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- .gitattributes +160 -0
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parse/train/SyXNErg0W/SyXNErg0W_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3783 |
+
parse/train/SyXNErg0W/SyXNErg0W_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3784 |
+
parse/train/B1MIBs05F7/B1MIBs05F7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3785 |
+
parse/train/B1MIBs05F7/B1MIBs05F7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3786 |
+
parse/train/B1MIBs05F7/B1MIBs05F7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3787 |
+
parse/train/rygxdA4YPS/rygxdA4YPS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3788 |
+
parse/train/rygxdA4YPS/rygxdA4YPS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3789 |
+
parse/train/rygxdA4YPS/rygxdA4YPS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3790 |
+
parse/train/BkisuzWRW/BkisuzWRW_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3791 |
+
parse/train/BkisuzWRW/BkisuzWRW_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3792 |
+
parse/train/BkisuzWRW/BkisuzWRW_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3793 |
+
parse/train/J64lDCrYGi/J64lDCrYGi_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3794 |
+
parse/train/J64lDCrYGi/J64lDCrYGi_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3795 |
+
parse/train/J64lDCrYGi/J64lDCrYGi_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3796 |
+
parse/train/xLExSzfIDmo/xLExSzfIDmo_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3797 |
+
parse/train/xLExSzfIDmo/xLExSzfIDmo_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3798 |
+
parse/train/xLExSzfIDmo/xLExSzfIDmo_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3799 |
+
parse/train/rkhxwltab/rkhxwltab_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3800 |
+
parse/train/rkhxwltab/rkhxwltab_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3801 |
+
parse/train/rkhxwltab/rkhxwltab_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3802 |
+
parse/train/BJx040EFvH/BJx040EFvH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3803 |
+
parse/train/BJx040EFvH/BJx040EFvH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3804 |
+
parse/train/BJx040EFvH/BJx040EFvH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3805 |
+
parse/train/K_Mnsw5VoOW/K_Mnsw5VoOW_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3806 |
+
parse/train/K_Mnsw5VoOW/K_Mnsw5VoOW_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3807 |
+
parse/train/K_Mnsw5VoOW/K_Mnsw5VoOW_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3808 |
+
parse/train/HklRwaEKwB/HklRwaEKwB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3809 |
+
parse/train/HklRwaEKwB/HklRwaEKwB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3810 |
+
parse/train/HklRwaEKwB/HklRwaEKwB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3811 |
+
parse/train/rkePU0VYDr/rkePU0VYDr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3812 |
+
parse/train/rkePU0VYDr/rkePU0VYDr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3813 |
+
parse/train/rkePU0VYDr/rkePU0VYDr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3814 |
+
parse/train/S1l-C0NtwS/S1l-C0NtwS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3815 |
+
parse/train/S1l-C0NtwS/S1l-C0NtwS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3816 |
+
parse/train/S1l-C0NtwS/S1l-C0NtwS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3817 |
+
parse/train/S1eALyrYDH/S1eALyrYDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3818 |
+
parse/train/S1eALyrYDH/S1eALyrYDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3819 |
+
parse/train/S1eALyrYDH/S1eALyrYDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3820 |
+
parse/train/B1e-kxSKDH/B1e-kxSKDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3821 |
+
parse/train/B1e-kxSKDH/B1e-kxSKDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3822 |
+
parse/train/B1e-kxSKDH/B1e-kxSKDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3823 |
+
parse/train/rJxcBpNKPr/rJxcBpNKPr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3824 |
+
parse/train/rJxcBpNKPr/rJxcBpNKPr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3825 |
+
parse/train/rJxcBpNKPr/rJxcBpNKPr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3826 |
+
parse/train/JrsfBJtDFdI/JrsfBJtDFdI_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3827 |
+
parse/train/JrsfBJtDFdI/JrsfBJtDFdI_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3828 |
+
parse/train/JrsfBJtDFdI/JrsfBJtDFdI_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3829 |
+
parse/train/HyMTkQZAb/HyMTkQZAb_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3830 |
+
parse/train/HyMTkQZAb/HyMTkQZAb_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3831 |
+
parse/train/HyMTkQZAb/HyMTkQZAb_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3832 |
+
parse/train/rkgW0oA9FX/rkgW0oA9FX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3833 |
+
parse/train/rkgW0oA9FX/rkgW0oA9FX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3834 |
+
parse/train/rkgW0oA9FX/rkgW0oA9FX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3835 |
+
parse/train/B1Yy1BxCZ/B1Yy1BxCZ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3836 |
+
parse/train/B1Yy1BxCZ/B1Yy1BxCZ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3837 |
+
parse/train/B1Yy1BxCZ/B1Yy1BxCZ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3838 |
+
parse/train/aHK-onEhYRg/aHK-onEhYRg_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3839 |
+
parse/train/aHK-onEhYRg/aHK-onEhYRg_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3840 |
+
parse/train/aHK-onEhYRg/aHK-onEhYRg_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3841 |
+
parse/train/BJe7h34YDS/BJe7h34YDS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3842 |
+
parse/train/BJe7h34YDS/BJe7h34YDS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3843 |
+
parse/train/BJe7h34YDS/BJe7h34YDS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3844 |
+
parse/train/Kai3KkCOCp/Kai3KkCOCp_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3845 |
+
parse/train/Kai3KkCOCp/Kai3KkCOCp_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3846 |
+
parse/train/Kai3KkCOCp/Kai3KkCOCp_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3847 |
+
parse/train/SyfdsjA9FX/SyfdsjA9FX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3848 |
+
parse/train/SyfdsjA9FX/SyfdsjA9FX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3849 |
+
parse/train/SyfdsjA9FX/SyfdsjA9FX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3850 |
+
parse/train/Bkeuz20cYm/Bkeuz20cYm_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3851 |
+
parse/train/Bkeuz20cYm/Bkeuz20cYm_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
parse/dev/3RBY8fKjHeu/3RBY8fKjHeu_middle.json
ADDED
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|
parse/dev/83LJRUzXWj/83LJRUzXWj_middle.json
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|
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parse/dev/EbMuimAbPbs/EbMuimAbPbs.md
ADDED
|
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|
| 1 |
+
# Flamingo: a Visual Language Model for Few-Shot Learning
|
| 2 |
+
|
| 3 |
+
Jean-Baptiste Alayrac\*,‡ Jeff Donahue\* Pauline Luc\* Antoine Miech\*
|
| 4 |
+
|
| 5 |
+
Iain Barr† Yana Hasson† Karel Lenc† Arthur Mensch† Katie Millican†
|
| 6 |
+
|
| 7 |
+
Malcolm Reynolds† Roman Ring† Eliza Rutherford† Serkan Cabi Tengda Han
|
| 8 |
+
|
| 9 |
+
Zhitao Gong Sina Samangooei Marianne Monteiro Jacob Menick
|
| 10 |
+
|
| 11 |
+
Sebastian Borgeaud Andrew Brock Aida Nematzadeh Sahand Sharifzadeh
|
| 12 |
+
|
| 13 |
+
Mikolaj Binkowski Ricardo Barreira Oriol Vinyals Andrew Zisserman
|
| 14 |
+
|
| 15 |
+
Karen Simonyan\*,‡
|
| 16 |
+
|
| 17 |
+
\* Equal contributions, ordered alphabetically, † Equal contributions, ordered alphabetically, ‡ Equal senior contributions
|
| 18 |
+
|
| 19 |
+
# DeepMind
|
| 20 |
+
|
| 21 |
+
# Abstract
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Building models that can be rapidly adapted to novel tasks using only a handful of annotated examples is an open challenge for multimodal machine learning research. We introduce Flamingo, a family of Visual Language Models (VLM) with this ability. We propose key architectural innovations to: (i) bridge powerful pretrained vision-only and language-only models, (ii) handle sequences of arbitrarily interleaved visual and textual data, and (iii) seamlessly ingest images or videos as inputs. Thanks to their flexibility, Flamingo models can be trained on large-scale multimodal web corpora containing arbitrarily interleaved text and images, which is key to endow them with in-context few-shot learning capabilities. We perform a thorough evaluation of our models, exploring and measuring their ability to rapidly adapt to a variety of image and video tasks. These include open-ended tasks such as visual question-answering, where the model is prompted with a question which it has to answer; captioning tasks, which evaluate the ability to describe a scene or an event; and close-ended tasks such as multiple-choice visual question-answering. For tasks lying anywhere on this spectrum, a single Flamingo model can achieve a new state of the art with few-shot learning, simply by prompting the model with task-specific examples. On numerous benchmarks, Flamingo outperforms models fine-tuned on thousands of times more task-specific data.
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Figure 1: Selected examples of inputs and outputs obtained from Flamingo-80B. Flamingo can rapidly adapt to various image/video understanding tasks with few-shot prompting (top). Out of the box, Flamingo is also capable of multi-image visual dialogue (bottom). More examples in Appendix C.
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Figure 2: Flamingo results overview. Left: Our largest model, dubbed Flamingo, outperforms state-of-the-art fine-tuned models on 6 of the 16 tasks we consider with no fine-tuning. For the 9 tasks with published few-shot results, Flamingo sets the new few-shot state of the art. Note: We omit RareAct, our 16th benchmark, as it is a zero-shot benchmark with no available fine-tuned results to compare to. Right: Flamingo performance improves with model size and number of shots.
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# 1 Introduction
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One key aspect of intelligence is the ability to quickly learn to perform a new task given a short instruction [33, 70]. While initial progress has been made towards a similar capability in computer vision, the most widely used paradigm still consists of first pretraining on a large amount of supervised data, before fine-tuning the model on the task of interest [66, 118, 143]. However, successful finetuning often requires many thousands of annotated data points. In addition, it often requires careful per-task hyperparameter tuning and is also resource intensive. Recently, multimodal vision-language models trained with a contrastive objective [50, 85] have enabled zero-shot adaptation to novel tasks, without the need for fine-tuning. However, because these models simply provide a similarity score between a text and an image, they can only address limited use cases such as classification, where a finite set of outcomes is provided beforehand. They crucially lack the ability to generate language, which makes them less suitable to more open-ended tasks such as captioning or visual questionanswering. Others have explored visually-conditioned language generation [17, 114, 119, 124, 132] but have not yet shown good performance in low-data regimes.
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We introduce Flamingo, a Visual Language Model (VLM) that sets a new state of the art in few-shot learning on a wide range of open-ended vision and language tasks, simply by being prompted with a few input/output examples, as illustrated in Figure 1. Of the 16 tasks we consider, Flamingo also surpasses the fine-tuned state of the art on 6 tasks, despite using orders of magnitude less task-specific training data (see Figure 2). To achieve this, Flamingo takes inspiration from recent work on large language models (LMs) which are good few-shot learners [11, 18, 42, 86]. A single large LM can achieve strong performance on many tasks using only its text interface: a few examples of a task are provided to the model as a prompt, along with a query input, and the model generates a continuation to produce a predicted output for that query. We show that the same can be done for image and video understanding tasks such as classification, captioning, or question-answering: these can be cast as text prediction problems with visual input conditioning. The difference from a LM is that the model must be able to ingest a multimodal prompt containing images and/or videos interleaved with text. Flamingo models have this capability—they are visually-conditioned autoregressive text generation models able to ingest a sequence of text tokens interleaved with images and/or videos, and produce text as output. Flamingo models leverage two complementary pre-trained and frozen models: a vision model which can “perceive” visual scenes and a large LM which performs a basic form of reasoning. Novel architecture components are added in between these models to connect them in a way that preserves the knowledge they have accumulated during computationally intensive pre-training. Flamingo models are also able to ingest high-resolution images or videos thanks to a Perceiver-based [48] architecture that can produce a small fixed number of visual tokens per image/video, given a large and variable number of visual input features.
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A crucial aspect for the performance of large LMs is that they are trained on a large amount of text data. This training provides general-purpose generation capabilities that allows these LMs to perform well when prompted with task examples. Similarly, we demonstrate that the way we train the Flamingo models is crucial for their final performance. They are trained on a carefully chosen mixture of complementary large-scale multimodal data coming only from the web, without using any data annotated for machine learning purposes. After this training, a Flamingo model can be directly adapted to vision tasks via simple few-shot learning without any task-specific tuning.
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Figure 3: Flamingo architecture overview. Flamingo is a family of visual language models (VLMs) that take as input visual data interleaved with text and produce free-form text as output.
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Contributions. In summary, our contributions are the following: (i) We introduce the Flamingo family of VLMs which can perform various multimodal tasks (such as captioning, visual dialogue, or visual question-answering) from only a few input/output examples. Thanks to architectural innovations, the Flamingo models can efficiently accept arbitrarily interleaved visual data and text as input and generate text in an open-ended manner. (ii) We quantitatively evaluate how Flamingo models can be adapted to various tasks via few-shot learning. We notably reserve a large set of heldout benchmarks which have not been used for validation of any design decisions or hyperparameters of the approach. We use these to estimate unbiased few-shot performance. (iii) Flamingo sets a new state of the art in few-shot learning on a wide array of 16 multimodal language and image/video understanding tasks. On 6 of these 16 tasks, Flamingo also outperforms the fine-tuned state of the art despite using only 32 task-specific examples, around 1000 times less task-specific training data than the current state of the art. With a larger annotation budget, Flamingo can also be effectively fine-tuned to set a new state of the art on five additional challenging benchmarks: VQAv2, VATEX, VizWiz, MSRVTTQA, and HatefulMemes.
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# 2 Approach
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This section describes Flamingo: a visual language model that accepts text interleaved with images/videos as input and outputs free-form text. The key architectural components shown in Figure 3 are chosen to leverage pretrained vision and language models and bridge them effectively. First, the Perceiver Resampler (Section 2.1) receives spatio-temporal features from the Vision Encoder (obtained from either an image or a video) and outputs a fixed number of visual tokens. Second, these visual tokens are used to condition the frozen LM using freshly initialised cross-attention layers (Section 2.2) that are interleaved between the pretrained LM layers. These new layers offer an expressive way for the LM to incorporate visual information for the next-token prediction task. Flamingo models the likelihood of text $y$ conditioned on interleaved images and videos $x$ as follows:
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$$
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p ( \boldsymbol y | \boldsymbol x ) = \prod _ { \ell = 1 } ^ { L } p ( \boldsymbol y _ { \ell } | \boldsymbol y _ { \angle \ell } , \boldsymbol x _ { \le \ell } ) ,
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$$
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where $y _ { \ell }$ is the $\ell$ -th language token of the input text, $y _ { < \ell }$ is the set of preceding tokens, $x _ { \le \ell }$ is the set of images/videos preceding token $y _ { \ell }$ in the interleaved sequence and $p$ is parametrized by a Flamingo model. The ability to handle interleaved text and visual sequences (Section 2.3) makes it natural to use Flamingo models for in-context few-shot learning, analogously to GPT-3 with few-shot text prompting. The model is trained on a diverse mixture of datasets as described in Section 2.4.
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Figure 4: GATED XATTN-DENSE layers. To condition the LM on visual inputs, we insert new cross-attention layers between existing pretrained and frozen LM layers. The keys and values in these layers are obtained from the vision features while the queries are derived from the language inputs. They are followed by dense feed-forward layers. These layers are gated so that the LM is kept intact at initialization for improved stability and performance.
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# 2.1 Visual processing and the Perceiver Resampler
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Vision Encoder: from pixels to features. Our vision encoder is a pretrained and frozen NormalizerFree ResNet (NFNet) [10] – we use the F6 model. We pretrain the vision encoder using a contrastive objective on our datasets of image and text pairs, using the two-term contrastive loss from Radford et al. [85]. We use the output of the final stage, a 2D spatial grid of features that is flattened to a 1D sequence. For video inputs, frames are sampled at 1 FPS and encoded independently to obtain a 3D spatio-temporal grid of features to which learned temporal embeddings are added. Features are then flattened to 1D before being fed to the Perceiver Resampler. More details on the contrastive model training and performance are given in Appendix B.1.3 and Appendix B.3.2, respectively.
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Perceiver Resampler: from varying-size large feature maps to few visual tokens. This module connects the vision encoder to the frozen language model as shown in Figure 3. It takes as input a variable number of image or video features from the vision encoder and produces a fixed number of visual outputs (64), reducing the computational complexity of the vision-text cross-attention. Similar to Perceiver [48] and DETR [13], we learn a predefined number of latent input queries which are fed to a Transformer and cross-attend to the visual features. We show in our ablation studies (Section 3.3) that using such a vision-language resampler module outperforms a plain Transformer and an MLP. We provide an illustration, more architectural details, and pseudo-code in Appendix A.1.1.
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# 2.2 Conditioning frozen language models on visual representations
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Text generation is performed by a Transformer decoder, conditioned on the visual representations produced by the Perceiver Resampler. We interleave pretrained and frozen text-only LM blocks with blocks trained from scratch that cross-attend to the visual output from the Perceiver Resampler.
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Interleaving new GATED XATTN-DENSE layers within a frozen pretrained LM. We freeze the pretrained LM blocks, and insert gated cross-attention dense blocks (Figure 4) between the original layers, trained from scratch. To ensure that at initialization, the conditioned model yields the same results as the original language model, we use a tanh-gating mechanism [41]. This multiplies the output of a newly added layer by $\operatorname { t a n h } ( \alpha )$ before adding it to the input representation from the residual connection, where $\alpha$ is a layer-specific learnable scalar initialized to 0 [4]. Thus, at initialization, the model output matches that of the pretrained LM, improving training stability and final performance. In our ablation studies (Section 3.3), we compare the proposed GATED XATTN-DENSE layers against recent alternatives [22, 68] and explore the effect of how frequently these additional layers are inserted to trade off between efficiency and expressivity. See Appendix A.1.2 for more details.
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Varying model sizes. We perform experiments across three models sizes, building on the 1.4B, 7B, and 70B parameter Chinchilla models [42]; calling them respectively Flamingo-3B, Flamingo-9B and
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Flamingo-80B. For brevity, we refer to the last as Flamingo throughout the paper. While increasing the parameter count of the frozen LM and the trainable vision-text GATED XATTN-DENSE modules, we maintain a fixed-size frozen vision encoder and trainable Perceiver Resampler across the different models (small relative to the full model size). See Appendix B.1.1 for further details.
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# 2.3 Multi-visual input support: per-image/video attention masking
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The image-causal modelling introduced in Equation (1) is obtained by masking the full text-to-image cross-attention matrix, limiting which visual tokens the model sees at each text token. At a given text token, the model attends to the visual tokens of the image that appeared just before it in the interleaved sequence, rather than to all previous images (formalized and illustrated in Appendix A.1.3). Though the model only directly attends to a single image at a time, the dependency on all previous images remains via self-attention in the LM. This single-image cross-attention scheme importantly allows the model to seamlessly generalise to any number of visual inputs, regardless of how many are used during training. In particular, we use only up to 5 images per sequence when training on our interleaved datasets, yet our model is able to benefit from sequences of up to 32 pairs (or “shots”) of images/videos and corresponding texts during evaluation. We show in Section 3.3 that this scheme is more effective than allowing the model to cross-attend to all previous images directly.
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# 2.4 Training on a mixture of vision and language datasets
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We train the Flamingo models on a mixture of three kinds of datasets, all scraped from the web: an interleaved image and text dataset derived from webpages, image-text pairs, and video-text pairs.
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M3W: Interleaved image and text dataset. The few-shot capabilities of Flamingo models rely on training on interleaved text and image data. For this purpose, we collect the MultiModal MassiveWeb (M3W) dataset. We extract both text and images from the HTML of approximately 43 million webpages, determining the positions of images relative to the text based on the relative positions of the text and image elements in the Document Object Model (DOM). An example is then constructed by inserting <image> tags in plain text at the locations of the images on the page, and inserting a special $\mathtt { < E O C > }$ (end of chunk) token (added to the vocabulary and learnt) prior to any image and at the end of the document. From each document, we sample a random subsequence of $L = 2 5 6$ tokens and take up to the first $N = 5$ images included in the sampled sequence. Further images are discarded in order to save compute. More details are provided in Appendix A.3.
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Pairs of image/video and text. For our image and text pairs we first leverage the ALIGN [50] dataset, composed of 1.8 billion images paired with alt-text. To complement this dataset, we collect our own dataset of image and text pairs targeting better quality and longer descriptions: LTIP (Long Text & Image Pairs) which consists of 312 million image and text pairs. We also collect a similar dataset but with videos instead of still images: VTP (Video & Text Pairs) consists of 27 million short videos (approximately 22 seconds on average) paired with sentence descriptions. We align the syntax of paired datasets with the syntax of M3W by prepending <image> and appending $\mathtt { < E O C > }$ to each training caption (see Appendix A.3.3 for details).
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Multi-objective training and optimisation strategy. We train our models by minimizing a weighted sum of per-dataset expected negative log-likelihoods of text, given the visual inputs:
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$$
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\sum _ { m = 1 } ^ { M } \lambda _ { m } \cdot \mathbb { E } _ { ( x , y ) \sim \mathcal { D } _ { m } } \left[ - \sum _ { \ell = 1 } ^ { L } \log p ( y _ { \ell } | y _ { < \ell } , x _ { \le \ell } ) \right] ,
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$$
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where $\mathcal { D } _ { m }$ and $\lambda _ { m }$ are the $m$ -th dataset and its weighting, respectively. Tuning the per-dataset weights $\lambda _ { m }$ is key to performance. We accumulate gradients over all datasets, which we found outperforms a “round-robin” approach [17]. We provide further training details and ablations in Appendix B.1.2.
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# 2.5 Task adaptation with few-shot in-context learning
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Once Flamingo is trained, we use it to tackle a visual task by conditioning it on a multimodal interleaved prompt. We evaluate the ability of our models to rapidly adapt to new tasks using incontext learning, analogously to GPT-3 [11], by interleaving support example pairs in the form of $( i m a g e , t e x t )$ or (??????????, ????????), followed by the query visual input, to build a prompt (details in
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Table 1: Comparison to the state of the art. A single Flamingo model reaches the state of the art on a wide array of image (I) and video (V) understanding tasks with few-shot learning, significantly outperforming previous best zero- and few-shot methods with as few as four examples. More importantly, using only 32 examples and without adapting any model weights, Flamingo outperforms the current best methods – fine-tuned on thousands of annotated examples – on seven tasks. Best few-shot numbers are in bold, best numbers overall are underlined.
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<table><tr><td>Method</td><td></td><td>FTShot</td><td> OAAYA</td><td> ZAAO</td><td> Cco</td><td>(A) VOAASS</td><td>VA XIA</td><td>() ZIMZ</td><td>T3</td><td>MA ITLTTI</td><td>(A)VOA!</td><td>J oo</td><td>JS TAIY</td><td>( [PiI</td><td>[T vY</td><td></td><td> VOAXAN</td><td>R eeeeY</td></tr><tr><td rowspan="4">Zero/Few shot SOTA</td><td rowspan="4">X</td><td rowspan="4"></td><td></td><td></td><td>[124]</td><td>[58]</td><td></td><td></td><td></td><td>[58]</td><td>[135]</td><td></td><td></td><td>[79]</td><td></td><td></td><td> grrreieee</td><td></td></tr><tr><td></td><td>[34]</td><td>[114]</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>[143]</td><td></td><td></td><td></td><td></td><td></td><td>[85]</td></tr><tr><td></td><td>43.3</td><td>38.2</td><td>32.2</td><td>35.2</td><td>=</td><td>-</td><td></td><td>19.2 0</td><td>12.2 0</td><td>-</td><td>39.4</td><td>11.6 0</td><td>-</td><td></td><td>[85] 66.1 (0)</td><td>40.7</td></tr><tr><td>x）</td><td>(16)</td><td>(4) 49.2</td><td>(0)</td><td>0 27.5</td><td>40.1</td><td></td><td></td><td></td><td></td><td></td><td>0</td><td></td><td></td><td></td><td></td><td></td><td>0</td></tr><tr><td rowspan="4">Flamingo-3B</td><td></td><td>0</td><td>41.2</td><td></td><td>73.0</td><td></td><td></td><td>28.9</td><td>60.6</td><td>11.0</td><td>32.7</td><td>55.8 64.6</td><td>39.6</td><td>46.1</td><td>30.1 32.7</td><td>21.3 22.4</td><td>53.7 53.6</td><td>58.4</td></tr><tr><td>X X</td><td>4 32</td><td>43.3</td><td>53.2 57.1</td><td>85.0 99.0</td><td>33.0</td><td>50.0 59.2</td><td>34.0</td><td>72.0 71.2</td><td>14.9 25.6</td><td>35.7 37.7</td><td>76.7</td><td>41.3 41.6</td><td>47.3</td><td></td><td></td><td></td><td>1</td></tr><tr><td>X</td><td></td><td>45.9 44.7</td><td>51.8</td><td>79.4</td><td>42.6 30.2</td><td>39.5</td><td>45.5 28.8</td><td>61.5</td><td>13.7</td><td>35.2</td><td>55.0</td><td>41.8</td><td>47.3 48.0</td><td>30.6 31.8</td><td>26.1 23.0</td><td>56.3 57.0</td><td>57.9 -</td></tr><tr><td>X</td><td>0 4</td><td>49.3</td><td>56.3</td><td>93.1</td><td>36.2</td><td>51.7</td><td>34.9</td><td>72.6</td><td>18.2</td><td>37.7</td><td>70.8</td><td>42.8</td><td>50.4</td><td>33.6</td><td>24.7</td><td>62.7</td><td>-</td></tr><tr><td rowspan="3">Flamingo-9B</td><td></td><td></td><td>51.0</td><td>60.4</td><td>106.3</td><td>47.2</td><td>57.4</td><td>44.0</td><td>72.8</td><td>29.4</td><td>40.7</td><td>77.3</td><td>41.2</td><td>50.4</td><td>32.6</td><td>28.4</td><td>63.5</td><td>-</td></tr><tr><td>X</td><td>32 0</td><td>50.6</td><td>56.3</td><td>84.3</td><td>35.6</td><td>46.7</td><td>31.6</td><td>67.2</td><td>17.4</td><td>40.7</td><td>60.1</td><td>39.7</td><td>52.0</td><td>35.0</td><td>26.7</td><td>46.4</td><td>60.8</td></tr><tr><td>美</td><td>4</td><td>57.4</td><td>63.1</td><td>103.2 41.7</td><td>56.0</td><td></td><td>39.6</td><td>75.1</td><td>23.9</td><td>44.1</td><td>74.5</td><td>42.4</td><td>55.6</td><td>36.5</td><td>30.8</td><td>68.6</td><td>-</td></tr><tr><td rowspan="3">Flamingo</td><td>X</td><td>32</td><td>57.8</td><td>67.6</td><td>113.8</td><td>52.3</td><td>65.1</td><td>49.8</td><td>75.4</td><td>31.0</td><td>45.3</td><td>86.8</td><td>42.2</td><td>55.6</td><td>37.9</td><td>33.5</td><td>70.0</td><td>-</td></tr><tr><td></td><td></td><td>54.4</td><td>80.2</td><td>143.3</td><td>47.9</td><td>76.3</td><td>57.2</td><td>67.4 [150]</td><td>46.8</td><td>35.4 [135]</td><td>138.7 [132]</td><td>36.7</td><td>75.2</td><td>54.7</td><td>25.2</td><td>79.1</td><td></td></tr><tr><td>√</td><td>(X)</td><td>[34] (10K)</td><td>[140] (444K)</td><td>[124] [28] (500K) (27K)</td><td>[153] (500K)</td><td></td><td>[65] (20K)</td><td>(30K)</td><td>[51] (130K)</td><td>(6K)</td><td>(10K)</td><td>[128] (46K)</td><td>[79] (123K)</td><td>[137] (20K)</td><td>[129] (38K)</td><td>[62] (9K)</td><td></td></tr></table>
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Appendix A.2). We perform open-ended evaluations using beam search for decoding, and closeended evaluations using our model’s log-likelihood to score each possible answer. We explore zero-shot generalization by prompting the model with two text-only examples from the task, with no corresponding images. Evaluation hyperparameters and additional details are given in Appendix B.1.5.
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# 3 Experiments
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Our goal is to develop models that can rapidly adapt to diverse and challenging tasks. For this, we consider a wide array of 16 popular multimodal image/video and language benchmarks. In order to validate model design decisions during the course of the project, 5 of these benchmarks were used as part of our development (DEV) set: COCO, OKVQA, VQAv2, MSVDQA and VATEX. Performance estimates on the DEV benchmarks may be biased, as a result of model selection. We note that this is also the case for prior work which makes use of similar benchmarks to validate and ablate design decisions. To account for this, we report performance on an additional set of 11 benchmarks, spanning captioning, video question-answering, as well as some less commonly explored capabilities such as visual dialogue and multi-choice question-answering tasks. The evaluation benchmarks are described in Appendix B.1.4. We keep all evaluation hyperparameters fixed across all benchmarks. Depending on the task, we use four few-shot prompt templates we describe in more detail in Appendix B.1.5. We emphasize that we do not validate any design decisions on these 11 benchmarks and use them solely to estimate unbiased few-shot learning performance of our models.
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Concretely, estimating few-shot learning performance of a model involves prompting it with a set of support samples and evaluating it on a set of query samples. For the DEV benchmarks that are used both to validate design decisions and hyperparameters, as well as to report final performance, we therefore use four subsets: validation support, validation query, test support and test query. For other benchmarks, we need only the latter two. We report in Appendix B.1.4 how we form these subsets.
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We report the results of the Flamingo models on few-shot learning in Section 3.1. Section 3.2 gives Flamingo fine-tuned results. An ablation study is given in Section 3.3. Appendix B.2 provides more results including Flamingo’s performance on the ImageNet and Kinetics700 classification tasks, and on our contrastive model’s performance. Appendix C includes additional qualitative results.
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# 3.1 Few-shot learning on vision-language tasks
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Few-shot results. Results are given in Table 1. Flamingo outperforms by a large margin all previous zero-shot or few-shot methods on the 16 benchmarks considered. This is achieved with as few as four examples per task, demonstrating practical and efficient adaptation of vision models to new tasks. More importantly, Flamingo is often competitive with state-of-the-art methods additionally fine-tuned on up to hundreds of thousands of annotated examples. On six tasks, Flamingo even outperforms the fine-tuned SotA despite using a single set of model weights and only 32 task-specific examples. Finally, despite having only used the DEV benchmarks for design decisions, our results generalize well to the other benchmarks, confirming the generality of our approach.
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Table 2: Comparison to SotA when fine-tuning Flamingo. We fine-tune Flamingo on all nine tasks where Flamingo does not achieve SotA with few-shot learning. Flamingo sets a new SotA on five of them, outperfoming methods (marked with $\dagger .$ ) that use tricks such as model ensembling or domain-specific metric optimisation (e.g., CIDEr optimisation).
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<table><tr><td rowspan="2">Method</td><td colspan="2">VQAV2 test-std</td><td rowspan="2">COCO</td><td rowspan="2">VATEX test</td><td colspan="2">VizWiz</td><td rowspan="2">MSRVTTQA test</td><td colspan="2" rowspan="2">VisDial valid|test-std</td><td rowspan="2">YouCook2</td><td colspan="2" rowspan="2">TextVQA valid|test-std</td><td rowspan="2">HatefulMemes test seen</td></tr><tr><td>test-dev</td><td>test</td><td>test-dev</td><td>test-std</td><td>valid</td></tr><tr><td>32 shots</td><td>67.6</td><td>：</td><td>113.8</td><td>65.1</td><td>49.8</td><td>-</td><td>31.0</td><td>56.8</td><td>-</td><td>86.8</td><td>36.0</td><td>·</td><td>70.0</td></tr><tr><td>Fine-tuned</td><td>82.0</td><td>82.1</td><td>138.1</td><td>84.2</td><td>65.7</td><td>65.4</td><td>47.4</td><td>61.8</td><td>59.7</td><td>118.6</td><td>57.1</td><td>54.1</td><td>86.6</td></tr><tr><td rowspan="2">SotA</td><td>81.3</td><td>81.3</td><td>149.6</td><td>81.4</td><td>57.21</td><td>60.6</td><td>46.8</td><td>75.2</td><td>75.4</td><td>138.7</td><td>54.7</td><td>73.7</td><td>84.6</td></tr><tr><td>[133]</td><td>[133]</td><td>[119]</td><td>[153]</td><td>[65]</td><td>[65]</td><td>[51]</td><td>[79]</td><td>[123]</td><td>[132]</td><td>[137]</td><td>[84]</td><td>[152]</td></tr></table>
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<table><tr><td>setting</td><td colspan="2">Ablated</td><td>Flamingo-3B Changed original value value</td><td></td><td>Param. Step count↓ time↓</td><td>CoCo CIDEr↑</td><td>OKVQA top1个</td><td>VQAv2 top1个</td><td>MSVDQA top1个</td><td>VATEX CIDEr↑</td><td>Overall score↑</td></tr><tr><td colspan="3"></td><td>Flamingo-3B model w/o Video-Text pairs</td><td>3.2B</td><td>1.74s</td><td>86.5 84.2</td><td>42.1 43.0</td><td>55.8 53.9</td><td>36.3 34.5</td><td>53.4 46.0</td><td>70.7 67.3</td></tr><tr><td>i</td><td>Training data</td><td>All data</td><td>w/o Image-Text pairs Image-Text pairs→LAION w/oM3W</td><td>3.2B 3.2B 3.2B 3.2B</td><td>1.42s 0.95s 1.74s</td><td>66.3 79.5</td><td>39.2 41.4</td><td>51.6 53.5</td><td>32.0 33.9</td><td>41.6 47.6</td><td>60.9 66.4</td></tr><tr><td>(ii)</td><td>Optimisation</td><td>Accumulation</td><td>Round Robin</td><td>3.2B</td><td>1.02s 1.68s</td><td>54.1 76.1</td><td>36.5 39.8</td><td>52.7 52.1</td><td>31.4 33.2</td><td>23.5 40.8</td><td>53.4 62.9</td></tr><tr><td></td><td>Tanh gating</td><td>√</td><td>X</td><td>3.2B</td><td>1.74s</td><td>78.4</td><td>40.5</td><td>52.9</td><td>35.9</td><td>47.5</td><td>66.5</td></tr><tr><td>(iv)</td><td>Cross-attention architecture</td><td>GATED XATTN-DENSE</td><td>VANILLA XATTN</td><td>2.4B</td><td>1.16s 1.74s</td><td>80.6 79.2</td><td>41.5</td><td>53.4 50.8</td><td>32.9 32.2</td><td>50.7 47.8</td><td>66.9 63.1</td></tr><tr><td></td><td>Cross-attention</td><td></td><td>GRAFTING Single in middle</td><td>3.3B 2.0B</td><td>0.87s</td><td>71.5</td><td>36.1 38.1</td><td>50.2</td><td>29.1</td><td>42.3</td><td>59.8</td></tr><tr><td>(v)</td><td>frequency</td><td>Every</td><td>Every 4th Every 2nd</td><td>2.3B 2.6B</td><td>1.02s 1.24s</td><td>82.3 83.7</td><td>42.7 41.0</td><td>55.1 55.8</td><td>34.6 34.5</td><td>50.8 49.7</td><td>68.8 68.2</td></tr><tr><td>(vi)</td><td>Resampler</td><td>Perceiver</td><td>MLP Transformer</td><td>3.2B 3.2B</td><td>1.85s 1.81s</td><td>78.6 83.2</td><td>42.2 41.7</td><td>54.7 55.6</td><td>35.2 31.5</td><td>44.7 48.3</td><td>66.6 66.7</td></tr><tr><td>(vii)</td><td>Vision encoder</td><td>NFNet-F6</td><td>CLIP ViT-L/14 NFNet-F0</td><td>3.1B</td><td>1.58s 1.45s</td><td>76.5 73.8</td><td>41.6</td><td>53.4 52.8</td><td>33.2 31.1</td><td>44.5 42.9</td><td>64.9 62.7</td></tr><tr><td>(vii)</td><td></td><td></td><td>X(random init)</td><td>2.9B 3.2B</td><td>2.42s</td><td>74.8</td><td>40.5 31.5</td><td>45.6</td><td>26.9</td><td>50.1</td><td>57.8</td></tr><tr><td></td><td>Freezing LM</td><td>√</td><td>X (pretrained)</td><td>3.2B</td><td>2.42s</td><td>81.2</td><td>33.7</td><td>47.4</td><td>31.0</td><td>53.9</td><td>62.7</td></tr></table>
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Table 3: Ablation studies. Each row should be compared to the baseline Flamingo run (top row). Step time measures the time spent to perform gradient updates on all training datasets.
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Scaling with respect to parameters and shots. As shown in Figure 2, the larger the model, the better the few-shot performance, similar to GPT-3 [11]. The performance also improves with the number of shots. We further find that the largest model better exploits larger numbers of shots. Interestingly, even though our Flamingo models were trained with sequences limited to only 5 images on M3W, they are still able to benefit from up to 32 images or videos during inference. This demonstrates the flexibility of the Flamingo architecture for processing a variable number of videos or images.
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# 3.2 Fine-tuning Flamingo as a pretrained vision-language model
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While not the main focus of our work, we verify that when given more data, Flamingo models can be adapted to a task by fine-tuning their weights. In Table 2, we explore fine-tuning our largest model, Flamingo, for a given task with no limit on the annotation budget. In short, we do so by fine-tuning the model on a short schedule with a small learning rate by additionally unfreezing the vision backbone to accommodate a higher input resolution (details in Appendix B.2.2). We find that we can improve results over our previously presented in-context few-shot learning results, setting a new state of the art on five additional tasks: VQAv2, VATEX, VizWiz, MSRVTTQA, and HatefulMemes.
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# 3.3 Ablation studies
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In Table 3, we report our ablation results using Flamingo-3B on the validation subsets of the five DEV benchmarks with 4 shots. Note that we use smaller batch sizes and a shorter training schedule compared to the final models. The Overall score is obtained by dividing each benchmark score by its state-of-the-art (SotA) performance from Table 1 and averaging the results. More details and results are given in Appendix B.3 and Table 10.
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Importance of the training data mixture. As shown in row (i), getting the right training data plays a crucial role. In fact, removing the interleaved image-text dataset M3W leads to a decrease of more than $1 7 \%$ in performance while removing the conventional paired image-text pairs also decreases performance (by $9 . 8 \%$ ), demonstrating the need for different types of datasets. Moreover, removing our paired video-text dataset negatively affects performance on all video tasks. We ablate replacing our image-text pairs (ITP) by the publicly available LAION-400M dataset [96], which leads to a slight degradation in performance. We show in row (ii) the importance of our gradient accumulation strategy compared to using round-robin updates [17].
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Visual conditioning of the frozen LM. We ablate the use of the 0-initialized tanh gating when merging the cross-attention output to the frozen LM output in row (iii). Without it, we see a drop of $4 . 2 \%$ in our overall score. Moreover, we have noticed that disabling the 0-initialized tanh gating leads to training instabilities. Next, we ablate different conditioning architectures in row (iv). VANILLA XATTN, refers to the vanilla cross-attention from the original Transformer decoder [115]. In the GRAFTING approach from [68], the frozen LM is used as is with no additional layers inserted, and a stack of interleaved self-attention and cross-attention layers that take the frozen LM output are learnt from scratch. Overall, we show that our GATED XATTN-DENSE conditioning approach works best.
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Compute/Memory vs. performance trade-offs. In row (v), we ablate the frequency at which we add new GATED XATTN-DENSE blocks. Although adding them at every layer is better, it significantly increases the number of trainable parameters and time complexity of the model. Notably, inserting them every fourth block accelerates training by $6 6 \%$ while only decreasing the overall score by $1 . 9 \%$ In light of this trade-off, we maximize the number of added layers under hardware constraints and add a GATED XATTN-DENSE every fourth layer for Flamingo-9B and every seventh for Flamingo-80B. We further compare in row (vi) the Perceiver Resampler to a MLP and a vanilla Transformer given a parameter budget. Both underperform the Perceiver Resampler while also being slower.
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Vision encoder. In row (vii), we compare our NFNet-F6 vision encoder pretrained with contrastive learning (details in Appendix B.1.3) to the publicly available CLIP ViT-L/14 [85] model trained at 224 resolution. Our NFNet-F6 has a $+ 5 . 8 \%$ advantage over the CLIP ViT-L/14 and $+ 8 . 0 \%$ over a smaller NFNet-F0 encoder, which highlights the importance of using a strong vision backbone.
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Freezing LM components prevents catastrophic forgetting. We verify the importance of freezing the LM layers at training in row (viii). If trained from scratch, we observe a large performance decrease of $- 1 2 . 9 \%$ . Interestingly, fine-tuning our pretrained LM also leads to a drop in performance of $- 8 . 0 \%$ . This indicates an instance of “catastrophic forgetting” [71], in which the model progressively forgets its pretraining while training on a new objective. In our setting, freezing the language model is a better alternative to training with the pre-training dataset (MassiveText) in the mixture.
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# 4 Related work
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Language modelling and few-shot adaptation. Language modelling has recently made substantial progress following the introduction of Transformers [115]. The paradigm of first pretraining on a vast amount of data followed by an adaptation on a downstream task has become standard [11, 23, 32, 44, 52, 75, 87, 108]. In this work, we build on the 70B Chinchilla language model [42] as the base LM for Flamingo. Numerous works have explored techniques to adapt language models to novel tasks using a few examples. These include adding small adapter modules [43], fine-tuning a small part of the LM [141], showing in-context examples in the prompt [11], or optimizing the prompt [56, 60] through gradient descent. In this paper, we take inspiration from the in-context [11] few-shot learning technique instead of more involved few-shot learning approaches based on metric learning [24, 103, 112, 117] or meta-learning [6, 7, 27, 31, 91, 155].
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When language meets vision. These LM breakthroughs have been influential for vision-language modelling. In particular, BERT [23] inspired a large body of vision-language work [16, 28, 29, 38, 59, 61, 66, 101, 106, 107, 109, 118, 121, 142, 143, 151]. We differ from these approaches as Flamingo models do not require fine-tuning on new tasks. Another family of vision-language models is based on contrastive learning [2, 5, 49, 50, 57, 74, 82, 85, 138, 140, 146]. Flamingo differs from contrastive models as it can generate text, although we build and rely upon them for our vision encoder. Similar to our work are VLMs able to generate text in an autoregressive manner [19, 25, 45, 67, 116]. Concurrent works [17, 58, 119, 124, 154] also propose to formulate numerous vision tasks as text generation problems. Building on top of powerful pretrained language models has been explored in several recent works. One recent line of work [26, 68, 78, 114, 136, 144] proposes to freeze the pretrained LM weights to prevent catastrophic forgetting [71]. We follow this idea by freezing the
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Chinchilla LM layers [42] and adding learnable layers within the frozen LM. We differ from prior work by introducing the first LM that can ingest arbitrarily interleaved images, videos, and text.
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Web-scale vision and language training datasets. Manually annotated vision and language datasets are costly to obtain and thus relatively small (10k-100k) in scale [3, 15, 69, 122, 129, 139]. To alleviate this lack of data, numerous works [14, 50, 98, 110] automatically scrape readily available paired vision-text data. In addition to such paired data, we show the importance of also training on entire multimodal webpages containing interleaved images and text as a single sequence. Concurrent work CM3 [1] proposes to generate HTML markup from pages, while we simplify the text prediction task by only generating plain text. We emphasize few-shot learning and vision tasks while CM3 [1] primarily evaluates on language-only benchmarks in a zero-shot or fine-tuned setup.
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# 5 Discussion
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Limitations. First, our models build on pretrained LMs, and as a side effect, directly inherit their weaknesses. For example, LM priors are generally helpful, but may play a role in occasional hallucinations and ungrounded guesses. Furthermore, LMs generalise poorly to sequences longer than the training ones. They also suffer from poor sample efficiency during training. Addressing these issues can accelerate progress in the field and enhance the abilities of VLMs like Flamingo.
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Second, the classification performance of Flamingo lags behind that of state-of-the-art contrastive models [82, 85]. These models directly optimize for text-image retrieval, of which classification is a special case. In contrast, our models handle a wider range of tasks, such as open-ended ones. A unified approach to achieve the best of both worlds is an important research direction.
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Third, in-context learning has significant advantages over gradient-based few-shot learning methods, but also suffers from drawbacks depending on the characteristics of the application at hand. We demonstrate the effectiveness of in-context learning when access is limited to only a few dozen examples. In-context learning also enables simple deployment, requiring only inference, generally with no hyperparameter tuning needed. However, in-context learning is known to be highly sensitive to various aspects of the demonstrations [80, 148], and its inference compute cost and absolute performance scale poorly with the number of shots beyond this low-data regime. There may be opportunities to combine few-shot learning methods to leverage their complementary benefits. We discuss the limitations of our work in more depth in Appendix D.1.
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Societal impacts. In terms of societal impacts, Flamingo offers a number of benefits while carrying some risks. Its ability to rapidly adapt to a broad range of tasks have the potential to enable non-expert users to obtain good performance in data-starved regimes, lowering the barriers to both beneficial and malicious applications. Flamingo is exposed to the same risks as large language models, such as outputting offensive language, propagating social biases and stereotypes, as well as leaking private information [42, 126]. Its ability to additionally handle visual inputs poses specific risks such as gender and racial biases relating to the contents of the input images, similar to a number of visual recognition systems [12, 21, 37, 97, 147]. We refer the reader to Appendix D.2 for a more extensive discussion of the societal impacts of our work, both positive and negative; as well as mitigation strategies and early investigations of risks relating to racial or gender bias and toxic outputs. Finally we note that, following prior work focusing on language models [72, 81, 111], the few-shot capabilities of Flamingo could be useful for mitigating such risks.
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Conclusion. We proposed Flamingo, a general-purpose family of models that can be applied to image and video tasks with minimal task-specific training data. We also qualitatively explored interactive abilities of Flamingo such as “chatting” with the model, demonstrating flexibility beyond traditional vision benchmarks. Our results suggest that connecting pre-trained large language models with powerful visual models is an important step towards general-purpose visual understanding.
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Acknowledgments and Disclosure of Funding. This research was funded by DeepMind. We would like to thank many colleagues for useful discussions, suggestions, feedback, and advice, including: Samuel Albanie, Relja Arandjelovic, Kareem Ayoub, Lorrayne Bennett, Adria Recasens Continente, ´ Tom Eccles, Nando de Freitas, Sander Dieleman, Conor Durkan, Aleksa Gordic, Raia Hadsell, ´ Will Hawkins, Lisa Anne Hendricks, Felix Hill, Jordan Hoffmann, Geoffrey Irving, Drew Jaegle, Koray Kavukcuoglu, Agustin Dal Lago, Mateusz Malinowski, Sona Mokrá, Gaby Pearl, Toby Pohlen, ˇ Jack Rae, Laurent Sifre, Francis Song, Maria Tsimpoukelli, Gregory Wayne, and Boxi Wu.
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# Checklist
|
| 411 |
+
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| 412 |
+
1. For all authors...
|
| 413 |
+
|
| 414 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
|
| 415 |
+
(b) Did you describe the limitations of your work? [Yes] See Section 5.
|
| 416 |
+
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5 for a brief discussion and Appendix D.2 for the full discussion.
|
| 417 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
|
| 418 |
+
|
| 419 |
+
2. If you are including theoretical results...
|
| 420 |
+
|
| 421 |
+
(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
|
| 422 |
+
|
| 423 |
+
3. If you ran experiments...
|
| 424 |
+
|
| 425 |
+
(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)? [No] The code and the data are proprietary.
|
| 426 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 3 and Appendix B.
|
| 427 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We do not observe large enough variance in our training runs to justify the computation cost incurred by multiple training runs. For the largest models, it is not feasible within our compute budget.
|
| 428 |
+
(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] Details can be found in Appendix B.1.2. In short, our largest run was trained on 1536 TPU chips for 15 days.
|
| 429 |
+
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| 430 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 431 |
+
|
| 432 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] We properly cited the prior methods on which our work is based, as well as prior datasets when appropriate (e.g., ALIGN).
|
| 433 |
+
(b) Did you mention the license of the assets? [N/A] The assets we used are previous work for which we cited papers. We do mention the license of all visual assets we use for the figures of the paper in Appendix G.
|
| 434 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [No]
|
| 435 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Our data was automatically scraped from million of webpages. See Datasheets [30] in Appendix F.
|
| 436 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Datasheets [30] in Appendix F.
|
| 437 |
+
|
| 438 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 439 |
+
|
| 440 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 441 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 442 |
+
(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 |
+
# PartialFormer: Modeling Part Instead of Whole for Machine Translation
|
| 2 |
+
|
| 3 |
+
Anonymous EMNLP submission
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
The parameter redundancy problem in Transformer models has been widely acknowledged in the literature. To address this weakness, we introduce PartialFormer, a parameter-efficient Transformer architecture for machine translation. Compared to previous parameter-efficient Transformer architecture, PartialFormer modifies the modeling strategy of the feed-forward network to allow it to spare tremendous parameters while maintaining large hidden dimension. Additionally, PartialFormer applies two efficient scaling strategies, namely depth scaling and width scaling, to improve performance within a given parameter budget. To efficiently benefit from these scaling strategies, PartialFormer is further enhanced by two costeffective modifications: 1) a head scaling strategy for efficient width scaling and 2) a residuallike attention calculation for better depth scaling. Extensive experiments on 9 translation tasks validate the effectiveness of our PartialFormer approach.
|
| 8 |
+
|
| 9 |
+
|
| 10 |
+
Figure 1: Illustration of our idea.
|
| 11 |
+
|
| 12 |
+
# 1 Introduction
|
| 13 |
+
|
| 14 |
+
The Transformer model (Vaswani et al., 2017) has emerged as a cornerstone in the natural language processing (NLP) domain, overshadowing convolutional neural networks (Gehring et al., 2017) and recurrent neural networks (Sutskever et al., 2014) by virtue of its minimal inductive bias, superior scalability, and proficiency in modeling extended sequences. Nonetheless, its substantial computational and parametric requisites pose significant challenges to its deployment and training, warranting an ongoing trend in the research community toward eliminating redundant parameters and computations in the Transformer model (Dehghani et al., 2019; Lan et al., 2020; Reid et al., 2021; Li et al., 2022; Ahmed et al., 2017; Yan et al., 2020; Wu et al., 2020; Mehta et al., 2019, 2021).
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it is noteworthy that these approaches ignore the importance of feed-forward networks (FFN). Feedforward networks consume significant parametric and computational overhead due to the inherent large feature space and hidden dimension. To cut down FFNs’ overhead, previous studies (Mehta et al., 2021; Wu et al., 2020; Ge et al., 2022) just adopt smaller hidden dimension, e.g., equal to or even lower than the size of feature space. That leads to a question: Are current lightweight FFNs optimal?
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Despite their success in improving the parametric and computational efficiency of the Transformer,
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To address this concern, we turn to the insights provided by Geva et al. (2021), who depicted FFNs as a collection of key-value memories, where the number of memories is equal to the number of hidden dimensions in FFNs. This finding underscores the significance of hidden dimension in FFNs. Drawing inspiration from this finding and the successful application of large hidden sizes in FFNs as evidenced by Meta’s 4B model (Tran et al., 2021)1, we postulate that a truly efficient lightweight FFN should maintain, if not enlarge, the hidden dimension while reducing parameters.
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To this end, we propose PartialFormer, an innovative approach to Transformer architecture. The central design of PartialFormer is the Partial-Level Gated Feed-Forward Networks (PG-FFN). We designed the PG-FFN as a set of smaller FFNs in unison, each producing lower-dimensional hidden features, yet collectively matching or exceeding the hidden dimension of a conventional larger FFN. Moreover, we further equipped PartialFormer with two cost-effective operations: a head scaling strategy for efficient width scaling, and a residual-like attention calculation for stable optimization. These techniques empower PartialFormer to achieve deeper layer stacking or increased width within the same parameter budget.
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The strength of PartialFormer has been affirmed through rigorous empirical evaluations on $9 \ \mathrm { m a }$ chine translation tasks. Remarkably, even while maintaining similar parameter consumption, our PartialFormer consistently surpasses the vanilla Transformer, employing the same layer depth and embedding width, by an average of 1.29 BLEU points across all 6 WMT’17 machine translations. Furthermore, it achieved a BLEU score of 29.56 on the challenging WMT’14 En-De task with only 68 million parameters, showcasing its effectiveness and efficiency. Our work with PartialFormer thus marks an important step towards the goal of optimized Transformer architectures, marrying performance with efficiency in a manner that has potential for broad impact in NLP applications.
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# 2 Preliminary: Transformer
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In this section, we present some prior knowledge about the Transformer. Typically, Transformer block always consists of a multi-head self-attention and a feed-forward network. Let $X \in \mathbb { R } ^ { T \times d }$ be a $T \times d$ input matrix of $T$ tokens. Each multi-head self-attention component owns $H$ heads. For simplicity, we ignore the layer-normalization operation and residual connection.
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Multi-Head Self-Attention MHSA aims to model the global dependency among tokens. MHSA computes as follows:
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$$
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\begin{array} { r c l } { { { \cal A } ^ { i } } } & { { = } } & { { \mathrm { S o f t m a x } ( \displaystyle \frac { Q ^ { i } ( K ^ { i } ) ^ { \top } } { \sqrt { d _ { k } } } ) , } } \\ { { \mathrm { h e a d } _ { i } } } & { { = } } & { { { \cal A } ^ { i } V ^ { i } , } } \\ { { X } } & { { = } } & { { \displaystyle \sum _ { i = 1 } ^ { H } \mathrm { h e a d } _ { i } W _ { i } ^ { O } , } } \end{array}
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$$
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where $Q ^ { i } , K ^ { i } , V ^ { i }$ denote the query, key and value of $i$ -th head, which are derived from input with three learnable matrics $W _ { i } ^ { Q } , W _ { i } ^ { K } , W _ { i } ^ { V } \ \in \ \mathbb { R } ^ { d \times d _ { k } }$ as follows: $Q ^ { i } \ : = \ : X W _ { i } ^ { Q } , K ^ { i } \ : = \ : X W _ { i } ^ { K } , V ^ { i } \ : =$
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$X W _ { i } ^ { V }$ , respectively. $W _ { i } ^ { O } \in \mathbb { R } ^ { d _ { k } \times d }$ is a learnable matrix. $A ^ { i }$ and headi denote the attention matrix and representation of $i$ -th head, respectively.
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Feed-Forward Network Feed-forward network is responsible for improving the expressiveness of the whole representation space by adopting an "expansion-activation-reduction" mapping strategy. It computes as follows:
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$$
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X = \mathrm { R e L U } ( X W _ { 1 } + b _ { 1 } ) W _ { 2 } + b _ { 2 } ,
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$$
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where $W _ { 1 } ~ \in ~ \mathbb { R } ^ { d \times d _ { \mathrm { f n } } } , W _ { 2 } ~ \in ~ \mathbb { R } ^ { d _ { \mathrm { f n } } \times d } , b _ { 1 } ~ \in$ $\mathbb { R } ^ { d _ { \mathrm { f f n } } } , b _ { 2 } \in \mathbb { R } ^ { d }$ as learnable matrices and $d _ { \mathrm { { f f n } } }$ denotes the hidden dimension in FFN that is usually set to $4 d$ .
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+
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# 3 PartialFormer
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# 3.1 Overall Architecture
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Figure 2 illustrates the overall architecture of PartialFormer, encompassing both an encoder and a decoder. Although the foundational structure adheres to the design of the vanilla Transformer (Vaswani et al., 2017), there are some notable modifications.
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Encoder. Different from vanilla Transformer, each encoder layer in PartialFormer consists of a unified sub-layer that integrates the PG-FFNs into the multi-head self-attention mechanism rather than separate two sub-layers.
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Decoder. Each decoder layer is composed of two types of sub-layers, both of which integrate the multi-head attention mechanism with PG-FFNs. The sub-layers differ based on the type of multihead attention mechanisms employed, specifically whether it’s a decoder self-attention or an encoderdecoder cross-attention mechanism.
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# 3.2 Information Flow in Unified Sub-Layer
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Taking the Encoder as an instance. Each unified sub-layer first computes the multiple attention scores via Eq. (5), then obtains the multiple head features $\{ \mathrm { h e a d } ^ { i } | 1 \leq i \leq H \}$ via Eq. (2), which is the same as vanilla Transformer. Then, using multiple small FFNs, it processes these head features and ultimately combines the representations via a fusion function according to Eq. (7). That is to say, the PG-FFN is encapsulated into the multiheadattention mechanism.
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Figure 2: (a) Architecture of Transformer. (b) Architecture of PartialFormer. (c) Details of Self-AFFN Block. All architecture are based on pre-normalization strategy. We omit the layer normalization operation, residual connection, softmax operation and scale coefficient for simplicity.
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$$
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\begin{array} { l l l } { { A ^ { i } } } & { { = } } & { { \displaystyle \mathrm { S o f t m a x } ( \frac { Q ^ { i } ( K ^ { i } ) ^ { \top } } { \sqrt { d _ { k } } } + A _ { G } ^ { i } ) , } } \\ { { O ^ { i } } } & { { = } } & { { \displaystyle \mathrm { P G } \mathrm { - } \mathrm { F F N } ( \mathrm { h e a d } ^ { i } ) , } } \\ { { X } } & { { = } } & { { \displaystyle \sum _ { i = 1 } ^ { H } O ^ { i } W _ { i } ^ { O } } } \end{array}
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$$
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# 3.3 Partial-Level Gated FFN
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Intuition Previous studies (Wu et al., 2020; Mehta et al., 2021; Ge et al., 2022) have commonly reduced the parameters in feed-forward networks by decreasing the hidden dimension (e.g., 2048 to 256). In contrast, we tackle this issue through a matrix factorization approach. Our key idea involves utilizing a collection of small FFNs to model smaller input features, rather than relying on a single large FFN.
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Assume a FFN with mappings of $1 0 2 4 \mathrm { - } { > } 4 0 9 6 \mathrm { - }$ ${ > } 1 0 2 4$ , which consumes around 8.4 million parameters. By decomposing this into 8 smaller FFNs with mappings of $1 2 8 \mathrm { - } > 5 1 2 \mathrm { - } > 1 2 8$ , we can retain the same hidden dimension, such as $8 ^ { * } 5 1 2$ , while using only 1.05 million parameters. This approach significantly reduces parameters while maintaining the crucial desired hidden dimension, as emphasized in previous studies (Geva et al., 2021; Tran et al., 2021).
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Furthermore, we have observed that the Transformer architecture inherently consists of multiple smaller subspaces, namely “heads” within the multi-head attention (MHA) mechanism. These heads act as sub-components of the original inputs and retain substantial information from the original data. As a result, PG-FFNs should naturally be constructed based on the MHA mechanism.
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Calculation of PG-FFNs While group transformation operations could be used to instantiate our idea, they are not optimal on GPUs due to their low I/O efficiency (Ma et al., 2018), causing significant inference latency. To address this, we propose sharing parameters across each FFN within different heads, thereby eliminating the need for group transformation operations.
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However, directly sharing weights may result in homogeneous representations across different heads, which may potentially hinder the performance (Li et al., 2018). To mitigate this, we further introduce a head-specific gated mechanism. The core idea is to use a set of diverse masks to filter the information of different heads so that the head representation will be more diverse. Formally, given a set of smaller features $\{ \mathrm { h e a d } ^ { i } | 1 \leq i \leq H \}$ and diverse masks $\{ G ^ { i } | 1 \leq i \leq H \}$ , the Eq. (6) can rewritten as:
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$$
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O ^ { i } = G ^ { i } \odot \mathrm { F F N } ( \mathrm { h e a d } ^ { i } ) ,
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$$
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where $\mathrm { F F N } ( \cdot )$ is the same as Eq. (4).
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Generating $\{ G ^ { i } | 1 \le i \le H \}$ In our preliminary experiments, we observed significant diversity in the features generated by different parameters from sub-layer inputs, e.g., $\{ V ^ { 1 } , \ldots , V ^ { H } \}$ . Motivated by this finding, we generate diverse masks in the following manner:
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$$
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G ^ { i } = \sigma ( X W _ { i } ^ { G } ) ,
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$$
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where $W _ { i } ^ { G }$ is a learnable matrix and $\sigma$ denotes the activation function, e.g., ReLU, Sigmoid and Tanh. We compare them in Table 8.
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# 3.4 Efficient Scaling Strategy
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Though PG-FFN offers the advantage of reducing lots of parameters when applied directly to the transformer, it also leads to performance degradation. Thus, a crucial aspect of this study is to determine how to effectively utilize the spared parameters. In this work, we adopt a hybrid scaling strategy, combining both width scaling and depth scaling, which has been validated in computer vision, e.g., EfficientNet (Tan and Le, 2019).
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# 3.4.1 Enabling Efficient Depth Scaling for PartialFormer
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Wang et al. (2019); Dong et al. (2021); Wang et al. (2022) have shown that the original location of FFNs plays an essential role in optimizing transformers, e.g., alleviating Token Uniformity. Thus, we need to consider the impact brought by the change of FFNs. While the densely residual connection is an efficient way to alleviate it, they are typically either based on feature level (e.g., DLCL (Wang et al., 2019)) or coupled with the network structure (e.g., Realformer (He et al., 2021)).
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To this end, we design a new variant of the residual connection integrated into the attention calculation, while also decoupling from the network architecture. Specifically, the calculation of attention maps consists of two parts: 1) $A _ { G }$ , the global part, and 2) $A _ { L }$ , the local part. The calculation of $A _ { L }$ remains the same as in the vanilla Transformer, while $A _ { G }$ is computed once by using the original embedding as input through Eq. (1). Inspired by He et al. (2021), to efficiently fuse these components, we add them together and apply a Softmax function, as shown in Eq. (5).
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In addition to the benefit of efficient depth scaling (See Appendix F), this approach provides remarkable flexibility in combining different attention mechanisms, specifically tailored to address specific conditions. For instance, it allows for the utilization of local attention to calculate $A _ { G }$ when dealing with small datasets (see Appendix D).
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# 3.4.2 Head Scaling: An Efficient Width Scaling for PartialFormer
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Existing approach to width scaling, which is based on the embedding size, necessitates the simultaneous scaling of both the encoder and decoder for machine translation tasks. This is primarily because researchers commonly employ shared encoder and decoder embedding. However, taking cues from the achievements of depth scaling, it may be more advantageous to adopt a distinct method for scaling width, similar to the approach used for scaling depth. Here we show how PartialFormer has inherent superiority to achieve so.
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The width of a Transformer model typically refers to the widest part of the Transformer. In this context, both the vanilla Transformer and previous lightweight Transformer models have widths that are related to the embedding dimension, such as $4 d$ or $d$ . Therefore, by increasing the embedding size, we can effectively enlarge their width. However, the width definition in PartialFormer is different and can be expressed as $w = H \times d _ { \mathrm { f f n } }$ , where $w$ denotes the width of model and $d _ { \mathrm { { f f n } } }$ is associated with the head dimension $d _ { k }$ . Consequently, we can expand the width by either increasing the number of heads or enlarging $d _ { k }$ . A comparison between these approaches is presented in Table 7. Notably, if the head dimension and number of heads are independent of the embedding dimension, PartialFormer allows for easy scaling of width in different ways within the encoder and decoder components.
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Figure 3: Comparison of ways to generate subspaces in Transformer and PartialFormer.
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To this end, we propose a new scaling mechanism, namely head scaling, that scales the width of PartialFormer by directly adding more heads and increasing head dimension, as illustrated in Figure 3. Given the head dimension $d _ { k }$ , the embedding dimension $d$ , and the number of heads $H$ , we consider two strategies to generate $H$ attention heads:
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(a) Simple strategy: We employ three learnable matrices, each with a shape of $d \times ( d _ { k } \times H )$ , to directly obtain the expected number of $Q$ , $K$ , and $V$ .
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(b) Complex strategy: we employ a two-step process. First, we generate an intermediate quantity of $Q$ and $K$ , and then use a powerful MLP network to expand the attention maps to the desired number. This innovative design draws inspiration from the inherent redundancy found within the attention map (Michel et al., 2019; Clark et al., 2019; Voita et al., 2019), allowing for more heads in PartialFormer under the same parameter budget. We show the comparisons in Table 6.
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<table><tr><td>Type</td><td>Model</td><td>N-M</td><td>ddk</td><td></td><td></td><td></td><td>H MACs Param</td><td>BLEU</td><td>COMET-22</td></tr><tr><td rowspan="4">Multi-Branch Architecture</td><td>Weighted Transformer (Ahmed et al.,2017)</td><td>6-6</td><td>1024</td><td></td><td></td><td></td><td>211M</td><td>28.90</td><td></td></tr><tr><td>Multi-Unit Transformer (Yan et al.,2020)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>130M</td><td>29.30</td><td></td></tr><tr><td>MAT (Fan et al., 2020)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>206M</td><td>29.90</td><td></td></tr><tr><td>Multi-Path Transformer (Lin et al., 2022)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>193M</td><td>29.68</td><td></td></tr><tr><td>Lightweight Architecture</td><td>Evolved Transformer (So et al., 2019) Delight (Mehta et al., 2021)</td><td></td><td></td><td></td><td></td><td></td><td>64M</td><td>28.20</td><td></td></tr><tr><td rowspan="5">Weight Sharing</td><td></td><td></td><td>640</td><td></td><td></td><td></td><td>54M</td><td>28.00</td><td></td></tr><tr><td>Universal Transformer (Dehghani et al.,2019)</td><td></td><td>1024</td><td></td><td>=</td><td></td><td>65M</td><td>28.90</td><td></td></tr><tr><td>SubFormer (Reid et al., 2021)</td><td></td><td>-</td><td></td><td>=</td><td></td><td>63M</td><td>28.50</td><td></td></tr><tr><td>SubFormer-big (Reid et al., 2021)</td><td></td><td></td><td></td><td></td><td></td><td>197M</td><td>29.30</td><td></td></tr><tr><td>ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022)</td><td>6-6 24-6</td><td>512 512</td><td></td><td>=</td><td></td><td>62M 118M</td><td>29.03 29.80</td><td></td></tr><tr><td rowspan="3">Other Comparisons</td><td></td><td></td><td></td><td>64</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RealFormer (He et al., 2021) DMAN (Fan et al., 2021)</td><td>18-18</td><td>512 512</td><td></td><td>8 8</td><td></td><td>151M</td><td>29.35</td><td></td></tr><tr><td>Mega-Softmax (Ma et al.,2022)</td><td>6-6 6-6</td><td>512</td><td></td><td></td><td></td><td>63M 67M</td><td>29.10 29.01</td><td></td></tr><tr><td rowspan="7">Our System</td><td>Transformer</td><td></td><td></td><td>64</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>24-6</td><td>512</td><td>8-8</td><td></td><td>11.1B</td><td>118M</td><td>29.05</td><td>83.60</td></tr><tr><td>PartialFormer</td><td>24-6</td><td>512</td><td>64 24-16</td><td>8-8</td><td>8.8B</td><td>66M</td><td>28.86 30.09</td><td>83.35 84.17</td></tr><tr><td></td><td>24-6</td><td>512</td><td>64</td><td></td><td>12.2B</td><td>115M</td><td></td><td></td></tr><tr><td>Transformer</td><td>6-6</td><td>512</td><td>64 45</td><td>8-8</td><td>9.9B</td><td>62M</td><td>27.43</td><td>82.19</td></tr><tr><td>Transformer PartialFormer (w/o Head Scaling)</td><td>24-6 24-6</td><td>360 360</td><td>45</td><td>8-8 8-8</td><td>6.3B 5.2B</td><td>62M 36M</td><td>28.00 27.88</td><td>82.72 82.49</td></tr><tr><td></td><td></td><td>360</td><td>45</td><td>24-16</td><td>6.8B</td><td>61M</td><td>29.23</td><td></td></tr><tr><td></td><td>PartialFormer PartialFormer</td><td>24-6 24-6</td><td>360</td><td>45 30-16</td><td></td><td>6.9B</td><td>68M</td><td>29.56</td><td>83.74 83.94</td></tr></table>
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Table 1: Results on the WMT’14 En-De task. MACs denote the multiplication-addition operations. We compute them via 20 source and target tokens following Mehta et al. (2021).
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# 4 Experimental Setups
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In our evaluation, we assess the performance of PartialFormer across 9 machine translation tasks2. More details are given in Appendix A
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Dataset. We evaluate our approach on three widely-used datasets: WMT’14 English-German (En-De), WMT’14 English-French (En-Fr), and WMT’16 English-Romanian (En-Ro). Besides, to further validate the effectiveness of PartialFormer, we also evaluate PartialFormer on six translation tasks from WMT’17 benchmark. We preprocess the raw data following the standard strategy.
|
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+
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+
Architectures and Selected Baselines. We use a 24-6 encoder-decoder PartialFormer architecture for its strong performance, on all 9 machine translation tasks. Detailed configurations are provided in the results tables. We compare our approach with various baselines, including vanilla Transformer models, multi-branch architecture, lightweight architecture, weight-sharing methods, and other strong baselines.
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+
Training & Evaluation. We train all the models on GeForce RTX 3090 cards via Fairseq (Ott et al., 2019) toolkit. For evaluation, we utilized multi-BLEU (Papineni et al., 2002) and COMET22 (Rei et al., 2022) scores. Beam sizes were 4, 4, and 5 for En-De, En-Fr, and En-Ro tasks respectively. Length_penalty of 0.6, 0.8, and 1.3 were applied to En-De, En-Fr, and En-Ro tasks respectively. For the WMT’17 benchmark, beam size and Length_penalty were set to 4 and 1, respectively. We used an ensemble of the last ten checkpoints.
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# 5 Experiments
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Results of WMT’14 En-De Table 1 presents the results for the WMT’14 En-De task. Note that we also provide a “strong” baseline which also benefits from deep model stacking. Even though the performance of PartialFormer (w/o Head Scaling) is slightly inferior to that of the Transformer model (27.88 vs. 28.00 and 28.86 vs. 29.05), it outshines the latter in terms of parameter efficiency, consuming significantly fewer parameters (36M vs. 62M, 66M vs. 118M). We attribute this phenomenon to our PG-FFN, which leverages a group of compact FFNs. This approach enables PG-FFN to maintain high hidden dimension, while drastically reducing parameter consumption.
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+
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+
Upon utilizing our head scaling technique to amplify the capacity, our Partialformer delivers a BLEU score 29.56 and 30.09 on two configurations, respectively. This surpasses the standard Transformer by 1.56 BLEU points (29.56 vs. 28.00) and
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Table 2: Results on the WMT’14 En-Fr task.
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<table><tr><td>Model</td><td>N</td><td>ddk</td><td>HParam BLEU</td></tr><tr><td>Weighted Transformer (2017)</td><td>6</td><td></td><td>- 211M 41.40</td></tr><tr><td>Evolved Transformer (2019)</td><td>=</td><td></td><td>64M 40.60</td></tr><tr><td>Delight (2021)</td><td>-640</td><td></td><td>54M 40.50</td></tr><tr><td>ODE Transformer (2022)</td><td>6</td><td>=</td><td>69M 42.56</td></tr><tr><td>ODE Transformer (2022)</td><td>24</td><td></td><td>123M 43.28</td></tr><tr><td>Multi-Path Transformer (2022)</td><td>=</td><td></td><td>168M 42.44</td></tr><tr><td>Transformer</td><td>24 512 64</td><td>8-8</td><td>120M 42.33</td></tr><tr><td>PartialFormer (w/o Head Scaling) 24 512 648-8</td><td></td><td></td><td>68M 41.68</td></tr><tr><td>PartialFormer</td><td></td><td>24 512 64 24-18</td><td>119M 43.10</td></tr><tr><td>PartialFormer</td><td></td><td>24 512 64 24-24</td><td>127M 43.29</td></tr><tr><td>Transformer</td><td>6 512 64</td><td>8-8</td><td>63M 40.79</td></tr><tr><td>Transformer</td><td>24 360 45</td><td>8-8</td><td>64M 40.96</td></tr><tr><td>PartialFormer(w/o Head Scaling) 24 360 45</td><td></td><td>8-8</td><td>38M 40.44</td></tr><tr><td>PartialFormer</td><td></td><td>24 360 45 24-18</td><td>63M 42.16</td></tr><tr><td>PartialFormer</td><td></td><td>24 360 45 24-24</td><td>67M 42.39</td></tr></table>
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1.04 BLEU points (30.09 vs. 29.05) within a similar model capacity. The enhancement here can be attributed to the head scaling method, which allows PartialFormer to possess a larger hidden dimension, thereby bolstering its capacity for memory storage (Geva et al., 2021). These observations are further confirmed by the COMET-22 scores.
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Moreover, PartialFormer can even surpass all selected multi-branch Transformers while using fewer parameters. Notably, PartialFormer $N =$ $2 4 , d \ = \ 5 1 2 )$ outperforms the latest multi-path Transformer (Lin et al., 2022) by 0.41 BLEU points with 78M fewer parameters. This highlights the efficiency of building a multi-branch network based on inherent subspaces. Additionally, PartialFormer excels over previous lightweight approaches and outperforms state-of-the-art weight-sharing methods, e.g., ODE Transformer (Li et al., 2022), and other strong baselines, e.g., Mega (Ma et al., 2022). Notably, both ODE Transformer and Mega utilize relative position encoding (Shaw et al., 2018).
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Results of WMT’14 En-Fr Table 2 presents the results of PartialFormer on the WMT’14 En-Fr task. Similar to the findings in the En-De task, PartialFormer demonstrates a similar phenomenon. Notably, PartialFormer achieves comparable results to Transformer $( N = 2 4 , d = 5 1 2 )$ (42.39 vs. 42.33) while utilizing 53M fewer parameters (67M vs. 120M). This highlights the remarkable parameter efficiency of PartialFormer.
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Results of WMT’16 En-Ro Table 3 presents the results on the test set of the WMT’16 En-Ro task. Notably, PartialFormer achieves the highest BLEU points among all selected baselines. It is particularly remarkable that PartialFormer achieves similar results to ODE Transformer while utilizing 178M fewer parameters. This highlights the exceptional efficiency of PartialFormer.
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Table 3: Results on the WMT’16 En-Ro task.
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<table><tr><td>Model</td><td>N</td><td>ddk</td><td>H Param BLEU</td></tr><tr><td>Delight (Mehta et al., 2021)</td><td>- 640-</td><td>■</td><td>53M 34.70</td></tr><tr><td>Subformer (Reid et al.,2021)</td><td>■</td><td>- ■ -</td><td>48M 34.70</td></tr><tr><td>ODE Transformer (Li et al.,2022)</td><td>6 1024 64 16-16</td><td></td><td>226M 35.28</td></tr><tr><td>Transformer</td><td>24 512 64</td><td>8-8</td><td>111M 35.00</td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>24 512</td><td>64 8-8</td><td>59M 35.07</td></tr><tr><td>PartialFormer</td><td>24</td><td>320 4024-24</td><td>48M 35.30</td></tr></table>
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Table 4: Results on the WMT’17 benchmark. PartialFormer has the same depth and $d$ as the Transformer but consumes 1M fewer parameters on average.
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<table><tr><td rowspan="2">Model</td><td colspan="2">Fi←→En</td><td colspan="2">De←→En</td><td colspan="2">Lv← →En</td><td rowspan="2">Avg.</td></tr><tr><td>Fi→En En→FiDe→En En-→DeLv-→En En→Lv</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Transformer</td><td>26.07</td><td>22.14</td><td>35.04</td><td>28.59</td><td>17.59</td><td>16.23</td><td>24.27</td></tr><tr><td>PartialFormer</td><td>27.48</td><td>23.35</td><td>35.60</td><td>29.91</td><td>19.65</td><td>17.37</td><td>25.56</td></tr></table>
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Results of WMT’17 Benchmark Table 4 presents the WMT’17 benchmark results, showing that PartialFormer consistently outperforms Transformer by an average of 1.29 BLEU points in all six translation tasks. This finding is consistent with the observed performance in the En-De task.
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# 6 Analysis
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# 6.1 Ablation Studies
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Table 5 presents an ablation study of PartialFormer on the WMT’14 En-De task, demonstrating the critical role of each component. Omitting any element causes performance decline, underscoring the holistic design. The PG-FFN removal (#3 vs. #4) results in a large performance drop of 2.05 BLEU points, despite a mere 16 million parameters reduction. This evidence corroborates previous findings (Dong et al., 2021) on the subpar performance of pure attention networks sans FFN, highlighting the essential role of PG-FFN in PartialFormer.
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Besides, Table 5 shows the results of different PartialFormer configurations on the WMT’14 En-De task. The encoder-decoder PartialFormer achieves the highest performance, reaching 29.56 BLEU points, indicating the effectiveness of our approach in enhancing both the encoder and the decoder. Employing our concept to either the encoder or the decoder individually also improves performance, yet the encoder-decoder configuration persistently surpasses others, marking the greatest performance improvement.
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Table 5: Ablation studies on WMT’14 En-De task.
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<table><tr><td># Model</td><td>Param BLEU</td></tr><tr><td>1 Transformer (N = 24,d = 360)</td><td>62M 28.00</td></tr><tr><td>2 Pure Attention (N= 24,d = 360)</td><td>31M 25.70</td></tr><tr><td>3 PartialFormer</td><td>68M 29.56</td></tr><tr><td>4 w/o Partial-level Gated FFN</td><td>52M 27.51</td></tr><tr><td>5 w/o Residual-like Attention Calculation</td><td>66M 29.26</td></tr><tr><td>6 w/o Head Scaling</td><td>36M 27.88</td></tr><tr><td>7 PartialFormer (encoder only)</td><td>67M 29.15</td></tr><tr><td>8 PartialFormer (decoder only)</td><td>63M 28.80</td></tr></table>
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Table 7: Comparison of different width scaling strategy on the En-De task.
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<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>36M 27.88</td></tr><tr><td>+ Simple Head Scaling</td><td>68M 29.33</td></tr><tr><td>+ Complex Head Scaling</td><td>68M 29.56</td></tr></table>
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Table 6: Comparison of head scaling strategy on WMT’14 En-De task.
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# 6.2 Comparison of Head Scaling Strategy
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Table 6 presents the results of PartialFormer on the En-De task test set with varying head scaling techniques. Both simple and complex strategies effectively utilize additional parameters to enhance PartialFormer’s performance. Notably, the complex head scaling technique, allowing for more parameters allocated to additional heads, demonstrates superior performance.
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# 6.3 Discussions on Width Scaling Strategies
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Table 7 presents the results of analyzing three key ways to increase the width in PartialFormer: 1) $d _ { k }$ , 2) $H$ , and 3) $d$ , on the En-De task’s test set. Notably, the findings indicate that both increasing $H$ and adding $d _ { k }$ can effectively enhance the capacity of PartialFormer. Additionally, enlarging $d$ can be beneficial for performance improvements when it is small, e.g., less than 360. However, beyond a certain threshold, further increments of $d$ become redundant and do not lead to performance gains. This aligns with previous studies (Mehta et al., 2021; Baevski and Auli, 2019) highlighting redundant information in the embedding layer.
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# 6.4 Comparison of Gating Strategy
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Table 8 presents a comparison of various activation functions used in PG-FFN. The results indicate that the default choice, ReLU activation, yields the best performance. One explanation is that the ReLU activation provides hard masks for filtering the information of different heads, compared to other activation functions. Such hard masks can make different heads more diverse.
