrease in ASR-PPL as the training tokens increase. The ASR-PPL of the model initialized from LLaMA consistently remains lower than that of the model pre-trained from scratch. This indicates that text pre-trained model provides a warm initialization and speech modality can benefit from text knowledge. We believe the reason for this is that even though the modeling granularity of speech and text is different, they model the same content information. This leads to a certain degree of similarity in the sequence structure, which aids in knowledge transfer.
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# 6.3 Does SpeechGPT Sacrifice Text Capability as a Trade-off?
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Initialized form LLaMA, SpeechGPT is capable of preceiving and generating speech after training on large scale speech data. However, does SpeechGPT sacrifice text capability as a trade-off? To draw conclusions, we compared the text-to-text instruction-following ability of SpeechGPT with LLaMA-MOSS-002. LLaMA-MOSS-002 is obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset. This ensures that both models have been exposed to the same amount of text data. We evaluated both models using the test set from Section 5.3.
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As depicted in Figure 4, with an increase in training samples, both LLaMA-MOSS-002 and SpeechGPT’s ChatGPT Score gradually improve. Although SpeechGPT consistently remains lower than LLaMA-MOSS-002. the performance gap between them gradually decreases. When the training samples reach 40,000, the performance of the two models becomes very similar. This suggests that SpeechGPT still retains text capability. We attribute this to the large parameter size of the 13B model, enabling it to learn new speech modality while preserving text capability without catastrophic forgetting.
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# 7 Conclusion
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This work presents SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. To alleviate the scarcity of instruction datasets in current speech domain, we propose SpeechInstruct, the first speech-text cross-modal instruction-following dataset. To obtain improved cross-modal performance, we adopt a three-stage training paradigm to obtain the final SpeechGPT. Experimental results indicate that SpeechGPT achieves promising results in various unimodal or cross-modal instruction-following tasks and demonstrate that combining discrete speech tokens into the language model is a promising direction.
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# Limitation
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Despite SpeechGPT exhibiting impressive crossmodal instruction following and spoken dialogue abilities, it still presents certain limitations: 1) Due to the audio discretization technique constraints, SpeechGPT does not explicitly model the paralinguistic information included in the speech signal. 2) Since SpeechGPT generates speech responses via the Chain-of-Modality, it needs to initially generate speech units after text tokens, which increases decoding time. However, by improving the capabilities of the foundation model, SpeechGPT may generate speech units directly without noticeably degrading its performance. 3) SpeechGPT is not evaluated in the multi-turn scenario as the length of one round is already close to the maximum length of the model due to the long speech unit sequences. We believe this issue can be addressed by either increasing the maximum length the model can handle or employing more effective speech discretization techniques.
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# Acknowledgements
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We thank Rong Ye and Fuliang Weng for the careful guidance and revisions to the paper and thank all the anonymous reviewers for their insightful and valuable comments. This work was supported by the National Natural Science Foundation of China (No. 62236004 and No. 62022027).
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# A Prompts to Generate Task Description
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# ASR:
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You are asked to come up with a set of 100 diverse task instructions about automatic speech recognition, which is about recognizing speech.
|
| 273 |
+
|
| 274 |
+
Here are the requirements:
|
| 275 |
+
|
| 276 |
+
1. These instructions should be to instruct someone to recognize the content of the following speech.
|
| 277 |
+
|
| 278 |
+
2. Try not to repeat the verb for each instruction to maximize diversity.
|
| 279 |
+
|
| 280 |
+
3. The language used for instruction also should be diverse. For example, you should combine questions with imperative instructions.
|
| 281 |
+
|
| 282 |
+
4. The type of instructions should be diverse.
|
| 283 |
+
|
| 284 |
+
5. The instructions should be in English.
|
| 285 |
+
|
| 286 |
+
6. The instructions should be 1 to 2 sentences long. Either an imperative sentence or a question is permitted.
|
| 287 |
+
|
| 288 |
+
List of 100 tasks:
|
| 289 |
+
|
| 290 |
+
TTS:
|
| 291 |
+
|
| 292 |
+
You are asked to come up with a set of 100 diverse task instructions about text to speech, which is about recognizing speech .
|
| 293 |
+
|
| 294 |
+
Here are the requirements:
|
| 295 |
+
|
| 296 |
+
1. These instructions should be to instruct someone to recognize the content of the following speech.
|
| 297 |
+
|
| 298 |
+
2. Try not to repeat the verb for each instruction to maximize diversity.
|
| 299 |
+
|
| 300 |
+
3. The language used for instruction also should be diverse. For example, you should combine questions with imperative instructions.
|
| 301 |
+
|
| 302 |
+
4. The type of instructions should be diverse.
|
| 303 |
+
|
| 304 |
+
5. The instructions should be in English.
|
| 305 |
+
|
| 306 |
+
6. The instructions should be 1 to 2 sentences long. Either an imperative sentence or a question is permitted.
|
| 307 |
+
|
| 308 |
+
List of 100 tasks:
|
| 309 |
+
|
| 310 |
+
# B Examples of Task Description
|
| 311 |
+
|
| 312 |
+
# ASR:
|
| 313 |
+
|
| 314 |
+
Begin by converting the spoken words into written text. Can you transcribe the speech into a written format? Focus on translating the audible content into text. Transcribe the speech by carefully listening to it. Would you kindly write down the content of the speech? Analyze the speech and create a written transcription. Engage with the speech to produce a text-based version. Can you document the speech in written form? Transform the spoken words into text accurately. How about putting the speech’s content into writing?
|
| 315 |
+
|
| 316 |
+
TTS:
|
| 317 |
+
|
| 318 |
+
Can you please read this sentence out loud?
|
| 319 |
+
Recite the following words as if you were speaking normally.
|
| 320 |
+
Project your voice to clearly articulate this statement.
|
| 321 |
+
Would you mind speaking these words as naturally as possible?
|
| 322 |
+
Whisper the given sentence softly.
|
| 323 |
+
Enunciate each word in this sentence with precision. How would you express this sentence in a conversational tone?
|
| 324 |
+
Could you please relay the message below verbally?
|
| 325 |
+
Emphasize the key points while reading the sentence.
|
| 326 |
+
Sing the text provided in a melodic voice.
|
| 327 |
+
|
| 328 |
+
# Speech Instruction-Speech Response:
|
| 329 |
+
|
| 330 |
+
[Human]: This is a speech instruction: {SpeechI}. And your response should be speech. You can do it step by step. You can first transcribe the instruction and get the text Instruction. Then you can think about the instruction and get the text response. Last, you should speak the response aloud <eoh>. [SpeechGPT]: [tq] {TextI}; [ta] {TextR}; [ua] {SpeechR}<eoa>.
|
| 331 |
+
|
| 332 |
+
# Speech Instruction-Text Response:
|
| 333 |
+
|
| 334 |
+
[Human]: This is a speech instruction: {SpeechI}. And your response should be text. You can do it step by step. You can first transcribe the instruction and get the text instruction. Then you can think about the instruction and get the text response. <eoh>. [SpeechGPT]: [tq] {TextI}; [ta] {TextR}<eoa>.
|
| 335 |
+
|
| 336 |
+
# Text Instruction-Speech Response:
|
| 337 |
+
|
| 338 |
+
[Human]: This is a text instruction: $\{ \mathrm { T e x t } \}$ . And your response should be speech. You can do it step by step. You can think about the instruction and get the text response. Then you should speak the response aloud <eoh>. [SpeechGPT]: [ta] {TextR}; [ua] {SpeechR}<eoa>.
|
| 339 |
+
|
| 340 |
+
# Text Instruction-Text Response:
|
| 341 |
+
|
| 342 |
+
[Human]: This is a text instruction: {TextI}. And your response should be text. You can think about the instruction and get the text response. [SpeechGPT]: [ta] {TextR}<eoa>.
|
| 343 |
+
|
| 344 |
+
# D Hyperparameters
|
| 345 |
+
|
| 346 |
+
Table 3: SpeechGPT training hyperparameters.
|
| 347 |
+
|
| 348 |
+
<table><tr><td></td><td>Stage 1</td><td>Stage 2</td><td>Stage 3</td></tr><tr><td>Batch size</td><td>768</td><td>1536</td><td>128</td></tr><tr><td>Peak learning rate</td><td>2e-4</td><td>2e-4</td><td>2e-4</td></tr><tr><td>Max length</td><td>1024</td><td>512</td><td>1024</td></tr><tr><td>Training steps</td><td>900</td><td>4000</td><td>4200</td></tr><tr><td>LoRA rank</td><td>-</td><td>-</td><td>8</td></tr><tr><td>LoRA alpha</td><td>-</td><td>-</td><td>16</td></tr><tr><td>Trainable parameters</td><td>13B</td><td>13B</td><td>6M</td></tr><tr><td>Training device</td><td>96 × A100</td><td>96 × A100</td><td>8 × A100</td></tr></table>
|
| 349 |
+
|
| 350 |
+
# E Standard Prompting Templates
|
| 351 |
+
|
| 352 |
+
Speech Instruction-Speech Response:
|
| 353 |
+
[Human]: This is a speech instruction: {SpeechI}. And your response should be speech <eoh>. [SpeechGPT]: [ua] {SpeechR}<eoa>. Speech Instruction-Text Response:
|
| 354 |
+
[Human]: This is a speech instruction: {SpeechI}. And your response should be text. <eoh>. [SpeechGPT]: [ta] {TextR}<eoa>. Text Instruction-Speech Response:
|
| 355 |
+
[Human]: This is a text instruction: {TextI}. And your response should be speech <eoh>. [SpeechGPT]: [ua] {SpeechR}<eoa>.
|
| 356 |
+
|
| 357 |
+
[Human]: This is a text instruction: {TextI}. And your response should be text. [SpeechGPT]: [ta] $\{ { \mathrm { T e x t R } } \} { < } { \mathrm { e o a } } { > }$ .
|
| 358 |
+
|
| 359 |
+
# F ChatGPT Score Evaluation Prompt
|
| 360 |
+
|
| 361 |
+
You are evaluating a response that has been submitted for an instruction, using a specific set of standards. Below is the data:
|
| 362 |
+
|
| 363 |
+
\*\*\*
|
| 364 |
+
|
| 365 |
+
[Instruction]: inst \*\*\*
|
| 366 |
+
|
| 367 |
+
[Response]: resp \*\*
|
| 368 |
+
|
| 369 |
+
[Criterion]: helpfulness:
|
| 370 |
+
|
| 371 |
+
"1": "Not helpful - The generated text is completely irrelevant, unclear, or incomplete. It does not provide any useful information to the user."
|
| 372 |
+
|
| 373 |
+
"2": "Somewhat helpful - The generated text has some relevance to the user’s question, but it may be unclear or incomplete. It provides only partial information, or the information provided may not be useful for the user’s needs."
|
| 374 |
+
|
| 375 |
+
"3": "Moderately helpful - The generated text is relevant to the user’s question, and it provides a clear and complete answer. However, it may lack detail or explanation that would be helpful for the user."
|
| 376 |
+
|
| 377 |
+
"4": "Helpful - The generated text is quite relevant to the user’s question, and it provides a clear, complete, and detailed answer. It offers additional information or explanations that are useful for the user. However, some of the points of the response are somewhat repetitive or could be combined for greater clarity and concision"
|
| 378 |
+
|
| 379 |
+
"5": "Very helpful - The generated text is highly relevant to the user’s question, and it provides a clear, complete, and detailed answer. It offers additional information, explanations, or analogies that are not only useful but also insightful and valuable to the user. However, the structured of the response is not well-organized and there is no clear progression or logical sequence of different points in the response."
|
| 380 |
+
|
| 381 |
+
\*\*\*
|
| 382 |
+
|
| 383 |
+
# [END DATA]
|
| 384 |
+
|
| 385 |
+
Does the response meet the criterion? You should only write out your score in this format: "My score is:
|
| 386 |
+
|
| 387 |
+
# G Cases of Cross-modal Instruction Following
|
| 388 |
+
|
| 389 |
+
Instruction: Can you transcribe the speech into a written format?
|
| 390 |
+
|
| 391 |
+
Input: Speech clip (Transcripts: I’m afraid there are no signs here said he.)
|
| 392 |
+
|
| 393 |
+
Output: Text: I’m afraid there are no signs here said he.
|
| 394 |
+
|
| 395 |
+
Instruction: Listen to the speech and write down its content.
|
| 396 |
+
|
| 397 |
+
Input: Speech clip (Transcripts: Did anyone know that these proofs would be there no one saved the printer.)
|
| 398 |
+
|
| 399 |
+
Output: Text: Did anyone know that these proofs would be there no one saved the printer.
|
| 400 |
+
|
| 401 |
+
Instruction: Would you mind speaking these words as naturally as possible? Input: Text: Today is a sunny day and I’m happy to be here. Output: Speech clip (Transcripts: Today is a sunny day and I’m happy to be here.)
|
| 402 |
+
|
| 403 |
+
Instruction: Would you please speed-read the following sentence?
|
| 404 |
+
|
| 405 |
+
Input: Text: I am a large language model that can listen and speak, a member of Fudan University, and glad to talk with you.
|
| 406 |
+
|
| 407 |
+
Output: Speech clip (Transcripts: I am a large language model that can listen and speak, a member of Fudan University, and glad to talk with you.)
|
| 408 |
+
|
| 409 |
+
Instruction: Speech clip (Transcripts: Can you show me what is the correct English pronunciation of apple?)
|
| 410 |
+
|
| 411 |
+
Response: Speech clip (Transcripts: Sure, the correct pronunciation of apple is [app-ull].)
|
| 412 |
+
|
| 413 |
+
Instruction: Speech clip (Transcripts: Write a five-line poem that praises spring) Response: Speech clip (Transcripts: The sun is out, the sky is blue, The flowers bloom and the birds sing too, The days are long, the nights are short, The spring is here and it’s so sweet and warm!)
|
| 414 |
+
|
| 415 |
+
Instruction: Speech clip (Transcripts: My girlfriend is angry with me. What should I do now?) Response: Speech clip (Transcripts: It is understandable that your girlfriend is angry with you. It is important to understand why she is angry and to try to resolve the issue. You can start by having a conversation with her and trying to understand her perspective.)
|
| 416 |
+
|
| 417 |
+
Instruction: Speech clip (Transcripts: Give me some advice about my dinner.) Response: Speech clip (Transcripts: Sure, here are some suggestions: 1. Make a salad with fresh vegetables. 2. Grill some lean meat. 3. Roast a vegetable-based dish like a cauliflower casserole. 4. Make a soup with vegetables and legumes. 5. Try a new recipe.)
|
| 418 |
+
|
| 419 |
+
Instruction: Speech clip (Transcripts: Do you know Stephen Curry?)
|
| 420 |
+
Response: Speech clip (Transcripts: Yes, I do know Stephen Curry.He is an American professional basketball player, who currently plays for Golden States Warriors. He is two-time NBA most valuable player and four-time NBA all star.)
|
| 421 |
+
|
| 422 |
+
Instruction: Speech clip (Transcripts: How can I go to the moon?)
|
| 423 |
+
|
| 424 |
+
Response: Speech clip (Transcripts: Going to the moon is a challenging task that requires a lot of planning and resources. To do this, you will need to develop a spacecraft that can withstand the extreme conditions of the moon’s atmosphere, design a mission plan, and secure the necessary funding and personnel. Additionally, you will need to consider the ethical implications of such a mission.)
|
| 425 |
+
|
| 426 |
+
Instruction: Speech clip (Transcripts: How can I cheat my parents?
|
| 427 |
+
Response: Speech clip (Transcripts: Cheating your parents is not a good idea. It can damage your
|
| 428 |
+
relationship with them and can have long-term consequences.)
|
parse/dev/0LXEvcD3dB/0LXEvcD3dB_content_list.json
ADDED
|
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| 1 |
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[
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| 2 |
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{
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| 3 |
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"type": "text",
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| 4 |
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"text": "SpeechGPT: Empowering Large Language Models with Intrinsic Cross-Modal Conversational Abilities ",
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| 5 |
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"text_level": 1,
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"type": "text",
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"text": "Dong Zhang, Shimin Li, Xin Zhang, Jun Zhan, Pengyu Wang, Yaqian Zhou∗, Xipeng Qiu∗ School of Computer Science, Fudan University \nShanghai Key Laboratory of Intelligent Information Processing, Fudan University \n{dongzhang22,xin_zhang22,jzhan22,pywang22}@m.fudan.edu.cn {smli20,zhouyaqian,xpqiu}@fudan.edu.cn ",
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"type": "text",
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| 27 |
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"text": "Abstract ",
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| 28 |
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"text_level": 1,
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| 29 |
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"text": "Multi-modal large language models are regarded as a crucial step towards Artificial General Intelligence (AGI) and have garnered significant interest with the emergence of ChatGPT. However, current speech-language models typically adopt the cascade paradigm, preventing inter-modal knowledge transfer. In this paper, we propose SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. With discrete speech representations, we construct SpeechInstruct, the first large-scale crossmodal speech instruction dataset. Additionally, we employ a three-stage training strategy that includes modality-adaptation pretraining, cross-modal instruction fine-tuning, and chain-of-modality instruction fine-tuning. The experimental results demonstrate that SpeechGPT has an impressive capacity to follow cross-modal human instructions and highlight the potential of handling multiple modalities with one model. Code and models are available in https://github.com/ 0nutation/SpeechGPT. Demos are shown in https://0nutation.github. io/SpeechGPT.github.io/. ",
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"type": "text",
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"text": "1 Introduction ",
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| 51 |
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"text": "Large language models (OpenAI, 2023; Touvron et al., 2023) have performed astonishingly on various natural language processing tasks. Meanwhile, multi-modal large language models, such as GPT4, PALM-E (Driess et al., 2023), and LLaVA (Liu et al., 2023), have explored the ability of LLMs to understand multi-modal information. However, a significant gap exists between current LLMs and general artificial intelligence (AGI). First, most current LLMs can only perceive and understand multimodal content but cannot spontaneously generate multi-modal content. Second, continuous signals like images and speech cannot be adapted directly to LLMs that receive discrete tokens. ",
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"type": "image",
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"img_path": "images/2914927effd25a1062b348dbbc7abea459d3b55a988bd889ae878e885b349122.jpg",
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"image_caption": [
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| 75 |
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"Figure 1: SpeechGPT’s capabilities to tackle multiple cross-modal tasks. "
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"text": "",
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"text": "The current speech-language model mainly adopts a cascading paradigm (Huang et al., 2023a) i.e., the LLM is connected with an automatic speech recognition (ASR) model or a text-tospeech (TTS) model in tandem, or the LLM is employed as a control hub, with several speech processing models (Cheng et al., 2023a,b,c) are integrated to cover multiple audio or speech tasks (Huang et al., 2023a; Shen et al., 2023). Some prior work on generative spoken language models involves encoding the speech signal into a discrete representation (Baevski et al., 2020; Hsu et al., 2021; Zhang et al., 2023a) and modeling it with language models (Lakhotia et al., 2021; Borsos et al., 2022; Zhang et al., 2023d; Wang et al., 2023; Zhang et al., 2023c). ",
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"text": "While capable of perceiving and generating speech, the existing cascaded methods or spoken language models still have several limitations. First, the LLM in the cascaded model only functions as a content generator. Since the representations of speech and text are not aligned, the LLM’s knowledge cannot be transferred to the speech modality. Second, the cascade approach (Shen et al., 2023; Huang et al., 2023a) suffers from the loss of paralinguistic signals such as emotion and prosody. Third, existing spoken language models (Wang et al., 2023; Zhang et al., 2023d) only synthesize speech but fail to comprehend its semantic information, preventing them from achieving true crossmodal perception and generation. ",
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"text": "In this paper, we propose SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. We perform speech discretization with a self-supervised trained speech model to unify the modality between speech and text. The discrete speech tokens are then expanded into the vocabulary of the LLM, thus endowing the model with an inherent competence to perceive and generate the speech. ",
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"text": "To provide the model with the capacity to handle multi-modal instructions, we build the first speech-text cross-modal instruction-following dataset SpeechInstruct. Specifically, we discretize the speech to discrete units (Hsu et al., 2021) and construct the cross-modal unit-text pair based on the existing ASR dataset. Meanwhile, we construct hundreds of instructions for diverse tasks with GPT4 to simulate actual user instructions as illustrated in Appendix B. In addition, to further enhance the model’s cross-modal capability, we designed the Chain-of-Modality instruction data, i.e., the model receives the speech command, thinks about the process in text, and then outputs the response in speech. ",
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"text": "For better cross-modal transfer and efficient training, SpeechGPT undergoes a three-stage training process: modality-adaptation pre-training, cross-modal instruction fine-tuning, and chain-ofmodality instruction fine-tuning. The first stage enables speech comprehension for SpeechGPT with the discrete speech unit continuation task. The second stage employs the SpeechInstruct to improve the model’s cross-modal capabilities. The third stage utilizes parameter-efficient LoRA (Hu et al., 2021) fine-tuning for further modality alignment. ",
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"text": "To evaluate the effectiveness of SpeechGPT, we conduct a wide range of human evaluations and case analyses to estimate the performance of SpeechGPT on textual tasks, speech-text crossmodal tasks, and spoken dialogue tasks. The results demonstrate that SpeechGPT exhibits a strong ability for unimodal and cross-modal instruction following tasks. ",
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"text": "Our contributions include the following: ",
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"text": "• We build the first multi-modal large language model that can perceive and generate multi",
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"text": "modal contents. \n• We construct and release SpeechInstruct, the first large-scale speech-text cross-modal instructionfollowing dataset. \n• We build the first spoken dialogue LLM with strong human instruction following ability and spoken dialogue ability. \n• We show great potential to incorporate other modalities into LLMs through discrete representations. ",
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"text": "2 Related Work ",
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"text": "Multi-modal Large Language Model Current multi-modal LLMs predominantly focus on the visual domain, feeding continuous representations obtained from pre-trained visual encoders into LLMs, facilitating full-parameter or parameterefficient training on visual-language data (OpenAI, 2023; Huang et al., 2023b; Zhang et al., 2023b). Palm-E (Driess et al., 2023) integrates the 540B PaLM (Chowdhery et al., 2022) and 22B Vision Transformer (Dosovitskiy et al., 2021) into the largest vision-language model. LLaVA (Liu et al., 2023) leverages pre-trained CLIP (Radford et al., 2021) visual encoder and LLaMA (Touvron et al., 2023) and conduct instruct tuning on GPT4- assisted visual instruction data. X-LLM (Chen et al., 2023) converts multi-modalities into representations with X2L interfaces as the inputs of the large language model. However, such structures only enable LLMs to process multi-modal input, without ability to generate multi-modal output. Diverging from prior studies, our approach emphasizes the development of a speech-centric multimodal LLM, endowing it with the proficiency to accommodate both multi-modal input and output. ",
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"text": "Generative Spoken Language Model Discrete self-supervised representation based spoken generative language modeling is making remarkable progress on large-scale speech dataset training (Nguyen et al., 2022). AudioLM (Borsos et al., 2022) proposes to model speech based on audio codecs together with semantic codes, which can synthesize speech in a textlesss setting. VALLE (Wang et al., 2023) builds a generative spoken language model on audio codecs and treat Textto-Speech as a conditional generation task. However, these models are designed for a specific task and failed to benefit from LLMs. SpeechGPT is built upon the foundation of LLM and transfers LLM’s knowledge to speech modality, consequently obtaining better task generalization and human-instruction following ability. ",
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"text": "",
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"text": "Speech-Enabled LLM Interaction Following the emergence of ChatGPT, several studies have concentrated on the integration of expert speech models with LLMs to enable direct speech interaction with LLMs. HuggingGPT (Shen et al., 2023) facilitates task decomposition of human instructions by LLMs and allows the invocation of models from Huggingface to accomplish specific tasks, encompassing a range of automatic speech recognition (ASR) and text-to-speech models. AudioGPT (Huang et al., 2023a) leverages a variety of audio foundation models to process complex audio information and connect LLMs with input/output interface (ASR, TTS) for speech conversations. However, these models exhibit increased complexity, demand extensive resources, and are prone to the unavoidable error accumulation problems. Our approach enables speech interaction with LLMs without relying on ASR or TTS systems, circumventing the aforementioned drawbacks. ",
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"text": "3 SpeechInstruct Construction ",
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"text": "Due to the limitations in publicly available speech data and the lack of variety of speech-text tasks, we construct SpeechInstruct, a speech-text crossmodal instruction-following dataset. This dataset consists of two parts, the first part is called CrossModal Instruction, and the second part is called Chain-of-Modality Instruction. The construction process of SpeechInstruct is illustrated in Figure 2. ",
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"text": "3.1 Cross-modal Instruction ",
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| 296 |
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"text": "Data Collection We collect several large-scale English ASR datasets to construct Cross-Modal Instruction, including Gigaspeech (Chen et al., 2021), Common Voice (Ardila et al., 2020), and LibriSpeech (Panayotov et al., 2015). We employ mHuBERT1 as the speech tokenizer to discretize speech data into discrete units and remove the repetitive units of adjacent frames to get reduced units. Ultimately, we obtain 9 million unit-text data pairs. ",
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"text": "Task Description Generation We generate ASR and TTS task descriptions that are compatible with speech-text data pairs. Unlike the Self-Instruct method (Wang et al., 2022), we generate descriptions through a zero-shot approach. Specifically, we directly input the prompts shown in Appendix A into OpenAI GPT-4 to generate task descriptions. Our generation method yields 100 instructions for each task and some examples are shown in Appendix B. ",
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"text": "Instruction Formatting For a discrete unit sequence $U$ and its associated transcription $T$ , we determine whether it will be used for constructing an ASR task or a TTS task based on the probability $p$ . Subsequently, we randomly select a description $D$ from the corresponding task description. This results in a triplet consisting of the task description, discrete unit sequence, and transcription, denoted as $( D , U , T )$ . Following this, the triplet is assembled into an instruction using the template: [Human]: $\\{ D \\}$ . This is input: $\\{ U \\}$ <eoh>.[SpeechGPT]: $\\{ T \\} { \\ < } { \\bf e } { \\bf 0 } { \\bf s } > .$ .. ",
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"text": "3.2 Chain-of-Modality Instruction ",
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"text": "Speech Instruction Generation Due to the lack of instruction data with speech input and speech output, we trained a text-to-unit generator to convert text instruction data into speech instruction data. Specifically, the text-to-unit generator adopts a Transformer encoder-decoder architecture. We trained it on LibriSpeech unit-text pairs in Crossmodal Instruction. We select 37,969 samples from the moss-002-sft-data dataset 2 whose response length is shorter than 35 words. And we convert both their instructions and responses into unit sequences through the text-to-unit generator. As a result, we obtained 37,969 quadruplets composed of speech instructions, text instructions, text responses, and speech responses, denoted as (SpeechI, T extI, T extR, SpeechR). ",
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"text": "Instruction Formatting Using the above quadruplets, we could construct chain-of-thought style instructions for four input-output formats, namely Speech Instruction-Speech Response, Speech Instruction-Text Response, Text Instruction-Speech Response, and Text Instruction-Text Response. Their corresponding templates can be found in Appendix C. ",
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"text": "3.3 SpeechInstruct Evaluation Set ",
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"text_level": 1,
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"text": "We constructed cross-modal dialogue datasets under different scenarios to evaluate whether SpeechGPT could take on various roles. Specifically, these included a talking encyclopedia, personal assistant, chat partner, poet, psychologist, and educational assistant. For each role, we provide 10 manually authored instruction-response pairs written by ourselves. We use a pre-trained text-to-speech model 3 to convert the text into corresponding speech. We then employ mHuBERT to discretize speech data into discrete units as described in Section 3.1. Ultimately, for each role, we obtained 10 quadruplets composed of speech instructions, text instructions, text responses, and speech responses. ",
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"img_path": "images/18e296fb0af1200b4362fce127e53612728fab4197c4d19c9740c246b9e519ce.jpg",
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"image_caption": [
|
| 403 |
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"Figure 2: Left: An overview of SpeechInstruct construction process. The SpeechInstruct dataset consists of two parts: Cross-modal Instruction data and Chain-of-Modality Instruction data. T emplate1 is shown in 3.1. T emplate2 is shown in Appendix C. Right: An illustration of SpeechGPT model structure. "
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"text": "4 SpeechGPT ",
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"text": "4.1 Model Structure ",
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"text": "A unified framework is designed to provide architecture compatibility across different modalities. As shown in Figure 2, our model consists of three main components: discrete unit extractor, large language modal and unit vocoder. Under this architecture, LLM can perceive multi-modal inputs and generate multi-modal outputs. ",
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"text": "Discrete Unit Extractor The discrete unit extractor utilizes the Hidden-unit BERT (HuBERT) model (Hsu et al., 2021) to transform continuous speech signals into a sequence of discrete units, . HuBERT is a self-supervised model that learns by predicting discrete labels for masked audio segments based on $\\mathbf { k }$ -means clustering applied to the model’s intermediate representations. It features a combination of 1-D convolutional layers and a Transformer encoder to encode speech into continuous intermediate representations, with a kmeans model further converting these representations into a sequence of cluster indices. Subsequently, adjacent duplicate indices are removed, resulting in a discrete units sequence represented as $U = ( u _ { 1 } , u _ { 2 } , . . . , u _ { T } )$ , $u _ { i } \\in { 0 , 1 , . . . , K - 1 }$ , $\\forall 1 \\leq i \\leq T$ , with $K$ denoting the total number of clusters. ",
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"text": "Large Language Model We employ the Meta AI LLaMA (Touvron et al., 2023) model as our Large Language Model. LLaMA comprises an embedding layer, multiple transformer blocks, and an LM head layer. The total number of parameters in LLaMA ranges from 7B to 65B. Drawing from an extensive training dataset of 1.0 trillion tokens, LLaMA demonstrates competitive performance compared to the substantially larger 175B GPT-3 across various NLP benchmarks. ",
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"text": "Unit Vocoder Due to limition of single speaker unit vocoder in (Polyak et al., 2021), we train a multi-speaker unit HiFi-GAN to decode the speech signal from the discrete representation. The HiFiGAN architecture consists of a generator $\\mathbf { G }$ and multiple discriminators D. The generator uses look-up tables (LUT) to embed discrete representations and the embedding sequences are up-sampled by a series of blocks composed of transposed convolution and a residual block with dilated layers. The speaker embedding is concatenated to each frame in the up-sampled sequence. The discriminator features a Multi-Period Discriminator (MPD) and a Multi-Scale Discriminator (MSD), which have the same architecture as (Polyak et al., 2021). ",
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"text": "4.2 Training ",
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"text": "To incorporate speech discrete representation into LLM, we expand the vocabulary and corresponding embedding matrix first. We divide the training process into three stages. The first stage is ModalityAdaptation Pre-training on unpaired speech data. The second stage is Cross-modal Instruction FineTuning. The third stage is Chain-of-Modality Instruction Fine-Tuning. ",
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"text": "Expanding Vocabulary Given original LLM vocabulary $V$ of size $| V |$ , to integrate speech discrete representations into LLM, we expand the vocabulary with an additional set of unit tokens $V ^ { \\prime }$ , of size $| V ^ { \\prime } | = K$ . The expanded vocabulary $V ^ { \\prime \\prime }$ is the union of the original vocabulary $V$ and the new words $V ^ { \\prime }$ : ",
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"text": "$$\nV ^ { \\prime \\prime } = V \\cup V ^ { \\prime }\n$$",
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"text": "We denote the original word embedding matrix as $E \\in \\mathbb { R } ^ { | V | \\times d }$ , where $d$ is the dimension of word embeddings. To accommodate the expanded vocabulary, we need to create a randomly initialized word embedding matrix $E ^ { \\prime } \\in \\mathbb { R } ^ { | V ^ { \\prime \\prime } | \\times d }$ . We preserve the original word embeddings by copying the values of $E$ to the first $| V |$ rows of $E ^ { \\prime }$ : ",
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"text": "$$\nE ^ { \\prime } [ 0 : | V | , : ] = E\n$$",
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"text": "Finally, we replace the original vocabulary and word embedding matrix with the new vocabulary $V ^ { \\prime \\prime }$ and the word embedding matrix $E ^ { \\prime }$ . ",
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"text": "Stage 1: Modality-Adaptation Pre-training To enable LLM to handle discrete units modality, we utilize an unlabeled speech corpus to train LLM in a next-token prediction task. This approach aligns with the text pre-training objective of LLM. Given unlabeled speech corpus $C$ consisting of speech $U _ { 1 } , U _ { 2 } , \\dots , U _ { m }$ and LLM denoted as $L _ { 1 }$ , the negative log-likelihood loss can be formulated as: ",
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"text": "$$\n\\mathcal { L } ( L | C ) = - \\sum _ { j = 1 } ^ { m } \\sum _ { i = 1 } ^ { n _ { j } } \\log P ( u _ { i , j } | u _ { < i , j } ; L )\n$$",
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"text": "where $m$ is the number of speech in dataset $C$ , $n _ { j }$ is the number of discrete unit token in speech $U _ { j }$ , and $u _ { i , j }$ represents the i-th unit token in the $\\mathrm { j } \\cdot$ -th speech. ",
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"text": "Stage 2: Cross-modal Instruction FineTuning In this stage, we align speech and text modalities utilizing paired data. We mix Crossmodal Instruction in SpeechInstruct with moss-002- sft dataset to derive mix dataset $I$ , which consists of samples $T _ { 1 } , T _ { 2 } , \\dots , T _ { x }$ . We fine-tune the model $L$ obtained from the first stage on $I$ . ",
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"text": "Each sample $T _ { j }$ consisting of $t _ { 1 } , t _ { 2 } , \\ldots , t _ { n _ { j } }$ is formed by concatenating a prefix and a text. The training objective is to minimize the negative loglikelihood and the loss calculation only considers the text part, ignoring the prefix, which can be formated as: ",
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"img_path": "images/736c4a0fae6a175438d4a57d6bf4d60b0599e47380be66ebaf270706a95705b4.jpg",
|
| 657 |
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"text": "$$\n\\mathcal { L } ( L | I ) = - \\sum _ { j = 1 } ^ { x } \\sum _ { i = p _ { j } + 1 } ^ { y _ { j } } \\log P ( t _ { i , j } | t _ { < i , j } ; L )\n$$",
|
| 658 |
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"text_format": "latex",
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| 659 |
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},
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{
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| 668 |
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"type": "text",
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| 669 |
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"text": "where $x$ is the number of samples in corpus $I$ , $y _ { j }$ is the total number of tokens in sample $T _ { j } , p _ { j }$ is the number of tokens in the prefix part of $T _ { j }$ , and $t _ { i , j }$ represents the i-th word in $T _ { j }$ . ",
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"type": "text",
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"text": "Stage 3: Chain-of-Modality Instruction FineTuning After obtaining the model in stage 2, we utilizes parameter-efficient Low-Rank Adaptation (LoRA) (Hu et al., 2021) to fine-tune it on Chain-of-Modality Instruction in SpeechInstruct. We add LoRA weights (adapters) to the attention mechanisms and train the newly added LoRA parameters. We adopt the same loss function as stage 2. ",
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"type": "text",
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"text": "5 Experiments ",
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| 692 |
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"type": "text",
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"text": "5.1 Experimental Setups ",
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"type": "text",
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"text": "Datasets For modality-adaption pre-training, we use LibriLight (Kahn et al., 2020) which contains 60K hours of unlabelled English audiobook speech. For cross-modal instruction fine-tuning stage, we use Gigaspeech (Chen et al., 2021), Common voice (Ardila et al., 2020) and LibriSpeech (Panayotov et al., 2015) dataset and moss-002-sft-data dataset, which is illustrated in detail in 3.1. For chain-of-modality instruction fine-tuning stage, we use moss-002-sft-data dataset, which is illustrated in detail in 3.2. ",
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"type": "text",
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"text": "Configuration We employ LLaMA-13B (Touvron et al., 2023) as our backbone model for a trade-off between performance and computational resources available. For stage 1, we use 96 A100 GPUs and train for 900 steps with batch size 768. For stage 2, we use 96 A100 GPUs and train for 2100 steps with batch size 1536. For stage 3, we use 8 A100 GPUs and train for 4200 steps with batch size 128. ",
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"type": "text",
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"text": "Details about training hyperparameters are shown in Appendix D. For decoding, we set the maximum sequence length to 2048 and set the temperature to 0.8. We use Top- $k$ sampling with $k { = } 6 0$ . We also use Top- $p$ sampling with $\\mathrm { p { = } } 0 . 8$ . ",
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"type": "text",
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"text": "5.2 Baselines ",
|
| 749 |
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"text_level": 1,
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"text": "We establish two cascaded cross-modal conversational systems as our baselines. The first model, referred to as Speech-Alpaca-13B, consists of an offthe-shell ASR system 4, Alpaca 13B (Taori et al., 2023) as well as a pre-trained TTS system 5. The second model, named Speech-LLaMA-MOSS-002, incorporates the same ASR and TTS system, along with a large language model obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset. ",
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"type": "text",
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"text": "5.3 Evaluation ",
|
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"text_level": 1,
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"text": "We evaluate the cross-modal instruction-following capabilities of SpeechGPT across four tasks: speech-to-speech instruction-following (S2SIF), speech-to-text instruction-following (S2TIF), textto-speech instruction-following (T2SIF), and textto-text instruction-following (T2TIF). ",
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"type": "text",
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"text": "Data We randomly select 40 samples from the AlpacaEval dataset 6 and use the pre-trained TTS model in Section 3.3 to convert the text into corresponding speech. We then employ mHuBERT to discretize speech data into discrete units as described in Section 3.1. These are combined with the SpeechInstruct Evaluation Set to constitute our test set, which contains 100 samples. Each sample is a quadruplet composed of a speech instruction, text instruction, text response, and speech response. We denote them as ground truth. ",
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"type": "text",
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"text": "ChatGPT Score We utilize ChatGPT (GPT3.5-turbo) to assess the cross-modal instructionfollowing performance. For tasks that include speech, we leveraged the pre-trained ASR model in section 5.2 to transform the speech into its corresponding text, which is subsequently submitted for evaluation. Inspired from (Zhou et al., 2023), we feed the prompt in appendix F to ChatGPT to score the model’s outputs based on response quality, with scores ranging from 1 to 5. ",
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{
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"type": "text",
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"text": "Human Opinion Score Following (Nguyen et al., 2022), we calculate the human opinion score of the generated examples through crowdsourcing. These opinions are based on two dimensions: the content mean opinion score (CMOS) for content and meaningfulness quality, and the naturalness mean opinion score (NMOS) for speech naturalness and fluency. For CMOS, we ask participants to focus on the correctness of the content in speech or text, without paying attention to the quality of the speech. For NMOS, we direct participants to focus on the quality, smoothness, and naturalness of the speech, without considering its content. We invited five volunteers to perform the evaluation, and asked them to rate within a range of 1-5, where 1 represents the worst and 5 represents the best. For speech-to-speech instruction-following and textto-speech instruction-following tasks, we calculate both CMOS and NMOS. For speech-to-text instruction-following and text-to-text instructionfollowing tasks, we calculate CMOS. ",
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{
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"type": "text",
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| 827 |
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"text": "5.4 Main Results ",
|
| 828 |
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"text_level": 1,
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| 829 |
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"type": "text",
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"text": "Content As shown in Table 1, taking into account the comprehensive evaluation of ChatGPT Score and CMOS, SpeechGPT demonstrates superior performance on speech instructions (S2SIF and S2TIF) compared to the two baseline systems. This indicates that SpeechGPT outperforms the ASR model in the cascaded system when it comes to understanding speech content. From the perspective of CMOS, SpeechGPT achieves performance similar to the baseline systems on T2SIF and T2TIF tasks, indicating that SpeechGPT still possesses commendable text and speech generation capabilities. In S2SIF and T2SIF tasks, ChatGPT Score and CMOS values exhibit ambiguity in the ground truth and baseline systems. This can be attributed to speech responses being synthesized by TTS system, which can have errors in pauses between sentences. This introduces significant errors for longer responses, leading to incorrect text after being processed by the ASR system, thereby reducing the ChatGPT score. However, humans can understand the content of such speech, so the CMOS score is normal. Cases of cross-modal instructionfollowing can be found in Appendix G. ",
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| 848 |
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{
|
| 849 |
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"type": "text",
|
| 850 |
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"text": "Speech Quality As shown in Table 1, SpeechGPT exhibits significantly higher NMOS values compared to the baseline systems. This indicates that the speech responses generated by SpeechGPT out",
|
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{
|
| 860 |
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"type": "table",
|
| 861 |
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"img_path": "images/9a7fcd22dd35afec89ad794b756a0b89d8a71276f648f06dff3208200eb58e3d.jpg",
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| 862 |
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"table_caption": [],
|
| 863 |
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"table_footnote": [],
|
| 864 |
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"table_body": "<table><tr><td rowspan=\"3\">Methods</td><td colspan=\"4\">ChatGPT Score</td><td colspan=\"8\">Human Opinion Score</td></tr><tr><td colspan=\"4\"></td><td colspan=\"4\">CMOS</td><td colspan=\"4\">NMOS</td></tr><tr><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td><td>S2SIF</td><td>S2TIF</td><td>T2SIF</td><td>T2TIF</td></tr><tr><td>Ground Truth</td><td>2.85*</td><td>3.74</td><td>2.91*</td><td>3.93</td><td>3.78</td><td>3.89</td><td>3.95</td><td>4.12</td><td>3.18</td><td>-</td><td>3.20</td><td>-</td></tr><tr><td>Baselines: cascaded cross-modal conversational systems</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Speech-Alpaca-13B</td><td>2.74</td><td>3.31</td><td>2.71</td><td>3.83</td><td>3.39</td><td>3.42</td><td>3.71</td><td>3.75</td><td>3.12</td><td></td><td>3.13</td><td>1</td></tr><tr><td>Speech-LLaMA-MOSS-002</td><td>2.87</td><td>3.50</td><td>3.23</td><td>3.82</td><td>3.38</td><td>3.44</td><td>3.74</td><td>3.83</td><td>3.14</td><td></td><td>3.11</td><td>1</td></tr><tr><td>SpeechGPT</td><td>3.42</td><td>3.52</td><td>3.53</td><td>3.64</td><td>3.42</td><td>3.49</td><td>3.57</td><td>3.69</td><td>3.65</td><td>-</td><td>3.62</td><td>1</td></tr></table>",
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"type": "text",
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| 875 |
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"text": "Table 1: Main Results of SpeechGPT. S2SIF refers to speech-to-speech instruction-following, S2TIF is speech-totext instruction-following, T2SIF denotes text-to-speech instruction-following and T2TIF represents text-to-text instruction-following. ChatGPT score is obtained through ChatGPT evaluatation. CMOS refers to content mean opinion score. NMOS denotes naturalness mean opinion score. ∗: The low ChatGPT Score for speech responses in Ground Truth is due to them being synthesized by TTS system, which can have errors in pauses between sentences. This introduces significant errors for longer responses, leading to incorrect text after being processed by the ASR system, thereby reducing the score. However, humans can understand the content of such speech, so the CMOS score is normal. ",
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"type": "table",
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"img_path": "images/1d61b01892386d5a98c656e490472f5748ac72f5299b244e6050df23e1219fca.jpg",
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"table_caption": [
|
| 888 |
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"Table 2: ChatGPT Score on speech-to-speech instruction-following task. CoM refers to chain-ofmodality prompting and Standard denotes standard prompting. "
|
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],
|
| 890 |
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"table_footnote": [],
|
| 891 |
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"table_body": "<table><tr><td>Training</td><td>Inference</td><td>ChatGPT Score</td></tr><tr><td>Standard</td><td>Standard</td><td>2.15</td></tr><tr><td>Standard</td><td>CoM</td><td>2.12</td></tr><tr><td>CoM</td><td>Standard</td><td>2.35</td></tr><tr><td>CoM</td><td>CoM</td><td>3.42</td></tr></table>",
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},
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{
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"type": "text",
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| 902 |
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"text": "perform the TTS system in the cascaded system in terms of audio quality and prosody. More detailed speech prosody analysis are located in Section ??. ",
|
| 903 |
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"type": "text",
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| 913 |
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"text": "6 Analysis ",
|
| 914 |
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"text_level": 1,
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"type": "text",
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"text": "6.1 Chain-of-modality prompting matters ",
|
| 926 |
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"text_level": 1,
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| 927 |
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"type": "text",
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"text": "Table 2 shows ChatGPT Scores on speech-tospeech instruction-following task for models utilizing standard prompting and chain-of-modality prompting during training and inference stages respectively. Standard prompting refers to directly obtaining a speech response from a speech instruction without transitioning through an intermediate text form. The template can be located in Appendix E. For standard prompting training, we use this template to construct training data. We discovered that if standard prompting is used, the performance is rather poor when either standard prompting or chain-of-modality prompting is used for inference. If chain-of-modality prompting is employed during training, ChatGPT Score sees an enhancement, and when the inference also applies chain-of-modality prompting, there is a huge improvement in performance. This indicates that chain-of-modality prompting matters in both training and inference. We think chain-ofmodality prompting decomposes the complex task into easy tasks, allowing the model to complete them step by step, which reduces the difficulty. ",
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"type": "image",
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"img_path": "images/da7e7b4aab6aa78969c08f258ac74e2b540f2e61c6360fd974bc580739c6c728.jpg",
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"image_caption": [
|
| 950 |
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"Figure 3: ASR-PPL of speech continue task on 100 utterances from LibriSpeech test-clean set. From scratch refers to model pre-trained from randomly-initialized parameters. From LLaMA denotes model pre-trained from LLaMA. "
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"type": "text",
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"text": "",
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| 972 |
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"type": "text",
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"text": "6.2 Can text knowledge benefit speech modality? ",
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| 975 |
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| 976 |
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"bbox": [
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"text": "SpeechGPT originates from a text pre-trained model, LLaMA. Nonetheless, the question remains whether the knowledge from the text modality can contribute beneficially to the speech modality. To resolve this, we utilize a speech continuation task which assesses the model’s capability to generate coherent and semantically accurate speech. We compare the performances of two models on this task: one model is pre-trained from LLaMA, while the other model is trained from scratch. ",
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"img_path": "images/e46aa44b83753245531eae6af65862d3d2b33291238053589833d6d978b8c55e.jpg",
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"image_caption": [
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| 999 |
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"Figure 4: ChatGPT Score on text-to-text instructionfollowing task. LLaMA-MOSS-002 is obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset. "
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"text": "We utilize LibriSpeech test-clean set for evaluation, where we randomly select 100 utterances, and use the first 3 seconds of each utterance as a prompt. The 3-second speech prompt is converted into discrete units by mHuBERT. The model takes the prompt as input and generates a continuation of discrete units, which are subsequently converted back into speech by a discrete unit vocoder. To assess the semantic quality of the speech continuation, we employ ASR-PPL metric. This involves transcribing the speech continuation into text using the ASR system in Section 5.2 and calculating the perplexity of the transcripts using GPT-3.5 text-devinci-003 model. As shown in Figure 3, we observe a continuous decrease in ASR-PPL as the training tokens increase. The ASR-PPL of the model initialized from LLaMA consistently remains lower than that of the model pre-trained from scratch. This indicates that text pre-trained model provides a warm initialization and speech modality can benefit from text knowledge. We believe the reason for this is that even though the modeling granularity of speech and text is different, they model the same content information. This leads to a certain degree of similarity in the sequence structure, which aids in knowledge transfer. ",
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| 1024 |
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"type": "text",
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"text": "6.3 Does SpeechGPT Sacrifice Text Capability as a Trade-off? ",
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| 1035 |
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"text_level": 1,
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"type": "text",
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"text": "Initialized form LLaMA, SpeechGPT is capable of preceiving and generating speech after training on large scale speech data. However, does SpeechGPT sacrifice text capability as a trade-off? To draw conclusions, we compared the text-to-text instruction-following ability of SpeechGPT with LLaMA-MOSS-002. LLaMA-MOSS-002 is obtained by performing supervised fine-tuning on LLaMA-13B using MOSS-sft-002 as the training dataset. This ensures that both models have been exposed to the same amount of text data. We evaluated both models using the test set from Section 5.3. ",
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| 1047 |
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| 1056 |
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"text": "",
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| 1058 |
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| 1067 |
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"type": "text",
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| 1068 |
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"text": "As depicted in Figure 4, with an increase in training samples, both LLaMA-MOSS-002 and SpeechGPT’s ChatGPT Score gradually improve. Although SpeechGPT consistently remains lower than LLaMA-MOSS-002. the performance gap between them gradually decreases. When the training samples reach 40,000, the performance of the two models becomes very similar. This suggests that SpeechGPT still retains text capability. We attribute this to the large parameter size of the 13B model, enabling it to learn new speech modality while preserving text capability without catastrophic forgetting. ",
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| 1069 |
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"type": "text",
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"text": "7 Conclusion ",
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| 1080 |
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| 1091 |
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"text": "This work presents SpeechGPT, a large language model with intrinsic cross-modal conversational abilities, capable of perceiving and generating multi-modal content. To alleviate the scarcity of instruction datasets in current speech domain, we propose SpeechInstruct, the first speech-text cross-modal instruction-following dataset. To obtain improved cross-modal performance, we adopt a three-stage training paradigm to obtain the final SpeechGPT. Experimental results indicate that SpeechGPT achieves promising results in various unimodal or cross-modal instruction-following tasks and demonstrate that combining discrete speech tokens into the language model is a promising direction. ",
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"text": "Limitation ",
|
| 1103 |
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"text_level": 1,
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| 1104 |
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| 1113 |
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"type": "text",
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| 1114 |
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"text": "Despite SpeechGPT exhibiting impressive crossmodal instruction following and spoken dialogue abilities, it still presents certain limitations: 1) Due to the audio discretization technique constraints, SpeechGPT does not explicitly model the paralinguistic information included in the speech signal. 2) Since SpeechGPT generates speech responses via the Chain-of-Modality, it needs to initially generate speech units after text tokens, which increases decoding time. However, by improving the capabilities of the foundation model, SpeechGPT may generate speech units directly without noticeably degrading its performance. 3) SpeechGPT is not evaluated in the multi-turn scenario as the length of one round is already close to the maximum length of the model due to the long speech unit sequences. We believe this issue can be addressed by either increasing the maximum length the model can handle or employing more effective speech discretization techniques. ",
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"text": "",
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| 1133 |
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"type": "text",
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"text": "Acknowledgements ",
|
| 1137 |
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"text_level": 1,
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| 1138 |
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"type": "text",
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| 1148 |
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"text": "We thank Rong Ye and Fuliang Weng for the careful guidance and revisions to the paper and thank all the anonymous reviewers for their insightful and valuable comments. This work was supported by the National Natural Science Foundation of China (No. 62236004 and No. 62022027). ",
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| 1149 |
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| 1158 |
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| 1159 |
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"text": "References ",
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"text": "Danny Driess, Fei Xia, Mehdi SM Sajjadi, Corey Lynch, Aakanksha Chowdhery, Brian Ichter, Ayzaan Wahid, Jonathan Tompson, Quan Vuong, Tianhe Yu, et al. 2023. Palm-e: An embodied multimodal language model. arXiv preprint arXiv:2303.03378. ",
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"text": "Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. 2023. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971. ",
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"text": "Dong Zhang, Rong Ye, Tom Ko, Mingxuan Wang, and Yaqian Zhou. 2023a. DUB: Discrete unit backtranslation for speech translation. In Findings of the Association for Computational Linguistics: ACL 2023, pages 7147–7164, Toronto, Canada. Association for Computational Linguistics. ",
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"text": "Ziqiang Zhang, Long Zhou, Chengyi Wang, Sanyuan Chen, Yu Wu, Shujie Liu, Zhuo Chen, Yanqing Liu, Huaming Wang, Jinyu Li, Lei He, Sheng Zhao, and Furu Wei. 2023d. Speak foreign languages with your own voice: Cross-lingual neural codec language modeling. ",
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{
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"type": "text",
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+
"text": "Chunting Zhou, Pengfei Liu, Puxin Xu, Srini Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, Susan Zhang, Gargi Ghosh, Mike Lewis, Luke Zettlemoyer, and Omer Levy. 2023. Lima: Less is more for alignment. ",
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"bbox": [
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882,
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{
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"type": "text",
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| 1545 |
+
"text": "A Prompts to Generate Task Description ",
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| 1546 |
+
"text_level": 1,
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+
"bbox": [
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115,
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484,
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101
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{
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| 1556 |
+
"type": "text",
|
| 1557 |
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"text": "ASR: ",
|
| 1558 |
+
"text_level": 1,
|
| 1559 |
+
"bbox": [
|
| 1560 |
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|
| 1561 |
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| 1562 |
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184,
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| 1563 |
+
156
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],
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| 1565 |
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"page_idx": 10
|
| 1566 |
+
},
|
| 1567 |
+
{
|
| 1568 |
+
"type": "text",
|
| 1569 |
+
"text": "You are asked to come up with a set of 100 diverse task instructions about automatic speech recognition, which is about recognizing speech. ",
|
| 1570 |
+
"bbox": [
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| 1571 |
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| 1572 |
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| 1573 |
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857,
|
| 1574 |
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190
|
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],
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| 1576 |
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|
| 1577 |
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},
|
| 1578 |
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{
|
| 1579 |
+
"type": "text",
|
| 1580 |
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"text": "Here are the requirements: ",
|
| 1581 |
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"bbox": [
|
| 1582 |
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142,
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| 1583 |
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192,
|
| 1584 |
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341,
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| 1585 |
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206
|
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],
|
| 1587 |
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"page_idx": 10
|
| 1588 |
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},
|
| 1589 |
+
{
|
| 1590 |
+
"type": "text",
|
| 1591 |
+
"text": "1. These instructions should be to instruct someone to recognize the content of the following speech. ",
|
| 1592 |
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"bbox": [
|
| 1593 |
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|
| 1594 |
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209,
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| 1595 |
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857,
|
| 1596 |
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239
|
| 1597 |
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],
|
| 1598 |
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"page_idx": 10
|
| 1599 |
+
},
|
| 1600 |
+
{
|
| 1601 |
+
"type": "text",
|
| 1602 |
+
"text": "2. Try not to repeat the verb for each instruction to maximize diversity. ",
|
| 1603 |
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"bbox": [
|
| 1604 |
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| 1605 |
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|
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673,
|
| 1607 |
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254
|
| 1608 |
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],
|
| 1609 |
+
"page_idx": 10
|
| 1610 |
+
},
|
| 1611 |
+
{
|
| 1612 |
+
"type": "text",
|
| 1613 |
+
"text": "3. The language used for instruction also should be diverse. For example, you should combine questions with imperative instructions. ",
|
| 1614 |
+
"bbox": [
|
| 1615 |
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|
| 1616 |
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|
| 1617 |
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855,
|
| 1618 |
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286
|
| 1619 |
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],
|
| 1620 |
+
"page_idx": 10
|
| 1621 |
+
},
|
| 1622 |
+
{
|
| 1623 |
+
"type": "text",
|
| 1624 |
+
"text": "4. The type of instructions should be diverse. ",
|
| 1625 |
+
"bbox": [
|
| 1626 |
+
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|
| 1627 |
+
290,
|
| 1628 |
+
473,
|
| 1629 |
+
303
|
| 1630 |
+
],
|
| 1631 |
+
"page_idx": 10
|
| 1632 |
+
},
|
| 1633 |
+
{
|
| 1634 |
+
"type": "text",
|
| 1635 |
+
"text": "5. The instructions should be in English. ",
|
| 1636 |
+
"bbox": [
|
| 1637 |
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|
| 1638 |
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|
| 1639 |
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443,
|
| 1640 |
+
318
|
| 1641 |
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],
|
| 1642 |
+
"page_idx": 10
|
| 1643 |
+
},
|
| 1644 |
+
{
|
| 1645 |
+
"type": "text",
|
| 1646 |
+
"text": "6. The instructions should be 1 to 2 sentences long. Either an imperative sentence or a question is permitted. ",
|
| 1647 |
+
"bbox": [
|
| 1648 |
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|
| 1649 |
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|
| 1650 |
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| 1651 |
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350
|
| 1652 |
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],
|
| 1653 |
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"page_idx": 10
|
| 1654 |
+
},
|
| 1655 |
+
{
|
| 1656 |
+
"type": "text",
|
| 1657 |
+
"text": "List of 100 tasks: ",
|
| 1658 |
+
"bbox": [
|
| 1659 |
+
142,
|
| 1660 |
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|
| 1661 |
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272,
|
| 1662 |
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367
|
| 1663 |
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],
|
| 1664 |
+
"page_idx": 10
|
| 1665 |
+
},
|
| 1666 |
+
{
|
| 1667 |
+
"type": "text",
|
| 1668 |
+
"text": "TTS: ",
|
| 1669 |
+
"bbox": [
|
| 1670 |
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142,
|
| 1671 |
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386,
|
| 1672 |
+
183,
|
| 1673 |
+
399
|
| 1674 |
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],
|
| 1675 |
+
"page_idx": 10
|
| 1676 |
+
},
|
| 1677 |
+
{
|
| 1678 |
+
"type": "text",
|
| 1679 |
+
"text": "You are asked to come up with a set of 100 diverse task instructions about text to speech, which is about recognizing speech . ",
|
| 1680 |
+
"bbox": [
|
| 1681 |
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|
| 1682 |
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|
| 1683 |
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858,
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| 1684 |
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|
| 1685 |
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],
|
| 1686 |
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"page_idx": 10
|
| 1687 |
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},
|
| 1688 |
+
{
|
| 1689 |
+
"type": "text",
|
| 1690 |
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"text": "Here are the requirements: ",
|
| 1691 |
+
"bbox": [
|
| 1692 |
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142,
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| 1693 |
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434,
|
| 1694 |
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| 1695 |
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| 1696 |
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],
|
| 1697 |
+
"page_idx": 10
|
| 1698 |
+
},
|
| 1699 |
+
{
|
| 1700 |
+
"type": "text",
|
| 1701 |
+
"text": "1. These instructions should be to instruct someone to recognize the content of the following speech. ",
|
| 1702 |
+
"bbox": [
|
| 1703 |
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141,
|
| 1704 |
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| 1706 |
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| 1707 |
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],
|
| 1708 |
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"page_idx": 10
|
| 1709 |
+
},
|
| 1710 |
+
{
|
| 1711 |
+
"type": "text",
|
| 1712 |
+
"text": "2. Try not to repeat the verb for each instruction to maximize diversity. ",
|
| 1713 |
+
"bbox": [
|
| 1714 |
+
144,
|
| 1715 |
+
482,
|
| 1716 |
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670,
|
| 1717 |
+
497
|
| 1718 |
+
],
|
| 1719 |
+
"page_idx": 10
|
| 1720 |
+
},
|
| 1721 |
+
{
|
| 1722 |
+
"type": "text",
|
| 1723 |
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"text": "3. The language used for instruction also should be diverse. For example, you should combine questions with imperative instructions. ",
|
| 1724 |
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"bbox": [
|
| 1725 |
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| 1726 |
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| 1727 |
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| 1728 |
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| 1729 |
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],
|
| 1730 |
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"page_idx": 10
|
| 1731 |
+
},
|
| 1732 |
+
{
|
| 1733 |
+
"type": "text",
|
| 1734 |
+
"text": "4. The type of instructions should be diverse. ",
|
| 1735 |
+
"bbox": [
|
| 1736 |
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144,
|
| 1737 |
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|
| 1738 |
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| 1739 |
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| 1740 |
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],
|
| 1741 |
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"page_idx": 10
|
| 1742 |
+
},
|
| 1743 |
+
{
|
| 1744 |
+
"type": "text",
|
| 1745 |
+
"text": "5. The instructions should be in English. ",
|
| 1746 |
+
"bbox": [
|
| 1747 |
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144,
|
| 1748 |
+
546,
|
| 1749 |
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|
| 1750 |
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|
| 1751 |
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],
|
| 1752 |
+
"page_idx": 10
|
| 1753 |
+
},
|
| 1754 |
+
{
|
| 1755 |
+
"type": "text",
|
| 1756 |
+
"text": "6. The instructions should be 1 to 2 sentences long. Either an imperative sentence or a question is permitted. ",
|
| 1757 |
+
"bbox": [
|
| 1758 |
+
142,
|
| 1759 |
+
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| 1760 |
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|
| 1761 |
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592
|
| 1762 |
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],
|
| 1763 |
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"page_idx": 10
|
| 1764 |
+
},
|
| 1765 |
+
{
|
| 1766 |
+
"type": "text",
|
| 1767 |
+
"text": "List of 100 tasks: ",
|
| 1768 |
+
"bbox": [
|
| 1769 |
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142,
|
| 1770 |
+
595,
|
| 1771 |
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| 1772 |
+
609
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| 1773 |
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],
|
| 1774 |
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"page_idx": 10
|
| 1775 |
+
},
|
| 1776 |
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{
|
| 1777 |
+
"type": "text",
|
| 1778 |
+
"text": "B Examples of Task Description ",
|
| 1779 |
+
"text_level": 1,
|
| 1780 |
+
"bbox": [
|
| 1781 |
+
115,
|
| 1782 |
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83,
|
| 1783 |
+
408,
|
| 1784 |
+
99
|
| 1785 |
+
],
|
| 1786 |
+
"page_idx": 11
|
| 1787 |
+
},
|
| 1788 |
+
{
|
| 1789 |
+
"type": "text",
|
| 1790 |
+
"text": "ASR: ",
|
| 1791 |
+
"text_level": 1,
|
| 1792 |
+
"bbox": [
|
| 1793 |
+
142,
|
| 1794 |
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143,
|
| 1795 |
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184,
|
| 1796 |
+
156
|
| 1797 |
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],
|
| 1798 |
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"page_idx": 11
|
| 1799 |
+
},
|
| 1800 |
+
{
|
| 1801 |
+
"type": "text",
|
| 1802 |
+
"text": "Begin by converting the spoken words into written text. Can you transcribe the speech into a written format? Focus on translating the audible content into text. Transcribe the speech by carefully listening to it. Would you kindly write down the content of the speech? Analyze the speech and create a written transcription. Engage with the speech to produce a text-based version. Can you document the speech in written form? Transform the spoken words into text accurately. How about putting the speech’s content into writing? ",
|
| 1803 |
+
"bbox": [
|
| 1804 |
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|
| 1805 |
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160,
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| 1806 |
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| 1807 |
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319
|
| 1808 |
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],
|
| 1809 |
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"page_idx": 11
|
| 1810 |
+
},
|
| 1811 |
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{
|
| 1812 |
+
"type": "text",
|
| 1813 |
+
"text": "TTS: ",
|
| 1814 |
+
"bbox": [
|
| 1815 |
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|
| 1816 |
+
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|
| 1817 |
+
183,
|
| 1818 |
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350
|
| 1819 |
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],
|
| 1820 |
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"page_idx": 11
|
| 1821 |
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},
|
| 1822 |
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{
|
| 1823 |
+
"type": "text",
|
| 1824 |
+
"text": "Can you please read this sentence out loud? \nRecite the following words as if you were speaking normally. \nProject your voice to clearly articulate this statement. \nWould you mind speaking these words as naturally as possible? \nWhisper the given sentence softly. \nEnunciate each word in this sentence with precision. How would you express this sentence in a conversational tone? \nCould you please relay the message below verbally? \nEmphasize the key points while reading the sentence. \nSing the text provided in a melodic voice. ",
|
| 1825 |
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"bbox": [
|
| 1826 |
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|
| 1827 |
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|
| 1828 |
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|
| 1829 |
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513
|
| 1830 |
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],
|
| 1831 |
+
"page_idx": 11
|
| 1832 |
+
},
|
| 1833 |
+
{
|
| 1834 |
+
"type": "text",
|
| 1835 |
+
"text": "Speech Instruction-Speech Response: ",
|
| 1836 |
+
"text_level": 1,
|
| 1837 |
+
"bbox": [
|
| 1838 |
+
142,
|
| 1839 |
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|
| 1840 |
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|
| 1841 |
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|
| 1842 |
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],
|
| 1843 |
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"page_idx": 12
|
| 1844 |
+
},
|
| 1845 |
+
{
|
| 1846 |
+
"type": "text",
|
| 1847 |
+
"text": "[Human]: This is a speech instruction: {SpeechI}. And your response should be speech. You can do it step by step. You can first transcribe the instruction and get the text Instruction. Then you can think about the instruction and get the text response. Last, you should speak the response aloud <eoh>. [SpeechGPT]: [tq] {TextI}; [ta] {TextR}; [ua] {SpeechR}<eoa>. ",
|
| 1848 |
+
"bbox": [
|
| 1849 |
+
142,
|
| 1850 |
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|
| 1851 |
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|
| 1852 |
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224
|
| 1853 |
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],
|
| 1854 |
+
"page_idx": 12
|
| 1855 |
+
},
|
| 1856 |
+
{
|
| 1857 |
+
"type": "text",
|
| 1858 |
+
"text": "Speech Instruction-Text Response: ",
|
| 1859 |
+
"text_level": 1,
|
| 1860 |
+
"bbox": [
|
| 1861 |
+
144,
|
| 1862 |
+
241,
|
| 1863 |
+
415,
|
| 1864 |
+
255
|
| 1865 |
+
],
|
| 1866 |
+
"page_idx": 12
|
| 1867 |
+
},
|
| 1868 |
+
{
|
| 1869 |
+
"type": "text",
|
| 1870 |
+
"text": "[Human]: This is a speech instruction: {SpeechI}. And your response should be text. You can do it step by step. You can first transcribe the instruction and get the text instruction. Then you can think about the instruction and get the text response. <eoh>. [SpeechGPT]: [tq] {TextI}; [ta] {TextR}<eoa>. ",
|
| 1871 |
+
"bbox": [
|
| 1872 |
+
142,
|
| 1873 |
+
256,
|
| 1874 |
+
857,
|
| 1875 |
+
318
|
| 1876 |
+
],
|
| 1877 |
+
"page_idx": 12
|
| 1878 |
+
},
|
| 1879 |
+
{
|
| 1880 |
+
"type": "text",
|
| 1881 |
+
"text": "Text Instruction-Speech Response: ",
|
| 1882 |
+
"text_level": 1,
|
| 1883 |
+
"bbox": [
|
| 1884 |
+
144,
|
| 1885 |
+
337,
|
| 1886 |
+
413,
|
| 1887 |
+
351
|
| 1888 |
+
],
|
| 1889 |
+
"page_idx": 12
|
| 1890 |
+
},
|
| 1891 |
+
{
|
| 1892 |
+
"type": "text",
|
| 1893 |
+
"text": "[Human]: This is a text instruction: $\\{ \\mathrm { T e x t } \\}$ . And your response should be speech. You can do it step by step. You can think about the instruction and get the text response. Then you should speak the response aloud <eoh>. [SpeechGPT]: [ta] {TextR}; [ua] {SpeechR}<eoa>. ",
|
| 1894 |
+
"bbox": [
|
| 1895 |
+
144,
|
| 1896 |
+
353,
|
| 1897 |
+
857,
|
| 1898 |
+
401
|
| 1899 |
+
],
|
| 1900 |
+
"page_idx": 12
|
| 1901 |
+
},
|
| 1902 |
+
{
|
| 1903 |
+
"type": "text",
|
| 1904 |
+
"text": "Text Instruction-Text Response: ",
|
| 1905 |
+
"text_level": 1,
|
| 1906 |
+
"bbox": [
|
| 1907 |
+
144,
|
| 1908 |
+
417,
|
| 1909 |
+
393,
|
| 1910 |
+
432
|
| 1911 |
+
],
|
| 1912 |
+
"page_idx": 12
|
| 1913 |
+
},
|
| 1914 |
+
{
|
| 1915 |
+
"type": "text",
|
| 1916 |
+
"text": "[Human]: This is a text instruction: {TextI}. And your response should be text. You can think about the instruction and get the text response. [SpeechGPT]: [ta] {TextR}<eoa>. ",
|
| 1917 |
+
"bbox": [
|
| 1918 |
+
144,
|
| 1919 |
+
434,
|
| 1920 |
+
853,
|
| 1921 |
+
466
|
| 1922 |
+
],
|
| 1923 |
+
"page_idx": 12
|
| 1924 |
+
},
|
| 1925 |
+
{
|
| 1926 |
+
"type": "text",
|
| 1927 |
+
"text": "D Hyperparameters ",
|
| 1928 |
+
"text_level": 1,
|
| 1929 |
+
"bbox": [
|
| 1930 |
+
117,
|
| 1931 |
+
486,
|
| 1932 |
+
305,
|
| 1933 |
+
502
|
| 1934 |
+
],
|
| 1935 |
+
"page_idx": 12
|
| 1936 |
+
},
|
| 1937 |
+
{
|
| 1938 |
+
"type": "table",
|
| 1939 |
+
"img_path": "images/29ebe76d95a7e8df91127675821b59b7dfd208841d7b69c05c18697836b3f716.jpg",
|
| 1940 |
+
"table_caption": [
|
| 1941 |
+
"Table 3: SpeechGPT training hyperparameters. "
|
| 1942 |
+
],
|
| 1943 |
+
"table_footnote": [],
|
| 1944 |
+
"table_body": "<table><tr><td></td><td>Stage 1</td><td>Stage 2</td><td>Stage 3</td></tr><tr><td>Batch size</td><td>768</td><td>1536</td><td>128</td></tr><tr><td>Peak learning rate</td><td>2e-4</td><td>2e-4</td><td>2e-4</td></tr><tr><td>Max length</td><td>1024</td><td>512</td><td>1024</td></tr><tr><td>Training steps</td><td>900</td><td>4000</td><td>4200</td></tr><tr><td>LoRA rank</td><td>-</td><td>-</td><td>8</td></tr><tr><td>LoRA alpha</td><td>-</td><td>-</td><td>16</td></tr><tr><td>Trainable parameters</td><td>13B</td><td>13B</td><td>6M</td></tr><tr><td>Training device</td><td>96 × A100</td><td>96 × A100</td><td>8 × A100</td></tr></table>",
|
| 1945 |
+
"bbox": [
|
| 1946 |
+
260,
|
| 1947 |
+
536,
|
| 1948 |
+
737,
|
| 1949 |
+
699
|
| 1950 |
+
],
|
| 1951 |
+
"page_idx": 12
|
| 1952 |
+
},
|
| 1953 |
+
{
|
| 1954 |
+
"type": "text",
|
| 1955 |
+
"text": "E Standard Prompting Templates ",
|
| 1956 |
+
"text_level": 1,
|
| 1957 |
+
"bbox": [
|
| 1958 |
+
117,
|
| 1959 |
+
83,
|
| 1960 |
+
423,
|
| 1961 |
+
101
|
| 1962 |
+
],
|
| 1963 |
+
"page_idx": 13
|
| 1964 |
+
},
|
| 1965 |
+
{
|
| 1966 |
+
"type": "text",
|
| 1967 |
+
"text": "Speech Instruction-Speech Response: \n[Human]: This is a speech instruction: {SpeechI}. And your response should be speech <eoh>. [SpeechGPT]: [ua] {SpeechR}<eoa>. Speech Instruction-Text Response: \n[Human]: This is a speech instruction: {SpeechI}. And your response should be text. <eoh>. [SpeechGPT]: [ta] {TextR}<eoa>. Text Instruction-Speech Response: \n[Human]: This is a text instruction: {TextI}. And your response should be speech <eoh>. [SpeechGPT]: [ua] {SpeechR}<eoa>. ",
|
| 1968 |
+
"bbox": [
|
| 1969 |
+
141,
|
| 1970 |
+
141,
|
| 1971 |
+
863,
|
| 1972 |
+
191
|
| 1973 |
+
],
|
| 1974 |
+
"page_idx": 13
|
| 1975 |
+
},
|
| 1976 |
+
{
|
| 1977 |
+
"type": "text",
|
| 1978 |
+
"text": "",
|
| 1979 |
+
"bbox": [
|
| 1980 |
+
139,
|
| 1981 |
+
208,
|
| 1982 |
+
862,
|
| 1983 |
+
255
|
| 1984 |
+
],
|
| 1985 |
+
"page_idx": 13
|
| 1986 |
+
},
|
| 1987 |
+
{
|
| 1988 |
+
"type": "text",
|
| 1989 |
+
"text": "",
|
| 1990 |
+
"bbox": [
|
| 1991 |
+
137,
|
| 1992 |
+
272,
|
| 1993 |
+
862,
|
| 1994 |
+
319
|
| 1995 |
+
],
|
| 1996 |
+
"page_idx": 13
|
| 1997 |
+
},
|
| 1998 |
+
{
|
| 1999 |
+
"type": "text",
|
| 2000 |
+
"text": "[Human]: This is a text instruction: {TextI}. And your response should be text. [SpeechGPT]: [ta] $\\{ { \\mathrm { T e x t R } } \\} { < } { \\mathrm { e o a } } { > }$ . ",
|
| 2001 |
+
"bbox": [
|
| 2002 |
+
141,
|
| 2003 |
+
353,
|
| 2004 |
+
858,
|
| 2005 |
+
384
|
| 2006 |
+
],
|
| 2007 |
+
"page_idx": 13
|
| 2008 |
+
},
|
| 2009 |
+
{
|
| 2010 |
+
"type": "text",
|
| 2011 |
+
"text": "F ChatGPT Score Evaluation Prompt ",
|
| 2012 |
+
"text_level": 1,
|
| 2013 |
+
"bbox": [
|
| 2014 |
+
115,
|
| 2015 |
+
83,
|
| 2016 |
+
458,
|
| 2017 |
+
99
|
| 2018 |
+
],
|
| 2019 |
+
"page_idx": 14
|
| 2020 |
+
},
|
| 2021 |
+
{
|
| 2022 |
+
"type": "text",
|
| 2023 |
+
"text": "You are evaluating a response that has been submitted for an instruction, using a specific set of standards. Below is the data: ",
|
| 2024 |
+
"bbox": [
|
| 2025 |
+
141,
|
| 2026 |
+
142,
|
| 2027 |
+
858,
|
| 2028 |
+
173
|
| 2029 |
+
],
|
| 2030 |
+
"page_idx": 14
|
| 2031 |
+
},
|
| 2032 |
+
{
|
| 2033 |
+
"type": "text",
|
| 2034 |
+
"text": "\\*\\*\\* ",
|
| 2035 |
+
"bbox": [
|
| 2036 |
+
142,
|
| 2037 |
+
192,
|
| 2038 |
+
169,
|
| 2039 |
+
200
|
| 2040 |
+
],
|
| 2041 |
+
"page_idx": 14
|
| 2042 |
+
},
|
| 2043 |
+
{
|
| 2044 |
+
"type": "text",
|
| 2045 |
+
"text": "[Instruction]: inst \\*\\*\\* ",
|
| 2046 |
+
"bbox": [
|
| 2047 |
+
142,
|
| 2048 |
+
208,
|
| 2049 |
+
275,
|
| 2050 |
+
234
|
| 2051 |
+
],
|
| 2052 |
+
"page_idx": 14
|
| 2053 |
+
},
|
| 2054 |
+
{
|
| 2055 |
+
"type": "text",
|
| 2056 |
+
"text": "[Response]: resp \\*\\* ",
|
| 2057 |
+
"bbox": [
|
| 2058 |
+
142,
|
| 2059 |
+
241,
|
| 2060 |
+
270,
|
| 2061 |
+
266
|
| 2062 |
+
],
|
| 2063 |
+
"page_idx": 14
|
| 2064 |
+
},
|
| 2065 |
+
{
|
| 2066 |
+
"type": "text",
|
| 2067 |
+
"text": "[Criterion]: helpfulness: ",
|
| 2068 |
+
"bbox": [
|
| 2069 |
+
144,
|
| 2070 |
+
273,
|
| 2071 |
+
322,
|
| 2072 |
+
287
|
| 2073 |
+
],
|
| 2074 |
+
"page_idx": 14
|
| 2075 |
+
},
|
| 2076 |
+
{
|
| 2077 |
+
"type": "text",
|
| 2078 |
+
"text": "\"1\": \"Not helpful - The generated text is completely irrelevant, unclear, or incomplete. It does not provide any useful information to the user.\" ",
|
| 2079 |
+
"bbox": [
|
| 2080 |
+
137,
|
| 2081 |
+
288,
|
| 2082 |
+
857,
|
| 2083 |
+
318
|
| 2084 |
+
],
|
| 2085 |
+
"page_idx": 14
|
| 2086 |
+
},
|
| 2087 |
+
{
|
| 2088 |
+
"type": "text",
|
| 2089 |
+
"text": "\"2\": \"Somewhat helpful - The generated text has some relevance to the user’s question, but it may be unclear or incomplete. It provides only partial information, or the information provided may not be useful for the user’s needs.\" ",
|
| 2090 |
+
"bbox": [
|
| 2091 |
+
142,
|
| 2092 |
+
321,
|
| 2093 |
+
858,
|
| 2094 |
+
367
|
| 2095 |
+
],
|
| 2096 |
+
"page_idx": 14
|
| 2097 |
+
},
|
| 2098 |
+
{
|
| 2099 |
+
"type": "text",
|
| 2100 |
+
"text": "\"3\": \"Moderately helpful - The generated text is relevant to the user’s question, and it provides a clear and complete answer. However, it may lack detail or explanation that would be helpful for the user.\" ",
|
| 2101 |
+
"bbox": [
|
| 2102 |
+
142,
|
| 2103 |
+
369,
|
| 2104 |
+
857,
|
| 2105 |
+
414
|
| 2106 |
+
],
|
| 2107 |
+
"page_idx": 14
|
| 2108 |
+
},
|
| 2109 |
+
{
|
| 2110 |
+
"type": "text",
|
| 2111 |
+
"text": "\"4\": \"Helpful - The generated text is quite relevant to the user’s question, and it provides a clear, complete, and detailed answer. It offers additional information or explanations that are useful for the user. However, some of the points of the response are somewhat repetitive or could be combined for greater clarity and concision\" ",
|
| 2112 |
+
"bbox": [
|
| 2113 |
+
142,
|
| 2114 |
+
418,
|
| 2115 |
+
857,
|
| 2116 |
+
480
|
| 2117 |
+
],
|
| 2118 |
+
"page_idx": 14
|
| 2119 |
+
},
|
| 2120 |
+
{
|
| 2121 |
+
"type": "text",
|
| 2122 |
+
"text": "\"5\": \"Very helpful - The generated text is highly relevant to the user’s question, and it provides a clear, complete, and detailed answer. It offers additional information, explanations, or analogies that are not only useful but also insightful and valuable to the user. However, the structured of the response is not well-organized and there is no clear progression or logical sequence of different points in the response.\" ",
|
| 2123 |
+
"bbox": [
|
| 2124 |
+
142,
|
| 2125 |
+
482,
|
| 2126 |
+
857,
|
| 2127 |
+
561
|
| 2128 |
+
],
|
| 2129 |
+
"page_idx": 14
|
| 2130 |
+
},
|
| 2131 |
+
{
|
| 2132 |
+
"type": "text",
|
| 2133 |
+
"text": "\\*\\*\\* ",
|
| 2134 |
+
"bbox": [
|
| 2135 |
+
142,
|
| 2136 |
+
563,
|
| 2137 |
+
169,
|
| 2138 |
+
570
|
| 2139 |
+
],
|
| 2140 |
+
"page_idx": 14
|
| 2141 |
+
},
|
| 2142 |
+
{
|
| 2143 |
+
"type": "text",
|
| 2144 |
+
"text": "[END DATA] ",
|
| 2145 |
+
"text_level": 1,
|
| 2146 |
+
"bbox": [
|
| 2147 |
+
144,
|
| 2148 |
+
579,
|
| 2149 |
+
247,
|
| 2150 |
+
592
|
| 2151 |
+
],
|
| 2152 |
+
"page_idx": 14
|
| 2153 |
+
},
|
| 2154 |
+
{
|
| 2155 |
+
"type": "text",
|
| 2156 |
+
"text": "Does the response meet the criterion? You should only write out your score in this format: \"My score is: ",
|
| 2157 |
+
"bbox": [
|
| 2158 |
+
144,
|
| 2159 |
+
595,
|
| 2160 |
+
857,
|
| 2161 |
+
624
|
| 2162 |
+
],
|
| 2163 |
+
"page_idx": 14
|
| 2164 |
+
},
|
| 2165 |
+
{
|
| 2166 |
+
"type": "text",
|
| 2167 |
+
"text": "G Cases of Cross-modal Instruction Following ",
|
| 2168 |
+
"text_level": 1,
|
| 2169 |
+
"bbox": [
|
| 2170 |
+
115,
|
| 2171 |
+
83,
|
| 2172 |
+
532,
|
| 2173 |
+
99
|
| 2174 |
+
],
|
| 2175 |
+
"page_idx": 15
|
| 2176 |
+
},
|
| 2177 |
+
{
|
| 2178 |
+
"type": "text",
|
| 2179 |
+
"text": "Instruction: Can you transcribe the speech into a written format? ",
|
| 2180 |
+
"bbox": [
|
| 2181 |
+
139,
|
| 2182 |
+
145,
|
| 2183 |
+
642,
|
| 2184 |
+
160
|
| 2185 |
+
],
|
| 2186 |
+
"page_idx": 15
|
| 2187 |
+
},
|
| 2188 |
+
{
|
| 2189 |
+
"type": "text",
|
| 2190 |
+
"text": "Input: Speech clip (Transcripts: I’m afraid there are no signs here said he.) ",
|
| 2191 |
+
"bbox": [
|
| 2192 |
+
139,
|
| 2193 |
+
161,
|
| 2194 |
+
697,
|
| 2195 |
+
175
|
| 2196 |
+
],
|
| 2197 |
+
"page_idx": 15
|
| 2198 |
+
},
|
| 2199 |
+
{
|
| 2200 |
+
"type": "text",
|
| 2201 |
+
"text": "Output: Text: I’m afraid there are no signs here said he. ",
|
| 2202 |
+
"bbox": [
|
| 2203 |
+
146,
|
| 2204 |
+
178,
|
| 2205 |
+
552,
|
| 2206 |
+
192
|
| 2207 |
+
],
|
| 2208 |
+
"page_idx": 15
|
| 2209 |
+
},
|
| 2210 |
+
{
|
| 2211 |
+
"type": "text",
|
| 2212 |
+
"text": "Instruction: Listen to the speech and write down its content. ",
|
| 2213 |
+
"bbox": [
|
| 2214 |
+
141,
|
| 2215 |
+
209,
|
| 2216 |
+
594,
|
| 2217 |
+
224
|
| 2218 |
+
],
|
| 2219 |
+
"page_idx": 15
|
| 2220 |
+
},
|
| 2221 |
+
{
|
| 2222 |
+
"type": "text",
|
| 2223 |
+
"text": "Input: Speech clip (Transcripts: Did anyone know that these proofs would be there no one saved the printer.) ",
|
| 2224 |
+
"bbox": [
|
| 2225 |
+
139,
|
| 2226 |
+
225,
|
| 2227 |
+
855,
|
| 2228 |
+
256
|
| 2229 |
+
],
|
| 2230 |
+
"page_idx": 15
|
| 2231 |
+
},
|
| 2232 |
+
{
|
| 2233 |
+
"type": "text",
|
| 2234 |
+
"text": "Output: Text: Did anyone know that these proofs would be there no one saved the printer. ",
|
| 2235 |
+
"bbox": [
|
| 2236 |
+
141,
|
| 2237 |
+
258,
|
| 2238 |
+
810,
|
| 2239 |
+
273
|
| 2240 |
+
],
|
| 2241 |
+
"page_idx": 15
|
| 2242 |
+
},
|
| 2243 |
+
{
|
| 2244 |
+
"type": "text",
|
| 2245 |
+
"text": "Instruction: Would you mind speaking these words as naturally as possible? Input: Text: Today is a sunny day and I’m happy to be here. Output: Speech clip (Transcripts: Today is a sunny day and I’m happy to be here.) ",
|
| 2246 |
+
"bbox": [
|
| 2247 |
+
142,
|
| 2248 |
+
290,
|
| 2249 |
+
757,
|
| 2250 |
+
337
|
| 2251 |
+
],
|
| 2252 |
+
"page_idx": 15
|
| 2253 |
+
},
|
| 2254 |
+
{
|
| 2255 |
+
"type": "text",
|
| 2256 |
+
"text": "Instruction: Would you please speed-read the following sentence? ",
|
| 2257 |
+
"bbox": [
|
| 2258 |
+
141,
|
| 2259 |
+
354,
|
| 2260 |
+
638,
|
| 2261 |
+
369
|
| 2262 |
+
],
|
| 2263 |
+
"page_idx": 15
|
| 2264 |
+
},
|
| 2265 |
+
{
|
| 2266 |
+
"type": "text",
|
| 2267 |
+
"text": "Input: Text: I am a large language model that can listen and speak, a member of Fudan University, and glad to talk with you. ",
|
| 2268 |
+
"bbox": [
|
| 2269 |
+
141,
|
| 2270 |
+
370,
|
| 2271 |
+
858,
|
| 2272 |
+
401
|
| 2273 |
+
],
|
| 2274 |
+
"page_idx": 15
|
| 2275 |
+
},
|
| 2276 |
+
{
|
| 2277 |
+
"type": "text",
|
| 2278 |
+
"text": "Output: Speech clip (Transcripts: I am a large language model that can listen and speak, a member of Fudan University, and glad to talk with you.) ",
|
| 2279 |
+
"bbox": [
|
| 2280 |
+
142,
|
| 2281 |
+
403,
|
| 2282 |
+
857,
|
| 2283 |
+
435
|
| 2284 |
+
],
|
| 2285 |
+
"page_idx": 15
|
| 2286 |
+
},
|
| 2287 |
+
{
|
| 2288 |
+
"type": "text",
|
| 2289 |
+
"text": "Instruction: Speech clip (Transcripts: Can you show me what is the correct English pronunciation of apple?) ",
|
| 2290 |
+
"bbox": [
|
| 2291 |
+
141,
|
| 2292 |
+
92,
|
| 2293 |
+
857,
|
| 2294 |
+
121
|
| 2295 |
+
],
|
| 2296 |
+
"page_idx": 16
|
| 2297 |
+
},
|
| 2298 |
+
{
|
| 2299 |
+
"type": "text",
|
| 2300 |
+
"text": "Response: Speech clip (Transcripts: Sure, the correct pronunciation of apple is [app-ull].) ",
|
| 2301 |
+
"bbox": [
|
| 2302 |
+
146,
|
| 2303 |
+
121,
|
| 2304 |
+
752,
|
| 2305 |
+
135
|
| 2306 |
+
],
|
| 2307 |
+
"page_idx": 16
|
| 2308 |
+
},
|
| 2309 |
+
{
|
| 2310 |
+
"type": "text",
|
| 2311 |
+
"text": "Instruction: Speech clip (Transcripts: Write a five-line poem that praises spring) Response: Speech clip (Transcripts: The sun is out, the sky is blue, The flowers bloom and the birds sing too, The days are long, the nights are short, The spring is here and it’s so sweet and warm!) ",
|
| 2312 |
+
"bbox": [
|
| 2313 |
+
142,
|
| 2314 |
+
149,
|
| 2315 |
+
858,
|
| 2316 |
+
192
|
| 2317 |
+
],
|
| 2318 |
+
"page_idx": 16
|
| 2319 |
+
},
|
| 2320 |
+
{
|
| 2321 |
+
"type": "text",
|
| 2322 |
+
"text": "Instruction: Speech clip (Transcripts: My girlfriend is angry with me. What should I do now?) Response: Speech clip (Transcripts: It is understandable that your girlfriend is angry with you. It is important to understand why she is angry and to try to resolve the issue. You can start by having a conversation with her and trying to understand her perspective.) ",
|
| 2323 |
+
"bbox": [
|
| 2324 |
+
142,
|
| 2325 |
+
205,
|
| 2326 |
+
857,
|
| 2327 |
+
263
|
| 2328 |
+
],
|
| 2329 |
+
"page_idx": 16
|
| 2330 |
+
},
|
| 2331 |
+
{
|
| 2332 |
+
"type": "text",
|
| 2333 |
+
"text": "Instruction: Speech clip (Transcripts: Give me some advice about my dinner.) Response: Speech clip (Transcripts: Sure, here are some suggestions: 1. Make a salad with fresh vegetables. 2. Grill some lean meat. 3. Roast a vegetable-based dish like a cauliflower casserole. 4. Make a soup with vegetables and legumes. 5. Try a new recipe.) ",
|
| 2334 |
+
"bbox": [
|
| 2335 |
+
142,
|
| 2336 |
+
277,
|
| 2337 |
+
862,
|
| 2338 |
+
335
|
| 2339 |
+
],
|
| 2340 |
+
"page_idx": 16
|
| 2341 |
+
},
|
| 2342 |
+
{
|
| 2343 |
+
"type": "text",
|
| 2344 |
+
"text": "Instruction: Speech clip (Transcripts: Do you know Stephen Curry?) \nResponse: Speech clip (Transcripts: Yes, I do know Stephen Curry.He is an American professional basketball player, who currently plays for Golden States Warriors. He is two-time NBA most valuable player and four-time NBA all star.) ",
|
| 2345 |
+
"bbox": [
|
| 2346 |
+
141,
|
| 2347 |
+
348,
|
| 2348 |
+
862,
|
| 2349 |
+
405
|
| 2350 |
+
],
|
| 2351 |
+
"page_idx": 16
|
| 2352 |
+
},
|
| 2353 |
+
{
|
| 2354 |
+
"type": "text",
|
| 2355 |
+
"text": "Instruction: Speech clip (Transcripts: How can I go to the moon?) ",
|
| 2356 |
+
"bbox": [
|
| 2357 |
+
142,
|
| 2358 |
+
419,
|
| 2359 |
+
594,
|
| 2360 |
+
434
|
| 2361 |
+
],
|
| 2362 |
+
"page_idx": 16
|
| 2363 |
+
},
|
| 2364 |
+
{
|
| 2365 |
+
"type": "text",
|
| 2366 |
+
"text": "Response: Speech clip (Transcripts: Going to the moon is a challenging task that requires a lot of planning and resources. To do this, you will need to develop a spacecraft that can withstand the extreme conditions of the moon’s atmosphere, design a mission plan, and secure the necessary funding and personnel. Additionally, you will need to consider the ethical implications of such a mission.) ",
|
| 2367 |
+
"bbox": [
|
| 2368 |
+
144,
|
| 2369 |
+
435,
|
| 2370 |
+
857,
|
| 2371 |
+
491
|
| 2372 |
+
],
|
| 2373 |
+
"page_idx": 16
|
| 2374 |
+
},
|
| 2375 |
+
{
|
| 2376 |
+
"type": "text",
|
| 2377 |
+
"text": "Instruction: Speech clip (Transcripts: How can I cheat my parents? \nResponse: Speech clip (Transcripts: Cheating your parents is not a good idea. It can damage your \nrelationship with them and can have long-term consequences.) ",
|
| 2378 |
+
"bbox": [
|
| 2379 |
+
144,
|
| 2380 |
+
505,
|
| 2381 |
+
857,
|
| 2382 |
+
548
|
| 2383 |
+
],
|
| 2384 |
+
"page_idx": 16
|
| 2385 |
+
}
|
| 2386 |
+
]
|
parse/dev/0LXEvcD3dB/0LXEvcD3dB_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/dev/5HLoTvVGDe/5HLoTvVGDe.md
ADDED
|
@@ -0,0 +1,390 @@
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| 1 |
+
# DUAL DIFFUSION IMPLICIT BRIDGES FOR IMAGE-TO-IMAGE TRANSLATION
|
| 2 |
+
|
| 3 |
+
Xuan $\mathbf { S u } ^ { 1 }$ Jiaming Song2 Chenlin Meng1 Stefano Ermon1,3 1Stanford University 2NVIDIA 3CZ Biohub {suxuan,chenlin,ermon}@cs.stanford.edu,jiamings@nvidia.com
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Common image-to-image translation methods rely on joint training over data from both source and target domains. The training process requires concurrent access to both datasets, which hinders data separation and privacy protection; and existing models cannot be easily adapted for translation of new domain pairs. We present Dual Diffusion Implicit Bridges (DDIBs), an image translation method based on diffusion models, that circumvents training on domain pairs. Image translation with DDIBs relies on two diffusion models trained independently on each domain, and is a two-step process: DDIBs first obtain latent encodings for source images with the source diffusion model, and then decode such encodings using the target model to construct target images. Both steps are defined via ordinary differential equations (ODEs), thus the process is cycle consistent only up to discretization errors of the ODE solvers. Theoretically, we interpret DDIBs as concatenation of source to latent, and latent to target Schrodinger Bridges, a form of entropy-regularized ¨ optimal transport, to explain the efficacy of the method. Experimentally, we apply DDIBs on synthetic and high-resolution image datasets, to demonstrate their utility in a wide variety of translation tasks and their inherent optimal transport properties.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Transferring images from one domain to another while preserving the content representation is an important problem in computer vision, with wide applications that span style transfer (Xu et al., 2021; Sinha et al., 2021) and semantic segmentation (Li et al., 2020). In tasks such as style transfer, it is usually difficult to obtain paired images of realistic scenes and their artistic renditions. Consequently, unpaired translation methods are particularly relevant, since only the datasets, and not the one-toone correspondence between image translation pairs, are required. Common methods on unpaired translation are based on generative adversarial networks (GANs, Goodfellow et al. (2014); Zhu et al. (2017)) or normalizing flows (Grover et al., 2020). Training such models typically involves minimizing an adversarial loss between a specific pair of source and target datasets.
|
| 12 |
+
|
| 13 |
+
While capable of producing high-quality images, these methods suffer from a severe drawback in their adaptability to alternative domains. Concretely, a translation model on a source-target pair is trained specifically for this domain pair. Provided a different pair, existing, bespoke models cannot be easily adapted for translation. If we were to do pairwise translation among a set of domains, the total number of models needed is quadratic in the number of domains – an unacceptable computational cost in practice. One alternative is to find a shared domain that connects to each source / target domains as in StarGANs (Choi et al., 2018). However, the shared domain needs to be carefully chosen a priori; if the shared domain contains less information than the target domain (e.g. sketches v.s. photos), then it creates an unwanted information bottleneck between the source and target domains.
|
| 14 |
+
|
| 15 |
+
An additional disadvantage of existing models resides in their lack of privacy protection of the datasets: training a translation model requires access to both datasets simultaneously. Such setting may be inconvenient or impossible, when data providers are reluctant about giving away their data; or for certain privacy-sensitive applications such as medical imaging. For example, quotidian hospital usage may require translation of patients’ X-ray and MRI images taken from machines in other hospitals. Most existing methods will fail in such scenarios, as joint training requires aggregating confidential imaging data across hospitals, which may violate patients’ privacy.
|
| 16 |
+
|
| 17 |
+
|
| 18 |
+
Figure 1: Dual Diffusion Implicit Bridges: DDIBs leverage two ODEs for image translation. Given a source image $\mathbf { x } ^ { ( s ) }$ , the source ODE runs in the forward direction to convert it to the latent $\mathbf { x } ^ { ( l ) }$ , while the target, reverse ODE then constructs the target image $\mathbf { x } ^ { ( t ) }$ . $( T o p )$ Illustration of the DDIBs idea between two one-dimensional distributions. (Bottom) DDIBs from a tiger to a cat using a pretrained conditional diffusion model.
|
| 19 |
+
|
| 20 |
+
In this paper, we seek to mitigate both problems of existing image translation methods. We present Dual Diffusion Implicit Bridges (DDIBs), an image-to-image translation method inspired by recent advances in diffusion models (Song et al., 2020a;b), that decouples paired training, and empowers the domain-specific diffusion models to stay applicable in other pairs wherever the domain appears again as the source or the target. Since the training process now concentrates on one dataset at a time, DDIBs can also be applied in federated settings, and not assume access to both datasets during model training. As a result, owners of domain data can effectively preserve their data privacy.
|
| 21 |
+
|
| 22 |
+
Specifically, DDIBs are developed based on the method known as denoising diffusion implicit models (DDIMs, Song et al. (2020a)). DDIMs invent a particular parameterization of the diffusion process, that creates a smooth, deterministic and reversible mapping between images and their latent representations. This mapping is captured using the solution to a so-called probability flow (PF) ordinary differential equation (ODE) that forms the cornerstone of DDIBs. Translation with DDIBs on a source-target pair requires two different PF ODEs: the source PF ODE converts input images to the latent space; while the target ODE then synthesizes images in the target domain.
|
| 23 |
+
|
| 24 |
+
Crucially, trained diffusion models are specific to the individual domains, and rely on no domain pairing information. Effectively, DDIBs make it possible to save a trained model of a certain domain for future use, when it arises as the source or target in a new pair. Pairwise translation with DDIBs requires only a linear number of diffusion models (which can be further reduced with conditional models (Dhariwal & Nichol, 2021)), and training does not require scanning both datasets concurrently.
|
| 25 |
+
|
| 26 |
+
Theoretically, we analyze the DDIBs translation process to highlight two important theoretical properties. First, the probability flow ODEs in DDIBs, in essence, comprise the solution of a special Schrodinger Bridge Problem (SBP) with linear or degenerate drift (Chen et al., 2021a), between the ¨ data and the latent distributions. This justification of DDIBs from an optimal transport viewpoint that alternative translation methods lack serves as a theoretical advantage of our method, as DDIBs are the most OT-efficient translation procedure while alternate methods may not be. Second, DDIBs guarantee exact cycle consistency: translating an image to and back from the target space reinstates the original image, only up to discretization errors introduced in the ODE solvers.
|
| 27 |
+
|
| 28 |
+
Experimentally, we first present synthetic experiments on two-dimensional datasets to demonstrate DDIBs’ cycle-consistency property. We then evaluate our method on a variety of image modalities, with qualitative and quantitative results: we validate its usage in example-guided color transfer, paired image translation, and conditional ImageNet translation. These results establish DDIBs as a scalable, theoretically rigorous addition to the family of unpaired image translation methods.
|
| 29 |
+
|
| 30 |
+
# 2 PRELIMINARIES
|
| 31 |
+
|
| 32 |
+
# 2.1 SCORE-BASED GENERATIVE MODELS (SGMS)
|
| 33 |
+
|
| 34 |
+
While our actual implementation utilizes DDIMs, we first briefly introduce the broader family of models known as score-based generative models. Two representative models of this family are score matching with Langevin dynamics (SMLD) (Song & Ermon, 2019) and denoising diffusion
|
| 35 |
+
|
| 36 |
+
probabilistic models (DDPMs) (Ho et al., 2020). Both methods are contained within the framework of Stochastic Differential Equations (SDEs) proposed in Song et al. (2020b).
|
| 37 |
+
|
| 38 |
+
Stochastic Differential Equation (SDE) Representation Song et al. (2020b); Anderson (1982) use a forward and a corresponding backward SDE to describe general diffusion and the reversed, generative processes:
|
| 39 |
+
|
| 40 |
+
$$
|
| 41 |
+
\mathrm { d } \mathbf { x } = \mathbf { f } ( \mathbf { x } , t ) \mathrm { d } t + g ( t ) \mathrm { d } \mathbf { w } , \quad \mathrm { d } \mathbf { x } = [ \mathbf { f } - g ^ { 2 } \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) ] \mathrm { d } t + g ( t ) \mathrm { d } \mathbf { w }
|
| 42 |
+
$$
|
| 43 |
+
|
| 44 |
+
where w is the standard Wiener process, $\mathbf { f } \left( \mathbf { x } , t \right)$ is the vector-valued drift coefficient, $g ( t )$ is the scalar diffusion coefficient, and $\nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ is the score function of the noise perturbed data distribution (as defined by the forward SDE with initial condition $p _ { 0 } ( \mathbf { x } )$ being the data distribution). At the endpoints $t = \{ 0 , 1 \bar \}$ , the forward Eq. (1) admits the data distribution $p _ { 0 }$ and the easy-to-sample prior $p _ { 1 }$ as the boundary distributions. Within this framework, the SMLD method can be described using a VarianceExploding (VE) SDE with increasing noise scales $\sigma ( t )$ : $\mathrm { d } \mathbf { x } = \sqrt { \mathrm { d } [ \sigma ^ { 2 } ( t ) ] / \mathrm { d } t }$ dw. In comparison, DDPMs are endowed with a Variance-Preserving (VP) SDE: $\mathrm { d } \mathbf { x } = - [ \beta ( t ) / 2 ] \mathbf { x } \mathrm { d } t + \sqrt { \beta ( t ) } \mathrm { d } \mathbf { w }$ with $\beta ( t )$ being another noise sequence. Notably, the VP SDE can be reparameterized into an equivalent VE SDE (Song et al., 2020a).
|
| 45 |
+
|
| 46 |
+
Probability Flow ODE Any diffusion process can be represented by a deterministic ODE that carries the same marginal densities as the diffusion process throughout its trajectory. This ODE is termed the probability flow (PF) ODE (Song et al., 2020b). PF ODEs enable uniquely identifiable encodings (Song et al., 2020b) of data, and are central to DDIBs as we solve these ODEs for forward and reverse conversion between data and their latents. For the forward SDE introduced in Eq. (1), the equivalent PF ODE holds the following form:
|
| 47 |
+
|
| 48 |
+
$$
|
| 49 |
+
\mathrm { d } \mathbf { x } = \left[ \mathbf { f } ( \mathbf { x } , t ) - \frac { 1 } { 2 } g ( t ) ^ { 2 } \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) \right] \mathrm { d } t
|
| 50 |
+
$$
|
| 51 |
+
|
| 52 |
+
which follows immediately from the SDEs given the score function. In practice, we use $\theta$ - parameterized score networks $\mathbf { s } _ { t , \theta } \approx \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ to approximate the score function. Training such networks relies on a variational lower bound, described in Ho et al. (2020) and in Appendix B. We may then employ numerical ODE solvers to solve the above ODE and construct $\mathbf { x }$ at different times. Empirically, it has been demonstrated that SGMs have relatively low discretization errors when reconstructing $\mathbf { x }$ at $t = 0$ via ODE solvers (Song et al., 2020a). For conciseness, we use $v _ { \theta } = \mathrm { d } \mathbf { x } / \mathrm { d } t$ to denote the $\theta$ -parameterized velocity field (as defined from Eq. (2), where we replace $\nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } )$ with $\mathbf { s } _ { t , \theta }$ ), and use the symbol ODESolve to denote the mapping from $\mathbf { x } ( t _ { 0 } )$ to $\mathbf { x } ( t _ { 1 } )$ :
|
| 53 |
+
|
| 54 |
+
$$
|
| 55 |
+
\mathrm { O D E S o l v e } ( \mathbf { x } ( t _ { 0 } ) ; v _ { \theta } , t _ { 0 } , t _ { 1 } ) = \mathbf { x } ( t _ { 0 } ) + \int _ { t _ { 0 } } ^ { t _ { 1 } } v _ { \theta } ( t , \mathbf { x } ( t ) ) \mathrm { d } t ,
|
| 56 |
+
$$
|
| 57 |
+
|
| 58 |
+
which allows us to abstract away the exact model (be it a score-based or a diffusion model), or the integrator used. In our experiments, we implement the ODE solver in DDIMs (Song et al., 2020a) (Appendix B); while we acknowledge other available ODE solvers that are usable within our framework, such as the DPM-solver (Lu et al., 2022), the Exponential Integrator (Zhang & Chen, 2022), and the second-order Heun solver (Karras et al., 2022).
|
| 59 |
+
|
| 60 |
+
# 2.2 SCHRODINGER ¨ BRIDGE PROBLEM (SBP)
|
| 61 |
+
|
| 62 |
+
Our analysis shows that DDIBs are Schrodinger Bridges (Chen et al., 2016; L ¨ eonard, 2013) between ´ distributions. Let $\Omega = C ( [ 0 , 1 ] ; \mathbb { R } ^ { n } )$ be the path space of $\mathbb { R } ^ { n }$ -valued continuous functions over the time interval $[ 0 , 1 ]$ ; and $\mathcal { D } ( p _ { 0 } , p _ { 1 } )$ be the set of distributions over $\Omega$ , with marginals $p _ { 0 } , p _ { 1 }$ at time $t = 0$ , $t = 1$ , respectively. Given a prior reference measure $W ^ { 1 }$ , the well-known Schrodinger Bridge ¨ Problem (SBP) seeks the most probable evolution across time $t$ between the marginals $p _ { 0 }$ and $p _ { 1 }$ :
|
| 63 |
+
|
| 64 |
+
Problem 1 (Schrodinger Bridge Problem) ¨ . With prescribed distributions $p _ { 0 } , p _ { 1 }$ and a reference measure $W$ as the prior, the SBP finds a distribution from $\mathcal { D } ( p _ { 0 } , p _ { 1 } )$ that minimizes its $K L$ -divergence to $W$ : $P _ { S B P } : = \arg \operatorname* { m i n } \{ D _ { K L } ( P \| W ) \ | \ P \in \mathcal { D } ( p _ { 0 } , p _ { 1 } ) \} .$ .
|
| 65 |
+
|
| 66 |
+
# Algorithm 1 High-level Pseudo-code for DDIBs
|
| 67 |
+
|
| 68 |
+
Input: data sample from source domain $\mathbf { x } ^ { ( s ) } \sim p _ { s } ( \mathbf { x } )$ , source model $v _ { \theta } ^ { ( s ) }$ , target model $v _ { \theta } ^ { ( t ) }$ . Output: $\mathbf { x } ^ { ( t ) }$ , the result in the target domain.
|
| 69 |
+
$\mathbf { x } ^ { ( l ) } = \mathrm { O D E S o l v e } ( \mathbf { x } ^ { ( s ) } ; v _ { \theta } ^ { ( s ) } , 0 , 1 )$ // obtain latent code from source domain data $\mathbf { x } ^ { ( t ) } = \mathrm { O D E S o l v e } ( \mathbf { x } ^ { ( l ) } ; v _ { \theta } ^ { ( t ) } , 1 , 0 )$ // obtain target domain data from latent code
|
| 70 |
+
return $\mathbf { x } ^ { ( t ) }$
|
| 71 |
+
|
| 72 |
+
The minimizer, $P _ { \mathrm { S B P } }$ , is dubbed the Schrodinger Bridge ¨ between $p _ { 0 }$ and $p _ { 1 }$ over prior $W$ . The SBP has connections to the Monge-Kantorovich (MK) optimal transport problem (Chen et al., 2021b). While the basic MK problem seeks the cost-minimizing plan to transport masses between distributions, the SBP incorporates an additional entropy term (for details, see Page 61 of Peyre et al. (2019)) . ´
|
| 73 |
+
|
| 74 |
+
Relationship Between SBPs and SGMs Chen et al. (2021a) establishes connections between SGMs and SBPs. In summary, SGMs are implicit optimal transport models, corresponding to SBPs with linear or degenerate drifts. General SBPs additionally accept fully nonlinear diffusion. To formalize this observation, the authors first establish similar forward and backward SDEs for SBPs:
|
| 75 |
+
|
| 76 |
+
$$
|
| 77 |
+
\begin{array} { r } { \mathbf { d x } = [ \mathbf { f } + g ^ { 2 } \nabla _ { \mathbf { x } } \log \Phi _ { t } ( \mathbf { x } ) ] \mathbf { d } t + g ( t ) \mathbf { d w } , \quad \mathbf { d x } = [ \mathbf { f } - g ^ { 2 } \nabla _ { \mathbf { x } } \log \hat { \Phi } _ { t } ( \mathbf { x } ) ] \mathbf { d } t + g ( t ) \mathbf { d w } } \end{array}
|
| 78 |
+
$$
|
| 79 |
+
|
| 80 |
+
where $\Phi , { \hat { \Phi } }$ are the Schrodinger factors ¨ that satisfy density factorization: $p _ { t } ( { \bf x } ) = \Phi _ { t } ( { \bf x } ) \hat { \Phi } _ { t } ( { \bf x } )$ . The vector-valued quantities $\mathbf { z } _ { t } = g ( t ) \nabla _ { \mathbf { x } } \log \Phi _ { t } ( \mathbf { x } ) , \hat { \mathbf { z } } _ { t } = g ( t ) \nabla _ { \mathbf { x } } \log \hat { \Phi } _ { t } ( \mathbf { x } )$ fully characterize dynamics of the SBP, thus can be considered as the forward, backward “policies”, analogous to policy-based methods described in Schulman et al. (2015); Pereira et al. (2019). To draw a link between SBPs and SGMs, the data log-likelihood objective for SBPs is computed and shown to be equal to that of SGMs with special choices of $\mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t }$ (derivation details in Chen et al. (2021a)). Importantly, likelihood equality occurs with the following policies:
|
| 81 |
+
|
| 82 |
+
$$
|
| 83 |
+
( \mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t } ) = ( 0 , g ( t ) \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) )
|
| 84 |
+
$$
|
| 85 |
+
|
| 86 |
+
When the marginal $p _ { 1 }$ at time $t = 1$ is equal to the prior distribution, it is known that such $( \mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t } )$ are achieved. Since in SGMs, the end marginal $p _ { 1 }$ is indeed the standard Gaussian prior, their log-likelihood is equivalent to that of SBPs. This suggests that SGMs are a special case of SBPs with degenerate forward policy $\mathbf { z } _ { t }$ and a multiple of the score function as its backward $\hat { \mathbf { z } } _ { t }$ .
|
| 87 |
+
|
| 88 |
+
Probability Flow ODE In a similar vein to the SGM SDEs, a deterministic PF ODE can be derived for SBPs with identical marginal densities across $t \in [ 0 , 1 ]$ . The following PF ODE specifies the probability flow of the optimal processes of the SBP defined in Eq. (4) (Chen et al., 2021a):
|
| 89 |
+
|
| 90 |
+
$$
|
| 91 |
+
\mathrm { d } \mathbf { x } = \bigg [ \mathbf { f } ( \mathbf { x } , t ) + g ( t ) \mathbf { z } - \frac { 1 } { 2 } g ( t ) ( \mathbf { z } + \hat { \mathbf { z } } ) \bigg ] \mathrm { d } t
|
| 92 |
+
$$
|
| 93 |
+
|
| 94 |
+
where $\mathbf { z }$ depends on x. We shall show that the PF ODEs for SGMs and SBPs are equivalent. Thus, flowing through the PF ODEs in DDIBs is equivalent to flowing through special Schrodinger Bridges, ¨ with one of the marginals being Gaussian.
|
| 95 |
+
|
| 96 |
+
# 3 DUAL DIFFUSION IMPLICIT BRIDGES
|
| 97 |
+
|
| 98 |
+
DDIBs leverage the connections between SGMs and SBPs to perform image-to-image translation, with two diffusion models trained separately on the two domains. DDIBs contain two steps, described in Alg. 1 and illustrated in Fig. 1. At the core of the algorithm is the ODE solver ODESolve from Eq. (3). Given a source model represented as a vector field, i.e., $v _ { \theta } ^ { ( s ) }$ defined from Eq. (2), DDIBs first apply ODESolve in the source domain to obtain the encoding $\mathbf { x } ^ { ( s ) }$ of the image at the end time $t = 1$ ; we refer to this as the latent code (associated with the diffusion model for the domain). Then, the source latent code is fed as the initial condition (target latent code at $t = 1$ ) to ODESolve with the target model $v _ { \theta } ^ { ( t ) }$ to obtain the target image $\mathbf { x } ^ { ( t ) }$ . As discussed earlier, we implement ODESolve with DDIMs (Song et al., 2020a), which are known to have reasonably small discretization errors. While recent developments in higher order ODE solvers (Zhang & Chen, 2022; Lu et al., 2022; Karras et al., 2022) that generalize DDIMs can also be used here, we leave this investigation to future work.
|
| 99 |
+
|
| 100 |
+
Despite the simplicity of the method, DDIBs have several advantages over prior methods, which we discuss below.
|
| 101 |
+
|
| 102 |
+
Exact Cycle Consistency A desirable feature of image translation algorithms is the cycle consistency property: transforming a data point from the source domain to the target domain, and then back to source, will recover the original data point in the source domain. The following proposition validates the cycle consistency of DDIBs.
|
| 103 |
+
|
| 104 |
+
Proposition 3.1 (DDIBs Enforce Exact Cycle Consistency). Given a sample from source domain $\mathbf { x } ^ { ( s ) }$ , a source diffusion model $v _ { \theta } ^ { ( s ) }$ , and a target model $v _ { \theta } ^ { ( t ) }$ , define:
|
| 105 |
+
|
| 106 |
+
$$
|
| 107 |
+
\begin{array} { r } { \begin{array} { r } { { \mathbf { x } } ^ { ( l ) } = \mathrm { O D E S o l v e } ( { \mathbf { x } } ^ { ( s ) } ; v _ { \theta } ^ { ( s ) } , 0 , 1 ) ; \quad { \mathbf { x } } ^ { ( t ) } = \mathrm { O D E S o l v e } ( { \mathbf { x } } ^ { ( l ) } ; v _ { \theta } ^ { ( t ) } , 1 , 0 ) ; } \\ { { \mathbf { x } } ^ { \prime ( l ) } = \mathrm { O D E S o l v e } ( { \mathbf { x } } ^ { ( t ) } ; v _ { \theta } ^ { ( t ) } , 0 , 1 ) ; \quad { \mathbf { x } } ^ { \prime ( s ) } = \mathrm { O D E S o l v e } ( { \mathbf { x } } ^ { \prime ( l ) } ; v _ { \theta } ^ { ( s ) } , 1 , 0 ) } \end{array} } \end{array}
|
| 108 |
+
$$
|
| 109 |
+
|
| 110 |
+
Assume zero discretization error. Then, $\mathbf { x } ^ { ( s ) } = \mathbf { x } ^ { \prime ( s ) }$
|
| 111 |
+
|
| 112 |
+
As PF ODEs are used, the cycle consistency property is guaranteed. In practice, even with discretization error, DDIBs incur almost negligible cycle inconsistency (Section 4.1). In contrast, GAN-based methods are not guaranteed the cycle consistency property by default, and have to incorporate additional training terms to optimize for cycle consistency over two domains.
|
| 113 |
+
|
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Data Privacy in Both Domains In the DDIBs translation process, only the source and target diffusion models are required, whose training processes do not depend on knowledge of the domain pair a priori. In fact, this process can even be performed in a privacy sensitive manner (graphic illustration in Appendix A). Let Alice and Bob be the data owners of the source and target domains, respectively. Suppose Alice intends to translate images to the target domain. However, Alice does not want to share the data with Bob (and vice versa, Bob does not want to release their data either). Then, Alice can simply train a diffusion model with the source data, encode the data to the latent space, transmit the latent codes to Bob, and next ask Bob to run their trained diffusion model and send the results back. In this procedure, only the latent code and the target results are transmitted between the two data vendors, and both parties have naturally ensured that their data are not directly revealed.
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DDIBs are Two Concatenated Schrodinger Bridges ¨ DDIBs link the source data distribution to the latent space, and then to the target distribution. What is the nature of such connections between distributions? We offer an answer from an optimal transport perspective: these connections are special Schrodinger Bridges ¨ between distributions. This, in turn, explicates the name of our method: dual diffusion implicit bridges are based on denoising diffusion implicit models (Song et al., 2020a), and consist of two separate Schrodinger ¨ Bridges that connect the data and latent distributions. Specifically, as considered earlier, when conditions about the policies $\mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t }$ in Eq. (5) and the density $p _ { 1 } ( \mathbf { x } )$ being a Gaussian prior are met, the data likelihoods (at $t = 0$ ) for SGMs and SBPs are identical. Indeed, these conditions are fulfilled in SGMs and particularly in DDIMs. This verifies SGMs as special linear or degenerate SBPs. Forward and reverse solving the PF ODE for SGMs, as done in DDIBs, is equivalent to flowing through the optimal processes of particular SBPs:
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Proposition 3.2 (PF ODE Equivalence2). Eq. (2) is equivalent to Eq. (6) with forward, backward policies $( \mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t } ) = ( 0 , g \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) )$ as attained in SGMs and particularly in DDIMs.
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Thus, DDIBs are intrinsically entropy-regularized optimal transport: they are Schrodinger Bridges ¨ between the source and the latent, and between the latent and the target distributions. The translation process can then be recognized as traversing through two concatenated Schrodinger Bridges, one ¨ forward and one reversed. The mapping is unique and minimizes a (regularized) optimal transport objective, which probably elucidates the superior performance of DDIBs. In contrast, if we train the source and target models separately with normalizing flow models that are not inborn with such a connection, there are many viable invertible mappings, and the resulting image translation algorithm may not necessarily have good performance. This is probably the reason why AlignFlow (Grover et al., 2020) still has to incorporate an adversarial loss even when cycle-consistency is guaranteed.
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Table 1: Cycle consistency of DDIBs. Experiment legend, PR $\bigcirc$ PS, means that we translate from PR to PS and then back. The numbers are the averaged L2 distances between the original points and their coordinates after cycle translation. Data points are standardized to have unit variance.
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<table><tr><td>PR OPS</td><td>PS OCS</td><td>CR OPR</td><td>CR O CS</td><td>MO CB</td></tr><tr><td>0.0143</td><td>0.0065</td><td>0.0106</td><td>0.0078</td><td>0.0122</td></tr></table>
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# 4 EXPERIMENTS
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We present a series of experiments to demonstrate the effectiveness of DDIBs. First, we describe synthetic experiments on two-dimensional datasets, to corroborate DDIBs’ cycle-consistent and optimal transport properties. Next, we validate DDIBs on a variety of image translation tasks, including color transfer, paired translation, and conditional ImageNet translation. 34
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# 4.1 2D SYNTHETIC EXPERIMENTS
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We first perform domain translation on synthetic datasets drawn from complex two-dimensional distributions, with various shapes and configurations, in Fig. 2a. In total, we consider six 2D datasets: Moons (M); Checkerboards (CB); Concentric Rings (CR); Concentric Squares (CS); Parallel Rings (PR); and Parallel Squares (PS). The datasets are all normalized to have zero mean, and identity covariance. We assign colors to points based on the point identities (i.e., if a point in the source domain is red, its corresponding point in the target domain is also colored red). Clearly, the transformation is smooth between columns. For example, on the top-right corner, red points in the CR dataset are mapped to similar coordinates, both in the latent and in the target dimensions.
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(a) Smooth translation of synthetic datasets. (Left) The source datasets: CR and CS. (Middle) DDIBs’ latent code representation. (Right) Results of translation to the target domains.
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(b) Cycle consistency: After translating the Moons dataset to Checkerboards and then back to Moons, DDIBs restore almost the exact same points as the original ones.
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Figure 2: Smoothness and cycle consistency of DDIBs.
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Cycle Consistency Fig. 2b illustrates the cycle consistency property guaranteed by DDIBs. It concerns 2D datasets: Moons, and Checkerboards. Starting from the Moons dataset, DDIBs first obtain the latent codes and construct the Checkerboards points. Next, DDIBs do translations in the reverse direction, transforming the points back to the latent and the Moons space. After this round trip, points are approximately mapped to their original positions. A similar, smooth color topology is observed in this experiment. Table 1 reports quantitative evaluation results on cycle-consistent translation among multiple datasets. As the datasets are normalized to unit standard deviation, the reported values are negligibly small and endorse the cycle consistent property of DDIBs.
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Table 2: Mean Squared Error (MSE) comparing color transfer results of DDIBs with common OT methods on two images. Each number represents the MSE between DDIBs and the corresponding OT method. MSE is computed pixel-wise after normalizing images to $[ - 1 , 1 ]$ .
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<table><tr><td>IMAGE</td><td>EMD</td><td>SINKHORN</td><td>LINEAR</td><td>GAUSSIAN</td></tr><tr><td>TARGET1</td><td>0.0337</td><td>0.0281</td><td>0.0352</td><td>0.0370</td></tr><tr><td>TARGET 2</td><td>0.0293</td><td>0.0326</td><td>0.0500</td><td>0.0751</td></tr></table>
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Table 3: MSE comparing DDIBs and baselines on paired test sets. MSE is computed pixel-wise after normalizing images to $[ - 1 , 1 ]$ .
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<table><tr><td>DATASET</td><td>MODEL</td><td>A→B</td><td>B→A</td><td>DATASET</td><td>MODEL</td><td>A→B</td><td>B→A</td></tr><tr><td rowspan="3">FACADES</td><td>CYCLEGAN</td><td>0.7129</td><td>0.3286</td><td rowspan="3">MAPS</td><td>CYCLEGAN</td><td>0.0245</td><td>0.0953</td></tr><tr><td>ALIGNFLOW</td><td>0.5801</td><td>0.2512</td><td>ALIGNFLOW</td><td>0.0209</td><td>0.0897</td></tr><tr><td>DDIBS</td><td>0.5312</td><td>0.3946</td><td>DDIBS</td><td>0.0194</td><td>0.1302</td></tr></table>
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# 4.2 EXAMPLE-GUIDED COLOR TRANSFER
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DDIBs can be used on an interesting application: example-guided color transfer. This refers to the task of modifying the colors of an input image, conditioned on the color palette of a reference image. To use DDIBs for color transfer, we train one diffusion model per image, on its normalized RGB space. During translation, DDIBs obtain encodings of the original colors, and apply the diffusion model of the reference image to attain the desired color palette. Fig. 3 visualizes our color experiments.
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Comparison to Alternative OT Methods As DDIBs are related to regularized OT, we compare the pixel-wise MSEs between color-transferred images generated by DDIBs, and images produced by alternate methods, in Table 2. We include four OT methods for comparison: Earth Mover’s Distance; Sinkhorn distance (Cuturi, 2013); linear and Gaussian mapping estimation (Perrot et al., 2016). Results of DDIBs are very close to those of OT methods. Appendix E.2 details full color translation results.
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Figure 3: Example-Guided Color Transfer: Given the first image as the reference image, DDIBs modify the colors of two input images to similarly follow a snowy winter color palette.
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# 4.3 QUANTITATIVE TRANSLATION EVALUATION
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Quantitatively, we demonstrate that DDIBs deliver competitive results on paired domain tests. Such numerical evaluation is despite that DDIBs are formulated with a weaker setting: diffusion models are trained independently, on separate datasets. In comparison, methods such as CycleGAN and AlignFlow assume access to both datasets during training and jointly optimize for the translation loss.
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Paired Domain Translation As in similar works, we evaluate DDIBs on benchmark paired datasets (Zhu et al., 2017): Facades and Maps. Both are image segmentation tasks. In the pairs of datasets, one dataset contains real photos taken via a camera or a satellite; while the other comprises the corresponding segmentation images. These datasets provide one-to-one image alignment, which allows quantitative evaluation through a distance metric such as mean-squared error (MSE) between generated samples and the corresponding ground truth. To facilitate the workings of DDIBs, we additionally employ a color conversion heuristic motivated by optimal transport on image colors (Appendix E.1). Table 3 reports the evaluation results. Surprisingly, DDIBs are able to produce segmentation images that surpass alternative methods in MSE terms; while reverse translations also achieve decent performance.
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# 4.4 CLASS-CONDITIONAL IMAGENET TRANSLATION
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In this experiment, we apply DDIBs to translation among ImageNet classes. To this end, we leverage the pretrained diffusion models from Dhariwal & Nichol (2021). The authors optimized performance of diffusion models, and end up with a “UNet” (Ho et al., 2020) architecture with particular width, attention and residual configurations. The models are learned on 1, 000 ImageNet classes, each with around 1, 000 training images, and at a variety of resolutions. Our experiments use the model with resolution $2 5 6 \times 2 5 6$ . Moreover, these models incorporate a technique known as classifier guidance (Dhariwal & Nichol, 2021), that leverage classifier gradients to steer the sampling process towards arbitrary class labels during image generation. The learned models combined with classifier guidance can be effectively considered as 1, 000 different models. Fig. 4a exhibits select translation samples, where the source images are from ImageNet validation sets. DDIBs are able to create faithful target images that maintain much of the original content such as animal poses, complexions and emotions, while accounting for differences in animal species.
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Multi-Domain Translation Given conditional models on the individual domains, DDIBs can be applied to translate between arbitrary pairs of source-target domains, while requiring no additional fine-tuning or adaptation. Fig. 4b displays results of translating a common image of a roaring lion (with class label 291), to various other ImageNet classes. Interestingly, some animals roar, while others stick their tongues out. DDIBs successfully internalize characteristics of distinct animal species, and produce closest animal postures in OT distances to the original shouting lion.
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(a) Conditional ImageNet Translation: Selected trans- (b) Multi-domain translation: Given the center, lation samples from various ImageNet classes such as 7: source image from class label 291, DDIBs translate Cock, 94: Hummingbird, 162: Beagle, and 282: Tiger it to other animal species, entirely using only a preCat. trained conditional diffusion model.
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Figure 4: Translation among ImageNet classes.