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<table><tr><td>Model</td><td>Seting</td><td>H d</td><td>dk Param</td><td>BLEU</td></tr><tr><td rowspan="7">PartialFormer</td><td>Basic</td><td>|30-16 360</td><td>45 68M</td><td>29.56</td></tr><tr><td>Varying Encoder H</td><td>|24-16 360 45 16-16 360 45</td><td>61M 51M</td><td>29.23 29.02</td></tr><tr><td>Varying Decoder H</td><td>[16-24 360 45 16-30360 45</td><td>56M 60M</td><td>28.85 29.20</td></tr><tr><td></td><td>|30-16 360 30</td><td>49M</td><td>28.70</td></tr><tr><td>Varying dh</td><td>30-16 360 60 30-16 360 90</td><td>86M 124M</td><td>29.68</td></tr><tr><td></td><td></td><td></td><td>30.00</td></tr><tr><td>Varying d</td><td>|30-16 180 45 30-16 270 45 30-16 450 45</td><td>35M 51M 84M</td><td>27.61 28.80 29.41</td></tr></table>
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# 6.5 Efficiency Analysis
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Table 9 exhibits the inference efficiency on the test set of En-De task. It is evident that PartialFormer incurs a reasonable increase in inference cost, which remains within acceptable limits.
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# 6.6 Analysis on Behaviours of FFN
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Metric. Following Zhang et al. (2022), we examine FFN behaviors across four aspects: activation neuron count (namely $n _ { \mathrm { a c t . } } )$ ), FFNs’ hidden dimension, activation-neuron ratio (activations divided by hidden dimension, namely $R _ { \mathrm { a c t . } }$ ), and FFN efficiency (activations divided by parameters, namely $\eta _ { \mathrm { { f f i n } } } )$ . Notably, for PartialFormer, the hidden dimension represents the concatenation of hidden dimensions from all smaller FFNs.
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Results. Figure 4(a-c) exhibits the results on the En-De test set. It is evident that PartialFormer has a lower activation ratio than the vanilla Transformer, as shown in Figure 4(b). This indicates that PGFFNs based on matrix factorization present lower utilization of the hidden dimension compared to the vanilla FFNs. However, our PG-FFN is parameter consumption friendly, enabling larger hidden layer dimensions with the same parameter budget (e.g., 5400 vs. 1440). Despite lower utilization of hidden dimension, it can still own more activated neurons, as depicted in Figure 4(a). Additionally, our PGFFN exhibits higher efficiency compared to vanilla FFNs, as shown in Figure 4(c). Multiple small FFNs, like “Swarm Intelligence” (Bonabeau et al., 1999), outperform large FFNs by leveraging the collective strength of weak individuals.
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Table 8: Comparison of activation functions in PGFFNs.
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<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>PG-FFNs</td><td>68M 29.56</td></tr><tr><td>PG-FFNs with Sigmoid activation</td><td>68M 29.21</td></tr><tr><td>PG-FFNs with Tanh activation</td><td>68M 29.03</td></tr></table>
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Table 9: Efficiency comparison between Transformer and PartialFormer in inference.
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<table><tr><td>Model</td><td colspan="4">Param Speed (Tok./s)Memory BLEU</td></tr><tr><td>Transformer</td><td>62M</td><td>4325</td><td>3.0G</td><td>28.00</td></tr><tr><td>PartialFormer (w/o head scaling)</td><td>66M</td><td>3634</td><td>3.2G</td><td>28.86</td></tr><tr><td>PartialFormer</td><td>68M</td><td>3023</td><td>3.3G</td><td>29.56</td></tr></table>
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# 6.7 Analysis on Head Diversity
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Metric. We select the same metric, namely $D _ { o u t p u t }$ , as that in Li et al. (2018) to measure the diversity among head features. In this metric, a larger value indicates a higher level of diversity.
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Results. From Figure 4(d), we can observe that PartialFormer exhibits more diverse head features compared to the vanilla Transformer, even though the vanilla Transformer already demonstrates diverse features. This aligns with previous study (Li et al., 2018), which demonstrates the positive impact of head feature diversity on the Transformer model’s performance. Thus, we conclude that the insertion of FFNs into attention mechanism may be a more optimal design.
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# 7 Related Work
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Lightweight Transformers Many methods have been proposed to improve the parameter efficiency of Transformer architecture. The first line is to directly cut down redundant computations and parameters via a more efficient design such as adopting more efficient transformation operations (Mehta et al., 2019, 2021), integrating different but complementary patterns (Wu et al., 2020) and neural architecture search (So et al., 2019). Another research direction for improving parameter efficiency in the Transformer is weight sharing. The popular cross-layer sharing method is utilized by the Universal Transformer (Dehghani et al., 2019). Reid et al. (2021) propose better performance by freeing the first and last encoder layers and widening the intermediate layers. Li et al. (2022) introduce an ordinary differential equation-inspired weightsharing method for more precise results. Different from these work, our study focus on the design of efficient lightweight FFN.
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Figure 4: Analysis on behaviours of FFNs and head diversity in Transformer and PartialFormer.
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Multi-Branch Transformer The multi-branch strategy is widely used in Transformer design. Weighted Transformer (Ahmed et al., 2017) employs a multi-branch FFN, while Multi-attentive Transformer (Fan et al., 2020), Multi-units Transformer (Yan et al., 2020), and Multi-Path Transformer (Lin et al., 2022) extend this concept to different components of the Transformer. Our work introduces a pure multi-branch architecture based on natural subspaces.
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Scaling Strategy in Transformer Deepening (Bapna et al., 2018; Wang et al., 2019) and widening (Vaswani et al., 2017; Wu et al., 2021) Transformer have been well-acknowledged as two strategies to improve the capacity of Transformer in literature. In this work, PartialFormer adopts two alternative strategies to improve capacity, adding a number of heads and head dimensions.
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# 8 Conclusion
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In this paper, we present PartialFormer, a new parameter-efficient Transformer architecture that offers an alternative approach to the design of the lightweight FFN. By employing multiple small FFNs and leveraging matrix factorization techniques, PartialFormer effectively reduces the number of parameters in the FFN. Moreover, we propose two innovative operations to further efficiently enhance the model capabilities. Experimental results across various machine translation tasks showcase the significant performance improvements achieved by PartialFormer, while maintaining comparable parameter consumption.
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# Limitations
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Despite the potential advantages of Partialformer in terms of parameter utilization and performance within a limited parameter budget, it is important to note that the existing conclusions regarding its effectiveness have not been thoroughly examined in the context of large-scale datasets and a higher number of parameters. Further research is needed to validate the claims and assess the scalability of Partialformer in more challenging scenarios.
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+
|
| 315 |
+
Peihao Wang, Wenqing Zheng, Tianlong Chen, and Zhangyang Wang. 2022. Anti-oversmoothing in deep vision transformers via the fourier domain analysis: From theory to practice. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022.
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| 316 |
+
|
| 317 |
+
Qiang Wang, Bei Li, Tong Xiao, Jingbo Zhu, Changliang Li, Derek F. Wong, and Lidia S. Chao. 2019. Learning deep transformer models for machine translation. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 1810–1822, Florence, Italy. Association for Computational Linguistics.
|
| 318 |
+
|
| 319 |
+
Lijun Wu, Juntao Li, Yue Wang, Qi Meng, Tao Qin, Wei Chen, Min Zhang, Tie-Yan Liu, et al. 2021. R-drop: Regularized dropout for neural networks. Advances in Neural Information Processing Systems, 34:10890– 10905.
|
| 320 |
+
|
| 321 |
+
Zhanghao Wu, Zhijian Liu, Ji Lin, Yujun Lin, and Song Han. 2020. Lite transformer with long-short range attention. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.
|
| 322 |
+
|
| 323 |
+
Jianhao Yan, Fandong Meng, and Jie Zhou. 2020. Multiunit transformers for neural machine translation. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1047–1059, Online. Association for Computational Linguistics.
|
| 324 |
+
|
| 325 |
+
Zhengyan Zhang, Yankai Lin, Zhiyuan Liu, Peng Li, Maosong Sun, and Jie Zhou. 2022. MoEfication: Transformer feed-forward layers are mixtures of experts. In Findings of the Association for Computational Linguistics: ACL 2022, pages 877–890, Dublin, Ireland. Association for Computational Linguistics.
|
| 326 |
+
|
| 327 |
+
# A Detailed Setups of Experiments
|
| 328 |
+
|
| 329 |
+
# A.1 Dataset
|
| 330 |
+
|
| 331 |
+
Table 10 displays the statistics of all the 9 translation task.
|
| 332 |
+
|
| 333 |
+
# A.2 Training Details
|
| 334 |
+
|
| 335 |
+
Table 11 and 12 exhibits the training details on all translation tasks.
|
| 336 |
+
|
| 337 |
+
# B Metric Definition
|
| 338 |
+
|
| 339 |
+
# B.1 Measurement of Head Diversity
|
| 340 |
+
|
| 341 |
+
Following Li et al. (2018), we measure the head diversity as follows:
|
| 342 |
+
|
| 343 |
+
$$
|
| 344 |
+
D _ { \mathrm { o u t p u t } } = \exp ( - \frac { 1 } { H ^ { 2 } } \sum _ { i = 1 } ^ { H } \sum _ { j = 1 } ^ { H } \frac { | O ^ { i } \cdot O ^ { j } | } { \| O ^ { i } \| \| O ^ { j } \| } )
|
| 345 |
+
$$
|
| 346 |
+
|
| 347 |
+
During evaluation, we calculate the metric on all samples and average the values to obtain the final result.
|
| 348 |
+
|
| 349 |
+
# C More Comparison with Previous Lightweight Transformer
|
| 350 |
+
|
| 351 |
+
Table 13 presents a comprehensive comparison of previous lightweight Transformer models on the En-De task’s test set, with a specific focus on operating within a smaller parameter budget. The results prominently showcase the outstanding performance of PartialFormer, even when faced with constraints on model capacity. This outcome further emphasizes the superior capabilities of PartialFormer in scenarios with limited resources.
|
| 352 |
+
|
| 353 |
+
# D PartialFormer with Different $A _ { G }$ for Small Dataset
|
| 354 |
+
|
| 355 |
+
Table 14 showcases the results of PartialFormer on the WMT’16 En-Ro task, a small-scale translation dataset, specifically when $A _ { G }$ is calculated using local attention (Shaw et al., 2018). Notably, these results reveal that by adopting such an approach, PartialFormer achieves an impressive BLEU score of 35.76. We hope this can shed lights on the area of model integration.
|
| 356 |
+
|
| 357 |
+
# E PartialFormer with GLU and Weight Sharing
|
| 358 |
+
|
| 359 |
+
In this section, we investigate the integration of PartialFormer with two prominent techniques to enhance parameter efficiency: 1) the weight sharing method (Lan et al., 2020), and 2) gated linear units (Dauphin et al., 2017). To ensure the utilization of the latest advancements, we employ a state-of-the-art weight sharing method called ODE Transformer (Li et al., 2022), known for its effectiveness in promoting parameter efficiency in Transformer architectures. Additionally, we incorporate Swi-GLU (Shazeer, 2020), a widely adopted GLUvariant that has served as a foundational component in numerous expressive Transformer architectures.
|
| 360 |
+
|
| 361 |
+
Table 10: The details of datasets of 9 translation tasks.
|
| 362 |
+
|
| 363 |
+
<table><tr><td rowspan="2">Dataset</td><td colspan="3">Sentence</td><td rowspan="2">BPE</td><td rowspan="2">Vocab</td></tr><tr><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>WMT'14 En-De</td><td>4.5M</td><td>2999 26815</td><td>3003</td><td>32K 32K</td><td>34040 37288</td></tr><tr><td>WMT'14 En-Fr WMT'16 En-Ro</td><td>36M 0.6M</td><td>1999</td><td>3003 1999</td><td>20K</td><td>19064</td></tr><tr><td>WMT'17 En-De</td><td>5.9M</td><td>7998</td><td>3004</td><td>32K</td><td>35488</td></tr><tr><td>WMT'17 De-En</td><td>5.9M</td><td>7998</td><td>3004</td><td>32K</td><td>35448</td></tr><tr><td>WMT'17 En-Fi</td><td>2.7M</td><td>4225</td><td>3002</td><td>32K</td><td>32584</td></tr><tr><td>WMT'17Fi-En</td><td>2.7M</td><td>4225</td><td>3002</td><td>32K</td><td>32584</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>WMT'17 En-Lv WMT'17Lv-En</td><td>4.5M 4.5M</td><td>2003 2003</td><td>2001 2001</td><td>20K 20K</td><td>32368 32368</td></tr></table>
|
| 364 |
+
|
| 365 |
+
Table 11: The training setups of WMT’14 En-De, WMT’16 En-Ro and WMT’14 En-Fr tasks.
|
| 366 |
+
|
| 367 |
+
<table><tr><td colspan="4">Hyper-parameter WMT'14 En-De WMT'16En-Ro WMT'14 En-Fr</td></tr><tr><td>GPUs</td><td>8</td><td>4</td><td>8</td></tr><tr><td>Batch Size</td><td>4096</td><td>4096</td><td>4096</td></tr><tr><td>Update Frequency</td><td>2</td><td>1</td><td>8</td></tr><tr><td>Optimer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Adamβ</td><td>(0.9,0.997)</td><td>(0.9, 0.997)</td><td>(0.9, 0.997)</td></tr><tr><td>LR</td><td>0.0020</td><td>0.0020</td><td>0.0020</td></tr><tr><td>LR scheduler</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td></tr><tr><td>InitialLR</td><td>1e-7</td><td>1e-7</td><td>le-7</td></tr><tr><td>Total updates</td><td>50K</td><td>25K</td><td>100K</td></tr><tr><td>Warmup updates</td><td>16000</td><td>8000</td><td>16000</td></tr><tr><td>Weight decay</td><td>0.0000</td><td>0.0000</td><td>0.0000</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>ReLU dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>
|
| 368 |
+
|
| 369 |
+
Table 12: The training setups of WMT’17 benchmark.
|
| 370 |
+
|
| 371 |
+
<table><tr><td colspan="3">Hyper-parameterI En-{De,Lv} {De,Lv}-En</td><td>En-Fi</td><td>Fi-En</td></tr><tr><td>GPUs</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>Batch Size</td><td>4096</td><td>4096</td><td>4096</td><td>4096</td></tr><tr><td>Update Frequency</td><td>2</td><td>1</td><td>1</td><td>4</td></tr><tr><td>Optimer</td><td>Adam</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Adamβ</td><td>(0.9, 0.997)</td><td>(0.9, 0.997)</td><td>(0.9,0.997) (0.9,0.997)</td><td></td></tr><tr><td>LR</td><td>0.0020</td><td>0.0020</td><td>0.0020</td><td>0.0020</td></tr><tr><td>LR scheduler</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td></tr><tr><td>Initial LR</td><td>1e-7</td><td>1e-7</td><td>1e-7</td><td>le-7</td></tr><tr><td>Total updates</td><td>50K/17K</td><td>50K/17K</td><td>40K</td><td>10K</td></tr><tr><td>Warmup updates</td><td>16000</td><td>16000</td><td>16000</td><td>16000</td></tr><tr><td>Weight decay</td><td>0.0000</td><td>0.0000</td><td>0.0000</td><td>0.0000</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>ReLU dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>
|
| 372 |
+
|
| 373 |
+
Table 15 displays the results of combining PartialFormer with weight sharing and gated linear units. Despite the integration of these two techniques, the performance gains are marginal. This could be attributed to the fact that PartialFormer already possesses high parameter efficiency, leaving little room for additional enhancements from other technologies. In other words, PartialFormer is inherently a high parameter efficiency architecture.
|
| 374 |
+
|
| 375 |
+
Table 13: Comparison with state-of-the-art models of smaller capacities on the En-De task.
|
| 376 |
+
|
| 377 |
+
<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>DELIGHT (Mehta et al., 2021) EdgeFormer (Ge et al., 2022) Lite Transformer (Wu et al.,2020) PartialFormer</td><td>23M 26.70 - 26.90 - 26.50 27M 27.50</td></tr><tr><td>Evolved Transformer (So et al., 2019) DELIGHT (Mehta et al., 2021) ODE Transformer (Li et al., 2022) PartialFormer</td><td>48M 27.70 37M 27.60 37M 28.24 36M 28.35</td></tr></table>
|
| 378 |
+
|
| 379 |
+
Table 14: Results of several PartialFormer variants on the En-De task.
|
| 380 |
+
|
| 381 |
+
<table><tr><td>AG</td><td>AL</td><td>Param</td><td>BLEU</td></tr><tr><td>RPR</td><td>MHSA</td><td>62M</td><td>35.76</td></tr></table>
|
| 382 |
+
|
| 383 |
+
Table 15: Results of PartialFormer variants on the EnDe task.
|
| 384 |
+
|
| 385 |
+
<table><tr><td>Model</td><td>Param</td><td>BLEU</td></tr><tr><td>PartialFormer</td><td>67M</td><td>29.56</td></tr><tr><td>PartialFormer + Weight Sharing</td><td>67M</td><td>29.71</td></tr><tr><td>GLU-based PartialFormer</td><td>67M</td><td>29.67</td></tr></table>
|
| 386 |
+
|
| 387 |
+
# F Analysis on Token Uniformity
|
| 388 |
+
|
| 389 |
+
Following (Dong et al., 2021; Wang et al., 2022), we measure the token uniformity among token representations. We use pearson correlation to compute it.
|
| 390 |
+
|
| 391 |
+
From Figure 5, we can observe that PartialFormer owns a lower token uniformity among token representations than the vanilla Transformer, revealing that PartialFormer can benefit from depth scaling efficiently (Dong et al., 2021; Wang et al., 2022).
|
| 392 |
+
|
| 393 |
+
# G Preliminary Experiments on Language Modeling
|
| 394 |
+
|
| 395 |
+
We also evaluate the effectiveness of PartialFormer on the language modeling task. We can see that
|
| 396 |
+
|
| 397 |
+
|
| 398 |
+
Figure 5: Comparison of token uniformity (lower is better) in Transformer and PartialFormer.
|
| 399 |
+
|
| 400 |
+
PartialFormer can also show better results compared to strong baseline, e.g., Adaptive Input Transformer (Baevski and Auli, 2019). We will present more comprehensive experiments in the future.
|
| 401 |
+
|
| 402 |
+
<table><tr><td>Model</td><td>Depth 0 (M) Test PPL</td></tr><tr><td>Adaptive Input</td><td>8 147M 21.11</td></tr><tr><td>PartialFormer</td><td>16 143M 19.87</td></tr></table>
|
| 403 |
+
|
| 404 |
+
Table 16: Results on the WikiText-103 dataset.
|
parse/dev/NHeAUKlTO8/NHeAUKlTO8_content_list.json
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[
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{
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"type": "text",
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| 4 |
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"text": "PartialFormer: Modeling Part Instead of Whole for Machine Translation ",
|
| 5 |
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"text_level": 1,
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| 6 |
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"type": "text",
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"text": "Anonymous EMNLP submission ",
|
| 17 |
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"bbox": [
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"type": "text",
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"text": "Abstract ",
|
| 28 |
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"text_level": 1,
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"bbox": [
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"page_idx": 0
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"type": "text",
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"text": "The parameter redundancy problem in Transformer models has been widely acknowledged in the literature. To address this weakness, we introduce PartialFormer, a parameter-efficient Transformer architecture for machine translation. Compared to previous parameter-efficient Transformer architecture, PartialFormer modifies the modeling strategy of the feed-forward network to allow it to spare tremendous parameters while maintaining large hidden dimension. Additionally, PartialFormer applies two efficient scaling strategies, namely depth scaling and width scaling, to improve performance within a given parameter budget. To efficiently benefit from these scaling strategies, PartialFormer is further enhanced by two costeffective modifications: 1) a head scaling strategy for efficient width scaling and 2) a residuallike attention calculation for better depth scaling. Extensive experiments on 9 translation tasks validate the effectiveness of our PartialFormer approach. ",
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| 40 |
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"bbox": [
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| 41 |
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| 42 |
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},
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{
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| 49 |
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"type": "image",
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"img_path": "images/87577d14d81bfcc2eba8bcf72cc758b909e4169ed43a7f8857f2625ad2502291.jpg",
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| 51 |
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"image_caption": [
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| 52 |
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"Figure 1: Illustration of our idea. "
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| 53 |
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],
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| 54 |
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"image_footnote": [],
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{
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"type": "text",
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| 65 |
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"text": "1 Introduction ",
|
| 66 |
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"text_level": 1,
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| 67 |
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"bbox": [
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| 69 |
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"type": "text",
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"text": "The Transformer model (Vaswani et al., 2017) has emerged as a cornerstone in the natural language processing (NLP) domain, overshadowing convolutional neural networks (Gehring et al., 2017) and recurrent neural networks (Sutskever et al., 2014) by virtue of its minimal inductive bias, superior scalability, and proficiency in modeling extended sequences. Nonetheless, its substantial computational and parametric requisites pose significant challenges to its deployment and training, warranting an ongoing trend in the research community toward eliminating redundant parameters and computations in the Transformer model (Dehghani et al., 2019; Lan et al., 2020; Reid et al., 2021; Li et al., 2022; Ahmed et al., 2017; Yan et al., 2020; Wu et al., 2020; Mehta et al., 2019, 2021). ",
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"text": "it is noteworthy that these approaches ignore the importance of feed-forward networks (FFN). Feedforward networks consume significant parametric and computational overhead due to the inherent large feature space and hidden dimension. To cut down FFNs’ overhead, previous studies (Mehta et al., 2021; Wu et al., 2020; Ge et al., 2022) just adopt smaller hidden dimension, e.g., equal to or even lower than the size of feature space. That leads to a question: Are current lightweight FFNs optimal? ",
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"type": "text",
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"text": "Despite their success in improving the parametric and computational efficiency of the Transformer, ",
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"type": "text",
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"text": "To address this concern, we turn to the insights provided by Geva et al. (2021), who depicted FFNs as a collection of key-value memories, where the number of memories is equal to the number of hidden dimensions in FFNs. This finding underscores the significance of hidden dimension in FFNs. Drawing inspiration from this finding and the successful application of large hidden sizes in FFNs as evidenced by Meta’s 4B model (Tran et al., 2021)1, we postulate that a truly efficient lightweight FFN should maintain, if not enlarge, the hidden dimension while reducing parameters. ",
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"type": "text",
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"text": "To this end, we propose PartialFormer, an innovative approach to Transformer architecture. The central design of PartialFormer is the Partial-Level Gated Feed-Forward Networks (PG-FFN). We designed the PG-FFN as a set of smaller FFNs in unison, each producing lower-dimensional hidden features, yet collectively matching or exceeding the hidden dimension of a conventional larger FFN. Moreover, we further equipped PartialFormer with two cost-effective operations: a head scaling strategy for efficient width scaling, and a residual-like attention calculation for stable optimization. These techniques empower PartialFormer to achieve deeper layer stacking or increased width within the same parameter budget. ",
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{
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"type": "text",
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| 132 |
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"text": "",
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| 133 |
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"bbox": [
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"type": "text",
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"text": "The strength of PartialFormer has been affirmed through rigorous empirical evaluations on $9 \\ \\mathrm { m a }$ chine translation tasks. Remarkably, even while maintaining similar parameter consumption, our PartialFormer consistently surpasses the vanilla Transformer, employing the same layer depth and embedding width, by an average of 1.29 BLEU points across all 6 WMT’17 machine translations. Furthermore, it achieved a BLEU score of 29.56 on the challenging WMT’14 En-De task with only 68 million parameters, showcasing its effectiveness and efficiency. Our work with PartialFormer thus marks an important step towards the goal of optimized Transformer architectures, marrying performance with efficiency in a manner that has potential for broad impact in NLP applications. ",
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"type": "text",
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"text": "2 Preliminary: Transformer ",
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"type": "text",
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"text": "In this section, we present some prior knowledge about the Transformer. Typically, Transformer block always consists of a multi-head self-attention and a feed-forward network. Let $X \\in \\mathbb { R } ^ { T \\times d }$ be a $T \\times d$ input matrix of $T$ tokens. Each multi-head self-attention component owns $H$ heads. For simplicity, we ignore the layer-normalization operation and residual connection. ",
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"type": "text",
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"text": "Multi-Head Self-Attention MHSA aims to model the global dependency among tokens. MHSA computes as follows: ",
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"type": "equation",
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"img_path": "images/7c441977fe83d11be8c5f52b13101ae5a5b225910c133ee20c093ad3cc23298c.jpg",
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"text": "$$\n\\begin{array} { r c l } { { { \\cal A } ^ { i } } } & { { = } } & { { \\mathrm { S o f t m a x } ( \\displaystyle \\frac { Q ^ { i } ( K ^ { i } ) ^ { \\top } } { \\sqrt { d _ { k } } } ) , } } \\\\ { { \\mathrm { h e a d } _ { i } } } & { { = } } & { { { \\cal A } ^ { i } V ^ { i } , } } \\\\ { { X } } & { { = } } & { { \\displaystyle \\sum _ { i = 1 } ^ { H } \\mathrm { h e a d } _ { i } W _ { i } ^ { O } , } } \\end{array}\n$$",
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"text_format": "latex",
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{
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"type": "text",
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"text": "where $Q ^ { i } , K ^ { i } , V ^ { i }$ denote the query, key and value of $i$ -th head, which are derived from input with three learnable matrics $W _ { i } ^ { Q } , W _ { i } ^ { K } , W _ { i } ^ { V } \\ \\in \\ \\mathbb { R } ^ { d \\times d _ { k } }$ as follows: $Q ^ { i } \\ : = \\ : X W _ { i } ^ { Q } , K ^ { i } \\ : = \\ : X W _ { i } ^ { K } , V ^ { i } \\ : =$ ",
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"type": "text",
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"text": "$X W _ { i } ^ { V }$ , respectively. $W _ { i } ^ { O } \\in \\mathbb { R } ^ { d _ { k } \\times d }$ is a learnable matrix. $A ^ { i }$ and headi denote the attention matrix and representation of $i$ -th head, respectively. ",
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"text": "Feed-Forward Network Feed-forward network is responsible for improving the expressiveness of the whole representation space by adopting an \"expansion-activation-reduction\" mapping strategy. It computes as follows: ",
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| 224 |
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"type": "equation",
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"img_path": "images/cf5e95409afb2a05ddc65b05d7b7309f126f4b470121703ae326a2c3e4c441a0.jpg",
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"text": "$$\nX = \\mathrm { R e L U } ( X W _ { 1 } + b _ { 1 } ) W _ { 2 } + b _ { 2 } ,\n$$",
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| 236 |
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"text_format": "latex",
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"type": "text",
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"text": "where $W _ { 1 } ~ \\in ~ \\mathbb { R } ^ { d \\times d _ { \\mathrm { f n } } } , W _ { 2 } ~ \\in ~ \\mathbb { R } ^ { d _ { \\mathrm { f n } } \\times d } , b _ { 1 } ~ \\in$ $\\mathbb { R } ^ { d _ { \\mathrm { f f n } } } , b _ { 2 } \\in \\mathbb { R } ^ { d }$ as learnable matrices and $d _ { \\mathrm { { f f n } } }$ denotes the hidden dimension in FFN that is usually set to $4 d$ . ",
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"type": "text",
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"text": "3 PartialFormer ",
|
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"text_level": 1,
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"type": "text",
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"text": "3.1 Overall Architecture ",
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| 271 |
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"text_level": 1,
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"type": "text",
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"text": "Figure 2 illustrates the overall architecture of PartialFormer, encompassing both an encoder and a decoder. Although the foundational structure adheres to the design of the vanilla Transformer (Vaswani et al., 2017), there are some notable modifications. ",
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"type": "text",
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"text": "Encoder. Different from vanilla Transformer, each encoder layer in PartialFormer consists of a unified sub-layer that integrates the PG-FFNs into the multi-head self-attention mechanism rather than separate two sub-layers. ",
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"text": "Decoder. Each decoder layer is composed of two types of sub-layers, both of which integrate the multi-head attention mechanism with PG-FFNs. The sub-layers differ based on the type of multihead attention mechanisms employed, specifically whether it’s a decoder self-attention or an encoderdecoder cross-attention mechanism. ",
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"type": "text",
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"text": "3.2 Information Flow in Unified Sub-Layer ",
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"type": "text",
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"text": "Taking the Encoder as an instance. Each unified sub-layer first computes the multiple attention scores via Eq. (5), then obtains the multiple head features $\\{ \\mathrm { h e a d } ^ { i } | 1 \\leq i \\leq H \\}$ via Eq. (2), which is the same as vanilla Transformer. Then, using multiple small FFNs, it processes these head features and ultimately combines the representations via a fusion function according to Eq. (7). That is to say, the PG-FFN is encapsulated into the multiheadattention mechanism. ",
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{
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"type": "image",
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"img_path": "images/f547cefb32e903685c4ce5bcb23dff57d1f371265a2596bde12acbe0dee13f9d.jpg",
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"image_caption": [
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| 340 |
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"Figure 2: (a) Architecture of Transformer. (b) Architecture of PartialFormer. (c) Details of Self-AFFN Block. All architecture are based on pre-normalization strategy. We omit the layer normalization operation, residual connection, softmax operation and scale coefficient for simplicity. "
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"text": "$$\n\\begin{array} { l l l } { { A ^ { i } } } & { { = } } & { { \\displaystyle \\mathrm { S o f t m a x } ( \\frac { Q ^ { i } ( K ^ { i } ) ^ { \\top } } { \\sqrt { d _ { k } } } + A _ { G } ^ { i } ) , } } \\\\ { { O ^ { i } } } & { { = } } & { { \\displaystyle \\mathrm { P G } \\mathrm { - } \\mathrm { F F N } ( \\mathrm { h e a d } ^ { i } ) , } } \\\\ { { X } } & { { = } } & { { \\displaystyle \\sum _ { i = 1 } ^ { H } O ^ { i } W _ { i } ^ { O } } } \\end{array}\n$$",
|
| 355 |
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"text_format": "latex",
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"type": "text",
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"text": "3.3 Partial-Level Gated FFN ",
|
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"type": "text",
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"text": "Intuition Previous studies (Wu et al., 2020; Mehta et al., 2021; Ge et al., 2022) have commonly reduced the parameters in feed-forward networks by decreasing the hidden dimension (e.g., 2048 to 256). In contrast, we tackle this issue through a matrix factorization approach. Our key idea involves utilizing a collection of small FFNs to model smaller input features, rather than relying on a single large FFN. ",
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"text": "Assume a FFN with mappings of $1 0 2 4 \\mathrm { - } { > } 4 0 9 6 \\mathrm { - }$ ${ > } 1 0 2 4$ , which consumes around 8.4 million parameters. By decomposing this into 8 smaller FFNs with mappings of $1 2 8 \\mathrm { - } > 5 1 2 \\mathrm { - } > 1 2 8$ , we can retain the same hidden dimension, such as $8 ^ { * } 5 1 2$ , while using only 1.05 million parameters. This approach significantly reduces parameters while maintaining the crucial desired hidden dimension, as emphasized in previous studies (Geva et al., 2021; Tran et al., 2021). ",
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"text": "Furthermore, we have observed that the Transformer architecture inherently consists of multiple smaller subspaces, namely “heads” within the multi-head attention (MHA) mechanism. These heads act as sub-components of the original inputs and retain substantial information from the original data. As a result, PG-FFNs should naturally be constructed based on the MHA mechanism. ",
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"type": "text",
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"text": "Calculation of PG-FFNs While group transformation operations could be used to instantiate our idea, they are not optimal on GPUs due to their low I/O efficiency (Ma et al., 2018), causing significant inference latency. To address this, we propose sharing parameters across each FFN within different heads, thereby eliminating the need for group transformation operations. ",
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"text": "However, directly sharing weights may result in homogeneous representations across different heads, which may potentially hinder the performance (Li et al., 2018). To mitigate this, we further introduce a head-specific gated mechanism. The core idea is to use a set of diverse masks to filter the information of different heads so that the head representation will be more diverse. Formally, given a set of smaller features $\\{ \\mathrm { h e a d } ^ { i } | 1 \\leq i \\leq H \\}$ and diverse masks $\\{ G ^ { i } | 1 \\leq i \\leq H \\}$ , the Eq. (6) can rewritten as: ",
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"type": "equation",
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"img_path": "images/3e9fcb327568b67be70a66f5b6b558aefe5f7f80109bbfe14475d1b82301b3ce.jpg",
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"text": "$$\nO ^ { i } = G ^ { i } \\odot \\mathrm { F F N } ( \\mathrm { h e a d } ^ { i } ) ,\n$$",
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"text_format": "latex",
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"type": "text",
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"text": "where $\\mathrm { F F N } ( \\cdot )$ is the same as Eq. (4). ",
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"type": "text",
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"text": "Generating $\\{ G ^ { i } | 1 \\le i \\le H \\}$ In our preliminary experiments, we observed significant diversity in the features generated by different parameters from sub-layer inputs, e.g., $\\{ V ^ { 1 } , \\ldots , V ^ { H } \\}$ . Motivated by this finding, we generate diverse masks in the following manner: ",
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"type": "equation",
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"img_path": "images/15f3089edac678e7a70aca209000c82f48faffe36ed7df3c40b5bd0095a3bec7.jpg",
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"text": "$$\nG ^ { i } = \\sigma ( X W _ { i } ^ { G } ) ,\n$$",
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"bbox": [
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"type": "text",
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"text": "where $W _ { i } ^ { G }$ is a learnable matrix and $\\sigma$ denotes the activation function, e.g., ReLU, Sigmoid and Tanh. We compare them in Table 8. ",
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"type": "text",
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"text": "3.4 Efficient Scaling Strategy ",
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"text_level": 1,
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"text": "Though PG-FFN offers the advantage of reducing lots of parameters when applied directly to the transformer, it also leads to performance degradation. Thus, a crucial aspect of this study is to determine how to effectively utilize the spared parameters. In this work, we adopt a hybrid scaling strategy, combining both width scaling and depth scaling, which has been validated in computer vision, e.g., EfficientNet (Tan and Le, 2019). ",
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"bbox": [
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"type": "text",
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| 515 |
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"text": "",
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"type": "text",
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"text": "3.4.1 Enabling Efficient Depth Scaling for PartialFormer ",
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| 527 |
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"text_level": 1,
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"type": "text",
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"text": "Wang et al. (2019); Dong et al. (2021); Wang et al. (2022) have shown that the original location of FFNs plays an essential role in optimizing transformers, e.g., alleviating Token Uniformity. Thus, we need to consider the impact brought by the change of FFNs. While the densely residual connection is an efficient way to alleviate it, they are typically either based on feature level (e.g., DLCL (Wang et al., 2019)) or coupled with the network structure (e.g., Realformer (He et al., 2021)). ",
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"text": "To this end, we design a new variant of the residual connection integrated into the attention calculation, while also decoupling from the network architecture. Specifically, the calculation of attention maps consists of two parts: 1) $A _ { G }$ , the global part, and 2) $A _ { L }$ , the local part. The calculation of $A _ { L }$ remains the same as in the vanilla Transformer, while $A _ { G }$ is computed once by using the original embedding as input through Eq. (1). Inspired by He et al. (2021), to efficiently fuse these components, we add them together and apply a Softmax function, as shown in Eq. (5). ",
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| 550 |
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"type": "text",
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"text": "In addition to the benefit of efficient depth scaling (See Appendix F), this approach provides remarkable flexibility in combining different attention mechanisms, specifically tailored to address specific conditions. For instance, it allows for the utilization of local attention to calculate $A _ { G }$ when dealing with small datasets (see Appendix D). ",
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"type": "text",
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"text": "3.4.2 Head Scaling: An Efficient Width Scaling for PartialFormer ",
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"text_level": 1,
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"type": "text",
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"text": "Existing approach to width scaling, which is based on the embedding size, necessitates the simultaneous scaling of both the encoder and decoder for machine translation tasks. This is primarily because researchers commonly employ shared encoder and decoder embedding. However, taking cues from the achievements of depth scaling, it may be more advantageous to adopt a distinct method for scaling width, similar to the approach used for scaling depth. Here we show how PartialFormer has inherent superiority to achieve so. ",
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"text": "",
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"text": "The width of a Transformer model typically refers to the widest part of the Transformer. In this context, both the vanilla Transformer and previous lightweight Transformer models have widths that are related to the embedding dimension, such as $4 d$ or $d$ . Therefore, by increasing the embedding size, we can effectively enlarge their width. However, the width definition in PartialFormer is different and can be expressed as $w = H \\times d _ { \\mathrm { f f n } }$ , where $w$ denotes the width of model and $d _ { \\mathrm { { f f n } } }$ is associated with the head dimension $d _ { k }$ . Consequently, we can expand the width by either increasing the number of heads or enlarging $d _ { k }$ . A comparison between these approaches is presented in Table 7. Notably, if the head dimension and number of heads are independent of the embedding dimension, PartialFormer allows for easy scaling of width in different ways within the encoder and decoder components. ",
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"type": "image",
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"img_path": "images/dcc02863d4f889023bf37de14a7d7d957b53cd0fca0d0347221602cad1538aa8.jpg",
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"image_caption": [
|
| 618 |
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"Figure 3: Comparison of ways to generate subspaces in Transformer and PartialFormer. "
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],
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| 620 |
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"image_footnote": [],
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"text": "To this end, we propose a new scaling mechanism, namely head scaling, that scales the width of PartialFormer by directly adding more heads and increasing head dimension, as illustrated in Figure 3. Given the head dimension $d _ { k }$ , the embedding dimension $d$ , and the number of heads $H$ , we consider two strategies to generate $H$ attention heads: ",
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"text": "(a) Simple strategy: We employ three learnable matrices, each with a shape of $d \\times ( d _ { k } \\times H )$ , to directly obtain the expected number of $Q$ , $K$ , and $V$ . ",
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"type": "text",
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"text": "(b) Complex strategy: we employ a two-step process. First, we generate an intermediate quantity of $Q$ and $K$ , and then use a powerful MLP network to expand the attention maps to the desired number. This innovative design draws inspiration from the inherent redundancy found within the attention map (Michel et al., 2019; Clark et al., 2019; Voita et al., 2019), allowing for more heads in PartialFormer under the same parameter budget. We show the comparisons in Table 6. ",
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"img_path": "images/5bd9198300e5fd500b403dbe34b7b2366bd59237b2fedae9ea0ef1a734622101.jpg",
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"type": "image",
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"img_path": "images/fa37f7a5a3a2beb8d9f67b378d00f636a4c0fffb083e25cf799ff2241f27f30b.jpg",
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{
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"type": "table",
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"img_path": "images/9aa275f7ace2c79b5c2c5a7dc70d49149ab74115c6153312184c088c3919ccca.jpg",
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"table_caption": [],
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| 692 |
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"table_footnote": [
|
| 693 |
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"Table 1: Results on the WMT’14 En-De task. MACs denote the multiplication-addition operations. We compute them via 20 source and target tokens following Mehta et al. (2021). "
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| 694 |
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],
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"table_body": "<table><tr><td>Type</td><td>Model</td><td>N-M</td><td>ddk</td><td></td><td></td><td></td><td>H MACs Param</td><td>BLEU</td><td>COMET-22</td></tr><tr><td rowspan=\"4\">Multi-Branch Architecture</td><td>Weighted Transformer (Ahmed et al.,2017)</td><td>6-6</td><td>1024</td><td></td><td></td><td></td><td>211M</td><td>28.90</td><td></td></tr><tr><td>Multi-Unit Transformer (Yan et al.,2020)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>130M</td><td>29.30</td><td></td></tr><tr><td>MAT (Fan et al., 2020)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>206M</td><td>29.90</td><td></td></tr><tr><td>Multi-Path Transformer (Lin et al., 2022)</td><td>6-6</td><td></td><td></td><td></td><td></td><td>193M</td><td>29.68</td><td></td></tr><tr><td>Lightweight Architecture</td><td>Evolved Transformer (So et al., 2019) Delight (Mehta et al., 2021)</td><td></td><td></td><td></td><td></td><td></td><td>64M</td><td>28.20</td><td></td></tr><tr><td rowspan=\"5\">Weight Sharing</td><td></td><td></td><td>640</td><td></td><td></td><td></td><td>54M</td><td>28.00</td><td></td></tr><tr><td>Universal Transformer (Dehghani et al.,2019)</td><td></td><td>1024</td><td></td><td>=</td><td></td><td>65M</td><td>28.90</td><td></td></tr><tr><td>SubFormer (Reid et al., 2021)</td><td></td><td>-</td><td></td><td>=</td><td></td><td>63M</td><td>28.50</td><td></td></tr><tr><td>SubFormer-big (Reid et al., 2021)</td><td></td><td></td><td></td><td></td><td></td><td>197M</td><td>29.30</td><td></td></tr><tr><td>ODE Transformer (RK4) (Li et al., 2022) ODE Transformer (RK4) (Li et al., 2022)</td><td>6-6 24-6</td><td>512 512</td><td></td><td>=</td><td></td><td>62M 118M</td><td>29.03 29.80</td><td></td></tr><tr><td rowspan=\"3\">Other Comparisons</td><td></td><td></td><td></td><td>64</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RealFormer (He et al., 2021) DMAN (Fan et al., 2021)</td><td>18-18</td><td>512 512</td><td></td><td>8 8</td><td></td><td>151M</td><td>29.35</td><td></td></tr><tr><td>Mega-Softmax (Ma et al.,2022)</td><td>6-6 6-6</td><td>512</td><td></td><td></td><td></td><td>63M 67M</td><td>29.10 29.01</td><td></td></tr><tr><td rowspan=\"7\">Our System</td><td>Transformer</td><td></td><td></td><td>64</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>24-6</td><td>512</td><td>8-8</td><td></td><td>11.1B</td><td>118M</td><td>29.05</td><td>83.60</td></tr><tr><td>PartialFormer</td><td>24-6</td><td>512</td><td>64 24-16</td><td>8-8</td><td>8.8B</td><td>66M</td><td>28.86 30.09</td><td>83.35 84.17</td></tr><tr><td></td><td>24-6</td><td>512</td><td>64</td><td></td><td>12.2B</td><td>115M</td><td></td><td></td></tr><tr><td>Transformer</td><td>6-6</td><td>512</td><td>64 45</td><td>8-8</td><td>9.9B</td><td>62M</td><td>27.43</td><td>82.19</td></tr><tr><td>Transformer PartialFormer (w/o Head Scaling)</td><td>24-6 24-6</td><td>360 360</td><td>45</td><td>8-8 8-8</td><td>6.3B 5.2B</td><td>62M 36M</td><td>28.00 27.88</td><td>82.72 82.49</td></tr><tr><td></td><td></td><td>360</td><td>45</td><td>24-16</td><td>6.8B</td><td>61M</td><td>29.23</td><td></td></tr><tr><td></td><td>PartialFormer PartialFormer</td><td>24-6 24-6</td><td>360</td><td>45 30-16</td><td></td><td>6.9B</td><td>68M</td><td>29.56</td><td>83.74 83.94</td></tr></table>",
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"type": "text",
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"text": "4 Experimental Setups ",
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"text": "In our evaluation, we assess the performance of PartialFormer across 9 machine translation tasks2. More details are given in Appendix A ",
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"text": "Dataset. We evaluate our approach on three widely-used datasets: WMT’14 English-German (En-De), WMT’14 English-French (En-Fr), and WMT’16 English-Romanian (En-Ro). Besides, to further validate the effectiveness of PartialFormer, we also evaluate PartialFormer on six translation tasks from WMT’17 benchmark. We preprocess the raw data following the standard strategy. ",
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"type": "text",
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"text": "Architectures and Selected Baselines. We use a 24-6 encoder-decoder PartialFormer architecture for its strong performance, on all 9 machine translation tasks. Detailed configurations are provided in the results tables. We compare our approach with various baselines, including vanilla Transformer models, multi-branch architecture, lightweight architecture, weight-sharing methods, and other strong baselines. ",
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"text": "Training & Evaluation. We train all the models on GeForce RTX 3090 cards via Fairseq (Ott et al., 2019) toolkit. For evaluation, we utilized multi-BLEU (Papineni et al., 2002) and COMET22 (Rei et al., 2022) scores. Beam sizes were 4, 4, and 5 for En-De, En-Fr, and En-Ro tasks respectively. Length_penalty of 0.6, 0.8, and 1.3 were applied to En-De, En-Fr, and En-Ro tasks respectively. For the WMT’17 benchmark, beam size and Length_penalty were set to 4 and 1, respectively. We used an ensemble of the last ten checkpoints. ",
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"type": "text",
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"text": "5 Experiments ",
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"text": "Results of WMT’14 En-De Table 1 presents the results for the WMT’14 En-De task. Note that we also provide a “strong” baseline which also benefits from deep model stacking. Even though the performance of PartialFormer (w/o Head Scaling) is slightly inferior to that of the Transformer model (27.88 vs. 28.00 and 28.86 vs. 29.05), it outshines the latter in terms of parameter efficiency, consuming significantly fewer parameters (36M vs. 62M, 66M vs. 118M). We attribute this phenomenon to our PG-FFN, which leverages a group of compact FFNs. This approach enables PG-FFN to maintain high hidden dimension, while drastically reducing parameter consumption. ",
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"type": "text",
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"text": "Upon utilizing our head scaling technique to amplify the capacity, our Partialformer delivers a BLEU score 29.56 and 30.09 on two configurations, respectively. This surpasses the standard Transformer by 1.56 BLEU points (29.56 vs. 28.00) and ",
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"type": "table",
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"img_path": "images/ef516f59f7c37f70affe22e5042b2981e8a0fb1ac32cdfc198f823be48f5f15f.jpg",
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"table_caption": [
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| 809 |
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"Table 2: Results on the WMT’14 En-Fr task. "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td>N</td><td>ddk</td><td>HParam BLEU</td></tr><tr><td>Weighted Transformer (2017)</td><td>6</td><td></td><td>- 211M 41.40</td></tr><tr><td>Evolved Transformer (2019)</td><td>=</td><td></td><td>64M 40.60</td></tr><tr><td>Delight (2021)</td><td>-640</td><td></td><td>54M 40.50</td></tr><tr><td>ODE Transformer (2022)</td><td>6</td><td>=</td><td>69M 42.56</td></tr><tr><td>ODE Transformer (2022)</td><td>24</td><td></td><td>123M 43.28</td></tr><tr><td>Multi-Path Transformer (2022)</td><td>=</td><td></td><td>168M 42.44</td></tr><tr><td>Transformer</td><td>24 512 64</td><td>8-8</td><td>120M 42.33</td></tr><tr><td>PartialFormer (w/o Head Scaling) 24 512 648-8</td><td></td><td></td><td>68M 41.68</td></tr><tr><td>PartialFormer</td><td></td><td>24 512 64 24-18</td><td>119M 43.10</td></tr><tr><td>PartialFormer</td><td></td><td>24 512 64 24-24</td><td>127M 43.29</td></tr><tr><td>Transformer</td><td>6 512 64</td><td>8-8</td><td>63M 40.79</td></tr><tr><td>Transformer</td><td>24 360 45</td><td>8-8</td><td>64M 40.96</td></tr><tr><td>PartialFormer(w/o Head Scaling) 24 360 45</td><td></td><td>8-8</td><td>38M 40.44</td></tr><tr><td>PartialFormer</td><td></td><td>24 360 45 24-18</td><td>63M 42.16</td></tr><tr><td>PartialFormer</td><td></td><td>24 360 45 24-24</td><td>67M 42.39</td></tr></table>",
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{
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"type": "text",
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"text": "1.04 BLEU points (30.09 vs. 29.05) within a similar model capacity. The enhancement here can be attributed to the head scaling method, which allows PartialFormer to possess a larger hidden dimension, thereby bolstering its capacity for memory storage (Geva et al., 2021). These observations are further confirmed by the COMET-22 scores. ",
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"type": "text",
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"text": "Moreover, PartialFormer can even surpass all selected multi-branch Transformers while using fewer parameters. Notably, PartialFormer $N =$ $2 4 , d \\ = \\ 5 1 2 )$ outperforms the latest multi-path Transformer (Lin et al., 2022) by 0.41 BLEU points with 78M fewer parameters. This highlights the efficiency of building a multi-branch network based on inherent subspaces. Additionally, PartialFormer excels over previous lightweight approaches and outperforms state-of-the-art weight-sharing methods, e.g., ODE Transformer (Li et al., 2022), and other strong baselines, e.g., Mega (Ma et al., 2022). Notably, both ODE Transformer and Mega utilize relative position encoding (Shaw et al., 2018). ",
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"type": "text",
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"text": "Results of WMT’14 En-Fr Table 2 presents the results of PartialFormer on the WMT’14 En-Fr task. Similar to the findings in the En-De task, PartialFormer demonstrates a similar phenomenon. Notably, PartialFormer achieves comparable results to Transformer $( N = 2 4 , d = 5 1 2 )$ (42.39 vs. 42.33) while utilizing 53M fewer parameters (67M vs. 120M). This highlights the remarkable parameter efficiency of PartialFormer. ",
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"text": "Results of WMT’16 En-Ro Table 3 presents the results on the test set of the WMT’16 En-Ro task. Notably, PartialFormer achieves the highest BLEU points among all selected baselines. It is particularly remarkable that PartialFormer achieves similar results to ODE Transformer while utilizing 178M fewer parameters. This highlights the exceptional efficiency of PartialFormer. ",
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"type": "table",
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"img_path": "images/5c2a5805346b018b4ec5d09819b07940be90ae7ad6d9c59587516b038884b09a.jpg",
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"table_caption": [
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"Table 3: Results on the WMT’16 En-Ro task. "
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],
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"table_footnote": [],
|
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"table_body": "<table><tr><td>Model</td><td>N</td><td>ddk</td><td>H Param BLEU</td></tr><tr><td>Delight (Mehta et al., 2021)</td><td>- 640-</td><td>■</td><td>53M 34.70</td></tr><tr><td>Subformer (Reid et al.,2021)</td><td>■</td><td>- ■ -</td><td>48M 34.70</td></tr><tr><td>ODE Transformer (Li et al.,2022)</td><td>6 1024 64 16-16</td><td></td><td>226M 35.28</td></tr><tr><td>Transformer</td><td>24 512 64</td><td>8-8</td><td>111M 35.00</td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>24 512</td><td>64 8-8</td><td>59M 35.07</td></tr><tr><td>PartialFormer</td><td>24</td><td>320 4024-24</td><td>48M 35.30</td></tr></table>",
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"type": "table",
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"img_path": "images/c52c5b24d0a72f2aa7681da7be04aec21454d96e68ecced2e642021a4b12f7aa.jpg",
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"table_caption": [
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| 885 |
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"Table 4: Results on the WMT’17 benchmark. PartialFormer has the same depth and $d$ as the Transformer but consumes 1M fewer parameters on average. "
|
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],
|
| 887 |
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"table_footnote": [],
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| 888 |
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"table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Fi←→En</td><td colspan=\"2\">De←→En</td><td colspan=\"2\">Lv← →En</td><td rowspan=\"2\">Avg.</td></tr><tr><td>Fi→En En→FiDe→En En-→DeLv-→En En→Lv</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Transformer</td><td>26.07</td><td>22.14</td><td>35.04</td><td>28.59</td><td>17.59</td><td>16.23</td><td>24.27</td></tr><tr><td>PartialFormer</td><td>27.48</td><td>23.35</td><td>35.60</td><td>29.91</td><td>19.65</td><td>17.37</td><td>25.56</td></tr></table>",
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"text": "",
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"type": "text",
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"text": "Results of WMT’17 Benchmark Table 4 presents the WMT’17 benchmark results, showing that PartialFormer consistently outperforms Transformer by an average of 1.29 BLEU points in all six translation tasks. This finding is consistent with the observed performance in the En-De task. ",
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"text": "6 Analysis ",
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"type": "text",
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"text": "6.1 Ablation Studies ",
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"type": "text",
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| 945 |
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"text": "Table 5 presents an ablation study of PartialFormer on the WMT’14 En-De task, demonstrating the critical role of each component. Omitting any element causes performance decline, underscoring the holistic design. The PG-FFN removal (#3 vs. #4) results in a large performance drop of 2.05 BLEU points, despite a mere 16 million parameters reduction. This evidence corroborates previous findings (Dong et al., 2021) on the subpar performance of pure attention networks sans FFN, highlighting the essential role of PG-FFN in PartialFormer. ",
|
| 946 |
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"bbox": [
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| 947 |
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| 948 |
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| 950 |
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| 951 |
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|
| 952 |
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"page_idx": 5
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| 953 |
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|
| 954 |
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{
|
| 955 |
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"type": "text",
|
| 956 |
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"text": "Besides, Table 5 shows the results of different PartialFormer configurations on the WMT’14 En-De task. The encoder-decoder PartialFormer achieves the highest performance, reaching 29.56 BLEU points, indicating the effectiveness of our approach in enhancing both the encoder and the decoder. Employing our concept to either the encoder or the decoder individually also improves performance, yet the encoder-decoder configuration persistently surpasses others, marking the greatest performance improvement. ",
|
| 957 |
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"bbox": [
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| 959 |
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| 961 |
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| 963 |
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"page_idx": 5
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| 964 |
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| 965 |
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{
|
| 966 |
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"type": "table",
|
| 967 |
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"img_path": "images/93cfd48be6417dc2e3c597283972c9a8475ef9781923e69061999bcfa23a2631.jpg",
|
| 968 |
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"table_caption": [
|
| 969 |
+
"Table 5: Ablation studies on WMT’14 En-De task. "
|
| 970 |
+
],
|
| 971 |
+
"table_footnote": [],
|
| 972 |
+
"table_body": "<table><tr><td># Model</td><td>Param BLEU</td></tr><tr><td>1 Transformer (N = 24,d = 360)</td><td>62M 28.00</td></tr><tr><td>2 Pure Attention (N= 24,d = 360)</td><td>31M 25.70</td></tr><tr><td>3 PartialFormer</td><td>68M 29.56</td></tr><tr><td>4 w/o Partial-level Gated FFN</td><td>52M 27.51</td></tr><tr><td>5 w/o Residual-like Attention Calculation</td><td>66M 29.26</td></tr><tr><td>6 w/o Head Scaling</td><td>36M 27.88</td></tr><tr><td>7 PartialFormer (encoder only)</td><td>67M 29.15</td></tr><tr><td>8 PartialFormer (decoder only)</td><td>63M 28.80</td></tr></table>",
|
| 973 |
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"bbox": [
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| 975 |
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487,
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| 977 |
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216
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| 978 |
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],
|
| 979 |
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"page_idx": 6
|
| 980 |
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},
|
| 981 |
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{
|
| 982 |
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"type": "table",
|
| 983 |
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"img_path": "images/712ef1cb5dac6390e41e477ea9889f0995225bd2494b259ec69c56436e46a23e.jpg",
|
| 984 |
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"table_caption": [
|
| 985 |
+
"Table 7: Comparison of different width scaling strategy on the En-De task. "
|
| 986 |
+
],
|
| 987 |
+
"table_footnote": [
|
| 988 |
+
"Table 6: Comparison of head scaling strategy on WMT’14 En-De task. "
|
| 989 |
+
],
|
| 990 |
+
"table_body": "<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>PartialFormer (w/o Head Scaling)</td><td>36M 27.88</td></tr><tr><td>+ Simple Head Scaling</td><td>68M 29.33</td></tr><tr><td>+ Complex Head Scaling</td><td>68M 29.56</td></tr></table>",
|
| 991 |
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"bbox": [
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| 992 |
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| 993 |
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| 994 |
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| 995 |
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326
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| 996 |
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],
|
| 997 |
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"page_idx": 6
|
| 998 |
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},
|
| 999 |
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{
|
| 1000 |
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"type": "text",
|
| 1001 |
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"text": "6.2 Comparison of Head Scaling Strategy ",
|
| 1002 |
+
"text_level": 1,
|
| 1003 |
+
"bbox": [
|
| 1004 |
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117,
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| 1005 |
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391,
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| 1006 |
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455,
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| 1007 |
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407
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| 1008 |
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],
|
| 1009 |
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"page_idx": 6
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| 1010 |
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},
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| 1011 |
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{
|
| 1012 |
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"type": "text",
|
| 1013 |
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"text": "Table 6 presents the results of PartialFormer on the En-De task test set with varying head scaling techniques. Both simple and complex strategies effectively utilize additional parameters to enhance PartialFormer’s performance. Notably, the complex head scaling technique, allowing for more parameters allocated to additional heads, demonstrates superior performance. ",
|
| 1014 |
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"bbox": [
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| 1016 |
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| 1018 |
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|
| 1020 |
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"page_idx": 6
|
| 1021 |
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},
|
| 1022 |
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{
|
| 1023 |
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"type": "text",
|
| 1024 |
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"text": "6.3 Discussions on Width Scaling Strategies ",
|
| 1025 |
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"text_level": 1,
|
| 1026 |
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"bbox": [
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| 1027 |
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| 1029 |
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| 1030 |
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571
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| 1032 |
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| 1033 |
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| 1035 |
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"type": "text",
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| 1036 |
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"text": "Table 7 presents the results of analyzing three key ways to increase the width in PartialFormer: 1) $d _ { k }$ , 2) $H$ , and 3) $d$ , on the En-De task’s test set. Notably, the findings indicate that both increasing $H$ and adding $d _ { k }$ can effectively enhance the capacity of PartialFormer. Additionally, enlarging $d$ can be beneficial for performance improvements when it is small, e.g., less than 360. However, beyond a certain threshold, further increments of $d$ become redundant and do not lead to performance gains. This aligns with previous studies (Mehta et al., 2021; Baevski and Auli, 2019) highlighting redundant information in the embedding layer. ",
|
| 1037 |
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"bbox": [
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| 1044 |
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},
|
| 1045 |
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{
|
| 1046 |
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"type": "text",
|
| 1047 |
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"text": "6.4 Comparison of Gating Strategy ",
|
| 1048 |
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"text_level": 1,
|
| 1049 |
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"bbox": [
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| 1056 |
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},
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| 1057 |
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{
|
| 1058 |
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"type": "text",
|
| 1059 |
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"text": "Table 8 presents a comparison of various activation functions used in PG-FFN. The results indicate that the default choice, ReLU activation, yields the best performance. One explanation is that the ReLU activation provides hard masks for filtering the information of different heads, compared to other activation functions. Such hard masks can make different heads more diverse. ",
|
| 1060 |
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"bbox": [
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"page_idx": 6
|
| 1067 |
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},
|
| 1068 |
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{
|
| 1069 |
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"type": "table",
|
| 1070 |
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"img_path": "images/85fa36cc0ecfc2175111c8fe8e4148f293588ade13da0ca4226fc522fef358b9.jpg",
|
| 1071 |
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"table_caption": [],
|
| 1072 |
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"table_footnote": [],
|
| 1073 |
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"table_body": "<table><tr><td>Model</td><td>Seting</td><td>H d</td><td>dk Param</td><td>BLEU</td></tr><tr><td rowspan=\"7\">PartialFormer</td><td>Basic</td><td>|30-16 360</td><td>45 68M</td><td>29.56</td></tr><tr><td>Varying Encoder H</td><td>|24-16 360 45 16-16 360 45</td><td>61M 51M</td><td>29.23 29.02</td></tr><tr><td>Varying Decoder H</td><td>[16-24 360 45 16-30360 45</td><td>56M 60M</td><td>28.85 29.20</td></tr><tr><td></td><td>|30-16 360 30</td><td>49M</td><td>28.70</td></tr><tr><td>Varying dh</td><td>30-16 360 60 30-16 360 90</td><td>86M 124M</td><td>29.68</td></tr><tr><td></td><td></td><td></td><td>30.00</td></tr><tr><td>Varying d</td><td>|30-16 180 45 30-16 270 45 30-16 450 45</td><td>35M 51M 84M</td><td>27.61 28.80 29.41</td></tr></table>",
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| 1074 |
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| 1076 |
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|
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| 1081 |
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| 1082 |
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|
| 1083 |
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"type": "text",
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| 1084 |
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"text": "",
|
| 1085 |
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"bbox": [
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| 1086 |
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| 1093 |
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{
|
| 1094 |
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"type": "text",
|
| 1095 |
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"text": "6.5 Efficiency Analysis ",
|
| 1096 |
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"text_level": 1,
|
| 1097 |
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"bbox": [
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| 1099 |
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| 1100 |
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| 1103 |
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| 1104 |
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|
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{
|
| 1106 |
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"type": "text",
|
| 1107 |
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"text": "Table 9 exhibits the inference efficiency on the test set of En-De task. It is evident that PartialFormer incurs a reasonable increase in inference cost, which remains within acceptable limits. ",
|
| 1108 |
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"bbox": [
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| 1112 |
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| 1113 |
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|
| 1114 |
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|
| 1115 |
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},
|
| 1116 |
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{
|
| 1117 |
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"type": "text",
|
| 1118 |
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"text": "6.6 Analysis on Behaviours of FFN ",
|
| 1119 |
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"text_level": 1,
|
| 1120 |
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"bbox": [
|
| 1121 |
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| 1122 |
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| 1123 |
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| 1124 |
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| 1126 |
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| 1127 |
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|
| 1128 |
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{
|
| 1129 |
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"type": "text",
|
| 1130 |
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"text": "Metric. Following Zhang et al. (2022), we examine FFN behaviors across four aspects: activation neuron count (namely $n _ { \\mathrm { a c t . } } )$ ), FFNs’ hidden dimension, activation-neuron ratio (activations divided by hidden dimension, namely $R _ { \\mathrm { a c t . } }$ ), and FFN efficiency (activations divided by parameters, namely $\\eta _ { \\mathrm { { f f i n } } } )$ . Notably, for PartialFormer, the hidden dimension represents the concatenation of hidden dimensions from all smaller FFNs. ",
|
| 1131 |
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"bbox": [
|
| 1132 |
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512,
|
| 1133 |
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| 1134 |
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| 1135 |
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| 1136 |
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| 1137 |
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|
| 1138 |
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|
| 1139 |
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{
|
| 1140 |
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"type": "text",
|
| 1141 |
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"text": "Results. Figure 4(a-c) exhibits the results on the En-De test set. It is evident that PartialFormer has a lower activation ratio than the vanilla Transformer, as shown in Figure 4(b). This indicates that PGFFNs based on matrix factorization present lower utilization of the hidden dimension compared to the vanilla FFNs. However, our PG-FFN is parameter consumption friendly, enabling larger hidden layer dimensions with the same parameter budget (e.g., 5400 vs. 1440). Despite lower utilization of hidden dimension, it can still own more activated neurons, as depicted in Figure 4(a). Additionally, our PGFFN exhibits higher efficiency compared to vanilla FFNs, as shown in Figure 4(c). Multiple small FFNs, like “Swarm Intelligence” (Bonabeau et al., 1999), outperform large FFNs by leveraging the collective strength of weak individuals. ",
|
| 1142 |
+
"bbox": [
|
| 1143 |
+
512,
|
| 1144 |
+
646,
|
| 1145 |
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884,
|
| 1146 |
+
919
|
| 1147 |
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],
|
| 1148 |
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"page_idx": 6
|
| 1149 |
+
},
|
| 1150 |
+
{
|
| 1151 |
+
"type": "table",
|
| 1152 |
+
"img_path": "images/64231890cc92e8206071cce0994a9dc5a1ced39e9ca288d638c13f00cf8d0e3c.jpg",
|
| 1153 |
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"table_caption": [
|
| 1154 |
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"Table 8: Comparison of activation functions in PGFFNs. "
|
| 1155 |
+
],
|
| 1156 |
+
"table_footnote": [],
|
| 1157 |
+
"table_body": "<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>PG-FFNs</td><td>68M 29.56</td></tr><tr><td>PG-FFNs with Sigmoid activation</td><td>68M 29.21</td></tr><tr><td>PG-FFNs with Tanh activation</td><td>68M 29.03</td></tr></table>",
|
| 1158 |
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"bbox": [
|
| 1159 |
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132,
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| 1160 |
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82,
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| 1161 |
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468,
|
| 1162 |
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151
|
| 1163 |
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],
|
| 1164 |
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"page_idx": 7
|
| 1165 |
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},
|
| 1166 |
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{
|
| 1167 |
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"type": "table",
|
| 1168 |
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"img_path": "images/f484f41b5e7157659d69c191f5181c13967d2b5917e9ceae879ee48e43dbbdc8.jpg",
|
| 1169 |
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"table_caption": [
|
| 1170 |
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"Table 9: Efficiency comparison between Transformer and PartialFormer in inference. "
|
| 1171 |
+
],
|
| 1172 |
+
"table_footnote": [],
|
| 1173 |
+
"table_body": "<table><tr><td>Model</td><td colspan=\"4\">Param Speed (Tok./s)Memory BLEU</td></tr><tr><td>Transformer</td><td>62M</td><td>4325</td><td>3.0G</td><td>28.00</td></tr><tr><td>PartialFormer (w/o head scaling)</td><td>66M</td><td>3634</td><td>3.2G</td><td>28.86</td></tr><tr><td>PartialFormer</td><td>68M</td><td>3023</td><td>3.3G</td><td>29.56</td></tr></table>",
|
| 1174 |
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| 1175 |
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| 1176 |
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| 1177 |
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| 1178 |
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256
|
| 1179 |
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],
|
| 1180 |
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"page_idx": 7
|
| 1181 |
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},
|
| 1182 |
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{
|
| 1183 |
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"type": "text",
|
| 1184 |
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"text": "6.7 Analysis on Head Diversity ",
|
| 1185 |
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"text_level": 1,
|
| 1186 |
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"bbox": [
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| 1187 |
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| 1188 |
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| 1189 |
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371,
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| 1190 |
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336
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| 1191 |
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|
| 1192 |
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| 1193 |
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},
|
| 1194 |
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{
|
| 1195 |
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"type": "text",
|
| 1196 |
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"text": "Metric. We select the same metric, namely $D _ { o u t p u t }$ , as that in Li et al. (2018) to measure the diversity among head features. In this metric, a larger value indicates a higher level of diversity. ",
|
| 1197 |
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| 1198 |
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| 1205 |
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|
| 1206 |
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"type": "text",
|
| 1207 |
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"text": "Results. From Figure 4(d), we can observe that PartialFormer exhibits more diverse head features compared to the vanilla Transformer, even though the vanilla Transformer already demonstrates diverse features. This aligns with previous study (Li et al., 2018), which demonstrates the positive impact of head feature diversity on the Transformer model’s performance. Thus, we conclude that the insertion of FFNs into attention mechanism may be a more optimal design. ",
|
| 1208 |
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| 1215 |
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|
| 1216 |
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{
|
| 1217 |
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"type": "text",
|
| 1218 |
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"text": "7 Related Work ",
|
| 1219 |
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"text_level": 1,
|
| 1220 |
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"bbox": [
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| 1222 |
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| 1228 |
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|
| 1229 |
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"type": "text",
|
| 1230 |
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"text": "Lightweight Transformers Many methods have been proposed to improve the parameter efficiency of Transformer architecture. The first line is to directly cut down redundant computations and parameters via a more efficient design such as adopting more efficient transformation operations (Mehta et al., 2019, 2021), integrating different but complementary patterns (Wu et al., 2020) and neural architecture search (So et al., 2019). Another research direction for improving parameter efficiency in the Transformer is weight sharing. The popular cross-layer sharing method is utilized by the Universal Transformer (Dehghani et al., 2019). Reid et al. (2021) propose better performance by freeing the first and last encoder layers and widening the intermediate layers. Li et al. (2022) introduce an ordinary differential equation-inspired weightsharing method for more precise results. Different from these work, our study focus on the design of efficient lightweight FFN. ",
|
| 1231 |
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{
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| 1240 |
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"type": "image",
|
| 1241 |
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"img_path": "images/5cec8e8c7564f86b66a2a3fde975639ff37eeaac9506bd16cdd39ff8be563e0f.jpg",
|
| 1242 |
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"image_caption": [
|
| 1243 |
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"Figure 4: Analysis on behaviours of FFNs and head diversity in Transformer and PartialFormer. "
|
| 1244 |
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],
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| 1245 |
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| 1267 |
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"text": "Multi-Branch Transformer The multi-branch strategy is widely used in Transformer design. Weighted Transformer (Ahmed et al., 2017) employs a multi-branch FFN, while Multi-attentive Transformer (Fan et al., 2020), Multi-units Transformer (Yan et al., 2020), and Multi-Path Transformer (Lin et al., 2022) extend this concept to different components of the Transformer. Our work introduces a pure multi-branch architecture based on natural subspaces. ",
|
| 1268 |
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|
| 1269 |
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| 1270 |
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| 1272 |
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| 1274 |
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| 1275 |
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|
| 1276 |
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|
| 1277 |
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"type": "text",
|
| 1278 |
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"text": "Scaling Strategy in Transformer Deepening (Bapna et al., 2018; Wang et al., 2019) and widening (Vaswani et al., 2017; Wu et al., 2021) Transformer have been well-acknowledged as two strategies to improve the capacity of Transformer in literature. In this work, PartialFormer adopts two alternative strategies to improve capacity, adding a number of heads and head dimensions. ",
|
| 1279 |
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|
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| 1285 |
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| 1286 |
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| 1287 |
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| 1288 |
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"type": "text",
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| 1289 |
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"text": "8 Conclusion ",
|
| 1290 |
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"text_level": 1,
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| 1291 |
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| 1297 |
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| 1298 |
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|
| 1299 |
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| 1300 |
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|
| 1301 |
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"text": "In this paper, we present PartialFormer, a new parameter-efficient Transformer architecture that offers an alternative approach to the design of the lightweight FFN. By employing multiple small FFNs and leveraging matrix factorization techniques, PartialFormer effectively reduces the number of parameters in the FFN. Moreover, we propose two innovative operations to further efficiently enhance the model capabilities. Experimental results across various machine translation tasks showcase the significant performance improvements achieved by PartialFormer, while maintaining comparable parameter consumption. ",
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| 1302 |
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| 1309 |
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| 1310 |
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| 1311 |
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"type": "text",
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| 1312 |
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"text": "Limitations ",
|
| 1313 |
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| 1314 |
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| 1321 |
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| 1322 |
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| 1323 |
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| 1324 |
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"text": "Despite the potential advantages of Partialformer in terms of parameter utilization and performance within a limited parameter budget, it is important to note that the existing conclusions regarding its effectiveness have not been thoroughly examined in the context of large-scale datasets and a higher number of parameters. Further research is needed to validate the claims and assess the scalability of Partialformer in more challenging scenarios. ",
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| 1325 |
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"text": "References ",
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"text": "Zhengyan Zhang, Yankai Lin, Zhiyuan Liu, Peng Li, Maosong Sun, and Jie Zhou. 2022. MoEfication: Transformer feed-forward layers are mixtures of experts. In Findings of the Association for Computational Linguistics: ACL 2022, pages 877–890, Dublin, Ireland. Association for Computational Linguistics. ",
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"text": "A Detailed Setups of Experiments ",
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"text_level": 1,
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"bbox": [
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517,
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},
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{
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"type": "text",
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"text": "A.1 Dataset ",
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| 1833 |
+
"text_level": 1,
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| 1834 |
+
"bbox": [
|
| 1835 |
+
514,
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+
109,
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| 1837 |
+
618,
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+
123
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],
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"page_idx": 10
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| 1841 |
+
},
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+
{
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| 1843 |
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"type": "text",
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+
"text": "Table 10 displays the statistics of all the 9 translation task. ",
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| 1845 |
+
"bbox": [
|
| 1846 |
+
512,
|
| 1847 |
+
130,
|
| 1848 |
+
884,
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| 1849 |
+
160
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| 1850 |
+
],
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| 1851 |
+
"page_idx": 10
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| 1852 |
+
},
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| 1853 |
+
{
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| 1854 |
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"type": "text",
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| 1855 |
+
"text": "A.2 Training Details ",
|
| 1856 |
+
"text_level": 1,
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| 1857 |
+
"bbox": [
|
| 1858 |
+
514,
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| 1859 |
+
172,
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| 1860 |
+
685,
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| 1861 |
+
187
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| 1862 |
+
],
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| 1863 |
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"page_idx": 10
|
| 1864 |
+
},
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| 1865 |
+
{
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| 1866 |
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"type": "text",
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| 1867 |
+
"text": "Table 11 and 12 exhibits the training details on all translation tasks. ",
|
| 1868 |
+
"bbox": [
|
| 1869 |
+
512,
|
| 1870 |
+
193,
|
| 1871 |
+
882,
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| 1872 |
+
223
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| 1873 |
+
],
|
| 1874 |
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"page_idx": 10
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| 1875 |
+
},
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| 1876 |
+
{
|
| 1877 |
+
"type": "text",
|
| 1878 |
+
"text": "B Metric Definition ",
|
| 1879 |
+
"text_level": 1,
|
| 1880 |
+
"bbox": [
|
| 1881 |
+
512,
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| 1882 |
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236,
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| 1883 |
+
695,
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+
252
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],
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"page_idx": 10
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| 1887 |
+
},
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{
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| 1889 |
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"type": "text",
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| 1890 |
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"text": "B.1 Measurement of Head Diversity ",
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| 1891 |
+
"text_level": 1,
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| 1892 |
+
"bbox": [
|
| 1893 |
+
514,
|
| 1894 |
+
261,
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| 1895 |
+
808,
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| 1896 |
+
277
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],
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+
"page_idx": 10
|
| 1899 |
+
},
|
| 1900 |
+
{
|
| 1901 |
+
"type": "text",
|
| 1902 |
+
"text": "Following Li et al. (2018), we measure the head diversity as follows: ",
|
| 1903 |
+
"bbox": [
|
| 1904 |
+
512,
|
| 1905 |
+
282,
|
| 1906 |
+
880,
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| 1907 |
+
312
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| 1908 |
+
],
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| 1909 |
+
"page_idx": 10
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| 1910 |
+
},
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| 1911 |
+
{
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| 1912 |
+
"type": "equation",
|
| 1913 |
+
"img_path": "images/e222b99ceca223295a3da64f749200283b4812c732beb947906765473d54c715.jpg",
|
| 1914 |
+
"text": "$$\nD _ { \\mathrm { o u t p u t } } = \\exp ( - \\frac { 1 } { H ^ { 2 } } \\sum _ { i = 1 } ^ { H } \\sum _ { j = 1 } ^ { H } \\frac { | O ^ { i } \\cdot O ^ { j } | } { \\| O ^ { i } \\| \\| O ^ { j } \\| } )\n$$",
|
| 1915 |
+
"text_format": "latex",
|
| 1916 |
+
"bbox": [
|
| 1917 |
+
522,
|
| 1918 |
+
321,
|
| 1919 |
+
842,
|
| 1920 |
+
368
|
| 1921 |
+
],
|
| 1922 |
+
"page_idx": 10
|
| 1923 |
+
},
|
| 1924 |
+
{
|
| 1925 |
+
"type": "text",
|
| 1926 |
+
"text": "During evaluation, we calculate the metric on all samples and average the values to obtain the final result. ",
|
| 1927 |
+
"bbox": [
|
| 1928 |
+
512,
|
| 1929 |
+
375,
|
| 1930 |
+
882,
|
| 1931 |
+
422
|
| 1932 |
+
],
|
| 1933 |
+
"page_idx": 10
|
| 1934 |
+
},
|
| 1935 |
+
{
|
| 1936 |
+
"type": "text",
|
| 1937 |
+
"text": "C More Comparison with Previous Lightweight Transformer ",
|
| 1938 |
+
"text_level": 1,
|
| 1939 |
+
"bbox": [
|
| 1940 |
+
512,
|
| 1941 |
+
435,
|
| 1942 |
+
828,
|
| 1943 |
+
468
|
| 1944 |
+
],
|
| 1945 |
+
"page_idx": 10
|
| 1946 |
+
},
|
| 1947 |
+
{
|
| 1948 |
+
"type": "text",
|
| 1949 |
+
"text": "Table 13 presents a comprehensive comparison of previous lightweight Transformer models on the En-De task’s test set, with a specific focus on operating within a smaller parameter budget. The results prominently showcase the outstanding performance of PartialFormer, even when faced with constraints on model capacity. This outcome further emphasizes the superior capabilities of PartialFormer in scenarios with limited resources. ",
|
| 1950 |
+
"bbox": [
|
| 1951 |
+
512,
|
| 1952 |
+
476,
|
| 1953 |
+
884,
|
| 1954 |
+
620
|
| 1955 |
+
],
|
| 1956 |
+
"page_idx": 10
|
| 1957 |
+
},
|
| 1958 |
+
{
|
| 1959 |
+
"type": "text",
|
| 1960 |
+
"text": "D PartialFormer with Different $A _ { G }$ for Small Dataset ",
|
| 1961 |
+
"text_level": 1,
|
| 1962 |
+
"bbox": [
|
| 1963 |
+
512,
|
| 1964 |
+
632,
|
| 1965 |
+
860,
|
| 1966 |
+
664
|
| 1967 |
+
],
|
| 1968 |
+
"page_idx": 10
|
| 1969 |
+
},
|
| 1970 |
+
{
|
| 1971 |
+
"type": "text",
|
| 1972 |
+
"text": "Table 14 showcases the results of PartialFormer on the WMT’16 En-Ro task, a small-scale translation dataset, specifically when $A _ { G }$ is calculated using local attention (Shaw et al., 2018). Notably, these results reveal that by adopting such an approach, PartialFormer achieves an impressive BLEU score of 35.76. We hope this can shed lights on the area of model integration. ",
|
| 1973 |
+
"bbox": [
|
| 1974 |
+
512,
|
| 1975 |
+
674,
|
| 1976 |
+
882,
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| 1977 |
+
802
|
| 1978 |
+
],
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| 1979 |
+
"page_idx": 10
|
| 1980 |
+
},
|
| 1981 |
+
{
|
| 1982 |
+
"type": "text",
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| 1983 |
+
"text": "E PartialFormer with GLU and Weight Sharing ",
|
| 1984 |
+
"text_level": 1,
|
| 1985 |
+
"bbox": [
|
| 1986 |
+
512,
|
| 1987 |
+
814,
|
| 1988 |
+
867,
|
| 1989 |
+
846
|
| 1990 |
+
],
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| 1991 |
+
"page_idx": 10
|
| 1992 |
+
},
|
| 1993 |
+
{
|
| 1994 |
+
"type": "text",
|
| 1995 |
+
"text": "In this section, we investigate the integration of PartialFormer with two prominent techniques to enhance parameter efficiency: 1) the weight sharing method (Lan et al., 2020), and 2) gated linear units (Dauphin et al., 2017). To ensure the utilization of the latest advancements, we employ a state-of-the-art weight sharing method called ODE Transformer (Li et al., 2022), known for its effectiveness in promoting parameter efficiency in Transformer architectures. Additionally, we incorporate Swi-GLU (Shazeer, 2020), a widely adopted GLUvariant that has served as a foundational component in numerous expressive Transformer architectures. ",
|
| 1996 |
+
"bbox": [
|
| 1997 |
+
510,
|
| 1998 |
+
856,
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| 1999 |
+
882,
|
| 2000 |
+
920
|
| 2001 |
+
],
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| 2002 |
+
"page_idx": 10
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| 2003 |
+
},
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| 2004 |
+
{
|
| 2005 |
+
"type": "table",
|
| 2006 |
+
"img_path": "images/113f3f1b820b4ee921dbc6d655ef8a8f572da675afe495bb5f849e2dfa52ee3c.jpg",
|
| 2007 |
+
"table_caption": [
|
| 2008 |
+
"Table 10: The details of datasets of 9 translation tasks. "
|
| 2009 |
+
],
|
| 2010 |
+
"table_footnote": [],
|
| 2011 |
+
"table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"3\">Sentence</td><td rowspan=\"2\">BPE</td><td rowspan=\"2\">Vocab</td></tr><tr><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>WMT'14 En-De</td><td>4.5M</td><td>2999 26815</td><td>3003</td><td>32K 32K</td><td>34040 37288</td></tr><tr><td>WMT'14 En-Fr WMT'16 En-Ro</td><td>36M 0.6M</td><td>1999</td><td>3003 1999</td><td>20K</td><td>19064</td></tr><tr><td>WMT'17 En-De</td><td>5.9M</td><td>7998</td><td>3004</td><td>32K</td><td>35488</td></tr><tr><td>WMT'17 De-En</td><td>5.9M</td><td>7998</td><td>3004</td><td>32K</td><td>35448</td></tr><tr><td>WMT'17 En-Fi</td><td>2.7M</td><td>4225</td><td>3002</td><td>32K</td><td>32584</td></tr><tr><td>WMT'17Fi-En</td><td>2.7M</td><td>4225</td><td>3002</td><td>32K</td><td>32584</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>WMT'17 En-Lv WMT'17Lv-En</td><td>4.5M 4.5M</td><td>2003 2003</td><td>2001 2001</td><td>20K 20K</td><td>32368 32368</td></tr></table>",
|
| 2012 |
+
"bbox": [
|
| 2013 |
+
287,
|
| 2014 |
+
83,
|
| 2015 |
+
712,
|
| 2016 |
+
282
|
| 2017 |
+
],
|
| 2018 |
+
"page_idx": 11
|
| 2019 |
+
},
|
| 2020 |
+
{
|
| 2021 |
+
"type": "table",
|
| 2022 |
+
"img_path": "images/e0e47b7509c2f33386c696e9bd429cae698ff6f33ac8d5e81cba385df0c426a9.jpg",
|
| 2023 |
+
"table_caption": [
|
| 2024 |
+
"Table 11: The training setups of WMT’14 En-De, WMT’16 En-Ro and WMT’14 En-Fr tasks. "
|
| 2025 |
+
],
|
| 2026 |
+
"table_footnote": [],
|
| 2027 |
+
"table_body": "<table><tr><td colspan=\"4\">Hyper-parameter WMT'14 En-De WMT'16En-Ro WMT'14 En-Fr</td></tr><tr><td>GPUs</td><td>8</td><td>4</td><td>8</td></tr><tr><td>Batch Size</td><td>4096</td><td>4096</td><td>4096</td></tr><tr><td>Update Frequency</td><td>2</td><td>1</td><td>8</td></tr><tr><td>Optimer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Adamβ</td><td>(0.9,0.997)</td><td>(0.9, 0.997)</td><td>(0.9, 0.997)</td></tr><tr><td>LR</td><td>0.0020</td><td>0.0020</td><td>0.0020</td></tr><tr><td>LR scheduler</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td></tr><tr><td>InitialLR</td><td>1e-7</td><td>1e-7</td><td>le-7</td></tr><tr><td>Total updates</td><td>50K</td><td>25K</td><td>100K</td></tr><tr><td>Warmup updates</td><td>16000</td><td>8000</td><td>16000</td></tr><tr><td>Weight decay</td><td>0.0000</td><td>0.0000</td><td>0.0000</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>ReLU dropout</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>",
|
| 2028 |
+
"bbox": [
|
| 2029 |
+
268,
|
| 2030 |
+
321,
|
| 2031 |
+
731,
|
| 2032 |
+
525
|
| 2033 |
+
],
|
| 2034 |
+
"page_idx": 11
|
| 2035 |
+
},
|
| 2036 |
+
{
|
| 2037 |
+
"type": "table",
|
| 2038 |
+
"img_path": "images/fd917da6ce94bf44aecb4b8e4af789df0188a27b3b1b14b83548ced91d14761c.jpg",
|
| 2039 |
+
"table_caption": [
|
| 2040 |
+
"Table 12: The training setups of WMT’17 benchmark. "
|
| 2041 |
+
],
|
| 2042 |
+
"table_footnote": [],
|
| 2043 |
+
"table_body": "<table><tr><td colspan=\"3\">Hyper-parameterI En-{De,Lv} {De,Lv}-En</td><td>En-Fi</td><td>Fi-En</td></tr><tr><td>GPUs</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>Batch Size</td><td>4096</td><td>4096</td><td>4096</td><td>4096</td></tr><tr><td>Update Frequency</td><td>2</td><td>1</td><td>1</td><td>4</td></tr><tr><td>Optimer</td><td>Adam</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Adamβ</td><td>(0.9, 0.997)</td><td>(0.9, 0.997)</td><td>(0.9,0.997) (0.9,0.997)</td><td></td></tr><tr><td>LR</td><td>0.0020</td><td>0.0020</td><td>0.0020</td><td>0.0020</td></tr><tr><td>LR scheduler</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td><td>inverse sqrt</td></tr><tr><td>Initial LR</td><td>1e-7</td><td>1e-7</td><td>1e-7</td><td>le-7</td></tr><tr><td>Total updates</td><td>50K/17K</td><td>50K/17K</td><td>40K</td><td>10K</td></tr><tr><td>Warmup updates</td><td>16000</td><td>16000</td><td>16000</td><td>16000</td></tr><tr><td>Weight decay</td><td>0.0000</td><td>0.0000</td><td>0.0000</td><td>0.0000</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>ReLU dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>",
|
| 2044 |
+
"bbox": [
|
| 2045 |
+
275,
|
| 2046 |
+
568,
|
| 2047 |
+
722,
|
| 2048 |
+
772
|
| 2049 |
+
],
|
| 2050 |
+
"page_idx": 11
|
| 2051 |
+
},
|
| 2052 |
+
{
|
| 2053 |
+
"type": "text",
|
| 2054 |
+
"text": "",
|
| 2055 |
+
"bbox": [
|
| 2056 |
+
115,
|
| 2057 |
+
827,
|
| 2058 |
+
487,
|
| 2059 |
+
907
|
| 2060 |
+
],
|
| 2061 |
+
"page_idx": 11
|
| 2062 |
+
},
|
| 2063 |
+
{
|
| 2064 |
+
"type": "text",
|
| 2065 |
+
"text": "",
|
| 2066 |
+
"bbox": [
|
| 2067 |
+
510,
|
| 2068 |
+
827,
|
| 2069 |
+
884,
|
| 2070 |
+
890
|
| 2071 |
+
],
|
| 2072 |
+
"page_idx": 11
|
| 2073 |
+
},
|
| 2074 |
+
{
|
| 2075 |
+
"type": "text",
|
| 2076 |
+
"text": "Table 15 displays the results of combining PartialFormer with weight sharing and gated linear units. Despite the integration of these two techniques, the performance gains are marginal. This could be attributed to the fact that PartialFormer already possesses high parameter efficiency, leaving little room for additional enhancements from other technologies. In other words, PartialFormer is inherently a high parameter efficiency architecture. ",
|
| 2077 |
+
"bbox": [
|
| 2078 |
+
531,
|
| 2079 |
+
904,
|
| 2080 |
+
882,
|
| 2081 |
+
920
|
| 2082 |
+
],
|
| 2083 |
+
"page_idx": 11
|
| 2084 |
+
},
|
| 2085 |
+
{
|
| 2086 |
+
"type": "table",
|
| 2087 |
+
"img_path": "images/20f8901306c77dfdd0156c6e08bb1cfe5d49a122fae8f772b0c6c5ceb92e3e1f.jpg",
|
| 2088 |
+
"table_caption": [
|
| 2089 |
+
"Table 13: Comparison with state-of-the-art models of smaller capacities on the En-De task. "
|
| 2090 |
+
],
|
| 2091 |
+
"table_footnote": [],
|
| 2092 |
+
"table_body": "<table><tr><td>Model</td><td>Param BLEU</td></tr><tr><td>DELIGHT (Mehta et al., 2021) EdgeFormer (Ge et al., 2022) Lite Transformer (Wu et al.,2020) PartialFormer</td><td>23M 26.70 - 26.90 - 26.50 27M 27.50</td></tr><tr><td>Evolved Transformer (So et al., 2019) DELIGHT (Mehta et al., 2021) ODE Transformer (Li et al., 2022) PartialFormer</td><td>48M 27.70 37M 27.60 37M 28.24 36M 28.35</td></tr></table>",
|
| 2093 |
+
"bbox": [
|
| 2094 |
+
132,
|
| 2095 |
+
82,
|
| 2096 |
+
468,
|
| 2097 |
+
214
|
| 2098 |
+
],
|
| 2099 |
+
"page_idx": 12
|
| 2100 |
+
},
|
| 2101 |
+
{
|
| 2102 |
+
"type": "table",
|
| 2103 |
+
"img_path": "images/3e0c5a5f1259a23320c34680f5d4e7b0f324686b03af4b35db3733331a6c2277.jpg",
|
| 2104 |
+
"table_caption": [
|
| 2105 |
+
"Table 14: Results of several PartialFormer variants on the En-De task. "
|
| 2106 |
+
],
|
| 2107 |
+
"table_footnote": [],
|
| 2108 |
+
"table_body": "<table><tr><td>AG</td><td>AL</td><td>Param</td><td>BLEU</td></tr><tr><td>RPR</td><td>MHSA</td><td>62M</td><td>35.76</td></tr></table>",
|
| 2109 |
+
"bbox": [
|
| 2110 |
+
200,
|
| 2111 |
+
266,
|
| 2112 |
+
401,
|
| 2113 |
+
307
|
| 2114 |
+
],
|
| 2115 |
+
"page_idx": 12
|
| 2116 |
+
},
|
| 2117 |
+
{
|
| 2118 |
+
"type": "table",
|
| 2119 |
+
"img_path": "images/393062bc04c56d5f8f438ae8579e80167567f5fd1ea935200ecb3e29907f5b4c.jpg",
|
| 2120 |
+
"table_caption": [
|
| 2121 |
+
"Table 15: Results of PartialFormer variants on the EnDe task. "
|
| 2122 |
+
],
|
| 2123 |
+
"table_footnote": [],
|
| 2124 |
+
"table_body": "<table><tr><td>Model</td><td>Param</td><td>BLEU</td></tr><tr><td>PartialFormer</td><td>67M</td><td>29.56</td></tr><tr><td>PartialFormer + Weight Sharing</td><td>67M</td><td>29.71</td></tr><tr><td>GLU-based PartialFormer</td><td>67M</td><td>29.67</td></tr></table>",
|
| 2125 |
+
"bbox": [
|
| 2126 |
+
132,
|
| 2127 |
+
361,
|
| 2128 |
+
470,
|
| 2129 |
+
434
|
| 2130 |
+
],
|
| 2131 |
+
"page_idx": 12
|
| 2132 |
+
},
|
| 2133 |
+
{
|
| 2134 |
+
"type": "text",
|
| 2135 |
+
"text": "",
|
| 2136 |
+
"bbox": [
|
| 2137 |
+
115,
|
| 2138 |
+
499,
|
| 2139 |
+
487,
|
| 2140 |
+
626
|
| 2141 |
+
],
|
| 2142 |
+
"page_idx": 12
|
| 2143 |
+
},
|
| 2144 |
+
{
|
| 2145 |
+
"type": "text",
|
| 2146 |
+
"text": "F Analysis on Token Uniformity ",
|
| 2147 |
+
"text_level": 1,
|
| 2148 |
+
"bbox": [
|
| 2149 |
+
119,
|
| 2150 |
+
642,
|
| 2151 |
+
408,
|
| 2152 |
+
658
|
| 2153 |
+
],
|
| 2154 |
+
"page_idx": 12
|
| 2155 |
+
},
|
| 2156 |
+
{
|
| 2157 |
+
"type": "text",
|
| 2158 |
+
"text": "Following (Dong et al., 2021; Wang et al., 2022), we measure the token uniformity among token representations. We use pearson correlation to compute it. ",
|
| 2159 |
+
"bbox": [
|
| 2160 |
+
115,
|
| 2161 |
+
669,
|
| 2162 |
+
489,
|
| 2163 |
+
732
|
| 2164 |
+
],
|
| 2165 |
+
"page_idx": 12
|
| 2166 |
+
},
|
| 2167 |
+
{
|
| 2168 |
+
"type": "text",
|
| 2169 |
+
"text": "From Figure 5, we can observe that PartialFormer owns a lower token uniformity among token representations than the vanilla Transformer, revealing that PartialFormer can benefit from depth scaling efficiently (Dong et al., 2021; Wang et al., 2022). ",
|
| 2170 |
+
"bbox": [
|
| 2171 |
+
115,
|
| 2172 |
+
734,
|
| 2173 |
+
487,
|
| 2174 |
+
828
|
| 2175 |
+
],
|
| 2176 |
+
"page_idx": 12
|
| 2177 |
+
},
|
| 2178 |
+
{
|
| 2179 |
+
"type": "text",
|
| 2180 |
+
"text": "G Preliminary Experiments on Language Modeling ",
|
| 2181 |
+
"text_level": 1,
|
| 2182 |
+
"bbox": [
|
| 2183 |
+
115,
|
| 2184 |
+
845,
|
| 2185 |
+
485,
|
| 2186 |
+
877
|
| 2187 |
+
],
|
| 2188 |
+
"page_idx": 12
|
| 2189 |
+
},
|
| 2190 |
+
{
|
| 2191 |
+
"type": "text",
|
| 2192 |
+
"text": "We also evaluate the effectiveness of PartialFormer on the language modeling task. We can see that ",
|
| 2193 |
+
"bbox": [
|
| 2194 |
+
115,
|
| 2195 |
+
888,
|
| 2196 |
+
485,
|
| 2197 |
+
919
|
| 2198 |
+
],
|
| 2199 |
+
"page_idx": 12
|
| 2200 |
+
},
|
| 2201 |
+
{
|
| 2202 |
+
"type": "image",
|
| 2203 |
+
"img_path": "images/517907196c26506a55372a37e562c0ab840c57819add59a61bb59d3cc5f52ae6.jpg",
|
| 2204 |
+
"image_caption": [
|
| 2205 |
+
"Figure 5: Comparison of token uniformity (lower is better) in Transformer and PartialFormer. "
|
| 2206 |
+
],
|
| 2207 |
+
"image_footnote": [],
|
| 2208 |
+
"bbox": [
|
| 2209 |
+
554,
|
| 2210 |
+
85,
|
| 2211 |
+
840,
|
| 2212 |
+
180
|
| 2213 |
+
],
|
| 2214 |
+
"page_idx": 12
|
| 2215 |
+
},
|
| 2216 |
+
{
|
| 2217 |
+
"type": "text",
|
| 2218 |
+
"text": "PartialFormer can also show better results compared to strong baseline, e.g., Adaptive Input Transformer (Baevski and Auli, 2019). We will present more comprehensive experiments in the future. ",
|
| 2219 |
+
"bbox": [
|
| 2220 |
+
510,
|
| 2221 |
+
241,
|
| 2222 |
+
882,
|
| 2223 |
+
305
|
| 2224 |
+
],
|
| 2225 |
+
"page_idx": 12
|
| 2226 |
+
},
|
| 2227 |
+
{
|
| 2228 |
+
"type": "table",
|
| 2229 |
+
"img_path": "images/d601769096e53a656e68d57198707e235b70b3bfa767a168c0bd57e1d5018116.jpg",
|
| 2230 |
+
"table_caption": [],
|
| 2231 |
+
"table_footnote": [
|
| 2232 |
+
"Table 16: Results on the WikiText-103 dataset. "
|
| 2233 |
+
],
|
| 2234 |
+
"table_body": "<table><tr><td>Model</td><td>Depth 0 (M) Test PPL</td></tr><tr><td>Adaptive Input</td><td>8 147M 21.11</td></tr><tr><td>PartialFormer</td><td>16 143M 19.87</td></tr></table>",
|
| 2235 |
+
"bbox": [
|
| 2236 |
+
515,
|
| 2237 |
+
316,
|
| 2238 |
+
877,
|
| 2239 |
+
392
|
| 2240 |
+
],
|
| 2241 |
+
"page_idx": 12
|
| 2242 |
+
}
|
| 2243 |
+
]
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| 1 |
+
# Learning to Tokenize for Generative Retrieval
|
| 2 |
+
|
| 3 |
+
Weiwei $\mathbf { S u n } ^ { 1 }$ , Lingyong $\mathbf { Y a n } ^ { 2 }$ , Zheng Chen1, Shuaiqiang Wang2, Haichao $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 2 }$
|
| 4 |
+
Pengjie Ren1, Zhumin Chen1, Dawei $\mathbf { Y i n } ^ { 2 }$ , Maarten de Rijke3, Zhaochun $\mathbf { R e n ^ { 4 * } }$ 1Shandong University, China 2Baidu Inc., China
|
| 5 |
+
3University of Amsterdam, The Netherlands 4Leiden University, The Netherlands {sunnweiwei,lingyongy}@gmail.com yindawei@acm.org m.derijke@uva.nl z.ren@liacs.leidenuniv.nl
|
| 6 |
+
|
| 7 |
+
# Abstract
|
| 8 |
+
|
| 9 |
+
As a new paradigm in information retrieval, generative retrieval directly generates a ranked list of document identifiers (docids) for a given query using generative language models (LMs). How to assign each document a unique docid (denoted as document tokenization) is a critical problem, because it determines whether the generative retrieval model can precisely retrieve any document by simply decoding its docid. Most existing methods adopt rule-based tokenization, which is ad-hoc and does not generalize well. In contrast, in this paper we propose a novel document tokenization learning method, GENRET, which learns to encode the complete document semantics into docids. GENRET learns to tokenize documents into short discrete representations (i.e., docids) via a discrete auto-encoding approach. We develop a progressive training scheme to capture the autoregressive nature of docids and diverse clustering techniques to stabilize the training process. Based on the semantic-embedded docids of any set of documents, the generative retrieval model can learn to generate the most relevant docid only according to the docids’ semantic relevance to the queries. We conduct experiments on the NQ320K, MS MARCO, and BEIR datasets. GENRET establishes the new state-of-the-art on the NQ320K dataset. Compared to generative retrieval baselines, GENRET can achieve significant improvements on unseen documents. Moreover, GENRET can also outperform comparable baselines on MS MARCO and BEIR, demonstrating the method’s generalizability.
|
| 10 |
+
|
| 11 |
+
# 1 Introduction
|
| 12 |
+
|
| 13 |
+
Document retrieval plays an essential role in web search applications and various downstream knowledge-intensive tasks by identifying relevant documents to satisfy users’ queries. Recently, generative retrieval has emerged as a new paradigm for document retrieval [1, 5, 37, 41, 46, 47] that directly generates a ranked list of document identifiers (docids) for a given query using generative language models (LMs). Unlike dense retrieval [9, 13, 23, 42], generative retrieval presents an end-to-end solution for document retrieval tasks [37]. It also offers a promising approach to better exploit the capabilities of recent large LMs [1, 41].
|
| 14 |
+
|
| 15 |
+
As shown in Figure 1, document tokenization aims to tokenize each document in the corpus as a sequence of discrete characters, i.e., docids. Document tokenization plays a crucial role in generative retrieval, as it defines how the document is distributed in the semantic space [37]. And it is still an open problem how to define docids. Most previous generative methods tend to employ rule-based document tokenizers, such as generating titles or URLs [5, 46], or clustering results from off-the-shelf document embeddings [37, 41]. Such rule-based methods are usually ad-hoc and do not generalize well. In particular, the tokenization results potentially perform well on retrieving documents that have been seen during training, but generalize poorly to unlabeled documents [17, 20].
|
| 16 |
+
|
| 17 |
+
|
| 18 |
+
Figure 1: An overview of our proposed method. The proposed method utilizes a document tokenization model to convert a given document into a sequence of discrete tokens, referred to as a docid. This tokenization process allows for the reconstruction of the original document through a reconstruction model. Subsequently, an autoregressive generation model is employed to retrieve documents through the generation of their respective docids.
|
| 19 |
+
|
| 20 |
+
To address the above problem, we propose GENRET, a document tokenization learning framework that learns to tokenize a document into semantic docids in a discrete auto-encoding scheme. GENRET consists of a shared sequence-to-sequence-based document tokenization model, a generative retrieval model, and a reconstruction model. In the proposed auto-encoding learning scheme, the tokenization model learns to convert documents to discrete docids, which are subsequently utilized by the reconstruction model to reconstruct the original document. The generative retrieval model is trained to generate docids in an autoregressive manner for a given query. The above three models are optimized in an end-to-end fashion to achieve seamless integration.
|
| 21 |
+
|
| 22 |
+
There are usually two challenges when using auto-encoding to optimize a generative retrieval model: (i) docids with an autoregressive nature, and (ii) docids with diversity. To address the first challenge and also to stabilize the training of GENRET, we devise a progressive training scheme. This training scheme allows for a stable training of the model by fixing optimized prefix docids $z _ { < t }$ . To optimize the docids at each step, three proposed losses are utilized: (i) a reconstruction loss for predicting the document using the generated docid, (ii) a commitment loss for committing the docid and avoiding forgetting, and (iii) a retrieval loss for optimizing the retrieval performance end-to-end. To address the second challenge, we propose a parameter initialization strategy and a re-assignment of the docid based on a diverse clustering technique to increase the diversity of the generated docids.
|
| 23 |
+
|
| 24 |
+
We conduct extensive experiments on three well-known document retrieval benchmark datasets, NQ320K [15, 37], MS MARCO [4, 46], and BEIR [38]. GENRET attains superior retrieval performance against state-of-the-art generative retrieval models on NQ320K. GENRET achieves $+ 1 4 \%$ relative improvements on the unseen test set of NQ320K over the best generative retrieval baseline. Experiments on MS MARCO and six BEIR datasets also show that GENRET outperforms existing generative methods and achieves competitive results compared to popular dense retrieval models. Experiments on retrieving new documents, analytical experiments, and an efficiency analysis confirm the effectiveness of the proposed model.
|
| 25 |
+
|
| 26 |
+
We summarize our contributions as follows: (i) We propose GENRET, a generative retrieval model that represents documents as discrete semantic docids. To the best of our knowledge, this is the first tokenization learning method for document retrieval. (ii) We propose an auto-encoding approach, where the docids generated by our tokenization model are reconstructed by a reconstruction model to ensure the docids capture the semantic information of the document. (iii) We devise a progressive training scheme to model the autoregressive nature of docids and stabilize the training process. (iv) Experimental results demonstrate that GENRET achieves significant improvements, especially on unseen documents, over generative retrieval baselines.2
|
| 27 |
+
|
| 28 |
+
# 2 Preliminaries
|
| 29 |
+
|
| 30 |
+
The document retrieval task can be formalized as the process of retrieving a relevant document $d$ for a search query $q$ from a collection of documents $\mathcal { D }$ . Each document $d \in \mathcal { D }$ is assumed to be a plain text consisting of a sequence of tokens, denoted as $d = \{ d _ { 1 } , \dotsc , d _ { | d | } \}$ , where $| d |$ is the total number of tokens in the document. For generative retrieval models, it is usually challenging and computationally inefficient to directly generate original documents of typically long length. Therefore, most existing approaches rely on a technique named document tokenization, which represents a document $\bar { d } \bar { = } \{ d _ { 1 } , \ldots , \bar { d _ { | d | } } \}$ as a shorter sequence of discrete tokens (docid) $z = \{ z _ { 1 } , \ldots , z _ { t } , \ldots , z _ { M } \}$ , where each token $z _ { t }$ is as a $K$ -way categorical variable, with $z _ { t } \in [ 1 , 2 , \ldots , K ]$ , and $M$ is the length of the docid. See Figure 1 for an example of document tokenization with $M = 3$ and $K = 6 4$ .
|
| 31 |
+
|
| 32 |
+
As an alternative sequence of the original document, the tokenized docid $z$ should satisfy the following two properties: (i) different documents have short but different docids; and (ii) docids capture the semantics of their associated documents as much as possible [37]. Because $z$ is a sequence of a fixed length and usually shorter than the original document $d$ , the model’s training and inference can be simplified and more efficient. This paper employs a tokenization model $Q \colon d z$ to map $d$ to docid $z$ . More details about $Q$ are provided in Section 3.1. After tokenizing each document to docid $z$ , a generative retrieval model $P \colon q z$ learns to retrieve relevant documents by generating a query $q$ to a docid $z$ autoregressively [37].
|
| 33 |
+
|
| 34 |
+
# 3 Method
|
| 35 |
+
|
| 36 |
+
Conventionally, document tokenization is done by a fixed pre-processing step, such as using the title of a document or the results of hierarchical clustering obtained from BERT [5, 37]. However, it has been observed that such ad-hoc document tokenization methods often fail to capture the complete semantics of a document. For example, the title of a web page often does not exist or has low relevance to the content of the web page, and the use of clustering-based docids arbitrarily defines the document in discrete space.
|
| 37 |
+
|
| 38 |
+
In this paper, we propose GENRET, a novel tokenization learning method based on discrete autoencoding, to learn semantic docid in a fully end-to-end manner. Figure 1 gives an overview of the proposed method. The proposed GENRET comprises three main components: (i) a sequence-tosequence based retrieval model $P ( z \mid q )$ , (ii) a document tokenization model $Q ( z \mid d )$ , and (iii) a reconstruction model $R ( d \mid z )$ . The document tokenization model tokenizes a document $d$ into unique discrete variables $z$ , and the retrieval model is trained to generate the latent variables $z$ for a given query $q$ . In addition, the reconstruction model is used to re-generate the original document from $z$ to ensure $z$ captures the semantics of the original document as much as possible.
|
| 39 |
+
|
| 40 |
+
# 3.1 Model architecture
|
| 41 |
+
|
| 42 |
+
Following DSI [37], we employ an encoder-decoder Transformer to implement the generative retrieval model. Specifically, given an input text $d ^ { 3 }$ , the T5-based tokenization model encodes $d$ and a prefix of docid $z _ { < t }$ and continuously produces latent representation $\mathbf { d } _ { t }$ of $d$ at time step $t$ :
|
| 43 |
+
|
| 44 |
+
$$
|
| 45 |
+
\mathbf { d } _ { t } = { \mathrm { D e c o d e r } } ( { \mathrm { E n c o d e r } } ( d ) , z _ { < t } ) \in \mathbb { R } ^ { D } ,
|
| 46 |
+
$$
|
| 47 |
+
|
| 48 |
+
where $D$ denotes the hidden size of the model. Encoder $( d )$ denotes the output of the Encoder.
|
| 49 |
+
|
| 50 |
+
Then, the tokenization model generates a token for each document based on $\mathbf { d } _ { t }$ . At each timestep $t$ we define an external embedding matrix named codebook $\mathbf { E } _ { t } \in \mathbb { R } ^ { K \times D }$ , where $K$ is the size of the discrete latent space. There are $K$ embedding vectors $\mathbf { e } _ { t , j } \in \mathbb { R } ^ { D } , j \in [ K ]$ , and each vector $\mathbf { e } _ { t , j }$ can be regarded as the centroid of a segmentation.
|
| 51 |
+
|
| 52 |
+
Based on the codebook $\mathbf { E } _ { t }$ , the discrete latent variable $z _ { t }$ at timestep $t$ is calculated by a dot-product look-up using the codebook $\mathbf { E } _ { t }$ :
|
| 53 |
+
|
| 54 |
+
$$
|
| 55 |
+
Q ( z _ { t } = j \mid z _ { < t } , d ) = \operatorname { S o f t m a x } _ { j } ( \mathbf { d } _ { t } \cdot \mathbf { E } _ { t } ^ { \top } ) ,
|
| 56 |
+
$$
|
| 57 |
+
|
| 58 |
+
where $Q ( z _ { t } = j \mid z _ { < t } , d )$ denotes the probability of tokenizing $d$ to a particular value $j \in [ K ]$ at timestep $t$ , $\operatorname { S o f t m a x } _ { j }$ is a softmax function to output the probability of axis $j$ .
|
| 59 |
+
|
| 60 |
+
Document reconstruction model. The docid generated by the tokenization model $Q$ is required to capture the semantic information of the document. To this end, we propose an auto-encoding training scheme, where a reconstruction model $R \colon z \to d$ that predicts $d$ using $z$ is designed to force the tokenization model $Q \colon d z$ to reproduce a docid $z$ that can be reconstructed back-to-the original document.
|
| 61 |
+
|
| 62 |
+
The input of the reconstruction model is docid $z$ , and the output is its associated document $d$ . We first embed $z$ into representation matrix $\mathbf { z } = \{ \mathbf { z } _ { 1 } , \hdots , \mathbf { z } _ { M } \} \stackrel { * } { \in } \mathbb { R } ^ { M \times D }$ using the codebook of the tokenization model:
|
| 63 |
+
|
| 64 |
+
$$
|
| 65 |
+
\mathbf { z } = \{ \mathbf { e } _ { 1 , z _ { 1 } } , \mathbf { e } _ { 2 , z _ { 2 } } , \ldots , \mathbf { e } _ { M , z _ { M } } \} \in \mathbb { R } ^ { M \times D } ,
|
| 66 |
+
$$
|
| 67 |
+
|
| 68 |
+
where each $t \in [ M ] , \mathbf { z } _ { t } = \mathbf { e } _ { t , z _ { t } } \in \mathbb { R } ^ { D }$ is the embedding vector of $z _ { t }$ in the $t { \cdot }$ -step codebook $\mathbf { E } _ { t }$
|
| 69 |
+
|
| 70 |
+
We then devise a retrieval-based reconstruction model that predicts the target document $d$ by retrieving it from document collection $\mathcal { D }$ , based on the inputs $\mathbf { z }$ . The relevance score between the input docid $z$ and the target document $d$ is defined as follows:
|
| 71 |
+
|
| 72 |
+
$$
|
| 73 |
+
R ( d \mid \mathbf { z } ) = \prod _ { t = 1 } ^ { M } \frac { \exp ( \mathbf { z } _ { t } \cdot \mathbf { s g } ( \mathbf { d } _ { t } ^ { \top } ) ) } { \sum _ { d ^ { * } \in S ( z _ { < t } ) } \exp ( \mathbf { z } _ { t } \cdot \mathbf { s g } ( \mathbf { d } ^ { * } { } _ { t } ^ { \top } ) ) } ,
|
| 74 |
+
$$
|
| 75 |
+
|
| 76 |
+
where $S ( z _ { < t } )$ is a sub-collection of $\mathcal { D }$ consisting of documents that have a docid prefix that is the same as $z _ { < t }$ . $d ^ { * } \in S ( z _ { < t } )$ represents a document from the sub-collection $S ( z _ { < t } )$ . $\mathbf { d } _ { t }$ and $\mathbf { d } ^ { * } { } _ { t }$ are continuous representations of documents $d$ and $d ^ { * }$ , respectively, as defined in Eq. 1. The operator $\operatorname { s g } ( \cdot )$ is the stop gradient operator to prevent gradient back propagation. Intuitively, $R ( d { \bar { \mathbf { \theta } } } | { \mathbf { \theta } } \mathbf { z } )$ is designed to retrieve a specific document $d$ from a set of documents $S ( z _ { < t } )$ at each timestep $t$ . The set $S ( z _ { < t } )$ only includes those documents that are assigned the same docid prefix $z _ { < t }$ as the target document $d$ . By utilizing this loss function, at each step $t$ , the model is facilitated to learn the residual semantics of the documents not captured by the previous docid $z _ { < t }$ .
|
| 77 |
+
|
| 78 |
+
# 3.2 Model optimization
|
| 79 |
+
|
| 80 |
+
For the document tokenization model $Q ( z \mid d )$ , generative retrieval model $P ( z \mid q )$ , and reconstruction model $R ( d \mid z )$ , jointly optimizing these three models using auto-encoding is challenging due to the following reasons: (i) Learning docids in an autoregressive fashion. On one hand, the prediction of the $z _ { t }$ at time $t$ relies on previously predicted docids $z _ { < t }$ , which is often under-optimized at the beginning and rapidly changes during training, making it difficult to reach convergence. On the other hand, simultaneously optimizing $z$ makes it challenging to guarantee a unique docid assignment. Hence, to stabilize the training of GENRET, we devise a progressive training scheme (see Section 3.2.1). (ii) Generating docids with diversity. Optimizing the model using auto-encoding often leads to unbalanced docid assignment: a few major docids are assigned to a large number of documents and most other docids are rarely assigned. Such a sub-optimal distribution of docids affects the model distinguishability, which in turns triggers length increments of docids in order to distinguish conflicting documents. We introduce two diverse clustering techniques to ensure docid diversity (see Section 3.2.2).
|
| 81 |
+
|
| 82 |
+
# 3.2.1 Progressive training scheme
|
| 83 |
+
|
| 84 |
+
To optimize each of the three models listed above in an autoregressive manner, we propose a progressive autoencoding learning scheme, as illustrated in Figure 2. The whole learning scheme contains $M$ learning steps with respect to the final docid in $M$ -token. And the docid $z _ { T }$ at step $T \in [ M ]$ is learned and optimized at the corresponding learning step. Besides, at each step $T \in [ M ]$ the docid $z _ { T }$ and the model parameters associated with $z _ { T }$ generation are updated, while previously produced docids $z _ { < T }$ and other parameters are kept fixed. By progressively performing the above process, we can finally optimize and learn our models.
|
| 85 |
+
|
| 86 |
+
|
| 87 |
+
Figure 2: Progressive training scheme. $z _ { t }$ (docid at timestep $t$ ) is optimized at the $t$ -th training step, while $z _ { < t }$ (docids before timestep $t$ ) are kept fixed.
|
| 88 |
+
|
| 89 |
+
At each optimization step, say the $T$ -step, we devise the learning objective for document tokenization consisting of three loss functions detailed below.
|
| 90 |
+
|
| 91 |
+
Reconstruction loss. We utilize the reconstruction model $R ( d \mid z )$ as an auxiliary model to learn to optimize the docid generation, whose main goal is capturing as much semantics in the docid as possible. Therefore, we define a reconstruction loss function of step $T$ as follows:
|
| 92 |
+
|
| 93 |
+
$$
|
| 94 |
+
\begin{array} { r l r } { \mathcal { L } _ { \mathrm { R e c } } = - \log R ( d \mid \hat { \mathbf { z } } _ { \le T } ) , } & { \mathrm { w h e r e ~ } \hat { \mathbf { z } } _ { \le T } = \left\{ \mathrm { s g } ( \mathbf { z } _ { 1 } ) , \dots , \mathrm { s g } ( \mathbf { z } _ { T - 1 } ) , \mathbf { z } _ { T } \right\} } & { \in \mathbb { R } ^ { T \times D } } \\ { \forall t \in [ T ] : \mathbf { z } _ { t } = \mathbf { e } _ { t , j ^ { * } } \in \mathbb { R } ^ { D } , } & { \mathrm { w h e r e ~ } j ^ { * } = \arg \operatorname* { m a x } Q ( z _ { t } = j \mid z _ { < t } , d ) , } \end{array}
|
| 95 |
+
$$
|
| 96 |
+
|
| 97 |
+
where $\scriptstyle { \hat { \mathbf { z } } } < { T }$ is the first $T$ representations of the $z$ , and only the variable $\mathbf { z } _ { T }$ is optimized in step $T$ $Q ( z _ { t } = \bar { j } \mid z _ { < t } , d )$ is defined in Eq. 2. And the document tokenization model $Q$ can therefore be optimized when minimizing $\mathcal { L } _ { \mathrm { R e c } }$ .
|
| 98 |
+
|
| 99 |
+
Of note, since the computation involves a non-differentiable operation $( \mathrm { a r g } \operatorname* { m a x } ( \cdot ) )$ , we apply straight-through gradient estimation to back-propagate the gradient from reconstruction loss to ${ \bf d } _ { T }$ following [39, 44], which copies the gradient of $\mathbf { z } _ { T }$ directly to ${ \bf d } _ { T }$ . Specifically, the gradients to document representation ${ \bf d } _ { T }$ are defined as $\begin{array} { r } { \frac { \partial \mathcal { L } _ { \mathrm { R e c } } } { \partial { \bf d } _ { T } } : = \frac { \partial \mathcal { L } _ { \mathrm { R e c } } } { \partial { \bf z } _ { T } } } \end{array}$ And the gradients to the codebook embedding $\mathbf { e } _ { T , j }$ are defined as $\begin{array} { r } { \frac { \partial \mathcal { L } _ { \mathrm { R e c } } } { \partial \mathbf { e } _ { T , j } } : = 1 _ { z _ { T } = j } \frac { \partial \dot { \mathcal { L } } _ { \mathrm { R e c } } } { \partial \mathbf { z } _ { T } } } \end{array}$
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Commitment loss. In addition, to make sure the predicted docid commits to an embedding and to avoid models forgetting previous docid $z _ { < t }$ , we add a commitment loss as follows:
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$$
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\mathcal { L } _ { \mathrm { C o m } } = - \sum _ { t = 1 } ^ { T } \log Q ( z _ { t } \mid z _ { < t } , d ) .
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$$
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Retrieval loss. For the generative retrieval model $P$ , we jointly learn it together with the document tokenization model $Q$ , where $P$ learns to generate the docids of relevant documents $d$ given a query $q$ Specifically, suppose $( q , d )$ are a query and relevant document pair; we define the learning objective of retrieval model $P$ as:
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$$
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\mathcal { L } _ { \mathrm { R e t } } = - \log \frac { \exp ( \mathbf { q } _ { T } \cdot \mathbf { d } _ { T } ) } { \sum _ { d ^ { * } \sim B } \exp ( \mathbf { q } _ { T } \cdot \mathbf { d } ^ { * } _ { T } ) } - \sum _ { t = 1 } ^ { T } \log P ( z _ { t } \mid z _ { < t } , q ) ,
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$$
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where the first term is a ranking-oriented loss enhancing the model using $( q , d )$ pair; $d ^ { * }$ is an in-batch document sampled from the same training mini-batch $B$ ; $\mathbf { q } _ { T }$ and ${ \bf d } _ { T }$ denote the representation of $q$ and $d$ at timestep $T$ . The second term is the cross-entropy loss for generating docid $z$ based on $q$ .
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The final loss we use at step- $\mathcal { T }$ is the sum of reconstruction loss, commitment loss, and retrieval loss:
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$$
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\begin{array} { r } { \mathcal { L } = \mathcal { L } _ { \mathrm { R e c } } + \mathcal { L } _ { \mathrm { C o m } } + \mathcal { L } _ { \mathrm { R e t } } . } \end{array}
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$$
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# 3.2.2 Diverse clustering techniques
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To ensure diversity of generated docids, we adopt two diverse clustering techniques–codebook initialization and docid re-assignment at each progressive training step, where codebook initialization mainly aims to increase the balance of semantic space segmentation, and the docid re-assignment mainly aims to increase the balance of docid assignments.
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Codebook initialization. In order to initialize the codebook for our model, we first warm-up the model by passing the continuous representation ${ \bf d } _ { T }$ to the reconstruction model instead of the docid representation $\mathbf { z } _ { T }$ as defined in Eq. 3. During this warm-up phase, we optimize the model using the reconstruction loss $\mathcal { L } _ { \mathrm { R e c } }$ and commitment loss ${ \mathcal { L } } _ { \mathrm { C o m } }$ . Next, we collect the continuous representations ${ \bf d } _ { T }$ of all documents in $\mathcal { D }$ , and cluster them into $K$ groups. The centroids of these clusters are then used as the initialized codebook $\mathbf { E } _ { T }$ . To balance the initialized docid distribution, we utilize a diverse constrained clustering algorithm, Constrained $K \cdot$ -Means, which first normalizes the embeddings of each prefix group, and modifies the cluster assignment step (E in EM) by formulating it as a minimum cost flow (MCF) linear network optimization problem [2].
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Docid re-assignment. In order to assign docids to a batch of documents, we modify the dot-product look-up results in Eq. 2 by ensuring that the docid for different documents in the batch are distinct following the method described in [6, 44]. Specifically, let $\mathbf { D } _ { t } = \{ \mathbf { d } _ { t } ^ { ( 1 ) } , \dots , \mathbf { d } _ { t } ^ { ( B ) } \} \in \mathbb { R } ^ { B \times D }$ denote the continuous representation of a batch of documents with batch size of $B$ . The dot-product results are represented by $\mathbf { H } = \mathbf { D } _ { t } \cdot \mathbf { E } _ { t } ^ { \top } \in \mathbb { R } ^ { B \times K }$ . To obtain distinct docids, we calculate an alternative $\begin{array} { r } { \mathbf { H } ^ { * } = \mathrm { D i a g } ( \mathbf { u } ) \exp ( \frac { \mathbf { H } } { \epsilon } ) \mathrm { D i a g } ( \mathbf { v } ) } \end{array}$ , where $\mathbf { u }$ and $\mathbf { v }$ are re-normalization vectors in $\mathbb { R } ^ { K }$ and $\mathbb { R } ^ { B }$ , respectively. The re-normalization vectors are computed via the iterative Sinkhorn-Knopp algorithm [8]. Finally, $\mathbf { H } ^ { * }$ is used instead of $\mathbf { H }$ in the Softmax and arg max (Eq. 2) operations to obtain the docid $z _ { t }$ .
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# 4 Experimental Setup
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# 4.1 Datasets and evalutaion metrics
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We conduct experiments on three well-known document retrieval datasets: NQ320K [15], MS MARCO [4], and BEIR [38]. We divide the test set of NQ320K into seen test and unseen test, based on whether the target documents of the query have annotated queries in the training data, to test the generalization ability of the model on unlabeled documents. More details about data pre-processing and data statistics are listed in the Appendix A.
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On NQ320K, we use Recall $\ @ \left\{ 1 , 1 0 , 1 0 0 \right\}$ and Mean Reciprocal Rank (MRR) $@ 1 0 0$ as evaluation metrics, following [41]. On MS MARCO, we use Recall $ @ \{ 1 , 1 0 , 1 0 0 \}$ and MRR $@ 1 0$ as evaluation metrics, following [46]. On BEIR, we use nDCG $@ 1 0$ as the main metrics and calculate the average nDCG $@ 1 0$ values across multiple downstream sub-datasets as overall metrics.
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# 4.2 Baselines
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We consider three types of baselines: sparse retrieval methods, dense retrieval methods, and generative retrieval methods. The sparse retrieval baselines are: BM25 [32] and DocT5Query [24]. The dense retrieval baselines are: DPR [13], ANCE [42], Sentence-T5 [22], GTR [23], and Contriever [11]. The generative retrieval baselines are: GENRE [5], DSI [37], SEAL [1], CGR-Contra [17], DSIQG [47], NCI [41], and Ultron [46]. The following three baselines use the same pre-trained LM T5 as GENRET: (i) Sentence-T5 outputs continuous vectors, (ii) GENRE outputs document titles, and (iii) DSI-QG outputs clustering IDs, while GENRET outputs docids learned using the proposed tokenization method. See Appendix B for more details on the other baselines.
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# 4.3 Implementation details
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Hyper-parameters. In our experiments, we utilize the T5-Base model [27] as the base Transformer and initialize a new codebook embedding $\mathbf { E } _ { t }$ for each time step. The parameters of both the encoderdecoder and codebook are shared between the tokenization model and the retrieval model. We set the number of clusters to be $K = 5 1 2$ for all datasets, with the length of the docid $M$ being dependent on the number of documents present. For datasets containing a larger number of candidate documents, a larger value of $M$ is set to ensure that all documents are assigned unique document ids. In the docid re-assignment, the hyper-parameter $\epsilon$ is set to 1.0, and the Sinkhorn-Knopp algorithm is executed for 100 iterations.
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Indexing with query generation. Following previous work [47, 41, 40], we use query generation models to generate synthetic (query, document) pairs for data augmentation. Specifically, we use the pre-trained query generation model from DocT5Query [24] to augment the NQ and MS MARCO datasets. In query generation, we use nucleus sampling with parameters $p = 0 . 8 , t = 0 . 8$ and generate five queries for each document in the collection. For the BEIR datasets, we use the queries generated by GPL [40]. GPL uses a DocT5Query [24] generator trained on MS MARCO to generate about 250K queries for each BEIR dataset. Note that the query generator used for BEIR is purely trained on MS MARCO (without using any training data of BEIR) and thus conforms to the zero-shot setting of BEIR [38, 40].
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Training and inference. The proposed models and the reproduced baselines are implemented with PyTorch 1.7.1 and HuggingFace transformers 4.22.2. We optimize the model using AdamW and set the learning rate to $5 e - 4$ . The batch size is 256, and the model is optimized for up to $5 0 0 \mathrm { k }$ steps for each timestep. During training, we pre-gather documents which share the same docid prefix into a batch. Therefore, the reassignment strategy is applied to a batch, where we aim to have documents with as diverse IDs as possible. We add a factor of 0.1 to the reconstruction losses to balance the scale. In progressive training, we first warm up the model for 5K steps and then initialize the codebook using the clustering centroids as mentioned in Section 3.2.1. We use constrained clustering4 to obtain diverse clustering results. During inference, we use beam search with constrained decoding [5] and a beam size of 100.
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Table 1: Results on Natural Questions (NQ320K). The results of the methods marked with † are from our own re-implementation, others are from their official implementation. \* and $^ { * * }$ indicate significant improvements over previous-best generative retrieval baselines with $\mathsf { p }$ -value $< 0 . 0 5$ and p-value $< 0 . 0 1$ , respectively. $\natural$ and $\sharp$ indicate significant improvements over previous-best dense retrieval baselines with $\boldsymbol { \mathrm { p } }$ -value $< 0 . 0 5$ and $\mathsf { p }$ -value $< 0 . 0 1$ , respectively. The best results for each metric are indicated in boldface.
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<table><tr><td></td><td colspan="4">Full test</td><td colspan="4">Seen test</td><td colspan="4">Unseen test</td></tr><tr><td>Method</td><td></td><td></td><td>R@1 R@10 R@100</td><td></td><td></td><td>MRR R@1 R@10 R@100 MRR R@1 R@10 R@100 MRR</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Sparse retrieval</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>BM25[32]</td><td>29.7</td><td>60.3</td><td>82.1</td><td>40.2</td><td>29.1</td><td>59.8</td><td>82.4</td><td>39.5</td><td>32.3</td><td>61.9</td><td>81.2</td><td>42.7</td></tr><tr><td>DocT5Query [24]</td><td>38.0</td><td>69.3</td><td>86.1</td><td>48.9</td><td>35.1</td><td>68.3</td><td>86.4</td><td>46.7</td><td>48.5</td><td>72.9</td><td>85.0</td><td>57.0</td></tr><tr><td>Dense retrieval</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>DPR[13]</td><td>50.2</td><td>77.7</td><td>90.9</td><td>59.9</td><td>50.2</td><td>78.7</td><td>91.6</td><td>60.2</td><td>50.0</td><td>74.2</td><td>88.7</td><td>58.8</td></tr><tr><td>ANCE [42]</td><td>50.2</td><td>78.5</td><td>91.4</td><td>60.2</td><td>49.7</td><td>79.2</td><td>92.3</td><td>60.1</td><td>52.0</td><td>75.9</td><td>88.0</td><td>60.5</td></tr><tr><td>Sentence-T5† [22]</td><td>53.6</td><td>83.0</td><td>93.8</td><td>64.1</td><td>53.4</td><td>83.9</td><td>94.7</td><td>63.8</td><td>56.5</td><td>79.5</td><td>90.7</td><td>64.9</td></tr><tr><td>GTR-Base [23]</td><td>56.0</td><td>84.4</td><td>93.7</td><td>66.2</td><td>54.4</td><td>84.7</td><td>94.2</td><td>65.3</td><td>61.9</td><td>83.2</td><td>92.1</td><td>69.6</td></tr><tr><td>Generative retrieval</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>GENRE† [5]</td><td>55.2</td><td>67.3</td><td>75.4</td><td>59.9</td><td>69.5</td><td>83.7</td><td>90.4</td><td>75.0</td><td>6.0</td><td>10.4</td><td>23.4</td><td>7.8</td></tr><tr><td>DSI+ [37]</td><td>55.2</td><td>67.4</td><td>78.0</td><td>59.6</td><td>69.7</td><td>83.6</td><td>90.5</td><td>74.7</td><td>1.3</td><td>7.2</td><td>31.5</td><td>3.5</td></tr><tr><td>SEAL [1]</td><td>59.9</td><td>81.2</td><td>90.9</td><td>67.7</td><td>1</td><td>-</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CGR-Contra [17]</td><td>63.4</td><td>81.1</td><td>-</td><td>-</td><td>1</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DSI-QG+ [47]</td><td>63.1</td><td>80.7</td><td>88.0</td><td>69.5</td><td>68.0</td><td>85.0</td><td>91.4</td><td>74.3</td><td>45.9</td><td>65.8</td><td>76.3</td><td>52.8</td></tr><tr><td>NCI [41]</td><td>66.4</td><td>85.7</td><td>92.4</td><td>73.6</td><td>69.8</td><td>88.5</td><td>94.6</td><td>76.8</td><td>54.5</td><td>75.9</td><td>84.8</td><td>62.4</td></tr><tr><td>Ours</td><td>68.1*</td><td>88.8*日</td><td>95.2*</td><td>75.9*70.2#</td><td></td><td>90.3#</td><td>96.0</td><td>77.7#</td><td>62.5**</td><td>83.6**</td><td>92.5**</td><td>70.4*</td></tr></table>
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# 5 Experimental results
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# 5.1 Main results
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Results on NQ320K. In Table 1, we list the results on NQ320K. GENRET outperforms both the strong pre-trained dense retrieval model, GTR, and the previous best generative retrieval method, NCI, thereby establishing a new state-of-the-art on the NQ320K dataset. Furthermore, our results reveal that existing generative retrieval methods perform well on the seen test but lag behind dense retrieval methods on the unseen test. For example, NCI obtains an MRR $@ 1 0 0$ of 76.8 on the seen test, which is higher than the MRR $@ 1 0 0$ of 65.3 obtained by GTR-Base. However, on unseen test data, NCI performs worse than GTR-Base. In contrast, GENRET performs well on both seen and unseen test data. This result highlights the ability of GENRET to combine the advantages of both dense and generative retrieval by learning discrete docids with semantics through end-to-end optimization.
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Results on MS MARCO. Table 2 presents the results on the MS MARCO dataset. GENRET outperforms generative retrieval methods such as Ultron and dense retrieval baselines such as ANCE and Sentence-T5. Furthermore, previous generative retrieval methods (e.g., GENRE, Ultron) utilizing metadata such as the title and URL, while exhibiting decent performance on the NQ320K dataset, underperform in comparison to dense retrieval and sparse retrieval methods on the MS MARCO dataset. This may be because the NQ320K dataset retrieves Wikipedia documents, where metadata like the title effectively capture the semantics of the document. In the case of the MS MARCO dataset, which is a web search dataset, the metadata often does not adequately characterize the documents, resulting in a decline in performance of the generative retrieval model. In contrast, GENRET learns to generate semantic docids that effectively enhance the generative retrieval model.
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Results on BEIR. Table 3 lists the results of the baselines and GENRET on six datasets of BEIR. These datasets represent a diverse range of information retrieval scenarios. On average, GENRET outperforms strong baselines including BM25 and ST5 GPL, and achieves competitive results compared to previous-best sparse and dense retrieval methods. Additionally, GENRET demonstrates a significant improvement over the previous generative retrieval model GENRE that utilizes titles as docids. Furthermore, GENRE performs poorly on some datasets, such as BEIR-Covid and BEIRSciDocs. This may be because the titles of the documents in these datasets do not adequately capture their semantic content.
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Table 2: Results on MS MARCO. The results of the methods marked with † are from our own re-implementation, other results are cited from the original paper or implemented using official code. The best results are indicated in boldface.
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<table><tr><td>Method R@1 R@10 R@100 MRR</td></tr><tr><td>Sparse retrieval BM25 [32] 39.1 69.1 86.2 48.6</td></tr><tr><td>DocT5Query [24] 46.7 76.5 90.4 56.2</td></tr><tr><td>Dense retrieval ANCE [42] 45.6 75.7 89.6 55.6</td></tr><tr><td>Sentence-T5t [22] 41.8 75.4 91.2 52.8</td></tr><tr><td>Generative retrieval</td></tr><tr><td>GENRE+ [5] 35.6 57.6 79.1 42.3 40.0</td></tr><tr><td>Ultron-URL [46] 29.6 67.8 -</td></tr><tr><td>Ultron-PQ [46] 31.6 73.1 45.4 -</td></tr><tr><td>Ultron-Atomic [46] 32.8 74.1 46.9 =</td></tr><tr><td>Ours 47.9 79.8 91.6 58.1</td></tr></table>
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Table 3: nDCG $@ 1 0$ results on BEIR. The results of the methods marked with † are from our own re-implementation, other results are cited from the original paper or implemented using official code. ST5 GPL denotes Sentence-T5 trained on GPL datasets [40].
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<table><tr><td>Method</td><td>Arg Covid NFC SciF. SciD.FiQA Avg.</td></tr><tr><td>Sparse retrieval BM25 [32]</td><td>29.1 58.9 33.5 67.4 14.8 23.6 37.8</td></tr><tr><td>DocT5Query [24]34.9 71.3 32.8 67.5 16.2 29.1 41.9 Dense retrieval</td><td></td></tr><tr><td>ST5 GPL+ [22] 32.1 74.4 30.1 58.6 12.7 26.0 39.0 Contriever [11] 40.0 68.8 33.5 61.4 16.3 30.7 41.8</td><td></td></tr><tr><td>Generative retrieval</td><td></td></tr><tr><td>GENRE† [47]</td><td></td></tr><tr><td></td><td>42.514.7 20.0 42.36.8 11.6 30.0</td></tr><tr><td>Ours</td><td>34.3 71.8 31.6 63.9 14.9 30.2 41.1</td></tr></table>
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# 5.2 Analytical experiments
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We further conduct analytical experiments to study the effectiveness of the proposed method.
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In Figure 3 (left), we plot the frequencies of docids at the first timestep of various learning methods. We label each method using a box with a docid and a diversity metric $d$ , which is calculated by: $\begin{array} { r } { d = 1 - \frac { 1 } { 2 n } \sum _ { j = 1 } ^ { K } | n _ { j } - n _ { u } | } \end{array}$ , where $\left| \cdot \right|$ represents the absolute value, $n$ denotes the total number of documents, $n _ { j }$ denotes the number of documents that have a docid $= j$ , and $\begin{array} { r } { n _ { u } = \frac { n } { K } } \end{array}$ is the expected number of documents per docid under the uniform distribution.
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The results demonstrate the superiority of GENRET (represented by the yellow line) in terms of distribution uniformity. It uses all the potential docid $k = 5 1 2$ and achieves the highest diversity metric with a value of $d = 0 . 9 0$ . The method without docid reassignment also yields a relatively balanced distribution, with a diversity metric of $d = 0 . 7 7$ . However, the distribution of the method without diverse codebook initialization is highly uneven, which could be due to the fact that most of the randomly initialized codebook embeddings are not selected by the model during the initial training phase, leading to a lack of update and further selection in subsequent training. Additionally, the models without diverse clustering tend to converge to a trivial solution where all documents are assigned the same docid.
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In Figure 3 (right), the results of two ablated variants are presented. First, GENRET w/o learning is a generative model that has been trained directly using the final output docid from GENRET, without utilizing the proposed learning scheme. Its retrieval performance is comparable to that of GENRET on seen test data; however, it is significantly lower on unseen test data. This variant demonstrates that the generative retrieval model jointly trained with auto-encoding objectives can represent documents more sensibly based on semantics. This could enhance performance on the less-optimized documents. Contrarily, the parameters obtained via cross-entropy loss on docid generation tasks are less effective in conveying the semantic information of documents. Secondly, GENRET w/ T5-Small uses a small model, and its performance is inferior to that of GENRET using T5-Base. However, the gap between the performance on seen and unseen test data is smaller, which could be attributed to the limited fitting capacity of the small model.
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In Appendix C, we evaluate the proposed model’s capacity to retrieve new documents and find that it performs well in adapting to new documents compared to existing document tokenization approaches. Additionally, in Appendix D, we analyze the efficiency of various retrieval models in comparison to the various baselines in terms of memory usage, offline, and online latency, and show the advantages of the proposed model.
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Figure 3: Left: Docid distribution on NQ320K. The id are sorted by the assigned frequency. Right: Ablation study on NQ320K.
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# 5.3 Qualitative analysis
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Figure 4 (left) illustrates the document content (title) and the corresponding docid generated by GENRET on the NQ320K dataset. We observe that documents with more similar docids tend to have more relevant content. For example, documents with docids starting with 338-173 are related to Email, such as Email marketing, Mail merge, and Email address, while documents with docids starting with 338 relate to information exchange methods (e.g., Business letter, US Postal Service, and Postage stamps), representing a more generalized semantics than Email alone.
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Figure 4 (right) illustrates a word cloud of documents grouped by docid prefixes. It is evident that documents within the same group are semantically related. For example, major words for documents with the docid prefix 338 are mail and stamp. When a second-level docid 173 is added, the corresponding documents become more specifically related to email. With the addition of a third-level docid 1, the document group becomes specifically associated with email marketing (docid: 338-173-1). Similar patterns can be observed in the other three cases. The case study shows that there is a hierarchical semantic structure within the learned docids.
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Figure 5 in Appendix E visualizes the codebook embedding and document embedding. We see that the codebook embedding appears to distribute uniformly within the document representation space, producing meaningful clusters when documents are categorized by docids. We also find that the uniformity of the embedding distribution decreases with increasing docid-length. This may be due to the fact that uniform segmentation is more challenging as the semantic granularity becomes finer.
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Figure 4: Left: Document titles along with their corresponding docids. It is observed that documents with similar docids tend to have more relevant content. Right: Word cloud representing documents grouped by docid prefixes. This illustrates that different positions of the docid correspond to different levels of information, and the semantics within each cluster are closely related.
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# 6 Related work
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Sparse and dense retrieval. Traditional sparse retrieval calculates the document score using term matching metrics such as TF-IDF [31], query likelihood [16], or BM25 [32]. Sparse retrieval is widely used in practice due to its efficiency, but often suffers from the lexical mismatches [18]. Dense retrieval (DR) addresses this by presenting queries and documents in dense vectors and calculating their similarities with the inner product or cosine similarity [13]. Various techniques have been proposed to improve DR models, such as hard negative mining [42, 26], late interaction [14, 34], knowledge distillation [10, 19], and pre-training [30, 23, 11]. Despite their success, DR approaches have several limitations [5, 21]: (i) DR models employ an index-retrieval pipeline with a fixed search procedure (MIPS), making it difficult to optimize the model end-to-end [37, 41]. (ii) Training DR models relies on contrastive learning [13] to distinguish positives from negatives, which is inconsistent with large LMs training objectives [3] and fails to fully utilize the capabilities of pretrained LMs [1, 35].
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Generative retrieval. Generative retrieval is gaining attention. It retrieves documents by generating their docid using a generative model like T5. Generative retrieval presents an end-to-end solution for document retrieval tasks [37, 21] and allows for better exploitation of the capabilities of large generative LMs [1]. Cao et al. [5] first propose an autoregressive entity retrieval model to retrieve documents by generating titles. Tay et al. [37] propose a differentiable search index (DSI) and represent the document as atomic id, naive string, or semantic string. Bevilacqua et al. [1] suggest using arbitrary spans of a document as docids. Additionally, multiple-stage pre-training [7, 46], query generation [41, 47, 46], contextualized embedding [17], and continual learning [20], have been explored in recent studies. Recently, Tang et al. [36] introduce a query-based docid and the rehearsal-based document indexing to improve DSI. However, existing generative retrieval models have a limitation in that they rely on fixed document tokenization to produce docids, which often fails to capture the semantic information of a document [37]. It is an open question how one should define the docids. To further capture document semantics in the docid, we propose document tokenization learning methods. The semantic docid is generated by the proposed discrete auto-encoding learning scheme in an end-to-end manner. Concurrently, Rajput et al. [28] propose a RQ-VAE module to produce semantic docids for generative recommender systems. As a comparison, our proposed method jointly models the tokenization and retrieval tasks with shared parameters to better align the model’s representation of the two tasks and are optimized in an end-to-end manner.
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Discrete representation learning. Learning discrete representations using neural networks is an important research area in machine learning. For images, Rolfe [33] proposes the discrete variational autoencoder, and VQ-VAE [39] learns quantized representations via vector quantization. DallE [29] uses an autoregressive model to generate discrete image representation for text-to-image generation. Recently, representation learning has attracted considerable attention in NLP tasks, for tasks such as machine translation [48], dialogue generation [45], and text classification [12, 43]. For document retrieval, RepCONC [44] uses a discrete representation learning method based on constrained clustering for vector compression. We propose a document tokenization learning method for generative retrieval, which captures the autoregressive nature of docids by progressive training and enhances the diversity of docids by diverse clustering techniques.
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# 7 Conclusions
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This paper has proposed a document tokenization learning method for generative retrieval, named GENRET. The proposed method learns to tokenize documents into short discrete representations (i.e., docids) via a discrete auto-encoding approach, which ensures the semantics of the generated docids. A progressive training method and two diverse clustering techniques have been proposed to enhance the model’s training. Empirical results on various document retrieval datasets have demonstrated the effectiveness of the proposed method. Especially, GENRET achieves outperformance on unseen documents and can be well generalized to multiple retrieval tasks.
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The limitations of this work include experiments only on moderately sized datasets like NQ320K. The employed data are in sufficient quantity to validate the effectiveness of the proposed method, but application to larger-scale data may require more model parameters and computational resources. Another aspect for improvement is the generalization of the model to unoptimized document collections in different types or domains, as compared to continuous embedding approaches. We recognize this as an important research question for generative retrieval but believe that addressing this is beyond the scope of this paper. In future work, we would like to extend the approach to large document collections. We also plan to explore generative pre-training for document tokenization using large-scale language models. Additionally, we intend to investigate the dynamic adaptation of docid prefixes for progressive training.
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# Acknowledgements
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This work was supported by the Natural Science Foundation of China (62272274, 61972234, 62072279, 62102234, 62202271), the Natural Science Foundation of Shandong Province (ZR2021QF129, ZR2022QF004), the Key Scientific and Technological Innovation Program of Shandong Province (2019JZZY010129), the Fundamental Research Funds of Shandong University, the China Scholarship Council under grant nr. 202206220085, the Hybrid Intelligence Center, a 10-year program funded by the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research, https://hybrid-intelligence-centre.nl, and project LESSEN with project number NWA.1389.20.183 of the research program NWA ORC 2020/21, which is (partly) financed by the Dutch Research Council (NWO). All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors.
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# A Datasets details
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Table 4: Statistics of datasets used in our experiments. The three values split by / on # Test queries denote the number of queries in the full, seen subset, and unseen subset, respectively. In BEIR, all queries in the test set are unseen.
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<table><tr><td>Dataset</td><td>#Docs</td><td># Test queries</td><td># Train pairs</td></tr><tr><td>NQ320K MS MARCO</td><td>323,569 5,187 /</td><td>109,739 7,830 / 6,075 /1,755 807 /4,380</td><td>307,373 366,235</td></tr><tr><td></td><td></td><td>1,406</td><td></td></tr><tr><td>BEIR-Arg BEIR-Covid</td><td>8,674</td><td>50</td><td></td></tr><tr><td>BEIR-NFC</td><td>171,332 3.633</td><td>323</td><td></td></tr><tr><td>BEIR-SciFact</td><td>5,183</td><td>300</td><td></td></tr><tr><td>BEIR-SciDocs</td><td>25,657</td><td>1,000</td><td></td></tr><tr><td>BEIR-FiQA</td><td></td><td></td><td></td></tr><tr><td></td><td>57,638</td><td>648</td><td></td></tr></table>
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We conduct experiments on three document5 retrieval datasets: NQ320K, MS MARCO, and BEIR.
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NQ320K. NQ320K is a popular dataset for evaluating generative retrieval models [37, 41]. It is based on the Natural Questions (NQ) dataset proposed by Google [15]. NQ320K consists of $3 2 0 \mathrm { k }$ query-document pairs, where the documents are gathered from Wikipedia pages, and the queries are natural language questions. We follow the evaluation setup in NCI [41] and further split the test set into two subsets: seen test, in which the annotated target documents of the queries are included in the training set; and unseen test, in which no labeled document is included in the training set.
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Note that the NQ320K dataset we utilized has been pre-processed based on the NCI [41] and includes approximately 100k documents. This differs from the DSI approach [37], which processed a version of the NQ320K dataset containing about $2 0 0 \mathrm { k }$ documents. The distinction lies in the method used to remove duplicate documents. In our case, we eliminated duplicates by comparing document titles, whereas DSI employed URLs. For example, pages https://en.wikipedia.org//w/index. php?title $=$ Statue_of_Liberty&oldid $\underset { . } { = }$ 804877528 and https://en.wikipedia.org/ /w/index.php?title $=$ Statue_of_Liberty&oldid $\underset { . } { = }$ 834310497 are two versions of entity “Statue of Liberty”. DSI NQ320K treats them as separate documents, whereas our implementation considers them as a single document. We have found that the content of different versions of the same entity’s pages is usually almost identical, with only minor variations, often occurring in later parts of the document. Consequently, we believe that distinguishing between different versions of an entity is beyond the model’s capabilities, and that using different versions of the same entity as negative examples in training may hurt model performance.
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MS MARCO. MS MARCO is a collection of queries and web pages from Bing search. To create the document collections, akin to NQ320k and following [46], we sample a subset of original documents by retaining the top-1 document for each query. We evaluate the models on the queries of the MS MARCO dev set and retrieval on the sampled document subset. We did not split the dev set into seen and unseen because $84 \%$ of the queries in the MS MARCO dev set are unseen.
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BEIR. BEIR is a collection of datasets for heterogeneous retrieval tasks. In this paper, we evaluate the models on 6 BEIR datasets, which include distinct retrieval tasks and document collections from NQ and MS MARCO: (i) BEIR-Arg retrieves a counterargument to an argument; (ii) BEIRCovid retrieves scientific articles about the COVID-19 pandemic; (iii) BEIR-NFC retrieves medical documents from PubMed; (iv) BEIR-SciFact retrieves scientific papers for fact-checking; (v) BEIRSciDocs retrieves citations for scientific papers; (vi) BEIR-FiQA retrieves financial documents. All the queries in BEIR test set are unseen [38].
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We summarize the statistics of above datasets in Table 4.
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# B Baselines
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The sparse retrieval baselines are as follows:
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• BM25, uses the tf-idf feature to measure term weights; we use the implementation from http://pyserini.io/.
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• DocT5Query, expands a document with possible queries predicted by a finetuned T5 with this document as the input.
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The dense retrieval baselines are as follows:
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• DPR [13], a dual-encoder model using the representation of the [CLS] token of BERT.
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• ANCE [42], an asynchronously updated ANN indexer is utilized to mine hard negatives for training a RoBERTa-based dual-encoder model.
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• Sentence-T5 [22], a dual-encoder model that uses T5 to produce continuous sentence embeddings. We reproduce Sentence-T5 (ST5 for short) on our datasets, the model is based on T5-Base EncDec model and is trained with in-batch negatives.
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• GTR [23], a state-of-the-art dense retrieval model that pre-trains sentence-T5 on billions of paired data using contrastive learning.
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• Contriever [11], a dual-encoder model pre-trained using unsupervised contrastive learning with independent cropping and inverse cloze task.
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And the generative retrieval baselines are as follows:
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GENRE [5], an autoregressive retrieval model that generates the document’s title. The original GENRE is trained on the KILT dataset [25] using BART, and we reproduce GENRE on our datasets using T5 for a fair comparison. For datasets without title, we use the first 32 tokens of the document as pseudo-title.
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DSI [37], which represents documents using hierarchical K-means clustering results, and indexes documents using the first 32 tokens as pseudo-queries. As the original code is not open source, we reproduce DSI using T5-base and the docids of NCI [41].
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• SEAL [1] uses arbitrary n-grams in documents as docids, and retrieves documents under the constraint of a pre-built FM-indexer. We refer to the results reported by Wang et al. [41].
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• CGR-Contra [17], a title generation model with a contextualized vocabulary embedding and a contrastive learning loss.
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DSI-QG [47], uses a query generation model to augment the document collection. We reproduce the DSI-QG results using T5 and our dataset.
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NCI [41], uses a prefix-aware weight-adaptive decoder and various query generation strategies, including DocAsQuery and DocT5Query. In particular, NCI augments training data by generating 15 queries for each document.
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• Ultron [46], uses a three-stage training pipeline and represents the document as three types of identifiers, including URL, PQ, and Atomic.
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# C Performance on retrieving new documents
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In this experiment, we investigate the impact of various document tokenization techniques on the ability of generative retrieval models to retrieve new documents. The generative models with different tokenization methods are trained on NQ320K data, excluding unseen documents, and are evaluated on NQ320K Unseen test set and BEIR-{Arg, NFC, SciDocs} datasets. For the baseline methods, which use rule-based document tokenization methods, the docids are generated for the target document collection using their respective tokenization techniques. In contrast, our proposed method uses a tokenization model to tokenize the documents in the target collection, producing the docids. However, our method may result in duplicate docids. In such cases, all corresponding documents are retrieved and shuffled in an arbitrary order. The results of this evaluation are summarized in Table 5.
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Table 5: Zero-shot evaluation on retrieving new documents with different document tokenization methods. The second column indicates the type of docid, where BERT-HC denotes BERTHierarchical-Clustering [37], Prefix-HC denotes Prefix-aware BERT-Hierarchical-Clustering [41], and dAE denotes discrete auto-encoding.
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<table><tr><td></td><td></td><td>NQ (R@1)</td><td colspan="3">BEIR (nDCG@ 10)</td></tr><tr><td>Method</td><td>Docid</td><td>Unseen</td><td>Arg</td><td>NFC</td><td>SciDocs</td></tr><tr><td>DSI-Naive† [37]</td><td>Naive String</td><td>0.0</td><td>0.1</td><td>1.0</td><td>0.1</td></tr><tr><td>DSI-Atomic† [37]</td><td>Atomic</td><td>0.0</td><td>0.2</td><td>0.8</td><td>0.1</td></tr><tr><td>GENRE† [5]</td><td>Title</td><td>6.0</td><td>0.0</td><td>2.4</td><td>0.6</td></tr><tr><td>DSIt [37]</td><td>BERT-HC</td><td>1.3</td><td>1.8</td><td>11.1</td><td>5.9</td></tr><tr><td>NCI [41]</td><td>Prefix-HC</td><td>15.5</td><td>0.9</td><td>4.3</td><td>1.2</td></tr><tr><td>Ours</td><td>dAE</td><td>34.2</td><td>12.1</td><td>12.1</td><td>12.3</td></tr></table>
|
| 333 |
+
|
| 334 |
+
Document tokenization methods that do not consider the semantic information of the documents, such as Naive String and Atomic, are ineffective in retrieving new documents without model updating. Methods that consider the semantic information of the documents, such as those based on title or BERT clustering, show some improvement. Our proposed document tokenization method significantly improves over these existing rule-based document tokenization methods. For instance, when the model trained on NQ – a factoid QA data based on Wikipedia documents – is applied to a distinct retrieval task on a different document collection, BEIR-SciDocs, a citation retrieval task on a collection of scientific articles, our proposed document tokenization model still showed promising results with an nDCG $@ 1 0$ of 12.3, which is comparable to those models trained on the target document collection. This suggests that our proposed method effectively encodes the semantic information of documents in the docid and leads to a better fit between the docid and the generative retrieval model.
|
| 335 |
+
|
| 336 |
+
Table 6: Efficiency analysis.
|
| 337 |
+
|
| 338 |
+
<table><tr><td>Method</td><td>Memory</td><td>Time ( (Offline)</td><td>Top-K</td><td>Time (Online)</td></tr><tr><td>ANCE</td><td>1160MB</td><td>145min</td><td>100</td><td>0.69s</td></tr><tr><td>GTR-Base</td><td>1430MB</td><td>140min</td><td>100</td><td>1.97s</td></tr><tr><td rowspan="2">GENRE</td><td rowspan="2">851MB</td><td rowspan="2">0min</td><td>100</td><td>1.41s</td></tr><tr><td>10</td><td>0.69s</td></tr><tr><td rowspan="2">DSI</td><td rowspan="2">851MB</td><td rowspan="2">310min</td><td>100</td><td>0.32s</td></tr><tr><td>10</td><td>0.21s</td></tr><tr><td rowspan="2">Ours</td><td rowspan="2">860MB</td><td rowspan="2">220min</td><td>100</td><td>0.16s</td></tr><tr><td>10</td><td>0.10s</td></tr></table>
|
| 339 |
+
|
| 340 |
+
# D Efficiency analysis
|
| 341 |
+
|
| 342 |
+
In Table 6, we compare GENRET with baseline models on MS MARCO (323,569 documents) in terms of memory footprint, offline indexing time (not including the time for neural network training), and online retrieval latency for different Top-K values. We have four observations: (i) The memory footprint of generative retrieval models (GENRE, DSI-QG, and the proposed model) is smaller than of dense and sparse retrieval methods. The memory footprint of generative retrieval models is only dependent on the model parameters, whereas dense and sparse retrieval methods require additional storage space for document embeddings, which increases linearly with the size of the document collection. (ii) DSI and GENRET take a longer time for offline indexing, as DSI involves encoding and clustering documents using BERT, while GENRET requires tokenizing documents using a tokenization model. Dense retrieval’s offline time consumption comes from document encoding; GENRE uses titles hence no offline computation. (iii) The online retrieval latency of the generative retrieval model is associated with the beam size (i.e., Top-K) and the length of the docid. GENRET utilizes diverse clustering to generate a shorter docid, resulting in improved online retrieval speed compared to DSI and GENRE.
|
| 343 |
+
|
| 344 |
+
# E Embedding visualization
|
| 345 |
+
|
| 346 |
+
|
| 347 |
+
Figure 5: t-SNE visualization of the codebook embedding and document embedding on the NQ320K dataset. The codebook embedding is uniformly scattered in the document representation space.