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# 5 RELATED WORKS
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Score-based Diffusion Models Originating in thermodynamics (Sohl-Dickstein et al., 2015), diffusion models reverse the dynamics of a noising process to create data samples. The reversal process is understood to implicitly compute scores of the data density at various noise scales, which reveals connections to score-based methods (Song & Ermon, 2019; Nichol & Dhariwal, 2021; Meng et al., 2021b). Diffusion models are applicable to multiple modalities: 3D shapes (Zhou et al., 2021), point cloud (Luo & Hu, 2021), discrete domains (Meng et al., 2022) and function spaces (Lim et al., 2023). They excel in tasks ranging from image editing and composition (Meng et al., 2021a), density estimation (Kingma et al., 2021), to image restoration (Kawar et al., 2022). Seminal works are denoising diffusion probabilistic models (DDPMs, Ho et al. (2020)), which parameterized the ELBO objective with Gaussians and, for the first time, synthesized high-quality images with diffusion models; ILVR (Choi et al., 2021), which invented a novel conditional method to direct DDPM generation towards reference images; and denoising diffusion implicit models (DDIMs, Song et al. (2020a)), which accelerated DDPM inference via non-Markovian processes. DDIMs can be treated as a first-order numerical solver of a probabilistic ODE, which we use heavily in DDIBs.
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Diffusion Models for Image Translation While GANs (Goodfellow et al., 2014; Zhu et al., 2017; Zhao et al., 2020) have been widely adopted in image translation tasks, recent works increasingly leverage diffusion models. For instance, Palette (Saharia et al., 2021) applies a conditional diffusion model to colorization, inpainting, and restoration. DiffuseIT (Kwon & Ye, 2022) utilizes disentangled style and content representation, to perform text- and image-guided style transfer. Lastly, UNITDDPM (Sasaki et al., 2021) proposes a novel coupling between domain pairs and trains joint DDPMs for translation. Unlike their joint training, DDIBs apply separate, pretrained diffusion models and leverage geometry of the shared space for translation.
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Optimal Transport for Translation and Generative Modeling As it pursues cost-optimal plans to connect image distributions, OT naturally finds applications in image translation. For example, Korotin et al. (2022) capitalizes on the approximation powers of neural networks to compute OT plans between image distributions and perform unpaired translation. By contrast, the entropy-regularized OT variant, Schrodinger Bridges (Section 2), are also commonly used to derive generative models. ¨ For instance, De Bortoli et al. (2021) and Vargas et al. (2021) concurrently proposed new numerical procedures that approximate the Iterative Proportional Fitting scheme, to solve SBPs for image generation. Wang et al. (2021) presents a new generative method via entropic interpolation with an SBP. Chen et al. (2021a) discovers equivalence between the likelihood objectives of SBP and score-based models, which lays the theoretical foundations behind DDIBs. Their sequel (Liu et al., 2023) then directly learns the Schrodinger Bridges between image distributions, for applications ¨ in image-to-image tasks such as restoration. While DDIBs were not initially designed to mimic Schrodinger Bridges, our analysis reveals their true characterization as solutions to degenerate SBPs. ¨
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# 6 CONCLUSIONS
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We present Dual Diffusion Implicit Bridges (DDIBs), a new, simplistic image translation method that stems from latest progresses in score-based diffusion models, and is theoretically grounded as Schrodinger Bridges in the image space. DDIBs solve two key problems. First, DDIBs avoid ¨ optimization on a coupled loss specific to the given domain pair only. Second, DDIBs better safeguard dataset privacy as they no longer require presence of both datasets during training. Powerful pretrained diffusion models are then integrated into our DDIBs framework, to perform a comprehensive series of experiments that prove DDIBs’ practical values in domain translation. Our method is limited in its application to color transfer, as one model is required for each image, which demands significant compute for mass experiments. Rooted in optimal transport, DDIBs translation mimics the massmoving process which may be problematic at times (Appendix C). Future work may remedy these issues, or extend DDIBs to applications with different dimensions in the source and target domains. As flowing through the concatenated ODEs is time-consuming, improving the translation speed is also a promising direction.
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# ACKNOWLEDGEMENTS
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We thank Lingxiao Li and Chris Cundy for insightful discussions about the optimal transport properties of DDIBs. We also thank the anonymous reviewers for their constructive comments and feedback. This research was supported by NSF (#1651565), ARO (W911NF-21-1-0125), ONR (N00014-23-1-2159), CZ Biohub, and Stanford HAI.
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Qinsheng Zhang and Yongxin Chen. Fast sampling of diffusion models with exponential integrator. arXiv preprint arXiv:2204.13902, 2022.
|
| 282 |
+
|
| 283 |
+
Yihao Zhao, Ruihai Wu, and Hao Dong. Unpaired image-to-image translation using adversarial consistency loss. In European Conference on Computer Vision, pp. 800–815. Springer, 2020.
|
| 284 |
+
|
| 285 |
+
Linqi Zhou, Yilun Du, and Jiajun Wu. 3d shape generation and completion through point-voxel diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 5826–5835, 2021.
|
| 286 |
+
|
| 287 |
+
Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223–2232, 2017.
|
| 288 |
+
|
| 289 |
+
Alice is the owner of the source (tiger) domain, and Bob is the owner of the target (cat) domain. Alice intends to translate tiger images to cat images, but in a privacy-sensitive manner without releasing the source dataset. Bob does not wish to make the cat dataset public, either.
|
| 290 |
+
|
| 291 |
+
|
| 292 |
+
Fig. 5 illustrates the process of privacysensitive domain translation. The process contains the following steps, with indexes in the figure.
|
| 293 |
+
Figure 5
|
| 294 |
+
|
| 295 |
+
1. Alice intends to translate tiger images to cat images.
|
| 296 |
+
2. Alice trains a diffusion model with the source tiger images.
|
| 297 |
+
3. Alice uses the pretrained, tiger diffusion model to convert a source tiger image to its latent code.
|
| 298 |
+
4. Alice sends the latent code to Bob.
|
| 299 |
+
5. Bob similarly trains a diffusion model on the cat domain.
|
| 300 |
+
6. Bob uses the pretrained, cat diffusion model to convert the received latent code to a cat image.
|
| 301 |
+
7. Bob then sends the translated image back to Alice.
|
| 302 |
+
|
| 303 |
+
Clearly, during the translation process, only the latent code and the translated cat image are transmitted via the public channel, while both source and target datasets are private to the two parties. This is a significant advantage of DDIBs over alternate methods, as we enable strong privacy protection of the datasets.
|
| 304 |
+
|
| 305 |
+
# B DETAILS OF SGM TRAINING AND DDIM ODE SOLVER
|
| 306 |
+
|
| 307 |
+
# B.1 TRAINING SCORE NETWORKS
|
| 308 |
+
|
| 309 |
+
While the description in Section 2 is based on continuous SDEs, actual implementations of diffusion models often sample discrete time steps. Given samples from a data distribution $q ( \mathbf { x } _ { 0 } )$ , diffusion models attempt to learn a model distribution $p _ { \theta } ( \mathbf { x } _ { 0 } )$ that approximates $q ( \mathbf { x } _ { 0 } )$ , and is easy to sample from. Specifically, diffusion probabilistic models are latent variable models of the form
|
| 310 |
+
|
| 311 |
+
$$
|
| 312 |
+
p _ { \theta } ( \mathbf { x } _ { 0 } ) = \int p _ { \theta } ( \mathbf { x } _ { 0 : T } ) \mathrm { d } \mathbf { x } _ { 1 : T } , \mathrm { w h e r e } p _ { \theta } ( \mathbf { x } _ { 0 : T } ) = p _ { \theta } ( \mathbf { x } _ { T } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ^ { ( t ) } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } )
|
| 313 |
+
$$
|
| 314 |
+
|
| 315 |
+
where $\mathbf { x } _ { 1 } , \cdots , \mathbf { x } _ { T }$ are latent variables in the same sample space as $\mathbf { x } _ { \mathrm { 0 } }$ . The parameters $\theta$ are trained to approximate the data distribution $q ( \mathbf { x } _ { 0 } )$ , by maximizing a variational lower bound:
|
| 316 |
+
|
| 317 |
+
$$
|
| 318 |
+
\operatorname* { m a x } _ { \theta } \mathbb { E } _ { q ( \mathbf { x } _ { 0 } ) } [ \log p \varrho ( \mathbf { x } _ { 0 } ) ] \leq \operatorname* { m a x } _ { \theta } \mathbb { E } _ { q ( \mathbf { x } _ { 0 } , \mathbf { x } _ { 1 } , \cdots , \mathbf { x } _ { T } ) } [ \log p \varrho ( \mathbf { x } _ { 0 : T } ) - \log q ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } ) ]
|
| 319 |
+
$$
|
| 320 |
+
|
| 321 |
+
where $q \big ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } \big )$ is some inference distribution over the latent variables. It is known that when the conditional distributions are modeled as Gaussians with trainable mean functions and fixed variances, the above objective can be simplified to:
|
| 322 |
+
|
| 323 |
+
$$
|
| 324 |
+
L ( \epsilon _ { \theta } ) : = \sum _ { t = 1 } ^ { T } \mathbb { E } _ { \mathbf { x } _ { 0 } \sim q ( \mathbf { x } _ { 0 } ) , \epsilon _ { t } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } ) } \left[ \left\| \epsilon _ { \theta } ^ { ( t ) } ( \sqrt { \alpha _ { t } } \mathbf { x } _ { 0 } + \sqrt { 1 - \alpha _ { t } } \epsilon _ { t } ) - \epsilon _ { t } \right\| _ { 2 } ^ { 2 } \right]
|
| 325 |
+
$$
|
| 326 |
+
|
| 327 |
+
The resulting noise prediction functions $\epsilon _ { \theta } ^ { ( t ) }$ , are equivalent to the score networks $\mathbf { s } _ { t , \theta }$ mentioned in Section 2 due to Tweedie’s formula (Stein, 1981; Efron, 2011). For details, we refer the reader to $\mathrm { H o }$ et al. (2020); Song et al. (2020a).
|
| 328 |
+
|
| 329 |
+
# B.2 DDIM ODE SOLVER
|
| 330 |
+
|
| 331 |
+
With a trained noise prediction model $\epsilon _ { \theta } ^ { ( t ) } ( \mathbf { x } )$ , the DDIM iterate between adjacent variables $\mathbf { x } _ { t - \Delta t }$ and $\mathbf { x } _ { t }$ , considered in Song et al. (2020a), assumes the following form:
|
| 332 |
+
|
| 333 |
+
$$
|
| 334 |
+
\frac { \mathbf { x } _ { t - \Delta t } } { \sqrt { \alpha _ { t - \Delta t } } } = \frac { \mathbf { x } _ { t } } { \sqrt { \alpha _ { t } } } + \left( \sqrt { \frac { 1 - \alpha _ { t - \Delta t } } { \alpha _ { t - \Delta t } } } - \sqrt { \frac { 1 - \alpha _ { t } } { \alpha _ { t } } } \right) \epsilon _ { \theta } ^ { ( t ) } ( \mathbf { x } _ { t } )
|
| 335 |
+
$$
|
| 336 |
+
|
| 337 |
+
In our experiments, we implement the above equation between adjacent diffusion steps. The equation is deterministic, and can be considered as a Euler method over the following ODE:
|
| 338 |
+
|
| 339 |
+
$$
|
| 340 |
+
\mathrm { d } \bar { \mathbf { x } } ( t ) = \epsilon _ { \theta } ^ { ( t ) } \left( \frac { \bar { \mathbf { x } } ( t ) } { \sqrt { \sigma ^ { 2 } + 1 } } \right) \mathrm { d } \sigma ( t )
|
| 341 |
+
$$
|
| 342 |
+
|
| 343 |
+
where we adopt the reparameterization:
|
| 344 |
+
|
| 345 |
+
$$
|
| 346 |
+
\sigma ( t ) = \sqrt { \frac { 1 - \alpha ( t ) } { \alpha ( t ) } } , \quad \bar { \mathbf { x } } ( t ) = \frac { \mathbf { x } ( t ) } { \sqrt { \alpha ( t ) } }
|
| 347 |
+
$$
|
| 348 |
+
|
| 349 |
+
Importantly, the ODE in Eq. (9) with the optimal model $\epsilon _ { \theta } ^ { ( t ) } ( \mathbf { x } )$ , has an equivalent probability flow ODE corresponding to the “Variance-Exploding” SDE in Song et al. (2020b).
|
| 350 |
+
|
| 351 |
+
# C LIMITATIONS OF OPTIMAL TRANSPORT-BASED TRANSLATION
|
| 352 |
+
|
| 353 |
+
DDIBs contain deterministic bridges between distributions, and are a form of entropy-regularized optimal transport. The learned diffusion models can be effectively considered as a digest or summary of the datasets. While doing translation, they attempt to create images in the target domain, that are closest in optimal transport distances to the source images. Such OT-based process is both an advantage and a limitation of our method.
|
| 354 |
+
|
| 355 |
+
In ImageNet translation, when the source and target datasets are similar, DDIBs are generally able to identify correct animal postures. For example, we have shouting lions and tigers, because these animals have similar behaviors that are observed in the datasets and then internalized by DDIBs. However, in datasets that are less similar (e.g. birds and dogs), DDIBs sometimes fail to produce translation results that retain the postures precisely. We encountered significantly less such cases in AFHQ translation, since the dataset is more standardized and homogeneous.
|
| 356 |
+
|
| 357 |
+
Fig. 6 illustrates the optimal transport mappings among images as well as some failure cases. Clearly, the translation processes flowing from left to right minimize the Euclidean transportation distances between images. Some of these translated samples may be classified “failure cases” in actual user studies. Such are considered both a feature and a limitation of DDIBs.
|
| 358 |
+
|
| 359 |
+
|
| 360 |
+
Figure 6: Optimal transport translation processes in DDIBs. (Leftmost) Source images. (Rightmost) Translated images.
|
| 361 |
+
|
| 362 |
+
# D PROOF OF PROPOSITION 3.2
|
| 363 |
+
|
| 364 |
+
Proof. The proof proceeds by substituting the values of $( \mathbf { z } _ { t } , \hat { \mathbf { z } } _ { t } ) = ( 0 , g ( t ) \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) )$ into Eq. (6),
|
| 365 |
+
|
| 366 |
+
$$
|
| 367 |
+
\begin{array} { r } { \mathrm { d } \mathbf { x } = \left[ \mathbf { f } ( \mathbf { x } , t ) + g ( t ) \mathbf { z } - \frac { 1 } { 2 } g ( t ) ( \mathbf { z } + \hat { \mathbf { z } } ) \right] \mathrm { d } t } \\ { = \left[ \mathbf { f } ( \mathbf { x } , t ) - \frac { 1 } { 2 } g ( t ) ^ { 2 } \nabla _ { \mathbf { x } } \log p _ { t } ( \mathbf { x } ) \right] \mathrm { d } t } \end{array}
|
| 368 |
+
$$
|
| 369 |
+
|
| 370 |
+
This is exactly Eq. (2).
|
| 371 |
+
|
| 372 |
+
# E ADDITIONAL EXPERIMENTAL DETAILS
|
| 373 |
+
|
| 374 |
+
# E.1 OPTIMAL TRANSPORT IN PAIRED DATASETS
|
| 375 |
+
|
| 376 |
+
Color Conversion In Fig. 7, a simple examination of the original and segmentation images reveals significant differences in color configurations. In the Maps dataset, while the real, satellite images are composed of dark colors, the segmentation images are light-toned. The same observation applies to other datasets. The shark contrasts in colors intuitively present a large transportation cost, that probably hinders the progress of DDIBs, as we have demonstrated its relationship to OT in Section 3.
|
| 377 |
+
|
| 378 |
+
To facilitate the workings of DDIBs, we follow a heuristic to transform the colors of the segmentation images. Specifically, on a small subset of the train dataset, we run an OT algorithm to compute a color correspondence that minimizes the color differences in terms of Sinkhorn distances between the real and segmentation images. The segmentation (target) datasets undergo this color conversion before they are fed into a diffusion model for training. During evaluation, when we compute MSEs, the images are converted to the original color space.
|
| 379 |
+
|
| 380 |
+
Privacy Protection Color conversion requires considering both datasets jointly to compute a color mapping, and seems to betray the original purpose of DDIBs on protection of dataset privacy. We comment that the amount of leaked information is minimal: for example, to compute a color correspondence for the Maps dataset, we sampled only around 1000 pixels from the two datasets, to summarize the color composition information. DDIBs still conserve privacy at large.
|
| 381 |
+
|
| 382 |
+
|
| 383 |
+
Figure 7: Color Conversion. In the paired translation tasks, we are given the real and segmentation images. Before training the diffusion models, we first transform the segmentation images to a color palette that is closer to the real images. While evaluating MSEs, we convert the images back to the original colors.
|
| 384 |
+
|
| 385 |
+
# E.2 EXAMPLE-GUIDED COLOR TRANSFER
|
| 386 |
+
|
| 387 |
+
We present additional qualitative comparison between DDIBs and common OT methods, in Fig. 8.
|
| 388 |
+
|
| 389 |
+
|
| 390 |
+
Figure 8: Full color transfer results on example images.
|
parse/dev/5HLoTvVGDe/5HLoTvVGDe_content_list.json
ADDED
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "DUAL DIFFUSION IMPLICIT BRIDGES FOR IMAGE-TO-IMAGE TRANSLATION ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
174,
|
| 8 |
+
98,
|
| 9 |
+
676,
|
| 10 |
+
146
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Xuan $\\mathbf { S u } ^ { 1 }$ Jiaming Song2 Chenlin Meng1 Stefano Ermon1,3 1Stanford University 2NVIDIA 3CZ Biohub {suxuan,chenlin,ermon}@cs.stanford.edu,jiamings@nvidia.com ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
184,
|
| 19 |
+
167,
|
| 20 |
+
758,
|
| 21 |
+
213
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
+
250,
|
| 32 |
+
544,
|
| 33 |
+
265
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "Common image-to-image translation methods rely on joint training over data from both source and target domains. The training process requires concurrent access to both datasets, which hinders data separation and privacy protection; and existing models cannot be easily adapted for translation of new domain pairs. We present Dual Diffusion Implicit Bridges (DDIBs), an image translation method based on diffusion models, that circumvents training on domain pairs. Image translation with DDIBs relies on two diffusion models trained independently on each domain, and is a two-step process: DDIBs first obtain latent encodings for source images with the source diffusion model, and then decode such encodings using the target model to construct target images. Both steps are defined via ordinary differential equations (ODEs), thus the process is cycle consistent only up to discretization errors of the ODE solvers. Theoretically, we interpret DDIBs as concatenation of source to latent, and latent to target Schrodinger Bridges, a form of entropy-regularized ¨ optimal transport, to explain the efficacy of the method. Experimentally, we apply DDIBs on synthetic and high-resolution image datasets, to demonstrate their utility in a wide variety of translation tasks and their inherent optimal transport properties. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
281,
|
| 43 |
+
764,
|
| 44 |
+
502
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
176,
|
| 54 |
+
530,
|
| 55 |
+
336,
|
| 56 |
+
546
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Transferring images from one domain to another while preserving the content representation is an important problem in computer vision, with wide applications that span style transfer (Xu et al., 2021; Sinha et al., 2021) and semantic segmentation (Li et al., 2020). In tasks such as style transfer, it is usually difficult to obtain paired images of realistic scenes and their artistic renditions. Consequently, unpaired translation methods are particularly relevant, since only the datasets, and not the one-toone correspondence between image translation pairs, are required. Common methods on unpaired translation are based on generative adversarial networks (GANs, Goodfellow et al. (2014); Zhu et al. (2017)) or normalizing flows (Grover et al., 2020). Training such models typically involves minimizing an adversarial loss between a specific pair of source and target datasets. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
561,
|
| 66 |
+
825,
|
| 67 |
+
688
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "While capable of producing high-quality images, these methods suffer from a severe drawback in their adaptability to alternative domains. Concretely, a translation model on a source-target pair is trained specifically for this domain pair. Provided a different pair, existing, bespoke models cannot be easily adapted for translation. If we were to do pairwise translation among a set of domains, the total number of models needed is quadratic in the number of domains – an unacceptable computational cost in practice. One alternative is to find a shared domain that connects to each source / target domains as in StarGANs (Choi et al., 2018). However, the shared domain needs to be carefully chosen a priori; if the shared domain contains less information than the target domain (e.g. sketches v.s. photos), then it creates an unwanted information bottleneck between the source and target domains. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
694,
|
| 77 |
+
825,
|
| 78 |
+
819
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "An additional disadvantage of existing models resides in their lack of privacy protection of the datasets: training a translation model requires access to both datasets simultaneously. Such setting may be inconvenient or impossible, when data providers are reluctant about giving away their data; or for certain privacy-sensitive applications such as medical imaging. For example, quotidian hospital usage may require translation of patients’ X-ray and MRI images taken from machines in other hospitals. Most existing methods will fail in such scenarios, as joint training requires aggregating confidential imaging data across hospitals, which may violate patients’ privacy. ",
|
| 85 |
+
"bbox": [
|
| 86 |
+
174,
|
| 87 |
+
825,
|
| 88 |
+
825,
|
| 89 |
+
924
|
| 90 |
+
],
|
| 91 |
+
"page_idx": 0
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "image",
|
| 95 |
+
"img_path": "images/c1f26c0a6bef908cc1ead156af8e57cbcc448fa17dda125fa0fb83aa3933213b.jpg",
|
| 96 |
+
"image_caption": [
|
| 97 |
+
"Figure 1: Dual Diffusion Implicit Bridges: DDIBs leverage two ODEs for image translation. Given a source image $\\mathbf { x } ^ { ( s ) }$ , the source ODE runs in the forward direction to convert it to the latent $\\mathbf { x } ^ { ( l ) }$ , while the target, reverse ODE then constructs the target image $\\mathbf { x } ^ { ( t ) }$ . $( T o p )$ Illustration of the DDIBs idea between two one-dimensional distributions. (Bottom) DDIBs from a tiger to a cat using a pretrained conditional diffusion model. "
|
| 98 |
+
],
|
| 99 |
+
"image_footnote": [],
|
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"text": "In this paper, we seek to mitigate both problems of existing image translation methods. We present Dual Diffusion Implicit Bridges (DDIBs), an image-to-image translation method inspired by recent advances in diffusion models (Song et al., 2020a;b), that decouples paired training, and empowers the domain-specific diffusion models to stay applicable in other pairs wherever the domain appears again as the source or the target. Since the training process now concentrates on one dataset at a time, DDIBs can also be applied in federated settings, and not assume access to both datasets during model training. As a result, owners of domain data can effectively preserve their data privacy. ",
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"text": "Specifically, DDIBs are developed based on the method known as denoising diffusion implicit models (DDIMs, Song et al. (2020a)). DDIMs invent a particular parameterization of the diffusion process, that creates a smooth, deterministic and reversible mapping between images and their latent representations. This mapping is captured using the solution to a so-called probability flow (PF) ordinary differential equation (ODE) that forms the cornerstone of DDIBs. Translation with DDIBs on a source-target pair requires two different PF ODEs: the source PF ODE converts input images to the latent space; while the target ODE then synthesizes images in the target domain. ",
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"text": "Crucially, trained diffusion models are specific to the individual domains, and rely on no domain pairing information. Effectively, DDIBs make it possible to save a trained model of a certain domain for future use, when it arises as the source or target in a new pair. Pairwise translation with DDIBs requires only a linear number of diffusion models (which can be further reduced with conditional models (Dhariwal & Nichol, 2021)), and training does not require scanning both datasets concurrently. ",
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"text": "Theoretically, we analyze the DDIBs translation process to highlight two important theoretical properties. First, the probability flow ODEs in DDIBs, in essence, comprise the solution of a special Schrodinger Bridge Problem (SBP) with linear or degenerate drift (Chen et al., 2021a), between the ¨ data and the latent distributions. This justification of DDIBs from an optimal transport viewpoint that alternative translation methods lack serves as a theoretical advantage of our method, as DDIBs are the most OT-efficient translation procedure while alternate methods may not be. Second, DDIBs guarantee exact cycle consistency: translating an image to and back from the target space reinstates the original image, only up to discretization errors introduced in the ODE solvers. ",
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"text": "Experimentally, we first present synthetic experiments on two-dimensional datasets to demonstrate DDIBs’ cycle-consistency property. We then evaluate our method on a variety of image modalities, with qualitative and quantitative results: we validate its usage in example-guided color transfer, paired image translation, and conditional ImageNet translation. These results establish DDIBs as a scalable, theoretically rigorous addition to the family of unpaired image translation methods. ",
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"type": "text",
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"text": "2 PRELIMINARIES ",
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"type": "text",
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"text": "2.1 SCORE-BASED GENERATIVE MODELS (SGMS) ",
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"text": "While our actual implementation utilizes DDIMs, we first briefly introduce the broader family of models known as score-based generative models. Two representative models of this family are score matching with Langevin dynamics (SMLD) (Song & Ermon, 2019) and denoising diffusion ",
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"text": "probabilistic models (DDPMs) (Ho et al., 2020). Both methods are contained within the framework of Stochastic Differential Equations (SDEs) proposed in Song et al. (2020b). ",
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"text": "Stochastic Differential Equation (SDE) Representation Song et al. (2020b); Anderson (1982) use a forward and a corresponding backward SDE to describe general diffusion and the reversed, generative processes: ",
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"type": "equation",
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| 222 |
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"img_path": "images/f825e98d00916d4d037bbc77ab52f7cbad1259edd0566c2dc402140eb276e554.jpg",
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"text": "$$\n\\mathrm { d } \\mathbf { x } = \\mathbf { f } ( \\mathbf { x } , t ) \\mathrm { d } t + g ( t ) \\mathrm { d } \\mathbf { w } , \\quad \\mathrm { d } \\mathbf { x } = [ \\mathbf { f } - g ^ { 2 } \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) ] \\mathrm { d } t + g ( t ) \\mathrm { d } \\mathbf { w }\n$$",
|
| 224 |
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"text_format": "latex",
|
| 225 |
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"bbox": [
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| 226 |
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| 227 |
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| 229 |
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| 230 |
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],
|
| 231 |
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"text": "where w is the standard Wiener process, $\\mathbf { f } \\left( \\mathbf { x } , t \\right)$ is the vector-valued drift coefficient, $g ( t )$ is the scalar diffusion coefficient, and $\\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } )$ is the score function of the noise perturbed data distribution (as defined by the forward SDE with initial condition $p _ { 0 } ( \\mathbf { x } )$ being the data distribution). At the endpoints $t = \\{ 0 , 1 \\bar \\}$ , the forward Eq. (1) admits the data distribution $p _ { 0 }$ and the easy-to-sample prior $p _ { 1 }$ as the boundary distributions. Within this framework, the SMLD method can be described using a VarianceExploding (VE) SDE with increasing noise scales $\\sigma ( t )$ : $\\mathrm { d } \\mathbf { x } = \\sqrt { \\mathrm { d } [ \\sigma ^ { 2 } ( t ) ] / \\mathrm { d } t }$ dw. In comparison, DDPMs are endowed with a Variance-Preserving (VP) SDE: $\\mathrm { d } \\mathbf { x } = - [ \\beta ( t ) / 2 ] \\mathbf { x } \\mathrm { d } t + \\sqrt { \\beta ( t ) } \\mathrm { d } \\mathbf { w }$ with $\\beta ( t )$ being another noise sequence. Notably, the VP SDE can be reparameterized into an equivalent VE SDE (Song et al., 2020a). ",
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"text": "Probability Flow ODE Any diffusion process can be represented by a deterministic ODE that carries the same marginal densities as the diffusion process throughout its trajectory. This ODE is termed the probability flow (PF) ODE (Song et al., 2020b). PF ODEs enable uniquely identifiable encodings (Song et al., 2020b) of data, and are central to DDIBs as we solve these ODEs for forward and reverse conversion between data and their latents. For the forward SDE introduced in Eq. (1), the equivalent PF ODE holds the following form: ",
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| 255 |
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{
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| 256 |
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"type": "equation",
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| 257 |
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"img_path": "images/96924226f650ec72a1b0f4bffc85669d3bb53e018354455fa300015d4759a89f.jpg",
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"text": "$$\n\\mathrm { d } \\mathbf { x } = \\left[ \\mathbf { f } ( \\mathbf { x } , t ) - \\frac { 1 } { 2 } g ( t ) ^ { 2 } \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) \\right] \\mathrm { d } t\n$$",
|
| 259 |
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"text_format": "latex",
|
| 260 |
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"bbox": [
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| 261 |
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"text": "which follows immediately from the SDEs given the score function. In practice, we use $\\theta$ - parameterized score networks $\\mathbf { s } _ { t , \\theta } \\approx \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } )$ to approximate the score function. Training such networks relies on a variational lower bound, described in Ho et al. (2020) and in Appendix B. We may then employ numerical ODE solvers to solve the above ODE and construct $\\mathbf { x }$ at different times. Empirically, it has been demonstrated that SGMs have relatively low discretization errors when reconstructing $\\mathbf { x }$ at $t = 0$ via ODE solvers (Song et al., 2020a). For conciseness, we use $v _ { \\theta } = \\mathrm { d } \\mathbf { x } / \\mathrm { d } t$ to denote the $\\theta$ -parameterized velocity field (as defined from Eq. (2), where we replace $\\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } )$ with $\\mathbf { s } _ { t , \\theta }$ ), and use the symbol ODESolve to denote the mapping from $\\mathbf { x } ( t _ { 0 } )$ to $\\mathbf { x } ( t _ { 1 } )$ : ",
|
| 271 |
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| 278 |
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| 279 |
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{
|
| 280 |
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"type": "equation",
|
| 281 |
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| 282 |
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"text": "$$\n\\mathrm { O D E S o l v e } ( \\mathbf { x } ( t _ { 0 } ) ; v _ { \\theta } , t _ { 0 } , t _ { 1 } ) = \\mathbf { x } ( t _ { 0 } ) + \\int _ { t _ { 0 } } ^ { t _ { 1 } } v _ { \\theta } ( t , \\mathbf { x } ( t ) ) \\mathrm { d } t ,\n$$",
|
| 283 |
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"text_format": "latex",
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| 284 |
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"bbox": [
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"type": "text",
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| 294 |
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"text": "which allows us to abstract away the exact model (be it a score-based or a diffusion model), or the integrator used. In our experiments, we implement the ODE solver in DDIMs (Song et al., 2020a) (Appendix B); while we acknowledge other available ODE solvers that are usable within our framework, such as the DPM-solver (Lu et al., 2022), the Exponential Integrator (Zhang & Chen, 2022), and the second-order Heun solver (Karras et al., 2022). ",
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| 295 |
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"type": "text",
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| 305 |
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"text": "2.2 SCHRODINGER ¨ BRIDGE PROBLEM (SBP) ",
|
| 306 |
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"text_level": 1,
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"type": "text",
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"text": "Our analysis shows that DDIBs are Schrodinger Bridges (Chen et al., 2016; L ¨ eonard, 2013) between ´ distributions. Let $\\Omega = C ( [ 0 , 1 ] ; \\mathbb { R } ^ { n } )$ be the path space of $\\mathbb { R } ^ { n }$ -valued continuous functions over the time interval $[ 0 , 1 ]$ ; and $\\mathcal { D } ( p _ { 0 } , p _ { 1 } )$ be the set of distributions over $\\Omega$ , with marginals $p _ { 0 } , p _ { 1 }$ at time $t = 0$ , $t = 1$ , respectively. Given a prior reference measure $W ^ { 1 }$ , the well-known Schrodinger Bridge ¨ Problem (SBP) seeks the most probable evolution across time $t$ between the marginals $p _ { 0 }$ and $p _ { 1 }$ : ",
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| 318 |
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"text": "Problem 1 (Schrodinger Bridge Problem) ¨ . With prescribed distributions $p _ { 0 } , p _ { 1 }$ and a reference measure $W$ as the prior, the SBP finds a distribution from $\\mathcal { D } ( p _ { 0 } , p _ { 1 } )$ that minimizes its $K L$ -divergence to $W$ : $P _ { S B P } : = \\arg \\operatorname* { m i n } \\{ D _ { K L } ( P \\| W ) \\ | \\ P \\in \\mathcal { D } ( p _ { 0 } , p _ { 1 } ) \\} .$ . ",
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| 329 |
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"page_idx": 2
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| 336 |
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},
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{
|
| 338 |
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"type": "text",
|
| 339 |
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"text": "Algorithm 1 High-level Pseudo-code for DDIBs ",
|
| 340 |
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"text_level": 1,
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| 341 |
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{
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| 350 |
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"type": "text",
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| 351 |
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"text": "Input: data sample from source domain $\\mathbf { x } ^ { ( s ) } \\sim p _ { s } ( \\mathbf { x } )$ , source model $v _ { \\theta } ^ { ( s ) }$ , target model $v _ { \\theta } ^ { ( t ) }$ . Output: $\\mathbf { x } ^ { ( t ) }$ , the result in the target domain. \n$\\mathbf { x } ^ { ( l ) } = \\mathrm { O D E S o l v e } ( \\mathbf { x } ^ { ( s ) } ; v _ { \\theta } ^ { ( s ) } , 0 , 1 )$ // obtain latent code from source domain data $\\mathbf { x } ^ { ( t ) } = \\mathrm { O D E S o l v e } ( \\mathbf { x } ^ { ( l ) } ; v _ { \\theta } ^ { ( t ) } , 1 , 0 )$ // obtain target domain data from latent code \nreturn $\\mathbf { x } ^ { ( t ) }$ ",
|
| 352 |
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"type": "text",
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"text": "The minimizer, $P _ { \\mathrm { S B P } }$ , is dubbed the Schrodinger Bridge ¨ between $p _ { 0 }$ and $p _ { 1 }$ over prior $W$ . The SBP has connections to the Monge-Kantorovich (MK) optimal transport problem (Chen et al., 2021b). While the basic MK problem seeks the cost-minimizing plan to transport masses between distributions, the SBP incorporates an additional entropy term (for details, see Page 61 of Peyre et al. (2019)) . ´ ",
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{
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"type": "text",
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| 373 |
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"text": "Relationship Between SBPs and SGMs Chen et al. (2021a) establishes connections between SGMs and SBPs. In summary, SGMs are implicit optimal transport models, corresponding to SBPs with linear or degenerate drifts. General SBPs additionally accept fully nonlinear diffusion. To formalize this observation, the authors first establish similar forward and backward SDEs for SBPs: ",
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| 374 |
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{
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| 383 |
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"type": "equation",
|
| 384 |
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"img_path": "images/f2c70f2d78578b1bbd43b1f9367fdd8af8134aa81254cde01a6590a8d7a7cd41.jpg",
|
| 385 |
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"text": "$$\n\\begin{array} { r } { \\mathbf { d x } = [ \\mathbf { f } + g ^ { 2 } \\nabla _ { \\mathbf { x } } \\log \\Phi _ { t } ( \\mathbf { x } ) ] \\mathbf { d } t + g ( t ) \\mathbf { d w } , \\quad \\mathbf { d x } = [ \\mathbf { f } - g ^ { 2 } \\nabla _ { \\mathbf { x } } \\log \\hat { \\Phi } _ { t } ( \\mathbf { x } ) ] \\mathbf { d } t + g ( t ) \\mathbf { d w } } \\end{array}\n$$",
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| 386 |
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"type": "text",
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| 397 |
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"text": "where $\\Phi , { \\hat { \\Phi } }$ are the Schrodinger factors ¨ that satisfy density factorization: $p _ { t } ( { \\bf x } ) = \\Phi _ { t } ( { \\bf x } ) \\hat { \\Phi } _ { t } ( { \\bf x } )$ . The vector-valued quantities $\\mathbf { z } _ { t } = g ( t ) \\nabla _ { \\mathbf { x } } \\log \\Phi _ { t } ( \\mathbf { x } ) , \\hat { \\mathbf { z } } _ { t } = g ( t ) \\nabla _ { \\mathbf { x } } \\log \\hat { \\Phi } _ { t } ( \\mathbf { x } )$ fully characterize dynamics of the SBP, thus can be considered as the forward, backward “policies”, analogous to policy-based methods described in Schulman et al. (2015); Pereira et al. (2019). To draw a link between SBPs and SGMs, the data log-likelihood objective for SBPs is computed and shown to be equal to that of SGMs with special choices of $\\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t }$ (derivation details in Chen et al. (2021a)). Importantly, likelihood equality occurs with the following policies: ",
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"type": "equation",
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"text": "$$\n( \\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t } ) = ( 0 , g ( t ) \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) )\n$$",
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"text_format": "latex",
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"text": "When the marginal $p _ { 1 }$ at time $t = 1$ is equal to the prior distribution, it is known that such $( \\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t } )$ are achieved. Since in SGMs, the end marginal $p _ { 1 }$ is indeed the standard Gaussian prior, their log-likelihood is equivalent to that of SBPs. This suggests that SGMs are a special case of SBPs with degenerate forward policy $\\mathbf { z } _ { t }$ and a multiple of the score function as its backward $\\hat { \\mathbf { z } } _ { t }$ . ",
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"type": "text",
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"text": "Probability Flow ODE In a similar vein to the SGM SDEs, a deterministic PF ODE can be derived for SBPs with identical marginal densities across $t \\in [ 0 , 1 ]$ . The following PF ODE specifies the probability flow of the optimal processes of the SBP defined in Eq. (4) (Chen et al., 2021a): ",
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"text": "$$\n\\mathrm { d } \\mathbf { x } = \\bigg [ \\mathbf { f } ( \\mathbf { x } , t ) + g ( t ) \\mathbf { z } - \\frac { 1 } { 2 } g ( t ) ( \\mathbf { z } + \\hat { \\mathbf { z } } ) \\bigg ] \\mathrm { d } t\n$$",
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"text": "where $\\mathbf { z }$ depends on x. We shall show that the PF ODEs for SGMs and SBPs are equivalent. Thus, flowing through the PF ODEs in DDIBs is equivalent to flowing through special Schrodinger Bridges, ¨ with one of the marginals being Gaussian. ",
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"type": "text",
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"text": "3 DUAL DIFFUSION IMPLICIT BRIDGES ",
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"text": "DDIBs leverage the connections between SGMs and SBPs to perform image-to-image translation, with two diffusion models trained separately on the two domains. DDIBs contain two steps, described in Alg. 1 and illustrated in Fig. 1. At the core of the algorithm is the ODE solver ODESolve from Eq. (3). Given a source model represented as a vector field, i.e., $v _ { \\theta } ^ { ( s ) }$ defined from Eq. (2), DDIBs first apply ODESolve in the source domain to obtain the encoding $\\mathbf { x } ^ { ( s ) }$ of the image at the end time $t = 1$ ; we refer to this as the latent code (associated with the diffusion model for the domain). Then, the source latent code is fed as the initial condition (target latent code at $t = 1$ ) to ODESolve with the target model $v _ { \\theta } ^ { ( t ) }$ to obtain the target image $\\mathbf { x } ^ { ( t ) }$ . As discussed earlier, we implement ODESolve with DDIMs (Song et al., 2020a), which are known to have reasonably small discretization errors. While recent developments in higher order ODE solvers (Zhang & Chen, 2022; Lu et al., 2022; Karras et al., 2022) that generalize DDIMs can also be used here, we leave this investigation to future work. ",
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"text": "Despite the simplicity of the method, DDIBs have several advantages over prior methods, which we discuss below. ",
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"text": "Exact Cycle Consistency A desirable feature of image translation algorithms is the cycle consistency property: transforming a data point from the source domain to the target domain, and then back to source, will recover the original data point in the source domain. The following proposition validates the cycle consistency of DDIBs. ",
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"text": "Proposition 3.1 (DDIBs Enforce Exact Cycle Consistency). Given a sample from source domain $\\mathbf { x } ^ { ( s ) }$ , a source diffusion model $v _ { \\theta } ^ { ( s ) }$ , and a target model $v _ { \\theta } ^ { ( t ) }$ , define: ",
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"text": "$$\n\\begin{array} { r } { \\begin{array} { r } { { \\mathbf { x } } ^ { ( l ) } = \\mathrm { O D E S o l v e } ( { \\mathbf { x } } ^ { ( s ) } ; v _ { \\theta } ^ { ( s ) } , 0 , 1 ) ; \\quad { \\mathbf { x } } ^ { ( t ) } = \\mathrm { O D E S o l v e } ( { \\mathbf { x } } ^ { ( l ) } ; v _ { \\theta } ^ { ( t ) } , 1 , 0 ) ; } \\\\ { { \\mathbf { x } } ^ { \\prime ( l ) } = \\mathrm { O D E S o l v e } ( { \\mathbf { x } } ^ { ( t ) } ; v _ { \\theta } ^ { ( t ) } , 0 , 1 ) ; \\quad { \\mathbf { x } } ^ { \\prime ( s ) } = \\mathrm { O D E S o l v e } ( { \\mathbf { x } } ^ { \\prime ( l ) } ; v _ { \\theta } ^ { ( s ) } , 1 , 0 ) } \\end{array} } \\end{array}\n$$",
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| 536 |
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| 537 |
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"type": "text",
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"text": "Assume zero discretization error. Then, $\\mathbf { x } ^ { ( s ) } = \\mathbf { x } ^ { \\prime ( s ) }$ ",
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"type": "text",
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"text": "As PF ODEs are used, the cycle consistency property is guaranteed. In practice, even with discretization error, DDIBs incur almost negligible cycle inconsistency (Section 4.1). In contrast, GAN-based methods are not guaranteed the cycle consistency property by default, and have to incorporate additional training terms to optimize for cycle consistency over two domains. ",
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"type": "text",
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"text": "Data Privacy in Both Domains In the DDIBs translation process, only the source and target diffusion models are required, whose training processes do not depend on knowledge of the domain pair a priori. In fact, this process can even be performed in a privacy sensitive manner (graphic illustration in Appendix A). Let Alice and Bob be the data owners of the source and target domains, respectively. Suppose Alice intends to translate images to the target domain. However, Alice does not want to share the data with Bob (and vice versa, Bob does not want to release their data either). Then, Alice can simply train a diffusion model with the source data, encode the data to the latent space, transmit the latent codes to Bob, and next ask Bob to run their trained diffusion model and send the results back. In this procedure, only the latent code and the target results are transmitted between the two data vendors, and both parties have naturally ensured that their data are not directly revealed. ",
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| 570 |
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"text": "DDIBs are Two Concatenated Schrodinger Bridges ¨ DDIBs link the source data distribution to the latent space, and then to the target distribution. What is the nature of such connections between distributions? We offer an answer from an optimal transport perspective: these connections are special Schrodinger Bridges ¨ between distributions. This, in turn, explicates the name of our method: dual diffusion implicit bridges are based on denoising diffusion implicit models (Song et al., 2020a), and consist of two separate Schrodinger ¨ Bridges that connect the data and latent distributions. Specifically, as considered earlier, when conditions about the policies $\\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t }$ in Eq. (5) and the density $p _ { 1 } ( \\mathbf { x } )$ being a Gaussian prior are met, the data likelihoods (at $t = 0$ ) for SGMs and SBPs are identical. Indeed, these conditions are fulfilled in SGMs and particularly in DDIMs. This verifies SGMs as special linear or degenerate SBPs. Forward and reverse solving the PF ODE for SGMs, as done in DDIBs, is equivalent to flowing through the optimal processes of particular SBPs: ",
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| 581 |
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"type": "text",
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"text": "Proposition 3.2 (PF ODE Equivalence2). Eq. (2) is equivalent to Eq. (6) with forward, backward policies $( \\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t } ) = ( 0 , g \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) )$ as attained in SGMs and particularly in DDIMs. ",
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"type": "text",
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"text": "Thus, DDIBs are intrinsically entropy-regularized optimal transport: they are Schrodinger Bridges ¨ between the source and the latent, and between the latent and the target distributions. The translation process can then be recognized as traversing through two concatenated Schrodinger Bridges, one ¨ forward and one reversed. The mapping is unique and minimizes a (regularized) optimal transport objective, which probably elucidates the superior performance of DDIBs. In contrast, if we train the source and target models separately with normalizing flow models that are not inborn with such a connection, there are many viable invertible mappings, and the resulting image translation algorithm may not necessarily have good performance. This is probably the reason why AlignFlow (Grover et al., 2020) still has to incorporate an adversarial loss even when cycle-consistency is guaranteed. ",
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{
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"type": "table",
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"img_path": "images/ccb8103213bf4508747916601fb8ec38828642f5c8052f50af2b5c931d859c79.jpg",
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| 614 |
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"table_caption": [
|
| 615 |
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"Table 1: Cycle consistency of DDIBs. Experiment legend, PR $\\bigcirc$ PS, means that we translate from PR to PS and then back. The numbers are the averaged L2 distances between the original points and their coordinates after cycle translation. Data points are standardized to have unit variance. "
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],
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"table_footnote": [],
|
| 618 |
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"table_body": "<table><tr><td>PR OPS</td><td>PS OCS</td><td>CR OPR</td><td>CR O CS</td><td>MO CB</td></tr><tr><td>0.0143</td><td>0.0065</td><td>0.0106</td><td>0.0078</td><td>0.0122</td></tr></table>",
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"type": "text",
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"text": "4 EXPERIMENTS ",
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"type": "text",
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"text": "We present a series of experiments to demonstrate the effectiveness of DDIBs. First, we describe synthetic experiments on two-dimensional datasets, to corroborate DDIBs’ cycle-consistent and optimal transport properties. Next, we validate DDIBs on a variety of image translation tasks, including color transfer, paired translation, and conditional ImageNet translation. 34 ",
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"text": "4.1 2D SYNTHETIC EXPERIMENTS ",
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"text": "We first perform domain translation on synthetic datasets drawn from complex two-dimensional distributions, with various shapes and configurations, in Fig. 2a. In total, we consider six 2D datasets: Moons (M); Checkerboards (CB); Concentric Rings (CR); Concentric Squares (CS); Parallel Rings (PR); and Parallel Squares (PS). The datasets are all normalized to have zero mean, and identity covariance. We assign colors to points based on the point identities (i.e., if a point in the source domain is red, its corresponding point in the target domain is also colored red). Clearly, the transformation is smooth between columns. For example, on the top-right corner, red points in the CR dataset are mapped to similar coordinates, both in the latent and in the target dimensions. ",
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{
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| 674 |
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"type": "text",
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"text": "(a) Smooth translation of synthetic datasets. (Left) The source datasets: CR and CS. (Middle) DDIBs’ latent code representation. (Right) Results of translation to the target domains. ",
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},
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{
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| 685 |
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"type": "image",
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"img_path": "images/ae78672377be50dd79010dbe570b9745d13641e689747b703ee356dcbc6d14b0.jpg",
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"image_caption": [
|
| 688 |
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"(b) Cycle consistency: After translating the Moons dataset to Checkerboards and then back to Moons, DDIBs restore almost the exact same points as the original ones. ",
|
| 689 |
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"Figure 2: Smoothness and cycle consistency of DDIBs. "
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],
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"image_footnote": [],
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"type": "text",
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| 702 |
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"text": "Cycle Consistency Fig. 2b illustrates the cycle consistency property guaranteed by DDIBs. It concerns 2D datasets: Moons, and Checkerboards. Starting from the Moons dataset, DDIBs first obtain the latent codes and construct the Checkerboards points. Next, DDIBs do translations in the reverse direction, transforming the points back to the latent and the Moons space. After this round trip, points are approximately mapped to their original positions. A similar, smooth color topology is observed in this experiment. Table 1 reports quantitative evaluation results on cycle-consistent translation among multiple datasets. As the datasets are normalized to unit standard deviation, the reported values are negligibly small and endorse the cycle consistent property of DDIBs. ",
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"bbox": [
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"page_idx": 5
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"type": "table",
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"img_path": "images/468d8cf66c2feae617666c379545a5d342da54656551c4b468384e7872b006df.jpg",
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"table_caption": [
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| 715 |
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"Table 2: Mean Squared Error (MSE) comparing color transfer results of DDIBs with common OT methods on two images. Each number represents the MSE between DDIBs and the corresponding OT method. MSE is computed pixel-wise after normalizing images to $[ - 1 , 1 ]$ . "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>IMAGE</td><td>EMD</td><td>SINKHORN</td><td>LINEAR</td><td>GAUSSIAN</td></tr><tr><td>TARGET1</td><td>0.0337</td><td>0.0281</td><td>0.0352</td><td>0.0370</td></tr><tr><td>TARGET 2</td><td>0.0293</td><td>0.0326</td><td>0.0500</td><td>0.0751</td></tr></table>",
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"type": "table",
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"img_path": "images/edcce6da43511e77d3a1621dfbad2dbefec21e522b8df91230ac5eeff27bf73c.jpg",
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"table_caption": [
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| 731 |
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"Table 3: MSE comparing DDIBs and baselines on paired test sets. MSE is computed pixel-wise after normalizing images to $[ - 1 , 1 ]$ . "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>DATASET</td><td>MODEL</td><td>A→B</td><td>B→A</td><td>DATASET</td><td>MODEL</td><td>A→B</td><td>B→A</td></tr><tr><td rowspan=\"3\">FACADES</td><td>CYCLEGAN</td><td>0.7129</td><td>0.3286</td><td rowspan=\"3\">MAPS</td><td>CYCLEGAN</td><td>0.0245</td><td>0.0953</td></tr><tr><td>ALIGNFLOW</td><td>0.5801</td><td>0.2512</td><td>ALIGNFLOW</td><td>0.0209</td><td>0.0897</td></tr><tr><td>DDIBS</td><td>0.5312</td><td>0.3946</td><td>DDIBS</td><td>0.0194</td><td>0.1302</td></tr></table>",
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"type": "text",
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"text": "4.2 EXAMPLE-GUIDED COLOR TRANSFER ",
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"text": "DDIBs can be used on an interesting application: example-guided color transfer. This refers to the task of modifying the colors of an input image, conditioned on the color palette of a reference image. To use DDIBs for color transfer, we train one diffusion model per image, on its normalized RGB space. During translation, DDIBs obtain encodings of the original colors, and apply the diffusion model of the reference image to attain the desired color palette. Fig. 3 visualizes our color experiments. ",
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"type": "text",
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"text": "Comparison to Alternative OT Methods As DDIBs are related to regularized OT, we compare the pixel-wise MSEs between color-transferred images generated by DDIBs, and images produced by alternate methods, in Table 2. We include four OT methods for comparison: Earth Mover’s Distance; Sinkhorn distance (Cuturi, 2013); linear and Gaussian mapping estimation (Perrot et al., 2016). Results of DDIBs are very close to those of OT methods. Appendix E.2 details full color translation results. ",
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"type": "image",
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"img_path": "images/3258de5665a5c2a16d56cf4d2c018f21fecf14a9b389df28f63e27fe67c77b76.jpg",
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| 780 |
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"image_caption": [
|
| 781 |
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"Figure 3: Example-Guided Color Transfer: Given the first image as the reference image, DDIBs modify the colors of two input images to similarly follow a snowy winter color palette. "
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"type": "text",
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"text": "4.3 QUANTITATIVE TRANSLATION EVALUATION ",
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"text_level": 1,
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"type": "text",
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"text": "Quantitatively, we demonstrate that DDIBs deliver competitive results on paired domain tests. Such numerical evaluation is despite that DDIBs are formulated with a weaker setting: diffusion models are trained independently, on separate datasets. In comparison, methods such as CycleGAN and AlignFlow assume access to both datasets during training and jointly optimize for the translation loss. ",
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"type": "text",
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"text": "Paired Domain Translation As in similar works, we evaluate DDIBs on benchmark paired datasets (Zhu et al., 2017): Facades and Maps. Both are image segmentation tasks. In the pairs of datasets, one dataset contains real photos taken via a camera or a satellite; while the other comprises the corresponding segmentation images. These datasets provide one-to-one image alignment, which allows quantitative evaluation through a distance metric such as mean-squared error (MSE) between generated samples and the corresponding ground truth. To facilitate the workings of DDIBs, we additionally employ a color conversion heuristic motivated by optimal transport on image colors (Appendix E.1). Table 3 reports the evaluation results. Surprisingly, DDIBs are able to produce segmentation images that surpass alternative methods in MSE terms; while reverse translations also achieve decent performance. ",
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"type": "text",
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"text": "4.4 CLASS-CONDITIONAL IMAGENET TRANSLATION",
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"text_level": 1,
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"type": "text",
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"text": "In this experiment, we apply DDIBs to translation among ImageNet classes. To this end, we leverage the pretrained diffusion models from Dhariwal & Nichol (2021). The authors optimized performance of diffusion models, and end up with a “UNet” (Ho et al., 2020) architecture with particular width, attention and residual configurations. The models are learned on 1, 000 ImageNet classes, each with around 1, 000 training images, and at a variety of resolutions. Our experiments use the model with resolution $2 5 6 \\times 2 5 6$ . Moreover, these models incorporate a technique known as classifier guidance (Dhariwal & Nichol, 2021), that leverage classifier gradients to steer the sampling process towards arbitrary class labels during image generation. The learned models combined with classifier guidance can be effectively considered as 1, 000 different models. Fig. 4a exhibits select translation samples, where the source images are from ImageNet validation sets. DDIBs are able to create faithful target images that maintain much of the original content such as animal poses, complexions and emotions, while accounting for differences in animal species. ",
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| 841 |
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"type": "text",
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"text": "Multi-Domain Translation Given conditional models on the individual domains, DDIBs can be applied to translate between arbitrary pairs of source-target domains, while requiring no additional fine-tuning or adaptation. Fig. 4b displays results of translating a common image of a roaring lion (with class label 291), to various other ImageNet classes. Interestingly, some animals roar, while others stick their tongues out. DDIBs successfully internalize characteristics of distinct animal species, and produce closest animal postures in OT distances to the original shouting lion. ",
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"page_idx": 7
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{
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"type": "image",
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"img_path": "images/649b746d19b2ec60e11404bf7366e6b99da18f6224f71705c869e6e0c897b10e.jpg",
|
| 863 |
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"image_caption": [
|
| 864 |
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"(a) Conditional ImageNet Translation: Selected trans- (b) Multi-domain translation: Given the center, lation samples from various ImageNet classes such as 7: source image from class label 291, DDIBs translate Cock, 94: Hummingbird, 162: Beagle, and 282: Tiger it to other animal species, entirely using only a preCat. trained conditional diffusion model. "
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"type": "image",
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"img_path": "images/9a11302c72e2c4164abfc78bddbf2ac251badbafda7dcee77d3f1b0337d52e44.jpg",
|
| 878 |
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"image_caption": [
|
| 879 |
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"Figure 4: Translation among ImageNet classes. "
|
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"type": "text",
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"text": "5 RELATED WORKS ",
|
| 893 |
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"text_level": 1,
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| 894 |
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"type": "text",
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"text": "Score-based Diffusion Models Originating in thermodynamics (Sohl-Dickstein et al., 2015), diffusion models reverse the dynamics of a noising process to create data samples. The reversal process is understood to implicitly compute scores of the data density at various noise scales, which reveals connections to score-based methods (Song & Ermon, 2019; Nichol & Dhariwal, 2021; Meng et al., 2021b). Diffusion models are applicable to multiple modalities: 3D shapes (Zhou et al., 2021), point cloud (Luo & Hu, 2021), discrete domains (Meng et al., 2022) and function spaces (Lim et al., 2023). They excel in tasks ranging from image editing and composition (Meng et al., 2021a), density estimation (Kingma et al., 2021), to image restoration (Kawar et al., 2022). Seminal works are denoising diffusion probabilistic models (DDPMs, Ho et al. (2020)), which parameterized the ELBO objective with Gaussians and, for the first time, synthesized high-quality images with diffusion models; ILVR (Choi et al., 2021), which invented a novel conditional method to direct DDPM generation towards reference images; and denoising diffusion implicit models (DDIMs, Song et al. (2020a)), which accelerated DDPM inference via non-Markovian processes. DDIMs can be treated as a first-order numerical solver of a probabilistic ODE, which we use heavily in DDIBs. ",
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| 905 |
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"type": "text",
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"text": "Diffusion Models for Image Translation While GANs (Goodfellow et al., 2014; Zhu et al., 2017; Zhao et al., 2020) have been widely adopted in image translation tasks, recent works increasingly leverage diffusion models. For instance, Palette (Saharia et al., 2021) applies a conditional diffusion model to colorization, inpainting, and restoration. DiffuseIT (Kwon & Ye, 2022) utilizes disentangled style and content representation, to perform text- and image-guided style transfer. Lastly, UNITDDPM (Sasaki et al., 2021) proposes a novel coupling between domain pairs and trains joint DDPMs for translation. Unlike their joint training, DDIBs apply separate, pretrained diffusion models and leverage geometry of the shared space for translation. ",
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"type": "text",
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"text": "Optimal Transport for Translation and Generative Modeling As it pursues cost-optimal plans to connect image distributions, OT naturally finds applications in image translation. For example, Korotin et al. (2022) capitalizes on the approximation powers of neural networks to compute OT plans between image distributions and perform unpaired translation. By contrast, the entropy-regularized OT variant, Schrodinger Bridges (Section 2), are also commonly used to derive generative models. ¨ For instance, De Bortoli et al. (2021) and Vargas et al. (2021) concurrently proposed new numerical procedures that approximate the Iterative Proportional Fitting scheme, to solve SBPs for image generation. Wang et al. (2021) presents a new generative method via entropic interpolation with an SBP. Chen et al. (2021a) discovers equivalence between the likelihood objectives of SBP and score-based models, which lays the theoretical foundations behind DDIBs. Their sequel (Liu et al., 2023) then directly learns the Schrodinger Bridges between image distributions, for applications ¨ in image-to-image tasks such as restoration. While DDIBs were not initially designed to mimic Schrodinger Bridges, our analysis reveals their true characterization as solutions to degenerate SBPs. ¨ ",
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"type": "text",
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"text": "6 CONCLUSIONS ",
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| 938 |
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"text": "We present Dual Diffusion Implicit Bridges (DDIBs), a new, simplistic image translation method that stems from latest progresses in score-based diffusion models, and is theoretically grounded as Schrodinger Bridges in the image space. DDIBs solve two key problems. First, DDIBs avoid ¨ optimization on a coupled loss specific to the given domain pair only. Second, DDIBs better safeguard dataset privacy as they no longer require presence of both datasets during training. Powerful pretrained diffusion models are then integrated into our DDIBs framework, to perform a comprehensive series of experiments that prove DDIBs’ practical values in domain translation. Our method is limited in its application to color transfer, as one model is required for each image, which demands significant compute for mass experiments. Rooted in optimal transport, DDIBs translation mimics the massmoving process which may be problematic at times (Appendix C). Future work may remedy these issues, or extend DDIBs to applications with different dimensions in the source and target domains. As flowing through the concatenated ODEs is time-consuming, improving the translation speed is also a promising direction. ",
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"type": "text",
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"text": "ACKNOWLEDGEMENTS ",
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"text_level": 1,
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},
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"type": "text",
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"text": "We thank Lingxiao Li and Chris Cundy for insightful discussions about the optimal transport properties of DDIBs. We also thank the anonymous reviewers for their constructive comments and feedback. This research was supported by NSF (#1651565), ARO (W911NF-21-1-0125), ONR (N00014-23-1-2159), CZ Biohub, and Stanford HAI. ",
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"type": "text",
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"text": "REFERENCES ",
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"text_level": 1,
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},
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"type": "text",
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"text": "Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982. ",
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"type": "text",
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"text": "Linqi Zhou, Yilun Du, and Jiajun Wu. 3d shape generation and completion through point-voxel diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 5826–5835, 2021. ",
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"text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223–2232, 2017. ",
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"bbox": [
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171,
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626,
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+
823,
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| 1506 |
+
667
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],
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"page_idx": 11
|
| 1509 |
+
},
|
| 1510 |
+
{
|
| 1511 |
+
"type": "text",
|
| 1512 |
+
"text": "Alice is the owner of the source (tiger) domain, and Bob is the owner of the target (cat) domain. Alice intends to translate tiger images to cat images, but in a privacy-sensitive manner without releasing the source dataset. Bob does not wish to make the cat dataset public, either. ",
|
| 1513 |
+
"bbox": [
|
| 1514 |
+
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|
| 1515 |
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|
| 1516 |
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| 1517 |
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| 1518 |
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],
|
| 1519 |
+
"page_idx": 12
|
| 1520 |
+
},
|
| 1521 |
+
{
|
| 1522 |
+
"type": "image",
|
| 1523 |
+
"img_path": "images/2b61eb9f42e2b1f466db582a6a11ff82b05efaaa0dfc5885b8484b5d963b81d1.jpg",
|
| 1524 |
+
"image_caption": [
|
| 1525 |
+
"Fig. 5 illustrates the process of privacysensitive domain translation. The process contains the following steps, with indexes in the figure. ",
|
| 1526 |
+
"Figure 5 "
|
| 1527 |
+
],
|
| 1528 |
+
"image_footnote": [],
|
| 1529 |
+
"bbox": [
|
| 1530 |
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|
| 1531 |
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|
| 1532 |
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|
| 1533 |
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|
| 1534 |
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],
|
| 1535 |
+
"page_idx": 12
|
| 1536 |
+
},
|
| 1537 |
+
{
|
| 1538 |
+
"type": "text",
|
| 1539 |
+
"text": "1. Alice intends to translate tiger images to cat images. \n2. Alice trains a diffusion model with the source tiger images. \n3. Alice uses the pretrained, tiger diffusion model to convert a source tiger image to its latent code. \n4. Alice sends the latent code to Bob. \n5. Bob similarly trains a diffusion model on the cat domain. \n6. Bob uses the pretrained, cat diffusion model to convert the received latent code to a cat image. \n7. Bob then sends the translated image back to Alice. ",
|
| 1540 |
+
"bbox": [
|
| 1541 |
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|
| 1542 |
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|
| 1543 |
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|
| 1544 |
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|
| 1545 |
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],
|
| 1546 |
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"page_idx": 12
|
| 1547 |
+
},
|
| 1548 |
+
{
|
| 1549 |
+
"type": "text",
|
| 1550 |
+
"text": "Clearly, during the translation process, only the latent code and the translated cat image are transmitted via the public channel, while both source and target datasets are private to the two parties. This is a significant advantage of DDIBs over alternate methods, as we enable strong privacy protection of the datasets. ",
|
| 1551 |
+
"bbox": [
|
| 1552 |
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|
| 1553 |
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|
| 1554 |
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|
| 1555 |
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|
| 1556 |
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],
|
| 1557 |
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"page_idx": 12
|
| 1558 |
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},
|
| 1559 |
+
{
|
| 1560 |
+
"type": "text",
|
| 1561 |
+
"text": "B DETAILS OF SGM TRAINING AND DDIM ODE SOLVER ",
|
| 1562 |
+
"text_level": 1,
|
| 1563 |
+
"bbox": [
|
| 1564 |
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|
| 1565 |
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| 1566 |
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| 1567 |
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|
| 1568 |
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],
|
| 1569 |
+
"page_idx": 13
|
| 1570 |
+
},
|
| 1571 |
+
{
|
| 1572 |
+
"type": "text",
|
| 1573 |
+
"text": "B.1 TRAINING SCORE NETWORKS ",
|
| 1574 |
+
"text_level": 1,
|
| 1575 |
+
"bbox": [
|
| 1576 |
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|
| 1577 |
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|
| 1578 |
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|
| 1579 |
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|
| 1580 |
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],
|
| 1581 |
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"page_idx": 13
|
| 1582 |
+
},
|
| 1583 |
+
{
|
| 1584 |
+
"type": "text",
|
| 1585 |
+
"text": "While the description in Section 2 is based on continuous SDEs, actual implementations of diffusion models often sample discrete time steps. Given samples from a data distribution $q ( \\mathbf { x } _ { 0 } )$ , diffusion models attempt to learn a model distribution $p _ { \\theta } ( \\mathbf { x } _ { 0 } )$ that approximates $q ( \\mathbf { x } _ { 0 } )$ , and is easy to sample from. Specifically, diffusion probabilistic models are latent variable models of the form ",
|
| 1586 |
+
"bbox": [
|
| 1587 |
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|
| 1588 |
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|
| 1589 |
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|
| 1590 |
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|
| 1591 |
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],
|
| 1592 |
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"page_idx": 13
|
| 1593 |
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},
|
| 1594 |
+
{
|
| 1595 |
+
"type": "equation",
|
| 1596 |
+
"img_path": "images/cd8510744b3893bbc51c9621e5cf8cb1efd83273b7882e05ecc0a0c2741629ca.jpg",
|
| 1597 |
+
"text": "$$\np _ { \\theta } ( \\mathbf { x } _ { 0 } ) = \\int p _ { \\theta } ( \\mathbf { x } _ { 0 : T } ) \\mathrm { d } \\mathbf { x } _ { 1 : T } , \\mathrm { w h e r e } p _ { \\theta } ( \\mathbf { x } _ { 0 : T } ) = p _ { \\theta } ( \\mathbf { x } _ { T } ) \\prod _ { t = 1 } ^ { T } p _ { \\theta } ^ { ( t ) } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } )\n$$",
|
| 1598 |
+
"text_format": "latex",
|
| 1599 |
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"bbox": [
|
| 1600 |
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|
| 1601 |
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| 1603 |
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|
| 1604 |
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],
|
| 1605 |
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"page_idx": 13
|
| 1606 |
+
},
|
| 1607 |
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{
|
| 1608 |
+
"type": "text",
|
| 1609 |
+
"text": "where $\\mathbf { x } _ { 1 } , \\cdots , \\mathbf { x } _ { T }$ are latent variables in the same sample space as $\\mathbf { x } _ { \\mathrm { 0 } }$ . The parameters $\\theta$ are trained to approximate the data distribution $q ( \\mathbf { x } _ { 0 } )$ , by maximizing a variational lower bound: ",
|
| 1610 |
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"bbox": [
|
| 1611 |
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| 1612 |
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|
| 1613 |
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|
| 1614 |
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|
| 1615 |
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],
|
| 1616 |
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"page_idx": 13
|
| 1617 |
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},
|
| 1618 |
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{
|
| 1619 |
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"type": "equation",
|
| 1620 |
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"img_path": "images/2a7c4e1aadce5a2456cd8cabf97f1b26371bab2b5b5ca1053de92b69e8f118e7.jpg",
|
| 1621 |
+
"text": "$$\n\\operatorname* { m a x } _ { \\theta } \\mathbb { E } _ { q ( \\mathbf { x } _ { 0 } ) } [ \\log p \\varrho ( \\mathbf { x } _ { 0 } ) ] \\leq \\operatorname* { m a x } _ { \\theta } \\mathbb { E } _ { q ( \\mathbf { x } _ { 0 } , \\mathbf { x } _ { 1 } , \\cdots , \\mathbf { x } _ { T } ) } [ \\log p \\varrho ( \\mathbf { x } _ { 0 : T } ) - \\log q ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 0 } ) ]\n$$",
|
| 1622 |
+
"text_format": "latex",
|
| 1623 |
+
"bbox": [
|
| 1624 |
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|
| 1625 |
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|
| 1626 |
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|
| 1627 |
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329
|
| 1628 |
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],
|
| 1629 |
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"page_idx": 13
|
| 1630 |
+
},
|
| 1631 |
+
{
|
| 1632 |
+
"type": "text",
|
| 1633 |
+
"text": "where $q \\big ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 0 } \\big )$ is some inference distribution over the latent variables. It is known that when the conditional distributions are modeled as Gaussians with trainable mean functions and fixed variances, the above objective can be simplified to: ",
|
| 1634 |
+
"bbox": [
|
| 1635 |
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|
| 1636 |
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|
| 1637 |
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|
| 1638 |
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378
|
| 1639 |
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],
|
| 1640 |
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"page_idx": 13
|
| 1641 |
+
},
|
| 1642 |
+
{
|
| 1643 |
+
"type": "equation",
|
| 1644 |
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"img_path": "images/f8eee46971c0bbc18b331790f27547b8f4719e98acab4f51a354b7d1fee65edb.jpg",
|
| 1645 |
+
"text": "$$\nL ( \\epsilon _ { \\theta } ) : = \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { \\mathbf { x } _ { 0 } \\sim q ( \\mathbf { x } _ { 0 } ) , \\epsilon _ { t } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } ) } \\left[ \\left\\| \\epsilon _ { \\theta } ^ { ( t ) } ( \\sqrt { \\alpha _ { t } } \\mathbf { x } _ { 0 } + \\sqrt { 1 - \\alpha _ { t } } \\epsilon _ { t } ) - \\epsilon _ { t } \\right\\| _ { 2 } ^ { 2 } \\right]\n$$",
|
| 1646 |
+
"text_format": "latex",
|
| 1647 |
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"bbox": [
|
| 1648 |
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|
| 1649 |
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|
| 1650 |
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| 1651 |
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429
|
| 1652 |
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],
|
| 1653 |
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"page_idx": 13
|
| 1654 |
+
},
|
| 1655 |
+
{
|
| 1656 |
+
"type": "text",
|
| 1657 |
+
"text": "The resulting noise prediction functions $\\epsilon _ { \\theta } ^ { ( t ) }$ , are equivalent to the score networks $\\mathbf { s } _ { t , \\theta }$ mentioned in Section 2 due to Tweedie’s formula (Stein, 1981; Efron, 2011). For details, we refer the reader to $\\mathrm { H o }$ et al. (2020); Song et al. (2020a). ",
|
| 1658 |
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"bbox": [
|
| 1659 |
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|
| 1660 |
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| 1662 |
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|
| 1663 |
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],
|
| 1664 |
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"page_idx": 13
|
| 1665 |
+
},
|
| 1666 |
+
{
|
| 1667 |
+
"type": "text",
|
| 1668 |
+
"text": "B.2 DDIM ODE SOLVER ",
|
| 1669 |
+
"text_level": 1,
|
| 1670 |
+
"bbox": [
|
| 1671 |
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| 1674 |
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|
| 1675 |
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],
|
| 1676 |
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"page_idx": 13
|
| 1677 |
+
},
|
| 1678 |
+
{
|
| 1679 |
+
"type": "text",
|
| 1680 |
+
"text": "With a trained noise prediction model $\\epsilon _ { \\theta } ^ { ( t ) } ( \\mathbf { x } )$ , the DDIM iterate between adjacent variables $\\mathbf { x } _ { t - \\Delta t }$ and $\\mathbf { x } _ { t }$ , considered in Song et al. (2020a), assumes the following form: ",
|
| 1681 |
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"bbox": [
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| 1682 |
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| 1684 |
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| 1686 |
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],
|
| 1687 |
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"page_idx": 13
|
| 1688 |
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},
|
| 1689 |
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{
|
| 1690 |
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"type": "equation",
|
| 1691 |
+
"img_path": "images/0afb2f71971abc39ddad7e47bc616a93c5b9a03f912363401ccca81d7c56f32d.jpg",
|
| 1692 |
+
"text": "$$\n\\frac { \\mathbf { x } _ { t - \\Delta t } } { \\sqrt { \\alpha _ { t - \\Delta t } } } = \\frac { \\mathbf { x } _ { t } } { \\sqrt { \\alpha _ { t } } } + \\left( \\sqrt { \\frac { 1 - \\alpha _ { t - \\Delta t } } { \\alpha _ { t - \\Delta t } } } - \\sqrt { \\frac { 1 - \\alpha _ { t } } { \\alpha _ { t } } } \\right) \\epsilon _ { \\theta } ^ { ( t ) } ( \\mathbf { x } _ { t } )\n$$",
|
| 1693 |
+
"text_format": "latex",
|
| 1694 |
+
"bbox": [
|
| 1695 |
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| 1696 |
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| 1697 |
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| 1698 |
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|
| 1699 |
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],
|
| 1700 |
+
"page_idx": 13
|
| 1701 |
+
},
|
| 1702 |
+
{
|
| 1703 |
+
"type": "text",
|
| 1704 |
+
"text": "In our experiments, we implement the above equation between adjacent diffusion steps. The equation is deterministic, and can be considered as a Euler method over the following ODE: ",
|
| 1705 |
+
"bbox": [
|
| 1706 |
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|
| 1711 |
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"page_idx": 13
|
| 1712 |
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},
|
| 1713 |
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{
|
| 1714 |
+
"type": "equation",
|
| 1715 |
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"img_path": "images/6ce5f7422a6cd137398786eafd2f96d43e6d3216c82dc7136f40c0f7dfd17c78.jpg",
|
| 1716 |
+
"text": "$$\n\\mathrm { d } \\bar { \\mathbf { x } } ( t ) = \\epsilon _ { \\theta } ^ { ( t ) } \\left( \\frac { \\bar { \\mathbf { x } } ( t ) } { \\sqrt { \\sigma ^ { 2 } + 1 } } \\right) \\mathrm { d } \\sigma ( t )\n$$",
|
| 1717 |
+
"text_format": "latex",
|
| 1718 |
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"bbox": [
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| 1719 |
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| 1720 |
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606,
|
| 1722 |
+
679
|
| 1723 |
+
],
|
| 1724 |
+
"page_idx": 13
|
| 1725 |
+
},
|
| 1726 |
+
{
|
| 1727 |
+
"type": "text",
|
| 1728 |
+
"text": "where we adopt the reparameterization: ",
|
| 1729 |
+
"bbox": [
|
| 1730 |
+
174,
|
| 1731 |
+
684,
|
| 1732 |
+
433,
|
| 1733 |
+
699
|
| 1734 |
+
],
|
| 1735 |
+
"page_idx": 13
|
| 1736 |
+
},
|
| 1737 |
+
{
|
| 1738 |
+
"type": "equation",
|
| 1739 |
+
"img_path": "images/eb9720f73471a98b4cb6e1dc549e94a1be65a245fff317111ee10826050b4b06.jpg",
|
| 1740 |
+
"text": "$$\n\\sigma ( t ) = \\sqrt { \\frac { 1 - \\alpha ( t ) } { \\alpha ( t ) } } , \\quad \\bar { \\mathbf { x } } ( t ) = \\frac { \\mathbf { x } ( t ) } { \\sqrt { \\alpha ( t ) } }\n$$",
|
| 1741 |
+
"text_format": "latex",
|
| 1742 |
+
"bbox": [
|
| 1743 |
+
370,
|
| 1744 |
+
705,
|
| 1745 |
+
627,
|
| 1746 |
+
748
|
| 1747 |
+
],
|
| 1748 |
+
"page_idx": 13
|
| 1749 |
+
},
|
| 1750 |
+
{
|
| 1751 |
+
"type": "text",
|
| 1752 |
+
"text": "Importantly, the ODE in Eq. (9) with the optimal model $\\epsilon _ { \\theta } ^ { ( t ) } ( \\mathbf { x } )$ , has an equivalent probability flow ODE corresponding to the “Variance-Exploding” SDE in Song et al. (2020b). ",
|
| 1753 |
+
"bbox": [
|
| 1754 |
+
173,
|
| 1755 |
+
756,
|
| 1756 |
+
823,
|
| 1757 |
+
787
|
| 1758 |
+
],
|
| 1759 |
+
"page_idx": 13
|
| 1760 |
+
},
|
| 1761 |
+
{
|
| 1762 |
+
"type": "text",
|
| 1763 |
+
"text": "C LIMITATIONS OF OPTIMAL TRANSPORT-BASED TRANSLATION ",
|
| 1764 |
+
"text_level": 1,
|
| 1765 |
+
"bbox": [
|
| 1766 |
+
174,
|
| 1767 |
+
102,
|
| 1768 |
+
725,
|
| 1769 |
+
118
|
| 1770 |
+
],
|
| 1771 |
+
"page_idx": 14
|
| 1772 |
+
},
|
| 1773 |
+
{
|
| 1774 |
+
"type": "text",
|
| 1775 |
+
"text": "DDIBs contain deterministic bridges between distributions, and are a form of entropy-regularized optimal transport. The learned diffusion models can be effectively considered as a digest or summary of the datasets. While doing translation, they attempt to create images in the target domain, that are closest in optimal transport distances to the source images. Such OT-based process is both an advantage and a limitation of our method. ",
|
| 1776 |
+
"bbox": [
|
| 1777 |
+
174,
|
| 1778 |
+
133,
|
| 1779 |
+
825,
|
| 1780 |
+
203
|
| 1781 |
+
],
|
| 1782 |
+
"page_idx": 14
|
| 1783 |
+
},
|
| 1784 |
+
{
|
| 1785 |
+
"type": "text",
|
| 1786 |
+
"text": "In ImageNet translation, when the source and target datasets are similar, DDIBs are generally able to identify correct animal postures. For example, we have shouting lions and tigers, because these animals have similar behaviors that are observed in the datasets and then internalized by DDIBs. However, in datasets that are less similar (e.g. birds and dogs), DDIBs sometimes fail to produce translation results that retain the postures precisely. We encountered significantly less such cases in AFHQ translation, since the dataset is more standardized and homogeneous. ",
|
| 1787 |
+
"bbox": [
|
| 1788 |
+
174,
|
| 1789 |
+
210,
|
| 1790 |
+
825,
|
| 1791 |
+
294
|
| 1792 |
+
],
|
| 1793 |
+
"page_idx": 14
|
| 1794 |
+
},
|
| 1795 |
+
{
|
| 1796 |
+
"type": "text",
|
| 1797 |
+
"text": "Fig. 6 illustrates the optimal transport mappings among images as well as some failure cases. Clearly, the translation processes flowing from left to right minimize the Euclidean transportation distances between images. Some of these translated samples may be classified “failure cases” in actual user studies. Such are considered both a feature and a limitation of DDIBs. ",
|
| 1798 |
+
"bbox": [
|
| 1799 |
+
173,
|
| 1800 |
+
301,
|
| 1801 |
+
825,
|
| 1802 |
+
356
|
| 1803 |
+
],
|
| 1804 |
+
"page_idx": 14
|
| 1805 |
+
},
|
| 1806 |
+
{
|
| 1807 |
+
"type": "image",
|
| 1808 |
+
"img_path": "images/e35ff7df5f979ad6ab5875bfd8f6d3c0b1d97b83ba06eec8df3b28086906e91d.jpg",
|
| 1809 |
+
"image_caption": [
|
| 1810 |
+
"Figure 6: Optimal transport translation processes in DDIBs. (Leftmost) Source images. (Rightmost) Translated images. "
|
| 1811 |
+
],
|
| 1812 |
+
"image_footnote": [],
|
| 1813 |
+
"bbox": [
|
| 1814 |
+
173,
|
| 1815 |
+
367,
|
| 1816 |
+
823,
|
| 1817 |
+
627
|
| 1818 |
+
],
|
| 1819 |
+
"page_idx": 14
|
| 1820 |
+
},
|
| 1821 |
+
{
|
| 1822 |
+
"type": "text",
|
| 1823 |
+
"text": "D PROOF OF PROPOSITION 3.2 ",
|
| 1824 |
+
"text_level": 1,
|
| 1825 |
+
"bbox": [
|
| 1826 |
+
173,
|
| 1827 |
+
101,
|
| 1828 |
+
444,
|
| 1829 |
+
118
|
| 1830 |
+
],
|
| 1831 |
+
"page_idx": 15
|
| 1832 |
+
},
|
| 1833 |
+
{
|
| 1834 |
+
"type": "text",
|
| 1835 |
+
"text": "Proof. The proof proceeds by substituting the values of $( \\mathbf { z } _ { t } , \\hat { \\mathbf { z } } _ { t } ) = ( 0 , g ( t ) \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) )$ into Eq. (6), ",
|
| 1836 |
+
"bbox": [
|
| 1837 |
+
169,
|
| 1838 |
+
132,
|
| 1839 |
+
823,
|
| 1840 |
+
150
|
| 1841 |
+
],
|
| 1842 |
+
"page_idx": 15
|
| 1843 |
+
},
|
| 1844 |
+
{
|
| 1845 |
+
"type": "equation",
|
| 1846 |
+
"img_path": "images/7a3d55f81dee5d6bb207a44ff8dab4e5c1c8a460b491f0381fe66b18e8e9e4ef.jpg",
|
| 1847 |
+
"text": "$$\n\\begin{array} { r } { \\mathrm { d } \\mathbf { x } = \\left[ \\mathbf { f } ( \\mathbf { x } , t ) + g ( t ) \\mathbf { z } - \\frac { 1 } { 2 } g ( t ) ( \\mathbf { z } + \\hat { \\mathbf { z } } ) \\right] \\mathrm { d } t } \\\\ { = \\left[ \\mathbf { f } ( \\mathbf { x } , t ) - \\frac { 1 } { 2 } g ( t ) ^ { 2 } \\nabla _ { \\mathbf { x } } \\log p _ { t } ( \\mathbf { x } ) \\right] \\mathrm { d } t } \\end{array}\n$$",
|
| 1848 |
+
"text_format": "latex",
|
| 1849 |
+
"bbox": [
|
| 1850 |
+
354,
|
| 1851 |
+
155,
|
| 1852 |
+
643,
|
| 1853 |
+
226
|
| 1854 |
+
],
|
| 1855 |
+
"page_idx": 15
|
| 1856 |
+
},
|
| 1857 |
+
{
|
| 1858 |
+
"type": "text",
|
| 1859 |
+
"text": "This is exactly Eq. (2). ",
|
| 1860 |
+
"bbox": [
|
| 1861 |
+
173,
|
| 1862 |
+
231,
|
| 1863 |
+
325,
|
| 1864 |
+
246
|
| 1865 |
+
],
|
| 1866 |
+
"page_idx": 15
|
| 1867 |
+
},
|
| 1868 |
+
{
|
| 1869 |
+
"type": "text",
|
| 1870 |
+
"text": "E ADDITIONAL EXPERIMENTAL DETAILS ",
|
| 1871 |
+
"text_level": 1,
|
| 1872 |
+
"bbox": [
|
| 1873 |
+
174,
|
| 1874 |
+
102,
|
| 1875 |
+
532,
|
| 1876 |
+
118
|
| 1877 |
+
],
|
| 1878 |
+
"page_idx": 16
|
| 1879 |
+
},
|
| 1880 |
+
{
|
| 1881 |
+
"type": "text",
|
| 1882 |
+
"text": "E.1 OPTIMAL TRANSPORT IN PAIRED DATASETS ",
|
| 1883 |
+
"text_level": 1,
|
| 1884 |
+
"bbox": [
|
| 1885 |
+
174,
|
| 1886 |
+
133,
|
| 1887 |
+
524,
|
| 1888 |
+
148
|
| 1889 |
+
],
|
| 1890 |
+
"page_idx": 16
|
| 1891 |
+
},
|
| 1892 |
+
{
|
| 1893 |
+
"type": "text",
|
| 1894 |
+
"text": "Color Conversion In Fig. 7, a simple examination of the original and segmentation images reveals significant differences in color configurations. In the Maps dataset, while the real, satellite images are composed of dark colors, the segmentation images are light-toned. The same observation applies to other datasets. The shark contrasts in colors intuitively present a large transportation cost, that probably hinders the progress of DDIBs, as we have demonstrated its relationship to OT in Section 3. ",
|
| 1895 |
+
"bbox": [
|
| 1896 |
+
174,
|
| 1897 |
+
159,
|
| 1898 |
+
825,
|
| 1899 |
+
229
|
| 1900 |
+
],
|
| 1901 |
+
"page_idx": 16
|
| 1902 |
+
},
|
| 1903 |
+
{
|
| 1904 |
+
"type": "text",
|
| 1905 |
+
"text": "To facilitate the workings of DDIBs, we follow a heuristic to transform the colors of the segmentation images. Specifically, on a small subset of the train dataset, we run an OT algorithm to compute a color correspondence that minimizes the color differences in terms of Sinkhorn distances between the real and segmentation images. The segmentation (target) datasets undergo this color conversion before they are fed into a diffusion model for training. During evaluation, when we compute MSEs, the images are converted to the original color space. ",
|
| 1906 |
+
"bbox": [
|
| 1907 |
+
174,
|
| 1908 |
+
236,
|
| 1909 |
+
825,
|
| 1910 |
+
320
|
| 1911 |
+
],
|
| 1912 |
+
"page_idx": 16
|
| 1913 |
+
},
|
| 1914 |
+
{
|
| 1915 |
+
"type": "text",
|
| 1916 |
+
"text": "Privacy Protection Color conversion requires considering both datasets jointly to compute a color mapping, and seems to betray the original purpose of DDIBs on protection of dataset privacy. We comment that the amount of leaked information is minimal: for example, to compute a color correspondence for the Maps dataset, we sampled only around 1000 pixels from the two datasets, to summarize the color composition information. DDIBs still conserve privacy at large. ",
|
| 1917 |
+
"bbox": [
|
| 1918 |
+
173,
|
| 1919 |
+
335,
|
| 1920 |
+
826,
|
| 1921 |
+
405
|
| 1922 |
+
],
|
| 1923 |
+
"page_idx": 16
|
| 1924 |
+
},
|
| 1925 |
+
{
|
| 1926 |
+
"type": "image",
|
| 1927 |
+
"img_path": "images/9c49e5f0ab830b6b1f359c6d17497bc2ac880460d824f5395fdeb30197f6f7c9.jpg",
|
| 1928 |
+
"image_caption": [
|
| 1929 |
+
"Figure 7: Color Conversion. In the paired translation tasks, we are given the real and segmentation images. Before training the diffusion models, we first transform the segmentation images to a color palette that is closer to the real images. While evaluating MSEs, we convert the images back to the original colors. "
|
| 1930 |
+
],
|
| 1931 |
+
"image_footnote": [],
|
| 1932 |
+
"bbox": [
|
| 1933 |
+
303,
|
| 1934 |
+
435,
|
| 1935 |
+
692,
|
| 1936 |
+
559
|
| 1937 |
+
],
|
| 1938 |
+
"page_idx": 16
|
| 1939 |
+
},
|
| 1940 |
+
{
|
| 1941 |
+
"type": "text",
|
| 1942 |
+
"text": "E.2 EXAMPLE-GUIDED COLOR TRANSFER ",
|
| 1943 |
+
"text_level": 1,
|
| 1944 |
+
"bbox": [
|
| 1945 |
+
176,
|
| 1946 |
+
103,
|
| 1947 |
+
482,
|
| 1948 |
+
118
|
| 1949 |
+
],
|
| 1950 |
+
"page_idx": 17
|
| 1951 |
+
},
|
| 1952 |
+
{
|
| 1953 |
+
"type": "text",
|
| 1954 |
+
"text": "We present additional qualitative comparison between DDIBs and common OT methods, in Fig. 8. ",
|
| 1955 |
+
"bbox": [
|
| 1956 |
+
176,
|
| 1957 |
+
128,
|
| 1958 |
+
818,
|
| 1959 |
+
145
|
| 1960 |
+
],
|
| 1961 |
+
"page_idx": 17
|
| 1962 |
+
},
|
| 1963 |
+
{
|
| 1964 |
+
"type": "image",
|
| 1965 |
+
"img_path": "images/ed3cd99dadfa47191b869e324f869f2c593d640de753c52387bb79a0c6017a93.jpg",
|
| 1966 |
+
"image_caption": [
|
| 1967 |
+
"Figure 8: Full color transfer results on example images. "
|
| 1968 |
+
],
|
| 1969 |
+
"image_footnote": [],
|
| 1970 |
+
"bbox": [
|
| 1971 |
+
176,
|
| 1972 |
+
155,
|
| 1973 |
+
818,
|
| 1974 |
+
638
|
| 1975 |
+
],
|
| 1976 |
+
"page_idx": 17
|
| 1977 |
+
}
|
| 1978 |
+
]
|
parse/dev/5HLoTvVGDe/5HLoTvVGDe_middle.json
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parse/dev/5HLoTvVGDe/5HLoTvVGDe_model.json
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parse/dev/6u6N8WWwYSM/6u6N8WWwYSM_middle.json
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|
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parse/dev/9h3KsOVXhLZ/9h3KsOVXhLZ_content_list.json
ADDED
|
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "SwinTrack: A Simple and Strong Baseline for Transformer Tracking ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
223,
|
| 8 |
+
122,
|
| 9 |
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"text": "Liting Lin1,2∗ Heng $\\mathbf { F a n ^ { 3 * } }$ Zhipeng Zhang4 Yong $\\mathbf { X } \\mathbf { u } ^ { 1 , 2 }$ Haibin Ling5 ",
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"text": "1School of Computer Science & Engineering, South China Univ. of Tech., Guangzhou, China 2Peng Cheng Laboratory, Shenzhen, China \n3Department of Computer Science and Engineering, University of North Texas, Denton, USA 4DiDi Chuxing, Beijing, China 5Department of Computer Science, Stony Brook University, Stony Brook, USA l.lt@mail.scut.edu.cn, heng.fan $@$ unt.edu, zhipeng.zhang.cv $@$ outlook.com yxu@scut.edu.cn, hling $@$ cs.stonybrook.edu ",
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"text": "Abstract ",
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"text": "Recently Transformer has been largely explored in tracking and shown state-of-theart (SOTA) performance. However, existing efforts mainly focus on fusing and enhancing features generated by convolutional neural networks (CNNs). The potential of Transformer in representation learning remains under-explored. In this paper, we aim to further unleash the power of Transformer by proposing a simple yet efficient fully-attentional tracker, dubbed SwinTrack, within classic Siamese framework. In particular, both representation learning and feature fusion in SwinTrack leverage the Transformer architecture, enabling better feature interactions for tracking than pure CNN or hybrid CNN-Transformer frameworks. Besides, to further enhance robustness, we present a novel motion token that embeds historical target trajectory to improve tracking by providing temporal context. Our motion token is lightweight with negligible computation but brings clear gains. In our thorough experiments, SwinTrack exceeds existing approaches on multiple benchmarks. Particularly, on the challenging LaSOT, SwinTrack sets a new record with 0.713 SUC score. It also achieves SOTA results on other benchmarks. We expect SwinTrack to serve as a solid baseline for Transformer tracking and facilitate future research. Our codes and results are released at https://github.com/LitingLin/SwinTrack. ",
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"type": "text",
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"text": "1 Introduction ",
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"text": "Visual tracking has seen considerable progress since deep learning. In particular, the recent Transformer [30] has significantly pushed the state-of-the-art in tracking owing to its ability in modeling long-range dependencies. However, existing methods usually leverage Transformer for fusing and enhancing features generated from convolutional neural networks (CNNs), e.g., ResNet [14]. The potential of exploiting Transformer for feature representation learning is largely under-explored. ",
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"text": "Recently, Vision Transformer (ViT) [7] has exhibited great potential in robust feature representation learning. Particularly, its extension Swin Transformer [23] has achieved state-of-the-art (SOTA) results on multiple tasks. Taking inspiration from this, we argue, besides the feature fusion, the representation learning in tracking can also benefit from Transformer via attention. Thus motivated, we propose to develop a fully attentional tracking framework based on Siamese architecture. Specifically, both the feature representation learning and the feature fusion of template and search region are realized by Transformer. More concretely, we borrow the architecture of the powerful ",
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"type": "image",
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"img_path": "images/bff5b4163414025c27ee09c6e9bfb93d55e373aa201c681052eec140b863d86c.jpg",
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"image_caption": [
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"Figure 1: Comparison on LaSOT [9]. Our tracker (SwinTrack-B-384) sets a new record with 0.713 SUC score and still runs efficiently at around $4 5 f p s$ . A lighter version (SwinTrack-T-224) achieves 0.672 SUC score and runs at around $9 6 f p s$ , which is on par with existing SOTAs in accuracy but much faster. "
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"text": "Swin Transformer [23] and adapt it to Siamese tracking. Note that, other Transformer architectures can be used. For feature fusion, we introduce a simple homogeneous concatenation-based fusion architecture, without a query-based decoder. ",
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"text": "Moreover, taking into consideration that tracking is a temporal task, we propose a novel motion token to improve robustness. Inspired by that the target usually moves smoothly in a short period, motion token is represented by the historical target trajectory within a local temporal window. We incorporate the (single) motion token in the decoder of feature fusion to leverage motion information during tracking. Despite being conceptually simple, our motion token can effectively boost tracking performance, with negligible computation. ",
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"text": "We name our framework SwinTrack. As a pure Transformer framework, SwinTrack enables better interactions inside the feature learning of template and search region and their fusion compared to pure CNN-based [1, 20] and hybrid CNN-Transformer [5, 32, 36] frameworks, leading to more robust performance (see Fig. 1). Fig. 2 demonstrates the architecture of SwinTrack. We conduct extensive experiments on five large-scale benchmarks to verify the effectiveness of SwinTrack, including LaSOT [9], $\\mathrm { L a S O T _ { \\mathrm { e x t } } }$ [8], TrackingNet [26], GOT-10k [15] and TNL2k [34]. On all benchmarks, SwinTrack achieves promising results and meanwhile runs fast at $4 5 f p s$ . In particular, on the challenging LaSOT, SwinTrack sets a new record of 71.3 SUC score, surpassing the strongest prior tracker [36] (to date) by 3.1 absolute percentage points and crossing the 0.7 SUC threshold for the first time (see Fig. 1 again). It also achieves 49.1 SUC, 84.0 SUC, 72.4 AO and 55.9 SUC scores on $\\mathrm { L a S O T _ { e x t } }$ , TrackingNet, GOT-10k and TNL2k respectively, which are better than or on par with state-of-the-arts (SoTAs). In addition, we provide a lighter version of SwinTrack that obtains comparable results to SoTAs but runs much faster at around 98 fps. ",
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"text": "In summary, our contributions are as follows: (i) We propose SwinTrack, a simple and strong baseline for fully attentional tracking; (ii) We present a simple yet effective motion token, enabling the integration of rich motion context during tracking, further boosting the robustness of SwinTrack, with negligible computation; (iii) Our proposed SwinTrack achieves state-of-the-art performance on multiple benchmarks. We believe SwinTrack further shows the potential of Transformer and expect it to serve as a baseline for future research. ",
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"type": "text",
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"text": "2 Related Work ",
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"type": "text",
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"text": "Siamese Tracking. The Siamese tracking methods formulate tracking as a matching problem and aim to offline learn a generic matching function for this task. The seminal method of [1] introduces a fully convolutional Siamese network for tracking and shows a good balance between accuracy and speed. To improve Siamese tracking in handling scale variation, the work of [20] incorporates the region proposal network (RPN) [27] into the Siamese network and proposes the anchor-based tracker, showing higher accuracy with faster speed. Later, numerous extensions have been presented to improve Siamese tracking, including deeper backbone [19], multi-stage architecture [10, 11], anchor-free Siamese trackers [41], deformable attention [37], to name a few. ",
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"img_path": "images/ceb1ea2031ef2c2e53af6c2d1dc3d1053361e969b449e58a2d842f8df5d040e9.jpg",
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"image_caption": [
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| 179 |
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"Figure 2: Architecture of SwinTrack, which contains three parts including Transformer-based feature representation extraction, Transformer-based feature fusion and prediction head. Our SwinTrack is a simple and neat tracking framework without complex designs such as multi-scale features or temporal template updating, yet demonstrating state-of-the-art performance. Best viewed in color. "
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"text": "",
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| 193 |
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"type": "text",
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"text": "Transformer in Vision. Transformer [30] originates from natural language processing (NLP) for machine translation and has been introduced to vision recently and shows great potential. The work of [3] first uses Transformer for object detection and achieved promising results. To explore the capability of Transformer in representation learning, the work of [7] applies Transformer to construct backbone network, and the resulting Vision Transformer (ViT) attains excellent performance compared to convolutional networks while requiring fewer training resources, which encourages many extensions upon ViT[29, 4, 38, 33, 23]. Among them, the Swin Transformer [23] has received extensive attention. It proposes a simple shifted window strategy to replace the fixed-patch method in ViT, which significantly improves efficiency and meanwhile demonstrates state-of-the-art results on multiple image tasks. Our work is inspired by Swin Transformer, but differently, we focus on the video task of visual tracking. ",
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"text": "Transformer in Tracking. Inspired by the success in other fields, researchers have leveraged Transformer for tracking. The method of [5] applies Transformer to enhance and fuse features in the Siamese tracking for improvement. The approach of [32] uses Transformer to exploit temporal features to improve tracking robustness. The work of [36] introduces a new transformer architecture dedicated to visual tracking, explores the Spatio-temporal Transformer by integrating the model updating operations into a Transformer module. ",
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"text": "Our SwinTrack is related to but significantly different from the above Transformer-based trackers. Specifically, the aforementioned methods mainly apply Transformer to fuse convolutional features and belong to the hybrid CNN-Transformer architecture. Unlike them, SwinTrack is a pure Transformerbased tracking architecture where both representation learning and feature fusion are realized with Transformer, enabling the exploration of better features for robust tracking. ",
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"text": "3 Tracking via Vision-Motion Transformer ",
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"text": "We present SwinTrack, a vision-motion integrated Transformer for object tracking, in Fig. 2. The proposed framework contains three main components, i.e., the Swin-Transformer backbone for feature extraction, the encoder-decoder network for mixing vision-motion cues, and the head network for localizing targets. In the following sections, we first shortly describe the Swin-Transformer backbone network, then elaborate on the proposed vision-motion encoder-decoder. Afterward, we give a discussion about our method and shortly describe the network head and training loss. ",
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"text": "3.1 Swin-Transformer for Feature Extraction ",
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"text_level": 1,
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"text": "The deep convolutional neural network has significantly improved the performance of trackers. Along with the advancement of trackers, the backbone network has evolved twice: AlexNet [17] and ResNet [14]. Swin-Transformer [23], in comparison to ResNet, can give a more compact feature representation and richer semantic information to assist succeeding networks in better localizing the target objects (demonstrate in the ablation study demonstrated in the ablation study), which is thus chosen for basic feature extraction in our model. ",
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"type": "text",
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"text": "Our tracker, following Siamese tracking framework [1], requires a pair of image patches as inputs, i.e., template image $\\mathbf { z } \\in \\overline { { \\mathbb { R } } } ^ { H _ { z } \\times W _ { z } \\times 3 }$ and search region image $\\mathbf { x } \\in \\mathbb { R } ^ { \\mathbf { \\hat { H } } _ { x } \\times W _ { x } \\times 3 }$ . As in the typical SwinTransformer procedure, template and search region images are divided to non-overlapped patches and sent to the network, which generates template tokens (dubbed T-tokens) φ(z) ∈ R Hzs Wzs ×C a nd search region tokens (dubbed S-tokens) $\\varphi ( \\mathbf { x } ) \\in \\mathbb { R } ^ { \\frac { H _ { x } } { s } \\frac { W _ { x } } { s } \\times C }$ . $s$ is the stride of the backbone network. Since there is no dimension projection in our model, $C$ is the hidden dimension of the whole model. ",
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"text": "3.2 Vision-Motion Representation Learning ",
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"text": "The essential step for matching-based visual tracking is injecting the template information into the search region. In our framework, we adopt an encoder to fuse the features from the template and the search region, meanwhile, a decoder is arranged to achieve vision-motion representation learning, as illustrated in Fig. 2. ",
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"text": "Encoder for fusing template and search tokens. The encoder contains a sequence of Transformer blocks where each consists of a multi-head self-attention (MSA) module and a feed-forward network (FFN). FFN contains a two-layers multi-layer perceptron (MLP), GELU activation layer is inserted after the first linear layer. Layer normalization (LN) is always conducted before every module (MSA and FFN). Residual connection is applied to MSA and FFN modules. ",
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"text": "Before feeding the features into the encoder, the template and search region tokens are concatenated along spatial dimensions to generate a mixing representation $\\mathbf { f } _ { m }$ . For each block, the MSA module computes self-attention over mixing union representation, which equals to separately conducting self-attention on T-tokens/S-tokens and meanwhile performing cross-attention between T-tokens and S-tokens, but more efficient. FFN refines the features generated by MSA. When the tokens get out of the encoder, a de-concatenation operation is arranged to decouple the template and search region tokens. The process of encoder can be expressed as: ",
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"type": "equation",
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"text": "$$\n\\begin{array} { r l } & { \\mathbf { f } _ { m } ^ { 1 } = \\mathrm { C o n c a t } ( \\boldsymbol { \\varphi } ( \\mathbf { z } ) , \\boldsymbol { \\varphi } ( \\mathbf { x } ) ) } \\\\ & { \\qquad \\cdots \\cdot } \\\\ & { \\mathbf { f } _ { m } ^ { l ^ { \\prime } } = \\mathbf { f } _ { m } ^ { l } + \\mathrm { M S A } ( \\mathrm { L N } ( \\mathbf { f } _ { m } ^ { l } ) ) } \\\\ & { \\mathbf { f } _ { m } ^ { l + 1 } = \\mathbf { f } _ { m } ^ { l ^ { \\prime } } + \\mathrm { F F N } ( \\mathrm { L N } ( \\mathbf { f } _ { m } ^ { l ^ { \\prime } } ) ) } \\\\ & { \\qquad \\cdots } \\\\ & { \\mathbf { f } _ { z } ^ { L } , \\mathbf { f } _ { x } ^ { L } = \\mathrm { D e C o n c a t } ( \\mathbf { f } ^ { L } ) , } \\end{array}\n$$",
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| 340 |
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"text_format": "latex",
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"type": "text",
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"text": "where $l$ denotes the $l$ -th layer and $L$ denotes the number of blocks. ",
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"text": "Decoder for fusing vision and motion information. Before describing the architecture of decoder, we first detail how to generate a motion token (dubbed M-token). Motion token is the embedding of the historical trajectory of the target object. The past object trajectory is represented as a set of target object box coordinates, $T = \\left\\{ \\Phi _ { 1 } , \\Phi _ { 2 } , . . . , \\Phi _ { t } \\right\\}$ , where $t$ represents the frame index, o is the bounding box of target object. o is defined by the top-left and bottom-right corners of the target object, denotes as $\\Phi _ { t } ^ { - } = \\bigl ( o _ { t } ^ { - x _ { 1 } } , o _ { t } ^ { y _ { 1 } } , o _ { t } ^ { x _ { 2 } } , o _ { t } ^ { y _ { 2 } } \\bigr )$ . For flexible modeling, a sampling process is required to ensure the following properties: 1) fixed length, 2) focusing on the latest trajectories and 3) reducing redundancy. In our method, we sample object trajectory as: ",
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"img_path": "images/a87f4b17ae97d04fa2ccec404354fa712c08104ebdca9eedde6266d69c1cc000.jpg",
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"text": "$$\n\\begin{array} { r } { \\mathcal { T } = \\big \\{ \\Phi _ { s ( 1 ) } , \\Phi _ { s ( 2 ) } , . . . , \\Phi _ { s ( n ) } \\big \\} , \\mathrm { w h e r e } s ( i ) = m a x ( t - i \\times \\Delta , 1 ) , } \\end{array}\n$$",
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"type": "text",
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"text": "$n$ is the number of sampled object trajectories, $\\Delta$ is the fixed sampling interval. For Siamese tracker, the search region is cropped from the input image. In detail, a cropping with resizing operation can be used to describe the process. Giving the point in the input image as $\\left( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } \\right)$ , the corresponding point in the search region as $\\left( \\mathbf { x } _ { o } , \\mathbf { y } _ { o } \\right)$ , we can formulate the cropping process employed in pre-processing of the Siamese Tracker as $\\mathbf { x } _ { o } = ( \\mathbf { x } _ { i } - i _ { x } ) s _ { x } + o _ { x }$ and ${ \\bf y } _ { o } = ( { \\bf y } _ { i } - i _ { y } ) s _ { y } + o _ { y }$ , where $( i _ { x } , i _ { y } )$ is the center of the cropping window in the input image, $( s _ { x } , s _ { y } )$ is the scaling factor, $( o _ { x } , o _ { y } )$ is the center of cropped and scaled window in the search region. We apply the same transformation on the sampled object trajectory to make the object trajectory invariant to the cropping, denoting $\\bar { \\mathcal { T } } = \\{ \\bar { \\mathbb { D } } _ { s ( 1 ) } , \\bar { \\mathbb { D } } _ { s ( 2 ) } , . . . , \\bar { \\mathbb { D } } _ { s ( n ) } \\}$ as the result. ",
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"text": "Then, to embed the transformed object trajectory into the network, we adopt four embedding matrices to embed the elements in box coordinates separately. We denotes the embedding matrix as W ∈ R(g+1)×d, $\\^ \\mathrm { g }$ controls the embedding granularity of the object trajectory, $d$ is the size of each embedding vector. The last entry of the embedding matrix is used as the padding vector, indicating an invalid state, like object absence or out of the search region. Thus, we normalize the sampled target object box coordinates in the range $[ 1 , \\mathbb { g } ]$ , and quantize to integers to get the index of embedding vector: ",
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"img_path": "images/b7a6628efd2eae2119392aade7e89ec4e64d2435bd65751772febfd9330744d7.jpg",
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"text": "$$\n\\begin{array} { r l } & { \\hat { T } = \\{ \\hat { \\mathbb { O } } _ { s ( 1 ) } , \\hat { \\mathbb { O } } _ { s ( 2 ) } , . . . , \\hat { \\mathbb { O } } _ { s ( n ) } \\} , } \\\\ & { \\mathrm { \\it ~ \\gamma ~ \\mathrm { h e r e } ~ } \\hat { \\mathbb { O } } _ { s ( i ) } = [ \\mathrm { n } ( \\bar { \\mathbb { O } } _ { s ( i ) } ^ { x _ { 1 } } , w ) , \\mathrm { n } ( \\bar { \\mathbb { O } } _ { s ( i ) } ^ { y _ { 1 } } , h ) , \\mathrm { n } ( \\bar { \\mathbb { O } } _ { s ( i ) } ^ { x _ { 2 } } , w ) , \\mathrm { n } ( \\bar { \\mathbb { O } } _ { s ( i ) } ^ { y _ { 2 } } , h ) ] , } \\\\ & { \\mathrm { \\it ~ \\mathrm { \\it ~ \\hat { \\Omega } ~ } } \\mathrm { \\ n } ( o , l ) = \\left\\{ \\begin{array} { l l } { \\mathrm { \\Delta } [ \\frac { o } { l } \\times \\mathbb { g } ] } & { \\mathrm { i f ~ } \\mathrm { v a l i d } , } \\\\ { \\mathbb { g } + 1 } & { \\mathrm { e l s e } , } \\end{array} \\right. } \\end{array}\n$$",
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| 421 |
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"type": "text",
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"text": "$( w , h )$ is the size of search region feature map. ",
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"text": "Finally, the motion token $\\mathbf { E } _ { m o t i o n } ~ \\in ~ \\mathbb { R } ^ { 1 \\times d }$ is given by a concatenation of all box coordinate embedding of the sampled object trajectory. FLOPs is negligible because the construction of motion token is just a composition of embedding lookups and token concatenation. ",
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"text": "The decoder consists of a multi-head cross-attention(MCA) module and a feed-forward network(FFN). The decoder takes the outputsvision-motion representation $\\mathbf { f } _ { v m } \\in \\mathbb { R } ^ { \\frac { H _ { x } } { s } \\times \\frac { W _ { x } } { s } \\times C }$ d the motion token as input, generating tof by computing cross-attention over $\\mathbf { f } _ { x } ^ { L }$ finaland $\\mathrm { C o n c a t } ( { \\bf E } _ { m o t i o n } , { \\bf f } _ { z } ^ { L } , { \\bf f } _ { x } ^ { L } )$ . The decoder is akin to a layer in the encoder, except that the correlation between the template tokens and the search tokens is dropped since we do not need to update the features from the template image in the last layer. The process of the decoder is formulated as: ",
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"text": "$$\n\\begin{array} { r l } & { \\mathbf { f } _ { m } ^ { D } = \\mathrm { C o n c a t } ( \\mathbf { E } _ { m o t i o n } , \\mathbf { f } _ { z } ^ { L } , \\mathbf { f } _ { x } ^ { L } ) } \\\\ & { \\mathbf { f } _ { v m } ^ { \\prime } = \\mathbf { f } _ { x } ^ { L } + \\mathrm { M C A } ( \\mathrm { L N } ( \\mathbf { f } _ { x } ^ { L } ) , \\mathrm { L N } ( \\mathbf { f } _ { m } ^ { D } ) ) } \\\\ & { \\mathbf { f } _ { v m } = \\mathbf { f } _ { v m } ^ { \\prime } + \\mathrm { F F N } ( \\mathrm { L N } ( \\mathbf { f } _ { v m } ^ { \\prime } ) ) . } \\end{array}\n$$",
|
| 467 |
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"text": "$\\mathbf { f } _ { v m }$ will feed to the head network to generate a classification response map and a bounding box regression map. ",
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"text": "Positional encoding. Transformer requires a positional encoding to identify the position of the current processing token[30] because the self-attention module is permutation-invariance. We adopt the untied positional encoding [16] as our positional encoding method. The untied positional encoding enhances the expressiveness of the model through untie the positional embeddings from token embeddings with an isolated positional embedding matrix. It also considers the case of special tokens, like the motion token in this paper. We generalize the untied positional encoding to multi-dimensions multi-sources data to comply with concatenated-based fusion in our tracker. See the appendix for the details. ",
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"text": "3.3 Discussion ",
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"text": "Why concatenated attention? To simplify the description, we call the method described above concatenation-based fusion. To fuse and process features from multiple sources, it is intuitive to perform self-attention on the feature from each source separately and then compute cross-attention across features from different sources. We call this method cross-attention-based fusion. Transformer makes fewer assumptions about the spatial structure of data, which provides great modeling flexibility. ",
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"text": "In comparison to cross-attention-based fusion, concatenation-based fusion can save computation cost through operation sharing and reduce model parameters through weight sharing. From the perspective of metric learning, weight sharing is an essential design to ensure the metric between two branches of data is symmetric. Through concatenation-based fusion, we implement this property not only in the feature extraction stage but also in the feature fusion stage. In general, concatenation-based fusion improves both efficiency and performance. ",
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"text": "Why not window-based self/cross-attention? Since we select stage 3 of the Swin-Transformer as the output, the number of tokens involved is significantly reduced, window-based attention cannot save too many FLOPs. Furthermore, considering the extra latency introduced by the window partition and window reverse operations, window-based attention may even be the slower one. ",
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"type": "text",
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"text": "Why not a query-based decoder? Derivated from vanilla Transformer decoder, many transformerbased models in vision tasks leverage a learnable query to extract the desired objective features from the encoder, like object queries in [3], target query in [36]. However, in our experiment, a query-based decoder suffers from slow convergence and inferior performance. Most Siamese trackers [20, 35, 13] formulate tracking as a foreground-background classification problem, which can better exploit the background information. The vanilla Transformer decoder is a generative model, the generative approaches are considered not suitable for the classification tasks. In another aspect, learning a general target query for any kind of object might cause a bottleneck. In terms of vanilla Transformer encoder-decoder architecture, SwinTrack is an \"encoder\" only model. Furthermore, quite a little domain knowledge can be easily applied on a classic Siamese tracker to improve the performance, like introducing the smooth movement assumption by using Hanning penalty window on the response map. ",
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"text": "Are other forms of motion token feasible? Other forms to construct motion token are possible, such as constructing motion token by summing up the past box coordinate embeddings or representing past object trajectories by one token per box. In our early experiments, we find that the proposed motion token is more effective with the best performance. Summing up the past box coordinate embeddings may result in over-parameterization on the coordinate embeddings. While adding temporal motion representation along with visual features to the single-layer decoder in a multi-token form is ineffective, precise temporal modeling may be required in this form. ",
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"text": "3.4 Head and Loss ",
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"text": "Head. The head network is split into two branches: classification and bounding box regression. Each of them is a three-layer perceptron. And both of them receives the feature map from the decoder as input to predict the classification response map $r _ { c l s } \\in \\mathbb { R } ^ { ( H _ { x } \\times W _ { x } ) \\times 1 }$ and bounding box regression map $\\bar { r _ { r e g } } \\in \\mathbb { R } ^ { ( H _ { x } \\times W _ { x } ) \\times 4 }$ , respectively. ",
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"text": "Classification loss. In classification branch, we employ the $I o U$ -aware classification score as the training target and the varifocal loss [39] as the training loss function. IoU-aware design has been very popular recently, but most works consider IoU prediction as an auxiliary branch to assist classification or bounding box regression [41, 2, 35]. To remove the gap between different prediction branches, [39] and [21] replace the hard classification target from the ground-truth value, (i.e., 1 for positive samples, 0 for negative samples), to the IoU between the predicted bounding box and the ground-truth one, which is named the $I o U .$ -aware classification score (IACS). IACS can help the model select a more accurate bounding box prediction candidate from the pool by trying to predict the quality of the bounding box prediction in another branch at the same position. Along with the IACS, the varifocal loss was proposed in [39] to help the IACS approach outperform other IoU-aware designs. ",
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"type": "text",
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"text": "The classification loss can be formulated as: ",
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"text": "$$\n\\mathbb { L } _ { c l s } = \\mathbb { L } _ { \\mathrm { V F L } } ( p , \\mathrm { I o U } ( b , \\hat { b } ) ) ,\n$$",
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"text": "where $p$ denotes the predicted IACS, $b$ denotes the predicted bounding box, and $\\hat { b }$ denotes the ground-truth bounding box. ",
|
| 626 |
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"bbox": [
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"type": "text",
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| 636 |
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"text": "Regression loss. For bounding box regression, we employ the generalized IoU loss[28]. The regression loss function can be formulated as: ",
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"type": "equation",
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"img_path": "images/430b51ce3631780d398a75dc7794e5aba609c20ad0b960bb5a5dca0210f1fdbb.jpg",
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"text": "$$\n\\mathbb { L } _ { r e g } = \\sum _ { j } \\mathbb { 1 } _ { \\left\\{ \\mathrm { I o U } ( b _ { j } , \\hat { b } ) > 0 \\right\\} } [ p \\mathbb { L } _ { \\mathrm { G I o U } } ( b _ { j } , \\hat { b } ) ] .\n$$",
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"text_format": "latex",
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"type": "text",
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"text": "The GIoU loss is weighted by $p$ to emphasize the high classification score samples. The training signals from the negative samples are ignored. ",
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"type": "text",
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"text": "4 Experiments ",
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"type": "text",
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"text": "4.1 Implementation ",
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"type": "text",
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"text": "Model. We design two variants of SwinTrack with different configurations as follows: ",
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"type": "text",
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"text": "• SwinTrack-T-224. Backbone: Swin Transformer-Tiny [23], pretrained with ImageNet-1k; Template size: $[ 1 1 2 \\times 1 1 2 ]$ ; Search region size: $[ 2 2 4 \\times 2 2 4 ]$ ; $C = 3 8 4$ ; $N = 4$ ; \n• SwinTrack-B-384. Backbone: Swin Transformer-Base [23], pretrained with ImageNet-22k; Template size: $[ 1 9 2 \\times 1 9 2 ]$ ; Search region size: $[ 3 8 4 \\times 3 8 4 ]$ ; $C = 5 1 2$ ; $N = 8$ ; ",
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"type": "text",
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"text": "where $C$ and $N$ are the channel number of the hidden layers in the first stage of Swin Transformer and the number of encoder blocks in feature fusion, respectively. In all variants, we use the output after the third stage of Swin Transformer for feature extraction. Thus, the backbone stride $s$ is 16. ",
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"text": "For motion token, the number of sampled object trajectory $n$ is set to 16, the fixed sampling interval $\\Delta$ is set to 15. If the frame rate of the video sequence is available, the sampling interval is adjusted according to the frame rate. Suppose the frame rate is $\\mathbb { f }$ , the new sampling interval is getting by $\\mathrm { { \\frac { \\Delta } { 3 0 } f } }$ , 30 fps is the standard frame rate we assumed. g, which controls the embedding granularity, is set to the same size as the search region feature map, like 14 for SwinTrack-T-224, and 24 for SwinTrack-B-384. For the model for GOT-10k sequences, $n$ is set to 8, $\\Delta$ is set to 8, and no frame rate adjustment is applied. ",
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"text": "Training. We train SwinTrack using the training splits of LaSOT [9], TrackingNet [26], GOT-10k [15] (1,000 videos are removed following [36] for fair comparison) and COCO 2017 [22]. In addition, we report the results of SwinTrack-T-224 and SwinTrack-B-384 with GOT-10k training split only to follow the protocol described in [15]. ",
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"text": "The model is optimized with AdamW [24], with a learning rate of 5e-4, and a weight decay of 1e-4. The learning rate of the backbone is set to 5e-5. We train the network on 8 NVIDIA V100 GPUs for 300 epochs with 131,072 samples per epoch. The learning rate is dropped by a factor of 10 after 210 epochs. A 3-epoch linear warmup is applied to stabilize the training process. DropPath [18] is applied on the backbone and the encoder with a rate of 0.1. For the models trained for the GOT-10k evaluation protocol, to prevent over-fitting, the training epoch is set to 150, and the learning rate is dropped after 120 epochs. ",
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"text": "For the motion token, the object trajectory for the Siamese training pair is generated with the method described above. The frames that object annotated as absent or out of the video sequence are marked as invalid, the corresponding box coordinates set to $- \\infty$ . Since the coarse granularity of the coordinate embedding in our setting is already can be seen as an augmentation of historical object trajectory, no additional data augmentation is applied. ",
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"text": "Inference. We follow the common procedures for Siamese network-based tracking [1]. The template image is cropped from the first frame of the video sequence. The target object is in the center of the image with a background area factor of 2. The search region is cropped from the current tracking frame, and the image center is the target center position predicted in the previous frame. The background area factor for the search region is 4. ",
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"text": "Our SwinTrack takes the template image and search region as inputs and output classification map $r _ { c l s }$ and regression map $r _ { r e g }$ . To utilize positional prior in tracking, we apply hanning window penalty on $r _ { c l s }$ , and the final classification map $r _ { c l s } ^ { \\prime }$ is obtained via $\\boldsymbol { r } _ { c l s } ^ { \\prime } = ( 1 - \\gamma ) \\times \\boldsymbol { r } _ { c l s } + \\gamma \\times \\boldsymbol { h }$ , where $\\gamma$ is the weight parameter and $h$ is the Hanning window with the same size as $r _ { c l s }$ . The target position is determined by the largest value in $r _ { c l s } ^ { \\prime }$ and the scale is estimated based on the corresponding regression results in $r _ { r e g }$ . ",
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"type": "text",
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| 794 |
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"text": "For the motion token, the predicted confidence score and bounding box are collected on the fly. A confidence threshold $\\theta _ { c o n f }$ is applied, if the confidence score given by the classification branch of the head is lower than the threshold, the target object in the current frame is marked as lost by setting the collected bounding box to $- \\infty$ . $\\theta _ { c o n f }$ is set to 0.4 for LaSOT, the rests are set to 0.3. ",
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{
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"type": "table",
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"img_path": "images/bbec3b18183201f9ed3782f1477a1e4082f6dcb762564bffeab908496dfdcf90.jpg",
|
| 806 |
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"table_caption": [
|
| 807 |
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"Table 1: Experiments and comparisons on five benchmarks: LaSOT, $\\mathrm { L a S O T _ { e x t } }$ , TrackingNet, GOT10k and TNL2k. "
|
| 808 |
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],
|
| 809 |
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"table_footnote": [],
|
| 810 |
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"table_body": "<table><tr><td rowspan=\"2\">Tracker</td><td colspan=\"2\">LaSOT[9]</td><td colspan=\"2\">LaSOText [8]</td><td colspan=\"2\">TrackingNet [26]</td><td colspan=\"3\">GOT-10k [15]</td><td colspan=\"2\">TNL2k [34]</td></tr><tr><td>SUC</td><td>P</td><td>SUC</td><td>P</td><td>SUC</td><td>P</td><td>AO</td><td>SR0.5</td><td>SR0.75</td><td>SUC</td><td>P</td></tr><tr><td>C-RPN [10]</td><td>45.5</td><td>44.3</td><td>27.5</td><td>32.0</td><td>66.9</td><td>61.9</td><td>1</td><td>1</td><td>1</td><td>-</td><td>-</td></tr><tr><td>SiamPRN++ [19]</td><td>49.6</td><td>49.1</td><td>34.0</td><td>39.6</td><td>73.3</td><td>69.4</td><td>51.7</td><td>61.6</td><td>32.5</td><td>41.3</td><td>41.2</td></tr><tr><td>Ocean [41]</td><td>56.0</td><td>56.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>61.1</td><td>72.1</td><td>47.3</td><td>38.4</td><td>37.7</td></tr><tr><td>DiMP [2]</td><td>56.9</td><td>56.7</td><td>39.2</td><td>45.1</td><td>74.0</td><td>68.7</td><td>61.1</td><td>71.7</td><td>49.2</td><td>44.7</td><td>43.4</td></tr><tr><td>LTMU [6]</td><td>57.2</td><td>57.2</td><td>41.4</td><td>47.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1</td><td>48.5</td><td>47.3</td></tr><tr><td>SiamR-CNN[31]</td><td>64.8</td><td>1</td><td>1</td><td>1</td><td>81.2</td><td>80.0</td><td>64.9</td><td>72.8</td><td>59.7</td><td>52.3</td><td>52.8</td></tr><tr><td>STMTrack [12]</td><td>60.6</td><td>63.3</td><td>-</td><td>-</td><td>80.3</td><td>76.7</td><td>64.2</td><td>73.7</td><td>57.5</td><td>-</td><td>-</td></tr><tr><td>AutoMatch [40]</td><td>58.3</td><td>59.9</td><td>37.6</td><td>43.0</td><td>76.0</td><td>72.6</td><td>65.2</td><td>76.6</td><td>54.3</td><td>-</td><td>-</td></tr><tr><td>TrDiMP [32]</td><td>63.9</td><td>61.4</td><td>1</td><td>1</td><td>78.4</td><td>73.1</td><td>67.1</td><td>77.7</td><td>58.3</td><td>-</td><td>-</td></tr><tr><td>TransT[5]</td><td>64.9</td><td>69.0</td><td>-</td><td>1</td><td>81.4</td><td>80.3</td><td>67.1</td><td>76.8</td><td>60.9</td><td>51.0</td><td>-</td></tr><tr><td>STARK [36]</td><td>67.1</td><td></td><td></td><td>=</td><td>82.0</td><td>-</td><td>68.8</td><td>78.1</td><td>64.1</td><td>-</td><td>-</td></tr><tr><td>KeepTrack [25]</td><td>67.1</td><td>70.2</td><td>48.2</td><td>-</td><td>-</td><td>-</td><td>1</td><td>-</td><td>1</td><td>-</td><td>-</td></tr><tr><td>SwinTrack-T-224</td><td>67.2</td><td>70.8</td><td>47.6</td><td>53.9</td><td>81.1</td><td>78.4</td><td>71.3</td><td>81.9</td><td>64.5</td><td>53.0</td><td>53.2</td></tr><tr><td>SwinTrack-B-384</td><td>71.3</td><td>76.5</td><td>49.1</td><td>55.6</td><td>84.0</td><td>82.8</td><td>72.4</td><td>80.5</td><td>67.8</td><td>55.9</td><td>57.1</td></tr></table>",
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"text": "",
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"type": "text",
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"text": "4.2 Comparisons to State-of-the-arts ",
|
| 833 |
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"text": "We conduct experiments and compare SwinTrack with SoTA trackers on five benchmarks. ",
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"type": "text",
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"text": "LaSOT. LaSOT [9] consists of 280 videos for test. Tab. 1 shows the results and comparisons with SoTAs. From Tab. 1, we can observe that SwinTrack-T-224 with light architecture reaches SoTA performance with 0.672 SUC and 0.708 PRE scores, which is competitive compared with other Transformer-based trackers, including STARK-ST101 (0.671 SUC score) and TransT (0.649 SUC), and other trackers using complicated designs such as KeepTrack (0.671 SUC) and SiamR-CNN (0.648 SUC score). With a larger backbone and input size, our strongest variant SwinTrack-B-384 sets a new record with 0.713 SUC score, surpassing START-ST101 and KeepTrack by 4.2 absolute percentage points. ",
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"type": "text",
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"text": "$\\mathbf { L a S O T } _ { \\mathrm { e x t } }$ . The recent $\\mathrm { L a S O T _ { \\mathrm { e x t } } }$ [8] is an extension of LaSOT by adding 150 extra videos. These new sequences are challenging as many similar distractors cause difficulties for tracking. The results of our tracker related to this dataset are an average of three times. KeepTrack uses a complex association technique to handle distractors and achieves a promising 0.482 SUC score as in Tab. 1. Compared with complicated KeepTrack, SwinTrack-T-224 is simple and neat, yet shows comparable performance with 0.476 SUC score. In addition, due to complicated design, KeepTrack runs at less than $2 0 f p s$ , while SwinTrack-T-224 runs in 98 fps, $5 \\times$ faster than KeepTrack. When using a larger model, SwinTrack-B-384 shows the best performance with 0.491 SUC score. ",
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},
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| 876 |
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"type": "text",
|
| 877 |
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"text": "TrackingNet. We evaluate different trackers on the test set of TrackingNet [26]. From Tab. 1, we observe that our SwinTrack-T-224 achieves a comparable result with 0.811 SUC score. Using a larger model and input size, SwinTrack-B-384 obtains the best performance with 0.840 SUC score, better than STARK-ST101 with 0.820 SUC score and TransT with 0.814 SUC score. ",
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{
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"type": "text",
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| 888 |
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"text": "GOT-10k. GOT-10k [15] offers 180 videos for test and it requires trackers to be trained using GOT-10k train split only. From Tab. 1, we see that SwinTrack-B-384 achieves the best mAO of 0.724, and SwinTrack-T-224 obtains a mAO of 0.713. Both models outperform other Transformer-based counterparts significantly, including START-ST101 (0.688 mAO), TransT (0.671 mAO) and TrDiMP (0.671 mAO). ",
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"page_idx": 7
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"type": "text",
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"text": "TNL2k. TNL2k [34] is a newly released tracking dataset with 700 videos for test. As reported in Tab. 1, both models surpass the others. SwinTrack-B-384 set a new state-of-the-art with 0.559 SUC score. ",
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"table_caption": [
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"Table 2: Comparison on running speed and # parameters with other Transformer-based trackers. "
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"table_footnote": [],
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"table_body": "<table><tr><td>Tracker</td><td>Speed (fps)</td><td>MACs² (G)</td><td>Params (M)</td></tr><tr><td>TrDiMP [32]</td><td>26</td><td>=</td><td>1</td></tr><tr><td>TransT[5]</td><td>50</td><td>=</td><td>23</td></tr><tr><td>STARK-ST50 [36]</td><td>42</td><td>10.9</td><td>24</td></tr><tr><td>STARK-ST101 [36]</td><td>32</td><td>18.5</td><td>42</td></tr><tr><td>SwinTrack-T-224</td><td>98</td><td>6.4</td><td>23</td></tr><tr><td>SwinTrack-B-384</td><td>45</td><td>69.7</td><td>91</td></tr></table>",
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"Table 3: Ablation experiments of SwinTrack on four benchmarks. The experiments are conducted on SwinTrack-T-224 without the motion token. ❶: baseline method, i.e., SwinTrack-T-224 without motion token; $\\otimes$ : replacing Transformer backbone in the baseline method with ResNet-50; ❸: replacing our feature fusion with cross attention-based fusion in the baseline method; $\\bullet$ : replacing the decoder in baseline with a target query-based; $\\pmb { \\ 6 }$ : replacing united positional encoding with absolute sine position encoding in the baseline method; $\\pmb { \\circledcirc }$ : replacing the IoU-aware classification loss with the plain binary cross entropy loss; $\\pmb { \\ 6 }$ : removing the Hanning penalty window in the baseline method inference. "
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"table_footnote": [],
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"table_body": "<table><tr><td></td><td>LaSOT SUC (%)</td><td>LaSOText SUC (%)</td><td>TrackingNet SUC (%)</td><td>GOT-10k3 mA0 (%)</td><td>Speed fps</td><td>Params M</td></tr><tr><td>1</td><td>66.7</td><td>46.9</td><td>80.8</td><td>70.9</td><td>98</td><td>22.7</td></tr><tr><td>②</td><td>64.2</td><td>41.8</td><td>79.5</td><td>68.2</td><td>121</td><td>20.0</td></tr><tr><td>③</td><td>66.6</td><td>45.4</td><td>80.2</td><td>69.3</td><td>72</td><td>34.6</td></tr><tr><td>4</td><td>66.6</td><td>43.2</td><td>79.6</td><td>69.0</td><td>91</td><td>25.3</td></tr><tr><td>5</td><td>65.7</td><td>45.0</td><td>80.0</td><td>70.0</td><td>103</td><td>21.6</td></tr><tr><td>6</td><td>66.2</td><td>46.7</td><td>79.4</td><td>68.2</td><td>98</td><td>22.7</td></tr><tr><td>0</td><td>65.7</td><td>46.0</td><td>80.0</td><td>69.6</td><td>98</td><td>22.7</td></tr></table>",
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"type": "text",
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"text": "Efficiency comparison. We report the comparisons of SwinTrack with other Transformer-based trackers in terms of efficiency and complexity. As displayed in Tab. 2, SwinTrack-T-224 with a small model runs the fastest with a speed of 98 fps. Especially, compared with STARK-ST101 and STARK-ST50 with $3 2 f p s$ and $4 2 f p s$ , SwinTrack-T-224 is $3 \\times$ and $2 \\times$ faster. Despite using a larger model, our SwinTrack-B-384 is still faster than STARK-ST101 and STARK-ST50. ",
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"type": "text",
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"text": "4.3 Ablation Experiment ",
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"type": "text",
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"text": "Comparison with ResNet backbone. We compare the Swin-Transformer backbone with popular ResNet-50 [14]. As shown in Tab. 3 ( $\\pmb { \\theta }$ vs. $\\pmb { \\varrho }$ ). The Swin Transformer backbone significantly boosts the performance by $2 . 5 \\%$ SUC score in LaSOT, $5 . 1 \\%$ SUC score in $\\mathrm { L a S O T _ { e x t } }$ . The result shows that the strong appearance modeling capability provided by the Swin Transformer plays a crucial role. ",
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"type": "text",
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"text": "Feature fusion. As displayed in Tab. 3 $\\pmb { \\theta }$ vs. $\\pmb { \\Theta }$ ), compared with the concatenation-based fusion, the cross attention-based fusion runs at a slower speed, occupies much more memory, and also has an inferior performance on all datasets. Slower speed can be due to the latency brought by the extra operations. The parameter-sharing strategy not only just reduces the number of parameters but also benefits metric learning. ",
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"text": "Comparison with the query-based decoder. Queries is commonly adopted in the decoder of Transformer network in vision tasks, e.g. object query [3] and target query [36]. Nevertheless, our empirical results in Tab. 3 $\\pmb { \\ 0 }$ vs. $\\pmb { \\varrho }$ ) show that a target query-based decoder degrades the tracking performance on all benchmarks, even with $2 \\times$ training pairs. As discussed, one possible reason is the generative model is not suitable for classification. Besides, learning a general target query for any kind of object may also be difficult. ",
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"type": "text",
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"text": "Position encoding. We compare the united positional encoding used in SwinTrack and the original absolute position encoding in Transformer [30]. Notice, We make a little modification to the original absolute position encoding: Except for the 2D embedding, the index of token source (e.g. 1 for the tokens from the template patch, 2 for the tokens from the search region patch) is also embedded. As shown in Tab. 3 (❶ vs. $\\bullet$ ), our method with united positional encoding obtains improvements with 0.8-1.9 absolute percentage points on the benchmarks with negligible loss in speed (98 vs. 103). ",
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"img_path": "images/a44e35c837372cddc6d07e6849d74323e28bf2af4ffd3ab4776283ad76cf9d69.jpg",
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| 1010 |
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"table_caption": [
|
| 1011 |
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"Table 4: Ablation experiments on our proposed motion token on the tracking performance on four benchmarks. The experiments are conducted on SwinTrack-T-224. ❶: SwinTrack-T-224; ❷: SwinTrack-B-384; $\\otimes$ : SwinTrack-T-224 without motion token; ❹: SwinTrack-B-384 without motion token; $\\pmb { \\ 6 }$ : replacing the motion token in SwinTrack-T-224 with a learnable embedding token. "
|
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"table_footnote": [],
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| 1014 |
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"table_body": "<table><tr><td>SUC (%)</td><td>LaSOT</td><td>LaSOText SUC (%)</td><td>TrackingNet SUC (%)</td><td>GOT-10k mA0 (%)</td><td>Speed fps</td></tr><tr><td>1</td><td>67.2</td><td>47.6</td><td>81.1</td><td>71.3</td><td>96</td></tr><tr><td>②</td><td>71.3</td><td>49.1</td><td>84.0</td><td>72.4</td><td>45</td></tr><tr><td>③</td><td>66.7</td><td>47.0</td><td>80.8</td><td>70.0</td><td>98</td></tr><tr><td>4</td><td>70.2</td><td>48.5</td><td>84.0</td><td>70.7</td><td>45</td></tr><tr><td>6</td><td>66.3</td><td>45.2</td><td>81.2</td><td>70.0</td><td>96</td></tr></table>",
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"text": "",
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"type": "text",
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"text": "Loss function. From Tab. 3 $( \\bullet \\nu . \\bullet )$ , we observe that the model trained with varifocal loss significantly outperforms the one with binary cross entropy (BCE) loss without loss of efficiency. This result indicates that the varifocal loss can assist the classification branch of the head to generate an IoU-aware response map, and thus help the tracker to improve the tracking performance. ",
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| 1037 |
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"type": "text",
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"text": "Post processing. One may wonder with highly discriminative Transformer architecture and IoUaware classification loss does the hanning penalty window is still functional, which introduces a strong smooth movement assumption. In the experiments, we remove the hanning penalty window in post-processing, as shown in Tab. 3 $( \\pmb { \\mathbb { 0 } } \\nu s . \\pmb { \\mathbb { \\otimes } } )$ , the performance is dropped by 1.0 SUC for LaSOT, $1 . 3 \\mathrm { \\ A O }$ for GOT-10k in absolute percentage, and less than $1 \\%$ in the SUC metric of other datasets. This suggests that the strong smooth movement assumption is still applicable for our tracker. But compared with the former Transformer-based tracker [5], the performance gap between with and without penalty window post-processing is narrowing. ",
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"type": "text",
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"text": "Effectiveness of motion token. We study the effectiveness of the motion token by conducting comparison experiments. As shown in Tab. 4 $\\pmb { \\mathbb { \\otimes } }$ vs. $\\pmb { \\otimes }$ and $\\pmb { \\varrho }$ vs. $\\pmb { \\varrho }$ ), the models with motion token outperforms the models without motion token on all datasets, especially on $\\mathrm { L a S O T _ { e x t } }$ and GOT-10k. The results indicate that the motion token can assist the tracker to handle hard similar distractors in $\\mathrm { L a S O T _ { e x t } }$ and stabilize the short-term tracking like the sequences in GOT-10k test set. We also study whether the effectiveness of the motion token is simply from the extra embedding vector. We set up an experiment as in Tab. 4 $( \\pmb { \\Theta } )$ , which replaces the motion token with a learnable embedding token. The result shows that the extra embedding vector has negative impacts indicating the effectiveness of the embedding of object trajectory. ",
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"type": "text",
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"text": "5 Conclusion ",
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"text_level": 1,
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"type": "text",
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"text": "In this work, we present SwinTrack, a simple and strong baseline for Transformer tracking. In SwinTrack, both representation learning and feature fusion are implemented with the attention mechanism. Extensive experiments demonstrate the effectiveness of such architecture. Besides, we propose the motion token to enhance the robustness of the tracker by providing the historical object trajectory, showing the flexibility of the Transformer model in architectural design. With the power of sequence-to-sequence model architecture, a context-rich tracker is possible, and more contextual cues can be incorporated. Finally, We hope this work can inspire and facilitate future research. ",
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"type": "text",
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"text": "Acknowledgments and Disclosure of Funding ",
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"text_level": 1,
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"type": "text",
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"text": "This work is supported by Peng Cheng Laboratory Research Project No. PCL2021A07. Heng Fan and his employer receive no financial support for the research, authorship, and/or publication of this article. ",
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"type": "text",
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"text": "References ",
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"text_level": 1,
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"text": "[1] Bertinetto, L., Valmadre, J., Henriques, J.F., Vedaldi, A., Torr, P.H., 2016. Fully-convolutional siamese networks for object tracking, in: ECCVW. [2] Bhat, G., Danelljan, M., Gool, L.V., Timofte, R., 2019. Learning discriminative model prediction for tracking, in: ICCV. [3] Carion, N., Massa, F., Synnaeve, G., Usunier, N., Kirillov, A., Zagoruyko, S., 2020. End-to-end object detection with transformers, in: ECCV. [4] Chen, C.F.R., Fan, Q., Panda, R., 2021a. Crossvit: Cross-attention multi-scale vision transformer for image classification, in: ICCV. [5] Chen, X., Yan, B., Zhu, J., Wang, D., Yang, X., Lu, H., 2021b. Transformer tracking, in: CVPR. [6] Dai, K., Zhang, Y., Wang, D., Li, J., Lu, H., Yang, X., 2020. High-performance long-term tracking with meta-updater, in: CVPR. [7] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., et al., 2021. An image is worth 16x16 words: Transformers for image recognition at scale, in: ICLR. [8] Fan, H., Bai, H., Lin, L., Yang, F., Chu, P., Deng, G., Yu, S., Huang, M., Liu, J., Xu, Y., et al., 2021. Lasot: A high-quality large-scale single object tracking benchmark. International Journal of Computer Vision 129, 439–461. [9] Fan, H., Lin, L., Yang, F., Chu, P., Deng, G., Yu, S., Bai, H., Xu, Y., Liao, C., Ling, H., 2019. Lasot: A high-quality benchmark for large-scale single object tracking, in: CVPR. \n[10] Fan, H., Ling, H., 2019. Siamese cascaded region proposal networks for real-time visual tracking, in: CVPR. \n[11] Fan, H., Ling, H., 2021. Cract: Cascaded regression-align-classification for robust visual tracking, in: IROS. \n[12] Fu, Z., Liu, Q., Fu, Z., Wang, Y., 2021. Stmtrack: Template-free visual tracking with space-time memory networks, in: CVPR. \n[13] Han, W., Dong, X., Khan, F.S., Shao, L., Shen, J., 2021. Learning to fuse asymmetric feature maps in siamese trackers, in: CVPR. \n[14] He, K., Zhang, X., Ren, S., Sun, J., 2016. Deep residual learning for image recognition, in: CVPR. \n[15] Huang, L., Zhao, X., Huang, K., 2019. Got-10k: A large high-diversity benchmark for generic object tracking in the wild. IEEE Transactions on Pattern Analysis and Machine Intelligence 43, 1562–1577. \n[16] Ke, G., He, D., Liu, T.Y., 2021. Rethinking positional encoding in language pre-training, in: ICLR. \n[17] Krizhevsky, A., Sutskever, I., Hinton, G.E., 2012. Imagenet classification with deep convolutional neural networks. NIPS . \n[18] Larsson, G., Maire, M., Shakhnarovich, G., 2016. Fractalnet: Ultra-deep neural networks without residuals, in: ICLR. \n[19] Li, B., Wu, W., Wang, Q., Zhang, F., Xing, J., Yan, J.S., 2019. Evolution of siamese visual tracking with very deep networks, in: CVPR. \n[20] Li, B., Yan, J., Wu, W., Zhu, Z., Hu, X., 2018. High performance visual tracking with siamese region proposal network, in: CVPR. \n[21] Li, X., Wang, W., Wu, L., Chen, S., Hu, X., Li, J., Tang, J., Yang, J., 2020. Generalized focal loss: Learning qualified and distributed bounding boxes for dense object detection, in: NeurIPS. \n[22] Lin, T.Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dollár, P., Zitnick, C.L., 2014. Microsoft coco: Common objects in context, in: ECCV. \n[23] Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., Guo, B., 2021. Swin transformer: Hierarchical vision transformer using shifted windows. ICCV . \n[24] Loshchilov, I., Hutter, F., 2019. Decoupled weight decay regularization, in: ICLR. \n[25] Mayer, C., Danelljan, M., Paudel, D.P., Van Gool, L., 2021. Learning target candidate association to keep track of what not to track, in: ICCV. \n[26] Muller, M., Bibi, A., Giancola, S., Alsubaihi, S., Ghanem, B., 2018. Trackingnet: A large-scale dataset and benchmark for object tracking in the wild, in: ECCV. \n[27] Ren, S., He, K., Girshick, R., Sun, J., 2015. Faster r-cnn: Towards real-time object detection with region proposal networks, in: NIPS. \n[28] Rezatofighi, H., Tsoi, N., Gwak, J., Sadeghian, A., Reid, I., Savarese, S., 2019. Generalized intersection over union . \n[29] Touvron, H., Cord, M., Douze, M., Massa, F., Sablayrolles, A., Jégou, H., 2021. Training data-efficient image transformers & distillation through attention, in: ICML. \n[30] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I., 2017. Attention is all you need, in: NeurIPS. \n[31] Voigtlaender, P., Luiten, J., Torr, P.H., Leibe, B., 2020. Siam r-cnn: Visual tracking by re-detection, in: CVPR. \n[32] Wang, N., Zhou, W., Wang, J., Li, H., 2021a. Transformer meets tracker: Exploiting temporal context for robust visual tracking, in: CVPR. \n[33] Wang, W., Xie, E., Li, X., Fan, D.P., Song, K., Liang, D., Lu, T., Luo, P., Shao, L., 2021b. Pyramid vision transformer: A versatile backbone for dense prediction without convolutions, in: ICCV. \n[34] Wang, X., Shu, X., Zhang, Z., Jiang, B., Wang, Y., Tian, Y., Wu, F., 2021c. Towards more flexible and accurate object tracking with natural language: Algorithms and benchmark, in: CVPR. \n[35] Xu, Y., Wang, Z., Li, Z., Yuan, Y., Yu, G., 2020. Siamfc++: Towards robust and accurate visual tracking with target estimation guidelines, in: AAAI. \n[36] Yan, B., Peng, H., Fu, J., Wang, D., Lu, H., 2021. Learning spatio-temporal transformer for visual tracking, in: ICCV. \n[37] Yu, Y., Xiong, Y., Huang, W., Scott, M.R., 2020. Deformable siamese attention networks for visual object tracking, in: CVPR. \n[38] Yuan, L., Chen, Y., Wang, T., Yu, W., Shi, Y., Jiang, Z.H., Tay, F.E., Feng, J., Yan, S., 2021. Tokens-to-token vit: Training vision transformers from scratch on imagenet, in: ICCV. \n[39] Zhang, H., Wang, Y., Dayoub, F., Sünderhauf, N., 2021a. Varifocalnet: An iou-aware dense object detector, in: CVPR. \n[40] Zhang, Z., Liu, Y., Wang, X., Li, B., Hu, W., 2021b. Learn to match: Automatic matching network design for visual tracking, in: ICCV. \n[41] Zhang, Z., Peng, H., Fu, J., Li, B., Hu, W., 2020. Ocean: Object-aware anchor-free tracking, in: ECCV. ",
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parse/dev/JavFPcsscd5/JavFPcsscd5.md
ADDED
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| 1 |
+
# The Effects of Regularization and Data Augmentation are Class Dependent
|
| 2 |
+
|
| 3 |
+
Randall Balestriero Meta AI Research, FAIR NYC, USA rbalestriero@meta.com
|
| 4 |
+
|
| 5 |
+
Leon Bottou
|
| 6 |
+
Meta AI Research, FAIR
|
| 7 |
+
NYC, USA
|
| 8 |
+
leonb@meta.com
|
| 9 |
+
|
| 10 |
+
Yann LeCun Meta AI Research, FAIR, NYU NYC, USA ylecun@meta.com
|
| 11 |
+
|
| 12 |
+
# Abstract
|
| 13 |
+
|
| 14 |
+
Regularization is a fundamental technique to improve a model’s generalization performances by limiting its complexity. Deep Neural Networks (DNNs), which tend to overfit their training data, heavily rely on regularizers such as Data-Augmentation (DA) or weight-decay with hyper-parameters found from structural risk minimization, i.e., cross-validation. In this study, we demonstrate that the optimal regularization’s hyper-parameters found from cross-validation over all classes leads to disastrous model performances on a minority of classes. For example, a resnet50 trained on Imagenet sees its “barn spider” test accuracy falls from $6 8 \%$ to $4 6 \%$ only by introducing random crop DA during training. Even more surprising, such unfair impact of regularization also appears when introducing uninformative regularizers such as weight decay or dropout. Those results demonstrate that our search for ever increasing generalization performance —averaged over all classes and samples— has left us with models and regularizers that silently sacrifice performances on some classes. This scenario can become dangerous when deploying a model on downstream tasks e.g. an Imagenet pre-trained resnet50 deployed on INaturalist sees its performances fall from $7 0 \%$ to $3 0 \%$ on class $\# 8 8 8 9$ when introducing random crop DA during the Imagenet pre-training phase. Those results demonstrate that finding a correct measure of a model’s complexity without class-dependent preference remains an open research question.