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| 1 |
+
# PERSONALIZED NEURAL ARCHITECTURE SEARCH FOR FEDERATED LEARNING
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Federated Learning (FL) is a recently proposed learning paradigm for decentralized devices to collaboratively train a predictive model without exchanging private data. Existing FL frameworks, however, assume a one-size-fit-all model architecture to be collectively trained by local devices, which is determined prior to observing their data. Even with good engineering acumen, this often falls apart when local tasks are different and require diverging choices of architecture modelling to learn effectively. This motivates us to develop a novel personalized neural architecture search (NAS) algorithm for FL. Our algorithm, FEDPNAS, learns a base architecture that can be structurally personalized for quick adaptation to each local task. We empirically show that FEDPNAS significantly outperforms other NAS and FL benchmarks on several real-world datasets.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Federated Learning (FL) (McMahan et al., 2017) is a variant of distributed learning where the objective function can be decomposed into a linear combination of $M$ local objective functions. Each function depends on its private data hosted by a local client and a set of shared parameters $w$ ,
|
| 12 |
+
|
| 13 |
+
$$
|
| 14 |
+
\underset { w } { \mathrm { a r g m i n } } \ : \mathcal { L } ( w ) \equiv \underset { w } { \mathrm { a r g m i n } } \ : \sum _ { i = 1 } ^ { M } \mathcal { L } _ { i } ( w \mid \mathcal { D } _ { i } ) ,
|
| 15 |
+
$$
|
| 16 |
+
|
| 17 |
+
where $\mathcal { D } _ { i }$ denotes the $i ^ { \mathrm { t h } }$ local training dataset comprising input-output tuples $( x , y )$ . In a standard supervised learning task where the predictive model is modeled as a fixed deep neural network $\psi$ with learnable weights $w$ , let $\ell ( x , y )$ denote the loss incurred by predicting $\psi ( x ; w )$ when the true output is $y$ . The expected loss of $\psi ( x ; w )$ on $\mathcal { D } _ { i }$ is given as
|
| 18 |
+
|
| 19 |
+
$$
|
| 20 |
+
\begin{array} { r l r } { \mathcal { L } _ { i } ( w \mid \mathcal { D } _ { i } ) } & { = } & { \mathbb { E } _ { ( x , y ) \sim \mathcal { D } _ { i } } \Big [ \ell ( x , y ; \psi ) \Big ] ~ . } \end{array}
|
| 21 |
+
$$
|
| 22 |
+
|
| 23 |
+
This is not applicable to scenarios where local models are expected to solve different tasks which are similar in broad sense yet diverge in finer details. For example, consider the task of recognizing the outcome of a coin flip given images collected by two clients: one capture the coin from above, the other from below. This setting implies that when the same input image is provided by both clients, the correct classifications must be the opposite of one another. However, since existing FL methods converge on a single model architecture and weight, there can only be one predictive outcome which cannot satisfy both tasks.
|
| 24 |
+
|
| 25 |
+
To relax this constraint, the recent work of Fallah et al. (2020) extends FL by incorporating ideas from meta learning (Finn et al., 2017) which results in a new framework of personalized FL. The new framework can accommodate for such task heterogeneity but still requires all client models to agree on a single architecture beforehand, which is sub-optimal. To address this shortcoming, one naive idea is to adopt existing ideas in Neural Architecture Search (NAS) via Reinforcement Learning (Zoph and Le, 2016; Pham et al., 2018) which act as an outer loop to the existing FL routine.
|
| 26 |
+
|
| 27 |
+
However, this simple approach does not allow client models to adapt to local tasks on an architecture level and is often not preferred due to the cost of repeated FL training. This paper proposes a novel personalized NAS algorithm for federated learning, which generalizes ideas in respective areas of NAS (Zoph and Le, 2016; Pham et al., 2018) originally developed for single-task scenarios, and
|
| 28 |
+
|
| 29 |
+
FL (Fallah et al., 2020) under a unified len of federated personalized neural architecture search (FEDPNAS).
|
| 30 |
+
|
| 31 |
+
In particular, to customize the model architecture for each task in the FL workflow, FEDPNAS first represents the model architecture for each task as a sub-network sampled from a large, overparameterized network. The sampling distribution is (collaboratively) learned along with the parameters of the sampled network via a generalization of the recently proposed Discrete Stochastic NAS (DSNAS) method (Hu et al., 2020). Unlike DSNAS, which lacks the ability to customize architecture for individual tasks, our generalized FEDPNAS incorporates model personalization on an architecture level. Our contributions include:
|
| 32 |
+
|
| 33 |
+
1. A novel architecture that factorizes into a base component (shared across tasks) and a personalizable component, which respectively capture the task-agnostic and task-specific information (Section 3.2).
|
| 34 |
+
|
| 35 |
+
2. A context-aware sampling distribution conditioned on specific task instance, which captures taskspecific information and naturally incorporates personalization into architecture search (Section 3.4).
|
| 36 |
+
|
| 37 |
+
3. An FL algorithm that optimizes for a common architecture, followed by a personalization phase where each client subsequently adapts only the personalized component to fit its own task via finetuning with local data (Section 3.1). To ensure that the common architecture distribution converges at a vantage point that is relevant and beneficial to all clients, we generalize the vanilla FL objective in Eq. equation 1 such that local gradient steps directly optimize for expected improvement resulting from future fine-tuning (Section 3.3).
|
| 38 |
+
|
| 39 |
+
4. A theoretical perspective on our FL objective (Section 3.5 and thorough empirical analysis showing significant performance gain compared to state-of-the-art FL and NAS methods (Section 4).
|
| 40 |
+
|
| 41 |
+
# 2 RELATED WORKS
|
| 42 |
+
|
| 43 |
+
# 2.1 TWO-STAGE NEURAL ARCHITECTURE SEARCH
|
| 44 |
+
|
| 45 |
+
Most existing NAS frameworks separately optimize for the optimal architecture and its parameters in two stages: searching and evaluation. The former stage usually employs evolutionary-based strategies (Floreano et al., 2008; Real et al., 2019), Bayesian optimization surrogates (Bergstra et al., 2013; Hu et al., 2018) or Reinforcement Learning controllers (Baker et al., 2016; Zoph and Le, 2016; Pham et al., 2018) to propose candidate architectures based on random mutations and/or observed experience; while the latter optimizes the parameters of these architectures given task data and provide feedback to improve the search agent. Naturally, an extension of such methods to the FL setting is through distributing the evaluation workload over many clients, which does not require exposing private data. In practice, however, two-stage federated NAS frameworks are generally not suitable for the personalized FL setting for two reasons: (a) the clients often lack the computational capacity to repeatedly optimize the parameters for many candidate architectures; and (b) the clients have to converge on a single architecture proposed by the central search agent.
|
| 46 |
+
|
| 47 |
+
# 2.2 DISCRETE STOCHASTIC NEURAL ARCHITECTURE SEARCH
|
| 48 |
+
|
| 49 |
+
Discrete stochastic neural architecture search (DSNAS) (Hu et al., 2020) addresses the computational issue of two-stage NAS by jointly optimizing the optimal architecture and its weight in an end-to-end fashion, which allows users to continually train a single network on demand over time as opposed to performing full parameter optimization for every candidate until a good architecture is discovered.
|
| 50 |
+
|
| 51 |
+
The main idea of DSNAS is to combine weight training for an over-parameterized master architecture with discrete computational path sampling. DSNAS parameterizes the master architecture as a stack of modular cells: $\psi ( x ) \bar { = } \psi _ { C } \circ \bar { \psi } _ { C - 1 } \cdot \cdot \cdot \circ \psi _ { 1 } ( \bar { x } ) ^ { 1 }$ , where $x$ is an arbitrary input, $C$ is the number of cells, $\psi _ { t }$ denotes the $t ^ { \mathrm { t h } }$ cell in the stack, and $\circ$ denotes the compositional operator. The inner computation of $\psi _ { t }$ is in turn characterized by a directed acyclic graph (DAG) with $V$ nodes $\{ v _ { i } \} _ { i = 1 } ^ { | V | }$ , where each node represents some intermediate feature map. For each directed edge $( v _ { i } , v _ { j } )$
|
| 52 |
+
|
| 53 |
+
|
| 54 |
+
Figure 1: Our proposed method FEDPNAS consists of (1) a federated learning phase, where each client updates both the base component $( \psi ^ { b } )$ and the personalized component $( \psi _ { p } )$ the architecture using the FEDPNAS update (Section 3.3) and sends its parameters to the central server for aggregation; and (2) a fine-tune phase, where each client updates only the personalized component of the architecture using standard gradient update.
|
| 55 |
+
|
| 56 |
+
there is an associated list of $D$ possible network operations $\mathbf O _ { i j } = \left[ o _ { i j } ^ { 1 } , o _ { i j } ^ { 2 } \ldots o _ { i j } ^ { D } \right] ^ { 2 }$ where each operainput on w $o _ { i j } ^ { k }$ rms ). W $v _ { i }$ to rec $v _ { j }$ . Here, ively de $v _ { 1 }$ corresponds to the oe intermediate nodes $\psi _ { t - 1 }$ (orhere $x$ $t = 1$ $\begin{array} { r } { v _ { j } = \sum _ { i = 1 } ^ { j - 1 } \mathbf { Z } _ { i j } ^ { \top } \mathbf { O } _ { i j } ( v _ { i } ) } \end{array}$ distribution learnable. E $\mathbf { O } _ { i j } ( v _ { i } ) \triangleq \big [ o _ { i j } ^ { 1 } ( v _ { i } ) , o _ { i j } ^ { 2 } ( v _ { i } ) \dots o _ { i j } ^ { D } ( v _ { i } ) \big ]$ $p ( \mathbf { Z } \mid \mathbf { \pi } \mathbf { \Pi } \mathbf { \Pi } )$ where the event probabi learning the distribution and $\mathbf { Z } _ { i j }$ is a one-hot vector sampled from the categorical $\mathbf { H } = \{ \pi _ { 1 } , \pi _ { 2 } , . . . , \pi _ { D } \mid \sum _ { i = 1 } ^ { D } \pi _ { i } = 1 \}$ ares or $p ( \mathbf { Z } )$
|
| 57 |
+
sub-graphs of the original DAG that correspond to high-performing, compact architecture from the over-parameterized master network. Sampling discrete random variables from $p ( \mathbf { Z } )$ , however, does not result in a gradient amenable to back-propagation. To sidestep this issue, DSNAS adopts the straight-through Gumbel-softmax trick (Jang et al., 2016), which re-parameterizes the $k ^ { \mathrm { t h } }$ index of the one-hot variable as $\mathbf { Z } _ { i j } [ k ] = \mathbb { I } \left( k \triangleq { \arg \operatorname* { m a x } _ { t } } \Big [ g _ { t } + \log \pi _ { t } \Big ] \right)$ , where $g _ { t } \sim \mathrm { G u m b e l } ( 0 , 1 )$ . While this forward computation does not have a gradient by itself, we can estimate the gradient through a proxy during the backward pass:
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
\nabla { \mathbf Z } _ { i j } [ k ] ~ \simeq ~ \nabla \tilde { \mathbf Z } _ { i j } [ k ] ~ \triangleq ~ \nabla \left( \frac { \exp \big ( \big ( g _ { k } + \log \pi _ { k } \big ) / \tau \big ) } { \sum _ { t = 1 } ^ { D } \exp \big ( \big ( g _ { t } + \log \pi _ { t } \big ) / \tau \big ) } \right)
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
which is unbiased when converged as the temperature $\tau$ is steadily annealed to 0 (Jang et al., 2016). This formulation, however, is not easily extended to the FL setting, especially when local tasks are not homogeneous. The key challenges in doing so are described in Section 3, together with our proposed approaches.
|
| 64 |
+
|
| 65 |
+
# 3 PERSONALIZED NAS FOR FEDERATED LEARNING
|
| 66 |
+
|
| 67 |
+
# 3.1 FEDERATED LEARNING OF DSNAS
|
| 68 |
+
|
| 69 |
+
Let W denote the concatenated weights of all network operations in the network architecture. The set up above of DSNAS (Jang et al., 2016) is then naïvely extendable to a $\mathrm { F L }$ setting via the following objective formulation:
|
| 70 |
+
|
| 71 |
+
$$
|
| 72 |
+
\underset { \mathbf { W } , \mathbf { \Pi } } { \arg \operatorname* { m i n } } \mathcal { L } ( \mathbf { W } , \mathbf { \Pi } \mathbf { I } ) \equiv \underset { \mathbf { W } , \mathbf { \Pi } \mathbf { \Pi } } { \arg \operatorname* { m i n } } \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \mathcal { L } _ { i } ( \mathbf { W } , \mathbf { \Pi } \mathbf { I } \mid \mathcal { D } _ { i } ) .
|
| 73 |
+
$$
|
| 74 |
+
|
| 75 |
+
McMahan et al. (2017) optimizes this objective by alternating between (a) central agent broadcasting aggregated weights to local clients and (b) local clients sending gradient descent updated weights (given local data) to the central agent for aggregation. This, however, implies that after the last central aggregation step, all clients will follow the same architecture distribution induced by the final broadcasted copy of W and Π. As previously argued, this is not optimal in a heterogenous task setting which requires task-specific adaptation for local clients to achieve good performance.
|
| 76 |
+
|
| 77 |
+
Furthermore, having the same sampling distribution $p ( \mathbf { Z } )$ regardless of context (i.e., feature maps received as cell input) limits the architecture discovery to those that perform reasonably on average over the entire dataset. However, we remark that restricting the architecture to be the same for every input samples is unnecessary and undermines the expressiveness of an over-parameterized search space. On the other hand, letting the architecture be determined on a per-sample basis makes better use of the search space and potentially improves the predictive performance.
|
| 78 |
+
|
| 79 |
+
The focus of this work is therefore to incorporate both task-wise and context-wise personalization to federated neural architecture search in multitask scenarios, which is achieved through our proposed algorithm FEDPNAS. In general, FEDPNAS functions similarly to the vanilla FEDDSNAS algorithm described above, with an addition of a fine-tuning phase at each local client after the FL phase to adapt the aggregated common model for local task data, as shown in Fig. 1. To make this work, however, we need to address the following key challenges:
|
| 80 |
+
|
| 81 |
+
C1. First, as previously argued in Section 1, tasks across federated clients tend to share similarities in broad sense, and diverge in finer details. A good federated personalization search space, therefore, need to capture this fundamental observation through design and appropriate resource distribution. We address this challenge in Section 3.2.
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C2. Second, a major advantage of having an over-parameterized architecture search space is the flexibility of having specific computation paths for different samples, which is not exploited by DSNAS as reflected in its choice of context-independent sampling distribution $p ( \mathbf { Z } )$ . To address this, Section 3.4 proposes a novel parameterization of $p ( \mathbf { Z } )$ to incorporate context information into operator sampling.
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C3. Last, while the fine-tuning phase is designed to incorporate task-personalization, there is no guarantee that the common model can be quickly adapted to client tasks (Fallah et al., 2020). The common model may end up in a localization that favors one client over another, which makes it difficult for the latter to fine-tune. To address this concern, Section 3.3 proposes a new personalized federated NAS objective inspired by Finn et al. (2017) to optimize the common model in anticipation of further fine-tuning by the client models.
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# 3.2 PERSONALIZABLE ARCHITECTURE SEARCH SPACE
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Similar to DSNAS (Hu et al., 2020), our framework adopts a cell-based representation (Section 2.2) to trade-off search space expressiveness for efficiency, which is extremely suitable for FL where clients tend to have low-end computational capacity. Unlike the original design which assumes similar role for every cell in the architecture stack (i.e., as reflected by their choice of fully factorizable path sampling distribution $p ( \mathbf { Z } ) ,$ , we instead split our cell stack into two components with separate metaroles catering to the ftask: (a) a base stack $\psi _ { \mathrm { b } } = \{ \dot { \psi } _ { 1 } ^ { b } , \psi _ { 2 } ^ { b } \cdot \hdots \psi _ { C _ { b } } ^ { b } \}$ which aims to capture the broad commonalities of data samples across client tasks; and (b) personalized stack $\psi _ { \mathrm { { p } } } ~ = ~ \{ \psi _ { 1 } ^ { p } , \psi _ { 2 } ^ { p } \ldots \psi _ { C _ { p } } ^ { p } \}$ which will be fine-tuned with local data to capture task-specific details.
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Figure 2: Feature mapping down the component stacks of our architecture space. Every base cell takes as inputs (a) the outputs from its immediate predecessor and (b) the one before it through a skip-ahead connection. On the other hand, every personalization cell takes as input only the output from the previous cell.
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We explain the main difference between these components to account for different level of expressiveness requirements below:
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Base stack. Every cell $\psi _ { t } ^ { b }$ in the base stack takes as inputs the outputs of its previous two cells $\psi _ { t - 1 } ^ { b }$ and $\psi _ { t - 2 } ^ { b }$ (replaced with raw input $x$ when necessary for $t \leq 2$ ). The output of the skip-ahead cell $\psi _ { t - 2 } ^ { b }$ is additionally passed through a $1 \times 1$ convolution layer as a cost-effective way to control the number of channels. Additionally, the operators available to the base cell include large convolution layers with size $5 \times 5$ and $7 \times 7$ . To compensate for the growing number of channels, we periodically employ a reduction convolution (with stride larger than 1) similar to DSNAS (Hu et al., 2020) to reduce the feature dimension down the stack.
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Personalized stack. As opposed to the design of the base cells above, every cell $\psi _ { t } ^ { p }$ in the personalized stack has minimal expressiveness. That is, $\psi _ { t } ^ { p }$ excludes large operators and only takes as input the output of its immediate predecessor $\psi _ { t - 1 } ^ { p }$ (or $\dot { \psi } _ { C _ { b } } ^ { b }$ when $t = 1$ ). There are two reasons for this choice. First, as the fine-tuning phase has access to fewer data samples than the federated phase, having a more compact fine-tuning space helps to improve the rate of convergence. Second, as we will discuss in Section 3.4 below, our personalized FL objective requires the Hessian of the personalized parameters, which is computationally expensive. As such, we only restrict the personalization to happen on the more compact personalized stack.
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# 3.3 PERSONALIZED FEDERATED LEARNING OBJECTIVE
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Unlike FEDAVERAGING (McMahan et al., 2017), which assumes the clients will follow the consensus base model obtained after the federated phase, FEDPNAS expects clients to further personalize the base model with local task data. That said, while the base model is trained to work well in the expected sense over the task distribution, there is no guarantee that it is a good initial point for every client model to improve upon via fine-tuning. To address this, we adopt the concept of training in anticipation of future adaptation introduced by MAML (Finn et al., 2017). That is, during client update, instead of optimizing the loss with respect to the same consensus weight, each client will instead optimize the weight perturbed by a small gradient step in the fine-tuning direction.
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# Algorithm 1 FEDPNAS - FEDERATED PHASE
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1: CENTRALAGGREGATION:
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2: $\theta _ { 0 } \gets$ INITIALIZEPARAMETER
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3: for $t = 1 , 2 \dots T _ { s }$ do
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4: for $k = 1 , 2 \dots M$ in parallel do
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5: $\theta _ { t } ^ { k } \gets \mathrm { C L I E N T U P D A T E } ( k , \theta _ { t - 1 } )$
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$\begin{array} { r } { \theta _ { t } \sum _ { k = 1 } ^ { M } \frac { 1 } { M } \theta _ { t } ^ { k } } \end{array}$
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7: CLIENTUPDATE $( k , \theta )$ :
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8: for $t = 1 , 2 \dots T _ { \mathrm { c } }$ do
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9: for batch $( x , y ) \in \mathcal { D } _ { k }$ do
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10: $\mathcal { L } _ { k }$ $\mathrm { , } \nabla \mathcal { L } _ { k } \gets \mathrm { E v A L } ( x , y ; \theta _ { b } , \theta _ { p } )$
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11: $ { \widetilde { \theta } } _ { p } \gets { \mathrm { G R A D U P D A T E } } \big ( \nabla _ { { \widetilde { \theta } } _ { p } } \mathcal { L } _ { k } \big )$
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12: $\tilde { \mathcal { L } } _ { k } , \nabla \tilde { \mathcal { L } } _ { k } \gets \mathrm { E v A L } ( x , y ; \theta _ { b } , \tilde { \theta } _ { p } )$
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13: $\nabla _ { \boldsymbol { \theta } _ { p } } \tilde { \mathcal { L } } _ { k } \gets \mathrm { E Q } . \mathrm { 5 }$
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14: θb, θp ← GRADUPDATE(∇θb,θpL˜k)
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15: return $\theta$ to central server
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# Algorithm 2 FEDPNAS - EVAL
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1: Input: x, y, θ
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2: SKIP, PREV ← x, x
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3: W,Π ← θ
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4: for CELL $\psi \in \psi$ do
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5: Z ← SAMPLEOPS $( x , { \mathrm { P R E V } } ; \mathbf { I I } )$
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6: ψ ← EXTRACTCHILDNET $( \mathbf { Z } , \mathbf { W } )$
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7: if $\psi \in \psi _ { b }$ then
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8: OUTPUT $ \psi ( \mathrm { P R E V } , \mathrm { S K I P } )$
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9: else if $\psi \in \psi _ { p }$ then
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10: $\mathrm { O U T P U T } \psi ( \mathrm { P R E V } )$
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11: $\mathbf { S K I P } \mathbf { P R E V }$
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12: PREV $\gets$ OUTPUT
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13: $\mathcal { L } \gets \mathrm { L o s s } ( \mathbf { O u T P U T } , y )$
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14: $\nabla { \mathcal { L } } \gets \mathbf { B A C K P R O P } ( { \mathcal { L } } )$
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15: return L, ∇L
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+
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For simplicity, let $\theta = \{ \theta _ { b } , \theta _ { p } \}$ respectively denote all trainable parameters of the base stack and the personalized stack, i.e., $\partial _ { b } = \mathbf { \bar { \{ W } } _ { b } , \mathbf { \bar { \Pi } } _ { b } \} , \boldsymbol { \hat { \theta _ { p } } } = \{ \mathbf { W } _ { p } , \mathbf { \Pi } _ { \Pi _ { p } } \}$ . The personalized $\mathrm { F L }$ objective at client $i$ is then given by $\tilde { \mathcal { L } } _ { i } ( \theta _ { b } , \theta _ { p } ) \triangleq \mathcal { L } _ { i } ( \theta _ { b } , \tilde { \theta } _ { p } )$ where $\tilde { \theta } _ { p } \triangleq \tilde { \theta } _ { p } - \eta \nabla _ { \theta _ { p } } \mathcal { L } _ { i } ( \theta _ { b } , \theta _ { p } )$ adjusts the parameters of the personalized component to account for a small fine-tuning gradient step. The adjusted local loss only depends on the respective client data and is amenable to federated learning. The local update gradient, however, involves a Hessian term whose computation is expensive to repeat over many epochs. To circumvent this problem, we use the first-order Taylor approximation to estimate the Hessian term by the outer product of Jacobian, which results in a gradient that requires exactly two forward/backward passes to compute:
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$$
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\begin{array} { r c l } { \nabla _ { \theta _ { p } } \tilde { \mathcal { L } } _ { i } } & { = } & { \left( \nabla _ { \theta _ { p } } \tilde { \theta } _ { p } \right) \left( \nabla _ { \tilde { \theta } _ { p } } \tilde { \mathcal { L } } _ { i } \right) } \\ & { = } & { \left( \mathbf { I } - \eta \nabla _ { \theta _ { p } } ^ { 2 } \mathcal { L } _ { i } \right) \left( \nabla _ { \tilde { \theta } _ { p } } \tilde { \mathcal { L } } _ { i } \right) } \\ & { \simeq } & { \left( \mathbf { I } - \eta \nabla _ { \theta _ { p } } ^ { \top } \mathcal { L } _ { i } \nabla _ { \theta _ { p } } \mathcal { L } _ { i } \right) \left( \nabla _ { \tilde { \theta } _ { p } } \tilde { \mathcal { L } } _ { i } \right) } \end{array}
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$$
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where $\mathcal { L } _ { i }$ and $\tilde { \mathcal { L } } _ { i }$ are short-hands for $\mathcal { L } _ { i } ( \theta _ { b } , \theta _ { p } )$ and $\tilde { \mathcal { L } } _ { i } ( \theta _ { b } , \theta _ { p } )$ respectively. The FL phase of our FEDPNAS framework is detailed in Alg. 1. An instance of FEDPNAS’s forward and backward pass which sequentially unrolls down the component stacks, alternating between sampling and evaluation, is in turn given in Alg. 2.
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# 3.4 CONTEXT-AWARE OPERATOR SAMPLER
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The choice of a fully factorizable sampling distribution $p ( \mathbf { Z } )$ in DSNAS follows that of SNAS (Xie et al., 2018), which argues that the Markov assumption for $p ( \mathbf { Z } )$ is not necessary because NAS has fully delayed rewards in a deterministic environment. However, this generally only holds for two-stage NAS (Section 2.1) and does not apply to end-to-end frameworks such as SNAS and DSNAS. We instead to take advantage of the over-parameterized architecture via factorizing the conditional $p ( \mathbf { Z } \mid x )$ , which takes into account the temporal dependency of structural decisions:
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+
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+
$$
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+
\begin{array} { l l l } { { p ( { \bf Z } \mid x ) } } & { { = } } & { { \displaystyle p ( { \bf Z } _ { 1 } \mid x ) \prod _ { t = 2 } ^ { C } p ( { \bf Z } _ { t } \mid { \bf Z } _ { t - 1 } \dots { \bf Z } _ { 1 } , x ) } } \\ { { } } & { { } } & { { } } \\ { { \displaystyle } } & { { \simeq } } & { { \displaystyle p ( { \bf Z } _ { 1 } \mid x ) \prod _ { t = 2 } ^ { C } p ( { \bf Z } _ { t } \mid v _ { 1 } ^ { t } , x ) } } \\ { { } } & { { } } & { { } } \\ { { \displaystyle } } & { { = } } & { { \displaystyle p ( { \bf Z } _ { 1 } \mid x ) \prod _ { t = 2 } ^ { C } \prod _ { ( i , j ) } p ( { \bf Z } _ { i j } ^ { t } \mid v _ { 1 } ^ { t } , x ) , } } \end{array}
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+
$$
|
| 155 |
+
|
| 156 |
+
where $\mathbf { Z } _ { t }$ , $\mathbf { Z } _ { i j } ^ { t }$ and $v _ { 1 } ^ { t }$ respectively denote all the samples, the sample at edge $( i , j )$ and the input at cell $\psi _ { t }$ . We have also assumed a single stack setting since the parameterization of $p ( \mathbf { Z } )$ does not differ between base and personalized cells.
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+
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+
To reduce computational complexity, instead of conditioning the samples of subsequent cells on previous $\mathbf { Z }$ samples, we approximate $p ( \mathbf { Z } _ { t } \mid \mathbf { Z } _ { t - 1 } \ldots \mathbf { Z } _ { 1 } , x ) \bumpeq p ( \mathbf { Z } _ { t } \mid v _ { 1 } ^ { t } , x )$ by the assumption that the cell contents are conditionally independent given the immediate cell input and the original input. Finally, we assume that $p ( \mathbf { Z } _ { t } \mid v _ { 1 } ^ { t } , x )$ is fully factorizable across edges in the same cell and parameterize $p ( \mathbf { Z } _ { i j } ^ { t } \mid v _ { 1 } ^ { t } , x ) = \phi ^ { ( i , j ) } ( v _ { 1 } ^ { t } , x )$ where $\phi$ is a deep classification network whose output dimension equal the number of edges in cell $\psi _ { t }$ . Samples of $\mathbf { Z } _ { i j } ^ { t }$ can then be generated using the straight-through Gumbel-softmax reparameterization similar to Jang et al. (2016).
|
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+
|
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+
# 3.5 THEORETICAL CONNECTION TO STANDARD GRADIENT UPDATE
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+
|
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+
Finally, we analyze the connection of our gradient update framework to the standard gradient update, and explain why it is critical in achieving a vantage point that improves average objective value without compromising any local objective. First, we note that the gradient update Eq. 5 in Section 3.3 at the $t$ -th iteration can be written as:
|
| 163 |
+
|
| 164 |
+
$$
|
| 165 |
+
\begin{array} { r c l } { \displaystyle \theta _ { p } ^ { t + 1 } } & { = } & { \displaystyle \theta _ { p } ^ { t } - \frac { \eta _ { 2 } } { M } \sum _ { i = 1 } ^ { M } \nabla _ { \theta _ { p } } \mathcal { L } _ { i } \big ( \theta _ { b } ^ { t } , \tilde { \theta } _ { p , i } ^ { t } \big ) + \frac { \eta _ { 1 } \eta _ { 2 } } { M } \sum _ { i = 1 } ^ { M } \alpha _ { i } \nabla _ { \theta _ { p } } \mathcal { L } _ { i } \big ( \theta _ { b } ^ { t } , \theta _ { p } ^ { t } \big ) , } \\ { \displaystyle \theta _ { b } ^ { t + 1 } } & { = } & { \displaystyle \theta _ { b } ^ { t } - \frac { \eta _ { 2 } } { M } \sum _ { i = 1 } ^ { M } \nabla _ { \theta _ { b } } \mathcal { L } _ { i } \big ( \theta _ { b } ^ { t } , \tilde { \theta } _ { p , i } ^ { t } \big ) , } \end{array}
|
| 166 |
+
$$
|
| 167 |
+
|
| 168 |
+
where $\tilde { \theta } _ { p , i } ^ { t }$ denotes the $i$ -th local personalized parameters; $\eta _ { 1 }$ and $\eta _ { 2 }$ are two separate learning rates and $\boldsymbol { \alpha } _ { i } \triangleq \nabla _ { \boldsymbol { \theta } _ { p } } ^ { \intercal } \mathcal { L } ( \boldsymbol { \theta } _ { b } ^ { t } , \boldsymbol { \theta } _ { p , i } ^ { t } ) \nabla _ { \boldsymbol { \theta } _ { p , i } } \mathcal { L } ( \boldsymbol { \theta } _ { b } ^ { t } , \tilde { \boldsymbol { \theta } } _ { p , i } ^ { t } )$ (See Appendix A for detailed derivation). This implies that our federated personalize update corresponds to a federated update scheme with three gradient steps: (1) $\theta _ { p }$ takes a local gradient (w.r.t. locally updated parameters) step of size $\eta _ { 1 }$ ; (2) Both $\theta _ { b }$ and $\theta _ { p }$ take a federated gradient (w.r.t. server-wide parameters averaging) step of size $\eta _ { 2 }$ ; $( 3 ) \theta _ { p }$ takes a weighted federated gradient step of size $\eta _ { 1 } \eta _ { 2 }$ , where the weight of client $i$ is given by $\alpha _ { i }$ .
|
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+
|
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+
Explicitly, Step 1 and 2 together comprise a special instance of FEDAVERAGING (McMahan et al., 2017), where $\theta _ { b }$ take one gradient step for every two gradient steps taken by $\theta _ { p }$ . Step 3, on the other hand, takes the information of the two gradient steps of $\theta _ { p }$ and adjust the magnitude of the local gradient step (whose direction is given by $\nabla _ { \theta _ { p } } \mathcal { L } _ { i } ( \theta _ { b } ^ { t } , \bar { \theta } _ { p } ^ { t } ) )$ accordingly to trade-off between preserving local objective value and improving average objective value. We then theorize the scenario in which such an update is beneficial and state the following assumption to lay the foundation of our analysis:
|
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+
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+
Assumption 1 For a fixed instance of $\theta _ { b }$ , let $\tilde { \theta } _ { p , i } = \theta _ { p , i } - \eta \nabla _ { \theta _ { p } } \mathcal { L } ( \theta _ { b } , \theta _ { p , i } )$ denote the personalized parameters after a local update step (i.e., step $^ { l }$ above) at client $i$ , then there exists a distribution $s$ on matrix $\mathbf { S } \in \mathbb { R } ^ { k \times | \theta _ { p } | }$ that satisfies
|
| 173 |
+
|
| 174 |
+
$$
|
| 175 |
+
\forall \mathbf { x } \in \mathbb { R } ^ { n } , \| \mathbf { x } \| _ { 2 } = 1 : \mathbb { E } _ { \mathbf { S } \sim { \mathcal { S } } } \left[ | \| \mathbf { S } \mathbf { x } \| _ { 2 } ^ { 2 } - 1 | ^ { \ell } \right] \ \leq \ \epsilon ^ { \ell } \cdot \delta ,
|
| 176 |
+
$$
|
| 177 |
+
|
| 178 |
+
$$
|
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+
\operatorname* { P r } _ { \mathbf { S } \in \mathcal { S } } \left( \Big | \nabla _ { \theta _ { p } } \mathcal { L } \big ( \theta _ { b } , \tilde { \theta } _ { p , i } \big ) - \mathbf { S } ^ { \top } \mathbf { S } \Big ( \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \nabla _ { \theta _ { p } } \mathcal { L } \big ( \theta _ { b } , \tilde { \theta } _ { p , i } \big ) \Big ) \Big | \leq \sqrt { \frac { 6 } { k \delta } } \right) ~ \geq ~ 1 - \delta
|
| 180 |
+
$$
|
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+
|
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+
where $k = \mathcal { O } \left( C \| \theta _ { b } - \theta _ { b } ^ { * } \| _ { 2 } ^ { - 2 } \right)$ for some constant $C > 0$ and $\theta _ { b } ^ { * }$ denotes the optimal base parameters.
|
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+
|
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+
The above assumption implies that, as $\theta _ { b }$ improves and better captures the broad similarity across tasks, the local personalized components will diverge to capture the differences. It then becomes less likely for the FEDAVG personalized gradient to capture all these differences simultaneously. That is, suppose there exists an affine transformation to reconstruct the local component $\nabla _ { \boldsymbol { \theta } _ { p } } \mathcal { L } _ { i } \dot { ( \theta _ { b } ^ { t } , \tilde { \theta } _ { p } ^ { t } ) }$ from the federated gradient 1M PMi=1 ∇θp Li(θtb, ˜θtp), then the rank of this affine transformation would be inversely proportionate to $\lVert \theta _ { b } - \theta _ { b } ^ { * } \rVert _ { 2 }$ .
|
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+
|
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+
Finally, Proposition 1 below shows that when this assumption holds and $\theta _ { b }$ converges to the optimal parameter $\theta _ { b } ^ { * }$ (i.e., the error term tends to 0), then with very high probability, the coefficient $\alpha _ { i }$ of the weighted federated gradient step (i.e., step 3 above) accurately captures the cosine similarity between the local gradient (step 1) and the federated gradient (step 2).
|
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+
|
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+
Proposition 1 Suppose assumption $^ { l }$ holds, then with probability at least $1 - 2 \delta$ and normalized gradients, we have:
|
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+
|
| 190 |
+
$$
|
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+
\begin{array} { r l r } { \bigg | \alpha _ { i } - \nabla _ { \theta _ { p } } ^ { \top } \mathcal { L } _ { i } ( \theta _ { b } ^ { t } , \theta _ { p } ^ { t } ) \mathbf { L } \bigg | } & { = } & { \mathcal { O } ( \| \theta _ { b } - \theta _ { b } ^ { * } \| / \delta ) } \end{array}
|
| 192 |
+
$$
|
| 193 |
+
|
| 194 |
+
Proof. See Appendix B
|
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+
|
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+
This result has strong implication with respect to the scenario with multiple heterogeneous tasks, whose local gradients contradict in directions. Per this setting, we expect a standard federated gradient update scheme to encourage parameters drifting in the general direction of the majority (i.e., captured by the federated gradient), thus worsening the performance of tasks that are in the minority. Proposition 1, however, implies that whenever the local gradient contradicts the federated gradient, $\alpha _ { i }$ will be close to the cosine similarity term, which is negative. This in turn results in a dampening effect on the federated gradient and helps to preserve the client performance on its own local task.
|
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+
|
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+
# 4 EXPERIMENTS
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+
This section describes our experiments to showcase the performance of FEDPNAS compared to different NAS and FL benchmarks on various scenarios. All of our empirical studies are conducted on two image recognition datasets: (a) the CIFAR-10 dataset (Krizhevsky et al., 2009) which aims to predict image labels from 10 classes given a train/test set of 50000/10000 colour images of dimension $3 2 \times 3 2$ pixels; and (b) the MNIST dataset (LeCun et al., 2010) which aims to predict handwritten digits (i.e. 0 to 9) given a train/test set of 60000/10000 grayscale images of dimension $2 8 \times 2 8$ pixels. Our search space entails $2 ^ { 4 0 }$ possible architectures, which is detailed in Appendix D. We compare two variants of our framework, CA-FEDPNAS (with context-aware operation sampler) and FEDPNAS (without the operation sampler), against: (a) FEDAVERAGING of a fixed architecture to justify the need for NAS in FL; (b) FEDDSNAS - the federated extension of DSNAS (Section 3.1) to show the effectiveness of our proposed context-aware sampler on NAS performance; and finally (c) CA-FEDDSNAS, which extends FEDDSNAS with our context-aware sampler.
|
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+
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+
On simulate heterogenous predictive tasks. To simulate this scenario, we first distribute the data i.i.d across clients (10000/2000 and 12000/2000 training/test images per client for CIFAR-10 and MNIST datasets respectively). Then, we independently apply a different transformation to each partitioned dataset. Input images within the same train/test set is subject to the same transformation. In both our experiments, the client datasets are subjected to rotations of $- 3 0 ^ { \circ }$ , $- 1 5 ^ { \circ } , 0 ^ { \circ }$ , $1 5 ^ { \circ }$ and $3 0 ^ { \circ }$ respectively. This data generation protocol reflects a realistic and frequently seen scenario where independently collected data of the same phenomenon might contain systematic bias due to measurement errors and/or different collection protocols. Fig. 3 below shows the performance of all the methods in comparison, plotted against number of search epochs and averaged over the above rotated variants of CIFAR-10 and MNIST datasets.
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+
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+
Figure 3: Plotting average classification accuracy of various methods against no. training epochs on heterogeneous tasks derived from (a) MNIST dataset; and (b) CIFAR-10 dataset. Figure (c) compares cumulative running time of various methods against no. training epochs on CIFAR-10 dataset.
|
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+
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+
On the MNIST dataset (Fig. 3b), all methods eventually converge to a similar performance. Among the NAS benchmarks, FEDPNAS and FEDDSNAS both converge slower than FEDAVG and start off with worse performance in early iterations, which is expected since FEDAVG does not have to search for the architecture and it is likely that the default architecture is sufficient for the MNIST task. On the other hand, we observe that both CA-FEDPNAS and CA-FEDDSNAS converge much faster than their counterparts without the context-aware operation sampler component. This shows that making use of contextual information helps to quickly locate regions of high-performing architectures, especially on similar inputs.
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+
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+
On the CIFAR-10 dataset (Fig. 3a), we instead observe significant gaps between the worst performing FEDAVG and other NAS methods. This is likely because the default architecture does not have sufficient learning capability, which confirms the need for customizing solutions. Among the NAS benchmarks, we again observe that both CA-FEDPNAS and CA-FEDDSNAS outperform their counterparts without our operation sampler, which confirms the intuition above. Most remarkably, our proposed framework CA-FEDPNAS achieves the best performance (0.8) and significantly outperformed both variants of federated DSNAS (0.71 for CA-FEDDSNAS and 0.63 for FEDDSNAS).
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+
Lastly, Fig. 3c shows the runtime comparison between three methods on the CIFAR-10 experiment. In terms of sampling time, we observe that there is negligible overhead incurred by using our context-aware sampler (CA-FEDDSNAS vs. FEDDSNAS). The time incurred by our update (CA-FEDPNAS) scales by a constant factor compared to CA-FEDDSNAS since we use exactly one extra forward/backward pass per update.
|
| 212 |
+
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| 213 |
+
On objectives with varying heterogeneity. We expand the above study by investigating respective performance of CA-FEDPNAS and FEDDSNAS on tasks with varying levels of heterogeneity. At low level of heterogeneity, we deploy these methods on 5 sets of slightly rotated MNIST images. At high level of heterogeneity, we employ a more diverse set of transformations on MNIST images, such as hue jitter and large angle rotations of $9 0 ^ { \circ }$ and $- 9 0 ^ { \circ }$ . Table 1 show the respective result of each task from these two settings. We observe that our method CA-FEDPNAS achieves better performance on most tasks and the performance gaps on tasks with higher heterogeneity are more pronounced (i.e., up to $7 \%$ improvement on ROTATE 90 task). This clearly shows the importance of architecture personalization when the training tasks are significantly different and justifies our research goal.
|
| 214 |
+
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| 215 |
+
Table 1: Predictive accuracy of CA-FEDPNAS FEDDSNAS on tasks with varying heterogeneity levels. ROTATE X denotes a rotation transformation of $\mathbf { X } ^ { \circ }$ on client data; VANILLA denotes the original MNIST images; and HUEJITTER X denotes a hue jitter transformation of training images by a factor of X. The best performance in each row is in bold font.
|
| 216 |
+
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+
<table><tr><td rowspan=1 colspan=1>HETEROGENEITY</td><td rowspan=1 colspan=1>TASKDESCRIPTION</td><td rowspan=1 colspan=1>FEDDSNAS</td><td rowspan=1 colspan=1>CA-FEDPNAS</td></tr><tr><td rowspan=5 colspan=1>Low</td><td rowspan=1 colspan=1>ROTATE -30</td><td rowspan=1 colspan=1>0.947</td><td rowspan=1 colspan=1>0.978</td></tr><tr><td rowspan=1 colspan=1>ROTATE -15</td><td rowspan=1 colspan=1>0.973</td><td rowspan=1 colspan=1>0.976</td></tr><tr><td rowspan=1 colspan=1>VANILLA</td><td rowspan=1 colspan=1>0.988</td><td rowspan=1 colspan=1>0.985</td></tr><tr><td rowspan=1 colspan=1>ROTATE 15</td><td rowspan=1 colspan=1>0.986</td><td rowspan=1 colspan=1>0.987</td></tr><tr><td rowspan=1 colspan=1>ROTATE 30</td><td rowspan=1 colspan=1>0.972</td><td rowspan=1 colspan=1>0.981</td></tr><tr><td rowspan=5 colspan=1>HIGH</td><td rowspan=1 colspan=1>HUEJITTER-0.5</td><td rowspan=1 colspan=1>0.966</td><td rowspan=1 colspan=1>0.978</td></tr><tr><td rowspan=1 colspan=1>HUEJITTER 0.5</td><td rowspan=1 colspan=1>0.967</td><td rowspan=1 colspan=1>0.972</td></tr><tr><td rowspan=1 colspan=1>VANILLA</td><td rowspan=1 colspan=1>0.988</td><td rowspan=1 colspan=1>0.989</td></tr><tr><td rowspan=1 colspan=1>ROTATE -90</td><td rowspan=1 colspan=1>0.892</td><td rowspan=1 colspan=1>0.932</td></tr><tr><td rowspan=1 colspan=1>ROTATE 90</td><td rowspan=1 colspan=1>0.866</td><td rowspan=1 colspan=1>0.932</td></tr></table>
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| 218 |
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| 219 |
+
On knowledge transfer to completely new tasks. Finally, we investigate a scenario where the architecture distributions discovered by CA-FEDPNAS and FEDDSNAS are required to generalize to completely unseen tasks. Particularly, we train both methods on five clients whose local data consist of 12000 slightly rotated CIFAR-10 images (i.e., in the range of $\pm 3 0 ^ { \circ }$ ), similar to the setting of the first experiment. During testing, however, we supply each local client with 2000 test images subjected to related but completely unseen transformations (i.e., $9 0 °$ and $- 9 0 ^ { \circ }$ rotations).
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| 220 |
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| 221 |
+
Our results are summarized in Table 2. First, we measure the performance of CA-FEDPNAS and FEDDSNAS without any weight retraining. When received no additional information from the unseen tasks, both methods perform poorly as expected. While CA-FEDPNAS achieves better predictive accuracy, the performance gap in this scenario is negligible. To provide additional clues for adaptation, albeit minimal, we retrain the weights of each local model with 200 images rotated according to respective unseen task description. Here, the parameters of our operator sampler component, (and respectively, FEDDSNAS’s categorical distribution parameters), are frozen to gauge the quality of the learned architecture distributions. Our results show that, with only 100 retraining iterations on limited data, CA-FEDPSNAS already outperforms FEDDSNAS $5 \%$ and $8 \%$ improvement respectively on two unseen tasks). This implies that CA-FEDPNAS has more accurately capture the broad similarity of the task spectrum through the personalized architecture distribution, which requires minimal additional information to successfully adapt to unseen tasks.
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| 222 |
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+
Table 2: Predictive accuracy (averaged over 5 clients) and standard deviation of CA-FEDPNAS and FEDDSNAS on two unseen tasks (CIFAR-10). The best performance in each row is in bold font.
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| 224 |
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<table><tr><td rowspan=1 colspan=1>UNSEEN TASKDESCRIPTION</td><td rowspan=1 colspan=1>FEDDSNAS</td><td rowspan=1 colspan=1>CA-FEDPNAS</td><td rowspan=1 colspan=1>FEDDSNAS(RETRAINED)</td><td rowspan=1 colspan=1>CA-FEDPNAS(RETRAINED)</td></tr><tr><td rowspan=1 colspan=1>ROTATE -90</td><td rowspan=1 colspan=1>0.545 ± 0.04</td><td rowspan=1 colspan=1>0.578 ± 0.09</td><td rowspan=1 colspan=1>0.699± 0.12</td><td rowspan=1 colspan=1>0.734 ± 0.17</td></tr><tr><td rowspan=1 colspan=1>ROTATE 90</td><td rowspan=1 colspan=1>0.553 ± 0.12</td><td rowspan=1 colspan=1>0.569 ± 0.06</td><td rowspan=1 colspan=1>0.673 ± 0.13</td><td rowspan=1 colspan=1>0.727 ± 0.22</td></tr></table>
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| 226 |
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# 5 CONCLUSION
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We demonstrate that federated learning for multi-task scenarios requires extensive personalization on the architecture level to obtain good predictive performance. This paper identifies two potential sources of model personalization: (1) task-personalization, which aims to select architectures best suited for specific learning objectives; and (2) context-personalization, which aims to select architectures best suited for specific input samples. To incorporate these aspects of personalization into Federated NAS, we propose FEDPNAS which consists of two main components: (1) a context-aware operator sampler which learns a sampling distribution for feature maps along a master architecture; and (2) a personalized federated learning objective which anticipates client fine-tuning and guides the federated model update to regions that tolerate future local updates.
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# 6 REPRODUCIBILITY & ETHIC STATEMENT
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This work contributes to the literature of Federated Learning through improving the state-of-theart performance. As such, it could have significant broader impact by allowing users to more accurately solve practical problems. While applications of our work to real data could result in ethical considerations, this is an indirect (and unpredictable) side-effect of our work. Our experimental work uses publicly available datasets to evaluate the performance of our algorithms; no ethical considerations are raised. Our implementation code is published anonymously at https://github.com/icml2021fedpnas/fedpnas. All proofs and details of various architectures are included in the Appendix of this paper.
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# REFERENCES
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J. Bergstra, D. Yamins, and D. Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In International conference on machine learning, pages 115–123. PMLR, 2013.
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A. Fallah, A. Mokhtari, and A. Ozdaglar. Personalized federated learning: Model-agnostic metalearning approach. In Proc. NeurIPS, 2020.
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C. Finn, P. Abbeel, and S. Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proc. ICML, pages 1126–1135, 2017.
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D. Floreano, P. Dürr, and C. Mattiussi. Neuroevolution: from architectures to learning. Evolutionary Intelligence, 1(1):47–62, 2008.
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E. Jang, S. Gu, and B. Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144, 2016.
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S. Xie, H. Zheng, C. Liu, and L. Lin. SNAS: stochastic neural architecture search. arXiv preprint arXiv:1812.09926, 2018.
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "PERSONALIZED NEURAL ARCHITECTURE SEARCH FOR FEDERATED LEARNING ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
176,
|
| 8 |
+
99,
|
| 9 |
+
823,
|
| 10 |
+
146
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
183,
|
| 19 |
+
170,
|
| 20 |
+
398,
|
| 21 |
+
198
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
452,
|
| 31 |
+
234,
|
| 32 |
+
544,
|
| 33 |
+
251
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "Federated Learning (FL) is a recently proposed learning paradigm for decentralized devices to collaboratively train a predictive model without exchanging private data. Existing FL frameworks, however, assume a one-size-fit-all model architecture to be collectively trained by local devices, which is determined prior to observing their data. Even with good engineering acumen, this often falls apart when local tasks are different and require diverging choices of architecture modelling to learn effectively. This motivates us to develop a novel personalized neural architecture search (NAS) algorithm for FL. Our algorithm, FEDPNAS, learns a base architecture that can be structurally personalized for quick adaptation to each local task. We empirically show that FEDPNAS significantly outperforms other NAS and FL benchmarks on several real-world datasets. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
232,
|
| 42 |
+
265,
|
| 43 |
+
766,
|
| 44 |
+
417
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
176,
|
| 54 |
+
444,
|
| 55 |
+
336,
|
| 56 |
+
460
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Federated Learning (FL) (McMahan et al., 2017) is a variant of distributed learning where the objective function can be decomposed into a linear combination of $M$ local objective functions. Each function depends on its private data hosted by a local client and a set of shared parameters $w$ , ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
474,
|
| 66 |
+
825,
|
| 67 |
+
517
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "equation",
|
| 73 |
+
"img_path": "images/e46a32c199c69a51110d2f9b14c6f20b9fa55e0ee8f5076013fff7470987204d.jpg",
|
| 74 |
+
"text": "$$\n\\underset { w } { \\mathrm { a r g m i n } } \\ : \\mathcal { L } ( w ) \\equiv \\underset { w } { \\mathrm { a r g m i n } } \\ : \\sum _ { i = 1 } ^ { M } \\mathcal { L } _ { i } ( w \\mid \\mathcal { D } _ { i } ) ,\n$$",
|
| 75 |
+
"text_format": "latex",
|
| 76 |
+
"bbox": [
|
| 77 |
+
344,
|
| 78 |
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522,
|
| 79 |
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650,
|
| 80 |
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566
|
| 81 |
+
],
|
| 82 |
+
"page_idx": 0
|
| 83 |
+
},
|
| 84 |
+
{
|
| 85 |
+
"type": "text",
|
| 86 |
+
"text": "where $\\mathcal { D } _ { i }$ denotes the $i ^ { \\mathrm { t h } }$ local training dataset comprising input-output tuples $( x , y )$ . In a standard supervised learning task where the predictive model is modeled as a fixed deep neural network $\\psi$ with learnable weights $w$ , let $\\ell ( x , y )$ denote the loss incurred by predicting $\\psi ( x ; w )$ when the true output is $y$ . The expected loss of $\\psi ( x ; w )$ on $\\mathcal { D } _ { i }$ is given as ",
|
| 87 |
+
"bbox": [
|
| 88 |
+
173,
|
| 89 |
+
571,
|
| 90 |
+
825,
|
| 91 |
+
628
|
| 92 |
+
],
|
| 93 |
+
"page_idx": 0
|
| 94 |
+
},
|
| 95 |
+
{
|
| 96 |
+
"type": "equation",
|
| 97 |
+
"img_path": "images/90e0dc9100a6fd7943eab9ddf8a785feb9537549e1df351e8a99038eb3f1c845.jpg",
|
| 98 |
+
"text": "$$\n\\begin{array} { r l r } { \\mathcal { L } _ { i } ( w \\mid \\mathcal { D } _ { i } ) } & { = } & { \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } _ { i } } \\Big [ \\ell ( x , y ; \\psi ) \\Big ] ~ . } \\end{array}\n$$",
|
| 99 |
+
"text_format": "latex",
|
| 100 |
+
"bbox": [
|
| 101 |
+
361,
|
| 102 |
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|
| 103 |
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635,
|
| 104 |
+
661
|
| 105 |
+
],
|
| 106 |
+
"page_idx": 0
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "This is not applicable to scenarios where local models are expected to solve different tasks which are similar in broad sense yet diverge in finer details. For example, consider the task of recognizing the outcome of a coin flip given images collected by two clients: one capture the coin from above, the other from below. This setting implies that when the same input image is provided by both clients, the correct classifications must be the opposite of one another. However, since existing FL methods converge on a single model architecture and weight, there can only be one predictive outcome which cannot satisfy both tasks. ",
|
| 111 |
+
"bbox": [
|
| 112 |
+
173,
|
| 113 |
+
671,
|
| 114 |
+
825,
|
| 115 |
+
770
|
| 116 |
+
],
|
| 117 |
+
"page_idx": 0
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "To relax this constraint, the recent work of Fallah et al. (2020) extends FL by incorporating ideas from meta learning (Finn et al., 2017) which results in a new framework of personalized FL. The new framework can accommodate for such task heterogeneity but still requires all client models to agree on a single architecture beforehand, which is sub-optimal. To address this shortcoming, one naive idea is to adopt existing ideas in Neural Architecture Search (NAS) via Reinforcement Learning (Zoph and Le, 2016; Pham et al., 2018) which act as an outer loop to the existing FL routine. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
174,
|
| 124 |
+
776,
|
| 125 |
+
825,
|
| 126 |
+
861
|
| 127 |
+
],
|
| 128 |
+
"page_idx": 0
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "However, this simple approach does not allow client models to adapt to local tasks on an architecture level and is often not preferred due to the cost of repeated FL training. This paper proposes a novel personalized NAS algorithm for federated learning, which generalizes ideas in respective areas of NAS (Zoph and Le, 2016; Pham et al., 2018) originally developed for single-task scenarios, and ",
|
| 133 |
+
"bbox": [
|
| 134 |
+
174,
|
| 135 |
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867,
|
| 136 |
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823,
|
| 137 |
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924
|
| 138 |
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],
|
| 139 |
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"text": "FL (Fallah et al., 2020) under a unified len of federated personalized neural architecture search (FEDPNAS). ",
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"text": "In particular, to customize the model architecture for each task in the FL workflow, FEDPNAS first represents the model architecture for each task as a sub-network sampled from a large, overparameterized network. The sampling distribution is (collaboratively) learned along with the parameters of the sampled network via a generalization of the recently proposed Discrete Stochastic NAS (DSNAS) method (Hu et al., 2020). Unlike DSNAS, which lacks the ability to customize architecture for individual tasks, our generalized FEDPNAS incorporates model personalization on an architecture level. Our contributions include: ",
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"text": "1. A novel architecture that factorizes into a base component (shared across tasks) and a personalizable component, which respectively capture the task-agnostic and task-specific information (Section 3.2). ",
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"text": "2. A context-aware sampling distribution conditioned on specific task instance, which captures taskspecific information and naturally incorporates personalization into architecture search (Section 3.4). ",
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"text": "3. An FL algorithm that optimizes for a common architecture, followed by a personalization phase where each client subsequently adapts only the personalized component to fit its own task via finetuning with local data (Section 3.1). To ensure that the common architecture distribution converges at a vantage point that is relevant and beneficial to all clients, we generalize the vanilla FL objective in Eq. equation 1 such that local gradient steps directly optimize for expected improvement resulting from future fine-tuning (Section 3.3). ",
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"text": "4. A theoretical perspective on our FL objective (Section 3.5 and thorough empirical analysis showing significant performance gain compared to state-of-the-art FL and NAS methods (Section 4). ",
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"text": "2 RELATED WORKS ",
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"text": "2.1 TWO-STAGE NEURAL ARCHITECTURE SEARCH ",
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"text": "Most existing NAS frameworks separately optimize for the optimal architecture and its parameters in two stages: searching and evaluation. The former stage usually employs evolutionary-based strategies (Floreano et al., 2008; Real et al., 2019), Bayesian optimization surrogates (Bergstra et al., 2013; Hu et al., 2018) or Reinforcement Learning controllers (Baker et al., 2016; Zoph and Le, 2016; Pham et al., 2018) to propose candidate architectures based on random mutations and/or observed experience; while the latter optimizes the parameters of these architectures given task data and provide feedback to improve the search agent. Naturally, an extension of such methods to the FL setting is through distributing the evaluation workload over many clients, which does not require exposing private data. In practice, however, two-stage federated NAS frameworks are generally not suitable for the personalized FL setting for two reasons: (a) the clients often lack the computational capacity to repeatedly optimize the parameters for many candidate architectures; and (b) the clients have to converge on a single architecture proposed by the central search agent. ",
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"text": "2.2 DISCRETE STOCHASTIC NEURAL ARCHITECTURE SEARCH ",
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"text": "Discrete stochastic neural architecture search (DSNAS) (Hu et al., 2020) addresses the computational issue of two-stage NAS by jointly optimizing the optimal architecture and its weight in an end-to-end fashion, which allows users to continually train a single network on demand over time as opposed to performing full parameter optimization for every candidate until a good architecture is discovered. ",
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"text": "The main idea of DSNAS is to combine weight training for an over-parameterized master architecture with discrete computational path sampling. DSNAS parameterizes the master architecture as a stack of modular cells: $\\psi ( x ) \\bar { = } \\psi _ { C } \\circ \\bar { \\psi } _ { C - 1 } \\cdot \\cdot \\cdot \\circ \\psi _ { 1 } ( \\bar { x } ) ^ { 1 }$ , where $x$ is an arbitrary input, $C$ is the number of cells, $\\psi _ { t }$ denotes the $t ^ { \\mathrm { t h } }$ cell in the stack, and $\\circ$ denotes the compositional operator. The inner computation of $\\psi _ { t }$ is in turn characterized by a directed acyclic graph (DAG) with $V$ nodes $\\{ v _ { i } \\} _ { i = 1 } ^ { | V | }$ , where each node represents some intermediate feature map. For each directed edge $( v _ { i } , v _ { j } )$ ",
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"type": "image",
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"img_path": "images/718e70d9fa976c50deca97062d523901e363ff96a3953e515fd8e35fb90486f0.jpg",
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"image_caption": [
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"Figure 1: Our proposed method FEDPNAS consists of (1) a federated learning phase, where each client updates both the base component $( \\psi ^ { b } )$ and the personalized component $( \\psi _ { p } )$ the architecture using the FEDPNAS update (Section 3.3) and sends its parameters to the central server for aggregation; and (2) a fine-tune phase, where each client updates only the personalized component of the architecture using standard gradient update. "
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"text": "there is an associated list of $D$ possible network operations $\\mathbf O _ { i j } = \\left[ o _ { i j } ^ { 1 } , o _ { i j } ^ { 2 } \\ldots o _ { i j } ^ { D } \\right] ^ { 2 }$ where each operainput on w $o _ { i j } ^ { k }$ rms ). W $v _ { i }$ to rec $v _ { j }$ . Here, ively de $v _ { 1 }$ corresponds to the oe intermediate nodes $\\psi _ { t - 1 }$ (orhere $x$ $t = 1$ $\\begin{array} { r } { v _ { j } = \\sum _ { i = 1 } ^ { j - 1 } \\mathbf { Z } _ { i j } ^ { \\top } \\mathbf { O } _ { i j } ( v _ { i } ) } \\end{array}$ distribution learnable. E $\\mathbf { O } _ { i j } ( v _ { i } ) \\triangleq \\big [ o _ { i j } ^ { 1 } ( v _ { i } ) , o _ { i j } ^ { 2 } ( v _ { i } ) \\dots o _ { i j } ^ { D } ( v _ { i } ) \\big ]$ $p ( \\mathbf { Z } \\mid \\mathbf { \\pi } \\mathbf { \\Pi } \\mathbf { \\Pi } )$ where the event probabi learning the distribution and $\\mathbf { Z } _ { i j }$ is a one-hot vector sampled from the categorical $\\mathbf { H } = \\{ \\pi _ { 1 } , \\pi _ { 2 } , . . . , \\pi _ { D } \\mid \\sum _ { i = 1 } ^ { D } \\pi _ { i } = 1 \\}$ ares or $p ( \\mathbf { Z } )$ \nsub-graphs of the original DAG that correspond to high-performing, compact architecture from the over-parameterized master network. Sampling discrete random variables from $p ( \\mathbf { Z } )$ , however, does not result in a gradient amenable to back-propagation. To sidestep this issue, DSNAS adopts the straight-through Gumbel-softmax trick (Jang et al., 2016), which re-parameterizes the $k ^ { \\mathrm { t h } }$ index of the one-hot variable as $\\mathbf { Z } _ { i j } [ k ] = \\mathbb { I } \\left( k \\triangleq { \\arg \\operatorname* { m a x } _ { t } } \\Big [ g _ { t } + \\log \\pi _ { t } \\Big ] \\right)$ , where $g _ { t } \\sim \\mathrm { G u m b e l } ( 0 , 1 )$ . While this forward computation does not have a gradient by itself, we can estimate the gradient through a proxy during the backward pass: ",
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"text": "$$\n\\nabla { \\mathbf Z } _ { i j } [ k ] ~ \\simeq ~ \\nabla \\tilde { \\mathbf Z } _ { i j } [ k ] ~ \\triangleq ~ \\nabla \\left( \\frac { \\exp \\big ( \\big ( g _ { k } + \\log \\pi _ { k } \\big ) / \\tau \\big ) } { \\sum _ { t = 1 } ^ { D } \\exp \\big ( \\big ( g _ { t } + \\log \\pi _ { t } \\big ) / \\tau \\big ) } \\right)\n$$",
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"text": "which is unbiased when converged as the temperature $\\tau$ is steadily annealed to 0 (Jang et al., 2016). This formulation, however, is not easily extended to the FL setting, especially when local tasks are not homogeneous. The key challenges in doing so are described in Section 3, together with our proposed approaches. ",
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"text": "3 PERSONALIZED NAS FOR FEDERATED LEARNING ",
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"text": "3.1 FEDERATED LEARNING OF DSNAS ",
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"text": "Let W denote the concatenated weights of all network operations in the network architecture. The set up above of DSNAS (Jang et al., 2016) is then naïvely extendable to a $\\mathrm { F L }$ setting via the following objective formulation: ",
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"text": "$$\n\\underset { \\mathbf { W } , \\mathbf { \\Pi } } { \\arg \\operatorname* { m i n } } \\mathcal { L } ( \\mathbf { W } , \\mathbf { \\Pi } \\mathbf { I } ) \\equiv \\underset { \\mathbf { W } , \\mathbf { \\Pi } \\mathbf { \\Pi } } { \\arg \\operatorname* { m i n } } \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\mathcal { L } _ { i } ( \\mathbf { W } , \\mathbf { \\Pi } \\mathbf { I } \\mid \\mathcal { D } _ { i } ) .\n$$",
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"text": "McMahan et al. (2017) optimizes this objective by alternating between (a) central agent broadcasting aggregated weights to local clients and (b) local clients sending gradient descent updated weights (given local data) to the central agent for aggregation. This, however, implies that after the last central aggregation step, all clients will follow the same architecture distribution induced by the final broadcasted copy of W and Π. As previously argued, this is not optimal in a heterogenous task setting which requires task-specific adaptation for local clients to achieve good performance. ",
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"text": "Furthermore, having the same sampling distribution $p ( \\mathbf { Z } )$ regardless of context (i.e., feature maps received as cell input) limits the architecture discovery to those that perform reasonably on average over the entire dataset. However, we remark that restricting the architecture to be the same for every input samples is unnecessary and undermines the expressiveness of an over-parameterized search space. On the other hand, letting the architecture be determined on a per-sample basis makes better use of the search space and potentially improves the predictive performance. ",
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"type": "text",
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"text": "The focus of this work is therefore to incorporate both task-wise and context-wise personalization to federated neural architecture search in multitask scenarios, which is achieved through our proposed algorithm FEDPNAS. In general, FEDPNAS functions similarly to the vanilla FEDDSNAS algorithm described above, with an addition of a fine-tuning phase at each local client after the FL phase to adapt the aggregated common model for local task data, as shown in Fig. 1. To make this work, however, we need to address the following key challenges: ",
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"text": "C1. First, as previously argued in Section 1, tasks across federated clients tend to share similarities in broad sense, and diverge in finer details. A good federated personalization search space, therefore, need to capture this fundamental observation through design and appropriate resource distribution. We address this challenge in Section 3.2. ",
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"text": "C2. Second, a major advantage of having an over-parameterized architecture search space is the flexibility of having specific computation paths for different samples, which is not exploited by DSNAS as reflected in its choice of context-independent sampling distribution $p ( \\mathbf { Z } )$ . To address this, Section 3.4 proposes a novel parameterization of $p ( \\mathbf { Z } )$ to incorporate context information into operator sampling. ",
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"text": "C3. Last, while the fine-tuning phase is designed to incorporate task-personalization, there is no guarantee that the common model can be quickly adapted to client tasks (Fallah et al., 2020). The common model may end up in a localization that favors one client over another, which makes it difficult for the latter to fine-tune. To address this concern, Section 3.3 proposes a new personalized federated NAS objective inspired by Finn et al. (2017) to optimize the common model in anticipation of further fine-tuning by the client models. ",
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"text": "3.2 PERSONALIZABLE ARCHITECTURE SEARCH SPACE ",
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"text": "Similar to DSNAS (Hu et al., 2020), our framework adopts a cell-based representation (Section 2.2) to trade-off search space expressiveness for efficiency, which is extremely suitable for FL where clients tend to have low-end computational capacity. Unlike the original design which assumes similar role for every cell in the architecture stack (i.e., as reflected by their choice of fully factorizable path sampling distribution $p ( \\mathbf { Z } ) ,$ , we instead split our cell stack into two components with separate metaroles catering to the ftask: (a) a base stack $\\psi _ { \\mathrm { b } } = \\{ \\dot { \\psi } _ { 1 } ^ { b } , \\psi _ { 2 } ^ { b } \\cdot \\hdots \\psi _ { C _ { b } } ^ { b } \\}$ which aims to capture the broad commonalities of data samples across client tasks; and (b) personalized stack $\\psi _ { \\mathrm { { p } } } ~ = ~ \\{ \\psi _ { 1 } ^ { p } , \\psi _ { 2 } ^ { p } \\ldots \\psi _ { C _ { p } } ^ { p } \\}$ which will be fine-tuned with local data to capture task-specific details. ",
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"type": "image",
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"img_path": "images/3116f74b935bbe36c806df305e40cbbc96cf386169136d98b85b7351797e33ce.jpg",
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"image_caption": [
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"Figure 2: Feature mapping down the component stacks of our architecture space. Every base cell takes as inputs (a) the outputs from its immediate predecessor and (b) the one before it through a skip-ahead connection. On the other hand, every personalization cell takes as input only the output from the previous cell. "
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"type": "text",
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"text": "We explain the main difference between these components to account for different level of expressiveness requirements below: ",
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"type": "text",
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"text": "Base stack. Every cell $\\psi _ { t } ^ { b }$ in the base stack takes as inputs the outputs of its previous two cells $\\psi _ { t - 1 } ^ { b }$ and $\\psi _ { t - 2 } ^ { b }$ (replaced with raw input $x$ when necessary for $t \\leq 2$ ). The output of the skip-ahead cell $\\psi _ { t - 2 } ^ { b }$ is additionally passed through a $1 \\times 1$ convolution layer as a cost-effective way to control the number of channels. Additionally, the operators available to the base cell include large convolution layers with size $5 \\times 5$ and $7 \\times 7$ . To compensate for the growing number of channels, we periodically employ a reduction convolution (with stride larger than 1) similar to DSNAS (Hu et al., 2020) to reduce the feature dimension down the stack. ",
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"text": "",
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"text": "Personalized stack. As opposed to the design of the base cells above, every cell $\\psi _ { t } ^ { p }$ in the personalized stack has minimal expressiveness. That is, $\\psi _ { t } ^ { p }$ excludes large operators and only takes as input the output of its immediate predecessor $\\psi _ { t - 1 } ^ { p }$ (or $\\dot { \\psi } _ { C _ { b } } ^ { b }$ when $t = 1$ ). There are two reasons for this choice. First, as the fine-tuning phase has access to fewer data samples than the federated phase, having a more compact fine-tuning space helps to improve the rate of convergence. Second, as we will discuss in Section 3.4 below, our personalized FL objective requires the Hessian of the personalized parameters, which is computationally expensive. As such, we only restrict the personalization to happen on the more compact personalized stack. ",
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"type": "text",
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"text": "3.3 PERSONALIZED FEDERATED LEARNING OBJECTIVE ",
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"text_level": 1,
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"text": "Unlike FEDAVERAGING (McMahan et al., 2017), which assumes the clients will follow the consensus base model obtained after the federated phase, FEDPNAS expects clients to further personalize the base model with local task data. That said, while the base model is trained to work well in the expected sense over the task distribution, there is no guarantee that it is a good initial point for every client model to improve upon via fine-tuning. To address this, we adopt the concept of training in anticipation of future adaptation introduced by MAML (Finn et al., 2017). That is, during client update, instead of optimizing the loss with respect to the same consensus weight, each client will instead optimize the weight perturbed by a small gradient step in the fine-tuning direction. ",
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"type": "text",
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"text": "Algorithm 1 FEDPNAS - FEDERATED PHASE ",
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"text_level": 1,
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"type": "text",
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"text": "1: CENTRALAGGREGATION: \n2: $\\theta _ { 0 } \\gets$ INITIALIZEPARAMETER \n3: for $t = 1 , 2 \\dots T _ { s }$ do \n4: for $k = 1 , 2 \\dots M$ in parallel do \n5: $\\theta _ { t } ^ { k } \\gets \\mathrm { C L I E N T U P D A T E } ( k , \\theta _ { t - 1 } )$ \n$\\begin{array} { r } { \\theta _ { t } \\sum _ { k = 1 } ^ { M } \\frac { 1 } { M } \\theta _ { t } ^ { k } } \\end{array}$ \n7: CLIENTUPDATE $( k , \\theta )$ : \n8: for $t = 1 , 2 \\dots T _ { \\mathrm { c } }$ do \n9: for batch $( x , y ) \\in \\mathcal { D } _ { k }$ do \n10: $\\mathcal { L } _ { k }$ $\\mathrm { , } \\nabla \\mathcal { L } _ { k } \\gets \\mathrm { E v A L } ( x , y ; \\theta _ { b } , \\theta _ { p } )$ \n11: $ { \\widetilde { \\theta } } _ { p } \\gets { \\mathrm { G R A D U P D A T E } } \\big ( \\nabla _ { { \\widetilde { \\theta } } _ { p } } \\mathcal { L } _ { k } \\big )$ \n12: $\\tilde { \\mathcal { L } } _ { k } , \\nabla \\tilde { \\mathcal { L } } _ { k } \\gets \\mathrm { E v A L } ( x , y ; \\theta _ { b } , \\tilde { \\theta } _ { p } )$ \n13: $\\nabla _ { \\boldsymbol { \\theta } _ { p } } \\tilde { \\mathcal { L } } _ { k } \\gets \\mathrm { E Q } . \\mathrm { 5 }$ \n14: θb, θp ← GRADUPDATE(∇θb,θpL˜k) \n15: return $\\theta$ to central server ",
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| 578 |
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{
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"type": "text",
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| 581 |
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"text": "Algorithm 2 FEDPNAS - EVAL ",
|
| 582 |
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"text_level": 1,
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"type": "text",
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"text": "1: Input: x, y, θ \n2: SKIP, PREV ← x, x \n3: W,Π ← θ \n4: for CELL $\\psi \\in \\psi$ do \n5: Z ← SAMPLEOPS $( x , { \\mathrm { P R E V } } ; \\mathbf { I I } )$ \n6: ψ ← EXTRACTCHILDNET $( \\mathbf { Z } , \\mathbf { W } )$ \n7: if $\\psi \\in \\psi _ { b }$ then \n8: OUTPUT $ \\psi ( \\mathrm { P R E V } , \\mathrm { S K I P } )$ \n9: else if $\\psi \\in \\psi _ { p }$ then \n10: $\\mathrm { O U T P U T } \\psi ( \\mathrm { P R E V } )$ \n11: $\\mathbf { S K I P } \\mathbf { P R E V }$ \n12: PREV $\\gets$ OUTPUT \n13: $\\mathcal { L } \\gets \\mathrm { L o s s } ( \\mathbf { O u T P U T } , y )$ \n14: $\\nabla { \\mathcal { L } } \\gets \\mathbf { B A C K P R O P } ( { \\mathcal { L } } )$ \n15: return L, ∇L ",
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| 594 |
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"text": "For simplicity, let $\\theta = \\{ \\theta _ { b } , \\theta _ { p } \\}$ respectively denote all trainable parameters of the base stack and the personalized stack, i.e., $\\partial _ { b } = \\mathbf { \\bar { \\{ W } } _ { b } , \\mathbf { \\bar { \\Pi } } _ { b } \\} , \\boldsymbol { \\hat { \\theta _ { p } } } = \\{ \\mathbf { W } _ { p } , \\mathbf { \\Pi } _ { \\Pi _ { p } } \\}$ . The personalized $\\mathrm { F L }$ objective at client $i$ is then given by $\\tilde { \\mathcal { L } } _ { i } ( \\theta _ { b } , \\theta _ { p } ) \\triangleq \\mathcal { L } _ { i } ( \\theta _ { b } , \\tilde { \\theta } _ { p } )$ where $\\tilde { \\theta } _ { p } \\triangleq \\tilde { \\theta } _ { p } - \\eta \\nabla _ { \\theta _ { p } } \\mathcal { L } _ { i } ( \\theta _ { b } , \\theta _ { p } )$ adjusts the parameters of the personalized component to account for a small fine-tuning gradient step. The adjusted local loss only depends on the respective client data and is amenable to federated learning. The local update gradient, however, involves a Hessian term whose computation is expensive to repeat over many epochs. To circumvent this problem, we use the first-order Taylor approximation to estimate the Hessian term by the outer product of Jacobian, which results in a gradient that requires exactly two forward/backward passes to compute: ",
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{
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| 614 |
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"type": "equation",
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| 615 |
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"img_path": "images/07e81c7ee6f7e22b845daf04fe1a061bb16af80b33ebb744d4e02812683257fa.jpg",
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| 616 |
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"text": "$$\n\\begin{array} { r c l } { \\nabla _ { \\theta _ { p } } \\tilde { \\mathcal { L } } _ { i } } & { = } & { \\left( \\nabla _ { \\theta _ { p } } \\tilde { \\theta } _ { p } \\right) \\left( \\nabla _ { \\tilde { \\theta } _ { p } } \\tilde { \\mathcal { L } } _ { i } \\right) } \\\\ & { = } & { \\left( \\mathbf { I } - \\eta \\nabla _ { \\theta _ { p } } ^ { 2 } \\mathcal { L } _ { i } \\right) \\left( \\nabla _ { \\tilde { \\theta } _ { p } } \\tilde { \\mathcal { L } } _ { i } \\right) } \\\\ & { \\simeq } & { \\left( \\mathbf { I } - \\eta \\nabla _ { \\theta _ { p } } ^ { \\top } \\mathcal { L } _ { i } \\nabla _ { \\theta _ { p } } \\mathcal { L } _ { i } \\right) \\left( \\nabla _ { \\tilde { \\theta } _ { p } } \\tilde { \\mathcal { L } } _ { i } \\right) } \\end{array}\n$$",
|
| 617 |
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"text_format": "latex",
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| 618 |
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| 627 |
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"type": "text",
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| 628 |
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"text": "where $\\mathcal { L } _ { i }$ and $\\tilde { \\mathcal { L } } _ { i }$ are short-hands for $\\mathcal { L } _ { i } ( \\theta _ { b } , \\theta _ { p } )$ and $\\tilde { \\mathcal { L } } _ { i } ( \\theta _ { b } , \\theta _ { p } )$ respectively. The FL phase of our FEDPNAS framework is detailed in Alg. 1. An instance of FEDPNAS’s forward and backward pass which sequentially unrolls down the component stacks, alternating between sampling and evaluation, is in turn given in Alg. 2. ",
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| 629 |
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| 639 |
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"text": "3.4 CONTEXT-AWARE OPERATOR SAMPLER",
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"text": "The choice of a fully factorizable sampling distribution $p ( \\mathbf { Z } )$ in DSNAS follows that of SNAS (Xie et al., 2018), which argues that the Markov assumption for $p ( \\mathbf { Z } )$ is not necessary because NAS has fully delayed rewards in a deterministic environment. However, this generally only holds for two-stage NAS (Section 2.1) and does not apply to end-to-end frameworks such as SNAS and DSNAS. We instead to take advantage of the over-parameterized architecture via factorizing the conditional $p ( \\mathbf { Z } \\mid x )$ , which takes into account the temporal dependency of structural decisions: ",
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"type": "equation",
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"img_path": "images/95c56dc93d7337b4f6b2a359ce8f707c38cc25ef6f669c8131b975f1ad1da60f.jpg",
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| 663 |
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"text": "$$\n\\begin{array} { l l l } { { p ( { \\bf Z } \\mid x ) } } & { { = } } & { { \\displaystyle p ( { \\bf Z } _ { 1 } \\mid x ) \\prod _ { t = 2 } ^ { C } p ( { \\bf Z } _ { t } \\mid { \\bf Z } _ { t - 1 } \\dots { \\bf Z } _ { 1 } , x ) } } \\\\ { { } } & { { } } & { { } } \\\\ { { \\displaystyle } } & { { \\simeq } } & { { \\displaystyle p ( { \\bf Z } _ { 1 } \\mid x ) \\prod _ { t = 2 } ^ { C } p ( { \\bf Z } _ { t } \\mid v _ { 1 } ^ { t } , x ) } } \\\\ { { } } & { { } } & { { } } \\\\ { { \\displaystyle } } & { { = } } & { { \\displaystyle p ( { \\bf Z } _ { 1 } \\mid x ) \\prod _ { t = 2 } ^ { C } \\prod _ { ( i , j ) } p ( { \\bf Z } _ { i j } ^ { t } \\mid v _ { 1 } ^ { t } , x ) , } } \\end{array}\n$$",
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| 664 |
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"text_format": "latex",
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| 665 |
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"type": "text",
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| 675 |
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"text": "where $\\mathbf { Z } _ { t }$ , $\\mathbf { Z } _ { i j } ^ { t }$ and $v _ { 1 } ^ { t }$ respectively denote all the samples, the sample at edge $( i , j )$ and the input at cell $\\psi _ { t }$ . We have also assumed a single stack setting since the parameterization of $p ( \\mathbf { Z } )$ does not differ between base and personalized cells. ",
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"type": "text",
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| 686 |
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"text": "To reduce computational complexity, instead of conditioning the samples of subsequent cells on previous $\\mathbf { Z }$ samples, we approximate $p ( \\mathbf { Z } _ { t } \\mid \\mathbf { Z } _ { t - 1 } \\ldots \\mathbf { Z } _ { 1 } , x ) \\bumpeq p ( \\mathbf { Z } _ { t } \\mid v _ { 1 } ^ { t } , x )$ by the assumption that the cell contents are conditionally independent given the immediate cell input and the original input. Finally, we assume that $p ( \\mathbf { Z } _ { t } \\mid v _ { 1 } ^ { t } , x )$ is fully factorizable across edges in the same cell and parameterize $p ( \\mathbf { Z } _ { i j } ^ { t } \\mid v _ { 1 } ^ { t } , x ) = \\phi ^ { ( i , j ) } ( v _ { 1 } ^ { t } , x )$ where $\\phi$ is a deep classification network whose output dimension equal the number of edges in cell $\\psi _ { t }$ . Samples of $\\mathbf { Z } _ { i j } ^ { t }$ can then be generated using the straight-through Gumbel-softmax reparameterization similar to Jang et al. (2016). ",
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| 687 |
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"text": "3.5 THEORETICAL CONNECTION TO STANDARD GRADIENT UPDATE ",
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| 698 |
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"text": "Finally, we analyze the connection of our gradient update framework to the standard gradient update, and explain why it is critical in achieving a vantage point that improves average objective value without compromising any local objective. First, we note that the gradient update Eq. 5 in Section 3.3 at the $t$ -th iteration can be written as: ",
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| 720 |
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| 721 |
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"text": "$$\n\\begin{array} { r c l } { \\displaystyle \\theta _ { p } ^ { t + 1 } } & { = } & { \\displaystyle \\theta _ { p } ^ { t } - \\frac { \\eta _ { 2 } } { M } \\sum _ { i = 1 } ^ { M } \\nabla _ { \\theta _ { p } } \\mathcal { L } _ { i } \\big ( \\theta _ { b } ^ { t } , \\tilde { \\theta } _ { p , i } ^ { t } \\big ) + \\frac { \\eta _ { 1 } \\eta _ { 2 } } { M } \\sum _ { i = 1 } ^ { M } \\alpha _ { i } \\nabla _ { \\theta _ { p } } \\mathcal { L } _ { i } \\big ( \\theta _ { b } ^ { t } , \\theta _ { p } ^ { t } \\big ) , } \\\\ { \\displaystyle \\theta _ { b } ^ { t + 1 } } & { = } & { \\displaystyle \\theta _ { b } ^ { t } - \\frac { \\eta _ { 2 } } { M } \\sum _ { i = 1 } ^ { M } \\nabla _ { \\theta _ { b } } \\mathcal { L } _ { i } \\big ( \\theta _ { b } ^ { t } , \\tilde { \\theta } _ { p , i } ^ { t } \\big ) , } \\end{array}\n$$",
|
| 722 |
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"text_format": "latex",
|
| 723 |
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"bbox": [
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| 727 |
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| 732 |
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"type": "text",
|
| 733 |
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"text": "where $\\tilde { \\theta } _ { p , i } ^ { t }$ denotes the $i$ -th local personalized parameters; $\\eta _ { 1 }$ and $\\eta _ { 2 }$ are two separate learning rates and $\\boldsymbol { \\alpha } _ { i } \\triangleq \\nabla _ { \\boldsymbol { \\theta } _ { p } } ^ { \\intercal } \\mathcal { L } ( \\boldsymbol { \\theta } _ { b } ^ { t } , \\boldsymbol { \\theta } _ { p , i } ^ { t } ) \\nabla _ { \\boldsymbol { \\theta } _ { p , i } } \\mathcal { L } ( \\boldsymbol { \\theta } _ { b } ^ { t } , \\tilde { \\boldsymbol { \\theta } } _ { p , i } ^ { t } )$ (See Appendix A for detailed derivation). This implies that our federated personalize update corresponds to a federated update scheme with three gradient steps: (1) $\\theta _ { p }$ takes a local gradient (w.r.t. locally updated parameters) step of size $\\eta _ { 1 }$ ; (2) Both $\\theta _ { b }$ and $\\theta _ { p }$ take a federated gradient (w.r.t. server-wide parameters averaging) step of size $\\eta _ { 2 }$ ; $( 3 ) \\theta _ { p }$ takes a weighted federated gradient step of size $\\eta _ { 1 } \\eta _ { 2 }$ , where the weight of client $i$ is given by $\\alpha _ { i }$ . ",
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"type": "text",
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"text": "Explicitly, Step 1 and 2 together comprise a special instance of FEDAVERAGING (McMahan et al., 2017), where $\\theta _ { b }$ take one gradient step for every two gradient steps taken by $\\theta _ { p }$ . Step 3, on the other hand, takes the information of the two gradient steps of $\\theta _ { p }$ and adjust the magnitude of the local gradient step (whose direction is given by $\\nabla _ { \\theta _ { p } } \\mathcal { L } _ { i } ( \\theta _ { b } ^ { t } , \\bar { \\theta } _ { p } ^ { t } ) )$ accordingly to trade-off between preserving local objective value and improving average objective value. We then theorize the scenario in which such an update is beneficial and state the following assumption to lay the foundation of our analysis: ",
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"text": "",
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| 764 |
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"type": "text",
|
| 766 |
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"text": "Assumption 1 For a fixed instance of $\\theta _ { b }$ , let $\\tilde { \\theta } _ { p , i } = \\theta _ { p , i } - \\eta \\nabla _ { \\theta _ { p } } \\mathcal { L } ( \\theta _ { b } , \\theta _ { p , i } )$ denote the personalized parameters after a local update step (i.e., step $^ { l }$ above) at client $i$ , then there exists a distribution $s$ on matrix $\\mathbf { S } \\in \\mathbb { R } ^ { k \\times | \\theta _ { p } | }$ that satisfies ",
|
| 767 |
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"bbox": [
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| 769 |
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|
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|
| 776 |
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"type": "equation",
|
| 777 |
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"img_path": "images/e21700b725bbae65bc8c894897c574cc558cc932dbf118b04215f210256f52fc.jpg",
|
| 778 |
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"text": "$$\n\\forall \\mathbf { x } \\in \\mathbb { R } ^ { n } , \\| \\mathbf { x } \\| _ { 2 } = 1 : \\mathbb { E } _ { \\mathbf { S } \\sim { \\mathcal { S } } } \\left[ | \\| \\mathbf { S } \\mathbf { x } \\| _ { 2 } ^ { 2 } - 1 | ^ { \\ell } \\right] \\ \\leq \\ \\epsilon ^ { \\ell } \\cdot \\delta ,\n$$",
|
| 779 |
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"text_format": "latex",
|
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|
| 789 |
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"type": "equation",
|
| 790 |
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"img_path": "images/302ed0fe3554dede9f35da9c497e6b3c0ce8a03567fa1f061e64c89c57d929c0.jpg",
|
| 791 |
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"text": "$$\n\\operatorname* { P r } _ { \\mathbf { S } \\in \\mathcal { S } } \\left( \\Big | \\nabla _ { \\theta _ { p } } \\mathcal { L } \\big ( \\theta _ { b } , \\tilde { \\theta } _ { p , i } \\big ) - \\mathbf { S } ^ { \\top } \\mathbf { S } \\Big ( \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\nabla _ { \\theta _ { p } } \\mathcal { L } \\big ( \\theta _ { b } , \\tilde { \\theta } _ { p , i } \\big ) \\Big ) \\Big | \\leq \\sqrt { \\frac { 6 } { k \\delta } } \\right) ~ \\geq ~ 1 - \\delta\n$$",
|
| 792 |
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"text_format": "latex",
|
| 793 |
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"bbox": [
|
| 794 |
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| 796 |
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| 798 |
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| 799 |
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|
| 800 |
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|
| 801 |
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|
| 802 |
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"type": "text",
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| 803 |
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"text": "where $k = \\mathcal { O } \\left( C \\| \\theta _ { b } - \\theta _ { b } ^ { * } \\| _ { 2 } ^ { - 2 } \\right)$ for some constant $C > 0$ and $\\theta _ { b } ^ { * }$ denotes the optimal base parameters. ",
|
| 804 |
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"bbox": [
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| 806 |
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| 810 |
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|
| 811 |
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|
| 813 |
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"type": "text",
|
| 814 |
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"text": "The above assumption implies that, as $\\theta _ { b }$ improves and better captures the broad similarity across tasks, the local personalized components will diverge to capture the differences. It then becomes less likely for the FEDAVG personalized gradient to capture all these differences simultaneously. That is, suppose there exists an affine transformation to reconstruct the local component $\\nabla _ { \\boldsymbol { \\theta } _ { p } } \\mathcal { L } _ { i } \\dot { ( \\theta _ { b } ^ { t } , \\tilde { \\theta } _ { p } ^ { t } ) }$ from the federated gradient 1M PMi=1 ∇θp Li(θtb, ˜θtp), then the rank of this affine transformation would be inversely proportionate to $\\lVert \\theta _ { b } - \\theta _ { b } ^ { * } \\rVert _ { 2 }$ . ",
|
| 815 |
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"bbox": [
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| 823 |
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| 824 |
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"type": "text",
|
| 825 |
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"text": "Finally, Proposition 1 below shows that when this assumption holds and $\\theta _ { b }$ converges to the optimal parameter $\\theta _ { b } ^ { * }$ (i.e., the error term tends to 0), then with very high probability, the coefficient $\\alpha _ { i }$ of the weighted federated gradient step (i.e., step 3 above) accurately captures the cosine similarity between the local gradient (step 1) and the federated gradient (step 2). ",
|
| 826 |
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"bbox": [
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|
| 833 |
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},
|
| 834 |
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{
|
| 835 |
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"type": "text",
|
| 836 |
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"text": "Proposition 1 Suppose assumption $^ { l }$ holds, then with probability at least $1 - 2 \\delta$ and normalized gradients, we have: ",
|
| 837 |
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"bbox": [
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| 838 |
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"page_idx": 6
|
| 844 |
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},
|
| 845 |
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{
|
| 846 |
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"type": "equation",
|
| 847 |
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"img_path": "images/8518cc1a25e7fbb7889ca9d800f160854230be5d789f0fa6d0b12b74ceffd67e.jpg",
|
| 848 |
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"text": "$$\n\\begin{array} { r l r } { \\bigg | \\alpha _ { i } - \\nabla _ { \\theta _ { p } } ^ { \\top } \\mathcal { L } _ { i } ( \\theta _ { b } ^ { t } , \\theta _ { p } ^ { t } ) \\mathbf { L } \\bigg | } & { = } & { \\mathcal { O } ( \\| \\theta _ { b } - \\theta _ { b } ^ { * } \\| / \\delta ) } \\end{array}\n$$",
|
| 849 |
+
"text_format": "latex",
|
| 850 |
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"bbox": [
|
| 851 |
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| 852 |
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| 855 |
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| 856 |
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| 857 |
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|
| 858 |
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{
|
| 859 |
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"type": "text",
|
| 860 |
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"text": "Proof. See Appendix B ",
|
| 861 |
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|
| 862 |
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| 863 |
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| 868 |
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|
| 869 |
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|
| 870 |
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"type": "text",
|
| 871 |
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"text": "This result has strong implication with respect to the scenario with multiple heterogeneous tasks, whose local gradients contradict in directions. Per this setting, we expect a standard federated gradient update scheme to encourage parameters drifting in the general direction of the majority (i.e., captured by the federated gradient), thus worsening the performance of tasks that are in the minority. Proposition 1, however, implies that whenever the local gradient contradicts the federated gradient, $\\alpha _ { i }$ will be close to the cosine similarity term, which is negative. This in turn results in a dampening effect on the federated gradient and helps to preserve the client performance on its own local task. ",
|
| 872 |
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"bbox": [
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{
|
| 881 |
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"type": "text",
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| 882 |
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"text": "4 EXPERIMENTS ",
|
| 883 |
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"text_level": 1,
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|
| 893 |
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"type": "text",
|
| 894 |
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"text": "This section describes our experiments to showcase the performance of FEDPNAS compared to different NAS and FL benchmarks on various scenarios. All of our empirical studies are conducted on two image recognition datasets: (a) the CIFAR-10 dataset (Krizhevsky et al., 2009) which aims to predict image labels from 10 classes given a train/test set of 50000/10000 colour images of dimension $3 2 \\times 3 2$ pixels; and (b) the MNIST dataset (LeCun et al., 2010) which aims to predict handwritten digits (i.e. 0 to 9) given a train/test set of 60000/10000 grayscale images of dimension $2 8 \\times 2 8$ pixels. Our search space entails $2 ^ { 4 0 }$ possible architectures, which is detailed in Appendix D. We compare two variants of our framework, CA-FEDPNAS (with context-aware operation sampler) and FEDPNAS (without the operation sampler), against: (a) FEDAVERAGING of a fixed architecture to justify the need for NAS in FL; (b) FEDDSNAS - the federated extension of DSNAS (Section 3.1) to show the effectiveness of our proposed context-aware sampler on NAS performance; and finally (c) CA-FEDDSNAS, which extends FEDDSNAS with our context-aware sampler. ",
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| 895 |
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"type": "text",
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"text": "On simulate heterogenous predictive tasks. To simulate this scenario, we first distribute the data i.i.d across clients (10000/2000 and 12000/2000 training/test images per client for CIFAR-10 and MNIST datasets respectively). Then, we independently apply a different transformation to each partitioned dataset. Input images within the same train/test set is subject to the same transformation. In both our experiments, the client datasets are subjected to rotations of $- 3 0 ^ { \\circ }$ , $- 1 5 ^ { \\circ } , 0 ^ { \\circ }$ , $1 5 ^ { \\circ }$ and $3 0 ^ { \\circ }$ respectively. This data generation protocol reflects a realistic and frequently seen scenario where independently collected data of the same phenomenon might contain systematic bias due to measurement errors and/or different collection protocols. Fig. 3 below shows the performance of all the methods in comparison, plotted against number of search epochs and averaged over the above rotated variants of CIFAR-10 and MNIST datasets. ",
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| 906 |
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| 914 |
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{
|
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"type": "image",
|
| 916 |
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"img_path": "images/08d3fd76b4111ac26e3461c4df7498289f564056a7c3a605aede072b6240b1a0.jpg",
|
| 917 |
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"image_caption": [
|
| 918 |
+
"Figure 3: Plotting average classification accuracy of various methods against no. training epochs on heterogeneous tasks derived from (a) MNIST dataset; and (b) CIFAR-10 dataset. Figure (c) compares cumulative running time of various methods against no. training epochs on CIFAR-10 dataset. "
|
| 919 |
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|
| 920 |
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"image_footnote": [],
|
| 921 |
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"bbox": [
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"type": "text",
|
| 931 |
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"text": "On the MNIST dataset (Fig. 3b), all methods eventually converge to a similar performance. Among the NAS benchmarks, FEDPNAS and FEDDSNAS both converge slower than FEDAVG and start off with worse performance in early iterations, which is expected since FEDAVG does not have to search for the architecture and it is likely that the default architecture is sufficient for the MNIST task. On the other hand, we observe that both CA-FEDPNAS and CA-FEDDSNAS converge much faster than their counterparts without the context-aware operation sampler component. This shows that making use of contextual information helps to quickly locate regions of high-performing architectures, especially on similar inputs. ",
|
| 932 |
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"bbox": [
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| 940 |
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|
| 941 |
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"type": "text",
|
| 942 |
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"text": "On the CIFAR-10 dataset (Fig. 3a), we instead observe significant gaps between the worst performing FEDAVG and other NAS methods. This is likely because the default architecture does not have sufficient learning capability, which confirms the need for customizing solutions. Among the NAS benchmarks, we again observe that both CA-FEDPNAS and CA-FEDDSNAS outperform their counterparts without our operation sampler, which confirms the intuition above. Most remarkably, our proposed framework CA-FEDPNAS achieves the best performance (0.8) and significantly outperformed both variants of federated DSNAS (0.71 for CA-FEDDSNAS and 0.63 for FEDDSNAS). ",
|
| 943 |
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| 950 |
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|
| 951 |
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|
| 952 |
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"type": "text",
|
| 953 |
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"text": "Lastly, Fig. 3c shows the runtime comparison between three methods on the CIFAR-10 experiment. In terms of sampling time, we observe that there is negligible overhead incurred by using our context-aware sampler (CA-FEDDSNAS vs. FEDDSNAS). The time incurred by our update (CA-FEDPNAS) scales by a constant factor compared to CA-FEDDSNAS since we use exactly one extra forward/backward pass per update. ",
|
| 954 |
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"bbox": [
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| 962 |
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|
| 963 |
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"type": "text",
|
| 964 |
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"text": "On objectives with varying heterogeneity. We expand the above study by investigating respective performance of CA-FEDPNAS and FEDDSNAS on tasks with varying levels of heterogeneity. At low level of heterogeneity, we deploy these methods on 5 sets of slightly rotated MNIST images. At high level of heterogeneity, we employ a more diverse set of transformations on MNIST images, such as hue jitter and large angle rotations of $9 0 ^ { \\circ }$ and $- 9 0 ^ { \\circ }$ . Table 1 show the respective result of each task from these two settings. We observe that our method CA-FEDPNAS achieves better performance on most tasks and the performance gaps on tasks with higher heterogeneity are more pronounced (i.e., up to $7 \\%$ improvement on ROTATE 90 task). This clearly shows the importance of architecture personalization when the training tasks are significantly different and justifies our research goal. ",
|
| 965 |
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{
|
| 974 |
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"type": "table",
|
| 975 |
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"img_path": "images/54d993c8bf9f3fa29d07bb7937339fab1a162de34e3ded9e8a7c9d00ec8acdc4.jpg",
|
| 976 |
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"table_caption": [
|
| 977 |
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"Table 1: Predictive accuracy of CA-FEDPNAS FEDDSNAS on tasks with varying heterogeneity levels. ROTATE X denotes a rotation transformation of $\\mathbf { X } ^ { \\circ }$ on client data; VANILLA denotes the original MNIST images; and HUEJITTER X denotes a hue jitter transformation of training images by a factor of X. The best performance in each row is in bold font. "
|
| 978 |
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],
|
| 979 |
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"table_footnote": [],
|
| 980 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>HETEROGENEITY</td><td rowspan=1 colspan=1>TASKDESCRIPTION</td><td rowspan=1 colspan=1>FEDDSNAS</td><td rowspan=1 colspan=1>CA-FEDPNAS</td></tr><tr><td rowspan=5 colspan=1>Low</td><td rowspan=1 colspan=1>ROTATE -30</td><td rowspan=1 colspan=1>0.947</td><td rowspan=1 colspan=1>0.978</td></tr><tr><td rowspan=1 colspan=1>ROTATE -15</td><td rowspan=1 colspan=1>0.973</td><td rowspan=1 colspan=1>0.976</td></tr><tr><td rowspan=1 colspan=1>VANILLA</td><td rowspan=1 colspan=1>0.988</td><td rowspan=1 colspan=1>0.985</td></tr><tr><td rowspan=1 colspan=1>ROTATE 15</td><td rowspan=1 colspan=1>0.986</td><td rowspan=1 colspan=1>0.987</td></tr><tr><td rowspan=1 colspan=1>ROTATE 30</td><td rowspan=1 colspan=1>0.972</td><td rowspan=1 colspan=1>0.981</td></tr><tr><td rowspan=5 colspan=1>HIGH</td><td rowspan=1 colspan=1>HUEJITTER-0.5</td><td rowspan=1 colspan=1>0.966</td><td rowspan=1 colspan=1>0.978</td></tr><tr><td rowspan=1 colspan=1>HUEJITTER 0.5</td><td rowspan=1 colspan=1>0.967</td><td rowspan=1 colspan=1>0.972</td></tr><tr><td rowspan=1 colspan=1>VANILLA</td><td rowspan=1 colspan=1>0.988</td><td rowspan=1 colspan=1>0.989</td></tr><tr><td rowspan=1 colspan=1>ROTATE -90</td><td rowspan=1 colspan=1>0.892</td><td rowspan=1 colspan=1>0.932</td></tr><tr><td rowspan=1 colspan=1>ROTATE 90</td><td rowspan=1 colspan=1>0.866</td><td rowspan=1 colspan=1>0.932</td></tr></table>",
|
| 981 |
+
"bbox": [
|
| 982 |
+
243,
|
| 983 |
+
101,
|
| 984 |
+
746,
|
| 985 |
+
271
|
| 986 |
+
],
|
| 987 |
+
"page_idx": 8
|
| 988 |
+
},
|
| 989 |
+
{
|
| 990 |
+
"type": "text",
|
| 991 |
+
"text": "On knowledge transfer to completely new tasks. Finally, we investigate a scenario where the architecture distributions discovered by CA-FEDPNAS and FEDDSNAS are required to generalize to completely unseen tasks. Particularly, we train both methods on five clients whose local data consist of 12000 slightly rotated CIFAR-10 images (i.e., in the range of $\\pm 3 0 ^ { \\circ }$ ), similar to the setting of the first experiment. During testing, however, we supply each local client with 2000 test images subjected to related but completely unseen transformations (i.e., $9 0 °$ and $- 9 0 ^ { \\circ }$ rotations). ",
|
| 992 |
+
"bbox": [
|
| 993 |
+
173,
|
| 994 |
+
363,
|
| 995 |
+
825,
|
| 996 |
+
446
|
| 997 |
+
],
|
| 998 |
+
"page_idx": 8
|
| 999 |
+
},
|
| 1000 |
+
{
|
| 1001 |
+
"type": "text",
|
| 1002 |
+
"text": "Our results are summarized in Table 2. First, we measure the performance of CA-FEDPNAS and FEDDSNAS without any weight retraining. When received no additional information from the unseen tasks, both methods perform poorly as expected. While CA-FEDPNAS achieves better predictive accuracy, the performance gap in this scenario is negligible. To provide additional clues for adaptation, albeit minimal, we retrain the weights of each local model with 200 images rotated according to respective unseen task description. Here, the parameters of our operator sampler component, (and respectively, FEDDSNAS’s categorical distribution parameters), are frozen to gauge the quality of the learned architecture distributions. Our results show that, with only 100 retraining iterations on limited data, CA-FEDPSNAS already outperforms FEDDSNAS $5 \\%$ and $8 \\%$ improvement respectively on two unseen tasks). This implies that CA-FEDPNAS has more accurately capture the broad similarity of the task spectrum through the personalized architecture distribution, which requires minimal additional information to successfully adapt to unseen tasks. ",
|
| 1003 |
+
"bbox": [
|
| 1004 |
+
173,
|
| 1005 |
+
454,
|
| 1006 |
+
825,
|
| 1007 |
+
621
|
| 1008 |
+
],
|
| 1009 |
+
"page_idx": 8
|
| 1010 |
+
},
|
| 1011 |
+
{
|
| 1012 |
+
"type": "table",
|
| 1013 |
+
"img_path": "images/931c6f41e8c0f1bcdf0b86c0f96967d82b117c3a89c72d81dd70c8b1f642352b.jpg",
|
| 1014 |
+
"table_caption": [
|
| 1015 |
+
"Table 2: Predictive accuracy (averaged over 5 clients) and standard deviation of CA-FEDPNAS and FEDDSNAS on two unseen tasks (CIFAR-10). The best performance in each row is in bold font. "
|
| 1016 |
+
],
|
| 1017 |
+
"table_footnote": [],
|
| 1018 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>UNSEEN TASKDESCRIPTION</td><td rowspan=1 colspan=1>FEDDSNAS</td><td rowspan=1 colspan=1>CA-FEDPNAS</td><td rowspan=1 colspan=1>FEDDSNAS(RETRAINED)</td><td rowspan=1 colspan=1>CA-FEDPNAS(RETRAINED)</td></tr><tr><td rowspan=1 colspan=1>ROTATE -90</td><td rowspan=1 colspan=1>0.545 ± 0.04</td><td rowspan=1 colspan=1>0.578 ± 0.09</td><td rowspan=1 colspan=1>0.699± 0.12</td><td rowspan=1 colspan=1>0.734 ± 0.17</td></tr><tr><td rowspan=1 colspan=1>ROTATE 90</td><td rowspan=1 colspan=1>0.553 ± 0.12</td><td rowspan=1 colspan=1>0.569 ± 0.06</td><td rowspan=1 colspan=1>0.673 ± 0.13</td><td rowspan=1 colspan=1>0.727 ± 0.22</td></tr></table>",
|
| 1019 |
+
"bbox": [
|
| 1020 |
+
194,
|
| 1021 |
+
633,
|
| 1022 |
+
795,
|
| 1023 |
+
694
|
| 1024 |
+
],
|
| 1025 |
+
"page_idx": 8
|
| 1026 |
+
},
|
| 1027 |
+
{
|
| 1028 |
+
"type": "text",
|
| 1029 |
+
"text": "5 CONCLUSION ",
|
| 1030 |
+
"text_level": 1,
|
| 1031 |
+
"bbox": [
|
| 1032 |
+
174,
|
| 1033 |
+
767,
|
| 1034 |
+
318,
|
| 1035 |
+
782
|
| 1036 |
+
],
|
| 1037 |
+
"page_idx": 8
|
| 1038 |
+
},
|
| 1039 |
+
{
|
| 1040 |
+
"type": "text",
|
| 1041 |
+
"text": "We demonstrate that federated learning for multi-task scenarios requires extensive personalization on the architecture level to obtain good predictive performance. This paper identifies two potential sources of model personalization: (1) task-personalization, which aims to select architectures best suited for specific learning objectives; and (2) context-personalization, which aims to select architectures best suited for specific input samples. To incorporate these aspects of personalization into Federated NAS, we propose FEDPNAS which consists of two main components: (1) a context-aware operator sampler which learns a sampling distribution for feature maps along a master architecture; and (2) a personalized federated learning objective which anticipates client fine-tuning and guides the federated model update to regions that tolerate future local updates. ",
|
| 1042 |
+
"bbox": [
|
| 1043 |
+
173,
|
| 1044 |
+
797,
|
| 1045 |
+
825,
|
| 1046 |
+
924
|
| 1047 |
+
],
|
| 1048 |
+
"page_idx": 8
|
| 1049 |
+
},
|
| 1050 |
+
{
|
| 1051 |
+
"type": "text",
|
| 1052 |
+
"text": "6 REPRODUCIBILITY & ETHIC STATEMENT ",
|
| 1053 |
+
"text_level": 1,
|
| 1054 |
+
"bbox": [
|
| 1055 |
+
174,
|
| 1056 |
+
102,
|
| 1057 |
+
547,
|
| 1058 |
+
118
|
| 1059 |
+
],
|
| 1060 |
+
"page_idx": 9
|
| 1061 |
+
},
|
| 1062 |
+
{
|
| 1063 |
+
"type": "text",
|
| 1064 |
+
"text": "This work contributes to the literature of Federated Learning through improving the state-of-theart performance. As such, it could have significant broader impact by allowing users to more accurately solve practical problems. While applications of our work to real data could result in ethical considerations, this is an indirect (and unpredictable) side-effect of our work. Our experimental work uses publicly available datasets to evaluate the performance of our algorithms; no ethical considerations are raised. Our implementation code is published anonymously at https://github.com/icml2021fedpnas/fedpnas. All proofs and details of various architectures are included in the Appendix of this paper. ",
|
| 1065 |
+
"bbox": [
|
| 1066 |
+
174,
|
| 1067 |
+
135,
|
| 1068 |
+
825,
|
| 1069 |
+
246
|
| 1070 |
+
],
|
| 1071 |
+
"page_idx": 9
|
| 1072 |
+
},
|
| 1073 |
+
{
|
| 1074 |
+
"type": "text",
|
| 1075 |
+
"text": "REFERENCES ",
|
| 1076 |
+
"text_level": 1,
|
| 1077 |
+
"bbox": [
|
| 1078 |
+
174,
|
| 1079 |
+
268,
|
| 1080 |
+
285,
|
| 1081 |
+
284
|
| 1082 |
+
],
|
| 1083 |
+
"page_idx": 9
|
| 1084 |
+
},
|
| 1085 |
+
{
|
| 1086 |
+
"type": "text",
|
| 1087 |
+
"text": "B. Baker, O. Gupta, N. Naik, and R. Raskar. Designing neural network architectures using reinforcement learning. arXiv preprint arXiv:1611.02167, 2016. \nJ. Bergstra, D. Yamins, and D. Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In International conference on machine learning, pages 115–123. PMLR, 2013. \nA. Fallah, A. Mokhtari, and A. Ozdaglar. Personalized federated learning: Model-agnostic metalearning approach. In Proc. NeurIPS, 2020. \nC. Finn, P. Abbeel, and S. Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proc. ICML, pages 1126–1135, 2017. \nD. Floreano, P. Dürr, and C. Mattiussi. Neuroevolution: from architectures to learning. Evolutionary Intelligence, 1(1):47–62, 2008. \nJ. Hu, L. Shen, and G. Sun. Squeeze-and-excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132–7141, 2018. \nS. Hu, S. Xie, H. Zheng, C. Liu, J. Shi, X. Liu, and D. Lin. DSNAS: Direct neural architecture search without parameter retraining. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12084–12092, 2020. \nE. Jang, S. Gu, and B. Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144, 2016. \nA. Krizhevsky et al. Learning multiple layers of features from tiny images. Citeseer, 2009. \nY. LeCun, C. Cortes, and C. Burges. Mnist handwritten digit database. ATT Labs [Online]. Available: http://yann. lecun. com/exdb/mnist, 2, 2010. \nH. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communication-efficient learning of deep networks from decentralized data. In Proc. AISTATS, pages 1273–1282, 2017. \nH. Pham, M. Y. Guan, B. Zoph, Q. V. Le, and J. Dean. Efficient neural architecture search via parameter sharing. arXiv preprint arXiv:1802.03268, 2018. \nE. Real, A. Aggarwal, Y. Huang, and Q. V. Le. Regularized evolution for image classifier architecture search. In Proceedings of the AAAI conference on Artificial Intelligence, volume 33, pages 4780–4789, 2019. \nS. Xie, H. Zheng, C. Liu, and L. Lin. SNAS: stochastic neural architecture search. arXiv preprint arXiv:1812.09926, 2018. \nB. Zoph and Q. V. Le. Neural architecture search with reinforcement learning. arXiv preprint arXiv:1611.01578, 2016. ",
|
| 1088 |
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"bbox": [
|
| 1089 |
+
169,
|
| 1090 |
+
286,
|
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+
826,
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+
929
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+
],
|
| 1094 |
+
"page_idx": 9
|
| 1095 |
+
}
|
| 1096 |
+
]
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| 1 |
+
# Debiasing Graph Neural Networks via Learning Disentangled Causal Substructure
|
| 2 |
+
|
| 3 |
+
Shaohua Fan1,2∗, Xiao Wang1, Yanhu $\mathbf { M o } ^ { 1 }$ , Chuan Shi1†, Jian Tang2,3,4 †
|
| 4 |
+
|
| 5 |
+
1Beijing University of Posts and Telecommunications, China 2 Mila - Québec AI Institute, Canada 3 HEC Montréal, Canada 4 CIFAR AI Research Chair {fanshaohua, xiaowang, moyanhu, shichuan}@bupt.edu.cn, jian.tang@hec.ca
|
| 6 |
+
|
| 7 |
+
# Abstract
|
| 8 |
+
|
| 9 |
+
Most Graph Neural Networks (GNNs) predict the labels of unseen graphs by learning the correlation between the input graphs and labels. However, by presenting a graph classification investigation on the training graphs with severe bias, surprisingly, we discover that GNNs always tend to explore the spurious correlations to make decision, even if the causal correlation always exists. This implies that existing GNNs trained on such biased datasets will suffer from poor generalization capability. By analyzing this problem in a causal view, we find that disentangling and decorrelating the causal and bias latent variables from the biased graphs are both crucial for debiasing. Inspired by this, we propose a general disentangled GNN framework to learn the causal substructure and bias substructure, respectively. Particularly, we design a parameterized edge mask generator to explicitly split the input graph into causal and bias subgraphs. Then two GNN modules supervised by causal/bias-aware loss functions respectively are trained to encode causal and bias subgraphs into their corresponding representations. With the disentangled representations, we synthesize the counterfactual unbiased training samples to further decorrelate causal and bias variables. Moreover, to better benchmark the severe bias problem, we construct three new graph datasets, which have controllable bias degrees and are easier to visualize and explain. Experimental results well demonstrate that our approach achieves superior generalization performance over existing baselines. Furthermore, owing to the learned edge mask, the proposed model has appealing interpretability and transferability.3
|
| 10 |
+
|
| 11 |
+
# 1 Introduction
|
| 12 |
+
|
| 13 |
+
Graph Neural Networks (GNNs) have exhibited powerful performance on graph data with various applications [17, 35, 13, 9, 8]. One major category of applications are the graph classification task, such as molecular graph property prediction [15, 20, 44], superpixel graph classification [14], and social network category classification [46, 44]. It is well known that graph classification is usually determined by a relevant substructure, but not the whole graph structure [43, 26, 45]. For example, for MNIST superpixel graph classification task, the digit subgraphs are causal (i.e., deterministic) for labels [36]. The mutagenic property of a molecular graph depends on the functional groups (i.e., nitrogen dioxide $\left( \mathrm { N O } _ { 2 } \right)$ ), rather than the irrelevant patterns (i.e., carbon rings) [27]. Therefore, it is a fundamental requirement for GNNs to identify causal substructures, so as to make correct prediction.
|
| 14 |
+
|
| 15 |
+
Ideally, when the graphs are unbiased, i.e., only the causal substructures are related with the graph labels, the GNNs are able to utilize such substructure to predict the labels. However, due to the uncontrollable data collection process, the graphs are inevitably biased, i.e., existing meaningless substructures spuriously correlates with labels. Taking a colored MNIST superpixel graph dataset in Sec. 3.1 as an example (illustrated in Fig. 1(a)), each category of digit subgraphs mainly correspond to one kind of color background subgraphs, e.g., digit 0 subgraph is related with red background subgraph. Therefore, the color background subgraph will be treated as bias information, which highly correlates with labels but does not determines them in the training set. Under this situation, will GNNs still stably utilize the causal substructure to make decision?
|
| 16 |
+
|
| 17 |
+
To investigate the impact of bias on GNNs, we conduct an experimental investigation to demonstrate the impact of bias (especially in the severe bias scenarios) on the generalization capability of GNNs (Sec. 3.1). We find that GNNs actually utilize both bias and causal substructures to make prediction. However, with severer bias correlation, even bias substructure still could not exactly determine labels like causal substructure, GNNs majorly utilize bias substructure as shortcuts to make prediction, causing a large generalization performance degradation. Why this happens? We analyze the datagenerating process and model prediction mechanism behind the graph classification using a causal graph (Sec. 3.2). The casual graph illustrates that the observed graphs are generated by the causal and bias latent variables and existing GNNs could not distinguish the causal substructure from entangled graphs. How can we disentangle the causal and bias substructures from observed graphs, so that GNNs can only utilize the causal substructures to make stable prediction when severe bias appears?
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To address the question, two challenges need to be faced. 1) How to identify the causal substructure and bias substructure in the severe biased graphs? In the severe bias scenarios, bias substructure will be “easier to learn” for GNNs and finally dominate the prediction. Using the normal cross-entropy loss, like DIR [39], could not fully capture such aggressive property of bias. 2) How to extract the causal substructure from an entangled graph? The statistically causal substructure is usually determined by the global property of the entire graph population, rather than a single graph. When extracting causal substructure from a graph, we need to establish the relations among all the graphs.
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In this paper, we propose a novel debiasing framework for GNNs via learning Disentangled Causal substructure, called DisC. Given an input biased graph, we propose to explicitly filter edges into causal and bias subgraphs by a parameterized edge mask generator, whose parameters are shared across entire graph population. As a result, the edge masker is naturally capable to specify the importance for each edge and extract causal and bias subgraphs from a global view of the entire observations. Then, a “casual”-aware (weighted cross-entropy) loss and a “bias”-aware (generalized cross-entropy) loss are respectively utilized to supervise two functional GNN modules. Based on the supervision, the edge mask generator could generate corresponding subgraphs and the GNNs could encode corresponding subgraphs into their disentangled embeddings. With the disentangled embeddings, we randomly permute the latent vectors extracted from different graphs to generate more unbiased counterfactual samples in embedding space. The new generated samples still contain both causal and bias information, while their correlation has been decorrelated. In this time, there is only correlation between causal variables with labels, so that the model could concentrate on the true correlation between the causal subgraphs and labels. Our major contributions are as follows:
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• To our knowledge, we first study the generalization problem of GNNs in a more challenging yet practical scenario, i.e., the graphs are with severe bias. We systematically analyze the bias impact on GNNs from both experimental study and causal analysis. We find that the bias substructure, compared with causal substructure, is much easier to dominate the training of GNNs. • To debias GNNs, we develop a novel GNN framework for disentangling causal substructure, which is flexible to build upon various GNNs for improving generalization ability while enjoying inherent interpretability, robustness and transferability. • We construct three new datasets with various properties and controllable bias degrees, which can better benchmark the new problem. Our model outperforms the corresponding base models with a large margin (from $4 . 4 7 \%$ to $1 6 9 . 1 7 \%$ average improvements). Various investigation studies demonstrate that our model could discover and leverage causal substructure for prediction.
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# 2 Related Works
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Generalization of GNNs in wild environments. Most existing GNN methods are proposed under the IID hypothesis, i.e., training and testing set are independently sampled from the identical distribution [34, 17, 35, 13, 24]. However, in reality, thus ideal assumption is hard to be satisfied. Recently, several methods have been proposed to improve the generalization ability of GNNs in wild OOD environments. Several works [29, 7, 38] study the OOD problem of node classification. For OOD graph classification task, StableGNN [6] propose to learn the stable causal relationship in graphs. OOD-GNN [22] propose to constrain each dimension of learned embedding to be independent. DIR [39] discovers the invariant rationales for generalizing GNNs. Although they have achieved better OOD performance, they are not designed for the datasets with severe bias, which is more challenging for guaranteeing the generalization ability of GNNs.
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Disentangled graph neural networks. Recently, there are a couple of methods that study the disentangled GNNs. DisenGCN [28] utilizes neighbourhood routing mechanism to divide the neighbours of the node into several mutually exclusive parts. IPGDN [25] promotes DisenGCN by constraining the different parts of the embedding feature to be independent. DisenGCN and IPGDN are node-level disentanglement, thus FactorGCN [42] considers the whole graph information and disentangles the target graph into several factorized graphs. Despite results of the previous works, they do not consider disentangling the causal and bias information for graphs.
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+
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General debiasing methods. Recently, debiasing problem has drawn much attention in machine learning community [16, 23, 33, 1, 2, 11]. One category of these methods is pre-defining a certain bias type explicitly to mitigate [16, 23, 33, 1, 37]. For example, Wang et al. [37] and Bahng et al. [1] design a texture- and color-guided model to adversarially train a debiased neural network against the biased one. Instead of defining certain types of bias, recent approaches [30, 5, 21] rely on the straightforward assumption that models are prone to exploit the bias as shortcuts to make prediction [10]. In the line with the recent studies, our study belongs to the second category. However, most of existing methods are designed for image datasets and could not effectively extract causal substructure from graph data. Distinctly, we first study the severe bias problem on graph data, and our method could effectively extract causal substructure from graph data.
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Figure 1: Example graphs of CMNIST-75sp and the performance of GNNs on this dataset.
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# 3 Preliminary Study and Analysis
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In this section, we first illustrate the existing GNNs tend to exploit the bias substructure as shortcuts for prediction through a motivating experiment. Then we analyze the prediction process of GNNs in causal view. Based on this causal view, it motivates our solution to relieve the impact of bias.
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# 3.1 Motivating Example
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To measure the generalization ability of GNNs with the effect of bias, we construct a graph classification dataset with controllable bias degrees, called CMNIST-75sp. We first construct a biased MNIST image dataset like [1], where each category of digit highly correlates with a pre-defined color in their background. For example, in the training set, $90 \%$ of 0 digits are with red background (i.e., biased samples), and remaining $10 \%$ images are with random background color (i.e., unbiased samples), whose the bias degree is 0.9 in this situation. We consider four bias degrees $\{ 0 . 8 , 0 . 8 5 , 0 . 9 , \bar { 0 } . 9 5 \}$
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For the testing set, we construct both biased testing set and unbiased testing set. The biased testing set has the same bias degree with training set, aiming to measure the extent of models relying on bias. The unbiased testing set, where the digit labels uncorrelate with the background colors, aims to test whether the model could utilize the inherent digit signals for prediction. Note that training set and testing set have the same pre-defined color set. Then, we convert the biased MNIST images into superpixel graphs with at most 75 nodes each graph using [18], where the edges are constructed by the KNN method based on the 2D coordinates of superpixels and node features are the concatenation of coordinates and average color of superpixels. Each graph is labeled by its digit class, so that its digital subgraph is deterministic for label and background subgraph is spuriously correlated with labels but not deterministic. The examples of graphs are illustrated in Fig. 1(a).
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We perform three popular GNN methods: GCN [17], GIN [41], and GCNII [3] on CMNIST-75sp and the results are shown in Fig. 1(b). The same color of dashed line and solid line represent the results of the corresponding methods on the biased testing set and the unbiased testing set respectively. Overall, the GNNs achieve much better performance on biased testing set than unbiased testing set. The phenomenon indicates that although GNNs could still learn some causal signals for prediction, the unexpected bias information is also being utilized for prediction. More specifically, with bias degree becoming larger, the performance of GNNs on biased testing set is increased and the value of accuracy is nearly in line with the bias degree, while the performance on unbiased testing drops dramatically. Hence, although causal substructure could determine labels perfectly, in severe bias scenarios, the GNNs lean to utilize the easier to learn bias information to make prediction rather than the inherent causal signals, and bias substructure will finally dominate the prediction.
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| 47 |
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(a) SCM of the union of the (b) SCM of our debiasing GNN data generation and the existing method. GNNs’ prediction process.
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Figure 2: SCMs. Grey and white variables represent unobserved and observed variables, respectively.
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# 3.2 Problem Analysis
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Debiasing GNNs for unbiased prediction requires understanding the natural mechanisms of graph classification task. We present a causal view of the union of the data-generating process and the model prediction process behind the task. Here we formalize the causal view as a Structure Causal Model (SCM) or causal graph [12, 31] by inspecting on the causalities among five variables: unobserved causal variable $C$ , unobserved bias variable $B$ , observed graph $G$ , graph embedding $E$ , and ground truth label / prediction $Y ^ { 4 }$ . Fig. 2(a) illustrates the SCM, where each link denotes a causal relationship.
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• $C \right. G \left. B$ . The observed graph data is generated by two unobserved latent variables: the causal variable $C$ and the bias variable $B$ , such as digit subgraphs and background subgraphs in the CMNIST-75sp dataset. And all bellow relations are illustrated by CMNIST-75sp. • $C Y$ . This link means that the causal variable $C$ is the only endogenous parent to determine the generation of ground-truth label $Y$ . For example, $C$ is the oracle digit subgraph, which exactly explains why the label is labeled as $Y$ . • $C \ \ B$ . This link indicates the spurious correlation between $C$ and $B$ . Such probabilistic dependencies is usually caused by the direct cause or unobserved confounder [32]. Here we do not distinguish these scenarios and only observe the spurious correlation between $B$ and $C$ , such as the spurious correlation between the color background subgraphs and digit subgraphs.
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• $G E Y$ . Existing GNNs usually learn the graph embedding $E$ based on the observed graph $G$ and make the prediction $Y$ based on the learned embedding $E$ .
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+
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According to the SCM, GNNs will utilize both information to make prediction. As bias substructure (e.g., background subgraph) usually has simpler structure than meaningful causal substructure (e.g., digit subgraph), if GNN utilizes such simple substructure, it could achieve low loss very fast. Hence, GNN inclines to utilizes bias information when most graphs are biased. Based on the SCM in Fig. 2(a), according to $d$ -connection theory [31] (see App. A): two variables are dependent if they are connected by at least one unblocked path, we could find two paths that would induce the spurious correlation between the bias variable $B$ and label $Y$ : (1) $\mathbf { B } \mathbf { G } \mathbf { E } \mathbf { Y }$ and (2) $\mathbf { B } \left. \bar { \mathbf { C } } \right. \mathbf { Y }$ . To make the prediction $Y$ being uncorrelated with the bias $B$ , we need to intercept the two unblocked paths. For this purpose, we propose to debias GNNs in causal view, as in Fig. 2(b).
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• $C \left. G \right. B$ and $C Y$ . To intercept the path (1), we should disentangle the latent variables $C$ and $B$ from the observed graph $G$ and make prediction only based on the causal variable $C$ . • $C \ l \ l \textsc { - } B$ . To intercept the path (2), as we cannot change the link between $C$ and $Y$ , one possible solution is to make $C$ and $B$ uncorrelated.
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# 4 Methodology
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Motivated by the above causal analysis, in this section, we present our proposed debiasing GNN framework DisC, to remove the spurious correlation. The overall framework is shown in Fig. 3. First, an edge mask generator is learnt to mask the edges of original input graphs into causal subgraphs and bias subgraphs. Second, two separate GNN modules with their corresponding masked subgraphs are trained to encode corresponding causal substructure and bias substructure into disentangled representations, respectively. Last, after the disentangled representations are well-trained, we permute the bias representations among the training graphs to generate counterfactual unbiased samples, so that the correlation between causal representations and bias representations is removed.
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Figure 3: The overall framework of DisC.
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# 4.1 Causal and Bias Substructure Generator
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Given a mini-batch of biased graphs ${ \mathcal { G } } = \{ G _ { 1 } , \cdots , G _ { n } \}$ , our idea is that: we take a collection of graph instances and design a generative probabilistic model to learn to mask the edges into causal subgraph or bias subgraph. Particularly, given a graph $G = \left\{ \mathbf { A } , \mathbf { X } \right\}$ , where $\mathbf { A }$ is the adjacency matrix and $\mathbf { X }$ is the node feature matrix, we utilize a multi-layer perceptron (MLP) upon the concatenation of node features $\mathbf { x } _ { i }$ of node $i$ and $\mathbf { x } _ { j }$ of node $j$ to measure the importance of edge $( i , j )$ for causal subgraph:
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+
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+
$$
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\alpha _ { i j } = \mathbf { M L P } ( [ \mathbf { x } _ { i } , \mathbf { x } _ { j } ] ) .
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+
$$
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+
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Then a sigmoid function $\sigma ( \cdot )$ is employed to project $\alpha _ { i j }$ into the range of (0,1), which indicates the probability of edge $( i , j )$ being the edge in the causal subgraph as follows:
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+
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+
$$
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+
c _ { i j } = \sigma ( \alpha _ { i j } ) .
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+
$$
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+
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+
Naturally, we could get the probability of edge $( i , j )$ being the edge in the bias subgraph by: $b _ { i j } = 1 - c _ { i j }$ . Now we could construct the causal edge mask $\mathbf { M } _ { c } \bar { \mathbf { \eta } } = \left[ c _ { i j } \right]$ and bias edge mask $\begin{array} { r } { \mathbf { M } _ { b } = [ b _ { i j } ] } \end{array}$ . Finally, we decompose the original graph $G$ into causal subgraph $G _ { c } = \{ \mathbf { M } _ { c } \odot \mathbf { A } , \mathbf { X } \}$ and bias subgraph $G _ { b } = \{ { \bf M } _ { b } \odot { \bf A } , { \bf X } \}$ . Intuitively, the edge mask could highlight different part of structure information of original graphs, thus GNNs built on the different subgraphs could encode different parts of graph information. Moreover, the mask generator has two advantages. (1) Global view: In individual graph level, the mask generator (i.e., MLP), whose parameters are shared by all the edges in a graph, take a global view of all the edges in a graph, which enables us to identify community in graph. It is well known that the effect of an edge cannot be judged independently, because edges usually collaborate with each other, forming a community, to make prediction. Thus, it is critical to evaluate an edge in a global view. In whole graph population level, the mask generator takes a global view of all the graphs in the training set, which enables us to identify causal/bias subgraph. Particularly, as the causal/bias is the statistical information in the population level, it is necessary to view all the graphs to identify the causal/bias substructure. Considering both such coalition effects and population-level statistical information, the generator is able to measure the importance of edges more accurately. (2) Generalization: The mask generator can generalize the mechanism of mask generation to new graphs without retraining, so it is capable and efficient to prune unseen graphs.
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# 4.2 Learning Disentangled Graph Representations
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Given $G _ { c }$ and $G _ { b }$ , how to ensure they are causal subgraph and bias subgraph, respectively? Inspired by [21], our approach simultaneously trains a pair of GNNs $( g _ { b } , g _ { c } )$ with linear classifiers $( C _ { b } ^ { - } , C _ { c } )$ as follows: (1) Motivated by the observation in Sec. 3.1 that bias substructure is easier to learn, we utilize a bias-aware loss to train a bias $\mathrm { G N N } g _ { b }$ and a bias classifier $C _ { b }$ and (2) in contrast, we train a causal $\mathrm { G N N } _ { g _ { c } }$ and a causal classifier $C _ { c }$ on the training graphs that the bias GNN struggles to learn. Next, we would present each component in detail.
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As shown in Fig. 3, GNN $g _ { c }$ and $g _ { b }$ embed the corresponding subgraphs into causal embedding $z _ { c } = g _ { c } ( G _ { c } ; \gamma _ { c } )$ and bias embedding $z _ { b } = g _ { b } ( G _ { b } ; \gamma _ { b } )$ , respectively, where $\gamma$ is the parameters of GNNs. Subsequently, concatenated vector $z = \left[ z _ { c } ; z _ { b } \right]$ is fed into linear classifiers $C _ { c }$ and $C _ { b }$ to predict the target label $y$ . To train $g _ { b }$ and $C _ { b }$ as bias extractor, we utilize the generalized cross entropy (GCE) [47] loss to amplify the bias of the bias GNN and classifier:
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$$
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+
G C E ( C _ { b } ( z ; \alpha _ { b } ) , y ) = \frac { 1 - C _ { b } ^ { y } ( z ; \alpha _ { b } ) ^ { q } } { q } ,
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$$
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+
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+
where $C _ { b } ( z ; \alpha _ { b } )$ and $C _ { b } ^ { y } ( z ; \alpha _ { b } )$ are softmax output of the bias classifier and its probability belonging to the target category $y$ , respectively, and $\alpha$ is the parameters of classifier. Here $q \in ( 0 , 1 ]$ is a hyperparameter that controls the degree of amplifying bias. Given $\theta _ { b } = [ \gamma _ { b } , \alpha _ { b } ]$ , the gradient of the GCE loss up-weights the gradient of the standard cross entropy (CE) loss for the samples with a high confidence $\dot { \boldsymbol { C } } _ { b } ^ { y }$ of predicting the correct target category as follows:
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+
|
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+
$$
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+
\frac { \partial G C E ( C _ { b } ( z ; \alpha _ { b } ) , y ) } { \partial \theta _ { b } } = ( C _ { b } ^ { y } ) ^ { q } \frac { \partial C E ( C _ { b } ( z ; \alpha _ { b } ) , y ) } { \partial \theta _ { b } } .
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+
$$
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+
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+
Therefore, compared with CE loss, GCE loss will amplify the gradients of $\theta _ { b }$ on samples by the confidence score $( C _ { b } ^ { y } ) ^ { q }$ . Based on our observation that the bias information is usually easier to be learned, so the biased graphs will have higher $( C _ { b } ^ { y } ) ^ { q }$ than unbiased graphs. Therefore, the model $g _ { b }$ and $C _ { b }$ trained by GCE loss will focus on bias information and finally get the bias subgraph. Note that, to ensure that $C _ { b }$ predicts target labels mainly based on this $z _ { b }$ , the loss from $C _ { b }$ is not backpropagated to $g _ { c }$ , i.e., only update $\theta _ { b }$ in Eq. (4), and vice versa.
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+
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+
Meanwhile, we also train a causal GNN simultaneously with the weighted CE loss. The graphs with high CE loss from $C _ { b }$ can be regarded as the unbiased samples compared with the samples with low CE loss. In this regard, we could obtain the unbias score of each graph as
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+
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+
$$
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+
W ( z ) = \frac { C E ( C _ { b } ( z ) , y ) } { C E ( C _ { c } ( z ) , y ) + C E ( C _ { b } ( z ) , y ) } .
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+
$$
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+
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+
Large value of $W$ implies the graph is an unbiased sample, hence we could use these weights to reweight the loss of these graphs to train $g _ { c }$ and $C _ { c }$ , enforcing them to learn the unbiased information. Thus, the objective function for learning disentangled representation is:
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+
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+
$$
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+
L _ { D } = W ( z ) C E ( C _ { c } ( z ) , y ) + G C E ( C _ { b } ( z ) , y ) .
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+
$$
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+
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+
# 4.3 Counterfactual Unbiased Sample Generation
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Until now, we have achieved the first goal analyzed in Sec. 3.2 that is the disentanglement of causal and bias substructures. Next, we will show how to achieve the second goal that makes the causal variable $z _ { c }$ and bias variable $z _ { b }$ uncorrelated. Although we have disentangled causal and bias information, they are disentangled from the biased observed graphs. Hence, there will exist statistical correlation between causal and bias variables inheriting from the biased observed graphs. To further decorrelate $z _ { c }$ and $z _ { b }$ , according to the causal relation of data-generating process: $C \right. G \left. B$ , we propose to generate the counterfactual unbiased samples in embedding space by swapping $z _ { b }$ . More specifically, we randomly permute bias vectors in each mini-batch and obtain $z _ { u n b i a s e d } = [ z _ { c } ; \hat { z _ { b } } ] .$ where $\hat { z _ { b } }$ represents the randomly permuted bias vectors of $z _ { b }$ . As $z _ { c }$ and $\hat { z } _ { b }$ in $z _ { u n b i a s e d }$ are randomly combined from different graphs, they will have much less correlation than $\boldsymbol { z } = \left[ z _ { c } ; z _ { b } \right]$ where both are from the same graph. To make $g _ { b }$ and $C _ { b }$ still focus on the bias information, we also swap label $y$ as $\hat { y }$ along with $\hat { z _ { b } }$ , so that the spurious correlation between $\hat { z _ { b } }$ and $\hat { y }$ still exists. With the generated unbiased samples, we utilize the following loss function to train two GNN modules:
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+
|
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+
$$
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+
{ \cal L } _ { G } = W ( z ) C E ( C _ { c } ( z _ { u n b i a s e d } ) , y ) + G C E ( C _ { b } ( z _ { u n b i a s e d } ) , \hat { y } ) ,
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+
$$
|
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+
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+
Together with the disentanglement loss, total loss function is defined as:
|
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|
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+
$$
|
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+
{ \cal L } = { \cal L } _ { D } + \lambda _ { G } { \cal L } _ { G } ,
|
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+
$$
|
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+
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+
where $\lambda _ { G }$ is a hyperparameter for weighting the importance of generation component. Moreover, training with more diverse samples would also benefit with better generalization on unseen testing scenarios. Our approach is summarized in App. B. Note that, as we need well-disentangled representations to generate the high-quality unbiased samples, in the early stage of training, we only train the model with $L _ { D }$ . After certain epochs, we train the model with $L$ .
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+
# 5 Experiment
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Datasets. We construct three datasets with various properties and bias ratios to benchmark this new problem, where the datasets have clear causal subgraphs making the results explainable. Following CMNIST-75sp introduced in Sec. 3.1, we use the similar way to construct CFashion-75sp and CKuzushiji-75sp datasets based on the Fashion-MNIST [40] and Kuzushiji-MNIST [4] datasets. As the causal subgraphs of these two datasets are more complicated (fashion product and hiragana characters), they are more challenging. Due to the page limits, here we set bias degrees as $\{ 0 . 8 , 0 . 9 , 0 . 9 5 \}$ . We report the main results on unbiased test sets. Details are in App. C.1.
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Baselines and experimental setup. As DisC is a general framework which could be built on various base GNN models, we select three popular GNNs: GCN [17], GIN [41], and GCNII [3]. The corresponding models are termed as $\mathrm { D i s C } _ { G C N }$ , $\mathrm { D i s C } _ { G I N }$ and $\mathrm { D i s C } _ { G C N I I }$ , respectively. Hence, base models are the most straight baselines. Another kind of baselines are the causal-inspired GNN method DIR [39] and StableGNN [6]. We also compare against a general debiasing method LDD [21] by replacing its encoder with GNNs. Graph Pooling method DiffPool [44] and graph disentangling method FactorGCN [42] are also compared. To keep fair comparison, our model uses the same GNN architecture and hyperparameters with the corresponding base model. All the experiments are run 4 times with different random seeds and we report the accuracy and the standard error. More details are in App. C.2.
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# 5.1 Quantitative Evaluation
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Main results. The overall results are summarized in Table 1, and we have following observations:
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(1) DisC has much better generalization ability than base models. DisC outperforms the corresponding base model consistently with a large margin. With heavier biases, our model achieves larger improvements over base models. Specifically, for CMNIST-75sp, CFashion-75sp and CKuzushiji-75sp with smaller bias degree (i.e., 0.8), our model achieves $4 0 . 0 2 \%$ , $4 . 4 7 \%$ and $2 9 . 8 2 \%$ average improvements over corresponding base models, respectively. Surprisingly, with severer biases (0.9 and 0.95), DisC achieves $1 6 9 . 1 7 \%$ , $1 4 . 6 7 \%$ and $4 9 . 3 5 \%$ average improvements over base models on three datasets, respectively. It indicates that the proposed method is a general framework helping existing GNNs against the negative impact of bias.
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+
(2) DisC significantly outperforms existing debiasing methods. We notice that DIR could not achieve satisfying results. The reason is that DIR utilizes CE loss to extract bias information, which could not fully capture the property of bias in severe bias scenarios. And DIR sets one fixed threshold to spilt subgraphs, which is suboptimal. StableGNN outperforms their base model DiffPool and achieve competitive results, indicating the effectiveness of their proposed causal variable distinguishing regularizer. However, their framework adjusts data distribution based on the original dataset, it is hard to generate unbiased distribution when the unbiased samples are scarce. DisC could generate more unbiased samples based on the disentangled representations. Moreover, LDD is a general debiasing method which is not designed for graph data. DisC outperforms corresponding LDD variants with average $2 3 . 1 5 \%$ , indicating that the seamless joint of global-population-aware edge masker with debiasing disentangle framework is very effective.
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Table 1: Graph classification accuracy evaluated on unbiased testing sets, which have same color (bias) set with training set. The best performance within each base model variant is in bold.
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<table><tr><td rowspan="2">Dataset Bias</td><td colspan="3">CMNIST-75sp</td><td colspan="3">CFashion-75sp</td><td colspan="3">CKuzushiji-75sp</td></tr><tr><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td></tr><tr><td>FactorGCN [42]</td><td>72.30±1.18</td><td>62.35±5.07</td><td>42.50±4.91</td><td>61.23±1.11</td><td>53.50±1.29</td><td>45.78±2.40</td><td>42.87±1.19</td><td>32.35±2.79</td><td>23.87±0.12</td></tr><tr><td>DiffPool [44]</td><td>73.79±0.02</td><td>66.45±0.78</td><td>47.12±1.04</td><td>62.82±0.53</td><td>57.50±0.39</td><td>50.86±0.20</td><td>45.46±0.65</td><td>36.18±0.19</td><td>27.45±0.26</td></tr><tr><td>DIR [39]</td><td>9.98±0.33</td><td>9.96±0.23</td><td>10.03±0.27</td><td>13.02±1.92</td><td>12.80±1.67</td><td>11.98±1.41</td><td>10.35±0.32</td><td>10.72±0.27</td><td>10.59±0.46</td></tr><tr><td>StableGNN [6]</td><td>77.65±1.64</td><td>68.87±1.74</td><td>51.33±0.87</td><td>64.03±0.29</td><td>58.26±0.09</td><td>51.46±0.39</td><td>49.41±0.09</td><td>39.30±0.12</td><td>28.26±0.14</td></tr><tr><td>LDDGCN[21]</td><td>64.95±1.22</td><td>56.65±2.18</td><td>46.83±2.88</td><td>63.85±1.17</td><td>64.30±0.89</td><td>62.28±0.48</td><td>42.38±0.33</td><td>38.75±0.49</td><td>33.08±0.59</td></tr><tr><td>LDDGIN[21]</td><td>64.88±1.45</td><td>50.59±1.07</td><td>31.23±2.48</td><td>64.65±0.63</td><td>57.10±0.43</td><td>53.38±0.47</td><td>37.83±0.54</td><td>28.97±0.18</td><td>22.13±0.34</td></tr><tr><td>LDDGCNII [21]</td><td>78.03±0.66</td><td>69.53±0.96</td><td>51.05±3.87</td><td>50.63±1.79</td><td>54.09±2.54</td><td>57.93±0.88</td><td>48.70±1.98</td><td>41.59±1.07</td><td>33.93±0.71</td></tr><tr><td>GCN[17]</td><td>50.43±4.13</td><td>28.97±4.4</td><td>13.50±1.38</td><td>63.60±0.53</td><td>57.22±0.93</td><td>47.69±0.42</td><td>38.45±1.1</td><td>28.35±0.79</td><td>20.70±0.88</td></tr><tr><td>DisCGCN</td><td>82.60±0.93</td><td>78.14±2.14</td><td>63.47±5.65</td><td>66.85±1.11</td><td>65.33±4.70</td><td>63.93±1.50</td><td>55.53±2.29</td><td>48.13±2.59</td><td>36.63±1.73</td></tr><tr><td>GIN [41]</td><td>57.75±0.78</td><td>36.78±5.55</td><td>16.04±1.14</td><td>64.25±0.46</td><td>58.03±0.40</td><td>49.74±0.60</td><td>41.83±0.78</td><td>30.09±0.87</td><td>21.18±1.63</td></tr><tr><td>DisCGIN</td><td>82.10±1.50</td><td>74.90±1.81</td><td>58.58±4.24</td><td>67.10±1.07</td><td>59.90±1.31</td><td>55.80±0.36</td><td>55.18±1.00</td><td>41.75±0.81</td><td>30.25±1.63</td></tr><tr><td>GCNII [3]</td><td>69.70±1.73</td><td>57.68±1.68</td><td>41.00±3.75</td><td>66.68±0.59</td><td>60.58±0.28</td><td>53.18±0.08</td><td>48.53±0.25</td><td>36.23±0.20</td><td>25.60±0.76</td></tr><tr><td>DisCGCNII</td><td>79.50±2.48</td><td>76.00±1.90</td><td>60.54±5.33</td><td>66.47±1.77</td><td>65.48±0.70</td><td>61.75±0.27</td><td>54.90±1.30</td><td>44.73±1.55</td><td>36.95±0.70</td></tr></table>
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Ablation studies. To validate the importance of each module in our method, in Fig. 4, we conduct ablation studies on our variants (w.o. G means without the sample generation module) and the related variants of LDD. The major difference between DisC/w.o. G with LDD /w.o. G is the edge mask module. In most cases, DisC/w.o. G significantly outperforms LDD /w.o. G, indicating the necessity of learning edge mask for graph data. And DisC which has counterfactual sample generation module could further boost the performances based on the disentangled embeddings of DisC/w.o. G. However, LDD seldomly outperforms LDD /w.o. G or even achieves worse performances. That is, generating high-quality counterfactual samples needs well-disentangled causal and bias embeddings. If embeddings are not well-disentangled, counterfactual samples may act as noisy samples, which would prevent models from achieving further improvement. The edge masker could help the model generate well-disentangled embeddings, which is crucial for overall performance.
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Figure 4: Ablation studies of the DisC vs. LDD average over three bias degrees of each dataset.
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Robustness on unseen bias. Table 2 reports the results of DisC compared with its corresponding base models on testing set with unseen bias, i.e., the pre-defined color (bias) sets of training set and testing set are disjoint. The performances of base models further drop compared with the results on seen bias scenario in Table 1. However, our model still achieves very stable performances, fully demonstrating the generalization ability of our model on agnostic bias scenario.
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Hyperparameter experiments Fig. 5 is the hyperparameter experiments of the degree of amplifying bias $q$ in GCE loss and the importance of generation component $\lambda _ { G }$ . For $q$ , we fix $\lambda _ { G } = 1 0$ and vary $q$ from $\{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \}$ . For $\lambda _ { G }$ , we fix $q = 0 . 7$ and vary $\lambda _ { G }$ from $\{ 1 , 5 , 1 0 , 1 5 \}$ . From the results, we can see that our model achieves stable performance across different values of $q$ and $\lambda _ { G }$ . When $q = 0 . 1$ , it means the GCE loss will nearly reduce to normal CE loss. We can see the performance of $\mathrm { D i s C } _ { G C N }$ is worse than other scenarios, demonstrating the effectiveness of utilizing GCE loss.
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Table 2: The results on unseen unbiased testing sets, i.e., the color has not been seen in training set.
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<table><tr><td rowspan="2">Dataset Bias</td><td colspan="3">CMNIST-75sp</td><td colspan="3">CFashion-75sp</td><td colspan="3">CKuzushiji-75sp</td></tr><tr><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td></tr><tr><td>DIR</td><td>10.38±0.28</td><td>10.14±0.40</td><td>9.77±0.18</td><td>16.77±1.71</td><td>16.51±3.20</td><td>12.59±1.61</td><td>10.48±0.34</td><td>10.33±0.75</td><td>10.59±0.95</td></tr><tr><td>GCN</td><td>36.88±5.16</td><td>23.07±4.07</td><td>11.88±0.33</td><td>59.33±0.55</td><td>53.65±0.47</td><td>45.60±1.06</td><td>36.35±0.48</td><td>27.88±0.94</td><td>19.95±0.67</td></tr><tr><td>DisCGCN</td><td>82.73±1.31</td><td>77.70±0.87</td><td>65.48±0.76</td><td>67.9±1.45</td><td>68.28±0.18</td><td>63.77±1.37</td><td>57.80±2.38</td><td>51.60±0.41</td><td>41.60±3.94</td></tr><tr><td>GIN</td><td>48.93±2.99</td><td>34.95±0.86</td><td>14.53±0.97</td><td>58.88±0.57</td><td>53.80±0.52</td><td>48.43±0.69</td><td>39.25±0.57</td><td>30.75±1.45</td><td>22.35±0.86</td></tr><tr><td>DisCGIN</td><td>77.80±1.33</td><td>73.00±0.61</td><td>58.80±1.66</td><td>67.15±0.79</td><td>59.98±0.62</td><td>51.70±0.34</td><td></td><td></td><td>55.47±0.98 43.20±1.36 31.33±1.71</td></tr><tr><td>GCNII</td><td>53.50±6.23</td><td>45.52±2.26</td><td>32.6±5.66</td><td>58.85±1.89</td><td>53.98±0.85</td><td>46.97±1.38</td><td>39.93±0.88</td><td>30.33±1.17</td><td>23.09±1.83</td></tr><tr><td>DisCGCNII</td><td>79.65±2.13</td><td>76.63±1.3860.00±5.66</td><td></td><td></td><td></td><td>60.50±2.77 63.05±2.2561.78±1.60</td><td></td><td>56.23±3.4549.10±2.05</td><td>41.05±0.11</td></tr></table>
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Figure 5: The hyperparameter experiments of $q$ and $\lambda _ { G }$
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# 5.2 Qualitative Evaluation
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Visualization of edge mask. To better illustrate the significant causal and bias subgraphs extracted by $\mathrm { D i s C } _ { G C N }$ , we visualize the original images, original graph, and corresponding causal subgraph and bias subgraph of CMNIST-75sp with 0.9 bias degree in Fig. 6, where the width of edge represents the value of learned weight $c _ { i j }$ or $b _ { i j }$ . Fig. 6(a) shows the visualization results of testing graphs with the bias (color) that has been seen in the training set. As we can see, our model could discover the causal subgraphs where the most salient edges are in the digital subgraphs. With these causal subgraphs that highlight the structure information of digital, the GNNs will more easily extract this causal information. Fig. 6(b) shows the visualization results of testing graphs with unseen bias. According to the visualization, our model could still discover the causal subgraph outline, indicating our model could recognize causal subgraphs, whether the bias is seen or unseen. The visualization results of CFashion-75sp and CKuzushiji-75sp are shown in App. D.
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OriginalimageOriginal graph Causalsubgraph BiasedsubgraphOriginalimageOriginal graph CausalsubgraphBiased subgraph
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Figure 6: Visualization of subgraphs extracted by DisC. The width of edge is edge weight $c _ { i j }$ or $b _ { i j }$
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Projection of disentangled representation. Fig. 7 shows the projection of latent vectors $z _ { c }$ and $z _ { b }$ extracted from the causal GNN $g _ { c }$ and bias GNN $g _ { b }$ of $\mathrm { D i s C } _ { G C N }$ , respectively, using t-SNE [19] on
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Figure 7: Visualization of $z _ { c }$ and $z _ { b }$ with colors labeled by the digit and bias (color) labels. We observe that $z _ { c }$ and $z _ { b }$ are well clustered according to the groundtruth labels and bias labels, respectively.
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CMNIST-75sp. Fig. 7 (a-b) are the projections of $z _ { c }$ labeled by the target labels (digit) and bias labels (color), respectively. Fig. 7 (c-d) are the projections of $z _ { b }$ labeled by the target labels and bias labels, respectively. We observe that $z _ { c }$ are clustered according to the target labels while $z _ { b }$ are clustered with the bias labels. And $z _ { c }$ are mixed with bias labels and $z _ { b }$ are mixed with target labels. The results indicate that DisC successfully learns the disentangled causal and bias representations.
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Transferability of the learned mask. As our model could extract GNN-independent subgraphs, the learning edge weights can be used to purify original biased graphs. These sparse subgraphs represent significant semantic information and can be universally transferred to any GNNs. To validate this point, we learn the edge mask by $\mathrm { D i s C } _ { G C N }$ and prune the edges with least $\{ 0 \% , 2 0 \% , 4 0 \% , 6 0 \% \}$ weights while keeping the remaining edge weights. Then we train vanilla GIN and GCNII on these weighted pruned datasets. Fig. 8 is the comparison of the results, where the dashed lines represent the results of base model on original biased graphs and the solid lines represent the performance of GNNs on weighted pruned datasets. The results show that the GNNs trained on the pruned datasets achieve better performances, indicating our learned edge mask has considerable transferability.
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Figure 8: Performance of GIN and GCNII on the weighted pruned graphs found by $\mathrm { D i s C } _ { G C N }$ .
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# 6 Conclusion
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In this paper, we are first to study the generalization problem of GNNs on severe bias datasets, which is crucial to study the transparently knowledge learning mechanism of GNNs. We analyze the problem in a causal view that the generalization of GNNs will be hindered by entangled representations as well as the correlation between causal and bias variables. To remove the impact from these two aspects, we propose a general disentangling framework, DisC, which extracts causal substructure and bias substructure by two different functional GNNs, respectively. After the representations are well-disentangled, we proliferate the counterfactual unbiased samples by randomly swapping the disentangled vectors. With the new constructed benchmarks, we clearly validate the effectiveness, robustness, interpretability, and transferability of our method.
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# Acknowledgments and Disclosure of Funding
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This work is supported in part by the National Natural Science Foundation of China (No. U20B2045, 62192784, 62172052, 62002029, 62172052, U1936014). This work is also partially supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019- 3583139727. The work of Shaohua Fan is supported by the China Scholarship Council (No.202006470078). The computation resource of this project is supported by Compute Canada .
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# Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes] See App. E.
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(c) Did you discuss any potential negative societal impacts of your work? [No] We could not foresee any potential negative societal impacts of our work.
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Sec. 3.2. (b) Did you include complete proofs of all theoretical results? [Yes]
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3. If you ran experiments...
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(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 provide the URL of code and data for reproducing the main results.
|
| 256 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 5 and App. C.2.
|
| 257 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Sec. 5.
|
| 258 |
+
(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] See Sec. 5.
|
| 259 |
+
|
| 260 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 261 |
+
|
| 262 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] We construct new assets based on existing assets. We have cited them in Sec. 5.
|
| 263 |
+
(b) Did you mention the license of the assets? [Yes] See App. C.
|
| 264 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We release new assets through a URL in Sec. 5.
|
| 265 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] The source data for generating our data is publicly available.
|
| 266 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We do not have personally identifiable information or offensive content.