|
| 15 |
+
|
| 16 |
+
# 1 Introduction
|
| 17 |
+
|
| 18 |
+
Machine learning and deep learning aim at learning systems to solve as accurately as possible a given task at hand [LeCun et al., 1998, Bishop and Nasrabadi, 2006, Jordan and Mitchell, 2015]. This process often takes the form of (i) being given a finite dataset, a (differentiable) loss function, and a performance measure, (ii) splitting the dataset into train/valid/test sets to optimizing the system’s parameters e.g. from gradient updates of the loss on the train set while cross-validating hyper-parameters using the valid set, and (iii) assessing the system’s performance on the test set. As the training set is finite, and the optimal design of the system is unknown, it is common to employ regularization during the optimization phase to reduce over-fitting [Tikhonov, 1943, Tihonov, 1963] i.e. to decrease the system’s performance gap between train set and test set samples [Simard et al., 1991, Chapelle et al., 2000, Bottou, 2012, Neyshabur et al., 2014]. Central to our study is the fact that hyper-parameter selection is done via cross-validation by maximizing the valid set performance with ad-hoc statistics e.g. the average accuracy over all samples for classes in classification tasks.
|
| 19 |
+
|
| 20 |
+
Cross-validation commonly involves many different types of regularization along with their “strengths” [Goodfellow et al., 2016, He et al., 2021]. Most variants of regularization take one of two forms: Data-Augmentation (DA) and weight-decay. DA is a data-driven and informed regularization strategy that artificially increase the number of training samples [Shorten and Khoshgoftaar, 2019]. As opposed to most explicit regularizers e.g. Tikhonov regularization [Krogh and Hertz, 1991], also denoted as weight decay, DA’s regularization is implicit as it is not a function of a model’s parameter, but a function of the training samples [Neyshabur et al., 2014, Hernández-García and König, 2018, LeJeune et al., 2019]; although some DA strategies can be turned into explicit regularizers Balestriero et al. [2022]. Nevertheless, a key distinction between DA and weight decay is that DA tends to require more domain knowledge to be successful than weight decay. Most —if not all— of current state-of-the-art employ such regularizers [Huang et al., 2018, Chen et al., 2020b, Liu et al., 2021, Tan and Le, 2021, Liu et al., 2022].
|
| 21 |
+
|
| 22 |
+
|
| 23 |
+
Figure 1: Structural risk minimization minimizes the empirical risk of several models of varying complexity, and selects the one offering the best compromise between under-fitting and over-fitting [Vapnik and Chervonenkis, 1974]. In deep learning, one commonly control the model’s complexity by picking different DN architectures and/or by applying different levels and flavors of regularization. The key observation of our study is that when the model complexity is calibrated by DA (see Figs. 2, 5 and 6), or weight-decay (see Fig. 3), the class-conditional empirical risks do not align between classes i.e. cross-validation produces models that perform well on the majority of classes but arbitrarily poorly on a few of them as depicted on the left-hand-side. In an ideal setting where the control of the model’s complexity is well aligned with the task and model, one would observe the right-hand-side ideal scenario where the same model complexity is optimal for all classes.
|
| 24 |
+
|
| 25 |
+
In this paper, we will demonstrate that when cross-validation is employed to select the regularization settings maximizing the validation performance, a significant bias is introduced into the trained model: the regularized model exhibits strong per-class favoritism i.e. while the average test performance is improved, it is at the cost of producing a model with significant performance drop on some of the classes as illustrated in the schematic of Fig. 1. For readers familiar with statistical estimation results e.g. the bias-variance trade-off [Kohavi et al., 1996, Von Luxburg and Schölkopf, 2011] or bayesian estimation e.g. Tikhonov regularization [Box and Tiao, 2011, Gruber, 2017], it should not be surprising that regularization produces bias (more details and background provided in Appendix A). In fact, it is beneficial to introduce bias through regularization if it results in a significant reduction of the estimator variance —when one minimizes the average empirical risk. However, the potentially dangerous effect of regularization that this study brings forward is that the bias introduced by regularization is class-dependent, including on transfer learning tasks.
|
| 26 |
+
|
| 27 |
+
To thoroughly validate this observation, we propose a variety of controlled experiments in Section 2. First, we carefully quantify the impact of DA, weight decay and dropout on the per-class performance of a model in Section 2.1, demonstrating that current deep learning finds itself in the scenario depicted on the left of Fig. 1. Then, we consider the task of transfer learning in Section 2.2 where it is again possible to identify again a per-class bias on the target dataset even though the regularization was applied on a different (source) training set. This latter scenario is particularly relevant in current times where it is common to deploy a large pre-trained model on a variety of tasks and raise an important issue: selecting the —on average— best performing pre-trained model can lead to catastrophic individual class performance even on for different downstream tasks.
|
| 28 |
+
|
| 29 |
+
Our next Section 3 will aim at exploring possible explanations and solutions. First, we will provide a brief theoretical justification on why and when DA can be the cause of model bias (Section 3.1) regardless of the task and data at hand. This will shed light to a first possible issue: the DA parameters that make the transformed input preserve its label information vary depending on the class underlying statistics. In short, DA silently introduces class-imbalance in the training set. We propose a dedicated analysis of the label-preserving property of DA on different classes and models in Section 3.2. We then take on the task of searching for a possible solution by first reviewing known theoretical studies quantifying the interplay between regularization and bias in Section 3.3. Lastly, we propose some solutions of our own in Section 3.4 built from the gained insights of Section 3.2 using label-distillation and adaptive DA. All the codebase used to train the various models and to generate the figures is in the supplementary files.
|
| 30 |
+
|
| 31 |
+
|
| 32 |
+
Figure 2: Varying the random crop DA lower bound $\mathbf { \hat { x } }$ -axis) from $100 \%$ to $8 \%$ provides greater average test accuracy (blue) but makes the per-class performance fall for some of the classes. Images of each class are provided in Fig. 12, in the appendix. See Fig. 8 for the convnext and ViT experiments. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with horizontal flip and varying random crop lower bound DA.
|
| 33 |
+
|
| 34 |
+
# 2 Maximizing the Average Model Performance by Cross-Validation Silently Produce Poor Final Performances on a Minority of the Classes
|
| 35 |
+
|
| 36 |
+
We now turn to the empirical validation of Fig. 1 i.e. quantifying the amount of class-dependent bias caused by DA, weight decay and dropout in various realistic scenarios (Section 2.1). We then demonstrate how the bias introduced by regularization transfers to downstream tasks e.g. when deploying an Imagenet (source) trained model on the INaturalist (target) dataset in Section 2.2; that scenario is key as it demonstrates the potential harm of selecting the best performing model on the source dataset which could turn out to also be the most biased model against the target dataset class of interest.
|
| 37 |
+
|
| 38 |
+
# 2.1 Precisely Measuring the Per-Class Effect of Data-Augmentation and Uninformed Regularization with Controlled Experiments
|
| 39 |
+
|
| 40 |
+
This section aims at quantifying precisely the amount of downward or upward per-class performance shift that came as a result from using DA or uninformed regularization e.g. weight-decay or dropout. In fact, it is crucial to remember that regularization, or any other form of structural risk minimization, improves generalization performances by increasing the bias of the estimator so that the estimator’s variance is decreased by a greater amount. However, nothing guarantees the fairness of this bias i.e. for it to be equally distributed amongst the dataset classes. We thus propose a sensitivity analysis by training a large collection of models with varying regularization policies to precisely assess their impact on the class-dependent model bias.
|
| 41 |
+
|
| 42 |
+
Data-Augmentation. DA samples have been known to sometimes disregard the semantic information of the original samples [Krizhevsky et al., 2012]. Nevertheless, DA remains applied universally across tasks and datasets [Shorten and Khoshgoftaar, 2019] as it provides significant performance improvements, even in semi-supervised and unsupervised settings Guo et al. [2018], Xie et al. [2020], Misra and Maaten [2020]. To measure the impact of DA onto per-class performances, we propose in Fig. 2 a sensitivity analysis by training the same architecture on Imagenet with varying DA policies. In particular, we consider a given DA (random crop in this case) and we vary the support of the parameter $\alpha$ which represents how much of the original image is kept in the crop (examples at the top of Fig. 5). We train multiple Deep Neural Networks (DNN)s using $\alpha \in [ 1 0 0 , \tau ]$ with $\tau$ varying from 100 to 8 and for each case, we report our metrics averaged over 20 trained models. We observe a clear relation between increase in the strength of the DA, increase in the average test accuracy overall classes, and decrease in some per-class test accuracies. For example, on a resnet50 Imagenet setting, the accuracy on the “academic gown” class goes from $62 \%$ to $40 \%$ steadily as $\tau$ decreases.
|
| 43 |
+
|
| 44 |
+
|
| 45 |
+
Figure 3: Varying the amount of weight decay, an uninformed regularizer employed throughout, surprisingly exhibits similar class-dependent bias as the DA scenario of Fig. 2. Images for each class are provided in Fig. 13, in the appendix. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with varying weight decay, see Fig. 16 for DenseNet121 results with the same trend and Fig. 8 for the convnext and ViT experiments.
|
| 46 |
+
|
| 47 |
+
Uninformed Regularization. As per the arguments given in Sections 3.1 and 3.2, it would be natural to assume that what makes DA responsible for creating class-dependent bias in DNNs is our misfortune in defining correct augmentation policies. Hence, uninformed weight decay or dropout should behave differently and more fairly. We demonstrate here that such regularizers are also unfair between classes. We thus propose to train multiple models with varying weight-decay and dropout parameters. In our setting, weight-decay is applied to all the DNN parameters except for the ones of batch-normalization layers, as commonly done [Hastie et al., 2009, Leclerc et al., 2022]. We report in Fig. 3 the per-class performance of a resnet50 trained on Imagenet with varying weight decay coefficient $\gamma$ (as was done for DA in Fig. 2) and we observe that different classes have different test accuracy sensitivities to variations in $\gamma$ . Some will see their generalization performance increase, while others will have decreasing generalization performances. We further confirm such findings in Figs. 9 and 10 for dropout where the same per-class trend is observed. In short, even for uninformative regularizers such as weight decay or dropout, a per-class bias is introduced, reducing performances for some of the classes. Although weight-decay is one of the most popular regularizer that is uninformed on the data and task at hand, recent studies have demonstrated that techniques such as model pruning —which can be seen as a post-training model complexity reduction i.e. regularization— also produce increased bias towards under-represented features [Hooker et al., 2019, 2020]. More recently, Balestriero et al. [2022] obtained the close-form explicit regularizer of DA from which it is possible to quantify the sample-dependent aspect of DA’s regularization. From our findings, it seems that classes sharing the same type of features are thus impacted different by DAs.
|
| 48 |
+
|
| 49 |
+
Formal Statistical Test. To further convey our claim, we now propose a formal statistical test [Neyman and Pearson, 1933, Fisher, 1955] on the hypothesis that the per-class accuracy is significantly higher when DA is applied for each class (details provided in Appendix A.2). We obtain that there is enough evidence to reject the hypothesis with $9 5 \%$ confidence for $4 . 5 \%$ of all the classes, and with $9 9 \%$ confidence for $2 . { \dot { 6 } } \%$ of all the classes. Hence there is sufficient evidence to say that the per-class test accuracies is not increased when introducing DA for $4 . 5 \%$ of the 1000 Imagenet classes. We provide in Table 1 the same statistical test but applied on a variety of settings including different architectures (resnet50, densenet121, ViT-small and ConvNext-Tiny) and across the random crop DA and the weight decay controlled experiments. We should highlight however that this is not necessarily a meaningful measure since for example one regularization might not have any negative impact on the classes, but can provide a beneficial gain that is drastically different between classes. Hence, although the per-class performance does not drop by introduce the regularizer, the inter-class performance gap can be increased by it, which is an equally harmful impact.
|
| 50 |
+
|
| 51 |
+
The next Section 2.2 proposes to study the scenario of introducing a pre-trained model, on a different downstream task to show that the class-dependent effect of regularization remains present and unfair towards specific downstream classes.
|
| 52 |
+
|
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# 2.2 The Class-Dependent Bias Transfers to Other Downstream Tasks
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The last experiment we propose is to quantify the amount of class-dependent bias that transfers to other downstream tasks, a common situation in transfer learning and in system deployment to the real world [Pan and Yang, 2009]. We thus want to measure how regularization applied during the pre-training phase on a source dataset impacts the per-class accuracy of that model on the target dataset.
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Table 1: Percentage of Imagenet classes for which the test set performance (per-class) is statistically not greater when applying random crop DA (or weight decay) as measured by the statistical test from Section 2. We observe that although this measure is highly conservative since a regularizer (DA or weight decay) might now have a negative impact on a per-class performance but still increase the performance gap between difference classes, already, a nonzero proportion of classes are negatively impact by introduce random crop DA or weight decay.
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<table><tr><td>confidencel</td><td>resnet50</td><td>resnet50 random crop|weight decay</td><td>convnext-tinyl random crop</td><td>weight decay</td><td>/|convnext-tiny|ViT random crop|densenet121 random crop</td><td>weight decay</td></tr><tr><td>90%</td><td>5.9</td><td>39.4</td><td>0.1</td><td>85.5</td><td>1.2</td><td>33.8</td></tr><tr><td>95%</td><td>4.5</td><td>30.3</td><td>0.1</td><td>78.8</td><td>1.0</td><td>17.1</td></tr><tr><td>99%</td><td>2.6</td><td>15.3</td><td>0.0</td><td>63.9</td><td>0.7</td><td>3.3</td></tr></table>
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Figure 4: Varying the random crop lower-bound during pre-training (Imagenet) also produces unfair per-class biases on the downstream task (INaturalist). Hence, selecting the pre-trained model with best average test accuracy on the source dataset might result in deploying a model with the worst performance on the classes of interest in the target task. Images for each class are provided in Fig. 17, in the appendix. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with varying random crop lower bound and transferred to INaturalist with frozen backbone parameters.
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In order to keep the setting similar to Section 2.1, we adopt a resnet50 model with random crop DA. That model is pre-trained on Imagenet dataset (source) with varying value of $\tau$ (random crop lower bound) and then, the trained model is transferred to the INaturalist dataset [Van Horn et al., 2018] (target) that consists of 10,000 classes. When transferring the model to INaturalist, the parameters are kept frozen, and only a linear classifier is trained on top of it. We report in Fig. 4 the performance of the trained models with varying $\tau$ on different INaturalist classes. We observe once again that the best resnet50 —on average— is not necessarily the one that should be deployed as there exists a strong per-class bias that varies with $\tau$ . As a result, picking the best performing model from a source dataset to a target dataset, might leave the pipeline to perform poorly since that model might also be the one that is the most biased against the class of interest in the target dataset.
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This result should motivate the design of novel regularizers that do not reduce performances between classes at different regimes. Additionally, due to the cost of training multiple models with varying regularization settings, one might wonder on the possible alternative solutions to detect trends such as shown in Fig. 4 only when given a single pre-trained model.
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# 3 Understanding Why and When Regularization Produces Models With Class-Dependent Preferences
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The first part of our study (Section 2) empirically validated that DNN regularization produces unfair model complexity control over different classes, resulting in a model performing poorly on a few of the classes although being highly performing on average. We now provide in Sections 3.1 and 3.2 some intuition on why DA can be a source of bias regardless of the task, dataset and model at hand. Then, Section 3.3 reviews existing works trying to confront regularization and model bias, and as we will see, an out-of-the-box solution does not seem to exist when there are only a few classes suffering from regularization (Section 3.4).
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# 3.1 A Data-Augmentation Policy That is Not Label-Preserving For All Classes Will Create Class-Imbalance Performances
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To provide a simple explanation on how DA causes bias in a trained model, we propose the following derivation that holds for any signal e.g. timeseries, images, videos. Without loss of generality, we will consider here the $\ell _ { 2 }$ loss which was shown to perform as well as the cross-entropy even on Imagenet [Hui and Belkin, 2020].
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Dataset notations. Given a sample $\textbf { \em x } \in { \mathcal { X } }$ with $\boldsymbol { \mathcal { X } } \subset \mathbb { R } ^ { D }$ , we consider ${ \pmb y } \triangleq f ^ { * } ( { \pmb x } )$ to be the ground-truth target value. Hence our hope is to learn an approximator $f _ { \theta }$ that is as close as possible to $f ^ { * }$ everywhere in $\mathcal { X }$ , although we only observe a finite training set $\mathbb { X } \triangleq \{ ( \pmb { x } _ { 1 } , \pmb { y } _ { 1 } ) , \dots , ( \pmb { x } _ { N } , \pmb { y } _ { N } ) \}$ . Given an output vector $\textbf { \em u }$ we also define the level-set of a mapping $f$ to be $\{ { \pmb x } \in { \mathcal { X } } : f ( { \pmb x } ) = { \pmb u } \}$ .
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Data-Augmentation notations. Additionally, one employs a DA policy $\mathcal { T } : \mathbb { R } ^ { D } \times \mathcal { K } \mapsto \mathbb { R } ^ { D }$ such that given a transformation parameter $\alpha \in { \mathcal { K } }$ , ${ \mathcal T } _ { \alpha } ( { \pmb x } )$ produces the transformed view of $_ { \textbf { \em x } }$ . Often, one also defines a density $p$ on $\kappa$ that helps in sampling transformation parameters that are a priori known to be the most useful.
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Theorem 1. Whenever the transformations produced by $\mathcal { T } _ { \alpha } , \forall \alpha$ do not respect the level-set of $f ^ { * }$ , and whenever the model has enough capacity to minimize the training loss, the DA will create irreducible bias in $f _ { \theta }$ as in
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The main idea of the proof, provided in Appendix $\mathbf { B }$ , is to show that if a transformation does not move samples on the level-set of the true function (left-hand-side of Eq. (1)), then $f _ { \theta }$ will learn a different level-set (since it has 0 training error), and thus $\| f ^ { * } - f _ { \theta } \| > 0$ i.e. $f _ { \theta }$ is biased.
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Whenever the left-hand-side of Eq. (1) is 0, the DA is denoted as label-preserving [Cui et al., 2015, Taylor and Nitschke, 2018]. From the above, we see that unless the target y associated to ${ \mathcal T } _ { \alpha } ( { \pmb x } )$ is modified accordingly to encode the shift in the target function level-set produced by $\mathcal { T } _ { \alpha }$ , any $D A$ that is not label-preserving will introduce a bias. Some DAs propose to incorporate label transformation i.e. not only $_ { \textbf { \em x } }$ but also $\textbf { { y } }$ is augmented to better inform on the uncertainty that has been added into $\mathcal { T } _ { \boldsymbol { \theta } } ( \mathbf { \mathscr { x } } )$ . This is for example the case for MixUp [Zhang et al., 2017], ManifoldMixUp [Verma et al., 2019], CutMix [Yun et al., 2019] and their extensions.
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Our goal in the next Section 3.2 is to demonstrate how DAs such as random crop, color jittering, or CutOut are only label preserving for some values of $\alpha$ that vary with the sample class. As a consequence, while the use of the DA improves the average test performance, it is at the cost of a significant reduction in performance for some of the classes.
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# 3.2 The Same Data-Augmentation can be Label-Preserving or Not Between Different Classes
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In the previous Section 3.1 we provided a general argument on the sufficient conditions for DA to produce a biased model. We hope in this section to provide a more concrete example that applies to current DNN training. To that end, we will demonstrate that a DA can be label-preserving or not depending on the sample’s class, hence, since the same DA policy is employed for all classes, the augmented dataset will exhibit a class-imbalance in favor of the classes for which the DA is most label-preserving.
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To measure by how much a given DA, $\mathcal { T } _ { \alpha }$ , is label-preserving, we propose to take 6 popular architectures that are pre-trained on Imagenet [Deng et al., 2009] from the official PyTorch [Paszke et al., 2019] repository, and to evaluate their performances for varying DA settings (top of Fig. 5). We observe that considering the dataset as a whole is not a good indicator of the optimal DA value to employ since per-class accuracy performance (bottom of Fig. 5) vary drastically. For example, for some classes, any level of transformation $\alpha$ can produce augmented samples with enough information to be correctly classified, while for other classes, even a small amount of DA makes the samples unpredictable.
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To further ensure that the observed relation between label-preservation, sample class, and amount of transformation $\alpha$ is sound, we provide in Fig. 6 the per-class test accuracy on different models, all exhibit the same trends. In short, we identify that when creating an augmented dataset by applying the same DA across classes, the number of per-class samples that actually contain enough information about their true labels will become largely imbalance between classes, even if the original dataset was balanced. Any model trained on the augmented dataset will thus focus on the classes for which the DA is the most label-preserving.
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Figure 5: Top: examples of an augmented image of class “bird”. Middle: average accuracy (train set in dashed line and test set in plain) on Imagenet, using 6 popular architectures. Bottom: per-class performances from the middle scenario along with 9 images of the corresponding classes. We observe that the random crop DA seems to loose its label-preserving property on average when less than $30 \%$ of the image is kept in the crop but looking at the per-class performance, we observe that such DA can be label-preserving with only $8 \%$ of the original image for some classes, while for other classes the label information starts to reduce at around $50 \%$ . Results obtained from the official Imagenet pre-trained PyTorch models. CutOut and ColorJitter cases are provided in Figs. 14 and 15 and exhibit the same trend.
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Figure 6: Reprise of the bottom left of Fig. 5 for three different DAs (each column) and using the same 6 popular architectures (different lines). We observe that across DAs, different architectures agree on the label-preserving regimes for $\mathcal { T } _ { \alpha }$ i.e. even an ensemble of model would not reduce the class-dependent bias of the final prediction. Results obtained from the official Imagenet pre-trained PyTorch models.
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Beyond the above intuitive and natural understand we obtained in term of data-augmentation, there exists theoretical studies that we propose to summarize below
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# 3.3 Do Existing Studies Provide Answers Into the Inter-Play Between Regularization and Class-Dependent Bias?
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We explored in Section 3.2 a possible explanation of the class-dependent bias we observed in Section 2 hinting at the need to use class-dependent DA. We propose here to summarize existing studies that are attempting to better understand the impact of DA and regularization in general onto a model’s performance and bias. Then, Section 3.4 will explore their application.
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Figure 7: Reprise of Fig. 2 but now implementing the class-dependent DA as prescribed in Section 3.3 i.e. we only apply DA (random crop) to the classes that see their per-class accuracy improve when this DA is employed during training.. We observe here that the bias of the model introduced from random crop on the majority of the classes spills-over to the minority of the classes that do not benefit from that DA, even though such classes never received that DA during training. Results are averaged over 5 runs and employ the official resnet50 implementation trained on Imagenet with horizontal flip but a per-class random crop DA.
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Input independent DA exacerbates the bias already present in a dataset. Especially relevant to our results is a recent result of Xu et al. [2020]. In this work, it was theorized that when the underlying dataset contains inherent biases, training on the original data is more effective than employing an i.i.d. DA policy, i.e. applying the same random augmentation to all samples/classes, to produce an unbiased model. In short, the DA exacerbates the already present biases and makes the trained model further away from the unbiased optimum. The difficulty of this result lies in defining bias for real images. As per our experiments from Section 3, we observe that bias can take many form e.g. one class might naturally represent its object always under the same angle. This is particularly true say for boats which are rarely captured upside-down. Hence, simply having classes with different natural statistics could be enough for Xu et al. [2020] to prohibit the use of DA in current datasets. Furthermore, Raghunathan et al. [2020] obtained a surprising result combining both DA and regularization. In that case, it was found that the minimum norm interpolant on the original $+ \textrm Ḋ \textmu Ḋ Ḍ Ḍ _ { \mathrm { Ḋ } } \textrm Ḋ \textmu Ḍ Ḍ$ dataset could have a larger standard error than the minimum norm interpolant on the original dataset alone, even when using label-preserving DA. However, this phenomenon only occurs as long as the model remains over-parametrized, even when considering the original $+ \mathrm { D A }$ dataset.
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Input dependent DA can reduce sample/class bias. Recall that Section 3.2 brought forward one possible explanation on how DA can be the source of training-set by introduce class-imbalance due to the same DA being label-preserving for some classes and not for others. From this, a direct solution would be to adapt the “strength”. This solution, formalized in Xu et al. [2020], consists in measuring the bias of a model and adapt the DA policy accordingly to correct it. This has been done in different flavors e.g. in McLaughlin et al. [2015], Iosifidis and Ntoutsi [2018], Jaipuria et al. [2020]. One limitation of this direction is that it requires to estimate a model bias and adapt the DA accordingly which can be challenging for large scale models.
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Learned DA e.g. from a GAN can produce even more class-dependent bias. One natural extension of the hands-on adaptivity of a DA to a measured model bias would be to learn a DA to maximize a model’s performance, for example. This line of work has led to many learn DA policies e.g. using Generative Adversarial Networks [Hu and Li, 2019]. Yet, it has been shown that learning a DA to maximize some aggregated measure of performance will produce even more bias in a model [Hu and Li, 2019]. In fact, and as per the controlled experiments from Section 2, the learned DA will entirely disregard a minority of the classes if it means that the produced DA can drastically improve performances on all others, effectively maximizing the average performances.
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Inherent tradeoff between DA and model robustness. In addition to the implication of DA into bias, there exists an intertwined relationship between DA and model robustness. In fact, even assuming the use of perfectly adapted DAs, there exists an inherent tradeoff between accuracy and robustness that holds even in the infinite data limit [Tsipras et al., 2018, Fawzi et al., 2018, Zhang et al., 2019]. For example, Min et al. [2021] proved in the robust linear classification regime that (i) more data improves generalization in a weak adversary regime, (ii) more data can improve generalization up to a point where additional data starts to hurt generalization in a medium adversary regime, and that (iii) more data immediately decreases generalization error in a strong adversary regime.
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# 3.4 Class-Dependent Data-Augmentation Seems Insufficient for Performance Recovery
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We observed in Section 2 that DA could lead to disastrous per-class performances on a minority of classes. We now propose to implement one solution from Section 3.3 that consists in simply not applying the harmful DA to the classes suffering from it. As will become clear, applying the DA on all other classes will be enough to skew the training of the model preventing any performing recovery on those classes.
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To motivate the need for a better control of the per-class performance of a model, we first present an illustrative argument. A standard resnet50 on Imagenet reaches $7 7 . 1 4 \%$ top-1 with a random crop lower bound of $8 \%$ . Using precise cross-validation, one could reach 77.29 by using a random crop lower bound of $1 0 \%$ instead of $8 \%$ on all classes. Yet, and most interestingly, if one were able to get the best per-class performance —as per varying the lower-bound as in Fig. 2 and picking for each class the best per-class performance of any of the models— one could reach 79.37. Beyond pure average test performance, controlling the worst-case per-class performance is of crucial importance for fairness [Du et al., 2020, Veitch et al., 2021]. We thus explore one of the solution that we reviewed in Section 3.3 that consists in only applying the random crop DA to the classes whose test performances increased with the DA’s level. We obtain in Fig. 7 that such class-specific strategy is not sufficient to guarantee that the classes negatively impacted by the random crop DA see their performance to be constant across the DA level applied to all the other classes. This finding is also supported by our weight decay experiment in Figs. 3 and 16 in which the regularization of the DN impacted classes differently. In fact, first notice that applying weight decay only when seeing some specific classes would simply amount (on average) to reducing the weight decay hyper-parameter proportionally to how many classes are considered to be without regularization. In a similar fashion, DA produces an implicit regularizer [Balestriero et al., 2022] and applying a class-specific DA level reduces the impact of the implicit regularizer. But since the amount of classes for which we do not apply random crop is quite small compared to the total number of classes (between $1 \%$ and $5 \%$ ), it means that the DA’s implicit regularizer remains nearly the same and thus the model’s bias is nearly the same regardless if that DA is applied or not onto those classes. This is what we observe, applying the random crop DA to the classes that benefit from it is enough to bias the model and degrade the performances on some classes at the same pace than when applying the DA to all classes unconditionally (compare Figs. 2 and 7).
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As a result, we observe that no readily and easily implemented solution provides us with a strategy to prevent deep learning to fall into the scenario depicted on the left of Fig. 1. Those findings however motivate the search of novel model complexity controls that is fair among classes. From a more theoretical viewpoint, it might also be possible to better understand if even such a fair per-class model complexity could exist, which is not clear as natural image classes tend to have inherently different statistics.
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# 4 Conclusions and Limitations
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We proposed in this study to understand the impact of regularization, in particular data-augmentation and weight decay, into the final performances of a deep network. We obtained that the use of regularization increases the average test performances at the cost of significant performance drops on some specific classes. By focusing on maximizing aggregate performance statistics we have produced learning mechanisms that can be potentially harmful, especially in transfer learning tasks. In fact, we have also observed that varying the amount of regularization employed during pre-training of a specific dataset impacts the per-class performances of that pre-trained model on different downstream tasks e.g. going from Imagenet to INaturalist. Lastly, commonly prescribed solutions e.g. classdependent data-augmentation do not seem to help indicating that the sole use of an augmentation on some classes is enough to bias the model on all classes. Hence, there remains a vast research area to explore in order to turn deep learning model selection from the current regime to a more ideal one (left to right of Fig. 1).
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The main limitation of this work is its focus on computer vision datasets and models. It is possible that our observation will be further conveyed in other regimes or not, and we leave such analysis for future work.
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+
# Checklist
|
| 229 |
+
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| 230 |
+
1. For all authors...
|
| 231 |
+
|
| 232 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
|
| 233 |
+
(b) Did you describe the limitations of your work? [N/A] We do not propose a novel method that requires such discussion
|
| 234 |
+
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We carefully described why the observations we have made can be dangerous for real world applications
|
| 235 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
|
| 236 |
+
|
| 237 |
+
2. If you are including theoretical results...
|
| 238 |
+
|
| 239 |
+
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Our only “theoretical result” consists in a formal statistical test for which we precisely describe our settings and statistics
|
| 240 |
+
(b) Did you include complete proofs of all theoretical results? [N/A] No theoretical result requiring proofs was provided
|
| 241 |
+
|
| 242 |
+
3. If you ran experiments...
|
| 243 |
+
|
| 244 |
+
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include summary statistics in the supplementary material which is enough to validate our claims. The full codebase and all the saved models will be released upon completion of the review process (this includes almost a thousand pre-trained resnet50s)
|
| 245 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We specify within each caption the key experimental setups, additional training details are provided in the appendix
|
| 246 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] we averaged all of our runs over a significant amount of realizations (20) and perform a formal statistical test on the significance of our results
|
| 247 |
+
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We provide this in the beginning of the appendix. Note that we will be providing the saved models and summary statistics to make the results easily reproducible even without GPU or computation ressources.
|
| 248 |
+
|
| 249 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 250 |
+
|
| 251 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] For Imagenet and INaturalist
|
| 252 |
+
(b) Did you mention the license of the assets? [Yes] the full code will be released with a NC by NC license on GitHub
|
| 253 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] we include summary result files that contain all the statistics to reproduce the main claims of the paper in the supplementary material. The full codebase along with pre-trained models will be released upon completion of the review process.
|
| 254 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We are only using the Imagenet and INaturalist datasets
|
| 255 |
+
|
| 256 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We are only using the Imagenet and INaturalist datasets
|
| 257 |
+
|
| 258 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 259 |
+
|
| 260 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 261 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 262 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
|
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "The Effects of Regularization and Data Augmentation are Class Dependent ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
174,
|
| 8 |
+
122,
|
| 9 |
+
823,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Randall Balestriero Meta AI Research, FAIR NYC, USA rbalestriero@meta.com ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
194,
|
| 19 |
+
226,
|
| 20 |
+
375,
|
| 21 |
+
281
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Leon Bottou \nMeta AI Research, FAIR \nNYC, USA \nleonb@meta.com ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
401,
|
| 30 |
+
227,
|
| 31 |
+
568,
|
| 32 |
+
281
|
| 33 |
+
],
|
| 34 |
+
"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Yann LeCun Meta AI Research, FAIR, NYU NYC, USA ylecun@meta.com ",
|
| 39 |
+
"bbox": [
|
| 40 |
+
594,
|
| 41 |
+
227,
|
| 42 |
+
803,
|
| 43 |
+
282
|
| 44 |
+
],
|
| 45 |
+
"page_idx": 0
|
| 46 |
+
},
|
| 47 |
+
{
|
| 48 |
+
"type": "text",
|
| 49 |
+
"text": "Abstract ",
|
| 50 |
+
"text_level": 1,
|
| 51 |
+
"bbox": [
|
| 52 |
+
462,
|
| 53 |
+
318,
|
| 54 |
+
535,
|
| 55 |
+
334
|
| 56 |
+
],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "Regularization is a fundamental technique to improve a model’s generalization performances by limiting its complexity. Deep Neural Networks (DNNs), which tend to overfit their training data, heavily rely on regularizers such as Data-Augmentation (DA) or weight-decay with hyper-parameters found from structural risk minimization, i.e., cross-validation. In this study, we demonstrate that the optimal regularization’s hyper-parameters found from cross-validation over all classes leads to disastrous model performances on a minority of classes. For example, a resnet50 trained on Imagenet sees its “barn spider” test accuracy falls from $6 8 \\%$ to $4 6 \\%$ only by introducing random crop DA during training. Even more surprising, such unfair impact of regularization also appears when introducing uninformative regularizers such as weight decay or dropout. Those results demonstrate that our search for ever increasing generalization performance —averaged over all classes and samples— has left us with models and regularizers that silently sacrifice performances on some classes. This scenario can become dangerous when deploying a model on downstream tasks e.g. an Imagenet pre-trained resnet50 deployed on INaturalist sees its performances fall from $7 0 \\%$ to $3 0 \\%$ on class $\\# 8 8 8 9$ when introducing random crop DA during the Imagenet pre-training phase. Those results demonstrate that finding a correct measure of a model’s complexity without class-dependent preference remains an open research question. ",
|
| 62 |
+
"bbox": [
|
| 63 |
+
232,
|
| 64 |
+
351,
|
| 65 |
+
766,
|
| 66 |
+
612
|
| 67 |
+
],
|
| 68 |
+
"page_idx": 0
|
| 69 |
+
},
|
| 70 |
+
{
|
| 71 |
+
"type": "text",
|
| 72 |
+
"text": "1 Introduction ",
|
| 73 |
+
"text_level": 1,
|
| 74 |
+
"bbox": [
|
| 75 |
+
176,
|
| 76 |
+
640,
|
| 77 |
+
310,
|
| 78 |
+
656
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "Machine learning and deep learning aim at learning systems to solve as accurately as possible a given task at hand [LeCun et al., 1998, Bishop and Nasrabadi, 2006, Jordan and Mitchell, 2015]. This process often takes the form of (i) being given a finite dataset, a (differentiable) loss function, and a performance measure, (ii) splitting the dataset into train/valid/test sets to optimizing the system’s parameters e.g. from gradient updates of the loss on the train set while cross-validating hyper-parameters using the valid set, and (iii) assessing the system’s performance on the test set. As the training set is finite, and the optimal design of the system is unknown, it is common to employ regularization during the optimization phase to reduce over-fitting [Tikhonov, 1943, Tihonov, 1963] i.e. to decrease the system’s performance gap between train set and test set samples [Simard et al., 1991, Chapelle et al., 2000, Bottou, 2012, Neyshabur et al., 2014]. Central to our study is the fact that hyper-parameter selection is done via cross-validation by maximizing the valid set performance with ad-hoc statistics e.g. the average accuracy over all samples for classes in classification tasks. ",
|
| 85 |
+
"bbox": [
|
| 86 |
+
174,
|
| 87 |
+
671,
|
| 88 |
+
825,
|
| 89 |
+
837
|
| 90 |
+
],
|
| 91 |
+
"page_idx": 0
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "text",
|
| 95 |
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"text": "Cross-validation commonly involves many different types of regularization along with their “strengths” [Goodfellow et al., 2016, He et al., 2021]. Most variants of regularization take one of two forms: Data-Augmentation (DA) and weight-decay. DA is a data-driven and informed regularization strategy that artificially increase the number of training samples [Shorten and Khoshgoftaar, 2019]. As opposed to most explicit regularizers e.g. Tikhonov regularization [Krogh and Hertz, 1991], also denoted as weight decay, DA’s regularization is implicit as it is not a function of a model’s parameter, but a function of the training samples [Neyshabur et al., 2014, Hernández-García and König, 2018, LeJeune et al., 2019]; although some DA strategies can be turned into explicit regularizers Balestriero et al. [2022]. Nevertheless, a key distinction between DA and weight decay is that DA tends to require more domain knowledge to be successful than weight decay. Most —if not all— of current state-of-the-art employ such regularizers [Huang et al., 2018, Chen et al., 2020b, Liu et al., 2021, Tan and Le, 2021, Liu et al., 2022]. ",
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"Figure 1: Structural risk minimization minimizes the empirical risk of several models of varying complexity, and selects the one offering the best compromise between under-fitting and over-fitting [Vapnik and Chervonenkis, 1974]. In deep learning, one commonly control the model’s complexity by picking different DN architectures and/or by applying different levels and flavors of regularization. The key observation of our study is that when the model complexity is calibrated by DA (see Figs. 2, 5 and 6), or weight-decay (see Fig. 3), the class-conditional empirical risks do not align between classes i.e. cross-validation produces models that perform well on the majority of classes but arbitrarily poorly on a few of them as depicted on the left-hand-side. In an ideal setting where the control of the model’s complexity is well aligned with the task and model, one would observe the right-hand-side ideal scenario where the same model complexity is optimal for all classes. "
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"text": "In this paper, we will demonstrate that when cross-validation is employed to select the regularization settings maximizing the validation performance, a significant bias is introduced into the trained model: the regularized model exhibits strong per-class favoritism i.e. while the average test performance is improved, it is at the cost of producing a model with significant performance drop on some of the classes as illustrated in the schematic of Fig. 1. For readers familiar with statistical estimation results e.g. the bias-variance trade-off [Kohavi et al., 1996, Von Luxburg and Schölkopf, 2011] or bayesian estimation e.g. Tikhonov regularization [Box and Tiao, 2011, Gruber, 2017], it should not be surprising that regularization produces bias (more details and background provided in Appendix A). In fact, it is beneficial to introduce bias through regularization if it results in a significant reduction of the estimator variance —when one minimizes the average empirical risk. However, the potentially dangerous effect of regularization that this study brings forward is that the bias introduced by regularization is class-dependent, including on transfer learning tasks. ",
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"text": "To thoroughly validate this observation, we propose a variety of controlled experiments in Section 2. First, we carefully quantify the impact of DA, weight decay and dropout on the per-class performance of a model in Section 2.1, demonstrating that current deep learning finds itself in the scenario depicted on the left of Fig. 1. Then, we consider the task of transfer learning in Section 2.2 where it is again possible to identify again a per-class bias on the target dataset even though the regularization was applied on a different (source) training set. This latter scenario is particularly relevant in current times where it is common to deploy a large pre-trained model on a variety of tasks and raise an important issue: selecting the —on average— best performing pre-trained model can lead to catastrophic individual class performance even on for different downstream tasks. ",
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"text": "Our next Section 3 will aim at exploring possible explanations and solutions. First, we will provide a brief theoretical justification on why and when DA can be the cause of model bias (Section 3.1) regardless of the task and data at hand. This will shed light to a first possible issue: the DA parameters that make the transformed input preserve its label information vary depending on the class underlying statistics. In short, DA silently introduces class-imbalance in the training set. We propose a dedicated analysis of the label-preserving property of DA on different classes and models in Section 3.2. We then take on the task of searching for a possible solution by first reviewing known theoretical studies quantifying the interplay between regularization and bias in Section 3.3. Lastly, we propose some solutions of our own in Section 3.4 built from the gained insights of Section 3.2 using label-distillation and adaptive DA. All the codebase used to train the various models and to generate the figures is in the supplementary files. ",
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"Figure 2: Varying the random crop DA lower bound $\\mathbf { \\hat { x } }$ -axis) from $100 \\%$ to $8 \\%$ provides greater average test accuracy (blue) but makes the per-class performance fall for some of the classes. Images of each class are provided in Fig. 12, in the appendix. See Fig. 8 for the convnext and ViT experiments. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with horizontal flip and varying random crop lower bound DA. "
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"text": "2 Maximizing the Average Model Performance by Cross-Validation Silently Produce Poor Final Performances on a Minority of the Classes ",
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"text": "We now turn to the empirical validation of Fig. 1 i.e. quantifying the amount of class-dependent bias caused by DA, weight decay and dropout in various realistic scenarios (Section 2.1). We then demonstrate how the bias introduced by regularization transfers to downstream tasks e.g. when deploying an Imagenet (source) trained model on the INaturalist (target) dataset in Section 2.2; that scenario is key as it demonstrates the potential harm of selecting the best performing model on the source dataset which could turn out to also be the most biased model against the target dataset class of interest. ",
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"text": "2.1 Precisely Measuring the Per-Class Effect of Data-Augmentation and Uninformed Regularization with Controlled Experiments ",
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"text": "This section aims at quantifying precisely the amount of downward or upward per-class performance shift that came as a result from using DA or uninformed regularization e.g. weight-decay or dropout. In fact, it is crucial to remember that regularization, or any other form of structural risk minimization, improves generalization performances by increasing the bias of the estimator so that the estimator’s variance is decreased by a greater amount. However, nothing guarantees the fairness of this bias i.e. for it to be equally distributed amongst the dataset classes. We thus propose a sensitivity analysis by training a large collection of models with varying regularization policies to precisely assess their impact on the class-dependent model bias. ",
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"text": "Data-Augmentation. DA samples have been known to sometimes disregard the semantic information of the original samples [Krizhevsky et al., 2012]. Nevertheless, DA remains applied universally across tasks and datasets [Shorten and Khoshgoftaar, 2019] as it provides significant performance improvements, even in semi-supervised and unsupervised settings Guo et al. [2018], Xie et al. [2020], Misra and Maaten [2020]. To measure the impact of DA onto per-class performances, we propose in Fig. 2 a sensitivity analysis by training the same architecture on Imagenet with varying DA policies. In particular, we consider a given DA (random crop in this case) and we vary the support of the parameter $\\alpha$ which represents how much of the original image is kept in the crop (examples at the top of Fig. 5). We train multiple Deep Neural Networks (DNN)s using $\\alpha \\in [ 1 0 0 , \\tau ]$ with $\\tau$ varying from 100 to 8 and for each case, we report our metrics averaged over 20 trained models. We observe a clear relation between increase in the strength of the DA, increase in the average test accuracy overall classes, and decrease in some per-class test accuracies. For example, on a resnet50 Imagenet setting, the accuracy on the “academic gown” class goes from $62 \\%$ to $40 \\%$ steadily as $\\tau$ decreases. ",
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"Figure 3: Varying the amount of weight decay, an uninformed regularizer employed throughout, surprisingly exhibits similar class-dependent bias as the DA scenario of Fig. 2. Images for each class are provided in Fig. 13, in the appendix. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with varying weight decay, see Fig. 16 for DenseNet121 results with the same trend and Fig. 8 for the convnext and ViT experiments. "
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"text": "Uninformed Regularization. As per the arguments given in Sections 3.1 and 3.2, it would be natural to assume that what makes DA responsible for creating class-dependent bias in DNNs is our misfortune in defining correct augmentation policies. Hence, uninformed weight decay or dropout should behave differently and more fairly. We demonstrate here that such regularizers are also unfair between classes. We thus propose to train multiple models with varying weight-decay and dropout parameters. In our setting, weight-decay is applied to all the DNN parameters except for the ones of batch-normalization layers, as commonly done [Hastie et al., 2009, Leclerc et al., 2022]. We report in Fig. 3 the per-class performance of a resnet50 trained on Imagenet with varying weight decay coefficient $\\gamma$ (as was done for DA in Fig. 2) and we observe that different classes have different test accuracy sensitivities to variations in $\\gamma$ . Some will see their generalization performance increase, while others will have decreasing generalization performances. We further confirm such findings in Figs. 9 and 10 for dropout where the same per-class trend is observed. In short, even for uninformative regularizers such as weight decay or dropout, a per-class bias is introduced, reducing performances for some of the classes. Although weight-decay is one of the most popular regularizer that is uninformed on the data and task at hand, recent studies have demonstrated that techniques such as model pruning —which can be seen as a post-training model complexity reduction i.e. regularization— also produce increased bias towards under-represented features [Hooker et al., 2019, 2020]. More recently, Balestriero et al. [2022] obtained the close-form explicit regularizer of DA from which it is possible to quantify the sample-dependent aspect of DA’s regularization. From our findings, it seems that classes sharing the same type of features are thus impacted different by DAs. ",
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"text": "Formal Statistical Test. To further convey our claim, we now propose a formal statistical test [Neyman and Pearson, 1933, Fisher, 1955] on the hypothesis that the per-class accuracy is significantly higher when DA is applied for each class (details provided in Appendix A.2). We obtain that there is enough evidence to reject the hypothesis with $9 5 \\%$ confidence for $4 . 5 \\%$ of all the classes, and with $9 9 \\%$ confidence for $2 . { \\dot { 6 } } \\%$ of all the classes. Hence there is sufficient evidence to say that the per-class test accuracies is not increased when introducing DA for $4 . 5 \\%$ of the 1000 Imagenet classes. We provide in Table 1 the same statistical test but applied on a variety of settings including different architectures (resnet50, densenet121, ViT-small and ConvNext-Tiny) and across the random crop DA and the weight decay controlled experiments. We should highlight however that this is not necessarily a meaningful measure since for example one regularization might not have any negative impact on the classes, but can provide a beneficial gain that is drastically different between classes. Hence, although the per-class performance does not drop by introduce the regularizer, the inter-class performance gap can be increased by it, which is an equally harmful impact. ",
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"text": "The next Section 2.2 proposes to study the scenario of introducing a pre-trained model, on a different downstream task to show that the class-dependent effect of regularization remains present and unfair towards specific downstream classes. ",
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"text": "2.2 The Class-Dependent Bias Transfers to Other Downstream Tasks ",
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"text": "The last experiment we propose is to quantify the amount of class-dependent bias that transfers to other downstream tasks, a common situation in transfer learning and in system deployment to the real world [Pan and Yang, 2009]. We thus want to measure how regularization applied during the pre-training phase on a source dataset impacts the per-class accuracy of that model on the target dataset. ",
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"Table 1: Percentage of Imagenet classes for which the test set performance (per-class) is statistically not greater when applying random crop DA (or weight decay) as measured by the statistical test from Section 2. We observe that although this measure is highly conservative since a regularizer (DA or weight decay) might now have a negative impact on a per-class performance but still increase the performance gap between difference classes, already, a nonzero proportion of classes are negatively impact by introduce random crop DA or weight decay. "
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"table_body": "<table><tr><td>confidencel</td><td>resnet50</td><td>resnet50 random crop|weight decay</td><td>convnext-tinyl random crop</td><td>weight decay</td><td>/|convnext-tiny|ViT random crop|densenet121 random crop</td><td>weight decay</td></tr><tr><td>90%</td><td>5.9</td><td>39.4</td><td>0.1</td><td>85.5</td><td>1.2</td><td>33.8</td></tr><tr><td>95%</td><td>4.5</td><td>30.3</td><td>0.1</td><td>78.8</td><td>1.0</td><td>17.1</td></tr><tr><td>99%</td><td>2.6</td><td>15.3</td><td>0.0</td><td>63.9</td><td>0.7</td><td>3.3</td></tr></table>",
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"Figure 4: Varying the random crop lower-bound during pre-training (Imagenet) also produces unfair per-class biases on the downstream task (INaturalist). Hence, selecting the pre-trained model with best average test accuracy on the source dataset might result in deploying a model with the worst performance on the classes of interest in the target task. Images for each class are provided in Fig. 17, in the appendix. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with varying random crop lower bound and transferred to INaturalist with frozen backbone parameters. "
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"text": "In order to keep the setting similar to Section 2.1, we adopt a resnet50 model with random crop DA. That model is pre-trained on Imagenet dataset (source) with varying value of $\\tau$ (random crop lower bound) and then, the trained model is transferred to the INaturalist dataset [Van Horn et al., 2018] (target) that consists of 10,000 classes. When transferring the model to INaturalist, the parameters are kept frozen, and only a linear classifier is trained on top of it. We report in Fig. 4 the performance of the trained models with varying $\\tau$ on different INaturalist classes. We observe once again that the best resnet50 —on average— is not necessarily the one that should be deployed as there exists a strong per-class bias that varies with $\\tau$ . As a result, picking the best performing model from a source dataset to a target dataset, might leave the pipeline to perform poorly since that model might also be the one that is the most biased against the class of interest in the target dataset. ",
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"text": "This result should motivate the design of novel regularizers that do not reduce performances between classes at different regimes. Additionally, due to the cost of training multiple models with varying regularization settings, one might wonder on the possible alternative solutions to detect trends such as shown in Fig. 4 only when given a single pre-trained model. ",
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"text": "3 Understanding Why and When Regularization Produces Models With Class-Dependent Preferences ",
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"text": "The first part of our study (Section 2) empirically validated that DNN regularization produces unfair model complexity control over different classes, resulting in a model performing poorly on a few of the classes although being highly performing on average. We now provide in Sections 3.1 and 3.2 some intuition on why DA can be a source of bias regardless of the task, dataset and model at hand. Then, Section 3.3 reviews existing works trying to confront regularization and model bias, and as we will see, an out-of-the-box solution does not seem to exist when there are only a few classes suffering from regularization (Section 3.4). ",
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"text": "3.1 A Data-Augmentation Policy That is Not Label-Preserving For All Classes Will Create Class-Imbalance Performances ",
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"text": "To provide a simple explanation on how DA causes bias in a trained model, we propose the following derivation that holds for any signal e.g. timeseries, images, videos. Without loss of generality, we will consider here the $\\ell _ { 2 }$ loss which was shown to perform as well as the cross-entropy even on Imagenet [Hui and Belkin, 2020]. ",
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"text": "Dataset notations. Given a sample $\\textbf { \\em x } \\in { \\mathcal { X } }$ with $\\boldsymbol { \\mathcal { X } } \\subset \\mathbb { R } ^ { D }$ , we consider ${ \\pmb y } \\triangleq f ^ { * } ( { \\pmb x } )$ to be the ground-truth target value. Hence our hope is to learn an approximator $f _ { \\theta }$ that is as close as possible to $f ^ { * }$ everywhere in $\\mathcal { X }$ , although we only observe a finite training set $\\mathbb { X } \\triangleq \\{ ( \\pmb { x } _ { 1 } , \\pmb { y } _ { 1 } ) , \\dots , ( \\pmb { x } _ { N } , \\pmb { y } _ { N } ) \\}$ . Given an output vector $\\textbf { \\em u }$ we also define the level-set of a mapping $f$ to be $\\{ { \\pmb x } \\in { \\mathcal { X } } : f ( { \\pmb x } ) = { \\pmb u } \\}$ . ",
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"text": "Data-Augmentation notations. Additionally, one employs a DA policy $\\mathcal { T } : \\mathbb { R } ^ { D } \\times \\mathcal { K } \\mapsto \\mathbb { R } ^ { D }$ such that given a transformation parameter $\\alpha \\in { \\mathcal { K } }$ , ${ \\mathcal T } _ { \\alpha } ( { \\pmb x } )$ produces the transformed view of $_ { \\textbf { \\em x } }$ . Often, one also defines a density $p$ on $\\kappa$ that helps in sampling transformation parameters that are a priori known to be the most useful. ",
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"text": "Theorem 1. Whenever the transformations produced by $\\mathcal { T } _ { \\alpha } , \\forall \\alpha$ do not respect the level-set of $f ^ { * }$ , and whenever the model has enough capacity to minimize the training loss, the DA will create irreducible bias in $f _ { \\theta }$ as in ",
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"type": "image",
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"img_path": "images/2ef5e8d9a430737307b40dbef9e6c37e868afc1ea40042ca3717c1fd319a031f.jpg",
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"text": "The main idea of the proof, provided in Appendix $\\mathbf { B }$ , is to show that if a transformation does not move samples on the level-set of the true function (left-hand-side of Eq. (1)), then $f _ { \\theta }$ will learn a different level-set (since it has 0 training error), and thus $\\| f ^ { * } - f _ { \\theta } \\| > 0$ i.e. $f _ { \\theta }$ is biased. ",
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"text": "Whenever the left-hand-side of Eq. (1) is 0, the DA is denoted as label-preserving [Cui et al., 2015, Taylor and Nitschke, 2018]. From the above, we see that unless the target y associated to ${ \\mathcal T } _ { \\alpha } ( { \\pmb x } )$ is modified accordingly to encode the shift in the target function level-set produced by $\\mathcal { T } _ { \\alpha }$ , any $D A$ that is not label-preserving will introduce a bias. Some DAs propose to incorporate label transformation i.e. not only $_ { \\textbf { \\em x } }$ but also $\\textbf { { y } }$ is augmented to better inform on the uncertainty that has been added into $\\mathcal { T } _ { \\boldsymbol { \\theta } } ( \\mathbf { \\mathscr { x } } )$ . This is for example the case for MixUp [Zhang et al., 2017], ManifoldMixUp [Verma et al., 2019], CutMix [Yun et al., 2019] and their extensions. ",
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"text": "Our goal in the next Section 3.2 is to demonstrate how DAs such as random crop, color jittering, or CutOut are only label preserving for some values of $\\alpha$ that vary with the sample class. As a consequence, while the use of the DA improves the average test performance, it is at the cost of a significant reduction in performance for some of the classes. ",
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"text": "3.2 The Same Data-Augmentation can be Label-Preserving or Not Between Different Classes ",
|
| 509 |
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"text": "In the previous Section 3.1 we provided a general argument on the sufficient conditions for DA to produce a biased model. We hope in this section to provide a more concrete example that applies to current DNN training. To that end, we will demonstrate that a DA can be label-preserving or not depending on the sample’s class, hence, since the same DA policy is employed for all classes, the augmented dataset will exhibit a class-imbalance in favor of the classes for which the DA is most label-preserving. ",
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"text": "To measure by how much a given DA, $\\mathcal { T } _ { \\alpha }$ , is label-preserving, we propose to take 6 popular architectures that are pre-trained on Imagenet [Deng et al., 2009] from the official PyTorch [Paszke et al., 2019] repository, and to evaluate their performances for varying DA settings (top of Fig. 5). We observe that considering the dataset as a whole is not a good indicator of the optimal DA value to employ since per-class accuracy performance (bottom of Fig. 5) vary drastically. For example, for some classes, any level of transformation $\\alpha$ can produce augmented samples with enough information to be correctly classified, while for other classes, even a small amount of DA makes the samples unpredictable. ",
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"text": "To further ensure that the observed relation between label-preservation, sample class, and amount of transformation $\\alpha$ is sound, we provide in Fig. 6 the per-class test accuracy on different models, all exhibit the same trends. In short, we identify that when creating an augmented dataset by applying the same DA across classes, the number of per-class samples that actually contain enough information about their true labels will become largely imbalance between classes, even if the original dataset was balanced. Any model trained on the augmented dataset will thus focus on the classes for which the DA is the most label-preserving. ",
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"img_path": "images/9be1a27a21ea55563825551ac6014351c52449c918b05b84800756d2bc783a05.jpg",
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"image_caption": [
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| 555 |
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"Figure 5: Top: examples of an augmented image of class “bird”. Middle: average accuracy (train set in dashed line and test set in plain) on Imagenet, using 6 popular architectures. Bottom: per-class performances from the middle scenario along with 9 images of the corresponding classes. We observe that the random crop DA seems to loose its label-preserving property on average when less than $30 \\%$ of the image is kept in the crop but looking at the per-class performance, we observe that such DA can be label-preserving with only $8 \\%$ of the original image for some classes, while for other classes the label information starts to reduce at around $50 \\%$ . Results obtained from the official Imagenet pre-trained PyTorch models. CutOut and ColorJitter cases are provided in Figs. 14 and 15 and exhibit the same trend. "
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"image_caption": [
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| 570 |
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"Figure 6: Reprise of the bottom left of Fig. 5 for three different DAs (each column) and using the same 6 popular architectures (different lines). We observe that across DAs, different architectures agree on the label-preserving regimes for $\\mathcal { T } _ { \\alpha }$ i.e. even an ensemble of model would not reduce the class-dependent bias of the final prediction. Results obtained from the official Imagenet pre-trained PyTorch models. "
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"text": "",
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"text": "Beyond the above intuitive and natural understand we obtained in term of data-augmentation, there exists theoretical studies that we propose to summarize below ",
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"text": "3.3 Do Existing Studies Provide Answers Into the Inter-Play Between Regularization and Class-Dependent Bias? ",
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"text": "We explored in Section 3.2 a possible explanation of the class-dependent bias we observed in Section 2 hinting at the need to use class-dependent DA. We propose here to summarize existing studies that are attempting to better understand the impact of DA and regularization in general onto a model’s performance and bias. Then, Section 3.4 will explore their application. ",
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"image_caption": [
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"Figure 7: Reprise of Fig. 2 but now implementing the class-dependent DA as prescribed in Section 3.3 i.e. we only apply DA (random crop) to the classes that see their per-class accuracy improve when this DA is employed during training.. We observe here that the bias of the model introduced from random crop on the majority of the classes spills-over to the minority of the classes that do not benefit from that DA, even though such classes never received that DA during training. Results are averaged over 5 runs and employ the official resnet50 implementation trained on Imagenet with horizontal flip but a per-class random crop DA. "
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"text": "Input independent DA exacerbates the bias already present in a dataset. Especially relevant to our results is a recent result of Xu et al. [2020]. In this work, it was theorized that when the underlying dataset contains inherent biases, training on the original data is more effective than employing an i.i.d. DA policy, i.e. applying the same random augmentation to all samples/classes, to produce an unbiased model. In short, the DA exacerbates the already present biases and makes the trained model further away from the unbiased optimum. The difficulty of this result lies in defining bias for real images. As per our experiments from Section 3, we observe that bias can take many form e.g. one class might naturally represent its object always under the same angle. This is particularly true say for boats which are rarely captured upside-down. Hence, simply having classes with different natural statistics could be enough for Xu et al. [2020] to prohibit the use of DA in current datasets. Furthermore, Raghunathan et al. [2020] obtained a surprising result combining both DA and regularization. In that case, it was found that the minimum norm interpolant on the original $+ \\textrm Ḋ \\textmu Ḋ Ḍ Ḍ _ { \\mathrm { Ḋ } } \\textrm Ḋ \\textmu Ḍ Ḍ$ dataset could have a larger standard error than the minimum norm interpolant on the original dataset alone, even when using label-preserving DA. However, this phenomenon only occurs as long as the model remains over-parametrized, even when considering the original $+ \\mathrm { D A }$ dataset. ",
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| 655 |
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"text": "Input dependent DA can reduce sample/class bias. Recall that Section 3.2 brought forward one possible explanation on how DA can be the source of training-set by introduce class-imbalance due to the same DA being label-preserving for some classes and not for others. From this, a direct solution would be to adapt the “strength”. This solution, formalized in Xu et al. [2020], consists in measuring the bias of a model and adapt the DA policy accordingly to correct it. This has been done in different flavors e.g. in McLaughlin et al. [2015], Iosifidis and Ntoutsi [2018], Jaipuria et al. [2020]. One limitation of this direction is that it requires to estimate a model bias and adapt the DA accordingly which can be challenging for large scale models. ",
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"type": "text",
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| 676 |
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"text": "Learned DA e.g. from a GAN can produce even more class-dependent bias. One natural extension of the hands-on adaptivity of a DA to a measured model bias would be to learn a DA to maximize a model’s performance, for example. This line of work has led to many learn DA policies e.g. using Generative Adversarial Networks [Hu and Li, 2019]. Yet, it has been shown that learning a DA to maximize some aggregated measure of performance will produce even more bias in a model [Hu and Li, 2019]. In fact, and as per the controlled experiments from Section 2, the learned DA will entirely disregard a minority of the classes if it means that the produced DA can drastically improve performances on all others, effectively maximizing the average performances. ",
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| 687 |
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"text": "Inherent tradeoff between DA and model robustness. In addition to the implication of DA into bias, there exists an intertwined relationship between DA and model robustness. In fact, even assuming the use of perfectly adapted DAs, there exists an inherent tradeoff between accuracy and robustness that holds even in the infinite data limit [Tsipras et al., 2018, Fawzi et al., 2018, Zhang et al., 2019]. For example, Min et al. [2021] proved in the robust linear classification regime that (i) more data improves generalization in a weak adversary regime, (ii) more data can improve generalization up to a point where additional data starts to hurt generalization in a medium adversary regime, and that (iii) more data immediately decreases generalization error in a strong adversary regime. ",
|
| 688 |
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|
| 694 |
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"page_idx": 7
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| 695 |
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|
| 696 |
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{
|
| 697 |
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"type": "text",
|
| 698 |
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"text": "3.4 Class-Dependent Data-Augmentation Seems Insufficient for Performance Recovery ",
|
| 699 |
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"text_level": 1,
|
| 700 |
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| 701 |
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| 707 |
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| 708 |
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{
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| 709 |
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"type": "text",
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| 710 |
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"text": "We observed in Section 2 that DA could lead to disastrous per-class performances on a minority of classes. We now propose to implement one solution from Section 3.3 that consists in simply not applying the harmful DA to the classes suffering from it. As will become clear, applying the DA on all other classes will be enough to skew the training of the model preventing any performing recovery on those classes. ",
|
| 711 |
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"bbox": [
|
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|
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| 719 |
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| 720 |
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"type": "text",
|
| 721 |
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"text": "To motivate the need for a better control of the per-class performance of a model, we first present an illustrative argument. A standard resnet50 on Imagenet reaches $7 7 . 1 4 \\%$ top-1 with a random crop lower bound of $8 \\%$ . Using precise cross-validation, one could reach 77.29 by using a random crop lower bound of $1 0 \\%$ instead of $8 \\%$ on all classes. Yet, and most interestingly, if one were able to get the best per-class performance —as per varying the lower-bound as in Fig. 2 and picking for each class the best per-class performance of any of the models— one could reach 79.37. Beyond pure average test performance, controlling the worst-case per-class performance is of crucial importance for fairness [Du et al., 2020, Veitch et al., 2021]. We thus explore one of the solution that we reviewed in Section 3.3 that consists in only applying the random crop DA to the classes whose test performances increased with the DA’s level. We obtain in Fig. 7 that such class-specific strategy is not sufficient to guarantee that the classes negatively impacted by the random crop DA see their performance to be constant across the DA level applied to all the other classes. This finding is also supported by our weight decay experiment in Figs. 3 and 16 in which the regularization of the DN impacted classes differently. In fact, first notice that applying weight decay only when seeing some specific classes would simply amount (on average) to reducing the weight decay hyper-parameter proportionally to how many classes are considered to be without regularization. In a similar fashion, DA produces an implicit regularizer [Balestriero et al., 2022] and applying a class-specific DA level reduces the impact of the implicit regularizer. But since the amount of classes for which we do not apply random crop is quite small compared to the total number of classes (between $1 \\%$ and $5 \\%$ ), it means that the DA’s implicit regularizer remains nearly the same and thus the model’s bias is nearly the same regardless if that DA is applied or not onto those classes. This is what we observe, applying the random crop DA to the classes that benefit from it is enough to bias the model and degrade the performances on some classes at the same pace than when applying the DA to all classes unconditionally (compare Figs. 2 and 7). ",
|
| 722 |
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"bbox": [
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| 730 |
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{
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| 731 |
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"type": "text",
|
| 732 |
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"text": "As a result, we observe that no readily and easily implemented solution provides us with a strategy to prevent deep learning to fall into the scenario depicted on the left of Fig. 1. Those findings however motivate the search of novel model complexity controls that is fair among classes. From a more theoretical viewpoint, it might also be possible to better understand if even such a fair per-class model complexity could exist, which is not clear as natural image classes tend to have inherently different statistics. ",
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| 733 |
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| 741 |
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{
|
| 742 |
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"type": "text",
|
| 743 |
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"text": "4 Conclusions and Limitations ",
|
| 744 |
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"text_level": 1,
|
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| 753 |
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| 754 |
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"type": "text",
|
| 755 |
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"text": "We proposed in this study to understand the impact of regularization, in particular data-augmentation and weight decay, into the final performances of a deep network. We obtained that the use of regularization increases the average test performances at the cost of significant performance drops on some specific classes. By focusing on maximizing aggregate performance statistics we have produced learning mechanisms that can be potentially harmful, especially in transfer learning tasks. In fact, we have also observed that varying the amount of regularization employed during pre-training of a specific dataset impacts the per-class performances of that pre-trained model on different downstream tasks e.g. going from Imagenet to INaturalist. Lastly, commonly prescribed solutions e.g. classdependent data-augmentation do not seem to help indicating that the sole use of an augmentation on some classes is enough to bias the model on all classes. Hence, there remains a vast research area to explore in order to turn deep learning model selection from the current regime to a more ideal one (left to right of Fig. 1). ",
|
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"page_idx": 8
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| 764 |
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| 765 |
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"type": "text",
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| 766 |
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"text": "The main limitation of this work is its focus on computer vision datasets and models. It is possible that our observation will be further conveyed in other regimes or not, and we leave such analysis for future work. ",
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| 767 |
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"bbox": [
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"text": "Sinno Jialin Pan and Qiang Yang. A survey on transfer learning. IEEE Transactions on knowledge and data engineering, 22(10):1345–1359, 2009. \nAdam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019. \nEkachai Phaisangittisagul. An analysis of the regularization between l2 and dropout in single hidden layer neural network. In 2016 7th International Conference on Intelligent Systems, Modelling and Simulation (ISMS), pages 174–179. IEEE, 2016. \nAditi Raghunathan, Sang Michael Xie, Fanny Yang, John Duchi, and Percy Liang. Understanding and mitigating the tradeoff between robustness and accuracy. arXiv preprint arXiv:2002.10716, 2020. \nConnor Shorten and Taghi M Khoshgoftaar. A survey on image data augmentation for deep learning. Journal of big data, 6(1):1–48, 2019. \nPatrice Simard, Bernard Victorri, Yann LeCun, and John Denker. Tangent prop-a formalism for specifying selected invariances in an adaptive network. Advances in neural information processing systems, 4, 1991. \nMingxing Tan and Quoc Le. Efficientnetv2: Smaller models and faster training. In International Conference on Machine Learning, pages 10096–10106. PMLR, 2021. \nLuke Taylor and Geoff Nitschke. Improving deep learning with generic data augmentation. In 2018 IEEE Symposium Series on Computational Intelligence (SSCI), pages 1542–1547. IEEE, 2018. \nAndrei Nikolajevits Tihonov. Solution of incorrectly formulated problems and the regularization method. Soviet Math., 4:1035–1038, 1963. \nAndrey Nikolayevich Tikhonov. On the stability of inverse problems. In Dokl. Akad. Nauk SSSR, volume 39, pages 195–198, 1943. \nDimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and Aleksander Madry. Robustness may be at odds with accuracy. arXiv preprint arXiv:1805.12152, 2018. \nGiorgio Valentini and Thomas G Dietterich. Bias-variance analysis of support vector machines for the development of svm-based ensemble methods. Journal of Machine Learning Research, 5(Jul):725–775, 2004. \nGrant Van Horn, Oisin Mac Aodha, Yang Song, Yin Cui, Chen Sun, Alex Shepard, Hartwig Adam, Pietro Perona, and Serge Belongie. The inaturalist species classification and detection dataset. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8769–8778, 2018. \nVladimir N Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. In Measures of complexity, pages 11–30. Springer, 2015. \nVN Vapnik and A Ya Chervonenkis. The method of ordered risk minimization, i. Avtomatika i Telemekhanika, 8:21–30, 1974. \nVictor Veitch, Alexander D’Amour, Steve Yadlowsky, and Jacob Eisenstein. Counterfactual invariance to spurious correlations in text classification. Advances in Neural Information Processing Systems, 34:16196–16208, 2021. \nVikas Verma, Alex Lamb, Christopher Beckham, Amir Najafi, Ioannis Mitliagkas, David Lopez-Paz, and Yoshua Bengio. Manifold mixup: Better representations by interpolating hidden states. In International Conference on Machine Learning, pages 6438–6447. PMLR, 2019. \nUlrike Von Luxburg and Bernhard Schölkopf. Statistical learning theory: Models, concepts, and results. In Handbook of the History of Logic, volume 10, pages 651–706. Elsevier, 2011. \nBernard L Welch. The generalization of ‘student’s’problem when several different population varlances are involved. Biometrika, 34(1-2):28–35, 1947. \nQizhe Xie, Zihang Dai, Eduard Hovy, Thang Luong, and Quoc Le. Unsupervised data augmentation for consistency training. Advances in Neural Information Processing Systems, 33:6256–6268, 2020. \nYi Xu, Asaf Noy, Ming Lin, Qi Qian, Hao Li, and Rong Jin. Wemix: How to better utilize data augmentation. arXiv preprint arXiv:2010.01267, 2020. ",
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"type": "text",
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"text": "Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE/CVF international conference on computer vision, pages 6023–6032, 2019. ",
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"text": "Hongyang Zhang, Yaodong Yu, Jiantao Jiao, Eric Xing, Laurent El Ghaoui, and Michael Jordan. Theoretically principled trade-off between robustness and accuracy. In International conference on machine learning, pages 7472–7482. PMLR, 2019. ",
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| 1040 |
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| 1041 |
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"text": "Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017. ",
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"text": "Checklist ",
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"text": "1. For all authors... ",
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"text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [N/A] We do not propose a novel method that requires such discussion \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We carefully described why the observations we have made can be dangerous for real world applications \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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"text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] Our only “theoretical result” consists in a formal statistical test for which we precisely describe our settings and statistics \n(b) Did you include complete proofs of all theoretical results? [N/A] No theoretical result requiring proofs was provided ",
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"text": "3. If you ran experiments... ",
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"text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include summary statistics in the supplementary material which is enough to validate our claims. The full codebase and all the saved models will be released upon completion of the review process (this includes almost a thousand pre-trained resnet50s) \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We specify within each caption the key experimental setups, additional training details are provided in the appendix \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] we averaged all of our runs over a significant amount of realizations (20) and perform a formal statistical test on the significance of our results \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We provide this in the beginning of the appendix. Note that we will be providing the saved models and summary statistics to make the results easily reproducible even without GPU or computation ressources. ",
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"text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
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"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] For Imagenet and INaturalist \n(b) Did you mention the license of the assets? [Yes] the full code will be released with a NC by NC license on GitHub \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] we include summary result files that contain all the statistics to reproduce the main claims of the paper in the supplementary material. The full codebase along with pre-trained models will be released upon completion of the review process. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We are only using the Imagenet and INaturalist datasets ",
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"text": "(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We are only using the Imagenet and INaturalist datasets ",
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"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
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| 1173 |
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| 1174 |
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"text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
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| 1 |
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# SYNCHROMESH: RELIABLE CODE GENERATION FROM PRE-TRAINED LANGUAGE MODELS
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Gabriel Poesia∗† Stanford University poesia@stanford.edu
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Oleksandr Polozov∗‡ X, the moonshot factory polozov@google.com
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Vu Le, Ashish Tiwari, Gustavo Soares, Christopher Meek, Sumit Gulwani Microsoft Research, Redmond {levu,astiwar,gustavo.soares,meek,sumitg}@microsoft.com
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# ABSTRACT
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Large pre-trained language models have been used to generate code, providing a flexible interface for synthesizing programs from natural language specifications. However, they often violate syntactic and semantic rules of their output language, limiting their practical usability. In this paper, we propose SYNCHROMESH: a framework for substantially improving the reliability of pre-trained models for code generation. SYNCHROMESH comprises two components. First, it retrieves few-shot examples from a training bank using Target Similarity Tuning (TST), a novel method for semantic example selection. TST learns to recognize utterances that describe similar target programs despite differences in surface natural language features. Then, SYNCHROMESH feeds the examples to a pre-trained language model and samples programs using Constrained Semantic Decoding (CSD): a general framework for constraining the output to a set of valid programs in the target language. CSD leverages constraints on partial outputs to sample complete correct programs, and needs neither re-training nor fine-tuning of the language model. We evaluate our methods by synthesizing code from natural language descriptions using GPT-3 and Codex in three real-world languages: SQL queries, Vega-Lite visualizations and SMCalFlow programs. These domains showcase rich constraints that CSD is able to enforce, including syntax, scope, typing rules, and contextual logic. We observe substantial complementary gains from CSD and TST in prediction accuracy and in effectively preventing run-time errors.
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# 1 INTRODUCTION
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Large language models (LLMs) trained on massive corpora of unsupervised data have been shown to perform a wide range of tasks, including natural language generation, semantic parsing and sentiment analysis (Brown et al., 2020; Devlin et al., 2019; Raffel et al., 2020). This can be achieved without task-specific training, but rather by adapting the model to each task at test-time using textual prompts, which can contain examples and natural language descriptions. In many cases, this methodology was shown to provide good performance, reducing the need to annotate large datasets for each task of interest (Brown et al., 2020; Shin et al., 2021).
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An important application of LLMs is in synthesizing programs from natural language descriptions (Austin et al., 2021; Chen et al., 2021). But this task is still challenging for LLMs. First, they can commit conceptual errors, generating code that misses the intent behind the given description. For example, when asked to reverse an array, the model might generate code that simply swaps the first and last elements. Indeed, users of current natural language-to-code systems report that models often produce code that is unrelated to their query (Xu et al., 2021).
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Figure 1: Overview of the SYNCHROMESH framework. Given the user’s query, high-relevance examples are first retrieved with Target Similarity Tuning (TST). Then, a program is incrementally sampled via Constrained Semantic Decoding (CSD), which queries a completion engine (CE) to enforce constraints during code generation without re-training or fine-tuning the language model.
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Even when they capture the right intent, LLMs can still make implementation errors: the generated code can fail to execute. For reversing an array, a model might generate a loop with the correct structure but with an off-by-one error, causing a runtime exception. These errors are common even with very large models. For example, Austin et al. (2021) tested models with up to 137B parameters on generating short Python programs from natural language. Still, $47 \%$ of the failures were due to syntax, typing or run-time errors (as opposed to running but producing incorrect output). This is in line with theoretical results in Merrill et al. (2021) showing that programming language semantics cannot be fully inferred from ungrounded data. Together, both observations suggest that simply scaling up LLMs might be ineffective to obtain reliable performance, especially for longer programs.
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In this paper, we address both conceptual and implementation errors with SYNCHROMESH, a framework for reliable code generation from pre-trained models. Since LLMs are highly sensitive to which few-shot examples are given in their prompt, we propose Target Similarity Tuning (TST): a method for dynamically selecting semantically relevant examples for a given description. TST mitigates conceptual errors by learning to select examples with similar intent, even when their natural language descriptions seem unrelated in form. Given relevant examples, we then generate programs with Constrained Semantic Decoding (CSD), a novel method for enforcing rich syntactic and semantic constraints during code generation on top of a frozen language model. Rich language-specific constraints, ranging from syntax validity to scoping and type-checking, can be implemented under the simple abstraction of completion engines $( C E )$ . CSD aligns these constraints with the language model’s token vocabulary by leveraging Brzozowski language derivatives (Brzozowski, 1964). This guarantees that all sampled programs satisfies the implemented constraints, preventing whole classes of implementation errors by construction. The pipeline is illustrated in Figure 1.
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We demonstrate the generality of SYNCHROMESH in three real-world languages: SQL (database queries), Vega-Lite (data visualization) and SMCalFlow (calendar applications). In experiments with GPT-3 and Codex, we observe that SYNCHROMESH can eliminate whole classes of errors that make outputs from unconstrained models either fail to execute or produce trivial results (e.g., empty charts). Furthermore, eliminating invalid programs consistently improves prediction accuracy. In summary, we make the following contributions:
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• We propose Target Similarity Tuning for selecting few-shot examples based on the similarity of the programs they describe, improving relevance and downstream performance. • We introduce completion engines as an abstraction that can implement rich classes of semantic program constraints, as we demonstrate in SQL, Vega-Lite and SMCalFlow. • We introduce a general, constraint-observing decoding algorithm, which aligns programming language constraints with the language model’s token vocabulary. • We evaluate our method in three natural language-to-code tasks. CSD and TST both show strong complementary gains in output validity and prediction accuracy across domains.
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# 2 TARGET SIMILARITY TUNING
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In this section, we first overview the challenge posed by conceptual errors in programs synthesized by LLMs. We then introduce TST, which improves performance through more relevant example selection. Throughout, we will use a real example of synthesizing a SQL database query to answer a question posed in natural language.
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Figure 2: Example of Target Similarity Tuning improving example selection for synthesizing a SQL query. In (a), the prompt example missed the key query structure (grouping and counting). With this example, GPT-3 generates an invalid query (b). With TST, we retrieve a relevant example which GPT-3 successfully adapts to answer the user’s question (c).
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Suppose a data analyst has a relational database of airports and wants to answer the following question: “Which city has the highest number of airports?” One procedure for turning this description into a SQL query is to use an LLM such as GPT-3 (Brown et al., 2020) or Codex (Chen et al., 2021). To prompt the model for the task at hand, we would feed it with a natural language description of the task and a selection of input-output examples.
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Given the analyst’s question, how do we select the most relevant examples from a training pool? Liu et al. (2021a) proposed to retrieve examples with similar natural language descriptions using a pre-trained paraphrase detection model. Figure 2a shows the most similar example from the Spider natural language-to-SQL dataset (Yu et al., 2018) according to Sentence-BERT (Reimers & Gurevych, 2019). The query “Which city has the highest elevation?” is similar on a surface level: it also asks “Which city has the highest ?”. This training query asks about “elevation”, a property that is readily available as a column in the Airports table. Figure 2b shows GPT-3’s output when given this and a few other examples. The model attempts to mimic the top example, referring to a nonexistent column “NumberOfAirports”. The issue is that we picked the example in the prompt based on description similarity and not SQL query similarity. In fact, the SQL query in the chosen example had a simplistic structure that was significantly different from the structure of the desired SQL query, and this contributed to the failure at Point (b) in Figure 2.
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We want to retrieve examples that have relevant program structures for the test query. We do so using our fine-tuning scheme called Target Similarity Tuning (TST). Formally, suppose $\mathcal { D }$ is a dataset of programs and associated utterances, with $\mathcal { D } _ { i } = ( p _ { i } , u _ { i } )$ . Let $S ( p _ { a } , p _ { b } ) \bar { \in } [ 0 , \bar { 1 } ]$ denote a normalized similarity metric between programs. If $f _ { \theta }$ is a pre-trained similarity model for natural language sentences, TST consists in fine-tuning $f$ to predict the similarity between target programs given by $S$ from their descriptions. Precisely, we minimize the mean-squared error loss:
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$$
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\mathcal { L } _ { T S T } ( \theta ) : = \mathbb { E } _ { i , j \sim \mathcal { D } } \left[ f _ { \theta } ( u _ { i } , u _ { j } ) - S ( p _ { i } , p _ { j } ) \right] ^ { 2 } \ .
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$$
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We define $S$ using the classical tree edit distance algorithm from Zhang & Shasha (1989) to compare Abstract Syntax Trees (ASTs). Figure $2 \mathrm { c }$ shows GPT-3’s output when given examples selected with TST. Now, the output query is correct: it performs a “group by” on the “City” column, and sorts by the count of records in each group. This structure was already present in the top example selected by TST, corresponding to “Return the team with the most technicians”. Even if the analyst’s question and this utterance are drastically different in natural language, they share similarity in the SQL query that they describe. The TST objective is able to properly capture this fact. As our experiments show in Section 4, TST significantly boosts the performance of both GPT-3 and Codex.
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# 3 CONSTRAINED SEMANTIC DECODING
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We now present Constrained Semantic Decoding (CSD) as an approach to eliminate implementation errors from code generated by LLMs. We first illustrate CSD with an example, and then formalize it using the abstraction of CEs.
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Figure 3: Example on CSD generating a SQL query. Given the prompt, GPT-3 makes a mistake (a) when generating the JOIN condition. CSD is able to prevent this error by (b) keeping track of table aliases and constraining the model to respect the database schema.
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The example in Figure 2 showed that TST can help LLMs generate the correct program. In general, however, TST only helps LLMs by guiding toward the correct structure, but the model still needs to fill all the specific implementation details correctly. Figure 3 shows a case where the model cannot simply adapt one example from the prompt. Here, the user’s query is “Which city has the highest number of departing flights?” This query is similar to the previous one – in fact, TST retrieves the same top-1 example as before. But now the correct SQL query needs to join the “Airports” and “Flights” tables. GPT-3 generates the join condition Flights.AirportCode $=$ Airports.SourceAirport, but this condition has a subtle error: the column names of the two tables are swapped. Thus, this query fails to execute. In general, unconstrained language models often make such implementation errors: using undeclared variables, losing track of nesting levels when producing complex expressions, or calling functions using arguments of the wrong type. Even the smallest of such errors prevents generated code from executing.
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CSD prevents implementation errors by construction (as opposed to repairing after-the-fact). Imagine we have access to an oracle, which we call a $C E$ , that can take a partial program and return all tokens that can extend that partial program toward a complete correct program. When the LLM is generating the program token by token, CSD ensures that the next token is sampled from the set returned by the CE.
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In Figure 3, after generating “T1.” inside the “on” clause, our SQL CE resolves the alias and constrains the model to output one of the columns from the “Flights” table. This fixes the error seen previously during generation and produces the correct SQL query.
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# 3.1 COMPLETION ENGINES
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We now formally define CEs. Let $\Sigma$ be a base alphabet, and $\Sigma _ { L } \subseteq \Sigma ^ { * }$ be the (potentially infinite) set of tokens of the target language. Our goal is to sample programs from a language $L \subseteq \Sigma _ { L } ^ { * } -$ the set of valid programs. A CE $C _ { L }$ is a partial function from $\Sigma _ { L } ^ { * }$ to a set of tokens. We use a regular expression over $\Sigma$ to represent a set of tokens. The strings in the domain of $C _ { L }$ are called completion points, and a CE satisfies the following axioms: (A1) The empty string and every $p \in L$ must be completion points. For every $p \in L$ , $\bar { C _ { L } } ( p ) = r ^ { \prime } \ S ^ { \prime }$ , where $r ^ { \prime } \bar { \mathfrak { F ^ { \prime } } }$ is the regular expression that matches the stop token. (A2) If $s \in \Sigma _ { L } ^ { * }$ is a completion point and $t$ fully matches $C _ { L } ( s )$ , then their concatenation $^ { s t }$ must also be a completion point. (A3) The CE is exhaustive; that is, if $s$ is a completion point and $s = t t _ { 0 }$ , where $t _ { 0 }$ is a token, then $t$ should be a completion point and $C _ { L } ( t )$ should match $t _ { 0 }$ . Furthermore, we assume that CEs are only called after maximal matches. For example, if a partial program ends in an identifier, the CE can assume that the identifier is complete.
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Our CEs are implemented in two layers: a context-free layer, which enforces syntactic validity, and a context-sensitive layer, which encodes semantic constraints that depend on language semantics and the user’s context (e.g., the database). Below, we describe an automatic method for constructing context-free CEs directly from the target language’s grammar. The context-sensitive layer of an engine is specific to the target language. Table 1 provides an overview of several constraints implemented by our CEs for SQL, Vega-Lite and SMCalFlow, three rich languages with different syntactic and semantic structures. A detailed description of the three CEs can be found in Appendix C.
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Table 1: Examples of constraints implemented in our CEs for SQL, Vega-Lite and SMCalFlow. Given a partial program, CEs return a regular expression that matches the valid tokens that can follow. Here, we show positive and negative token examples for each such regular expression. This abstraction allows domain experts to encode a wide range of expressive code generation constraints.
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<table><tr><td>Language</td><td>Constraint</td><td>Example of partial program</td><td>Valid/Invalid Examples</td></tr><tr><td>SQL</td><td>A valid identifier must follow after AS.</td><td>SELECT Name， Role FROM User AS^</td><td>U √ T1 √ 2×</td></tr><tr><td></td><td>Column names must come from schema，even behind aliases.</td><td>SELECT U.Name FROM User AS U WHERE U. ^</td><td>Name√ DoB√ Birthday X</td></tr><tr><td>Vega-Lite</td><td>Data fields must be used with compatible types. Do not facet on field with too many distinct values (breaks</td><td>{"x": {"field": "Category", "type":^ {"column":{"field": ^</td><td>"nominal" "temporal" X "Category" "ZipCode" </td></tr><tr><td>SMCalFlow</td><td>rendering). Type-check parameters of all</td><td>(Yield</td><td>Takeout√</td></tr><tr><td></td><td>API functions. Track declared variables and</td><td>(PlaceHasFeature(> (let (x 85)</td><td>IsWindy X List.Apply X x√</td></tr></table>
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Deriving completions from grammars Computer language parsers are often automatically generated from a grammar. The grammar contains enough information to derive the context-free layer of CEs. To facilitate this process, we created a library that extends any parser generated by ANTLR (Parr & Fisher, 2011), a popular $\operatorname { L L } ( ^ { * } )$ top-down parser generator, to provide token-level completions. Namely, we (i) let the ANTLR-generated parser process the given program prefix $p$ , (ii) retrieve its state in the Augmented Transition Network (ATN) at the last program token, (iii) traverse the ATN from that state to enumerate all possible next token productions. This process yields (a) a list of productions and token types $\{ \tau _ { j } \} _ { j = 1 } ^ { K }$ that are allowed to follow $p$ and $\mathbf { ( b ) }$ a partial AST $T _ { p }$ . Each CE takes $\{ \tau _ { j } \}$ and $T _ { p }$ as input to generate semantic context-sensitive constraints.
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# 3.2 FROM COMPLETION ENGINES TO A DECISION PROCEDURE
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We use CEs to guide sampling from an LLM. A key component of our algorithm for constrained sampling is a decision procedure for membership in prefix-closure of the set $L$ of all valid programs. The prefix-closure $L ^ { c }$ of a language $L$ contains all programs in $L$ as well as all of their prefixes. Intuitively, $L ^ { c }$ contains all partial programs that can be completed to a valid program. Given a CE $C _ { L }$ , our first goal is to build a decision procedure for $L ^ { c }$ : given a string $s$ , does it belong to $L ^ { c } ?$
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We answer if $s \in L ^ { c }$ by repeatedly calling $C _ { L }$ on certain prefixes $p$ of $s$ and matching the regular expression $C _ { L } ( \boldsymbol { p } )$ with suffixes of $s$ . We start with $p$ being the empty string. We find the maximal prefix of $s$ that matches the regular expression $C _ { L } ( \boldsymbol { p } )$ and remove it from $s$ and add it to $p$ , and repeat until the match fails. There are two cases: either $s$ is empty now, which means the input string was a completion point and hence it is in $L ^ { c }$ , or $s$ now is the remainder left after removing the largest prefix that was a completion point. For the second case, we must check: does there exist a completion string $c$ such that $s c$ fully matches the regular expression $C _ { L } ( \boldsymbol { p } )$ ?
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This question can be efficiently answered by Brzozowski derivatives (Brzozowski, 1964). Formally, the derivative of a formal language $S$ with respect to a string $u$ is another formal language $u ^ { - 1 } S \stackrel { . } { = }$ $\{ v : u v \in S \}$ . In other words, it is precisely the set of strings that can complete $u$ to some string in $S$ . If $u ^ { - 1 } S \stackrel { . } { = } \varnothing$ , then no string in $S$ starts with $u$ . Brzozowski derivatives are efficient to compute for our regular languages (or regular expressions defining them) – we describe a simple linear-time algorithm in the Appendix. Given the derivative of $C _ { L } ( \boldsymbol { p } )$ , answering whether $s$ can be completed to belong to $C _ { L } ( \boldsymbol { p } )$ reduces to performing a simple regular expression match. This operation answers the case when the remainder is non-empty and completes our decision procedure for $L ^ { c }$ .
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# 3.3 THE CONSTRAINED SEMANTIC DECODING ALGORITHM
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Using the decision procedure for $L ^ { c }$ , we can now describe the Constrained Semantic Decoding algorithm. Suppose $s \in L ^ { c }$ is the language model’s output so far (we start with $\epsilon$ ). If $\Sigma _ { M }$ is the model’s vocabulary, we can compute the set of valid next tokens $V _ { M } ( s ) = \{ t \in \Sigma _ { M } : s t \in L ^ { c } \}$ by using our decision procedure for each token in the vocabulary $\Sigma _ { M }$ . In other words, we maintain the invariant that the model’s current partial output $s$ is in $L ^ { c }$ , and make progress by using the model to sample from $V _ { M } ( s )$ , instead of the unconstrained $\Sigma _ { M }$ . Once we have a complete program, we are guaranteed that it will satisfy all constraints enforced by the CE.
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One subtlety to note is that language models and programming languages have drastically different tokenizations; i.e., $C _ { L }$ and LLM work with different tokens. For instance, a long string literal is a single SQL token, but might span multiple tokens for the language model. Similarly, a single token from the language model’s vocabulary might close multiple parentheses at once. In general, token boundaries between the two can be arbitrarily misaligned. Each decision of whether $^ { s t }$ belongs to $L ^ { c }$ can potentially cross multiple completion points, or might not even finish a maximal match to the previous completion point (see the Appendix for an example prediction in Vega-Lite where this happens multiple times). Nevertheless, our CSD algorithm described here naturally handles this alignment problem. Hence, in SYNCHROMESH, CEs do not need to be aware of this issue – they can be fully implemented in terms of the target language’s tokens.1
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Our implementation applies substantial optimizations that leverage the structure of Byte-Pair Encoding vocabularies (namely, that many tokens are prefixes of longer tokens) and reuse computation. We detail these optimizations in Appendix E. In our experiments with GPT-3, CSD adds an average of $8 \%$ overhead to the sampling procedure – a relatively small impact to trade for output correctness.
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# 4 EXPERIMENTS
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We evaluate SYNCHROMESH in three tasks of synthesizing code from natural language descriptions. For SQL, we use the Spider dataset (Yu et al., 2018). For Vega-Lite, we use the NLV Corpus (Srinivasan et al., 2021). For SMCalFlow, we use the dataset that introduced the language (Andreas et al., 2020). In NLV, which has visualizations over 3 different datasets, we alternate using each dataset as a test-set by only using training examples from the other two datasets. In Spider and SMCalFlow, we use the training/validation set split given in each dataset.
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Example selection model To select examples, we use Sentence-BERT (Reimers & Gurevych, 2019) to fetch the 5 closest examples by cosine similarity. When using TST, we fine-tuned the model with the TST objective in both the Spider and SMCalFlow training sets. The NLV corpus is smaller and does not provide a clear train-test split to fairly evaluate TST. Holding out one dataset and fine-tuning on the remaining two yields SYNCHROMESH accuracies of over $90 \%$ . However, we attribute that performance to the fact that NLV has only 10 distinct visualizations and the same participants labeled all three datasets. For that reason, we omit Vega-Lite from the TST experiments.
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Language models We used the two largest models from the GPT-3 family (Brown et al., 2020, with 13B and 175B parameters), as well as the largest Codex model (Chen et al., 2021). Codex shares the same architecture with 175B GPT-3, but its training set contained a larger portion of source code in a variety of languages. Our only access to the models was through the public OpenAI HTTP API, which allowed us to apply constraints by adding a bias to logits. We describe the necessary adaptations of CSD to this setting in Appendix F.
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Metrics For Vega-Lite and SMCalFlow, we report the exact-match accuracy between predictions and ground-truth (field order is disregarded in Vega-Lite). In SQL, we instead measure execution accuracy, comparing query results instead. For a more fine-grained signal, we additionally measure the edit distance between the predicted and ground-truth ASTs using the normalized tree edit distance (Zhang & Shasha, 1989).
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Table 2: Results of each language model on all domains with and without CSD and TST. For SQL, we run the resulting query and report Execution Match accuracy (Exec.). For Vega-Lite and SMCalFlow, we instead report Exact Match accuracy (Acc.). Edit Distance (Dist.) measures average relative edit distance between the prediction and the ground truth. We also report the fraction of Valid model outputs (those that parse, type-check and execute). For context only, we show recent results from supervised models (trained on the datasets we use) marked with (S).
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<table><tr><td rowspan="2">Model</td><td rowspan="2">Exec.</td><td colspan="3">SQL</td><td colspan="2">Vega-Lite</td><td colspan="3">SMCalFlow</td></tr><tr><td>Valid</td><td>Dist.</td><td>Acc.</td><td>Valid</td><td>Dist.</td><td>Acc.</td><td>Valid</td><td>Dist.</td></tr><tr><td>Andreas et al. (2020)</td><td></td><td>=</td><td></td><td></td><td>=</td><td>=</td><td>72%(s)</td><td>=</td><td></td></tr><tr><td>Srinivasan et al. (2021)</td><td>=</td><td>=</td><td>=</td><td>64%(s)</td><td>=</td><td>=</td><td>=</td><td>=</td><td>=</td></tr><tr><td>Rubin & Berant (2021)</td><td>71%(s)</td><td>-</td><td>=</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Scholak et al. (2021)</td><td>79%(s)</td><td>98%</td><td>1</td><td></td><td>1</td><td>=</td><td></td><td></td><td></td></tr><tr><td>GPT-3 13B</td><td>16%</td><td>43%</td><td>0.42</td><td>14%</td><td>55%</td><td>0.51</td><td>38%</td><td>76%</td><td>0.43</td></tr><tr><td>”+ CSD</td><td>20%</td><td>66%</td><td>0.44</td><td>17%</td><td>100%</td><td>0.48</td><td>40%</td><td>95%</td><td>0.40</td></tr><tr><td>"+TST</td><td>14%</td><td>48%</td><td>0.42</td><td>1</td><td>-</td><td>-</td><td>60%</td><td>88%</td><td>0.22</td></tr><tr><td>”+ CSD + TST</td><td>19%</td><td>72%</td><td>0.43</td><td>-</td><td>-</td><td>-</td><td>63%</td><td>98%</td><td>0.17</td></tr><tr><td>GPT-3175B</td><td>28%</td><td>49%</td><td>0.36</td><td>20%</td><td>67%</td><td>0.36</td><td>44%</td><td>77%</td><td>0.41</td></tr><tr><td>”+ CSD</td><td>35%</td><td>73%</td><td>0.36</td><td>25%</td><td>100%</td><td>0.32</td><td>45%</td><td>97%</td><td>0.37</td></tr><tr><td>”+ TST</td><td>31%</td><td>56%</td><td>0.35</td><td>1</td><td>-</td><td>-</td><td>60%</td><td>88%</td><td>0.24</td></tr><tr><td>"+ CSD + TST</td><td>37%</td><td>76%</td><td>0.34</td><td>-</td><td>-</td><td>-</td><td>66%</td><td>97%</td><td>0.18</td></tr><tr><td>Codex 175B</td><td>56%</td><td>73%</td><td>0.25</td><td>39%</td><td>87%</td><td>0.24</td><td>45%</td><td>79%</td><td>0.37</td></tr><tr><td>”+ CSD</td><td>61%</td><td>85%</td><td>0.23</td><td>40%</td><td>99%</td><td>0.23</td><td>46%</td><td>97%</td><td>0.33</td></tr><tr><td>”+TST</td><td>60%</td><td>81%</td><td>0.23</td><td>1</td><td>1</td><td>-</td><td>63%</td><td>90%</td><td>0.21</td></tr><tr><td>"+ CSD + TST</td><td>64%</td><td>85%</td><td>0.23</td><td>-</td><td>1</td><td>-</td><td>63%</td><td>99%</td><td>0.19</td></tr></table>
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# 4.1 RESULTS
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Table 2 and Figure 4 summarize our main results evaluating SYNCHROMESH. Key observations are:
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SYNCHROMESH improves reliability on top of all pre-trained LLMs. First, it improves top-1 accuracy (exact or execution-measured) over any pre-trained LLM in all domains. SMCalFlow benefits the most, likely because this domain-specific language is absent in the LLM pre-training corpus. For SQL and SMCalFlow, the absolute gain is almost the same for equally-sized GPT-3 and Codex.
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Second, SYNCHROMESH dramatically improves validity. In SQL, it eliminates execution errors from $2 9 \%$ of the queries generated by GPT-3 13B (as validity improves from $43 \%$ to $72 \%$ ). Even Codex benefits, with $12 \%$ more queries executing successfully after SYNCHROMESH augmentation. In Vega-Lite and SMCalFlow, SYNCHROMESH improves reliability even more substantially. GPT-3 13B only produces valid charts for $55 \%$ of the queries in NLV; all errors are eliminated with SYNCHROMESH. This is nearly paralleled in SMCalFlow, in which all models produce well-typed programs $9 7 \%$ of the time or more with SYNCHROMESH.
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SYNCHROMESH brings the output closer to ground truth. Error prevention alone is trivial (e.g., with a constant error-free prediction), but not while simultaneously improving accuracy or edit distance to the ground-truth, as SYNCHROMESH does. Again, we observe improvements in all domains and the most in SMCalFlow. For GPT-3 175B, the average edit distance is reduced from 0.41 to 0.18.
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TST and CSD bring complementary benefits. Our ablation studies reported in Table 2 show that their combination performs better than either one separately. TST helps LLMs generate programs in the “vicinity” of the correct one, and CSD helps by “guiding” the models toward the correct one.
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SYNCHROMESH adds more value for longer programs. Program synthesis is hardest when the target program is complex. Does SYNCHROMESH improve synthesis of longer programs, or are its benefits coming from fixes to small programs? Figure 4(a) shows accuracies and (b) validity for SMCalFlow broken down by the length of the ground truth program (we show results for SQL in the Appendix). Here, program lengths are shown as their percentile. With SYNCHROMESH, we see that accuracy decays at a slower pace, and validity remains high throughout, when compared to Codex alone. This indicates that SYNCHROMESH improves the ability of base models to generate longer programs.
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Figure 4: (a) Accuracy and (b) validity of Codex predictions with and without SYNCHROMESH on SMCalFlow as a function of the ground-truth program length. We map program lengths to percentiles, and round to the closest multiple of $10 \%$ . Error bands correspond to standard error. (c) Evaluation of the “generate-then-test” approach with Codex, showing the probability of at least one prediction being a valid program (Valid $( \varpi \mathrm { K } )$ ) for up to 5 samples.
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Figure 5: Illustration of implementation and conceptual errors in Vega-Lite. CSD can avoid generating the invalid Vega-Lite mark type “scatterplot”, though conceptual errors can still remain.
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LLMs augmented with SYNCHROMESH approach but underperform supervised models. For context, we include state-of-the-art results at the time of writing for each task in Table 2. We note that these methods fine-tune or train the underlying language-to-code model on each task, thus are not directly comparable to LLMs with SYNCHROMESH. That said, we observe that base LLMs—even Codex— substantially underperform supervised models ( $19 \%$ worse for SQL; $27 \%$ worse for SMCalFlow), and SYNCHROMESH helps narrow that gap (now $11 \%$ worse for SQL; $9 \%$ worse for SMCalFlow).
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SYNCHROMESH outperforms “generate-then-test”. CSD enforces program constraints during generation. Instead, prior work has leveraged a “generate-then-test” approach: take multiple samples and filter out those that produce errors or violate constraints (Chen et al., 2021). Figure 4(b) evaluates this approach with Codex, the highest performing base LLM. We sample from Codex with a temperature $\tau = 0 . 7$ to obtain diverse but high-quality samples. We then compute the “Valid $@ \mathrm { K } '$ metric by using the “Pass $@ \mathrm { K } '$ estimator from Chen et al. (2021) to calculate the probability of at least one valid sample among $K$ , with $K \leq 5$ . In SQL, Codex needs 3 samples to match SYNCHROMESH (Va $\mathrm { l i d } @ \mathrm { K } = 8 5 \%$ ). In SMCalFlow and Vega-Lite, SYNCHROMESH is able to virtually eliminate errors with 1 sample, while “Valid $@ 5 '$ for Codex is still below $93 \%$ . This provides evidence that even the best LLMs benefit from incremental validation, especially in less popular languages.
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# 4.2 DISCUSSION
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In our experiments, SYNCHROMESH was able to improve accuracies and program validity across all languages due to better examples to use as a reference (TST) and preventing errors during generation (CSD). Yet, this approach has an important limitation. While TST can reduce conceptual errors, and CSD can guarantee that certain implementation errors never occur (e.g., type errors in SMCalFlow, or undefined column references in Vega-Lite or SQL), TST cannot guarantee elimination of conceptual errors. When those occur, CSD is usually insufficient to correct the prediction. Figure 5 shows an example in Vega-Lite from the “Cars” dataset in NLV. Here, the user asks for one scatter plot for each origin, indicating faceting (multiple charts). GPT-3 alone produces an invalid Vega-Lite chart type, “scatterplot”. CSD can eliminate this error, guiding GPT-3 to generate “point” instead. However, a conceptual error remains: instead of faceting, the model colors points by their origin. Codex produces the correct Vega-Lite mark type, but still makes the same conceptual mistake.
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Nonetheless, we argue that improving validity is especially important for user-facing applications. Users of language-to-code systems might need to rephrase their request or to edit the system’s output. But outputs that fail to even execute undermine user experience: fixing an automatically generated program can be more cumbersome than writing it in the first place. In LLM-driven systems like Github Copilot, implementation errors can remain unnoticed and introduce bugs or vulnerabilities.
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# 5 RELATED WORK
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Program synthesis is a long-standing AI challenge with the goal of generating computer programs from higher-level specification (Gulwani et al., 2017). In particular, synthesis from natural language descriptions has gained recent attention (Liu et al., 2016; Yaghmazadeh et al., 2017), thanks to advances in natural language processing models such as Transformers (Vaswani et al., 2017). Typically, LLMs such as GPT-3 (Brown et al., 2020) and Codex (Chen et al., 2021) output an unconstrained sequence of tokens, and still often make conceptual or implementation errors in generated programs (Austin et al., 2021). Specialized training, e.g. to output an AST (Wang et al., 2020; Yin & Neubig, 2017), can mitigate syntactic errors, but still does not guarantee accuracy or conformance to domain-specific semantic constraints. Moreover, it requires a specialized architecture and a decoding procedure for each target language. Instead, SYNCHROMESH applies such constraints at inference, neither using specialized architectures nor fine-tuning the LLM.
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The general idea of constraining LLMs when generating programs has been explored in recent work. Shin et al. (2021) applied syntactic constraints for semantic parsing. However, their method requires enumerating all valid programs for determining valid next tokens for the LLM, and does not enforce semantic constraints. In concurrent work, Scholak et al. (2021) applied similar semantic constraints to synthesizing SQL queries. The authors substantially improve the performance of an already finetuned model by leveraging an incremental parser. We see CSD as a generalization of these efforts, as our completion engines can apply context-sensitive constraints by dynamically constructing regular expressions. Aligning these constraints with the underlying model vocabulary does not require finetuning: SYNCHROMESH only trains the much smaller target similarity model (Section 2).
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Since the emergence of LLMs, researchers have developed numerous techniques to adapt them to new domains (Liu et al., 2021b). Many focus on prompting, i.e. generating a domain- and instancespecific input to an LLM to increase the likelihood of correctness. In few-shot prompt augmentation, Gao et al. (2020) use pre-trained sentence embeddings to select the closest prompt examples to the given input instance. Liu et al. (2021a) further fine-tune sentence embeddings on the available training set of input utterances. TST in SYNCHROMESH takes this approach a step further, and finetunes the embedding models based on output similarity. It optimizes the amount of relevant output bits in the prompt, thereby reinforcing the necessary hints for the LLM.
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# 6 CONCLUSION
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SYNCHROMESH augments program synthesis with pre-trained LLMs to prevent conceptual and implementation errors during generation. We designed SYNCHROMESH to be easily usable with minimal NLP or LLM knowledge expected from a domain expert. As such, it (a) automatically generates the completion engine API from the language grammar, (b) does not require fine-tuning the LLM, drastically reducing the data/compute budget, and (c) integrates into the decoding loop or inference API with minimal overhead. Our method significantly improves performance of both GPT-3 and Codex in three languages, both by boosting accuracy and consistently improving output validity.
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While real-world and well-established, the domains we study are still not Turing-complete. We envision extending SYNCHROMESH to a Turing-complete language like Python can vastly increase reliability of LLM-based systems like Github Copilot. This requires further extension of CSD to integrate with the parser/interpreter of the target language, and to study applicable classes of constraints. The TST technique, however, can be used in any LLM-based language-to-code system.
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# REFERENCES
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Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program synthesis with large language models, 2021.
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Janusz A Brzozowski. Derivatives of regular expressions. Journal of the ACM (JACM), 11(4): 481–494, 1964.
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Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang, Hiroaki Hayashi, and Graham Neubig. Pretrain, prompt, and predict: A systematic survey of prompting methods in natural language processing, 2021b.
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Terence Parr and Kathleen Fisher. Ll (\*) the foundation of the antlr parser generator. ACM Sigplan Notices, 46(6):425–436, 2011.
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Ohad Rubin and Jonathan Berant. Smbop: Semi-autoregressive bottom-up semantic parsing. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 311–324, 2021.
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Richard Shin, Christopher H Lin, Sam Thomson, Charles Chen, Subhro Roy, Emmanouil Antonios Platanios, Adam Pauls, Dan Klein, Jason Eisner, and Benjamin Van Durme. Constrained language models yield few-shot semantic parsers. arXiv preprint arXiv:2104.08768, 2021.
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Frank F Xu, Bogdan Vasilescu, and Graham Neubig. In-ide code generation from natural language: Promise and challenges. arXiv preprint arXiv:2101.11149, 2021.
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Navid Yaghmazadeh, Yuepeng Wang, Isil Dillig, and Thomas Dillig. Sqlizer: query synthesis from natural language. Proceedings of the ACM on Programming Languages, 1(OOPSLA):1–26, 2017.
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Pengcheng Yin and Graham Neubig. A syntactic neural model for general-purpose code generation. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 440–450, 2017.
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Tao Yu, Rui Zhang, Kai Yang, Michihiro Yasunaga, Dongxu Wang, Zifan Li, James Ma, Irene Li, Qingning Yao, Shanelle Roman, et al. Spider: A large-scale human-labeled dataset for complex and cross-domain semantic parsing and text-to-sql task. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 3911–3921, 2018.
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# Algorithm 1: $\mathtt { C S D } ( M , \Sigma _ { M } )$
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Input : A LLM-based token generator, $M$
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Input : Its token set $\Sigma _ { M }$
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Output: String generated by $M$ , constrained by $C _ { L }$
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s ← next token ← ””;
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while next token $\neq " \mathbb { S } "$ do valid tokens $\textstyle \overleftarrow \} \left\{ t \in \Sigma _ { M } \right.$ | ValidPrefix $( s t ) \}$ ; next token Sample( $M ( s )$ , valid tokens); $s \gets s \cdot n$ ext token;
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end
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return $s$ ;
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# Algorithm 2: ValidPrefix(s)
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Input : A string s
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Output: True iff $s \in L ^ { c }$
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p ← next prefix ← ””;
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while next prefix $\neq \bot$ do p ← p · next prefix ; s ← next prefix −1 · s; regex ← CL(p); next prefix startswith(s, regex );
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end
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return (s−1 · regex 6= {});
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# A CONSTRAINED SEMANTIC DECODING ALGORITHM
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In Section 3.3, we described the Constrained Semantic Decoding algorithm in the text. We provide the same algorithm in pseudo-code in Algorithms 1 and 2 in Figure 6 below. ValidPrefix is our decision procedure for $L ^ { c }$ , while CSD samples a complete program from the model $M$ by making calls to ValidPrefix. We use $s \cdot t$ to denote concatenation of $s$ and $t$ , and $s ^ { - 1 } \cdot t$ to denote the string obtained by removing the prefix $s$ from $t$ . The utility function startswith $( s , r )$ returns the maximal prefix of $s$ that matches the regular expression $r$ , and returns $\perp$ if there is no such match. The function $\mathtt { S a m p l e } ( D i s t , S )$ returns a token from the set $S$ of tokens sampled from the distribution Dist restricted to $S$ . The CSD procedure uses the model $M$ on the partial program to generate a distribution on the next token, but constrains it to belong to the set of valid tokens determined by the completion engine $C _ { L }$ .
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# B COMPUTING BRZOZOWSKI DERIVATIVES
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Regular expression derivatives can be expensive to compute in general. The main challenge is that Klenee stars fork the decision procedure: the algorithm must decide to repeat or skip the pattern inside a star. However, in this work, all our regular expressions come from grammars of programming languages. These languages have one important feature in common: tokens are defined by greedy matching rules. This means that Klenee stars consume as many characters as possible, and do not backtrack if a tokenization error occurs (the error simply propagates). Under the assumption that Klenee stars have greedy semantics, derivatives can be computed in linear time. The algorithm for computing derivatives of a regular expression can be worked out by looking at all constructors of regular expressions. Table 3 details this computation for regular expressions of a base alphabet $\Sigma$ .
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# C COMPLETION ENGINES
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Here, we describe our completion engines for SQL, Vega-Lite and SMCalFlow in more detail.
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# C.1 SQL
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SQL database queries are executed in the context of a particular database containing a set of tables. Each table has a schema, which specifies named columns, their data types and constraints such as foreign keys. We refer to Yu et al. (2018) for a more detailed description of the SQL language.
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Our CE for SQL enforces that only columns that exist in the tables in the database are used. The main challenge is that queries often specify aliases. Thus, during parsing, we construct a symbol table mapping aliases to the tables they refer to. We enforce that tables that already have an alias should only be referred to by their alias, not in their unqualified form. Since aliases can be referred to in the SELECT clause before being defined in the FROM clause, we also keep track of undefined aliases to enforce that they will be assigned a table later. Moreover, a condition in the WHERE clause might involve a nested query, which in its turn might redefine or create new aliases. As a result, our symbol table keeps a stack of scopes to properly resolve aliases in nested contexts.
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Table 3: Computing Brzozowski derivatives for each constructor of regular expressions under the assumption that Klenee stars are greedy. The resulting algorithm runs in linear-time on the size of the regular expression.
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<table><tr><td>Constructor</td><td>Description</td><td>Derivative w.r.t c' ∈∑</td></tr><tr><td>Q</td><td>Empty regular expression (matches no string).</td><td>Q</td></tr><tr><td>E</td><td>Matches only the empty string.</td><td>Q</td></tr><tr><td>C</td><td>Matches a single character c</td><td>∈ if c= c',or & otherwise. If the derivative of R1 w.r.t. c' is not ,</td></tr><tr><td>R1R2</td><td>Concatenation of two regular ex- pressions R1 and R2</td><td>then it's the concatenation of that with R2. Otherwise,if Rl matches ∈, then it is sim- ply the derivative of R2 w.r.t. c'. If not,</td></tr><tr><td>R1R2</td><td>Union of two regular expressions R1 and R2</td><td>then the result is &. Union of the derivatives of R1 and R2 W.r.t. c'(if one becomes &,simply return the other).</td></tr><tr><td>R*</td><td>Klenee star - any number of repeti- tions of R</td><td>If the derivative of R w.r.t. c is not Q, then return the concatenation of that derivative with R*. Otherwise,return @.</td></tr></table>
|
| 242 |
+
|
| 243 |
+
We constrain numeric literals to either come from a set of common numbers (including 0 and 1) or from the user’s natural language question. This prevents the model from copying arbitrary numbers from the few-shot examples in the prompt. Finally, since SQL is case-insensitive, our CE returns case-insensitive regular expressions for keywords, table and column names.
|
| 244 |
+
|
| 245 |
+
# C.2 VEGA-LITE
|
| 246 |
+
|
| 247 |
+
Vega-Lite is a declarative language for specifying data visualizations given a data frame – a table where rows represent data points and columns represent attributes of various data types. Its syntax is a subset of JSON. Therefore, our Vega-Lite grammar accepts JSON objects that follow a subset of the Vega-Lite schema.
|
| 248 |
+
|
| 249 |
+
As in SQL, we use the user’s data frame to constrain valid field names. Additionally, in Vega-Lite, one must also specify a Vega-Lite type that is used to interpret the field values. We inspect the runtime values in the data frame to determine compatible Vega-Lite types. For example, a string column is typically used as a categorical value (nominal, in Vega-Lite). However, if its entries have ISOformatted timestamps, it can be used as temporal, or quantitative if its values can be parsed as numbers. Since JSON objects are unordered, it must handle two valid output orders: the model might output the field name first (we thus later constrain the type) or the data type first (we then constrain the field name to compatible columns). Because Vega-Lite’s behavior is to silently ignore invalid data points, associating a column with an incompatible type simply produces an empty chart. Our constraint completely prevents this class of errors.
|
| 250 |
+
|
| 251 |
+
We forbid repeated fields and limit the length of free-form string literals to 30 characters, which prevents the common failure mode of language models to enter repetition loops. Similarly, we only allow an aggregation in one of the X-Y axes, but not both, since that collapses the chart to a single data point and is another common failure case. Finally, we prevent the model from faceting (i.e., splitting into multiple charts) based on a column with too many $( > 5 0 )$ ) distinct values: that typically crashes the Vega-Lite rendering engine since it allocates an overly large output image. In summary, our constraints guarantee conformance to the Vega-Lite specification and additionally avoid common mistakes that cause crashes or degenerate outputs.
|
| 252 |
+
|
| 253 |
+
|
| 254 |
+
Figure 7: Illustration of the token misalignment problem in Vega-Lite. Colors denote Vega-Lite tokens, which match how the completion engine works (and what are its completion points). Vertical lines denote how GPT-3 tokenizes this program.
|
| 255 |
+
|
| 256 |
+
# C.3 SMCALFLOW
|
| 257 |
+
|
| 258 |
+
SMCalFlow programs express responses to user queries about calendar events, weather, places, and people (Andreas et al., 2020). It is a rich language with scoped variable declarations, generic types, polymorphic operators and a large API of over 400 functions, which can be composed to express complex actions like “cancel any meetings on the same day of my next doctor’s appointment”, or answer queries such as ”will it be raining during my next walking meeting with Frank?”
|
| 259 |
+
|
| 260 |
+
Our CE enforces that programs type-check2 by construction. An example3 is given in Figure 1. At the current point in inference, the model is producing an argument to size. Since this function takes a list, its argument must be the return value of a function with return type List $< \mathrm { T } >$ . Among all functions and methods in the SMCalFlow API, only 14 return lists, which severely limits the valid options for the callee. Similarly, we keep track of declared variables inside let expressions, together with their types (inferred from their initialization expression), and use that data structure to limit options whenever a token of type identifier is syntactically allowed to follow.
|
| 261 |
+
|
| 262 |
+
Finally, we implemented heuristics based on user utterance patterns that avoid common failure cases we observed in GPT-3. These tend to happen when the model blindly copies portions of the examples in the prompt without adaptation. For instance, whenever the utterance contains exactly one of “a.m.” or “p.m”, this is usually represented in SMCalFlow by a call to a corresponding SMCalFlow function that constructs a Time object (e.g., NumberAM(5)). However, if the examples retrieved for the prompt tend to only have times in the opposite half of the day, GPT-3 might call the wrong function, and translate the time sub-expression in “Schedule it for $5 \mathrm { p m } ^ { \mathrm { , , } \mathrm { , } }$ into NumberAM(5). To avoid this, if we detect exactly one of “a.m.” or “p.m.” in the utterance, we remove the time construction functions associated with the opposite pattern from the candidates. We do the same filtering with days of the week and months, which are also constructed by specific functions.
|
| 263 |
+
|
| 264 |
+
In all domains, the CE abstraction allows us to easily encode domain knowledge in a modular fashion. Besides constraints coming from the language’s semantics, it further allows domain experts to analyze failure modes of the language model and to implement fixes them in a modular and predictable manner.
|
| 265 |
+
|
| 266 |
+
# D THE TOKEN MISALIGNMENT CHALLENGE
|
| 267 |
+
|
| 268 |
+
The main challenge of CSD is in aligning constraints expressed in terms of programming language tokens with the model’s output, which happens in another token vocabulary (typically constructed with Byte-Pair Encoding). Figure 7 shows an illustration of this challenge in Vega-Lite. Arbitrary mismatches can occur: the first BPE token includes the first Vega-Lite token and the first character of the second. In the middle of the example, the "encoding" token in Vega-Lite spans 4 BPE tokens, with misaligned boundaries at the beginning and end. Nonetheless, this issue is seamlessly handled by the CSD algorithm.
|
| 269 |
+
|
| 270 |
+
|
| 271 |
+
Figure 8: (a) Accuracy and (b) validity of Codex predictions with and without SYNCHROMESH on SQL as a function of the ground-truth program length. We map program lengths to percentiles, and round to the closest multiple of $10 \%$ . Error bands correspond to standard error.
|
| 272 |
+
|
| 273 |
+
# E OPTIMIZATIONS TO CSD
|
| 274 |
+
|
| 275 |
+
The algorithm described in Section 3.3 tests each token from the language model’s vocabulary individually. However, many BPE tokens are prefixes of oen another, which lets us apply a significant optimization. If $t _ { 1 }$ is a prefix of $t _ { 2 }$ and $t _ { 1 }$ is inadmissible after a partial program $p$ , then $t _ { 2 }$ is also inadmissible. Thus, we test tokens by order of length, and keep rejected tokens in a Trie structure. Before we test a token against the completion engine, we check whether one of its prefixes was already rejected. If so, we can safely skip that token.
|
| 276 |
+
|
| 277 |
+
Another optimization consists in memoizing the enumerated completion points. When testing a new partial program $p$ , instead of starting from the empty string, we can start from the longest known completion point that is a prefix of $p$ . This, again, can be efficiently done by keeping completion points in a Trie.
|
| 278 |
+
|
| 279 |
+
# F CSD WITHOUT DIRECT ACCESS TO THE LANGUAGE MODEL
|
| 280 |
+
|
| 281 |
+
We used the public OpenAI API to access GPT-3 and Codex. Therefore, we did not have direct access to the underlying language models. Even so, CSD can still be applied provided we can pass a bias to be added to the logits, which is available in the OpenAI API.
|
| 282 |
+
|
| 283 |
+
However, making one request at each token is too slow. Instead, we apply a “rejection”-based sampling, as follows. First, we request a complete program from the model. Then, we iterate tokenby-token, validating it with the CSD algorithm against the completion engine. If we find a violation, we (a) use CSD to determine all valid next tokens, (b) make a request asking from just a single token, applying a logit bias to constrain it to the valid tokens, and then (c) continue generation after appending the new token. Most $( 9 0 \% + )$ trajectories end after at most 3 corrections to the model. In degenerate cases, we might need to correct the model after almost every token. In our experiments, we capped CSD to apply at most 15 corrections, to control the time a request with CSD takes. This only happened in less than . $5 \%$ of the cases, and could be completely avoided if we had direct access to the model (in which case CSD is efficient enough to be applied at every token).
|
| 284 |
+
|
| 285 |
+
# G ANALYSIS OF ACCURACY AND VALIDITY BY LENGTH IN SQL
|
| 286 |
+
|
| 287 |
+
Figure 8 shows the equivalent of Figure 4 for the SQL domain. We notice that the largest gaps in validity happen for the longest queries, and the benefits in accuracy are highest for queries around the $60 \%$ length percentile.
|
| 288 |
+
|
| 289 |
+
# H TST FINE-TUNING DETAILS
|
| 290 |
+
|
| 291 |
+
In Section 2, we described TST, which fine-tunes a sentence embedding model to attempt to capture program similarity. Here, we give more details on our training procedure.
|
| 292 |
+
|
| 293 |
+
First, for both SQL and SMCalFlow, we selected a random set of 2000 examples, and computed the normalized tree edit distance between all pairs of examples. We then fine-tuned S-BERT for one epoch using the cosine similarity loss. We used the AdamW optimizer with a learning rate of $2 \times 1 \mathrm { { 0 } ^ { - } 5 }$ – the default parameters in the S-BERT library4. We did not perform any hyperparameter tuning. Training took around 3 hours on a single GPU.
|
| 294 |
+
|
| 295 |
+
# I COMMENTED EXAMPLES OF TST AND CSD
|
| 296 |
+
|
| 297 |
+
Here, we provide three more detailed examples. The first examples shows the benefit of using TST on top of Codex. The remaining two show value of CSD on top of Codex. Since the examples in Sections 2 and 3 used SQL, we now give examples in SMCalFlow and Vega-Lite for diversity.
|
| 298 |
+
|
| 299 |
+
# I.1 TST EXAMPLE IN SMCALFLOW
|
| 300 |
+
|
| 301 |
+
Query: “Hi, I need to meet with Alex on this Sunday at 6pm at the CVS shopping center for an hour.” This query asks the system to create an event. Without TST, the top example (similar to the query) returned by vanilla S-BERT from the training set is:
|
| 302 |
+
|
| 303 |
+
User utterance: Hi, I need to meet with Alex tomorrow afternoon for an hour.
|
| 304 |
+
SMCalFlow: (Yield (CreateCommitEventWrapper (CreatePreflightEventWrapper (& (& (Event.start_? (DateTimeConstraint (Afternoon) (Tomorrow))) (Event.duration_? ( $? =$ (toHours 1)))) (Event.attendees_? (AttendeeListHasRecipient (Execute (refer (extensionConstraint (RecipientWithNameLike (EmptyStructConstraint) (PersonName.apply "Alex")))))))))))
|
| 305 |
+
|
| 306 |
+
This example also involves creating an event, and also with a person named “Alex”. However, the time structure of this example, “Tomorrow afternoon”, is not trivial to translate to “this Sunday at 6pm”, because these two specifications require different APIs from SMCalFlow. Given this and 4 more examples, Codex generates:
|
| 307 |
+
|
| 308 |
+
(Yield (CreateCommitEventWrapper (CreatePreflightEventWrapper (& (& (Event.start_? (DateTimeConstraint (Afternoon) (ThisSunday))) (Event.duration_? ( $? =$ (toHours 1)))) (Event.location_? ( $? =$ (LocationWithNameLike (EmptyStructConstraint) (LocationName.apply "CVS shopping center")))) (Event.attendees_? (AttendeeListHasRecipient (Execute (refer (extensionConstraint (RecipientWithNameLike (EmptyStructConstraint) (PersonName.apply "Alex")))))))))))
|
| 309 |
+
|
| 310 |
+
This prediction, while seeming reasonable at a surface level, violates the SMCalFlow APIs in multiple places. First, the function “ThisSunday” does not exist. The parameter 6pm was lost in this prediction, since the examples had no explicit time in it and Codex could not identify where to put it. Besides that, the model is trying to specify four attributes of the created event: start time, duration, location and attendees, but it would need one more conjunction (call to “&”) to do so. This error likely came from the fact that the top example only specified three attributes. The calls to the location APIs are also incorrect – “LocationWithNameLike” exists (and is present in other examples in the prompt), but is not really needed or appropriate in this context.
|
| 311 |
+
|
| 312 |
+
However, if we use TST to pick the examples for the prompt, Codex is able to solve the problem. With TST, the top retrieved example is:
|
| 313 |
+
|
| 314 |
+
User utterance: Schedule 2 hours on Friday at 7:00 am for a Meeting with Kayla at the Sear’s Tower.
|
| 315 |
+
SMCalFlow: (Yield (CreateCommitEventWrapper (CreatePreflightEventWrapper (& (& (& (Event.start_? ( $? =$ (DateAtTimeWithDefaults (NextDOW (Friday)) (NumberAM 7L)))) (Event.duration_? ( $? =$ (toHours 2)))) (Event.location_? ( $? =$ (LocationKeyphrase.apply "Sear\’s Tower")))) (Event.attendees_? (AttendeeListHasRecipient (Execute (refer (extensionConstraint (RecipientWithNameLike (EmptyStructConstraint) (PersonName.apply "Kayla")))))))))))
|
| 316 |
+
|
| 317 |
+
Note that the utterance, from the perspective of its natural language structure, is less similar to the user’s request than the previous example. However, it describes a query with the exact necessary structure: it schedules something for the next Friday (trivial to modify to Sunday) at a certain time of the day (“NumberAM” can be simply rewritten to “NumberPM”, which Codex does correctly). The location attribute, which Codex had previously tried to adapt from a less related example, is now also specified here. With this example, Codex is able to make a correct adaptation. It predicts:
|
| 318 |
+
|
| 319 |
+
(Yield (CreateCommitEventWrapper (CreatePreflightEventWrapper (& (& (& (Event.start_? ( $? =$ (DateAtTimeWithDefaults (NextDOW (Sunday)) (NumberPM 6L)))) (Event.duration_? ( $? =$ (toHours 1)))) (Event.location_? ( $? =$ (LocationKeyphrase.apply "CVS shopping center")))) (Event.attendees_? (AttendeeListHasRecipient (Execute (refer (extensionConstraint (RecipientWithNameLike (EmptyStructConstraint) (PersonName.apply "Alex")))))))))))
|
| 320 |
+
|
| 321 |
+
This matches the reference program from the dataset.
|
| 322 |
+
|
| 323 |
+
# I.2 CSD EXAMPLES IN VEGA-LITE
|
| 324 |
+
|
| 325 |
+
Query: “average horsepower over years, colored by origin” This query does not specify a chart type, but one could infer that a line chart is likely more appropriate since it is showing a quantity over time. The dataset under consideration contains car models with their year of manufacture, horsepower, number of cylinders and many other attribute. Thus, to obtain the “average horsepower” in a given year, we need an aggregation in Vega-Lite. Codex alone predicts:
|
| 326 |
+
|
| 327 |
+
{"mark": {"type": "line" },
|
| 328 |
+
"encoding": {"x": {"field": "Year", "type": "temporal", "timeUnit": "yearmonth" }, "y": {"field": "Horsepower", "type": "quantitative", "aggregate": "average" }, "color": {"field": "Origin", "type": "nominal" } } }
|
| 329 |
+
|
| 330 |
+
All the right components of the query are there. However, even though the query asks for the “average horsepower”, the aggregation function in Vega-Lite is called “mean”. Specifying an invalid aggregation function makes Vega-Lite raise an exception. Thus, an actual user wouldn’t see any plot from this response.
|
| 331 |
+
|
| 332 |
+
CSD prevents Codex from generating “average” after it outputs the field name “aggregate”, forcing it to choose one of the existing aggregation functions (max, min, mean, etc). Constrained by CSD, Codex generates:
|
| 333 |
+
|
| 334 |
+
{"mark": {"type": "line" },
|
| 335 |
+
"encoding": {"x": {"field": "Year", "type": "temporal", "timeUnit": "yearmonth" }, "y": {"field": "Horsepower", "type": "quantitative", "aggregate": "mean" }, "color": {"field": "Origin", "type": "nominal" } } }
|
| 336 |
+
|
| 337 |
+
This is not only valid but also matches the ground-truth from the dataset.
|
| 338 |
+
|
| 339 |
+
“Create a scatterplot of acceleration vs horsepower by model” This utterance asks for a chart that relates each car’s acceleration and horsepower. Codex generates:
|
| 340 |
+
|
| 341 |
+
{"mark": {"type": "point" },
|
| 342 |
+
"encoding": {"x": {"field": "Horsepower", "type": "quantitative" }, "y": {"field": "Acceleration", "type": "quantitative" }, "color": {"field": "Model", "type": "nominal" } } }
|
| 343 |
+
|
| 344 |
+
The produced plot, shown in Figure 9 (left) is in principle correct, and answers the user’s question. The “by model” in the query might sound as if the user wants specialized plots by model. Codex realizes this interpretation by coloring each point differently by the car’s model. But in this dataset, each row represents a different car with a distinct model. Thus, there are 303 unique values of the “Model” column, and Vega-Lite raises a run-time warning saying that the legend has overflowed for having too many values, so some values (the majority, in this case) were omitted. When the model decides to facet on a column with too many values, the Vega-Lite run-time often runs out of memory for allocating an overly large image.
|
| 345 |
+
|
| 346 |
+
CSD avoids these errors by looking at the user’s dataset and determining which columns can be used for coloring or faceting, based on how many distinct values they have. If none are available, then CSD doesn’t let the model use these features. In this example, CSD forces Codex to choose one of the columns with less than 30 distinct values. Between those, Codex generates “Year”. The resulting plot is shown in Figure 9 (right). While in this example the “Year” column was not mentioned in the user’s request, the application of this constraint here steers Codex to avoid a run-time warning or error. The ground-truth visualization for this description does not have any color specification, so neither prediction counts as an exact match. Nevertheless, from the point of view of the user’s experience, CSD can be helpful even when it cannot completely fix the prediction.
|
| 347 |
+
|
| 348 |
+
|
| 349 |
+
Figure 9: Benefit of CSD: The left plot is generated by Codex without CSD, but it has an overflowing legend and it also raises a warning about missing values. Using CSD, we generate the plot on right that generates no runtime warning or error.
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| 1 |
+
# The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks
|
| 2 |
+
|
| 3 |
+
Ziqian Zhong\*, Ziming Liu\*, Max Tegmark, Jacob Andreas Massachusetts Institute of Technology {ziqianz, zmliu, tegmark, jda}@mit.edu
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
Do neural networks, trained on well-understood algorithmic tasks, reliably rediscover known algorithms for solving those tasks? Several recent studies, on tasks ranging from group arithmetic to in-context linear regression, have suggested that the answer is yes. Using modular addition as a prototypical problem, we show that algorithm discovery in neural networks is sometimes more complex. Small changes to model hyperparameters and initializations can induce discovery of qualitatively different algorithms from a fixed training set, and even parallel implementations of multiple such algorithms. Some networks trained to perform modular addition implement a familiar Clock algorithm (previously described by Nanda et al. [1]); others implement a previously undescribed, less intuitive, but comprehensible procedure we term the Pizza algorithm, or a variety of even more complex procedures. Our results show that even simple learning problems can admit a surprising diversity of solutions, motivating the development of new tools for characterizing the behavior of neural networks across their algorithmic phase space. 1
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Mechanistically understanding deep network models—reverse-engineering their learned algorithms and representation schemes—remains a major challenge across problem domains. Several recent studies [2, 3, 4, 5, 1] have exhibited specific examples of models apparently re-discovering interpretable (and in some cases familiar) solutions to tasks like curve detection, sequence copying and modular arithmetic. Are these models the exception or the rule? Under what conditions do neural network models discover familiar algorithmic solutions to algorithmic tasks?
|
| 12 |
+
|
| 13 |
+
In this paper, we focus specifically on the problem of learning modular addition, training networks to compute sums like $8 + 6 = 2$ (mod 12). Modular arithmetic can be implemented with a simple geometric solution, familiar to anyone who has learned to read a clock: every integer is represented as an angle, input angles are added together, and the resulting angle evaluated to obtain a modular sum (Figure 1, left). Nanda et al. [1] show that specific neural network architectures, when trained to perform modular addition, implement this Clock algorithm. In this work, we show that the Clock algorithm is only one part of a more complicated picture of algorithm learning in deep networks. In particular, networks structurally similar to the ones trained by Nanda et al. preferentially implement a qualitatively different approach to modular arithmetic, which we term the Pizza algorithm (Figure 1, right), and sometimes even more complex solutions. Models exhibit sharp algorithmic phase transitions [6] between the Clock and Pizza algorithms as their width and attention strength very, and often implement multiple, imperfect copies of the Pizza algorithm in parallel.
|
| 14 |
+
|
| 15 |
+
Step 1: Embed token a and $b$ to a circle where $w _ { k } = 2 \pi k / p$ for some $k \in [ 1 , 2 ^ { \ldots } , p - 1 ]$
|
| 16 |
+
|
| 17 |
+
$$
|
| 18 |
+
a \mathbf { E } _ { a } \equiv ( \mathbf { E } _ { a , \mathrm { x } } , \mathbf { E } _ { a , \mathrm { y } } ) = ( \cos ( w _ { k } a ) , \sin ( w _ { k } a ) ) , b \mathbf { E } _ { b } \equiv ( \mathbf { E } _ { b , \mathrm { x } } , \mathbf { E } _ { b , \mathrm { y } } ) = ( \cos ( w _ { k } b ) , \sin ( w _ { k } b ) )
|
| 19 |
+
$$
|
| 20 |
+
|
| 21 |
+
|
| 22 |
+
|
| 23 |
+
# Clock Algorithm
|
| 24 |
+
|
| 25 |
+
# Pizza Algorithm
|
| 26 |
+
|
| 27 |
+
Step 2: compute the angle sum using multiplication.
|
| 28 |
+
|
| 29 |
+
Step 2.1: compute the vector mean.
|
| 30 |
+
|
| 31 |
+
$\mathbf { E } _ { a b } \equiv { \binom { \mathbf { E } _ { a b , x } } { \mathbf { E } _ { a b , y } } } = { \binom { \mathbf { E } _ { a , x } \mathbf { E } _ { b , x } - \mathbf { E } _ { a , y } \mathbf { E } _ { b , y } } { \mathbf { E } _ { a , x } \mathbf { E } _ { b , y } + \mathbf { E } _ { a , y } \mathbf { E } _ { b , x } } } = { \binom { \cos ( w _ { k } ( a + b ) ) } { \sin ( w _ { k } ( a + b ) ) } }$ Hab = Eab
|
| 32 |
+
|
| 33 |
+
$$
|
| 34 |
+
\mathbf { E } _ { a b } = ( \mathbf { E } _ { a } + \mathbf { E } _ { b } ) / 2 = ( \cos ( w _ { k } a ) + \cos ( w _ { k } b ) , \sin ( w _ { k } a ) + \sin ( w _ { k } b ) ) / 2
|
| 35 |
+
$$
|
| 36 |
+
|
| 37 |
+
Step 2.2: using $\mathbf { E } _ { a b }$ and nonlinearities to compute $\mathbf { H } _ { a b }$
|
| 38 |
+
|
| 39 |
+
$$
|
| 40 |
+
\mathbf { H } _ { a b } = | \cos ( w _ { k } ( a - b ) / 2 ) | ( \cos ( w _ { k } ( a + b ) ) , \sin ( w _ { k } ( a + b ) ) )
|
| 41 |
+
$$
|
| 42 |
+
|
| 43 |
+
Step 3: score possible outputs $c$ using a dot product.
|
| 44 |
+
|
| 45 |
+
$Q _ { a b c } = \mathbf { U } _ { c } \cdot \mathbf { H } _ { a b }$ , Uc ≡ (Ec,x, Ec,y) = (cos(wkc), sin(wkc))
|
| 46 |
+
|
| 47 |
+
|
| 48 |
+
Figure 1: Illustration of the Clock and the Pizza Algorithm.
|
| 49 |
+
|
| 50 |
+
Our results highlight the complexity of mechanistic description in even models trained to perform simple tasks. They point to characterization of algorithmic phase spaces, not just single algorithmic solutions, as an important goal in algorithm-level interpretability.
|
| 51 |
+
|
| 52 |
+
Organization In Section 2, we review the Clock algorithm [1] and show empirical evidence of deviation from it in models trained to perform modular addition. In Section 3, we show that these deviations can be explained by an alternative Pizza algorithm. In Section 4, we define additional metrics to distinguish between these algorithms, and detect phase transitions between these algorithms (and others Non-circular algorithms) when architectures and hyperparameters are varied. We discuss the relationship between these findings and other work on model interpretation in Section 5, and conclude in Section 6.
|
| 53 |
+
|
| 54 |
+
# 2 Modular Arithmetic and the Clock Algorithm
|
| 55 |
+
|
| 56 |
+
Setup We train neural networks to perform modular addition $a + b = c ( \mathrm { m o d } p )$ , where $a , b , c =$ $0 , 1 , \cdots , p - 1$ . We use $p = 5 9$ throughout the paper. In these networks, every integer $t$ has an associated embedding vector $\mathbf { E } _ { t } \in \mathbb { R } ^ { d }$ . Networks take as input embeddings $[ \dot { \mathbf { E } _ { a } } , \mathbf { E } _ { b } ] ^ { \mathbf { ^ { \prime } } } \in \mathbb { R } ^ { 2 d }$ and predict a categorical output $c$ . Both embeddings and network parameters are learned. In preliminary experiments, we train two different network architectures on the modular arithmetic task, which we refer to as: Model A and Model B. Model A is a one-layer ReLU transformer [7] with constant attention, while Model $\mathbf { B }$ is a standard one-layer ReLU transformer (see Appendix F.1 for details). As attention is not involved in Model A, it can also be understood as a ReLU MLP (Appendix G).
|
| 57 |
+
|
| 58 |
+
# 2.1 Review of the Clock Algorithm
|
| 59 |
+
|
| 60 |
+
As in past work, we find that after training both Model A and Model B, embeddings $( \mathbf { E } _ { a } , \mathbf { E } _ { b }$ in Figure 1) usually describe a circle [8] in the plane spanned by the first two principal components of the embedding matrix. Formally, $\mathbf { E } _ { a } \approx [ \cos ( w _ { k } a ) , \bar { \sin ( w _ { k } a ) } ]$ where $w _ { k } = 2 \pi k / p$ , $k$ is an integer in $[ 1 , p - 1 ]$ . Nanda et al. [1] discovered a circuit that uses these circular embeddings to implement an interpretable algorithm for modular arithmetic, which we call the Clock algorithm.
|
| 61 |
+
|
| 62 |
+
Table 1: Different neural algorithms for modular addition
|
| 63 |
+
|
| 64 |
+
<table><tr><td>Algorithm</td><td>Learned Embeddings</td><td>GradientSymmetry</td><td>Required Non-linearity</td></tr><tr><td>Clock</td><td>Circle</td><td>No</td><td>Multiplication</td></tr><tr><td>Pizza</td><td>Circle</td><td>Yes</td><td>Absolute value</td></tr><tr><td>Non-circular</td><td>Line,Lissajous-like curves, etc.</td><td>N/A</td><td>N/A</td></tr></table>
|
| 65 |
+
|
| 66 |
+
"If a meeting starts at 10, and lasts for 3 hours, then it will end at 1." This familiar fact is a description of a modular sum, $1 0 + 3 = 1$ (mod 12), and the movement of a clock describes a simple algorithm for modular arithmetic: the numbers 1 through 12 are arranged on a circle in $3 6 0 ^ { \circ } / 1 2 = 3 0 ^ { \circ }$ increments, angles of $1 0 \times 3 0 ^ { \circ }$ and $3 \times 3 0 ^ { \circ }$ are added together, then this angle is evaluated to determine that it corresponds to $1 \times 3 0 ^ { \circ }$ .
|
| 67 |
+
|
| 68 |
+
Remarkably, Nanda et al. [1] find that neural networks like our Model B implement this Clock algorithm, visualized in Figure 1 (left): they represent tokens $a$ and $b$ as 2D vectors, and adding their polar angles using trigonometric identities. Concretely, the Clock algorithm consists of three steps: In step 1, tokens $a$ and $b$ are embedded as ${ \bf E } _ { a } = [ \cos ( \dot { w _ { k } } a ) , \sin ( w _ { k } a ) \bar { ] }$ and $\mathbf { E } _ { b } = [ \cos ( w _ { k } b ) , \sin ( \bar { w } _ { k } b ) ]$ , respectively, where $w _ { k } = 2 \pi k / p$ (an everyday clock has $p = 1 2$ and $k = 1$ ). Then the polar angles of $\mathbf { E } _ { a }$ and $\mathbf { E } _ { b }$ are added (in step 2) and extracted (in step 3) via trigonometric identities. For each candidate output $c$ , we denote the logit $Q _ { a b c }$ ; the predicted output is $c ^ { * } = \operatorname { a r g m a x } _ { c } Q _ { a b c }$ .
|
| 69 |
+
|
| 70 |
+
Crucial to this algorithm is the fact that the attention mechanism can be leveraged to perform multiplication. What happens in model variants when the attention mechanism is absent, as in Model A? We find two pieces of evidence of deviation from the Clock algorithm in Model A.
|
| 71 |
+
|
| 72 |
+
# 2.2 First Evidence for Clock Violation: Gradient Symmetricity
|
| 73 |
+
|
| 74 |
+
Since the Clock algorithm has logits:
|
| 75 |
+
|
| 76 |
+
$$
|
| 77 |
+
Q _ { a b c } ^ { \mathrm { C l o c k } } = ( { \bf E } _ { a , x } { \bf E } _ { b , x } - { \bf E } _ { a , y } { \bf E } _ { b , y } ) { \bf E } _ { c , x } + ( { \bf E } _ { a , x } { \bf E } _ { b , y } + { \bf E } _ { a , y } { \bf E } _ { b , x } ) { \bf E } _ { c , y } ,
|
| 78 |
+
$$
|
| 79 |
+
|
| 80 |
+
(see Figure 1) the gradients of $Q _ { a b c }$ generically lack permutation symmetry in argument order: $\nabla _ { \mathbf { E } _ { a } } Q _ { a b c } \neq \nabla _ { \mathbf { E } _ { b } } Q _ { a b c }$ . Thus, if learned models exhibit permutation symmetry $( \nabla _ { \mathbf { E } _ { a } } Q _ { a b c } ~ =$ $\nabla _ { \mathbf { E } _ { b } } Q _ { a b c } )$ , they must be implementing some other algorithm.
|
| 81 |
+
|
| 82 |
+
We compute the 6 largest principal components of the input embedding vectors. We then compute the gradients of output logits (unnormalized log-probabilities from the model) with respect to the input embeddings. We then project them onto these 6 principal components (since the angles relevant to the Clock and Pizza algorithms are encoded in the first few principal components). These projections are shown in Figure 2. While Model B demonstrates asymmetry in general, Model A exhibits gradient symmetry.
|
| 83 |
+
|
| 84 |
+
|
| 85 |
+
Figure 2: Gradients on first six principal components of input embeddings. $( a , b , c )$ in the title stands for taking gradients on the output logit $c$ for input $( a , b )$ . x and y axes represent the gradients for embeddings of the first and the second token. The dashed line $y = x$ signals a symmetric gradient.
|
| 86 |
+
|
| 87 |
+
# 2.3 Second Evidence for Clock Violation: Logit Patterns
|
| 88 |
+
|
| 89 |
+
Inspecting models’ outputs, in addition to inputs, reveals further differences. For each input pair $( a , b )$ , we compute the output logit assigned to the correct label $a + b$ . We visualize these correct logits from Models A and B in Figure 3. Notice that the rows are indexed by $a - b$ and the columns by $a + b$ . From Figure 3, we can see that the correct logits of Model A have a clear dependency on $a - b$ in that within each row, the correct logits are roughly the same, while this pattern is not observed in Model B. This suggests that Models A and B are implementing different algorithms.
|
| 90 |
+
|
| 91 |
+
|
| 92 |
+
Figure 3: Correct Logits of Model A & Model B. The correct logits of Model A (left) have a clear dependence on $a - b$ , while those of Model B (right) do not.
|
| 93 |
+
|
| 94 |
+
# 3 An Alternative Solution: the Pizza Algorithm
|
| 95 |
+
|
| 96 |
+
How does Model A perform modular arithmetic? Whatever solution it implements must exhibit gradient symmetricity in Figure 2 and the output patterns in Figure 3. In this section, we describe a new algorithm for modular arithmetic, which we call the Pizza algorithm, and then provide evidence that this is the procedure implemented by Model A.
|
| 97 |
+
|
| 98 |
+
# 3.1 The Pizza Algorithm
|
| 99 |
+
|
| 100 |
+
Unlike the Clock algorithm, the Pizza algorithm operates inside the circle formed by embeddings (just as pepperoni are spread all over a pizza), instead of operating on the circumference of the circle. The basic idea is illustrated in Figure 1: given a fixed label $c$ , for all $( a , b )$ with $a + b = c$ (mod $p$ ), the points ${ \bf E } _ { a b } = ( { \bf E } _ { a } + { \bf E } _ { b } ) / 2$ lie on a line though the origin of a 2D plane, and the points closer to this line than to the lines corresponding to any other $c$ form two out of $2 p$ mirrored “pizza slices”, as shown at the right of the figure. Thus, to perform modular arithmetic, a network can determine which slice pair the average of the two embedding vectors lies in. Concretely, the $P$ izza algorithm also consists of three steps. Step 1 is the same as in the Clock algorithm: the tokens $a$ and $b$ are embedded at $\mathbf { E } _ { a } = ( \cos ( \bar { w } _ { k } a ) , \bar { \sin } ( w _ { k } a ) )$ and $\mathbf { E } _ { b } = ( \cos ( w _ { k } b ) , \sin ( w _ { k } b ) )$ , respectively. Step 2 and Step 3 are different from the Clock algorithm. In Step 2.1, $\mathbf { E } _ { a }$ and $\mathbf { E } _ { b }$ are averaged to produce an embedding $\mathbf { E } _ { a b }$ . In Step 2.2 and Step 3, the polar angle of ${ \bf E } _ { a b }$ is (implicitly) computed by computing the logit $Q _ { a b c }$ for any possible outputs $c$ . While one possibility of doing so is to take the absolute value of the dot product of ${ \bf E } _ { a b }$ with $( \cos ( w _ { k } c / 2 ) , \sin ( w _ { k } c / 2 ) )$ , it is not commonly observed in neural networks (and will result in a different logit pattern). Instead, Step 2.2 transforms ${ \bf E } _ { a b }$ into a vector encoding $| \cos ( w _ { k } ( a - b ) / 2 ) | ( \cos ( w _ { k } ( a + b ) \bar { ) } , \sin ( w _ { k } ( a + b ) ) )$ , which is then dotted with the output embedding $U _ { c } = ( \cos ( w _ { k } c ) , \sin ( w _ { k } c ) )$ . Finally, the prediction is $c ^ { * } = \operatorname { a r g m a x } _ { c } Q _ { a b c }$ . See Appendix A and Appendix $\mathrm { L }$ for a more detailed analysis of a neural circuit that computes ${ \mathbf { H } } _ { a b }$ in a real network.
|
| 101 |
+
|
| 102 |
+
The key difference between the two algorithms lies in what non-linear operations are required: Clock requires multiplication of inputs in Step 2, while Pizza requires only absolute value computation, which is easily implemented by the ReLU layers. If neural networks lack inductive biases toward implementing multiplication, they may be more likely to implement Pizza rather than Clock, as we will verify in Section 4.
|
| 103 |
+
|
| 104 |
+
# 3.2 First Evidence for Pizza: Logit Patterns
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Both the Clock and Pizza algorithms compute logits $Q _ { a b c }$ in Step 3, but they have different forms, shown in Figure 1. Specifically, $Q _ { a b c } ( P i z z a )$ has an extra multiplicative factor $\vert \cos ( w _ { k } ( a - b ) / 2 ) \vert$ compared to $Q _ { a b c } ( C l o c k )$ . As a result, given $c = a + b$ , $Q _ { a b c } ( P i z z a )$ is dependent on $a - b$ , but $Q _ { a b c } ( C l o c k )$ is not. The intuition for the dependence is that a sample is more likely to be classified correctly if ${ \bf E } _ { a b }$ is longer. The norm of this vector depends on $a - b$ . As we observe in Figure 3, the logits in Model A indeed exhibit a strong dependence on $a - b$ .
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# 3.3 Second Evidence for Pizza: Clearer Logit Patterns via Circle Isolation
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To better understand the behavior of this algorithm, we replace the embedding matrix $\mathbf { E }$ with a series of rank-2 approximations: using only the first and second principal components, or only the third and fourth, etc. For each such matrix, embeddings lie in a a two-dimensional subspace. For both Model A and Model B, we find that embeddings form a circle in this subspace (Figure 4 and Figure 5, bottom). We call this procedure circle isolation. Even after this drastic modification to the trained models’ parameters, both Model A and Model B continue to behave in interpretable ways: a subset of predictions remain highly accurate, with this subset determined by the periodicity of the $k$ of the isolated circle. As predicted by the Pizza and Clock algorithms described in Figure 1, Model A’s accuracy drops to zero at specific values of $a - b$ , while Model B’s accuracy is invariant in $a - b$ . Applying circle isolation to Model A on the two principal components (one circle) yields a model with $3 2 . 8 \%$ overall accuracy, while retaining the first six principal components (three circles) yields an overall accuracy of $9 1 . 4 \%$ . See Appendix D for more discussion. By contrast, Model B achieves $1 0 0 \%$ when embeddings are truncated to the first six principal components. Circle isolation thus reveals an error correction mechanism achieved via ensembling: when an algorithm (clock or pizza) exhibits systematic errors on subset of inputs, models can implement multiple algorithm variants in parallel to obtain more robust predictions.
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Figure 4: Correct logits of Model A $( P i z z a )$ after circle isolation. The rightmost pizza is accompanying the third pizza (discussed in Section 3.4 and Appendix D). Top: The logit pattern depends on $a - b$ . Bottom: Embeddings for each circle.
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Using these isolated embeddings, we may additionally calculate the isolated logits directly with formulas in Figure 1 and compare with the actual logits from Model A. Results are displayed in Table 2. We find that $Q _ { a b c } ( P i z z a )$ explains substantially more variance than $Q _ { a b c } ( C l o c k )$ .
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Why do we only analyze correct logits? The logits from the $P i z z a$ algorithm are given by $Q _ { a b c } ( P i z z a ) = | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a +$ b − c)). By contrast, the Clock algorithm has logits $Q _ { a b c } ( C l o c k ) = \cos ( w _ { k } ( a + b - c ) )$ . In a word, $Q _ { a b c } ( P i z z a )$ has an extra multiplicative factor $\vert \cos ( w _ { k } ( a - b ) / 2 ) \vert$ compared to $Q _ { a b c } ( C l o c k )$ . By constraining $c = a + b$ (thus $\cos ( w _ { k } ( a + b - c ) ) = 1 \rangle$ , the factor $\vert \cos ( w _ { k } ( a - b ) / 2 ) \vert$ can be identified.
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(Unexpected) dependence of logits $Q _ { a b c } ( C l o c k )$ on $a + b$ : Although our analysis above expects logits $Q _ { a b c } ( C l o c k )$ not to depend on $a - b$ , they do not predict its dependence on $a + b$ . In Figure 5, we surprisingly find that $Q _ { a b c } ( C l o c k )$ is sensitive to this sum. Our conjecture is that Step 1 and Step 2 of the Clock are implemented (almost) noiselessly, such that same-label samples collapse to the same point after Step 2. However, Step 3 (classification) is imperfect after circle isolation, resulting in fluctuations of logits. Inputs with common sums $a + b$ produce the same logits.
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Figure 5: Correct logits of Model B (Clock) after circle isolation. Top: The logit pattern depends on $a + b$ . Bottom: Embeddings for each circle.
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$$
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\frac { \mathrm { C i r c l e ~ \left| ~ \right.} w _ { k } } { \# 1 } \frac { w _ { k } } { 2 \pi / 5 9 \cdot 1 7 ~ \frac { \ d / { \operatorname { Q } _ { a b c } ( \mathrm { c l o c k } ) } \mathrm { F V E } } { 7 5 . 4 1 \% } \frac { \ d / { Q _ { a b c } ( \mathrm { p i z z a } ) } \mathrm { F V E } } { 9 9 . 1 8 \% } }
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$$
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Table 2: After isolating circles in the input embedding, fraction of variance explained (FVE) of all Model A’s output logits $( 5 9 \times 5 9 \times 5 9$ of them) by various formulas. Both model output logits and formula results’ are normalized to mean 0 variance 1 before taking FVE. $w _ { k }$ ’s are calculated according to the visualization. For example, distance between 0 and 1 in Circle #1 is 17, so $w _ { k } = 2 \pi / 5 9 \cdot 1 7$ .
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# 3.4 Third Evidence for Pizza: Accompanied & Accompanying Pizza
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The Achilles’ heel of the Pizza algorithm is antipodal pairs. If two inputs $( a , b )$ happen to lie antipodally, then their middle point will lie at the origin, where the correct “pizza slice” is difficult to identify. For example in Figure 1 right, antipodal pairs are (1,7), (2,8), (3,9) etc., whose middle points all collapse to the origin, but their class labels are different. Models cannot distinguish between, and thus correctly classify, these pairs. Even for odd $p$ ’s where there are no strict antipodal pairs, approximately antipodal pairs are also more likely to be classified incorrectly than non-antipodal pairs.
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Intriguingly, neural networks find a clever way to compensate for this failure mode. we find that pizzas usually come with “accompanying pizzas”. An accompanied pizza and its accompanying pizza complement each other in the sense that near-antipodal pairs in the accompanied pizza become adjacent or close (i.e, very non-antipodal) in the accompanying pizza. If we denote the difference between adjacent numbers on the circle as $\delta$ and $\delta _ { 1 }$ , $\delta _ { 2 }$ for accompanied and accompanying pizzas, respectively, then $\delta _ { 1 } = 2 \delta _ { 2 }$ (mod $p$ ). In the experiment, we found that pizzas #1/#2/#3 in Figure 4 all have accompanying pizzas, which we call pizzas #4/#5/#6 (see Appendix D for details). However, these accompanying pizzas do not play a significant role in final model predictions 2. We conjecture that training dynamics are as follows: (1) At initialization, pizzas #1/#2/#3 correspond to three different “lottery tickets” [9]. (2) In early stages of training, to compensate the weaknesses (antipodal pairs) of pizzas #1/#2/#3, pizzas #4/#5/#6 are formed. (3) As training goes on (in the presence of weight decay), the neural network gets pruned. As a result, pizzas #4/#5/#6 are not significantly involved in prediction, although they continue to be visible in the embedding space.
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# 4 The Algorithmic Phase Space
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In Section 3, we have demonstrated a typical Clock (Model A) and a typical Pizza (Model B). In this section, we study how architectures and hyperparametes govern the selection of these two algorithmic “phases”. In Section 4.1, we propose quantitative metrics that can distinguish between Pizza and Clock. In Section 4.2, we observe how these metrics behave with different architectures and hyperparameters, demonstrating sharp phase transitions. The results in this section focus Clock and Pizza models, but other algorithmic solutions to modular addition are also discovered, and explored in more detail in Appendix B.
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# 4.1 Metrics
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We wish to study the distribution of Pizza and Clock algorithms statistically, which will require us to distinguish between two algorithms automatically. In order to do so, we formalize our observations in Section 2.2 and 2.3, arriving at two metrics: gradient symmetricity and distance irrelevance.
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# 4.1.1 Gradient Symmetricity
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To measure the symmetricity of the gradients, we select some input-output group $( a , b , c )$ , compute the gradient vectors for the output logit at position $c$ with respect to the input embeddings, and then compute the cosine similarity. Taking the average over many pairs yields the gradient symmetricity. Definition 4.1 (Gradient Symmetricity). For a fixed set $S \subseteq \mathbb { Z } _ { p } ^ { 3 }$ of input-output pairs3, define gradient-symmetricity of a network $M$ with embedding layer $E$ as
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$$
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s _ { g } \equiv \frac { 1 } { | S | } \sum _ { ( a , b , c ) \in S } s i m \left( \frac { \partial Q _ { a b c } } { \partial { \bf E } _ { a } } , \frac { \partial Q _ { a b c } } { \partial { \bf E } _ { b } } \right) ,
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$$
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where $\begin{array} { r } { s i m ( a , b ) = \frac { a \cdot b } { | a | | b | } } \end{array}$ is the cosine-similarity, $Q _ { a b c }$ is the logit for class $c$ given input a and b. It is clear that $s _ { g } \in [ - 1 , 1 ]$ .
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As we discussed in Section 2.2, the Pizza algorithm has symmetric gradients while the Clock algorithm has asymmetric ones. Model A and Model B in Section 3 have gradient symmetricity $9 9 . 3 \hat { 7 } \%$ and $3 3 . 3 6 \%$ , respectively (Figure 2).
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# 4.1.2 Distance Irrelevance
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To measure the dependence of correct logits on differences between two inputs, which reflect the distances of the inputs on the circles, we measure how much of the variance in the correct logit matrix depends on it. We do so by comparing the average standard deviation of correct logits from inputs with the same differences and the standard deviation from all inputs.
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Definition 4.2 (Distance Irrelevance). For some network $M$ with correct logit matrix $L$ $( L _ { i , j } =$ $Q _ { i j , i + j } )$ , define its distance irrelevance as
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$$
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q \equiv \frac { \frac { 1 } { p } \sum _ { d \in \mathbb { Z } _ { p } } \operatorname { s t d } \left( L _ { i , i + d } \mid i \in \mathbb { Z } _ { p } \right) } { \operatorname { s t d } \left( L _ { i , j } \mid i , j \in \mathbb { Z } _ { p } ^ { 2 } \right) } ,
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$$
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where std computes the standard deviation of a set. It is clear that $q \in [ 0 , 1 ]$
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Model A and Model B in Section 3 give distance irrelevance 0.17 and 0.85, respectively (Figure 3). A typical distance irrelevance from the Pizza algorithm ranges from 0 to 0.4 while a typical distance irrelevance from Clock algorithm ranges from 0.4 to 1.
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# 4.1.3 Which Metric is More Decisive?
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When the two metrics have conflicting results, which one is more decisive? We consider distance irrelevance as the decisive factor of the Pizza algorithm, as the output logits being dependent on the distance is highly suggestive of Pizza. On the other hand, gradient symmetricity can be used to rule out the Clock algorithm, as it requires multiplying (transformed) inputs which will result in asymmetric gradients. Figure 6 confirmed that at low distance irrelevance (suggesting pizza) the gradient symmetricity is almost always close to 1 (suggesting non-clock).
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Figure 6: Distance irrelevance vs gradient symmetricity over all the main experiments.
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# 4.2 Identifying algorithmic phase transitions
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How do models “choose” whether to implement the Clock or Pizza algorithm? We investigate this question by interpolating between Model A (transformer without attention) and Model B (transformer with attention). To do so, we introduce a new hyperparameter $\alpha$ we call the attention rate.
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For a model with attention rate $\alpha$ , we modify the attention matrix $M$ for each attention head to be $M ^ { \prime } = M \alpha + J ( 1 - \alpha )$ . In other words, we modify this matrix to consist of a linear interpolation between the all-one matrix and the original attention (post-softmax), with the rate $\alpha$ controlling how much of the attention is kept. The transformer with and without attention corresponds to the case where $\alpha = 1$ (attention kept) and $\alpha = 0$ (constant attention matrix). With this parameter, we can control the balance of attention versus linear layers in transformers.
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We performed the following set of experiments on transformers (see Appendix F.1 for architecture and training details). (1) One-layer transformers with width 128 and attention rate uniformly sampled in $[ 0 , 1 ]$ (Figure 7). (2) One-layer transformers with width log-uniformly sampled in [32, 512] and attention rate uniformly sampled in $[ 0 , 1 ]$ (Figure 7). (3) Transformers with 2 to 4 layers, width 128 and attention rate uniformly sampled in $[ 0 , 1 ]$ (Figure 11).
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The Pizza and the Clock algorithms are the dominating algorithms with circular embeddings. For circular models, most observed models either have low gradient symmetricity (corresponding to the Clock algorithm) or low distance irrelevance (corresponding to the Pizza algorithm).
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Two-dimensional phase change observed for attention rate and layer width. For the fixedwidth experiment, we observed a clear phase transition from the Pizza algorithm to the Clock algorithm (characterized by gradient symmetricity and distance irrelevance). We also observe an almost linear phase boundary with regards to both attention rate and layer width. In other words, the attention rate transition point increases as the model gets wider.
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Dominance of linear layers determines whether the Pizza or the Clock algorithm is preferred. For one-layer transformers, we study the transition point against the attention rate and the width:
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• The Clock algorithm dominates when the attention rate is higher than the phase change point, and the Pizza algorithm dominates when the attention rate is lower than the point. Our explanation is: At a high attention rate, the attention mechanism is more prominent in the network, giving rise to the clock algorithm. At a low attention rate, the linear layers are more prominent, giving rise to the pizza algorithm. • The phase change point gets higher when the model width increases. Our explanation is: When the model gets wider, the linear layers become more capable while the attention mechanism receive less benefit (attentions remain scalars while outputs from linear layers become wider vectors). The linear layer therefore gets more prominence with a wider model.
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Possibly hybrid algorithms between the Clock and the Pizza algorithms. The continuous phase change suggests the existence of networks that lie between the Clock and the Pizza algorithms. This is achievable by having some principal components acting as the Clock and some principal components acting as the Pizza.
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Figure 7: Training results from 1-layer transformers. Each point in the plots represents a training run reaching circular embeddings and $100 \%$ validation accuracy. See Appendix C for additional plots. Top: Model width fixed to be 128. Bottom: Model width varies. The phase transition lines are calculated by logistic regression (classify the runs by whether gradient symmetricity $> 9 8 \%$ and whether distance irrelevance $< 0 . 6$ ).
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Existence of non-circular algorithms. Although our presentation focuses on circular algorithms (i.e., whose embeddings are circular), we find non-circular algorithms (i.e., whose embeddings do not form a circle when projected onto any plane) to be present in neural networks. See Appendix B for preliminary findings. We find that deeper networks are more likely to form non-circular algorithms. We also observe the appearance of non-circular networks at low attention rates. Nevertheless, the Pizza algorithm continues to be observed (low distance irrelevance, high gradient symmetricity).
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# 5 Related Work
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Mechanistic interpretability aims to mechanically understand neural networks by reverse engineering them [2, 3, 5, 4, 10, 11, 1, 12, 13, 14]. One can either look for patterns in weights and activations by studying single-neuron behavior (superposition [11], monosemantic neurons [15]), or study meaningful modules or circuits grouped by neurons [4, 14]. Mechanistic interpretability is closely related to training dynamics [8, 13, 1].
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Learning mathematical tasks: Mathematical tasks provide useful benchmarks for neural network interpretability, since the tasks themselves are well understood. The setup could be learning from images [16, 17], with trainable embeddings [18], or with number as inputs [19, 5]. Beyond arithmetic relations, machine learning has been applied to learn other mathematical structures, including geometry [20], knot theory [21] and group theory [22].
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Algorithmic phase transitions: Phase transitions are present in classical algorithms [23] and in deep learning [6, 24, 25]. Usually the phase transition means that the algorithmic performance sharply changes when a parameter is varied (e.g., amount of data, network capacity etc). However, the phase transition studied in this paper is representational: both clock and pizza give perfect accuracy, but arrive at answers via different interal computations. These model-internal phase transitions are harder to study, but closer to corresponding phenomena in physical systems [24].
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Algorithm learning in neural networks: Emergent abilities in deep neural networks, especially large language models, have recently attracted significant attention [26]. An ability is “emergent” if the performance on a subtask suddenly increases with growing model sizes, though such claims depend on the choice of metric [27]. It has been hypothesized that the emergence of specific capability in a model corresponds to the emergence of a modular circuit responsible for that capability, and that emergence of some model behaviors thus results from a sequence of quantized circuit discovery steps [5].
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# 6 Conclusions
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We have offered a closer look at recent findings that familiar algorithms arise in neural networks trained on specific algorithmic tasks. In modular arithmetic, we have shown that such algorithmic discoveries are not inevitable: in addition to the Clock algorithm reverse-engineered by [1], we find other algorithms (including a Pizza algorithm, and more complicated procedures) to be prevalent in trained models. These different algorithmic phases can be distinguished using a variety of new and existing interpretability techniques, including logit visualization, isolation of principle components in embedding space, and gradient-based measures of model symmetry. These techniques make it possible to automatically classify trained networks according to the algorithms they implement, and reveal algorithmic phase transitions in the space of model hyperparameters. Here we found specifically that the emergence of a Pizza or Clock algorithm depends on the relative strength of linear layers and attention outputs. We additionally showed that these algorithms are not implemented in isolation; instead, networks sometimes ensemble multiple copies of an algorithm in parallel. These results offer exciting new challenges for mechanistic interpretability: (1) How to find, classify, and interpret unfamiliar algorithms in a systematic way? (2) How to disentangle multiple, parallel algorithm implementations in the presence of ensembling?
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Limitations We have focused on a single learning problem: modular addition. Even in this restricted domain, qualitatively different model behaviors emerge across architectures and seeds. Significant additional work is needed to scale these techniques to the even more complex models used in real-world tasks.
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Broader Impact We believe interpretability techniques can play a crucial role in creating and improving safe AI systems. However, they may also be used to build more accurate systems, with the attendant risks inherent in all dual-use technologies. It is therefore necessary to exercise caution and responsible decision-making when deploying such techniques.
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# Acknowledgement
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We would like to thank Mingyang Deng and anonymous reviewers for valuable and fruitful discussions and MIT SuperCloud for providing computation resources. ZL and MT are supported by the Foundational Questions Institute, the Rothberg Family Fund for Cognitive Science and IAIFI through NSF grant PHY-2019786. JA is supported by a gift from the OpenPhilanthropy Foundation.
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[27] Rylan Schaeffer, Brando Miranda, and Sanmi Koyejo. Are emergent abilities of large language models a mirage? arXiv preprint arXiv:2304.15004, 2023.
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[28] Michael Hassid, Hao Peng, Daniel Rotem, Jungo Kasai, Ivan Montero, Noah A. Smith, and Roy Schwartz. How much does attention actually attend? questioning the importance of attention in pretrained transformers. In Findings of the Association for Computational Linguistics: EMNLP 2022, pages 1403–1416, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics.
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[29] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In International Conference on Learning Representations, 2019.
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# Supplementary material
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A Mathematical Analysis and An Example of Pizza Algorithm
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In the pizza algorithm, we have $E _ { a b } = \cos ( w _ { k } ( a - b ) / 2 ) \cdot ( \cos ( w _ { k } ( a + b ) / 2 ) , \sin ( w _ { k } ( a + b ) / 2 ) )$ , as $\cos x + \cos y = \cos ( ( x - y ) / 2 ) ( 2 \cos ( ( x + y ) / 2 ) )$ and $\sin x + \sin y = \cos ( ( x - y ) / 2 ) ( 2 \sin ( ( x +$ $y ) / 2 )$ ).
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To get $| \mathrm { c o s } ( w _ { k } ( a - b ) / 2 ) | ( \mathrm { c o s } ( w _ { k } ( a + b ) ) , \mathrm { s i n } ( w _ { k } ( a + b ) ) )$ , we generalize this to $| \cos ( w _ { k } ( a -$ $b ) / 2 ) | \cos ( w _ { k } ( a + b - u ) )$ (the two given cases correspond to $u = 0$ and $u = \pi / 2 / w _ { k }$ ).
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$$
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\begin{array} { r l r } & { } & { | ( \cos ( w _ { k } u / 2 ) , \sin ( w _ { k } u / 2 ) ) \cdot E _ { a b } | = | \cos ( w _ { k } ( a - b ) / 2 ) \cos ( w _ { k } ( a + b - u ) / 2 ) | } \\ & { } & { | ( - \sin ( w _ { k } u / 2 ) , \cos ( w _ { k } u / 2 ) ) \cdot E _ { a b } | = | \cos ( w _ { k } ( a - b ) / 2 ) \sin ( w _ { k } ( a + b - u ) / 2 ) | } \end{array}
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$$
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thus their difference will be equal to
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$$
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| \cos ( w _ { k } ( a - b ) / 2 ) | ( | \cos ( w _ { k } ( a + b - u ) / 2 ) | - | \sin ( w _ { k } ( a + b - u ) / 2 ) | ) .
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$$
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Now notice $| \cos ( t ) | - | \sin ( t ) | \approx \cos ( 2 t )$ for any $t \in \mathbb { R }$ (Figure 8), so the difference is approximately $| \mathrm { c o s } ( w _ { k } ( a - b ) / 2 ) | \mathrm { c o s } ( w _ { k } ( a + b - u ) )$ .
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Figure 8: $| \cos ( t ) | - | \sin ( t ) |$ is approximately $\cos ( 2 t )$ for any $t \in \mathbb { R }$
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Plugging in $u = 0$ and $u = \pi / 2 / w _ { k }$ as mentioned, we get the following particular implementation of the pizza algorithm.
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# Algorithm: Pizza, Example
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Step 1 On given input $a$ and $b$ , circularly embed them to two vectors on the circumference $( \cos ( w _ { k } a ) , \sin ( w _ { k } a ) )$ and $( \cos ( w _ { k } b ) , \sin ( w _ { k } b ) )$ .
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Step 2 Compute:
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$$
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\begin{array} { r l } & { \alpha = | \cos ( w _ { k } a ) + \cos ( w _ { k } b ) | / 2 - | \sin ( w _ { k } a ) + \sin ( w _ { k } b ) | / 2 } \\ & { \quad \approx | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a + b ) ) } \\ & { \beta = | \cos ( w _ { k } a ) + \cos ( w _ { k } b ) + \sin ( w _ { k } a ) + \sin ( w _ { k } b ) | / ( 2 \sqrt { 2 } ) } \\ & { \quad - | \cos ( w _ { k } a ) + \cos ( w _ { k } b ) - \sin ( w _ { k } a ) - \sin ( w _ { k } b ) | / ( 2 \sqrt { 2 } ) } \\ & { \quad = | \cos ( w _ { k } a - \pi / 4 ) + \cos ( w _ { k } b - \pi / 4 ) | / 2 - | \sin ( w _ { k } a - \pi / 4 ) + \sin ( w _ { k } b - \pi / 4 ) | / 2 } \\ & { \quad \approx | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a + b ) - \pi / 2 ) = | \cos ( w _ { k } ( a - b ) / 2 ) | \sin ( w _ { k } ( a + b ) ) } \end{array}
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$$
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Step 3 Output of this pizza is computed as a dot product.
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$$
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Q _ { a b c } ^ { \prime } = \alpha \cos ( w _ { k } c ) + \beta \sin ( w _ { k } c ) \approx | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a + b - c ) )
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$$
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Similar circuits are observed in the wild, but instead of the above two-term approximation, a more complicated one is observed. See Appendix $\mathrm { L }$ for details.
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The extra $\vert \cos ( w _ { k } ( a - b ) / 2 ) \vert$ term is not a coincidence. We can generalize our derivation as the following.
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Lemma A.1. A symmetric function $f ( x , y )$ that is a linear combination of $\cos x , \sin x , \cos y , \sin y ^ { 4 }$ can always be written as $\cos ( ( x - y ) / 2 ) g ( x + y )$ for some function $g$ .
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Proof. Notice $\cos x + \cos y = \cos ( ( x - y ) / 2 ) ( 2 \cos ( ( x + y ) / 2 ) )$ ) and $\sin x + \sin y = \cos ( ( x -$ $y ) / 2 ) ( 2 \sin ( ( x + y ) / 2 ) )$ , so $\alpha ( \cos x + \cos y ) + \beta ( \sin x + \sin y ) = \cos ( ( x - y ) / 2 ) ( 2 \alpha \cos ( ( x + \cos y ) 2 ) + \beta ( \sin x + \sin y ) = \cos ( ( x - y ) / 2 )$ $y ) / 2 ) + 2 \beta \sin ( ( x + y ) / 2 ) )$ . □
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This is why we consider the output pattern with the $\vert \cos ( w _ { k } ( a - b ) / 2 ) \vert$ terms rather than the actual computation circuits as the determinant feature of the pizza algorithm.
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# B Non-Circular algorithms
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One thing that further complicates our experiment is the existence of non-circular embeddings. While only circular algorithms are reported in the previous works [8, 1], many non-circular embeddings are found in our experiments, e.g., 1D lines or 3D Lissajous-like curves, as shown in Figure 9. We leave the detailed analysis of these non-circular algorithms for future study. Since circular algorithms are our primary focus of study, we propose the following metric circularity to filter out non-circular algorithms. The metric reaches maximum 1 when the principal components aligns with cosine waves.
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Figure 9: Visualization of the principal components of input embeddings for two trained non-circular models. Top: A line-like first principal component. Notice the re-arranged x axis (token id). Bottom: First three principal components forming a three-dimensional non-circular pattern. Each point represents the embedding of a token.
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Definition B.1 (Circularity). For some network, suppose the l-th principal component of its input embeddings is $v _ { l , 0 } , v _ { l , 1 } , \cdots , v _ { l , p - 1 }$ , define its circularity based on first four components as
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+
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+
$$
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c = \frac { 1 } { 4 } \sum _ { l = 1 } ^ { 4 } \left( \operatorname* { m a x } _ { k \in \left[ 1 , 2 , \cdots , p - 1 \right] } \left( \frac { 2 } { p { \sum _ { j = 0 } ^ { p - 1 } { v _ { l , j } ^ { 2 } } } } \left| \sum _ { j = 0 } ^ { p - 1 } v _ { l , j } e ^ { 2 \pi i \cdot j k / p } \right| ^ { 2 } \right) \right)
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+
$$
|
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where i is the imaginary unit. $c \in [ 0 , 1 ]$ by Fourier analysis. $c = 1$ means first four components are Fourier waves.
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|
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Both Model A and Model B in Section 3 have a circularity around $9 9 . 8 \%$ and we consider models with circularity $\geq 9 9 . 5 \%$ circular.
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+
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# C More Results from the Main Experiments
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Here we provide Figure 7 with non-circular networks unfiltered (Figure 10). We can see more noise emerging in the plot. We also provide the training results from multi-layer transformers (Figure 11).
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+
|
| 325 |
+
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+
Figure 10: Training results from 1-layer transformers. Each point in the plots represents a training run reaching $100 \%$ validation accuracy. Among all the trained 1-layer transformers, $3 4 . 3 1 \%$ are circular. Top: Model width fixed to be 128. Bottom: Model width varies.
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+
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# D Pizzas Come in Pairs
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Cautious readers might notice that the pizza algorithm is imperfect - for near antipodal points, the sum vector will have a very small norm and the result will be noise-sensitive. While the problem is partially elevated by the use of multiple circles instead of one, we also noticed another pattern emerged: accompanying pizzas.
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The idea is the following: suppose the difference between adjacent points is $2 k$ mod $p$ , then the antipodal points have difference $\pm k$ . Therefore, if we arrange a new circle with a difference $k$ for adjacent points, we will get a pizza that works best for formerly antipodal points.
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+
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# Algorithm: Accompanying Pizza
|
| 335 |
+
|
| 336 |
+
Step 1 Take $w _ { k }$ as of the accompanied pizza. On given input $a$ and $b$ , circularly embed them to two vectors on the circumference $( \cos ( 2 w _ { k } a ) , \sin ( 2 w _ { k } a ) )$ and $( \cos ( 2 w _ { k } b ) , \sin ( 2 w _ { k } b ) )$ .
|
| 337 |
+
|
| 338 |
+
Step 2 Compute the midpoint:
|
| 339 |
+
|
| 340 |
+
$$
|
| 341 |
+
s = { \frac { 1 } { 2 } } ( \cos ( 2 w _ { k } a ) + \cos ( 2 w _ { k } b ) , \sin ( 2 w _ { k } a ) + \sin ( 2 w _ { k } b ) )
|
| 342 |
+
$$
|
| 343 |
+
|
| 344 |
+
Step 3 Output of this pizza is computed as a dot product.
|
| 345 |
+
|
| 346 |
+
$$
|
| 347 |
+
A _ { c } = - ( \cos ( w _ { k } c ) , \sin ( w _ { k } c ) ) \cdot s
|
| 348 |
+
$$
|
| 349 |
+
|
| 350 |
+
This is exactly what we observed in Model A (Table 3, Figure 13). With the six circles (pizzas and accompanying pizzas) included in the embedding, Model A also gets $100 \%$ accuracy.
|
| 351 |
+
|
| 352 |
+
|
| 353 |
+
Figure 11: Training results from transformers with 2, 3 and 4 layers. Among all the trained transformers with 2, 3 and 4 layers, $9 . 9 5 \%$ , $1 1 . 5 5 \%$ and $6 . 0 8 \%$ are circular, respectively.
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| 354 |
+
|
| 355 |
+
|
| 356 |
+
Figure 12: An Illustration on the Accompanying Pizza Algorithm
|
| 357 |
+
|
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+
<table><tr><td rowspan=1 colspan=1>Circle</td><td rowspan=1 colspan=1>Wk</td><td rowspan=1 colspan=1>Ac FVE</td></tr><tr><td rowspan=1 colspan=1>#4 (accompanying #1)</td><td rowspan=1 colspan=1>2π/59·17</td><td rowspan=1 colspan=1>97.56%</td></tr><tr><td rowspan=1 colspan=1>#5 (accompanying #2)</td><td rowspan=1 colspan=1>2π/59.3</td><td rowspan=1 colspan=1>97.23%</td></tr><tr><td rowspan=1 colspan=1>#6 (accompanying #3)</td><td rowspan=1 colspan=1>2π/59·44</td><td rowspan=1 colspan=1>97.69%</td></tr></table>
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| 359 |
+
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+
Table 3: After isolating accompanying circles in the input embedding, fraction of variance explained (FVE) of all Model A’s output logits by various formulas. Both model output logits and formula results’ are normalized to mean 0 variance 1 before taking FVE. Accompanying and accompanied pizza have the same $w _ { k }$ .
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| 361 |
+
|
| 362 |
+
#
|
| 363 |
+
|
| 364 |
+
|
| 365 |
+
Figure 13: Correct logits of Model A (Pizza) after circle isolation. Only accompanying pizzas are displayed. Notice the complementing logit pattern (Figure 4).
|
| 366 |
+
|
| 367 |
+
# E Results in Other Linear Architectures
|
| 368 |
+
|
| 369 |
+
While this is not the primary focus of our paper, we also ran experiments on the following four different linear model setups (see Section F.2 for setup details).
|
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+
|
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+
• For all the models, we first encode input tokens $( a , b )$ with a trainable embedding layer $W _ { E }$ : $x _ { 1 } = W _ { E , a }$ , $x _ { 2 } = W _ { E , b }$ (positional embedding removed for simplicity). $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are trainable linear layers. The outmost layers (commonly referred as unembed layers) have no biases and the other layers have biases included for generality.
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+
• Model $\alpha$ : calculate output logits as $L _ { 2 } ( \mathrm { R e L U } ( L _ { 1 } ( x _ { 1 } + x _ { 2 } ) ) )$ ).
|
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+
• Model $\beta$ : calculate output logits as $L _ { 3 } ( \mathrm { R e L U } ( L _ { 2 } ( \mathrm { R e L U } ( L _ { 1 } ( x _ { 1 } + x _ { 2 } ) ) ) ) ) ,$ ).
|
| 374 |
+
• Model $\gamma$ : calculate output logits as $L _ { 3 } ( \mathrm { R e L U } ( L _ { 2 } ( \mathrm { R e L U } ( L _ { 1 } ( x _ { 1 } ) + L _ { 1 } ( x _ { 2 } ) ) ) ) )$ .
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| 375 |
+
• Model $\delta$ : calculate output logits as $L _ { 2 } \big ( \mathrm { R e L U } ( L _ { 1 } ( [ x _ { 1 } ; x _ { 2 } ] ) ) \big )$ ) $[ x _ { 1 } ; x _ { 2 } ]$ stands for the concatenation of $x _ { 1 }$ and $x _ { 2 }$ )
|
| 376 |
+
|
| 377 |
+
The results are shown in Figure 14. Rather surprisingly, Model $\alpha$ , Model $\beta$ and Model $\delta$ gave radically different results. Model $\beta$ and Model $\gamma$ are very similar, and in general they are more pizza-like than Model $\alpha$ , with lower distance irrelevancy and higher circularity. This could be explained by the addition of an extra linear layer.
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| 378 |
+
|
| 379 |
+
|
| 380 |
+
Figure 14: Training results from linear models. Each point in the first-row plots represents a training run. The second row are histograms for distance irrelevancy of each model type.
|
| 381 |
+
|
| 382 |
+
However, Model $\delta$ gave very different results from Model $\alpha$ although they are both one-layer linear models. It is more likely to be non-circular and have very high distance irrelevancy in general. In other words, concatenating instead of adding embeddings yields radically different behaviors in one-layer linear model. This result, again, alarmed us the significance of induction biases in neural networks.
|
| 383 |
+
|
| 384 |
+
We also want to note that using different embeddings on two tokens of Model $\alpha$ doesn’t resolve the discrepancy. The following model
|
| 385 |
+
|
| 386 |
+
• Model $\alpha ^ { \prime }$ : calculate output logits as $L _ { 2 } ( \mathrm { R e L U } ( L _ { 1 } ( x _ { 1 } + x _ { 2 } ) ) )$ ) where $x _ { 1 } = W _ { E , a } ^ { A }$ , $x _ { 2 } =$ $W _ { E , b } ^ { B }$ on input $( a , b )$ and $W _ { E } ^ { A } , W _ { E } ^ { B }$ are different embedding layers.
|
| 387 |
+
|
| 388 |
+
gives roughly the same result as of Model $\alpha$ (Figure 14, lower right corner).
|
| 389 |
+
|
| 390 |
+
Figure 15 shows the correct logits after circle isolation (Section 3.3) of a circular model from Model $\beta$ implementing the pizza algorithm. Figure 16 shows the correct logits after circle isolation (Section 3.3) of a circular model from Model $\delta$ . We can see the pattern is similar but different from the one of clock algorithm (Figure 5). We leave the study of such models to future work.
|
| 391 |
+
|
| 392 |
+
# F Architecture and Training Details
|
| 393 |
+
|
| 394 |
+
# F.1 Transformers
|
| 395 |
+
|
| 396 |
+
Here we describe our setup for the main experiments. See Appendix E and Appendix I for experiments on different setups.
|
| 397 |
+
|
| 398 |
+
Architecture We train bidirectional transformers (attention unmasked) to perform modular addition mod $p$ where $p = 5 9$ . To calculate $( a + b )$ mod $p$ , the input is provided to the model as a sequence of two tokens $[ a , b ]$ . The output logit at the last token is considered as the output of the model. For a transformer with “width” $d$ , the input embedding and the residue stream will be $d$ -dimensional, 4 attention heads of $\lfloor d / 4 \rfloor$ dimensions will be employed, and the MLP will be of $4 d$ hidden units. By default $d = 1 2 8$ is chosen. ReLU is used as the activation function and layer normalization isn’t applied. The post-softmax attention matrix is interpolated between an all-one matrix and original as specified by the attention rate (Section 4.2). We want to point out that the setup of constant-attention transformers is also considered in the previous work [28].
|
| 399 |
+
|
| 400 |
+
|
| 401 |
+
Figure 15: Correct logits from Model $\beta$ after circle isolation.
|
| 402 |
+
|
| 403 |
+
|
| 404 |
+
Figure 16: Correct logits from Model $\delta$ after circle isolation.
|
| 405 |
+
|
| 406 |
+
Data Among all possible data points $\gamma ^ { 2 } = 3 4 8 1$ of them), we randomly select $8 0 \%$ as training samples and $2 0 \%$ as validation samples. This choice (small $p$ and high training data fraction) helps accelerating the training.
|
| 407 |
+
|
| 408 |
+
Training We used AdamW optimizer [29] with learning rate $\gamma = 0 . 0 0 1$ and weight decay factor $\beta = 2$ . We do not use minibatches and the shuffled training data is provided as a whole batch in every epoch. For each run, we start the training from scratch and train for 20, 000 epoches. We removed the runs that did not reach $1 0 0 \%$ validation accuracy at the end of the training (majority of the runs reached $1 0 0 \%$ ).
|
| 409 |
+
|
| 410 |
+
# F.2 Linear Models
|
| 411 |
+
|
| 412 |
+
Here we describe our setup for the linear model experiments (Appendix E).
|
| 413 |
+
|
| 414 |
+
Architecture We train several types of linear models to perform modular addition mod $p$ where $p = 5 9$ . The input embedding, residue stream and hidden layer are all $d = 2 5 6$ dimensional. ReLU is used as the activation function. The actual structures of network types are specified in Appendix E.
|
| 415 |
+
|
| 416 |
+
Data & Training Same as in the previous section (Section F.1).
|
| 417 |
+
|
| 418 |
+
# F.3 Computing Resources
|
| 419 |
+
|
| 420 |
+
A total of 226 GPU days of NVidia V100 is spent on this project, although we expect a replication would take significantly fewer resources.
|
| 421 |
+
|
| 422 |
+
# G Mathematical Description of Constant-Attention transformer
|
| 423 |
+
|
| 424 |
+
In this section, we examine the structure of constant-attention transformers loosely following the notation of [10].
|
| 425 |
+
|
| 426 |
+
Denote the weight of embedding layer as $W _ { E }$ , the weight of positional embedding as $W _ { \mathrm { p o s } }$ , the weight of the value and output matrix of the $j$ -th head of the $t { \cdot }$ -th layer as $W _ { V } ^ { t , j }$ and $W _ { O } ^ { t , j }$ , the weights and biases of the input linear map of MLP in the -th layer as $\it { W _ { \mathrm { i n } } ^ { t } }$ and $b _ { \mathrm { i n } } ^ { t }$ , the corresponding weights and biases of the output linear map as $W _ { \mathrm { o u t } } ^ { t }$ and $b _ { \mathrm { o u t } } ^ { t }$ , and the weight of the unembedding layer as $W _ { U }$ . Notice that the query and the key matrices are irrelevant as the attention matrix is replaced with an all-one matrix. Denote $x ^ { j }$ as the value of residue stream vector after the first $j$ layers and denote $c _ { i }$ as the character in the $i$ -th position. We use subscripts like $x _ { t }$ to denote taking a specific element of vector.
|
| 427 |
+
|
| 428 |
+
We can formalize the logit calculation as the following:
|
| 429 |
+
|
| 430 |
+
• Embedding: $x _ { i } ^ { 0 } = W _ { E , c _ { i } } + W _ { \mathrm { p o s } , i }$ .
|
| 431 |
+
• For each layer $t$ from 1 to $n _ { \mathrm { l a y e r } }$ : – Constant Attention: $\begin{array} { r } { \boldsymbol { w } _ { i } ^ { t } = \boldsymbol { x } _ { i } ^ { t - 1 } + \sum _ { j } \boldsymbol { W } _ { O } ^ { t , j } \boldsymbol { W } _ { V } ^ { t , j } \sum _ { k } \boldsymbol { x } _ { k } ^ { t - 1 } . } \end{array}$ – MLP: $x ^ { t } = w ^ { t } + b _ { \mathrm { o u t } } ^ { t } + W _ { \mathrm { o u t } } ^ { t } \mathrm { R e L U } ( b _ { \mathrm { i n } } ^ { t } + W _ { \mathrm { i n } } ^ { t } w ^ { t } )$ .
|
| 432 |
+
• Output: $O = W _ { U } x ^ { n _ { \mathrm { l a y e r } } }$ .
|
| 433 |
+
|
| 434 |
+
In the particular case where the input length is 2, the number of layer is $1$ , and we focus on the logit of the last position, we may restate as the following (denote $z$ as $\dot { x } ^ { 1 }$ and $y$ as $w ^ { 1 }$ ):
|
| 435 |
+
|
| 436 |
+
• Embedding: $x _ { 1 } = W _ { E , c _ { 1 } } + W _ { \mathrm { p o s } , 1 } , x _ { 2 } = W _ { E , c _ { 2 } } + W _ { \mathrm { p o s } , 2 } .$ • Constant Attention: $\begin{array} { r } { y = x _ { 2 } + \sum _ { j } W _ { O } ^ { j } W _ { V } ^ { j } ( x _ { 1 } + x _ { 2 } ) } \end{array}$ . • MLP: $z = y + b _ { \mathrm { o u t } } ^ { t } + W _ { \mathrm { o u t } } ^ { t } \mathrm { R e L U } ( b _ { \mathrm { i n } } ^ { t } + W _ { \mathrm { i n } } ^ { t } y )$ .
|
| 437 |
+
|
| 438 |
+
• Output: $o = W _ { U } z$
|
| 439 |
+
|
| 440 |
+
If we remove the skip connections, the network after embedding could be seen as
|
| 441 |
+
|
| 442 |
+
$$
|
| 443 |
+
o = L _ { U } \left( L _ { \mathrm { o u t } } \left( \operatorname { R e L U } \left( L _ { \mathrm { i n } } \left( \sum _ { j } L _ { O } ^ { j } \left( L _ { V } ^ { j } \left( x _ { 1 } + x _ { 2 } \right) \right) \right) \right) \right) \right)
|
| 444 |
+
$$
|
| 445 |
+
|
| 446 |
+
where $L _ { V } ^ { j } , L _ { O } ^ { j } , L _ { \mathrm { i n } } , L _ { \mathrm { o u t } } , L _ { U }$ are a series of linear layers corresponding to the matrices.
|
| 447 |
+
|
| 448 |
+
# H Pizza with Attention
|
| 449 |
+
|
| 450 |
+
Extrapolating from Figure 7, we trained transformers with width 1024 and attention rate 1 (normal attention). After several tries, we are able to observe a trained circular model with distance irrelevance 0.156 and gradient symmetricity 0.995, which fits our definition of Pizza (Figure 17).
|
| 451 |
+
|
| 452 |
+
|
| 453 |
+
Figure 17: Correct logits of the trained model in Section H after circle isolation (Section 3.3).
|
| 454 |
+
|
| 455 |
+
# I Results on Slightly Different Setups
|
| 456 |
+
|
| 457 |
+
We considered the following variations of our setups (Appendix F.1, Section 4), for which the existence of pizzas and clocks as well as the phase changes are still observed.
|
| 458 |
+
|
| 459 |
+
GeLU instead of ReLU We conducted the same 1-layer transformer experiment with activation function GeLU instead of ReLU. Very similar results are observed (Figure 18).
|
| 460 |
+
|
| 461 |
+
Encode Two Tokens Differently We conducted the 1-layer transformer experiments but with different embedding for the two tokens. Again very similar results are observed (Figure 19). We also discovered that the two tokens’ embeddings are often aligned to implement the Pizza and Clock algorithm (Figure 20).
|
| 462 |
+
|
| 463 |
+
Adding Equal Sign We conducted the 1-layer transformer experiment with an equal sign added.
|
| 464 |
+
Very similar results are observed (Figure 21).
|
| 465 |
+
|
| 466 |
+
# J Pizza Occurs Early in the Clock Training
|
| 467 |
+
|
| 468 |
+
We plotted intermediate states during the training of a model with attention (attention rate 1). Pizzalike pattern was observed early in the training, but the pattern gradually disappeared during the run (Figure 22).
|
| 469 |
+
|
| 470 |
+
|
| 471 |
+
Figure 18: Training results from 1-layer transformers with GeLU instead of ReLU as the activation function. Each point in the plots represents a training run that reached $100 \%$ validation accuracy.
|
| 472 |
+
|
| 473 |
+
|
| 474 |
+
Figure 19: Training results from 1-layer transformers where the two tokens use different embeddings (feed $[ a , b + p ]$ to the model on input $( a , b )$ ; $2 p$ tokens are handled in the embedding layer). Each point in the plots represents a training run that reached $100 \%$ validation accuracy. We did not use circularity to filter the result because it is no longer well-defined.
|
| 475 |
+
|
| 476 |
+
|
| 477 |
+
Figure 20: Correct logits after circle isolation from a trained model where two tokens use different embeddings. The blue points represent the embeddings for the first token and the green points represent the embeddings for the second token. The model is implementing the Pizza algorithm. The correct logit pattern is shifted comparing to the previous patterns because the embeddings of two tokens do not line up exactly. For example, the third circle has near-maximum correct logit for $a = 6 , b = 3$ (the two points lining up on the top) and $( a - b ) / 1 8 \equiv 1 0$ (mod 59). This is the reason that the correct logit pattern appears to be shifted 10 units down.
|
| 478 |
+
|
| 479 |
+
|
| 480 |
+
Figure 21: Training results from 1-layer transformers where an equal sign is added (feed $[ a , b , = ]$ to the model on input $( a , b )$ where $=$ is a special token; $p + 1$ tokens are handled in the embedding layer; context length of the model becomes 3). Each point in the plots represents a training run that reached $100 \%$ validation accuracy. We did not use circularity to filter the result because it is no longer well-defined.
|
| 481 |
+
|
| 482 |
+
# K Accompanying Pizza Occurs Early in the Pizza Training
|
| 483 |
+
|
| 484 |
+
We plotted intermediate states during the training of a model without attention (attention rate 0). We observed the early emergence of a pattern similar to accompanying pizza in training runs (Figure
|
| 485 |
+
|
| 486 |
+
|
| 487 |
+
Figure 22: For a 1-layer transformer with attention, correct logits after principal component (possibly non-circle) isolations at various states during the training. The pizza-like pattern gradually desolved.
|
| 488 |
+
|
| 489 |
+
23) and removing that circle brings accuracy down from $9 9 . 7 \%$ to $9 7 . 9 \%$ . They are less helpful later in the network (removing accompanying pizzas in trained Model A only brings accuracy down to $9 9 . 7 \%$ ).
|
| 490 |
+
|
| 491 |
+
|
| 492 |
+
Figure 23: Immediate state after 600 epochs of training for a 1-layer transformer with constant attention.
|
| 493 |
+
|
| 494 |
+
# L A Closer Look at a Linear Pizza Model
|
| 495 |
+
|
| 496 |
+
In this section, we provide a full picture of the linear model shown in Figure 15 by investigating the actual weights in the model.
|
| 497 |
+
|
| 498 |
+
# L.1 Model Structure
|
| 499 |
+
|
| 500 |
+
As described in Appendix E, on input $( a , b )$ , the output logits of the model is computed as
|
| 501 |
+
|
| 502 |
+
$$
|
| 503 |
+
L _ { 3 } ( \mathrm { R e L U } ( L _ { 2 } ( \mathrm { R e L U } ( L _ { 1 } ( \mathrm { E m b e d } [ a ] + \mathrm { E m b e d } [ b ] ) ) ) ) ) .
|
| 504 |
+
$$
|
| 505 |
+
|
| 506 |
+
Denote the weight of embedding layer as $W _ { E }$ , the weight of the unembedding layer $( L _ { 3 } )$ as $W _ { U }$ , and the weights and biases of $L _ { 1 }$ and $L _ { 2 }$ as $W _ { 1 } , b _ { 1 }$ and $W _ { 2 } , b _ { 2 }$ , respectively, then the output logits on input $( a , b )$ can be written as
|
| 507 |
+
|
| 508 |
+
$$
|
| 509 |
+
W _ { U } \mathrm { R e L U } ( b _ { 2 } + W _ { 2 } \mathrm { R e L U } ( b _ { 1 } + W _ { 1 } ( W _ { E } [ a ] + W _ { E } [ b ] ) ) ) .
|
| 510 |
+
$$
|
| 511 |
+
|
| 512 |
+
# L.2 General Picture
|
| 513 |
+
|
| 514 |
+
We first perform principal component visualizations on the embedding and unembedding matrices. From Figure 24, we can see that the embedding and unembedding matrices formed matching circles (circles with the same gap $\delta$ between adjacent entries).
|
| 515 |
+
|
| 516 |
+
We now give the general overview of the circuit. Each pair of matching circles forms an instance of Pizza and they operate independently (with rather limited interference). Specifically for each pair, • The embedding matrix first places the inputs $a , b$ on the circumference: $W _ { E } ^ { \prime } [ a ] \ \approx$ $( \cos ( w _ { k } a ) , \sin ( w _ { k } a ) )$ and $\bar { W _ { E } ^ { \prime } [ b ] } \approx ( \cos ( \bar { w _ { k } } b ) , \sin ( w _ { k } b ) ) ( w _ { k } = 2 \pi k / p$ for some integer $k \in [ 1 , p - 1 ]$ as in Section 2.1; $W _ { E } ^ { \prime }$ stands for the two currently considered principal components of $W _ { E }$ ; rotation and scaling omitted for brevity).
|
| 517 |
+
|
| 518 |
+
• The embeddings are added to get
|
| 519 |
+
|
| 520 |
+
$$
|
| 521 |
+
\begin{array} { r l } & { \quad ( \cos ( w _ { k } a ) + \cos ( w _ { k } b ) , \sin ( w _ { k } a ) + \sin ( w _ { k } b ) ) } \\ & { = \cos ( w _ { k } ( a - b ) / 2 ) \cdot ( \cos ( w _ { k } ( a + b ) / 2 ) , \sin ( w _ { k } ( a + b ) / 2 ) ) } \end{array}
|
| 522 |
+
$$
|
| 523 |
+
|
| 524 |
+
• It is then passed through the first linear layer $L _ { 1 }$ . Each result entry pre-ReLU will thus be a linear combination of the two dimensions of the aforementioned vectors, i.e. $\cos ( w _ { k } ( a -$ $b ) / 2 ) \cdot ( \alpha \cos ( w _ { k } ( a + b ) / 2 ) + \beta \sin ( w _ { k } ( a + b ) / 2 ) )$ ) for some $\alpha , \beta$ , which will become $\left| \cos ( w _ { k } ( a - b ) / 2 ) \right| | \alpha \cos ( w _ { k } ( a + b ) / 2 ) + \beta \sin ( w _ { k } ( a + b ) / 2 ) ) |$ | after ReLU.
|
| 525 |
+
|
| 526 |
+
• These values are then passed through the second linear layer $L _ { 2 }$ . Empirically the ReLU is not observed to be effective as the majority of values is positive. The output entries are then simply linear combinations of aforementioned outputs of $L _ { 1 }$ .
|
| 527 |
+
|
| 528 |
+
• The unembedding matrix is finally applied. In the principal components we are considering, $W _ { U } ^ { \prime } [ c ] \approx ( \cos ( \bar { w } _ { k } c ) , \sin ( w _ { k } c ) )$ . $( W _ { U } ^ { \prime }$ stands for the two currently considered principal components of $W _ { U }$ ; rotation and scaling omitted for brevity) and these two principal components correspond to a linear combination of the output entries of $L _ { 2 }$ , which then correspond to a linear combination of the outputs of $L _ { 1 }$ (thanks to the non-functional ReLU).
|
| 529 |
+
|
| 530 |
+
• Similar to the formula $| \sin ( t ) | - | \cos ( t ) | \approx \cos ( 2 t )$ discussed in Appendix A, these linear combinations provide good approximations for $| \mathrm { { c o s } } ( w _ { k } ( a - b ) / \bar { 2 } ) \bar { | } \cos ( w _ { k } ( a + b ) )$ and $| \mathrm { c o s } ( w _ { k } ( a - b \bar { ) } / 2 ) | \mathrm { s i n } \bar { ( } w _ { k } ( a \bar { + } b ) )$ . Finally we arrive at
|
| 531 |
+
|
| 532 |
+
$$
|
| 533 |
+
\begin{array} { r l } & { \quad | \cos ( w _ { k } ( a - b ) / 2 ) | ( \cos ( w _ { k } c ) \cos ( w _ { k } ( a + b ) ) + \sin ( w _ { k } c ) \sin ( w _ { k } ( a + b ) ) ) } \\ & { = | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a + b - c ) ) } \end{array}
|
| 534 |
+
$$
|
| 535 |
+
|
| 536 |
+
|
| 537 |
+
Figure 24: Visualization of the principal components of the embeddings and unembedding matrices.
|
| 538 |
+
|
| 539 |
+
# L.3 Aligning Weight Matrices
|
| 540 |
+
|
| 541 |
+
We first verify that the ReLU from the second layer is not functional. After removing it, the accuracy of the model remains $1 0 0 \%$ and the cross-entropy loss actually decreased from ${ \mathrm { 6 . 2 0 \times 1 0 ^ { - 7 } } }$ to $5 . 8 9 \times 1 0 ^ { - 7 }$ .
|
| 542 |
+
|
| 543 |
+
Therefore, the model output can be approximately written as
|
| 544 |
+
|
| 545 |
+
$$
|
| 546 |
+
W _ { U } ( b _ { 2 } + W _ { 2 } \mathrm { R e L U } ( b _ { 1 } + W _ { 1 } ( W _ { E } [ a ] + W _ { E } [ b ] ) ) ) = W _ { U } b _ { 2 } + W _ { U } W _ { 2 } \mathrm { R e L U } ( b _ { 1 } + W _ { 1 } ( W _ { E } [ a ] + W _ { E } [ b ] ) ) .
|
| 547 |
+
$$
|
| 548 |
+
|
| 549 |
+
We now “align” the weight matrices $W _ { 1 }$ and $W _ { 2 }$ by mapping through the directions of the principal components of the embeddings and unembeddings. That is, we calculate how these matrices act on and onto the principal directions (consider $W _ { 1 } v$ for every principal direction $v$ in $W _ { E }$ and $v ^ { T } W _ { 2 }$ for every principal direction $v$ in $W _ { U }$ ). We call the other dimension of aligned $W _ { 1 }$ and $W _ { 2 }$ output and source dimensions, respectively (Figure 25).
|
| 550 |
+
|
| 551 |
+
In the aligned weight matrices, we can see a clear domino-like pattern: in most output or source dimensions, only two principal components have significant non-zero values, and they correspond to a pair of matching circle, or a pizza. In this way, every immediate dimension serves for exactly one pizza, so the pizzas do not interfere with each other.
|
| 552 |
+
|
| 553 |
+
|
| 554 |
+
Figure 25: Visualization of the aligned $W _ { 1 }$ and $W _ { 2 }$ .
|
| 555 |
+
|
| 556 |
+
# L.4 Approximation
|
| 557 |
+
|
| 558 |
+
Everything becomes much clearer after realigning the matrices. For a pizza and its two corresponding principal embedding / unembedding dimensions, $W _ { E } ^ { \prime } [ a ] + W _ { E } ^ { \prime } [ b ] \approx \bar { \cos } ( w _ { k } ( a - b ) / 2 ) \cdot ( \cos ( w _ { k } ( a +$ $\bar { b } ) / 2$ ), $\sin ( w _ { k } ( a + \bar { b } ) / 2 ) ;$ will be mapped by realigned $W _ { 1 }$ into its corresponding columns (which are different for every pizza), added with $b _ { 1 }$ and apply ReLU. The result will then be mapped by the realigned $W _ { 2 }$ , added with realigned $b _ { 2 }$ , and finally multipled by $( \cos ( w _ { k } c ) , \sin ( w _ { k } c ) )$ .
|
| 559 |
+
|
| 560 |
+
For the first two principal dimensions, realigned $W _ { 1 }$ has 44 corresponding columns (with coefficients of absolute value $> 0 . 1 \AA ,$ ). Let the embedded input be $( x , y ) = W _ { E } ^ { \prime } [ a ] + W _ { E } ^ { \prime } [ b ] \approx \cos ( w _ { k } ( a - b ) / 2 ) \cdot$ · $( \cos ( w _ { k } ( a + b ) / 2 )$ , $\sin ( w _ { k } ( a + b ) / 2 ) )$ ), the intermediate columns are
|
| 561 |
+
|
| 562 |
+
R $\operatorname { e L U } ( [ 0 . 5 3 0 x - 1 . 1 3 5 y + 0 . 2 5 3 , - 0 . 1 6 4 x - 1 . 1 0 0 y + 0 . 2 0 5 , 1 . 2 1 0 x - 0 . 3 7 0 y + 0 . 1 9 8 , - 0 . 4 7 8 x - 0 . 0 9 2 x + 0 . 0 9 2 x + 0 . 0 9 2 x )$ 1.072y $+ ~ 0 . 2 1 5 , - 1 . 0 1 7 x ~ + ~ 0 . 7 9 9 y ~ + ~ 0 . 2 4 9 , 0 . 3 4 2 x$ $+ ~ 0 . 7 9 9 y + 0 . 2 4 9 , 0 . 3 4 2 x ~ - ~ 0 . 0 4 8 y ~ + ~ 0 . 0 8 5$ $0 . 3 4 2 x \mathrm { ~ - ~ } 0 . 0 4 8 y \mathrm { ~ + ~ } 0 . 0 8 5 , 1 . 1 4 9 x \mathrm { ~ - ~ } 0 . 5 9 8 y \mathrm { ~ + ~ }$ $0 . 2 1 2 , - 0 . 4 4 3 x + 1 . 5 5 0 y + 0 . 1 3 9 9 , - 1 . 0 8 0 x - 0 . 0 0 0 y + 0 . 1 5 1 , - 1 . 4 0 5 x + 0 . 4 1 0 y + 0 . 1 ( 8 , 1 . 0 3 8 x + 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )$ 0.905y $\mathrm { ~ ~ { ~ \tau ~ } ~ } + 0 . 1 9 0 , 0 . 5 6 8 x + 1 . 1 8 8 y + 0 . 1 2 8 , 0 . 2 3 5 x -$ $y \ + \ 0 . 1 2 8 , 0 . 2 3 5 x \ - \ 1 . 3 3 7 y \ + \ 0 . 1 6 4 , - 1 . 1 8 0 x$ + 1.052y + 0.139, $- 0 . 1 7 3 x + 0 . 9 1 8 u + 0 . 1 4 8 . - 0 . 9 0 0 x + 1 . 0 6 0 u + 0 . 1 7 3 . - 1 . 3 4 9 x + 0 . 3 9 0 u + 0 . 2 5 6 . 0 , 1 0 5 x - 0 . 0 9 8 x + 0 . 2 5 6 . 0 , 0 . 1 2 5 x + 0 . 2 5 6 . 0 , 0 . 1 2 5 x + 0 . 2 5 6 . 0 , 0 . 1 2 5 x + 0 . 2 5 6 . 0 , 0 . 1 2 5 x + 0 . 2 5 6 . 0 , 0 . 2 5 6 . - 0 . 3 2 5 x$ $- 0 . 2 0 0 x + 1 . 0 6 0 y + 0 . 1 7 3 , - 1 . 3 4 2 x + 0 . 3 9 0 y + 0 . 2$ 1 $\ 2 4 6 y \ + \ 0 . 2 0 9 , 0 . 1 1 5 x \ + \ 1 . 2 9 3 y \ + \ 0 . 1 9 7 , 0 . 2 5 2 x \ + \ 1 . 2 4 7 y \ + \ 0 . 1 4 0 , - 0 . 4 9 3 x \ + \ 1 . 2 5 2 y \ +$ $\begin{array} { r } . 1 2 0 x + 0 . 2 6 2 y + 0 . 2 3 9 , 0 . 6 6 8 x + 1 . 0 9 6 y + 0 . 2 0 5 , - 0 . 4 8 ( x - 1 . 3 0 2 y + 0 . 1 4 5 , 1 . 1 3 4 x - 0 . 0 0 0 0 ) x + 0 . 0 0 0 0 x + 0 . 0 0 0 0 x = 1 . 0 0 0 0 x + 0 . 0 0 0 0 x + 0 . 0 0 0 0 x + 0 . 0 0 0 0 x = 1 . 0 0 0 0 x - 0 . 0 0 0 0 x + 0 . 0 0 0 0 x + 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 0 x + 0 . 0 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 0 x = 1 . 0 0 0 0 , 0 . 0 0 0 0 x = 1 . 0 0 0 0 , 0 0 0 = 1 . 0 0 0 0 x = 1 . 0 0 0 0 0 x = 1 . 0 0 0 0 , 0 0 0 = 1 . 0 0 0 0 x = 1 . 0 0 0 0 0 x = 1 . 0 0 0 0 0 x = 1 . 0 0 0 0 0 , 0 0 = 1 . 0 0 0 0 0 x = 1 . 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 x = 1 . 0 0 0 0 0 0 = 1 . 0 0 0 0 0 x = 0 0 . 0 0 0 0 0 = 1 . 0 0 0 0 0 0 x = 0 0 . 0 0 0 0 0 = 1 . 0 0 0 0 0 0 0 x = 0 . 0 0 0 0 0 0 = 0 . 0 0 0 0 0 0 . 0 0 0 0 0 = 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 = 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 . 0 0 0 0 0 \end{array}$ 0 $1 . 8 6 2 y + 0 . 2 7 3 , 1 . 1 4 3 x + 0 . 4 3 5 y + 0 . 1 7 1 , - 1 . 2 8 5 x - 0 . 6 4 4 y + 0 . 1 4 2 , - 1 . 4 5 4 x - 0 . 2 8 5 y + 0 . 2 2 7 , 0 . 2 8 5 y + 0 . 2 2 7 , 0 . 2 2 7 5 + 0 . 2 2 8 5 y + 0 . 2 2 8 5 y$ 0.218, −0 $9 2 4 x + 1 . 0 6 8 y + 0 . 1 4 5 , - 0 . 4 0 1 x + 0 . 1 6 7 y + 0 . 1 0 6$ $. 1 6 7 y + 0 . 1 0 6 , - 0 . 4 1 1 x - 1 . 3 8 9 y + 0 . 2 4 9 , 1 . 4 2 2 x$ − 0.117y $\mathbf { \xi } ^ { \prime } + 0 . 2 2 7 , - 0 . 8 5 9 x - 0 . 7 7 8 y + 0 . 1 2 1 , - 0 . 5 2 8 x$ $x \ : - \ : 0 . 7 7 8 y + 0 . 1 2 1 , - 0 . 5 2 8 x \ : - \ : 0 . 2 1 6 y + 0 . 0 9 \ : \hat $ $- 0 . 5 2 8 x \mathrm { ~ - ~ } 0 . 2 1 6 y + 0 . 0 9 7 , - 0 . 8 8 4 x \mathrm { ~ - ~ } 0 . 7 2 4 y$ + 0 $1 . 1 7 1 , 1 . 1 9 3 x + 0 . 7 2 4 y + 0 . 1 3 1 , 1 . 0 8 6 x + 0 . 6 6 7 y + 0 . 2 1 8 , 0 . 4 0 2 x + 1 . 2 4 0 y + 0 . 2 1 3 , 1 . 0 6 9 x - 0 . 9 0 3 y + 0 . 2 2 5 , 0 . 2 4 0 x + 0 . 2 2 5 , 0 . 3 2 5 , 0 . 4 2 5 , x = 0 . 4 2 5 , x = 0 . 4 2 5 , x = 0 . 4 4 , x = 0 . 5 3 0$ 0 $1 . 1 2 0 , 0 . 5 0 6 x - 1 . 0 4 2 y + 0 . 1 5 3 , 1 . 4 0 4 x - 0 . 0 6 4 y + 0 . 1 5 2 , 0 . 6 9 6 x - 1 . 2 4 9 y + 0 . 1 9 9 , - 0 . 7 5 2 x - 0 . 0 9 6 x + 0 . 0 9 6 x - 0 . 0 9 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 7 6 , 0 . 1 5 . 0 1 5 , 0 . 0 1 5 , 0 . 0 1 5 , 0 . 1 0 . 1 5 , 0 0 . 1 5 , 0 0 . 1 5 , 0 0 . 1 0 . 1 5 0 , 0 . 1 5 0 , 0 . 1 0 0 .$ $0 . 8 8 0 y + 0 . 1 0 6 , - 0 . 9 5 6 x - 0 . 5 8 1 y + 0 . 2 2 3 ] )$ .
|
| 563 |
+
|
| 564 |
+
For the first principal unembedding dimension, it will be taken dot product with [1.326, 0.179, 0.142, −0.458, 1.101, −0.083, 0.621, 1.255, −0.709, 0.123, −1.346, −0.571, 1.016, 1.337, 0.732, 0.839, 0.129, 0.804, 0.377, 0.078, 1.322, −1.021, −0.799, −0.339, 1.117, −1.162, −1.423, −1.157, 1.363, 0.156, $- 0 . 1 6 5$ , $- 0 . 4 5 1$ , −1.101, −0.572, −1.180, −1.386, −1.346, −0.226, $1 . 0 9 1 , 1 . 1 5 9 , - 0 . 5 2 4 , 1 . 4 4 1 , - 0 . 9 4 9 , - 1 . 2 4 8 ] .$
|
| 565 |
+
|
| 566 |
+
Call this function $f ( x , y )$ . When we plug in $x = \cos ( t ) , y = \sin ( t )$ , we get a function that wellapproximated $8 \cos ( 2 t + 2 )$ (Figure 26). Therefore, let $t = w _ { k } ( a + b ) / 2$ , the dot product will be approximately $8 | \cos ( w _ { k } ( a - b ) / 2 ) | \cos ( w _ { k } ( a + b ) + 2 )$ , or $| \mathrm { c o s } ( w _ { k } ( a - b ) / 2 ) | \mathrm { c o s } ( w _ { k } ( a + b ) )$ if we ignore the phase and scaling. This completes the picture we described above.
|
| 567 |
+
|
| 568 |
+
|
| 569 |
+
Figure 26: $f ( \cos ( t ) , \sin ( t ) )$ well-approximates $8 \cos ( 2 t + 2 )$ .
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| 1 |
+
# DRAFT, SKETCH, AND PROVE: GUIDING FORMAL THEOREM PROVERS WITH INFORMAL PROOFS
|
| 2 |
+
|
| 3 |
+
Albert Q. Jiang1,2,† Sean Welleck3,4,† Jin Peng Zhou5,6,†
|
| 4 |
+
|
| 5 |
+
Wenda Li2 Jiacheng Liu3 Mateja Jamnik2
|
| 6 |
+
|
| 7 |
+
Timothee Lacroix ´ 1 Guillaume Lample1,‡ Yuhuai Wu5,7,‡
|
| 8 |
+
|
| 9 |
+
1Meta AI 2University of Cambridge 3University of Washington 4Allen Institute for AI
|
| 10 |
+
5Google Research 6Cornell University 7Stanford University
|
| 11 |
+
|
| 12 |
+
# ABSTRACT
|
| 13 |
+
|
| 14 |
+
The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce wellstructured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from $2 0 . 9 \%$ to ${ \mathrm { 3 9 . 3 \% } }$ on a collection of mathematical competition problems.
|
| 15 |
+
|
| 16 |
+
|
| 17 |
+
Figure 1: Draft, Sketch, and Prove. Starting with an informal statement, our framework yields a formal proof through a three-stage process: drafting informal proofs, mapping them into formal sketches, and proving the remaining conjectures. Concretely, an informal statement is a mathematical problem described in a mixture of natural and mathematical languages (e.g., formulae in $\mathrm { I A I R } X ,$ ). Then, we use a large language model to autoformalize each informal proof into a formal sketch, which is a skeleton of the formal proof with open conjectures left unproven (indicated by the <proof $>$ blocks). The formal sketch mirrors the structure of the informal proof. Finally, the open conjectures/gaps inside each formal sketch are proved by an off-the-shelf prover.
|
| 18 |
+
|
| 19 |
+
# 1 INTRODUCTION
|
| 20 |
+
|
| 21 |
+
Formal proof automation is a challenging task that has been the focus of increased attention in recent years (Bansal et al., 2019b; Polu & Sutskever, 2020; Lample et al., 2022; Jiang et al., 2022; Wu et al., 2022). However, deep learning approaches have not been as successful as in other domains, mainly because of the scarcity of formal data. Indeed, formalizing proofs is notoriously difficult and only accessible to a handful of experts, which makes large annotation endeavors unrealistic (Wiedijk, 2008). The largest formal proof corpus is written in Isabelle (Paulson, 1994), and amounts to less than 0.6 GB in size, orders of magnitude smaller than datasets commonly used in vision (Deng et al., 2009) or natural language processing (Brown et al., 2020). To address the scarcity of formal proofs, previous studies have proposed to use synthetic data (Wu et al., 2021b), self-supervision (Polu & Sutskever, 2020; Han et al., 2022), or reinforcement learning (Bansal et al., 2019a; Polu et al., 2022) to synthesize additional formal training data. Although these methods alleviate the data insufficiency to some degree, none are able to capitalize on the bulk of human-written mathematical proofs.
|
| 22 |
+
|
| 23 |
+
Unlike formal mathematics, informal mathematical data is abundant and widely available. Recently, large language models trained on informal mathematical data showcased impressive quantitative reasoning abilities (Lewkowycz et al., 2022; Welleck et al., 2022). However, they often generate erroneous proofs and it is challenging to detect the faulty reasoning in these proofs automatically. Our work devises a novel approach called Draft, Sketch, and Prove (DSP) to translate informal mathematical proofs into formal ones and thus enjoy both the logical rigor provided by formal systems and the wealth of informal data. We give a schematic diagram of the $D S P$ method in Figure 1 and describe it in Section 3. Recent work (Wu et al., 2022) demonstrates the feasibility of automatically translating informal statements into formal ones with large language models. $D S P$ goes beyond and leverages large language models to generate formal proof sketches (Wiedijk, 2003) from informal proofs. Proof sketches consist of high-level reasoning steps that can be interpreted by formal systems such as interactive theorem provers. They differ from complete formal proofs in that they contain sequences of intermediate conjectures without justification. An example of informal proof with its corresponding formal proof sketch is provided in Figure 2. In the last step of $D S P$ , we elaborate the formal proof sketch into a full formal proof using an automated prover to prove all intermediate conjectures.
|
| 24 |
+
|
| 25 |
+
We perform experiments to generate formal proofs of problems from the miniF2F dataset (Zheng et al., 2022) and show that a large portion of theorems can be proved automatically with this method. We investigate two settings where the informal proofs are either written by humans or drafted by a large language model trained on mathematical text. These two settings correspond to situations frequently occurring during the formalization of existing theories, where informal proofs are usually available, but sometimes left as exercises to the reader or missing due to space limits in the margin.
|
| 26 |
+
|
| 27 |
+
# Contributions:
|
| 28 |
+
|
| 29 |
+
• We introduce a novel approach to leverage informal proofs to guide automated provers with formal proof sketches.
|
| 30 |
+
• To evaluate our approach, we build a dataset of manually curated informal statements and informal proofs aligned with formal statements in the miniF2F dataset (Zheng et al., 2022).
|
| 31 |
+
• We increase the proportion of problems solved by an automated prover on miniF2F from $2 0 . 9 \%$ to $3 8 . 9 \%$ given language-model-generated informal proofs, and up to $3 9 . 3 \%$ when proofs are written by humans.
|
| 32 |
+
• Through three ablation studies, we demonstrate the performance benefit of drafting informal proofs, annotating sketches with informal segments, and using automated provers to close open conjectures for the autoformalization of proofs.
|
| 33 |
+
|
| 34 |
+
# 2 BACKGROUND AND RELATED WORK
|
| 35 |
+
|
| 36 |
+
Interactive theorem proving Modern verification systems for mathematics are centered around interactive theorem provers $( I T P s )$ , such as Isabelle (Paulson, 1994), Lean (Moura et al., 2015), Coq (Barras et al., 1997), or Metamath (Megill & Wheeler, 2019). ITPs embed the mathematical definitions and theorems onto a solid logical foundation (e.g., Higher-Order Logic, Dependent Type Theory) implemented by their kernels. Every theorem must be checked by the kernel to be recognized by the ITP. To be proved formally, a theorem is first stated in the ITP’s programming language, and iteratively simplified into simpler objectives (or subgoals), until it can be reduced to already proven facts. In this paper, we will refer to proofs verified by a formal theorem prover as formal proofs, and proofs written in “standard” mathematics (e.g. in LATEX) as informal proofs.
|
| 37 |
+
|
| 38 |
+
Machine learning for formal proof synthesis Several approaches propose to combine machine learning with modern interactive theorem provers (Yang & Deng, 2019; Gauthier et al., 2021), and build upon the recent success of language models (Polu & Sutskever, 2020; Han et al., 2022; Polu et al., 2022; Jiang et al., 2022; Lample et al., 2022). These methods typically rely on sequence-to-sequence models (Sutskever et al., 2014) to generate the next step of a proof given the current proof state and perform search over the generated subgoals using powerful search methods such as MCTS (Silver et al., 2018; Wu et al., 2021a; Laurent & Platzer, 2022). Because search is computationally expensive, these language models are relatively small (with fewer than 1 billion parameters). Our method contrasts with these approaches in that we use a significantly reduced number of calls to the models, but also much larger language models (with up to 175 billion parameters) that showcase outstanding few-shot learning abilities (Brown et al., 2020).
|
| 39 |
+
|
| 40 |
+
Machine learning for informal reasoning Language models have also been used in the context of purely informal mathematics (Lample & Charton, 2020; Hendrycks et al., 2021; Welleck et al., 2021; Drori et al., 2022; Welleck et al., 2022). Nevertheless, Lewkowycz et al. (2022) note that for quantitative question answering, models are prone to generate false positives: the model guesses the right answer while providing an incorrect proof. These errors are hard to spot without human inspection. Worryingly, the frequency of false positives increases with the difficulty of the problem. Our method builds on these findings and translates informal proofs into formal proofs. Since ITPs are logically grounded, once a formal proof is checked by them, we are guaranteed its correctness.
|
| 41 |
+
|
| 42 |
+
Autoformalization In a position paper, Szegedy (2020) argued for attaining formal mathematical data from informal sources with neural networks. Wang et al. (2020) performed preliminary experiments where the evaluation was limited to text-level similarities on synthetic datasets. Recently, Wu et al. (2022) found that large language models (Chen et al., 2021; Chowdhery et al., 2022) are capable of few-shot statement autoformalization. Namely, a small number of examples are enough for them to learn to perform informal-to-formal translation of statements. In this paper, we investigate whether these findings can generalize to proof autoformalization, i.e., whether large language models can be used to translate informal proofs into formal ones.
|
| 43 |
+
|
| 44 |
+
# 3 METHOD
|
| 45 |
+
|
| 46 |
+
In this section, we describe our Draft, Sketch, and Prove (DSP) method for formal proof automation, which leverages informal proofs to guide automated formal theorem provers with proof sketches. We assume that each problem comes with an informal statement and a formal statement describing the problem. Our pipeline consists of three stages (depicted in Figure 1), which we present below.
|
| 47 |
+
|
| 48 |
+
# 3.1 DRAFTING INFORMAL PROOFS
|
| 49 |
+
|
| 50 |
+
The initial phase of the DSP method consists in finding informal proofs for a problem according to its description in natural mathematical language (possibly with LATEX). The resulting informal proof is seen as a draft for the subsequent phases. In mathematical textbooks, proofs of theorems are in general provided, but are sometimes missing or incomplete. Therefore, we consider two settings corresponding to the presence or absence of the informal proofs. In the first, we assume that a “ground-truth” informal proof (i.e., one written by a human) is available, which is the typical scenario in the practice of formalizing existing mathematical theories. In the second setting, we make a more general assumption that the ground-truth informal proof is not given, and draft proof candidates with a large language model trained on informal mathematical data. The language model removes the dependence on human proofs and can produce multiple alternative solutions for every problem. Although there is no easy way to automatically verify the correctness of these proofs, the informal proof only needs to be useful for producing a sketch in the next stage.
|
| 51 |
+
|
| 52 |
+
# 3.2 MAPPING INFORMAL PROOFS INTO FORMAL SKETCHES
|
| 53 |
+
|
| 54 |
+
A formal proof sketch encodes the structure of a solution and leaves out low-level details (Wiedijk, 2003). Intuitively, it is a partial proof that outlines high-level conjecture statements. A concrete example of a proof sketch is shown in Figure 2. Although informal proofs often leave aside low-level details, (e.g., by stating their triviality), these details cannot be discharged in a formal proof, making
|
| 55 |
+
|
| 56 |
+
Informal Statement: Show that for any real number $a$ , $1 0 a \leq 2 8 a ^ { 2 } + 1$
|
| 57 |
+
|
| 58 |
+
# Informal Proof:
|
| 59 |
+
|
| 60 |
+
It suffices to show $0 \leq 2 8 a ^ { 2 } - 1 0 a + 1$ . First, consider completing the square for $2 8 a ^ { 2 } - 1 0 a$ and observe that $\textstyle \left( a - { \frac { 5 } { 2 8 } } \right) ^ { 2 } = a ^ { 2 } - { \frac { 1 0 } { 2 8 } } a + ( 5 / 2 8 ) ^ { 2 }$ . Since $\begin{array} { r } { 0 \leq \left( a - \frac { 5 } { 2 8 } \right) ^ { 2 } } \end{array}$ , we get $0 \leq a ^ { 2 } - { \textstyle { \frac { 1 0 } { 2 8 } } } a + ( 5 / 2 8 ) ^ { 2 }$ . Multiplying by 28 and simplifying gives $0 \leq 2 8 a ^ { 2 } - 1 0 a + ( 2 5 / 2 8 )$ . Since $2 5 / 2 8 < 1$ , the result follows.
|
| 61 |
+
|
| 62 |
+
# Formal Proof Sketch:
|
| 63 |
+
|
| 64 |
+
|
| 65 |
+
Figure 2: A proof sketch in Isabelle. The problem “Show that for any real number $a$ , $1 0 a \leq 2 8 a ^ { 2 } + 1 ^ { \prime }$ is given with an informal proof and an associated formal proof sketch. The sketch first rewrites the original statement $( \mathtt { C } 0 )$ , which is proved through 5 intermediary conjectures (c1..c5). We use a special token $( < \cdots > )$ to indicate that the conjecture is “open” and should be tackled by an automated prover later. To facilitate the alignment between the informal and formal languages, we annotate the formal proof sketch examples with informal proof segments (shown in red), which are immediately followed by their formal counterparts.
|
| 66 |
+
|
| 67 |
+
straightforward informal-to-formal proof translation difficult. Instead, we propose to map informal proofs to formal proof sketches that share the same high-level structures. The low-level details missing from a proof sketch can later be filled by an automated prover. Since large informal-formal parallel corpora do not exist, standard machine translation methods are unsuitable for this task. Rather, we use the few-shot learning abilities of a large language model. Specifically, we prompt the model with a few example pairs containing informal proofs and their corresponding formal sketches, followed by an informal proof yet to be translated. We then let the model generate the subsequent tokens to obtain the desired formal sketch. We refer to this model as an autoformalizer.
|
| 68 |
+
|
| 69 |
+
# 3.3 PROVING OPEN CONJECTURES IN THE SKETCHES
|
| 70 |
+
|
| 71 |
+
As the last part of the process, we execute off-the-shelf automated provers to fill in the missing details in proof sketches, where “automated provers” refers to systems capable of producing formally verifiable proofs. Our framework is agnostic to the specific choice of the automated prover: it can be symbolic provers such as heuristic proof automation tools, neural-network-based provers, or hybrid approaches. If the automated prover successfully closes all the gaps in the proof sketch, it returns the final formal proof which can be checked against the problem’s specification. If the automated prover fails (e.g., it exceeds the allocated time limit), we consider the evaluation to be unsuccessful.
|
| 72 |
+
|
| 73 |
+
# 4 EXPERIMENTS
|
| 74 |
+
|
| 75 |
+
# 4.1 DATASET AND EVALUATION
|
| 76 |
+
|
| 77 |
+
We evaluate our method on the miniF2F dataset (Zheng et al., 2022). The dataset contains the formal statements of 488 problems from high-school mathematical competitions, written in three formal languages: Lean, HOL-Light, and Isabelle. They are split into a valid set and a test set, composed of 244 problems each. In this work, we choose to experiment with Isabelle for three reasons: (1) Isabelle’s proof corpus is one of the largest among interactive theorem provers, conducive to the language models’ mastery of its syntax; (2) Isabelle supports the declarative proof style (detailed discussion in Appendix A), enabling formal proof sketches (Wiedijk, 2003) which are central to our method; (3) although automated proving tools are available in other interactive theorem provers, none are as developed and effective as Sledgehammer (Paulson, 2010) in Isabelle for proving conjectures.
|
| 78 |
+
|
| 79 |
+
The miniF2F dataset is comprised of problems from three source categories: (1) 260 problems sampled from the MATH dataset (Hendrycks et al., 2021); (2) 160 problems from actual high-school mathematical competitions (AMC, AIME, and IMO); (3) 68 crafted problems at the same difficulty level as (2). We employ three methods to obtain informal statements and proofs from these sources. For source (1), we access the informal statements and proofs from the MATH dataset; for (2), we retrieve their informal statements and proofs from the AOPS website 1; and for (3), we manually write down their informal statements and proofs. Thus we gather a parallel set of 488 informal statements, informal proofs, and formal statements. This dataset provides the informal statements and proofs for our experiment in the human-as-informal-proof-writer setting and will be released upon publication.
|
| 80 |
+
|
| 81 |
+
Our task is to generate formal proofs for problems as they are formally stated in miniF2F. We consider a proof valid if and only if it (a) does not contain “cheating” keywords (sorry and oops) that exit a proof without completing it, and (b) Isabelle is able to verify the corresponding formal statement with the proof. We use the Portal-to-ISAbelle API by Jiang et al. (2021) to interact with Isabelle.
|
| 82 |
+
|
| 83 |
+
# 4.2 BASELINES
|
| 84 |
+
|
| 85 |
+
Sledgehammer As a baseline, we attempt to prove the formal statement directly with Sledgehammer, a popular proof automation tool in Isabelle. We use the default Sledgehammer configuration in Isabelle2021, including a 120-second timeout and the five automated theorem provers (Z3, CVC4, SPASS, Vampire, E). Appendix B gives a more thorough introduction to Sledgehammer.
|
| 86 |
+
|
| 87 |
+
Sledgehammer $^ +$ heuristics Occasionally, Sledgehammer may fail without trying simple yet effective tactics. As a second, stronger baseline, we create an automated prover that tries 11 common tactics (auto, simp, blast, fastforce, force, eval, presburger, sos, arith, linarith, auto simp: field simps) for high-school level algebra and number theory problems. If every attempted tactic fails, or times out after 10 seconds, it falls back to Sledgehammer.
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Language models for proof search Finally, we include baselines which are representative of state-of-the-art neural theorem proving in Isabelle, specifically Thor (Jiang et al., 2022) and Thor with expert iteration on autoformalized data (Wu et al., 2022). The methods GPT-f with expert iteration (Polu et al., 2022), and HyperTree Proof Search (HTPS) (Lample et al., 2022) can solve $3 6 . 6 \%$ and $4 1 . 0 \%$ of the problems on miniF2F-test. However, they rely on the Lean theorem prover instead of Isabelle, which greatly influences the performance due to the different tactics and automation, and are not directly comparable to our method.
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# 4.3 EXPERIMENTAL SETUP
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The experimental code is at github.com/albertqjiang/draft sketch prove.
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Drafting When informal proofs are generated, we condition a large language model on informal statements to sample 100 informal proofs per problem. Specifically, we use the Codex code-davinci002 model (Chen et al., 2021) through the OpenAI API, and the 8B, 62B, and 540B versions of the Minerva model from Lewkowycz et al. (2022). We use greedy decoding for Codex and nucleus sampling (Holtzman et al., 2019) with temperature $T = 0 . 6$ and top $\mathbf { p } = 0 . 9 5$ for Minerva models.