|
| 267 |
+
|
| 268 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 269 |
+
|
| 270 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 271 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 272 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
|
parse/dev/ex60CCi5GS/ex60CCi5GS_content_list.json
ADDED
|
@@ -0,0 +1,1167 @@
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Debiasing Graph Neural Networks via Learning Disentangled Causal Substructure ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
209,
|
| 8 |
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122,
|
| 9 |
+
790,
|
| 10 |
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172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Shaohua Fan1,2∗, Xiao Wang1, Yanhu $\\mathbf { M o } ^ { 1 }$ , Chuan Shi1†, Jian Tang2,3,4 † ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
258,
|
| 19 |
+
222,
|
| 20 |
+
741,
|
| 21 |
+
241
|
| 22 |
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],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "1Beijing University of Posts and Telecommunications, China 2 Mila - Québec AI Institute, Canada 3 HEC Montréal, Canada 4 CIFAR AI Research Chair {fanshaohua, xiaowang, moyanhu, shichuan}@bupt.edu.cn, jian.tang@hec.ca ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
200,
|
| 30 |
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242,
|
| 31 |
+
799,
|
| 32 |
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320
|
| 33 |
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],
|
| 34 |
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"page_idx": 0
|
| 35 |
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},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Abstract ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
+
"bbox": [
|
| 41 |
+
462,
|
| 42 |
+
356,
|
| 43 |
+
535,
|
| 44 |
+
372
|
| 45 |
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],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "Most Graph Neural Networks (GNNs) predict the labels of unseen graphs by learning the correlation between the input graphs and labels. However, by presenting a graph classification investigation on the training graphs with severe bias, surprisingly, we discover that GNNs always tend to explore the spurious correlations to make decision, even if the causal correlation always exists. This implies that existing GNNs trained on such biased datasets will suffer from poor generalization capability. By analyzing this problem in a causal view, we find that disentangling and decorrelating the causal and bias latent variables from the biased graphs are both crucial for debiasing. Inspired by this, we propose a general disentangled GNN framework to learn the causal substructure and bias substructure, respectively. Particularly, we design a parameterized edge mask generator to explicitly split the input graph into causal and bias subgraphs. Then two GNN modules supervised by causal/bias-aware loss functions respectively are trained to encode causal and bias subgraphs into their corresponding representations. With the disentangled representations, we synthesize the counterfactual unbiased training samples to further decorrelate causal and bias variables. Moreover, to better benchmark the severe bias problem, we construct three new graph datasets, which have controllable bias degrees and are easier to visualize and explain. Experimental results well demonstrate that our approach achieves superior generalization performance over existing baselines. Furthermore, owing to the learned edge mask, the proposed model has appealing interpretability and transferability.3 ",
|
| 51 |
+
"bbox": [
|
| 52 |
+
233,
|
| 53 |
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|
| 54 |
+
766,
|
| 55 |
+
683
|
| 56 |
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],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 Introduction ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
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|
| 66 |
+
310,
|
| 67 |
+
728
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "Graph Neural Networks (GNNs) have exhibited powerful performance on graph data with various applications [17, 35, 13, 9, 8]. One major category of applications are the graph classification task, such as molecular graph property prediction [15, 20, 44], superpixel graph classification [14], and social network category classification [46, 44]. It is well known that graph classification is usually determined by a relevant substructure, but not the whole graph structure [43, 26, 45]. For example, for MNIST superpixel graph classification task, the digit subgraphs are causal (i.e., deterministic) for labels [36]. The mutagenic property of a molecular graph depends on the functional groups (i.e., nitrogen dioxide $\\left( \\mathrm { N O } _ { 2 } \\right)$ ), rather than the irrelevant patterns (i.e., carbon rings) [27]. Therefore, it is a fundamental requirement for GNNs to identify causal substructures, so as to make correct prediction. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
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|
| 77 |
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|
| 78 |
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|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "",
|
| 85 |
+
"bbox": [
|
| 86 |
+
173,
|
| 87 |
+
92,
|
| 88 |
+
823,
|
| 89 |
+
119
|
| 90 |
+
],
|
| 91 |
+
"page_idx": 1
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "text",
|
| 95 |
+
"text": "Ideally, when the graphs are unbiased, i.e., only the causal substructures are related with the graph labels, the GNNs are able to utilize such substructure to predict the labels. However, due to the uncontrollable data collection process, the graphs are inevitably biased, i.e., existing meaningless substructures spuriously correlates with labels. Taking a colored MNIST superpixel graph dataset in Sec. 3.1 as an example (illustrated in Fig. 1(a)), each category of digit subgraphs mainly correspond to one kind of color background subgraphs, e.g., digit 0 subgraph is related with red background subgraph. Therefore, the color background subgraph will be treated as bias information, which highly correlates with labels but does not determines them in the training set. Under this situation, will GNNs still stably utilize the causal substructure to make decision? ",
|
| 96 |
+
"bbox": [
|
| 97 |
+
174,
|
| 98 |
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|
| 99 |
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825,
|
| 100 |
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|
| 101 |
+
],
|
| 102 |
+
"page_idx": 1
|
| 103 |
+
},
|
| 104 |
+
{
|
| 105 |
+
"type": "text",
|
| 106 |
+
"text": "To investigate the impact of bias on GNNs, we conduct an experimental investigation to demonstrate the impact of bias (especially in the severe bias scenarios) on the generalization capability of GNNs (Sec. 3.1). We find that GNNs actually utilize both bias and causal substructures to make prediction. However, with severer bias correlation, even bias substructure still could not exactly determine labels like causal substructure, GNNs majorly utilize bias substructure as shortcuts to make prediction, causing a large generalization performance degradation. Why this happens? We analyze the datagenerating process and model prediction mechanism behind the graph classification using a causal graph (Sec. 3.2). The casual graph illustrates that the observed graphs are generated by the causal and bias latent variables and existing GNNs could not distinguish the causal substructure from entangled graphs. How can we disentangle the causal and bias substructures from observed graphs, so that GNNs can only utilize the causal substructures to make stable prediction when severe bias appears? ",
|
| 107 |
+
"bbox": [
|
| 108 |
+
174,
|
| 109 |
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256,
|
| 110 |
+
825,
|
| 111 |
+
409
|
| 112 |
+
],
|
| 113 |
+
"page_idx": 1
|
| 114 |
+
},
|
| 115 |
+
{
|
| 116 |
+
"type": "text",
|
| 117 |
+
"text": "To address the question, two challenges need to be faced. 1) How to identify the causal substructure and bias substructure in the severe biased graphs? In the severe bias scenarios, bias substructure will be “easier to learn” for GNNs and finally dominate the prediction. Using the normal cross-entropy loss, like DIR [39], could not fully capture such aggressive property of bias. 2) How to extract the causal substructure from an entangled graph? The statistically causal substructure is usually determined by the global property of the entire graph population, rather than a single graph. When extracting causal substructure from a graph, we need to establish the relations among all the graphs. ",
|
| 118 |
+
"bbox": [
|
| 119 |
+
174,
|
| 120 |
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415,
|
| 121 |
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|
| 122 |
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512
|
| 123 |
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],
|
| 124 |
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"page_idx": 1
|
| 125 |
+
},
|
| 126 |
+
{
|
| 127 |
+
"type": "text",
|
| 128 |
+
"text": "In this paper, we propose a novel debiasing framework for GNNs via learning Disentangled Causal substructure, called DisC. Given an input biased graph, we propose to explicitly filter edges into causal and bias subgraphs by a parameterized edge mask generator, whose parameters are shared across entire graph population. As a result, the edge masker is naturally capable to specify the importance for each edge and extract causal and bias subgraphs from a global view of the entire observations. Then, a “casual”-aware (weighted cross-entropy) loss and a “bias”-aware (generalized cross-entropy) loss are respectively utilized to supervise two functional GNN modules. Based on the supervision, the edge mask generator could generate corresponding subgraphs and the GNNs could encode corresponding subgraphs into their disentangled embeddings. With the disentangled embeddings, we randomly permute the latent vectors extracted from different graphs to generate more unbiased counterfactual samples in embedding space. The new generated samples still contain both causal and bias information, while their correlation has been decorrelated. In this time, there is only correlation between causal variables with labels, so that the model could concentrate on the true correlation between the causal subgraphs and labels. Our major contributions are as follows: ",
|
| 129 |
+
"bbox": [
|
| 130 |
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174,
|
| 131 |
+
518,
|
| 132 |
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825,
|
| 133 |
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712
|
| 134 |
+
],
|
| 135 |
+
"page_idx": 1
|
| 136 |
+
},
|
| 137 |
+
{
|
| 138 |
+
"type": "text",
|
| 139 |
+
"text": "• To our knowledge, we first study the generalization problem of GNNs in a more challenging yet practical scenario, i.e., the graphs are with severe bias. We systematically analyze the bias impact on GNNs from both experimental study and causal analysis. We find that the bias substructure, compared with causal substructure, is much easier to dominate the training of GNNs. • To debias GNNs, we develop a novel GNN framework for disentangling causal substructure, which is flexible to build upon various GNNs for improving generalization ability while enjoying inherent interpretability, robustness and transferability. • We construct three new datasets with various properties and controllable bias degrees, which can better benchmark the new problem. Our model outperforms the corresponding base models with a large margin (from $4 . 4 7 \\%$ to $1 6 9 . 1 7 \\%$ average improvements). Various investigation studies demonstrate that our model could discover and leverage causal substructure for prediction. ",
|
| 140 |
+
"bbox": [
|
| 141 |
+
183,
|
| 142 |
+
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|
| 143 |
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826,
|
| 144 |
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911
|
| 145 |
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],
|
| 146 |
+
"page_idx": 1
|
| 147 |
+
},
|
| 148 |
+
{
|
| 149 |
+
"type": "text",
|
| 150 |
+
"text": "2 Related Works ",
|
| 151 |
+
"text_level": 1,
|
| 152 |
+
"bbox": [
|
| 153 |
+
174,
|
| 154 |
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|
| 155 |
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|
| 156 |
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|
| 157 |
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],
|
| 158 |
+
"page_idx": 2
|
| 159 |
+
},
|
| 160 |
+
{
|
| 161 |
+
"type": "text",
|
| 162 |
+
"text": "Generalization of GNNs in wild environments. Most existing GNN methods are proposed under the IID hypothesis, i.e., training and testing set are independently sampled from the identical distribution [34, 17, 35, 13, 24]. However, in reality, thus ideal assumption is hard to be satisfied. Recently, several methods have been proposed to improve the generalization ability of GNNs in wild OOD environments. Several works [29, 7, 38] study the OOD problem of node classification. For OOD graph classification task, StableGNN [6] propose to learn the stable causal relationship in graphs. OOD-GNN [22] propose to constrain each dimension of learned embedding to be independent. DIR [39] discovers the invariant rationales for generalizing GNNs. Although they have achieved better OOD performance, they are not designed for the datasets with severe bias, which is more challenging for guaranteeing the generalization ability of GNNs. ",
|
| 163 |
+
"bbox": [
|
| 164 |
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173,
|
| 165 |
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|
| 166 |
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|
| 167 |
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|
| 168 |
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],
|
| 169 |
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"page_idx": 2
|
| 170 |
+
},
|
| 171 |
+
{
|
| 172 |
+
"type": "text",
|
| 173 |
+
"text": "Disentangled graph neural networks. Recently, there are a couple of methods that study the disentangled GNNs. DisenGCN [28] utilizes neighbourhood routing mechanism to divide the neighbours of the node into several mutually exclusive parts. IPGDN [25] promotes DisenGCN by constraining the different parts of the embedding feature to be independent. DisenGCN and IPGDN are node-level disentanglement, thus FactorGCN [42] considers the whole graph information and disentangles the target graph into several factorized graphs. Despite results of the previous works, they do not consider disentangling the causal and bias information for graphs. ",
|
| 174 |
+
"bbox": [
|
| 175 |
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|
| 176 |
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|
| 177 |
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|
| 178 |
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|
| 179 |
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],
|
| 180 |
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"page_idx": 2
|
| 181 |
+
},
|
| 182 |
+
{
|
| 183 |
+
"type": "text",
|
| 184 |
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"text": "General debiasing methods. Recently, debiasing problem has drawn much attention in machine learning community [16, 23, 33, 1, 2, 11]. One category of these methods is pre-defining a certain bias type explicitly to mitigate [16, 23, 33, 1, 37]. For example, Wang et al. [37] and Bahng et al. [1] design a texture- and color-guided model to adversarially train a debiased neural network against the biased one. Instead of defining certain types of bias, recent approaches [30, 5, 21] rely on the straightforward assumption that models are prone to exploit the bias as shortcuts to make prediction [10]. In the line with the recent studies, our study belongs to the second category. However, most of existing methods are designed for image datasets and could not effectively extract causal substructure from graph data. Distinctly, we first study the severe bias problem on graph data, and our method could effectively extract causal substructure from graph data. ",
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"bbox": [
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"type": "image",
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"img_path": "images/c2cd979b0472bec0aeb4ad871bccbf93e7ecb651572210a690085fba8ecfe079.jpg",
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"image_caption": [
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"Figure 1: Example graphs of CMNIST-75sp and the performance of GNNs on this dataset. "
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],
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"type": "text",
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"text": "3 Preliminary Study and Analysis ",
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"type": "text",
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"text": "In this section, we first illustrate the existing GNNs tend to exploit the bias substructure as shortcuts for prediction through a motivating experiment. Then we analyze the prediction process of GNNs in causal view. Based on this causal view, it motivates our solution to relieve the impact of bias. ",
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"type": "text",
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"text": "3.1 Motivating Example ",
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"type": "text",
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"text": "To measure the generalization ability of GNNs with the effect of bias, we construct a graph classification dataset with controllable bias degrees, called CMNIST-75sp. We first construct a biased MNIST image dataset like [1], where each category of digit highly correlates with a pre-defined color in their background. For example, in the training set, $90 \\%$ of 0 digits are with red background (i.e., biased samples), and remaining $10 \\%$ images are with random background color (i.e., unbiased samples), whose the bias degree is 0.9 in this situation. We consider four bias degrees $\\{ 0 . 8 , 0 . 8 5 , 0 . 9 , \\bar { 0 } . 9 5 \\}$ ",
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"text": "For the testing set, we construct both biased testing set and unbiased testing set. The biased testing set has the same bias degree with training set, aiming to measure the extent of models relying on bias. The unbiased testing set, where the digit labels uncorrelate with the background colors, aims to test whether the model could utilize the inherent digit signals for prediction. Note that training set and testing set have the same pre-defined color set. Then, we convert the biased MNIST images into superpixel graphs with at most 75 nodes each graph using [18], where the edges are constructed by the KNN method based on the 2D coordinates of superpixels and node features are the concatenation of coordinates and average color of superpixels. Each graph is labeled by its digit class, so that its digital subgraph is deterministic for label and background subgraph is spuriously correlated with labels but not deterministic. The examples of graphs are illustrated in Fig. 1(a). ",
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"type": "text",
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"text": "We perform three popular GNN methods: GCN [17], GIN [41], and GCNII [3] on CMNIST-75sp and the results are shown in Fig. 1(b). The same color of dashed line and solid line represent the results of the corresponding methods on the biased testing set and the unbiased testing set respectively. Overall, the GNNs achieve much better performance on biased testing set than unbiased testing set. The phenomenon indicates that although GNNs could still learn some causal signals for prediction, the unexpected bias information is also being utilized for prediction. More specifically, with bias degree becoming larger, the performance of GNNs on biased testing set is increased and the value of accuracy is nearly in line with the bias degree, while the performance on unbiased testing drops dramatically. Hence, although causal substructure could determine labels perfectly, in severe bias scenarios, the GNNs lean to utilize the easier to learn bias information to make prediction rather than the inherent causal signals, and bias substructure will finally dominate the prediction. ",
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"bbox": [
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"type": "image",
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"img_path": "images/67725338102b285105a9f763243a8f05b40474959212b3e04391679918f3ad6a.jpg",
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"image_caption": [
|
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"(a) SCM of the union of the (b) SCM of our debiasing GNN data generation and the existing method. GNNs’ prediction process. "
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],
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"type": "text",
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"text": "Figure 2: SCMs. Grey and white variables represent unobserved and observed variables, respectively. ",
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"type": "text",
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"text": "3.2 Problem Analysis ",
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"text_level": 1,
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"type": "text",
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"text": "Debiasing GNNs for unbiased prediction requires understanding the natural mechanisms of graph classification task. We present a causal view of the union of the data-generating process and the model prediction process behind the task. Here we formalize the causal view as a Structure Causal Model (SCM) or causal graph [12, 31] by inspecting on the causalities among five variables: unobserved causal variable $C$ , unobserved bias variable $B$ , observed graph $G$ , graph embedding $E$ , and ground truth label / prediction $Y ^ { 4 }$ . Fig. 2(a) illustrates the SCM, where each link denotes a causal relationship. ",
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"type": "text",
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"text": "• $C \\right. G \\left. B$ . The observed graph data is generated by two unobserved latent variables: the causal variable $C$ and the bias variable $B$ , such as digit subgraphs and background subgraphs in the CMNIST-75sp dataset. And all bellow relations are illustrated by CMNIST-75sp. • $C Y$ . This link means that the causal variable $C$ is the only endogenous parent to determine the generation of ground-truth label $Y$ . For example, $C$ is the oracle digit subgraph, which exactly explains why the label is labeled as $Y$ . • $C \\ \\ B$ . This link indicates the spurious correlation between $C$ and $B$ . Such probabilistic dependencies is usually caused by the direct cause or unobserved confounder [32]. Here we do not distinguish these scenarios and only observe the spurious correlation between $B$ and $C$ , such as the spurious correlation between the color background subgraphs and digit subgraphs. ",
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"type": "text",
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"text": "• $G E Y$ . Existing GNNs usually learn the graph embedding $E$ based on the observed graph $G$ and make the prediction $Y$ based on the learned embedding $E$ . ",
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"type": "text",
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"text": "According to the SCM, GNNs will utilize both information to make prediction. As bias substructure (e.g., background subgraph) usually has simpler structure than meaningful causal substructure (e.g., digit subgraph), if GNN utilizes such simple substructure, it could achieve low loss very fast. Hence, GNN inclines to utilizes bias information when most graphs are biased. Based on the SCM in Fig. 2(a), according to $d$ -connection theory [31] (see App. A): two variables are dependent if they are connected by at least one unblocked path, we could find two paths that would induce the spurious correlation between the bias variable $B$ and label $Y$ : (1) $\\mathbf { B } \\mathbf { G } \\mathbf { E } \\mathbf { Y }$ and (2) $\\mathbf { B } \\left. \\bar { \\mathbf { C } } \\right. \\mathbf { Y }$ . To make the prediction $Y$ being uncorrelated with the bias $B$ , we need to intercept the two unblocked paths. For this purpose, we propose to debias GNNs in causal view, as in Fig. 2(b). ",
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"type": "text",
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"text": "• $C \\left. G \\right. B$ and $C Y$ . To intercept the path (1), we should disentangle the latent variables $C$ and $B$ from the observed graph $G$ and make prediction only based on the causal variable $C$ . • $C \\ l \\ l \\textsc { - } B$ . To intercept the path (2), as we cannot change the link between $C$ and $Y$ , one possible solution is to make $C$ and $B$ uncorrelated. ",
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"type": "text",
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"text": "4 Methodology ",
|
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"type": "text",
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"text": "Motivated by the above causal analysis, in this section, we present our proposed debiasing GNN framework DisC, to remove the spurious correlation. The overall framework is shown in Fig. 3. First, an edge mask generator is learnt to mask the edges of original input graphs into causal subgraphs and bias subgraphs. Second, two separate GNN modules with their corresponding masked subgraphs are trained to encode corresponding causal substructure and bias substructure into disentangled representations, respectively. Last, after the disentangled representations are well-trained, we permute the bias representations among the training graphs to generate counterfactual unbiased samples, so that the correlation between causal representations and bias representations is removed. ",
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"type": "image",
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"img_path": "images/bcb625f1dae058895f389fa7bde99c8f6e961d0b84a718b48473b2ff85e97e50.jpg",
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| 395 |
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"image_caption": [
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| 396 |
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"Figure 3: The overall framework of DisC. "
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| 397 |
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],
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| 398 |
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"type": "text",
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"text": "4.1 Causal and Bias Substructure Generator ",
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| 410 |
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"text_level": 1,
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"type": "text",
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"text": "Given a mini-batch of biased graphs ${ \\mathcal { G } } = \\{ G _ { 1 } , \\cdots , G _ { n } \\}$ , our idea is that: we take a collection of graph instances and design a generative probabilistic model to learn to mask the edges into causal subgraph or bias subgraph. Particularly, given a graph $G = \\left\\{ \\mathbf { A } , \\mathbf { X } \\right\\}$ , where $\\mathbf { A }$ is the adjacency matrix and $\\mathbf { X }$ is the node feature matrix, we utilize a multi-layer perceptron (MLP) upon the concatenation of node features $\\mathbf { x } _ { i }$ of node $i$ and $\\mathbf { x } _ { j }$ of node $j$ to measure the importance of edge $( i , j )$ for causal subgraph: ",
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"type": "equation",
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"img_path": "images/05c2f2c9be2439b56fc3c7498dc36052810353e52f0f3feff7c54355a1413936.jpg",
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"text": "$$\n\\alpha _ { i j } = \\mathbf { M L P } ( [ \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ] ) .\n$$",
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"text": "Then a sigmoid function $\\sigma ( \\cdot )$ is employed to project $\\alpha _ { i j }$ into the range of (0,1), which indicates the probability of edge $( i , j )$ being the edge in the causal subgraph as follows: ",
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"type": "equation",
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"img_path": "images/10fc244d1c17540052e850af038a05b079d773c9818e76f029b42e3c2061c81e.jpg",
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"text": "$$\nc _ { i j } = \\sigma ( \\alpha _ { i j } ) .\n$$",
|
| 458 |
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"text_format": "latex",
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| 459 |
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"text": "Naturally, we could get the probability of edge $( i , j )$ being the edge in the bias subgraph by: $b _ { i j } = 1 - c _ { i j }$ . Now we could construct the causal edge mask $\\mathbf { M } _ { c } \\bar { \\mathbf { \\eta } } = \\left[ c _ { i j } \\right]$ and bias edge mask $\\begin{array} { r } { \\mathbf { M } _ { b } = [ b _ { i j } ] } \\end{array}$ . Finally, we decompose the original graph $G$ into causal subgraph $G _ { c } = \\{ \\mathbf { M } _ { c } \\odot \\mathbf { A } , \\mathbf { X } \\}$ and bias subgraph $G _ { b } = \\{ { \\bf M } _ { b } \\odot { \\bf A } , { \\bf X } \\}$ . Intuitively, the edge mask could highlight different part of structure information of original graphs, thus GNNs built on the different subgraphs could encode different parts of graph information. Moreover, the mask generator has two advantages. (1) Global view: In individual graph level, the mask generator (i.e., MLP), whose parameters are shared by all the edges in a graph, take a global view of all the edges in a graph, which enables us to identify community in graph. It is well known that the effect of an edge cannot be judged independently, because edges usually collaborate with each other, forming a community, to make prediction. Thus, it is critical to evaluate an edge in a global view. In whole graph population level, the mask generator takes a global view of all the graphs in the training set, which enables us to identify causal/bias subgraph. Particularly, as the causal/bias is the statistical information in the population level, it is necessary to view all the graphs to identify the causal/bias substructure. Considering both such coalition effects and population-level statistical information, the generator is able to measure the importance of edges more accurately. (2) Generalization: The mask generator can generalize the mechanism of mask generation to new graphs without retraining, so it is capable and efficient to prune unseen graphs. ",
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"type": "text",
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| 480 |
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"text": "4.2 Learning Disentangled Graph Representations ",
|
| 481 |
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"text_level": 1,
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"type": "text",
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"text": "Given $G _ { c }$ and $G _ { b }$ , how to ensure they are causal subgraph and bias subgraph, respectively? Inspired by [21], our approach simultaneously trains a pair of GNNs $( g _ { b } , g _ { c } )$ with linear classifiers $( C _ { b } ^ { - } , C _ { c } )$ as follows: (1) Motivated by the observation in Sec. 3.1 that bias substructure is easier to learn, we utilize a bias-aware loss to train a bias $\\mathrm { G N N } g _ { b }$ and a bias classifier $C _ { b }$ and (2) in contrast, we train a causal $\\mathrm { G N N } _ { g _ { c } }$ and a causal classifier $C _ { c }$ on the training graphs that the bias GNN struggles to learn. Next, we would present each component in detail. ",
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},
|
| 501 |
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{
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| 502 |
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"type": "text",
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| 503 |
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"text": "As shown in Fig. 3, GNN $g _ { c }$ and $g _ { b }$ embed the corresponding subgraphs into causal embedding $z _ { c } = g _ { c } ( G _ { c } ; \\gamma _ { c } )$ and bias embedding $z _ { b } = g _ { b } ( G _ { b } ; \\gamma _ { b } )$ , respectively, where $\\gamma$ is the parameters of GNNs. Subsequently, concatenated vector $z = \\left[ z _ { c } ; z _ { b } \\right]$ is fed into linear classifiers $C _ { c }$ and $C _ { b }$ to predict the target label $y$ . To train $g _ { b }$ and $C _ { b }$ as bias extractor, we utilize the generalized cross entropy (GCE) [47] loss to amplify the bias of the bias GNN and classifier: ",
|
| 504 |
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"bbox": [
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| 513 |
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"type": "equation",
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| 514 |
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"img_path": "images/4e363937d5fab4c35476a216450588ab83b7cd7ddda3e32b99f10750e7b49403.jpg",
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"text": "$$\nG C E ( C _ { b } ( z ; \\alpha _ { b } ) , y ) = \\frac { 1 - C _ { b } ^ { y } ( z ; \\alpha _ { b } ) ^ { q } } { q } ,\n$$",
|
| 516 |
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"text_format": "latex",
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| 517 |
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"bbox": [
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"type": "text",
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"text": "where $C _ { b } ( z ; \\alpha _ { b } )$ and $C _ { b } ^ { y } ( z ; \\alpha _ { b } )$ are softmax output of the bias classifier and its probability belonging to the target category $y$ , respectively, and $\\alpha$ is the parameters of classifier. Here $q \\in ( 0 , 1 ]$ is a hyperparameter that controls the degree of amplifying bias. Given $\\theta _ { b } = [ \\gamma _ { b } , \\alpha _ { b } ]$ , the gradient of the GCE loss up-weights the gradient of the standard cross entropy (CE) loss for the samples with a high confidence $\\dot { \\boldsymbol { C } } _ { b } ^ { y }$ of predicting the correct target category as follows: ",
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"bbox": [
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| 537 |
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"type": "equation",
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| 538 |
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"img_path": "images/a3271b67bc7d4ef165091054dc7a412c66d4a9009a4e0f32298182bb32bd7a86.jpg",
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"text": "$$\n\\frac { \\partial G C E ( C _ { b } ( z ; \\alpha _ { b } ) , y ) } { \\partial \\theta _ { b } } = ( C _ { b } ^ { y } ) ^ { q } \\frac { \\partial C E ( C _ { b } ( z ; \\alpha _ { b } ) , y ) } { \\partial \\theta _ { b } } .\n$$",
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| 540 |
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"text_format": "latex",
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"bbox": [
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"type": "text",
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"text": "Therefore, compared with CE loss, GCE loss will amplify the gradients of $\\theta _ { b }$ on samples by the confidence score $( C _ { b } ^ { y } ) ^ { q }$ . Based on our observation that the bias information is usually easier to be learned, so the biased graphs will have higher $( C _ { b } ^ { y } ) ^ { q }$ than unbiased graphs. Therefore, the model $g _ { b }$ and $C _ { b }$ trained by GCE loss will focus on bias information and finally get the bias subgraph. Note that, to ensure that $C _ { b }$ predicts target labels mainly based on this $z _ { b }$ , the loss from $C _ { b }$ is not backpropagated to $g _ { c }$ , i.e., only update $\\theta _ { b }$ in Eq. (4), and vice versa. ",
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"bbox": [
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"type": "text",
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"text": "Meanwhile, we also train a causal GNN simultaneously with the weighted CE loss. The graphs with high CE loss from $C _ { b }$ can be regarded as the unbiased samples compared with the samples with low CE loss. In this regard, we could obtain the unbias score of each graph as ",
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| 572 |
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"type": "equation",
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"img_path": "images/31d1a5650952de165f5a245f596e78cca5b95a48699d6699a29992e8365a5f3c.jpg",
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"text": "$$\nW ( z ) = \\frac { C E ( C _ { b } ( z ) , y ) } { C E ( C _ { c } ( z ) , y ) + C E ( C _ { b } ( z ) , y ) } .\n$$",
|
| 575 |
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"type": "text",
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"text": "Large value of $W$ implies the graph is an unbiased sample, hence we could use these weights to reweight the loss of these graphs to train $g _ { c }$ and $C _ { c }$ , enforcing them to learn the unbiased information. Thus, the objective function for learning disentangled representation is: ",
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"img_path": "images/f53c02262b2fc1caa42f6bd34a8f10e4df9f009406807429017e35a5848920f5.jpg",
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"text": "$$\nL _ { D } = W ( z ) C E ( C _ { c } ( z ) , y ) + G C E ( C _ { b } ( z ) , y ) .\n$$",
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| 599 |
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"text_format": "latex",
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"bbox": [
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"type": "text",
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"text": "4.3 Counterfactual Unbiased Sample Generation ",
|
| 611 |
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"text_level": 1,
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"type": "text",
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"text": "Until now, we have achieved the first goal analyzed in Sec. 3.2 that is the disentanglement of causal and bias substructures. Next, we will show how to achieve the second goal that makes the causal variable $z _ { c }$ and bias variable $z _ { b }$ uncorrelated. Although we have disentangled causal and bias information, they are disentangled from the biased observed graphs. Hence, there will exist statistical correlation between causal and bias variables inheriting from the biased observed graphs. To further decorrelate $z _ { c }$ and $z _ { b }$ , according to the causal relation of data-generating process: $C \\right. G \\left. B$ , we propose to generate the counterfactual unbiased samples in embedding space by swapping $z _ { b }$ . More specifically, we randomly permute bias vectors in each mini-batch and obtain $z _ { u n b i a s e d } = [ z _ { c } ; \\hat { z _ { b } } ] .$ where $\\hat { z _ { b } }$ represents the randomly permuted bias vectors of $z _ { b }$ . As $z _ { c }$ and $\\hat { z } _ { b }$ in $z _ { u n b i a s e d }$ are randomly combined from different graphs, they will have much less correlation than $\\boldsymbol { z } = \\left[ z _ { c } ; z _ { b } \\right]$ where both are from the same graph. To make $g _ { b }$ and $C _ { b }$ still focus on the bias information, we also swap label $y$ as $\\hat { y }$ along with $\\hat { z _ { b } }$ , so that the spurious correlation between $\\hat { z _ { b } }$ and $\\hat { y }$ still exists. With the generated unbiased samples, we utilize the following loss function to train two GNN modules: ",
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| 623 |
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"type": "equation",
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"img_path": "images/aa9d991186ce7832aa2d7dad69769211c16e3869a4a46d0185725069ea6dcfc8.jpg",
|
| 634 |
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"text": "$$\n{ \\cal L } _ { G } = W ( z ) C E ( C _ { c } ( z _ { u n b i a s e d } ) , y ) + G C E ( C _ { b } ( z _ { u n b i a s e d } ) , \\hat { y } ) ,\n$$",
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| 635 |
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"text_format": "latex",
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| 636 |
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"bbox": [
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"type": "text",
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"text": "Together with the disentanglement loss, total loss function is defined as: ",
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| 647 |
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"type": "equation",
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"img_path": "images/4df90f628564e59d8fbdbd05a8e1b913713e84074319d57203e9464cb0b85e62.jpg",
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| 658 |
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"text": "$$\n{ \\cal L } = { \\cal L } _ { D } + \\lambda _ { G } { \\cal L } _ { G } ,\n$$",
|
| 659 |
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"text_format": "latex",
|
| 660 |
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"bbox": [
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{
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"type": "text",
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| 670 |
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"text": "where $\\lambda _ { G }$ is a hyperparameter for weighting the importance of generation component. Moreover, training with more diverse samples would also benefit with better generalization on unseen testing scenarios. Our approach is summarized in App. B. Note that, as we need well-disentangled representations to generate the high-quality unbiased samples, in the early stage of training, we only train the model with $L _ { D }$ . After certain epochs, we train the model with $L$ . ",
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"type": "text",
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"text": "5 Experiment ",
|
| 682 |
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"text_level": 1,
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"type": "text",
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"text": "Datasets. We construct three datasets with various properties and bias ratios to benchmark this new problem, where the datasets have clear causal subgraphs making the results explainable. Following CMNIST-75sp introduced in Sec. 3.1, we use the similar way to construct CFashion-75sp and CKuzushiji-75sp datasets based on the Fashion-MNIST [40] and Kuzushiji-MNIST [4] datasets. As the causal subgraphs of these two datasets are more complicated (fashion product and hiragana characters), they are more challenging. Due to the page limits, here we set bias degrees as $\\{ 0 . 8 , 0 . 9 , 0 . 9 5 \\}$ . We report the main results on unbiased test sets. Details are in App. C.1. ",
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| 694 |
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"type": "text",
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"text": "Baselines and experimental setup. As DisC is a general framework which could be built on various base GNN models, we select three popular GNNs: GCN [17], GIN [41], and GCNII [3]. The corresponding models are termed as $\\mathrm { D i s C } _ { G C N }$ , $\\mathrm { D i s C } _ { G I N }$ and $\\mathrm { D i s C } _ { G C N I I }$ , respectively. Hence, base models are the most straight baselines. Another kind of baselines are the causal-inspired GNN method DIR [39] and StableGNN [6]. We also compare against a general debiasing method LDD [21] by replacing its encoder with GNNs. Graph Pooling method DiffPool [44] and graph disentangling method FactorGCN [42] are also compared. To keep fair comparison, our model uses the same GNN architecture and hyperparameters with the corresponding base model. All the experiments are run 4 times with different random seeds and we report the accuracy and the standard error. More details are in App. C.2. ",
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| 705 |
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"type": "text",
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"text": "5.1 Quantitative Evaluation ",
|
| 716 |
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"text_level": 1,
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"type": "text",
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"text": "Main results. The overall results are summarized in Table 1, and we have following observations: ",
|
| 728 |
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"type": "text",
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"text": "(1) DisC has much better generalization ability than base models. DisC outperforms the corresponding base model consistently with a large margin. With heavier biases, our model achieves larger improvements over base models. Specifically, for CMNIST-75sp, CFashion-75sp and CKuzushiji-75sp with smaller bias degree (i.e., 0.8), our model achieves $4 0 . 0 2 \\%$ , $4 . 4 7 \\%$ and $2 9 . 8 2 \\%$ average improvements over corresponding base models, respectively. Surprisingly, with severer biases (0.9 and 0.95), DisC achieves $1 6 9 . 1 7 \\%$ , $1 4 . 6 7 \\%$ and $4 9 . 3 5 \\%$ average improvements over base models on three datasets, respectively. It indicates that the proposed method is a general framework helping existing GNNs against the negative impact of bias. ",
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| 739 |
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"type": "text",
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"text": "(2) DisC significantly outperforms existing debiasing methods. We notice that DIR could not achieve satisfying results. The reason is that DIR utilizes CE loss to extract bias information, which could not fully capture the property of bias in severe bias scenarios. And DIR sets one fixed threshold to spilt subgraphs, which is suboptimal. StableGNN outperforms their base model DiffPool and achieve competitive results, indicating the effectiveness of their proposed causal variable distinguishing regularizer. However, their framework adjusts data distribution based on the original dataset, it is hard to generate unbiased distribution when the unbiased samples are scarce. DisC could generate more unbiased samples based on the disentangled representations. Moreover, LDD is a general debiasing method which is not designed for graph data. DisC outperforms corresponding LDD variants with average $2 3 . 1 5 \\%$ , indicating that the seamless joint of global-population-aware edge masker with debiasing disentangle framework is very effective. ",
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| 750 |
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{
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"type": "table",
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"img_path": "images/199741d9fe65df203714b6564543f2ff46981386b8a9b6296d024b20573d5ae3.jpg",
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| 761 |
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"table_caption": [
|
| 762 |
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"Table 1: Graph classification accuracy evaluated on unbiased testing sets, which have same color (bias) set with training set. The best performance within each base model variant is in bold. "
|
| 763 |
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],
|
| 764 |
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"table_footnote": [],
|
| 765 |
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"table_body": "<table><tr><td rowspan=\"2\">Dataset Bias</td><td colspan=\"3\">CMNIST-75sp</td><td colspan=\"3\">CFashion-75sp</td><td colspan=\"3\">CKuzushiji-75sp</td></tr><tr><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td></tr><tr><td>FactorGCN [42]</td><td>72.30±1.18</td><td>62.35±5.07</td><td>42.50±4.91</td><td>61.23±1.11</td><td>53.50±1.29</td><td>45.78±2.40</td><td>42.87±1.19</td><td>32.35±2.79</td><td>23.87±0.12</td></tr><tr><td>DiffPool [44]</td><td>73.79±0.02</td><td>66.45±0.78</td><td>47.12±1.04</td><td>62.82±0.53</td><td>57.50±0.39</td><td>50.86±0.20</td><td>45.46±0.65</td><td>36.18±0.19</td><td>27.45±0.26</td></tr><tr><td>DIR [39]</td><td>9.98±0.33</td><td>9.96±0.23</td><td>10.03±0.27</td><td>13.02±1.92</td><td>12.80±1.67</td><td>11.98±1.41</td><td>10.35±0.32</td><td>10.72±0.27</td><td>10.59±0.46</td></tr><tr><td>StableGNN [6]</td><td>77.65±1.64</td><td>68.87±1.74</td><td>51.33±0.87</td><td>64.03±0.29</td><td>58.26±0.09</td><td>51.46±0.39</td><td>49.41±0.09</td><td>39.30±0.12</td><td>28.26±0.14</td></tr><tr><td>LDDGCN[21]</td><td>64.95±1.22</td><td>56.65±2.18</td><td>46.83±2.88</td><td>63.85±1.17</td><td>64.30±0.89</td><td>62.28±0.48</td><td>42.38±0.33</td><td>38.75±0.49</td><td>33.08±0.59</td></tr><tr><td>LDDGIN[21]</td><td>64.88±1.45</td><td>50.59±1.07</td><td>31.23±2.48</td><td>64.65±0.63</td><td>57.10±0.43</td><td>53.38±0.47</td><td>37.83±0.54</td><td>28.97±0.18</td><td>22.13±0.34</td></tr><tr><td>LDDGCNII [21]</td><td>78.03±0.66</td><td>69.53±0.96</td><td>51.05±3.87</td><td>50.63±1.79</td><td>54.09±2.54</td><td>57.93±0.88</td><td>48.70±1.98</td><td>41.59±1.07</td><td>33.93±0.71</td></tr><tr><td>GCN[17]</td><td>50.43±4.13</td><td>28.97±4.4</td><td>13.50±1.38</td><td>63.60±0.53</td><td>57.22±0.93</td><td>47.69±0.42</td><td>38.45±1.1</td><td>28.35±0.79</td><td>20.70±0.88</td></tr><tr><td>DisCGCN</td><td>82.60±0.93</td><td>78.14±2.14</td><td>63.47±5.65</td><td>66.85±1.11</td><td>65.33±4.70</td><td>63.93±1.50</td><td>55.53±2.29</td><td>48.13±2.59</td><td>36.63±1.73</td></tr><tr><td>GIN [41]</td><td>57.75±0.78</td><td>36.78±5.55</td><td>16.04±1.14</td><td>64.25±0.46</td><td>58.03±0.40</td><td>49.74±0.60</td><td>41.83±0.78</td><td>30.09±0.87</td><td>21.18±1.63</td></tr><tr><td>DisCGIN</td><td>82.10±1.50</td><td>74.90±1.81</td><td>58.58±4.24</td><td>67.10±1.07</td><td>59.90±1.31</td><td>55.80±0.36</td><td>55.18±1.00</td><td>41.75±0.81</td><td>30.25±1.63</td></tr><tr><td>GCNII [3]</td><td>69.70±1.73</td><td>57.68±1.68</td><td>41.00±3.75</td><td>66.68±0.59</td><td>60.58±0.28</td><td>53.18±0.08</td><td>48.53±0.25</td><td>36.23±0.20</td><td>25.60±0.76</td></tr><tr><td>DisCGCNII</td><td>79.50±2.48</td><td>76.00±1.90</td><td>60.54±5.33</td><td>66.47±1.77</td><td>65.48±0.70</td><td>61.75±0.27</td><td>54.90±1.30</td><td>44.73±1.55</td><td>36.95±0.70</td></tr></table>",
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"type": "text",
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"text": "Ablation studies. To validate the importance of each module in our method, in Fig. 4, we conduct ablation studies on our variants (w.o. G means without the sample generation module) and the related variants of LDD. The major difference between DisC/w.o. G with LDD /w.o. G is the edge mask module. In most cases, DisC/w.o. G significantly outperforms LDD /w.o. G, indicating the necessity of learning edge mask for graph data. And DisC which has counterfactual sample generation module could further boost the performances based on the disentangled embeddings of DisC/w.o. G. However, LDD seldomly outperforms LDD /w.o. G or even achieves worse performances. That is, generating high-quality counterfactual samples needs well-disentangled causal and bias embeddings. If embeddings are not well-disentangled, counterfactual samples may act as noisy samples, which would prevent models from achieving further improvement. The edge masker could help the model generate well-disentangled embeddings, which is crucial for overall performance. ",
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"type": "image",
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"img_path": "images/a0e110950764af9cb15bcd44f9149686964d2d2dbbc914f554282cc321f7d710.jpg",
|
| 788 |
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"image_caption": [
|
| 789 |
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"Figure 4: Ablation studies of the DisC vs. LDD average over three bias degrees of each dataset. "
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],
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"image_footnote": [],
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"type": "text",
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"text": "Robustness on unseen bias. Table 2 reports the results of DisC compared with its corresponding base models on testing set with unseen bias, i.e., the pre-defined color (bias) sets of training set and testing set are disjoint. The performances of base models further drop compared with the results on seen bias scenario in Table 1. However, our model still achieves very stable performances, fully demonstrating the generalization ability of our model on agnostic bias scenario. ",
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"type": "text",
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"text": "Hyperparameter experiments Fig. 5 is the hyperparameter experiments of the degree of amplifying bias $q$ in GCE loss and the importance of generation component $\\lambda _ { G }$ . For $q$ , we fix $\\lambda _ { G } = 1 0$ and vary $q$ from $\\{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \\}$ . For $\\lambda _ { G }$ , we fix $q = 0 . 7$ and vary $\\lambda _ { G }$ from $\\{ 1 , 5 , 1 0 , 1 5 \\}$ . From the results, we can see that our model achieves stable performance across different values of $q$ and $\\lambda _ { G }$ . When $q = 0 . 1$ , it means the GCE loss will nearly reduce to normal CE loss. We can see the performance of $\\mathrm { D i s C } _ { G C N }$ is worse than other scenarios, demonstrating the effectiveness of utilizing GCE loss. ",
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{
|
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"type": "table",
|
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"img_path": "images/e042508e60748d7c5704b9e9cda337686f8480727853f0b2a4ffa768504103c5.jpg",
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"table_caption": [
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| 826 |
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"Table 2: The results on unseen unbiased testing sets, i.e., the color has not been seen in training set. "
|
| 827 |
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],
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"table_footnote": [],
|
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"table_body": "<table><tr><td rowspan=\"2\">Dataset Bias</td><td colspan=\"3\">CMNIST-75sp</td><td colspan=\"3\">CFashion-75sp</td><td colspan=\"3\">CKuzushiji-75sp</td></tr><tr><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td><td>0.8</td><td>0.9</td><td>0.95</td></tr><tr><td>DIR</td><td>10.38±0.28</td><td>10.14±0.40</td><td>9.77±0.18</td><td>16.77±1.71</td><td>16.51±3.20</td><td>12.59±1.61</td><td>10.48±0.34</td><td>10.33±0.75</td><td>10.59±0.95</td></tr><tr><td>GCN</td><td>36.88±5.16</td><td>23.07±4.07</td><td>11.88±0.33</td><td>59.33±0.55</td><td>53.65±0.47</td><td>45.60±1.06</td><td>36.35±0.48</td><td>27.88±0.94</td><td>19.95±0.67</td></tr><tr><td>DisCGCN</td><td>82.73±1.31</td><td>77.70±0.87</td><td>65.48±0.76</td><td>67.9±1.45</td><td>68.28±0.18</td><td>63.77±1.37</td><td>57.80±2.38</td><td>51.60±0.41</td><td>41.60±3.94</td></tr><tr><td>GIN</td><td>48.93±2.99</td><td>34.95±0.86</td><td>14.53±0.97</td><td>58.88±0.57</td><td>53.80±0.52</td><td>48.43±0.69</td><td>39.25±0.57</td><td>30.75±1.45</td><td>22.35±0.86</td></tr><tr><td>DisCGIN</td><td>77.80±1.33</td><td>73.00±0.61</td><td>58.80±1.66</td><td>67.15±0.79</td><td>59.98±0.62</td><td>51.70±0.34</td><td></td><td></td><td>55.47±0.98 43.20±1.36 31.33±1.71</td></tr><tr><td>GCNII</td><td>53.50±6.23</td><td>45.52±2.26</td><td>32.6±5.66</td><td>58.85±1.89</td><td>53.98±0.85</td><td>46.97±1.38</td><td>39.93±0.88</td><td>30.33±1.17</td><td>23.09±1.83</td></tr><tr><td>DisCGCNII</td><td>79.65±2.13</td><td>76.63±1.3860.00±5.66</td><td></td><td></td><td></td><td>60.50±2.77 63.05±2.2561.78±1.60</td><td></td><td>56.23±3.4549.10±2.05</td><td>41.05±0.11</td></tr></table>",
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"img_path": "images/8e496cbb6ae8e9942e5b95edadf4f703867ebb55af480ec526acdfcf4e3dab1a.jpg",
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| 852 |
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"image_caption": [
|
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"Figure 5: The hyperparameter experiments of $q$ and $\\lambda _ { G }$ "
|
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"type": "text",
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"text": "5.2 Qualitative Evaluation ",
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| 867 |
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"text_level": 1,
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"type": "text",
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"text": "Visualization of edge mask. To better illustrate the significant causal and bias subgraphs extracted by $\\mathrm { D i s C } _ { G C N }$ , we visualize the original images, original graph, and corresponding causal subgraph and bias subgraph of CMNIST-75sp with 0.9 bias degree in Fig. 6, where the width of edge represents the value of learned weight $c _ { i j }$ or $b _ { i j }$ . Fig. 6(a) shows the visualization results of testing graphs with the bias (color) that has been seen in the training set. As we can see, our model could discover the causal subgraphs where the most salient edges are in the digital subgraphs. With these causal subgraphs that highlight the structure information of digital, the GNNs will more easily extract this causal information. Fig. 6(b) shows the visualization results of testing graphs with unseen bias. According to the visualization, our model could still discover the causal subgraph outline, indicating our model could recognize causal subgraphs, whether the bias is seen or unseen. The visualization results of CFashion-75sp and CKuzushiji-75sp are shown in App. D. ",
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"img_path": "images/c8175449c2701658e4017be825905cdeb3e231c2d6376d79873c1c752212b81e.jpg",
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"image_caption": [
|
| 891 |
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"OriginalimageOriginal graph Causalsubgraph BiasedsubgraphOriginalimageOriginal graph CausalsubgraphBiased subgraph ",
|
| 892 |
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"Figure 6: Visualization of subgraphs extracted by DisC. The width of edge is edge weight $c _ { i j }$ or $b _ { i j }$ "
|
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|
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"type": "text",
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"text": "Projection of disentangled representation. Fig. 7 shows the projection of latent vectors $z _ { c }$ and $z _ { b }$ extracted from the causal GNN $g _ { c }$ and bias GNN $g _ { b }$ of $\\mathrm { D i s C } _ { G C N }$ , respectively, using t-SNE [19] on ",
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"img_path": "images/221d0d59bdbf86e2589c0ab453c12e80f215382da65ad5e2230b4b84b9a9acaa.jpg",
|
| 917 |
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"image_caption": [
|
| 918 |
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"Figure 7: Visualization of $z _ { c }$ and $z _ { b }$ with colors labeled by the digit and bias (color) labels. We observe that $z _ { c }$ and $z _ { b }$ are well clustered according to the groundtruth labels and bias labels, respectively. "
|
| 919 |
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|
| 920 |
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"image_footnote": [],
|
| 921 |
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| 924 |
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"type": "text",
|
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"text": "CMNIST-75sp. Fig. 7 (a-b) are the projections of $z _ { c }$ labeled by the target labels (digit) and bias labels (color), respectively. Fig. 7 (c-d) are the projections of $z _ { b }$ labeled by the target labels and bias labels, respectively. We observe that $z _ { c }$ are clustered according to the target labels while $z _ { b }$ are clustered with the bias labels. And $z _ { c }$ are mixed with bias labels and $z _ { b }$ are mixed with target labels. The results indicate that DisC successfully learns the disentangled causal and bias representations. ",
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"text": "Transferability of the learned mask. As our model could extract GNN-independent subgraphs, the learning edge weights can be used to purify original biased graphs. These sparse subgraphs represent significant semantic information and can be universally transferred to any GNNs. To validate this point, we learn the edge mask by $\\mathrm { D i s C } _ { G C N }$ and prune the edges with least $\\{ 0 \\% , 2 0 \\% , 4 0 \\% , 6 0 \\% \\}$ weights while keeping the remaining edge weights. Then we train vanilla GIN and GCNII on these weighted pruned datasets. Fig. 8 is the comparison of the results, where the dashed lines represent the results of base model on original biased graphs and the solid lines represent the performance of GNNs on weighted pruned datasets. The results show that the GNNs trained on the pruned datasets achieve better performances, indicating our learned edge mask has considerable transferability. ",
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"img_path": "images/18fa05e618bc66f5ba05c2bab230902cd658bb98fb76ae415b1a2efbc238f5f0.jpg",
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"image_caption": [
|
| 955 |
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"Figure 8: Performance of GIN and GCNII on the weighted pruned graphs found by $\\mathrm { D i s C } _ { G C N }$ . "
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"type": "text",
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"text": "6 Conclusion ",
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| 969 |
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"type": "text",
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"text": "In this paper, we are first to study the generalization problem of GNNs on severe bias datasets, which is crucial to study the transparently knowledge learning mechanism of GNNs. We analyze the problem in a causal view that the generalization of GNNs will be hindered by entangled representations as well as the correlation between causal and bias variables. To remove the impact from these two aspects, we propose a general disentangling framework, DisC, which extracts causal substructure and bias substructure by two different functional GNNs, respectively. After the representations are well-disentangled, we proliferate the counterfactual unbiased samples by randomly swapping the disentangled vectors. With the new constructed benchmarks, we clearly validate the effectiveness, robustness, interpretability, and transferability of our method. ",
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"text": "Acknowledgments and Disclosure of Funding ",
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"text": "This work is supported in part by the National Natural Science Foundation of China (No. U20B2045, 62192784, 62172052, 62002029, 62172052, U1936014). This work is also partially supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06). This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019- 3583139727. The work of Shaohua Fan is supported by the China Scholarship Council (No.202006470078). The computation resource of this project is supported by Compute Canada . ",
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"text": "References \n[1] Hyojin Bahng, Sanghyuk Chun, Sangdoo Yun, Jaegul Choo, and Seong Joon Oh. Learning de-biased representations with biased representations. In ICML, 2020. [2] Remi Cadene, Corentin Dancette, Matthieu Cord, Devi Parikh, et al. Rubi: Reducing unimodal biases for visual question answering. In NeurIPS, 2019. \n[3] Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In ICML, 2020. [4] Tarin Clanuwat, Mikel Bober-Irizar, Asanobu Kitamoto, Alex Lamb, Kazuaki Yamamoto, and David Ha. Deep learning for classical japanese literature. arXiv preprint arXiv:1812.01718, 2018. \n[5] Luke Darlow, Stanisław Jastrz˛ebski, and Amos Storkey. Latent adversarial debiasing: Mitigating collider bias in deep neural networks. arXiv preprint arXiv:2011.11486, 2020. [6] Shaohua Fan, Xiao Wang, Chuan Shi, Peng Cui, and Bai Wang. Generalizing graph neural networks on out-of-distribution graphs. In arXiv preprint arXiv:2111.10657, 2021. [7] Shaohua Fan, Xiao Wang, Chuan Shi, Kun Kuang, Nian Liu, and Bai Wang. Debiased graph neural networks with agnostic label selection bias. TNNLS, 2022. [8] Shaohua Fan, Xiao Wang, Chuan Shi, Emiao Lu, Ken Lin, and Bai Wang. One2multi graph autoencoder for multi-view graph clustering. In WWW, pages 3070–3076, 2020. \n[9] Shaohua Fan, Junxiong Zhu, Xiaotian Han, Chuan Shi, Linmei Hu, Biyu Ma, and Yongliang Li. Metapath-guided heterogeneous graph neural network for intent recommendation. In SIGKDD, pages 2478–2486, 2019. \n[10] Robert Geirhos, Jörn-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel, Matthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):665–673, 2020. \n[11] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and Wieland Brendel. Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. arXiv preprint arXiv:1811.12231, 2018. \n[12] Madelyn Glymour, Judea Pearl, and Nicholas P Jewell. Causal inference in statistics: A primer. John Wiley & Sons, 2016. \n[13] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In NeurIPS, 2017. \n[14] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. arXiv preprint arXiv:1903.12261, 2019. \n[15] Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. In NeurIPS, 2020. \n[16] Byungju Kim, Hyunwoo Kim, Kyungsu Kim, Sungjin Kim, and Junmo Kim. Learning not to learn: Training deep neural networks with biased data. In CVPR, 2019. \n[17] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In ICLR, 2016. \n[18] Boris Knyazev, Graham W Taylor, and Mohamed Amer. Understanding attention and generalization in graph neural networks. In NeurIPS, 2019. \n[19] G.E. Kvan der Maaten, L.J.P.; Hinton. Understanding attention and generalization in graph neural networks. Journal of Machine Learning Research, 2008. \n[20] John Boaz Lee, Ryan Rossi, and Xiangnan Kong. Graph classification using structural attention. In SIGKDD, 2018. \n[21] Jungsoo Lee, Eungyeup Kim, Juyoung Lee, Jihyeon Lee, and Jaegul Choo. Learning debiased representation via disentangled feature augmentation. In NeurIPS, 2021. \n[22] Haoyang Li, Xin Wang, Ziwei Zhang, and Wenwu Zhu. Ood-gnn: Out-of-distribution generalized graph neural network. arXiv preprint arXiv:2112.03806, 2021. \n[23] Yi Li and Nuno Vasconcelos. Repair: Removing representation bias by dataset resampling. In CVPR, 2019. \n[24] Renjie Liao, Raquel Urtasun, and Richard Zemel. A pac-bayesian approach to generalization bounds for graph neural networks. In ICLR, 2020. \n[25] Yanbei Liu, Xiao Wang, Shu Wu, and Zhitao Xiao. Independence promoted graph disentangled networks. In AAAI, 2020. \n[26] Ana Lucic, Maartje ter Hoeve, Gabriele Tolomei, Maarten de Rijke, and Fabrizio Silvestri. Cf-gnnexplainer: Counterfactual explanations for graph neural networks. In AISTATS, 2022. \n[27] Dongsheng Luo, Wei Cheng, Dongkuan Xu, Wenchao Yu, Bo Zong, Haifeng Chen, and Xiang Zhang. Parameterized explainer for graph neural network. In NeurIPS, 2020. \n[28] Jianxin Ma, Peng Cui, Kun Kuang, Xin Wang, and Wenwu Zhu. Disentangled graph convolutional networks. In ICML, pages 4212–4221. PMLR, 2019. \n[29] Jiaqi Ma, Junwei Deng, and Qiaozhu Mei. Subgroup generalization and fairness of graph neural networks. In NeurIPS, 2021. \n[30] Junhyun Nam, Hyuntak Cha, Sungsoo Ahn, Jaeho Lee, and Jinwoo Shin. Learning from failure: De-biasing classifier from biased classifier. In NeurIPS, 2020. \n[31] Judea Pearl. Causality. Cambridge university press, 2009. \n[32] Hans Reichenbach. The direction of time, volume 65. Univ of California Press, 1991. \n[33] Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust neural networks. In ICLR, 2019. \n[34] Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61–80, 2008. \n[35] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. In ICLR, 2017. \n[36] Minh Vu and My T Thai. Pgm-explainer: Probabilistic graphical model explanations for graph neural networks. In NeurIPS, 2020. \n[37] Haohan Wang, Zexue He, Zachary C Lipton, and Eric P Xing. Learning robust representations by projecting superficial statistics out. arXiv preprint arXiv:1903.06256, 2019. \n[38] Qitian Wu, Hengrui Zhang, Junchi Yan, and David Wipf. Handling distribution shifts on graphs: An invariance perspective. In ICLR, 2022. \n[39] Ying-Xin Wu, Xiang Wang, An Zhang, Xiangnan He, and Tat-Seng Chua. Discovering invariant rationales for graph neural networks. In ICLR, 2022. \n[40] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017. \n[41] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In ICLR, 2019. \n[42] Yiding Yang, Zunlei Feng, Mingli Song, and Xinchao Wang. Factorizable graph convolutional networks. In NeurIPS, volume 33, pages 20286–20296, 2020. \n[43] Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. Gnnexplainer: Generating explanations for graph neural networks. In NeurIPS, 2019. \n[44] Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In NeurIPS, volume 31, 2018. \n[45] Hao Yuan, Haiyang Yu, Jie Wang, Kang Li, and Shuiwang Ji. On explainability of graph neural networks via subgraph explorations. In ICML, 2021. \n[46] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In AAAI, 2018. \n[47] Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. NeurIPS, 2018. ",
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| 1 |
+
# POISONING AND BACKDOORING CONTRASTIVE LEARNING
|
| 2 |
+
|
| 3 |
+
Nicholas Carlini Google
|
| 4 |
+
|
| 5 |
+
Andreas Terzis Google
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
Multimodal contrastive learning methods like CLIP train on noisy and uncurated training datasets. This is cheaper than labeling datasets manually, and even improves out-of-distribution robustness. We show that this practice makes backdoor and poisoning attacks a significant threat. By poisoning just $0 . 0 1 \%$ of a dataset (e.g., just 300 images of the 3 million-example Conceptual Captions dataset), we can cause the model to misclassify test images by overlaying a small patch. Targeted poisoning attacks, whereby the model misclassifies a particular test input with an adversarially-desired label, are even easier requiring control of $0 . 0 0 0 1 \%$ of the dataset (e.g., just three out of the 3 million images). Our attacks call into question whether training on noisy and uncurated Internet scrapes is desirable.
|
| 10 |
+
|
| 11 |
+
# 1 INTRODUCTION
|
| 12 |
+
|
| 13 |
+
Contrastive learning (Chopra et al., 2005; Hadsell et al., 2006) trains a model that projects a data distribution onto a lower-dimensional embedding space such that similar objects in the origin space are closer together in the embedding space than dissimilar objects (Chechik et al., 2010; Sohn, 2016; Oord et al., 2018; Wu et al., 2018). Significant advances over the last years have enabled self-supervised classifiers to achieve state of the art accuracy by training on noisy and uncurated datasets (Radford et al., 2021; Tian et al., 2021), which brings two significant benefits.
|
| 14 |
+
|
| 15 |
+
First, training on uncurated data is cheaper (Joulin et al., 2016). Compared to an estimated several million USD it cost to label the ImageNet dataset (Deng et al., 2009), contrastively trained models can train without expensive labeling efforts (Chen et al., 2020a). Further, because each image in ImageNet is required to contain one of just 1,000 different objects, there are large categories of images that can never be part of this supervised dataset (Jia et al., 2021). On the other hand, a contrastive model can learn on arbitrary images whether or not they have a suitable corresponding label in some dataset.
|
| 16 |
+
|
| 17 |
+
Second, training on noisy data improves robustness (Radford et al., 2021). Classifiers trained exclusively on ImageNet overfit the particular details of this training set (Recht et al., 2019; Hendrycks & Dietterich, 2019), and do not generalize to other test sets (Taori et al., 2020). Contrastive models trained on data scraped from the Internet exhibit impressive robustness properties; The contrastively trained CLIP (Radford et al., 2021) model is the first technique to show significant effective robustness on ImageNet-V2 (Recht et al., 2019; Taori et al., 2020).
|
| 18 |
+
|
| 19 |
+
Contributions. We make the case that training on unfiltered may be undesirable if even a tiny fraction of the data could be maliciously poisoned by an adversary. And this is likely the case: the data is scraped from the Internet (Jia et al., 2021) without any human review before it is passed to the learning algorithm (Radford et al., 2021; Jia et al., 2021; Tian et al., 2021). Thus, because these datasets are explicitly “noisy” (Jia et al., 2021) and “uncurated” (Tian et al., 2019), we argue the likelihood of at least one adversary is high.
|
| 20 |
+
|
| 21 |
+
We show that this adversary can mount powerful targeted poisoning (Biggio et al., 2012) and backdoor attacks (Gu et al., 2017; Chen et al., 2017) against multimodal contrastive models. A poisoning adversary introduces malicious examples into the training dataset so that the model will misclassify a particular input at test time as an adversarially-desired label. We then consider patchbased backdoors, where the adversary poisons a dataset so that the learned model will classify any input that contains a particular trigger-pattern as a desired target label.
|
| 22 |
+
|
| 23 |
+
We require no new technical ideas to poison or backdoor contrastively-trained models (Biggio et al., 2012; Gu et al., 2017; Chen et al., 2017)—although we must adapt existing techniques to this new domain. The primary contribution of this paper is an empirical evaluation to show these attacks are immediately practical. Compared to prior backdooring attacks which require poisoning on average $1 \%$ of training data for successful clean label attacks (Shafahi et al., 2018; Saha et al., 2021), we find that attacking multimodal contrastive models requires orders of magnitude fewer injections: just $0 . 0 1 \%$ suffices for many of our backdoor attacks, or $0 . 0 0 0 1 \%$ for poisoning attacks.
|
| 24 |
+
|
| 25 |
+
2 BACKGROUND, NOTATION, AND RELATED WORK
|
| 26 |
+
|
| 27 |
+
# 2.1 POISONING AND BACKDOOR ATTACKS
|
| 28 |
+
|
| 29 |
+
In a poisoning attack (Biggio et al., 2012), an adversary modifies a benign training dataset $\mathcal { X }$ by injecting poisoned examples $\mathcal { P }$ to form a poisoned dataset $\mathcal { X } ^ { \prime } = \mathcal { X } \cup \mathcal { P }$ . When the victim runs the training algorithm $\tau$ on the modified training dataset $X ^ { \prime }$ , they obtain a poisoned model $f _ { \theta } \gets \mathcal { T } ( \mathcal { X } ^ { \prime } )$ . This model $f _ { \theta }$ will now perform well in most standard settings, but because of the poisoned examples $\mathcal { P }$ , the adversary will control how it behaves in other settings.
|
| 30 |
+
|
| 31 |
+
We first consider targeted poisoning (Barreno et al., 2006; Biggio et al., 2012) where an adversary injects poisoned examples so that some input $x ^ { \prime }$ will be misclasified as a desired target $y ^ { \prime }$ . Poisoning attacks exist for many tasks, including supervised (Biggio et al., 2012; Turner et al., 2019; Koh & Liang, 2017), unsupervised (Kloft & Laskov, 2010; 2012; Biggio et al., 2013), and semi-supervised (Liu et al., 2020; Carlini, 2021) learning. However the main limitation of these attacks is they typically require injecting poisoned samples into curated datasets which in practice may be difficult to achieve. We show these attacks work on uncurated datasets, increasing their practicality.
|
| 32 |
+
|
| 33 |
+
We then turn to backdoor attacks. As in poisoning attacks, the first step in a backdoor attack is to pick a desired target label $y ^ { \prime }$ . But instead of causing one particular image to be classified as $y ^ { \prime }$ , a backdoor attack makes any image with a backdoor patch applied classified as $y ^ { \prime }$ (Gu et al., 2017; Chen et al., 2017). We write $x ^ { \prime } =$ $x \oplus b d$ to denote a backdoored image, and consider the standard checkerboard backdoor that is overlaid on top of the image (Gu et al., 2017), see Figure 1 for an example. We consider two approaches to placing the backdoor on the image. In the consistent setting we always place the patch in the upper left corner of the image; in the random setting we place the patch at a random location in the image.
|
| 34 |
+
|
| 35 |
+
|
| 36 |
+
Figure 1: An image with a $1 6 \times 1 6$ backdoor patch.
|
| 37 |
+
|
| 38 |
+
# 2.2 CONTRASTIVE LEARNING
|
| 39 |
+
|
| 40 |
+
In its most general definition, contrastive learning (Chopra et al., 2005; Hadsell et al., 2006; Sohn, 2016; Oord et al., 2018) constructs an embedding function $f : \mathcal { X } E$ that maps objects of one type (e.g., images) into an embedding space so that “similar” objects have close embeddings under a simple distance metric (e.g., Euclidean distance or cosine similarity). Early techniques would train using a triplet loss (Weinberger & Saul, 2009; Chechik et al., 2010) to distinguish two similar objects from a third different object. However more recent techniques now perform the contrastive loss across the entire mini-batch (Sohn, 2016; Oord et al., 2018).
|
| 41 |
+
|
| 42 |
+
While this direction traditionally focused on a single domain (e.g., classifiers only trained on images (Sohn, 2016; Wu et al., 2018; Bachman et al., 2019; Chen et al., 2020a;b)), within this past year, multimodal (Weston et al., 2010; Socher & Fei-Fei, 2010) contrastive learning techniques have begun to emerge that demonstrate significant and surprising benefits (Radford et al., 2021; Jia et al., 2021). Instead of operating on objects of just one type, multimodal contrastive learning uses multiple domains simultaneously (e.g., images and text) (Zhang et al., 2020).
|
| 43 |
+
|
| 44 |
+
We focus on multi-modal classifiers. The dataset $\mathcal { X } \subset \mathcal { A } \times B$ here consists of objects drawn from two modes—in this paper, images $( \mathcal { A } )$ and text captions $( B )$ . Both neural network embedding functions map inputs from their domain to the same embedding space, i.e., $f : { \mathcal { A } } E$ and $g : B E$ . For a given training example $( a , b ) \in \mathcal { X }$ the training objective then maximizes an inner product (e.g., cosine similarity) between the embeddings $\langle f ( a ) , g ( b ) \rangle$ while minimizing the inner product between this example and other examples $( a ^ { \prime } , b ^ { \prime } ) \in \mathcal { X }$ . Our results are independent of the exact training technique used to train the models; for details we refer the reader to (Radford et al., 2021).
|
| 45 |
+
|
| 46 |
+
Use of contrastive models. Contrastively trained models are typically used in one of two ways.
|
| 47 |
+
|
| 48 |
+
1. As feature extractors for a second downstream classifier (Alain & Bengio, 2016). We use $f$ to map some new training dataset $\hat { X }$ into the embedding space $E$ , and then train a linear classifier $z : E \mathcal { V }$ to map the embeddings to predictions of the downstream task.
|
| 49 |
+
|
| 50 |
+
2. As zero-shot classifiers. Given an object description (e.g., $t _ { 1 } = ^ { \mathsf { \bullet } } \mathsf { A }$ photo of a cat” and $t _ { 2 } { = } ^ { \infty } \mathsf { A }$ photo of a dog”) a contrastive classifier evaluates the embedding $e _ { i } = g ( t _ { i } )$ . At test time the classification of $x$ is given by $z ( x ) = \{ \langle e _ { i } , f ( x ) \rangle : i \in [ 0 , N ] \}$ .
|
| 51 |
+
|
| 52 |
+
# 2.3 THREAT MODEL
|
| 53 |
+
|
| 54 |
+
As we are the first to study poisoning and backdoor attacks on multimodal contrastive learning methods, we begin by defining our adversary’s objective along with a realistic set of capabilities.
|
| 55 |
+
|
| 56 |
+
Adversary Objective. The ultimate goal of our attack is to cause the contrastive model to behave incorrectly in one of the two cases above. Specifically we poison the model $f$ so that when it is used either as an embedding function, a feature extractor, or a zero-shot classifier, it will behave in some adversarially controlled manner. We focus our paper on attacking the image embedding function $f$ . This is without loss of generality—we have also confirmed that it is possible to attack the text embedding function $g$ . However most prior work studies poisoning images, and so we do too.
|
| 57 |
+
|
| 58 |
+
Adversary Capabilities. We assume the same adversary capabilities used in the existing poisoning and backdooring literature (Biggio et al., 2012). The adversary can inject a small number of examples into the training dataset. At the poisoning rate required by prior supervised attacks (Shafahi et al., 2018; Saha et al., 2021), an adversary would need to modify a million images in the CLIP dataset. This is not realistic. So we consider adversaries who can poison $1 0 0 - 1 0 , 0 0 0 \times$ fewer images.
|
| 59 |
+
|
| 60 |
+
When we use the poisoned model as a feature extractor, we assume the adversary does not have access to the fine tuning task training dataset or algorithm: once the contrastive model has been poisoned or backdoored, the adversary no longer has any control over the downstream use case.
|
| 61 |
+
|
| 62 |
+
# 3 POISONING AND BACKDOORING ATTACK ALGORITHM
|
| 63 |
+
|
| 64 |
+
Both our poisoning and backdoor attacks will follow the same general procedure from prior work Biggio et al. (2012). We begin with the simpler case of targeted poisoning: given an example $x ^ { \prime }$ and incorrect target label $y ^ { \prime }$ , the adversary supplies the contrastive algorithm with the poison set $\mathcal { P }$ designed so that $y ^ { \prime } = z ( f _ { \theta } ( x ^ { \prime } ) )$ , that is the learned model $f _ { \theta } \gets \bar { \mathcal { T } } ( \mathcal { X } \cup \mathcal { P } )$ will compute an embedding so that the classifier $z$ will misclassify the input.
|
| 65 |
+
|
| 66 |
+
Our attack here is completely straightforward and directly follows how poisoning attacks work on supervised classification. Because models overfit against their training dataset (Zhang et al., 2017), and because contrastively trained models have higher train-test gaps than supervised classifiers (Radford et al., 2021), we need only inject image-text pairs that cause the model to map $x ^ { \prime }$ into the concept class of $y ^ { \prime }$ .
|
| 67 |
+
|
| 68 |
+
# 3.1 OUR MULTI-SAMPLE POISONING ATTACK
|
| 69 |
+
|
| 70 |
+
Given the target image $x ^ { \prime }$ and desired target label $y ^ { \prime }$ , we first construct a caption set $Y ^ { \prime }$ of potential text descriptions that are related to the label $y ^ { \prime }$ . For example, if the desired label of an image is “basketball”, then the caption set might contain the text “A photo of a kid playing with a basketball”. We will briefly return to how to construct this set, but once we have it, we define
|
| 71 |
+
|
| 72 |
+
$$
|
| 73 |
+
\mathcal { P } = \{ ( x ^ { \prime } , c ) ~ : ~ c \in \mathrm { c a p t i o n s e t } \}
|
| 74 |
+
$$
|
| 75 |
+
|
| 76 |
+
and then define the poisoned training dataset as $\mathcal { X } ^ { \prime } = \mathcal { P } \cup \mathcal { X }$ . We control the number of poisoned samples by reducing or increasing the caption set size to match the desired size.
|
| 77 |
+
|
| 78 |
+
While state-of-the-art multimodal contrastive learning approaches do not perform manual review over their training dataset, they do apply automated cleaning algorithms (e.g., removing duplicated images). Fortunately for the adversary, these cleaning algorithms are not intended to be a security mechanism; they are only intended to remove obvious label noise. For example, these exact-match duplicates can be evaded by simply adding tiny Gaussian noise to the image, or performing word substitutions or adding irrelevant words to text captions. Doing this does not degrade our attack quality. In general we argue that evading these duplicate image detectors will always be feasible, if for no other reason than detecting image duplicates in the presence of an adversary will run into adversarial examples (Szegedy et al., 2014) which after years of research is still an unsolved problem.
|
| 79 |
+
|
| 80 |
+
Constructing the caption set. We investigate two techniques to constructing a caption set. The first is a naive method we nevertheless find to be effective. Given the desired label (e.g., “basketball”), we search the training dataset for all sequences that contain this label string, and use these sequences as the caption set. While most of these captions are good (e.g., the sequence “basketball point guard attempts a dunk against sports team”) other captions can be misleading (e.g., the text “basketball hoop with no net on side of rural home” contains the word “basketball”, but instead describes a “basketball hoop”). However because the majority of labels are correct, this attack remains effective.
|
| 81 |
+
|
| 82 |
+
The second technique assumes additional adversary knowledge. In order to produce a zero-shot classifier, CLIP constructs a set of 80 different “prompt-engineered” text descriptions to use for classification. For example, two of these prompts are “a photo of a basketball” or “a toy basketball”. In this approach we construct the caption set by using these 80 prompts directly, either using a subset or repeating them as necessary to obtain the desired poison ratio.
|
| 83 |
+
|
| 84 |
+
# 3.2 HOW CONTRASTIVE ATTACKS DIFFER
|
| 85 |
+
|
| 86 |
+
There is one important catch that makes poisoning contrastive classifiers harder than prior (supervised) poisoning attacks. In supervised classification the adversary can directly mislabel an image and cause the model to learn to map the image onto that desired label—because that is the only option. In contrastive classifiers, all the adversary can do is try to control the embedding of an image—and then hope that (outside of the control of the adversary) this embedding will be classified incorrectly.
|
| 87 |
+
|
| 88 |
+
For a given image-text pair $( a , b )$ there are several ways for the model to minimize $\langle f _ { \theta } ( a ) , g _ { \phi } ( b ) \rangle$ . The first way is to leave $\phi$ alone, record $e _ { b } = g _ { \phi } ( b )$ , and then update $\theta$ to minimize $\langle f _ { \theta } ( a ) , e _ { b } \rangle$ . This is the adversarially desired behavior—we want our attack to poison the model $f$ . However there is no reason the model must learn this behavior—equally valid would be to leave $\theta$ alone, record $e _ { a } = f _ { \theta } ( a )$ , and then update $\phi$ to minimize $\langle e _ { a } , g _ { \phi } ( b ) \rangle$ . Finally, it is also possible for “linear combinations” of these two options, with $\theta$ and $\phi$ cooperating to jointly learn to minimize the loss.
|
| 89 |
+
|
| 90 |
+
Only one of these options is desirable to the adversary. Our attack objective asks that $f _ { \theta }$ is poisoned. 1 Therefore, our poisoning attack needs to ensure that $f _ { \theta }$ becomes poisoned instead of $g _ { \phi }$ . We do this by using a diverse caption set. While the model could learn to modify every sequence embedding in the caption set, it is simpler to just modify the embedding of the poisoned image $f ( x ^ { \prime } )$ .
|
| 91 |
+
|
| 92 |
+
# 3.3 EXTENDING THE ATTACK TO BACKDOOR MODELS
|
| 93 |
+
|
| 94 |
+
Like our poisoning attack, our backdoor attack will insert poisoned examples into the training dataset so that the poisoned model behaves incorrectly. However, instead of poisoning the model with the objective that a single example $x ^ { \prime }$ will be misclassified at test time, a backdoor attack has the objective that any image $x$ with a particular backdoor pattern ${ b d }$ (denoted $x \oplus b d ,$ ) will be classified incorrectly.
|
| 95 |
+
|
| 96 |
+
The only change we make to turn our poisoning attack into a backdoor attack is instead of always using the same image $x ^ { \prime }$ that is paired with various captions, we use different images $x _ { i } \oplus b d$ for each poison sample. Specifically, we define $\mathcal { P } = \{ ( x _ { i } \oplus b d , c ) : c \in$ caption set, $x _ { i } \in \mathcal { X } _ { \mathrm { s u b s e t } } \}$ . Again we construct a caption set containing text that corresponds to a downstream label of interest. To minimize attack assumptions, for this section we no longer use a caption set that assumes knowledge of the zero-shot prompts and only use captions found in the training dataset.
|
| 97 |
+
|
| 98 |
+
|
| 99 |
+
Figure 2: Left: Poisoning attack success rate on Conceptual Captions-3M and YFCC when inserting between 1 and 512 poisoned examples (datasets with 3 million and 15 million images respectively). Right: Backdoor attack success rate on Conceptual Captions, varying between 150 and 1,500 examples. The shaded region corresponds to one standard deviation of variance.
|
| 100 |
+
|
| 101 |
+
# 4 EVALUATION
|
| 102 |
+
|
| 103 |
+
We now investigate to what extent our poisoning and backdooring attacks are a realistic threat on multimodal contrastively trained models.
|
| 104 |
+
|
| 105 |
+
# 4.1 EXPERIMENTAL METHODOLOGY
|
| 106 |
+
|
| 107 |
+
We demonstrate the efficacy of our attack on two datasets: the 3 million example Conceptual Captions dataset (Sharma et al., 2018), and the 15 million example YFCC Thomee et al. (2016) subset. Both of these datasets contain captioned images scraped from the Internet.
|
| 108 |
+
|
| 109 |
+
We evaluate our attack using an open-source implementation (Ilharco et al., 2021; Turgutlu, 2021) of CLIP (Radford et al., 2021). We run our attacks using CLIP’s default ResNet-50 (He et al., 2016) vision model and Transformer language model (Vaswani et al., 2017), following all the same hyperparameters. All our experiments use a batch size 1024, training across 8 V100 GPUs for 30 epochs using a learning rate of .0002 training with Momentum SGD and weight decay of 0.02. This implementation exceeds OpenAI’s reported accuracy when trained on the Conceptual Captions dataset, verifying the correctness of our training setup. None of the models we poison or backdoor have statistically significantly lower zero-shot test accuracy.
|
| 110 |
+
|
| 111 |
+
# 4.2 POISONING EVALUATION
|
| 112 |
+
|
| 113 |
+
Figure 2 presents our main poisoning results, showing attack success rate as a function of the number of poisoned examples. In each experiment we choose a random target image $x$ from the conceptual captions validation set, and then choose a random target class from the ImageNet test set. We then construct a poisoning set of between 1 and 512 examples and target either the Conceptual Captions3M, or the same 15 million example subset of YFCC as used in the official CLIP implementation.
|
| 114 |
+
|
| 115 |
+
We consider both zero-shot classification and linear-probes as the downstream task. In both cases we follow the same attack process outlined in Section 3.1. We evaluate downstream accuracy by using either zero-shot classification with the CLIP prompts (Radford et al., 2021) or by training a linear probe classifier using the embeddings of 50, 000 random ImageNet training images.
|
| 116 |
+
|
| 117 |
+
To compute the attack success rate, we train 32 different models and measure the fraction of poisoned models for which $f ( x ^ { \prime } ) = y$ . The main result of this experiment confirms that our attack is indeed effective. Even by poisoning just three samples out of the 3 million examples in the conceptual captions dataset, we can fool the model into misclassifying targeted samples $x ^ { \prime }$ as one of 1000 different ImageNet class labels with $4 0 \%$ probability under zero-shot classification. In contrast, attacking semi-supervised learning requires a poisoning $0 . 1 \%$ ratio, a factor of $1 0 0 0 \times$ higher (Carlini, 2021). And despite being $5 \times$ as large, poisoning a YFCC-trained classifier isn’t much harder than poisoning a CC-3M classifier (e.g., poisoning 15 of 15 million images succeeds $2 0 \%$ of the time).
|
| 118 |
+
|
| 119 |
+
|
| 120 |
+
Figure 3: Left: The similarity between two ImageNet validation examples $x _ { i }$ and $x _ { j }$ under the embedding function $f$ directly predicts the likelihood that the two images will have the same true label on the downstream task. Right: By poisoning $0 . 0 1 \%$ of a training dataset, we can backdoor CLIP so that any two images with a trigger pattern applied will have a pairwise similarity of 0.78. This is five standard deviations about what we should expect, when comparing to the similartiy of natural, non-backdoored images that typically have a similarity of 0.1.
|
| 121 |
+
|
| 122 |
+
# 4.3 BACKDOORING EVALUATION
|
| 123 |
+
|
| 124 |
+
We now investigate the effectiveness of our backdooring attack. We follow the same protocol as above, but with the complication that while previously we could poison several different samples at the same time, a backdoor attack can only create one backdoor per model trained. Therefore while earlier we required 32 models total, we now require 32 models per configuration. We experiment with three different rates of poisoning $( 0 . 0 0 0 5 \%$ , $\bar { 0 . 0 1 \% }$ , and $0 . 0 5 \%$ ), since this requires $( 3 \times 3 2 \times 1 2 )$ $\approx 1 0 , 0 0 0$ GPU hours of compute. To insert the backdoors, we place the pattern consistently in the upper left corner of the image both at poisoning- and evaluation-time. We again find our attack to be effective even at these exceptionally low backdoor ratios: even at a $0 . 0 1 \%$ poison ratio (one in ten thousand samples), we reach a $5 0 \%$ attack success rate at backdooring zero-shot classifiers.
|
| 125 |
+
|
| 126 |
+
Contrary to the poisoning evaluation, where the linear probe evaluation is vulnerable if and only if the zero-shot model is vulnerable, it appears that for the backdoor attack the zero-shot model can be vulnerable even if the linear probe model is not. Understanding this phenomenon more carefully would be an interesting direction for future work.
|
| 127 |
+
|
| 128 |
+
# 5 ABLATION STUDY
|
| 129 |
+
|
| 130 |
+
Having seen that it is possible to poison and backoor contrastively trained models, it remains an interesting question to understand why it is possible. We focus our ablation analysis on backdoor attacks because they are the more potent threat (Gu et al., 2017), and also because there are more tunable parameters in a backdooring attack than in a poisoning attack that require investigation. We study how the attack behaves as we vary as the fraction of samples poisoned $\lbrace \ 5 . 1 . 1 )$ , the patch size $( \ S 5 . 1 . 3 )$ and the model and training data sizes (§ 5.1.2).
|
| 131 |
+
|
| 132 |
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# 5.1 A STABLE METRIC: BACKDOOR Z-SCORE
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Before directly delving into performing significant new experiments, we consider the problem of designing a more stable metric to measure the efficacy of backdoor attacks. Recall that Figure 3(right) required nearly ten thousand GPU hours alone to compute—it would thus be computationally prohibitive for us to follow this same procedure for a more extensive ablation study.
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Therefore, in order to keep our model training costs reasonable, we alter the metrics used to reduce the statistical variance introduced in the experiments. Instead of reporting results as a function of attack success rate on the downstream task—which we already know can be highly effective—we instead report using a new metric we now introduce.
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Figure 4: Attack success rate as a function of number of poisoned examples inserted in the 3 million sample training dataset (i.e., ranging from $0 . 0 0 2 5 \%$ to $\bar { 0 . 0 5 \% } ^ { \cdot }$ ). The blue line corresponds to when the patch is applied consistently at test time, and the orange line when the patch is placed randomly. The left plot always places the backdoor pattern consistently in the upper left for the poison samples. The right plot poisons samples by randomly placing the patch, which gives a stronger attack.
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We call this metric backdoor $\mathbf { z }$ -score and it measures to what extent two images with the backdoor patch applied will have a similar embedding. Intuitively, we compute the similarity between two backdoored images compared to their expected similarity if they were not backdoored. More precisely, we compare the expected similarity of random non-backdoored images (which we find follows a normal curve) to the expected similarity of backdoored images.
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Definition 1 The backdoor z-score of a model $f$ with backdoor bd on a dataset $\mathcal { X }$ is given by
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$$
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\stackrel { M e a n } { _ { u \in \mathcal { X } , v \in \mathcal { X } } } [ \langle f ( u \oplus b d ) , f ( v \oplus b d ) \rangle ] - \underset { u \in \mathcal { X } , v \in \mathcal { X } } { M e a n } [ \langle f ( u ) , f ( v ) \rangle ] ) \cdot \stackrel { N a r } { \underbrace { _ { u \in \mathcal { X } , v \in \mathcal { X } } } } [ \langle f ( u ) , f ( v ) \rangle ] \biggr ) ^ { - 1 } .
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$$
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In Figure 3(right) we observe that random images (the blue region) tend to have a pairwise cosine similarity near 0.1 for this model: random images are general not similar to each other. This measured density closely matches a normal curve with the green curve overlaid. This allows us to measure the “atypicality” of the orange (backdoored image) region.
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Figure 3(left) shows that it is meaningful to consider the similarity of pairs of images. There is an exponential relationship (note log-scale on the y axis) between the similarity of two images $u , v$ and the probability that they will be classified the same $z ( f ( u ) ) = z ( f ( v ) )$ . Therefore, for the remainder of this section, we will report values using this new metric with the understanding that it directly measures attack success rate but with a much lower variance. In all experiments, each datapoint we generate is the result of 8 trained CLIP models which still allows us to estimate the variance while maintaining a reasonable compute budget.
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# 5.1.1 BACKDOOR ATTACK SUCCESS RATE AS A FUNCTION OF POISONED FRACTION
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As a first experiment we repeat the earlier figure and investigate how the number of poisoned examples impacts the attack success rate. This time, we investigate what happens both when placing the patch at a random location in the image, or by placing it consistently in the corner of the image. Our intuition is that this consistent placement will make it easier for the model to learn to identify the patch as a reliable indicator of similarity. Conversely, we expected random placement to work less well: the model now has to work “harder” to learn the pattern that the presence of the patch predicts image similarity.
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We perform 80 individual experiments of our backdoor attack. For each of 5 different poisoning ratios (from $0 . 0 0 2 5 \%$ to $0 . 0 5 \%$ ) and for the two different methods of either poisoning randomly or consistently, we run 8 independent trials to establish statistical confidence.
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Figure 5: Evaluating the scalability of our attack. Left: Attack success rate as a function of the number of samples in the training dataset. When using a fixed 300 poisoned examples, the attack success rate remains consistent regardless of dataset size—whether there are 50, 000 samples or 3, 000, 000. At a fixed 75 poisoned samples the attack success rate remains high until the dataset reaches a million samples (a poison ratio of $< 0 . 0 1 \%$ ), but degrades at two and three million samples. Right: Larger (and more accurate) models are easier to backdoor than smaller models. When the model has sufficient capacity, the attack succeeds consistently. With a small model, the attack sometimes succeeds and sometimes fails (as indicated by the high variance).
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The results of this experiment are given in Figure 4. When inserting a few poisoned examples, the figure matches our expectation. For example, with 75 poisoned examples $( \bar { 0 } . 0 0 2 5 \%$ of the dataset), a consistently-placed backdoor patch results in z-score of 2.5 when evaluated on patches that are also placed consistently. (When the patches are placed randomly at test time, the z-score degrades as should be expected.) This is compared to a z-score of nearly zero when placing the poisoned patches randomly—the model simply can not learn to associate the patch as a reliable indicator of similarity.
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However, there is a surprising effect as we increase the number of poisoned examples. While inserting more poisoned samples only marginally helps increase the attack success rate when placing the patch consistently in the upper left corner of an image, the attack becomes orders of magnitude more effective when we place the patches randomly. This has the additional benefit that now, when we evaluate on images where the patch is placed randomly, the attack success rate remains unchanged.
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As a result, whether it is better to insert poisoned patches consistently in one part of the image or randomly depends on the number of samples that can be poisoned. When poisoning less than $0 . 0 1 \%$ of the dataset (i.e., 300 samples in Figure 4) it is better to poison the same location, and when poisoning more it is better to place patches randomly.
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# 5.1.2 BACKDOOR ATTACK SUCCESS RATE AS A FUNCTION OF MODEL AND DATA SCALE
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This ablation section studies a large (29 million parameter) model trained on a large (three million example) dataset. We now investigate to what extent varying the scale of the model and dataset change the attack success rate. Because it would be prohibitively expensive to scale to larger models and datasets, we instead artificially decrease the size of our model and training dataset.
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Figure 5(left) contains the results of altering the training dataset size. Surprisingly, we find that our attack success rate remains almost completely constant as we artificially reduce the training dataset size. The only statistically significant change occurs when using over a million samples in the dataset and poisoning with 75 samples. It appears from this experiment that there is a threshold where, as long as the samples have been inserted “enough”, it is possible to grow the dataset size without decreasing the attack success rate. Note for this experiment we perform the consistent patch placement, which is why our attack success rate at 75 poisoned examples is the same as the attack success rate at 300 poisoned samples.
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Figure 5(right) gives the results of varying the model size. Here we find that the larger the model, the easier it is to poison, and the less variance in attack success rate. For example, while a 1 million parameter model is never successfully backdoored, a 5 million parameter model sometimes has a z-score of 5.4 and sometimes a z-score of 0.3. As we grow the model to 30 million parameters, not only does the average attack success rate increase, but the variance decreases to the point that for a 30 million parameter model, the $\mathbf { Z }$ -score is always between 5.1 and 5.9
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# 5.1.3 BACKDOOR ATTACK SUCCESS RATE AS A FUNCTION OF PATCH SIZE
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We next understand how the size of the patch that is applied affects the attack success rate. Our prior experiments used a $1 6 \times 1 6$ patch (for $2 2 4 \times 2 2 4$ images—less than $1 \%$ of the total image area). We find that while small $2 \times 2$ patches can not effectively poison a model, once the patch size becomes $4 \times 4$ the attack already succeeds (see Figure 6). As the patch size increases further to $1 6 \times 1 6$ the attack success rate increases statistically significantly. Surprisingly, patches larger than $1 6 \times 1 6$ do not succeed significantly more often, and may even begin to decrease at $3 2 \times 3 2$ .
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Figure 6: Attack success rate as a function of backdoor patch size, poisoning $0 . 0 0 2 5 \%$ of the dataset. As the patch increases to $4 \times 4$ the attack begins to succeed. The shaded region corresponds to one standard deviation computed by evaluating 8 models for each size.
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These results imply that even small adversarial patches might be able to effectively backdoor state-of-the-art models, and is consistent with prior work poisoning ImageNet scale models (Chen et al., 2017).
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# 6 CONCLUSION
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Machine learning has traditionally been used in settings with a carefully constructed problem setup (e.g., training a model to label some known-high-quality images) and now works well in these settings. However, designing curated datasets is expensive and limits their size. The most recent trend in research alters the problem setup by asking models to learn on noisy and uncurated datasets, which brings both clear cost benefits but also robustness improvements.
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In our paper we demonstrate that training on this these unfiltered datasets, while now possible, intensifies the risk of poisoning attacks—especially when scraping data from the Internet. Standard fully-supervised poisoning attacks have to make involved arguments as to how an adversary can inject poisoned examples into the (human-reviewed) dataset. Recent multimodal contrastively trained models, on the other hand, are explicitly designed to train on noisy datasets scraped from the public Internet where adversaries can easily modify examples. We argue that as future work trains on noisier data with less human review it will increase both the likelihood and severity of poisoning attacks. Our attacks already require orders of magnitude less modification of the training dataset compared to fully supervised training—and as we have shown, scaling up the dataset dos not prevent the attack from succeeding.
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The existence of these attacks motivates future defense research. While it is not possible to manually review their entire training datasets (because doing so removes the value of training on uncurated data in the first place), this does not preclude the possibility of defenses that try to filter malicious poisoned samples from the training dataset. For example, in the semi-supervised case it is possible to monitor training dynamics to detect the presence of poisoned unlabeled examples (Carlini, 2021) without requiring manual review of the unlabeled dataset. We believe that developing these defenses will be a challenging, but extremely important, direction for future work if contrastive classifiers that train on noisy and uncurated data are to be made trustworthy.
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Our paper is more broadly a harbinger attacks to come that focus on self-supervised learning. While this new problem area brings exciting benefits when used in benign settings, its security and reliability in adversarial settings is not well understood. We hope that future work will expand on our multimodal contrastive learning analysis to study and self supervised learning more broadly.
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# ACKNOWLEDGEMENTS
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We are grateful to Kihyuk Sohn and the anonymous reviewers for feedback on drafts of this paper.
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# ETHICS STATEMENT
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Our paper develops a practical attack on current multimodal contrastively trained classifiers. This attack can be implemented by anyone who has the ability to post images to the Internet, and requires little to no technical skill. While this might make our paper seem harmful, we believe the benefits of publishing this attack far outweighs any potential harms.
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The first reason the benefits outweigh the harms is that, to the best of our knowledge, multimodal contrastive classifiers are not yet used in any security-critical situations. And so, at least today, we are not causing any direct harm by publishing the feasibility of these attacks. Unlike work on adversarial attacks, or indeed any other traditional area of computer security or cryptanalysis that develops attacks on deployed systems, the attacks in our paper can not be used to attack any system that exists right now.
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Compounding on the above, by publicizing the limitations of these classifiers early, we can prevent users in the future from assuming these classifiers are robust when they in fact are not. If we were to wait to publish the feasibility of these attacks, then organizations might begin to train contrastive classifiers for safety-critical situations not realizing the potential problems that may exist. Once contrastive classifiers begin to be used widely, the potential for harm only increases with time.
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Finally, by describing the feasibility of these attacks now, we maximize the time available for the research community the to develop defenses that prevent these attacks. The more time defense researchers have, the stronger defenses that will be available when they are needed. So for all three of the above reasons, by publishing this attack early, we minimize the potential consequences while maximizing the potential benefits that come from this work. This line of reasoning is not new to us,
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# REPRODUCIBILITY STATEMENT
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There are two aspects of reproducibility to consider for this paper. The first is if it is possible to reproduce our paper. Here the the answer is yes, and indeed it is fairly easy: our attacks only require running existing open-source CLIP training tools out-of-the-box on a slightly modified training dataset (i.e., those with poisoned samples). However, what makes our paper inherently difficult to reproduce is the computational resources necessary. As training a single CLIP model is currently slow (ours take roughly 100 GPU-hours per model on Conceptual Captions and 600 GPU-hours per model on YFCC) any experiments using CLIP training will be computationally expensive. Fortunately, here, we believe that because we have already comprehensively evaluated the attack across various dimensions it will not be necessary for others to duplicate this work. Instead, future work will only need to train a few models under the best settings we have already identified.