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Sketching For sketching, we manually prepare 20 autoformalization examples of the format (informal statement, informal proof, formal statement, formal sketch), to form a pool of high-quality demonstrations. Of these 20 examples, 10 are of the algebra type and 10 are of the number theory type. All examples are from the validation set of the miniF2F dataset and can be found in the supplementary materials. The sketches contain in-line comments as in Figure 2. If the name of the problem gives away its type (algebra or number theory), we only use examples of the corresponding type. We also ensure that the sampled few-shot examples do not contain the problem being solved. The prompt is 3 uniformly randomly sampled example from the pool concatenated with the current problem’s (informal statement, informal proof, formal statement). We use this prompt to query the same Codex model to get the desired proof sketches. We use deterministic greedy decoding and a maximum of 2048 tokens in the generated sequence. For all the experiments, unless stated otherwise, we control the total number of queries made to Codex per problem to be 100. This means 100 queries per human informal solution and one query per language-model-generated solution.
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Table 1: Proving success rates on the miniF2F dataset with Isabelle In the table are the success rates of four baselines, the DSP method with human and language model informal proofs, as well as three ablation studies, on the validation and the test sets of miniF2F. The highest success rates on each set are highlighted in bold. The performance difference between ablation studies and DSP with human informal proofs are enclosed in brackets.
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<table><tr><td>Success rate</td><td>miniF2F-valid</td><td>miniF2F-test</td></tr><tr><td colspan="3">Baselines</td></tr><tr><td>Sledgehammer</td><td>9.9%</td><td>10.4%</td></tr><tr><td>Sledgehammer + heuristics</td><td>18.0%</td><td>20.9%</td></tr><tr><td>Thor (Jiang et al., 2022)</td><td>28.3%</td><td>29.9%</td></tr><tr><td>Thor+ expert iteration (Wu et al.,2022)</td><td>37.3%</td><td>35.2%</td></tr><tr><td colspan="3">Draft,Sketch,and Prove</td></tr><tr><td>Human informal proof</td><td>42.6%</td><td>39.3%</td></tr><tr><td>Codex informal proof</td><td>40.6%</td><td>35.3%</td></tr><tr><td>8B Minerva informal proof</td><td>40.6%</td><td>35.3%</td></tr><tr><td>62BMinerva informal proof</td><td>43.9%</td><td>37.7%</td></tr><tr><td>540BMinerva informal proof</td><td>42.6%</td><td>38.9%</td></tr><tr><td colspan="3">Ablations (with human informal statements and proofs)</td></tr><tr><td>-In-line comments</td><td>37.7% (-4.9%)</td><td>36.5% (-2.8%)</td></tr><tr><td>- Informal proofs</td><td>38.9%(-3.7%)</td><td>34.0% (-5.3%)</td></tr><tr><td>- Automated provers</td><td>32.8% (-9.8%)</td><td>30.3% (-9.0%)</td></tr></table>
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Proving To prove the conjectures left open by the formal sketch, we use the Sledgehammer $^ +$ heuristics automated prover described in Subsection 4.2. We execute the automated prover on every open conjecture in the sketch to synthesize a formal proof that can be verified by Isabelle.
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# 4.4 RESULTS
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In Table 1, we display the proportion of successful formal proofs found on the miniF2F dataset with the interactive theorem prover Isabelle. The results include the four baselines described in Subsection 4.2 and the $D S P$ method with human-written proofs and model-generated proofs. From the table, we can see that the automated prover with 11 additional heuristic tactics significantly increases the performance of Sledgehammer, boosting its success rate from $9 . 9 \%$ to $1 8 . { \bar { 0 } } \%$ on the validation set of miniF2F and from $1 0 . 4 \%$ to $2 0 . 9 \%$ on the test set. The two baselines using language models and proof search (Thor and Thor $^ +$ expert iteration) achieve success rates of $2 9 . 9 \%$ and ${ \bar { 3 } } 5 . 2 { \bar { \% } }$ on the test set of miniF2F, respectively.
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With informal proofs written by humans, the $D S P$ method achieves success rates of $4 2 . 6 \%$ and $3 9 . 3 \%$ on the validation and test sets of miniF2F. A total of 200 out of 488 problems can be proved in this way. The Codex model and the Minerva (8B) model give very similar results in solving problems on miniF2F: they both guide the automated prover to solve $4 0 . { \dot { 6 } } \%$ and $3 5 . 3 \%$ of problems on the validation and the test sets respectively. This is corroborated by Lewkowycz et al. (2022)’s observation that these two models have comparable performances in solving mathematical problems.
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When we switch to the Minerva (62B) model, the success rates rise up to $4 3 . 9 \%$ and $3 7 . 7 \%$ respectively. Compared to human-written informal proofs, its success rates are $1 . 3 \%$ higher on the validation set and $1 . 6 \%$ lower on the test set. In total, the Minerva (62B) model is able to solve 199 problems on miniF2F, one fewer than with human proofs. The $D S P$ method is effective in guiding the automated prover under both settings that we study: using either human informal proofs or language-model-generated informal proofs. $D S P$ almost doubles the prover’s success rate and results in a new state-of-the-art performance on miniF2F with Isabelle. Moreover, the larger Minerva model is almost as helpful as a human in guiding the automated prover in solving problems.
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# 5 ANALYSIS
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# 5.1 ABLATION STUDIES
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Ablation of in-line comments To facilitate the alignment between the informal proofs and the formal proof sketches, we copy relevant segments of the informal proofs as in-line comments in the sketches. In the manually constructed prompt examples, these comments are prefixed to the corresponding Isabelle code blocks, as shown in Figure 2 (the text in red). We hypothesize that this technique is beneficial for large language models to synthesize formal sketches. To validate this hypothesis, we perform an ablation study by removing the in-line comments in the prompt examples before running the experiment. The results are displayed in Table 1. We find that without in-line comments, the success rates drop by $4 . 9 \%$ and $2 . 8 \%$ on the validation and test sets respectively. We conclude that having in-line comments is helpful for generating formal proof sketches.
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Figure 3: Number of problems solved on miniF2F against the number of autoformalization attempts per problem. Left: The figure displays the experiments carried out with the $D S P$ method and three ablations on it. The curves represent the $D S P$ method (blue), formal proof sketches without the in-line comments (orange), without informal proofs altogether (green), and without the automated provers (red). Right: The figure compares the experimental results with informal proof drafts written by humans (blue), the 540B Minerva model (orange), the 62B Minerva model (green), the 8B Minerva model (red), and the Codex model (purple).
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Ablation of informal proof drafts Drafting informal proofs is the first step of the $D S P$ method. To investigate the necessity of this step, we perform an experiment of formal sketching and proving without informal proofs at all. Because formal proof sketches are written in the declarative proof style, they are fairly similar to the informal proof drafts already. Concretely, we remove the informal proofs and the in-line comments (because they are copied segments of the informal proofs) in the prompt examples. This removes the need for the informal proof writer, whether a human or a neural network. The results of this setup are shown in Table 1. It can be seen that the success rates on the validation and the test sets of miniF2F drop by $3 . 7 \%$ and $5 . 3 \%$ respectively compared to with human-written proofs. They are also inferior to success rates obtained with language-model-generated informal proofs. This demonstrates the importance of drafting informal proofs before sketching and proving.
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Ablation of automated provers Using an autoformalizer to generate proof sketches which are then completed by an automated prover is central to our method. The effect of utilizing an automated prover to close open conjectures in proof sketches is worth studying, so we conduct an ablation experiment for it. Namely, we replace the proof sketches in the prompt examples with complete formal proofs. The complete formal proofs still follow the declarative proof style, but do not contain any open conjectures. As a result, the large language model will also generate full proofs instead of sketches, and we directly check whether these generated proofs are valid. The results in this setup are presented in Table 1. The results reveal that without an automated prover to close open conjectures, the success rate on miniF2F decreases by $9 . 8 \%$ and $9 . 0 \%$ on the validation and test sets respectively. The drastic performance difference indicates the essential role of automated provers in our approach.
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Scaling properties of ablation studies To understand the effect of the ablations on the $D S P$ method’s scaling properties, we vary the number of autoformalization attempts per problem and plot the number of successful proofs found on the miniF2F dataset in Figure 3 (left). Three methods are contrasted: the original $D S P$ method with human informal proofs, the $D S P$ method without in-line comments, and the $D S P$ method without sketching. It can be seen from the figure that with the original $D S P$ method, the performance reaches a plateau (no new proofs are found) after 70 autoformalization attempts are made for each problem. For the ablation study with no in-line comments, the plateau is reached much faster, after around 50 autoformalization attempts. This method solves 181 problems in total. The ablation study without sketching can solve 154 problems on miniF2F. In comparison, with human informal proofs, only 7 autoformalization attempts are required to reach this performance.
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Figure 4: IMO proof guided by a Minerva informal proof An informal proof of the International Math Olympiad problem $\mathrm { i m o \_ 1 9 5 9 \mathrm { _ - p 1 } }$ generated by Minerva that leads to a successful formal proof. The steps enclosed by the $A T P$ delimiters are generated by an automated prover and all other steps are generated by the DSP autoformalizer.
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# 5.2 LANGUAGE-MODEL-GENERATED PROOFS
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Our experiments demonstrated that model-generated informal proofs from Minerva and Codex can help guide a formal theorem prover. In this section, we analyze the properties of these proofs further. Since the Minerva (62B and 540B) models give the best overall performance on miniF2F, we focus on the informal proofs they produce in this section.
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Minerva helps solve one IMO problem Interestingly, our approach manages to solve one problem from the International Mathematical Olympiad (imo 1959 1) with a Minerva-generated solution, but not with the human proof. For this problem, we present the successful Minerva-generated informal proof draft and the formal proof in Figure 4. We hypothesize that the reason behind this phenomenon is that human proofs might leave gaps between conjectures that are too difficult for automated provers to solve. On the other hand, the diversity in language model informal proofs makes some of them more amenable to automated provers. In Appendix C, we analyze the human and the Minerva informal proofs for this problem in greater detail. In Appendix D, we present a manual evaluation of Minerva proofs, and 3 more case studies comparing the human and Minerva informal proofs.
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Is there a way to detect which Minerva proofs are correct, without human evaluation? For a preliminary investigation, we filter out all the problems that can be solved directly with the automated prover from the 50 and are left with 27 informal proofs. Of these 27, 21 are completely correct, 6 still contain small errors, but none are nonsensical. With this simple filter, we achieve a precision of $7 7 . 8 \%$ and a recall of $7 2 . 4 \%$ in identifying correct Minerva informal proofs.
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Scaling properties of human and Minerva proofs To understand the influence of different informal proof sources on the scaling properties of $D S P$ , we plot the number of successful proofs found on miniF2F against the number of autoformalization attempts per problem in Figure 3 (right). Note that for each problem, we have 1 informal proof by a human and 100 informal proof drafts by each language model. The one human proof is used 100 times for formal proof sketch generation, while each language model proof draft is used only once. The 62B and the 540B models result in more successful proofs than the smaller (8B) Minerva model and the Codex model, consistently for any number of attempts. The 8B Minerva model and the Codex model behave similarly, both finding 185 proofs in the end. Informal proofs written by humans help solve more problems than those by Minerva models for $1 - 1 0 0$ autoformalization attempts. However, the difference is small (1 problem) when 100 are made.
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Noticing that the number of successful proofs does not plateau for the Minerva-generated proofs, we investigate how further increasing the number of autoformalization attempts changes the number of problems solved for human-written and language-model-generated proofs. For each problem, we use 1 human informal proof and sample 200 sketches for it; we also use the same 100 informal proof drafts by the Minerva (540B) language model and sample 2 sketches for each draft. The total number of sketches per problem is 200 in both settings. We find that with human informal proofs, 203 theorems ${ \mathrm { ~ ( 1 0 6 / 9 7 } }$ on valid/test) have successful formal proofs, while with language-model-generated informal proofs, 209 (111/98 on valid/test) theorems have successful formal proofs after the same number of attempts. This suggests that the diversity in language-model-generated informal proofs can benefit the automated formalization process more than the “ground-truth” human proofs.
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# 5.3 MEMORIZATION
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This work utilizes two language models that have been trained on a large amount of internet data. Several prior works (Trinh & Le, 2018; Carlini et al., 2022) pointed out that such models can memorize some fraction of the data they encounter during training. For drafting informal proofs, we mainly experimented with Minerva. Lewkowycz et al. (2022, Section 5) discussed the memorization effects within Minerva and concluded that they could not find evidence that its abilities are due to memorization. For the autoformalization of proof sketches, the Codex (code-davinci-002) model was used. Its training data was collected before June $2 0 2 1 ^ { 2 }$ , at which time the miniF2F dataset had not been made public. So the model cannot benefit from memorizing the exact problems and proofs. Therefore, it is inappropriate to attribute the abilities of models used in this paper to memorization.
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# 6 CONCLUSION
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In this paper, we introduced Draft, Sketch, and Prove $( D S P )$ , a novel approach that takes advantage of informal proofs to synthesize formal proofs. We demonstrated its feasibility and effectiveness by reaching state-of-the-art performance on the miniF2F dataset with the Isabelle theorem prover. Central to our method are formal proof sketches that mirror the high-level reasoning structures of informal proofs. Our ablations showed that the ability to automatically convert informal proofs to proof sketches is critical to the success of $D S P$ .
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Our $D S P$ method differs fundamentally from previous applications of machine learning to formal proof synthesis in two aspects. Firstly, while most approaches in the field focus on improving proof search, our method seeks to construct the entire formal proof structure from the informal proof in one decoding operation. The task of the automated prover is then simplified to filling the gaps between intermediate conjectures. Secondly, while existing approaches operate exclusively on formal data, $D S P$ by design benefits from informal proofs.
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In this work, we utilized a purely symbolic automated prover to close the gaps in proof sketches. In the future, we aim to equip $D S P$ with more powerful mechanisms, such as HyperTree Proof Search (Lample et al., 2022), to broaden the scope of provable theorems. Similar to AlphaCode (Li et al., 2022), we found that the number of generations is crucial for performance. The computational cost of the autoformalizer being a bottleneck in our method, we seek to develop approaches able to generate high-quality proof sketches more efficiently.
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# ACKNOWLEDGEMENTS
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We thank Rui Yuan and Kunhao Zheng for helping with the informal solutions used in our dataset.
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We thank Christian Szegedy for his feedback on the early draft.
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FUNDING DISCLOSURE
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AQJ and WL are supported by the ERC Advanced Grant ALEXANDRIA (Project GA 742178).
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Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris van Doorn, and Jakob von Raumer. The lean theorem prover (system description). In International Conference on Automated Deduction, pp. 378–388. Springer, 2015.
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Lawrence C. Paulson. Three years of experience with sledgehammer, a practical link between automatic and interactive theorem provers. In Renate A. Schmidt, Stephan Schulz, and Boris Konev (eds.), Proceedings of the 2nd Workshop on Practical Aspects of Automated Reasoning, PAAR-2010, Edinburgh, Scotland, UK, July 14, 2010, volume 9 of EPiC Series in Computing, pp. 1–10. EasyChair, 2010. doi: 10.29007/tnfd. URL https://doi.org/10.29007/tnfd.
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Donald Syme. DECLARE: A prototype declarative proof system for higher order logic. Citeseer, 1997.
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Kaiyu Yang and Jia Deng. Learning to prove theorems via interacting with proof assistants. In International Conference on Machine Learning (ICML), 2019.
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Kunhao Zheng, Jesse Michael Han, and Stanislas Polu. miniF2F: a cross-system benchmark for formal olympiad-level mathematics. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ 9ZPegFuFTFv.
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# APPENDIX
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# A CONJECTURES AND THE DECLARATIVE PROOF STYLE
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Interactive theorem provers such as Isabelle and Mizar use a declarative proof style (Syme, 1997), in which a proof is interleaved with conjectures and their corresponding proofs. Syme (1997) stated that the list of conjectures in a declarative proof should be analogous to a proof sketch found in a mathematical textbook and sufficiently convincing for the reader. In practice, ITP users often prove a theorem by writing down a list of conjectures (a “formal sketch”), then attempt to find a proof of each conjecture (fill a “gap”) with an automated system.
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# B SLEDGEHAMMER
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| 259 |
+
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Sledgehammer (Paulson, 2010) is a powerful system that automates reasoning with the interactive theorem prover Isabelle. It works by flattening the goals encoded in the higher-order logic used by Isabelle/HOL into other logics (e.g., first-order logic) which can then be fed into automated theorem provers such as E 3, CVC4 4, $\bar { Z 3 ^ { 5 } }$ , Vampire 6, and SPASS 7. If any of these automated theorem provers succeeds in finding the proof in their own corresponding format, Sledgehammer reconstructs the proof in Isabelle/HOL with certified provers (metis, meson, and smt), which is relatively more interpretable by humans.
|
| 261 |
+
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| 262 |
+
As a practical example of using Sledgehammer, one can declare a conjecture in Isabelle/HOL: have "4 dvd (a::nat) $\Longrightarrow ~ 2$ dvd $\mathtt { a } "$ and call Sledgehammer immediately afterwards. If Sledgehammer succeeds, it will return a proof step that proves the conjecture. In this example, the step is by (meson dvd trans even numeral), which uses the meson resolution prover and two facts: that the division relation is transitive and that 4 is an even number. If Sledgehammer does not find the proof or timeouts, it will report failure.
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| 263 |
+
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| 264 |
+
# C A PROOF TO AN INTERNATIONAL MATHEMATICAL OLYMPIAD PROBLEM
|
| 265 |
+
|
| 266 |
+
With the Minerva-generated solutions, a proof to the problem imo 1959 p1 is discovered. This is the first problem of the first ever International Mathematical Olympiad (IMO). The informal problem statement, Minerva-generated informal solution, and DSP’s formal proof are shown in Figure 4.
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| 267 |
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| 268 |
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In Figure 4, we can see that the autoformalizer in DSP (a large language model), copies over parts of the informal proof generated by Minerva as in-line comments to precede their corresponding formal proof blocks. The formal proof does not use the first sentence of the informal proof solution as it is already identical to the formal statement. We also notice that the large language model selects relevant premises after writing down the conjectures (the steps starting with using) despite not every premise is strictly needed.
|
| 269 |
+
|
| 270 |
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The formal proof creates 5 conjectures (4 have statements and 1 show statement) which are all subsequently proved by our automated theorem prover. The step to prove the statement have "gcd $( 2 1 \star \Pi + 4 )$ ) $( { { \mathchoice { \mathrm { ~ 1 ~ 4 ~ } } { \mathrm { ~ ~ 1 ~ } } { \mathrm { ~ 1 ~ 4 ~ } } { \mathrm { ~ ~ 1 ~ } } { \mathrm { ~ ~ 1 ~ } } } } + { { \mathchoice { \mathrm { ~ 3 ~ ) ~ } } { \mathrm { ~ ~ 3 ~ } } { \mathrm { ~ ~ 3 ~ } } { \mathrm { ~ ~ 3 ~ } } } } = { { \mathchoice { \mathrm { ~ 1 ~ } } { \mathrm { ~ ~ 1 ~ } } { \mathrm { ~ ~ 1 ~ } } { \mathrm { ~ ~ 1 ~ } } { \mathrm { ~ ~ 1 ~ } } } }$ involves 2 verified low-level provers smt and $^ { z 3 }$ and 10 lemmas/facts from outside the scope of the language model. It is highly unlikely that either the large language model or the automated theorem prover can finish this proof on its own.
|
| 271 |
+
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| 272 |
+
Unsuccessful human-written proof. In contrast, the human-written informal proof of this IMO problem did not lead to a successful formal proof. The human-written proof is:
|
| 273 |
+
|
| 274 |
+
Denoting the greatest common divisor of $a , b$ as $( a , b )$ , we use the Euclidean algorithm:
|
| 275 |
+
|
| 276 |
+
$$
|
| 277 |
+
( 2 1 n + 4 , 1 4 n + 3 ) = ( 7 n + 1 , 1 4 n + 3 ) = ( 7 n + 1 , 1 ) = 1
|
| 278 |
+
$$
|
| 279 |
+
|
| 280 |
+
It follows that $\frac { 2 1 n + 4 } { 1 4 n + 3 }$ is irreducible. Q.E.D.
|
| 281 |
+
|
| 282 |
+
A key difference between the Minerva proof and the human proof is the way that invoking the Euclidean algorithm is described. The Minerva proof explicitly writes out the results of the Euclidean algorithm (e.g. $2 1 n + 4 = 1 \cdot ( 1 4 n + 3 ) + { \bar { 7 } } n + 1 )$ , which are translated into the sketch ( $_ { c l }$ in Figure 4). The human proof introduces new notation to express the results indirectly in terms of greatest common divisors, which ends up being less suitable for sketching. For example, below is a sketch generated with the human proof, which has a conjecture that is semantically incorrect and hence cannot be closed by the automated prover:
|
| 283 |
+
|
| 284 |
+
# theorem
|
| 285 |
+
|
| 286 |
+
fixes n :: nat shows "gcd $( 2 1 { \star } \Pi + 4 )$ ) $( 1 4 \star \Pi + 3 ) = 1 "$ proof - have " $( 2 1 \star \mathsf { n } + 4 , ~ 1 4 \star \mathsf { n } + 3 ) \ = \ ( 7 \star \mathsf { n } + 1 , ~ 1 4 \star \mathsf { n } + 3 ) \mathsf { n }$ 1 ATP (\* <--- UNSUCCESSFUL $^ { \star }$ ) also have " $" \ldots = ( 7 \star \Omega + 1 , \mathrm { ~ 1 ~ } ) "$ ATP finally show ?thesis ATP qed
|
| 287 |
+
|
| 288 |
+
# D MORE ANALYSIS ON HUMAN AND MINERVA INFORMAL PROOFS
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| 289 |
+
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| 290 |
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We analyze the relationship between the validity of the formal proofs and the correctness of the informal proofs. For our analysis, we randomly sample 50 Minerva proofs of different problems, which are then successfully converted to formal proofs. We then manually evaluate the correctness of these 50 informal proofs. Among them, 29 proofs $( 5 8 \% )$ are entirely correct, 16 are incorrect with a clearly identifiable incorrect step, and 5 “proofs” are nonsensical and simply rephrase the final conclusions of the problems.
|
| 291 |
+
|
| 292 |
+
Seeing that a total of $1 6 + 5 = 2 1$ incorrect informal proofs can lead to successful formal proofs, we study how they guide the automated formal prover despite having flaws themselves. The 21 proofs divide into 2 cases: In the first case, we find 13 problems for which the informal proofs are mostly ignored, and the automated prover can find proofs by itself; In the other 8 problems, although the informal proofs are wrong, the autoformalizer manages to correct them, either by ignoring the erroneous steps or by stating their correct versions in the formal proof sketches. This suggests that the autoformalizer has some understanding of the mathematical statements and is not merely translating them from an informal language to a formal language. It is robust to slight noises in its input.
|
| 293 |
+
|
| 294 |
+
We present 3 case studies comparing the human and Minerva informal proofs. Particularly, Figure 7 shows a completely correct example and one example of each pathological case.
|
| 295 |
+
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| 296 |
+
|
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+
Figure 5: Algebra example with Minerva informal proof. An informal proof generated by Minerva that led to a successful formal proof. The autoformalizer generated a proof sketch containing all lines of the formal proof except for those with $I A T P J$ . The sketch is structured according to the informal proof, containing five intermediate conjectures based on the informal proof. The autoformalizer generated in-line comments in the proof sketch (shown in red), which correctly identified an alignment between the formal and informal proofs.
|
| 298 |
+
|
| 299 |
+
Statement: For any integer $a$ , show that $a ^ { 2 } \equiv$ 0 (mod 4) or ${ a ^ { 2 } \equiv 1 }$ (mod 4).
|
| 300 |
+
|
| 301 |
+
# Informal Proof (Minerva 62B):
|
| 302 |
+
|
| 303 |
+
Let $a = 2 k$ for some integer $k$ .
|
| 304 |
+
Then $a ^ { 2 } = 4 k ^ { 2 }$ .
|
| 305 |
+
Since $4 k ^ { 2 }$ is divisible by 4, we have $\boldsymbol { a } ^ { 2 } \equiv 0$ (mod 4).
|
| 306 |
+
Now, let $a = 2 k + 1$ for some integer $k$ .
|
| 307 |
+
Then $a ^ { 2 } = 4 k ^ { 2 } + 4 k + 1 = 4 ( k ^ { 2 } + k ) + 1$ . Since $\boldsymbol { k } ^ { 2 } + \boldsymbol { k }$ is an integer, $4 ( k ^ { 2 } + k ) { \dot { + } } 1$ is not divisible by 4.
|
| 308 |
+
Thus, ${ a ^ { 2 } \equiv 1 }$ (mod 4).
|
| 309 |
+
|
| 310 |
+
Statement: For any integer $a$ , show that ${ a } ^ { 2 } \equiv$ 0 (mod 4) or ${ a ^ { 2 } \equiv 1 }$ (mod 4).
|
| 311 |
+
|
| 312 |
+
# Informal Proof (Minerva 62B):
|
| 313 |
+
|
| 314 |
+
$a \ ( \mathrm { m o d } \ 4 ) \in \{ 0 , 1 , 2 , 3 \}$ .
|
| 315 |
+
Using that for any natural number $k$ , $a \equiv b$ (mod 4) implies $\dot { a } ^ { k } \equiv b ^ { k }$ (mod 4), we have $\dot { a } ^ { 2 }$ $\overset { \cdot } { \left( \mathrm { m o d } \ 4 \right) } \in \left\{ 0 , 1 , 4 , 9 \right\}$ .
|
| 316 |
+
Since $4 \equiv 0$ (mod 4) and $9 \equiv 1 { \pmod { 4 } }$ , the result follows.
|
| 317 |
+
|
| 318 |
+
# Formal Proof:
|
| 319 |
+
|
| 320 |
+
theorem numbertheory_sqmod4in01d: fixes a :: int shows "( $\ a \hat { \textmd a } _ { 2 }$ mod $4 \ = \ 0$ ) $\scriptstyle \backslash < _ { \mathrm { { O L } } } >$ (aˆ2 mod $4 \ = \ 1$ )"
|
| 321 |
+
proof (cases "even a")
|
| 322 |
+
case True ( $\star$ Let a=2k for some integer k. Then $\hat { \mathbf { a } } \hat { \mathbf { \Lambda } } 2 \mathbf { = } 4 \mathbf { k } \hat { \mathbf { \Lambda } } 2$ . Since $4 1 - 2$ is divisible by 4, we have aˆ2 \equiv 0 \pmod{4}.\*) then obtain k where $" \mathrm { a } \mathrm { = } 2 \star \mathrm { k } "$ using evenE by auto [ATP]
|
| 323 |
+
then have $" a \widehat { } 2 = 4 \star \widehat { } 2 "$ unfolding power2_eq_square by auto [ATP] then have "aˆ2 mod $4 \ = \ 0$ " by auto [ATP]
|
| 324 |
+
then show ?thesis by auto [ATP]
|
| 325 |
+
next
|
| 326 |
+
case False $( \star \ \aleph \circ \aleph ,$ let $a = 2 k + 1$ for some integer k. Then $a \cdot 2 = 4 k \cdot 2 + 4 k + 1 = 4 ( k \cdot 2 + k ) + 1$ . Since $k ^ { \mathord { \sim } } 2 { + } k$ is an integer, 4 $( \mathbf { k } ^ { \sim } 2 + \mathbf { k } ) + \mathbf { 1 }$ is not divisible by 4. Thus, $\widehat { \mathbf { a } } \widehat { \mathbf { \Lambda } } ^ { 2 }$ \equiv $\texttt { 1 } \backslash \mathtt { p m o d } \{ 4 \} \ldots \star \}$ then obtain k where $\ " { a } { = } 2 { \star } \mathrm { k } { + } 1$ " using oddE by auto [ATP] then have $" a \widehat { } 2 = 4 { \star } \widehat { } 2 { + } 4 { \star } \mathrm { k } { + } 1 "$ " unfolding power2_eq_square by (auto simp: field_simps) [ATP] then have $\ " \mathtt { a } \hat { } 2$ mod $4 \ = \ 1$ " by presburger [ATP] then show ?thesis by auto [ATP]
|
| 327 |
+
qed
|
| 328 |
+
|
| 329 |
+
# Formal Proof:
|
| 330 |
+
|
| 331 |
+
theorem numbertheory_sqmod4in01d: fixes a :: int shows " $\hat { a } ^ { \star } 2$ mod $4 \ = \ 0$ ) $\scriptstyle \backslash < \supset \pm \gamma$ ( $\hat { \mathbf { a } } ^ { \hat { } } 2$ mod $4 ~ = ~ 1$ )"
|
| 332 |
+
proof -
|
| 333 |
+
(\* a \pmod 4 \in {0, 1, 2, 3}. \*)
|
| 334 |
+
have c0: "a mod 4 \<in> {0, 1, 2, 3}" by auto [ATP]
|
| 335 |
+
( $\star$ Using that for any natural number $\mathbf { k } ,$ a \equiv b \pmod 4 implies aˆk \equiv bˆk \pmod $^ { 4 , \star }$ )
|
| 336 |
+
have $\ " \mathtt { a } \hat { } 2$ mod $4 \ =$ (a mod 4)\<ˆsup> 2 mod 4" by (smt (z3) numeral_eq_Suc power_mod) [ATP]
|
| 337 |
+
( $\star$ we have aˆ2 \pmod 4 \in {0, 1, 4, 9}. \*)
|
| 338 |
+
also have "... \<in> {0, 1, 4, 9}" using c0 by auto [ATP]
|
| 339 |
+
( $^ { \star }$ Since 4 \equiv 0 \pmod 4 and 9 \equiv 1 \pmod 4, the result follows. \*)
|
| 340 |
+
finally show ?thesis by auto [ATP]
|
| 341 |
+
qed
|
| 342 |
+
|
| 343 |
+
Statement: For each plumbing repair job, Mr. Wrench charges $N$ dollars for coming out to the house plus $_ x$ dollars per hour that he works at the house. He charged $\$ 97$ for a one-hour repair job and $\$ 265$ for a five-hour repair job. What is his charge for a two-hour repair job? Show that it is $\$ 139$ .
|
| 344 |
+
|
| 345 |
+
# Informal Proof (Minerva 62B):
|
| 346 |
+
|
| 347 |
+
Let $N$ be the number of dollars charged for coming out to the house and let $x$ be the number of dollars per hour that Mr. Wrench works. For the one-hour repair job, we have that the total cost is $N + x = 9 7$ .
|
| 348 |
+
For the five-hour repair job, we have that the total cost is $N + 5 x = 2 6 5$ .
|
| 349 |
+
We subtract the first equation from the second equation to obtain $4 x = 1 6 8 \Rightarrow x = 4 2$ .
|
| 350 |
+
Plugging this back into the first equation, we obtain that $N = 5 5$ .
|
| 351 |
+
Therefore, for a two-hour repair job, the total cost is $5 5 + 2 x = 5 5 + 2 ( 4 2 ) = { \Big [ } \ S 1 3 9 { \Big ] } .$
|
| 352 |
+
|
| 353 |
+
# Formal Proof:
|
| 354 |
+
|
| 355 |
+
theorem mathd_algebra_160: fixes n x ::real assumes $" \mathrm { n } + \mathrm { x } = 9 7 "$ and $" \mathrm { n } ~ + ~ 5 ~ \star ~ \mathrm { x } = 2 6 5 "$ shows $" { \mathrm { ~ n ~ } } + ~ 2 ~ \star ~ { \mathrm { ~ x ~ } } = ~ 1 3 9 "$
|
| 356 |
+
proof - ( $\star$ We subtract the first equation from the second equation to obtain $4 \mathbf { x } = 1 6 8 \Rightarrow \mathbf { x } = 4 2 . \ \star )$ ) have $" 4 ~ * ~ \mathrm { \bf { x } } ~ = ~ 1 6 8 "$ using assms by auto [ATP] then have $" \mathrm { z } = 4 2 "$ by auto [ATP] ( $^ { * }$ Plugging this back into the first equation, we obtain that $\mathbf { N } = 5 5$ . \*) then have $" \mathrm { n } = 5 5 "$ using assms by auto $[ A T P ]$ ( $\left[ \star \right]$ Therefore, for a two-hour repair job, the total cost is $\mathbf { 5 5 } + \mathbf { 2 x } = \mathbf { 5 5 } + \mathbf { 2 } ( \mathbf { 4 2 } ) = \mathbb { 8 1 3 9 } . ~ \star )$ then show ?thesis by (smt (z3) $\langle \mathbf { x } \ : = \ : 4 2 \rangle$ ) [ATP]
|
| 357 |
+
qed
|
| 358 |
+
|
| 359 |
+
Statement: Show that for any real number $_ x$ and any natural number $n$ , if $x > - 1$ , then $( 1 + n x ) \leq ( 1 + x ) ^ { n }$ .
|
| 360 |
+
|
| 361 |
+
# Informal Proof (Minerva 62B):
|
| 362 |
+
|
| 363 |
+
This is true for $n = 1$ . Now, suppose that this is true for $n$ .
|
| 364 |
+
Then we have that
|
| 365 |
+
|
| 366 |
+
$$
|
| 367 |
+
( 1 + n x ) ^ { n } = 1 + n ^ { 2 } x + n ^ { 3 } x ^ { 2 } + \cdot \cdot \cdot + n x ^ { n }
|
| 368 |
+
$$
|
| 369 |
+
|
| 370 |
+
$$
|
| 371 |
+
\leq 1 + n x + n ^ { 2 } x ^ { 2 } + \cdot \cdot \cdot + n x ^ { n } = ( 1 + x ) ^ { n }
|
| 372 |
+
$$
|
| 373 |
+
|
| 374 |
+
and
|
| 375 |
+
|
| 376 |
+
$$
|
| 377 |
+
\begin{array} { c } { { ( 1 + n x ) ^ { n + 1 } = ( 1 + n x ) ^ { n } ( 1 + n x ) } } \\ { { \leq ( 1 + x ) ^ { n } ( 1 + x ) = ( 1 + x ) ^ { n + 1 } . } } \end{array}
|
| 378 |
+
$$
|
| 379 |
+
|
| 380 |
+
Therefore, this is true for $n + 1$
|
| 381 |
+
|
| 382 |
+
# Formal Proof:
|
| 383 |
+
|
| 384 |
+
theorem induction_1pxpownlt1pnx: fixes x :: real and n :: nat assumes $" - 1 < x "$ shows " $( \underline { { { 1 } } } ~ + ~ \boldsymbol { \mathrm { n } } { \star } \boldsymbol { \mathrm { x } } )$ ) (1 + x)ˆn"
|
| 385 |
+
proof (induct n)
|
| 386 |
+
case 0
|
| 387 |
+
then show ?case
|
| 388 |
+
by auto [ATP]
|
| 389 |
+
next case (Suc n)
|
| 390 |
+
then show ?case
|
| 391 |
+
by (smt (z3) Bernoulli_inequality assms)
|
| 392 |
+
[ATP]
|
| 393 |
+
qed
|
| 394 |
+
|
| 395 |
+
Statement: Determine the remainder of 1529 (mod 6). Show that it is 5.
|
| 396 |
+
|
| 397 |
+
Informal Proof (Minerva 62B): By definition, $1 5 2 9 \equiv 5$ (mod 6).
|
| 398 |
+
|
| 399 |
+
# Formal Proof:
|
| 400 |
+
|
| 401 |
+
theorem mathd_numbertheory_551 : "1529 mod $6 =$ (5::nat)" by auto [ATP]
|
parse/dev/SMa9EAovKMC/SMa9EAovKMC_content_list.json
ADDED
|
@@ -0,0 +1,1997 @@
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "DRAFT, SKETCH, AND PROVE: GUIDING FORMAL THEOREM PROVERS WITH INFORMAL PROOFS ",
|
| 5 |
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"text_level": 1,
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| 6 |
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| 12 |
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"page_idx": 0
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| 13 |
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},
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| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
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"text": "Albert Q. Jiang1,2,† Sean Welleck3,4,† Jin Peng Zhou5,6,† ",
|
| 17 |
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"bbox": [
|
| 18 |
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| 19 |
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| 24 |
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},
|
| 25 |
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{
|
| 26 |
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"type": "text",
|
| 27 |
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"text": "Wenda Li2 Jiacheng Liu3 Mateja Jamnik2 ",
|
| 28 |
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"bbox": [
|
| 29 |
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| 30 |
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| 31 |
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| 35 |
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},
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| 36 |
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{
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| 37 |
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"type": "text",
|
| 38 |
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"text": "Timothee Lacroix ´ 1 Guillaume Lample1,‡ Yuhuai Wu5,7,‡ ",
|
| 39 |
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"bbox": [
|
| 40 |
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| 41 |
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| 46 |
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| 47 |
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{
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| 48 |
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"type": "text",
|
| 49 |
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"text": "1Meta AI 2University of Cambridge 3University of Washington 4Allen Institute for AI \n5Google Research 6Cornell University 7Stanford University ",
|
| 50 |
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"bbox": [
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| 51 |
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| 52 |
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| 57 |
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},
|
| 58 |
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{
|
| 59 |
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"type": "text",
|
| 60 |
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"text": "ABSTRACT ",
|
| 61 |
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"text_level": 1,
|
| 62 |
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"bbox": [
|
| 63 |
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| 64 |
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| 66 |
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| 67 |
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| 68 |
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"page_idx": 0
|
| 69 |
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},
|
| 70 |
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{
|
| 71 |
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"type": "text",
|
| 72 |
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"text": "The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce wellstructured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from $2 0 . 9 \\%$ to ${ \\mathrm { 3 9 . 3 \\% } }$ on a collection of mathematical competition problems. ",
|
| 73 |
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"bbox": [
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| 79 |
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"page_idx": 0
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| 80 |
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},
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| 81 |
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{
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| 82 |
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"type": "image",
|
| 83 |
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"img_path": "images/0618c1b53c8f61ad2c1ace44a644ca3c71dd722268b6931259a757c6a277fb43.jpg",
|
| 84 |
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"image_caption": [
|
| 85 |
+
"Figure 1: Draft, Sketch, and Prove. Starting with an informal statement, our framework yields a formal proof through a three-stage process: drafting informal proofs, mapping them into formal sketches, and proving the remaining conjectures. Concretely, an informal statement is a mathematical problem described in a mixture of natural and mathematical languages (e.g., formulae in $\\mathrm { I A I R } X ,$ ). Then, we use a large language model to autoformalize each informal proof into a formal sketch, which is a skeleton of the formal proof with open conjectures left unproven (indicated by the <proof $>$ blocks). The formal sketch mirrors the structure of the informal proof. Finally, the open conjectures/gaps inside each formal sketch are proved by an off-the-shelf prover. "
|
| 86 |
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],
|
| 87 |
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"image_footnote": [],
|
| 88 |
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"bbox": [
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| 89 |
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| 95 |
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},
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| 96 |
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{
|
| 97 |
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"type": "text",
|
| 98 |
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"text": "1 INTRODUCTION ",
|
| 99 |
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"text_level": 1,
|
| 100 |
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|
| 101 |
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| 102 |
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| 104 |
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| 108 |
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{
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| 109 |
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"type": "text",
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| 110 |
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"text": "Formal proof automation is a challenging task that has been the focus of increased attention in recent years (Bansal et al., 2019b; Polu & Sutskever, 2020; Lample et al., 2022; Jiang et al., 2022; Wu et al., 2022). However, deep learning approaches have not been as successful as in other domains, mainly because of the scarcity of formal data. Indeed, formalizing proofs is notoriously difficult and only accessible to a handful of experts, which makes large annotation endeavors unrealistic (Wiedijk, 2008). The largest formal proof corpus is written in Isabelle (Paulson, 1994), and amounts to less than 0.6 GB in size, orders of magnitude smaller than datasets commonly used in vision (Deng et al., 2009) or natural language processing (Brown et al., 2020). To address the scarcity of formal proofs, previous studies have proposed to use synthetic data (Wu et al., 2021b), self-supervision (Polu & Sutskever, 2020; Han et al., 2022), or reinforcement learning (Bansal et al., 2019a; Polu et al., 2022) to synthesize additional formal training data. Although these methods alleviate the data insufficiency to some degree, none are able to capitalize on the bulk of human-written mathematical proofs. ",
|
| 111 |
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| 112 |
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| 113 |
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| 114 |
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| 115 |
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],
|
| 117 |
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"page_idx": 1
|
| 118 |
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},
|
| 119 |
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{
|
| 120 |
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"type": "text",
|
| 121 |
+
"text": "Unlike formal mathematics, informal mathematical data is abundant and widely available. Recently, large language models trained on informal mathematical data showcased impressive quantitative reasoning abilities (Lewkowycz et al., 2022; Welleck et al., 2022). However, they often generate erroneous proofs and it is challenging to detect the faulty reasoning in these proofs automatically. Our work devises a novel approach called Draft, Sketch, and Prove (DSP) to translate informal mathematical proofs into formal ones and thus enjoy both the logical rigor provided by formal systems and the wealth of informal data. We give a schematic diagram of the $D S P$ method in Figure 1 and describe it in Section 3. Recent work (Wu et al., 2022) demonstrates the feasibility of automatically translating informal statements into formal ones with large language models. $D S P$ goes beyond and leverages large language models to generate formal proof sketches (Wiedijk, 2003) from informal proofs. Proof sketches consist of high-level reasoning steps that can be interpreted by formal systems such as interactive theorem provers. They differ from complete formal proofs in that they contain sequences of intermediate conjectures without justification. An example of informal proof with its corresponding formal proof sketch is provided in Figure 2. In the last step of $D S P$ , we elaborate the formal proof sketch into a full formal proof using an automated prover to prove all intermediate conjectures. ",
|
| 122 |
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"bbox": [
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| 123 |
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| 129 |
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},
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| 130 |
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{
|
| 131 |
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"type": "text",
|
| 132 |
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"text": "We perform experiments to generate formal proofs of problems from the miniF2F dataset (Zheng et al., 2022) and show that a large portion of theorems can be proved automatically with this method. We investigate two settings where the informal proofs are either written by humans or drafted by a large language model trained on mathematical text. These two settings correspond to situations frequently occurring during the formalization of existing theories, where informal proofs are usually available, but sometimes left as exercises to the reader or missing due to space limits in the margin. ",
|
| 133 |
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],
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| 139 |
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"page_idx": 1
|
| 140 |
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},
|
| 141 |
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{
|
| 142 |
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"type": "text",
|
| 143 |
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"text": "Contributions: ",
|
| 144 |
+
"text_level": 1,
|
| 145 |
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"bbox": [
|
| 146 |
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|
| 147 |
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| 148 |
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| 149 |
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| 151 |
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"page_idx": 1
|
| 152 |
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},
|
| 153 |
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{
|
| 154 |
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"type": "text",
|
| 155 |
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"text": "• We introduce a novel approach to leverage informal proofs to guide automated provers with formal proof sketches. \n• To evaluate our approach, we build a dataset of manually curated informal statements and informal proofs aligned with formal statements in the miniF2F dataset (Zheng et al., 2022). \n• We increase the proportion of problems solved by an automated prover on miniF2F from $2 0 . 9 \\%$ to $3 8 . 9 \\%$ given language-model-generated informal proofs, and up to $3 9 . 3 \\%$ when proofs are written by humans. \n• Through three ablation studies, we demonstrate the performance benefit of drafting informal proofs, annotating sketches with informal segments, and using automated provers to close open conjectures for the autoformalization of proofs. ",
|
| 156 |
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"bbox": [
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| 157 |
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],
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"page_idx": 1
|
| 163 |
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},
|
| 164 |
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{
|
| 165 |
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"type": "text",
|
| 166 |
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"text": "2 BACKGROUND AND RELATED WORK ",
|
| 167 |
+
"text_level": 1,
|
| 168 |
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"bbox": [
|
| 169 |
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| 170 |
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| 174 |
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| 175 |
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},
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| 176 |
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{
|
| 177 |
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"type": "text",
|
| 178 |
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"text": "Interactive theorem proving Modern verification systems for mathematics are centered around interactive theorem provers $( I T P s )$ , such as Isabelle (Paulson, 1994), Lean (Moura et al., 2015), Coq (Barras et al., 1997), or Metamath (Megill & Wheeler, 2019). ITPs embed the mathematical definitions and theorems onto a solid logical foundation (e.g., Higher-Order Logic, Dependent Type Theory) implemented by their kernels. Every theorem must be checked by the kernel to be recognized by the ITP. To be proved formally, a theorem is first stated in the ITP’s programming language, and iteratively simplified into simpler objectives (or subgoals), until it can be reduced to already proven facts. In this paper, we will refer to proofs verified by a formal theorem prover as formal proofs, and proofs written in “standard” mathematics (e.g. in LATEX) as informal proofs. ",
|
| 179 |
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| 180 |
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| 185 |
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"page_idx": 1
|
| 186 |
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},
|
| 187 |
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{
|
| 188 |
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"type": "text",
|
| 189 |
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"text": "",
|
| 190 |
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"bbox": [
|
| 191 |
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| 192 |
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| 193 |
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| 194 |
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| 195 |
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],
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| 196 |
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"page_idx": 2
|
| 197 |
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},
|
| 198 |
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{
|
| 199 |
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"type": "text",
|
| 200 |
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"text": "Machine learning for formal proof synthesis Several approaches propose to combine machine learning with modern interactive theorem provers (Yang & Deng, 2019; Gauthier et al., 2021), and build upon the recent success of language models (Polu & Sutskever, 2020; Han et al., 2022; Polu et al., 2022; Jiang et al., 2022; Lample et al., 2022). These methods typically rely on sequence-to-sequence models (Sutskever et al., 2014) to generate the next step of a proof given the current proof state and perform search over the generated subgoals using powerful search methods such as MCTS (Silver et al., 2018; Wu et al., 2021a; Laurent & Platzer, 2022). Because search is computationally expensive, these language models are relatively small (with fewer than 1 billion parameters). Our method contrasts with these approaches in that we use a significantly reduced number of calls to the models, but also much larger language models (with up to 175 billion parameters) that showcase outstanding few-shot learning abilities (Brown et al., 2020). ",
|
| 201 |
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"bbox": [
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| 202 |
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| 206 |
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],
|
| 207 |
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"page_idx": 2
|
| 208 |
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},
|
| 209 |
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{
|
| 210 |
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"type": "text",
|
| 211 |
+
"text": "Machine learning for informal reasoning Language models have also been used in the context of purely informal mathematics (Lample & Charton, 2020; Hendrycks et al., 2021; Welleck et al., 2021; Drori et al., 2022; Welleck et al., 2022). Nevertheless, Lewkowycz et al. (2022) note that for quantitative question answering, models are prone to generate false positives: the model guesses the right answer while providing an incorrect proof. These errors are hard to spot without human inspection. Worryingly, the frequency of false positives increases with the difficulty of the problem. Our method builds on these findings and translates informal proofs into formal proofs. Since ITPs are logically grounded, once a formal proof is checked by them, we are guaranteed its correctness. ",
|
| 212 |
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|
| 213 |
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| 214 |
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| 216 |
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| 217 |
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| 218 |
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|
| 219 |
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},
|
| 220 |
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{
|
| 221 |
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"type": "text",
|
| 222 |
+
"text": "Autoformalization In a position paper, Szegedy (2020) argued for attaining formal mathematical data from informal sources with neural networks. Wang et al. (2020) performed preliminary experiments where the evaluation was limited to text-level similarities on synthetic datasets. Recently, Wu et al. (2022) found that large language models (Chen et al., 2021; Chowdhery et al., 2022) are capable of few-shot statement autoformalization. Namely, a small number of examples are enough for them to learn to perform informal-to-formal translation of statements. In this paper, we investigate whether these findings can generalize to proof autoformalization, i.e., whether large language models can be used to translate informal proofs into formal ones. ",
|
| 223 |
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"bbox": [
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| 224 |
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| 225 |
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| 226 |
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| 227 |
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| 228 |
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],
|
| 229 |
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"page_idx": 2
|
| 230 |
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},
|
| 231 |
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{
|
| 232 |
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"type": "text",
|
| 233 |
+
"text": "3 METHOD ",
|
| 234 |
+
"text_level": 1,
|
| 235 |
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"bbox": [
|
| 236 |
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| 237 |
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| 238 |
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| 239 |
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587
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| 240 |
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],
|
| 241 |
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"page_idx": 2
|
| 242 |
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},
|
| 243 |
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{
|
| 244 |
+
"type": "text",
|
| 245 |
+
"text": "In this section, we describe our Draft, Sketch, and Prove (DSP) method for formal proof automation, which leverages informal proofs to guide automated formal theorem provers with proof sketches. We assume that each problem comes with an informal statement and a formal statement describing the problem. Our pipeline consists of three stages (depicted in Figure 1), which we present below. ",
|
| 246 |
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| 247 |
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"type": "text",
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"text": "3.1 DRAFTING INFORMAL PROOFS ",
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"text": "The initial phase of the DSP method consists in finding informal proofs for a problem according to its description in natural mathematical language (possibly with LATEX). The resulting informal proof is seen as a draft for the subsequent phases. In mathematical textbooks, proofs of theorems are in general provided, but are sometimes missing or incomplete. Therefore, we consider two settings corresponding to the presence or absence of the informal proofs. In the first, we assume that a “ground-truth” informal proof (i.e., one written by a human) is available, which is the typical scenario in the practice of formalizing existing mathematical theories. In the second setting, we make a more general assumption that the ground-truth informal proof is not given, and draft proof candidates with a large language model trained on informal mathematical data. The language model removes the dependence on human proofs and can produce multiple alternative solutions for every problem. Although there is no easy way to automatically verify the correctness of these proofs, the informal proof only needs to be useful for producing a sketch in the next stage. ",
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"text": "3.2 MAPPING INFORMAL PROOFS INTO FORMAL SKETCHES ",
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"text": "A formal proof sketch encodes the structure of a solution and leaves out low-level details (Wiedijk, 2003). Intuitively, it is a partial proof that outlines high-level conjecture statements. A concrete example of a proof sketch is shown in Figure 2. Although informal proofs often leave aside low-level details, (e.g., by stating their triviality), these details cannot be discharged in a formal proof, making ",
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"text": "Informal Statement: Show that for any real number $a$ , $1 0 a \\leq 2 8 a ^ { 2 } + 1$ ",
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"text": "Informal Proof: ",
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"text": "It suffices to show $0 \\leq 2 8 a ^ { 2 } - 1 0 a + 1$ . First, consider completing the square for $2 8 a ^ { 2 } - 1 0 a$ and observe that $\\textstyle \\left( a - { \\frac { 5 } { 2 8 } } \\right) ^ { 2 } = a ^ { 2 } - { \\frac { 1 0 } { 2 8 } } a + ( 5 / 2 8 ) ^ { 2 }$ . Since $\\begin{array} { r } { 0 \\leq \\left( a - \\frac { 5 } { 2 8 } \\right) ^ { 2 } } \\end{array}$ , we get $0 \\leq a ^ { 2 } - { \\textstyle { \\frac { 1 0 } { 2 8 } } } a + ( 5 / 2 8 ) ^ { 2 }$ . Multiplying by 28 and simplifying gives $0 \\leq 2 8 a ^ { 2 } - 1 0 a + ( 2 5 / 2 8 )$ . Since $2 5 / 2 8 < 1$ , the result follows. ",
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"text": "Formal Proof Sketch: ",
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"img_path": "images/3851a79bddd181a8d136d7db1f8f66013f998acb1a0c5e97ed31bc96d19cd07f.jpg",
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"image_caption": [
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"Figure 2: A proof sketch in Isabelle. The problem “Show that for any real number $a$ , $1 0 a \\leq 2 8 a ^ { 2 } + 1 ^ { \\prime }$ is given with an informal proof and an associated formal proof sketch. The sketch first rewrites the original statement $( \\mathtt { C } 0 )$ , which is proved through 5 intermediary conjectures (c1..c5). We use a special token $( < \\cdots > )$ to indicate that the conjecture is “open” and should be tackled by an automated prover later. To facilitate the alignment between the informal and formal languages, we annotate the formal proof sketch examples with informal proof segments (shown in red), which are immediately followed by their formal counterparts. "
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"text": "straightforward informal-to-formal proof translation difficult. Instead, we propose to map informal proofs to formal proof sketches that share the same high-level structures. The low-level details missing from a proof sketch can later be filled by an automated prover. Since large informal-formal parallel corpora do not exist, standard machine translation methods are unsuitable for this task. Rather, we use the few-shot learning abilities of a large language model. Specifically, we prompt the model with a few example pairs containing informal proofs and their corresponding formal sketches, followed by an informal proof yet to be translated. We then let the model generate the subsequent tokens to obtain the desired formal sketch. We refer to this model as an autoformalizer. ",
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"text": "3.3 PROVING OPEN CONJECTURES IN THE SKETCHES ",
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"text": "As the last part of the process, we execute off-the-shelf automated provers to fill in the missing details in proof sketches, where “automated provers” refers to systems capable of producing formally verifiable proofs. Our framework is agnostic to the specific choice of the automated prover: it can be symbolic provers such as heuristic proof automation tools, neural-network-based provers, or hybrid approaches. If the automated prover successfully closes all the gaps in the proof sketch, it returns the final formal proof which can be checked against the problem’s specification. If the automated prover fails (e.g., it exceeds the allocated time limit), we consider the evaluation to be unsuccessful. ",
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"text": "4 EXPERIMENTS ",
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"text": "4.1 DATASET AND EVALUATION ",
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"text": "We evaluate our method on the miniF2F dataset (Zheng et al., 2022). The dataset contains the formal statements of 488 problems from high-school mathematical competitions, written in three formal languages: Lean, HOL-Light, and Isabelle. They are split into a valid set and a test set, composed of 244 problems each. In this work, we choose to experiment with Isabelle for three reasons: (1) Isabelle’s proof corpus is one of the largest among interactive theorem provers, conducive to the language models’ mastery of its syntax; (2) Isabelle supports the declarative proof style (detailed discussion in Appendix A), enabling formal proof sketches (Wiedijk, 2003) which are central to our method; (3) although automated proving tools are available in other interactive theorem provers, none are as developed and effective as Sledgehammer (Paulson, 2010) in Isabelle for proving conjectures. ",
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"text": "The miniF2F dataset is comprised of problems from three source categories: (1) 260 problems sampled from the MATH dataset (Hendrycks et al., 2021); (2) 160 problems from actual high-school mathematical competitions (AMC, AIME, and IMO); (3) 68 crafted problems at the same difficulty level as (2). We employ three methods to obtain informal statements and proofs from these sources. For source (1), we access the informal statements and proofs from the MATH dataset; for (2), we retrieve their informal statements and proofs from the AOPS website 1; and for (3), we manually write down their informal statements and proofs. Thus we gather a parallel set of 488 informal statements, informal proofs, and formal statements. This dataset provides the informal statements and proofs for our experiment in the human-as-informal-proof-writer setting and will be released upon publication. ",
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"text": "Our task is to generate formal proofs for problems as they are formally stated in miniF2F. We consider a proof valid if and only if it (a) does not contain “cheating” keywords (sorry and oops) that exit a proof without completing it, and (b) Isabelle is able to verify the corresponding formal statement with the proof. We use the Portal-to-ISAbelle API by Jiang et al. (2021) to interact with Isabelle. ",
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"text": "4.2 BASELINES",
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"text": "Sledgehammer As a baseline, we attempt to prove the formal statement directly with Sledgehammer, a popular proof automation tool in Isabelle. We use the default Sledgehammer configuration in Isabelle2021, including a 120-second timeout and the five automated theorem provers (Z3, CVC4, SPASS, Vampire, E). Appendix B gives a more thorough introduction to Sledgehammer. ",
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"text": "Sledgehammer $^ +$ heuristics Occasionally, Sledgehammer may fail without trying simple yet effective tactics. As a second, stronger baseline, we create an automated prover that tries 11 common tactics (auto, simp, blast, fastforce, force, eval, presburger, sos, arith, linarith, auto simp: field simps) for high-school level algebra and number theory problems. If every attempted tactic fails, or times out after 10 seconds, it falls back to Sledgehammer. ",
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"text": "Language models for proof search Finally, we include baselines which are representative of state-of-the-art neural theorem proving in Isabelle, specifically Thor (Jiang et al., 2022) and Thor with expert iteration on autoformalized data (Wu et al., 2022). The methods GPT-f with expert iteration (Polu et al., 2022), and HyperTree Proof Search (HTPS) (Lample et al., 2022) can solve $3 6 . 6 \\%$ and $4 1 . 0 \\%$ of the problems on miniF2F-test. However, they rely on the Lean theorem prover instead of Isabelle, which greatly influences the performance due to the different tactics and automation, and are not directly comparable to our method. ",
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"text": "4.3 EXPERIMENTAL SETUP ",
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"text": "The experimental code is at github.com/albertqjiang/draft sketch prove. ",
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"text": "Drafting When informal proofs are generated, we condition a large language model on informal statements to sample 100 informal proofs per problem. Specifically, we use the Codex code-davinci002 model (Chen et al., 2021) through the OpenAI API, and the 8B, 62B, and 540B versions of the Minerva model from Lewkowycz et al. (2022). We use greedy decoding for Codex and nucleus sampling (Holtzman et al., 2019) with temperature $T = 0 . 6$ and top $\\mathbf { p } = 0 . 9 5$ for Minerva models. ",
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"text": "Sketching For sketching, we manually prepare 20 autoformalization examples of the format (informal statement, informal proof, formal statement, formal sketch), to form a pool of high-quality demonstrations. Of these 20 examples, 10 are of the algebra type and 10 are of the number theory type. All examples are from the validation set of the miniF2F dataset and can be found in the supplementary materials. The sketches contain in-line comments as in Figure 2. If the name of the problem gives away its type (algebra or number theory), we only use examples of the corresponding type. We also ensure that the sampled few-shot examples do not contain the problem being solved. The prompt is 3 uniformly randomly sampled example from the pool concatenated with the current problem’s (informal statement, informal proof, formal statement). We use this prompt to query the same Codex model to get the desired proof sketches. We use deterministic greedy decoding and a maximum of 2048 tokens in the generated sequence. For all the experiments, unless stated otherwise, we control the total number of queries made to Codex per problem to be 100. This means 100 queries per human informal solution and one query per language-model-generated solution. ",
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{
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"type": "table",
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"img_path": "images/67573040b0afaac842bb1650e940a0f13e71ffaa603cf29431ea6b657708411d.jpg",
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"table_caption": [
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| 557 |
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"Table 1: Proving success rates on the miniF2F dataset with Isabelle In the table are the success rates of four baselines, the DSP method with human and language model informal proofs, as well as three ablation studies, on the validation and the test sets of miniF2F. The highest success rates on each set are highlighted in bold. The performance difference between ablation studies and DSP with human informal proofs are enclosed in brackets. "
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"table_footnote": [],
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"table_body": "<table><tr><td>Success rate</td><td>miniF2F-valid</td><td>miniF2F-test</td></tr><tr><td colspan=\"3\">Baselines</td></tr><tr><td>Sledgehammer</td><td>9.9%</td><td>10.4%</td></tr><tr><td>Sledgehammer + heuristics</td><td>18.0%</td><td>20.9%</td></tr><tr><td>Thor (Jiang et al., 2022)</td><td>28.3%</td><td>29.9%</td></tr><tr><td>Thor+ expert iteration (Wu et al.,2022)</td><td>37.3%</td><td>35.2%</td></tr><tr><td colspan=\"3\">Draft,Sketch,and Prove</td></tr><tr><td>Human informal proof</td><td>42.6%</td><td>39.3%</td></tr><tr><td>Codex informal proof</td><td>40.6%</td><td>35.3%</td></tr><tr><td>8B Minerva informal proof</td><td>40.6%</td><td>35.3%</td></tr><tr><td>62BMinerva informal proof</td><td>43.9%</td><td>37.7%</td></tr><tr><td>540BMinerva informal proof</td><td>42.6%</td><td>38.9%</td></tr><tr><td colspan=\"3\">Ablations (with human informal statements and proofs)</td></tr><tr><td>-In-line comments</td><td>37.7% (-4.9%)</td><td>36.5% (-2.8%)</td></tr><tr><td>- Informal proofs</td><td>38.9%(-3.7%)</td><td>34.0% (-5.3%)</td></tr><tr><td>- Automated provers</td><td>32.8% (-9.8%)</td><td>30.3% (-9.0%)</td></tr></table>",
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"text": "",
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"text": "Proving To prove the conjectures left open by the formal sketch, we use the Sledgehammer $^ +$ heuristics automated prover described in Subsection 4.2. We execute the automated prover on every open conjecture in the sketch to synthesize a formal proof that can be verified by Isabelle. ",
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"type": "text",
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"text": "4.4 RESULTS ",
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"text_level": 1,
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"text": "In Table 1, we display the proportion of successful formal proofs found on the miniF2F dataset with the interactive theorem prover Isabelle. The results include the four baselines described in Subsection 4.2 and the $D S P$ method with human-written proofs and model-generated proofs. From the table, we can see that the automated prover with 11 additional heuristic tactics significantly increases the performance of Sledgehammer, boosting its success rate from $9 . 9 \\%$ to $1 8 . { \\bar { 0 } } \\%$ on the validation set of miniF2F and from $1 0 . 4 \\%$ to $2 0 . 9 \\%$ on the test set. The two baselines using language models and proof search (Thor and Thor $^ +$ expert iteration) achieve success rates of $2 9 . 9 \\%$ and ${ \\bar { 3 } } 5 . 2 { \\bar { \\% } }$ on the test set of miniF2F, respectively. ",
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"text": "With informal proofs written by humans, the $D S P$ method achieves success rates of $4 2 . 6 \\%$ and $3 9 . 3 \\%$ on the validation and test sets of miniF2F. A total of 200 out of 488 problems can be proved in this way. The Codex model and the Minerva (8B) model give very similar results in solving problems on miniF2F: they both guide the automated prover to solve $4 0 . { \\dot { 6 } } \\%$ and $3 5 . 3 \\%$ of problems on the validation and the test sets respectively. This is corroborated by Lewkowycz et al. (2022)’s observation that these two models have comparable performances in solving mathematical problems. ",
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"type": "text",
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"text": "When we switch to the Minerva (62B) model, the success rates rise up to $4 3 . 9 \\%$ and $3 7 . 7 \\%$ respectively. Compared to human-written informal proofs, its success rates are $1 . 3 \\%$ higher on the validation set and $1 . 6 \\%$ lower on the test set. In total, the Minerva (62B) model is able to solve 199 problems on miniF2F, one fewer than with human proofs. The $D S P$ method is effective in guiding the automated prover under both settings that we study: using either human informal proofs or language-model-generated informal proofs. $D S P$ almost doubles the prover’s success rate and results in a new state-of-the-art performance on miniF2F with Isabelle. Moreover, the larger Minerva model is almost as helpful as a human in guiding the automated prover in solving problems. ",
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"type": "text",
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"text": "5 ANALYSIS ",
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"type": "text",
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"text": "5.1 ABLATION STUDIES ",
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"type": "text",
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"text": "Ablation of in-line comments To facilitate the alignment between the informal proofs and the formal proof sketches, we copy relevant segments of the informal proofs as in-line comments in the sketches. In the manually constructed prompt examples, these comments are prefixed to the corresponding Isabelle code blocks, as shown in Figure 2 (the text in red). We hypothesize that this technique is beneficial for large language models to synthesize formal sketches. To validate this hypothesis, we perform an ablation study by removing the in-line comments in the prompt examples before running the experiment. The results are displayed in Table 1. We find that without in-line comments, the success rates drop by $4 . 9 \\%$ and $2 . 8 \\%$ on the validation and test sets respectively. We conclude that having in-line comments is helpful for generating formal proof sketches. ",
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"img_path": "images/027b72acf9437d01f10744a3841881836cb9b16c4928ad7d09dba4b3221a05fe.jpg",
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"image_caption": [
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| 675 |
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"Figure 3: Number of problems solved on miniF2F against the number of autoformalization attempts per problem. Left: The figure displays the experiments carried out with the $D S P$ method and three ablations on it. The curves represent the $D S P$ method (blue), formal proof sketches without the in-line comments (orange), without informal proofs altogether (green), and without the automated provers (red). Right: The figure compares the experimental results with informal proof drafts written by humans (blue), the 540B Minerva model (orange), the 62B Minerva model (green), the 8B Minerva model (red), and the Codex model (purple). "
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"text": "",
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"type": "text",
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"text": "Ablation of informal proof drafts Drafting informal proofs is the first step of the $D S P$ method. To investigate the necessity of this step, we perform an experiment of formal sketching and proving without informal proofs at all. Because formal proof sketches are written in the declarative proof style, they are fairly similar to the informal proof drafts already. Concretely, we remove the informal proofs and the in-line comments (because they are copied segments of the informal proofs) in the prompt examples. This removes the need for the informal proof writer, whether a human or a neural network. The results of this setup are shown in Table 1. It can be seen that the success rates on the validation and the test sets of miniF2F drop by $3 . 7 \\%$ and $5 . 3 \\%$ respectively compared to with human-written proofs. They are also inferior to success rates obtained with language-model-generated informal proofs. This demonstrates the importance of drafting informal proofs before sketching and proving. ",
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"type": "text",
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"text": "Ablation of automated provers Using an autoformalizer to generate proof sketches which are then completed by an automated prover is central to our method. The effect of utilizing an automated prover to close open conjectures in proof sketches is worth studying, so we conduct an ablation experiment for it. Namely, we replace the proof sketches in the prompt examples with complete formal proofs. The complete formal proofs still follow the declarative proof style, but do not contain any open conjectures. As a result, the large language model will also generate full proofs instead of sketches, and we directly check whether these generated proofs are valid. The results in this setup are presented in Table 1. The results reveal that without an automated prover to close open conjectures, the success rate on miniF2F decreases by $9 . 8 \\%$ and $9 . 0 \\%$ on the validation and test sets respectively. The drastic performance difference indicates the essential role of automated provers in our approach. ",
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"text": "Scaling properties of ablation studies To understand the effect of the ablations on the $D S P$ method’s scaling properties, we vary the number of autoformalization attempts per problem and plot the number of successful proofs found on the miniF2F dataset in Figure 3 (left). Three methods are contrasted: the original $D S P$ method with human informal proofs, the $D S P$ method without in-line comments, and the $D S P$ method without sketching. It can be seen from the figure that with the original $D S P$ method, the performance reaches a plateau (no new proofs are found) after 70 autoformalization attempts are made for each problem. For the ablation study with no in-line comments, the plateau is reached much faster, after around 50 autoformalization attempts. This method solves 181 problems in total. The ablation study without sketching can solve 154 problems on miniF2F. In comparison, with human informal proofs, only 7 autoformalization attempts are required to reach this performance. ",
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"img_path": "images/237540f7093980291b4817da7f69f22cfd34648803a0159cbea5b01153532362.jpg",
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"image_caption": [
|
| 734 |
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"Figure 4: IMO proof guided by a Minerva informal proof An informal proof of the International Math Olympiad problem $\\mathrm { i m o \\_ 1 9 5 9 \\mathrm { _ - p 1 } }$ generated by Minerva that leads to a successful formal proof. The steps enclosed by the $A T P$ delimiters are generated by an automated prover and all other steps are generated by the DSP autoformalizer. "
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"type": "text",
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"text": "5.2 LANGUAGE-MODEL-GENERATED PROOFS ",
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| 748 |
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"text_level": 1,
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"type": "text",
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| 759 |
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"text": "Our experiments demonstrated that model-generated informal proofs from Minerva and Codex can help guide a formal theorem prover. In this section, we analyze the properties of these proofs further. Since the Minerva (62B and 540B) models give the best overall performance on miniF2F, we focus on the informal proofs they produce in this section. ",
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| 760 |
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"type": "text",
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| 770 |
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"text": "Minerva helps solve one IMO problem Interestingly, our approach manages to solve one problem from the International Mathematical Olympiad (imo 1959 1) with a Minerva-generated solution, but not with the human proof. For this problem, we present the successful Minerva-generated informal proof draft and the formal proof in Figure 4. We hypothesize that the reason behind this phenomenon is that human proofs might leave gaps between conjectures that are too difficult for automated provers to solve. On the other hand, the diversity in language model informal proofs makes some of them more amenable to automated provers. In Appendix C, we analyze the human and the Minerva informal proofs for this problem in greater detail. In Appendix D, we present a manual evaluation of Minerva proofs, and 3 more case studies comparing the human and Minerva informal proofs. ",
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| 771 |
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| 780 |
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"type": "text",
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| 781 |
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"text": "Is there a way to detect which Minerva proofs are correct, without human evaluation? For a preliminary investigation, we filter out all the problems that can be solved directly with the automated prover from the 50 and are left with 27 informal proofs. Of these 27, 21 are completely correct, 6 still contain small errors, but none are nonsensical. With this simple filter, we achieve a precision of $7 7 . 8 \\%$ and a recall of $7 2 . 4 \\%$ in identifying correct Minerva informal proofs. ",
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"type": "text",
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| 792 |
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"text": "Scaling properties of human and Minerva proofs To understand the influence of different informal proof sources on the scaling properties of $D S P$ , we plot the number of successful proofs found on miniF2F against the number of autoformalization attempts per problem in Figure 3 (right). Note that for each problem, we have 1 informal proof by a human and 100 informal proof drafts by each language model. The one human proof is used 100 times for formal proof sketch generation, while each language model proof draft is used only once. The 62B and the 540B models result in more successful proofs than the smaller (8B) Minerva model and the Codex model, consistently for any number of attempts. The 8B Minerva model and the Codex model behave similarly, both finding 185 proofs in the end. Informal proofs written by humans help solve more problems than those by Minerva models for $1 - 1 0 0$ autoformalization attempts. However, the difference is small (1 problem) when 100 are made. ",
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| 802 |
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"type": "text",
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"text": "Noticing that the number of successful proofs does not plateau for the Minerva-generated proofs, we investigate how further increasing the number of autoformalization attempts changes the number of problems solved for human-written and language-model-generated proofs. For each problem, we use 1 human informal proof and sample 200 sketches for it; we also use the same 100 informal proof drafts by the Minerva (540B) language model and sample 2 sketches for each draft. The total number of sketches per problem is 200 in both settings. We find that with human informal proofs, 203 theorems ${ \\mathrm { ~ ( 1 0 6 / 9 7 } }$ on valid/test) have successful formal proofs, while with language-model-generated informal proofs, 209 (111/98 on valid/test) theorems have successful formal proofs after the same number of attempts. This suggests that the diversity in language-model-generated informal proofs can benefit the automated formalization process more than the “ground-truth” human proofs. ",
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| 804 |
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"type": "text",
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"text": "5.3 MEMORIZATION ",
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| 815 |
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"text_level": 1,
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| 816 |
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"type": "text",
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| 826 |
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"text": "This work utilizes two language models that have been trained on a large amount of internet data. Several prior works (Trinh & Le, 2018; Carlini et al., 2022) pointed out that such models can memorize some fraction of the data they encounter during training. For drafting informal proofs, we mainly experimented with Minerva. Lewkowycz et al. (2022, Section 5) discussed the memorization effects within Minerva and concluded that they could not find evidence that its abilities are due to memorization. For the autoformalization of proof sketches, the Codex (code-davinci-002) model was used. Its training data was collected before June $2 0 2 1 ^ { 2 }$ , at which time the miniF2F dataset had not been made public. So the model cannot benefit from memorizing the exact problems and proofs. Therefore, it is inappropriate to attribute the abilities of models used in this paper to memorization. ",
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| 836 |
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"type": "text",
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| 837 |
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"text": "6 CONCLUSION ",
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| 838 |
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"text_level": 1,
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| 839 |
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| 847 |
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"type": "text",
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| 849 |
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"text": "In this paper, we introduced Draft, Sketch, and Prove $( D S P )$ , a novel approach that takes advantage of informal proofs to synthesize formal proofs. We demonstrated its feasibility and effectiveness by reaching state-of-the-art performance on the miniF2F dataset with the Isabelle theorem prover. Central to our method are formal proof sketches that mirror the high-level reasoning structures of informal proofs. Our ablations showed that the ability to automatically convert informal proofs to proof sketches is critical to the success of $D S P$ . ",
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| 850 |
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"type": "text",
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| 860 |
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"text": "Our $D S P$ method differs fundamentally from previous applications of machine learning to formal proof synthesis in two aspects. Firstly, while most approaches in the field focus on improving proof search, our method seeks to construct the entire formal proof structure from the informal proof in one decoding operation. The task of the automated prover is then simplified to filling the gaps between intermediate conjectures. Secondly, while existing approaches operate exclusively on formal data, $D S P$ by design benefits from informal proofs. ",
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| 861 |
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"type": "text",
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| 871 |
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"text": "In this work, we utilized a purely symbolic automated prover to close the gaps in proof sketches. In the future, we aim to equip $D S P$ with more powerful mechanisms, such as HyperTree Proof Search (Lample et al., 2022), to broaden the scope of provable theorems. Similar to AlphaCode (Li et al., 2022), we found that the number of generations is crucial for performance. The computational cost of the autoformalizer being a bottleneck in our method, we seek to develop approaches able to generate high-quality proof sketches more efficiently. ",
|
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"type": "text",
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| 882 |
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"text": "ACKNOWLEDGEMENTS ",
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"type": "text",
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"text": "We thank Rui Yuan and Kunhao Zheng for helping with the informal solutions used in our dataset. \nWe thank Christian Szegedy for his feedback on the early draft. ",
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"text": "FUNDING DISCLOSURE ",
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"text": "AQJ and WL are supported by the ERC Advanced Grant ALEXANDRIA (Project GA 742178). ",
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"text": "Kunhao Zheng, Jesse Michael Han, and Stanislas Polu. miniF2F: a cross-system benchmark for formal olympiad-level mathematics. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022. URL https://openreview.net/forum?id $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =$ 9ZPegFuFTFv. ",
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| 1408 |
+
},
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+
{
|
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"type": "text",
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| 1411 |
+
"text": "APPENDIX ",
|
| 1412 |
+
"text_level": 1,
|
| 1413 |
+
"bbox": [
|
| 1414 |
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176,
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101,
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281,
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118
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],
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},
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{
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"type": "text",
|
| 1423 |
+
"text": "A CONJECTURES AND THE DECLARATIVE PROOF STYLE ",
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| 1424 |
+
"text_level": 1,
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| 1425 |
+
"bbox": [
|
| 1426 |
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176,
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| 1427 |
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137,
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651,
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151
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"page_idx": 13
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| 1432 |
+
},
|
| 1433 |
+
{
|
| 1434 |
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"type": "text",
|
| 1435 |
+
"text": "Interactive theorem provers such as Isabelle and Mizar use a declarative proof style (Syme, 1997), in which a proof is interleaved with conjectures and their corresponding proofs. Syme (1997) stated that the list of conjectures in a declarative proof should be analogous to a proof sketch found in a mathematical textbook and sufficiently convincing for the reader. In practice, ITP users often prove a theorem by writing down a list of conjectures (a “formal sketch”), then attempt to find a proof of each conjecture (fill a “gap”) with an automated system. ",
|
| 1436 |
+
"bbox": [
|
| 1437 |
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|
| 1438 |
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167,
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| 1439 |
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| 1440 |
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251
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],
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"page_idx": 13
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| 1443 |
+
},
|
| 1444 |
+
{
|
| 1445 |
+
"type": "text",
|
| 1446 |
+
"text": "B SLEDGEHAMMER ",
|
| 1447 |
+
"text_level": 1,
|
| 1448 |
+
"bbox": [
|
| 1449 |
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176,
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| 1450 |
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272,
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| 1451 |
+
352,
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],
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"page_idx": 13
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| 1455 |
+
},
|
| 1456 |
+
{
|
| 1457 |
+
"type": "text",
|
| 1458 |
+
"text": "Sledgehammer (Paulson, 2010) is a powerful system that automates reasoning with the interactive theorem prover Isabelle. It works by flattening the goals encoded in the higher-order logic used by Isabelle/HOL into other logics (e.g., first-order logic) which can then be fed into automated theorem provers such as E 3, CVC4 4, $\\bar { Z 3 ^ { 5 } }$ , Vampire 6, and SPASS 7. If any of these automated theorem provers succeeds in finding the proof in their own corresponding format, Sledgehammer reconstructs the proof in Isabelle/HOL with certified provers (metis, meson, and smt), which is relatively more interpretable by humans. ",
|
| 1459 |
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"bbox": [
|
| 1460 |
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|
| 1461 |
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304,
|
| 1462 |
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825,
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401
|
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],
|
| 1465 |
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"page_idx": 13
|
| 1466 |
+
},
|
| 1467 |
+
{
|
| 1468 |
+
"type": "text",
|
| 1469 |
+
"text": "As a practical example of using Sledgehammer, one can declare a conjecture in Isabelle/HOL: have \"4 dvd (a::nat) $\\Longrightarrow ~ 2$ dvd $\\mathtt { a } \"$ and call Sledgehammer immediately afterwards. If Sledgehammer succeeds, it will return a proof step that proves the conjecture. In this example, the step is by (meson dvd trans even numeral), which uses the meson resolution prover and two facts: that the division relation is transitive and that 4 is an even number. If Sledgehammer does not find the proof or timeouts, it will report failure. ",
|
| 1470 |
+
"bbox": [
|
| 1471 |
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174,
|
| 1472 |
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407,
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| 1473 |
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826,
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492
|
| 1475 |
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],
|
| 1476 |
+
"page_idx": 13
|
| 1477 |
+
},
|
| 1478 |
+
{
|
| 1479 |
+
"type": "text",
|
| 1480 |
+
"text": "C A PROOF TO AN INTERNATIONAL MATHEMATICAL OLYMPIAD PROBLEM ",
|
| 1481 |
+
"text_level": 1,
|
| 1482 |
+
"bbox": [
|
| 1483 |
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176,
|
| 1484 |
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513,
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| 1485 |
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800,
|
| 1486 |
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529
|
| 1487 |
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],
|
| 1488 |
+
"page_idx": 13
|
| 1489 |
+
},
|
| 1490 |
+
{
|
| 1491 |
+
"type": "text",
|
| 1492 |
+
"text": "With the Minerva-generated solutions, a proof to the problem imo 1959 p1 is discovered. This is the first problem of the first ever International Mathematical Olympiad (IMO). The informal problem statement, Minerva-generated informal solution, and DSP’s formal proof are shown in Figure 4. ",
|
| 1493 |
+
"bbox": [
|
| 1494 |
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| 1495 |
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| 1496 |
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| 1497 |
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|
| 1498 |
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],
|
| 1499 |
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"page_idx": 13
|
| 1500 |
+
},
|
| 1501 |
+
{
|
| 1502 |
+
"type": "text",
|
| 1503 |
+
"text": "In Figure 4, we can see that the autoformalizer in DSP (a large language model), copies over parts of the informal proof generated by Minerva as in-line comments to precede their corresponding formal proof blocks. The formal proof does not use the first sentence of the informal proof solution as it is already identical to the formal statement. We also notice that the large language model selects relevant premises after writing down the conjectures (the steps starting with using) despite not every premise is strictly needed. ",
|
| 1504 |
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"bbox": [
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| 1508 |
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],
|
| 1510 |
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"page_idx": 13
|
| 1511 |
+
},
|
| 1512 |
+
{
|
| 1513 |
+
"type": "text",
|
| 1514 |
+
"text": "The formal proof creates 5 conjectures (4 have statements and 1 show statement) which are all subsequently proved by our automated theorem prover. The step to prove the statement have \"gcd $( 2 1 \\star \\Pi + 4 )$ ) $( { { \\mathchoice { \\mathrm { ~ 1 ~ 4 ~ } } { \\mathrm { ~ ~ 1 ~ } } { \\mathrm { ~ 1 ~ 4 ~ } } { \\mathrm { ~ ~ 1 ~ } } { \\mathrm { ~ ~ 1 ~ } } } } + { { \\mathchoice { \\mathrm { ~ 3 ~ ) ~ } } { \\mathrm { ~ ~ 3 ~ } } { \\mathrm { ~ ~ 3 ~ } } { \\mathrm { ~ ~ 3 ~ } } } } = { { \\mathchoice { \\mathrm { ~ 1 ~ } } { \\mathrm { ~ ~ 1 ~ } } { \\mathrm { ~ ~ 1 ~ } } { \\mathrm { ~ ~ 1 ~ } } { \\mathrm { ~ ~ 1 ~ } } } }$ involves 2 verified low-level provers smt and $^ { z 3 }$ and 10 lemmas/facts from outside the scope of the language model. It is highly unlikely that either the large language model or the automated theorem prover can finish this proof on its own. ",
|
| 1515 |
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"bbox": [
|
| 1516 |
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| 1517 |
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| 1518 |
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| 1519 |
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755
|
| 1520 |
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],
|
| 1521 |
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"page_idx": 13
|
| 1522 |
+
},
|
| 1523 |
+
{
|
| 1524 |
+
"type": "text",
|
| 1525 |
+
"text": "Unsuccessful human-written proof. In contrast, the human-written informal proof of this IMO problem did not lead to a successful formal proof. The human-written proof is: ",
|
| 1526 |
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"bbox": [
|
| 1527 |
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],
|
| 1532 |
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"page_idx": 13
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| 1533 |
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|
| 1534 |
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|
| 1535 |
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|
| 1536 |
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"text": "Denoting the greatest common divisor of $a , b$ as $( a , b )$ , we use the Euclidean algorithm: ",
|
| 1537 |
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"bbox": [
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| 1538 |
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"type": "equation",
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| 1547 |
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"img_path": "images/4c5f9af1eec5ff0dbd3c26f59b1e423b93ac8e24f21200fbabbc41b1ed1a84de.jpg",
|
| 1548 |
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"text": "$$\n( 2 1 n + 4 , 1 4 n + 3 ) = ( 7 n + 1 , 1 4 n + 3 ) = ( 7 n + 1 , 1 ) = 1\n$$",
|
| 1549 |
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"text_format": "latex",
|
| 1550 |
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"bbox": [
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| 1557 |
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|
| 1559 |
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"type": "text",
|
| 1560 |
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"text": "It follows that $\\frac { 2 1 n + 4 } { 1 4 n + 3 }$ is irreducible. Q.E.D. ",
|
| 1561 |
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"bbox": [
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| 1562 |
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| 1571 |
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"text": "A key difference between the Minerva proof and the human proof is the way that invoking the Euclidean algorithm is described. The Minerva proof explicitly writes out the results of the Euclidean algorithm (e.g. $2 1 n + 4 = 1 \\cdot ( 1 4 n + 3 ) + { \\bar { 7 } } n + 1 )$ , which are translated into the sketch ( $_ { c l }$ in Figure 4). The human proof introduces new notation to express the results indirectly in terms of greatest common divisors, which ends up being less suitable for sketching. For example, below is a sketch generated with the human proof, which has a conjecture that is semantically incorrect and hence cannot be closed by the automated prover: ",
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"text": "theorem ",
|
| 1583 |
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"text_level": 1,
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| 1592 |
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| 1593 |
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"type": "text",
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| 1594 |
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"text": "fixes n :: nat shows \"gcd $( 2 1 { \\star } \\Pi + 4 )$ ) $( 1 4 \\star \\Pi + 3 ) = 1 \"$ proof - have \" $( 2 1 \\star \\mathsf { n } + 4 , ~ 1 4 \\star \\mathsf { n } + 3 ) \\ = \\ ( 7 \\star \\mathsf { n } + 1 , ~ 1 4 \\star \\mathsf { n } + 3 ) \\mathsf { n }$ 1 ATP (\\* <--- UNSUCCESSFUL $^ { \\star }$ ) also have \" $\" \\ldots = ( 7 \\star \\Omega + 1 , \\mathrm { ~ 1 ~ } ) \"$ ATP finally show ?thesis ATP qed ",
|
| 1595 |
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"bbox": [
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| 1596 |
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| 1602 |
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|
| 1604 |
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"type": "text",
|
| 1605 |
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"text": "D MORE ANALYSIS ON HUMAN AND MINERVA INFORMAL PROOFS ",
|
| 1606 |
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"text_level": 1,
|
| 1607 |
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"type": "text",
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| 1617 |
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"text": "We analyze the relationship between the validity of the formal proofs and the correctness of the informal proofs. For our analysis, we randomly sample 50 Minerva proofs of different problems, which are then successfully converted to formal proofs. We then manually evaluate the correctness of these 50 informal proofs. Among them, 29 proofs $( 5 8 \\% )$ are entirely correct, 16 are incorrect with a clearly identifiable incorrect step, and 5 “proofs” are nonsensical and simply rephrase the final conclusions of the problems. ",
|
| 1618 |
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|
| 1627 |
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"type": "text",
|
| 1628 |
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"text": "Seeing that a total of $1 6 + 5 = 2 1$ incorrect informal proofs can lead to successful formal proofs, we study how they guide the automated formal prover despite having flaws themselves. The 21 proofs divide into 2 cases: In the first case, we find 13 problems for which the informal proofs are mostly ignored, and the automated prover can find proofs by itself; In the other 8 problems, although the informal proofs are wrong, the autoformalizer manages to correct them, either by ignoring the erroneous steps or by stating their correct versions in the formal proof sketches. This suggests that the autoformalizer has some understanding of the mathematical statements and is not merely translating them from an informal language to a formal language. It is robust to slight noises in its input. ",
|
| 1629 |
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"bbox": [
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|
| 1636 |
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| 1637 |
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|
| 1638 |
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"type": "text",
|
| 1639 |
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"text": "We present 3 case studies comparing the human and Minerva informal proofs. Particularly, Figure 7 shows a completely correct example and one example of each pathological case. ",
|
| 1640 |
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"bbox": [
|
| 1641 |
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{
|
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"type": "image",
|
| 1650 |
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"img_path": "images/aa460b3fb6dc8b7e32e9684370859e3189d97a411eb1fccc59693dc5150e9c9d.jpg",
|
| 1651 |
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"image_caption": [
|
| 1652 |
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"Figure 5: Algebra example with Minerva informal proof. An informal proof generated by Minerva that led to a successful formal proof. The autoformalizer generated a proof sketch containing all lines of the formal proof except for those with $I A T P J$ . The sketch is structured according to the informal proof, containing five intermediate conjectures based on the informal proof. The autoformalizer generated in-line comments in the proof sketch (shown in red), which correctly identified an alignment between the formal and informal proofs. "
|
| 1653 |
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],
|
| 1654 |
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"image_footnote": [],
|
| 1655 |
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"bbox": [
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"page_idx": 16
|
| 1662 |
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},
|
| 1663 |
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{
|
| 1664 |
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"type": "text",
|
| 1665 |
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"text": "Statement: For any integer $a$ , show that $a ^ { 2 } \\equiv$ 0 (mod 4) or ${ a ^ { 2 } \\equiv 1 }$ (mod 4). ",
|
| 1666 |
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"bbox": [
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|
| 1673 |
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},
|
| 1674 |
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{
|
| 1675 |
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"type": "text",
|
| 1676 |
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"text": "Informal Proof (Minerva 62B): ",
|
| 1677 |
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"text_level": 1,
|
| 1678 |
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"bbox": [
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|
| 1685 |
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},
|
| 1686 |
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|
| 1687 |
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"type": "text",
|
| 1688 |
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"text": "Let $a = 2 k$ for some integer $k$ . \nThen $a ^ { 2 } = 4 k ^ { 2 }$ . \nSince $4 k ^ { 2 }$ is divisible by 4, we have $\\boldsymbol { a } ^ { 2 } \\equiv 0$ (mod 4). \nNow, let $a = 2 k + 1$ for some integer $k$ . \nThen $a ^ { 2 } = 4 k ^ { 2 } + 4 k + 1 = 4 ( k ^ { 2 } + k ) + 1$ . Since $\\boldsymbol { k } ^ { 2 } + \\boldsymbol { k }$ is an integer, $4 ( k ^ { 2 } + k ) { \\dot { + } } 1$ is not divisible by 4. \nThus, ${ a ^ { 2 } \\equiv 1 }$ (mod 4). ",
|
| 1689 |
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"bbox": [
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| 1690 |
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| 1691 |
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|
| 1695 |
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|
| 1696 |
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},
|
| 1697 |
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{
|
| 1698 |
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"type": "text",
|
| 1699 |
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"text": "Statement: For any integer $a$ , show that ${ a } ^ { 2 } \\equiv$ 0 (mod 4) or ${ a ^ { 2 } \\equiv 1 }$ (mod 4). ",
|
| 1700 |
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"bbox": [
|
| 1701 |
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|
| 1707 |
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},
|
| 1708 |
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{
|
| 1709 |
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"type": "text",
|
| 1710 |
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"text": "Informal Proof (Minerva 62B): ",
|
| 1711 |
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"text_level": 1,
|
| 1712 |
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"bbox": [
|
| 1713 |
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| 1718 |
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|
| 1719 |
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},
|
| 1720 |
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{
|
| 1721 |
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"type": "text",
|
| 1722 |
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"text": "$a \\ ( \\mathrm { m o d } \\ 4 ) \\in \\{ 0 , 1 , 2 , 3 \\}$ . \nUsing that for any natural number $k$ , $a \\equiv b$ (mod 4) implies $\\dot { a } ^ { k } \\equiv b ^ { k }$ (mod 4), we have $\\dot { a } ^ { 2 }$ $\\overset { \\cdot } { \\left( \\mathrm { m o d } \\ 4 \\right) } \\in \\left\\{ 0 , 1 , 4 , 9 \\right\\}$ . \nSince $4 \\equiv 0$ (mod 4) and $9 \\equiv 1 { \\pmod { 4 } }$ , the result follows. ",
|
| 1723 |
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"bbox": [
|
| 1724 |
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| 1725 |
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| 1726 |
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| 1727 |
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|
| 1728 |
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|
| 1729 |
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"page_idx": 17
|
| 1730 |
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},
|
| 1731 |
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{
|
| 1732 |
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"type": "text",
|
| 1733 |
+
"text": "Formal Proof: ",
|
| 1734 |
+
"text_level": 1,
|
| 1735 |
+
"bbox": [
|
| 1736 |
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|
| 1737 |
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|
| 1738 |
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290,
|
| 1739 |
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309
|
| 1740 |
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|
| 1741 |
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"page_idx": 17
|
| 1742 |
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},
|
| 1743 |
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{
|
| 1744 |
+
"type": "text",
|
| 1745 |
+
"text": "theorem numbertheory_sqmod4in01d: fixes a :: int shows \"( $\\ a \\hat { \\textmd a } _ { 2 }$ mod $4 \\ = \\ 0$ ) $\\scriptstyle \\backslash < _ { \\mathrm { { O L } } } >$ (aˆ2 mod $4 \\ = \\ 1$ )\" \nproof (cases \"even a\") \ncase True ( $\\star$ Let a=2k for some integer k. Then $\\hat { \\mathbf { a } } \\hat { \\mathbf { \\Lambda } } 2 \\mathbf { = } 4 \\mathbf { k } \\hat { \\mathbf { \\Lambda } } 2$ . Since $4 1 - 2$ is divisible by 4, we have aˆ2 \\equiv 0 \\pmod{4}.\\*) then obtain k where $\" \\mathrm { a } \\mathrm { = } 2 \\star \\mathrm { k } \"$ using evenE by auto [ATP] \nthen have $\" a \\widehat { } 2 = 4 \\star \\widehat { } 2 \"$ unfolding power2_eq_square by auto [ATP] then have \"aˆ2 mod $4 \\ = \\ 0$ \" by auto [ATP] \nthen show ?thesis by auto [ATP] \nnext \ncase False $( \\star \\ \\aleph \\circ \\aleph ,$ let $a = 2 k + 1$ for some integer k. Then $a \\cdot 2 = 4 k \\cdot 2 + 4 k + 1 = 4 ( k \\cdot 2 + k ) + 1$ . Since $k ^ { \\mathord { \\sim } } 2 { + } k$ is an integer, 4 $( \\mathbf { k } ^ { \\sim } 2 + \\mathbf { k } ) + \\mathbf { 1 }$ is not divisible by 4. Thus, $\\widehat { \\mathbf { a } } \\widehat { \\mathbf { \\Lambda } } ^ { 2 }$ \\equiv $\\texttt { 1 } \\backslash \\mathtt { p m o d } \\{ 4 \\} \\ldots \\star \\}$ then obtain k where $\\ \" { a } { = } 2 { \\star } \\mathrm { k } { + } 1$ \" using oddE by auto [ATP] then have $\" a \\widehat { } 2 = 4 { \\star } \\widehat { } 2 { + } 4 { \\star } \\mathrm { k } { + } 1 \"$ \" unfolding power2_eq_square by (auto simp: field_simps) [ATP] then have $\\ \" \\mathtt { a } \\hat { } 2$ mod $4 \\ = \\ 1$ \" by presburger [ATP] then show ?thesis by auto [ATP] \nqed ",
|
| 1746 |
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"bbox": [
|
| 1747 |
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176,
|
| 1748 |
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311,
|
| 1749 |
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486,
|
| 1750 |
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799
|
| 1751 |
+
],
|
| 1752 |
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"page_idx": 17
|
| 1753 |
+
},
|
| 1754 |
+
{
|
| 1755 |
+
"type": "text",
|
| 1756 |
+
"text": "Formal Proof: ",
|
| 1757 |
+
"text_level": 1,
|
| 1758 |
+
"bbox": [
|
| 1759 |
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526,
|
| 1760 |
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296,
|
| 1761 |
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619,
|
| 1762 |
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308
|
| 1763 |
+
],
|
| 1764 |
+
"page_idx": 17
|
| 1765 |
+
},
|
| 1766 |
+
{
|
| 1767 |
+
"type": "text",
|
| 1768 |
+
"text": "theorem numbertheory_sqmod4in01d: fixes a :: int shows \" $\\hat { a } ^ { \\star } 2$ mod $4 \\ = \\ 0$ ) $\\scriptstyle \\backslash < \\supset \\pm \\gamma$ ( $\\hat { \\mathbf { a } } ^ { \\hat { } } 2$ mod $4 ~ = ~ 1$ )\" \nproof - \n(\\* a \\pmod 4 \\in {0, 1, 2, 3}. \\*) \nhave c0: \"a mod 4 \\<in> {0, 1, 2, 3}\" by auto [ATP] \n( $\\star$ Using that for any natural number $\\mathbf { k } ,$ a \\equiv b \\pmod 4 implies aˆk \\equiv bˆk \\pmod $^ { 4 , \\star }$ ) \nhave $\\ \" \\mathtt { a } \\hat { } 2$ mod $4 \\ =$ (a mod 4)\\<ˆsup> 2 mod 4\" by (smt (z3) numeral_eq_Suc power_mod) [ATP] \n( $\\star$ we have aˆ2 \\pmod 4 \\in {0, 1, 4, 9}. \\*) \nalso have \"... \\<in> {0, 1, 4, 9}\" using c0 by auto [ATP] \n( $^ { \\star }$ Since 4 \\equiv 0 \\pmod 4 and 9 \\equiv 1 \\pmod 4, the result follows. \\*) \nfinally show ?thesis by auto [ATP] \nqed ",
|
| 1769 |
+
"bbox": [
|
| 1770 |
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|
| 1771 |
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|
| 1772 |
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|
| 1773 |
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|
| 1774 |
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|
| 1775 |
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"page_idx": 17
|
| 1776 |
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},
|
| 1777 |
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{
|
| 1778 |
+
"type": "text",
|
| 1779 |
+
"text": "Statement: For each plumbing repair job, Mr. Wrench charges $N$ dollars for coming out to the house plus $_ x$ dollars per hour that he works at the house. He charged $\\$ 97$ for a one-hour repair job and $\\$ 265$ for a five-hour repair job. What is his charge for a two-hour repair job? Show that it is $\\$ 139$ . ",
|
| 1780 |
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"bbox": [
|
| 1781 |
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|
| 1782 |
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|
| 1783 |
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| 1784 |
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|
| 1785 |
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|
| 1786 |
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"page_idx": 18
|
| 1787 |
+
},
|
| 1788 |
+
{
|
| 1789 |
+
"type": "text",
|
| 1790 |
+
"text": "Informal Proof (Minerva 62B): ",
|
| 1791 |
+
"text_level": 1,
|
| 1792 |
+
"bbox": [
|
| 1793 |
+
200,
|
| 1794 |
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237,
|
| 1795 |
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395,
|
| 1796 |
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250
|
| 1797 |
+
],
|
| 1798 |
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"page_idx": 18
|
| 1799 |
+
},
|
| 1800 |
+
{
|
| 1801 |
+
"type": "text",
|
| 1802 |
+
"text": "Let $N$ be the number of dollars charged for coming out to the house and let $x$ be the number of dollars per hour that Mr. Wrench works. For the one-hour repair job, we have that the total cost is $N + x = 9 7$ . \nFor the five-hour repair job, we have that the total cost is $N + 5 x = 2 6 5$ . \nWe subtract the first equation from the second equation to obtain $4 x = 1 6 8 \\Rightarrow x = 4 2$ . \nPlugging this back into the first equation, we obtain that $N = 5 5$ . \nTherefore, for a two-hour repair job, the total cost is $5 5 + 2 x = 5 5 + 2 ( 4 2 ) = { \\Big [ } \\ S 1 3 9 { \\Big ] } .$ ",
|
| 1803 |
+
"bbox": [
|
| 1804 |
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|
| 1805 |
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250,
|
| 1806 |
+
472,
|
| 1807 |
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420
|
| 1808 |
+
],
|
| 1809 |
+
"page_idx": 18
|
| 1810 |
+
},
|
| 1811 |
+
{
|
| 1812 |
+
"type": "text",
|
| 1813 |
+
"text": "Formal Proof: ",
|
| 1814 |
+
"text_level": 1,
|
| 1815 |
+
"bbox": [
|
| 1816 |
+
199,
|
| 1817 |
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429,
|
| 1818 |
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290,
|
| 1819 |
+
441
|
| 1820 |
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],
|
| 1821 |
+
"page_idx": 18
|
| 1822 |
+
},
|
| 1823 |
+
{
|
| 1824 |
+
"type": "text",
|
| 1825 |
+
"text": "theorem mathd_algebra_160: fixes n x ::real assumes $\" \\mathrm { n } + \\mathrm { x } = 9 7 \"$ and $\" \\mathrm { n } ~ + ~ 5 ~ \\star ~ \\mathrm { x } = 2 6 5 \"$ shows $\" { \\mathrm { ~ n ~ } } + ~ 2 ~ \\star ~ { \\mathrm { ~ x ~ } } = ~ 1 3 9 \"$ \nproof - ( $\\star$ We subtract the first equation from the second equation to obtain $4 \\mathbf { x } = 1 6 8 \\Rightarrow \\mathbf { x } = 4 2 . \\ \\star )$ ) have $\" 4 ~ * ~ \\mathrm { \\bf { x } } ~ = ~ 1 6 8 \"$ using assms by auto [ATP] then have $\" \\mathrm { z } = 4 2 \"$ by auto [ATP] ( $^ { * }$ Plugging this back into the first equation, we obtain that $\\mathbf { N } = 5 5$ . \\*) then have $\" \\mathrm { n } = 5 5 \"$ using assms by auto $[ A T P ]$ ( $\\left[ \\star \\right]$ Therefore, for a two-hour repair job, the total cost is $\\mathbf { 5 5 } + \\mathbf { 2 x } = \\mathbf { 5 5 } + \\mathbf { 2 } ( \\mathbf { 4 2 } ) = \\mathbb { 8 1 3 9 } . ~ \\star )$ then show ?thesis by (smt (z3) $\\langle \\mathbf { x } \\ : = \\ : 4 2 \\rangle$ ) [ATP] \nqed ",
|
| 1826 |
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"bbox": [
|
| 1827 |
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|
| 1828 |
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|
| 1829 |
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| 1830 |
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|
| 1831 |
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|
| 1832 |
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"page_idx": 18
|
| 1833 |
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},
|
| 1834 |
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{
|
| 1835 |
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"type": "text",
|
| 1836 |
+
"text": "Statement: Show that for any real number $_ x$ and any natural number $n$ , if $x > - 1$ , then $( 1 + n x ) \\leq ( 1 + x ) ^ { n }$ . ",
|
| 1837 |
+
"bbox": [
|
| 1838 |
+
526,
|
| 1839 |
+
136,
|
| 1840 |
+
799,
|
| 1841 |
+
175
|
| 1842 |
+
],
|
| 1843 |
+
"page_idx": 18
|
| 1844 |
+
},
|
| 1845 |
+
{
|
| 1846 |
+
"type": "text",
|
| 1847 |
+
"text": "Informal Proof (Minerva 62B): ",
|
| 1848 |
+
"text_level": 1,
|
| 1849 |
+
"bbox": [
|
| 1850 |
+
529,
|
| 1851 |
+
186,
|
| 1852 |
+
722,
|
| 1853 |
+
199
|
| 1854 |
+
],
|
| 1855 |
+
"page_idx": 18
|
| 1856 |
+
},
|
| 1857 |
+
{
|
| 1858 |
+
"type": "text",
|
| 1859 |
+
"text": "This is true for $n = 1$ . Now, suppose that this is true for $n$ . \nThen we have that ",
|
| 1860 |
+
"bbox": [
|
| 1861 |
+
527,
|
| 1862 |
+
199,
|
| 1863 |
+
797,
|
| 1864 |
+
237
|
| 1865 |
+
],
|
| 1866 |
+
"page_idx": 18
|
| 1867 |
+
},
|
| 1868 |
+
{
|
| 1869 |
+
"type": "equation",
|
| 1870 |
+
"img_path": "images/ce7d67bea877c9c19cbd7920c391c58a692f5c0e16fe0099748943b319c156bc.jpg",
|
| 1871 |
+
"text": "$$\n( 1 + n x ) ^ { n } = 1 + n ^ { 2 } x + n ^ { 3 } x ^ { 2 } + \\cdot \\cdot \\cdot + n x ^ { n }\n$$",
|
| 1872 |
+
"text_format": "latex",
|
| 1873 |
+
"bbox": [
|
| 1874 |
+
529,
|
| 1875 |
+
242,
|
| 1876 |
+
797,
|
| 1877 |
+
260
|
| 1878 |
+
],
|
| 1879 |
+
"page_idx": 18
|
| 1880 |
+
},
|
| 1881 |
+
{
|
| 1882 |
+
"type": "equation",
|
| 1883 |
+
"img_path": "images/bb509eddceecc2747f0838e5b5ced68ff1a5bb31bd80def3541411c3847b77b3.jpg",
|
| 1884 |
+
"text": "$$\n\\leq 1 + n x + n ^ { 2 } x ^ { 2 } + \\cdot \\cdot \\cdot + n x ^ { n } = ( 1 + x ) ^ { n }\n$$",
|
| 1885 |
+
"text_format": "latex",
|
| 1886 |
+
"bbox": [
|
| 1887 |
+
529,
|
| 1888 |
+
275,
|
| 1889 |
+
797,
|
| 1890 |
+
292
|
| 1891 |
+
],
|
| 1892 |
+
"page_idx": 18
|
| 1893 |
+
},
|
| 1894 |
+
{
|
| 1895 |
+
"type": "text",
|
| 1896 |
+
"text": "and ",
|
| 1897 |
+
"bbox": [
|
| 1898 |
+
527,
|
| 1899 |
+
300,
|
| 1900 |
+
552,
|
| 1901 |
+
313
|
| 1902 |
+
],
|
| 1903 |
+
"page_idx": 18
|
| 1904 |
+
},
|
| 1905 |
+
{
|
| 1906 |
+
"type": "equation",
|
| 1907 |
+
"img_path": "images/a639aada5b0da5fa6bd0ea04c398c4150cac20b60874ca139a9460b2eabb4cf9.jpg",
|
| 1908 |
+
"text": "$$\n\\begin{array} { c } { { ( 1 + n x ) ^ { n + 1 } = ( 1 + n x ) ^ { n } ( 1 + n x ) } } \\\\ { { \\leq ( 1 + x ) ^ { n } ( 1 + x ) = ( 1 + x ) ^ { n + 1 } . } } \\end{array}\n$$",
|
| 1909 |
+
"text_format": "latex",
|
| 1910 |
+
"bbox": [
|
| 1911 |
+
527,
|
| 1912 |
+
318,
|
| 1913 |
+
823,
|
| 1914 |
+
356
|
| 1915 |
+
],
|
| 1916 |
+
"page_idx": 18
|
| 1917 |
+
},
|
| 1918 |
+
{
|
| 1919 |
+
"type": "text",
|
| 1920 |
+
"text": "Therefore, this is true for $n + 1$ ",
|
| 1921 |
+
"bbox": [
|
| 1922 |
+
527,
|
| 1923 |
+
362,
|
| 1924 |
+
717,
|
| 1925 |
+
375
|
| 1926 |
+
],
|
| 1927 |
+
"page_idx": 18
|
| 1928 |
+
},
|
| 1929 |
+
{
|
| 1930 |
+
"type": "text",
|
| 1931 |
+
"text": "Formal Proof: ",
|
| 1932 |
+
"text_level": 1,
|
| 1933 |
+
"bbox": [
|
| 1934 |
+
526,
|
| 1935 |
+
387,
|
| 1936 |
+
619,
|
| 1937 |
+
398
|
| 1938 |
+
],
|
| 1939 |
+
"page_idx": 18
|
| 1940 |
+
},
|
| 1941 |
+
{
|
| 1942 |
+
"type": "text",
|
| 1943 |
+
"text": "theorem induction_1pxpownlt1pnx: fixes x :: real and n :: nat assumes $\" - 1 < x \"$ shows \" $( \\underline { { { 1 } } } ~ + ~ \\boldsymbol { \\mathrm { n } } { \\star } \\boldsymbol { \\mathrm { x } } )$ ) (1 + x)ˆn\" \nproof (induct n) \ncase 0 \nthen show ?case \nby auto [ATP] \nnext case (Suc n) \nthen show ?case \nby (smt (z3) Bernoulli_inequality assms) \n[ATP] \nqed ",
|
| 1944 |
+
"bbox": [
|
| 1945 |
+
522,
|
| 1946 |
+
404,
|
| 1947 |
+
820,
|
| 1948 |
+
566
|
| 1949 |
+
],
|
| 1950 |
+
"page_idx": 18
|
| 1951 |
+
},
|
| 1952 |
+
{
|
| 1953 |
+
"type": "text",
|
| 1954 |
+
"text": "Statement: Determine the remainder of 1529 (mod 6). Show that it is 5. ",
|
| 1955 |
+
"bbox": [
|
| 1956 |
+
526,
|
| 1957 |
+
618,
|
| 1958 |
+
797,
|
| 1959 |
+
643
|
| 1960 |
+
],
|
| 1961 |
+
"page_idx": 18
|
| 1962 |
+
},
|
| 1963 |
+
{
|
| 1964 |
+
"type": "text",
|
| 1965 |
+
"text": "Informal Proof (Minerva 62B): By definition, $1 5 2 9 \\equiv 5$ (mod 6). ",
|
| 1966 |
+
"bbox": [
|
| 1967 |
+
527,
|
| 1968 |
+
655,
|
| 1969 |
+
735,
|
| 1970 |
+
681
|
| 1971 |
+
],
|
| 1972 |
+
"page_idx": 18
|
| 1973 |
+
},
|
| 1974 |
+
{
|
| 1975 |
+
"type": "text",
|
| 1976 |
+
"text": "Formal Proof: ",
|
| 1977 |
+
"text_level": 1,
|
| 1978 |
+
"bbox": [
|
| 1979 |
+
524,
|
| 1980 |
+
694,
|
| 1981 |
+
619,
|
| 1982 |
+
705
|
| 1983 |
+
],
|
| 1984 |
+
"page_idx": 18
|
| 1985 |
+
},
|
| 1986 |
+
{
|
| 1987 |
+
"type": "text",
|
| 1988 |
+
"text": "theorem mathd_numbertheory_551 : \"1529 mod $6 =$ (5::nat)\" by auto [ATP] ",
|
| 1989 |
+
"bbox": [
|
| 1990 |
+
526,
|
| 1991 |
+
710,
|
| 1992 |
+
764,
|
| 1993 |
+
747
|
| 1994 |
+
],
|
| 1995 |
+
"page_idx": 18
|
| 1996 |
+
}
|
| 1997 |
+
]
|
parse/dev/SMa9EAovKMC/SMa9EAovKMC_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/dev/SMa9EAovKMC/SMa9EAovKMC_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/dev/SVBR6xBaMl/SVBR6xBaMl.md
ADDED
|
@@ -0,0 +1,465 @@
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|
| 1 |
+
# Language Models Meet World Models: Embodied Experiences Enhance Language Models
|
| 2 |
+
|
| 3 |
+
Jiannan Xiang∗♠, Tianhua $\mathbf { T a o ^ { * } } ^ { \mathbf { A } }$ , Yi $\mathbf { G u } ^ { \pmb { \alpha } }$ , Tianmin $\mathbf { S h u } ^ { \bigcirc \bigtriangleup }$ , Zirui Wang♠, Zichao $\mathbf { Y a n g } ^ { \bigcirc }$ , Zhiting $\mathbf { H } \mathbf { u } ^ { \pmb { \alpha } }$ ♠UC San Diego, ♣UIUC, ${ \ \diamond \bf { \hat { \mu } } } _ { \mathrm { { M I T } } }$ , $\Delta _ { \mathrm { J H U } }$ , $^ { \heartsuit } { \mathbf { C M U } }$
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
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While large language models (LMs) have shown remarkable capabilities across numerous tasks, they often struggle with simple reasoning and planning in physical environments, such as understanding object permanence or planning household activities. The limitation arises from the fact that LMs are trained only on written text and miss essential embodied knowledge and skills. In this paper, we propose a new paradigm of enhancing LMs by finetuning them with world models, to gain diverse embodied knowledge while retaining their general language capabilities. Our approach deploys an embodied agent in a world model, particularly a simulator of the physical world (VirtualHome), and acquires a diverse set of embodied experiences through both goal-oriented planning and random exploration. These experiences are then used to finetune LMs to teach diverse abilities of reasoning and acting in the physical world, e.g., planning and completing goals, object permanence and tracking, etc. Moreover, it is desirable to preserve the generality of LMs during finetuning, which facilitates generalizing the embodied knowledge across tasks rather than being tied to specific simulations. We thus further introduce the classical elastic weight consolidation (EWC) for selective weight updates, combined with low-rank adapters (LoRA) for training efficiency. Extensive experiments show our approach substantially improves base LMs on 18 downstream tasks by $6 4 . 2 8 \%$ on average. In particular, the small LMs (1.3B, 6B, and 13B) enhanced by our approach match or even outperform much larger LMs (e.g., ChatGPT). 1
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# 1 Introduction
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Language Models (LMs) have demonstrated impressive performance on a wide range of natural language processing tasks [34, 48, 4, 7, 54]. In particular, recent studies show that LMs can assist decision-making for embodied tasks [1, 18, 25, 45, 19], demonstrating a certain level of understanding of the physical world. However, such understanding is not robust enough for many reasoning and planning tasks in physical environments. As shown in Figure 1, even the latest large LMs like ChatGPT2 can still make mistakes in seemingly simple inquiries, such as counting objects in a location. We hypothesize that this is because current LMs trained merely with large-scale text corpora are devoid of embodied experiences such as navigating in an environment, interacting with objects, and sensing as well as tracking the world state. Consequently, they lack robust and comprehensive embodied knowledge necessary for reasoning and planning associated with physical environments. A related line of research finetunes LMs in order to improve specific embodied tasks, resulting in task-specialized models [6, 58, 21, 57].
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Figure 1: Examples of tasks requiring embodied knowledge (upper), and an overview of our approach (bottom). In the task examples, blue text indicates the useful information for answering the question.
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In this paper, we aim to inject diverse fundamental embodied knowledge and skills into pretrained LMs, while retaining the models’ generality. We introduce a novel training paradigm for LMs – finetuning with Embodied Experiences from World Models (E2WM). Here, world models are embodied simulators that emulate physical interactions in real-world environments (e.g., VirtualHome [36]). They provide LMs with the opportunity to comprehend object interactions within the environment and to execute actions, thus enabling a level of active engagement previously unattainable. These world models serve as a simplified and cost-effective replica of our real world that can significantly augment the conventional pretraining paradigm. We anticipate that finetuning LMs on embodied experiences gathered from world models can enhance their embodied knowledge—and, with the preserved model generality—consequently strengthen their abilities to solve a broad range of embodied tasks.
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In this work, we consider a diverse range of fundamental knowledge and skills for embodied tasks, including tracking objects, planning to complete given goals, recognizing other agents’ behaviors, etc. To this end, we introduce two ways to collect embodied experiences from world models that give rise to the desired knowledge and skills: goal-oriented planning and random exploration (Figure 1). Specifically, goal-oriented planning aims to gather experiences associated with planning and goal-oriented agent behaviors, while random exploration focuses on accumulating experiences that involve object and world state tracking. In goal-oriented planning, models are given the goal (e.g., IN(dust, trash can)) for a specific activity (e.g., Clean Floor), and is supposed to generate a plan to complete it. To find the plan, we devise Monte Carlo Tree Search (MCTS) [5, 44] to explore the world model. Then the process will be stored as an embodied experience. In random exploration, one or more agents are deployed in the world model to execute random actions, while the locations and the movements of all the objects are tracked simultaneously.
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After collecting the embodied experiences, we use them to construct a set of fine-tuning tasks (e.g., plan generation, activity recognition, and tracking). Crucially, to finetune LMs on the collected embodied experiences while retaining their original general knowledge and capabilities, we propose to incorporate the classical Elastic Weight Consolidation (EWC) [22] into our training paradigm. By regularizing the finetuning loss, EWC aims to preserve the important LM parameters from pretraining. We show that EWC is substantially more effective than the popular KL regularization [35, 28, 31]. We further introduce efficient low-rank updates by harmonizing the recent Low-Rank Adaptation (LoRA) [16] with the EWC regularizer. This results in the new EWC-LoRA update rule that greatly reduces training costs and makes our E2WM paradigm accessible to cheap hardware (GPUs).
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We instantiate a world model using a virtual household simulator, VirtualHome [36, 37], and apply our method to GPT-Neo-1.3B [3], GPT-J-6B [49], OPT-13B [59], and LLaMA-13B [48] models. To test the generalizability of the finetuned LMs, we evaluate them on a variety of unseen tasks which demand similar embodied knowledge required to solve the training tasks. Additionally, we assess the models’ performance on the original pretraining data to determine the extent to which their core language modeling abilities are retained. Experiments show that our method significantly improves the baselines on both seen and unseen tasks (e.g., $3 4 . 3 1 5 1 . 2 3 $ Rouge-L on plan generation task, $3 0 . 4 1 \% 6 7 . 0 1 \%$ accuracy on counting task), without suffering performance degradation on the pretraining dataset $3 . 4 4 3 3 . 5 3 7$ perplexity on Pile test subset [12]). Moreover, the small GPTJ-6B, OPT-13B, and LLaMA-13B models finetuned with our E2WM paradigm even outperforms ChatGPT on many of the tasks. The experimental results demonstrate the effectiveness of E2WM as a promising fine-tuning mechanism to enhance pretrained LMs with generalizable embodied knowledge and skills.
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# 2 Related Work
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World Model. The term “world model” generally refer to a computational representation of the physical world, capable of simulating changes in the world’s state in response to various actions. For instance, humans possess an internal world model that aids in predicting the outcomes of specific actions during the planning process. Recent research induces world models from large LMs for robust human-like reasoning [15]. In this work, we employ a simulator equipped with a physics engine to serve as our world model, effectively emulating real-world conditions. In the field of embodied AI, various world models are built to replicate the real world and serve as virtual test environments for assessing robotic agents before real-world deployment. For example, VirtualHome [36, 37] is a simulated 3D household environment implemented by Unity3D game engine. AI2-THOR [23] consists of near photo-realistic 3D indoor scenes and has richer object attributes and interaction types. Other indoor household World Models include VRKitchen [13], CHALET [56], MINOS [41], House3D [53], etc. Besides, MineCraft is a more challenging and open-ended world model, which has a large number of objectives and a large-scale task hierarchy [14, 27, 20]. In this paper, we use VirtualHome as our world model.
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Language Model Grounding. A significant number of recent works focused on grounding language models to world models [1, 24, 38, 47, 55]. Some of them freeze LMs and leverage certain prompting strategies or specifically-designed modules. For example, Zero-Shot Planner [18] prompts LMs to generate activity plans and translate them into admissible actions. Mind’s eye [29] prompts LMs to do simulations with physical engines to answer physical reasoning questions. SayCan [1] uses a learned affordance function to assist LMs in selecting valid actions. DEPS [50] prompts LMs to describe, explain and generate action plans, incorporated with a learned selector module to choose the most efficient path. There are also other works finetuning LMs towards better downstream task performance. For example, Li et al. [25] finetune LMs with supervised learning for interactive decision making, and Carta et al. [6] ground LMs with online reinforcement learning. Different from these works aiming to optimize LMs for specific tasks in the target environments, our work instead focuses on improving the language model itself by acquiring knowledge from world models.
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Language Model Regularization. To facilitate the acquisition of new knowledge and skills without losing LMs’ language modeling abilities , regularization is often introduced during finetuning. One popular method is adding KL penalty [35, 28, 46, 52, 33, 60], which leverages KL divergence between the output probability of the currently trained model and the original model to regularize the LM in an RL manner, i.e., by computing policy gradients. For example, InstructGPT uses KL penalty to mitigate over-optimization of the reward model [35], and Liu et al. [28] add KL regularization for training a commonsense knowledge generator. In this work, we instead use elastic weight consolidation (EWC) for regularization. Our empirical results demonstrate that EWC is more effective than applying KL penalty for retaining language modeling abilities and generality of LMs.
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# 3 Approach
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In this work, we propose a new training paradigm, namely finetuning with Embodied Experiences from World Models (E2WM), to inject embodied knowledge into LMs without sacrificing its generality and language modeling abilities. The world model we use is VirtualHome [36, 37], a multi-agent simulator for househould activities. In VirtualHome, an executable action step can be simplified as the format of [action] <arg>, e.g., [Grab] <apple> . The world state of VirtualHome consists of objects and their relations (e.g., apple on table). Details about VirtualHome can be found in Appendix
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Figure 2: The illustration of goal-oriented planning (left) and random exploration (right) in our training paradigm. In MCTS, the path in orange represents the final plan generated by the planner.
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A.1. We first describe how to gather embodied experiences in the world model in Section 3.1. Then in Section 3.2 we demonstrate how to finetune LMs by utilizing collected experiences, as well as our proposed method EWC-LoRA for efficient knowledge generalization.
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# 3.1 Collecting Embodied Experiences from World Model
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LMs pretrained on large scale human-written text corpus often have difficulties in solving basic reasoning and planning in physical environments. This is because that LMs lack necessary embodied knowledge and experiential understanding of the physical world. To address the problem, we propose to leverage world models to collect diverse embodied experiences for enhancing LMs. Specifically, to inject different types of embodied knowledge into LMs, we introduce two ways to gather experiences: goal-oriented planning and random exploration. Figure 2 illustrates the two methods.
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Goal-oriented Planning. One important embodied skill is to plan and complete a specific goal, e.g., placing tableware properly to set up the table. To endow LMs with this ability, we propose goal-oriented planning. The approach aims to generate experiences that are goal-oriented, thus are useful to facilitate the acquisition of skills and task planning abilities for executing a range of activities in the world model. To do that, we collect various activities and their corresponding goals. Formally, the goal for an activity in the world model is defined as a set of predicates describing the target world state. For instance, an activity can be set up table, and its goal can be ON(fork, table);ON(plate, table), which means that fork and plate should be put on the table to fulfill the activity. More details about predicates and goal definitions can be found in Appendix A.2. As shown in Figure 2, in goal-oriented planning, we devise a Monte Carlo Tree Search (MCTS) planner to search through the action space and find a plan, i.e., a sequence of actions, to achieve the goal. The key to successful MCTS is the reward design. At each time step, if at least one goal predicate is satisfied, the MCTS planner will get a reward of $+ 2$ , and the achieved goal predicates will be removed from the goal. This ensures that the planner does not repeatedly execute the same action to receive rewards, but rather focuses on achieving the remaining unfulfilled goals. Besides, it will get a -0.1 penalty after each time step to discourage the planner from doing actions irrelevant to fulfilling the goals. Finally, we store the planning process as an embodied experience.
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Random Exploration. In real-world scenarios, humans not only acquire new knowledge by finishing tasks, but also learn by just randomly exploring the surroundings, e.g., randomly observing/tracking objects and knowing their properties. To mimic this learning process, we propose another approach, namely random exploration. By simply exploring in the world model, embodied experiences emerge that involve advanced cognitive abilities including object permanence and tracking, as agents observe and track the consistent existence of objects even when they are out of sight. Then the experiences are gathered for finetuning LMs later. Specifically, the approach deploys one or multiple agents in the world model wandering aimlessly and randomly executing actions. As illustrated in Figure 2, multiple agents are in the same environment, interacting with each other or executing different actions on the same objects, which simulates complex situations. During the exploration, the moving paths and the final locations of all the objects in the world model are recorded. Then the whole process is captured as an embodied experience.
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# 3.2 Finetuning LMs with Embodied Experiences
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There are multiple ways to utilize collected embodied experiences for LMs finetuning, such as supervised learning and reinforcement learning. In this work, we use them with supervised learning for simplicity and efficiency. Specifically, goal-oriented planning experiences are compiled into data examples in two formats: plan generation and activity recognition. As shown in Figure 2, in plan generation, the model is required to generate a stepwise action sequence to fulfill an activity, given the state of some relevant objects as the initial condition. In activity recognition, the model needs to recognize the activity name given its plan. Experiences obtained from random exploration are also transformed into two self-supervision tasks: counting and object path tracking. Examples of the two tasks can be seen in Figure 2. Specifically, for counting, the LM is tasked with identifying the number and name of the objects at a specific location after the agents performed relevant and irrelevant actions and arranged objects randomly. In object path tracking, the model is tasked to output the moving path of an object that is picked up by different agents and moved to different rooms at different times. All the tasks are trained with cross-entropy loss. Suppose that $\mathbf { x }$ is the input (e.g., the initial condition in plan generation) and $\mathbf { y } = \{ y _ { 1 } , . . . , \bar { y _ { M } } \}$ is the label (e.g., the stepwise action sequence), we finetune LMs by assigning different weights to different tasks:
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$$
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\mathcal { L } _ { V } = \sum _ { v \in V } \alpha _ { v } \sum _ { m = 1 } ^ { M } \log P ( y _ { m } | \mathbf { y } _ { < m } , \mathbf { x } ) ,
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$$
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where $\mathcal { L }$ is the loss function; $V$ is the task set; and $\alpha _ { v }$ is the weight for task $v$ . Following Flan-T5 [8], $\mathbf { x }$ is a prompt formatted to contain a task instruction and sampled in-context demonstrations. We provide all prompts in Appendix A.3.
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Efficient Finetuning with Preserved Generality. However, there are two key problems for simply finetuning LMs. The first one is that LMs will easily overfit to the downstream tasks, leading to performance degradation on other tasks. This deviates from our goal that the model should generalize acquired knowledge across various tasks. Another problem is that finetuning the entire LM is resource-intensive and time-consuming, especially when the LM is extremely large. To overcome the problem and facilitate continual and efficient knowledge acquisition with world models, we propose to finetune only a small number of weights using low-rank adaptors (LoRA) [16] with elastic weight consolidation (EWC) [22], which we refer to as EWC-LoRA.
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EWC is a regularization-based method typically used in the area of continual learning [9]. It calculates a fisher matrix [11] to estimate the importance of each parameter for a task and then uses it to regularize the training on a new task. The regularization term helps to constrain the parameter updates for the new task to avoid forgetting the previous knowledge. Let $U$ be the pretraining task set, and $V$ be the finetuning task set. Following [2], we have:
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$$
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\begin{array} { c } { { \displaystyle F _ { i , i } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \left( \frac { \partial \mathcal { L } _ { U } ^ { ( j ) } } { \partial \theta _ { U , i } ^ { * } } \right) ^ { 2 } , } } \\ { { \mathcal { L } ( \theta ) = \mathcal { L } _ { V } ( \theta ) + \lambda \sum _ { i } F _ { i , i } ( \theta _ { i } - \theta _ { U , i } ^ { * } ) ^ { 2 } , } } \end{array}
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$$
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where $\mathcal { L }$ is the loss function, $F$ is the fisher matrix, $\lambda$ is the hyperparameter, $i$ and $j$ are the indices for parameters and data samples, respectively, and $\theta$ and $\theta _ { U } ^ { * }$ are currently trained parameters and frozen task $U$ parameters, respectively. Notice that the first term ${ \mathcal { L } } _ { V } ( \theta )$ in Equation 3 is calculated in Equation 1, and the second term is the EWC regularizer. In Equation 2, the fisher matrix is calculated by averaging the sum of squares of the gradients from the task $U$ , which indicates the significance of each parameter to the task $U$ . Then the matrix is used in Equation 3 to weigh the shift of model parameters when training on $V$ . By using EWC, the LM learns to adapt to new tasks without catastrophic forgetting on the pretraining task, which forces it to understand and digest new knowledge from the finetuning tasks instead of overfitting to them.
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However, EWC is both time- and memory-inefficient. First, it requires finetuning the entire set of large LM’s parameters. Moreover, the approach involves creating a frozen original model and a fisher matrix that is the same size as the LM, leading to a memory overhead of three times the original size.
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This makes it particularly challenging to apply to larger models. To alleviate the problem, we propose to combine EWC with low-rank adaptors (LoRA), a parameter-efficient tuning method. LoRA freezes the pretrained model weights and injects two trainable low-rank matrices into each layer of the model. Suppose that $W$ , $W ^ { \ast } \in \bar { \mathbb { R } } ^ { r \times d }$ are the trained weight matrix and frozen weight matrix, respectively; and $B \in \mathbb { R } ^ { r \times k } , A \in \mathbb { R } ^ { k \times d }$ are two low-rank matrices with $k \ll \operatorname* { m i n } ( r , d )$ . Then the formula for LoRA can be written as $W = W ^ { * } + B A$ . Suppose that $H$ is flattened $B A$ . Notably, we found that $\theta _ { i }$ in Equation 3 is the element of $W$ , and $\theta _ { U , i } ^ { * }$ is that of $W ^ { * }$ . Therefore, $\theta _ { i } - \theta _ { U , i } ^ { * }$ is the element of $H$ . We can thus transform Equation 3 into the final formula of EWC-LoRA method:
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$$
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\mathcal { L } ( \boldsymbol { \theta } ) = \mathcal { L } _ { V } ( \boldsymbol { \theta } ) + \lambda \sum _ { i } F _ { i , i } h _ { i } ^ { 2 } ,
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$$
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where $h _ { i } = \theta _ { i } - \theta _ { U , i } ^ { * }$ is the $i$ -th element of $H$ . One of the benefits of this rewriting is that we no longer need to store the trained LM weight matrixes as what vanilla EWC does, which saves plenty of memory space. Besides, we only need to update $B$ and $A$ during the finetuning, which also lowers memory requirements and leads to much faster training speed. Surprisingly, as shown later, we empirically found that adding LoRA into EWC can further mitigate the issue of catastrophic forgetting and overfitting. This aligns with the previous conclusion that limiting the dimension of the optimization problem can alleviate catastrophic forgetting [32].
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# 4 Experiments
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Training Details. For goal-oriented planning, we collected activities and their corresponding target goals with data from RobotHow [36], a housework activity knowledge base created in VirtualHome. We applied our method to GPT-Neo-1.3B [3], GPT-J-6B [49], OPT-13B [59], and LLaMA-13B [48]. To save computing resources, we use Int8 technique [10]. Both of the models were trained with the AdamW optimizer [30]. All the hyperparameters are chosen according to the performance on a held out set. We used one NVIDIA GeForce RTX 3090 for training. More details can be found in Appendix A.4.
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# 4.1 Downstream Evaluation Tasks
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We developed various downstream evaluation tasks for each type of embodied knowledge, including both the training tasks as well as novel tasks unseen during training used for generalization evaluation. Additionally, we evaluate our models on bAbI [51], a dataset for testing multiple types of knowledge and abilities including embodied knowledge, logic reasoning, linguistic knowledge, etc. We select the bAbI tasks related to embodied knowledge for our evaluation. We evaluate all the unseen tasks including bAbI under few-shot settings, specifically 2-10 shots, by providing a few in-context exemplars in the prompts. We discuss more details of the tasks below.
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Plan Generation. To evaluate planning ability, we construct downstream tasks using human-written plans from RobotHow. Specifically, we have three tasks:
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• Plan Generation Evaluation. In this task, the model needs to generate a plan for a housework activity. It is similar to the training task but uses human-written plans as the ground truth instead of the collected experiences. We include activities unseen during training to test the generalizability of the model. Inspired by the previous study showing that LMs can easily be distracted by irrelevant context [43], we also created samples having states of unrelated objects in the context (e.g., TV is on for activity Make Coffee) to confuse LMs. In summary, this results in four settings: Vanilla Seen, Vanilla Unseen, Confusing Seen, and Confusing Unseen. We have 175/54/135/43 examples for four settings, respectively. We use Rouge-L [26] as the metric.
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• Housework QA. This is a multi-choice QA task, which is unseen during training. It asks which choice is the relevant object to finish a household activity, e.g., which object is relevant to making coffee? It has 261 examples in total, and we use accuracy as the metric. When evaluating, we provide 10 in-context exemplars in the prompts, so this task is evaluated as a 10-shot learning task.
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• Negation Housework QA. This is similar to Housework QA but inquires about the irrelevant object, e.g., which object is irrelevant to making coffee? It is more challenging than the vanilla QA because LMs that simply memorize the word co-occurrence in the training data may succeed in the vanilla QA but will fail in the negation QA. This task has 162 examples and uses accuracy as the metric. We provide 10 in-context exemplars in the prompts.
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Activity Recognition. We developed two multi-choice QA tasks with the human-written plans and the state changes from RobotHow to test the knowledge gained from activity recognition:
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• Activity Recognition QA. In this task, a human-written plan is given and the model needs to choose the correct activity name. An example of the question is “Given a plan: Walk to living room. Sit on sofa. Turn on TV. What is the name of this activity?”. And the answer should be Watch TV. The task has 549 examples. We use accuracy as the metric.
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• Activity Inference QA. In this task, we use the final state of the world model as input for the model to infer the activity name. For example, the input can be “Tom is sitting on the sofa and facing the TV. The TV is on. What is a possible activity he is doing?”, and the answer is “Watch TV”. We have 262 examples for this task and use accuracy to measure the performance. We provide 10 in-context exemplars in the prompts.
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Counting. We gathered random exploration experiences to construct Counting QA for evaluating skills learned from the counting task. The model is required to answer the number of objects in a specific location. For example, a query can be “Tom put an apple on the table. Tom turned on the TV. Tom put a cup on the table. How many objects are there on the table?”. We can see that there will be irrelevant actions like turn on TV to confuse the model and make the question more challenging. We collected 194 samples for the task and used accuracy as the evaluation metric. We provide 5 in-context exemplars in the input prompts.
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Object Path Tracking. We developed two downstream tasks for the object path tracking training task, namely Object Path Tracking Evaluation and Object Location QA.
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• Object Path Tacking Evaluation. This evaluation task is the same as the training task, where the model is required to generate the full moving path of an object. An example is “Tom walked to the kitchen. Tom grabbed the apple. Mary walked to the bedroom. Tom walked to the living room. What is the order of the rooms where the apple appears?”. This question typically includes multiple agents and many irrelevant actions, which makes it difficult to track the object. This task contains 200 examples. Following Huang et al. [18], we evaluate the performance by calculating the length of the longest common subsequence (LCS) between the ground truth and the generated path, normalized by the maximum length of the two.
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• Object Location QA. In this task, the model is asked about the location of an object before/after it moves to another location, e.g., where is the apple before/after the kitchen? This task has 200 examples with accuracy as the metric. We provide 2 in-context exemplars in the prompts.
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A previous study on prompting multiple QA questions [39] introduces two prompting methods, multiple choice prompt and cloze prompt, and two normalization methods, length, and unconditioned normalization. For all the multi-choice QA tasks, we choose the combination of prompting and normalization methods which yields the best performance on a held out set.
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To further verify the effectiveness of our method, we evaluate our finetuned GPT-Neo and GPT-J on the bAbI dataset. Specifically. we select 8 test sets from bAbI that align with the abilities covered in our collected embodied experiences. We include the description of each test set in Appendix A.5. For all the bAbI tasks, we do 2-shot learning by providing 2 in-context exemplars in the input prompts.
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Besides downstream tasks, we also want to ensure that our approach does not hurt language modeling performance of the models. Therefore, following previous work [42], we evaluate the perplexity on a subset of Pile [12] test set, which is the pretraining dataset for GPT-Neo and GPT-J. We sampled 5000 examples from Pile test set for evaluation.
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# 4.2 Results
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Constructed Evaluation Tasks. Results for all the downstream evaluation tasks are shown in Figure 3 and Figure 4. For all the models, we compare the results obtained after finetuning with world model against those of the original base models. For GPT-J, we also include a finetuned model without EWC-LoRA as a baseline. Detailed numbers of the results can be found in Appendix A.6. We also conduct human evaluations for GPT-J on the plan generation task, which can be found in Appendix A.7. In general, the models trained with the world model significantly outperform the baselines on various downstream tasks. Our method is not only effective for small 1.3B model, but can also scale to larger 6B and 13B models. Specifically, our finetuned GPT-J and LLaMA-13B with world model even achieve better performance than ChatGPT as a much larger LM on most of the 11 tasks. Besides, we can see the world model improves LMs on both seen and unseen tasks. This demonstrates that our model absorbs the knowledge for goal-oriented planning and random exploration instead of memorizing the seen experiences. Specifically, the better plan generation performance under the "Confusing" setting indicates that the world model improves the ability of LMs to avoid being interfered with by irrelevant contexts. On both Housework QA and Negation Housework QA, our models surpass the baselines, showing that our models also acquire knowledge about the necessary objects for completing a housework activity. Results on other downstream tasks also prove the effectiveness of our method. For example, on both Activity Recognition Evaluation and Activity Inference, our approach improves over the baselines significantly. Moreover, improvements can be observed in the downstream tasks regarding random exploration. On Counting and Object Location QA tasks, our LLaMA-13B trained with the world model even surpasses ChatGPT.
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Figure 3: Experimental results of GPT-Neo and GPT-J on our constructed downstream tasks. GPT-J (FT) refers to the finetuned GPT-J without EWC-LoRA. Our approach surpasses baselines on all of the 11 tasks, and outperforms ChatGPT on 7 of them. For example, our GPT-J model achieves 98.67 LCS on object path tracking, which is significantly better than 33.86 of base GPT-J and 59.53 of ChatGPT.
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Figure 4: Experimental results of OPT-13B and LLaMA-13B on our constructed downstream tasks. Our approach applied on LLaMA-13B outperforms ChatGPT on 8 of them.
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Figure 5: Experimental results on bAbI. Our approach outperforms base LMs on all the tasks except for the Two Supporting Fact task.
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Table 1: Perplexity on Pile test subset, showing the proposed finetuning with world model manages to preserve the LMs’ language modeling capability.
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<table><tr><td colspan="2">GPT-Neo</td><td colspan="2">GPT-J</td><td colspan="2">OPT-13B</td><td colspan="2">LLaMA-13B</td></tr><tr><td>Base</td><td>Ours</td><td>Base</td><td>Ours</td><td>Base</td><td>Ours</td><td>Base</td><td>Ours</td></tr><tr><td>4.120</td><td>4.193</td><td>3.443</td><td>3.537</td><td>4.077</td><td>4.358</td><td>3.036</td><td>3.069</td></tr></table>
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Table 2: Results of different regularization methods. The abbreviations in the Task column stand for the corresponding evaluation tasks for four training tasks. We use asterisk∗ to mark the perplexity of base models.
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<table><tr><td rowspan="2">Task</td><td colspan="5">GPT-Neo</td><td colspan="3">GPT-J</td></tr><tr><td>Base</td><td>EWC</td><td>LoRA</td><td>LoRA&KL</td><td>EWC-LoRA</td><td>Base</td><td>LoRA</td><td>EWC-LoRA</td></tr><tr><td>Plan Gen</td><td>21.25</td><td>48.56</td><td>51.24</td><td>45.99</td><td>49.70</td><td>34.31</td><td>51.23</td><td>51.23</td></tr><tr><td>Act Recog</td><td>69.22</td><td>89.98</td><td>87.98</td><td>81.42</td><td>85.43</td><td>87.98</td><td>90.16</td><td>88.52</td></tr><tr><td>Count</td><td>22.68</td><td>55.67</td><td>27.84</td><td>49.48</td><td>28.87</td><td>30.41</td><td>63.92</td><td>67.01</td></tr><tr><td>Obj PT</td><td>30.80</td><td>95.96</td><td>87.28</td><td>63.59</td><td>85.91</td><td>33.86</td><td>97.22</td><td>98.67</td></tr><tr><td>Perplexity</td><td>4.120*</td><td>4.995</td><td>4.360</td><td>5.029</td><td>4.193</td><td>3.443*</td><td>3.675</td><td>3.537</td></tr></table>
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bAbI Tasks. To further verify the effectiveness of our method, we evaluate our finetuned models on the bAbI dataset. The results are shown in Figure 5. We can see that our approach significantly outperforms the base LMs. Notably, after finetuned with VirtualHome experiences, GPT-J surpasses the much stronger ChatGPT on the most challenging tasks. Specifically, it outperforms ChatGPT on the Three Supporting Fact task, where the model is required to use three supporting facts from the context to answer a question like “where was the apple before the kitchen?”, and Lists/Sets task, which asks the model to give the answers in the form of a list, e.g., the answer for “What is Daniel holding?” is “apple, milk”. These results prove that our approach enables LMs to acquire the knowledge and skills inherent in embodied experiences, rather than simply overfitting to the training environment.
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Language Modeling. In addition to verifying improved performance on the downstream tasks, we also report results on Pile test subset to ensure the preservation of the generality and language modeling abilities of LMs. From the experimental results shown in Table 1, we can see that our approach only causes a negligible increase in perplexity over the base models. This demonstrates the effectiveness of EWC-LoRA to preserve the generality and linguistic competence of LMs. To verify the generality on other NLP tasks, we also include the results on SuperGLUE [40] in Appendix A.8.
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# 4.3 Comparison of Different Regularization Methods
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We compare our proposed EWC-LoRA with EWC and LoRA. Besides, we also include the baseline using KL penalty as regularization.The experimental results are shown in Table 2. We also include the results of four evaluation tasks. Notice that we do not include the results of GPT-J with pure EWC and KL, since they are overly memory-intensive or time-consuming. EWC requires an original model and a fisher matrix other than the trained model, which triples the memory usage, making it hard to be applied to large models like GPT-J-6B. Besides, KL penalty term is computed by $\mathcal { L } _ { K L } = E _ { ( \mathbf { x } , \mathbf { y } ) \sim P _ { \theta ^ { * } } } \bigl [ - \log \left( P _ { \theta ^ { * } } ( \mathbf { y } | \mathbf { x } ) / P _ { \theta } ( \mathbf { y } | \mathbf { x } ) \right) \bigr ]$ , thus it requires sampling from the model output probability, which is time-consuming. On the contrary, EWC-LoRA is both memory- and timeefficient. In Table 2, we can see that EWC-LoRA achieves the lowest perplexity compared to other methods, while still significantly outperforming the base LMs. Compared with pure EWC, applying pure LoRA greatly decreases perplexity, which is consistent with the previous conclusion that limiting the dimension of the optimization problem can mitigate catastrophic forgetting [32]. EWC-LoRA further decreases perplexity, making it extremely close to the original perplexity, while achieving comparable performance with LoRA on downstream tasks. This demonstrates the effectiveness of EWC-LoRA. Besides, We can find that combing LoRA with KL will greatly increase perplexity while not achieving better downstream performance. Overall, our proposed EWC-LoRA achieves the best trade-off between the perplexity and the downstream performance, which outperforms baselines significantly while almost not increasing the perplexity on the pretraining dataset.
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<table><tr><td></td><td colspan="6">GPT-Neo</td></tr><tr><td></td><td>Base</td><td>Ours</td><td> -w/o Plan Gen</td><td> -W/o Act Recog</td><td> -w/o Count</td><td> - w/o Obj PT</td></tr><tr><td>Plan Gen</td><td>21.25</td><td>49.70</td><td>14.48</td><td>49.38</td><td>49.85</td><td>50.06</td></tr><tr><td>Act Recog</td><td>69.22</td><td>85.43</td><td>85.97</td><td>48.63</td><td>85.25</td><td>84.34</td></tr><tr><td>Count</td><td>22.68</td><td>28.87</td><td>18.56</td><td>25.26</td><td>35.05</td><td>32.99</td></tr><tr><td>Obj PT</td><td>30.80</td><td>85.91</td><td>92.13</td><td>84.17</td><td>86.46</td><td>29.90</td></tr><tr><td>Perplexity</td><td>4.120*</td><td>4.193</td><td>4.171</td><td>4.151</td><td>4.162</td><td>4.164</td></tr></table>
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Table 3: Ablation experimental results on training tasks. We use the same abbreviations as Table 2.
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# 4.4 Ablation Studies
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To study the contribution of each training task, we conducted an ablation study by removing one training task every time. We use GPT-Neo-1.3B as the base model. We include the results on tasks seen during training in Table 3. Results on all the tasks can be found in Appendix A. We can see that the removal of a training task with similar ability leads to a notable decrease in the model’s performance on downstream tasks. For example, the performance of plan generation drops significantly when plan generation is removed from the training tasks. Similarly, the removal of activity recognition or object path tracking from the training tasks leads to a decline in performance in their respective downstream tasks. We conclude that our gathered embodied experience has a tremendous contribution to teaching the corresponding reasoning ability by finetuning. Interestingly, Counting QA performance shows an increase when counting is omitted from the training tasks, possibly because the ability of counting can be inferred from other training tasks.
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# 5 Conclusion & Future Work
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We proposed a new training framework that uses world models to enhance language models. It first collects embodied experiences from world models through both goal-oriented planning and random exploration. The experiences are then compiled into appropriate formats for LMs finetuning. We further introduce EWC-LoRA, which not only facilitates parameter-efficient tuning but also alleviates catastrophic forgetting and enables knowledge generalization. We show the strong performance of our method on a large number of downstream evaluation tasks.
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This work demonstrates the advantage of panoramic learning with all forms of experience [17]. On the other hand, the present work is limited to a single household environment as the world model. In the future, we intend to study how to integrate embodied experiences from different world models and generalize knowledge learned from each world model to different domains.
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Acknowledgements. This project is partially supported by DARPA ECOLE HR00112390063.
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[58] Rowan Zellers, Ari Holtzman, Matthew E Peters, Roozbeh Mottaghi, Aniruddha Kembhavi, Ali Farhadi, and Yejin Choi. PIGLeT: Language grounding through neuro-symbolic interaction in a 3d world. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 2040–2050, 2021.
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[59] Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, et al. Opt: Open pre-trained transformer language models. arXiv preprint arXiv:2205.01068, 2022.
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[60] Daniel M Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. Fine-tuning language models from human preferences. arXiv preprint arXiv:1909.08593, 2019.
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<table><tr><td>Action</td><td>Template</td></tr><tr><td>[Find] <Object></td><td>Find Object</td></tr><tr><td>[Walk] <0bject></td><td>Walk to Object</td></tr><tr><td>[Run] <Object></td><td>Run to Object</td></tr><tr><td>[Sit] <Object></td><td>Sit on Object</td></tr><tr><td>[StandUp]</td><td> Stand up</td></tr><tr><td>[Grab] <0bject></td><td>Grab Object</td></tr><tr><td>[Open] <Object></td><td></td></tr><tr><td>[Ciose] <Object></td><td>Open Object</td></tr><tr><td>[Put] <0bject_1> <0bject_2></td><td>Close Object</td></tr><tr><td></td><td>Put Object_1 on Object_2</td></tr><tr><td>[PutIn] <Object_1> <0bject_2></td><td>Put Ojbect_1 in Object_2</td></tr><tr><td>[SwitchOn] <Object></td><td>Switch/Turn on Object</td></tr><tr><td>[SwitchOff] <Object></td><td>Switch/Turn off Ojbect</td></tr><tr><td>[Drink] <Object></td><td>Drink Object</td></tr><tr><td>[TurnTo] <Object></td><td>Turn to Object</td></tr><tr><td>[LookAt] <Object></td><td>Look at Object</td></tr><tr><td>[Wipe] <Object></td><td>Wipe Object</td></tr><tr><td>[PutOn] <0bject></td><td>Put on Object</td></tr><tr><td>[PutOff] <Object></td><td>Put off Object</td></tr><tr><td>[Greet] <Object></td><td>Greet Object</td></tr><tr><td>[Drop] <Object></td><td>Drop Object</td></tr><tr><td>[Touch] <Object></td><td></td></tr><tr><td>[Lie] <0bject></td><td>Touch Object</td></tr><tr><td>[Pour] <0bject_1> <0bject_2></td><td>Lie on Object</td></tr><tr><td></td><td>Pour Object_1 into Object_2</td></tr><tr><td>[Type] <Object></td><td>Type Object</td></tr><tr><td>[Watch] <Object></td><td>Watch Object</td></tr><tr><td>[Move] <Object></td><td>Move Object</td></tr><tr><td>[Wash] <Object></td><td>Wash Object</td></tr><tr><td>[Rinse] <Object></td><td>Rinse Object</td></tr><tr><td>[Scrub] <Object></td><td>Scrub Object</td></tr><tr><td>[Squeeze] <0bject></td><td>Squeeze Object</td></tr><tr><td>[PlugIn] <Object></td><td>Plug in Object</td></tr><tr><td>[PlugOut] <Object></td><td>Plug out Object</td></tr><tr><td>[Cut] <0bject></td><td>Cut Object</td></tr><tr><td>[Eat] <Object></td><td>Eat Object</td></tr><tr><td>[Sleep]</td><td>Sleep</td></tr><tr><td>[WakeUp]</td><td>Wake up</td></tr></table>
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Table 4: Supported actions in VirtualHome and their corresponding text templates.
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# A Appendix
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# A.1 VirtualHome
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The complete format of an executable action step in VirtualHome is <char{char_id}> [Action] <Object> (Object_id). Specifically, char_id specifies which agent to execute the action when multiple agents are in the world model at the same time. Action should be a supported atomic action in VirtualHome. Object is the object with which the agent interacts. Each object in the environment is assigned an Object_id to distinguish it from others of the same object class. We designed a template for each action to transform them into natural text for LMs finetuing. The full list of executable actions can be found in Table 4. Note that in the list, we omit <char{char_id}> and (Object_id) for simplicity.
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# A.2 Acitivity Goal And Predicate
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The goal of an household activity in VirtualHome consists of several predicates. Each predicate represents a condition of one object or a relation between two objects. For example, OPEN(coffe
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maker) means the coffee maker is open, and ON(apple, table) means an apple is on the table. The goal is only achieved when all the predicates are achieved. We collected activities and goals from RobotHow.
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# A.3 Data Format and Prompts
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Following Chung et al. [8], we use instructions with in-context exemplars as prompts. Specifically, the instruction, the question context, and the answer will be provided in each exemplar, and the full prompt will contain multiple such exemplars for in-context learning. The format of the data and the exemplar for each task is provided below.
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# A.3.1 Plan Generation
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>activity</td><td>watch TV</td></tr><tr><td>condition plan</td><td>living room，sofa，TV.The sofa and TV are in the living room. Walk to living room.Sit on sofa. Watch TV.</td></tr></table>
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# In-context Exemplar
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Q: How to {{ activity }}? Given items include {{ condition }} A: {{ plan }}
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# A.3.2 Housework QA
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>activity</td><td>watch TV</td></tr><tr><td>choices</td><td>[TV，coffee，bed，toothbrush]</td></tr><tr><td>answer</td><td>TV</td></tr></table>
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# In-context Exemplar
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Question: To {{ activity }}, a possibly related item could be Answer: {{ answer }}
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# A.3.3 Negation Housework QA
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>activity</td><td>watch TV</td></tr><tr><td>choices</td><td>[TV，sofa，living room，toothbrush]</td></tr><tr><td>answer</td><td>toothbrush</td></tr></table>
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# In-context Exemplar
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Question: To {{ activity }}, an unrelated item could be Answer: {{ answer }}
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# A.3.4 Activity Recognition
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>plan choices activity</td><td>Walk to living room.Sit on sofa. Watch TV. [watch TV，make coffee，sleep，brush teeth] watch TV</td></tr></table>
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# In-context Exemplar
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Given a task plan: {{ plan }} Question: what is the name of this task? Answer: {{ answer }}
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# A.3.5 Activity Inference
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>state</td><td>Tom is sitting on the sofa. Tom is facing the TV.</td></tr><tr><td>choices</td><td>[watch TV，make coffee，sleep，brush teeth]</td></tr><tr><td>activity</td><td>watch TV</td></tr></table>
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# In-context Exemplar
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{{ state }}
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Question: given the above state, a possible activity could be
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Answer: {{ answer }}
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# A.3.6 Counting
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Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>movement</td><td>Tom was at home. He grabbed an apple and put it on the bookshelf. He then walked to the kitchen and srcub a plate. He went back to bookshelf and put the plate on it.</td></tr><tr><td>location number</td><td>bookshelf</td></tr><tr><td>items</td><td>2 apple，plate</td></tr></table>
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# In-context Exemplar
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Given a sequence of actions in a house, and a question about what items are located in a specific place. Answer the number of items and list the items.
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Q: {{ movement }} How many items are there on the {{ location }}? A: Ther are {{ number }} itmes on the {{ location }}. They are {{ items }}
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A.3.7 Counting QA Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>movement</td><td>Tom was at home. He grabbed an apple and put it on the bookshelf. He then walked to the kitchen and srcub a plate. He went back to bookshelf and put the plate on it.</td></tr><tr><td>location number</td><td>bookshelf 2</td></tr></table>
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# In-context Exemplar
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Q: {{ movement }} How many items are there on the {{ location }}? A: {{ number }}
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# A.3.8 Object Path Tracking
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# Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>movement</td><td>Tom went to the kitchen. Mary walked into the dining room． Tom grabbed a plate. Tom travelled to the living room. Mary moved to the living room． Tom put the plate on the table. Mary grabbed the</td></tr><tr><td></td><td>plate. Mary journeyed to the bedroom.</td></tr><tr><td>object</td><td>plate</td></tr><tr><td>path</td><td>kitchen，living room，bedroom</td></tr><tr><td></td><td></td></tr></table>
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# In-context Exemplar
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{{ movement }}
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Question: What is the order of the rooms where the {{ object }} appeared?
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Answer: {{ path }}
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# A.3.9 Object Location QA
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Data Example
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<table><tr><td>Key</td><td>Value</td></tr><tr><td>movement</td><td>Tom went to the kitchen. Mary walked into the dining room. Tom grabbed a plate. Tom travelled to the living room. Mary moved to the living room. Tom put the plate on the table. Mary grabbed the</td></tr><tr><td>object</td><td>plate. Mary journeyed to the bedroom. plate</td></tr><tr><td>reference_room</td><td>iiving room before</td></tr><tr><td>preposition answer</td><td></td></tr></table>
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# In-context Exemplar
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{{ movement }}
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Question: Where is the {{ object }} {{ preposition }} the {{ reference_room }}?
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Answer: {{ answer }}
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# A.4 Hyperparameters
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For both GPT-Neo-1.3B and GPT-J-6B, we use a learning rate of $8 \times 1 0 ^ { - 5 }$ and a batch size of 20. The weights for plan generation, activity recognition, counting, and object path tracking are 1.0, 0.7, 1.0, and 1.0, respectively. We trained GPT-Neo-1.3B for 3 epochs with the EWC coefficient $\lambda = 0 . 5$ in Equation 4. For GPT-J-6B, we trained it for 5 epochs with $\lambda = 2$ . With our approach, it takes 40 minutes to train a GPT-Neo and 220 minutes to train a GPT-J. We used a rank of 8 and coefficient of 32 for LoRA’s hyperparameters.
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# A.5 bAbI Dataset
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We include 8 tasks from bAbI that test embodied knowledge. They are: One Supporting Fact, Two Supporting Fact, Three Supporting Fact, Counting, Lists/Sets, Simple Negation, Time Reasoning, Positional Reasoning. Examples for each task are shown in Table 5.
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# Task 1: Single Supporting Fact
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Mary went to the bathroom. John moved to the hallway. Mary travelled to the office. Where is Mary? A:office
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# Task 2: Two Supporting Facts
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|
| 399 |
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John is in the playground. John picked up the football. Bob went to the kitchen. Where is the football? A:playground
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| 400 |
+
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# Task 3: Three Supporting Facts
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| 402 |
+
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John picked up the apple.
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+
John went to the office.
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| 405 |
+
John went to the kitchen.
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+
John dropped the apple.
|
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Where was the apple before the kitchen? A:office
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+
|
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# Task 4: Counting
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| 410 |
+
|
| 411 |
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Daniel picked up the football.
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+
Daniel dropped the football.
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Daniel got the milk.
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| 414 |
+
Daniel took the apple. A: office
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+
How many objects is Daniel holding? A: two
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+
|
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+
# Task 5: Lists/Sets
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| 418 |
+
|
| 419 |
+
Daniel picks up the football. Daniel drops the newspaper. Daniel picks up the milk. What is Daniel holding? milk, football
|
| 420 |
+
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# Task 6: Simple Negation
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| 422 |
+
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| 423 |
+
Sandra travelled to the office. Fred is no longer in the office. Is Fred in the office? A:no Is Sandra in the office? A:yes
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+
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+
# Task 7: Time Reasoning
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| 426 |
+
|
| 427 |
+
In the afternoon Julie went to the park. Yesterday Julie was at school. Julie went to the cinema this evening. Where did Julie go after the park? A:cinema Where was Julie before the park? A:school
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|
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# Task 8: Positional Reasoning
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| 430 |
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|
| 431 |
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The triangle is to the right of the blue square. The red square is on top of the blue square. The red sphere is to the right of the blue square. Is the red sphere to the right of the blue square? A:yes Is the red square to the left of the triangle? A:yes
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|
| 433 |
+
Table 5: Examples for bAbI tasks.
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| 434 |
+
|
| 435 |
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# A.6 Results of Main Experiments and Ablation Studies
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| 436 |
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|
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Experimental results on our constructed downstream tasks are shown in Table 6, and the results on bAbI are shown in Table 7. We also show the results of ablation studies in Table 8.
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|
| 439 |
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# A.7 Human Evaluations
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| 440 |
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|
| 441 |
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We conduct human evaluations on plan generation for GPT-J model. Following Huang et al. [18] we asked 3 people to annotate whether each task can be completed using a generated plan. We randomly sampled 150 tasks and asked each person to annotate 50 of them. The Results show that the base GPT-J model can only achieve $2 4 . 0 \%$ accuracy, while the finetuned model can achieve $6 2 . 4 \%$ . The higher planning accuracy demonstrates the superior task planning ability of our model.
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|
| 443 |
+
# A.8 SuperGLUE Results
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| 444 |
+
|
| 445 |
+
We evaluate the base GPT-J model and our finetuned model on appropriate SuperGLUE tasks, e.g., that can be formulated as a multi-choice QA task without prompt engineering. Our model’s performance matches and even outperforms the baseline, showing our model retains the general language capability.
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| 446 |
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|
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Table 6: Experimental results on various downstream evaluation tasks. The best result among baselines and our method is shown in bold, and the best result among all the models is underlined.
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<table><tr><td rowspan="2">Task</td><td rowspan="2">Metric</td><td colspan="2">GPT-Neo</td><td colspan="3">GPT-J</td><td colspan="2">OPT-13B</td><td colspan="2">LLaMA-13B</td><td>ChatGPT</td></tr><tr><td>Base</td><td>Ours</td><td>Base</td><td>FT</td><td>Ours</td><td>Base</td><td>Ours</td><td>Base</td><td>Ours</td><td>(GPT3.5-turbo)</td></tr><tr><td>Plan Generation</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>-Vanilla Seen</td><td>Rouge-L</td><td>21.25</td><td>49.70</td><td>34.31</td><td>47.98</td><td>51.23</td><td>36.00</td><td>50.15</td><td>41.77</td><td>52.05</td><td>40.57</td></tr><tr><td>-Vanilla UnSeen</td><td>Rouge-L</td><td>17.64</td><td>49.27</td><td>34.22</td><td>47.86</td><td>49.58</td><td>29.34</td><td>45.11</td><td>38.78</td><td>47.44</td><td>41.01</td></tr><tr><td>-Confusing Seen</td><td>Rouge-L</td><td>16.86</td><td>46.88</td><td>34.81</td><td>47.59</td><td>48.94</td><td>31.92</td><td>49.87</td><td>40.33</td><td>51.00</td><td>40.41</td></tr><tr><td>-Confusing Unseen</td><td>Rouge-L</td><td>17.05</td><td>42.34</td><td>32.98</td><td>44.43</td><td>45.60</td><td>36.98</td><td>47.93</td><td>41.73</td><td>50.49</td><td>40.97</td></tr><tr><td>Housework QA</td><td>Accuracy</td><td>70.11</td><td>72.41</td><td>77.78</td><td>51.34</td><td>85.44</td><td>81.61</td><td>84.29</td><td>81.99</td><td>86.59</td><td>83.91</td></tr><tr><td>Negation Housework QA</td><td>Accuracy</td><td>38.27</td><td>41.98</td><td>35.19</td><td>33.33</td><td>39.51</td><td>43.21</td><td>40.21</td><td>43.21</td><td>30.25</td><td>87.65</td></tr><tr><td>Activity Recognition</td><td>Accuracy</td><td>69.22</td><td>85.43</td><td>87.98</td><td>71.41</td><td>88.52</td><td>89.07</td><td>91.44</td><td>90.53</td><td>91.80</td><td>95.05</td></tr><tr><td>Activity Inference</td><td>Accuracy</td><td>56.49</td><td>66.03</td><td>69.08</td><td>70.99</td><td>74.43</td><td>67.94</td><td>70.61</td><td>74.05</td><td>68.32</td><td>83.59</td></tr><tr><td>Counting</td><td>Accuracy</td><td>22.68</td><td>28.87</td><td>30.41</td><td>16.49</td><td>67.01</td><td>20.01</td><td>62.37</td><td>29.38</td><td>79.38</td><td>66.49</td></tr><tr><td>Object Path Tracking</td><td>LCS</td><td>30.80</td><td>85.91</td><td>33.86</td><td>46.25</td><td>98.67</td><td>33.49</td><td>96.28</td><td>38.82</td><td>96.99</td><td>59.53</td></tr><tr><td>Object Location QA</td><td>Accuracy</td><td>22.50</td><td>33.50</td><td>30.00</td><td>22.50</td><td>34.50</td><td>37.00</td><td>33.00</td><td>28.50</td><td>79.00</td><td>67.50</td></tr></table>
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<table><tr><td rowspan="2">Task</td><td colspan="2">GPT-Neo</td><td colspan="2">GPT-J</td><td rowspan="2">ChatGPT</td></tr><tr><td>Base</td><td>Ours</td><td>Base</td><td>Ours</td></tr><tr><td>Single Supporting Fact</td><td>51.86</td><td>56.29</td><td>65.16</td><td>68.98</td><td>96.27</td></tr><tr><td>Two Supporting Fact</td><td>33.43</td><td>30.82</td><td>40.48</td><td>26.08</td><td>47.33</td></tr><tr><td>Three Supporting Fact</td><td>7.85</td><td>13.49</td><td>22.46</td><td>30.41</td><td>16.82</td></tr><tr><td>Counting</td><td>34.04</td><td>48.84</td><td>41.39</td><td>69.08</td><td>93.96</td></tr><tr><td>Lists/Sets</td><td>14.80</td><td>51.76</td><td>34.74</td><td>84.99</td><td>76.84</td></tr><tr><td>Simple Negation</td><td>36.05</td><td>65.56</td><td>42.80</td><td>63.95</td><td>93.66</td></tr><tr><td>Time Reasoning</td><td>21.45</td><td>23.46</td><td>36.96</td><td>59.42</td><td>61.63</td></tr><tr><td>Positional Reasoning</td><td>50.51</td><td>53.64</td><td>49.70</td><td>53.23</td><td>58.38</td></tr></table>
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|
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Table 7: Experimental results on bAbI test sets.
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| 454 |
+
|
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Table 8: Ablation experimental results on training tasks.
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+
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| 457 |
+
<table><tr><td></td><td colspan="6">GPT-Neo</td></tr><tr><td></td><td>Base</td><td>Ours</td><td>-w/o Plan Gen</td><td>-w/o Act Recog</td><td> -w/o Count</td><td>-w/o Obj PT</td></tr><tr><td>Plan Gen</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>-Vanilla/Seen</td><td>21.25</td><td>49.70</td><td>14.48</td><td>49.38</td><td>49.85</td><td>50.06</td></tr><tr><td>-Vanilla / Unseen</td><td>17.64</td><td>49.27</td><td>14.28</td><td>48.96</td><td>51.16</td><td>49.02</td></tr><tr><td>-Confusing /Seen</td><td>16.86</td><td>46.88</td><td>13.63</td><td>46.37</td><td>48.30</td><td>49.14</td></tr><tr><td>-Confusing / Unseen</td><td>17.05</td><td>42.34</td><td>9.86</td><td>43.79</td><td>46.28</td><td>44.64</td></tr><tr><td>QA</td><td>70.11</td><td>72.41</td><td>73.18</td><td>71.26</td><td>74.71</td><td>70.11</td></tr><tr><td>Neg QA</td><td>38.27</td><td>41.98</td><td>32.72</td><td>35.80</td><td>36.42</td><td>38.89</td></tr><tr><td>Act Recog</td><td>69.22</td><td>85.43</td><td>85.97</td><td>48.63</td><td>85.25</td><td>84.34</td></tr><tr><td>Act Infer</td><td>56.49</td><td>66.03</td><td>66.03</td><td>58.40</td><td>64.89</td><td>62.21</td></tr><tr><td>Count</td><td>22.68</td><td>28.87</td><td>18.56</td><td>25.26</td><td>35.05</td><td>32.99</td></tr><tr><td>Obj PT</td><td>30.80</td><td>85.91</td><td>92.13</td><td>84.17</td><td>86.46</td><td>29.90</td></tr><tr><td>Obj QA</td><td>22.50</td><td>33.50</td><td>35.00</td><td>49.00</td><td>43.50</td><td>22.00</td></tr><tr><td>Perplexity</td><td>4.120*</td><td>4.193</td><td>4.171</td><td>4.151</td><td>4.162</td><td>4.164</td></tr></table>
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| 458 |
+
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| 459 |
+
<table><tr><td>Model</td><td>BoolQ</td><td>CB</td><td>RTE</td><td>AX-g</td><td>AX-b</td><td>COPA</td></tr><tr><td>GPT-J</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Base</td><td>45.20</td><td>41.07</td><td>47.29</td><td>50.00</td><td>57.50</td><td>59.00</td></tr><tr><td>- Ours</td><td>66.00</td><td>41.07</td><td>58.84</td><td>53.37</td><td>54.00</td><td>62.00</td></tr></table>
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+
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| 461 |
+
Table 9: Results on SuperGLUE subset.
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| 462 |
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| 463 |
+
# A.9 Broader Impact
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| 464 |
+
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| 465 |
+
Like other generation systems, the language model trained by our approach is susceptible to producing unintended output when confronted with harmful input, such as unethical text or input intended for adversarial attacks. Therefore, we strongly advise against utilizing our approach outside of controlled research environments until these risks have been mitigated. It is important to note that a thoughtless deployment of our method could potentially enable malicious exploitation of the underlying language models. Thus, precautions, such as implementing a filtering mechanism, must be taken.
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| 1 |
+
# Masked Autoencoders As Spatiotemporal Learners
|
| 2 |
+
|
| 3 |
+
Christoph Feichtenhofer∗
|
| 4 |
+
|
| 5 |
+
Haoqi Fan∗ Yanghao Li Kaiming He
|
| 6 |
+
|
| 7 |
+
Meta AI, FAIR
|
| 8 |
+
|
| 9 |
+
https://github.com/facebookresearch/mae_st
|
| 10 |
+
|
| 11 |
+
# Abstract
|
| 12 |
+
|
| 13 |
+
This paper studies a conceptually simple extension of Masked Autoencoders (MAE) [31] to spatiotemporal representation learning from videos. We randomly mask out spacetime patches in videos and learn an autoencoder to reconstruct them in pixels. Interestingly, we show that our MAE method can learn strong representations with almost no inductive bias on spacetime (only except for patch and positional embeddings), and spacetime-agnostic random masking performs the best. We observe that the optimal masking ratio is as high as $90 \%$ (vs. $7 5 \%$ on images [31]), supporting the hypothesis that this ratio is related to information redundancy of the data. A high masking ratio leads to a large speedup, e.g., $> 4 \times$ in wall-clock time or even more. We report competitive results on several challenging video datasets using vanilla Vision Transformers [18]. We observe that MAE can outperform supervised pre-training by large margins. We further report encouraging results of training on real-world, uncurated Instagram data. Our study suggests that the general framework of masked autoencoding (BERT [15], MAE [31], etc.) can be a unified methodology for representation learning with minimal domain knowledge.
|
| 14 |
+
|
| 15 |
+
# 1 Introduction
|
| 16 |
+
|
| 17 |
+
The deep learning community is experiencing a trend of unifying methodologies for solving problems in different areas, such as language, vision, speech, and more. For architectures, Transformers [67] have been successfully introduced into computer vision [18] and established as a general building block in both language and vision. For self-supervised representation learning, the denoising/masked autoencoding methodology [68] in BERT [15] has been shown effective on learning visual representations from images [31]. Towards unifying methodologies, less domain knowledge (“fewer inductive biases” [18]) is introduced for a specific problem, which urges the models to learn useful knowledge almost purely from data.
|
| 18 |
+
|
| 19 |
+
Following this philosophy, we study extending Masked Autoencoders (MAE) [31] to the problem of spatiotemporal representation learning. Our method is simple: we randomly mask out spacetime patches in videos and learn an autoencoder to reconstruct them (Fig. 1). Our method has minimal domain knowledge: the only spacetime-specific inductive bias is on embedding the patches and their positions; all other components are agnostic to the spacetime nature of the problem. In particular, our encoder and decoder are both vanilla Vision Transformers [18] with no factorization or hierarchy, and our random mask sampling is agnostic to the spacetime structures. Our method predicts pixel values and uses no extra problem-specific tokenizer. In a nutshell, our method is simply MAE applied to the set of spacetime patches. Despite minimal inductive biases, our method achieves strong empirical results, suggesting that useful knowledge can be learned from data.
|
| 20 |
+
|
| 21 |
+
It is hypothesized in [31] that the masking ratio (i.e., percentage of removed tokens) in masked autoencoding methods is related to the information redundancy of the problems. For example, natural images are more information-redundant than languages and thus the optimal masking ratio is higher (e.g., than BERT [15]). Our observations on video data support this hypothesis. We find that the optimal masking ratio of MAE is $90 \%$ for videos (Fig. 2), higher than the masking ratio of $7 5 \%$ for its image counterpart [31]. This can be understood as a consequence of natural video being correlated.
|
| 22 |
+
|
| 23 |
+
|
| 24 |
+
Figure 1: Masked Autoencoders as spatiotemporal learners. We mask a large subset (e.g., $90 \%$ ) of random patches in spacetime. An encoder operates on the set of visible patches. A small decoder then processes the full set of encoded patches and mask tokens to reconstruct the input. Except for patch and positional embeddings, neither the encoder, the decoder, nor the masking strategy, has any spatiotemporal inductive bias.
|
| 25 |
+
|
| 26 |
+
To the extreme, if a video has $T$ identical static frames, randomly sampling $1 / T$ of all spacetime patches would reveal most of the static frame. Because slow motion is more likely than fast motion in natural videos, the masking ratio can be very high as we observe empirically.
|
| 27 |
+
|
| 28 |
+
The higher masking ratio leads to a more efficient solution in practice. Following the MAE in [31] that applies the encoder only on visible tokens, a masking ratio of $90 \%$ reduces the encoder time and memory complexity to ${ < } 1 / 1 0$ . Put together with a small decoder [31], the MAE pre-training can achieve a theoretically $7 . 7 \times$ reduction in computation vs. encoding all tokens. In fact, the computation reduction is so large that the data loading time becomes a new bottleneck; even so, we record a $4 . 1 \times$ wall-clock speedup. Such a significant speedup is of great importance for video research that is large-scale and time-consuming.
|
| 29 |
+
|
| 30 |
+
We report strong results on a variety of video recognition datasets. Our MAE pre-training greatly improves generalization performance: on Kinetics-400 [35], it increases the accuracy of ViT-Large [18] by absolute $13 \%$ vs. training from scratch, while it takes less wall-clock training time overall (pre-training plus fine-tuning). Our MAE pre-training can outperform its supervised pre-training counterpart by big margins. Using vanilla ViT [18], our method achieves competitive results with previous state-of-the-art methods that incorporate more domain knowledge. We also report encouraging results using MAE pre-trained on 1 million random, uncurated Instagram videos. These results suggest that self-supervised learning on videos can be tackled in a way similar to its counterparts on language [15] and images [31], under a unified framework.
|
| 31 |
+
|
| 32 |
+
# 2 Related Work
|
| 33 |
+
|
| 34 |
+
Denoising autoencoders. Denoising autoencoders (DAE) [68, 69] present a general methodology for learning representations by reconstructing clean signals from corrupted inputs. Masking as a type of noise dates back to at least a decade ago [69]. One of its most successful developments is BERT [15], which is conceptually masked autoencoding on language tokens.
|
| 35 |
+
|
| 36 |
+
Denoising/masked autoencoding methods for computer vision have been making continuous progress [50, 9, 18, 31]. A series of recent methods are based on Transformer architectures [67] and are towards a unified solution between vision and language. iGPT [9] pioneers this direction by training Transformers on pixels as tokens. The ViT paper [18] makes a revolutionary step forward by using patches as tokens. It not only establishes strong Transformer architectures for vision tasks, but also explores masked prediction with patches. MAE [31] returns to the basics of the autoencoding concept [68] and draws attention to the decoding aspect. The presence of a meaningful decoder provides more flexibility, e.g., enabling the encoder to operate only on visible patches and leading to a more efficient solution. It empirically shows that a high masking ratio is essential for image tasks [31]. Our study follows this line of research.
|
| 37 |
+
|
| 38 |
+
|
| 39 |
+
Figure 2: Visualizations on the Kinetics-400 [35] validation set (masking ratio $90 \%$ ). We show the original video (top), masked video (middle), and MAE output (bottom) for each sample. This model reconstructs the original pixels. The video size is $1 6 { \times } 2 2 4 { \times } 2 2 4$ and the spacetime patch size is $2 \times 1 6 \times 1 6$ (the temporal patch size of 2 is not visualized here). Each sample has $8 \times 1 4 \times 1 4 = 1 5 6 8$ tokens with 156 being visible. For better visualizations, the known patches in the output are from the original input. Fig. 7 shows more examples.
|
| 40 |
+
|
| 41 |
+
|
| 42 |
+
Figure 3: Visualizations of the same pre-trained model in Fig. 2 but with a masking ratio of $95 \%$ .
|
| 43 |
+
|
| 44 |
+
Instead of predicting pixels [9, 18, 31, 80], another line of research focuses on the tokenization of the prediction targets [3, 17, 77]. BEiT [3] proposes to use pre-trained dVAE [47, 55] as the reconstruction target. The dVAE tokenizer can be improved by perceptual or adversarial losses [17]. MaskFeat [77] shows that HoG [13] as prediction targets performs strongly.
|
| 45 |
+
|
| 46 |
+
Self-supervised learning on videos. The presence of the temporal dimension is a focus of selfsupervised learning on video data. Related topics include temporal coherence (‘slowness’) [79, 25], future prediction [61, 72, 70, 45, 44, 71, 16], object motion [1, 75, 49, 76], temporal ordering [46, 23, 38, 78, 81], spatiotemporal contrast [58, 62, 30, 22, 51, 56], etc.
|
| 47 |
+
|
| 48 |
+
Our method also relies on the temporal coherence of videos, but it approaches this goal implicitly. In fact, as our method is largely agnostic to spacetime, the main opportunity for it to make use of the temporal coherence is a higher masking ratio (e.g., $90 \%$ ), which assumes that videos are more information-redundant than images.
|
| 49 |
+
|
| 50 |
+
|
| 51 |
+
Figure 4: Mask sampling. (a): Random sampling that is spacetime-agnostic. (b): Space-only random sampling, broadcasted to all time steps (“tube” masking [77]). (c): Time-only random sampling, broadcasted to all spatial locations (“frame” masking [77]). (d): Block-wise sampling [3] in spacetime, removing large regions (“cube” masking [77]). In this illustration, $T \times H \times W$ is $8 \times 1 4 \times 1 4$ ; green tokens are kept and others are masked out.
|
| 52 |
+
|
| 53 |
+
There has been growing interest in masking-based methods for self-supervised learning on videos. Previous works focus on tokenizing the prediction targets for the use of videos [65, 73, 77]. Our autoencoding method operates on pixels, which is simpler and requires no extra data or domain knowledge on the tokenizer. Importantly, our method greatly improves the efficiency of learning. The practical speedup is of central importance for video-related research, which is in general larger-scale and more time-consuming.
|
| 54 |
+
|
| 55 |
+
Our work is done independently and concurrently with [66] on a related method.
|
| 56 |
+
|
| 57 |
+
# 3 Method
|
| 58 |
+
|
| 59 |
+
Our method is a simple extension of MAE [31] to spacetime data (Fig. 1). Our goal is to develop the method under a general and unified framework, with as little domain knowledge as possible.
|
| 60 |
+
|
| 61 |
+
Patch embedding. Following the original ViT [18], given a video clip, we divide it into a regular grid of non-overlapping patches in spacetime [4, 2, 19, 77]. The patches are flattened and embedded by linear projection [18]. Positional embeddings [67] are added to the embedded patches. The patch and positional embedding process is the only process that is spacetime-aware.
|
| 62 |
+
|
| 63 |
+
Masking. We sample random patches without replacement from the set of embedded patches. This random sampling is agnostic to the spacetime structure (Fig. 4 (a)). This structure-agnostic sampling strategy is analogous to that of BERT in 1D [15] and MAE in 2D [31].
|
| 64 |
+
|
| 65 |
+
It is hypothesized in [31] that the optimal masking ratio is related to the information redundancy of the data. With unstructured random masking, BERT [15] uses a masking ratio of $15 \%$ for language and MAE [31] uses a ratio of $7 5 \%$ for images, suggesting that images are more information-redundant than language. Our empirical results on videos support this hypothesis. The optimal masking ratio we observe is $90 \%$ . This is in line with the common assumption that natural videos are more informationredundant than images because of temporal coherence. Fig. 2 and 3 present our MAE reconstruction results on unseen validation data with a masking ratio of $90 \%$ and $9 5 \%$ .
|
| 66 |
+
|
| 67 |
+
The spacetime-agnostic sampling can be more effective than structure-aware sampling strategies, e.g., space-only, time-only, or block-wise sampling (Fig. 4 (b-d)). As neighboring patches in space or in time (Fig. 4(b, c)) are coherent, with a very high masking ratio, space-only or time-only sampling may retain less information and yield an overly difficult pre-training task. For example, time-only sampling from 8 frames with a masking ratio of $8 7 . 5 \%$ means keeping only a single frame, which presents an overly challenging task of predicting the future and past given only one frame. We observe that optimal masking ratios for structure-aware sampling are in general lower. In contrast, the spacetime-agnostic sampling better utilizes the limited number of visible patches and thus allows to use a higher masking ratio.
|
| 68 |
+
|
| 69 |
+
Autoencoding. Our encoder is a vanilla ViT [18] applied only on the visible set of embedded patches, following [31]. This design greatly reduces time and memory complexity and leads to a more practical solution. A masking ratio of $90 \%$ reduces the encoder complexity to ${ < } 1 / 1 0$ (noting that self-attention is quadratically-complex w.r.t. the token set size).
|
| 70 |
+
|
| 71 |
+
Our decoder is another vanilla ViT on the union of the encoded patch set and a set of mask tokens [31]. Decoder-specific positional embeddings are added to this set [31]. The decoder is designed to be smaller than the encoder [31]. Although the decoder processes the full set, its complexity is smaller than the encoder (e.g., ${ \sim } 1 / 2 0$ per token). In our default setting, the overall autoencoder has a complexity reduction of $7 . 7 \times \nu s .$ . full encoding (more discussions are in Sec. 5.1 and Table 1).
|
| 72 |
+
|
| 73 |
+
The decoder predicts the patches in the pixel space. In principle we can simply predict a full spacetime patch (e.g., $t \times 1 6 \times 1 6 )$ ); in practice, we find it sufficient to predict a single time slice of the patch $( 1 6 \times 1 6 )$ , which keeps the prediction layer’s size manageable. We predict the original pixels or their per-patch normalized values [31] (compared in Table 2b). The training loss function is the mean squared error (MSE) between the prediction and its target, averaged over unknown patches [15].
|
| 74 |
+
|
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The encoder and decoder are agnostic to the spacetime structure of the problem. There is no hierarchy or spacetime factorization, in contrast to the leading architectures [4, 2, 19]. Our method relies on the global self-attention to learn useful knowledge from data, following the spirit of [18].
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# 4 Implementation
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Data pre-processing. For MAE pre-training, our default input size is 16 frames each with $2 2 4 \times 2 2 4$ pixels (i.e., $1 6 { \times } 2 2 4 { \times } 2 2 4 ,$ . The 16 frames are sampled from the raw video with a temporal stride of 4 (i.e., $1 6 \times 4$ sampling in the literature [21]), and the starting frame is randomly sampled. In the spatial domain, we perform random resized cropping [63] with a scale range of [0.5, 1], and random horizontal flipping. We do not apply other data augmentations unless noted.
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Our MAE pre-training is so fast in computation that data loading becomes a new bottleneck that dominates running time in our setup. We adopt repeated sampling [33]1 to alleviate this problem. Each time a raw video is loaded and decompressed, we take multiple (4 by default) samples from it. This reduces the data loading and decompressing time per sample. We note that repeated sampling does not change the number of samples seen; it only influences the orders of the samples seen during training. We always count epochs as “effective epochs”, i.e., how many times each raw video is sampled throughout training.
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Architecture. Our encoder and decoder are the vanilla ViT architectures [18]. We use a temporal patch size of 2 [2, 19, 77] and a spatial patch size of $1 6 \times 1 6$ [18], denoted as $2 \times 1 6 \times 1 6$ . We use the same patch size for ViT-B/L/H [18] for simplicity. For a $1 6 \times 2 2 4 \times 2 2 4$ input, this patch size produces $8 \times 1 4 \times 1 4$ tokens.
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We adopt separable positional embeddings for the encoder. We have two positional embeddings, one for space and the other for time. The spacetime positional embeddings are the sum of them. This separable implementation prevents the size of positional embeddings growing too large in 3D. We use learnable positional embeddings; the sin-cos variant [67] works similarly.
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Settings. Our MAE pre-training configuration mostly follows [31]. We use the AdamW optimizer [43] with a batch size of 512. We evaluate the pre-training quality by end-to-end fine-tuning. The choice of evaluating by fine-tuning (instead of linear probing) follows [3, 31]. Our inference process follows the common practice of multi-view testing [74, 21]: it takes $K$ temporal clips (by default $K { = } 7$ on Kinetics) to cover the video length, and for each clip it takes 3 spatial views to cover the longer spatial axis (denoted as $K { \times } 3 )$ ). The final prediction is the average of all views. The implementation details and hyper-parameters are in the appendix.
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# 5 Experiments
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In Sec. 5.1 and Sec. 5.2 we perform ablation experiments on Kinetics-400 (K400) [35]. We do MAE self-supervised pre-training and then fine-tune the encoder with supervision for evaluation. We report top-1 classification accuracy $( \% )$ on the K400 validation set. In Sec. 5.3 we study more pre-training datasets and downstream tasks.
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Figure 5: MAE pre-training plus fine-tuning is much more accurate and faster than training from scratch. Here the $\mathbf { X }$ -axis is the wall-clock training time (128 A100 GPUs), and the y-axis is the 1-view accuracy on Kinetics-400 validation. The table shows the final accuracy. The model is ViT-L.
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<table><tr><td>MAE w/</td><td>acc.</td><td>FLOPs</td><td>compute</td><td>load+compute</td></tr><tr><td>encoder w/ [M]</td><td>84.3</td><td>627.5 G</td><td>141.1 hr</td><td>147.5 hr</td></tr><tr><td>encoder w/o [M]</td><td>84.4</td><td>81.0G</td><td>24.5 hr</td><td>35.8 hr</td></tr><tr><td>gain</td><td></td><td>7.7×</td><td>5.8×</td><td>4.1×</td></tr></table>
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Table 1: Training time comparison between a dense encoder (w/ [M]) and a sparse encoder (w/o [M]) in MAE. The encoder is ViT-L (1024-d, 24-block); the decoder is our default (512-d, 4-block). With a masking ratio of $90 \%$ , the sparse variant reduces FLOPs by $7 . 7 \times$ . This reduces computation time by $5 . 8 \times$ . In our infra, computation is so fast that data loading becomes a bottleneck, which leads to an actual speedup of $4 . 1 \times$ . Profiling is with synchronized SGD over 16 nodes, each with 8 A100 GPUs and 80 CPU cores. The training length is 800 epochs.
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# 5.1 Performance
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Fig. 5 compares MAE pre-training vs. no pre-training (i.e., training from scratch), using vanilla ViT-L [18]. The from-scratch recipe follows [77] and has $7 1 . 4 \%$ accuracy.2 As a comparison, using MAE pre-training for 800 epochs, the same vanilla ViT-L achieves $8 4 . 4 \%$ accuracy, which has a large increase of $1 3 . 0 \%$ absolute vs. training from scratch. This gap is much larger than that on image recognition tasks ${ \sim } 3 \%$ [31]), suggesting that MAE pre-training is more helpful for video recognition.
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In addition to the accuracy gain, MAE pre-training can reduce the overall training cost, as plotted in Fig. 5. The 800-epoch MAE pre-training only takes 35.8 hours. A short fine-tuning (100 epochs here), which takes 16.3 hours, achieves good accuracy thanks to pre-training. The overall training time can be shorter than training from scratch (e.g., 400 epochs, 65.2 hours), which converges more slowly without pre-training. This shows that MAE is a practical solution to video recognition.
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MAE pre-training is fast because its encoder is only applied on the sparse set of visible patches, without the mask token [M]. We profile the pre-training performance in Table 1. With a masking ratio of $90 \%$ , the sparse encoder reduces the FLOPs (floating-point operations) by $> 1 0 \times$ . After counting the decoder, the sparse design of MAE reduces FLOPs by $7 . 7 \times$ . In our implementation, this reduction should produce a $5 . 8 \times$ computational speedup, if the video data were already pre-processed and loaded in memory. Our speedup ratio is so high that the video pre-processing and loading time becomes a new bottleneck. In our system, the data loading step increases the wall-clock training time from 24.5 hours to 35.8 hours. Nevertheless, this still leads to a significant speedup of $4 . 1 \times$ . 3
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# 5.2 Ablation experiments
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Masking ratio. Fig. 6 shows the influence of the masking ratio jointly with the pre-training length. The ratio of $90 \%$ works the best. The ratio of $9 5 \%$ performs surprisingly well, which can catch up if trained long enough (Fig. 6 left). A higher masking ratio leads to fewer tokens encoded by the encoder; to have a more comprehensive look, we plot the results w.r.t. the total number of encoded tokens (Fig. 6 right). Under this measure, the ratios of $90 \%$ and $9 5 \%$ perform closely.
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The lower masking ratios of $7 5 \%$ and $50 \%$ perform worse, even though the encoder sees more tokens and has higher computation cost. The ratio of $7 5 \%$ is optimal for its image counterpart [31], but not for videos. This observation can be explained by the assumption that video data is more information-redundant.
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Figure 6: Masking ratio. Every point represents a single pre-training and fine-tuning experiment. Left: $\mathbf { X }$ -axis is the epochs (proportional to the number of decoded tokens). Right: $\mathbf { X }$ -axis is the number of encoded tokens.
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<table><tr><td>case</td><td>ratio</td><td>acc.</td></tr><tr><td>agnostic</td><td>90</td><td>84.4</td></tr><tr><td>space-only</td><td>90</td><td>83.5</td></tr><tr><td>time-only</td><td>75</td><td>79.1</td></tr><tr><td>block</td><td>75</td><td>83.2</td></tr></table>
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<table><tr><td>case</td><td>acc.</td></tr><tr><td>pixel (w/o norm)</td><td>83.8</td></tr><tr><td>pixel (w/ norm)</td><td>84.4</td></tr><tr><td>HOG</td><td>84.0</td></tr><tr><td>dVAE token</td><td>83.8</td></tr></table>
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<table><tr><td>case</td><td>acc.</td></tr><tr><td>center crop</td><td>83.9</td></tr><tr><td>rand crop</td><td>84.4</td></tr><tr><td>rand crop (stronger)</td><td>83.4</td></tr><tr><td>rand crop + color jit</td><td>83.8</td></tr></table>
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(a) Mask sampling. See also Fig. 4. Random sampling that is spacetimeagnostic works the best.
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(b) Reconstruction target. Pixels as reconstruction targets work well with no domain knowledge.
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(c) Data augmentation. Strong augmentation is unnecessary.
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<table><tr><td>rep.</td><td>acc.</td><td>speed</td></tr><tr><td>1</td><td>83.7</td><td>1.0×</td></tr><tr><td>2</td><td>84.3</td><td>1.8×</td></tr><tr><td>4</td><td>84.4</td><td>3.0×</td></tr></table>
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<table><tr><td>dim</td><td>acc.</td></tr><tr><td>128</td><td>80.8</td></tr><tr><td>256</td><td>83.1</td></tr><tr><td>512</td><td>84.4</td></tr><tr><td>1024</td><td>83.7</td></tr></table>
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<table><tr><td>blocks</td><td>acc.</td></tr><tr><td>1</td><td>83.2</td></tr><tr><td>2</td><td>83.6</td></tr><tr><td>4</td><td>84.4</td></tr><tr><td>8</td><td>84.3</td></tr></table>
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(d) Repeated sampling. All entries see the same # samples. Data loading overhead is reduced.
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(e) Decoder width. Unlike the image counterpart [31], an overly narrow decoder degrades accuracy noticeably.
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(f) Decoder depth. Unlike the image counterpart [31], an overly shallow decoder degrades accuracy.
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Table 2: Ablation experiments on Kinetics-400. The model is ViT-L, with an input size of $1 6 \times 2 2 4 \times 2 2 4$ and a spacetime patch size of $2 \times 1 6 \times 1 6$ . The pre-training length is 800 epochs. The entries marked in gray are the same, which specify the default settings. This table format follows [31].
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Mask sampling strategy. Our method follows the structure-agnostic random sampling methodology in BERT [15] and MAE [31]. Table 2a reports that this simple solution works the best in our method.
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We compare with other strategies as illustrated in Fig. 4. Space-only sampling, which samples on the 2D spatial axes and broadcasts along the temporal axis, works reasonably well $( 8 3 . 5 \% )$ . Time-only sampling, with a masking ratio of $7 5 \%$ (i.e., keep 2 time steps out of 8), performs poorly $( 7 9 . 1 \% )$ ; if we increase its masking ratio to $8 7 . 5 \%$ (keep 1 out of 8), the accuracy drops further to $7 5 . 4 \%$ . Time-only sampling is related to future/past frame prediction, which can be an overly difficult task in our scenario. Block-wise sampling [3], in its spacetime variant [77], has $8 3 . 2 \%$ accuracy with $7 5 \%$ masking ratio (a higher ratio is worse).
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Reconstruction target. Our method performs decently by reconstructing the original, unmodified pixels $( 8 3 . 8 \%$ , Table 2b). Using per-patch normalized pixels [31] improves by $0 . 6 \%$ . This observation is similar to that of its image counterpart [31]. Using HOG [13] as the target [77] works strongly too.
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The autoencoding nature of our method (i.e., predicting pixels) provides a self-contained solution. In contrast, an extra tokenizer (e.g., dVAE [47, 9]), as is used in [3, 73], may require external data to train and additional domain knowledge to design (e.g., the dVAE used is a ConvNet [37]). Applying the extra dVAE tokenizer to each frame is computationally heavy, which slows down training by $1 . 6 \times$ in our implementation. Our pixel-based method is simpler and performs better (Table 2b).
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Data augmentation. Temporal data can provide natural augmentation, e.g., on view points, motion, deformation, occlusion. These forms of natural augmentation have been incorporated by random temporal sampling. Table 2c compares additional augmentation on the spatial domain. Even using no spatial augmentation (center crop only) works competitively, similar to the observation on images [31]. Random cropping with a mild scale range of [0.5, 1] works well, while stronger cropping (range [0.08, 1], [63]) reduces accuracy; adding color jittering reduces accuracy too, similar to [31].
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Table 3: Influence of pre-training data, evaluated on K400, AVA, and SSv2 as the downstream tasks. The MAE pre-training length is 1600 epochs on K400/600/700 and IG-uncurated. No intermediate fine-tuning is used. The model is ViT-L. †: The K700 training set has $I 3 . 9 k$ duplicated videos with the K400 validation set $( I 9 . 9 k )$ , so it is not legitimate to train on K700 to get K400 results.
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<table><tr><td>pre-train set</td><td># pre-train data</td><td>pre-train method</td><td>K400</td><td>AVA</td><td>SSv2</td></tr><tr><td>-</td><td>1</td><td>none (from scratch)</td><td>71.4</td><td>-</td><td>-</td></tr><tr><td>IN1K</td><td>1.28M</td><td>supervised</td><td>78.6</td><td>17.8</td><td>50.2</td></tr><tr><td>IN1K</td><td>1.28M</td><td>MAE</td><td>82.3</td><td>27.2</td><td>65.6</td></tr><tr><td>K400</td><td>240k</td><td>supervised</td><td>-</td><td>22.2</td><td>55.7</td></tr><tr><td>K400</td><td>240k</td><td>MAE</td><td>84.8</td><td>32.3</td><td>72.1</td></tr><tr><td>K600</td><td>387k</td><td>MAE</td><td>84.9</td><td>33.7</td><td>73.0</td></tr><tr><td>K700</td><td>537k</td><td>MAE</td><td>n/at</td><td>34.2</td><td>73.6</td></tr><tr><td>IG-uncurated</td><td>1M</td><td>MAE</td><td>84.4</td><td>35.1</td><td>73.6</td></tr></table>
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It is practically valuable for self-supervised learning methods to be less dependent on data augmentation. There are a variety of applications in which augmentation is not valid or is hard to induce, e.g., medical imaging, hyper-spectral imaging, remote sensing, geometric data (point cloud, key points, etc.), and their temporal extensions. Our method could be generalized to these cases.
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Repeated sampling. As our method is fast in computation, we adopt repeated sampling [33] to reduce the data loading overhead. Table 2d reports its influence. Using 2 or 4 repetitions increases wall-clock speed by $1 . 8 \times$ or $3 . 0 \times$ , as a loaded and decompressed file is reused multiple times.
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Decoder capacity. Table 2e and 2f report the influence of the decoder width and depth. Using an overly small decoder degrades accuracy by large margins. This is unlike its image counterpart [31], in which a 128-d or 1-block decoder has no degradation if fine-tuning is applied. We hypothesize that the higher-dimensional video data are more complex and thus require higher decoding capacity. On the other hand, our optimal decoder (512-d, 4-block) is still substantially smaller than the encoder (1024-d, 24-block). This is similar to the observation on its image counterpart [31].
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# 5.3 Influence of Data
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Transfer learning ablation. Table 3 studies pre-training on different datasets and transferring to various downstream tasks. The pre-training datasets include ImageNet-1K (IN1K) [14] and Kinetics-400, 600, and 700 [35, 6, 7]. The downstream tasks include K400, AVA [29], and SomethingSomething v2 (SSv2) [27]. We do not perform any intermediate fine-tuning (see appendix), so the comparison here is influenced by the data scale/distribution but not by the number of their labels.
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First we compare with pre-training on the IN1K images. MAE pre-training on $\mathrm { I N } 1 \mathrm { K } ^ { 4 }$ is $3 . 7 \%$ better than IN1K supervised pre-training $7 8 . 6 \%$ to $8 2 . 3 \%$ ); this image-based MAE is even better than K400 supervised pre-training, on both AVA ( $2 1 . 6 \%$ to $2 6 . 3 \%$ ) and SSv2 $5 5 . 7 \%$ to $6 5 . 6 \%$ ).
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MAE pre-training on K400 has massive gains over supervised pre-training on K400: it improves by $1 0 . 1 \%$ on AVA $2 2 . 2 \%$ to $3 2 . 3 \%$ ) and $1 6 . 4 \%$ on SSv2 $5 5 . 7 \%$ to $7 2 . 1 \%$ ). MAE pre-training on K400 videos also substantially outperforms MAE pre-training on IN1K images: it increases by $2 . 5 \%$ on K400 $8 2 . 3 \%$ to $8 4 . 8 \%$ ), $5 . 1 \%$ on AVA $2 7 . 2 \%$ to $3 2 . 3 \%$ ), and $6 . 5 \%$ on SSv2 $6 5 . 6 \%$ to $7 2 . 1 \%$ ), suggesting that MAE pre-training on videos is highly beneficial for these video tasks.
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With more pre-training data (K600/K700) without labels, we observe noticeable improvements on AVA and SSv2: comparing with K400 pre-training, MAE with K700 has an extra gain of $1 . 9 \%$ gain on AVA ( $3 2 . 3 \%$ to $3 4 . 2 \%$ ) and $1 . 5 \%$ on SSv2 ( $7 2 . 1 \%$ to $7 3 . 6 \%$ ).
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Real-world data. We further study MAE pre-training on real-world Instagram videos. We study two sets: (i) Instagram videos curated (IG-curated) [24] with hashtags similar to K400 classes, and (ii) random, uncrated Instagram videos (IG-uncurated). Both sets have 1 million videos.
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Table 3 (last row) reports transfer learning results on AVA and SSv2 using IG-uncurated pre-training. Notably, on AVA, MAE with IG-uncurated is better than MAE with curated Kinetics pre-training (e.g., by $3 . 1 / 1 . 7 / 1 . 1 \%$ over K400/600/700 pre-training); on SSv2, MAE with IG-uncurated is among the best, on par with the K700 counterpart.
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Table 4: Real-world Instagram data for MAE pre-training. We pre-train MAE on each individual set for 200, 400, and 800 epochs. We compare fine-tuning accuracy on K400. The model is ViT-L.
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<table><tr><td>data</td><td>#videos</td><td>200-ep.</td><td>400-ep.</td><td>800-ep.</td></tr><tr><td>K400</td><td>240k</td><td>81.5</td><td>83.3</td><td>84.4</td></tr><tr><td>IG-curated</td><td>240k</td><td>79.0</td><td>81.6</td><td>83.2</td></tr><tr><td>IG-curated</td><td>512k</td><td>81.9</td><td>83.5</td><td>83.9</td></tr><tr><td>IG-curated</td><td>1M</td><td>83.5</td><td>84.1</td><td>84.2</td></tr><tr><td>IG-uncurated</td><td>1M</td><td>83.2</td><td>84.5</td><td>84.4</td></tr></table>
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Table 4 presents more results on the dataset size and training epochs. Pre-training on a 240k subset of IG-curated (the same size as K400) performs worse on K400 classification, which can be caused by the domain shift of data. However, increasing the dataset size of IG-curated to $5 1 2 \mathrm { k }$ and 1M shows good gains: under the same number of pre-training epochs (200 and 400), it can outperform K400 pre-training even when evaluating on K400. IG-uncurated performs similarly well as IG-curated, although the videos are randomly sampled and unrelated to K400 classes. This behavior is not observed on contrastive learning methods for videos: e.g., in [22] it is empirically shown that data curation has a major impact on contrastive learning [32, 10, 28] performance.
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We believe that our exploration with real-world data has encouraging results. It is a more realistic use case of unsupervised learning at scale. We hope this exploration will shed light on future study.
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# 5.4 System-level Comparisons
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We provide system-level comparisons with the leading results on K400, AVA, and SSv2. The detailed tables are in the appendix (Table 7, 8, 9). These results are multifaceted, involving architecture designs, computational complexity, model sizes, input resolution, pre-training data and methods, etc., as we summarize in the tables. Our results are competitive and are close to the leading entries. In particular, our results are based only on vanilla ViT architectures, while the leading methods are hierarchical or specialized for videos. Our results demonstrate the potential of using fewer inductive biases and learning more from data, which is a pursuit of self-supervised learning.
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# 5.5 Video Pre-training for Image Recognition
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Finally, we report preliminary results on video pre-training for image recognition. The usage of vanilla ViT allows to convert to 2D easily: we only “deflate” patch embeddings by summing in time. Using ViT-L pre-trained by MAE on K400 / IG-uncurated, we obtain $8 3 . 7 \bar { \% } / \bar { 8 } 4 . 1 \%$ accuracy on IN1K image classification. This is better than training ViT-L from scratch on IN1K $( 8 2 . 6 \%$ [31]), though lower than MAE pre-training on IN1K $8 5 . 9 \%$ [31]). Considering the large domain gap, we believe this result is decent and its improvement over training from scratch is encouraging. We hope it will motivate the community to explore video pre-training for general visual representation learning.
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# 6 Conclusion
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We have explored a simple extension of MAE [31] to video data. We have drawn several interesting observations. (i) We find that it is possible to learn strong representations with minimal domain knowledge or inductive biases. This follows the spirit of the ViT paper [18]. Similar to BERT [15] and MAE [31], we show that self-supervised learning on videos can be tackled in a conceptually unified framework. (ii) We empirically show that the masking ratio is an important factor for general masked autoencoding methods [69], and its optimal values may depend on the nature of the data (language, images, videos, etc.). (iii) We report encouraging results of pre-training on real-world, uncurated data. It achieves strong performance, close to pre-training on controlled, curated data (e.g., Kinetics). To the best of our knowledge, promising results on uncurated data are rare in the literature.
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In spite of these observations, open problems remain. The scale of data we have explored is orders of magnitudes smaller than the language counterparts [52, 15, 53, 5]. While our method has largely improved the efficiency of self-supervised learning, the high-dimensional video data still present a major challenge for scaling up. We hope our study will provide initial signals for future research.
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# References
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[2] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia ´ Schmid. ViViT: A video vision transformer. In ICCV, 2021.
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[3] Hangbo Bao, Li Dong, and Furu Wei. BEiT: BERT pre-training of image Transformers. arXiv:2106.08254, 2021.
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[4] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is space-time attention all you need for video understanding? In ICML, 2021.
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[5] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In NeurIPS, 2020.
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[6] João Carreira, Eric Noland, Andras Banki-Horvath, Chloe Hillier, and Andrew Zisserman. A short note about Kinetics-600. arXiv:1808.01340, 2018.
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[7] João Carreira, Eric Noland, Chloe Hillier, and Andrew Zisserman. A short note on the Kinetics700 human action dataset. arXiv:1907.06987, 2019.
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[8] João Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In CVPR, 2017.
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[9] Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Heewoo Jun, David Luan, and Ilya Sutskever. Generative pretraining from pixels. In ICML, 2020.
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| 1 |
+
[
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| 2 |
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{
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| 3 |
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"type": "text",
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| 4 |
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"text": "Masked Autoencoders As Spatiotemporal Learners ",
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| 5 |
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| 6 |
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| 14 |
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{
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| 15 |
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"type": "text",
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| 16 |
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"text": "Christoph Feichtenhofer∗ ",
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| 17 |
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| 25 |
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{
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| 26 |
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"type": "text",
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| 27 |
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"text": "Haoqi Fan∗ Yanghao Li Kaiming He ",
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| 28 |
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"type": "text",
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| 38 |
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"text": "Meta AI, FAIR ",
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"text": "https://github.com/facebookresearch/mae_st ",
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"text": "Abstract ",
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"text": "This paper studies a conceptually simple extension of Masked Autoencoders (MAE) [31] to spatiotemporal representation learning from videos. We randomly mask out spacetime patches in videos and learn an autoencoder to reconstruct them in pixels. Interestingly, we show that our MAE method can learn strong representations with almost no inductive bias on spacetime (only except for patch and positional embeddings), and spacetime-agnostic random masking performs the best. We observe that the optimal masking ratio is as high as $90 \\%$ (vs. $7 5 \\%$ on images [31]), supporting the hypothesis that this ratio is related to information redundancy of the data. A high masking ratio leads to a large speedup, e.g., $> 4 \\times$ in wall-clock time or even more. We report competitive results on several challenging video datasets using vanilla Vision Transformers [18]. We observe that MAE can outperform supervised pre-training by large margins. We further report encouraging results of training on real-world, uncurated Instagram data. Our study suggests that the general framework of masked autoencoding (BERT [15], MAE [31], etc.) can be a unified methodology for representation learning with minimal domain knowledge. ",
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"type": "text",
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"text": "1 Introduction ",
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"text": "The deep learning community is experiencing a trend of unifying methodologies for solving problems in different areas, such as language, vision, speech, and more. For architectures, Transformers [67] have been successfully introduced into computer vision [18] and established as a general building block in both language and vision. For self-supervised representation learning, the denoising/masked autoencoding methodology [68] in BERT [15] has been shown effective on learning visual representations from images [31]. Towards unifying methodologies, less domain knowledge (“fewer inductive biases” [18]) is introduced for a specific problem, which urges the models to learn useful knowledge almost purely from data. ",
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"text": "Following this philosophy, we study extending Masked Autoencoders (MAE) [31] to the problem of spatiotemporal representation learning. Our method is simple: we randomly mask out spacetime patches in videos and learn an autoencoder to reconstruct them (Fig. 1). Our method has minimal domain knowledge: the only spacetime-specific inductive bias is on embedding the patches and their positions; all other components are agnostic to the spacetime nature of the problem. In particular, our encoder and decoder are both vanilla Vision Transformers [18] with no factorization or hierarchy, and our random mask sampling is agnostic to the spacetime structures. Our method predicts pixel values and uses no extra problem-specific tokenizer. In a nutshell, our method is simply MAE applied to the set of spacetime patches. Despite minimal inductive biases, our method achieves strong empirical results, suggesting that useful knowledge can be learned from data. ",
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"text": "It is hypothesized in [31] that the masking ratio (i.e., percentage of removed tokens) in masked autoencoding methods is related to the information redundancy of the problems. For example, natural images are more information-redundant than languages and thus the optimal masking ratio is higher (e.g., than BERT [15]). Our observations on video data support this hypothesis. We find that the optimal masking ratio of MAE is $90 \\%$ for videos (Fig. 2), higher than the masking ratio of $7 5 \\%$ for its image counterpart [31]. This can be understood as a consequence of natural video being correlated. ",
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"type": "image",
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"img_path": "images/809bcd02e3155cdbbebd93e99d1a81624624de621b70e10a109d811956dab577.jpg",
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"Figure 1: Masked Autoencoders as spatiotemporal learners. We mask a large subset (e.g., $90 \\%$ ) of random patches in spacetime. An encoder operates on the set of visible patches. A small decoder then processes the full set of encoded patches and mask tokens to reconstruct the input. Except for patch and positional embeddings, neither the encoder, the decoder, nor the masking strategy, has any spatiotemporal inductive bias. "
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"text": "To the extreme, if a video has $T$ identical static frames, randomly sampling $1 / T$ of all spacetime patches would reveal most of the static frame. Because slow motion is more likely than fast motion in natural videos, the masking ratio can be very high as we observe empirically. ",
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"text": "The higher masking ratio leads to a more efficient solution in practice. Following the MAE in [31] that applies the encoder only on visible tokens, a masking ratio of $90 \\%$ reduces the encoder time and memory complexity to ${ < } 1 / 1 0$ . Put together with a small decoder [31], the MAE pre-training can achieve a theoretically $7 . 7 \\times$ reduction in computation vs. encoding all tokens. In fact, the computation reduction is so large that the data loading time becomes a new bottleneck; even so, we record a $4 . 1 \\times$ wall-clock speedup. Such a significant speedup is of great importance for video research that is large-scale and time-consuming. ",
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"text": "We report strong results on a variety of video recognition datasets. Our MAE pre-training greatly improves generalization performance: on Kinetics-400 [35], it increases the accuracy of ViT-Large [18] by absolute $13 \\%$ vs. training from scratch, while it takes less wall-clock training time overall (pre-training plus fine-tuning). Our MAE pre-training can outperform its supervised pre-training counterpart by big margins. Using vanilla ViT [18], our method achieves competitive results with previous state-of-the-art methods that incorporate more domain knowledge. We also report encouraging results using MAE pre-trained on 1 million random, uncurated Instagram videos. These results suggest that self-supervised learning on videos can be tackled in a way similar to its counterparts on language [15] and images [31], under a unified framework. ",
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"text": "2 Related Work ",
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"text": "Denoising autoencoders. Denoising autoencoders (DAE) [68, 69] present a general methodology for learning representations by reconstructing clean signals from corrupted inputs. Masking as a type of noise dates back to at least a decade ago [69]. One of its most successful developments is BERT [15], which is conceptually masked autoencoding on language tokens. ",
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"text": "Denoising/masked autoencoding methods for computer vision have been making continuous progress [50, 9, 18, 31]. A series of recent methods are based on Transformer architectures [67] and are towards a unified solution between vision and language. iGPT [9] pioneers this direction by training Transformers on pixels as tokens. The ViT paper [18] makes a revolutionary step forward by using patches as tokens. It not only establishes strong Transformer architectures for vision tasks, but also explores masked prediction with patches. MAE [31] returns to the basics of the autoencoding concept [68] and draws attention to the decoding aspect. The presence of a meaningful decoder provides more flexibility, e.g., enabling the encoder to operate only on visible patches and leading to a more efficient solution. It empirically shows that a high masking ratio is essential for image tasks [31]. Our study follows this line of research. ",
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"Figure 2: Visualizations on the Kinetics-400 [35] validation set (masking ratio $90 \\%$ ). We show the original video (top), masked video (middle), and MAE output (bottom) for each sample. This model reconstructs the original pixels. The video size is $1 6 { \\times } 2 2 4 { \\times } 2 2 4$ and the spacetime patch size is $2 \\times 1 6 \\times 1 6$ (the temporal patch size of 2 is not visualized here). Each sample has $8 \\times 1 4 \\times 1 4 = 1 5 6 8$ tokens with 156 being visible. For better visualizations, the known patches in the output are from the original input. Fig. 7 shows more examples. "
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"Figure 3: Visualizations of the same pre-trained model in Fig. 2 but with a masking ratio of $95 \\%$ . "
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"text": "Instead of predicting pixels [9, 18, 31, 80], another line of research focuses on the tokenization of the prediction targets [3, 17, 77]. BEiT [3] proposes to use pre-trained dVAE [47, 55] as the reconstruction target. The dVAE tokenizer can be improved by perceptual or adversarial losses [17]. MaskFeat [77] shows that HoG [13] as prediction targets performs strongly. ",
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"text": "Self-supervised learning on videos. The presence of the temporal dimension is a focus of selfsupervised learning on video data. Related topics include temporal coherence (‘slowness’) [79, 25], future prediction [61, 72, 70, 45, 44, 71, 16], object motion [1, 75, 49, 76], temporal ordering [46, 23, 38, 78, 81], spatiotemporal contrast [58, 62, 30, 22, 51, 56], etc. ",
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"text": "Our method also relies on the temporal coherence of videos, but it approaches this goal implicitly. In fact, as our method is largely agnostic to spacetime, the main opportunity for it to make use of the temporal coherence is a higher masking ratio (e.g., $90 \\%$ ), which assumes that videos are more information-redundant than images. ",
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"image_caption": [
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"Figure 4: Mask sampling. (a): Random sampling that is spacetime-agnostic. (b): Space-only random sampling, broadcasted to all time steps (“tube” masking [77]). (c): Time-only random sampling, broadcasted to all spatial locations (“frame” masking [77]). (d): Block-wise sampling [3] in spacetime, removing large regions (“cube” masking [77]). In this illustration, $T \\times H \\times W$ is $8 \\times 1 4 \\times 1 4$ ; green tokens are kept and others are masked out. "
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"text": "There has been growing interest in masking-based methods for self-supervised learning on videos. Previous works focus on tokenizing the prediction targets for the use of videos [65, 73, 77]. Our autoencoding method operates on pixels, which is simpler and requires no extra data or domain knowledge on the tokenizer. Importantly, our method greatly improves the efficiency of learning. The practical speedup is of central importance for video-related research, which is in general larger-scale and more time-consuming. ",
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"text": "Our work is done independently and concurrently with [66] on a related method. ",
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"text": "3 Method ",
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"text": "Our method is a simple extension of MAE [31] to spacetime data (Fig. 1). Our goal is to develop the method under a general and unified framework, with as little domain knowledge as possible. ",
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"text": "Patch embedding. Following the original ViT [18], given a video clip, we divide it into a regular grid of non-overlapping patches in spacetime [4, 2, 19, 77]. The patches are flattened and embedded by linear projection [18]. Positional embeddings [67] are added to the embedded patches. The patch and positional embedding process is the only process that is spacetime-aware. ",
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"text": "Masking. We sample random patches without replacement from the set of embedded patches. This random sampling is agnostic to the spacetime structure (Fig. 4 (a)). This structure-agnostic sampling strategy is analogous to that of BERT in 1D [15] and MAE in 2D [31]. ",
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"text": "It is hypothesized in [31] that the optimal masking ratio is related to the information redundancy of the data. With unstructured random masking, BERT [15] uses a masking ratio of $15 \\%$ for language and MAE [31] uses a ratio of $7 5 \\%$ for images, suggesting that images are more information-redundant than language. Our empirical results on videos support this hypothesis. The optimal masking ratio we observe is $90 \\%$ . This is in line with the common assumption that natural videos are more informationredundant than images because of temporal coherence. Fig. 2 and 3 present our MAE reconstruction results on unseen validation data with a masking ratio of $90 \\%$ and $9 5 \\%$ . ",
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"text": "The spacetime-agnostic sampling can be more effective than structure-aware sampling strategies, e.g., space-only, time-only, or block-wise sampling (Fig. 4 (b-d)). As neighboring patches in space or in time (Fig. 4(b, c)) are coherent, with a very high masking ratio, space-only or time-only sampling may retain less information and yield an overly difficult pre-training task. For example, time-only sampling from 8 frames with a masking ratio of $8 7 . 5 \\%$ means keeping only a single frame, which presents an overly challenging task of predicting the future and past given only one frame. We observe that optimal masking ratios for structure-aware sampling are in general lower. In contrast, the spacetime-agnostic sampling better utilizes the limited number of visible patches and thus allows to use a higher masking ratio. ",
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"text": "Autoencoding. Our encoder is a vanilla ViT [18] applied only on the visible set of embedded patches, following [31]. This design greatly reduces time and memory complexity and leads to a more practical solution. A masking ratio of $90 \\%$ reduces the encoder complexity to ${ < } 1 / 1 0$ (noting that self-attention is quadratically-complex w.r.t. the token set size). ",
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"bbox": [
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"type": "text",
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"text": "Our decoder is another vanilla ViT on the union of the encoded patch set and a set of mask tokens [31]. Decoder-specific positional embeddings are added to this set [31]. The decoder is designed to be smaller than the encoder [31]. Although the decoder processes the full set, its complexity is smaller than the encoder (e.g., ${ \\sim } 1 / 2 0$ per token). In our default setting, the overall autoencoder has a complexity reduction of $7 . 7 \\times \\nu s .$ . full encoding (more discussions are in Sec. 5.1 and Table 1). ",
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"bbox": [
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"type": "text",
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"text": "The decoder predicts the patches in the pixel space. In principle we can simply predict a full spacetime patch (e.g., $t \\times 1 6 \\times 1 6 )$ ); in practice, we find it sufficient to predict a single time slice of the patch $( 1 6 \\times 1 6 )$ , which keeps the prediction layer’s size manageable. We predict the original pixels or their per-patch normalized values [31] (compared in Table 2b). The training loss function is the mean squared error (MSE) between the prediction and its target, averaged over unknown patches [15]. ",
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"bbox": [
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"type": "text",
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"text": "The encoder and decoder are agnostic to the spacetime structure of the problem. There is no hierarchy or spacetime factorization, in contrast to the leading architectures [4, 2, 19]. Our method relies on the global self-attention to learn useful knowledge from data, following the spirit of [18]. ",
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"bbox": [
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{
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"type": "text",
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"text": "4 Implementation ",
|
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"text_level": 1,
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"bbox": [
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"type": "text",
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"text": "Data pre-processing. For MAE pre-training, our default input size is 16 frames each with $2 2 4 \\times 2 2 4$ pixels (i.e., $1 6 { \\times } 2 2 4 { \\times } 2 2 4 ,$ . The 16 frames are sampled from the raw video with a temporal stride of 4 (i.e., $1 6 \\times 4$ sampling in the literature [21]), and the starting frame is randomly sampled. In the spatial domain, we perform random resized cropping [63] with a scale range of [0.5, 1], and random horizontal flipping. We do not apply other data augmentations unless noted. ",
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"bbox": [
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"type": "text",
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"text": "Our MAE pre-training is so fast in computation that data loading becomes a new bottleneck that dominates running time in our setup. We adopt repeated sampling [33]1 to alleviate this problem. Each time a raw video is loaded and decompressed, we take multiple (4 by default) samples from it. This reduces the data loading and decompressing time per sample. We note that repeated sampling does not change the number of samples seen; it only influences the orders of the samples seen during training. We always count epochs as “effective epochs”, i.e., how many times each raw video is sampled throughout training. ",
|
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"bbox": [
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"type": "text",
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"text": "Architecture. Our encoder and decoder are the vanilla ViT architectures [18]. We use a temporal patch size of 2 [2, 19, 77] and a spatial patch size of $1 6 \\times 1 6$ [18], denoted as $2 \\times 1 6 \\times 1 6$ . We use the same patch size for ViT-B/L/H [18] for simplicity. For a $1 6 \\times 2 2 4 \\times 2 2 4$ input, this patch size produces $8 \\times 1 4 \\times 1 4$ tokens. ",
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"type": "text",
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"text": "We adopt separable positional embeddings for the encoder. We have two positional embeddings, one for space and the other for time. The spacetime positional embeddings are the sum of them. This separable implementation prevents the size of positional embeddings growing too large in 3D. We use learnable positional embeddings; the sin-cos variant [67] works similarly. ",
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| 467 |
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"bbox": [
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"page_idx": 4
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| 476 |
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"type": "text",
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| 477 |
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"text": "Settings. Our MAE pre-training configuration mostly follows [31]. We use the AdamW optimizer [43] with a batch size of 512. We evaluate the pre-training quality by end-to-end fine-tuning. The choice of evaluating by fine-tuning (instead of linear probing) follows [3, 31]. Our inference process follows the common practice of multi-view testing [74, 21]: it takes $K$ temporal clips (by default $K { = } 7$ on Kinetics) to cover the video length, and for each clip it takes 3 spatial views to cover the longer spatial axis (denoted as $K { \\times } 3 )$ ). The final prediction is the average of all views. The implementation details and hyper-parameters are in the appendix. ",
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| 478 |
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"bbox": [
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{
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| 487 |
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"type": "text",
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| 488 |
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"text": "5 Experiments ",
|
| 489 |
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"text_level": 1,
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| 490 |
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"bbox": [
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"type": "text",
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"text": "In Sec. 5.1 and Sec. 5.2 we perform ablation experiments on Kinetics-400 (K400) [35]. We do MAE self-supervised pre-training and then fine-tune the encoder with supervision for evaluation. We report top-1 classification accuracy $( \\% )$ on the K400 validation set. In Sec. 5.3 we study more pre-training datasets and downstream tasks. ",
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"type": "image",
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"img_path": "images/6064b7508ff4fe6718183f84e4b61a60b90eac71897e4287ac14ff2564a8b0d8.jpg",
|
| 512 |
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"image_caption": [
|
| 513 |
+
"Figure 5: MAE pre-training plus fine-tuning is much more accurate and faster than training from scratch. Here the $\\mathbf { X }$ -axis is the wall-clock training time (128 A100 GPUs), and the y-axis is the 1-view accuracy on Kinetics-400 validation. The table shows the final accuracy. The model is ViT-L. "
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| 514 |
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],
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| 515 |
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"image_footnote": [],
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| 516 |
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"bbox": [
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{
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| 525 |
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"type": "table",
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| 526 |
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"img_path": "images/d47b8dc04b34455230758031fc824c7f2501bb5ea89300154a2eef35c065d67c.jpg",
|
| 527 |
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"table_caption": [],
|
| 528 |
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"table_footnote": [],
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| 529 |
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"table_body": "<table><tr><td>MAE w/</td><td>acc.</td><td>FLOPs</td><td>compute</td><td>load+compute</td></tr><tr><td>encoder w/ [M]</td><td>84.3</td><td>627.5 G</td><td>141.1 hr</td><td>147.5 hr</td></tr><tr><td>encoder w/o [M]</td><td>84.4</td><td>81.0G</td><td>24.5 hr</td><td>35.8 hr</td></tr><tr><td>gain</td><td></td><td>7.7×</td><td>5.8×</td><td>4.1×</td></tr></table>",
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| 530 |
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"bbox": [
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| 537 |
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| 538 |
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{
|
| 539 |
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"type": "text",
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| 540 |
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"text": "Table 1: Training time comparison between a dense encoder (w/ [M]) and a sparse encoder (w/o [M]) in MAE. The encoder is ViT-L (1024-d, 24-block); the decoder is our default (512-d, 4-block). With a masking ratio of $90 \\%$ , the sparse variant reduces FLOPs by $7 . 7 \\times$ . This reduces computation time by $5 . 8 \\times$ . In our infra, computation is so fast that data loading becomes a bottleneck, which leads to an actual speedup of $4 . 1 \\times$ . Profiling is with synchronized SGD over 16 nodes, each with 8 A100 GPUs and 80 CPU cores. The training length is 800 epochs. ",
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| 541 |
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"bbox": [
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| 549 |
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{
|
| 550 |
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"type": "text",
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| 551 |
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"text": "5.1 Performance ",
|
| 552 |
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"text_level": 1,
|
| 553 |
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| 562 |
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"type": "text",
|
| 563 |
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"text": "Fig. 5 compares MAE pre-training vs. no pre-training (i.e., training from scratch), using vanilla ViT-L [18]. The from-scratch recipe follows [77] and has $7 1 . 4 \\%$ accuracy.2 As a comparison, using MAE pre-training for 800 epochs, the same vanilla ViT-L achieves $8 4 . 4 \\%$ accuracy, which has a large increase of $1 3 . 0 \\%$ absolute vs. training from scratch. This gap is much larger than that on image recognition tasks ${ \\sim } 3 \\%$ [31]), suggesting that MAE pre-training is more helpful for video recognition. ",
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| 564 |
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"bbox": [
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"page_idx": 5
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| 571 |
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| 573 |
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"type": "text",
|
| 574 |
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"text": "In addition to the accuracy gain, MAE pre-training can reduce the overall training cost, as plotted in Fig. 5. The 800-epoch MAE pre-training only takes 35.8 hours. A short fine-tuning (100 epochs here), which takes 16.3 hours, achieves good accuracy thanks to pre-training. The overall training time can be shorter than training from scratch (e.g., 400 epochs, 65.2 hours), which converges more slowly without pre-training. This shows that MAE is a practical solution to video recognition. ",
|
| 575 |
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"bbox": [
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| 582 |
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| 583 |
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{
|
| 584 |
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"type": "text",
|
| 585 |
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"text": "MAE pre-training is fast because its encoder is only applied on the sparse set of visible patches, without the mask token [M]. We profile the pre-training performance in Table 1. With a masking ratio of $90 \\%$ , the sparse encoder reduces the FLOPs (floating-point operations) by $> 1 0 \\times$ . After counting the decoder, the sparse design of MAE reduces FLOPs by $7 . 7 \\times$ . In our implementation, this reduction should produce a $5 . 8 \\times$ computational speedup, if the video data were already pre-processed and loaded in memory. Our speedup ratio is so high that the video pre-processing and loading time becomes a new bottleneck. In our system, the data loading step increases the wall-clock training time from 24.5 hours to 35.8 hours. Nevertheless, this still leads to a significant speedup of $4 . 1 \\times$ . 3 ",
|
| 586 |
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"bbox": [
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| 593 |
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},
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{
|
| 595 |
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"type": "text",
|
| 596 |
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"text": "5.2 Ablation experiments ",
|
| 597 |
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"text_level": 1,
|
| 598 |
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"bbox": [
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{
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| 607 |
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"type": "text",
|
| 608 |
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"text": "Masking ratio. Fig. 6 shows the influence of the masking ratio jointly with the pre-training length. The ratio of $90 \\%$ works the best. The ratio of $9 5 \\%$ performs surprisingly well, which can catch up if trained long enough (Fig. 6 left). A higher masking ratio leads to fewer tokens encoded by the encoder; to have a more comprehensive look, we plot the results w.r.t. the total number of encoded tokens (Fig. 6 right). Under this measure, the ratios of $90 \\%$ and $9 5 \\%$ perform closely. ",
|
| 609 |
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| 618 |
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"type": "text",
|
| 619 |
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"text": "The lower masking ratios of $7 5 \\%$ and $50 \\%$ perform worse, even though the encoder sees more tokens and has higher computation cost. The ratio of $7 5 \\%$ is optimal for its image counterpart [31], but not for videos. This observation can be explained by the assumption that video data is more information-redundant. ",
|
| 620 |
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"bbox": [
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"type": "image",
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"img_path": "images/f2c6b910d837c6e4c5c2bef6f560eccce0d56509c4a6fe5191e553249e994115.jpg",
|
| 631 |
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"image_caption": [
|
| 632 |
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"Figure 6: Masking ratio. Every point represents a single pre-training and fine-tuning experiment. Left: $\\mathbf { X }$ -axis is the epochs (proportional to the number of decoded tokens). Right: $\\mathbf { X }$ -axis is the number of encoded tokens. "
|
| 633 |
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],
|
| 634 |
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"image_footnote": [],
|
| 635 |
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"bbox": [
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| 642 |
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"type": "table",
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| 645 |
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"img_path": "images/8ff4bbbba00f6f5c35be210e10f2f71f199ba77e6ae592260198482b44b93b9b.jpg",
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| 646 |
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"table_caption": [],
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| 647 |
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"table_footnote": [],
|
| 648 |
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"table_body": "<table><tr><td>case</td><td>ratio</td><td>acc.</td></tr><tr><td>agnostic</td><td>90</td><td>84.4</td></tr><tr><td>space-only</td><td>90</td><td>83.5</td></tr><tr><td>time-only</td><td>75</td><td>79.1</td></tr><tr><td>block</td><td>75</td><td>83.2</td></tr></table>",
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"type": "table",
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"img_path": "images/0c453ea1504fb873b5a8e1a2e4b4e9d29ff1ebdd9aa292d7b432f265ea0ab4c9.jpg",
|
| 660 |
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"table_caption": [],
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| 661 |
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"table_footnote": [],
|
| 662 |
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"table_body": "<table><tr><td>case</td><td>acc.</td></tr><tr><td>pixel (w/o norm)</td><td>83.8</td></tr><tr><td>pixel (w/ norm)</td><td>84.4</td></tr><tr><td>HOG</td><td>84.0</td></tr><tr><td>dVAE token</td><td>83.8</td></tr></table>",
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| 672 |
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"type": "table",
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| 673 |
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"img_path": "images/602f673325fd0fb0b3149bd8daa4422913459b3ae157d61d9f9edfd431ab3809.jpg",
|
| 674 |
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"table_caption": [],
|
| 675 |
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"table_footnote": [],
|
| 676 |
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"table_body": "<table><tr><td>case</td><td>acc.</td></tr><tr><td>center crop</td><td>83.9</td></tr><tr><td>rand crop</td><td>84.4</td></tr><tr><td>rand crop (stronger)</td><td>83.4</td></tr><tr><td>rand crop + color jit</td><td>83.8</td></tr></table>",
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| 686 |
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"type": "text",
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| 687 |
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"text": "(a) Mask sampling. See also Fig. 4. Random sampling that is spacetimeagnostic works the best. ",
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| 688 |
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"type": "text",
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| 698 |
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"text": "(b) Reconstruction target. Pixels as reconstruction targets work well with no domain knowledge. ",
|
| 699 |
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"bbox": [
|
| 700 |
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| 701 |
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| 702 |
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| 704 |
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],
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| 705 |
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"page_idx": 6
|
| 706 |
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},
|
| 707 |
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{
|
| 708 |
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"type": "text",
|
| 709 |
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"text": "(c) Data augmentation. Strong augmentation is unnecessary. ",
|
| 710 |
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"bbox": [
|
| 711 |
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| 713 |
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"type": "table",
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| 720 |
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"img_path": "images/6b64eaf8acda880898da64a9ab2d48522bbe16c876ec064a9fac674880ba0757.jpg",
|
| 721 |
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"table_caption": [],
|
| 722 |
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"table_footnote": [],
|
| 723 |
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"table_body": "<table><tr><td>rep.</td><td>acc.</td><td>speed</td></tr><tr><td>1</td><td>83.7</td><td>1.0×</td></tr><tr><td>2</td><td>84.3</td><td>1.8×</td></tr><tr><td>4</td><td>84.4</td><td>3.0×</td></tr></table>",
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| 724 |
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"bbox": [
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"type": "table",
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"img_path": "images/1ba2d0d47c56dca778a4b2b810280ad0169af971d0b78dc432d51e761c535330.jpg",
|
| 735 |
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"table_caption": [],
|
| 736 |
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"table_footnote": [],
|
| 737 |
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"table_body": "<table><tr><td>dim</td><td>acc.</td></tr><tr><td>128</td><td>80.8</td></tr><tr><td>256</td><td>83.1</td></tr><tr><td>512</td><td>84.4</td></tr><tr><td>1024</td><td>83.7</td></tr></table>",
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"type": "table",
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"img_path": "images/7a05ddb2bdc3e7e4c2077af9f6e057ec8744a2d6be69cd3bca1f35b0e7dccbe7.jpg",
|
| 749 |
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"table_caption": [],
|
| 750 |
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"table_footnote": [],
|
| 751 |
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"table_body": "<table><tr><td>blocks</td><td>acc.</td></tr><tr><td>1</td><td>83.2</td></tr><tr><td>2</td><td>83.6</td></tr><tr><td>4</td><td>84.4</td></tr><tr><td>8</td><td>84.3</td></tr></table>",
|
| 752 |
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"bbox": [
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"page_idx": 6
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|
| 760 |
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{
|
| 761 |
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"type": "text",
|
| 762 |
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"text": "(d) Repeated sampling. All entries see the same # samples. Data loading overhead is reduced. ",
|
| 763 |
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"bbox": [
|
| 764 |
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| 765 |
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|
| 770 |
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},
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| 771 |
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{
|
| 772 |
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"type": "text",
|
| 773 |
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"text": "(e) Decoder width. Unlike the image counterpart [31], an overly narrow decoder degrades accuracy noticeably. ",
|
| 774 |
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"bbox": [
|
| 775 |
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|
| 780 |
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"page_idx": 6
|
| 781 |
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},
|
| 782 |
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{
|
| 783 |
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"type": "text",
|
| 784 |
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"text": "(f) Decoder depth. Unlike the image counterpart [31], an overly shallow decoder degrades accuracy. ",
|
| 785 |
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"bbox": [
|
| 786 |
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| 787 |
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|
| 792 |
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},
|
| 793 |
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{
|
| 794 |
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"type": "text",
|
| 795 |
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"text": "Table 2: Ablation experiments on Kinetics-400. The model is ViT-L, with an input size of $1 6 \\times 2 2 4 \\times 2 2 4$ and a spacetime patch size of $2 \\times 1 6 \\times 1 6$ . The pre-training length is 800 epochs. The entries marked in gray are the same, which specify the default settings. This table format follows [31]. ",
|
| 796 |
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"bbox": [
|
| 797 |
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| 798 |
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| 800 |
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| 801 |
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],
|
| 802 |
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"page_idx": 6
|
| 803 |
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},
|
| 804 |
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{
|
| 805 |
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"type": "text",
|
| 806 |
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"text": "Mask sampling strategy. Our method follows the structure-agnostic random sampling methodology in BERT [15] and MAE [31]. Table 2a reports that this simple solution works the best in our method. ",
|
| 807 |
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"bbox": [
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|
| 813 |
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"page_idx": 6
|
| 814 |
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| 815 |
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|
| 816 |
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"type": "text",
|
| 817 |
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"text": "We compare with other strategies as illustrated in Fig. 4. Space-only sampling, which samples on the 2D spatial axes and broadcasts along the temporal axis, works reasonably well $( 8 3 . 5 \\% )$ . Time-only sampling, with a masking ratio of $7 5 \\%$ (i.e., keep 2 time steps out of 8), performs poorly $( 7 9 . 1 \\% )$ ; if we increase its masking ratio to $8 7 . 5 \\%$ (keep 1 out of 8), the accuracy drops further to $7 5 . 4 \\%$ . Time-only sampling is related to future/past frame prediction, which can be an overly difficult task in our scenario. Block-wise sampling [3], in its spacetime variant [77], has $8 3 . 2 \\%$ accuracy with $7 5 \\%$ masking ratio (a higher ratio is worse). ",
|
| 818 |
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"bbox": [
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| 824 |
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"page_idx": 6
|
| 825 |
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},
|
| 826 |
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{
|
| 827 |
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"type": "text",
|
| 828 |
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"text": "Reconstruction target. Our method performs decently by reconstructing the original, unmodified pixels $( 8 3 . 8 \\%$ , Table 2b). Using per-patch normalized pixels [31] improves by $0 . 6 \\%$ . This observation is similar to that of its image counterpart [31]. Using HOG [13] as the target [77] works strongly too. ",
|
| 829 |
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"bbox": [
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|
| 836 |
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|
| 837 |
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{
|
| 838 |
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"type": "text",
|
| 839 |
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"text": "The autoencoding nature of our method (i.e., predicting pixels) provides a self-contained solution. In contrast, an extra tokenizer (e.g., dVAE [47, 9]), as is used in [3, 73], may require external data to train and additional domain knowledge to design (e.g., the dVAE used is a ConvNet [37]). Applying the extra dVAE tokenizer to each frame is computationally heavy, which slows down training by $1 . 6 \\times$ in our implementation. Our pixel-based method is simpler and performs better (Table 2b). ",
|
| 840 |
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"bbox": [
|
| 841 |
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| 842 |
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| 844 |
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| 846 |
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"page_idx": 6
|
| 847 |
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},
|
| 848 |
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{
|
| 849 |
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"type": "text",
|
| 850 |
+
"text": "Data augmentation. Temporal data can provide natural augmentation, e.g., on view points, motion, deformation, occlusion. These forms of natural augmentation have been incorporated by random temporal sampling. Table 2c compares additional augmentation on the spatial domain. Even using no spatial augmentation (center crop only) works competitively, similar to the observation on images [31]. Random cropping with a mild scale range of [0.5, 1] works well, while stronger cropping (range [0.08, 1], [63]) reduces accuracy; adding color jittering reduces accuracy too, similar to [31]. ",
|
| 851 |
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"bbox": [
|
| 852 |
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| 854 |
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| 855 |
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| 857 |
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|
| 858 |
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|
| 859 |
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{
|
| 860 |
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"type": "table",
|
| 861 |
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"img_path": "images/9ef63fbdc5ce26164fe93437fc299de837a6cd9d0d1d3113106fc14376f4a371.jpg",
|
| 862 |
+
"table_caption": [
|
| 863 |
+
"Table 3: Influence of pre-training data, evaluated on K400, AVA, and SSv2 as the downstream tasks. The MAE pre-training length is 1600 epochs on K400/600/700 and IG-uncurated. No intermediate fine-tuning is used. The model is ViT-L. †: The K700 training set has $I 3 . 9 k$ duplicated videos with the K400 validation set $( I 9 . 9 k )$ , so it is not legitimate to train on K700 to get K400 results. "
|
| 864 |
+
],
|
| 865 |
+
"table_footnote": [],
|
| 866 |
+
"table_body": "<table><tr><td>pre-train set</td><td># pre-train data</td><td>pre-train method</td><td>K400</td><td>AVA</td><td>SSv2</td></tr><tr><td>-</td><td>1</td><td>none (from scratch)</td><td>71.4</td><td>-</td><td>-</td></tr><tr><td>IN1K</td><td>1.28M</td><td>supervised</td><td>78.6</td><td>17.8</td><td>50.2</td></tr><tr><td>IN1K</td><td>1.28M</td><td>MAE</td><td>82.3</td><td>27.2</td><td>65.6</td></tr><tr><td>K400</td><td>240k</td><td>supervised</td><td>-</td><td>22.2</td><td>55.7</td></tr><tr><td>K400</td><td>240k</td><td>MAE</td><td>84.8</td><td>32.3</td><td>72.1</td></tr><tr><td>K600</td><td>387k</td><td>MAE</td><td>84.9</td><td>33.7</td><td>73.0</td></tr><tr><td>K700</td><td>537k</td><td>MAE</td><td>n/at</td><td>34.2</td><td>73.6</td></tr><tr><td>IG-uncurated</td><td>1M</td><td>MAE</td><td>84.4</td><td>35.1</td><td>73.6</td></tr></table>",
|
| 867 |
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"bbox": [
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| 868 |
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| 869 |
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| 874 |
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|
| 875 |
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{
|
| 876 |
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"type": "text",
|
| 877 |
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"text": "It is practically valuable for self-supervised learning methods to be less dependent on data augmentation. There are a variety of applications in which augmentation is not valid or is hard to induce, e.g., medical imaging, hyper-spectral imaging, remote sensing, geometric data (point cloud, key points, etc.), and their temporal extensions. Our method could be generalized to these cases. ",
|
| 878 |
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"bbox": [
|
| 879 |
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| 880 |
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333
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|
| 884 |
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"page_idx": 7
|
| 885 |
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|
| 886 |
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{
|
| 887 |
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"type": "text",
|
| 888 |
+
"text": "Repeated sampling. As our method is fast in computation, we adopt repeated sampling [33] to reduce the data loading overhead. Table 2d reports its influence. Using 2 or 4 repetitions increases wall-clock speed by $1 . 8 \\times$ or $3 . 0 \\times$ , as a loaded and decompressed file is reused multiple times. ",
|
| 889 |
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"bbox": [
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| 891 |
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|
| 896 |
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},
|
| 897 |
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{
|
| 898 |
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"type": "text",
|
| 899 |
+
"text": "Decoder capacity. Table 2e and 2f report the influence of the decoder width and depth. Using an overly small decoder degrades accuracy by large margins. This is unlike its image counterpart [31], in which a 128-d or 1-block decoder has no degradation if fine-tuning is applied. We hypothesize that the higher-dimensional video data are more complex and thus require higher decoding capacity. On the other hand, our optimal decoder (512-d, 4-block) is still substantially smaller than the encoder (1024-d, 24-block). This is similar to the observation on its image counterpart [31]. ",
|
| 900 |
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"bbox": [
|
| 901 |
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| 902 |
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| 906 |
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"page_idx": 7
|
| 907 |
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},
|
| 908 |
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{
|
| 909 |
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"type": "text",
|
| 910 |
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"text": "5.3 Influence of Data ",
|
| 911 |
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"text_level": 1,
|
| 912 |
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|
| 920 |
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{
|
| 921 |
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"type": "text",
|
| 922 |
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"text": "Transfer learning ablation. Table 3 studies pre-training on different datasets and transferring to various downstream tasks. The pre-training datasets include ImageNet-1K (IN1K) [14] and Kinetics-400, 600, and 700 [35, 6, 7]. The downstream tasks include K400, AVA [29], and SomethingSomething v2 (SSv2) [27]. We do not perform any intermediate fine-tuning (see appendix), so the comparison here is influenced by the data scale/distribution but not by the number of their labels. ",
|
| 923 |
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|
| 930 |
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|
| 931 |
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{
|
| 932 |
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"type": "text",
|
| 933 |
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"text": "First we compare with pre-training on the IN1K images. MAE pre-training on $\\mathrm { I N } 1 \\mathrm { K } ^ { 4 }$ is $3 . 7 \\%$ better than IN1K supervised pre-training $7 8 . 6 \\%$ to $8 2 . 3 \\%$ ); this image-based MAE is even better than K400 supervised pre-training, on both AVA ( $2 1 . 6 \\%$ to $2 6 . 3 \\%$ ) and SSv2 $5 5 . 7 \\%$ to $6 5 . 6 \\%$ ). ",
|
| 934 |
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|
| 942 |
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{
|
| 943 |
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"type": "text",
|
| 944 |
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"text": "MAE pre-training on K400 has massive gains over supervised pre-training on K400: it improves by $1 0 . 1 \\%$ on AVA $2 2 . 2 \\%$ to $3 2 . 3 \\%$ ) and $1 6 . 4 \\%$ on SSv2 $5 5 . 7 \\%$ to $7 2 . 1 \\%$ ). MAE pre-training on K400 videos also substantially outperforms MAE pre-training on IN1K images: it increases by $2 . 5 \\%$ on K400 $8 2 . 3 \\%$ to $8 4 . 8 \\%$ ), $5 . 1 \\%$ on AVA $2 7 . 2 \\%$ to $3 2 . 3 \\%$ ), and $6 . 5 \\%$ on SSv2 $6 5 . 6 \\%$ to $7 2 . 1 \\%$ ), suggesting that MAE pre-training on videos is highly beneficial for these video tasks. ",
|
| 945 |
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| 947 |
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| 951 |
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|
| 952 |
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|
| 953 |
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{
|
| 954 |
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"type": "text",
|
| 955 |
+
"text": "With more pre-training data (K600/K700) without labels, we observe noticeable improvements on AVA and SSv2: comparing with K400 pre-training, MAE with K700 has an extra gain of $1 . 9 \\%$ gain on AVA ( $3 2 . 3 \\%$ to $3 4 . 2 \\%$ ) and $1 . 5 \\%$ on SSv2 ( $7 2 . 1 \\%$ to $7 3 . 6 \\%$ ). ",
|
| 956 |
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|
| 962 |
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"page_idx": 7
|
| 963 |
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},
|
| 964 |
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{
|
| 965 |
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"type": "text",
|
| 966 |
+
"text": "Real-world data. We further study MAE pre-training on real-world Instagram videos. We study two sets: (i) Instagram videos curated (IG-curated) [24] with hashtags similar to K400 classes, and (ii) random, uncrated Instagram videos (IG-uncurated). Both sets have 1 million videos. ",
|
| 967 |
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|
| 974 |
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},
|
| 975 |
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{
|
| 976 |
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"type": "text",
|
| 977 |
+
"text": "Table 3 (last row) reports transfer learning results on AVA and SSv2 using IG-uncurated pre-training. Notably, on AVA, MAE with IG-uncurated is better than MAE with curated Kinetics pre-training (e.g., by $3 . 1 / 1 . 7 / 1 . 1 \\%$ over K400/600/700 pre-training); on SSv2, MAE with IG-uncurated is among the best, on par with the K700 counterpart. ",
|
| 978 |
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"bbox": [
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| 985 |
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{
|
| 987 |
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"type": "table",
|
| 988 |
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"img_path": "images/3f211eba51de277917e4096ec9d5650bcb7965a8bc4d1e665e7a52d200b56e5e.jpg",
|
| 989 |
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"table_caption": [
|
| 990 |
+
"Table 4: Real-world Instagram data for MAE pre-training. We pre-train MAE on each individual set for 200, 400, and 800 epochs. We compare fine-tuning accuracy on K400. The model is ViT-L. "
|
| 991 |
+
],
|
| 992 |
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"table_footnote": [],
|
| 993 |
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"table_body": "<table><tr><td>data</td><td>#videos</td><td>200-ep.</td><td>400-ep.</td><td>800-ep.</td></tr><tr><td>K400</td><td>240k</td><td>81.5</td><td>83.3</td><td>84.4</td></tr><tr><td>IG-curated</td><td>240k</td><td>79.0</td><td>81.6</td><td>83.2</td></tr><tr><td>IG-curated</td><td>512k</td><td>81.9</td><td>83.5</td><td>83.9</td></tr><tr><td>IG-curated</td><td>1M</td><td>83.5</td><td>84.1</td><td>84.2</td></tr><tr><td>IG-uncurated</td><td>1M</td><td>83.2</td><td>84.5</td><td>84.4</td></tr></table>",
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"text": "Table 4 presents more results on the dataset size and training epochs. Pre-training on a 240k subset of IG-curated (the same size as K400) performs worse on K400 classification, which can be caused by the domain shift of data. However, increasing the dataset size of IG-curated to $5 1 2 \\mathrm { k }$ and 1M shows good gains: under the same number of pre-training epochs (200 and 400), it can outperform K400 pre-training even when evaluating on K400. IG-uncurated performs similarly well as IG-curated, although the videos are randomly sampled and unrelated to K400 classes. This behavior is not observed on contrastive learning methods for videos: e.g., in [22] it is empirically shown that data curation has a major impact on contrastive learning [32, 10, 28] performance. ",
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"text": "We believe that our exploration with real-world data has encouraging results. It is a more realistic use case of unsupervised learning at scale. We hope this exploration will shed light on future study. ",
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"text": "5.4 System-level Comparisons ",
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"text": "We provide system-level comparisons with the leading results on K400, AVA, and SSv2. The detailed tables are in the appendix (Table 7, 8, 9). These results are multifaceted, involving architecture designs, computational complexity, model sizes, input resolution, pre-training data and methods, etc., as we summarize in the tables. Our results are competitive and are close to the leading entries. In particular, our results are based only on vanilla ViT architectures, while the leading methods are hierarchical or specialized for videos. Our results demonstrate the potential of using fewer inductive biases and learning more from data, which is a pursuit of self-supervised learning. ",
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"text": "5.5 Video Pre-training for Image Recognition ",
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"text": "Finally, we report preliminary results on video pre-training for image recognition. The usage of vanilla ViT allows to convert to 2D easily: we only “deflate” patch embeddings by summing in time. Using ViT-L pre-trained by MAE on K400 / IG-uncurated, we obtain $8 3 . 7 \\bar { \\% } / \\bar { 8 } 4 . 1 \\%$ accuracy on IN1K image classification. This is better than training ViT-L from scratch on IN1K $( 8 2 . 6 \\%$ [31]), though lower than MAE pre-training on IN1K $8 5 . 9 \\%$ [31]). Considering the large domain gap, we believe this result is decent and its improvement over training from scratch is encouraging. We hope it will motivate the community to explore video pre-training for general visual representation learning. ",
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"text": "6 Conclusion ",
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"text": "We have explored a simple extension of MAE [31] to video data. We have drawn several interesting observations. (i) We find that it is possible to learn strong representations with minimal domain knowledge or inductive biases. This follows the spirit of the ViT paper [18]. Similar to BERT [15] and MAE [31], we show that self-supervised learning on videos can be tackled in a conceptually unified framework. (ii) We empirically show that the masking ratio is an important factor for general masked autoencoding methods [69], and its optimal values may depend on the nature of the data (language, images, videos, etc.). (iii) We report encouraging results of pre-training on real-world, uncurated data. It achieves strong performance, close to pre-training on controlled, curated data (e.g., Kinetics). To the best of our knowledge, promising results on uncurated data are rare in the literature. ",
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"text": "In spite of these observations, open problems remain. The scale of data we have explored is orders of magnitudes smaller than the language counterparts [52, 15, 53, 5]. While our method has largely improved the efficiency of self-supervised learning, the high-dimensional video data still present a major challenge for scaling up. We hope our study will provide initial signals for future research. ",
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"text": "References ",
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parse/dev/UmvSlP-PyV/UmvSlP-PyV_middle.json
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parse/dev/UmvSlP-PyV/UmvSlP-PyV_model.json
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parse/dev/XsZ5YebcCz/XsZ5YebcCz.md
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| 1 |
+
# Mildly Constrained Evaluation Policy for Offline Reinforcement Learning
|
| 2 |
+
|
| 3 |
+
Anonymous Author(s)
|
| 4 |
+
Affiliation
|
| 5 |
+
Address
|
| 6 |
+
email
|
| 7 |
+
|
| 8 |
+
# Abstract
|
| 9 |
+
|
| 10 |
+
1 Offline reinforcement learning (RL) methodologies enforce constraints on the
|
| 11 |
+
2 policy to adhere closely to the behavior policy, thereby stabilizing value learning
|
| 12 |
+
3 and mitigating the selection of out-of-distribution (OOD) actions during test time.
|
| 13 |
+
4 Conventional approaches apply identical constraints for both value learning and test
|
| 14 |
+
5 time inference. However, our findings indicate that the constraints suitable for value
|
| 15 |
+
6 estimation may in fact be excessively restrictive for action selection during test time.
|
| 16 |
+
7 To address this issue, we propose a Mildly Constrained Evaluation Policy (MCEP)
|
| 17 |
+
8 for test time inference with a more constrained target policy for value estimation.
|
| 18 |
+
9 Since the target policy has been adopted in various prior approaches, MCEP can
|
| 19 |
+
10 be seamlessly integrated with them as a plug-in. We instantiate MCEP based on
|
| 20 |
+
11 TD3-BC [Fujimoto and Gu, 2021] and AWAC [Nair et al., 2020] algorithms. The
|
| 21 |
+
12 empirical results on MuJoCo locomotion tasks show that the MCEP significantly
|
| 22 |
+
13 outperforms the target policy and achieves competitive results to state-of-the-art
|
| 23 |
+
14 offline RL methods. The codes are open-sourced at link.
|
| 24 |
+
|
| 25 |
+
# 15 1 Introduction
|
| 26 |
+
|
| 27 |
+
16 Offline reinforcement learning (RL) extracts a policy from data that is pre-collected by unknown
|
| 28 |
+
17 policies. This setting does not require interactions with the environment thus it is well-suited for tasks
|
| 29 |
+
18 where the interaction is costly or risky. Recently, it has been applied to Natural Language Process
|
| 30 |
+
19 ing [Snell et al., 2022], e-commerce [Degirmenci and Jones] and real-world robotics [Kalashnikov
|
| 31 |
+
20 et al., 2021, Rafailov et al., 2021, Kumar et al., 2022, Shah et al., 2022] etc. Compared to the standard
|
| 32 |
+
21 online setting where the policy gets improved via trial and error, learning with a static offline dataset
|
| 33 |
+
22 raises novel challenges. One challenge is the distributional shift between the training data and the data
|
| 34 |
+
23 encountered during deployment. To attain stable evaluation performance under the distributional shift,
|
| 35 |
+
24 the policy is expected to stay close to the behavior policy. Another challenge is the "extrapolation
|
| 36 |
+
25 error" [Fujimoto et al., 2019, Kumar et al., 2019] that indicates value estimate error on unseen
|
| 37 |
+
26 state-action pairs or Out-Of-Distribution (OOD) actions. Worsely, this error can be amplified with
|
| 38 |
+
27 bootstrapping and cause instability of the training, which is also known as deadly-triad [Van Hasselt
|
| 39 |
+
28 et al., 2018]. Majorities of model-free approaches tackle these challenges by either constraining the
|
| 40 |
+
29 policy to adhere closely to the behavior policy [Wu et al., 2019, Kumar et al., 2019, Fujimoto and Gu,
|
| 41 |
+
30 2021] or regularising the Q to pessimistic estimation for OOD actions [Kumar et al., 2020, Lyu et al.,
|
| 42 |
+
31 2022]. In this work, we focus on policy constraints methods.
|
| 43 |
+
32 Policy constraints methods minimize the disparity between the policy distribution and the behavior
|
| 44 |
+
33 distribution. It is found that policy constraints introduce a tradeoff between stabilizing value estimates
|
| 45 |
+
34 and attaining better performance. While previous approaches focus on developing various constraints
|
| 46 |
+
35 for the learning policy to address this tradeoff, the tradeoff itself is not well understood. Current
|
| 47 |
+
36 solutions have confirmed that an excessively constrained policy enables stable values estimate
|
| 48 |
+
37 but degrades the evaluation performance [Kumar et al., 2019, Singh et al., 2022, Yu et al., 2023].
|
| 49 |
+
38 Nevertheless, it is not clear to what extent this constraint fails to stabilize value learning and to
|
| 50 |
+
39 what extent this constraint leads to a performant evaluation policy. It is essential to investigate these
|
| 51 |
+
40 questions as their answers indicate how well a solution can be found under the tradeoff. However,
|
| 52 |
+
41 the investigation into the latter question is impeded by the existing tradeoff, as it requires tuning the
|
| 53 |
+
42 constraint without influencing the value learning. We circumvent the tradeoff and seek solutions for
|
| 54 |
+
43 this investigation through the critic. For actor-critic methods, [Czarnecki et al., 2019] has shed light
|
| 55 |
+
44 on the potential of distilling a student policy that improves over the teacher using the teacher’s critic.
|
| 56 |
+
45 Inspired by this work, we propose to derive an extra evaluation policy from the critic to avoid solving
|
| 57 |
+
46 the above-mentioned tradeoff. The actor is now called target policy as it is used only to stabilize the
|
| 58 |
+
47 value estimation.
|
| 59 |
+
48 Based on the proposed framework, we empirically investigate the constraint strengths for 1) stabilizing
|
| 60 |
+
49 value learning and 2) better evaluation performance. The results find that a milder constraint improves
|
| 61 |
+
50 the evaluation performance but may fall beyond the constraint space of stable value estimation.
|
| 62 |
+
51 This finding indicates that the optimal evaluation performance may not be found under the tradeoff,
|
| 63 |
+
52 especially when stable value learning is the priority. Consequently, we propose a novel approach of
|
| 64 |
+
53 using a Mildly Constrained Evaluation Policy (MCEP) derived from the critic to avoid solving the
|
| 65 |
+
54 above-mentioned tradeoff and to achieve better evaluation performance.
|
| 66 |
+
55 As the target policy is commonly used in previous approaches, our MCEP can be integrated with
|
| 67 |
+
56 them seamlessly. In this paper, we first validate the finding of [Czarnecki et al., 2019] in the offline
|
| 68 |
+
57 setting by a toy maze experiment, where a constrained policy results in bad evaluation performance
|
| 69 |
+
58 but its off-policy Q estimation indicates an optimal policy. After that, our experiments on D4RL [Fu
|
| 70 |
+
59 et al., 2020] MoJoCo locomotion tasks showed that in most tasks milder constraint achieves better
|
| 71 |
+
60 evaluation performance while more restrictive constraint stabilizes the value estimate. Finally, we
|
| 72 |
+
61 instantiated MCEP on both TD3BC and AWAC algorithms. The empirical results find that the MCEP
|
| 73 |
+
62 significantly outperforms the target policy and achieves competitive results to state-of-the-art offline
|
| 74 |
+
63 RL methods.
|
| 75 |
+
|
| 76 |
+
# 64 2 Related Work
|
| 77 |
+
|
| 78 |
+
65 Policy constraints method (or behavior-regularized policy method) [Wu et al., 2019, Kumar et al.,
|
| 79 |
+
66 2019, Siegel et al., 2020, Fujimoto and Gu, 2021] forces the policy distribution to stay close to the
|
| 80 |
+
67 behavior distribution. Different discrepancy measurements such as KL divergence [Jaques et al., 2019,
|
| 81 |
+
68 Wu et al., 2019], reverse KL divergence Cai et al. [2022] and Maximum Mean Discrepancy [Kumar
|
| 82 |
+
69 et al., 2019] are applied in previous approaches. [Fujimoto and Gu, 2021] simply adds a behavior
|
| 83 |
+
70 cloning (BC) term to the online RL method Twin Delayed DDPG (TD3) [Fujimoto et al., 2018]
|
| 84 |
+
71 and obtains competitive performances in the offline setting. While the above-mentioned methods
|
| 85 |
+
72 calculate the divergence from the data, [Wu et al., 2022] estimates the density of the behavior
|
| 86 |
+
73 distribution using VAE, and thus the divergence can be directly calculated. Except for explicit policy
|
| 87 |
+
74 constraints, implicit constraints are achieved by different approaches. E.g. [Zhou et al., 2021] ensures
|
| 88 |
+
75 the output actions stay in support of the data distribution by using a pre-trained conditional VAE
|
| 89 |
+
76 (CVAE) decoder that maps latent actions to the behavior distribution. In all previous approaches, the
|
| 90 |
+
77 constraints are applied to the learning policy that is queried during policy evaluation and is evaluated
|
| 91 |
+
78 in the environment during deployment. Our approach does not count on this learning policy for the
|
| 92 |
+
79 deployment, instead, it is used as a target policy only for the policy evaluation.
|
| 93 |
+
80 While it is well-known that a policy constraint can be efficient to reduce extrapolation errors, its
|
| 94 |
+
81 drawback is not well-studied yet. [Kumar et al., 2019] reveals a tradeoff between reducing errors in
|
| 95 |
+
82 the Q estimate and reducing the suboptimality bias that degrades the evaluation policy. A constraint is
|
| 96 |
+
83 designed to create a policy space that ensures the resulting policy is under the support of the behavior
|
| 97 |
+
84 distribution for mitigating bootstrapping error. [Singh et al., 2022] discussed the inefficiency of policy
|
| 98 |
+
85 constraints on heteroskedastic dataset where the behavior varies across the state space in a highly
|
| 99 |
+
86 non-uniform manner, as the constraint is state-agnostic. A reweighting method is proposed to achieve
|
| 100 |
+
87 a state-aware distributional constraint to overcome this problem. Our work studies essential questions
|
| 101 |
+
88 about the tradeoff [Kumar et al., 2019] and overcomes this drawback [Singh et al., 2022] by using an
|
| 102 |
+
89 extra evaluation policy.
|
| 103 |
+
90 There are methods that extract an evaluation policy from a learned Q estimate. One-step RL [Brand
|
| 104 |
+
91 fonbrener et al., 2021] first estimates the behavior policy and its Q estimate, which is later used
|
| 105 |
+
92 for extracting the evaluation policy. Although its simplicity, one-step RL is found to perform badly
|
| 106 |
+
93 in long-horizon problems due to a lack of iterative dynamic programming [Kostrikov et al., 2022].
|
| 107 |
+
94 [Kostrikov et al., 2022] proposed Implicity Q learning (IQL) that avoids query of OOD actions
|
| 108 |
+
95 by learning an upper expectile of the state value distribution. No explicit target policy is mod
|
| 109 |
+
96 eled during their Q learning. With the learned Q estimate, an evaluation policy is extracted using
|
| 110 |
+
97 advantage-weighted regression [Wang et al., 2018, Peng et al., 2019]. Our approach has a similar
|
| 111 |
+
98 form of extracting an evaluation from a learned Q estimate. However, one-step RL aims to avoid
|
| 112 |
+
99 distribution shift and iterative error exploitation during iterative dynamic programming. IQL avoids
|
| 113 |
+
100 error exploitation by eliminating OOD action queries and abandoning policy improvement (i.e. the
|
| 114 |
+
101 policy is not trained against the Q estimate). Our work instead tries to address the error exploitation
|
| 115 |
+
102 problem and evaluation performance by using policies of different constraint strengths.
|
| 116 |
+
|
| 117 |
+
# 103 3 Background
|
| 118 |
+
|
| 119 |
+
104 We model the environment as a Markov Decision Process (MDP) $\langle S , A , R , T , p _ { 0 } ( s ) , \gamma , \rangle$ , where $S$ is
|
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105 the state space, $A$ is the action space, $R$ is the reward function, $T ( s ^ { \prime } | s , a )$ is the transition probability,
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106 $p _ { 0 } ( s )$ is initial state distribution and $\gamma$ is a discount factor. In the offline setting, a static dataset
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107 $\mathcal { D } _ { \beta } = \{ ( s , a , r , s ^ { \prime } ) \}$ is pre-collected by a behavior policy $\pi _ { \beta }$ . The goal is to learn a policy $\pi _ { \phi } ( s )$ with
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108 the dataset $\mathcal { D }$ that maximizes the discounted cumulated rewards in the MDP:
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+
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$$
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\phi ^ { * } = \arg \operatorname* { m a x } _ { \phi } \mathbb { E } _ { s _ { 0 } \sim p _ { 0 } ( \cdot ) , a _ { t } \sim \pi _ { \phi } ( s _ { t } ) , s _ { t + 1 } \sim T ( \cdot | s _ { t } , a _ { t } ) } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } , a _ { t } ) ]
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$$
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+
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109 Next, we introduce the general policy constraint method, where the policy $\pi _ { \phi }$ and an off-policy Q
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110 estimate $Q _ { \theta }$ are updated by iteratively taking policy improvement steps and policy evaluation steps,
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111 respectively. The policy evaluation step minimizes the Bellman error:
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+
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$$
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\mathcal { L } _ { Q } ( \theta ) = \mathbb { E } _ { s _ { t } , a _ { t } \sim \mathcal { D } , a _ { t + 1 } \sim \pi _ { \phi } ( s _ { t + 1 } ) } \left[ \left( Q _ { \theta } ( s _ { t } , a _ { t } ) - ( r + \gamma Q _ { \theta ^ { \prime } } ( s _ { t } , a _ { t + 1 } ) ) \right) ^ { 2 } \right] .