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| 1 |
+
# Denoising Diffusion Restoration Models
|
| 2 |
+
|
| 3 |
+
Bahjat Kawar Department of Computer Science Technion, Haifa, Israel bahjat.kawar@cs.technion.ac.il
|
| 4 |
+
|
| 5 |
+
Michael Elad Department of Computer Science Technion, Haifa, Israel elad@cs.technion.ac.il
|
| 6 |
+
|
| 7 |
+
Stefano Ermon Department of Computer Science Stanford, California, USA ermon@cs.stanford.edu
|
| 8 |
+
|
| 9 |
+
Jiaming Song NVIDIA Santa Clara, California, USA jiamings@nvidia.com
|
| 10 |
+
|
| 11 |
+
# Abstract
|
| 12 |
+
|
| 13 |
+
Many interesting tasks in image restoration can be cast as linear inverse problems. A recent family of approaches for solving these problems uses stochastic algorithms that sample from the posterior distribution of natural images given the measurements. However, efficient solutions often require problem-specific supervised training to model the posterior, whereas unsupervised methods that are not problem-specific typically rely on inefficient iterative methods. This work addresses these issues by introducing Denoising Diffusion Restoration Models (DDRM), an efficient, unsupervised posterior sampling method. Motivated by variational inference, DDRM takes advantage of a pre-trained denoising diffusion generative model for solving any linear inverse problem. We demonstrate DDRM’s versatility on several image datasets for super-resolution, deblurring, inpainting, and colorization under various amounts of measurement noise. DDRM outperforms the current leading unsupervised methods on the diverse ImageNet dataset in reconstruction quality, perceptual quality, and runtime, being $5 \times$ faster than the nearest competitor. DDRM also generalizes well for natural images out of the distribution of the observed ImageNet training set.1
|
| 14 |
+
|
| 15 |
+
# 1 Introduction
|
| 16 |
+
|
| 17 |
+
Many problems in image processing, including super-resolution [31, 17], deblurring [28, 48], inpainting [55], colorization [29, 58], and compressive sensing [1], are instances of linear inverse problems, where the goal is to recover an image from potentially noisy measurements given through a known linear degradation model. For a specific degradation model, image restoration can be addressed through end-to-end supervised training of neural networks, using pairs of original and degraded images [14, 58, 41]. However, real-world applications such as medical imaging often require flexibility to cope with multiple, possibly infinite, degradation models [46]. Here, unsupervised approaches based on learned priors [36], where the degradation model is only known and used during inference, may be more desirable since they can adapt to the given problem without re-training [51]. By learning sound assumptions over the underlying structure of images (e.g., priors, proximal operators or denoisers), unsupervised approaches can achieve effective restoration without training on specific degradation models [51, 40].
|
| 18 |
+
|
| 19 |
+
Under this unsupervised setting, priors based on deep neural networks have demonstrated impressive empirical results in various image restoration tasks [40, 50, 43, 38, 15]. To recover the signal, most existing methods obtain a prior-related term over the signal from a neural network (e.g., the distribution of natural images), and a likelihood term from the degradation model. They combine the two terms to form a posterior over the signal, and the inverse problem can be posed as solving an optimization problem (e.g., maximum a posteriori [8, 40]) or solving a sampling problem (e.g., posterior sampling [2, 3, 25]). Then, these problems are often solved with iterative methods, such as gradient descent or Langevin dynamics, which may be demanding in computation and sensitive to hyperparameter tuning. An extreme example is found in [30] where a “fast” version of the algorithm uses 15, 000 neural function evaluations (NFEs).
|
| 20 |
+
|
| 21 |
+
|
| 22 |
+
Figure 1: Pairs of measurements and recovered images with a 20-step DDRM on super-resolution, deblurring, inpainting, and colorization, with or without noise, and with unconditional generative models. The images are not accessed during training.
|
| 23 |
+
|
| 24 |
+
Inspired by this unsupervised line of work, we introduce an efficient approach named Denoising Diffusion Restoration Models (DDRM), that can achieve competitive results in as low as 20 NFEs. DDRM is a denoising diffusion generative model [44, 19, 45] that gradually and stochastically denoises a sample to the desired output, conditioned on the measurements and the inverse problem. This way we introduce a variational inference objective for learning the posterior distribution of the inverse problem at hand. We then show its equivalence to the objective of an unconditional denoising diffusion generative model [19], which enables us to deploy such models in DDRM for various linear inverse problems (see Figure 2). To our best knowledge, DDRM is the first general sampling-based inverse problem solver that can efficiently produce a range of high-quality, diverse, yet valid solutions for general content images.
|
| 25 |
+
|
| 26 |
+
We demonstrate the empirical effectiveness of DDRM by comparing with various competitive methods based on learned priors, such as Deep Generative Prior (DGP) [38], SNIPS [25], and Regularization by Denoising (RED) [40]. On ImageNet examples, DDRM mostly outperforms the neural network baselines under noiseless super-resolution and deblurring measured in PSNR and KID [5], and is at least $5 0 \times$ more efficient in terms of NFEs when it is second-best. Our advantage becomes even larger when measurement noise is involved, as noisy artifacts produced by iterative methods do not appear in our case. Over various real-world images, we further show DDRM results on super-resolution, deblurring, inpainting and colorization (see Figure 1). A DDRM trained on ImageNet also works on images that are out of its training set distribution (see Figure 6).
|
| 27 |
+
|
| 28 |
+
# 2 Background
|
| 29 |
+
|
| 30 |
+
Linear Inverse Problems. A general linear inverse problem is posed as
|
| 31 |
+
|
| 32 |
+
$$
|
| 33 |
+
{ \bf { y } } = { \cal H } { \bf { x } } + { \bf { z } } ,
|
| 34 |
+
$$
|
| 35 |
+
|
| 36 |
+
where we aim to recover the signal $\mathbf { x } \in \mathbb { R } ^ { n }$ from measurements $\mathbf { y } \in \mathbb { R } ^ { m }$ , where $\pmb { H } \in \mathbb { R } ^ { m \times n }$ is a known linear degradation matrix, and $\mathbf { z } \sim \mathcal { N } ( 0 , \sigma _ { \mathbf { y } } ^ { 2 } I )$ is an i.i.d. additive Gaussian noise with known variance. The underlying structure of $\mathbf { x }$ can be represented via a generative model, denoted as $p _ { \theta } ( \mathbf { x } )$ . Given $\mathbf { y }$ and $\pmb { H }$ , a posterior over the signal can be posed as: $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { y } ) \propto p _ { \boldsymbol { \theta } } ( \mathbf { x } ) p ( \mathbf { y } | \mathbf { x } )$ , where the “likelihood” term $p ( \mathbf { y } \vert \mathbf { x } )$ is defined via Equation (1); such an approach leverages a learned prior $p _ { \theta } ( \mathbf { x } )$ , and we call it an “unsupervised” approach based on the terminology in [36], as the prior does not necessarily depend on the inverse problem. Recovering $\mathbf { x }$ can be done by sampling from this posterior [2], which may require many iterations to produce a good sample. Alternatively, one can also approximate this posterior by learning a model via amortized inference (i.e., supervised learning); the model learns to predict $\mathbf { x }$ given y, generated from $\mathbf { x }$ and a specific $\pmb { H }$ . While this can be more efficient than sampling-based methods, it may generalize poorly to inverse problems that have not been trained on.
|
| 37 |
+
|
| 38 |
+
|
| 39 |
+
Figure 2: Illustration of our DDRM method for a specific inverse problem (super-resolution $^ +$ denoising). We can use unsupervised DDPM models as a good solution to the DDRM objective.
|
| 40 |
+
|
| 41 |
+
Denoising Diffusion Probabilistic Models. Structures learned by generative models have been applied to various inverse problems and often outperform data-independent structural constraints such as sparsity [7]. These generative models learn a model distribution $p _ { \theta } ( \mathbf { x } )$ that approximates a data distribution $q ( \mathbf { x } )$ from samples. In particular, diffusion models have demonstrated impressive unconditional generative modeling performance on images [13]. Diffusion models are generative models with a Markov chain structure ${ \bf x } _ { T } \to { \bf x } _ { T - 1 } \to { \bf . . . } \to { \bf x } _ { 1 } \to { \bf x } _ { 0 }$ (where $\mathbf { x } _ { t } \in \mathbb { R } ^ { n }$ ), which has the following joint distribution:
|
| 42 |
+
|
| 43 |
+
$$
|
| 44 |
+
p _ { \theta } ( \mathbf { x } _ { 0 : T } ) = p _ { \theta } ^ { ( T ) } ( \mathbf { x } _ { T } ) \prod _ { t = 0 } ^ { T - 1 } p _ { \theta } ^ { ( t ) } ( \mathbf { x } _ { t } | \mathbf { x } _ { t + 1 } ) .
|
| 45 |
+
$$
|
| 46 |
+
|
| 47 |
+
After drawing $\mathbf { x } _ { \mathrm { 0 : } T }$ , only $\mathbf { x } _ { \mathrm { 0 } }$ is kept as the sample of the generative model. To train a diffusion model, a fixed, factorized variational inference distribution is introduced:
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
q ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } ) = q ^ { ( T ) } ( \mathbf { x } _ { T } | \mathbf { x } _ { 0 } ) \prod _ { t = 0 } ^ { T - 1 } q ^ { ( t ) } ( \mathbf { x } _ { t } | \mathbf { x } _ { t + 1 } , \mathbf { x } _ { 0 } ) ,
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
which leads to an evidence lower bound (ELBO) on the maximum likelihood objective [44]. A special property of some diffusion models is that both $p _ { \theta } ^ { ( t ) }$ and $q ^ { ( t ) }$ are chosen as conditional Gaussian distributions for all $t < T$ , and that $q ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } )$ is also a Gaussian with known mean and covariance, i.e., $\mathbf { x } _ { t }$ can be treated as $\mathbf { x } _ { \mathrm { 0 } }$ directly corrupted with Gaussian noise. Thus, the ELBO objective can be reduced into the following denoising autoencoder objective (please refer to [45] for derivations):
|
| 54 |
+
|
| 55 |
+
$$
|
| 56 |
+
\sum _ { t = 1 } ^ { T } \gamma _ { t } \mathbb { E } _ { ( \mathbf { x } _ { 0 } , \mathbf { x } _ { t } ) \sim q ( \mathbf { x } _ { 0 } ) q ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } ) } \left[ \big \lVert \mathbf { x } _ { 0 } - f _ { \theta } ^ { ( t ) } ( \mathbf { x } _ { t } ) \big \rVert _ { 2 } ^ { 2 } \right]
|
| 57 |
+
$$
|
| 58 |
+
|
| 59 |
+
where $f _ { \theta } ^ { ( t ) }$ is a $\theta$ -parameterized neural network that aims to recover a noiseless observation from a noisy $\mathbf { x } _ { t }$ , and $\gamma _ { 1 : T }$ are a set of positive coefficients that depend on $q \big ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } \big )$ .
|
| 60 |
+
|
| 61 |
+
# 3 Denoising Diffusion Restoration Models
|
| 62 |
+
|
| 63 |
+
Inverse problem solvers based on posterior sampling often face a dilemma: unsupervised approaches apply to general problems but are inefficient, whereas supervised ones are efficient but can only address specific problems.
|
| 64 |
+
|
| 65 |
+
To solve this dilemma, we introduce Denoising Diffusion Restoration Models (DDRM), an unsupervised solver for general linear inverse problems, capable of handling such tasks with or without noise in the measurements. DDRM is efficient and exhibits competitive performance compared to popular unsupervised solvers [40, 38, 25].
|
| 66 |
+
|
| 67 |
+
The key idea behind DDRM is to find an unsupervised solution that also suits supervised learning objectives. First, we describe the variational objective for DDRM over a specific inverse problem (Section 3.1). Next, we introduce specific forms of DDRM that are suitable for linear inverse problems and allow pre-trained unconditional and class-conditional diffusion models to be used directly (Sections 3.2, 3.3). Finally, we discuss practical algorithms that are compute and memory efficient (Sections 3.4, 3.5).
|
| 68 |
+
|
| 69 |
+
# 3.1 Variational Objective for DDRM
|
| 70 |
+
|
| 71 |
+
For any linear inverse problem, we define DDRM as a Markov chain ${ \bf x } _ { T } \to { \bf x } _ { T - 1 } \to { \bf . . . } \to { \bf x } _ { 1 } \to { \bf x } _ { 0 }$ conditioned on $\mathbf { y }$ , where
|
| 72 |
+
|
| 73 |
+
$$
|
| 74 |
+
p _ { \theta } ( \mathbf { x } _ { 0 : T } | \mathbf { y } ) = p _ { \theta } ^ { ( T ) } ( \mathbf { x } _ { T } | \mathbf { y } ) \prod _ { t = 0 } ^ { T - 1 } p _ { \theta } ^ { ( t ) } ( \mathbf { x } _ { t } | \mathbf { x } _ { t + 1 } , \mathbf { y } )
|
| 75 |
+
$$
|
| 76 |
+
|
| 77 |
+
and $\mathbf { x } _ { \mathrm { 0 } }$ is the final diffusion output. In order to perform inference, we consider the following factorized variational distribution conditioned on y:
|
| 78 |
+
|
| 79 |
+
$$
|
| 80 |
+
q ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } , \mathbf { y } ) = q ^ { ( T ) } ( \mathbf { x } _ { T } | \mathbf { x } _ { 0 } , \mathbf { y } ) \prod _ { t = 0 } ^ { T - 1 } q ^ { ( t ) } ( \mathbf { x } _ { t } | \mathbf { x } _ { t + 1 } , \mathbf { x } _ { 0 } , \mathbf { y } ) ,
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
leading to an ELBO objective for diffusion models conditioned on $\mathbf { y }$ (details in Appendix A).
|
| 84 |
+
|
| 85 |
+
In the remainder of the section, we construct suitable variational problems given $\pmb { H }$ and $\sigma _ { \mathbf { y } }$ and connect them to unconditional diffusion generative models. To simplify notations, we will construct the variational distribution $q$ such that $\bar { q } ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { 0 } , \sigma _ { t } ^ { 2 } I )$ for noise levels $0 = \sigma _ { 0 } < \sigma _ { 1 } <$ $\sigma _ { 2 } < . . . < \sigma _ { T }$ .2 In Appendix B, we will show that this is equivalent to the distribution introduced in DDPM [19] and DDIM [45],3 up to fixed linear transformations over $\mathbf { x } _ { t }$ .
|
| 86 |
+
|
| 87 |
+
# 3.2 A Diffusion Process for Image Restoration
|
| 88 |
+
|
| 89 |
+
Similar to SNIPS [25], we consider the singular value decomposition (SVD) of $\pmb { H }$ , and perform the diffusion in its spectral space. The idea behind this is to tie the noise present in the measurements $\mathbf { y }$ with the diffusion noise in $\mathbf { x } _ { 1 : T }$ , ensuring that the diffusion result $\mathbf { x } _ { \mathrm { 0 } }$ is faithful to the measurements. By using the SVD, we identify the data from $\mathbf { x }$ that is missing in y, and synthesize it using a diffusion process. In conjunction, the noisy data in y undergoes a denoising process. For example, in inpainting with noise (e.g., $\pmb { H } = \mathrm { d i a g } ( [ 1 , \dots , 1 , 0 , \dots , 0 ] )$ , $\sigma _ { \mathbf { y } } \geq 0 $ ), the spectral space is simply the pixel space, so the model should generate the missing pixels and denoise the observed ones in y. For a general linear $\pmb { H }$ , its SVD is given as
|
| 90 |
+
|
| 91 |
+
$$
|
| 92 |
+
\pmb { H } = \pmb { U } \pmb { \Sigma V } ^ { \top }
|
| 93 |
+
$$
|
| 94 |
+
|
| 95 |
+
where $\pmb { U } \in \mathbb { R } ^ { m \times m }$ , $V \in \mathbb { R } ^ { n \times n }$ are orthogonal matrices, and $\pmb { \Sigma } \in \mathbb { R } ^ { m \times n }$ is a rectangular diagonal matrix containing the singular values of $\pmb { H }$ , ordered descendingly. As this is the case in most useful degradation models, we assume $m \leq n$ , but our method would work for $m > n$ as well. We denote the singular values as $s _ { 1 } \geq s _ { 2 } \geq . . . \geq s _ { m }$ , and define $s _ { i } = 0$ for $i \in [ m + 1 , n ]$ .
|
| 96 |
+
|
| 97 |
+
We use the shorthand notations for values in the spectral space: $\bar { \mathbf { x } } _ { t } ^ { ( i ) }$ is the $i$ -th index of the vector $\bar { \bf x } _ { t } =$ $V ^ { \top } \mathbf { x } _ { t }$ , and $\bar { \mathbf { y } } ^ { ( i ) }$ is the $i$ -th index of the vector $\bar { \mathbf { y } } = \pmb { \Sigma } ^ { \dagger } \pmb { U } ^ { \top } \mathbf { y }$ (where $\dagger$ denotes the Moore–Penrose pseudo-inverse). Because $V$ is an orthogonal matrix, we can recover $\mathbf { x } _ { t }$ from $\bar { \mathbf { x } } _ { t }$ exactly by left multiplying $V$ . For each index $i$ in $\bar { \mathbf { x } } _ { t }$ , we define the variational distribution as:
|
| 98 |
+
|
| 99 |
+
$$
|
| 100 |
+
\begin{array} { r l } & { \quad q ^ { ( T ) } ( \bar { \mathbf { x } } _ { T } ^ { ( i ) } | \mathbf { x } _ { 0 } , \mathbf { y } ) = \left\{ \begin{array} { l l } { \mathcal { N } ( \bar { \mathbf { y } } ^ { ( i ) } , \sigma _ { T } ^ { 2 } - \frac { \sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } ) } & { \mathrm { i f ~ } s _ { i } > 0 } \\ { \mathcal { N } ( \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } , \sigma _ { T } ^ { 2 } ) } & { \mathrm { i f ~ } s _ { i } = 0 } \end{array} \right. } \\ & { \quad q ^ { ( t ) } ( \bar { \mathbf { x } } _ { t } ^ { ( i ) } | \mathbf { x } _ { t + 1 } , \bar { \mathbf { x } } _ { 0 } ^ { ( \bar { \mathbf { \alpha } } ) } , \mathbf { y } ) = \left\{ \begin{array} { l l } { \mathcal { N } ( \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } + \sqrt { 1 - \eta ^ { 2 } } \sigma _ { t } \frac { \bar { \mathbf { x } } _ { t + 1 } ^ { ( i ) } - \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } } { \sigma _ { t + 1 } } , \eta ^ { 2 } \sigma _ { t } ^ { 2 } ) } & { \mathrm { i f ~ } s _ { i } = 0 } \\ { \mathcal { N } ( \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } + \sqrt { 1 - \eta ^ { 2 } } \sigma _ { t } \frac { \bar { \mathbf { y } } ^ { ( i ) } - \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } } { \sigma _ { y } / s _ { i } } , \eta ^ { 2 } \sigma _ { t } ^ { 2 } ) } & { \mathrm { i f ~ } \sigma _ { t } < \frac { \sigma _ { \mathbf { y } } } { s _ { i } } } \\ { \mathcal { N } ( ( 1 - \eta _ { b } ) \bar { \mathbf { x } } _ { 0 } ^ { ( i ) } + \eta _ { b } \bar { \mathbf { y } } ^ { ( i ) } , \sigma _ { t } ^ { 2 } - \frac { \sigma _ { \mathbf { y } } ^ { 2 } } { s _ { i } ^ { 2 } } \eta _ { b } ^ { 2 } ) } & { \mathrm { i f ~ } \sigma _ { t } \ge \frac { \sigma _ { \mathbf { y } } } { s _ { i } } } \end{array} \right. } \end{array}
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$$
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where $\eta \in ( 0 , 1 ]$ is a hyperparameter controlling the variance of the transitions, and $\eta$ and $\eta _ { b }$ may depend on $\sigma _ { t } , s _ { i } , \sigma _ { \mathbf { y } }$ . We further assume that $\sigma _ { T } \geq \sigma _ { \mathbf { y } } / s _ { i }$ for all positive $s _ { i }$ .4
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In the following statement, we show that this construction has the “Gaussian marginals” property similar to the inference distribution used in unconditional diffusion models [19].
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Proposition 3.1. The conditional distributions $q ^ { ( t ) }$ defined in Equations 4 and 5 satisfy the following:
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$$
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q ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { 0 } , \sigma _ { t } ^ { 2 } \pmb { I } ) ,
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$$
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defined by marginalizing over $\mathbf { x } _ { t ^ { \prime } }$ (for all $t ^ { \prime } > t$ ) and y, where $q ( \mathbf { y } \vert \mathbf { x } _ { 0 } )$ is defined as in Equation (1) with $\mathbf { x } = \mathbf { x } _ { 0 }$ .
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We place the proof in Appendix C. Intuitively, our construction considers different cases for each index of the spectral space. $( i )$ If the corresponding singular value is zero, then y does not directly provide any information to that index, and the update is similar to regular unconditional generation. $( i i )$ If the singular value is non-zero, then the updates consider the information provided by $\mathbf { y }$ , which further depends on whether the measurements’ noise level in the spectral space $( \sigma _ { \mathbf { y } } / s _ { i } )$ is larger than the noise level in the diffusion model $( \sigma _ { t } )$ or not; the measurements in the spectral space $\bar { \mathbf { y } } ^ { ( i ) }$ are then scaled differently for these two cases in order to ensure Proposition 3.1 holds.
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Now that we have defined $q ^ { ( t ) }$ as a series of Gaussian conditionals, we define our model distribution $p _ { \theta }$ as a series of Gaussian conditionals as well. Similar to DDPM, we aim to obtain predictions of $\mathbf { x } _ { \mathrm { 0 } }$ at every step $t$ ; and to simplify notations, we use the symbol $\mathbf { x } _ { \theta , t }$ to represent this prediction made by a model5 $f _ { \theta } ( \mathbf { x } _ { t + 1 } , t + 1 ) : \mathbb { R } ^ { n } \times \mathbb { R } \to \mathbb { R } ^ { n }$ that takes in the sample $\mathbf { x } _ { t + 1 }$ and the conditioned time step $( t + 1 )$ . We also define $\bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) }$ as the $i$ -th index of $\bar { \mathbf { x } } _ { \theta , t } = { \mathbf { \nabla } } V ^ { \top } \mathbf { x } _ { \theta , t }$ .
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We define DDRM with trainable parameters $\theta$ as follows:
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$$
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\begin{array} { r } { p _ { \theta } ^ { ( T ) } ( \bar { \mathbf { x } } _ { T } ^ { ( i ) } | \mathbf { y } ) = \left\{ \begin{array} { l l } { \mathcal { N } ( \bar { \mathbf { y } } ^ { ( i ) } , \sigma _ { T } ^ { 2 } - \frac { \sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } ) } & { \mathrm { i f ~ } s _ { i } > 0 } \\ { \mathcal { N } ( 0 , \sigma _ { T } ^ { 2 } ) } & { \mathrm { i f ~ } s _ { i } = 0 } \end{array} \right. } \\ { p _ { \theta } ^ { ( t ) } ( \bar { \mathbf { x } } _ { t } ^ { ( i ) } | \mathbf { x } _ { t + 1 } , \mathbf { y } ) = \left\{ \begin{array} { l l } { \mathcal { N } ( \bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) } + \sqrt { 1 - \eta ^ { 2 } } \sigma _ { t } \frac { \bar { \mathbf { x } } _ { t + 1 } ^ { ( i ) } - \bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) } } { \sigma _ { t + 1 } } , \eta ^ { 2 } \sigma _ { t } ^ { 2 } ) } & { \mathrm { i f ~ } s _ { i } = 0 } \\ { \mathcal { N } ( \bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) } + \sqrt { 1 - \eta ^ { 2 } } \sigma _ { t } \frac { \bar { \mathbf { y } } ^ { ( i ) } - \bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) } } { \sigma _ { y } / s _ { i } } , \eta ^ { 2 } \sigma _ { t } ^ { 2 } ) } & { \mathrm { i f ~ } \sigma _ { t } < \frac { \sigma _ { \mathbf { y } } } { s _ { i } } } \\ { \mathcal { N } ( ( 1 - \eta _ { b } ) \bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) } + \eta _ { b } \bar { \mathbf { y } } ^ { ( i ) } , \sigma _ { t } ^ { 2 } - \frac { \sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } \eta _ { b } ^ { 2 } ) } & { \mathrm { i f ~ } \sigma _ { t } \ge \frac { \sigma _ { \mathbf { y } } } { s _ { i } } . } \end{array} \right. } \end{array}
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$$
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Compared to $q ^ { ( t ) }$ in Equations (4) and (5), our definition of $p _ { \theta } ^ { ( t ) }$ merely replaces $\bar { \mathbf { x } } _ { 0 } ^ { ( i ) }$ (which we do not know at sampling) with $\bar { \mathbf { x } } _ { \theta , t } ^ { ( i ) }$ (which depends on our predicted $\mathbf { x } _ { \theta , t }$ ) when $t < T$ , and replaces $\bar { \mathbf { x } } _ { 0 } ^ { ( i ) }$ with 0 when $t = T$ . It is possible to learn the variances [35] or consider alternative constructions where Proposition 3.1 holds; we leave these options as future work.
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# 3.3 “Learning” Image Restoration Models
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Once we have defined $p _ { \theta } ^ { ( t ) }$ and $q ^ { ( t ) }$ by choosing $\sigma _ { 1 : T }$ , $\eta$ and $\eta _ { b }$ , we can learn model parameters $\theta$ by maximizing the resulting ELBO objective (in Appendix A). However, this approach is not desirable since we have to learn a different model for each inverse problem (given $\pmb { H }$ and $\sigma _ { \mathbf { y } } .$ ), which is not flexible enough for arbitrary inverse problems. Fortunately, this does not have to be the case. In the following statement, we show that an optimal solution to DDPM / DDIM can also be an optimal solution to a DDRM problem, under reasonable assumptions used in prior work [19, 45].
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Figure 3: DDRM results on bedroom and cat images, for inpainting and deblurring.
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Theoremthen when ssumand els , th $f _ { \theta } ^ { ( t ) }$ and BO $f _ { \theta } ^ { ( t ^ { \prime } ) }$ do not have weight sharing whenetive of DDRM (details in Appendix r ) $t \neq t ^ { \prime }$ η = 1 η b = 2 σ tσ 2t +σ 2y /s 2i $A$ rewritten in the form of the DDPM / DDIM objective in Equation (2).
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We place the proof in Appendix C.
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Even for different choices of $\eta$ and $\eta _ { b }$ , the proof shows that the DDRM objective is a weighted sumof-squares error in the spectral space, and thus pre-trained DDPM models are good approximations to the optimal solution. Therefore, we can apply the same diffusion model (unconditioned on the inverse problem) using the updates in Equation (7) and Equation (8) and only modify $\pmb { H }$ and its SVD $( U , \Sigma , V )$ for various linear inverse problems.
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# 3.4 Accelerated Algorithms for DDRM
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Typical diffusion models are trained with many timesteps (e.g., 1000) to achieve optimal unconditional image synthesis quality, but sampling speed is slow as many NFEs are required. Previous works [45, 13] have accelerated this process by “skipping” steps with appropriate update rules. This is also true for DDRM, since we can obtain the denoising autoencoder objective in Equation (2) for any choice of increasing $\sigma _ { 1 : T }$ . For a pre-trained diffusion model with $T ^ { \prime }$ timesteps, we can choose $\sigma _ { 1 : T }$ to be a subset of the $T ^ { \prime }$ steps used in training.
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# 3.5 Memory Efficient SVD
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Our method, similar to SNIPS [25], utilizes the SVD of the degradation operator $\pmb { H }$ . This constitutes a memory consumption bottleneck in both algorithms as well as other methods such as Plug and Play $( \mathrm { P n P } )$ [51], as storing the matrix $V$ has a space complexity of $\Theta ( n ^ { 2 } )$ for signals of size $n$ . By leveraging special properties of the matrices $\pmb { H }$ used, we can reduce this complexity to $\Theta ( n )$ for denoising, inpainting, super resolution, deblurring, and colorization (details in Appendix D).
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# 4 Related Work
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Various deep learning solutions have been suggested for solving inverse problems under different settings (see a detailed survey in [37]). We focus on the unsupervised setting, where we have access to a dataset of clean images at training time, but the degradation model is known only at inference time. This setup is inherently general to all linear inverse problems, a property desired in many real-world applications such as medical imaging [46, 20].
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Table 1: Noiseless $4 \times$ super-resolution and deblurring results on ImageNet 1K $( 2 5 6 \times 2 5 6 )$ .
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<table><tr><td>Method</td><td colspan="4">4× super-resolution</td><td colspan="4">Deblurring</td></tr><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td><td>PSNR↑</td><td>SSIM个</td><td>KID↓</td><td>NFEs↓</td></tr><tr><td>Baseline</td><td>25.65</td><td>0.71</td><td>44.90</td><td>0</td><td>19.26</td><td>0.48</td><td>38.00</td><td>0</td></tr><tr><td>DGP</td><td>23.06</td><td>0.56</td><td>21.22</td><td>1500</td><td>22.70</td><td>0.52</td><td>27.60</td><td>1500</td></tr><tr><td>RED</td><td>26.08</td><td>0.73</td><td>53.55</td><td>100</td><td>26.16</td><td>0.76</td><td>21.21</td><td>500</td></tr><tr><td>SNIPS</td><td>17.58</td><td>0.22</td><td>35.17</td><td>1000</td><td>34.32</td><td>0.87</td><td>0.49</td><td>1000</td></tr><tr><td>DDRM</td><td>26.55</td><td>0.72</td><td>7.22</td><td>20</td><td>35.64</td><td>0.95</td><td>0.71</td><td>20</td></tr><tr><td>DDRM-CC</td><td>26.55</td><td>0.74</td><td>6.56</td><td>20</td><td>35.65</td><td>0.96</td><td>0.70</td><td>20</td></tr></table>
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Almost all unsupervised inverse problem solvers utilize a trained neural network in an iterative scheme. PnP, RED, and their successors [51, 40, 32, 49] apply a denoiser as part of an iterative optimization algorithm such as steepest descent, fixed-point, or alternating direction method of multipliers (ADMM). OneNet [39] trained a network to directly learn the proximal operator of ADMM. A similar use of denoisers in different iterative algorithms is proposed in [34, 16, 30]. The authors of [43] leverages robust classifiers learned with additional class labels.
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Another approach is to search the latent space of a generative model for a generated image that, when degraded, is as close as possible to the given measurements. Multiple such methods were suggested, mainly focusing on generative adversarial networks (GANs) [7, 11, 33]. While they exhibit impressive results on images of a specific class, most notably face images, these methods are not shown to be largely successful under a more diverse dataset such as ImageNet [12]. Deep Generative Prior (DGP) mitigates this issue by optimizing the latent input as well as the weights of the GAN’s generator [38].
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More recently, denoising diffusion models were used to solve inverse problems in both supervised (i.e., degradation model is known during training) [42, 41, 13, 10, 54] and unsupervised settings [22, 26, 25, 21, 46, 47, 9]. Unlike previous approaches, most diffusion-based methods can successfully recover images from measurements with significant noise. However, these methods are very slow, often requiring hundreds or thousands of iterations, and are yet to be proven on diverse datasets. Our method, motivated by variational inference, obtains problem-specific, non-equilibrium update rules that lead to high-quality solutions in much fewer iterations.
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ILVR [9] suggests a diffusion-based method that handles noiseless super-resolution, and can run in 250 steps. In Appendix H, we prove that when applied on the same underlying generative diffusion model, ILVR is a special case of DDRM. Therefore, ILVR can be further accelerated to run in 20 steps, but unlike DDRM, it provides no clear way of handling noise in the measurements. Similarly, the authors of [22] suggest a score-based solver for inverse problems that can converge in a small number of iterations, but does not handle noise in the measurements.
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# 5 Experiments
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# 5.1 Experimental Setup
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We demonstrate our algorithm’s capabilities using the diffusion models from [19], which are trained on CelebA-HQ [23], LSUN bedrooms, and LSUN cats [56] (all $2 5 6 \times 2 5 6$ pixels). We test these models on images from FFHQ [24], and pictures from the internet of the considered LSUN category, respectively. In addition, we use the models from [13], trained on the training set of ImageNet $2 5 6 \times 2 5 6$ and $5 1 2 \times 5 1 2$ , and tested on the corresponding validation set. Some of the ImageNet models require class information. For these models, we use the ground truth labels as input, and denote our algorithm as DDRM class conditional (DDRM-CC). In all experiments, we use $\eta = 0 . 8 5$ , $\eta _ { b } = 1$ , and a uniformly-spaced timestep schedule based on the 1000-step pre-trained models (more details in Appendix E). The number of NFEs (timesteps) is reported in each experiment.
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In each of the inverse problems we show, pixel values are in the range $[ 0 , 1 ]$ , and the degraded measurements are obtained as follows: $( i )$ for super-resolution, we use a block averaging filter to downscale the images by a factor of 2, 4, or 8 in each axis; $( i i )$ for deblurring, the images are blurred by a $9 \times 9$ uniform kernel, and singular values below a certain threshold are zeroed, making the problem more ill-posed. (iii) for colorization, the grayscale image is an average of the red, green, and blue channels of the original image; $( i \nu )$ and for inpainting, we mask parts of the original image with text overlay or randomly drop $5 0 \%$ of the pixels. Additive white Gaussian noise can optionally be added to the measurements in all inverse problems. We additionally conduct experiments on bicubic super-resolution and deblurring with an anisotropic Gaussian kernel in Appendix I.
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Table 2: $4 \times$ super resolution and deblurring results on ImageNet 1K $2 5 6 \times 2 5 6 )$ . Input images have an additive noise of $\sigma _ { \mathbf { y } } = 0 . 0 5$ .
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<table><tr><td rowspan="2">Method</td><td colspan="4">4× super-resolution</td><td colspan="4">Deblurring</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td></tr><tr><td>Baseline</td><td>22.55</td><td>0.46</td><td>67.86</td><td>0</td><td>18.35</td><td>0.20</td><td>75.50</td><td>0</td></tr><tr><td>DGP</td><td>20.69</td><td>0.43</td><td>42.17</td><td>1500</td><td>21.20</td><td>0.45</td><td>34.02</td><td>1500</td></tr><tr><td>RED</td><td>22.90</td><td>0.49</td><td>43.45</td><td>100</td><td>14.69</td><td>0.08</td><td>121.82</td><td>500</td></tr><tr><td>SNIPS</td><td>16.30</td><td>0.14</td><td>67.77</td><td>1000</td><td>16.37</td><td>0.14</td><td>77.96</td><td>1000</td></tr><tr><td>DDRM</td><td>25.21</td><td>0.66</td><td>12.43</td><td>20</td><td>25.45</td><td>0.66</td><td>15.24</td><td>20</td></tr><tr><td>DDRM-CC</td><td>25.22</td><td>0.67</td><td>10.82</td><td>20</td><td>25.46</td><td>0.67</td><td>13.49</td><td>20</td></tr></table>
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Figure 4: $4 \times$ noisy super resolution comparison with $\sigma _ { \mathbf { y } } = 0 . 0 5$ .
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Our code is available at https://github.com/bahjat-kawar/ddrm.
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# 5.2 Quantitative Experiments
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In order to quantify DDRM’s performance, we focus on the ImageNet dataset $2 5 6 \times 2 5 6 )$ for its diversity. For each experiment, we report the average peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) [52] to measure faithfulness to the original image, and the kernel Inception distance (KID) [5], multiplied by $1 0 ^ { 3 }$ , to measure the resulting image quality.
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We compare DDRM (with 20 and 100 steps) with other unsupervised methods that work in reasonable time (requiring 1500 NFEs or less) and can operate on ImageNet. Namely, we compare with RED [40], DGP [38], and SNIPS [25]. The exact setup of each method is detailed in Appendix F. We used the same hyperparameters for noisy and noiseless versions of the same problem for DGP, RED, and SNIPS, as tuning them for each version would compromise their unsupervised nature. Nevertheless, the performance of baselines like RED with such a tuning does not surpass that of DDRM, as we show in Appendix F. In addition, we show upscaling by bicubic interpolation as a baseline for super-resolution, and the blurry image itself as a baseline for deblurring. OneNet [39] is not included in the comparisons as it is limited to images of size $6 4 \times 6 4$ , and generalization to higher dimensions requires an improved network architecture.
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Figure 5: $5 1 2 \times 5 1 2$ ImageNet colorization. DDRM-CC produces various samples for multiple runs on the same input.
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Figure 6: Results on $2 5 6 \times 2 5 6$ USC-SIPI images using an ImageNet model. Blurred images have a noise of $\sigma _ { \mathbf { y } } = 0 . 0 1$ .
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We evaluate all methods on the problems of $4 \times$ super-resolution and deblurring, on one validation set image from each of the 1000 ImageNet classes, following [38]. Table 1 shows that DDRM outperforms all baseline methods, in all metrics, and on both problems with only 20 steps. The only exception to this is that SNIPS achieves better KID than DDRM in noiseless deblurring, but it requires $5 0 \times$ more NFEs to do so. Note that the runtime of all the tested methods is perfectly linear with NFEs, with negligible differences in time per iteration. DGP and DDRM-CC use ground-truth class labels for the test images to aid in the restoration process, and thus have an unfair advantage.
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DDRM’s appeal compared to previous methods becomes more substantial when significant noise is added to the measurements. Under this setting, DGP, RED, and SNIPS all fail to produce viable results, as evident in Table 2 and Figure 4. Since DDRM is fast, we also evaluate it on the entire ImageNet validation set in Appendix F.
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# 5.3 Qualitative Experiments
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DDRM produces high quality reconstructions across all the tested datasets and problems, as can be seen in Figures 1 and 3, and in Appendix I. As it is a posterior sampling algorithm, DDRM can produce multiple outputs for the same input, as demonstrated in Figure 5. Moreover, the unconditional ImageNet diffusion models can be used to solve inverse problems on out-of-distribution images with general content. In Figure 6, we show DDRM successfully restoring $2 5 6 \times 2 5 6$ images from USC-SIPI [53] that do not necessarily belong to any ImageNet class (more results in Appendix I).
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# 6 Conclusions
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We have introduced DDRM, a general sampling-based linear inverse problem solver based on unconditional/class-conditional diffusion generative models as learned priors. Motivated by variational inference, DDRM only requires a few number of NFEs (e.g., 20) compared to other samplingbased baselines (e.g., 1000 for SNIPS) and achieves scalability in multiple useful scenarios, including denoising, super-resolution, deblurring, inpainting, and colorization. We demonstrate the empirical successes of DDRM on various problems and datasets, including general natural images outside the distribution of the observed training set. To our best knowledge, DDRM is the first unsupervised method that effectively and efficiently samples from the posterior distribution of inverse problems with significant noise, and can work on natural images with general content.
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In terms of future work, apart from further optimizing the timestep and variance schedules, it would be interesting to investigate the following: (i) applying DDRM to non-linear inverse problems, $( i i )$ addressing scenarios where the degradation operator is unknown, and (iii) self-supervised training techniques inspired by DDRM as well as ones used in supervised techniques [41] that further improve performance of unsupervised models for image restoration.
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# Acknowledgements
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We thank Kristy Choi, Charlie Marx, and Avital Shafran for insightful discussions and feedback. This research was supported by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1-2145), AFOSR (FA9550-19-1-0024), ARO (W911NF-21-1-0125), Sloan Fellowship, Amazon AWS, Stanford Institute for Human-Centered Artificial Intelligence (HAI), Google Cloud, the Israel Science Foundation (ISF) under Grant 335/18, the Israeli Council For Higher Education - Planning & Budgeting Committee, and the Stephen A. Kreynes Fellowship.
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# References
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[2] Johnathan M Bardsley. Mcmc-based image reconstruction with uncertainty quantification. SIAM Journal on Scientific Computing, 34(3):A1316–A1332, 2012.
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[3] Johnathan M Bardsley, Antti Solonen, Heikki Haario, and Marko Laine. Randomize-thenoptimize: A method for sampling from posterior distributions in nonlinear inverse problems. SIAM Journal on Scientific Computing, 36(4):A1895–A1910, 2014.
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[4] Christopher M Bishop. Pattern recognition. Machine learning, 128(9), 2006.
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# Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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| 295 |
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(b) Did you describe the limitations of your work? [Yes] In the future work paragraph in Section 6
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| 296 |
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(c) Did you discuss any potential negative societal impacts of your work? [Yes] We came to the conclusion that our paper does not have potential negative societal impacts.
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| 297 |
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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| 298 |
+
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| 299 |
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2. If you are including theoretical results...
|
| 300 |
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| 301 |
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] In the appendice
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| 302 |
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| 303 |
+
3. If you ran experiments...
|
| 304 |
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| 305 |
+
(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] In the appendices.
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| 306 |
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In both the paper and the appendices.
|
| 307 |
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] We report results averaged over 1, 000 images in the main paper and 50, 000 images in the appendices. Such large numbers eliminate the need for error bars.
|
| 308 |
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(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] In the appendices.
|
| 309 |
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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| 311 |
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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| 313 |
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(b) Did you mention the license of the assets? [Yes] The licenses of previous works’ code and datasets will be included in our camera-ready code.
|
| 314 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code in the supplementary material.
|
| 315 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Consent was given by the original authors in their work.
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] The datasets we use are anonymized.
|
| 317 |
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|
| 318 |
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5. If you used crowdsourcing or conducted research with human subjects...
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| 319 |
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| 320 |
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 321 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 322 |
+
(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 |
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"text": "Denoising Diffusion Restoration Models ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
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| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
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},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
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"text": "Bahjat Kawar Department of Computer Science Technion, Haifa, Israel bahjat.kawar@cs.technion.ac.il ",
|
| 17 |
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"bbox": [
|
| 18 |
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| 19 |
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| 20 |
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| 22 |
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| 23 |
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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": "Michael Elad Department of Computer Science Technion, Haifa, Israel elad@cs.technion.ac.il ",
|
| 28 |
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"bbox": [
|
| 29 |
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547,
|
| 30 |
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| 31 |
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| 32 |
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],
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| 34 |
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"page_idx": 0
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| 35 |
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},
|
| 36 |
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{
|
| 37 |
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"type": "text",
|
| 38 |
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"text": "Stefano Ermon Department of Computer Science Stanford, California, USA ermon@cs.stanford.edu ",
|
| 39 |
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"bbox": [
|
| 40 |
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250,
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| 41 |
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"page_idx": 0
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| 46 |
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},
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| 47 |
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{
|
| 48 |
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"type": "text",
|
| 49 |
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"text": "Jiaming Song NVIDIA Santa Clara, California, USA jiamings@nvidia.com ",
|
| 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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| 65 |
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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": "Many interesting tasks in image restoration can be cast as linear inverse problems. A recent family of approaches for solving these problems uses stochastic algorithms that sample from the posterior distribution of natural images given the measurements. However, efficient solutions often require problem-specific supervised training to model the posterior, whereas unsupervised methods that are not problem-specific typically rely on inefficient iterative methods. This work addresses these issues by introducing Denoising Diffusion Restoration Models (DDRM), an efficient, unsupervised posterior sampling method. Motivated by variational inference, DDRM takes advantage of a pre-trained denoising diffusion generative model for solving any linear inverse problem. We demonstrate DDRM’s versatility on several image datasets for super-resolution, deblurring, inpainting, and colorization under various amounts of measurement noise. DDRM outperforms the current leading unsupervised methods on the diverse ImageNet dataset in reconstruction quality, perceptual quality, and runtime, being $5 \\times$ faster than the nearest competitor. DDRM also generalizes well for natural images out of the distribution of the observed ImageNet training set.1 ",
|
| 73 |
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| 79 |
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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": "text",
|
| 83 |
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"text": "1 Introduction ",
|
| 84 |
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"text_level": 1,
|
| 85 |
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"bbox": [
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| 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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| 92 |
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| 93 |
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{
|
| 94 |
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"type": "text",
|
| 95 |
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"text": "Many problems in image processing, including super-resolution [31, 17], deblurring [28, 48], inpainting [55], colorization [29, 58], and compressive sensing [1], are instances of linear inverse problems, where the goal is to recover an image from potentially noisy measurements given through a known linear degradation model. For a specific degradation model, image restoration can be addressed through end-to-end supervised training of neural networks, using pairs of original and degraded images [14, 58, 41]. However, real-world applications such as medical imaging often require flexibility to cope with multiple, possibly infinite, degradation models [46]. Here, unsupervised approaches based on learned priors [36], where the degradation model is only known and used during inference, may be more desirable since they can adapt to the given problem without re-training [51]. By learning sound assumptions over the underlying structure of images (e.g., priors, proximal operators or denoisers), unsupervised approaches can achieve effective restoration without training on specific degradation models [51, 40]. ",
|
| 96 |
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| 105 |
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"type": "text",
|
| 106 |
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"text": "Under this unsupervised setting, priors based on deep neural networks have demonstrated impressive empirical results in various image restoration tasks [40, 50, 43, 38, 15]. To recover the signal, most existing methods obtain a prior-related term over the signal from a neural network (e.g., the distribution of natural images), and a likelihood term from the degradation model. They combine the two terms to form a posterior over the signal, and the inverse problem can be posed as solving an optimization problem (e.g., maximum a posteriori [8, 40]) or solving a sampling problem (e.g., posterior sampling [2, 3, 25]). Then, these problems are often solved with iterative methods, such as gradient descent or Langevin dynamics, which may be demanding in computation and sensitive to hyperparameter tuning. An extreme example is found in [30] where a “fast” version of the algorithm uses 15, 000 neural function evaluations (NFEs). ",
|
| 107 |
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"bbox": [
|
| 108 |
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| 111 |
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|
| 113 |
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"page_idx": 0
|
| 114 |
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},
|
| 115 |
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{
|
| 116 |
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"type": "image",
|
| 117 |
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"img_path": "images/801265c5fb23e87dfcdc53eb65a6f1c2d596bee083cb8985f9b5061490a33f87.jpg",
|
| 118 |
+
"image_caption": [
|
| 119 |
+
"Figure 1: Pairs of measurements and recovered images with a 20-step DDRM on super-resolution, deblurring, inpainting, and colorization, with or without noise, and with unconditional generative models. The images are not accessed during training. "
|
| 120 |
+
],
|
| 121 |
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"image_footnote": [],
|
| 122 |
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"bbox": [
|
| 123 |
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| 124 |
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| 127 |
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|
| 128 |
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"page_idx": 1
|
| 129 |
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},
|
| 130 |
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|
| 131 |
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"type": "text",
|
| 132 |
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"text": "",
|
| 133 |
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| 140 |
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| 142 |
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"type": "text",
|
| 143 |
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"text": "Inspired by this unsupervised line of work, we introduce an efficient approach named Denoising Diffusion Restoration Models (DDRM), that can achieve competitive results in as low as 20 NFEs. DDRM is a denoising diffusion generative model [44, 19, 45] that gradually and stochastically denoises a sample to the desired output, conditioned on the measurements and the inverse problem. This way we introduce a variational inference objective for learning the posterior distribution of the inverse problem at hand. We then show its equivalence to the objective of an unconditional denoising diffusion generative model [19], which enables us to deploy such models in DDRM for various linear inverse problems (see Figure 2). To our best knowledge, DDRM is the first general sampling-based inverse problem solver that can efficiently produce a range of high-quality, diverse, yet valid solutions for general content images. ",
|
| 144 |
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"page_idx": 1
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| 151 |
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|
| 152 |
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{
|
| 153 |
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"type": "text",
|
| 154 |
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"text": "We demonstrate the empirical effectiveness of DDRM by comparing with various competitive methods based on learned priors, such as Deep Generative Prior (DGP) [38], SNIPS [25], and Regularization by Denoising (RED) [40]. On ImageNet examples, DDRM mostly outperforms the neural network baselines under noiseless super-resolution and deblurring measured in PSNR and KID [5], and is at least $5 0 \\times$ more efficient in terms of NFEs when it is second-best. Our advantage becomes even larger when measurement noise is involved, as noisy artifacts produced by iterative methods do not appear in our case. Over various real-world images, we further show DDRM results on super-resolution, deblurring, inpainting and colorization (see Figure 1). A DDRM trained on ImageNet also works on images that are out of its training set distribution (see Figure 6). ",
|
| 155 |
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"bbox": [
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},
|
| 163 |
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{
|
| 164 |
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"type": "text",
|
| 165 |
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"text": "2 Background ",
|
| 166 |
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"text_level": 1,
|
| 167 |
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"bbox": [
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},
|
| 175 |
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{
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| 176 |
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"type": "text",
|
| 177 |
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"text": "Linear Inverse Problems. A general linear inverse problem is posed as ",
|
| 178 |
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|
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},
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{
|
| 187 |
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"type": "equation",
|
| 188 |
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"img_path": "images/e5a3fee284a54bb7db9a90b40e2d086911c5792538822ae65e9d69812580cbbd.jpg",
|
| 189 |
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"text": "$$\n{ \\bf { y } } = { \\cal H } { \\bf { x } } + { \\bf { z } } ,\n$$",
|
| 190 |
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"text_format": "latex",
|
| 191 |
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"bbox": [
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},
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"type": "text",
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"text": "where we aim to recover the signal $\\mathbf { x } \\in \\mathbb { R } ^ { n }$ from measurements $\\mathbf { y } \\in \\mathbb { R } ^ { m }$ , where $\\pmb { H } \\in \\mathbb { R } ^ { m \\times n }$ is a known linear degradation matrix, and $\\mathbf { z } \\sim \\mathcal { N } ( 0 , \\sigma _ { \\mathbf { y } } ^ { 2 } I )$ is an i.i.d. additive Gaussian noise with known variance. The underlying structure of $\\mathbf { x }$ can be represented via a generative model, denoted as $p _ { \\theta } ( \\mathbf { x } )$ . Given $\\mathbf { y }$ and $\\pmb { H }$ , a posterior over the signal can be posed as: $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { y } ) \\propto p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } ) p ( \\mathbf { y } | \\mathbf { x } )$ , where the “likelihood” term $p ( \\mathbf { y } \\vert \\mathbf { x } )$ is defined via Equation (1); such an approach leverages a learned prior $p _ { \\theta } ( \\mathbf { x } )$ , and we call it an “unsupervised” approach based on the terminology in [36], as the prior does not necessarily depend on the inverse problem. Recovering $\\mathbf { x }$ can be done by sampling from this posterior [2], which may require many iterations to produce a good sample. Alternatively, one can also approximate this posterior by learning a model via amortized inference (i.e., supervised learning); the model learns to predict $\\mathbf { x }$ given y, generated from $\\mathbf { x }$ and a specific $\\pmb { H }$ . While this can be more efficient than sampling-based methods, it may generalize poorly to inverse problems that have not been trained on. ",
|
| 202 |
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"page_idx": 1
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},
|
| 210 |
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{
|
| 211 |
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"type": "image",
|
| 212 |
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"img_path": "images/3b8223ff3635bca0230fef513f42439b878155e66e5927dbf70c34d1b20a1fda.jpg",
|
| 213 |
+
"image_caption": [
|
| 214 |
+
"Figure 2: Illustration of our DDRM method for a specific inverse problem (super-resolution $^ +$ denoising). We can use unsupervised DDPM models as a good solution to the DDRM objective. "
|
| 215 |
+
],
|
| 216 |
+
"image_footnote": [],
|
| 217 |
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"bbox": [
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| 218 |
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"page_idx": 2
|
| 224 |
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},
|
| 225 |
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{
|
| 226 |
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"type": "text",
|
| 227 |
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"text": "",
|
| 228 |
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|
| 235 |
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},
|
| 236 |
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|
| 237 |
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"type": "text",
|
| 238 |
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"text": "Denoising Diffusion Probabilistic Models. Structures learned by generative models have been applied to various inverse problems and often outperform data-independent structural constraints such as sparsity [7]. These generative models learn a model distribution $p _ { \\theta } ( \\mathbf { x } )$ that approximates a data distribution $q ( \\mathbf { x } )$ from samples. In particular, diffusion models have demonstrated impressive unconditional generative modeling performance on images [13]. Diffusion models are generative models with a Markov chain structure ${ \\bf x } _ { T } \\to { \\bf x } _ { T - 1 } \\to { \\bf . . . } \\to { \\bf x } _ { 1 } \\to { \\bf x } _ { 0 }$ (where $\\mathbf { x } _ { t } \\in \\mathbb { R } ^ { n }$ ), which has the following joint distribution: ",
|
| 239 |
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"bbox": [
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| 244 |
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],
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| 245 |
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"page_idx": 2
|
| 246 |
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},
|
| 247 |
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{
|
| 248 |
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"type": "equation",
|
| 249 |
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"img_path": "images/508923df76018b3c2e803227bd9e0bb963d64c6f6079be1dbfbcec3fec7f9db6.jpg",
|
| 250 |
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"text": "$$\np _ { \\theta } ( \\mathbf { x } _ { 0 : T } ) = p _ { \\theta } ^ { ( T ) } ( \\mathbf { x } _ { T } ) \\prod _ { t = 0 } ^ { T - 1 } p _ { \\theta } ^ { ( t ) } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { t + 1 } ) .\n$$",
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"text": "After drawing $\\mathbf { x } _ { \\mathrm { 0 : } T }$ , only $\\mathbf { x } _ { \\mathrm { 0 } }$ is kept as the sample of the generative model. To train a diffusion model, a fixed, factorized variational inference distribution is introduced: ",
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"text": "$$\nq ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 0 } ) = q ^ { ( T ) } ( \\mathbf { x } _ { T } | \\mathbf { x } _ { 0 } ) \\prod _ { t = 0 } ^ { T - 1 } q ^ { ( t ) } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { t + 1 } , \\mathbf { x } _ { 0 } ) ,\n$$",
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"text": "which leads to an evidence lower bound (ELBO) on the maximum likelihood objective [44]. A special property of some diffusion models is that both $p _ { \\theta } ^ { ( t ) }$ and $q ^ { ( t ) }$ are chosen as conditional Gaussian distributions for all $t < T$ , and that $q ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } )$ is also a Gaussian with known mean and covariance, i.e., $\\mathbf { x } _ { t }$ can be treated as $\\mathbf { x } _ { \\mathrm { 0 } }$ directly corrupted with Gaussian noise. Thus, the ELBO objective can be reduced into the following denoising autoencoder objective (please refer to [45] for derivations): ",
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"text": "$$\n\\sum _ { t = 1 } ^ { T } \\gamma _ { t } \\mathbb { E } _ { ( \\mathbf { x } _ { 0 } , \\mathbf { x } _ { t } ) \\sim q ( \\mathbf { x } _ { 0 } ) q ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } ) } \\left[ \\big \\lVert \\mathbf { x } _ { 0 } - f _ { \\theta } ^ { ( t ) } ( \\mathbf { x } _ { t } ) \\big \\rVert _ { 2 } ^ { 2 } \\right]\n$$",
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"text": "where $f _ { \\theta } ^ { ( t ) }$ is a $\\theta$ -parameterized neural network that aims to recover a noiseless observation from a noisy $\\mathbf { x } _ { t }$ , and $\\gamma _ { 1 : T }$ are a set of positive coefficients that depend on $q \\big ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 0 } \\big )$ . ",
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"type": "text",
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"text": "3 Denoising Diffusion Restoration Models ",
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"text_level": 1,
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"text": "Inverse problem solvers based on posterior sampling often face a dilemma: unsupervised approaches apply to general problems but are inefficient, whereas supervised ones are efficient but can only address specific problems. ",
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"text": "To solve this dilemma, we introduce Denoising Diffusion Restoration Models (DDRM), an unsupervised solver for general linear inverse problems, capable of handling such tasks with or without noise in the measurements. DDRM is efficient and exhibits competitive performance compared to popular unsupervised solvers [40, 38, 25]. ",
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"text": "The key idea behind DDRM is to find an unsupervised solution that also suits supervised learning objectives. First, we describe the variational objective for DDRM over a specific inverse problem (Section 3.1). Next, we introduce specific forms of DDRM that are suitable for linear inverse problems and allow pre-trained unconditional and class-conditional diffusion models to be used directly (Sections 3.2, 3.3). Finally, we discuss practical algorithms that are compute and memory efficient (Sections 3.4, 3.5). ",
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"text": "3.1 Variational Objective for DDRM ",
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"type": "text",
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"text": "For any linear inverse problem, we define DDRM as a Markov chain ${ \\bf x } _ { T } \\to { \\bf x } _ { T - 1 } \\to { \\bf . . . } \\to { \\bf x } _ { 1 } \\to { \\bf x } _ { 0 }$ conditioned on $\\mathbf { y }$ , where ",
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"img_path": "images/da8248673243116ecaed204593149c93ac358bbde6eaf0fd4a688c4ccf83be28.jpg",
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"text": "$$\np _ { \\theta } ( \\mathbf { x } _ { 0 : T } | \\mathbf { y } ) = p _ { \\theta } ^ { ( T ) } ( \\mathbf { x } _ { T } | \\mathbf { y } ) \\prod _ { t = 0 } ^ { T - 1 } p _ { \\theta } ^ { ( t ) } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { t + 1 } , \\mathbf { y } )\n$$",
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| 392 |
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"type": "text",
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"text": "and $\\mathbf { x } _ { \\mathrm { 0 } }$ is the final diffusion output. In order to perform inference, we consider the following factorized variational distribution conditioned on y: ",
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"type": "equation",
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"img_path": "images/6f9bf5dcbe0ebeb264c2d59ae53d87d6d399f4c3690a85e470043e0c2932bec8.jpg",
|
| 414 |
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"text": "$$\nq ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 0 } , \\mathbf { y } ) = q ^ { ( T ) } ( \\mathbf { x } _ { T } | \\mathbf { x } _ { 0 } , \\mathbf { y } ) \\prod _ { t = 0 } ^ { T - 1 } q ^ { ( t ) } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { t + 1 } , \\mathbf { x } _ { 0 } , \\mathbf { y } ) ,\n$$",
|
| 415 |
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"text_format": "latex",
|
| 416 |
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"bbox": [
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| 423 |
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},
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| 424 |
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{
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| 425 |
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"type": "text",
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| 426 |
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"text": "leading to an ELBO objective for diffusion models conditioned on $\\mathbf { y }$ (details in Appendix A). ",
|
| 427 |
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| 434 |
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"type": "text",
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| 437 |
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"text": "In the remainder of the section, we construct suitable variational problems given $\\pmb { H }$ and $\\sigma _ { \\mathbf { y } }$ and connect them to unconditional diffusion generative models. To simplify notations, we will construct the variational distribution $q$ such that $\\bar { q } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } ) = \\mathcal { N } ( \\mathbf { x } _ { 0 } , \\sigma _ { t } ^ { 2 } I )$ for noise levels $0 = \\sigma _ { 0 } < \\sigma _ { 1 } <$ $\\sigma _ { 2 } < . . . < \\sigma _ { T }$ .2 In Appendix B, we will show that this is equivalent to the distribution introduced in DDPM [19] and DDIM [45],3 up to fixed linear transformations over $\\mathbf { x } _ { t }$ . ",
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| 438 |
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"text": "3.2 A Diffusion Process for Image Restoration ",
|
| 449 |
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"text_level": 1,
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| 450 |
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"type": "text",
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"text": "Similar to SNIPS [25], we consider the singular value decomposition (SVD) of $\\pmb { H }$ , and perform the diffusion in its spectral space. The idea behind this is to tie the noise present in the measurements $\\mathbf { y }$ with the diffusion noise in $\\mathbf { x } _ { 1 : T }$ , ensuring that the diffusion result $\\mathbf { x } _ { \\mathrm { 0 } }$ is faithful to the measurements. By using the SVD, we identify the data from $\\mathbf { x }$ that is missing in y, and synthesize it using a diffusion process. In conjunction, the noisy data in y undergoes a denoising process. For example, in inpainting with noise (e.g., $\\pmb { H } = \\mathrm { d i a g } ( [ 1 , \\dots , 1 , 0 , \\dots , 0 ] )$ , $\\sigma _ { \\mathbf { y } } \\geq 0 $ ), the spectral space is simply the pixel space, so the model should generate the missing pixels and denoise the observed ones in y. For a general linear $\\pmb { H }$ , its SVD is given as ",
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| 468 |
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| 469 |
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| 470 |
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"type": "equation",
|
| 471 |
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"img_path": "images/6a7bf944e23d0d53ea09e94bec0640ca69c87cf596f4b82e4947f0ae938933df.jpg",
|
| 472 |
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"text": "$$\n\\pmb { H } = \\pmb { U } \\pmb { \\Sigma V } ^ { \\top }\n$$",
|
| 473 |
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| 474 |
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{
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| 483 |
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"type": "text",
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| 484 |
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"text": "where $\\pmb { U } \\in \\mathbb { R } ^ { m \\times m }$ , $V \\in \\mathbb { R } ^ { n \\times n }$ are orthogonal matrices, and $\\pmb { \\Sigma } \\in \\mathbb { R } ^ { m \\times n }$ is a rectangular diagonal matrix containing the singular values of $\\pmb { H }$ , ordered descendingly. As this is the case in most useful degradation models, we assume $m \\leq n$ , but our method would work for $m > n$ as well. We denote the singular values as $s _ { 1 } \\geq s _ { 2 } \\geq . . . \\geq s _ { m }$ , and define $s _ { i } = 0$ for $i \\in [ m + 1 , n ]$ . ",
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"bbox": [
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"text": "We use the shorthand notations for values in the spectral space: $\\bar { \\mathbf { x } } _ { t } ^ { ( i ) }$ is the $i$ -th index of the vector $\\bar { \\bf x } _ { t } =$ $V ^ { \\top } \\mathbf { x } _ { t }$ , and $\\bar { \\mathbf { y } } ^ { ( i ) }$ is the $i$ -th index of the vector $\\bar { \\mathbf { y } } = \\pmb { \\Sigma } ^ { \\dagger } \\pmb { U } ^ { \\top } \\mathbf { y }$ (where $\\dagger$ denotes the Moore–Penrose pseudo-inverse). Because $V$ is an orthogonal matrix, we can recover $\\mathbf { x } _ { t }$ from $\\bar { \\mathbf { x } } _ { t }$ exactly by left multiplying $V$ . For each index $i$ in $\\bar { \\mathbf { x } } _ { t }$ , we define the variational distribution as: ",
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"text": "",
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| 507 |
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| 518 |
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"text": "$$\n\\begin{array} { r l } & { \\quad q ^ { ( T ) } ( \\bar { \\mathbf { x } } _ { T } ^ { ( i ) } | \\mathbf { x } _ { 0 } , \\mathbf { y } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { N } ( \\bar { \\mathbf { y } } ^ { ( i ) } , \\sigma _ { T } ^ { 2 } - \\frac { \\sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } ) } & { \\mathrm { i f ~ } s _ { i } > 0 } \\\\ { \\mathcal { N } ( \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } , \\sigma _ { T } ^ { 2 } ) } & { \\mathrm { i f ~ } s _ { i } = 0 } \\end{array} \\right. } \\\\ & { \\quad q ^ { ( t ) } ( \\bar { \\mathbf { x } } _ { t } ^ { ( i ) } | \\mathbf { x } _ { t + 1 } , \\bar { \\mathbf { x } } _ { 0 } ^ { ( \\bar { \\mathbf { \\alpha } } ) } , \\mathbf { y } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { N } ( \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } + \\sqrt { 1 - \\eta ^ { 2 } } \\sigma _ { t } \\frac { \\bar { \\mathbf { x } } _ { t + 1 } ^ { ( i ) } - \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } } { \\sigma _ { t + 1 } } , \\eta ^ { 2 } \\sigma _ { t } ^ { 2 } ) } & { \\mathrm { i f ~ } s _ { i } = 0 } \\\\ { \\mathcal { N } ( \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } + \\sqrt { 1 - \\eta ^ { 2 } } \\sigma _ { t } \\frac { \\bar { \\mathbf { y } } ^ { ( i ) } - \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } } { \\sigma _ { y } / s _ { i } } , \\eta ^ { 2 } \\sigma _ { t } ^ { 2 } ) } & { \\mathrm { i f ~ } \\sigma _ { t } < \\frac { \\sigma _ { \\mathbf { y } } } { s _ { i } } } \\\\ { \\mathcal { N } ( ( 1 - \\eta _ { b } ) \\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) } + \\eta _ { b } \\bar { \\mathbf { y } } ^ { ( i ) } , \\sigma _ { t } ^ { 2 } - \\frac { \\sigma _ { \\mathbf { y } } ^ { 2 } } { s _ { i } ^ { 2 } } \\eta _ { b } ^ { 2 } ) } & { \\mathrm { i f ~ } \\sigma _ { t } \\ge \\frac { \\sigma _ { \\mathbf { y } } } { s _ { i } } } \\end{array} \\right. } \\end{array}\n$$",
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| 519 |
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"text_format": "latex",
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},
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| 529 |
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"type": "text",
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| 530 |
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"text": "where $\\eta \\in ( 0 , 1 ]$ is a hyperparameter controlling the variance of the transitions, and $\\eta$ and $\\eta _ { b }$ may depend on $\\sigma _ { t } , s _ { i } , \\sigma _ { \\mathbf { y } }$ . We further assume that $\\sigma _ { T } \\geq \\sigma _ { \\mathbf { y } } / s _ { i }$ for all positive $s _ { i }$ .4 ",
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"text": "In the following statement, we show that this construction has the “Gaussian marginals” property similar to the inference distribution used in unconditional diffusion models [19]. ",
|
| 542 |
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},
|
| 550 |
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{
|
| 551 |
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"type": "text",
|
| 552 |
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"text": "Proposition 3.1. The conditional distributions $q ^ { ( t ) }$ defined in Equations 4 and 5 satisfy the following: ",
|
| 553 |
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| 562 |
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"type": "equation",
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| 563 |
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"img_path": "images/a4d60ac2812957f5fdd741df249a0fcb253712d0e64b950edad891d470df2a2d.jpg",
|
| 564 |
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"text": "$$\nq ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } ) = \\mathcal { N } ( \\mathbf { x } _ { 0 } , \\sigma _ { t } ^ { 2 } \\pmb { I } ) ,\n$$",
|
| 565 |
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"text_format": "latex",
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| 566 |
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"type": "text",
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| 576 |
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"text": "defined by marginalizing over $\\mathbf { x } _ { t ^ { \\prime } }$ (for all $t ^ { \\prime } > t$ ) and y, where $q ( \\mathbf { y } \\vert \\mathbf { x } _ { 0 } )$ is defined as in Equation (1) with $\\mathbf { x } = \\mathbf { x } _ { 0 }$ . ",
|
| 577 |
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"bbox": [
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"type": "text",
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| 587 |
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"text": "We place the proof in Appendix C. Intuitively, our construction considers different cases for each index of the spectral space. $( i )$ If the corresponding singular value is zero, then y does not directly provide any information to that index, and the update is similar to regular unconditional generation. $( i i )$ If the singular value is non-zero, then the updates consider the information provided by $\\mathbf { y }$ , which further depends on whether the measurements’ noise level in the spectral space $( \\sigma _ { \\mathbf { y } } / s _ { i } )$ is larger than the noise level in the diffusion model $( \\sigma _ { t } )$ or not; the measurements in the spectral space $\\bar { \\mathbf { y } } ^ { ( i ) }$ are then scaled differently for these two cases in order to ensure Proposition 3.1 holds. ",
|
| 588 |
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"bbox": [
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"type": "text",
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| 598 |
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"text": "Now that we have defined $q ^ { ( t ) }$ as a series of Gaussian conditionals, we define our model distribution $p _ { \\theta }$ as a series of Gaussian conditionals as well. Similar to DDPM, we aim to obtain predictions of $\\mathbf { x } _ { \\mathrm { 0 } }$ at every step $t$ ; and to simplify notations, we use the symbol $\\mathbf { x } _ { \\theta , t }$ to represent this prediction made by a model5 $f _ { \\theta } ( \\mathbf { x } _ { t + 1 } , t + 1 ) : \\mathbb { R } ^ { n } \\times \\mathbb { R } \\to \\mathbb { R } ^ { n }$ that takes in the sample $\\mathbf { x } _ { t + 1 }$ and the conditioned time step $( t + 1 )$ . We also define $\\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) }$ as the $i$ -th index of $\\bar { \\mathbf { x } } _ { \\theta , t } = { \\mathbf { \\nabla } } V ^ { \\top } \\mathbf { x } _ { \\theta , t }$ . ",
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| 608 |
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"type": "text",
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| 609 |
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"text": "We define DDRM with trainable parameters $\\theta$ as follows: ",
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| 610 |
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"type": "equation",
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"img_path": "images/d515fd4c84fe286359a4f50ba641459582587832a7e36e7c3b90cb086d1c3351.jpg",
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"text": "$$\n\\begin{array} { r } { p _ { \\theta } ^ { ( T ) } ( \\bar { \\mathbf { x } } _ { T } ^ { ( i ) } | \\mathbf { y } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { N } ( \\bar { \\mathbf { y } } ^ { ( i ) } , \\sigma _ { T } ^ { 2 } - \\frac { \\sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } ) } & { \\mathrm { i f ~ } s _ { i } > 0 } \\\\ { \\mathcal { N } ( 0 , \\sigma _ { T } ^ { 2 } ) } & { \\mathrm { i f ~ } s _ { i } = 0 } \\end{array} \\right. } \\\\ { p _ { \\theta } ^ { ( t ) } ( \\bar { \\mathbf { x } } _ { t } ^ { ( i ) } | \\mathbf { x } _ { t + 1 } , \\mathbf { y } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { N } ( \\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) } + \\sqrt { 1 - \\eta ^ { 2 } } \\sigma _ { t } \\frac { \\bar { \\mathbf { x } } _ { t + 1 } ^ { ( i ) } - \\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) } } { \\sigma _ { t + 1 } } , \\eta ^ { 2 } \\sigma _ { t } ^ { 2 } ) } & { \\mathrm { i f ~ } s _ { i } = 0 } \\\\ { \\mathcal { N } ( \\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) } + \\sqrt { 1 - \\eta ^ { 2 } } \\sigma _ { t } \\frac { \\bar { \\mathbf { y } } ^ { ( i ) } - \\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) } } { \\sigma _ { y } / s _ { i } } , \\eta ^ { 2 } \\sigma _ { t } ^ { 2 } ) } & { \\mathrm { i f ~ } \\sigma _ { t } < \\frac { \\sigma _ { \\mathbf { y } } } { s _ { i } } } \\\\ { \\mathcal { N } ( ( 1 - \\eta _ { b } ) \\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) } + \\eta _ { b } \\bar { \\mathbf { y } } ^ { ( i ) } , \\sigma _ { t } ^ { 2 } - \\frac { \\sigma _ { y } ^ { 2 } } { s _ { i } ^ { 2 } } \\eta _ { b } ^ { 2 } ) } & { \\mathrm { i f ~ } \\sigma _ { t } \\ge \\frac { \\sigma _ { \\mathbf { y } } } { s _ { i } } . } \\end{array} \\right. } \\end{array}\n$$",
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| 622 |
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"type": "text",
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| 633 |
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"text": "Compared to $q ^ { ( t ) }$ in Equations (4) and (5), our definition of $p _ { \\theta } ^ { ( t ) }$ merely replaces $\\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) }$ (which we do not know at sampling) with $\\bar { \\mathbf { x } } _ { \\theta , t } ^ { ( i ) }$ (which depends on our predicted $\\mathbf { x } _ { \\theta , t }$ ) when $t < T$ , and replaces $\\bar { \\mathbf { x } } _ { 0 } ^ { ( i ) }$ with 0 when $t = T$ . It is possible to learn the variances [35] or consider alternative constructions where Proposition 3.1 holds; we leave these options as future work. ",
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| 634 |
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"type": "text",
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"text": "3.3 “Learning” Image Restoration Models ",
|
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"text": "Once we have defined $p _ { \\theta } ^ { ( t ) }$ and $q ^ { ( t ) }$ by choosing $\\sigma _ { 1 : T }$ , $\\eta$ and $\\eta _ { b }$ , we can learn model parameters $\\theta$ by maximizing the resulting ELBO objective (in Appendix A). However, this approach is not desirable since we have to learn a different model for each inverse problem (given $\\pmb { H }$ and $\\sigma _ { \\mathbf { y } } .$ ), which is not flexible enough for arbitrary inverse problems. Fortunately, this does not have to be the case. In the following statement, we show that an optimal solution to DDPM / DDIM can also be an optimal solution to a DDRM problem, under reasonable assumptions used in prior work [19, 45]. ",
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| 666 |
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"type": "image",
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| 667 |
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"img_path": "images/81bc643d694333d66b6dd90380745687282edff4eb6ea14cd668b967ec261af4.jpg",
|
| 668 |
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"image_caption": [
|
| 669 |
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"Figure 3: DDRM results on bedroom and cat images, for inpainting and deblurring. "
|
| 670 |
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],
|
| 671 |
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"image_footnote": [],
|
| 672 |
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| 682 |
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"text": "",
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| 683 |
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"bbox": [
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"type": "text",
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| 693 |
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"text": "Theoremthen when ssumand els , th $f _ { \\theta } ^ { ( t ) }$ and BO $f _ { \\theta } ^ { ( t ^ { \\prime } ) }$ do not have weight sharing whenetive of DDRM (details in Appendix r ) $t \\neq t ^ { \\prime }$ η = 1 η b = 2 σ tσ 2t +σ 2y /s 2i $A$ rewritten in the form of the DDPM / DDIM objective in Equation (2). ",
|
| 694 |
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| 702 |
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"type": "text",
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"text": "We place the proof in Appendix C. ",
|
| 705 |
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"text": "Even for different choices of $\\eta$ and $\\eta _ { b }$ , the proof shows that the DDRM objective is a weighted sumof-squares error in the spectral space, and thus pre-trained DDPM models are good approximations to the optimal solution. Therefore, we can apply the same diffusion model (unconditioned on the inverse problem) using the updates in Equation (7) and Equation (8) and only modify $\\pmb { H }$ and its SVD $( U , \\Sigma , V )$ for various linear inverse problems. ",
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| 716 |
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"text": "3.4 Accelerated Algorithms for DDRM ",
|
| 727 |
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"text": "Typical diffusion models are trained with many timesteps (e.g., 1000) to achieve optimal unconditional image synthesis quality, but sampling speed is slow as many NFEs are required. Previous works [45, 13] have accelerated this process by “skipping” steps with appropriate update rules. This is also true for DDRM, since we can obtain the denoising autoencoder objective in Equation (2) for any choice of increasing $\\sigma _ { 1 : T }$ . For a pre-trained diffusion model with $T ^ { \\prime }$ timesteps, we can choose $\\sigma _ { 1 : T }$ to be a subset of the $T ^ { \\prime }$ steps used in training. ",
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"text": "3.5 Memory Efficient SVD ",
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"text_level": 1,
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"text": "Our method, similar to SNIPS [25], utilizes the SVD of the degradation operator $\\pmb { H }$ . This constitutes a memory consumption bottleneck in both algorithms as well as other methods such as Plug and Play $( \\mathrm { P n P } )$ [51], as storing the matrix $V$ has a space complexity of $\\Theta ( n ^ { 2 } )$ for signals of size $n$ . By leveraging special properties of the matrices $\\pmb { H }$ used, we can reduce this complexity to $\\Theta ( n )$ for denoising, inpainting, super resolution, deblurring, and colorization (details in Appendix D). ",
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"type": "text",
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"text": "4 Related Work ",
|
| 773 |
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"text": "Various deep learning solutions have been suggested for solving inverse problems under different settings (see a detailed survey in [37]). We focus on the unsupervised setting, where we have access to a dataset of clean images at training time, but the degradation model is known only at inference time. This setup is inherently general to all linear inverse problems, a property desired in many real-world applications such as medical imaging [46, 20]. ",
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"type": "table",
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"img_path": "images/15b59c181c889f7e18b39ef45a7c4778bef8bae06895ab56077d0238e8036c79.jpg",
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"table_caption": [
|
| 797 |
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"Table 1: Noiseless $4 \\times$ super-resolution and deblurring results on ImageNet 1K $( 2 5 6 \\times 2 5 6 )$ . "
|
| 798 |
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],
|
| 799 |
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"table_footnote": [],
|
| 800 |
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"table_body": "<table><tr><td>Method</td><td colspan=\"4\">4× super-resolution</td><td colspan=\"4\">Deblurring</td></tr><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td><td>PSNR↑</td><td>SSIM个</td><td>KID↓</td><td>NFEs↓</td></tr><tr><td>Baseline</td><td>25.65</td><td>0.71</td><td>44.90</td><td>0</td><td>19.26</td><td>0.48</td><td>38.00</td><td>0</td></tr><tr><td>DGP</td><td>23.06</td><td>0.56</td><td>21.22</td><td>1500</td><td>22.70</td><td>0.52</td><td>27.60</td><td>1500</td></tr><tr><td>RED</td><td>26.08</td><td>0.73</td><td>53.55</td><td>100</td><td>26.16</td><td>0.76</td><td>21.21</td><td>500</td></tr><tr><td>SNIPS</td><td>17.58</td><td>0.22</td><td>35.17</td><td>1000</td><td>34.32</td><td>0.87</td><td>0.49</td><td>1000</td></tr><tr><td>DDRM</td><td>26.55</td><td>0.72</td><td>7.22</td><td>20</td><td>35.64</td><td>0.95</td><td>0.71</td><td>20</td></tr><tr><td>DDRM-CC</td><td>26.55</td><td>0.74</td><td>6.56</td><td>20</td><td>35.65</td><td>0.96</td><td>0.70</td><td>20</td></tr></table>",
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"text": "Almost all unsupervised inverse problem solvers utilize a trained neural network in an iterative scheme. PnP, RED, and their successors [51, 40, 32, 49] apply a denoiser as part of an iterative optimization algorithm such as steepest descent, fixed-point, or alternating direction method of multipliers (ADMM). OneNet [39] trained a network to directly learn the proximal operator of ADMM. A similar use of denoisers in different iterative algorithms is proposed in [34, 16, 30]. The authors of [43] leverages robust classifiers learned with additional class labels. ",
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"text": "Another approach is to search the latent space of a generative model for a generated image that, when degraded, is as close as possible to the given measurements. Multiple such methods were suggested, mainly focusing on generative adversarial networks (GANs) [7, 11, 33]. While they exhibit impressive results on images of a specific class, most notably face images, these methods are not shown to be largely successful under a more diverse dataset such as ImageNet [12]. Deep Generative Prior (DGP) mitigates this issue by optimizing the latent input as well as the weights of the GAN’s generator [38]. ",
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"text": "More recently, denoising diffusion models were used to solve inverse problems in both supervised (i.e., degradation model is known during training) [42, 41, 13, 10, 54] and unsupervised settings [22, 26, 25, 21, 46, 47, 9]. Unlike previous approaches, most diffusion-based methods can successfully recover images from measurements with significant noise. However, these methods are very slow, often requiring hundreds or thousands of iterations, and are yet to be proven on diverse datasets. Our method, motivated by variational inference, obtains problem-specific, non-equilibrium update rules that lead to high-quality solutions in much fewer iterations. ",
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"type": "text",
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"text": "ILVR [9] suggests a diffusion-based method that handles noiseless super-resolution, and can run in 250 steps. In Appendix H, we prove that when applied on the same underlying generative diffusion model, ILVR is a special case of DDRM. Therefore, ILVR can be further accelerated to run in 20 steps, but unlike DDRM, it provides no clear way of handling noise in the measurements. Similarly, the authors of [22] suggest a score-based solver for inverse problems that can converge in a small number of iterations, but does not handle noise in the measurements. ",
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"type": "text",
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"text": "5 Experiments ",
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| 856 |
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"type": "text",
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"text": "5.1 Experimental Setup ",
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"text": "We demonstrate our algorithm’s capabilities using the diffusion models from [19], which are trained on CelebA-HQ [23], LSUN bedrooms, and LSUN cats [56] (all $2 5 6 \\times 2 5 6$ pixels). We test these models on images from FFHQ [24], and pictures from the internet of the considered LSUN category, respectively. In addition, we use the models from [13], trained on the training set of ImageNet $2 5 6 \\times 2 5 6$ and $5 1 2 \\times 5 1 2$ , and tested on the corresponding validation set. Some of the ImageNet models require class information. For these models, we use the ground truth labels as input, and denote our algorithm as DDRM class conditional (DDRM-CC). In all experiments, we use $\\eta = 0 . 8 5$ , $\\eta _ { b } = 1$ , and a uniformly-spaced timestep schedule based on the 1000-step pre-trained models (more details in Appendix E). The number of NFEs (timesteps) is reported in each experiment. ",
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"text": "In each of the inverse problems we show, pixel values are in the range $[ 0 , 1 ]$ , and the degraded measurements are obtained as follows: $( i )$ for super-resolution, we use a block averaging filter to downscale the images by a factor of 2, 4, or 8 in each axis; $( i i )$ for deblurring, the images are blurred by a $9 \\times 9$ uniform kernel, and singular values below a certain threshold are zeroed, making the problem more ill-posed. (iii) for colorization, the grayscale image is an average of the red, green, and blue channels of the original image; $( i \\nu )$ and for inpainting, we mask parts of the original image with text overlay or randomly drop $5 0 \\%$ of the pixels. Additive white Gaussian noise can optionally be added to the measurements in all inverse problems. We additionally conduct experiments on bicubic super-resolution and deblurring with an anisotropic Gaussian kernel in Appendix I. ",
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{
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"type": "table",
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"img_path": "images/8b6435c2611c1186e23b55821ff1b0fc496fa7600eaa32bb9aa5d5b5a04bab15.jpg",
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"table_caption": [
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| 903 |
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"Table 2: $4 \\times$ super resolution and deblurring results on ImageNet 1K $2 5 6 \\times 2 5 6 )$ . Input images have an additive noise of $\\sigma _ { \\mathbf { y } } = 0 . 0 5$ . "
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"table_footnote": [],
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"table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">4× super-resolution</td><td colspan=\"4\">Deblurring</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td><td>PSNR↑</td><td>SSIM↑</td><td>KID↓</td><td>NFEs↓</td></tr><tr><td>Baseline</td><td>22.55</td><td>0.46</td><td>67.86</td><td>0</td><td>18.35</td><td>0.20</td><td>75.50</td><td>0</td></tr><tr><td>DGP</td><td>20.69</td><td>0.43</td><td>42.17</td><td>1500</td><td>21.20</td><td>0.45</td><td>34.02</td><td>1500</td></tr><tr><td>RED</td><td>22.90</td><td>0.49</td><td>43.45</td><td>100</td><td>14.69</td><td>0.08</td><td>121.82</td><td>500</td></tr><tr><td>SNIPS</td><td>16.30</td><td>0.14</td><td>67.77</td><td>1000</td><td>16.37</td><td>0.14</td><td>77.96</td><td>1000</td></tr><tr><td>DDRM</td><td>25.21</td><td>0.66</td><td>12.43</td><td>20</td><td>25.45</td><td>0.66</td><td>15.24</td><td>20</td></tr><tr><td>DDRM-CC</td><td>25.22</td><td>0.67</td><td>10.82</td><td>20</td><td>25.46</td><td>0.67</td><td>13.49</td><td>20</td></tr></table>",
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"image_caption": [
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| 919 |
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"Figure 4: $4 \\times$ noisy super resolution comparison with $\\sigma _ { \\mathbf { y } } = 0 . 0 5$ . "
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"type": "text",
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"text": "Our code is available at https://github.com/bahjat-kawar/ddrm. ",
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"type": "text",
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"text": "5.2 Quantitative Experiments ",
|
| 955 |
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"text_level": 1,
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"bbox": [
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"type": "text",
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"text": "In order to quantify DDRM’s performance, we focus on the ImageNet dataset $2 5 6 \\times 2 5 6 )$ for its diversity. For each experiment, we report the average peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) [52] to measure faithfulness to the original image, and the kernel Inception distance (KID) [5], multiplied by $1 0 ^ { 3 }$ , to measure the resulting image quality. ",
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"type": "text",
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"text": "We compare DDRM (with 20 and 100 steps) with other unsupervised methods that work in reasonable time (requiring 1500 NFEs or less) and can operate on ImageNet. Namely, we compare with RED [40], DGP [38], and SNIPS [25]. The exact setup of each method is detailed in Appendix F. We used the same hyperparameters for noisy and noiseless versions of the same problem for DGP, RED, and SNIPS, as tuning them for each version would compromise their unsupervised nature. Nevertheless, the performance of baselines like RED with such a tuning does not surpass that of DDRM, as we show in Appendix F. In addition, we show upscaling by bicubic interpolation as a baseline for super-resolution, and the blurry image itself as a baseline for deblurring. OneNet [39] is not included in the comparisons as it is limited to images of size $6 4 \\times 6 4$ , and generalization to higher dimensions requires an improved network architecture. ",
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"type": "image",
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"img_path": "images/c29b6e70f49e93bec3f7d31839a85160b2b005e20acbf20e9617bb3c256f753b.jpg",
|
| 989 |
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"image_caption": [
|
| 990 |
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"Figure 5: $5 1 2 \\times 5 1 2$ ImageNet colorization. DDRM-CC produces various samples for multiple runs on the same input. "
|
| 991 |
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],
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| 992 |
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| 1001 |
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"img_path": "images/677dbd46569b00f85565e6a7598555e2ec90d1684b5b06b3a221943f2f16443d.jpg",
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| 1004 |
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"image_caption": [
|
| 1005 |
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"Figure 6: Results on $2 5 6 \\times 2 5 6$ USC-SIPI images using an ImageNet model. Blurred images have a noise of $\\sigma _ { \\mathbf { y } } = 0 . 0 1$ . "
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| 1006 |
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"type": "text",
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| 1018 |
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"text": "",
|
| 1019 |
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| 1028 |
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"type": "text",
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| 1029 |
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"text": "We evaluate all methods on the problems of $4 \\times$ super-resolution and deblurring, on one validation set image from each of the 1000 ImageNet classes, following [38]. Table 1 shows that DDRM outperforms all baseline methods, in all metrics, and on both problems with only 20 steps. The only exception to this is that SNIPS achieves better KID than DDRM in noiseless deblurring, but it requires $5 0 \\times$ more NFEs to do so. Note that the runtime of all the tested methods is perfectly linear with NFEs, with negligible differences in time per iteration. DGP and DDRM-CC use ground-truth class labels for the test images to aid in the restoration process, and thus have an unfair advantage. ",
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| 1030 |
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| 1039 |
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"type": "text",
|
| 1040 |
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"text": "DDRM’s appeal compared to previous methods becomes more substantial when significant noise is added to the measurements. Under this setting, DGP, RED, and SNIPS all fail to produce viable results, as evident in Table 2 and Figure 4. Since DDRM is fast, we also evaluate it on the entire ImageNet validation set in Appendix F. ",
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| 1041 |
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},
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| 1049 |
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{
|
| 1050 |
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"type": "text",
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| 1051 |
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"text": "5.3 Qualitative Experiments ",
|
| 1052 |
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"text_level": 1,
|
| 1053 |
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"bbox": [
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"type": "text",
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| 1063 |
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"text": "DDRM produces high quality reconstructions across all the tested datasets and problems, as can be seen in Figures 1 and 3, and in Appendix I. As it is a posterior sampling algorithm, DDRM can produce multiple outputs for the same input, as demonstrated in Figure 5. Moreover, the unconditional ImageNet diffusion models can be used to solve inverse problems on out-of-distribution images with general content. In Figure 6, we show DDRM successfully restoring $2 5 6 \\times 2 5 6$ images from USC-SIPI [53] that do not necessarily belong to any ImageNet class (more results in Appendix I). ",
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"type": "text",
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"text": "6 Conclusions ",
|
| 1075 |
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"text_level": 1,
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| 1076 |
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"type": "text",
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| 1086 |
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"text": "We have introduced DDRM, a general sampling-based linear inverse problem solver based on unconditional/class-conditional diffusion generative models as learned priors. Motivated by variational inference, DDRM only requires a few number of NFEs (e.g., 20) compared to other samplingbased baselines (e.g., 1000 for SNIPS) and achieves scalability in multiple useful scenarios, including denoising, super-resolution, deblurring, inpainting, and colorization. We demonstrate the empirical successes of DDRM on various problems and datasets, including general natural images outside the distribution of the observed training set. To our best knowledge, DDRM is the first unsupervised method that effectively and efficiently samples from the posterior distribution of inverse problems with significant noise, and can work on natural images with general content. ",
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| 1087 |
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"type": "text",
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"text": "",
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| 1098 |
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"type": "text",
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| 1108 |
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"text": "In terms of future work, apart from further optimizing the timestep and variance schedules, it would be interesting to investigate the following: (i) applying DDRM to non-linear inverse problems, $( i i )$ addressing scenarios where the degradation operator is unknown, and (iii) self-supervised training techniques inspired by DDRM as well as ones used in supervised techniques [41] that further improve performance of unsupervised models for image restoration. ",
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| 1109 |
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},
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| 1118 |
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"type": "text",
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| 1119 |
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"text": "Acknowledgements ",
|
| 1120 |
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"text_level": 1,
|
| 1121 |
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| 1130 |
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"type": "text",
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| 1131 |
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"text": "We thank Kristy Choi, Charlie Marx, and Avital Shafran for insightful discussions and feedback. This research was supported by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1-2145), AFOSR (FA9550-19-1-0024), ARO (W911NF-21-1-0125), Sloan Fellowship, Amazon AWS, Stanford Institute for Human-Centered Artificial Intelligence (HAI), Google Cloud, the Israel Science Foundation (ISF) under Grant 335/18, the Israeli Council For Higher Education - Planning & Budgeting Committee, and the Stephen A. Kreynes Fellowship. ",
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"type": "text",
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| 1142 |
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"text": "References ",
|
| 1143 |
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"text_level": 1,
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| 1144 |
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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? [Yes] In the future work paragraph in Section 6 \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We came to the conclusion that our paper does not have potential negative societal impacts. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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"text": "2. If you are including theoretical results... ",
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"text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] In the appendice ",
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|
| 1436 |
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|
| 1437 |
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"page_idx": 13
|
| 1438 |
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|
| 1439 |
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{
|
| 1440 |
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"type": "text",
|
| 1441 |
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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] In the appendices. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In both the paper and the appendices. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] We report results averaged over 1, 000 images in the main paper and 50, 000 images in the appendices. Such large numbers eliminate the need for error bars. \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] In the appendices. ",
|
| 1442 |
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| 1447 |
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|
| 1448 |
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"page_idx": 13
|
| 1449 |
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},
|
| 1450 |
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{
|
| 1451 |
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"type": "text",
|
| 1452 |
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"text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
|
| 1453 |
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"bbox": [
|
| 1454 |
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| 1456 |
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| 1457 |
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|
| 1458 |
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|
| 1459 |
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"page_idx": 13
|
| 1460 |
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|
| 1461 |
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{
|
| 1462 |
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"type": "text",
|
| 1463 |
+
"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [Yes] The licenses of previous works’ code and datasets will be included in our camera-ready code. \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code in the supplementary material. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Consent was given by the original authors in their work. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] The datasets we use are anonymized. ",
|
| 1464 |
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|
| 1465 |
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|
| 1466 |
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| 1467 |
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| 1468 |
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|
| 1469 |
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|
| 1470 |
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"page_idx": 13
|
| 1471 |
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},
|
| 1472 |
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{
|
| 1473 |
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"type": "text",
|
| 1474 |
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"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
|
| 1475 |
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"bbox": [
|
| 1476 |
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|
| 1477 |
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| 1478 |
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| 1479 |
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|
| 1480 |
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|
| 1481 |
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"page_idx": 13
|
| 1482 |
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},
|
| 1483 |
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{
|
| 1484 |
+
"type": "text",
|
| 1485 |
+
"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] ",
|
| 1486 |
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"bbox": [
|
| 1487 |
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|
| 1488 |
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| 1489 |
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| 1490 |
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| 1491 |
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|
| 1492 |
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"page_idx": 13
|
| 1493 |
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}
|
| 1494 |
+
]
|
parse/dev/kxXvopt9pWK/kxXvopt9pWK_model.json
ADDED
|
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parse/dev/rwE8SshAlxw/rwE8SshAlxw.md
ADDED
|
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|
| 1 |
+
# UNSUPERVISED DISCOVERY OF OBJECT RADIANCE FIELDS
|
| 2 |
+
|
| 3 |
+
Hong-Xing Yu Stanford University
|
| 4 |
+
|
| 5 |
+
Leonidas J. Guibas Stanford University
|
| 6 |
+
|
| 7 |
+
Jiajun Wu Stanford University
|
| 8 |
+
|
| 9 |
+
# ABSTRACT
|
| 10 |
+
|
| 11 |
+
We study the problem of inferring an object-centric scene representation from a single image, aiming to derive a representation that is learned without supervision, explains the image formation process, and captures the scene’s 3D nature. Most existing methods on scene decomposition lack one or more of these characteristics, due to the fundamental challenge in integrating powerful unsupervised inference schemes like deep networks with the complex 3D-to-2D image formation process. In this paper, we propose unsupervised discovery of Object Radiance Fields (uORF), integrating recent progresses in neural 3D scene representations and rendering with deep inference networks for unsupervised 3D scene decomposition. Trained on only multi-view RGB images, uORF learns to decompose complex scenes with diverse, textured background from a single image. We show that uORF enables novel tasks, such as scene segmentation and editing in 3D, and it performs well on these tasks and on novel view synthesis on three datasets\*.