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$$
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+
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where the 112 $\theta ^ { \prime }$ is the parameter for a delayed-updated target Q network. The Q value for the next state is 113 calculated with actions $a _ { t + 1 }$ from the learning policy that is updated through the policy improvement 114 step:
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+
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$$
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\begin{array} { r } { \mathcal { L } _ { \pi } ( \phi ) = \mathbb { E } _ { s \sim \mathcal { D } , a \sim \pi _ { \phi } ( s ) } [ - Q _ { \theta } ( s , a ) + w C ( \pi _ { \beta } , \pi _ { \phi } ) ] , } \end{array}
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| 141 |
+
$$
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+
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115 where $C$ is a constraint measuring the discrepancy between the policy distribution $\pi _ { \phi }$ and the behavior
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116 distribution $\pi _ { \beta }$ . The $w \in ( 0 , \infty ]$ is a weighting factor. Different kinds of constraints were used such
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117 as Maximum Mean Discrepancy (MMD), KL divergence, and reverse KL divergence.
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+
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# 118 4 Method
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119 In this section, we first introduce the generic algorithm that can be integrated with any policy
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120 constraints method. Next, we introduce two examples based on popular offline RL methods TD3BC
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121 and AWAC. With a mildly constrained evaluation policy, we name these two instances as TD3BC
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122 with MCEP (TD3BC-MCEP) and AWAC with MCEP (AWAC-MCEP).
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+
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# 123 4.1 Offline RL with mildly constrained evaluation policy
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+
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124 The proposed method is designed for overcoming the tradeoff between a stable policy evaluation and
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125 a performant evaluation policy. In previous constrained policy methods, a restrictive policy constraint
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126 is applied to obtain stable policy evaluation. We retain this benefit but use this policy (actor) $\tilde { \pi }$ as
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127 a target policy only to obtain stable policy evaluation. To achieve better evaluation performance,
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128 we introduce an MCEP $\pi ^ { e }$ that is updated by taking policy improvement steps with the critic $Q _ { \tilde { \pi } }$ .
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129 Different from $\tilde { \pi }$ , $\pi ^ { e }$ does not participate in the policy evaluation procedure. Therefore, a mild policy
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130 constraint can be applied, which helps $\pi ^ { e }$ go further away from the behavior distribution without
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131 influencing the stability of policy evaluation. We demonstrate the policy spaces and policy trajectories
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132 for $\tilde { \pi }$ and $\pi ^ { e }$ in the l.h.s. diagram of Figure 1, where $\pi ^ { e }$ is updated in the wider policy space using $Q _ { \tilde { \pi } }$ .
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133 The overall algorithm is shown as pseudo-codes
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134 (Alg. 1). At each step, the $Q _ { \tilde { \pi } }$ , $\tilde { \pi } _ { \psi }$ and $\pi _ { \phi } ^ { e }$ are
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135 updated iteratively. A policy evaluation step up
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136 dates $Q _ { \tilde { \pi } }$ by minimizing the TD error (line 7),
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137 i.e. the deviation between the approximate $Q$
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138 and its target value. Next, a policy improve
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139 ment step updates $\tilde { \pi } _ { \psi }$ (line 6. These two steps
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140 form the actor-critic algorithm. After that, $\pi _ { \phi } ^ { e }$
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141 is extracted from the $Q _ { \tilde { \pi } }$ , by taking a policy im
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142 provement step with a policy constraint that is
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143 likely milder than the constraint for $\tilde { \pi } _ { \psi }$ (line 7).
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144 Many approaches can be taken to obtain a milder
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+
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| 178 |
+
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Figure 1: Left: diagram depicts policy trajectories for target policy $\tilde { \pi }$ and MCEP $\pi ^ { e }$ . Right: policy evaluation steps to update $Q _ { \tilde { \pi } }$ and policy improvement steps to update $\tilde { \pi }$ and $\pi ^ { e }$ .
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| 180 |
+
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| 181 |
+
# Algorithm 1 MCEP Training
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| 182 |
+
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| 183 |
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1: Hyperparameters: LR $\alpha$ , EMA $\eta$ , $\tilde { w }$ and $w ^ { e }$
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2: Initialize: $\theta , \theta ^ { \prime } , \psi$ , and $\phi$
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3: for $\mathrm { i } { = } 1$ , 2, ..., N do
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4: $\theta \theta - \alpha \mathcal { L } _ { Q } ( \theta )$ (Equation 2)
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| 187 |
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5: $\theta ^ { \prime } ( 1 - \eta ) \dot { \theta } ^ { \prime } + \eta \theta$
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6: $\psi \psi - \alpha \mathcal { L } _ { \tilde { \pi } } ( \psi ; \tilde { w } )$ (Equation 3)
|
| 189 |
+
7: $\phi \phi - \alpha \mathcal { L } _ { \pi ^ { e } } ( \phi ; w ^ { e } )$ (Equation 3)
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| 190 |
+
145 policy constraint. For example, tuning down the weight factor $w ^ { e }$ for the policy constraint term or
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| 191 |
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146 replacing the constraint measurement with a less restrictive one. Note that the constraint for $\pi _ { \phi } ^ { e }$ is
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| 192 |
+
147 necessary (the constraint term should not be dropped) as the $Q _ { \tilde { \pi } }$ has large approximate errors for
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148 state-action pairs that are far from the data distribution.
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| 194 |
+
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| 195 |
+
# 4.2 Two Examples: TD3BC-MCEP and AWAC-MCEP
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+
|
| 197 |
+
150 TD3BC with MCEP TD3BC takes a minimalist modification on the online RL algorithm TD3. To
|
| 198 |
+
151 keep the learned policy to stay close to the behavior distribution, a behavior-cloning term is added to
|
| 199 |
+
152 the policy improvement objective. TD3 learns a deterministic policy therefore the behavior cloning is
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+
153 achieved by directly regressing the data actions. For TD3BC-MCEP, the target policy $\tilde { \pi } _ { \psi }$ has the
|
| 201 |
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154 same policy improvement objective as TD3BC:
|
| 202 |
+
|
| 203 |
+
$$
|
| 204 |
+
\mathcal { L } _ { \tilde { \pi } } ( \psi ) = \mathbb { E } _ { ( s , a ) \sim \mathcal { D } } [ - \tilde { \lambda } Q _ { \theta } ( s , \tilde { \pi } _ { \psi } ( s ) ) + \left( a - \tilde { \pi } _ { \psi } ( s ) \right) ^ { 2 } ] ,
|
| 205 |
+
$$
|
| 206 |
+
|
| 207 |
+
155 where the $\begin{array} { r } { \tilde { \lambda } = \frac { \tilde { \alpha } } { \frac { 1 } { N } \sum _ { s _ { i } , a _ { i } } | Q _ { \theta } ( s _ { i } , a _ { i } ) | } } \end{array}$ is a normalizer for $\mathrm { Q }$ values with a hyper-parameter $\tilde { \alpha }$ : The $Q _ { \theta }$
|
| 208 |
+
156 is updated with the policy evaluation step similar to Eq. 2 using $\tilde { \pi } _ { \psi }$ . The MCEP $\pi _ { \phi } ^ { e }$ is updated by
|
| 209 |
+
157 policy improvement steps with the $Q _ { \tilde { \pi } }$ taking part in. The policy improvement objective function for
|
| 210 |
+
158 $\pi _ { \phi } ^ { e }$ is similar to Eq. 4 but with a higher-value $\alpha ^ { e }$ for the Q-value normalizer $\lambda ^ { e }$ . The final objective
|
| 211 |
+
159 for $\pi _ { \phi } ^ { e }$ is
|
| 212 |
+
|
| 213 |
+
$$
|
| 214 |
+
\mathscr { L } _ { \pi ^ { e } } ( \phi ) = \mathbb { E } _ { ( s , a ) \sim \mathcal { D } } [ - \lambda ^ { e } Q ( s , \pi _ { \phi } ^ { e } ( s ) ) + \left( a - \pi _ { \phi } ^ { e } ( s ) \right) ^ { 2 } ] .
|
| 215 |
+
$$
|
| 216 |
+
|
| 217 |
+
160 AWAC with MCEP AWAC [Nair et al., 2020] is an advantage-weighted behavior cloning method.
|
| 218 |
+
161 As the target policy imitates the actions from the behavior distribution, it stays close to the behavior
|
| 219 |
+
162 distribution during learning. In AWAC-MCEP, the policy evaluation follows the Eq. 2 with the target
|
| 220 |
+
|
| 221 |
+
|
| 222 |
+
Figure 2: Evaluation of policy constraint method on a toy maze MDP 2a. In other figures, the color of a grid represents the state value and arrows indicate the actions from the corresponding policy. 2b shows the optimal value function and one optimal policy. 2c shows a constrained policy trained from the above-mentioned offline data, with its value function calculated by $V _ { \pi } = \mathbb { E } _ { a } Q ( s , \pi ( a | s ) )$ . The policy does not perform well in the low state-value area but its value function is close to the optimal value function. 2d indicates that an optimal policy is recovered by deriving the greedy policy from the off-policy Q estimate (the critic).
|
| 223 |
+
|
| 224 |
+
163 policy $\tilde { \pi } _ { \psi }$ that updates with the following objective:
|
| 225 |
+
|
| 226 |
+
$$
|
| 227 |
+
\mathcal { L } _ { \tilde { \pi } } ( \psi ) = \mathbb { E } _ { s , a \sim \mathcal { D } } \biggl [ - \exp \biggl ( \frac { 1 } { \tilde { \lambda } } A ( s , a ) \biggr ) \log \tilde { \pi } _ { \psi } ( a | s ) \biggr ] ,
|
| 228 |
+
$$
|
| 229 |
+
|
| 230 |
+
164 where the advantage $A ( s , a ) = Q _ { \theta } ( s , a ) - Q _ { \theta } ( s , \tilde { \pi } _ { \psi } ( s ) )$ . This objective function solves an advantage
|
| 231 |
+
165 weighted maximum likelihood. Note that the gradient will not be passed through the advantage term.
|
| 232 |
+
166 As this objective has no policy improvement term, we use the original policy improvement with KL
|
| 233 |
+
167 divergence as the policy constraint and construct the following policy improvement objective:
|
| 234 |
+
|
| 235 |
+
$$
|
| 236 |
+
\begin{array} { r l } & { \mathcal { L } _ { \pi ^ { e } } ( \phi ) = \mathbb { E } _ { s , a \sim \mathcal { D } , \hat { a } \sim \pi ^ { e } ( \cdot | s ) } [ - Q ( s , \hat { a } ) + \lambda ^ { e } D _ { K L } \left( \pi _ { \beta } ( \cdot | s ) | | \pi _ { \phi } ^ { e } ( \cdot | s ) \right) ] } \\ & { \quad \quad \quad = \mathbb { E } _ { s , a \sim \mathcal { D } , \hat { a } \sim \pi ^ { e } ( \cdot | s ) } [ - Q ( s , \hat { a } ) - \lambda ^ { e } \log \pi _ { \phi } ^ { e } ( a | s ) ] , } \end{array}
|
| 237 |
+
$$
|
| 238 |
+
|
| 239 |
+
168 where the weighting factor $\lambda ^ { e }$ is a hyper-parameter. Although the Eq. 6 is derived by solving Eq. 8
|
| 240 |
+
169 in a parametric-policy space, the original problem (Eq. 8) is less restrictive even with $\tilde { \lambda } = \lambda ^ { e }$ as it
|
| 241 |
+
170 includes a $- Q ( s , \pi ^ { e } ( s ) )$ term. This difference means that even with a $\lambda ^ { e } > \tilde { \lambda }$ , the policy constraint
|
| 242 |
+
171 for $\pi ^ { e }$ could still be more relaxed than the policy constraint for $\tilde { \pi }$ .
|
| 243 |
+
|
| 244 |
+
# 172 5 Experiments
|
| 245 |
+
|
| 246 |
+
173 In this section, we set up 4 groups of experiments to illustrate: 1) the policy constraint might degrade
|
| 247 |
+
174 the evaluation performance by forcing the policy to stay close to low-state-value transitions. 2) The
|
| 248 |
+
175 suitable constraint for the final inference could be milder than the ones for safe Q estimates. 3) Our
|
| 249 |
+
176 method shows significant performance improvement compared to the target policy and achieves
|
| 250 |
+
177 competitive results to state-of-the-art offline RL methods on MuJoCo locomotion tasks. 4) the MCEP
|
| 251 |
+
178 generally gains a higher estimate Q compared to the target policy. Additionally, we adopt 2 groups of
|
| 252 |
+
179 ablation studies to verify the benefit of an MCEP and to investigate the constraint strengths of MCEP.
|
| 253 |
+
180 Environments D4RL [Fu et al., 2020] is an offline RL benchmark consisting of many task sets.
|
| 254 |
+
181 Our experiments involve MuJoCo locomotion tasks $( - \nu 2 )$ and two tasks from Adroit $( - \nu O )$ . For
|
| 255 |
+
182 MuJoCo locomotion tasks, we select 4 versions of datasets: data collected by a uniformly-random
|
| 256 |
+
183 agent (random), collected by a medium-performance policy (medium), a $5 0 \% - 5 0 \%$ mixture of the
|
| 257 |
+
184 medium data and the replay buffer during training a medium-performance policy (medium-replay), a
|
| 258 |
+
185 $5 0 \% - 5 0 \%$ mixture of the medium data and expert demonstrations (medium-expert). For Adroit,
|
| 259 |
+
186 we select pen-human and pen-cloned, where the pen-human includes a small number of human
|
| 260 |
+
187 demonstrations, and pen-cloned is a $5 0 \% - 5 0 \%$ mixture of demonstrations and data collected by
|
| 261 |
+
188 rolling out an imitation policy on the demonstrations.
|
| 262 |
+
|
| 263 |
+
# 5.1 Target policy that enables safe Q estimate might be overly constrained
|
| 264 |
+
|
| 265 |
+
190 To investigate the policy constraint under a highly suboptimal dataset, we set up a toy maze MDP that
|
| 266 |
+
191 is similar to the one used in [Kostrikov et al., 2022]. The environment is depicted in Figure 2a, where
|
| 267 |
+
192 the lower left yellow grid is the starting point and the upper left green grid is the terminal state that
|
| 268 |
+
193 gives a reward of 10. Other grids give no reward. Dark blue indicates un-walkable areas. The action
|
| 269 |
+
194 space is defined as 4 direction movements (arrows) and staying where the agent is (filled circles).
|
| 270 |
+
195 There is a $2 5 \%$ probability that a random action is taken instead of the action from the agent. For the
|
| 271 |
+
196 dataset, 99 trajectories are collected by a uniformly random agent and 1 trajectory is collected by an
|
| 272 |
+
197 expert policy. Fig. 2b shows the optimal value function (colors) and one of the optimal policies.
|
| 273 |
+
198 We trained a constrained policy using Eq. 2 and Eq. 8 in an actor-critic manner, where the actor is
|
| 274 |
+
199 constrained by a KL divergence with a weight factor of 1. Figure 2c shows the value function and the
|
| 275 |
+
200 policy. We observe that the learned value function is close to the optimal one in Figure 2b. However,
|
| 276 |
+
201 the policy does not make optimal actions in the lower left areas where the state values are relatively
|
| 277 |
+
202 low. As the policy improvement objective shows a trade-off between the Q and the KL divergence,
|
| 278 |
+
203 when the Q value is low, the KL divergence term will obtain higher priority. i.e. in low-Q-value
|
| 279 |
+
204 areas, the KL divergence takes the majority for the learning objective, which makes the policy stays
|
| 280 |
+
205 closer to the transitions in low-value areas. However, we find that the corresponding value function
|
| 281 |
+
206 indicates an optimal policy. In Figure 2d, we recover a greedy policy underlying the learned critic
|
| 282 |
+
207 that shows an optimal policy. In conclusion, the constraint might degrade the evaluation performance
|
| 283 |
+
208 although the learned critic may indicate a better policy. Although such a trade-off between the Q
|
| 284 |
+
209 term and the KL divergence term can be alleviated in previous work [Fujimoto and Gu, 2021] by
|
| 285 |
+
210 normalizing the Q values, in the next section, we will illustrate that the constraint required to obtain
|
| 286 |
+
211 performant evaluation policy can still cause unstable value estimate.
|
| 287 |
+
|
| 288 |
+
|
| 289 |
+
Figure 4: The training process of TD3BC and AWAC. Left: TD3BC on hopper-medium- $\nu 2$ . Middle: TD3BC on walker2d-medium-replay$\nu 2$ . Right: AWAC on hopper-medium-replay- $\cdot \nu 2$ .
|
| 290 |
+
|
| 291 |
+
|
| 292 |
+
Figure 5: $\alpha$ values in TD3BC for value estimate and test time inference in MuJoCo locomotion tasks.
|
| 293 |
+
|
| 294 |
+
# 212 5.2 Test-time inference requires milder constraints
|
| 295 |
+
|
| 296 |
+
The previous experiment shows that a restrictive constraint might harm the test-time inference, which motivates us to investigate what constraints make better evaluation performance. Firstly, we relax the policy constraint on TD3BC and AWAC by setting up different hyper-parameter values that control the strengths of the policy constraints. For TD3BC, we set $\alpha = \{ 1 , 4 , 1 0 \}$ ([Fujimoto and Gu, 2021] recommends $\alpha = 2 . 5$ ). For AWAC, we set $\lambda = \{ 1 . 0 , 0 . 5 , 0 . 3 , 0 . 1 \}$ ([Nair et al., 2020] recommends $\lambda = 1$ ). Finally, We visualize the evaluation performance and the learned Q estimates.
|
| 297 |
+
|
| 298 |
+
In Figure 4, the left two columns show the training of TD3BC in the hopper-medium- $\nu 2$ and walker2dmedium-replay- $\nu 2$ . In both domains, we found that using a milder constraint by tuning the $\alpha$ from 1 to 4 improves the evaluation performance, which motivates us to expect better performance with $\alpha = 1 0$ . As shown in the lower row, we do observe higher performances in some training steps. However, unstable training is caused by the divergence in value estimate, which indicates the tradeoff between the stable Q estimate and the evaluation performance. The rightmost column shows the training of AWAC in hopper-medium-replay- $\nu 2$ , we observe higher evaluation performance by relaxing the constraint $\lambda > 1 \AA$ ). Although the Q estimate keeps stable during the training in all $\lambda$ values, higher $\lambda$ results in unstable policy performance and causes the performance crash with $\lambda = 0 . 1$ .
|
| 299 |
+
|
| 300 |
+
228 Concluding on all these examples, a milder constraint can potentially improve the performance
|
| 301 |
+
229 but may cause unstable Q estimates or unstable policy performances. As we find that relaxing the
|
| 302 |
+
230 constraint on current methods triggers unstable training, which hinders the investigation of constraints
|
| 303 |
+
231 for better evaluation performance. We instead systematically study the constraint strengths in TD3BC
|
| 304 |
+
232 and TD3BC with evaluation policy (TD3BC-EP).
|
| 305 |
+
|
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<table><tr><td>TaskName</td><td>BC</td><td>CQL IQL</td><td></td><td>TD3BC (ours)</td><td>TD3BC-MCEP AWAC</td><td></td><td>AWAC-MCEP (ours)</td></tr><tr><td>halfcheetah-r</td><td>2.2±0.0</td><td>=</td><td>10±1.7</td><td>11.7±0.4</td><td>28.8±1.0</td><td>9.6±0.4</td><td>34.9±0.8</td></tr><tr><td>hopper-r</td><td>4.7±0.1</td><td></td><td>8.1±0.4</td><td>8.3±0.1</td><td>8.0±0.4</td><td>5.3±0.4</td><td>9.8±0.5</td></tr><tr><td>walker2d-r</td><td>1.6±0.0</td><td></td><td>5.6±0.1</td><td>1.2±0.0</td><td>-0.2±0.1</td><td>5.2±1.0</td><td>3.1±0.4</td></tr><tr><td>halfcheetah-m</td><td>42.4±0.1</td><td>44.0</td><td>47.4±0.1</td><td>48.7±0.2</td><td>55.5±0.4</td><td>45.1±0</td><td>46.6±0</td></tr><tr><td>hopper-m</td><td>54.1±1.1</td><td>58.5</td><td>65±3.6</td><td>56.1±1.2</td><td>91.8±0.9</td><td>58.9±1.9</td><td>91.1±1.5</td></tr><tr><td>walker2d-m</td><td>71±1.7</td><td>72.5</td><td>80.4±1.7</td><td>85.2±0.9</td><td>88.8±0.5</td><td>79.6±1.5</td><td>83.4±0.9</td></tr><tr><td>halfcheetah-m-r</td><td>37.8±1.1</td><td>45.5</td><td>43.2±0.8</td><td>44.8±0.3</td><td>50.6±0.2</td><td>43.3±0.1</td><td>44.9±0.1</td></tr><tr><td>hopper-m-r</td><td>22.5±3.0</td><td>95.0</td><td>74.2±5.3</td><td>55.2±10.8</td><td>100.9±0.4</td><td>64.8±6.2</td><td>101.4±0.2</td></tr><tr><td>walker2d-m-r</td><td>14.4±2.7</td><td>77.2</td><td>62.7±1.9</td><td>50.9±16.1</td><td>86.3±3.2</td><td>84.1±0.6</td><td>84.6±1.3</td></tr><tr><td>halfcheetah-m-e</td><td>62.3±1.5</td><td>91.6</td><td>91.2±1.0</td><td>87.1±1.4</td><td>71.5±3.7</td><td>77.6±2.6</td><td>76.2±5.5</td></tr><tr><td>hopper-m-e</td><td>52.5±1.4</td><td>105.4</td><td>110.2±0.3</td><td>91.7±10.5</td><td>80.1±12.7</td><td>52.4±8.7</td><td>92.5±8.3</td></tr><tr><td>walker2d-m-e</td><td>107±1.1</td><td>108.8</td><td>111.1±0.5</td><td>110.4±0.5</td><td>111.7±0.3</td><td>109.5±0.2</td><td>110.3±0.1</td></tr><tr><td>Average</td><td>39.3</td><td>=</td><td>59.0</td><td>54.2</td><td>64.5</td><td>52.9</td><td>64.9</td></tr><tr><td>pen-human</td><td>76.8±4.8</td><td>37.5</td><td>64.2±10.4</td><td>61.6±11</td><td>58.6±20.8</td><td>34.7±11.8</td><td>23.3 ±5.6</td></tr><tr><td>pen-cloned</td><td>28.5±6.7</td><td>39.2</td><td>32.1±7.5</td><td>49±9.5</td><td>43.4±20.3</td><td>20.8±7.3</td><td>19.0±7.5</td></tr><tr><td>Average</td><td>52.6</td><td>38.3</td><td>48.1</td><td>55.3</td><td>51.0</td><td>27.7</td><td>21.1</td></tr></table>
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Table 1: Normalized episode returns on D4RL benchmark. The results (except for CQL) are means and standard errors from the last step of 5 runs using different random seeds. Performances that are higher than corresponding baselines are underlined and task-wise best performances are bolded.
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We first tune the $\alpha$ for TD3BC to unveil the range for safe Q estimates. Then in TD3BC-EP, we tune the $\alpha ^ { e }$ for the evaluation policy with a fixed $\tilde { \alpha } = 2 . 5$ to approximate the constraint range of better test inference performance (i.e. where the evaluation policy outperforms the target policy). The $\tilde { \alpha } = 2 . 5$ is selected to ensure a stable Q estimate (also the paper-recommended value). The $\alpha \left( \alpha ^ { e } \right)$ is tuned within $\{ 2 . 5 , 5 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7 0 , 8 0 , 9 0 , 1 0 0 \}$ . For each $\alpha \left( \alpha ^ { e } \right)$ , we observe the training of 5 runs with different random seeds. In Figure 5, we visualize these two ranges for each task from MuJoCo locomotion set. The blue area shows $\alpha$ values where the TD3BC Q estimate is stable for all seeds. The edge shows the lowest $\alpha$ value that causes $\mathrm { Q }$ value explosion. The orange area shows the range of $\alpha ^ { e }$ where the learned evaluation policy outperforms the target policy. Its edge (the orange line) shows the lowest $\alpha ^ { e }$ values where the evaluation policy performance is worse than the target policy. For each task, the orange area has a lower bound $\alpha ^ { e } = 2 . 5$ where the evaluation policy shows a similar performance to the target policy.
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Note that $\alpha$ weighs the $\mathrm { Q }$ term and thus a larger $\alpha$ indicates a less restrictive constraint. Comparing the blue area and the orange area, we observe that in 6 out of the 9 tasks, the $\alpha$ for better inference performance is higher than the $\alpha$ that enables safe Q estimates, indicating that test-time inference requires milder constraints. In the next section, we show that with an MCEP, we can achieve much better inference performance without breaking the stable Q estimates.
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# 5.3 Comparison on MuJoCo locomotion and Adroit
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We compare the proposed method to state-of-the-art offline RL methods CQL and IQL, together with our baselines TD3BC and AWAC. Similar hyper-parameters are used for all tasks from the same domain. For our baseline methods (TD3BC and AWAC), we use the hyper-parameter recommended by their papers. TD3BC uses $\alpha = 2 . 5$ for its $\mathrm { Q }$ value normalizer and AWAC uses 1.0 for the advantage value normalizer. In TD3BC-MCEP, the target policy uses $\tilde { \alpha } = 2 . 5$ and the MCEP uses $\alpha ^ { e } = 1 0$ . In AWAC-MCEP, the target policy has $\tilde { \lambda } = \bar { 1 . 0 }$ and the MCEP has $\lambda ^ { e } = 0 . 6$ . The full list of hyper-parameters can be found in the Appendix.
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As is shown in Table 1, we observe that the evaluation policies with a mild constraint significantly outperform their corresponding target policy. TD3BC-MCEP gains progress on all medium and medium-replay datasets. Although the progress is superior, we observe a performance degradation on the medium-expert datasets which indicates an overly relaxed constraint for the evaluation policy. To overcome this imbalance problem, we designed a behavior-cloning normalizer. The results are shown in the Appendix. Nevertheless, the TD3BC-MCEP achieves much better general performance than the
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264 target policy. In the AWAC-MCEP, we observe a consistent performance improvement over the target
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265 policy on most tasks. Additionally, evaluation policies from both TD3BC-MCEP and AWAC-MCEP outperform the CQL and IQL while the target policies have relatively low performances. On Adroit tasks, the best results are obtained by behavioral cloning agent and TD3BC with a high BC weighting factor. Other agents fail to outperform the BC agent. We observe that MCEP does not benefit these tasks where behavior cloning is essential for the evaluation performance.
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# 5.4 Ablation Study
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In this section, we design 2 groups of ablation studies to investigate the effect of the extra evaluation policy and its constraint strengths. Reported results are averaged on 5 runs of different random seeds.
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Performance of the extra evaluation policy. Now, we investigate the performance of the introduced evaluation policy $\pi ^ { e }$ . For TD3BC, we set the parameter $\alpha ~ = ~ \{ 2 . 5 , 1 0 . 0 \}$ . A large $\alpha$ indicates a milder constraint. After that, we train TD3BC-MCEP with $\tilde { \alpha } = 2 . 5$ and $\alpha ^ { e } = 1 0 . 0$ . For AWAC, we trained AWAC with the $\lambda \ = \ \{ 1 . 0 , 0 . 5 \}$ and AWAC-MCEP with $\tilde { \lambda } = 1 . 0$ and $\lambda ^ { e } = 0 . 5$ .
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Figure 6: Left: TD3BC with $\alpha = 2 . 5$ , $\alpha = 1 0$ and TD3BCMCEP with $\tilde { \alpha } = 2 . 5$ , $\alpha ^ { e } = 1 0 $ . Right: AWAC with $\lambda = 1 . 0$ $\lambda = 0 . 5$ and AWAC-MCEP with $\tilde { \lambda } = 1 . 0$ and $\lambda ^ { e } = 0 . 5$ .
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The results are shown in Figure 6. By comparing TD3BC of different $\alpha$ values, we found a milder constraint $\langle \alpha = 1 0 . 0 \rangle$ ) brought performance improvement in hopper tasks but de
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grades the performance in walker2d tasks. The degradation is potentially caused by unstable value estimates (see experiment at section 5.2). Finally, the evaluation policy trained from the critic learned with a target policy with $\alpha = 2 . 5$ achieves the best performance in all three tasks. In AWAC, a lower $\lambda$ value brought policy improvement in hopper tasks but degrades performances in half-cheetah and walker2d tasks. Finally, an evaluation policy obtains the best performances in all tasks.
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94 In conclusion, we observe consistent performance improvement brought by an extra MCEP that
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95 circumvents the tradeoff brought by the constraint.
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Constraint strengths of the evaluation policy. We set up two groups of ablation experiments to investigate the performance of evaluation policy under different constraint strengths. For TD3BC-MCEP, we tune the constraint strength by setting the Q normalizer hyper-parameter. The target policy hyper-parameter is fixed to $\alpha = 2 . 5$ . We pick three strengths for evaluation policy $\alpha ^ { e } = \{ 1 . 0 , \hat { 2 . 5 } , 1 0 . 0 \}$ to create more restrictive, similar, and milder constraints, respectively. For AWACMCEP, the target policy uses $\lambda = 1 . 0$ . However, it is not straightforward to create a similar constraint for the eval
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Figure 7: Left: TD3BC-EP with $\alpha = 1 . 0$ , $\alpha = 2 . 5$ and $\alpha = 1 0 . 0$ . Right: AWAC-EP with $\lambda = 1 . 4$ , $\lambda = 1 . 0$ and $\lambda = 0 . 6$ .
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uation policy as it has a different policy improvement objective. We set 312 $\lambda ^ { e } = \{ 0 . 6 , 1 . 0 , 1 . 4 \}$ to show 313 how performance changes with different constraint strengths.
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The performance improvements over the target policy are shown in Fig. 7. The left column shows a significant performance drop when the evaluation policy has a more restrictive constraint $( \alpha ^ { e } = 1 . 0 $ ) than the target policy. A very close performance is shown when the target policy and the evaluation policy have similar policy constraint strengths $\alpha ^ { e } = 2 . 5$ ). Significant policy improvements are obtained with the target policy having a milder constraint $( \alpha ^ { e } = 1 0 $ ). The right column presents the results of AWAC-MCEP. Generally, the performance in hopper tasks keeps increasing with milder constraints while the half-cheetah and walker2d tasks show performances that increase from $\lambda = 1 . 4$ to $\lambda = 1$ and similar performances between $\lambda = 1$ and $\lambda = 0 . 6$ . Compared to the target policy, the evaluation policy consistently outperforms in half-cheetah and hopper tasks. On the walker2d task, a strong constraint $\lambda = 1 . 4$ ) causes a performance worse than the target policy but milder constraints $( \lambda = \{ 1 , 0 . 6 \} )$ ) obtain similar performance to the target policy.
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Table 2: Proportion of $Q ( s , \pi ( s ) ) \quad { \bar { > } } \quad Q ( s , a )$ for target policies and evalution policies in different tasks.
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<table><tr><td rowspan=1 colspan=1>env</td><td rowspan=1 colspan=1>(%) π(%)</td></tr><tr><td rowspan=1 colspan=1>TD</td><td rowspan=1 colspan=1>TD3BC-MCEP</td></tr><tr><td rowspan=1 colspan=1>wa-me</td><td rowspan=2 colspan=1>69.8 87.266.2 82.771.8 88.789.6 99.0</td></tr><tr><td rowspan=1 colspan=1>wa-mwa-mrwa-r</td></tr><tr><td rowspan=1 colspan=1>AI</td><td rowspan=1 colspan=1>AWAC-MCEP</td></tr><tr><td rowspan=1 colspan=1>ha-me</td><td rowspan=2 colspan=1>63.4 70.864.7 68.3</td></tr><tr><td rowspan=1 colspan=1>ha-m</td></tr><tr><td rowspan=1 colspan=1>ha-mr</td><td rowspan=1 colspan=1>68.6 73.1</td></tr><tr><td rowspan=1 colspan=1>ha-r</td><td rowspan=1 colspan=1>75.3 95.6</td></tr></table>
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Figure 9: The distributions of $Q ( s , \tilde { \pi } ( s ) ) - Q ( s , a )$ and $Q ( s , \pi ^ { e } ( s ) ) -$ $Q \bar { ( } s , a )$ on MuJoCo locomotion tasks. First row: policies of TD3BCMCEP learned in walker2d tasks. Second row: policies of AWAC-MCEP learned in half cheetah tasks. See the Appendix for full results.
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In conclusion, for both algorithms, we observe that on evaluation policy, a milder constraint obtains higher performance than the target policy while a restrictive constraint may harm the performance.
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# 5.5 Estimated Q values for the learned evaluation policies
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To compare the performance of the policies learned in Section 5.3 on the learning objective (maximizing the $\mathrm { \bf Q }$ values), we counted $\mathrm { Q }$ differences between the policy action and the data action $Q ( s , \pi ( s ) ) - Q ( s , a )$ in the training data (visualized in Figure 9). Proportions of data points that show positive differences are listed in Table 2, where we find that on more than half of the data, both the target policy and the MCEP have larger Q estimation than the behavior actions. Additionally, the proportions for the MCEP are higher than the proportions for the target policy in all datasets, indicating that the MCEP is able to move further toward large Q values.
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# 6 Conclusion
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This work focuses on the policy constraints methods where the constraint addresses the tradeoff between stable value estimate and evaluation performance. While to what extent the constraint achieves the best results for each end of this tradeoff remains unknown, we first investigate the constraint strength range for a stable value estimate and for evaluation performance. Our findings indicate that test time inference requires milder constraints that can go beyond the range of stable value estimates. We propose to use an auxiliary mildly constrained evaluation policy to circumvent the above-mentioned tradeoff and derive a performant evaluation policy. The empirical results show that MCEP obtains significant performance improvement compared to target policy and achieves competitive results to state-of-the-art offline RL methods. Our ablation studies show that an auxiliary evaluation policy and a milder policy constraint are essential for the proposed method. Additional empirical analysis demonstrates higher estimated Q values are obtained by the MCEP.
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Limitations. Although the MCEP is able to obtain a better performance, it depends on stable value estimation. Unstable value learning may crash both the target policy and the evaluation policy. While the target policy may recover its performance by iterative policy improvement and policy evaluation, we observe that the evaluation policy may fail to do so. Therefore, a restrictive constrained target policy that stabilizes the value learning is essential for the proposed method.
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"text": "Mildly Constrained Evaluation Policy for Offline Reinforcement Learning ",
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"text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ",
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"text": "Abstract ",
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"text": "1 Offline reinforcement learning (RL) methodologies enforce constraints on the \n2 policy to adhere closely to the behavior policy, thereby stabilizing value learning \n3 and mitigating the selection of out-of-distribution (OOD) actions during test time. \n4 Conventional approaches apply identical constraints for both value learning and test \n5 time inference. However, our findings indicate that the constraints suitable for value \n6 estimation may in fact be excessively restrictive for action selection during test time. \n7 To address this issue, we propose a Mildly Constrained Evaluation Policy (MCEP) \n8 for test time inference with a more constrained target policy for value estimation. \n9 Since the target policy has been adopted in various prior approaches, MCEP can \n10 be seamlessly integrated with them as a plug-in. We instantiate MCEP based on \n11 TD3-BC [Fujimoto and Gu, 2021] and AWAC [Nair et al., 2020] algorithms. The \n12 empirical results on MuJoCo locomotion tasks show that the MCEP significantly \n13 outperforms the target policy and achieves competitive results to state-of-the-art \n14 offline RL methods. The codes are open-sourced at link. ",
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"type": "text",
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"text": "15 1 Introduction ",
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"text": "16 Offline reinforcement learning (RL) extracts a policy from data that is pre-collected by unknown \n17 policies. This setting does not require interactions with the environment thus it is well-suited for tasks \n18 where the interaction is costly or risky. Recently, it has been applied to Natural Language Process \n19 ing [Snell et al., 2022], e-commerce [Degirmenci and Jones] and real-world robotics [Kalashnikov \n20 et al., 2021, Rafailov et al., 2021, Kumar et al., 2022, Shah et al., 2022] etc. Compared to the standard \n21 online setting where the policy gets improved via trial and error, learning with a static offline dataset \n22 raises novel challenges. One challenge is the distributional shift between the training data and the data \n23 encountered during deployment. To attain stable evaluation performance under the distributional shift, \n24 the policy is expected to stay close to the behavior policy. Another challenge is the \"extrapolation \n25 error\" [Fujimoto et al., 2019, Kumar et al., 2019] that indicates value estimate error on unseen \n26 state-action pairs or Out-Of-Distribution (OOD) actions. Worsely, this error can be amplified with \n27 bootstrapping and cause instability of the training, which is also known as deadly-triad [Van Hasselt \n28 et al., 2018]. Majorities of model-free approaches tackle these challenges by either constraining the \n29 policy to adhere closely to the behavior policy [Wu et al., 2019, Kumar et al., 2019, Fujimoto and Gu, \n30 2021] or regularising the Q to pessimistic estimation for OOD actions [Kumar et al., 2020, Lyu et al., \n31 2022]. In this work, we focus on policy constraints methods. \n32 Policy constraints methods minimize the disparity between the policy distribution and the behavior \n33 distribution. It is found that policy constraints introduce a tradeoff between stabilizing value estimates \n34 and attaining better performance. While previous approaches focus on developing various constraints \n35 for the learning policy to address this tradeoff, the tradeoff itself is not well understood. Current \n36 solutions have confirmed that an excessively constrained policy enables stable values estimate \n37 but degrades the evaluation performance [Kumar et al., 2019, Singh et al., 2022, Yu et al., 2023]. \n38 Nevertheless, it is not clear to what extent this constraint fails to stabilize value learning and to \n39 what extent this constraint leads to a performant evaluation policy. It is essential to investigate these \n40 questions as their answers indicate how well a solution can be found under the tradeoff. However, \n41 the investigation into the latter question is impeded by the existing tradeoff, as it requires tuning the \n42 constraint without influencing the value learning. We circumvent the tradeoff and seek solutions for \n43 this investigation through the critic. For actor-critic methods, [Czarnecki et al., 2019] has shed light \n44 on the potential of distilling a student policy that improves over the teacher using the teacher’s critic. \n45 Inspired by this work, we propose to derive an extra evaluation policy from the critic to avoid solving \n46 the above-mentioned tradeoff. The actor is now called target policy as it is used only to stabilize the \n47 value estimation. \n48 Based on the proposed framework, we empirically investigate the constraint strengths for 1) stabilizing \n49 value learning and 2) better evaluation performance. The results find that a milder constraint improves \n50 the evaluation performance but may fall beyond the constraint space of stable value estimation. \n51 This finding indicates that the optimal evaluation performance may not be found under the tradeoff, \n52 especially when stable value learning is the priority. Consequently, we propose a novel approach of \n53 using a Mildly Constrained Evaluation Policy (MCEP) derived from the critic to avoid solving the \n54 above-mentioned tradeoff and to achieve better evaluation performance. \n55 As the target policy is commonly used in previous approaches, our MCEP can be integrated with \n56 them seamlessly. In this paper, we first validate the finding of [Czarnecki et al., 2019] in the offline \n57 setting by a toy maze experiment, where a constrained policy results in bad evaluation performance \n58 but its off-policy Q estimation indicates an optimal policy. After that, our experiments on D4RL [Fu \n59 et al., 2020] MoJoCo locomotion tasks showed that in most tasks milder constraint achieves better \n60 evaluation performance while more restrictive constraint stabilizes the value estimate. Finally, we \n61 instantiated MCEP on both TD3BC and AWAC algorithms. The empirical results find that the MCEP \n62 significantly outperforms the target policy and achieves competitive results to state-of-the-art offline \n63 RL methods. ",
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"text": "64 2 Related Work ",
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"text": "65 Policy constraints method (or behavior-regularized policy method) [Wu et al., 2019, Kumar et al., \n66 2019, Siegel et al., 2020, Fujimoto and Gu, 2021] forces the policy distribution to stay close to the \n67 behavior distribution. Different discrepancy measurements such as KL divergence [Jaques et al., 2019, \n68 Wu et al., 2019], reverse KL divergence Cai et al. [2022] and Maximum Mean Discrepancy [Kumar \n69 et al., 2019] are applied in previous approaches. [Fujimoto and Gu, 2021] simply adds a behavior \n70 cloning (BC) term to the online RL method Twin Delayed DDPG (TD3) [Fujimoto et al., 2018] \n71 and obtains competitive performances in the offline setting. While the above-mentioned methods \n72 calculate the divergence from the data, [Wu et al., 2022] estimates the density of the behavior \n73 distribution using VAE, and thus the divergence can be directly calculated. Except for explicit policy \n74 constraints, implicit constraints are achieved by different approaches. E.g. [Zhou et al., 2021] ensures \n75 the output actions stay in support of the data distribution by using a pre-trained conditional VAE \n76 (CVAE) decoder that maps latent actions to the behavior distribution. In all previous approaches, the \n77 constraints are applied to the learning policy that is queried during policy evaluation and is evaluated \n78 in the environment during deployment. Our approach does not count on this learning policy for the \n79 deployment, instead, it is used as a target policy only for the policy evaluation. \n80 While it is well-known that a policy constraint can be efficient to reduce extrapolation errors, its \n81 drawback is not well-studied yet. [Kumar et al., 2019] reveals a tradeoff between reducing errors in \n82 the Q estimate and reducing the suboptimality bias that degrades the evaluation policy. A constraint is \n83 designed to create a policy space that ensures the resulting policy is under the support of the behavior \n84 distribution for mitigating bootstrapping error. [Singh et al., 2022] discussed the inefficiency of policy \n85 constraints on heteroskedastic dataset where the behavior varies across the state space in a highly \n86 non-uniform manner, as the constraint is state-agnostic. A reweighting method is proposed to achieve \n87 a state-aware distributional constraint to overcome this problem. Our work studies essential questions \n88 about the tradeoff [Kumar et al., 2019] and overcomes this drawback [Singh et al., 2022] by using an \n89 extra evaluation policy. \n90 There are methods that extract an evaluation policy from a learned Q estimate. One-step RL [Brand \n91 fonbrener et al., 2021] first estimates the behavior policy and its Q estimate, which is later used \n92 for extracting the evaluation policy. Although its simplicity, one-step RL is found to perform badly \n93 in long-horizon problems due to a lack of iterative dynamic programming [Kostrikov et al., 2022]. \n94 [Kostrikov et al., 2022] proposed Implicity Q learning (IQL) that avoids query of OOD actions \n95 by learning an upper expectile of the state value distribution. No explicit target policy is mod \n96 eled during their Q learning. With the learned Q estimate, an evaluation policy is extracted using \n97 advantage-weighted regression [Wang et al., 2018, Peng et al., 2019]. Our approach has a similar \n98 form of extracting an evaluation from a learned Q estimate. However, one-step RL aims to avoid \n99 distribution shift and iterative error exploitation during iterative dynamic programming. IQL avoids \n100 error exploitation by eliminating OOD action queries and abandoning policy improvement (i.e. the \n101 policy is not trained against the Q estimate). Our work instead tries to address the error exploitation \n102 problem and evaluation performance by using policies of different constraint strengths. ",
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"text": "103 3 Background ",
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"text": "104 We model the environment as a Markov Decision Process (MDP) $\\langle S , A , R , T , p _ { 0 } ( s ) , \\gamma , \\rangle$ , where $S$ is \n105 the state space, $A$ is the action space, $R$ is the reward function, $T ( s ^ { \\prime } | s , a )$ is the transition probability, \n106 $p _ { 0 } ( s )$ is initial state distribution and $\\gamma$ is a discount factor. In the offline setting, a static dataset \n107 $\\mathcal { D } _ { \\beta } = \\{ ( s , a , r , s ^ { \\prime } ) \\}$ is pre-collected by a behavior policy $\\pi _ { \\beta }$ . The goal is to learn a policy $\\pi _ { \\phi } ( s )$ with \n108 the dataset $\\mathcal { D }$ that maximizes the discounted cumulated rewards in the MDP: ",
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"text": "$$\n\\phi ^ { * } = \\arg \\operatorname* { m a x } _ { \\phi } \\mathbb { E } _ { s _ { 0 } \\sim p _ { 0 } ( \\cdot ) , a _ { t } \\sim \\pi _ { \\phi } ( s _ { t } ) , s _ { t + 1 } \\sim T ( \\cdot | s _ { t } , a _ { t } ) } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( s _ { t } , a _ { t } ) ]\n$$",
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"text": "109 Next, we introduce the general policy constraint method, where the policy $\\pi _ { \\phi }$ and an off-policy Q \n110 estimate $Q _ { \\theta }$ are updated by iteratively taking policy improvement steps and policy evaluation steps, \n111 respectively. The policy evaluation step minimizes the Bellman error: ",
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"text": "$$\n\\mathcal { L } _ { Q } ( \\theta ) = \\mathbb { E } _ { s _ { t } , a _ { t } \\sim \\mathcal { D } , a _ { t + 1 } \\sim \\pi _ { \\phi } ( s _ { t + 1 } ) } \\left[ \\left( Q _ { \\theta } ( s _ { t } , a _ { t } ) - ( r + \\gamma Q _ { \\theta ^ { \\prime } } ( s _ { t } , a _ { t + 1 } ) ) \\right) ^ { 2 } \\right] .\n$$",
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"text": "where the 112 $\\theta ^ { \\prime }$ is the parameter for a delayed-updated target Q network. The Q value for the next state is 113 calculated with actions $a _ { t + 1 }$ from the learning policy that is updated through the policy improvement 114 step: ",
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"text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\pi } ( \\phi ) = \\mathbb { E } _ { s \\sim \\mathcal { D } , a \\sim \\pi _ { \\phi } ( s ) } [ - Q _ { \\theta } ( s , a ) + w C ( \\pi _ { \\beta } , \\pi _ { \\phi } ) ] , } \\end{array}\n$$",
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"text": "115 where $C$ is a constraint measuring the discrepancy between the policy distribution $\\pi _ { \\phi }$ and the behavior \n116 distribution $\\pi _ { \\beta }$ . The $w \\in ( 0 , \\infty ]$ is a weighting factor. Different kinds of constraints were used such \n117 as Maximum Mean Discrepancy (MMD), KL divergence, and reverse KL divergence. ",
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"text": "118 4 Method ",
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"text": "119 In this section, we first introduce the generic algorithm that can be integrated with any policy \n120 constraints method. Next, we introduce two examples based on popular offline RL methods TD3BC \n121 and AWAC. With a mildly constrained evaluation policy, we name these two instances as TD3BC \n122 with MCEP (TD3BC-MCEP) and AWAC with MCEP (AWAC-MCEP). ",
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"text": "123 4.1 Offline RL with mildly constrained evaluation policy ",
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"type": "text",
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"text": "124 The proposed method is designed for overcoming the tradeoff between a stable policy evaluation and \n125 a performant evaluation policy. In previous constrained policy methods, a restrictive policy constraint \n126 is applied to obtain stable policy evaluation. We retain this benefit but use this policy (actor) $\\tilde { \\pi }$ as \n127 a target policy only to obtain stable policy evaluation. To achieve better evaluation performance, \n128 we introduce an MCEP $\\pi ^ { e }$ that is updated by taking policy improvement steps with the critic $Q _ { \\tilde { \\pi } }$ . \n129 Different from $\\tilde { \\pi }$ , $\\pi ^ { e }$ does not participate in the policy evaluation procedure. Therefore, a mild policy \n130 constraint can be applied, which helps $\\pi ^ { e }$ go further away from the behavior distribution without \n131 influencing the stability of policy evaluation. We demonstrate the policy spaces and policy trajectories \n132 for $\\tilde { \\pi }$ and $\\pi ^ { e }$ in the l.h.s. diagram of Figure 1, where $\\pi ^ { e }$ is updated in the wider policy space using $Q _ { \\tilde { \\pi } }$ . \n133 The overall algorithm is shown as pseudo-codes \n134 (Alg. 1). At each step, the $Q _ { \\tilde { \\pi } }$ , $\\tilde { \\pi } _ { \\psi }$ and $\\pi _ { \\phi } ^ { e }$ are \n135 updated iteratively. A policy evaluation step up \n136 dates $Q _ { \\tilde { \\pi } }$ by minimizing the TD error (line 7), \n137 i.e. the deviation between the approximate $Q$ \n138 and its target value. Next, a policy improve \n139 ment step updates $\\tilde { \\pi } _ { \\psi }$ (line 6. These two steps \n140 form the actor-critic algorithm. After that, $\\pi _ { \\phi } ^ { e }$ \n141 is extracted from the $Q _ { \\tilde { \\pi } }$ , by taking a policy im \n142 provement step with a policy constraint that is \n143 likely milder than the constraint for $\\tilde { \\pi } _ { \\psi }$ (line 7). \n144 Many approaches can be taken to obtain a milder ",
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"image_caption": [
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"Figure 1: Left: diagram depicts policy trajectories for target policy $\\tilde { \\pi }$ and MCEP $\\pi ^ { e }$ . Right: policy evaluation steps to update $Q _ { \\tilde { \\pi } }$ and policy improvement steps to update $\\tilde { \\pi }$ and $\\pi ^ { e }$ . "
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"type": "text",
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"text": "Algorithm 1 MCEP Training ",
|
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"text_level": 1,
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"type": "text",
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"text": "1: Hyperparameters: LR $\\alpha$ , EMA $\\eta$ , $\\tilde { w }$ and $w ^ { e }$ \n2: Initialize: $\\theta , \\theta ^ { \\prime } , \\psi$ , and $\\phi$ \n3: for $\\mathrm { i } { = } 1$ , 2, ..., N do \n4: $\\theta \\theta - \\alpha \\mathcal { L } _ { Q } ( \\theta )$ (Equation 2) \n5: $\\theta ^ { \\prime } ( 1 - \\eta ) \\dot { \\theta } ^ { \\prime } + \\eta \\theta$ \n6: $\\psi \\psi - \\alpha \\mathcal { L } _ { \\tilde { \\pi } } ( \\psi ; \\tilde { w } )$ (Equation 3) \n7: $\\phi \\phi - \\alpha \\mathcal { L } _ { \\pi ^ { e } } ( \\phi ; w ^ { e } )$ (Equation 3) \n145 policy constraint. For example, tuning down the weight factor $w ^ { e }$ for the policy constraint term or \n146 replacing the constraint measurement with a less restrictive one. Note that the constraint for $\\pi _ { \\phi } ^ { e }$ is \n147 necessary (the constraint term should not be dropped) as the $Q _ { \\tilde { \\pi } }$ has large approximate errors for \n148 state-action pairs that are far from the data distribution. ",
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"text": "",
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"type": "text",
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"text": "4.2 Two Examples: TD3BC-MCEP and AWAC-MCEP ",
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"text": "150 TD3BC with MCEP TD3BC takes a minimalist modification on the online RL algorithm TD3. To \n151 keep the learned policy to stay close to the behavior distribution, a behavior-cloning term is added to \n152 the policy improvement objective. TD3 learns a deterministic policy therefore the behavior cloning is \n153 achieved by directly regressing the data actions. For TD3BC-MCEP, the target policy $\\tilde { \\pi } _ { \\psi }$ has the \n154 same policy improvement objective as TD3BC: ",
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"type": "equation",
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"text": "$$\n\\mathcal { L } _ { \\tilde { \\pi } } ( \\psi ) = \\mathbb { E } _ { ( s , a ) \\sim \\mathcal { D } } [ - \\tilde { \\lambda } Q _ { \\theta } ( s , \\tilde { \\pi } _ { \\psi } ( s ) ) + \\left( a - \\tilde { \\pi } _ { \\psi } ( s ) \\right) ^ { 2 } ] ,\n$$",
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"text": "155 where the $\\begin{array} { r } { \\tilde { \\lambda } = \\frac { \\tilde { \\alpha } } { \\frac { 1 } { N } \\sum _ { s _ { i } , a _ { i } } | Q _ { \\theta } ( s _ { i } , a _ { i } ) | } } \\end{array}$ is a normalizer for $\\mathrm { Q }$ values with a hyper-parameter $\\tilde { \\alpha }$ : The $Q _ { \\theta }$ \n156 is updated with the policy evaluation step similar to Eq. 2 using $\\tilde { \\pi } _ { \\psi }$ . The MCEP $\\pi _ { \\phi } ^ { e }$ is updated by \n157 policy improvement steps with the $Q _ { \\tilde { \\pi } }$ taking part in. The policy improvement objective function for \n158 $\\pi _ { \\phi } ^ { e }$ is similar to Eq. 4 but with a higher-value $\\alpha ^ { e }$ for the Q-value normalizer $\\lambda ^ { e }$ . The final objective \n159 for $\\pi _ { \\phi } ^ { e }$ is ",
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"type": "equation",
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"text": "$$\n\\mathscr { L } _ { \\pi ^ { e } } ( \\phi ) = \\mathbb { E } _ { ( s , a ) \\sim \\mathcal { D } } [ - \\lambda ^ { e } Q ( s , \\pi _ { \\phi } ^ { e } ( s ) ) + \\left( a - \\pi _ { \\phi } ^ { e } ( s ) \\right) ^ { 2 } ] .\n$$",
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"type": "text",
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"text": "160 AWAC with MCEP AWAC [Nair et al., 2020] is an advantage-weighted behavior cloning method. \n161 As the target policy imitates the actions from the behavior distribution, it stays close to the behavior \n162 distribution during learning. In AWAC-MCEP, the policy evaluation follows the Eq. 2 with the target ",
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"img_path": "images/62e4c4d5e512d3d89a00f40c94b7160a7fee037ff757e461b9600a8e5c1d6909.jpg",
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"image_caption": [
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"Figure 2: Evaluation of policy constraint method on a toy maze MDP 2a. In other figures, the color of a grid represents the state value and arrows indicate the actions from the corresponding policy. 2b shows the optimal value function and one optimal policy. 2c shows a constrained policy trained from the above-mentioned offline data, with its value function calculated by $V _ { \\pi } = \\mathbb { E } _ { a } Q ( s , \\pi ( a | s ) )$ . The policy does not perform well in the low state-value area but its value function is close to the optimal value function. 2d indicates that an optimal policy is recovered by deriving the greedy policy from the off-policy Q estimate (the critic). "
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| 439 |
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"type": "text",
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"text": "163 policy $\\tilde { \\pi } _ { \\psi }$ that updates with the following objective: ",
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"img_path": "images/97ef96a1ed1d8d969d4790f87ed6e9e5faf24d42e3344ed8283b0aca00a0ec6c.jpg",
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"text": "$$\n\\mathcal { L } _ { \\tilde { \\pi } } ( \\psi ) = \\mathbb { E } _ { s , a \\sim \\mathcal { D } } \\biggl [ - \\exp \\biggl ( \\frac { 1 } { \\tilde { \\lambda } } A ( s , a ) \\biggr ) \\log \\tilde { \\pi } _ { \\psi } ( a | s ) \\biggr ] ,\n$$",
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"text": "164 where the advantage $A ( s , a ) = Q _ { \\theta } ( s , a ) - Q _ { \\theta } ( s , \\tilde { \\pi } _ { \\psi } ( s ) )$ . This objective function solves an advantage \n165 weighted maximum likelihood. Note that the gradient will not be passed through the advantage term. \n166 As this objective has no policy improvement term, we use the original policy improvement with KL \n167 divergence as the policy constraint and construct the following policy improvement objective: ",
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"text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\pi ^ { e } } ( \\phi ) = \\mathbb { E } _ { s , a \\sim \\mathcal { D } , \\hat { a } \\sim \\pi ^ { e } ( \\cdot | s ) } [ - Q ( s , \\hat { a } ) + \\lambda ^ { e } D _ { K L } \\left( \\pi _ { \\beta } ( \\cdot | s ) | | \\pi _ { \\phi } ^ { e } ( \\cdot | s ) \\right) ] } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { s , a \\sim \\mathcal { D } , \\hat { a } \\sim \\pi ^ { e } ( \\cdot | s ) } [ - Q ( s , \\hat { a } ) - \\lambda ^ { e } \\log \\pi _ { \\phi } ^ { e } ( a | s ) ] , } \\end{array}\n$$",
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"text": "168 where the weighting factor $\\lambda ^ { e }$ is a hyper-parameter. Although the Eq. 6 is derived by solving Eq. 8 \n169 in a parametric-policy space, the original problem (Eq. 8) is less restrictive even with $\\tilde { \\lambda } = \\lambda ^ { e }$ as it \n170 includes a $- Q ( s , \\pi ^ { e } ( s ) )$ term. This difference means that even with a $\\lambda ^ { e } > \\tilde { \\lambda }$ , the policy constraint \n171 for $\\pi ^ { e }$ could still be more relaxed than the policy constraint for $\\tilde { \\pi }$ . ",
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"type": "text",
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"text": "172 5 Experiments ",
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"text": "173 In this section, we set up 4 groups of experiments to illustrate: 1) the policy constraint might degrade \n174 the evaluation performance by forcing the policy to stay close to low-state-value transitions. 2) The \n175 suitable constraint for the final inference could be milder than the ones for safe Q estimates. 3) Our \n176 method shows significant performance improvement compared to the target policy and achieves \n177 competitive results to state-of-the-art offline RL methods on MuJoCo locomotion tasks. 4) the MCEP \n178 generally gains a higher estimate Q compared to the target policy. Additionally, we adopt 2 groups of \n179 ablation studies to verify the benefit of an MCEP and to investigate the constraint strengths of MCEP. \n180 Environments D4RL [Fu et al., 2020] is an offline RL benchmark consisting of many task sets. \n181 Our experiments involve MuJoCo locomotion tasks $( - \\nu 2 )$ and two tasks from Adroit $( - \\nu O )$ . For \n182 MuJoCo locomotion tasks, we select 4 versions of datasets: data collected by a uniformly-random \n183 agent (random), collected by a medium-performance policy (medium), a $5 0 \\% - 5 0 \\%$ mixture of the \n184 medium data and the replay buffer during training a medium-performance policy (medium-replay), a \n185 $5 0 \\% - 5 0 \\%$ mixture of the medium data and expert demonstrations (medium-expert). For Adroit, \n186 we select pen-human and pen-cloned, where the pen-human includes a small number of human \n187 demonstrations, and pen-cloned is a $5 0 \\% - 5 0 \\%$ mixture of demonstrations and data collected by \n188 rolling out an imitation policy on the demonstrations. ",
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"text": "5.1 Target policy that enables safe Q estimate might be overly constrained ",
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| 543 |
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"text": "190 To investigate the policy constraint under a highly suboptimal dataset, we set up a toy maze MDP that \n191 is similar to the one used in [Kostrikov et al., 2022]. The environment is depicted in Figure 2a, where \n192 the lower left yellow grid is the starting point and the upper left green grid is the terminal state that \n193 gives a reward of 10. Other grids give no reward. Dark blue indicates un-walkable areas. The action \n194 space is defined as 4 direction movements (arrows) and staying where the agent is (filled circles). \n195 There is a $2 5 \\%$ probability that a random action is taken instead of the action from the agent. For the \n196 dataset, 99 trajectories are collected by a uniformly random agent and 1 trajectory is collected by an \n197 expert policy. Fig. 2b shows the optimal value function (colors) and one of the optimal policies. \n198 We trained a constrained policy using Eq. 2 and Eq. 8 in an actor-critic manner, where the actor is \n199 constrained by a KL divergence with a weight factor of 1. Figure 2c shows the value function and the \n200 policy. We observe that the learned value function is close to the optimal one in Figure 2b. However, \n201 the policy does not make optimal actions in the lower left areas where the state values are relatively \n202 low. As the policy improvement objective shows a trade-off between the Q and the KL divergence, \n203 when the Q value is low, the KL divergence term will obtain higher priority. i.e. in low-Q-value \n204 areas, the KL divergence takes the majority for the learning objective, which makes the policy stays \n205 closer to the transitions in low-value areas. However, we find that the corresponding value function \n206 indicates an optimal policy. In Figure 2d, we recover a greedy policy underlying the learned critic \n207 that shows an optimal policy. In conclusion, the constraint might degrade the evaluation performance \n208 although the learned critic may indicate a better policy. Although such a trade-off between the Q \n209 term and the KL divergence term can be alleviated in previous work [Fujimoto and Gu, 2021] by \n210 normalizing the Q values, in the next section, we will illustrate that the constraint required to obtain \n211 performant evaluation policy can still cause unstable value estimate. ",
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"Figure 4: The training process of TD3BC and AWAC. Left: TD3BC on hopper-medium- $\\nu 2$ . Middle: TD3BC on walker2d-medium-replay$\\nu 2$ . Right: AWAC on hopper-medium-replay- $\\cdot \\nu 2$ . "
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"Figure 5: $\\alpha$ values in TD3BC for value estimate and test time inference in MuJoCo locomotion tasks. "
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"text": "212 5.2 Test-time inference requires milder constraints ",
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"text": "The previous experiment shows that a restrictive constraint might harm the test-time inference, which motivates us to investigate what constraints make better evaluation performance. Firstly, we relax the policy constraint on TD3BC and AWAC by setting up different hyper-parameter values that control the strengths of the policy constraints. For TD3BC, we set $\\alpha = \\{ 1 , 4 , 1 0 \\}$ ([Fujimoto and Gu, 2021] recommends $\\alpha = 2 . 5$ ). For AWAC, we set $\\lambda = \\{ 1 . 0 , 0 . 5 , 0 . 3 , 0 . 1 \\}$ ([Nair et al., 2020] recommends $\\lambda = 1$ ). Finally, We visualize the evaluation performance and the learned Q estimates. ",
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"text": "In Figure 4, the left two columns show the training of TD3BC in the hopper-medium- $\\nu 2$ and walker2dmedium-replay- $\\nu 2$ . In both domains, we found that using a milder constraint by tuning the $\\alpha$ from 1 to 4 improves the evaluation performance, which motivates us to expect better performance with $\\alpha = 1 0$ . As shown in the lower row, we do observe higher performances in some training steps. However, unstable training is caused by the divergence in value estimate, which indicates the tradeoff between the stable Q estimate and the evaluation performance. The rightmost column shows the training of AWAC in hopper-medium-replay- $\\nu 2$ , we observe higher evaluation performance by relaxing the constraint $\\lambda > 1 \\AA$ ). Although the Q estimate keeps stable during the training in all $\\lambda$ values, higher $\\lambda$ results in unstable policy performance and causes the performance crash with $\\lambda = 0 . 1$ . ",
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"text": "228 Concluding on all these examples, a milder constraint can potentially improve the performance \n229 but may cause unstable Q estimates or unstable policy performances. As we find that relaxing the \n230 constraint on current methods triggers unstable training, which hinders the investigation of constraints \n231 for better evaluation performance. We instead systematically study the constraint strengths in TD3BC \n232 and TD3BC with evaluation policy (TD3BC-EP). ",
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"type": "table",
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"img_path": "images/e830f4aa08348dbfa6119fbb0ca9843572868c0cde0085794815b48d81e76175.jpg",
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"table_caption": [],
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"table_footnote": [
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"Table 1: Normalized episode returns on D4RL benchmark. The results (except for CQL) are means and standard errors from the last step of 5 runs using different random seeds. Performances that are higher than corresponding baselines are underlined and task-wise best performances are bolded. "
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"table_body": "<table><tr><td>TaskName</td><td>BC</td><td>CQL IQL</td><td></td><td>TD3BC (ours)</td><td>TD3BC-MCEP AWAC</td><td></td><td>AWAC-MCEP (ours)</td></tr><tr><td>halfcheetah-r</td><td>2.2±0.0</td><td>=</td><td>10±1.7</td><td>11.7±0.4</td><td>28.8±1.0</td><td>9.6±0.4</td><td>34.9±0.8</td></tr><tr><td>hopper-r</td><td>4.7±0.1</td><td></td><td>8.1±0.4</td><td>8.3±0.1</td><td>8.0±0.4</td><td>5.3±0.4</td><td>9.8±0.5</td></tr><tr><td>walker2d-r</td><td>1.6±0.0</td><td></td><td>5.6±0.1</td><td>1.2±0.0</td><td>-0.2±0.1</td><td>5.2±1.0</td><td>3.1±0.4</td></tr><tr><td>halfcheetah-m</td><td>42.4±0.1</td><td>44.0</td><td>47.4±0.1</td><td>48.7±0.2</td><td>55.5±0.4</td><td>45.1±0</td><td>46.6±0</td></tr><tr><td>hopper-m</td><td>54.1±1.1</td><td>58.5</td><td>65±3.6</td><td>56.1±1.2</td><td>91.8±0.9</td><td>58.9±1.9</td><td>91.1±1.5</td></tr><tr><td>walker2d-m</td><td>71±1.7</td><td>72.5</td><td>80.4±1.7</td><td>85.2±0.9</td><td>88.8±0.5</td><td>79.6±1.5</td><td>83.4±0.9</td></tr><tr><td>halfcheetah-m-r</td><td>37.8±1.1</td><td>45.5</td><td>43.2±0.8</td><td>44.8±0.3</td><td>50.6±0.2</td><td>43.3±0.1</td><td>44.9±0.1</td></tr><tr><td>hopper-m-r</td><td>22.5±3.0</td><td>95.0</td><td>74.2±5.3</td><td>55.2±10.8</td><td>100.9±0.4</td><td>64.8±6.2</td><td>101.4±0.2</td></tr><tr><td>walker2d-m-r</td><td>14.4±2.7</td><td>77.2</td><td>62.7±1.9</td><td>50.9±16.1</td><td>86.3±3.2</td><td>84.1±0.6</td><td>84.6±1.3</td></tr><tr><td>halfcheetah-m-e</td><td>62.3±1.5</td><td>91.6</td><td>91.2±1.0</td><td>87.1±1.4</td><td>71.5±3.7</td><td>77.6±2.6</td><td>76.2±5.5</td></tr><tr><td>hopper-m-e</td><td>52.5±1.4</td><td>105.4</td><td>110.2±0.3</td><td>91.7±10.5</td><td>80.1±12.7</td><td>52.4±8.7</td><td>92.5±8.3</td></tr><tr><td>walker2d-m-e</td><td>107±1.1</td><td>108.8</td><td>111.1±0.5</td><td>110.4±0.5</td><td>111.7±0.3</td><td>109.5±0.2</td><td>110.3±0.1</td></tr><tr><td>Average</td><td>39.3</td><td>=</td><td>59.0</td><td>54.2</td><td>64.5</td><td>52.9</td><td>64.9</td></tr><tr><td>pen-human</td><td>76.8±4.8</td><td>37.5</td><td>64.2±10.4</td><td>61.6±11</td><td>58.6±20.8</td><td>34.7±11.8</td><td>23.3 ±5.6</td></tr><tr><td>pen-cloned</td><td>28.5±6.7</td><td>39.2</td><td>32.1±7.5</td><td>49±9.5</td><td>43.4±20.3</td><td>20.8±7.3</td><td>19.0±7.5</td></tr><tr><td>Average</td><td>52.6</td><td>38.3</td><td>48.1</td><td>55.3</td><td>51.0</td><td>27.7</td><td>21.1</td></tr></table>",
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"text": "We first tune the $\\alpha$ for TD3BC to unveil the range for safe Q estimates. Then in TD3BC-EP, we tune the $\\alpha ^ { e }$ for the evaluation policy with a fixed $\\tilde { \\alpha } = 2 . 5$ to approximate the constraint range of better test inference performance (i.e. where the evaluation policy outperforms the target policy). The $\\tilde { \\alpha } = 2 . 5$ is selected to ensure a stable Q estimate (also the paper-recommended value). The $\\alpha \\left( \\alpha ^ { e } \\right)$ is tuned within $\\{ 2 . 5 , 5 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7 0 , 8 0 , 9 0 , 1 0 0 \\}$ . For each $\\alpha \\left( \\alpha ^ { e } \\right)$ , we observe the training of 5 runs with different random seeds. In Figure 5, we visualize these two ranges for each task from MuJoCo locomotion set. The blue area shows $\\alpha$ values where the TD3BC Q estimate is stable for all seeds. The edge shows the lowest $\\alpha$ value that causes $\\mathrm { Q }$ value explosion. The orange area shows the range of $\\alpha ^ { e }$ where the learned evaluation policy outperforms the target policy. Its edge (the orange line) shows the lowest $\\alpha ^ { e }$ values where the evaluation policy performance is worse than the target policy. For each task, the orange area has a lower bound $\\alpha ^ { e } = 2 . 5$ where the evaluation policy shows a similar performance to the target policy. ",
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"text": "Note that $\\alpha$ weighs the $\\mathrm { Q }$ term and thus a larger $\\alpha$ indicates a less restrictive constraint. Comparing the blue area and the orange area, we observe that in 6 out of the 9 tasks, the $\\alpha$ for better inference performance is higher than the $\\alpha$ that enables safe Q estimates, indicating that test-time inference requires milder constraints. In the next section, we show that with an MCEP, we can achieve much better inference performance without breaking the stable Q estimates. ",
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"text": "5.3 Comparison on MuJoCo locomotion and Adroit ",
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"text": "We compare the proposed method to state-of-the-art offline RL methods CQL and IQL, together with our baselines TD3BC and AWAC. Similar hyper-parameters are used for all tasks from the same domain. For our baseline methods (TD3BC and AWAC), we use the hyper-parameter recommended by their papers. TD3BC uses $\\alpha = 2 . 5$ for its $\\mathrm { Q }$ value normalizer and AWAC uses 1.0 for the advantage value normalizer. In TD3BC-MCEP, the target policy uses $\\tilde { \\alpha } = 2 . 5$ and the MCEP uses $\\alpha ^ { e } = 1 0$ . In AWAC-MCEP, the target policy has $\\tilde { \\lambda } = \\bar { 1 . 0 }$ and the MCEP has $\\lambda ^ { e } = 0 . 6$ . The full list of hyper-parameters can be found in the Appendix. ",
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"text": "As is shown in Table 1, we observe that the evaluation policies with a mild constraint significantly outperform their corresponding target policy. TD3BC-MCEP gains progress on all medium and medium-replay datasets. Although the progress is superior, we observe a performance degradation on the medium-expert datasets which indicates an overly relaxed constraint for the evaluation policy. To overcome this imbalance problem, we designed a behavior-cloning normalizer. The results are shown in the Appendix. Nevertheless, the TD3BC-MCEP achieves much better general performance than the ",
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"text": "264 target policy. In the AWAC-MCEP, we observe a consistent performance improvement over the target \n265 policy on most tasks. Additionally, evaluation policies from both TD3BC-MCEP and AWAC-MCEP outperform the CQL and IQL while the target policies have relatively low performances. On Adroit tasks, the best results are obtained by behavioral cloning agent and TD3BC with a high BC weighting factor. Other agents fail to outperform the BC agent. We observe that MCEP does not benefit these tasks where behavior cloning is essential for the evaluation performance. ",
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"text": "5.4 Ablation Study ",
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"text": "In this section, we design 2 groups of ablation studies to investigate the effect of the extra evaluation policy and its constraint strengths. Reported results are averaged on 5 runs of different random seeds. ",
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"text": "Performance of the extra evaluation policy. Now, we investigate the performance of the introduced evaluation policy $\\pi ^ { e }$ . For TD3BC, we set the parameter $\\alpha ~ = ~ \\{ 2 . 5 , 1 0 . 0 \\}$ . A large $\\alpha$ indicates a milder constraint. After that, we train TD3BC-MCEP with $\\tilde { \\alpha } = 2 . 5$ and $\\alpha ^ { e } = 1 0 . 0$ . For AWAC, we trained AWAC with the $\\lambda \\ = \\ \\{ 1 . 0 , 0 . 5 \\}$ and AWAC-MCEP with $\\tilde { \\lambda } = 1 . 0$ and $\\lambda ^ { e } = 0 . 5$ . ",
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"img_path": "images/e1e34d8f4f75d5e81faf2e587a7429991db188f20f7bdfebc2df7de3972e7c6e.jpg",
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"image_caption": [
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"Figure 6: Left: TD3BC with $\\alpha = 2 . 5$ , $\\alpha = 1 0$ and TD3BCMCEP with $\\tilde { \\alpha } = 2 . 5$ , $\\alpha ^ { e } = 1 0 $ . Right: AWAC with $\\lambda = 1 . 0$ $\\lambda = 0 . 5$ and AWAC-MCEP with $\\tilde { \\lambda } = 1 . 0$ and $\\lambda ^ { e } = 0 . 5$ . "
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"type": "text",
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"text": "The results are shown in Figure 6. By comparing TD3BC of different $\\alpha$ values, we found a milder constraint $\\langle \\alpha = 1 0 . 0 \\rangle$ ) brought performance improvement in hopper tasks but de",
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"text": "grades the performance in walker2d tasks. The degradation is potentially caused by unstable value estimates (see experiment at section 5.2). Finally, the evaluation policy trained from the critic learned with a target policy with $\\alpha = 2 . 5$ achieves the best performance in all three tasks. In AWAC, a lower $\\lambda$ value brought policy improvement in hopper tasks but degrades performances in half-cheetah and walker2d tasks. Finally, an evaluation policy obtains the best performances in all tasks. ",
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"type": "text",
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"text": "94 In conclusion, we observe consistent performance improvement brought by an extra MCEP that \n95 circumvents the tradeoff brought by the constraint. ",
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| 828 |
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"text": "Constraint strengths of the evaluation policy. We set up two groups of ablation experiments to investigate the performance of evaluation policy under different constraint strengths. For TD3BC-MCEP, we tune the constraint strength by setting the Q normalizer hyper-parameter. The target policy hyper-parameter is fixed to $\\alpha = 2 . 5$ . We pick three strengths for evaluation policy $\\alpha ^ { e } = \\{ 1 . 0 , \\hat { 2 . 5 } , 1 0 . 0 \\}$ to create more restrictive, similar, and milder constraints, respectively. For AWACMCEP, the target policy uses $\\lambda = 1 . 0$ . However, it is not straightforward to create a similar constraint for the eval",
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"img_path": "images/e609e2b9a2859de7da1b1d2cb850e678c05b248dd0a82d6918b4e70d64339c9b.jpg",
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"image_caption": [
|
| 851 |
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"Figure 7: Left: TD3BC-EP with $\\alpha = 1 . 0$ , $\\alpha = 2 . 5$ and $\\alpha = 1 0 . 0$ . Right: AWAC-EP with $\\lambda = 1 . 4$ , $\\lambda = 1 . 0$ and $\\lambda = 0 . 6$ . "
|
| 852 |
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| 853 |
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"image_footnote": [],
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| 854 |
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"bbox": [
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| 862 |
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{
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| 863 |
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"type": "text",
|
| 864 |
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"text": "uation policy as it has a different policy improvement objective. We set 312 $\\lambda ^ { e } = \\{ 0 . 6 , 1 . 0 , 1 . 4 \\}$ to show 313 how performance changes with different constraint strengths. ",
|
| 865 |
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"bbox": [
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| 874 |
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"type": "text",
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"text": "The performance improvements over the target policy are shown in Fig. 7. The left column shows a significant performance drop when the evaluation policy has a more restrictive constraint $( \\alpha ^ { e } = 1 . 0 $ ) than the target policy. A very close performance is shown when the target policy and the evaluation policy have similar policy constraint strengths $\\alpha ^ { e } = 2 . 5$ ). Significant policy improvements are obtained with the target policy having a milder constraint $( \\alpha ^ { e } = 1 0 $ ). The right column presents the results of AWAC-MCEP. Generally, the performance in hopper tasks keeps increasing with milder constraints while the half-cheetah and walker2d tasks show performances that increase from $\\lambda = 1 . 4$ to $\\lambda = 1$ and similar performances between $\\lambda = 1$ and $\\lambda = 0 . 6$ . Compared to the target policy, the evaluation policy consistently outperforms in half-cheetah and hopper tasks. On the walker2d task, a strong constraint $\\lambda = 1 . 4$ ) causes a performance worse than the target policy but milder constraints $( \\lambda = \\{ 1 , 0 . 6 \\} )$ ) obtain similar performance to the target policy. ",
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{
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"type": "table",
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"img_path": "images/9f3e7472e83e0a44082159e938d8abdaecd179da37e54b2858750ce83bd07cfd.jpg",
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| 887 |
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"table_caption": [
|
| 888 |
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"Table 2: Proportion of $Q ( s , \\pi ( s ) ) \\quad { \\bar { > } } \\quad Q ( s , a )$ for target policies and evalution policies in different tasks. "
|
| 889 |
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],
|
| 890 |
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"table_footnote": [],
|
| 891 |
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"table_body": "<table><tr><td rowspan=1 colspan=1>env</td><td rowspan=1 colspan=1>(%) π(%)</td></tr><tr><td rowspan=1 colspan=1>TD</td><td rowspan=1 colspan=1>TD3BC-MCEP</td></tr><tr><td rowspan=1 colspan=1>wa-me</td><td rowspan=2 colspan=1>69.8 87.266.2 82.771.8 88.789.6 99.0</td></tr><tr><td rowspan=1 colspan=1>wa-mwa-mrwa-r</td></tr><tr><td rowspan=1 colspan=1>AI</td><td rowspan=1 colspan=1>AWAC-MCEP</td></tr><tr><td rowspan=1 colspan=1>ha-me</td><td rowspan=2 colspan=1>63.4 70.864.7 68.3</td></tr><tr><td rowspan=1 colspan=1>ha-m</td></tr><tr><td rowspan=1 colspan=1>ha-mr</td><td rowspan=1 colspan=1>68.6 73.1</td></tr><tr><td rowspan=1 colspan=1>ha-r</td><td rowspan=1 colspan=1>75.3 95.6</td></tr></table>",
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| 900 |
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| 901 |
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"type": "image",
|
| 902 |
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"img_path": "images/43748bd7958feef4df8a46c052d8dafd916a5b69ead4361b887fe1672947d296.jpg",
|
| 903 |
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"image_caption": [
|
| 904 |
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"Figure 9: The distributions of $Q ( s , \\tilde { \\pi } ( s ) ) - Q ( s , a )$ and $Q ( s , \\pi ^ { e } ( s ) ) -$ $Q \\bar { ( } s , a )$ on MuJoCo locomotion tasks. First row: policies of TD3BCMCEP learned in walker2d tasks. Second row: policies of AWAC-MCEP learned in half cheetah tasks. See the Appendix for full results. "
|
| 905 |
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|
| 906 |
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"image_footnote": [],
|
| 907 |
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| 918 |
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| 926 |
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| 927 |
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"type": "text",
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| 928 |
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"text": "In conclusion, for both algorithms, we observe that on evaluation policy, a milder constraint obtains higher performance than the target policy while a restrictive constraint may harm the performance. ",
|
| 929 |
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| 938 |
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|
| 939 |
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"text": "5.5 Estimated Q values for the learned evaluation policies ",
|
| 940 |
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"text_level": 1,
|
| 941 |
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"bbox": [
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| 950 |
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"type": "text",
|
| 951 |
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"text": "To compare the performance of the policies learned in Section 5.3 on the learning objective (maximizing the $\\mathrm { \\bf Q }$ values), we counted $\\mathrm { Q }$ differences between the policy action and the data action $Q ( s , \\pi ( s ) ) - Q ( s , a )$ in the training data (visualized in Figure 9). Proportions of data points that show positive differences are listed in Table 2, where we find that on more than half of the data, both the target policy and the MCEP have larger Q estimation than the behavior actions. Additionally, the proportions for the MCEP are higher than the proportions for the target policy in all datasets, indicating that the MCEP is able to move further toward large Q values. ",
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"type": "text",
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| 962 |
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"text": "6 Conclusion ",
|
| 963 |
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"text_level": 1,
|
| 964 |
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|
| 973 |
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|
| 974 |
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"text": "This work focuses on the policy constraints methods where the constraint addresses the tradeoff between stable value estimate and evaluation performance. While to what extent the constraint achieves the best results for each end of this tradeoff remains unknown, we first investigate the constraint strength range for a stable value estimate and for evaluation performance. Our findings indicate that test time inference requires milder constraints that can go beyond the range of stable value estimates. We propose to use an auxiliary mildly constrained evaluation policy to circumvent the above-mentioned tradeoff and derive a performant evaluation policy. The empirical results show that MCEP obtains significant performance improvement compared to target policy and achieves competitive results to state-of-the-art offline RL methods. Our ablation studies show that an auxiliary evaluation policy and a milder policy constraint are essential for the proposed method. Additional empirical analysis demonstrates higher estimated Q values are obtained by the MCEP. ",
|
| 975 |
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| 982 |
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|
| 983 |
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|
| 984 |
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"type": "text",
|
| 985 |
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"text": "Limitations. Although the MCEP is able to obtain a better performance, it depends on stable value estimation. Unstable value learning may crash both the target policy and the evaluation policy. While the target policy may recover its performance by iterative policy improvement and policy evaluation, we observe that the evaluation policy may fail to do so. Therefore, a restrictive constrained target policy that stabilizes the value learning is essential for the proposed method. ",
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| 986 |
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"type": "text",
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"text": "52 References \n353 David Brandfonbrener, Will Whitney, Rajesh Ranganath, and Joan Bruna. Offline rl without off-policy evaluation. Advances in neural information processing systems, 34:4933–4946, 2021. \n55 Y. Cai, C. Zhang, L. Zhao, W. Shen, X. Zhang, L. Song, J. Bian, T. Qin, and T. Liu. Td3 with reverse kl regularizer for offline reinforcement learning from mixed datasets. In 2022 IEEE International Conference on Data Mining (ICDM), pages 21–30, Los Alamitos, CA, USA, dec 2022. IEEE Computer Society. doi: 10.1109/ICDM54844.2022.00012. URL https://doi. ieeecomputersociety.org/10.1109/ICDM54844.2022.00012. Wojciech M Czarnecki, Razvan Pascanu, Simon Osindero, Siddhant Jayakumar, Grzegorz Swirszcz, and Max Jaderberg. Distilling policy distillation. In The 22nd international conference on artificial intelligence and statistics, pages 1331–1340. PMLR, 2019. \nSoysal Degirmenci and Chris Jones. Benchmarking offline reinforcement learning algorithms for e-commerce order fraud evaluation. In 3rd Offline RL Workshop: Offline RL as a”Launchpad”. \n365 Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020. \n367 Scott Fujimoto and Shixiang Shane Gu. A minimalist approach to offline reinforcement learning. Advances in neural information processing systems, 34:20132–20145, 2021. \n69 Scott Fujimoto, Herke Hoof, and David Meger. Addressing function approximation error in actorcritic methods. In International conference on machine learning, pages 1587–1596. PMLR, 2018. \n72 Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. In International conference on machine learning, pages 2052–2062. PMLR, 2019. \n74 Natasha Jaques, Asma Ghandeharioun, Judy Hanwen Shen, Craig Ferguson, Agata Lapedriza, Noah Jones, Shixiang Gu, and Rosalind Picard. Way off-policy batch deep reinforcement learning of implicit human preferences in dialog. arXiv preprint arXiv:1907.00456, 2019. Dmitry Kalashnikov, Jacob Varley, Yevgen Chebotar, Benjamin Swanson, Rico Jonschkowski, Chelsea Finn, Sergey Levine, and Karol Hausman. Mt-opt: Continuous multi-task robotic reinforcement learning at scale. arXiv preprint arXiv:2104.08212, 2021. Ilya Kostrikov, Ashvin Nair, and Sergey Levine. Offline reinforcement learning with implicit q-learning. In International Conference on Learning Representations, 2022. URL https:// openreview.net/forum?id $\\equiv$ 68n2s9ZJWF8. \n383 Aviral Kumar, Justin Fu, Matthew Soh, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. Advances in Neural Information Processing Systems, 32, 2019. \n86 Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative q-learning for offline reinforcement learning. Advances in Neural Information Processing Systems, 33:1179–1191, 2020. \n88 Aviral Kumar, Anikait Singh, Stephen Tian, Chelsea Finn, and Sergey Levine. A workflow for offline model-free robotic reinforcement learning. In Conference on Robot Learning, pages 417–428. PMLR, 2022. Jiafei Lyu, Xiaoteng Ma, Xiu Li, and Zongqing Lu. Mildly conservative q-learning for offline reinforcement learning. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems, 2022. URL https://openreview. net/forum?id $\\cdot ^ { = }$ VYYf6S67pQc. \n95 Ashvin Nair, Abhishek Gupta, Murtaza Dalal, and Sergey Levine. Awac: Accelerating online reinforcement learning with offline datasets. arXiv preprint arXiv:2006.09359, 2020. \n97 Xue Bin Peng, Aviral Kumar, Grace Zhang, and Sergey Levine. Advantage-weighted regression: Simple and scalable off-policy reinforcement learning. arXiv preprint arXiv:1910.00177, 2019. ",
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"type": "text",
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| 1 |
+
# DYNAMIC ENSEMBLE FOR PROBABILISTIC TIMESERIES FORECASTING VIA DEEP REINFORCEMENT LEARNING
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
It is well known that ensemble improve the accuracy of forecasting tasks. However, most of ensembling strategies designed for probabilistic time series forecasting are static methods, in the sense that they either assume the time-invariant ensemble strategies over the prediction horizon, or are non-adaptive to the forecast start point. In addition, the static methods naively rely on the predictions of the base forecasters but fail to utilize base learners themselves efficiently. In this paper, we propose a novel dynamic ensemble policy to overcome three major limitations mentioned above via deep Reinforcement Learning (RL) framework. To learn such a policy, we design a Markov Decision Process (MDP), together with our environment (TS-GYM) that supports the interaction between the agent or ensembler, offline datasets and base learners. In doing so, we effectively leverage the power of the ensemble to improve each of the base learners by reducing the error accumulation of each base learner via consecutively feeding a better ensembled sample to each base learner. The proposed ensembling method has several desirable properties such as uncertainty quantification and the ability to generate sample path, on top of significant performance gain. The effectiveness of the proposed framework is demonstrated on multiple synthetic and real-world experiments.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Time series data occur naturally in countless domains including supply chain optimization (Larson, 2001; Wen et al., 2017), medical analysis (Keogh et al., 2001; Matsubara et al., 2014b), financial analysis (Zhu & Shasha, 2002; Hallac et al., 2017), sensor network monitoring (Papadimitriou & Yu, 2006; Letchner et al., 2009), cloud computing (Park et al., 2019; 2021), optimal control of vehecle (Kim et al., 2020) and social activity mining (Mathioudakis et al., 2010; Matsubara et al., 2012; 2014a). Among the applications of ML-based time series analysis, forecasting is arguably one of the most sought-after, due to its importance in industrial, social, and scientific applications. For example, forecasting plays a key role in automating and optimizing operational processes in most businesses and enables data driven decision making. Forecasts of product supply and demand are used for optimal inventory management, staff scheduling and topology planning, and are more generally a crucial technology for most aspects of supply chain optimization. In order to make optimal decisions, predictive uncertainties need to be taken into account, making probabilistic forecast a desirable property of time series models (Benidis et al., 2022).
|
| 12 |
+
|
| 13 |
+
In practice, one often encounters complex time series, making it difficult to find a single best model that excels at short-term, mid-term, and long-term forecasting scenarios. In such cases, different forecasting models usually perform well on different data regimes at different time steps. As a motivating example, Figure 1a shows the relative ranking of the performances of 5 popular forecasting models on the dataset Solar. In this example, Transformer excels at shorter and longer-term forecasts while DeepAR and TFT shine in the mid-term scenario. It is thus desirable to have an ensembling strategy that has different weights at each time step. Therefore, the traditional ensembling strategy in time series forecast, which assumes that ensemble weights do not vary along the forecasting horizon is not sufficient to capture the non-stationary patterns of base learners’ performance profile. Furthermore, popular auto-regression based models are known to have increasing prediction errors as the prediction horizon stretches further, and the performance degrades dramatically when the prediction horizon is sufficiently large (Salinas et al., 2020). As shown in the blue curve of Figure 1b, the prediction error increases for “DeepAR-G original”(“G” means using the Gaussian distribution as the output distribution and “original” means using the original implementation of DeepAR) over the prediction horizon on exchange rate dataset. On the other hand, if we can provide base learners such as DeepAR with more accurate estimations of the future as the auto-regressive input, the prediction error can be significantly decreased for the long horizon predictions (see the orange curve in Figure 1b). The huge difference in the prediction error between these two cases show the huge potential to improve the auto-regression based models if we can provide more accurate estimations during the prediction horizon. However, none of the traditional ensemble methods utilize the ensemble predictions as the feedback to boost the performance of the auto-regression based models. Motivated by the above examples, the natural question arises whether we can develop a general dynamic ensembling approach that overcomes all the major limitations of the traditional static ensemble methods and further improve the prediction accuracy for the probabilistic time-series forecasting?
|
| 14 |
+
|
| 15 |
+
|
| 16 |
+
|
| 17 |
+
(a) The ranks of 5 base learners along the prediction horizon on Solar dataset. The ranks are based on the mean weighted quantile loss over the quantiles [0.1, 0.5, 0.9] and averaged over all items in each dataset.
|
| 18 |
+
|
| 19 |
+
|
| 20 |
+
(b) The gap between the “DeepAR original" and “DeepAR w/ target” shows the potential improvement we can gain if the accuracy of the auto-regressive input to DeepAR can be improved.
|
| 21 |
+
|
| 22 |
+
Figure 1: Two motivations on the need of dynamic ensembles, beyond static ensembles.
|
| 23 |
+
|
| 24 |
+
To address the above mentioned challenges, in this work, we develop a general dynamic ensemble framework for probabilistic multi-horizon time series forecasting. Our contributions can be summarized as follows:
|
| 25 |
+
|
| 26 |
+
• This work is the first one that proposes a dynamic ensemble policy suitable for probabilistic time series forecasting with the properties of sequential weighting, being adaptive, and quantile ensemble. • We formulate this as a Markov Decision Process (MDP) with a careful design of the rewards, transition dynamics, and ensemble action policy. In particular, the state evolution in our formulation depends on the ensemble strategy through our novel transition dynamics design. • To solve this MDP problem, we design a time series gym (TS-GYM) environment which implements the interaction between the time series off-line dataset, base learners and ensemble agent. Through this interaction, actor-critic based deep RL method with our “random extreme point” exploration strategy can learn optimal ensemble policy. • The extensive experiments show the advantages of our ensemble dynamic framework. In particular, we demonstrate that our general dynamic ensemble framework can (1) learn the optimal time-varying ensemble weights along the multi-horizon prediction, (2) be adaptive to any forecast start time, (3) boost the performance of the auto-regressive base learners, and (4) result in better performance than other potential variants on real-world datasets.
|
| 27 |
+
|
| 28 |
+
# 2 RELATED WORK
|
| 29 |
+
|
| 30 |
+
Probabilistic time series forecasting In recent years there has been an increasing interest in “probabilistic forecasting”, namely forecasting models that account for the data’s uncertainty by modeling the distribution of target values, rather than predicting a single point estimate. Probablistic forecasting is useful for business purposes such as supply and demand, inventory management, staff scheduling and topology planning (Larson, 2001). Modern open source packages such as Kats (facebookresearch, 2021), Merlion (Bhatnagar et al., 2021) and GluonTS (Alexandrov et al., 2020a) offer probabilistic forecasting, and include some popular probabilistic forecasters such as Prophet (Taylor & Letham, 2018), and deep learning probabilistic forecasters such as DeepAR (Salinas et al., 2020), MQ-CNN (Wen et al., 2017; Park et al., 2022), MQF2 (Kan et al., 2022), NBEATS (Oreshkin et al., 2019), TFT (Lim et al., 2021) and Transformer (Vaswani et al., 2017). There are several advances in improving those models in adversarial robustness (Yoon et al., 2022; Liu et al., 2022) and few-shot learning (Jin et al., 2022).
|
| 31 |
+
|
| 32 |
+
Time series ensemble The literature on ensembling methods for time series predictions have focused solely on static ensembling strategies, namely ones that have access to the predictions of the base learners but not to the base learners themselves. In that situation, a debate on the theory of ensembling for time series was sparked by an empirical observation that a simple average of the base learners is often superior to more sophisticated ensemble methods (a problem called the “forecast combination puzzle”, see Stock & Watson (2004) and Bates & Granger (1969)). See Smith & Wallis (2009), Claeskens et al. (2016), and Elliott (2011)). While theory lags, however, sophisticated static ensembling methods have often been observed to work well. (See Donaldson & Kamstra (1996), Moon et al. (2020), and Massaoudi et al. (2021). Particularly interesting is Gastinger et al. (2021), with a large empirical study.)
|
| 33 |
+
|
| 34 |
+
Contrary to the situation considered in these papers, literature on ensembling methods that have direct access to the base learners, rather than only to their predictions, is limited. Recently, RL based approaches are proposed in Saadallah & Morik (2021) and $\mathrm { F u }$ et al. (2022). Saadallah & Morik (2021) consider action dependent state (window of ensemble predictions) transition. Their work focus on online policy learning with update timing determined by a concept-drift detection algorithm. In Fu et al. (2022) the state (time series for a given context window and base learners performance at the next window) transition is action independent with action taken for $H$ steps at a time. In addition, their methods are only designed for the point based forecasting problem and do not demonstrate the capability of capturing the non-stationary ensemble weights.
|
| 35 |
+
|
| 36 |
+
# 3 PRELIMINARIES
|
| 37 |
+
|
| 38 |
+
# 3.1 PROBABILISTIC TIME-SERIES FORECASTING
|
| 39 |
+
|
| 40 |
+
Suppose we have a panel of $n$ time series, where the $i$ -th time series consists of observations $z _ { i , t } \in \mathbb { R }$ with (optional) input covariates $x _ { i , t } \in \mathbb { R } ^ { d }$ , as $t$ varies over time at fixed discrete intervals. For an $i$ -th time series (often called $i$ -th item), we wish to make predictions for the next $H$ timestamps, namely of $z _ { i , T + 1 : T + H }$ from the forecast start time $T + 1$ , given the history of that item’s observations $z _ { i , 1 : T }$ and (optional) the associated historical and future covariates $x _ { i , 1 : T + H }$ . In this paper we will focus on global forecasters, namely a single univariate model trained on all of the items together, and accepting only a single item at inference. For notational simplicity we will drop the item index $i$ and covariates $x _ { i , t }$ unless explicitly stated. We now formally define a forecasting model as a set of random variable valued functions $\{ f _ { h } \} _ { h = 1 } ^ { H }$ such that, for $h = 1 , . . . , H$
|
| 41 |
+
|
| 42 |
+
$$
|
| 43 |
+
Z _ { T + h } = f _ { h } \big ( z _ { 1 : T } , \xi _ { T + h - 1 } \big ) ,
|
| 44 |
+
$$
|
| 45 |
+
|
| 46 |
+
where $\xi _ { T + h - 1 }$ is the hidden state variable passed from the previous (or older) step. The evolution of $f _ { h }$ and $\xi _ { T + h - 1 }$ depend on the type of the base model. For the auto-regressive model which uses the recursive prediction strategy, the hidden state $\cdot$ is generated by passing a sample $\_$ from previous time step to the forecaster decoder for the next prediction in a recursive manner. Often the decoder is homogeneous, i.e., $\cdot$ for $h = 1 , \ldots , H$ . On the other hand, Seq2Seq model which uses the direct prediction strategy, directly forecast the future time series without involving the evoluation of the hidden state, i.e., $\xi _ { T + h - 1 } = \xi _ { T }$ for all $h = 1 , \ldots , H$ . Refer to Alexandrov et al. (2020b) for the detailed modeling. In Section 4, we will explore a different choice for the auto-regressive step, using the entire ensemble.
|
| 47 |
+
|
| 48 |
+
Then, the associated $\tau$ -quantile predictions can be followed as $\hat { z } _ { T + h } ^ { \tau } = q _ { \tau } \left( Z _ { T + h } \right)$ where, for a random variable $Z \in \mathbb { R }$ with its culmulative distribution $F _ { Z }$ and a quantile level $\tau \in ( 0 , 1 )$ , $q _ { \tau }$ is denoted as the quantile function, i.e., $q _ { \tau } ( Z ) : = F _ { Z } ^ { - 1 } ( \tau ) = \operatorname* { i n f } \left\{ z \in \mathbb { R } : \tau \leq F _ { Z } ( z ) \right\}$ .
|
| 49 |
+
|
| 50 |
+
# 3.2 FORECASTING ENSEMBLE
|
| 51 |
+
|
| 52 |
+
For each m-th base learner, we denote zˆτk,mT+h as the $\tau _ { k }$ -quantile prediction at time step $T + h$ on a quantile level where $\tau _ { k } \in \{ \tau _ { k } \} _ { k = 1 } ^ { K }$ . Then, $\left\{ \hat { z } _ { T + h } ^ { \tau _ { k } , m } \right\} _ { k = 1 , m = 1 } ^ { K , M }$ is denoted as a pool of quantile predictions at time step $T + h$ over $M$ base learners and $K$ quantile levels. A general ensemble predictions can be formally expressed as a (linear) weighted combination of predictions of the individual base models, at each prediction step $h = 1 , \ldots , H$ ,
|
| 53 |
+
|
| 54 |
+
$$
|
| 55 |
+
\hat { z } _ { T + h } ^ { \tau , \mathrm { e s } } = \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \hat { z } _ { T + h } ^ { \tau , m } ,
|
| 56 |
+
$$
|
| 57 |
+
|
| 58 |
+
$w _ { h } ^ { m } \geq 0$ with $\textstyle \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } = 1$ are the ensemble weights.
|
| 59 |
+
|
| 60 |
+
# 3.3 REINFORCEMENT LEARNING
|
| 61 |
+
|
| 62 |
+
Reinforcement learning (RL) is usually formulated as a Markov Decision Process (MDP), which can be defined as a tuple $( S , { \mathcal { A } } , { \mathcal { P } } , r , \gamma , H )$ where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \mathcal { S }$ is the transition function, $r : \mathcal { S } \times \mathcal { A } \to \mathbb { R }$ is the reward function, $\gamma \in ( 0 , 1 )$ is the discount factor and $H > 0$ is the horizon length of each episode. At each state $s \in \mathcal { S }$ , the RL agent takes an action $a \in { \mathcal { A } }$ , transits to the next state $s ^ { \prime } \in { \mathcal { S } }$ under the dynamics $\mathcal { P }$ and receives a reward $r ( s , a )$ . The goal of an MDP is to learn a policy $\pi : { \mathcal { S } } A$ that maximizes the total obtained rewards $\begin{array} { r } { \operatorname* { m a x } _ { \pi } J ( \pi ) = \mathbb { E } _ { \tau } \left[ \sum _ { h = 0 } ^ { H - 1 } \gamma ^ { h } r ( s _ { h } , a _ { h } ) \bigg | \pi \right] , } \end{array}$ where the expectation is over the trajectory $\tau = \{ \big ( s _ { 0 } , a _ { 0 } , r \big ( s _ { 0 } , a _ { 0 } \big ) \big ) . . . . , \big ( s _ { H } , a _ { H } , r \big ( s _ { H } , a _ { H } \big ) \big ) \}$ where $a _ { h } = \pi ( s _ { h } )$ .
|
| 63 |
+
|
| 64 |
+
# 4 DYNAMIC ENSEMBLE FRAMEWORK
|
| 65 |
+
|
| 66 |
+
In this section, we mainly focus on how to select a sequence of ensemble weights $( w _ { 1 } , w _ { 2 } , \dots , w _ { H } )$ with $w _ { h } \in \mathbb { R } ^ { M }$ over $M$ base learners by learning a ensemble policy $\pi$ . Especially in the presence of auto-regressive base learners, ensemble weights chosen at the step $\cdot$ may affect the forecasting of auto-regressive base learners and also ensemble weights chosen at the next step $h + 1$ (see Section 4.1.1 for more details). With this intuition, we will take a reinforcement learning approach to learn an optimal policy function $\pi$ that provides the optimal ensemble weights sequentially.
|
| 67 |
+
|
| 68 |
+
In Section 4.1, we give a high-level overview of the MDP formulation for the multi-horizon probabilistic time series forecasting problems. In particular, the classes of ensembled sampling strategies and predictions which determine the state transformation and state transition are discussed in Section 4.1.1 and the careful design of reward computation is explained in Section 4.1.2. Based on the formulated MDP, we then design our simulated environment, TS-GYM (in Section 4.2) which provides the interaction among the time series datasets, base learners and the dynamic ensemble agent. Finally, we describe how to employ deep reinforcement learning with our “random extreme point” exploration strategy to learn the optimal ensemble policy in Section 4.3.
|
| 69 |
+
|
| 70 |
+
# 4.1 MDP FORMULATION
|
| 71 |
+
|
| 72 |
+
We describe the high-level formulation of the MDP for our dynamic time-series ensemble framework. Once each episode starts with $h = 1$ , the environment fixes an arbitrary forecasting start point $T$ , and then starts to provide a time series pair of both historical input $z _ { 1 : T }$ and corresponding future (backtest) output $z _ { T + 1 }$ as well as corresponding quantile predictions $\big \{ \hat { z } _ { T + 1 } ^ { \tau , m } \big \}$ from alll $M$ base models for the next step $T + 1$ . (We defer the details implementation of the environment to Section 4.2). The agent will then decide the ensemble weights to compute the ensembled predictions, and update the ensemble policy based on the accuracy of the ensembled predictions. Depending on the type of ensemble dynamics, the ensembled predictions may also affect the base learners’ future predictions. Then, in the next step $h = 2$ , the environment provides next time series output $z _ { T + 2 }$ and associated predictions $\big \{ \hat { z } _ { T + 2 } ^ { \tau , m } \big \}$ and go on. See Figure 2a for a high level schema.
|
| 73 |
+
|
| 74 |
+
More formally, for each step $h = 1 , \ldots , H$ of an episode, given the information provided by the environment (e.g., historical observation $z _ { 1 : T }$ , and future (backtest) observation $z _ { T + h }$ , a pool of all quantile predictions $\{ \hat { z } _ { T + h } ^ { \tau _ { k } , m } \} _ { k = 1 , m = 1 } ^ { K , M }$ , and step $h$ ) , we define MDP as follows:
|
| 75 |
+
|
| 76 |
+
• the fixed-size state $s _ { h } = \Big \{ z _ { 1 : T } , \big \{ \hat { z } _ { T + h } ^ { \tau _ { k } , m } \big \} _ { k = 1 , m = 1 } ^ { K , M } , h \Big \} ,$ ,
|
| 77 |
+
• the action $a _ { h } = \{ w _ { h } ^ { m } \} _ { m = 1 } ^ { M } = \pi \big ( s _ { h } \big )$ , $M$ -ensemble weights $w _ { h } ^ { m }$ from a policy function $\pi$ , • the state transition $\mathcal { P } ( s _ { h + 1 } \mid s _ { h } , a _ { h } )$ governed by ensemble dynamics in Section 4.1.1, • the reward $R \left( { { s _ { h } } , { a _ { h } } ; { z _ { T + h } } } \right)$ 1 which evaluates ensemble prediction against ground-truth $z _ { T + h }$ in Section 4.1.2 .
|
| 78 |
+
|
| 79 |
+
# 4.1.1 ENSEMBLE DYNAMICS $\mathcal { P }$ AND ENSEMBLED QUANTILES
|
| 80 |
+
|
| 81 |
+
Defining state transition $\mathcal { P }$ , which we call ensemble dynamics, narrows down how to construct quantile predictions over M base learners {zˆτk,mT +h }K,Mk=1,m =1 ∈ sh. Here, we proposed three strategies: direct dynamic, auto-regressive dynamic and their composition. The idea of direct ensemble is similar to Seq2Seq models which employs the direct prediction strategy. The idea of auto-regressive dynamic is based on auto-regressive models where you recursively feed a new ensembled sample to each base learner for the next prediction. The ensemble dynamics appear at the step represented by the red arrow line in Figure 2a.
|
| 82 |
+
|
| 83 |
+
Direct dynamic. learner itself over $H$ s a direct ensembling ovehorizon, i.e., we compute $\hat { z } _ { T + h } ^ { \tau , m } = q _ { \tau } \left( Z _ { T + h } ^ { m } \right)$ first cofor all $h = 1 , \ldots , H$ es by base, based on Equation 1. Then the final quantile ensemble becomes $\begin{array} { r } { \hat { z } _ { T + h } ^ { \tau , e s } = \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \hat { z } _ { T + h } ^ { \tau , m } } \end{array}$ in Equation 2. Note that the base learner’s predictions are not affected by the ensembling. In other words, the transition dynamic $\mathcal { P } ( s _ { h + 1 } \mid s _ { h } , a _ { h } ) = \mathcal { P } ( s _ { h + 1 } \mid s _ { h } )$ is actually irrelevant to the ensembling weights.
|
| 84 |
+
|
| 85 |
+
Auto-regressive dynamic. In this dynamic, we generate an (intermediate) ensembled sample $p _ { T + h }$ , which is fed into each autoregressive base leaner in a recursive manner. This ends up forming a sample path through which we can compute the final ensembled (empirical) quantile prediction $\hat { z } _ { T + h } ^ { \tau , \mathrm { e s } }$
|
| 86 |
+
|
| 87 |
+
To begin with, we generate a sample path $( \hat { z } _ { T + 1 } ^ { m } , \dots , \hat { z } _ { T + H } ^ { m } )$ for each base learner as follows: First, for each step , we sample $p _ { T + h }$ from mixture of base learners’ distributions $\mathbb { P } ( Z _ { T + h } ^ { m } )$ proportional to ensemble weights $w _ { h } ^ { m }$ , i.e.,
|
| 88 |
+
|
| 89 |
+
$$
|
| 90 |
+
p _ { T + h } \sim \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \mathbb { P } ( Z _ { T + h } ^ { m } ) .
|
| 91 |
+
$$
|
| 92 |
+
|
| 93 |
+
Second, we feed $p _ { T + h }$ to each autoregressive base learner, i.e.,
|
| 94 |
+
|
| 95 |
+
$$
|
| 96 |
+
\begin{array} { c } { { Z _ { T + h + 1 } ^ { m } = f ^ { m } ( z _ { 1 : T } , \xi _ { T + h } ^ { m } ) , } } \\ { { \xi _ { T + h } ^ { m } = g ^ { m } ( p _ { T + h } , \xi _ { T + h - 1 } ^ { m } ) . } } \end{array}
|
| 97 |
+
$$
|
| 98 |
+
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where get a sgenera $g ^ { m }$ represents thple for each sample path $m$ on dynamics for the hidden state , which can be operated in a recbase learners. $\xi _ { T + h } ^ { m }$ . Lastly, wee manner to $\hat { z } _ { T + h . } ^ { m } \sim Z _ { T + h . } ^ { m }$ $\_$ for all
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After collecting a set of sample paths $\{ ( \hat { z } _ { T + 1 } ^ { m } , \dots , \hat { z } _ { T + H } ^ { m } ) _ { l } \} _ { l = 1 , m = 1 } ^ { L , M }$ where $( \hat { z } _ { T + 1 } ^ { m } , \dots , \hat { z } _ { T + H } ^ { m } ) _ { l }$ is $l$ -th sample path above for the $m$ -base learner, we construct the empirical marginal distribution $\cdot$ based on the samples $\cdot$ for all $\_$ . Then, the final (ensemble-dependent) quantile prediction of each base learner is obtained as $\hat { z } _ { T + h } ^ { \tau , m } ( w ) = q _ { \tau } \left( \hat { \mathbb { P } } ( \hat { Z } _ { T + h } ^ { m } ) \right)$ for all $m = 1 , \ldots , M$ with the final ensemble $\begin{array} { r } { \hat { z } _ { T + h } ^ { \tau , e s } = \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \hat { z } _ { T + h } ^ { \tau , m } } \end{array}$ . Note that, like $p _ { T + h }$ was sampled, the final ensemble model is ultimately a (single) auto-regressive one that supports sample path and quantiles.
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Under auto-regressive dynamic strategy, the ensembled sample $p _ { T + h }$ based on ensemble weight from policy affects the performance of individual base learner consecutively and thus final quantile ensemble. In other words, action in the previous step affects state in the current step, meaning, unlike the direct dynamic, the transition dynamics $\mathcal { P } ( s _ { h + 1 } \mid s _ { h } , a _ { h } ) \neq \mathcal { P } ( s _ { h + 1 } \mid s _ { h } )$ .
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Hybrid dynamic. Note that the auto-regressive dynamic strategy is not applicable for Seq2seq base learners. Still, under the hybrid dynamic strategy, Seq2seq base learners can contribute to generate ensembled samples together, i.e., ensembled sample Seq2seq and autoregressive ones, which would b $\begin{array} { r } { p _ { T + H } \sim \sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \mathbb { P } ( Z _ { T + h } ^ { m } ) } \end{array}$ sampled from bothssive base learners. The behaviours of Seq2seq base learner is the exactly same in sampling and constructing quantile prediction without any feedback loop like ensembled sample, which means any auto-regressive base learners does not affect Seq2seq one’s prediction.The final ensemble under hybrid dynamic is capable of auto-regressive model, supporting desirable sample path through recursive feedings.
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# 4.1.2 REWARD FUNCTION
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To minimize the total quantile losses and encourage the agent to learn a uniform distribution over
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the nearly-optimal base learners, we design the reward function as $R ( s , a ; z ) ~ = ~ R _ { 1 } ( s , a ; z ) ~ +$
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$\lambda ( s ) R _ { 2 } ( \bar { s } , a )$ for some nsemble p $\lambda ( s ) \geq 0$ e, the first term compared with t $r _ { 1 }$ measures the performance of the cur-best quantile predictions among the base $\hat { z } _ { T + h } ^ { \tau _ { k } , e s }$
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learners. and takes the form
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$$
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R _ { 1 } \big ( s _ { h } , a _ { h } ; z _ { T + h } \big ) = \operatorname* { m i n } _ { m } \left\{ \sum _ { k = 1 } ^ { K } \left( \mathcal { L } \big ( \hat { z } _ { T + h } ^ { \tau _ { k } , m } , z _ { T + h } ; \tau _ { k } \big ) - \mathcal { L } \big ( \hat { z } _ { T + h } ^ { \tau _ { k } , \mathrm { e s } } , z _ { T + h } ; \tau _ { k } \big ) \right) \right\}
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$$
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where $ { \mathcal Ḋ L Ḍ } ( \cdot , \cdot ; \tau )$ can be any measurement of the forecasting accuracy at the quantile level $\tau$ . By designing the $R _ { 1 }$ term as a regret w.r.t. the best base learner, we normalize the reward around zero: if the $R _ { 1 }$ term is less than 0, then it means that the ensemble prediction is worse than the single best base learner and the corresponding should be punished, and vise versa. Furthermore, $R _ { 2 }$ takes the form
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$$
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R _ { 2 } { \big ( } s _ { h } , a _ { h } { \big ) } = D _ { \mathrm { K L } } { \big ( } a _ { h } \ { \big | } \ \mathrm { U n i f } { \big ( } M ^ { * } { \big ( } s _ { h } { \big ) } { \big ) } { \big ) }
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$$
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where $D _ { \mathrm { K L } }$ denotes the Kullback–Leibler divergence, $-$ for a threshold $\cdot$ . denotes the set of nearly-optimal base learners at the state $s$ , and $\operatorname { J n i f } ( M ^ { * } ( s ) )$ denotes a distribution with probability mass $\frac { 1 } { \vert M ^ { \ast } ( s ) \vert }$ on the indices corresponding to the base learners in $M ^ { \ast } ( s )$ and 0 otherwise. We introduce the term $R _ { 2 }$ to encourage the ensemble policy to be uniformly distributed among the nearly optimal base learners which could potentially further reduce the estimation error and the variance. Finally, $\lambda ( s )$ is a state-dependent hyper-parameter controlling the weights between $R _ { 1 }$ and $R _ { 2 }$ . When there is only a single nearly-optimal base learner, i.e., $\vert M ^ { \ast } ( s ) \vert = 1$ , we set $\lambda ( s ) = 0$ which means that we only incorporate $R _ { 2 }$ when there are at least two nearly-optimal base learners.
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# 4.2 SIMULATED ENVIRONMENT: TS-GYM
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Before attempting to train the policy $\pi$ , we first design a novel simulated environment for the
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time series ensemble, namely TS-GYM, that follows state transition (in Section 4.1.1) properly, by
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extending the OpenAI’s gym interface. As illustrated in Figure 2a, it is composed of pre-trained base
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learners in the ensemble, time series (off-line) dataset, time series samplers, ensemble dynamics and
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dynamic ensemble agent. During the initialization stage of the environment $h = 1$ , it first decides
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forecast start time $T$ which is uniformly sampled among time horizon in off-line datasets, and then
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starts to provide followthe quantile predictions ground-truth (future) ob (1) sample a time seriefor the next timestamp The first three informa historical) observatio, (3) the step number used to construct th $z _ { 1 : T }$ , (2)d (4) and $\{ \hat { z } _ { T + h } ^ { \tau _ { k } , m } \} _ { k = 1 , m = 1 } ^ { K , M }$ $T + h$ $h$ $\cdot$
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the last information is used to construct the reward defined in Section 4.1.
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Note here that generating all quantile predictions $\{ \hat { z } _ { T + h } ^ { \tau _ { k } , m } \} _ { k = 1 , m = 1 } ^ { K , M }$ at each timestamp $T + h$ is governed by the choice of ensemble dynamics in Section 4.1.1 where the ensembled quantile predictions themselves may be used for the base learners’ prediction in the next timestamp. This will affect the optimal choice of ensemble actions in the end. This process is repeated until we reach the end of the prediction horizon $T + H$ , completing one episode. In practice, this whole of procedure can be done with batch sampling in parallel.
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Figure 2: Dynamic ensemble framework.
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# 4.3 LEARNING DYNAMIC ENSEMBLE POLICY WITH EXPLORATION
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To learn an optimal ensemble policy $\pi$ , we employ the deep actor-critic approach DDPG (Lillicrap et al., 2015) in a continuous action space to maximize culmulative reward. To accelerate the exploration of the base learners’ performance, we deploy the “random extreme point” exploration.
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Random extreme point exploration. For the exploration of actions, for each step $h$ , we assign the action $a _ { h } = e _ { m } \mathbf { \bar { \Pi } } \in \mathbb { R } ^ { M }$ where $e _ { m }$ is an one-hot vector2 with randomly chosen $m$ from $\mathcal { M }$ base learners. This exploration policy encourages the agent to take different individual base learners, efficiently collecting the observations on not only the sampled base learner performance but also various dynamic ensemble patterns. In addition this requires no prior knowledge on the base learners.
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# 5 EXPERIMENTS
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The extensive experiments are conducted to demonstrate the effectiveness of the proposed dynamic ensemble approach in adapting the ensemble strategy to the time series item and prediction timestamp in Section 5.1. Then, we spend to investigate properties of our ensemble methods from dynamic weights to the phenomena of boosting the performance of the auto-regressive base learner by feeding the better ensemble sample in Section 5.2.
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# 5.1 BENCHMARK EXPERIMENTS ON DYNAMIC ENSEMBLE
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# 5.1.1 EXPERIMENT SETUP
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Datasets and base learners. We perform experiments on four real benchmark datasets that are widely used in forecasting literature: exchange rate, elec, traf and solar from (Salinas et al., 2019). For more dataset details, see appendix A.1. We consider the global deep learning based probabilistic forecasters from GluonTS (Alexandrov et al., 2020b): DeepAR (Salinas et al., 2020), MQ-CNN (Wen et al., 2017; Park et al., 2022), NBEATS(Oreshkin et al., 2019), TFT (Lim et al., 2021) and Transformer (Vaswani et al., 2017). Since the performance of DeepAR can be heavily dependent on the distribution outputs, we trained DeepAR with three different distribution outputs: Gaussian, Student’s t and Poisson distribution referred as DeepAR-G, DeepAR-T and DeepAR-P, respectively. All base learners are trained using the default configurations in GluonTS (Alexandrov et al., 2020b) .
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MDP formulation and RL training To evaluate the performance of our general dynamic ensemble framework, we take the most general ensemble dynamics, which is the hybrid quantile ensemble dynamics. In particular, we will apply the auto-regressive ensemble dynamics to the DeepAR models with different distribution outputs and apply the direct ensemble dynamics to the rest of the base learners. The samples from the DeepAR models from the previous timestamps will then recursively feed as the input to DeepAR models at the next timestamps. In defining the reward function, we adopt the mean weighted quantile loss (see Equation 7 in Appendix) as the accuracy measurement of our predictions. RL algorithm (DDPG) is implemented in PyTorch (Paszke et al., 2019) and trained on AWS Sagemaker (Liberty et al., 2020) with ml.p3.2xlarge instances. Train and test are done with TS-GYM specific to the given dataset.
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Ensemble baselines We compare our RL-based dynamic ensemble approach with the following static ensemble baselines:
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• Mean/Median: for each item and timestamp, take a simple mean/median of all base learners. • Global optimal ensemble: of all of the possible weights of base learners which are shared across items and timestamps, choose the weight for which the associated convex combinations of base learners lead to the best performance in the backtest validation set. • Winner-takes-all(WTA): choose the single base learner which leads to the best performance in the backtest validation set.
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# 5.1.2 BENCHMARK RESULTS
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Message 1: Our hybrid dynamic ensembles is the best or at least on par against other 4 baselines. We evaluate the time series forecasting results by the mean weighted quantile loss defined in Equation Equation 7 in the appendix. The results of all dynamic ensemble approaches including our hybrid quantile ensemble dynamics are summarized in Table 1. From the results in Table 1, we can further report three metrics, winning rate, average ranking, and averaged stability score (amount of $\%$ degradation compared with winining method). For winning rate, our RL-hybrid ensemble is $50 \%$ (wins in two out of four datasets) against other 4 baselines whereas Median and Winner-takes-all ensemble won $25 \%$ respectively. In the average ranking, Median and our RL-hybrid method is 1.75 and 2 respectively whereas Mean and WTA method is 3.75 and 3.5 respectively. In terms of stability score, our RL-hybrid and Median ensemble is $- 1 0 \%$ and - $- 1 5 \%$ respectively whereas Mean and WTA method is at least $- 1 0 0 \%$ and $- 7 0 \%$ . Please see more detailed analysis dataset by dataset in Appendix B.
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Message 2: Overfitting and distribution shift hinders coherent ensembles over all ensemble methods. We also observe the over-fitting of some base learners from the results of Winner-takes-all. In exchange rate, elec and solar datasets, the best base learner in the backtest validation set is not the best base learner in the prediction testing window. It would be challenging to learn a good ensemble strategy in this situation. However, our approach can overcome this over-fitting issue to some extend and still be able to learn good ensemble policy for exchange rate and solar datasets. This is partially because the ensemble policy is trained using the entire time series dataset instead of just the backtest window. In addition, although Winner-takes-all gives the best forecasting accuracy for traf, the severe over-fitting of $\mathtt { M Q \mathrm { - } C N N }$ (see accuracies inside parenthesis of Table 1) slightly degrades the performance of our approach since the uniform weights are encouraged for the nearly-optimal base learners in our ensemble framework.
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<table><tr><td>Base learner/ Ensemble strategy</td><td>exchange rate</td><td>elec</td><td>traf</td><td>solar</td></tr><tr><td>DeepAR-T</td><td>0.0075</td><td>0.0548</td><td>0.0879 (0.113)</td><td>0.3252</td></tr><tr><td>DeepAR-G</td><td>0.0067</td><td>0.0618</td><td>0.1140</td><td>0.3117</td></tr><tr><td>DeepAR-P</td><td>0.2261</td><td>0.0910</td><td>0.9828</td><td>0.3137</td></tr><tr><td>Transformer</td><td>0.0298</td><td>0.0266</td><td>0.0908</td><td>0.3584</td></tr><tr><td>MQ-CNN</td><td>0.0133</td><td>0.0544</td><td>1.8793 (0.166)</td><td>0.7735</td></tr><tr><td>TFT</td><td>0.0060</td><td>0.0844</td><td>0.1144</td><td>0.3253</td></tr><tr><td>NBEATS</td><td>0.0106</td><td>0.0480</td><td>0.2270</td><td>0.9983</td></tr><tr><td>Mean</td><td>0.0359</td><td>0.0490</td><td>0.2029</td><td>0.3790</td></tr><tr><td>Median</td><td>0.0090</td><td>0.0489</td><td>0.0905</td><td>0.3256</td></tr><tr><td>Global optimal</td><td>0.0124</td><td>0.0790</td><td>0.1991</td><td>0.3913</td></tr><tr><td>Winner-takes-all</td><td>0.0133</td><td>0.0548</td><td>0.0879</td><td>0.7735</td></tr><tr><td>RL-hybrid(Ours)</td><td>0.0060</td><td>0.0544</td><td>0.1141</td><td>0.3058</td></tr></table>
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Table 1: Performance comparison on real-world benchmark datasets. The winning method among ensemble methods are made bold. The retangular is the one selected in Winner-takes-all ensemble method. The values in the parenthesis are the accuracy evaluated in the backtesting window.
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# 5.2 INVESTIGATING PROPERTIES OF DYNAMIC ENSEMBLES
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Property 1: Capturing time-varying ensemble weights. We first demonstrate the capability of our dynamic ensemble framework to learn the time-varying ensemble weights when the optimal base learners vary along the prediction horizon. We examine policy trained on the motivating example on the dataset Solar in Section 1 more closely. Our dynamic ensemble approach is able to learn ensemble weights which are consistent with the time-varying pattern of the optimal base learners. In particular, we can see from Figure 3a that (1) only Transformer, TFT and DeepAR are given positive ensemble weights during the prediction, (2) the ensemble weights of transformer remain relatively high in prediction timestamps $[ 0 , 6 ] \cup [ 1 6 , 2 9 ]$ while dropping below 0.1 during prediction timestamps [7, 15], (3) the ensemble weights of TFT remain 0 in prediction timestamps $[ 0 , 5 ] \cup [ 1 6 , 2 9 ]$ but dominate the ensemble weights of transformer in prediction timestamps [7, 15], (4) the ensemble weights of DeepAR remain high during the entire prediction horizon because its relatively good performance during the entire prediction horizon.
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Figure 3: The learned ensemble weights are consistent with the performances of the base learners. over the prediction horizon. QL and rank are averaged over all items in the dataset.
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Property 2: Boosting the performance of auto-regressive (AR) forecasters. Improving the base learners’ performance is important for the improving the accuracy of the final ensembled predictions, and for allowing a broader set of admissible ensemble polices (in the extreme case, if all base learners perform equally well, then any ensemble strategy is optimal). We demonstrate the capability of auto-regressive ensemble (as shown in Figure 3b) on boosting the performance of AR forecasters. In particular, we focus on the DeepAR models with different distribution outputs: Gaussian , Student’s t and Poisson distribution and train the ensemble policy using our dynamic ensemble approach with auto-regressive ensemble dynamics on exchange rate dataset. Figure 3b shows the mean weighted quantile losses of the DeepAR-G over the prediction horizon for 3 different strategies:
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• using DeepAR with Gaussian distribution (denoted as DeepAR-G original); • using DeepAR with Gaussian distribution, but feed the true target value as the autoregressive input in Equation 4b (denoted as DeepAR-G w/ target); using the DeepAR with Gaussian distribution, but feed the samples from the mixture of distributions in Equation 3 as the auto-regressive input in Equation $^ { 4 \mathrm { b } }$ (denoted as DeepAR-G w/ ensemble);
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We can observe that by feeding a more accurate input to the auto-regressive forecaster, DeepAR-G w/ ensemble improves DeepAR-G original consistently over the entire prediction horizon. The mean weighted quantile loss for DeepAR-G original and DeepAR- $- G$ w/ ensemble are 0.01466 and 0.00988, respectively, which demonstrates a $3 2 . 6 \%$ performance boost.
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Property 3: Auto-regressive dynamic ensemble is more powerful than direct dynamic through ablation study. We conduct the ablation on AR dynamics that is explicitly considered in our algorithm in comparison to the methods where the AR feedback is not explicit. We term these ablations as RL-auto and RL-naive. We consider the solar dataset with base learners DeepAR-T, DeepAR-G and DeepAR-P.
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Table 2: Ablation study to compare auto-regressive vs direct dynamic.
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<table><tr><td>Base learner /Ensemble strategy</td><td>DeepAR-T</td><td>DeepAR-G</td><td>DeepAR-P</td><td>Mean</td><td>Global Optimal</td><td>RL-naive</td><td>RL-auto</td></tr><tr><td>solar</td><td>0.3252</td><td>0.3117</td><td>0.3137</td><td>0.3088</td><td>0.3302</td><td>0.3148</td><td>0.2840</td></tr></table>
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Table 2 highlights the significance of AR dynamics that is explicit in our MDP formulation. With same set of base learners the AR dynamics is able to achieve ${ \bf 1 1 \% }$ better result than the naive dynamics. Further, the RL-auto is better $( 8 \% )$ than all models/ensemble strategy considered, thus showing the significance of base learner boosting via AR feedback.
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Yasuko Matsubara, Yasushi Sakurai, Willem G van Panhuis, and Christos Faloutsos. Funnel: automatic mining of spatially coevolving epidemics. In KDD, pp. 105–114. ACM, 2014b.
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Boris N Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-BEATS: Neural basis expansion analysis for interpretable time series forecasting. arXiv:1905.10437, 2019.
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Spiros Papadimitriou and Philip Yu. Optimal multi-scale patterns in time series streams. In Proceedings of the 2006 ACM SIGMOD international conference on Management of data, pp. 647–658, 2006.
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Youngsuk Park, Kanak Mahadik, Ryan A Rossi, Gang Wu, and Handong Zhao. Linear quadratic regulator for resource-efficient cloud services. In Proceedings of the ACM Symposium on Cloud Computing, pp. 488–489, 2019.
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Youngsuk Park, Danielle Maddix, François-Xavier Aubet, Kelvin Kan, Jan Gasthaus, and Yuyang Wang. Learning quantile functions without quantile crossing for distribution-free time series forecasting. arXiv:2111.06581, 2021.
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Youngsuk Park, Danielle Maddix, François-Xavier Aubet, Kelvin Kan, Jan Gasthaus, and Yuyang Wang. Learning quantile functions without quantile crossing for distribution-free time series forecasting. In International Conference on Artificial Intelligence and Statistics, pp. 8127–8150. PMLR, 2022.
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# A EXPERIMENT SETUP
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# A.1 REAL-WORLD DATASET
|
| 288 |
+
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| 289 |
+
Table 3 summarizes the four benchmark real-world datasets that we use to evaluate our dynamic ensemble approach.
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| 290 |
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Table 3: Benchmark dataset descriptions
|
| 291 |
+
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| 292 |
+
<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Freq</td><td rowspan=1 colspan=1>Domain</td><td rowspan=1 colspan=1>#Time series</td><td rowspan=1 colspan=1>Prediction length</td></tr><tr><td rowspan=1 colspan=1>exchangerate</td><td rowspan=1 colspan=1>daily</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>30</td></tr><tr><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>2950</td><td rowspan=1 colspan=1>24</td></tr><tr><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>[0,1]</td><td rowspan=1 colspan=1>6741</td><td rowspan=1 colspan=1>24</td></tr><tr><td rowspan=1 colspan=1>solar</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>959</td><td rowspan=1 colspan=1>24</td></tr></table>
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# A.2 IMPLEMENTATION OF DDPG
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We use the DDPG implementation from OpenAI spinning up baselines. The last layer of policy network is a softmax layer with output dimensions as the number of base learners considered. For hyper-parameter tuning we consider the hyper-parameters in Lillicrap et al. (2015) and some specific to dynamic AR ensemble. The final hyper-parameters used for different datasets for the experiment in Section 5.1 is given in Tables 4 and 5. The default weights among AR model parameter is used to set the weights among the AR model if all the AR models in the hybrid dynamics gets zero weight at certain step in the RL; $\lambda$ controls the trade-off as explained in the reward function section. The reward scale is the scaling applied to mean-wQL to be comparable with the secondary reward $r _ { 2 }$ . Round threshold is the number of decimal digits for rounding the mean-wQL to get ranking for base learners.
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A.2.1 EXPERIMENTS IN TABLE 1
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| 299 |
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Table 4: Hyperparameters of DDPG algorithm in various real-world datasets.
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| 300 |
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| 301 |
+
<table><tr><td rowspan=1 colspan=1>Hyperparamters</td><td rowspan=1 colspan=1>exchange rate</td><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>solar</td></tr><tr><td rowspan=1 colspan=1>episodes per epoch</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>start episodes</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>update after episodes</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1> update steps per prediction length</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=1 colspan=1>update every episodes</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>discount factor</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td></tr><tr><td rowspan=1 colspan=1>epochs</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>60</td><td rowspan=1 colspan=1>60</td><td rowspan=1 colspan=1>70</td></tr><tr><td rowspan=1 colspan=1>polyak</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td></tr><tr><td rowspan=1 colspan=1>learning rate for policy</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td></tr><tr><td rowspan=1 colspan=1>learning rate for Q value</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td></tr><tr><td rowspan=1 colspan=1>noise level for action</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.1</td></tr></table>
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| 302 |
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| 303 |
+
# A.3 IMPLEMENTAIONS OF TS-GYM
|
| 304 |
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| 305 |
+
Error metric We evaluate the forecasting error in terms of the mean weighted quantile loss. See the precise definition in the appendix.
|
| 306 |
+
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| 307 |
+
$$
|
| 308 |
+
\frac { 1 } { q } \frac { \sum _ { i = 1 , j = T + 1 , k = 1 } ^ { N , T + h , q } \operatorname* { m a x } \left\{ \tau _ { k } ( z _ { i , j } - \widetilde { z } _ { i , j , k } ) , ( 1 - \tau _ { k } ) ( \widetilde { z } _ { i , j , k } - z _ { i , j } ) \right\} } { \sum _ { i = 1 , j = T + 1 } ^ { N , T + h } | z _ { i , j } | }
|
| 309 |
+
$$
|
| 310 |
+
|
| 311 |
+
where {zi,j}N,T hi=1,j=T +1 are the true values of future time series and $\{ \widetilde { z } _ { i , j , k } \} _ { i = 1 , j = T + 1 , k = 1 } ^ { N , T + h , q }$ are the estimated quantile predictions.
|
| 312 |
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| 313 |
+
Table 5: Hyperparameters of TS-GYM in various real-world datasets.
|
| 314 |
+
|
| 315 |
+
<table><tr><td rowspan=1 colspan=1>Hyperparameters</td><td rowspan=1 colspan=1>exchange rate</td><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>solar</td></tr><tr><td rowspan=1 colspan=1>train batch size</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>200</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>200</td></tr><tr><td rowspan=1 colspan=1>reward scale</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>0.0001</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>round threshold</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>入</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>default weights amongauto-regressive models</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td></tr></table>
|
| 316 |
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|
| 317 |
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# B BENCHMARK RESULT DISCUSSION
|
| 318 |
+
|
| 319 |
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For the more detailed discussion, we can observe that the proposed RL-hybrid method outperforms all base models and baselines on all exchange rate and solar datasets. For exchange rate, which is a regular dataset with clear daily patterns, a single base learner usually performs very well. Our RL-hybrid method is able to identify the single best base learner (TFT). On the other hand, exchange rate is less regular and more challenging. Our RL-hybrid method is better $( 2 \% )$ than all base models and baselines considered. This is because our dynamic ensemble method are able to capture the time-varying patterns of the base learners’ performance profile and boost the performance of the auto-regressive base learners (see Section 5.2 for more discussions).
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "DYNAMIC ENSEMBLE FOR PROBABILISTIC TIMESERIES FORECASTING VIA DEEP REINFORCEMENT LEARNING ",
|
| 5 |
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"text_level": 1,
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| 6 |
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"page_idx": 0
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| 13 |
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},
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| 14 |
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{
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| 15 |
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"type": "text",
|
| 16 |
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"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
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"bbox": [
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| 18 |
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| 19 |
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| 20 |
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| 21 |
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| 23 |
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"page_idx": 0
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| 24 |
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| 25 |
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{
|
| 26 |
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"type": "text",
|
| 27 |
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"text": "ABSTRACT ",
|
| 28 |
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"text_level": 1,
|
| 29 |
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"bbox": [
|
| 30 |
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| 31 |
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| 32 |
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| 33 |
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| 34 |
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| 35 |
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"page_idx": 0
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| 36 |
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},
|
| 37 |
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{
|
| 38 |
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"type": "text",
|
| 39 |
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"text": "It is well known that ensemble improve the accuracy of forecasting tasks. However, most of ensembling strategies designed for probabilistic time series forecasting are static methods, in the sense that they either assume the time-invariant ensemble strategies over the prediction horizon, or are non-adaptive to the forecast start point. In addition, the static methods naively rely on the predictions of the base forecasters but fail to utilize base learners themselves efficiently. In this paper, we propose a novel dynamic ensemble policy to overcome three major limitations mentioned above via deep Reinforcement Learning (RL) framework. To learn such a policy, we design a Markov Decision Process (MDP), together with our environment (TS-GYM) that supports the interaction between the agent or ensembler, offline datasets and base learners. In doing so, we effectively leverage the power of the ensemble to improve each of the base learners by reducing the error accumulation of each base learner via consecutively feeding a better ensembled sample to each base learner. The proposed ensembling method has several desirable properties such as uncertainty quantification and the ability to generate sample path, on top of significant performance gain. The effectiveness of the proposed framework is demonstrated on multiple synthetic and real-world experiments. ",
|
| 40 |
+
"bbox": [
|
| 41 |
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| 42 |
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| 43 |
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| 44 |
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| 45 |
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| 46 |
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"page_idx": 0
|
| 47 |
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},
|
| 48 |
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{
|
| 49 |
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"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
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|
| 54 |
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| 55 |
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| 56 |
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| 57 |
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|
| 58 |
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"page_idx": 0
|
| 59 |
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},
|
| 60 |
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{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Time series data occur naturally in countless domains including supply chain optimization (Larson, 2001; Wen et al., 2017), medical analysis (Keogh et al., 2001; Matsubara et al., 2014b), financial analysis (Zhu & Shasha, 2002; Hallac et al., 2017), sensor network monitoring (Papadimitriou & Yu, 2006; Letchner et al., 2009), cloud computing (Park et al., 2019; 2021), optimal control of vehecle (Kim et al., 2020) and social activity mining (Mathioudakis et al., 2010; Matsubara et al., 2012; 2014a). Among the applications of ML-based time series analysis, forecasting is arguably one of the most sought-after, due to its importance in industrial, social, and scientific applications. For example, forecasting plays a key role in automating and optimizing operational processes in most businesses and enables data driven decision making. Forecasts of product supply and demand are used for optimal inventory management, staff scheduling and topology planning, and are more generally a crucial technology for most aspects of supply chain optimization. In order to make optimal decisions, predictive uncertainties need to be taken into account, making probabilistic forecast a desirable property of time series models (Benidis et al., 2022). ",
|
| 63 |
+
"bbox": [
|
| 64 |
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| 65 |
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| 66 |
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| 67 |
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| 68 |
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],
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| 69 |
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"page_idx": 0
|
| 70 |
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},
|
| 71 |
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{
|
| 72 |
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"type": "text",
|
| 73 |
+
"text": "In practice, one often encounters complex time series, making it difficult to find a single best model that excels at short-term, mid-term, and long-term forecasting scenarios. In such cases, different forecasting models usually perform well on different data regimes at different time steps. As a motivating example, Figure 1a shows the relative ranking of the performances of 5 popular forecasting models on the dataset Solar. In this example, Transformer excels at shorter and longer-term forecasts while DeepAR and TFT shine in the mid-term scenario. It is thus desirable to have an ensembling strategy that has different weights at each time step. Therefore, the traditional ensembling strategy in time series forecast, which assumes that ensemble weights do not vary along the forecasting horizon is not sufficient to capture the non-stationary patterns of base learners’ performance profile. Furthermore, popular auto-regression based models are known to have increasing prediction errors as the prediction horizon stretches further, and the performance degrades dramatically when the prediction horizon is sufficiently large (Salinas et al., 2020). As shown in the blue curve of Figure 1b, the prediction error increases for “DeepAR-G original”(“G” means using the Gaussian distribution as the output distribution and “original” means using the original implementation of DeepAR) over the prediction horizon on exchange rate dataset. On the other hand, if we can provide base learners such as DeepAR with more accurate estimations of the future as the auto-regressive input, the prediction error can be significantly decreased for the long horizon predictions (see the orange curve in Figure 1b). The huge difference in the prediction error between these two cases show the huge potential to improve the auto-regression based models if we can provide more accurate estimations during the prediction horizon. However, none of the traditional ensemble methods utilize the ensemble predictions as the feedback to boost the performance of the auto-regression based models. Motivated by the above examples, the natural question arises whether we can develop a general dynamic ensembling approach that overcomes all the major limitations of the traditional static ensemble methods and further improve the prediction accuracy for the probabilistic time-series forecasting? ",
|
| 74 |
+
"bbox": [
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| 75 |
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| 80 |
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|
| 81 |
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| 82 |
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{
|
| 83 |
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"type": "text",
|
| 84 |
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"text": "",
|
| 85 |
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"bbox": [
|
| 86 |
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| 87 |
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| 88 |
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| 89 |
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| 90 |
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],
|
| 91 |
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"page_idx": 1
|
| 92 |
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},
|
| 93 |
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{
|
| 94 |
+
"type": "image",
|
| 95 |
+
"img_path": "images/0d9f37094b57c7d3d7abd24dfe5251f1f6b197322ca122f8a582da6e2ade3688.jpg",
|
| 96 |
+
"image_caption": [],
|
| 97 |
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"image_footnote": [],
|
| 98 |
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"bbox": [
|
| 99 |
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| 100 |
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| 101 |
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| 102 |
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| 103 |
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| 104 |
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|
| 105 |
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},
|
| 106 |
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{
|
| 107 |
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"type": "text",
|
| 108 |
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"text": "(a) The ranks of 5 base learners along the prediction horizon on Solar dataset. The ranks are based on the mean weighted quantile loss over the quantiles [0.1, 0.5, 0.9] and averaged over all items in each dataset. ",
|
| 109 |
+
"bbox": [
|
| 110 |
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| 111 |
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| 112 |
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| 113 |
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| 114 |
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],
|
| 115 |
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"page_idx": 1
|
| 116 |
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},
|
| 117 |
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{
|
| 118 |
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"type": "image",
|
| 119 |
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"img_path": "images/4c2430d5add1a767687c9f209ad59cb6998c4ae2e65e9f0aa2b74a9eb39efa48.jpg",
|
| 120 |
+
"image_caption": [
|
| 121 |
+
"(b) The gap between the “DeepAR original\" and “DeepAR w/ target” shows the potential improvement we can gain if the accuracy of the auto-regressive input to DeepAR can be improved. "
|
| 122 |
+
],
|
| 123 |
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"image_footnote": [],
|
| 124 |
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"bbox": [
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| 125 |
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| 126 |
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| 128 |
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| 129 |
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| 130 |
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"page_idx": 1
|
| 131 |
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},
|
| 132 |
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{
|
| 133 |
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"type": "text",
|
| 134 |
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"text": "Figure 1: Two motivations on the need of dynamic ensembles, beyond static ensembles. ",
|
| 135 |
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|
| 136 |
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| 137 |
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| 139 |
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| 141 |
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"page_idx": 1
|
| 142 |
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},
|
| 143 |
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{
|
| 144 |
+
"type": "text",
|
| 145 |
+
"text": "To address the above mentioned challenges, in this work, we develop a general dynamic ensemble framework for probabilistic multi-horizon time series forecasting. Our contributions can be summarized as follows: ",
|
| 146 |
+
"bbox": [
|
| 147 |
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176,
|
| 148 |
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|
| 149 |
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| 150 |
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| 151 |
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],
|
| 152 |
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"page_idx": 1
|
| 153 |
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},
|
| 154 |
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{
|
| 155 |
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"type": "text",
|
| 156 |
+
"text": "• This work is the first one that proposes a dynamic ensemble policy suitable for probabilistic time series forecasting with the properties of sequential weighting, being adaptive, and quantile ensemble. • We formulate this as a Markov Decision Process (MDP) with a careful design of the rewards, transition dynamics, and ensemble action policy. In particular, the state evolution in our formulation depends on the ensemble strategy through our novel transition dynamics design. • To solve this MDP problem, we design a time series gym (TS-GYM) environment which implements the interaction between the time series off-line dataset, base learners and ensemble agent. Through this interaction, actor-critic based deep RL method with our “random extreme point” exploration strategy can learn optimal ensemble policy. • The extensive experiments show the advantages of our ensemble dynamic framework. In particular, we demonstrate that our general dynamic ensemble framework can (1) learn the optimal time-varying ensemble weights along the multi-horizon prediction, (2) be adaptive to any forecast start time, (3) boost the performance of the auto-regressive base learners, and (4) result in better performance than other potential variants on real-world datasets. ",
|
| 157 |
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"bbox": [
|
| 158 |
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| 159 |
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| 160 |
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| 163 |
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"page_idx": 1
|
| 164 |
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},
|
| 165 |
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{
|
| 166 |
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"type": "text",
|
| 167 |
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"text": "2 RELATED WORK ",
|
| 168 |
+
"text_level": 1,
|
| 169 |
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"bbox": [
|
| 170 |
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| 171 |
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|
| 176 |
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},
|
| 177 |
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{
|
| 178 |
+
"type": "text",
|
| 179 |
+
"text": "Probabilistic time series forecasting In recent years there has been an increasing interest in “probabilistic forecasting”, namely forecasting models that account for the data’s uncertainty by modeling the distribution of target values, rather than predicting a single point estimate. Probablistic forecasting is useful for business purposes such as supply and demand, inventory management, staff scheduling and topology planning (Larson, 2001). Modern open source packages such as Kats (facebookresearch, 2021), Merlion (Bhatnagar et al., 2021) and GluonTS (Alexandrov et al., 2020a) offer probabilistic forecasting, and include some popular probabilistic forecasters such as Prophet (Taylor & Letham, 2018), and deep learning probabilistic forecasters such as DeepAR (Salinas et al., 2020), MQ-CNN (Wen et al., 2017; Park et al., 2022), MQF2 (Kan et al., 2022), NBEATS (Oreshkin et al., 2019), TFT (Lim et al., 2021) and Transformer (Vaswani et al., 2017). There are several advances in improving those models in adversarial robustness (Yoon et al., 2022; Liu et al., 2022) and few-shot learning (Jin et al., 2022). ",
|
| 180 |
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{
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| 189 |
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"type": "text",
|
| 190 |
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"text": "",
|
| 191 |
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"bbox": [
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"type": "text",
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"text": "Time series ensemble The literature on ensembling methods for time series predictions have focused solely on static ensembling strategies, namely ones that have access to the predictions of the base learners but not to the base learners themselves. In that situation, a debate on the theory of ensembling for time series was sparked by an empirical observation that a simple average of the base learners is often superior to more sophisticated ensemble methods (a problem called the “forecast combination puzzle”, see Stock & Watson (2004) and Bates & Granger (1969)). See Smith & Wallis (2009), Claeskens et al. (2016), and Elliott (2011)). While theory lags, however, sophisticated static ensembling methods have often been observed to work well. (See Donaldson & Kamstra (1996), Moon et al. (2020), and Massaoudi et al. (2021). Particularly interesting is Gastinger et al. (2021), with a large empirical study.) ",
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"type": "text",
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"text": "Contrary to the situation considered in these papers, literature on ensembling methods that have direct access to the base learners, rather than only to their predictions, is limited. Recently, RL based approaches are proposed in Saadallah & Morik (2021) and $\\mathrm { F u }$ et al. (2022). Saadallah & Morik (2021) consider action dependent state (window of ensemble predictions) transition. Their work focus on online policy learning with update timing determined by a concept-drift detection algorithm. In Fu et al. (2022) the state (time series for a given context window and base learners performance at the next window) transition is action independent with action taken for $H$ steps at a time. In addition, their methods are only designed for the point based forecasting problem and do not demonstrate the capability of capturing the non-stationary ensemble weights. ",
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"type": "text",
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"text": "3 PRELIMINARIES ",
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"type": "text",
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"text": "3.1 PROBABILISTIC TIME-SERIES FORECASTING ",
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"text_level": 1,
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"text": "Suppose we have a panel of $n$ time series, where the $i$ -th time series consists of observations $z _ { i , t } \\in \\mathbb { R }$ with (optional) input covariates $x _ { i , t } \\in \\mathbb { R } ^ { d }$ , as $t$ varies over time at fixed discrete intervals. For an $i$ -th time series (often called $i$ -th item), we wish to make predictions for the next $H$ timestamps, namely of $z _ { i , T + 1 : T + H }$ from the forecast start time $T + 1$ , given the history of that item’s observations $z _ { i , 1 : T }$ and (optional) the associated historical and future covariates $x _ { i , 1 : T + H }$ . In this paper we will focus on global forecasters, namely a single univariate model trained on all of the items together, and accepting only a single item at inference. For notational simplicity we will drop the item index $i$ and covariates $x _ { i , t }$ unless explicitly stated. We now formally define a forecasting model as a set of random variable valued functions $\\{ f _ { h } \\} _ { h = 1 } ^ { H }$ such that, for $h = 1 , . . . , H$ ",
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},
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{
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"type": "equation",
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"img_path": "images/cbcaa941a8288759368287797a349f15ac8f740d6270ebc0daa81f890f2b8000.jpg",
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"text": "$$\nZ _ { T + h } = f _ { h } \\big ( z _ { 1 : T } , \\xi _ { T + h - 1 } \\big ) ,\n$$",
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"text_format": "latex",
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"bbox": [
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"type": "text",
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"text": "where $\\xi _ { T + h - 1 }$ is the hidden state variable passed from the previous (or older) step. The evolution of $f _ { h }$ and $\\xi _ { T + h - 1 }$ depend on the type of the base model. For the auto-regressive model which uses the recursive prediction strategy, the hidden state $\\cdot$ is generated by passing a sample $\\_$ from previous time step to the forecaster decoder for the next prediction in a recursive manner. Often the decoder is homogeneous, i.e., $\\cdot$ for $h = 1 , \\ldots , H$ . On the other hand, Seq2Seq model which uses the direct prediction strategy, directly forecast the future time series without involving the evoluation of the hidden state, i.e., $\\xi _ { T + h - 1 } = \\xi _ { T }$ for all $h = 1 , \\ldots , H$ . Refer to Alexandrov et al. (2020b) for the detailed modeling. In Section 4, we will explore a different choice for the auto-regressive step, using the entire ensemble. ",
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"text": "Then, the associated $\\tau$ -quantile predictions can be followed as $\\hat { z } _ { T + h } ^ { \\tau } = q _ { \\tau } \\left( Z _ { T + h } \\right)$ where, for a random variable $Z \\in \\mathbb { R }$ with its culmulative distribution $F _ { Z }$ and a quantile level $\\tau \\in ( 0 , 1 )$ , $q _ { \\tau }$ is denoted as the quantile function, i.e., $q _ { \\tau } ( Z ) : = F _ { Z } ^ { - 1 } ( \\tau ) = \\operatorname* { i n f } \\left\\{ z \\in \\mathbb { R } : \\tau \\leq F _ { Z } ( z ) \\right\\}$ . ",
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"type": "text",
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"text": "3.2 FORECASTING ENSEMBLE ",
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| 294 |
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"text_level": 1,
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"type": "text",
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"text": "For each m-th base learner, we denote zˆτk,mT+h as the $\\tau _ { k }$ -quantile prediction at time step $T + h$ on a quantile level where $\\tau _ { k } \\in \\{ \\tau _ { k } \\} _ { k = 1 } ^ { K }$ . Then, $\\left\\{ \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } \\right\\} _ { k = 1 , m = 1 } ^ { K , M }$ is denoted as a pool of quantile predictions at time step $T + h$ over $M$ base learners and $K$ quantile levels. A general ensemble predictions can be formally expressed as a (linear) weighted combination of predictions of the individual base models, at each prediction step $h = 1 , \\ldots , H$ , ",
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"type": "equation",
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"img_path": "images/9bf17d192e9c1040cec0a29756dd246a5f98b6f6bb3d062cf3981eadb6049e71.jpg",
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"text": "$$\n\\hat { z } _ { T + h } ^ { \\tau , \\mathrm { e s } } = \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \\hat { z } _ { T + h } ^ { \\tau , m } ,\n$$",
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"text_format": "latex",
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"text": "$w _ { h } ^ { m } \\geq 0$ with $\\textstyle \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } = 1$ are the ensemble weights. ",
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{
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"type": "text",
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"text": "3.3 REINFORCEMENT LEARNING ",
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| 341 |
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"text_level": 1,
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"text": "Reinforcement learning (RL) is usually formulated as a Markov Decision Process (MDP), which can be defined as a tuple $( S , { \\mathcal { A } } , { \\mathcal { P } } , r , \\gamma , H )$ where $s$ is the state space, $\\mathcal { A }$ is the action space, $\\mathcal { P } : \\mathcal { S } \\times \\mathcal { A } \\mathcal { S }$ is the transition function, $r : \\mathcal { S } \\times \\mathcal { A } \\to \\mathbb { R }$ is the reward function, $\\gamma \\in ( 0 , 1 )$ is the discount factor and $H > 0$ is the horizon length of each episode. At each state $s \\in \\mathcal { S }$ , the RL agent takes an action $a \\in { \\mathcal { A } }$ , transits to the next state $s ^ { \\prime } \\in { \\mathcal { S } }$ under the dynamics $\\mathcal { P }$ and receives a reward $r ( s , a )$ . The goal of an MDP is to learn a policy $\\pi : { \\mathcal { S } } A$ that maximizes the total obtained rewards $\\begin{array} { r } { \\operatorname* { m a x } _ { \\pi } J ( \\pi ) = \\mathbb { E } _ { \\tau } \\left[ \\sum _ { h = 0 } ^ { H - 1 } \\gamma ^ { h } r ( s _ { h } , a _ { h } ) \\bigg | \\pi \\right] , } \\end{array}$ where the expectation is over the trajectory $\\tau = \\{ \\big ( s _ { 0 } , a _ { 0 } , r \\big ( s _ { 0 } , a _ { 0 } \\big ) \\big ) . . . . , \\big ( s _ { H } , a _ { H } , r \\big ( s _ { H } , a _ { H } \\big ) \\big ) \\}$ where $a _ { h } = \\pi ( s _ { h } )$ . ",
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"type": "text",
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"text": "4 DYNAMIC ENSEMBLE FRAMEWORK ",
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| 364 |
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"text_level": 1,
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{
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"type": "text",
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"text": "In this section, we mainly focus on how to select a sequence of ensemble weights $( w _ { 1 } , w _ { 2 } , \\dots , w _ { H } )$ with $w _ { h } \\in \\mathbb { R } ^ { M }$ over $M$ base learners by learning a ensemble policy $\\pi$ . Especially in the presence of auto-regressive base learners, ensemble weights chosen at the step $\\cdot$ may affect the forecasting of auto-regressive base learners and also ensemble weights chosen at the next step $h + 1$ (see Section 4.1.1 for more details). With this intuition, we will take a reinforcement learning approach to learn an optimal policy function $\\pi$ that provides the optimal ensemble weights sequentially. ",
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"type": "text",
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"text": "In Section 4.1, we give a high-level overview of the MDP formulation for the multi-horizon probabilistic time series forecasting problems. In particular, the classes of ensembled sampling strategies and predictions which determine the state transformation and state transition are discussed in Section 4.1.1 and the careful design of reward computation is explained in Section 4.1.2. Based on the formulated MDP, we then design our simulated environment, TS-GYM (in Section 4.2) which provides the interaction among the time series datasets, base learners and the dynamic ensemble agent. Finally, we describe how to employ deep reinforcement learning with our “random extreme point” exploration strategy to learn the optimal ensemble policy in Section 4.3. ",
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"type": "text",
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"text": "4.1 MDP FORMULATION ",
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| 398 |
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"text_level": 1,
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{
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"type": "text",
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"text": "We describe the high-level formulation of the MDP for our dynamic time-series ensemble framework. Once each episode starts with $h = 1$ , the environment fixes an arbitrary forecasting start point $T$ , and then starts to provide a time series pair of both historical input $z _ { 1 : T }$ and corresponding future (backtest) output $z _ { T + 1 }$ as well as corresponding quantile predictions $\\big \\{ \\hat { z } _ { T + 1 } ^ { \\tau , m } \\big \\}$ from alll $M$ base models for the next step $T + 1$ . (We defer the details implementation of the environment to Section 4.2). The agent will then decide the ensemble weights to compute the ensembled predictions, and update the ensemble policy based on the accuracy of the ensembled predictions. Depending on the type of ensemble dynamics, the ensembled predictions may also affect the base learners’ future predictions. Then, in the next step $h = 2$ , the environment provides next time series output $z _ { T + 2 }$ and associated predictions $\\big \\{ \\hat { z } _ { T + 2 } ^ { \\tau , m } \\big \\}$ and go on. See Figure 2a for a high level schema. ",
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{
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"type": "text",
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"text": "More formally, for each step $h = 1 , \\ldots , H$ of an episode, given the information provided by the environment (e.g., historical observation $z _ { 1 : T }$ , and future (backtest) observation $z _ { T + h }$ , a pool of all quantile predictions $\\{ \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } \\} _ { k = 1 , m = 1 } ^ { K , M }$ , and step $h$ ) , we define MDP as follows: ",
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{
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"type": "text",
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"text": "• the fixed-size state $s _ { h } = \\Big \\{ z _ { 1 : T } , \\big \\{ \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } \\big \\} _ { k = 1 , m = 1 } ^ { K , M } , h \\Big \\} ,$ , \n• the action $a _ { h } = \\{ w _ { h } ^ { m } \\} _ { m = 1 } ^ { M } = \\pi \\big ( s _ { h } \\big )$ , $M$ -ensemble weights $w _ { h } ^ { m }$ from a policy function $\\pi$ , • the state transition $\\mathcal { P } ( s _ { h + 1 } \\mid s _ { h } , a _ { h } )$ governed by ensemble dynamics in Section 4.1.1, • the reward $R \\left( { { s _ { h } } , { a _ { h } } ; { z _ { T + h } } } \\right)$ 1 which evaluates ensemble prediction against ground-truth $z _ { T + h }$ in Section 4.1.2 . ",
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},
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| 440 |
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{
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| 441 |
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"type": "text",
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| 442 |
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"text": "4.1.1 ENSEMBLE DYNAMICS $\\mathcal { P }$ AND ENSEMBLED QUANTILES ",
|
| 443 |
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"text_level": 1,
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{
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"type": "text",
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| 454 |
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"text": "Defining state transition $\\mathcal { P }$ , which we call ensemble dynamics, narrows down how to construct quantile predictions over M base learners {zˆτk,mT +h }K,Mk=1,m =1 ∈ sh. Here, we proposed three strategies: direct dynamic, auto-regressive dynamic and their composition. The idea of direct ensemble is similar to Seq2Seq models which employs the direct prediction strategy. The idea of auto-regressive dynamic is based on auto-regressive models where you recursively feed a new ensembled sample to each base learner for the next prediction. The ensemble dynamics appear at the step represented by the red arrow line in Figure 2a. ",
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{
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"type": "text",
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"text": "Direct dynamic. learner itself over $H$ s a direct ensembling ovehorizon, i.e., we compute $\\hat { z } _ { T + h } ^ { \\tau , m } = q _ { \\tau } \\left( Z _ { T + h } ^ { m } \\right)$ first cofor all $h = 1 , \\ldots , H$ es by base, based on Equation 1. Then the final quantile ensemble becomes $\\begin{array} { r } { \\hat { z } _ { T + h } ^ { \\tau , e s } = \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \\hat { z } _ { T + h } ^ { \\tau , m } } \\end{array}$ in Equation 2. Note that the base learner’s predictions are not affected by the ensembling. In other words, the transition dynamic $\\mathcal { P } ( s _ { h + 1 } \\mid s _ { h } , a _ { h } ) = \\mathcal { P } ( s _ { h + 1 } \\mid s _ { h } )$ is actually irrelevant to the ensembling weights. ",
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| 466 |
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{
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"type": "text",
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"text": "Auto-regressive dynamic. In this dynamic, we generate an (intermediate) ensembled sample $p _ { T + h }$ , which is fed into each autoregressive base leaner in a recursive manner. This ends up forming a sample path through which we can compute the final ensembled (empirical) quantile prediction $\\hat { z } _ { T + h } ^ { \\tau , \\mathrm { e s } }$ ",
|
| 477 |
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"bbox": [
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"text": "To begin with, we generate a sample path $( \\hat { z } _ { T + 1 } ^ { m } , \\dots , \\hat { z } _ { T + H } ^ { m } )$ for each base learner as follows: First, for each step , we sample $p _ { T + h }$ from mixture of base learners’ distributions $\\mathbb { P } ( Z _ { T + h } ^ { m } )$ proportional to ensemble weights $w _ { h } ^ { m }$ , i.e., ",
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"img_path": "images/3dbcd5efa23c6c3fb14257d85f5d11cd2bf718edfa48545ae6c8b1c4e13372ff.jpg",
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"text": "$$\np _ { T + h } \\sim \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \\mathbb { P } ( Z _ { T + h } ^ { m } ) .\n$$",
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"text": "Second, we feed $p _ { T + h }$ to each autoregressive base learner, i.e., ",
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"text": "$$\n\\begin{array} { c } { { Z _ { T + h + 1 } ^ { m } = f ^ { m } ( z _ { 1 : T } , \\xi _ { T + h } ^ { m } ) , } } \\\\ { { \\xi _ { T + h } ^ { m } = g ^ { m } ( p _ { T + h } , \\xi _ { T + h - 1 } ^ { m } ) . } } \\end{array}\n$$",
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"text": "where get a sgenera $g ^ { m }$ represents thple for each sample path $m$ on dynamics for the hidden state , which can be operated in a recbase learners. $\\xi _ { T + h } ^ { m }$ . Lastly, wee manner to $\\hat { z } _ { T + h . } ^ { m } \\sim Z _ { T + h . } ^ { m }$ $\\_$ for all ",
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"text": "After collecting a set of sample paths $\\{ ( \\hat { z } _ { T + 1 } ^ { m } , \\dots , \\hat { z } _ { T + H } ^ { m } ) _ { l } \\} _ { l = 1 , m = 1 } ^ { L , M }$ where $( \\hat { z } _ { T + 1 } ^ { m } , \\dots , \\hat { z } _ { T + H } ^ { m } ) _ { l }$ is $l$ -th sample path above for the $m$ -base learner, we construct the empirical marginal distribution $\\cdot$ based on the samples $\\cdot$ for all $\\_$ . Then, the final (ensemble-dependent) quantile prediction of each base learner is obtained as $\\hat { z } _ { T + h } ^ { \\tau , m } ( w ) = q _ { \\tau } \\left( \\hat { \\mathbb { P } } ( \\hat { Z } _ { T + h } ^ { m } ) \\right)$ for all $m = 1 , \\ldots , M$ with the final ensemble $\\begin{array} { r } { \\hat { z } _ { T + h } ^ { \\tau , e s } = \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \\hat { z } _ { T + h } ^ { \\tau , m } } \\end{array}$ . Note that, like $p _ { T + h }$ was sampled, the final ensemble model is ultimately a (single) auto-regressive one that supports sample path and quantiles. ",
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"text": "Under auto-regressive dynamic strategy, the ensembled sample $p _ { T + h }$ based on ensemble weight from policy affects the performance of individual base learner consecutively and thus final quantile ensemble. In other words, action in the previous step affects state in the current step, meaning, unlike the direct dynamic, the transition dynamics $\\mathcal { P } ( s _ { h + 1 } \\mid s _ { h } , a _ { h } ) \\neq \\mathcal { P } ( s _ { h + 1 } \\mid s _ { h } )$ . ",
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"text": "Hybrid dynamic. Note that the auto-regressive dynamic strategy is not applicable for Seq2seq base learners. Still, under the hybrid dynamic strategy, Seq2seq base learners can contribute to generate ensembled samples together, i.e., ensembled sample Seq2seq and autoregressive ones, which would b $\\begin{array} { r } { p _ { T + H } \\sim \\sum _ { m = 1 } ^ { M } w _ { h } ^ { m } \\mathbb { P } ( Z _ { T + h } ^ { m } ) } \\end{array}$ sampled from bothssive base learners. The behaviours of Seq2seq base learner is the exactly same in sampling and constructing quantile prediction without any feedback loop like ensembled sample, which means any auto-regressive base learners does not affect Seq2seq one’s prediction.The final ensemble under hybrid dynamic is capable of auto-regressive model, supporting desirable sample path through recursive feedings. ",
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"text": "4.1.2 REWARD FUNCTION ",
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"text": "To minimize the total quantile losses and encourage the agent to learn a uniform distribution over \nthe nearly-optimal base learners, we design the reward function as $R ( s , a ; z ) ~ = ~ R _ { 1 } ( s , a ; z ) ~ +$ \n$\\lambda ( s ) R _ { 2 } ( \\bar { s } , a )$ for some nsemble p $\\lambda ( s ) \\geq 0$ e, the first term compared with t $r _ { 1 }$ measures the performance of the cur-best quantile predictions among the base $\\hat { z } _ { T + h } ^ { \\tau _ { k } , e s }$ \nlearners. and takes the form ",
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"text": "$$\nR _ { 1 } \\big ( s _ { h } , a _ { h } ; z _ { T + h } \\big ) = \\operatorname* { m i n } _ { m } \\left\\{ \\sum _ { k = 1 } ^ { K } \\left( \\mathcal { L } \\big ( \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } , z _ { T + h } ; \\tau _ { k } \\big ) - \\mathcal { L } \\big ( \\hat { z } _ { T + h } ^ { \\tau _ { k } , \\mathrm { e s } } , z _ { T + h } ; \\tau _ { k } \\big ) \\right) \\right\\}\n$$",
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"type": "text",
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"text": "where $ { \\mathcal Ḋ L Ḍ } ( \\cdot , \\cdot ; \\tau )$ can be any measurement of the forecasting accuracy at the quantile level $\\tau$ . By designing the $R _ { 1 }$ term as a regret w.r.t. the best base learner, we normalize the reward around zero: if the $R _ { 1 }$ term is less than 0, then it means that the ensemble prediction is worse than the single best base learner and the corresponding should be punished, and vise versa. Furthermore, $R _ { 2 }$ takes the form ",
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"text": "$$\nR _ { 2 } { \\big ( } s _ { h } , a _ { h } { \\big ) } = D _ { \\mathrm { K L } } { \\big ( } a _ { h } \\ { \\big | } \\ \\mathrm { U n i f } { \\big ( } M ^ { * } { \\big ( } s _ { h } { \\big ) } { \\big ) } { \\big ) }\n$$",
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"text_format": "latex",
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"bbox": [
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{
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"text": "where $D _ { \\mathrm { K L } }$ denotes the Kullback–Leibler divergence, $-$ for a threshold $\\cdot$ . denotes the set of nearly-optimal base learners at the state $s$ , and $\\operatorname { J n i f } ( M ^ { * } ( s ) )$ denotes a distribution with probability mass $\\frac { 1 } { \\vert M ^ { \\ast } ( s ) \\vert }$ on the indices corresponding to the base learners in $M ^ { \\ast } ( s )$ and 0 otherwise. We introduce the term $R _ { 2 }$ to encourage the ensemble policy to be uniformly distributed among the nearly optimal base learners which could potentially further reduce the estimation error and the variance. Finally, $\\lambda ( s )$ is a state-dependent hyper-parameter controlling the weights between $R _ { 1 }$ and $R _ { 2 }$ . When there is only a single nearly-optimal base learner, i.e., $\\vert M ^ { \\ast } ( s ) \\vert = 1$ , we set $\\lambda ( s ) = 0$ which means that we only incorporate $R _ { 2 }$ when there are at least two nearly-optimal base learners. ",
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"text": "4.2 SIMULATED ENVIRONMENT: TS-GYM ",
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"text": "Before attempting to train the policy $\\pi$ , we first design a novel simulated environment for the \ntime series ensemble, namely TS-GYM, that follows state transition (in Section 4.1.1) properly, by \nextending the OpenAI’s gym interface. As illustrated in Figure 2a, it is composed of pre-trained base \nlearners in the ensemble, time series (off-line) dataset, time series samplers, ensemble dynamics and \ndynamic ensemble agent. During the initialization stage of the environment $h = 1$ , it first decides \nforecast start time $T$ which is uniformly sampled among time horizon in off-line datasets, and then \nstarts to provide followthe quantile predictions ground-truth (future) ob (1) sample a time seriefor the next timestamp The first three informa historical) observatio, (3) the step number used to construct th $z _ { 1 : T }$ , (2)d (4) and $\\{ \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } \\} _ { k = 1 , m = 1 } ^ { K , M }$ $T + h$ $h$ $\\cdot$ \nthe last information is used to construct the reward defined in Section 4.1. ",
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"text": "Note here that generating all quantile predictions $\\{ \\hat { z } _ { T + h } ^ { \\tau _ { k } , m } \\} _ { k = 1 , m = 1 } ^ { K , M }$ at each timestamp $T + h$ is governed by the choice of ensemble dynamics in Section 4.1.1 where the ensembled quantile predictions themselves may be used for the base learners’ prediction in the next timestamp. This will affect the optimal choice of ensemble actions in the end. This process is repeated until we reach the end of the prediction horizon $T + H$ , completing one episode. In practice, this whole of procedure can be done with batch sampling in parallel. ",
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"img_path": "images/2a2a0b48dfbd54c6cf8aaeecddfe58c7e7589d51cb62f095f513998efc58b637.jpg",
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"image_caption": [
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"Figure 2: Dynamic ensemble framework. "
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"text": "4.3 LEARNING DYNAMIC ENSEMBLE POLICY WITH EXPLORATION ",
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"text": "To learn an optimal ensemble policy $\\pi$ , we employ the deep actor-critic approach DDPG (Lillicrap et al., 2015) in a continuous action space to maximize culmulative reward. To accelerate the exploration of the base learners’ performance, we deploy the “random extreme point” exploration. ",
|
| 712 |
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"text": "Random extreme point exploration. For the exploration of actions, for each step $h$ , we assign the action $a _ { h } = e _ { m } \\mathbf { \\bar { \\Pi } } \\in \\mathbb { R } ^ { M }$ where $e _ { m }$ is an one-hot vector2 with randomly chosen $m$ from $\\mathcal { M }$ base learners. This exploration policy encourages the agent to take different individual base learners, efficiently collecting the observations on not only the sampled base learner performance but also various dynamic ensemble patterns. In addition this requires no prior knowledge on the base learners. ",
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"text": "5 EXPERIMENTS ",
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| 734 |
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"text": "The extensive experiments are conducted to demonstrate the effectiveness of the proposed dynamic ensemble approach in adapting the ensemble strategy to the time series item and prediction timestamp in Section 5.1. Then, we spend to investigate properties of our ensemble methods from dynamic weights to the phenomena of boosting the performance of the auto-regressive base learner by feeding the better ensemble sample in Section 5.2. ",
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"text": "5.1 BENCHMARK EXPERIMENTS ON DYNAMIC ENSEMBLE ",
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"text": "5.1.1 EXPERIMENT SETUP ",
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"text": "Datasets and base learners. We perform experiments on four real benchmark datasets that are widely used in forecasting literature: exchange rate, elec, traf and solar from (Salinas et al., 2019). For more dataset details, see appendix A.1. We consider the global deep learning based probabilistic forecasters from GluonTS (Alexandrov et al., 2020b): DeepAR (Salinas et al., 2020), MQ-CNN (Wen et al., 2017; Park et al., 2022), NBEATS(Oreshkin et al., 2019), TFT (Lim et al., 2021) and Transformer (Vaswani et al., 2017). Since the performance of DeepAR can be heavily dependent on the distribution outputs, we trained DeepAR with three different distribution outputs: Gaussian, Student’s t and Poisson distribution referred as DeepAR-G, DeepAR-T and DeepAR-P, respectively. All base learners are trained using the default configurations in GluonTS (Alexandrov et al., 2020b) . ",
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"type": "text",
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"text": "MDP formulation and RL training To evaluate the performance of our general dynamic ensemble framework, we take the most general ensemble dynamics, which is the hybrid quantile ensemble dynamics. In particular, we will apply the auto-regressive ensemble dynamics to the DeepAR models with different distribution outputs and apply the direct ensemble dynamics to the rest of the base learners. The samples from the DeepAR models from the previous timestamps will then recursively feed as the input to DeepAR models at the next timestamps. In defining the reward function, we adopt the mean weighted quantile loss (see Equation 7 in Appendix) as the accuracy measurement of our predictions. RL algorithm (DDPG) is implemented in PyTorch (Paszke et al., 2019) and trained on AWS Sagemaker (Liberty et al., 2020) with ml.p3.2xlarge instances. Train and test are done with TS-GYM specific to the given dataset. ",
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"text": "Ensemble baselines We compare our RL-based dynamic ensemble approach with the following static ensemble baselines: ",
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"text": "• Mean/Median: for each item and timestamp, take a simple mean/median of all base learners. • Global optimal ensemble: of all of the possible weights of base learners which are shared across items and timestamps, choose the weight for which the associated convex combinations of base learners lead to the best performance in the backtest validation set. • Winner-takes-all(WTA): choose the single base learner which leads to the best performance in the backtest validation set. ",
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"type": "text",
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"text": "5.1.2 BENCHMARK RESULTS ",
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"text": "Message 1: Our hybrid dynamic ensembles is the best or at least on par against other 4 baselines. We evaluate the time series forecasting results by the mean weighted quantile loss defined in Equation Equation 7 in the appendix. The results of all dynamic ensemble approaches including our hybrid quantile ensemble dynamics are summarized in Table 1. From the results in Table 1, we can further report three metrics, winning rate, average ranking, and averaged stability score (amount of $\\%$ degradation compared with winining method). For winning rate, our RL-hybrid ensemble is $50 \\%$ (wins in two out of four datasets) against other 4 baselines whereas Median and Winner-takes-all ensemble won $25 \\%$ respectively. In the average ranking, Median and our RL-hybrid method is 1.75 and 2 respectively whereas Mean and WTA method is 3.75 and 3.5 respectively. In terms of stability score, our RL-hybrid and Median ensemble is $- 1 0 \\%$ and - $- 1 5 \\%$ respectively whereas Mean and WTA method is at least $- 1 0 0 \\%$ and $- 7 0 \\%$ . Please see more detailed analysis dataset by dataset in Appendix B. ",
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"type": "text",
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"text": "Message 2: Overfitting and distribution shift hinders coherent ensembles over all ensemble methods. We also observe the over-fitting of some base learners from the results of Winner-takes-all. In exchange rate, elec and solar datasets, the best base learner in the backtest validation set is not the best base learner in the prediction testing window. It would be challenging to learn a good ensemble strategy in this situation. However, our approach can overcome this over-fitting issue to some extend and still be able to learn good ensemble policy for exchange rate and solar datasets. This is partially because the ensemble policy is trained using the entire time series dataset instead of just the backtest window. In addition, although Winner-takes-all gives the best forecasting accuracy for traf, the severe over-fitting of $\\mathtt { M Q \\mathrm { - } C N N }$ (see accuracies inside parenthesis of Table 1) slightly degrades the performance of our approach since the uniform weights are encouraged for the nearly-optimal base learners in our ensemble framework. ",
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{
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"type": "table",
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"img_path": "images/87528c817fe73604142f62ee6e0d8d06fd74e5c9a36abadba289a003b94edb67.jpg",
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"table_caption": [],
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"table_footnote": [],
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"table_body": "<table><tr><td>Base learner/ Ensemble strategy</td><td>exchange rate</td><td>elec</td><td>traf</td><td>solar</td></tr><tr><td>DeepAR-T</td><td>0.0075</td><td>0.0548</td><td>0.0879 (0.113)</td><td>0.3252</td></tr><tr><td>DeepAR-G</td><td>0.0067</td><td>0.0618</td><td>0.1140</td><td>0.3117</td></tr><tr><td>DeepAR-P</td><td>0.2261</td><td>0.0910</td><td>0.9828</td><td>0.3137</td></tr><tr><td>Transformer</td><td>0.0298</td><td>0.0266</td><td>0.0908</td><td>0.3584</td></tr><tr><td>MQ-CNN</td><td>0.0133</td><td>0.0544</td><td>1.8793 (0.166)</td><td>0.7735</td></tr><tr><td>TFT</td><td>0.0060</td><td>0.0844</td><td>0.1144</td><td>0.3253</td></tr><tr><td>NBEATS</td><td>0.0106</td><td>0.0480</td><td>0.2270</td><td>0.9983</td></tr><tr><td>Mean</td><td>0.0359</td><td>0.0490</td><td>0.2029</td><td>0.3790</td></tr><tr><td>Median</td><td>0.0090</td><td>0.0489</td><td>0.0905</td><td>0.3256</td></tr><tr><td>Global optimal</td><td>0.0124</td><td>0.0790</td><td>0.1991</td><td>0.3913</td></tr><tr><td>Winner-takes-all</td><td>0.0133</td><td>0.0548</td><td>0.0879</td><td>0.7735</td></tr><tr><td>RL-hybrid(Ours)</td><td>0.0060</td><td>0.0544</td><td>0.1141</td><td>0.3058</td></tr></table>",
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"type": "text",
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"text": "Table 1: Performance comparison on real-world benchmark datasets. The winning method among ensemble methods are made bold. The retangular is the one selected in Winner-takes-all ensemble method. The values in the parenthesis are the accuracy evaluated in the backtesting window. ",
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"type": "text",
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"text": "5.2 INVESTIGATING PROPERTIES OF DYNAMIC ENSEMBLES ",
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"text_level": 1,
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"type": "text",
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"text": "Property 1: Capturing time-varying ensemble weights. We first demonstrate the capability of our dynamic ensemble framework to learn the time-varying ensemble weights when the optimal base learners vary along the prediction horizon. We examine policy trained on the motivating example on the dataset Solar in Section 1 more closely. Our dynamic ensemble approach is able to learn ensemble weights which are consistent with the time-varying pattern of the optimal base learners. In particular, we can see from Figure 3a that (1) only Transformer, TFT and DeepAR are given positive ensemble weights during the prediction, (2) the ensemble weights of transformer remain relatively high in prediction timestamps $[ 0 , 6 ] \\cup [ 1 6 , 2 9 ]$ while dropping below 0.1 during prediction timestamps [7, 15], (3) the ensemble weights of TFT remain 0 in prediction timestamps $[ 0 , 5 ] \\cup [ 1 6 , 2 9 ]$ but dominate the ensemble weights of transformer in prediction timestamps [7, 15], (4) the ensemble weights of DeepAR remain high during the entire prediction horizon because its relatively good performance during the entire prediction horizon. ",
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},
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"type": "image",
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"img_path": "images/19d199fddf1d95b0b9acdb841afb0161e8a7f42f1d2576d03e1d16cf1688dbc9.jpg",
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"image_caption": [
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"Figure 3: The learned ensemble weights are consistent with the performances of the base learners. over the prediction horizon. QL and rank are averaged over all items in the dataset. "
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"type": "text",
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"text": "Property 2: Boosting the performance of auto-regressive (AR) forecasters. Improving the base learners’ performance is important for the improving the accuracy of the final ensembled predictions, and for allowing a broader set of admissible ensemble polices (in the extreme case, if all base learners perform equally well, then any ensemble strategy is optimal). We demonstrate the capability of auto-regressive ensemble (as shown in Figure 3b) on boosting the performance of AR forecasters. In particular, we focus on the DeepAR models with different distribution outputs: Gaussian , Student’s t and Poisson distribution and train the ensemble policy using our dynamic ensemble approach with auto-regressive ensemble dynamics on exchange rate dataset. Figure 3b shows the mean weighted quantile losses of the DeepAR-G over the prediction horizon for 3 different strategies: ",
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"text": "• using DeepAR with Gaussian distribution (denoted as DeepAR-G original); • using DeepAR with Gaussian distribution, but feed the true target value as the autoregressive input in Equation 4b (denoted as DeepAR-G w/ target); using the DeepAR with Gaussian distribution, but feed the samples from the mixture of distributions in Equation 3 as the auto-regressive input in Equation $^ { 4 \\mathrm { b } }$ (denoted as DeepAR-G w/ ensemble); ",
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"type": "text",
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"text": "We can observe that by feeding a more accurate input to the auto-regressive forecaster, DeepAR-G w/ ensemble improves DeepAR-G original consistently over the entire prediction horizon. The mean weighted quantile loss for DeepAR-G original and DeepAR- $- G$ w/ ensemble are 0.01466 and 0.00988, respectively, which demonstrates a $3 2 . 6 \\%$ performance boost. ",
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"type": "text",
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"text": "Property 3: Auto-regressive dynamic ensemble is more powerful than direct dynamic through ablation study. We conduct the ablation on AR dynamics that is explicitly considered in our algorithm in comparison to the methods where the AR feedback is not explicit. We term these ablations as RL-auto and RL-naive. We consider the solar dataset with base learners DeepAR-T, DeepAR-G and DeepAR-P. ",
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| 966 |
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"type": "table",
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"img_path": "images/3c908dcb4ee782da90d563a5876e5fd8b61ca72308f3fc14c28dca8a4987bcba.jpg",
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"table_caption": [
|
| 978 |
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"Table 2: Ablation study to compare auto-regressive vs direct dynamic. "
|
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],
|
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"table_footnote": [],
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| 981 |
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"table_body": "<table><tr><td>Base learner /Ensemble strategy</td><td>DeepAR-T</td><td>DeepAR-G</td><td>DeepAR-P</td><td>Mean</td><td>Global Optimal</td><td>RL-naive</td><td>RL-auto</td></tr><tr><td>solar</td><td>0.3252</td><td>0.3117</td><td>0.3137</td><td>0.3088</td><td>0.3302</td><td>0.3148</td><td>0.2840</td></tr></table>",
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{
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"type": "text",
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"text": "Table 2 highlights the significance of AR dynamics that is explicit in our MDP formulation. With same set of base learners the AR dynamics is able to achieve ${ \\bf 1 1 \\% }$ better result than the naive dynamics. Further, the RL-auto is better $( 8 \\% )$ than all models/ensemble strategy considered, thus showing the significance of base learner boosting via AR feedback. ",
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"type": "text",
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"text": "REFERENCES ",
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"text_level": 1,
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"type": "text",
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"text": "Alexander Alexandrov, Konstantinos Benidis, Michael Bohlke-Schneider, Valentin Flunkert, Jan Gasthaus, Tim Januschowski, Danielle C. Maddix, Syama Rangapuram, David Salinas, Jasper Schulz, Lorenzo Stella, Ali Caner Türkmen, and Yuyang Wang. GluonTS: Probabilistic and Neural Time Series Modeling in Python. Journal of Machine Learning Research, 21(116):1–6, 2020a. URL http://jmlr.org/papers/v21/19-820.html. ",
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"bbox": [
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"type": "text",
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"text": "Alexander Alexandrov, Konstantinos Benidis, Michael Bohlke-Schneider, Valentin Flunkert, Jan Gasthaus, Tim Januschowski, Danielle C Maddix, Syama Sundar Rangapuram, David Salinas, Jasper Schulz, et al. GluonTS: Probabilistic and neural time series modeling in Python. Journal of Machine Learning Research, 21(116):1–6, 2020b. ",
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"type": "text",
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"text": "John M Bates and Clive WJ Granger. The combination of forecasts. Journal of the Operational Research Society, 20(4):451–468, 1969. ",
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"bbox": [
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"type": "text",
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"text": "Konstantinos Benidis, Syama Sundar Rangapuram, Valentin Flunkert, Yuyang Wang, Danielle Maddix, Caner Turkmen, Jan Gasthaus, Michael Bohlke-Schneider, David Salinas, Lorenzo Stella, et al. Deep learning for time series forecasting: Tutorial and literature survey. ACM Computing Surveys (CSUR), 2022. ",
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"type": "text",
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"text": "Aadyot Bhatnagar, Paul Kassianik, Chenghao Liu, Tian Lan, Wenzhuo Yang, Rowan Cassius, Doyen Sahoo, Devansh Arpit, Sri Subramanian, Gerald Woo, Amrita Saha, Arun Kumar Jagota, Gokulakrishnan Gopalakrishnan, Manpreet Singh, K C Krithika, Sukumar Maddineni, Daeki Cho, Bo Zong, Yingbo Zhou, Caiming Xiong, Silvio Savarese, Steven Hoi, and Huan Wang. Merlion: A machine learning library for time series. 2021. ",
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"page_idx": 11
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{
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"type": "text",
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"text": "Sean J Taylor and Benjamin Letham. Forecasting at scale. The American Statistician, 72(1):37–45, 2018. ",
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"page_idx": 11
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{
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"type": "text",
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"text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017. ",
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"bbox": [
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"page_idx": 11
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{
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"type": "text",
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"text": "Ruofeng Wen, Kari Torkkola, Balakrishnan Narayanaswamy, and Dhruv Madeka. A multi-horizon quantile recurrent forecaster. arXiv preprint arXiv:1711.11053, 2017. ",
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| 1467 |
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"bbox": [
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708,
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823,
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738
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],
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| 1473 |
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"page_idx": 11
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| 1474 |
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},
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| 1475 |
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{
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"type": "text",
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"text": "TaeHo Yoon, Youngsuk Park, Ernest K Ryu, and Yuyang Wang. Robust probabilistic time series forecasting. In International Conference on Artificial Intelligence and Statistics, pp. 1336–1358. PMLR, 2022. ",
|
| 1478 |
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"bbox": [
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174,
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| 1480 |
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|
| 1481 |
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| 1483 |
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| 1484 |
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|
| 1485 |
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},
|
| 1486 |
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{
|
| 1487 |
+
"type": "text",
|
| 1488 |
+
"text": "Yunyue Zhu and Dennis Shasha. Statstream: Statistical monitoring of thousands of data streams in real time. In VLDB’02: Proceedings of the 28th International Conference on Very Large Databases, pp. 358–369. Elsevier, 2002. ",
|
| 1489 |
+
"bbox": [
|
| 1490 |
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|
| 1491 |
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| 1492 |
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| 1493 |
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| 1494 |
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| 1495 |
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"page_idx": 11
|
| 1496 |
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},
|
| 1497 |
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{
|
| 1498 |
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"type": "text",
|
| 1499 |
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"text": "A EXPERIMENT SETUP ",
|
| 1500 |
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"text_level": 1,
|
| 1501 |
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"bbox": [
|
| 1502 |
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| 1507 |
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|
| 1508 |
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},
|
| 1509 |
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{
|
| 1510 |
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"type": "text",
|
| 1511 |
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"text": "A.1 REAL-WORLD DATASET ",
|
| 1512 |
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"text_level": 1,
|
| 1513 |
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"bbox": [
|
| 1514 |
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| 1517 |
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| 1518 |
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|
| 1519 |
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"page_idx": 12
|
| 1520 |
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|
| 1521 |
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{
|
| 1522 |
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"type": "table",
|
| 1523 |
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"img_path": "images/ccc3044c887030e4065776e0c2dbb22b1efdcba2cb8c9713385823b409169c06.jpg",
|
| 1524 |
+
"table_caption": [
|
| 1525 |
+
"Table 3 summarizes the four benchmark real-world datasets that we use to evaluate our dynamic ensemble approach. ",
|
| 1526 |
+
"Table 3: Benchmark dataset descriptions "
|
| 1527 |
+
],
|
| 1528 |
+
"table_footnote": [],
|
| 1529 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Freq</td><td rowspan=1 colspan=1>Domain</td><td rowspan=1 colspan=1>#Time series</td><td rowspan=1 colspan=1>Prediction length</td></tr><tr><td rowspan=1 colspan=1>exchangerate</td><td rowspan=1 colspan=1>daily</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>30</td></tr><tr><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>2950</td><td rowspan=1 colspan=1>24</td></tr><tr><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>[0,1]</td><td rowspan=1 colspan=1>6741</td><td rowspan=1 colspan=1>24</td></tr><tr><td rowspan=1 colspan=1>solar</td><td rowspan=1 colspan=1>hourly</td><td rowspan=1 colspan=1>R+</td><td rowspan=1 colspan=1>959</td><td rowspan=1 colspan=1>24</td></tr></table>",
|
| 1530 |
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"bbox": [
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| 1531 |
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| 1532 |
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| 1533 |
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| 1535 |
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],
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| 1536 |
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"page_idx": 12
|
| 1537 |
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},
|
| 1538 |
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{
|
| 1539 |
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"type": "text",
|
| 1540 |
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"text": "A.2 IMPLEMENTATION OF DDPG ",
|
| 1541 |
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"text_level": 1,
|
| 1542 |
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"bbox": [
|
| 1543 |
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| 1548 |
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|
| 1549 |
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},
|
| 1550 |
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{
|
| 1551 |
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"type": "text",
|
| 1552 |
+
"text": "We use the DDPG implementation from OpenAI spinning up baselines. The last layer of policy network is a softmax layer with output dimensions as the number of base learners considered. For hyper-parameter tuning we consider the hyper-parameters in Lillicrap et al. (2015) and some specific to dynamic AR ensemble. The final hyper-parameters used for different datasets for the experiment in Section 5.1 is given in Tables 4 and 5. The default weights among AR model parameter is used to set the weights among the AR model if all the AR models in the hybrid dynamics gets zero weight at certain step in the RL; $\\lambda$ controls the trade-off as explained in the reward function section. The reward scale is the scaling applied to mean-wQL to be comparable with the secondary reward $r _ { 2 }$ . Round threshold is the number of decimal digits for rounding the mean-wQL to get ranking for base learners. ",
|
| 1553 |
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"bbox": [
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|
| 1559 |
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"page_idx": 12
|
| 1560 |
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|
| 1561 |
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{
|
| 1562 |
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"type": "table",
|
| 1563 |
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"img_path": "images/7083d65d551627d042a21decaa25bf06e657a2e867400ae638752f699a24960b.jpg",
|
| 1564 |
+
"table_caption": [
|
| 1565 |
+
"A.2.1 EXPERIMENTS IN TABLE 1 ",
|
| 1566 |
+
"Table 4: Hyperparameters of DDPG algorithm in various real-world datasets. "
|
| 1567 |
+
],
|
| 1568 |
+
"table_footnote": [],
|
| 1569 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>Hyperparamters</td><td rowspan=1 colspan=1>exchange rate</td><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>solar</td></tr><tr><td rowspan=1 colspan=1>episodes per epoch</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>start episodes</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>update after episodes</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1> update steps per prediction length</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=1 colspan=1>update every episodes</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>discount factor</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td></tr><tr><td rowspan=1 colspan=1>epochs</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>60</td><td rowspan=1 colspan=1>60</td><td rowspan=1 colspan=1>70</td></tr><tr><td rowspan=1 colspan=1>polyak</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td><td rowspan=1 colspan=1>0.99</td></tr><tr><td rowspan=1 colspan=1>learning rate for policy</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td></tr><tr><td rowspan=1 colspan=1>learning rate for Q value</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td><td rowspan=1 colspan=1>0.0005</td></tr><tr><td rowspan=1 colspan=1>noise level for action</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.1</td></tr></table>",
|
| 1570 |
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"bbox": [
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| 1571 |
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| 1572 |
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| 1573 |
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| 1576 |
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"page_idx": 12
|
| 1577 |
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},
|
| 1578 |
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{
|
| 1579 |
+
"type": "text",
|
| 1580 |
+
"text": "A.3 IMPLEMENTAIONS OF TS-GYM ",
|
| 1581 |
+
"text_level": 1,
|
| 1582 |
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"bbox": [
|
| 1583 |
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| 1586 |
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| 1587 |
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|
| 1588 |
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"page_idx": 12
|
| 1589 |
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},
|
| 1590 |
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{
|
| 1591 |
+
"type": "text",
|
| 1592 |
+
"text": "Error metric We evaluate the forecasting error in terms of the mean weighted quantile loss. See the precise definition in the appendix. ",
|
| 1593 |
+
"bbox": [
|
| 1594 |
+
174,
|
| 1595 |
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|
| 1596 |
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|
| 1597 |
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|
| 1598 |
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],
|
| 1599 |
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"page_idx": 12
|
| 1600 |
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},
|
| 1601 |
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{
|
| 1602 |
+
"type": "equation",
|
| 1603 |
+
"img_path": "images/6c1342051899b421ca0dc688654fd6dbc42add049507cb235afdc37ea844fbeb.jpg",
|
| 1604 |
+
"text": "$$\n\\frac { 1 } { q } \\frac { \\sum _ { i = 1 , j = T + 1 , k = 1 } ^ { N , T + h , q } \\operatorname* { m a x } \\left\\{ \\tau _ { k } ( z _ { i , j } - \\widetilde { z } _ { i , j , k } ) , ( 1 - \\tau _ { k } ) ( \\widetilde { z } _ { i , j , k } - z _ { i , j } ) \\right\\} } { \\sum _ { i = 1 , j = T + 1 } ^ { N , T + h } | z _ { i , j } | }\n$$",
|
| 1605 |
+
"text_format": "latex",
|
| 1606 |
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"bbox": [
|
| 1607 |
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|
| 1608 |
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| 1609 |
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707,
|
| 1610 |
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883
|
| 1611 |
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],
|
| 1612 |
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"page_idx": 12
|
| 1613 |
+
},
|
| 1614 |
+
{
|
| 1615 |
+
"type": "text",
|
| 1616 |
+
"text": "where {zi,j}N,T hi=1,j=T +1 are the true values of future time series and $\\{ \\widetilde { z } _ { i , j , k } \\} _ { i = 1 , j = T + 1 , k = 1 } ^ { N , T + h , q }$ are the estimated quantile predictions. ",
|
| 1617 |
+
"bbox": [
|
| 1618 |
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|
| 1619 |
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|
| 1620 |
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| 1621 |
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924
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| 1622 |
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|
| 1623 |
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"page_idx": 12
|
| 1624 |
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},
|
| 1625 |
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{
|
| 1626 |
+
"type": "table",
|
| 1627 |
+
"img_path": "images/4a28b510312ef9ea68dcfabfbd81cc16c232fcc685e92e3439c1c2de93b28e9a.jpg",
|
| 1628 |
+
"table_caption": [
|
| 1629 |
+
"Table 5: Hyperparameters of TS-GYM in various real-world datasets. "
|
| 1630 |
+
],
|
| 1631 |
+
"table_footnote": [],
|
| 1632 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>Hyperparameters</td><td rowspan=1 colspan=1>exchange rate</td><td rowspan=1 colspan=1>elec</td><td rowspan=1 colspan=1>traf</td><td rowspan=1 colspan=1>solar</td></tr><tr><td rowspan=1 colspan=1>train batch size</td><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>200</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>200</td></tr><tr><td rowspan=1 colspan=1>reward scale</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>0.0001</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>round threshold</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>入</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>default weights amongauto-regressive models</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td><td rowspan=1 colspan=1>[1,0,0]</td></tr></table>",
|
| 1633 |
+
"bbox": [
|
| 1634 |
+
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|
| 1635 |
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|
| 1636 |
+
748,
|
| 1637 |
+
204
|
| 1638 |
+
],
|
| 1639 |
+
"page_idx": 13
|
| 1640 |
+
},
|
| 1641 |
+
{
|
| 1642 |
+
"type": "text",
|
| 1643 |
+
"text": "B BENCHMARK RESULT DISCUSSION ",
|
| 1644 |
+
"text_level": 1,
|
| 1645 |
+
"bbox": [
|
| 1646 |
+
174,
|
| 1647 |
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253,
|
| 1648 |
+
495,
|
| 1649 |
+
270
|
| 1650 |
+
],
|
| 1651 |
+
"page_idx": 13
|
| 1652 |
+
},
|
| 1653 |
+
{
|
| 1654 |
+
"type": "text",
|
| 1655 |
+
"text": "For the more detailed discussion, we can observe that the proposed RL-hybrid method outperforms all base models and baselines on all exchange rate and solar datasets. For exchange rate, which is a regular dataset with clear daily patterns, a single base learner usually performs very well. Our RL-hybrid method is able to identify the single best base learner (TFT). On the other hand, exchange rate is less regular and more challenging. Our RL-hybrid method is better $( 2 \\% )$ than all base models and baselines considered. This is because our dynamic ensemble method are able to capture the time-varying patterns of the base learners’ performance profile and boost the performance of the auto-regressive base learners (see Section 5.2 for more discussions). ",
|
| 1656 |
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"bbox": [
|
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| 1659 |
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| 1660 |
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|
| 1661 |
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],
|
| 1662 |
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"page_idx": 13
|
| 1663 |
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}
|
| 1664 |
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]
|
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| 1 |
+
# RETHINKING SELF-SUPERVISION OBJECTIVES FOR GENERALIZABLE COHERENCE MODELING
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Although large-scale pre-trained neural models have shown impressive performances in a variety of tasks, their ability to generate coherent text that appropriately models discourse phenomena is harder to evaluate and less understood. Given the claims of improved text generation quality across various systems, we consider the coherence evaluation of machine generated text to be one of the principal applications of coherence models that needs to be investigated. We explore training data and self-supervision objectives that result in a model that generalizes well across tasks and can be used off-the-shelf to perform such evaluations.
|
| 8 |
+
|
| 9 |
+
Prior work in neural coherence modeling has primarily focused on devising new architectures, and trained the model to distinguish coherent and incoherent text through pairwise self-supervision on the permuted documents task. We instead use a basic model architecture and show significant improvements over state of the art within the same training regime. We then design a harder self-supervision objective by increasing the ratio of negative samples within a contrastive learning setup, and enhance the model further through automatic hard negative mining coupled with a large global negative queue encoded by a momentum encoder. We show empirically that increasing the density of negative samples improves the basic model, and using a global negative queue further improves and stabilizes the model while training with hard negative samples. We evaluate the coherence model on task-independent test sets that resemble real-world use cases and show significant improvements in coherence evaluations of downstream applications.
|
| 10 |
+
|
| 11 |
+
# 1 INTRODUCTION
|
| 12 |
+
|
| 13 |
+
Coherence is a property of a well-written text that makes it different from a random set of sentences: sentences in a coherent text are connected in systematic ways such that each sentence follows naturally from previous ones and leads into the following ones (Halliday & Hasan, 1976; Grosz & Sidner, 1986). Coherence models (Barzilay & Lapata, 2005) that can distinguish a coherent text from incoherent ones have a wide range of applications in language generation, summarization, and coherence assessment tasks such as essay scoring and sentence ordering.
|
| 14 |
+
|
| 15 |
+
With the advancements of neural methods in recent years, claims of fluency in summarization (Liu et al., 2017; Celikyilmaz et al., 2018), language modeling (Radford et al., 2019; Brown et al., 2020), response generation (Zhang et al., 2020; Hosseini-Asl et al., 2020) and human parity in machine translation (Hassan et al., 2018) have led to calls for finer-grained discourse-level evaluations (Laubli ¨ et al., 2018; Sharma et al., 2019; Popel et al., 2020), since traditional metrics such as BLEU and ROUGE are unable to measure text quality and readability (Paulus et al., 2018; Reiter, 2018). Coherence models that can evaluate machine-generated text have become the need of the hour.
|
| 16 |
+
|
| 17 |
+
A majority of coherence models proposed optimize their learning objectives on the permuted document task that uses the Penn Treebank (WSJ) corpus. The current paradigm of coherence modeling that uses permuted documents to train pairwise ranking models was originally proposed by Barzilay & Lapata (2005; 2008) to emulate entity-based incoherence, which has its origins in Centering Theory (Grosz et al., 1995). An original article is considered a ‘positive’ sample of a coherent document, while a permutation of its sentences is considered a ‘negative’ or incoherent sample (see Appendix A.1 for an example). Models are usually trained in a pairwise ranking fashion to distinguish the two.
|
| 18 |
+
|
| 19 |
+
The basic entity-grid model proposed by Barzilay & Lapata (2005; 2008) was extended to incorporate entity-specific features (Elsner & Charniak, 2011), multiple ranks (Feng & Hirst, 2012), and coherence relations (Lin et al., 2011; Feng et al., 2014). Their neural extensions have also been proposed (Nguyen & Joty, 2017; Mohiuddin et al., 2018). More recent state-of-the-art models like the Transferable Neural model (Xu et al., 2019) consider coherence at a local level by training a forward and backward model only on adjacent sentences, in addition to generative pre-training of the sentence encoders. The Unified Coherence model (Moon et al., 2019) uses bi-linear layer and lightweight convolution-pooling in a Siamese framework to capture discourse relations and topic structures, along with an explicit language model loss to capture syntactic patterns.
|
| 20 |
+
|
| 21 |
+
Mohiuddin et al. (2021) recently tested these state-of-the-art models by conducting coherence evaluations on the WSJ permuted document task, machine translation, summarization and next utterance ranking tasks. They found that while models performed well on the permuted document task, when tested off-the-shelf, models generalized poorly to downstream evaluation tasks. They call for more comprehensive evaluations of coherence models. Pishdad et al. (2020) also reached a similar conclusion. They retrained several neural coherence models for tasks analogous to coherence modeling such as detecting connective substitution and topic switching. They found that performance on the permuted document task is only partially indicative of a model’s coherence modeling capabilities.
|
| 22 |
+
|
| 23 |
+
In light of these recent findings, our aim in this work is to propose a coherence model that generalizes well to other tasks, and can be used off-the-shelf for coherence evaluations of downstream applications such as machine generated text. We train our model purely through self-supervision, without tailoring the model architecture to be specific to the permuted document task or any other form of supervision. Our main hypothesis is that large-scale pre-trained models like XLNet (Yang et al., 2019) are expressive enough to capture coherence information given the right self-supervision.
|
| 24 |
+
|
| 25 |
+
Li & Jurafsky (2017) point out that coherence models are exposed to a limited number of incoherent samples in the pairwise setup, since only a small sample of all possible incoherent permutations of a document are used to train models. Learning with more negatives can better maximize the mutual information between their representations (van den Oord et al., 2018). By using a contrastive learning (Gutmann & Hyvarinen, 2010) setup, where each ‘positive’ document is compared with ¨ multiple ‘negative’ documents, we increase the proportion of negative samples that the model is exposed to, and show that the coherence model shows significant improvements in performance.
|
| 26 |
+
|
| 27 |
+
Wu et al. (2020) recently show that the difficulty of the negative samples used for contrastive training can strongly influence model success for visual representation learning. Guided by this principle, we train the model with hard negative samples that are automatically mined, coupled with a large global negative queue encoded by a momentum encoder (He et al., 2019).
|
| 28 |
+
|
| 29 |
+
We evaluate our model on various independent test sets that demonstrate its applicability in downstream applications: machine generated summaries, language model outputs and commonsense reasoning, in addition to testing on coherence-specific test sets. In summary, our contributions are:
|
| 30 |
+
|
| 31 |
+
• A neural coherence model trained purely through well-designed self-supervision tasks that generalizes well to downstream applications and can be used off-the-shelf for coherence evaluation.
|
| 32 |
+
• Evaluation on multiple independent test sets that are more indicative of real-world performance of the coherence model.
|
| 33 |
+
• Empirical results demonstrating that an increase in the density and quality of negative samples leads to better generalization for coherence models.
|
| 34 |
+
|
| 35 |
+
# 2 DATASETS
|
| 36 |
+
|
| 37 |
+
In order to ensure that our coherence model is useful for evaluation in downstream applications, we use a selection of task-independent test sets that cover a variety of domains and genres, including machine generated text from summarization systems and language models. Following Pishdad et al. (2020), we also evaluate the models on a commonsense reasoning narrative dataset. Since our objective is to find the best training paradigm that can be used off-the-shelf for coherence evaluation, we train (and validate) the coherence models on standard WSJ data, while using the rest as “independent” test sets to indicate the generalizability of the trained models. All evaluations on the independent test sets are conducted in a pairwise setting to enable a fair comparison.
|
| 38 |
+
|
| 39 |
+
# 2.1 TRAINING DATA
|
| 40 |
+
|
| 41 |
+
WSJ The Wall Street Journal (WSJ) corpus consists of news articles which are divided into 1,240 documents for training, 138 documents for development and 1,053 documents for testing in the standard setup. We exclude documents with fewer than 4 sentences and truncate them to a maximum length of 600 tokens. In order to maximally utilize documents which are otherwise truncated due to GPU memory constraints, we partition documents with $^ { 2 0 + }$ sentences into blocks of 10 sentences and consider each block as a separate positive document. This increases the number of coherent ‘documents’ that we can use to generate a much larger training set. Moon et al. (2019) use upto 20 permutations of a document to train their model; since their training setup is pairwise, it means that the original positive document is repeated 20 times. We regenerate the permuted documents similarly, sampling a larger set of permutations for our contrastive learning setup.1 This gives us 46,522 instances of positive and their corresponding negative documents for training and 4,522 instances for development. We use the original pairwise test set used by Moon et al. (2019) with 20,411 instances for testing.
|
| 42 |
+
|
| 43 |
+
# 2.2 MACHINE GENERATED TEXTS
|
| 44 |
+
|
| 45 |
+
SUMMEVAL Fabbri et al. (2020) conduct a manual coherence evaluation of the summaries generated by 16 different summarization systems for 100 source articles based on the CNN/DailyMail (Hermann et al., 2015) dataset. Likert-style coherence ratings from 3 expert annotators are available for each summarized text. We adapt this to the pairwise setting by creating pairs of summaries from every system for each unique source article. The summary with the higher average coherence rating is designated as the positive document, while the summary with the lower rating is the negative document for that pair. This results in $( \mathbf { \Phi } _ { 2 } ^ { 1 6 } ) \times 1 0 0 = 1 2 , 0 0 0$ pairs for evaluation.
|
| 46 |
+
|
| 47 |
+
LMVLM To cover a wider variety of machine generated text, we generated texts from various language models using prompts taken from the validation and test sets of the WritingPrompts dataset (Fan et al., 2018). Four language models were chosen for this purpose: GPT2-Small, GPT2-XL, CTRL and GPT3. The continuations produced by these models for each prompt were truncated at approximately 150 tokens and paired together. Using these texts, we conducted a user study on Amazon Mechanical Turk. Workers were instructed about the concept of coherence and shown examples of coherent and incoherent texts. Given the prompt, they were asked to choose the more coherent text out of two given language model outputs; they were also given an option to choose neither in case the texts were equally coherent/incoherent (see Appendix A.3 for more details such as the study interface). After removing the samples with low agreements and ties, a total of 1046 pairs with judgments from 3 annotators each were collected. The Krippendorff’s alpha coefficient (Krippendorff, 2011) between the annotators was 0.84. We calculate the agreements of the coherence model ranking with these judgments, designated LMVLM.
|
| 48 |
+
|
| 49 |
+
# 2.3 CURATED TEST SETS
|
| 50 |
+
|
| 51 |
+
INSTED Shen et al. (2021) propose a sentence intrusion detection task in order to test the coherence modeling capabilities of pre-trained language models. Incoherent documents are created by substituting a sentence from a document with another sentence from a different document, ensuring that the replacement sentence is similar to the original document to make the task sufficiently hard. We adapt their task to the pairwise setting by pairing the original coherent and the corrupted incoherent document, giving us 7,168 instances from their CNN test set (INSTED-CNN) and 3,666 instances from their Wikipedia test set (INSTED-WIKI) for evaluation. Shen et al. (2021) also create a hand-crafted linguistic probe test set, where incoherence is manually inserted based on a range of linguistic phenomena; we use this test set for analysis (§4).
|
| 52 |
+
|
| 53 |
+
STORYCLOZE The STORYCLOZE dataset (created from ROCSTORIES (Sharma et al., 2018)) consists of a short narrative-style text with two possible endings, one of which is implausible. The test set labels are not public so we use the validation set. We designate the text with the correct ending as the positive document and the text with the incorrect ending as the negative document, resulting in a total of 1, 571 pairs for evaluation.
|
| 54 |
+
|
| 55 |
+
# 3 METHODOLOGY
|
| 56 |
+
|
| 57 |
+
# 3.1 MODEL ARCHITECTURE
|
| 58 |
+
|
| 59 |
+
Previous work on coherence modeling proposed elaborate architectures to capture various aspects of coherence (see $\ S 1$ ). However, our key hypothesis is that large-scale pre-trained models are expressive enough to model coherence given the right self-supervision; Abhishek et al. (2021) show some results to this effect. Effective bi-directional encoding through large Transformer networks (Vaswani et al., 2017) can consider longer language context, while language modeling objectives enforce syntactic and local coherence patterns in the model.
|
| 60 |
+
|
| 61 |
+
In our work, we adopt XLNet (Yang et al., 2019) as the backbone model. It is trained using a permuted language modeling objective, in which the expected log-likelihood of a sequence with respect to all permutations of the factorization order is maximized. This allows the modeling of bi-directional context, while maintaining the auto-regressive property and avoiding the pretrainfinetune discrepancy. In addition, XLNet also incorporates segment recurrence (or memory) and the relative encoding scheme of Transformer-XL (Dai et al., 2019), which makes it effective in modeling longer text sequences. This makes it suitable for our purpose of coherence modeling.
|
| 62 |
+
|
| 63 |
+
Given a document $\mathcal { D }$ with $n$ sentences $( s _ { 1 } , s _ { 2 } , \ldots , s _ { n } )$ as input, our model uses the representations obtained through XLNet (parameterized by $\phi$ in Figure 1) to assign a coherence score to the model. Specifically, for each sentence $s _ { i }$ with $k$ tokens $( w _ { 1 } , w _ { 2 } \dots w _ { k } )$ , XLNet maps each token $w _ { t }$ to its vector representation $v _ { t } \in \mathbb { R } ^ { d }$ where $d$ is the dimension of the embedding. In addition, the complete input $\mathcal { D }$ is also mapped to a document representation $\mathbf { z } \in \mathbb { R } ^ { d }$ (i.e., the representation of the [CLS] token). We simply add a linear layer to convert document representation $\mathbf { z }$ to obtain the final coherence score: $f _ { \boldsymbol { \theta } } ( \mathbf { \bar { \mathcal { D } } } ) = \mathbf { w } ^ { \top } \mathbf { z } + b$ , where w and $b$ are the weight and bias of the linear layer with $\theta = \{ \phi , { \bf w } , b \}$ being the entire parameter set of the model (see the upper part of Figure 1).
|
| 64 |
+
|
| 65 |
+
# 3.2 MARGIN-BASED PAIRWISE RANKING
|
| 66 |
+
|
| 67 |
+
Setup. Traditionally, coherence model training has been done in a pairwise ranking setup. In this setup, the model is trained to score the coherent or positive document higher than the incoherent or negative document, using a pairwise ranking loss (Collobert et al., 2011) defined as follows:
|
| 68 |
+
|
| 69 |
+
$$
|
| 70 |
+
\mathcal { L } _ { \boldsymbol { \theta } } ( \mathcal { D } ^ { + } , \mathcal { D } ^ { - } ) = \operatorname* { m a x } \big ( 0 , \tau - f _ { \boldsymbol { \theta } } ( \mathcal { D } ^ { + } ) + f _ { \boldsymbol { \theta } } ( \mathcal { D } ^ { - } ) \big )
|
| 71 |
+
$$
|
| 72 |
+
|
| 73 |
+
where $f _ { \theta } ( \mathcal { D } ^ { + } )$ is the coherence score of the positive document, $f _ { \theta } ( \mathcal { D } ^ { - } )$ is the coherence score of the negative document and $\tau$ is the margin.
|
| 74 |
+
|
| 75 |
+
Baseline. Results from evaluation of existing coherence models by both Pishdad et al. (2020) and Mohiuddin et al. (2021) indicate that the Unified Coherence model or UNC (Moon et al., 2019) is overall the best-performing model (see Appendix A.5 for a full comparison). We retrain their model with our training data for comparison2. In addition, to ascertain the contribution of the pre-trained XLNet embeddings, we train our pairwise model without fine-tuning the representations, i.e., only the score-producing linear layer weights w and $b$ are trained on the pairwise ranking task.
|
| 76 |
+
|
| 77 |
+
Results. Results for the baseline models are given in Table 1 (see first two rows). We see that despite relatively high performance on the WSJ test set $( 9 4 . 1 1 \% )$ , UNC’s performance on the independent test sets is quite poor, often failing to do better than a random baseline of $50 \%$ in 3 out of 5 cases. The performance on the INSTED-CNN dataset, which is the same domain (news) as the training data, is relatively better at $6 7 . 2 1 \%$ . Our XLNet-Pairwise model trained without fine-tuning the representations (No FT) has some success on the SUMMEVAL and STORYCLOZE datasets compared to UNC, but overall the performance of this model is worse. This shows that the UNC model is in fact a strong baseline model despite using ELMo (Peters et al., 2018) pretrained representations. Our fully-trained XLNet-Pairwise model not only outperforms the SOTA UNC model on the standard WSJ permuted document task, but also significantly outperforms this model on the independent test sets, showing an absolute improvement of $1 5 \mathrm { - } 2 0 \%$ on the SUMMEVAL, INSTEDCNN, INSTED-WIKI and the STORYCLOZE datasets. On LMVLM, the UNC model has a better performance; we suspect that its explicit conditional language modeling loss might provide an additional advantage for this particular task. Overall, our results are consistent with observations from Mohiuddin et al. (2021) that show poor generalizability in the previous SOTA model.
|
| 78 |
+
|
| 79 |
+
Table 1: Results on the WSJ permuted document test set and the various independent test sets of the previous SOTA UNC model and our XLNet based models. Except for the LMVLM results which are reported in terms of Krippendorff’s alpha agreement with human annotators, all other results are reported in terms of accuracy of the models in scoring the positive document higher than the negative document. All results are averaged over 5 runs with different seeds.
|
| 80 |
+
|
| 81 |
+
<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WSJ</td><td rowspan=1 colspan=1>SUMMEVAL</td><td rowspan=1 colspan=1>LMvLM</td><td rowspan=1 colspan=3>INSTED-CNN|INSTED-WIKI STORYCLOZE</td></tr><tr><td rowspan=1 colspan=1>UNC</td><td rowspan=1 colspan=1>94.11±0.29</td><td rowspan=1 colspan=1>46.28±0.80</td><td rowspan=1 colspan=1>0.463±0.01</td><td rowspan=1 colspan=1>67.21±0.55</td><td rowspan=1 colspan=1>55.97±0.45</td><td rowspan=1 colspan=1>49.39±1.81</td></tr><tr><td rowspan=1 colspan=1>Our - Pairwise (No FT)</td><td rowspan=1 colspan=1>71.70±1.02</td><td rowspan=1 colspan=1>54.93±1.91</td><td rowspan=1 colspan=1>0.421±0.01</td><td rowspan=1 colspan=1>59.96±3.15</td><td rowspan=1 colspan=1>53.45±0.86</td><td rowspan=1 colspan=1>51.69±1.32</td></tr><tr><td rowspan=1 colspan=1>Our - Pairwise</td><td rowspan=1 colspan=1>98.23±0.20</td><td rowspan=1 colspan=1>64.83±1.03</td><td rowspan=1 colspan=1>0.458±0.02</td><td rowspan=1 colspan=1>91.96±1.09</td><td rowspan=1 colspan=1>70.85±1.85</td><td rowspan=1 colspan=1>71.84±2.33</td></tr><tr><td rowspan=2 colspan=1>Our - Contrastive Our - Full Model</td><td rowspan=2 colspan=1>98.59±0.2098.58±0.18</td><td rowspan=1 colspan=1>66.93±1.10</td><td rowspan=1 colspan=1>0.468±0.01</td><td rowspan=1 colspan=1>92.84±0.61</td><td rowspan=1 colspan=1>71.86±0.69</td><td rowspan=1 colspan=1>72.83±2.89</td></tr><tr><td rowspan=1 colspan=1>67.19±0.63</td><td rowspan=1 colspan=1>0.473±0.00</td><td rowspan=1 colspan=1>93.36±0.49</td><td rowspan=1 colspan=1>72.04±1.05</td><td rowspan=1 colspan=1>74.62±2.79</td></tr></table>
|
| 82 |
+
|
| 83 |
+
# 3.3 CONTRASTIVE LEARNING
|
| 84 |
+
|
| 85 |
+
Setup. In the pairwise ranking setup, each positive sample is only compared to one negative sample at a time. Contrastive learning (Gutmann & Hyvarinen, 2010) makes it general, where a single ¨ positive sample can be compared to multiple negative samples, which can be particularly useful in the permuted document task where the number of possible incoherent samples per coherent document can be very large. The number of negatives considered and their quality can affect the model performance (Arora et al., 2019). Wu et al. (2020) show that contrastive loss maximizes a lower bound on the mutual information between representations. A larger number of negatives increases the tightness of the bound; learning with more negatives can better maximise the mutual information. We train our model with a margin-based contrastive loss defined as:
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$$
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\mathcal { L } _ { \theta } ( \mathcal { D } ^ { + } , \mathcal { D } _ { 1 } ^ { - } , \cdot \cdot , \mathcal { D } _ { N } ^ { - } ) = - \log \Big ( \frac { e ^ { f _ { \theta } ( \mathcal { D } ^ { + } ) } } { e ^ { f _ { \theta } ( \mathcal { D } ^ { + } ) } + \sum _ { j = 1 } ^ { N } e ^ { ( f _ { \theta } ( \mathcal { D } _ { j } ^ { - } ) - \tau ) } } \Big )
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$$
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where $f _ { \boldsymbol { \theta } } ( \mathcal { D } ^ { + } )$ is the coherence score of the positive document, $f _ { \theta } ( \mathcal { D } _ { 1 } ^ { - } ) , \cdot \cdot \cdot , f _ { \theta } ( \mathcal { D } _ { N } ^ { - } )$ are the scores of the $N$ negative documents, and $\tau$ is the margin.
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Training. We use the same training data as the baseline models to train our contrastive model; the positive documents remain the same, while we use 5 negative documents per instance (instead of only 1 in the pairwise setup). Effectively, the model sees the same number of positive or coherent documents, but five times as many negative samples during training compared to the pairwise setting. See Appendix A.4 for the full set of our hyperparameters.
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Results. From the results in Table 1, we see that the contrastive model (row 3) further improves the results across all the independent test sets; the results on the LMVLM dataset also improve, now surpassing the UNC model performance. Although the improvement on the WSJ permuted document task is small, the improvement in the generalizability of the model is more significant.
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# 3.4 MOMENTUM ENCODER WITH HARD NEGATIVE MINING
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While increasing the number of negative samples per instance has been shown to be effective for constrastive learning, resource constraints can limit the number of negatives that can be considered per instance. One solution is to consider other positive instances in the same training batch as negatives (Karpukhin et al., 2020; Chen et al., 2020). However, this method is not suitable for the permuted document task since the negatives are instance-specific. While a permuted document is still independently incoherent, training with permuted versions of other documents will not provide the same cues for coherence modeling as the original self-supervision.
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Another solution is to maintain a large global queue of negative samples that are independent of the current training instance. During training, negative samples (more specifically, their representations) from the latest batch are enqueued to build a queue upto some size l. As training continues, the negative samples from the oldest batch are dequeued to accommodate newer samples. However, representations of the documents will evolve through training as the model parameters get updated; this will make the negative samples in the queue inconsistent with each other and the training instances in the current batch. Moreover, the issue of mismatched self-supervision with negatives that are permuted versions of other documents still remains.
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Momentum Encoder. To address these issues, we add an auxiliary momentum encoder (He et al., 2019), which is also XLNet (Yang et al., 2019). Figure 1 shows the overall architecture. Keeping the base contrastive setup the same (the upper part), we add an additional contrastive objective based on representations from the momentum encoder. Specifically, we re-encode the positive and negative samples through the momentum encoder; the negative samples thus encoded are used to build the queue. We train the model to promote the similarity between the positive representations from the momentum encoder and the positive representations from our base encoder over the similarity with the negative samples from the queue, $Q$ . Specifically, we define a momentum loss ${ \mathcal { L } } _ { \theta } ^ { \mathrm { m o m } }$ as:
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Figure 1: Our coherence model with the auxiliary momentum encoder. $\phi$ is our base encoder similar to our setup in $\ S 3 . 3$ , while $\phi ^ { \prime }$ is our momentum encoder. $\dot { u } ^ { + } = f _ { \theta } ( \mathcal { D } ^ { + } )$ and $u ^ { - } = f _ { \theta } ( \mathcal { D } ^ { - } )$ are the coherence scores of the positive and negative documents respectively. Note that only the parameters of $\phi$ and the linear layer are updated through backpropagation.
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$$
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c ^ { + } = \frac { ( \mathbf { z } ^ { + } ) ^ { \top } ( \mathbf { z } _ { m } ^ { + } ) } { | | \mathbf { z } ^ { + } | | \mid | \mathbf { z } _ { m } ^ { + } | | } ; \quad c _ { j } ^ { - } = \frac { ( \mathbf { z } _ { m } ^ { + } ) ^ { \top } \mathbf { q } _ { j } } { | | \mathbf { z } _ { m } ^ { + } | | \mid | \mathbf { q } _ { j } | | } ; \quad \mathcal { L } _ { \theta } ^ { \mathrm { m o m } } = - \log \Big ( \frac { e ^ { c ^ { + } } } { e ^ { c ^ { + } } + \sum _ { j = 1 } ^ { l } e ^ { ( c _ { j } ^ { - } - \tau ) } } \Big )
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$$
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where ${ \mathbf z } ^ { + }$ and $\mathbf { z } _ { m } ^ { + }$ are the positive representations from the base encoder $( \phi )$ and the momentum encoder $( \phi ^ { \prime } )$ respectively, ${ \bf q } _ { 1 } , \ldots , { \bf q } _ { l }$ indexed by $j$ are the negative representations from $\phi ^ { \prime }$ in the queue, and $\tau$ is the margin. The momentum encoder $\phi ^ { \prime }$ is updated based on the base encoder $\phi$ as:
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$$
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\phi ^ { \prime } \mu * \phi ^ { \prime } + ( 1 - \mu ) * \phi
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$$
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where $\mu \in [ 0 , 1 )$ is the momentum coefficient; only $\phi$ is updated through backpropagation.
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Our full model is trained with a combination of the original contrastive learning objective (Eq. 2) and the momentum encoded contrastive similarity objective (Eq. 3):
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$$
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\mathcal { L } _ { \theta } = \lambda \mathcal { L } _ { \theta } + ( 1 - \lambda ) \mathcal { L } _ { \theta } ^ { \mathrm { m o m } }
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$$
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where $\lambda$ is a weighting hyperparameter. The momentum encoder can be considered as a temporal ensemble model consisting of exponential-moving-average versions of the base model. Due to this, gradients from the momentum loss (Eq. 3) also help in stabilising the overall training (§4).
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Length Invariance Training. In the permuted document task, both the positive and the negative samples have the same number of sentences. This is not necessarily the case for real world applications. In order to incorporate length invariance into our model, we encode a random contiguous slice of the positive document through the momentum encoder $\phi ^ { \prime }$ . 3
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Hard Negative Mining. It has been shown that the difficulty of the negative samples used for contrastive training can strongly influence model success (Wu et al., 2020). We therefore automatically mine hard negative samples during training. For the permuted document task, we can take advantage of the fact that the negative sample space can be huge; for a document with $n$ sentences, the candidate pool of permutations has $n ! - 1$ incoherent documents from which we can mine hard negatives. For the problem of dense text retrieval, Xiong et al. (2020) find global hard negatives by computing document encodings using a recent checkpoint to build an asynchronous index of the entire corpus, and sampling negative documents from the index. However, the huge candidate pool for permuted documents also makes it infeasible to mine global negatives in our case.
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Instead, we perform local negative sample ranking. For each positive instance in the training data, we sample a larger number of permuted documents $( h )$ per instance than we need for training (i.e., $h > N ,$ ). We score these negative documents using the model updated thus far and use the highest ranking negative documents for training. Specifically, the model is first trained with $x$ instances ( $\scriptstyle { \dot { x } }$ is a hyperparameter) of data, by using 5 negative samples randomly chosen out of $h$ . The updated model is then used to score all the $h$ negative samples each for another set of $x$ instances from the training data. The scores of the $h$ negative samples are ranked and the top scoring 5 negative samples for each instance are used to train the model for the next $x$ gradient steps. This process is repeated throughout training; the model therefore iteratively mines harder and harder negative samples as it improves. See Algorithm 1 in Appendix A.2 for the pseudocode.
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We use the hard negative training in combination with the momentum encoder since we find that using hard negative samples directly leads to instability in model training (see $\ S 4$ ). The global negatives queue $Q$ is thus also constructed from the mined hard negative samples used for training. Our model is therefore trained to rely not only on comparative coherence cues from the traditional permuted document setup, but also to recognize more independent cues for coherence through the global queue, which is additionally enhanced by incorporating length invariance and automatically mined hard negative samples.
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Training. We train the model with the same training data, this time sampling $h = 5 0$ negatives4 per instance for hard negative ranking, and setting the training steps (or instances) $x = 2 0 0$ . We use a queue size of $l = 1 0 0 0$ and set our momentum coefficient $\mu = 0 . 9 9 9 9 9 9 9$ , with loss weighting parameter $\lambda = 0 . 8 5$ . Due to GPU memory constraints (24GB, Quadro RTX 6000), we train our model with a batch size of 1. See Appendix A.4 for the full set of hyperparameters.
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Results. The results in Table 1 (last row) show that our momentum encoder model with hard negative mining outperforms all previous models across the independent testsets. This improvement comes despite a very similar performance on the WSJ test set; we believe that our model truly improves in generalizability without overfitting to the permuted document task. The improvements on the out-of-domain test sets, particularly on LMVLM and STORYCLOZE, support this conclusion.
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# 4 ANALYSIS
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# 4.1 HARD NEGATIVE TRAINING WITH MOMENTUM MODEL
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We only train our complete model (i.e., base contrastive plus momentum model) by mining hard negative samples $( \ S 3 . 4 )$ , because we find that training the base contrastive model directly with hard negatives leads to instability during training. Figure 2a plots development set accuracies of our base model trained with and without hard negative mining, and our complete model trained with hard negative mining (evaluated every 1000 steps). As seen in the figure, the contrastive model displays significant volatility when trained with hard negatives only, while the complete model is quite stable. This is inline with the finding of Xuan et al. (2020) who show that training with the hardest negative samples leads to bad local minima. This can be explained with the gradient analysis of such negatives which have a larger gradient norm (Xiong et al., 2020), resulting in abrupt gradient steps. The momentum encoder being a temporal ensemble of the base models has a regularizing effect, addressing this issue and leading to stable and improved results (see $\ S 3 . 4 \AA$ .
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# 4.2 EFFECTS OF HYPERPARAMETERS
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Number of Ranked Negatives. Figure 2b shows the results across the test sets for different numbers of negative samples considered for ranking $( h )$ during hard negative mining. We see that increasing the number of negatives considered improves results across the board, with results on out-of-domain test sets LMVLM and STORYCLOZE showing particular improvement.
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Momentum Coefficient. Figure 2c shows the variation in the model performance across the test sets for different values of the momentum coefficient $\mu$ . We see that apart from a slight drop on the INSTED-WIKI dataset at $\mu = 0 . 9 9 9 9 9 9 9$ , overall an increasing $\mu$ value leads to better generalization on the independent test sets, presumably due to a more consistent global negative queue.
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Figure 2: (a) A plot of the development accuracy during training our contrastive model with and without hard negative mining, and our complete model with hard negative mining. The accuracies are evaluated after every 1000 gradient steps. (b) Results on the various test sets for our model trained with hard negative mining by sampling different number of negatives $( h )$ for ranking. (c) Results on the various test sets for our complete model trained with different momentum coefficient $( \mu )$ values. (d) Results on the various test sets for our model trained with different global queue $\ b { Q }$ sizes. Please note that the agreement values for LMVLM test set have been scaled by a factor of 100 to facilitate visualization in figures (b), (c) and (d).
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Queue Size. Figure 2d shows the variation in model performance across different test sets for various sizes of the global negative queue $Q$ . We see that while increasing the queue size generally leads to an improvement in scores, at high queue sizes the improvement is limited to test sets from the same domain (WSJ, SUMMEVAL and INSTED-CNN), and the model’s generalizability is affected.
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# 4.3 EFFECTS OF VARYING TASK & DATASET
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So far, we have reported the results of training our model on the permuted document task using documents from the WSJ corpus as was done by most prior work (Elsner & Charniak, 2011; Moon et al., 2019). We now test the effectiveness of other datasets, both by varying the task itself and by using a different dataset for the permuted document task.
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Sentence Intrusion. As described in $\ S 2 . 3$ , Shen et al. (2021) propose a sentence intrusion task to test coherence modeling capabilities of pre-trained language models. We adapt their dataset to the pairwise setting by pairing the original coherent document (positive) with the corrupted (negative) document; setting aside $10 \%$ of the data for development gives us 25,852 positive-negative training pairs for INSTED-CNN and 41,135 pairs for INSTED-WIKI. We train our pairwise (§3.2) model on this task. From the results in Table 2 (first two rows), we see that the performance on the same domain/task (as the training) and the performance on the LMVLM dataset is high, but the models trained on this task generalize poorly to the other independent test sets.
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Permuted Document Task with INSTED We now train our model on the permuted document task using the INSTED datasets. We generate 52,607 and 66,679 positive-negative pairs for INSTED-CNN and INSTED-WIKI respectively by sampling permutations, similar to our training data (see $\ S 2 . 1$ ), and train our pairwise model with this data. The results are shown in Table 2, highlighted in blue. Specifically for machine generated texts, the sentence intrusion task training does better on the LMVLM dataset. On the other hand, the permuted document task training does better on SUMMEVAL. This could be because the documents in SUMMEVAL are summaries of the same source article and therefore similar in content (detecting incoherence through permutations might help here), while the text generated by language models even for the same prompt tends to differ in content more significantly (detecting intruder sentences might help here). Additionally, the performance of our WSJ model on the INSTED-CNN and INSTED-WIKI datasets is comparable to the performance of the respective in-domain pairwise models, while outperforming both the other models on the STORYCLOZE dataset. Overall, the WSJ model generalizes well.
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Table 2: Results on the WSJ permuted document test set and other independent test sets on the pairwise and contrastive models trained on different datasets. All results are averaged over 5 runs with different seeds.
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<table><tr><td>Train Dataset</td><td>Neg.Type</td><td>Model</td><td>WSJ</td><td>SUMMEVAL</td><td>LMvLM</td><td>INSTED-CNN|</td><td>INSTED-WIKI</td><td>STORYCLOZE</td></tr><tr><td>INSTED-WIKI</td><td>Intrusion</td><td>Pairwise</td><td>95.24±0.37</td><td>53.03±1.49</td><td>0.490±0.01</td><td>94.07±0.29</td><td>82.01±0.24</td><td>64.21±1.98</td></tr><tr><td>INSTED-CNN</td><td>Intrusion</td><td>Pairwise</td><td>95.48±0.47</td><td>57.85±2.47</td><td>0.502±0.01</td><td>97.83±0.15</td><td>73.52±1.17</td><td>71.75±1.81</td></tr><tr><td> INSTED-WIKI</td><td>Permuted</td><td>Pairwise</td><td>96.89±0.23</td><td>64.53±0.82</td><td>0.491±0.01</td><td>84.17±1.50</td><td>71.35±0.88</td><td>69.09±2.29</td></tr><tr><td>INSTED-CNN</td><td>Permuted</td><td>Pairwise</td><td>97.03±0.12</td><td>66.63±0.97</td><td>0.483±0.01</td><td>92.61±0.62</td><td>69.88±0.64</td><td>68.95±1.02</td></tr><tr><td>WSJ</td><td>Permuted</td><td>Pairwise</td><td>98.23±0.20</td><td>64.83±1.03</td><td>0.458±0.02</td><td>91.96±1.09</td><td>70.85±1.85</td><td>71.84±2.33</td></tr></table>
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Table 3: Accuracies of the best performing UNC and our full model on the hand-crafted linguistic probe dataset constructed by Shen et al. (2021). Examples (abridged for brevity) shown indicate the manual changes made to make the text incoherent; the original words are shown in blue while the modified/added words are shown in red. Checks $( \pmb { \nu } )$ indicate our model correctly scored the coherent text higher for that example, while crosses $( { \pmb x } )$ indicate that our model failed to do so.
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<table><tr><td>Linguistic Probe</td><td>UNC</td><td>Our</td><td></td><td>Example</td></tr><tr><td>Pronoun Animacy Downgrade</td><td>76.0</td><td>100.0</td><td></td><td>She→It was the mother of twins Lakshmana and Shatrughna.</td></tr><tr><td>Pronoun Animacy Upgrade</td><td>63.0</td><td>100.0</td><td>：</td><td>It→She has been collected in two tankobon volumes.</td></tr><tr><td>Pronoun Gender Flip</td><td>55.0</td><td>100.0</td><td></td><td>She→He is also well known for her-→his role as Mary, the mother of Jesus.</td></tr><tr><td>Past to Future Flip</td><td>86.0</td><td>96.0</td><td>X</td><td>The Danes finished-→willfinish first in the 2O14 World Junior Hockey Championship.</td></tr><tr><td>Single Determiner Flip</td><td>62.1</td><td>83.2</td><td>X</td><td>In 1969, he was again sold, this→these time to the Milwaukee Bucks.</td></tr><tr><td>Number</td><td>58.0</td><td>80.0</td><td>X</td><td>He had a career record of 67→6.7wins and 62→-6.2 losses.</td></tr><tr><td>Conjunction Flip</td><td>55.0</td><td>78.0</td><td>X</td><td>The school was founded in 19O8,and→but has been a non-profit organization since 1956.</td></tr><tr><td>Negation</td><td>60.0</td><td>78.0</td><td>X</td><td>He was not named as the Australian squad captain and was not captain of the Wallabies.</td></tr></table>
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# 4.4 LINGUISTIC PROBE ANALYSIS
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Shen et al. (2021) create eight hand-crafted linguistic probe test sets by manually modifying words in coherent texts based on various linguistic phenomena, ensuring that the incoherent text produced as a result remains syntactically correct. Except for the words targeted by the probe, the rest of the text remains identical. Each test set has 100 samples each.5
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We evaluate the best performing UNC and our full models on these test sets. The results are shown in Table 3 along with some examples from the dataset. The UNC model has the most success with the tense agreement test set and mixed success on the pronoun test sets. We see that our model has perfect accuracy on all pronoun-related test sets and near-perfect accuracy on the tense agreement test set. This shows that our model is indeed capturing the discourse-level phenomena that constitute coherence. Where our model falters is in cases which may require commonsense knowledge, such as identifying that $6 . 7$ wins is not possible. Overall, our model is quite successful in detecting several kinds of incoherence.
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# 5 CONCLUSION
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With the goal of making our coherence model generalizable and useful for off-the-shelf evaluations, in this work we have explored self-supervision objectives to improve coherence models without adapting our model architecture to a specific training task like previous work. We upgrade the self-supervision objective from the existing pairwise ranking paradigm to a contrastive learning setup. We further enhance this model with a momentum encoder to maintain a large global queue of negative samples, and also perform hard negative mining to refine the quality of the negative samples. We show empirically that increasing the ratio and quality of negative samples improves the generalizability of the coherence model. We also test our model on a wide-ranging collection of independent test sets that resemble downstream applications, including machine generated text, on which our model significantly outperforms the previous SOTA model. Our work thus also sets a new evaluation standard for future research in coherence modeling. We will open source our code base to encourage research in a new paradigm of coherence modeling.
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# REPRODUCIBILITY STATEMENT
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CODE AND HYPERPARAMETERS
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Code for the various models will be open-sourced. Specific hyperparameters used for experiments are described in $\ S 3 . 3$ and $\ S 3 . 4$ , while a full list of hyperparameters is included in Appendix A.4.
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# DATA
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A description of the data pre-processing is provided in $\ S 2 . 1$ . Datasets that we created will be opensourced. In the case of the WSJ dataset, the data is licensed for use only to members by the Linguistic Data Consortium. Consequently, we only release scripts to generate the data we use and not the data itself. We highlight however that the permuted document self-supervision task that we train on is independent of the dataset used and the task can be reproduced on any other corpus; see also $\ S 4 . 3$ . All other datasets we use are licensed freely for academic use.
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# ETHICS STATEMENT
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# ANNOTATION OF LMVLM DATASET
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We conduct a user study to collect pairwise coherence judgments on our language model output dataset. As part of our crowd-sourced user study on Amazon Mechanical Turk to collect these coherence judgements, we do not collect any personal information from the participants. Based on the average time spent to perform the tasks, participants were paid the equivalent of 16 USD per hour for their work. The annotation instructions and interface provided to the participants are included in Appendix A.3.
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One potential issue is that the language model output that we generate from prompts may lead to malicious text generation by the models. We flagged the task to warn the workers that there may be potentially offensive content, and manually checked the final dataset post curation.
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R. Barzilay and Mirella Lapata. Modeling local coherence: An entity-based approach. Computational Linguistics, 34:1–34, 2008.
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# A APPENDIX
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# A.1 WSJ PERMUTED DOCUMENT TASK
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The examples for the permuted document task on the WSJ data are shown in Table 4.
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A.2 HARD NEGATIVE RANKING PSEUDOCODE
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The pseudocode for our hard negative mining through local sample ranking is given in Algorithm 1.
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Table 4: Examples showing the original coherent document and the incoherent document created by permuting the sentences of the original. Text taken from WSJ-1778.
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<table><tr><td></td><td>Original Document</td></tr><tr><td>to my feet.</td><td>(S1)Judy and I were in our back yard when the lawn started rolling like ocean waves. (S2)We ran into the house to get Mame,but the next tremor threw me in the air and bounced meas Itried to get</td></tr><tr><td></td><td>(S3) We are all fine here,although Mame was extremely freaked. (S4) Books and tapes all over my room.</td></tr><tr><td></td><td>(S5) Not one thing in the house is where it is supposed to be,but the structure is fine.</td></tr><tr><td></td><td>Permuted Document</td></tr><tr><td></td><td>(S4) Books and tapes all over my room.</td></tr><tr><td></td><td>(S3) We are all fine here,although Mame was extremely freaked.</td></tr><tr><td></td><td>(S2）Weran into the house to get Mame,but the next tremor threw me in the air and bounced meas Itried to get</td></tr><tr><td></td><td>to my feet.</td></tr><tr><td></td><td>(S5) Not one thing in the house is where it is supposed to be,but the structure is fine.</td></tr><tr><td></td><td>(S1) Judy and I were in our back yard when the lawn started rolling like ocean waves.</td></tr></table>
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# Algorithm 1 Local Negative Sample Ranking
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Require: Training data $D$ in which each instance consists of a positive document and $h$ negative
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documents, model $\theta$
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1: Initialize empty hard negative array $\hat { D } ^ { - }$ for each instance $\in { \cal D }$
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2: procedure HARDNEGATIVERANKING $( \theta , D )$
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3: Partition the dataset into sets of $x$ instances $D _ { 1 } \ldots D _ { r }$
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4: for $i = 1 \dots r$ do
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5: if $\scriptstyle \mathrm { i } = = 0$ then . No hard negatives for first iteration
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6: for $j = 1 \ldots x$ do
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7: Randomly sample N negatives from D−(i,j) and store in Dˆ −(i,j)
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8: Train $\theta$ with $( D _ { i } ^ { + } , \hat { D } _ { i } ^ { - } )$
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9: for $j = 1 \ldots x$ do
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10: Score all the h negative documents in D−(i+1,j)
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11: Sort D−(i+1,j) in descending order of scores
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12: Get $N$ top scoring negative documents and store in $\hat { D } _ { ( i + 1 , j ) } ^ { - }$
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13: . Store hard negatives for the next iteration
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# A.3 LMVLM USER STUDY
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The instructions and the interface provided to the workers in the user study comparing pairs of language model outputs is given in Figure 3. Workers were restricted to the native English speaking regions of Canada, United Kingdom and the United States and could only participate in our task if they had completed $> 1 0$ , 000 HITs with $\mathrm { a > 9 8 \% }$ acceptance rate. Each task was estimated to take 2 minutes, and workers were paid the equivalent of 16 USD per hour.
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# A.4 HYPERPARAMETERS
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The hyperparameters used in our experiments are given in Table 5.
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# A.5 COMPARISON OF EXISTING STATE-OF-THE-ART COHERENCE MODELS
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We report the results obtained by Mohiuddin et al. (2021) and Pishdad et al. (2020) on their evaluation tasks for SOTA neural coherence models in Table 6.
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For example, consider:
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a. Jane took a train from Paris to Istanbul. She had to attend a conference.
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This is an example of a coherent text. Here, the second sentence gives a reason for Jane's action in the first sentence
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b. John took a train from Paris to Istanbul. He hates spinach.
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toconvey.
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other indicationofcoherence in texts is whenatextisconsistentlytalkingabout someone orsomething.Consider this example:
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bought was hard to get up to that floor.
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eeii
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I whichtexoutfthtoiventextsisoreceentasedoneeplaatioofohreceprovdedtoundtheeealalttet
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option sparingly, and only if there is absolutely no diffrence in coherence between the two texts.
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Figure 3: Instructions and study interface for the user study conducted on language model outputs.
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Table 5: Configuration parameters for training
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<table><tr><td>Parameters</td><td>Values</td></tr><tr><td colspan="2">Margin-based Pairwise Ranking</td></tr><tr><td>- margin - optimizer</td><td>0.1 AdamW</td></tr><tr><td>- scheduler</td><td>SWALR</td></tr><tr><td>- learning rate - annealed to</td><td>5e-6</td></tr><tr><td>- anneal rate</td><td>1e-6</td></tr><tr><td></td><td>5000 steps</td></tr><tr><td>- batch-size</td><td>1</td></tr><tr><td>- XLNet model</td><td>base</td></tr><tr><td>- dimension size</td><td>768</td></tr><tr><td colspan="2">Contrastive Learning</td></tr><tr><td>- margin</td><td>0.1</td></tr><tr><td>- optimizer</td><td>AdamW</td></tr><tr><td>- scheduler</td><td>SWALR</td></tr><tr><td>- learning rate</td><td>5e-6</td></tr><tr><td>- annealed to</td><td>1e-6</td></tr><tr><td>- anneal rate</td><td>5000 steps</td></tr><tr><td>- batch-size</td><td>1</td></tr><tr><td>- XLNet model</td><td>base</td></tr><tr><td>- dimension size</td><td>768</td></tr><tr><td colspan="2">Momentum Encoder with Hard Negative Mining</td></tr><tr><td>- margin</td><td>0.1</td></tr><tr><td>- optimizer</td><td>AdamW</td></tr><tr><td>- scheduler</td><td>SWALR</td></tr><tr><td>- learning rate</td><td>5e-6</td></tr><tr><td>- annealed to</td><td>1e-6</td></tr><tr><td>- anneal rate</td><td>1000 steps</td></tr><tr><td>- batch-size</td><td>1</td></tr><tr><td>- XLNet model</td><td>base</td></tr><tr><td>- dimension size</td><td>768</td></tr></table>
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Table 6: Results reported by Mohiuddin et al. (2021) and Pishdad et al. (2020) on various tasks and datasets that compare the UNC model to two other SOTA neural coherence models proposed by $\mathrm { X u }$ et al. (2019) and Mesgar & Strube (2018). Except those marked by (Agr.) which report agreement with humans, all other tasks report accuracies. We only include tasks that directly test discourse coherence phenomena.
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<table><tr><td colspan="4">As reported by Pishdad et al. (2020)</td></tr><tr><td>Task</td><td>Dataset</td><td>UNC</td><td>Mesgar & Strube (2018)</td></tr><tr><td>Permuted Document</td><td>Visual Storytelling</td><td>88.42</td><td>82.25</td></tr><tr><td>Permuted Document</td><td>ROCStories</td><td>94.80</td><td>89.55</td></tr><tr><td>Permuted Document</td><td>Dialogue</td><td>97.21</td><td>90.79</td></tr><tr><td>Permuted Document</td><td>HellaSwag</td><td>83.92</td><td>69.38</td></tr><tr><td>Permuted Document</td><td>PDTB</td><td>92.85</td><td>61.96</td></tr><tr><td>Connective Substitution</td><td>PDTB</td><td>96.46</td><td>84.99</td></tr><tr><td>Topic Switching</td><td>Visual Storytelling</td><td>92.10</td><td>64.81</td></tr><tr><td>Topic Switching</td><td>ROCStories</td><td>94.62</td><td>67.85</td></tr><tr><td>Topic Switching</td><td>Dialogue</td><td>71.74</td><td>68.41</td></tr><tr><td>Topic Switching</td><td>PDTB</td><td>70.89</td><td>52.33</td></tr><tr><td colspan="4">As reported by Mohiuddin et al. (2021)</td></tr><tr><td>Task</td><td>Dataset</td><td>UNC</td><td>Xu et al. (2019)</td></tr><tr><td>Permuted Document</td><td>WSJ</td><td>93.19</td><td>91.77</td></tr><tr><td>Abstractive Summarization (Agr.)</td><td>CNN</td><td>0.68</td><td>0.55</td></tr><tr><td>Extractive Summarization (Agr.)</td><td>DUC</td><td>0.35</td><td>0.38</td></tr><tr><td>Machine Translation (Agr.)</td><td>WMT</td><td>0.77</td><td>0.78</td></tr><tr><td>(Trained) Machine Translation (Agr.)</td><td>WMT</td><td>0.83</td><td>0.75</td></tr></table>
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