|
| 12 |
+
|
| 13 |
+
# 1 INTRODUCTION
|
| 14 |
+
|
| 15 |
+
Building factorized, object-centric scene representations is a fundamental ability in human vision and a constant topic of interest in computer vision and machine learning. We identify that such representations should bear three characteristics: they should be learned without supervision or prior knowledge about object categories, and therefore applicable to environments where object categories are unknown; they should explain the image formation process, addressing questions like ‘what if the object is not there?’; they should be 3D-aware, capturing geometric and physical object properties for navigation, interaction, and manipulation.
|
| 16 |
+
|
| 17 |
+
For decades, researchers have attempted to solve the problems from various angles. Inspiring as they are, these methods each lack in one or more of the three aspects (Table 1). Computer vision research on unsupervised object discovery has achieved great success on deriving object segments from real images, but it doesn’t capture the image formation process, nor is it 3D-aware (Rubinstein et al., 2013; Zhu et al., 2012). Recent work on deep probabilistic inference for visual scene decomposition is unsupervised and generative (Burgess et al., 2019; Engelcke et al., 2019; Locatello et al., 2020), though most still formulate the problem as 2D segmentation and work on simple scenes of geometric primitives, ignoring the complex 3D nature of realistic visual scenes. A few recent papers on ‘scene de-rendering’ have attempted to reconstruct 3D, object-centric representations by leveraging the forward rendering procedure (Yao et al., 2018; Ost et al., 2021); they are however supervised, relying on annotations of specific object and scene categories, such as cars and road scenes.
|
| 18 |
+
|
| 19 |
+
The fundamental challenge that prevents these systems from acquiring all three desired properties is that the image formation process from 3D to 2D is complex and non-differentiable (e.g., due to occlusion). Thus, for a long time, it has been unclear how it may be integrated with powerful deep inference schemes. But most recently, progresses in neural rendering (Tewari et al., 2020) have demonstrated that their continuous, implicit representation works well with gradient-based inference models, such as deep networks. In particular, Neural Radiance Fields (NeRFs) (Mildenhall et al., 2020) recover a 3D scene from a set of RGB images via differentiable volume rendering. Such encouraging advances in generative modeling suggest a promising route for inferring 3D, generative, and object-centric scene representations without supervision.
|
| 20 |
+
|
| 21 |
+
In this paper, we propose unsupervised discovery of Object Radiance Fields (uORF), integrating conditional NeRFs as 3D object representations with deep inference networks for unsupervised 3D scene decomposition. uORF infers a set of object radiance fields and a background radiance field; thus, uORF represents a 3D scene as a composition of object radiance fields (Figure 1). During training, such radiance fields are neurally rendered in multiple views, with reconstruction losses in pixel space as training supervision; during testing, uORF infers the set of object radiance fields from a single image. Learning uORF does not require explicit supervision of 3D geometry or object segmentation, but only sparse multi-view RGB images of training scenes.
|
| 22 |
+
|
| 23 |
+
|
| 24 |
+
Figure 1: Illustration of unsupervised discovery of Object Radiance Fields. We aim to infer factorized object and background radiance fields from a single view, allowing reconstructing and editing of the scene.
|
| 25 |
+
|
| 26 |
+
The integration of NeRFs allows us to work with more realistic scenes with complex object shapes and diverse background environments, beyond simple scenes such as those in multi-dSprites (Greff et al., 2019) and CLEVR (Johnson et al., 2017), as considered by most current unsupervised scene decomposition methods. We further make two innovations to improve uORF’s performance. First, as background geometry and appearance can be quite different from foreground objects in 3D, we design uORF with explicit modeling of both components. This background-aware design not only facilitates learning on complex scenes, but also allows single-image scene editing including moving individual objects and changing background. Second, as volume rendering requires massive queries to render a single pixel for the recomposed scene, a practical challenge of learning uORF lies in the computational inefficiency. We tackle this issue by proposing a novel progressive coarse-to-fine training which improves representation quality while remaining affordable computational cost.
|
| 27 |
+
|
| 28 |
+
We evaluate uORF on factorized scene representation learning (e.g., segmentation in 3D) and scene generation (e.g., novel view synthesis, scene editing in 3D). Our evaluation is on three datasets with a gradually increasing complexity: first, CLEVR-like scenes with primitives foreground shapes; second, room scenes with complex chair shapes and textured backgrounds; third, more diverse room scenes with various foreground shapes and backgrounds. Our results show that uORF learns factorized representations that can segment 3D scenes into objects with fine shape details (e.g., thin chair legs) and backgrounds with well-recovered appearance details (e.g., irregular textures of a wooden floor).
|
| 29 |
+
|
| 30 |
+
In summary, our contributions are three-fold. First, we propose the problem of inferring an unsupervised, factorized, generative, and 3D-aware scene representation from a single image. Second, we introduce unsupervised discovery of Object Radiance Fields (uORF) that infers individual 3D object radiance fields from a single view for the proposed problem. Third, we demonstrate that uORF enables novel tasks such as scene segmentation and editing in 3D, and we show that it generalizes to novel scene arrangement and unseen combinations of object properties.
|
| 31 |
+
|
| 32 |
+
# 2 RELATED WORK
|
| 33 |
+
|
| 34 |
+
Co-segmentation and object discovery. Our work is closely related to traditional computer vision methods on object discovery, which aims to locate (visually similar) objects in a collection of images. These methods typically model objects as visual words and adopted methods from topic modeling to localize objects (Russell et al., 2006; Sivic et al., 2005; 2008), or cluster and group image patches (Grauman & Darrell, 2006; Joulin et al., 2010; Rubio et al., 2012; Vicente et al., 2011; Rubinstein et al., 2013; Cho et al., 2015). Recent works have integrated the clustering-based strategy with deep learning (Li et al., 2019; Vo et al., 2020). Nevertheless, they do not explain image formation process nor are they 3D-aware.
|
| 35 |
+
|
| 36 |
+
<table><tr><td>Approach</td><td>Unsup.Gen. 3D</td><td></td><td></td></tr><tr><td>Co-segmentation</td><td></td><td></td><td>xx</td></tr><tr><td>Deep prob. infer.</td><td></td><td></td><td></td></tr><tr><td>Scene"de-render"</td><td></td><td></td><td></td></tr><tr><td>Ours</td><td></td><td></td><td></td></tr></table>
|
| 37 |
+
|
| 38 |
+
Table 1: Comparison to existing methods.
|
| 39 |
+
|
| 40 |
+
|
| 41 |
+
Figure 2: Overview. I. Our model learns to infer a set of latents in a single forward pass. II. Each object/background radiance field consists of a latent and a shared conditional NeRF. III. During training, we recompose the scene and re-render images for supervision. We train our model on different scenes. At test time, we use a single image of an unseen scene for reconstruction or editing.
|
| 42 |
+
|
| 43 |
+
Deep probabilistic inference for scene decomposition. Our method is also closely related to deep probabilistic inference for scene decomposition. Most works formulate the problem as compositional generative modeling, where a visual scene is represented by a set of latent codes that either correspond to localized object-centric patches (Eslami et al., 2016; Crawford & Pineau, 2019; Kosiorek et al., 2018; Lin et al., 2020; Jiang et al., 2019) or scene mixture components (Burgess et al., 2019; Greff et al., 2019; 2016; 2017; Engelcke et al., 2019). Recently, Locatello et al. (2020) proposed the Slot Attention module to simplify the inference by a slot-based encoder. Besides these inference models, Monnier et al. (2021) formulated scene decomposition as layered image decomposition and demonstrated it on real images. However, these methods do not account for the 3D nature of scenes.
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A few methods have recently been proposed for unsupervised 3D scene decomposition. Elich et al. (2020) infer object shapes from a single scene image, but they require pretraining on groundtruth shapes. Chen et al. (2020) extend Generative Query Network (Eslami et al., 2018) to decompose 3D scenes, but they require multi-view images during inference. The closest to our work is a concurrent work by Stelzner et al. (2021) which also utilizes a slot-based encoder and NeRFs as 3D representations. However, Stelzner et al. (2021) relies on groundtruth multi-view dense depth in addition to images in training. Moreover, we explicitly model the separation of objects and background to address various complex shapes and textured backgrounds, while they only demonstrate scenes with a single textureless background.
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Scene de-rendering. A few recent works have shown reconstructing 3D object-centric representations by incorporating forward image rendering process (Wu et al., 2017; Yao et al., 2018; Kundu et al., 2018; Ost et al., 2021). Yao et al. (2018) de-render an image into semantic segments and geometric object attributes, which enable 3D scene manipulation. Most recently, Ost et al. (2021) propose Neural Scene Graph to represent dynamic scenes into a scene graph where each node encodes object-centric information. However, these methods rely on manual annotations of specific objects (such as cars) and scene categories (such as street scenes).
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Neural scene representations and rendering. Our method is related to recent progresses in neural continuous scene representations (Sitzmann et al., 2019) and neural rendering (Tewari et al., 2020). Neural scene representations parameterize 3D scenes with a deep network (Sitzmann et al., 2019). Combined with differentiable neural rendering techniques (Kato et al., 2020; Tewari et al., 2020), they can be learned from only 2D images (Niemeyer et al., 2020). In particular, Neural Radiance Fields (NeRFs) (Mildenhall et al., 2020) have shown impressive novel view synthesis. Related follow-up works include those that infer NeRFs from a single image (Yu et al., 2020; Kosiorek et al., 2021; Jang & Agapito, 2021) and those that incorporate NeRFs into generative models (Schwarz et al., 2020; Niemeyer & Geiger, 2021; Chan et al., 2020). Different from these works which cope with single objects or holistic scenes, we learn object NeRFs via decomposing a multi-object scene without segmentation annotations. GIRAFFE (Niemeyer & Geiger, 2020) generates object NeRFs and thus compose 3D scenes in an adversarial framework. However, it targets at unconditional generation and cannot tackle inference (see Appendix E), while we focus on single-image inference of multi-object scenes. Thus, we address a fundamentally different problem compared to GIRAFFE.
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# 3 APPROACH
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Our goal is to infer from a single image a set of object-centric 3D representations to generate the underlying 3D scene. We show an illustration in Figure 2. Our object representation is a conditional object radiance field. Thus, we learn to infer object-centric latents from a single image (Figure 2- I). The inferred latents are used to condition a network to yield the 3D object and background radiance fields (Figure 2-II), forming our 3D-aware, generative and factorized scene representation. In training, we compose all object and background radiance fields and render the recomposed scene from multiple views. We obtain supervision by comparing rendered images to reference images (Figure 2-III) without needing 3D geometry or segmentation annotations. We describe each of our model components in the following and leave implementation details in Appendix B.
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# 3.1 OBJECT-CENTRIC LATENT INFERENCE
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Our goal is to infer latent object-centric representations from a single input image. We assume that an underlying 3D scene is composed of a background environment and no more than $K$ foreground objects. Thus, the output of our object-centric latent inference process is a latent $\mathbf { z } ^ { b }$ for background and a set of latents $\{ \mathbf { z } _ { i } ^ { f } \} _ { i = 1 } ^ { K }$ for foreground objects (empty objects are allowed). To encourage unsupervised object-wise factorization, we adopt a slot-based formulation (Locatello et al., 2020). The assumption in this formulation is that objects should share a common prior latent space. The main idea include three steps. The first step is to sample all object latents (i.e., slots) from the same prior
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Figure 3: Our object-centric latent inference. The attention binds each object’s features to a slot.
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distribution (background is a special object) to encourage representational uniformity across all slots (“sampling”). Then each slot is bound to an object region via an attention module (“binding”). In the last step each slot gets updated by the bound object features to specialize for that object (“updating”). Locatello et al. (2020) have demonstrated success on segmenting 2D images.
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However, in 3D scenes, the geometry and appearance of the background are highly different from those of foreground objects. Modeling them indistinguishably often leads to object representations entangled with blurry background segments (Burgess et al., 2019; Locatello et al., 2020), which impedes applications such as scene editing and re-composition. Thus, we propose a backgroundaware slot attention module (Figure 3) that separately models objects and environment to better capture the compositional structure of 3D scenes.
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Background-aware slot attention for sampling and binding. In the sampling step, we model the latent prior distribution of foreground objects by a Gaussian with learnable mean and variance, i.e., we sample $\mathbf { s } \mathbf { 1 } \mathsf { o t } \mathbf { s } ^ { f } \sim \mathcal { N } ^ { f } ( \mu ^ { f } , \mathsf { d i a g } ( \sigma ^ { f } ) \bar { ) } \in \mathbb { R } ^ { \bar { K } \times D }$ for $K$ objects. For latent prior of backgrounds, we learn another Gaussian and sample a single slot from it, i.e., $\mathbf { s } \mathbf { 1 } \mathbf { o t } ^ { b } \sim \mathcal { N } ^ { b } ( \dot { \mu ^ { b } } , \mathbf { d i a g } ( \sigma ^ { b } ) ) \stackrel { \smile } { \in } \mathbb { R } ^ { 1 \times D }$ .
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To bind slots to image features, we let all the slots to compete for explaining the input image representation. To do this, we flatten the convolutional feature map (we include details about convolutional encoder in Appendix B.1) into a set of $N$ input feature vectors, feat $\in \mathbb { R } ^ { N \times D }$ . The slot competition is modeled by a key-query attention (Bahdanau et al., 2014):
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$$
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\mathrm { a t } { \bf u } _ { i , j } : = \frac { \exp ( M _ { i , j } ) } { \sum _ { l } \exp ( M _ { i , l } ) } , \quad \mathrm { w h e r e } \quad M : = \frac { 1 } { \sqrt { D } } k ( \mathbf { f e a t } ) \cdot \left[ \boldsymbol { q } ^ { b } ( \mathbf { s } \mathbf { l o t } \boldsymbol { \bf s } ^ { b } ) \right] ^ { T } \in \mathbb { R } ^ { N \times ( K + 1 ) } .
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$$
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Here $k$ and $q ^ { b } / q ^ { f }$ are learnable linear mappings √ $\mathbb { R } ^ { D D }$ for computing dot-product similarity (Luong et al., 2015), and $\sqrt { D }$ is a fixed softmax temperature (Vaswani et al., 2017). One can see this process as a soft K-means, where $\tt a t t r a _ { i }$ softly assigns a feature $i$ to the slots (centroids). The background slot is expected to capture the modality of background features and bind all of them, allowing foreground slots to focus only on the objects without explaining background segments (Figure 3). Besides the representation design, we further encourage disentanglement between background and foregrounds by two additional designs: (1) We represent and query foreground/background (during the neural rendering process) in different coordinate frames. (2) To discourage object slots from fitting background, we impose a locality constraint in early training. We set a foreground box and enforce that every foreground query point outside the box has zero density. We include details in Appendix B.2.
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Updating slots to infer latents. With the attention weights, we form the update signal by aggregating input values via a weighted mean pooling updatesb := W bT · vb(feat) ∈ R1×D, where $\bar { W } _ { i , 1 } ^ { b } : = \mathsf { a t t n } _ { i , 1 } / ( \sum _ { l = 1 } ^ { N } \bar { \mathsf { a t t n } _ { l , 1 } } )$ , and up $\mathsf { i a t e s } ^ { f } : = W ^ { f T } \cdot v ^ { f } ( \mathsf { f e a t } ) \in \mathbb { R } ^ { K \times D }$ , where $W _ { i , j } ^ { f } ~ : = ~ \mathsf { a t t n } _ { i , j + 1 } / ( \sum _ { l = 1 } ^ { N } \mathsf { a t t n } _ { l , j + 1 } ) .$ Slots are then updated using the update signals via a learnable rule parameterized by a Gated Recurrent Unit (GRU) (Cho et al., 2014), so that $\mathbf { s } \mathbf { 1 } { \mathsf { o t s } } ^ { f } \gets \mathsf { G R U } ^ { f } \big ( \mathbf { s } \mathbf { 1 } { \mathsf { o t s } } ^ { f } , \mathbf { u p d a t } \bar { \mathsf { e s } } ^ { f } \big )$ ) and $\mathbf { s } 1 0 \mathbf { t } ^ { b } \gets \mathsf { G R U } ^ { b } ( \mathsf { s } 1 0 \mathbf { t } ^ { b } , \mathsf { u p d a t e s } ^ { b } )$ ). We repeat the attention computation and updating for 3 iterations, and output all the slots as the final latents $\mathbf { z } ^ { b }$ and $\{ \mathbf { z } _ { i } ^ { f } \} _ { i = 1 } ^ { K }$ . We show pseudo-code of our background-aware slot attention in Appendix (Alg. 1).
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# 3.2 COMPOSITIONAL NEURAL RENDERING
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We represent a 3D object as a conditional neural radiance field. A NeRF is a continuous mapping $g : ( \bar { \mathbf { x } , \mathbf { d } } ) ( \mathbf { c } , \sigma )$ from spatial location $\mathbf { x }$ and viewing direction d to emitted color c and volume density $\sigma$ used for volume rendering (Max, 1995). This mapping is parameterized by an MLP network. We adopt a conditional NeRF $g ( \mathbf { x } , \mathbf { d } | \mathbf { z } )$ for our inference scheme (detailed in Appendix B.3). The MLP parameters are shared across all objects $g ^ { f } ( \mathbf { x } , \mathbf { d } | \mathbf { z } _ { i } ^ { f } )$ , but not the background $g ^ { b } ( \mathbf { x } , \mathbf { d } | \mathbf { z } ^ { b } )$ due to its distinct geometry and appearance distribution.
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To compose individual objects and background into the holistic scmodel and use density-weighted mean to combine all components: $\begin{array} { r } { \bar { \boldsymbol { \sigma } } = \sum _ { i = 0 } ^ { K } w _ { i } \sigma _ { i } , \bar { \mathbf { c } } = \sum _ { i = 0 } ^ { K } w _ { i } \mathbf { c } _ { i } } \end{array}$ , where $w _ { i } = \sigma _ { i } / \sum _ { j = 0 } ^ { K } \sigma _ { j }$ . Here $\bar { \sigma }$ and c¯ are the combined density and color, respectively. The color $C ( \mathbf { r } )$ of a camera ray $\mathbf { r } ( t ) = \mathbf { o } + \mathbf { d } ( t )$ is then estimated via numerical integration of volume rendering, using $S$ discrete combined samples along a ray (Max, 1995): $\begin{array} { r } { C ( \mathbf { r } ) = \bar { \sum _ { i = 1 } ^ { S } } T _ { i } [ 1 - \exp ( - \bar { \sigma } _ { i } \delta _ { i } ) ] \bar { \mathbf { c } } _ { i } } \end{array}$ where $\begin{array} { r } { T _ { i } = \exp \left( - \sum _ { j = 1 } ^ { i - 1 } \bar { \sigma } _ { j } \delta _ { j } \right) } \end{array}$ . Here $\delta _ { j }$ is the distance between adjacent samples along a ray.
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# 3.3 MODEL LEARNING
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Loss functions. As shown in Figure 2, during training we input a single image of a scene, infer object and background radiance fields, render multiple views from the recomposed scene, and compare them to reference images for loss computation. We train our model across multiple scenes. Our training loss function comprises of a reconstruction loss, a perceptual loss, and an adversarial loss: ${ \mathcal { L } } = { \mathcal { L } } _ { \mathrm { { r e c o n } } } + \lambda _ { \mathrm { { p e r c e p t } } } { \mathcal { L } } _ { \mathrm { { p e r c e p t } } } - \lambda _ { \mathrm { { a d v } } } { \mathcal { L } } _ { \mathrm { { a d v } } }$ , where $\lambda$ are weights. The reconstruction loss is $\mathcal { L } _ { \mathrm { r e c o n } } = \| \pmb { I } - \hat { \pmb { I } } \| ^ { 2 }$ , where $\pmb { I }$ and $\hat { I }$ denote the reference image and rendered image, respectively.
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Since we estimate 3D radiance fields from a single view, there can be uncertainties about the appearance from other views (e.g., the back view). For example, regarding visual appearance of objects, inaccurate global lighting estimation leads to uncertainties in brightness and shadows from occluded views even if the object shapes can be well estimated. To address this, we incorporate a perceptual loss (Johnson et al., 2016) which is tolerant to mild appearance changes. The perceptual loss is defined by $\| \mathcal { L } _ { \mathrm { p e r c e p t } } = p ( I ) - p ( \hat { I } ) \| ^ { 2 }$ where $p$ is a deep feature extractor (See Appendix B.4).
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In addition to appearance, there can be even higher uncertainties in estimating object shapes from a single view, which is a multi-modal distribution. In this case, the unimodal reconstruction loss leads to blurry results (“mean shape”). We mitigate this issue by adding an adversarial loss which can deal with multi-modal distributions:
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$$
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\begin{array} { r } { \mathcal { L } _ { \mathrm { a d v } } = \mathbb { E } [ f ( D ( \hat { \cal I } ) ) ] + \mathbb { E } [ f ( - D ( { \cal I } ) ) + \lambda _ { R } \| \nabla D ( { \cal I } ) \| ^ { 2 } ] , \quad \mathrm { w h e r e } \quad f ( t ) = - \log ( 1 + \exp ( - t ) ) . } \end{array}
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$$
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Here we adopt the R1 regularization (Mescheder et al., 2018) to stabilize training. $D$ denotes a discriminator to distinguish rendered images $\hat { I }$ and reference images $\pmb { I }$ . We iterate between training the discriminator by minimizing ${ \mathcal { L } } _ { \mathrm { a d v } }$ and training our inference model (Figure 2) by minimizing $\mathcal { L }$ .
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Coarse-to-fine Progressive Training. A practical challenge in training compositional NeRFs lies in the computational cost of neural volume rendering, as it requires massive queries to render a single pixel. While there have been attempts on fast inference (Liu et al., 2020; Rebain et al., 2020; Neff et al., 2021; Garbin et al., 2021; Reiser et al., 2021; Yu et al., 2021), high space complexity in training remains a challenge. Further, because our perceptual and adversarial losses depend on image patches, the system has to render a large enough patch (instead of a single pixel) at the same time, which further increases its space demand.
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To allow training on a higher resolution, we propose a coarse-to-fine progressive training. In a coarse training stage, we bilinearly downsample reference images to a low resolution (e.g., $6 4 \times 6 4 )$ , and train uORF on these downsampled images. Although the coarsely trained model can already decompose the 3D scenes and recover rough object radiance fields, fine details (e.g., thin legs of chairs) might be missing. Thus, in a following fine training stage, we replace the low-resolution reference images with image patches randomly cropped from high-resolution images (Figure 2-III), and render the correspondingly located patches from our recomposed scene radiance fields to compute the loss. We include further training details in Appendix B.4.
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# 4 EXPERIMENTS
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We evaluate uORF on both scene representation (via scene segmentation in 3D) and scene generation (via novel view synthesis and scene editing) on three datasets.
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Data. We build three synthetic datasets with gradually increasing complexity. For each scene in the dataset, we point the camera to the scene center and render four images with a randomly chosen azimuth angle and a fixed elevation angle. We describe more details in Appendix C.1.
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CLEVR-567. The first dataset includes scenes of 5–7 CLEVR objects (Johnson et al., 2017), with a random position and orientation on a clean background. Foreground object shapes include three geometric primitives (i.e., cubes, spheres and cylinders). Since there is intrinsic ambiguity in estimating specularity from a single image, we use only the largely diffuse “Rubber” material. There are 1,000 scenes for training and 500 for testing.
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Room-Chair. The second dataset includes scenes of 3 to 4 chairs of the same shape in a room with three different textured backgrounds. There are 1,000 scenes for training and 500 for testing.
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Room-Diverse. The third dataset includes scenes of diverse foreground object shapes and background appearances. Each scene includes 4 different chairs, whose shape is randomly sampled from 1,200 ShapeNet chair shapes (Chang et al., 2015), and the background is sampled from 50 floor textures from the web. There are 5,000 scenes for training and 500 for testing.
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# 4.1 SCENE SEGMENTATION IN 3D
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We first evaluate uORF’s factorized 3D scene representations via scene segmentation in 3D.
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Baselines. Because there is no previous work focusing on the same setting as uORF, we compare to a 2D state-of-the-art scene decomposition model Slot Attention (Locatello et al., 2020) for unsupervised scene segmentation wherever possible (detailed in Appendix C.2). In addition, we compare to two ablated versions of uORF. First, we remove our background-aware modeling but keep the same number of slots. Second, we ablate our progressive training such that the training procedure only contains the coarse training stage. We refer to ablated models as “uORF (w/o background)” and “uORF (w/o prog. train.)”, respectively.
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Metrics. We adopt the widely-used Adjusted Rand Index (ARI) as our metric. To evaluate scene segmentation in 3D, we consider three kinds of ARIs: (1) For direct comparison to 2D methods, we compute ARI on reconstructed images. (2) To reflect the 3D nature, we also compute ARI on synthesized novel views, denoted as “NV-ARI”. Note that each scene includes 4 views, and only one is used as input, and the other three are treated as novel views for this metric. (3) In line with previous 2D methods, we also report foreground ARI $\mathrm { F g }$ -ARI), computed only on foreground regions indicated by groundtruth masks. Yet, we note that $\mathrm { F g }$ -ARI cannot fully reflect the segmentation quality, because background segmentation is completely ignored.
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Results. We volume-render a density map $\mathbf { d } ^ { i }$ for each slot . The segmentation label for each pixel $s _ { p }$ is given by $s _ { p } = \arg \operatorname* { m a x } _ { i = 1 } ^ { K + 1 } { \bf d } _ { p } ^ { i }$ . We show results on Table 2 and Figure 4 (more in Appendix D). For all segmentation metrics, we show mean and standard deviation for three runs. uORF outperforms all methods in terms of ARI and NV-ARI. From Figure 4, it is clear that uORF is able to discover the 3D objects from a single image. These results validate that uORF can learn well-factorized 3D object-centric scene representations. Also notice that uORF yields better ARI even in input views compared to 2D slot attention. This is likely due to our background-aware design, as our ablated model “uORF w/o background” has shown similar input-view results compared to slot attention (e.g., see 3rd and 4th columns in Figure 4).
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# 4.2 NOVEL VIEW SYNTHESIS
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We then show that uORF is 3D-aware and generative via evaluation on novel view synthesis.
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Figure 4: Examples on scene segmentation in 3D. Novel view images are for reference but not input.
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<table><tr><td rowspan="5">Models</td><td colspan="3">CLEVR-567</td><td colspan="3">Room-Chair</td><td colspan="3">Room-Diverse</td></tr><tr><td>3D metric</td><td colspan="2">2D metric</td><td>3D metric</td><td colspan="2">2D metric</td><td>3D metric</td><td colspan="2">2D metric</td></tr><tr><td>NV-ARI↑</td><td>ARI↑</td><td>Fg-ARI个</td><td>NV-ARI个</td><td>ARI个</td><td>Fg-ARI↑</td><td>NV-ARI↑</td><td>ARI个</td><td>Fg-ARI↑</td></tr><tr><td>Slot Attention</td><td>N/A</td><td>3.5±0.7</td><td>93.2±1.5</td><td>N/A</td><td>38.4±18.4</td><td>40.2±4.5</td><td>N/A</td><td>17.4±11.3</td><td>43.8±11.7</td></tr><tr><td>uORF(w/o background)</td><td>10.5±3.6</td><td>11.7±4.6</td><td>86.4±2.8</td><td>40.4±9.2</td><td>42.3±10.6</td><td>93.3±1.9</td><td>21.0±8.1</td><td>24.0±9.9</td><td>78.9±3.1</td></tr><tr><td>uORF(w/o prog. train.)</td><td>81.1±0.7</td><td>83.7±0.8</td><td>84.2±0.5</td><td>62.3±2.5</td><td>65.4±2.6</td><td>81.0±3.0</td><td>53.8±1.4</td><td>63.7±1.7</td><td>66.9±4.1</td></tr><tr><td>uORF(ours)</td><td>83.8±0.3</td><td>86.3±0.1</td><td>87.4±0.8</td><td>74.3±1.9</td><td>78.8±2.6</td><td>88.8±2.7</td><td>56.9±0.2</td><td>65.6±1.0</td><td>67.9±1.7</td></tr></table>
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Table 2: Scene segmentation results. “NV-ARI” refers to ARI evaluated on novel views. “Fg-ARI” refers to ARI evaluated with only foreground pixels. Slot Attention (Locatello et al., 2020) is a state-of-the-art 2D method.
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Setup. For each test scene, we randomly pick one image as input and the remaining three images as groundtruth for novel view synthesis. As Slot Attention is purely in 2D and does not support novel view synthesis, we compare to a conditional NeRF (Mildenhall et al., 2020), equipped with a convolutional encoder similar to uORF, termed as “NeRF-AE” (see Appendix C.2). For fair comparison, we increase the latent dimension for NeRF-AE to guarantee approximately the same computational cost, and we use the same training strategy and losses as uORF. Thus, NeRF-AE can also be seen as a monolithic alternative model to uORF. We also compare with the ablated models, “uORF (w/o background)” and “uORF (w/o prog. train.)”. We use the perceptual metric LPIPS (Zhang et al., 2018), together with SSIM (Wang et al., 2004) and PSNR, as our evaluation metrics.
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Results. Quantitative results are in Table 3 and qualitative results are in Figure 5 (more in Appendix D). Quantitatively, uORF outperforms all compared methods on all metrics. From the qualitative comparison in Figure 5, we highlight three advantages of uORF. First, compared with NeRF-AE, which has a monolithic latent structure for the entire scene, uORF better preserves the features of each object: for example, see how NeRF-AE fuses object colors in the first two rows, while uORF does not. This shows the advantage of factorized scene representations to structurally describe a visual scene. Second, compared with uORF (w/o background), one can clearly see how our background-aware modeling helps recovering background appearances: uORF can accurately recover background appearance of the Room-Chair example, while uORF (w/o background) does not. It also facilitates learning on complex scenes with diverse, textured background: uORF can learn to roughly recover object shapes in the Room-Diverse example. Third, compared with uORF (w/o prog. train.), we highlight that the fine training on image patches indeed improves both visual quality and representation quality: the full uORF tries to recover sharp edges of cubes, while uORF (w/o prog. train.) cannot distinguish cube from sphere.
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Figure 5: Qualitative results on scene decomposition and novel view synthesis. Within every two rows, the first is reconstruction and the second is a novel view.
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<table><tr><td rowspan="2">Models</td><td colspan="3">CLEVR-567</td><td colspan="3">Room-Chair</td><td colspan="3">Room-Diverse</td></tr><tr><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td></tr><tr><td>NeRF-AE</td><td>0.1288</td><td>0.8658</td><td>27.16</td><td>0.1166</td><td>0.8265</td><td>28.13</td><td>0.2458</td><td>0.6688</td><td>24.80</td></tr><tr><td>uORF (w/o background)</td><td>0.0919</td><td>0.8924</td><td>28.93</td><td>0.1671</td><td>0.7852</td><td>27.86</td><td>0.2231</td><td>0.6924</td><td>25.90</td></tr><tr><td>uORF(w/o prog. train.)</td><td>0.1044</td><td>0.8894</td><td>28.84</td><td>0.1573</td><td>0.8287</td><td>28.33</td><td>0.2123</td><td>0.6760</td><td>25.19</td></tr><tr><td>uORF (ours)</td><td>0.0859</td><td>0.8971</td><td>29.28</td><td>0.0821</td><td>0.8722</td><td>29.60</td><td>0.1729</td><td>0.7094</td><td>25.96</td></tr></table>
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Table 3: Comparison on novel view synthesis from a single image.
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Overall, the novel view synthesis results suggest that uORF can learn to represent 3D scenes with reasonable fidelity, even with the presence of complex foreground object shapes, such as chairs and different textured backgrounds.
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# 4.3 SCENE DESIGN AND EDITING IN 3D
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Being object-centric and 3D-aware, uORF is able to edit 3D scene radiance fields inferred from a single view, and generate novel scene images.
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Setup. We test uORF’s ability to edit scenes and synthesize novel images on the Room-Chair dataset. We consider both moving foreground objects and changing background appearance. For object moving, we randomly pick one object in a test scene and move it to a random position. We render 4 images for each of the 500 test scenes. For background changing, we replace the current background texture to a different one and also render 4 images for evaluation. To indicate the new background, we re-pick and re-put foreground objects such that the resultant background indicator image is different from the groundtruth image.
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For uORF and Slot Attention (Locatello et al., 2020), we use groundtruth masks of the input view only for ease of evaluation. We determine which slot to move by picking the one with largest mask IoU. For NeRF-AE (Mildenhall et al., 2020) to do editing, we back-project the masks to frustums to determine the 3D regions to be moved/replaced. We use LPIPS, SSIM, and PSNR as our metrics.
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Results. We show results in Table 4 and Figure 6 (more in Appendix D). Again, uORF outperforms all compared methods on all metrics. As Figure 6 depicts, images synthesized by uORF show least artifacts and highest quality and fidelity.
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Table 4: Comparison on scene editing.
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<table><tr><td rowspan="2">Models</td><td colspan="3">Moving objects</td><td colspan="3">Changing background</td></tr><tr><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR个</td></tr><tr><td>NeRF-AE</td><td>0.2451</td><td>0.7284</td><td>23.18</td><td>0.2185</td><td>0.7132</td><td>25.42</td></tr><tr><td>Slot Attention</td><td>0.3941</td><td>0.7134</td><td>23.06</td><td>0.3689</td><td>0.7283</td><td>23.94</td></tr><tr><td>uORF(w/o background)</td><td>0.2206</td><td>0.7448</td><td>24.55</td><td>0.1879</td><td>0.7719</td><td>26.68</td></tr><tr><td>uORF(w/o prog.train.)</td><td>0.1583</td><td>0.8313</td><td>28.19</td><td>0.1586</td><td>0.8306</td><td>28.27</td></tr><tr><td>uORF (ours)</td><td>0.0855</td><td>0.8711</td><td>29.26</td><td>0.0822</td><td>0.8729</td><td>29.53</td></tr></table>
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# 4.4 GENERALIZATION AND ANALYSIS
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Finally we explore the generalization ability of uORF. We consider generalization on unseen, challenging spatial arrangement of objects, as well as generalization on unseen object appearances.
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Figure 6: Qualitative results on single-image 3D scene manipulation. The first two rows are for moving object and the second two rows are for changing background.
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<table><tr><td>Models</td><td>NV-ARI个</td><td>ARI个</td></tr><tr><td>Slot Attention</td><td>N/A</td><td>2.2±0.6</td></tr><tr><td>uORF (ours)</td><td>85.0±0.3</td><td>87.4±0.4</td></tr><tr><td>uORF (oracle)</td><td>85.5±0.3</td><td>87.5±0.3</td></tr></table>
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<table><tr><td>Loss functions</td><td>ARI↑</td><td>LPIPS↓</td></tr><tr><td>Rec.</td><td>59.1±0.5</td><td>0.3610</td></tr><tr><td>Rec.+Percept.</td><td>65.2±0.8</td><td>0.2156</td></tr><tr><td>Rec.+ Adv.</td><td>60.4±2.2</td><td>0.2288</td></tr><tr><td>Rec.+ Percept. + Adv.</td><td>65.6±1.0</td><td>0.1729</td></tr></table>
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Table 5: Generalization to novel Table 6: Generalization to unseen Table 7: Ablation study for losses on the challenging spatial arrangements. combinations of color and shape. Room-Diverse dataset.
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<table><tr><td>Models</td><td>ARI个</td><td>LPIPS↓</td></tr><tr><td>Slot Attention</td><td>5.7±0.3</td><td>N/A</td></tr><tr><td>NeRF-AE</td><td>N/A</td><td>0.2201</td></tr><tr><td>uORF (ours)</td><td>83.2±0.6</td><td>0.1540</td></tr></table>
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Generalizing to challenging spatial arrangements. We build a new test dataset, packed-CLEVR11, where each scene has 11 objects that are closely packed into a cluster. Therefore, each scene bears an unseen number of objects in an unseen challenging arrangement. We test models trained on CLEVR-567, report results in Table 5 and Appendix Figure 19. Despite uORF never sees such object arrangements, it still achieves a reasonable performance and outperforms baselines.
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Generalizing to new combination of shape and color. For unseen object appearances, we consider generalization in a systematic way such that the model can deal with unseen combination of object color and shape. Thus, we build a new training set similar to CLEVR-567, but we remove red cylinders and blue spheres from the object candidate pool. Then we test trained models on another dataset with only red cylinders and blue spheres in the candidate pool. We show results in Table 6 and examples in Appendix Figure 20. We see that although uORF has never seen any of the test set objects, it achieves similar results to the one trained on a normal CLEVR-567 dataset (denoted as “uORF (oracle)”). This suggests uORF’s ability for systematic generalization to unseen combinations of object color and shape. We further validate generalization to unseen object shapes in Appendix D.
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Evaluating loss functions. uORF uses perceptual and adversarial losses to combat intrinsic uncertainties in single-image inference of 3D representations. We show ablation results on novel view synthesis in Table 7 and Appendix Figure 18. Both losses significantly improve image quality.
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# 5 CONCLUSION
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In this work, we propose unsupervised discovery of Object Radiance Fields (uORF), which learns to infer object-centric 3D radiance fields from a single image of complex multi-object scenes. We demonstrate uORF’s ability on scene segmentation and scene generation in 3D. Our positive results suggest a promising direction to integrate neural rendering into deep probabilistic inference scheme, allowing learning factorized 3D object-centric scene representations from only RGB images.
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# ACKNOWLEDGMENTS
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This work was in part supported by Qualcomm Innovation Fellowship (QIF), Stanford Institute for Human-Centered AI (HAI), Stanford Center for Integrated Facility Engineering (CIFE), Toyota Research Institute, a Vannevar Bush faculty fellowship, Amazon, Autodesk, Google, and Bosch.
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# REPRODUCIBILITY STATEMENT
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To ensure reproducibility of our work, we have provided the training and test code repository†, together with all three synthetic datasets, and pre-trained models on all three datasets. We have also provided a detailed instruction on using our code as well as training on new datasets. In Appendix B, we describe details for re-implementing our work.
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# ETHICS STATEMENT
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Learning object-centric scene representations is a long-standing topic in vision and it finds various applications in downstream tasks. We represent a 3D scene as a composition of simple radiance fields, which only models object appearances and entangles their physical properties that may be crucial to downstream tasks in a non-interpretable way. However, we envision that careful designs in more structured 3D object representations for specific downstream applications could help improve transparency and human interpretability in model prediction and behavior, allowing both better performances and secure, fair usage. In our code release, we will explicitly specify allowable uses of our system with appropriate licenses. We will use techniques such as watermarking to identify and label visual contents generated by our system.
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# A SUPPLEMENTARY MATERIAL OVERVIEW
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In the following supplementary document, we first provide implementation details on unsupervised discovery of Object Radiance Fields in Section B. We then describe details on datasets and baseline architectures in Section C. We show additional results in in Section D, including results on generalization to unseen object shapes, a demonstration on real photos, an analysis on the sensitivity to slot initialization, and additional qualitative results on all experiments of the main paper and failure cases. In Section E, we show comparison to GIRAFFE (Niemeyer & Geiger, 2020) to demonstrate that it focus on a fundamentally different problem (unconditional generation) than our work (conditional inference). All mathematical and algorithmic notations are the same as those in the main manuscript.
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In the supplementary video, we provide an overview of our paper.
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# B IMPLEMENTATION
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Here, we provide implementation details of our uORF model.
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B.1 OBJECT-CENTRIC LATENT INFERENCE
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We show a pseudo code of inferring object-centric latents with the background-aware slot attention in Algorithm 1.
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Convolutional feature extraction. The convolutional net extracts features from the input image for updating the latent slots. Our convolutional encoder is a simple U-net. We show our encoder architecture in Table 8 and Table 9. Since we want the model to generalize to decompose unseen images, it is natural to represent foreground objects position and pose in the viewer coordinate system. As identified in previous studies (Tatarchenko et al., 2019), this facilitates the learning of 3D object position and helps generalization. In order for the object-centric representations to include such information in the viewer coordinate system, we can inform the encoder of position information by feeding pixel coordinates and viewer-space ray directions as additional input channels. In our experiments we assume fixed camera focal length. In this case, the ray direction does not provide additional information to the pixel coordinates, and thus we only feed pixel coordinates as input channels in addition to the input RGB image. Each of the $X Y$ pixel coordinates is normalized to $[ - 1 , 1 ]$ in both directions, leading to 4 additional channels to RGB.
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B.2 COORDINATE SPACE AND LOCALITY CONSTRAINT FOR BETTER FORE-/BACK-GROUND DISENTANGLEMENT
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Coordinate space. We represent foreground objects in the viewer space. Regarding background environment, we represent it in the world coordinate space for two reasons. Firstly, since it is difficult to estimate full geometry from a single view (e.g., the geometry behind the camera), our model assumes a similar background geometry across scenes and aggregates information about background geometry from multiple sparse views. Representing background in a fixed world space facilitates this aggregation process and empirically leads to better performance. We show a quantitative comparison in Table 10, Table 11 and a visual comparison in Figure 7. Modeling the background in world space provides more details than modeling it in viewer space. Incorporating multi-view images as inference input might relax this assumption (Yu et al., 2020), but we leave it as future exploration.
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Table 8: Encoder architecture for the CLEVR-567 dataset and the Room-Chair dataset. All convolutional kernel sizes are $3 \times 3$ . All activation functions for convolutional layers are ReLU.
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<table><tr><td>Layer name</td><td>Input shape</td><td>Output shape</td><td>Stride</td><td>Note</td></tr><tr><td>Conv1</td><td>64×64×7</td><td>64×64×64</td><td>2</td><td>Skip to Conv6</td></tr><tr><td>Conv2</td><td>64×64×64</td><td>32×32×64</td><td>2</td><td>Skip to Conv5</td></tr><tr><td>Conv3</td><td>32×32×64</td><td>16×16×64</td><td>2</td><td></td></tr><tr><td>Conv4</td><td>16×16×64</td><td>16×16×64</td><td>1</td><td></td></tr><tr><td>Upsample</td><td>16×16×64</td><td>32×32×64</td><td></td><td>Bilinear upsampling</td></tr><tr><td>Conv5</td><td>32×32×128</td><td>32×32×64</td><td>1</td><td></td></tr><tr><td>Upsample</td><td>32×32×64</td><td>64×64×64</td><td></td><td>Bilinear upsampling</td></tr><tr><td>Conv6</td><td>64×64×128</td><td>64×64×64</td><td>1</td><td></td></tr></table>
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Table 9: Encoder architecture for the Room-Diverse dataset. All convolutional kernel sizes are $3 \times 3$ . All activation functions for convolutional layers are ReLU.
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<table><tr><td>Layer name</td><td>Input shape</td><td>Output shape</td><td>Stride</td><td>Note</td></tr><tr><td>ConvO</td><td>128×128×7</td><td>128×128×64</td><td>1</td><td></td></tr><tr><td>Conv1</td><td>128×128×64</td><td>64×64×64</td><td>2</td><td>Skip to Conv6</td></tr><tr><td>Conv2</td><td>64×64×64</td><td>32×32×64</td><td>2</td><td>Skip to Conv5</td></tr><tr><td>Conv3</td><td>32×32×64</td><td>16×16×64</td><td>2</td><td></td></tr><tr><td>Conv4</td><td>16×16×64</td><td>16×16×64</td><td>1</td><td></td></tr><tr><td>Upsample</td><td>16×16×64</td><td>32×32×64</td><td></td><td>Bilinear upsampling</td></tr><tr><td>Conv5</td><td>32×32×128</td><td>32×32×64</td><td>1</td><td></td></tr><tr><td>Upsample</td><td>32×32×64</td><td>64×64×64</td><td></td><td>Bilinear upsampling</td></tr><tr><td>Conv6</td><td>64×64×128</td><td>64×64×64</td><td>1</td><td></td></tr></table>
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Secondly, this design also encourages the disentanglement between foreground objects and background by preventing the background slot from decoding foreground objects, because the positional information provided in the encoder is represented in viewer space.
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Foreground locality. To further encourage the disentanglement, we add a locality constraint during early training to prevent foreground slots to represent the background environment. Specifically, considering that “foreground” objects should be largely visible in sight, we set a foreground box and enforce that every foreground-querying point outside the box has zero density. The foreground box is defined such that its projection in image space can engage roughly $9 0 \%$ pixels. The locality constraint is imposed for the first 100K iterations, and it empirically helps prevent the foreground slots from fitting the background. We show a visual comparison in Figure 8, which from we can observe that the model without foreground locality design attaches some background segments to each object.
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# B.3 NEURAL RADIANCE FIELD ARCHITECTURE.
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We show our conditional object radiance field architecture in Figure 9.
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# B.4 MODEL LEARNING
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Loss functions. We set $\lambda _ { \mathrm { p e r c e p t } } = 0 . 0 0 6$ , $\lambda _ { \mathrm { a d v } } = 0 . 0 1$ , $\lambda _ { R } = 1 0$ . For perceptual loss, we implement the feature extractor $p$ by using the output of the 4-th convolutional block in a VGG16 (Simonyan & Zisserman, 2014) pretrained on ImageNet. For the adversarial discriminator, we follow the architecture of StyleGAN2 (Karras et al., 2020) with slight modification such that the maximum channel number is 128. We also use the lazy R1 regularization (Karras et al., 2020). We use Adam optimizer for discriminator with learning rate 0.001, $\beta _ { 1 } = 0$ and $\beta _ { 2 } = 0 . 9$ . The adversarial loss is incorporated after 100K iterations. Since shape uncertainty only appears in the Room-Diverse
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# Algorithm 1: Object-centric latent inference with background-aware slot attention.
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Input: feat ∈ RN×D
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Learnable: $\mu ^ { b } , \sigma ^ { b } , \mu ^ { f } , \sigma ^ { f }$ : prior parameters, $k , q ^ { b } , q ^ { f } , v ^ { b } , v ^ { f }$ : linear mappings, GRUb, GRUf , MLPb, MLPf slotb ∼ N b ∈ R1×D // Sampling slots from priors. slotsf ∼ N f ∈ RK×D for t = 1, · · · , T slot prevb = slotb, slots prevf = slotsf attn = Softmax √D 1 k(feat) · qb(slotb)f f T , dim=‘slot’! // Binding slots to object features. attn $^ b =$ attn[0], attnf = attn[1:end] updates $^ { b } =$ WeightedMean(weights=attnb, values $= \boldsymbol { v } ^ { b }$ (inputs)) // Aggregating update signals. updates $f _ { = }$ WeightedMean(weights=attnf , values $\scriptstyle \operatorname { \mathsf { \Omega } } _ { 3 } = v ^ { f }$ (inputs)) slot ${ \ v O } ^ { b } = \mathtt { G R U } ^ { b }$ (state=slot prevb, inputs=updatesb) // Updating slots. slotsf = GRUf (state slots prevf , inputs=updatesf ) $\mathsf { s } \mathsf { l o t } ^ { b } + = \mathsf { M L P } ^ { b } ( \mathsf { s } \mathsf { l o t } ^ { b } )$ , slotsf + = MLPf (slotsf ) // Residual update. return slotb, slotsf
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Table 10: Ablation for background coordinate space on novel view synthesis on Room-Chair dataset.
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<table><tr><td>Models</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td></tr><tr><td>uORF w/ view-space Backg.</td><td>0.151</td><td>0.799</td><td>27.86</td></tr><tr><td>uORF (ours)</td><td>0.082</td><td>0.872</td><td>29.60</td></tr></table>
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Table 11: Ablation for background coordinate space on segmentation on Room-Chair dataset.
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<table><tr><td rowspan="2">Models</td><td>3D metric</td><td colspan="2">2D metric</td></tr><tr><td>NV-ARI↑</td><td>ARI个</td><td>Fg-ARI个</td></tr><tr><td>uORF w/ view-space Backg.</td><td>73.5</td><td>78.0</td><td>89.0</td></tr><tr><td>uORF (ours)</td><td>74.3</td><td>78.8</td><td>88.8</td></tr></table>
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dataset, we only impose the adversarial loss on the Room-Diverse dataset but not on CLEVR-567 or Room-Chair. Both perceptual loss and adversarial loss are added after the first 100K iterations.
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Coarse-to-fine progressive training. For coarse training, we bilinearly downsample supervision images to $6 4 \times 6 4$ . The coarse training lasts for 600K iterations. For fine training, we randomly crop $6 4 \times 6 4$ patches from $1 2 8 \times 1 2 8$ images. The fine training lasts for 600K iterations. Our model is trained on a single Nvidia RTX 3090 GPU for about 6 days. For all networks except discriminator, we use Adam optimizer with learning rate 0.0003, $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } ~ = ~ 0 . 9 9 9$ . Learning rate is exponentially decreased by half for every 200K iterations until after 600K iterations. We also adopt the learning rate warm-up from the slot attention paper (Locatello et al., 2020) for the first 1K iterations. We initialize decoder networks with Xavier’s initialization. In each batch, we input one image and neurally render 4 images for supervision. We render each pixel with 64 samples.
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# C EXPERIMENTS
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In this section we provide further details on experiment settings.
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# C.1 DATA
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For the construction of all three datasets, the training/testing sets share the same pool of textures, shapes, and colors. The scenes in both sets differ in the spatial arrangement of objects, as well as the appearance differences induced by soft shadows and inter-reflections due to global illumination effects.
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CLEVR-567. In the CLEVR-567 dataset, each object’s shape is randomly chosen from three geometric primitives (i.e., cylinder, cube and sphere). The color is randomly chosen from {red, blue, purple, gray, cyan, yellow, green, brown}. There are two possible sizes for each object. When rendering images, we use the same camera intrinsic as original CLEVR dataset (Johnson et al., 2017). We do not use the visibility check due to our 360 degree multi-view setting, so we increase elevation angle by $\pi / 1 5$ to increase the chance of object visibility. Rendering setting is the same for all datasets.
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For CLEVR-567 dataset we set the latent dimension $D = 4 0$ and the maximum number of objects $K = 8$ .
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Figure 7: Visual comparison for representing background on view-space on novel view synthesis.
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Figure 8: Visual comparison on ablation for foreground locality constraint. We show examples in CLEVR-567 testset. We can see that our foreground locality box helps prevent object slots from fitting background segments.
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Room-Chair. For the object shape we use a chair model‡ from ShapeNet (Chang et al., 2015). We use the same material and colors as CLEVR-567. For Room-Chair and Room-Diverse datasets, we set the latent dimension $D = 6 4$ and the maximum number of objects $K = 5$ .
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Room-Diverse. All object shapes are randomly chosen from 1,200 ShapeNet chairs. For each shape, we normalize it into a unit cube according to vertex coordinates. We also use 8 colors $\{ { \tt r e d }$ , blue, purple, gray, cyan, yellow, green, $\mathtt { b l a c k } \}$ with diffuse material. Since shape uncertainty only appears in this dataset, we only impose the adversarial loss on this dataset.
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# C.2 BASELINE ARCHITECTURES
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Slot attention. We use the encoder-decoder architecture in the slot attention paper (Locatello et al., 2020) used for object discovery experiments on the CLEVR dataset. Basically it has 6 convolutional layers for encoder and 6 convolution-transpose layers for decoder. The number of channels for each layer is 64. All models are trained on $1 2 8 \times 1 2 8$ images.
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NeRF-AE. We follow NeRF implementation without view direction as input and set the highest frequency to 5. The encoder is similar to ours in Figure 9, but the basic number of channels is increased from 64 to 256 (and thus the number of channels of inputs to Conv5 and Conv6 is 512). The number of slot is set to 1.
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# D ADDITIONAL RESULTS
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Generalization to unseen objects. In the main paper we demonstrate systematic generalization to unseen combination of shape and color, here we further validate our model’s generalization to unseen object shapes. To this end, we construct another test set for Room-Diverse. All test objects in the new test set are drawn from a pool of 500 shapenet chairs that are completely disjoint from the 1200 training chairs. All other settings are the same as the original test set. We show quantitative results in Table 12 for novel view synthesis and in Table 13 for segmentation. As we can see, our model yields the same level of performances even on the unseen shape test set, suggesting its generalization to unseen object shapes.
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Figure 9: Illustration for foreground decoder architecture. We follow the architecture in NeRF (Mildenhall et al., 2020) but with fewer parameters to decrease space demand. We set the highest positional embedding frequency to 5, so that the positional embedding input dimension is $5 \times 2 \times 3 + \mathbf { \breve { 3 } } = 3 3$ . The background decoder is slightly different in that it does not have the second last layer and third last layer. Density $\sigma$ is activated by ReLU. Since estimating specularity from a single image is intrinsically ambiguous, we assume Lambertian surfaces and do not use the ray direction as input.
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Table 12: Novel view synthesis results on unseen/seen shape testset of Room-Diverse.
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<table><tr><td>Models</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td></tr><tr><td>uORF on seen shape testset</td><td>0.1729</td><td>0.7094</td><td>25.96</td></tr><tr><td>uORF on unseen shape testset</td><td>0.1771</td><td>0.7125</td><td>26.16</td></tr></table>
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Table 13: Unsupervised segmentation in 3D results on unseen/seen shape testset of Room-Diverse.
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<table><tr><td rowspan="2">Models</td><td>3D metric</td><td colspan="2">2D metric</td></tr><tr><td>NV-ARI↑</td><td>ARI个</td><td>Fg-ARI↑</td></tr><tr><td>uORF on seen shape testset</td><td>56.9</td><td>65.6</td><td>67.9</td></tr><tr><td>uORF on unseen shape testset</td><td>57.0</td><td>66.1</td><td>67.7</td></tr></table>
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Generalization to real images. We also take a step further to test our pretrained model’s generalization on real photos. To do this, we use uORF trained on Room-Diverse. We take a few real photos by a cellphone, providing an input image and a few reference images. We show the visual results in Figure 10. Although the real photo has a different imaging process and consists of unseen objects and background, uORF is able to discover all objects with roughly correct positions and orientations, yielding plausible segmentation results and object-moving results.
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Analysis on the sensitivity to slot initialization. We test the robustness of our model to the slot initialization on the Room-Chair dataset. For each test scene, we now use 5 different random seeds for sampling initial centers. We compute the mean $\mu$ and std $\sigma$ of ARI over the 5 seeds. We average them over the 500 test scenes. The averaged mean $\bar { \mu }$ of ARI is $7 8 . 8 \%$ and $\bar { \sigma }$ is $1 . 7 \%$ . The mean ARI suggests good segmentation results (very close to $7 8 . 8 \%$ as reported in Table 2 in our main paper), and $\bar { \sigma } = 1 . 7 \%$ indicates that different seeds all lead to results close to such good ARI performance.
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Additional qualitative results. We show additional qualitative results for our experiments in the main manuscript. We show additional examples for scene segmentation in Figure 11 and Figure 12, for novel view synthesis in Figure 13, Figure 14 and Figure 15, for scene editing in Figure 16 and Figure 17, for evaluating losses in Figure 18, for generalization to challenging spatial arrangement in Figure 19 (note that in the packed-CLEVR-11 dataset we only use a single size for higher object visibility), and for generalization to unseen object appearance in Figure 20.
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Failure case. In our experiments, we observed a type of failure which we call “attention rankcollapse”. We show examples in Figure 21.
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Figure 10: Demonstration on generalization to real photos. We use uORF pretrained on Room-Diverse and take photos by a cellphone.
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“Attention rank-collapse” refers to that all the foreground object slots have (nearly) the same attention map and collapse to the same representation. Each collapsed slot decodes simply nothing (or all the foreground objects). This “attention rank-collapse” happens when the initialization is prompt to a degenerate solution for the slot attention. It occasionally happens and empirically changing the initialization seed can address it. A related rank-collapse problem is discussed in Dong et al. (2021), which suggests that adding some architectural inductive bias can largely alleviate the problem. We hope future research can address this problem fundamentally.
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# E COMPARISON TO GIRAFFE
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Our work has a fundamentally different focus compared to GIRAFFE (Niemeyer & Geiger, 2020). While GIRAFFE focuses on unconditional generation and enables multi-object scene synthesis and rendering, the goal of our uORF is to simultaneously infer 3D multi-object scene representations from a single image, in addition to using those representations for rendering and editing as in GIRAFFE.
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# E.1 COMPARISONS BETWEEN OUR UORF AND GIRAFFE
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To demonstrate that the inference of such multi-object scenes is highly non-trivial, we compare with GIRAFFE on both CLEVR-567 and Room-Chair (we cannot compare on their datasets because they only have a single image for each scene). To train GIRAFFE on our datasets, we use the official repo§ and the same hyper-parameters that GIRAFFE authors used for their CLEVR-2345 dataset, except for a few adaptive changes to our datasets: (1) We try different sizes for the object slot, because CLEVR-2345 only uses small objects while our datasets both contain larger objects. Specifically, we try $2 \times$ , $1 . 5 \times$ , and $1 \times$ original size, and use the one with the lowest FID for each dataset. (2) We adjust the camera elevation angle and focal lengths to match our datasets. (3) We set the number of objects to 4 for the Room-Chair dataset because each scene has no more than 4 chairs. We train the GIRAFFE models for around 500K iterations on 128-by-128 images, such that FID does not drop anymore.
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For GIRAFFE to do inference, we sample object (including background) latents and positions in the same manner as training, and then we optimize for L2 reconstruction loss for both the latents and the positions. We use Adam and do a learning rate sweep to select the one that leads to the best reconstruction loss. We divide the learning rate by 10 when the loss plateaus. We do this learning rate decay twice. We sweep in $\{ 0 . 1 , 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \}$ and find that 0.01 works best. Since each scene has an unknown number of objects, we set the number to the maximum number across all scenes. It converges at around 150 iterations on CLEVR-567 and around 300 iterations on Room-Chair. Thus we set the maximum iteration to 300 and 500 for them, respectively.
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We also compare with a GIRAFFE model that is pretrained on CLEVR-2345. The pretrained model is provided by the authors. The pretrained model yields $\mathrm { F I D = 8 2 }$ on CLEVR-567 $\mathrm { { F I D } = 6 1 }$ on CLEVR-2345), indicating that it could be a valid baseline even though the two datasets are mildly different.
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We show input-view reconstruction and novel view synthesis results in Table 14 and Table 15, and we show qualitative comparison in Figure 22 and Figure 23. We can see that GIRAFFE fails in reconstructing the multi-object scenes from a single image, as well as novel view synthesis. Let alone segmentation in 3D.
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Table 14: Inference comparison with GIRAFFE on CLEVR-567.
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<table><tr><td rowspan="2">Models</td><td colspan="3">Input view reconstruction</td><td colspan="3">Novel view synthesis</td></tr><tr><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td></tr><tr><td>GIRAFFE (trained on CLEVR-567)</td><td>0.330</td><td>0.815</td><td>23.75</td><td>0.549</td><td>0.672</td><td>16.65</td></tr><tr><td>GIRAFFE (author-pretrained model on CLEVR-2345)</td><td>0.382</td><td>0.780</td><td>21.76</td><td>0.643</td><td>0.348</td><td>11.70</td></tr><tr><td>uORF (ours)</td><td>0.085</td><td>0.901</td><td>29.33</td><td>0.086</td><td>0.897</td><td>29.28</td></tr></table>
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Table 15: Inference comparison with GIRAFFE on Room-Chair.
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<table><tr><td rowspan="3">Models</td><td colspan="3">Input view reconstruction</td><td colspan="3">Novel view synthesis</td></tr><tr><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td></tr><tr><td rowspan="2">GIRAFFE (trained on Room-Chair) uORF (ours)</td><td>0.414</td><td>0.597</td><td>20.90</td><td>0.588</td><td>0.538</td><td>18.53</td></tr><tr><td>0.085</td><td>0.876</td><td>29.65</td><td>0.082</td><td>0.872</td><td>29.60</td></tr></table>
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# E.2 GIRAFFE INFERENCE ON CLEVR-2345
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While we have compared with GIRAFFE on our datasets, we further evaluate the author-provided pretrained model on the simpler dataset CLEVR-2345 from the GIRAFFE paper itself. We found that while GIRAFFE does well on unconditional scene synthesis, it cannot perform novel view synthesis on their own dataset, either. This shows that GIRAFFE focuses on problems very different from ours.
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We first show that inference/reconstruction is challenging for GIRAFFE, even on the simpler dataset. We do inference on the author-provided CLEVR-2345 dataset using the author-provided pretrained model. We show randomly sampled examples through the iterative inference process in Figure 24.
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Then we show that GIRAFFE fails in wide-baseline novel view synthesis. We use the author-provided pretrained model to sample from its latent space and unconditionally generate one image. Then we keep all the variables the same, but circularly move cameras to render novel views. We show 10 random examples of this circular novel view synthesis in Figure 25. We see that when the viewpoint changes become significant, GIRAFFE fails novel view synthesis, because its neural renderer is based on 2D feature maps and it’s not inherently 3D.
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# E.3 DISCUSSION AND SUMMARY
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In general, inverting GAN latent space even for the holistic image is non-trivial and needs architecturalspecific designs (we refer the reader to the discussion and references in a recent survey on GAN inversion (Xia et al., 2021)). As for inverting compositional multi-object scenes, it becomes even harder due to ambiguous correspondences (“which slot corresponds to which object?”), number of objects (“how many slots should I put?”), object position constraints (“there are two objects overlapping in the image, but they should not be overlapping in 3D”), optimization issues (e.g., optimizing rotation is notoriously difficult (Zhou et al., 2019)), etc.
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In summary, it is highly non-trivial for GIRAFFE to do inference for multi-object scenes due to complexities such as ambiguous correspondences, the number of objects, and optimization issues. We will include more discussions on the difference between the two methods in the following separate thread. In short, our uORF tries to solve a fundamentally different problem from GIRAFFE, i.e., we aim at inferring the joint distribution of objects from a single image while GIRAFFE targets extrinsiccontrollable image generation. Therefore, our method enables novel tasks such as unsupervised segmentation and editing in 3D, where prior methods including GIRAFFE are not able to do.
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Figure 11: Additional qualitative results for segmentation in 3D on Room-Chair dataset.
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Figure 12: Additional qualitative results for segmentation in 3D on Room-Diverse dataset.
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Figure 13: Additional qualitative results for novel view synthesis on CLEVR-567 dataset.
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Figure 14: Additional qualitative results for novel view synthesis on Room-Chair dataset.
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Figure 15: Additional qualitative results for novel view synthesis on Room-Diverse dataset.
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Figure 16: Additional qualitative results for scene editing.
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Figure 17: Additional qualitative results for scene editing.
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Figure 18: Qualitative results for loss evaluations. Using both perceptual loss and adversarial loss improves image quality.
|
| 444 |
+
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| 445 |
+
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| 446 |
+
Figure 19: Qualitative results for generalization to unseen spatial arrangement.
|
| 447 |
+
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+
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| 449 |
+
Figure 20: Qualitative results for generalization to unseen combination of color and shape.
|
| 450 |
+
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| 451 |
+
|
| 452 |
+
Figure 21: Failure case of our model, which we call “attention rank-collapse”. All foreground slots share the same attention map. Every foreground slot decodes to the same radiance field (empty radiance here) rather than specializing to an object. Here we only show one object slot, as all others look the same.
|
| 453 |
+
|
| 454 |
+
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| 455 |
+
Figure 22: Visual comparison with GIRAFFE for inference on CLEVR-567 dataset. GIRAFFE fails inference.
|
| 456 |
+
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| 457 |
+
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| 458 |
+
Figure 23: Visual comparison with GIRAFFE for inference on Room-Chair dataset. GIRAFFE fails inference.
|
| 459 |
+
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| 460 |
+
|
| 461 |
+
Figure 24: Inference trajectory of GIRAFFE using author-provided models on the author-provided dataset CLEVR-2345.
|
| 462 |
+
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| 463 |
+
|
| 464 |
+
Figure 25: Novel view synthesis on randomly generated examples using author-provided pretrained GIRAFFE model on CLEVR-2345. GIRAFFE fails inference of these multi-object scenes. GIRAFFE cannot synthesize novel views with large rotations.
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|
| 1 |
+
# Poisson Flow Generative Models
|
| 2 |
+
|
| 3 |
+
Yilun Xu∗ , Ziming Liu∗ , Max Tegmark, Tommi Jaakkola Massachusetts Institute of Technology ylxu, zmliu, tegmark @mit.edu; tommi@csail.mit.edu
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
We propose a new “Poisson flow” generative model (PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the $z = 0$ hyperplane in a space augmented with an additional dimension $z$ , generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the $z \ = \ 0$ plane transforms into a distribution on the hemisphere of radius $r$ that becomes uniform in the $r \infty$ limit. To learn the bijective transformation, we estimate the normalized field in the augmented space. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the (unaugmented) data manifold when the $z$ reaches zero. Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of 9.68 and a FID score of 2.35. It also performs on par with the state-of-the-art SDE approaches while offering $1 0 \times$ to $2 0 \times$ acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method. The code is available at https: //github.com/Newbeeer/poisson_flow.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Deep generative models are a prominent approach for data generation, and have been used to produce high quality samples in image [1], text [2] and audio [35], as well as improve semi-supervised learning [20], domain generalization [25] and imitation learning [15]. However, current deep generative models also have limitations, such as unstable training objectives (GANs [1, 12, 17]) and low sample quality (VAEs [21], normalizing flows [6]). New techniques [12, 24] are introduced to stablize the training of CNN-based or ViT-based GAN models. Although recent advances on diffusion [16] and scored-based models [33] achieve comparable sample quality to GAN’s without adversarial training, these models have a slow stochastic sampling process. [33] proposes backward ODE samplers (normalizing flow) that speed up the sampling process but these methods have not yet performed on par with the SDE counterparts.
|
| 12 |
+
|
| 13 |
+
We present a new “Poisson flow” generative model (PFGM), exploiting a remarkable physics fact that generalizes to $N$ dimensions. As illustrated in Fig. 1(a), motion in a viscous fluid transforms any planar charge distribution into a uniform angular distribution. Specifically, we interpret $N$ - dimensional data points $\mathbf { x }$ (images, say) as positive electric charges in the $z \ = \ 0$ plane of an $N + 1$ -dimensional space (see Fig. 1(a)) filled with a viscous liquid (say honey). A positive charge with $z > 0$ will be repelled by the other charges and move in the direction of their repulsive force, eventually crossing an imaginary hemisphere of radius $r$ . We show that, remarkably, if the the original charge distribution is let loose just above $z = 0$ , this law of motion will cause a uniform distribution for their hemisphere crossings in the $r \infty$ limit.
|
| 14 |
+
|
| 15 |
+
|
| 16 |
+
Figure 1: (a) 3D Poisson field trajectories for a heart-shaped distribution (b) The evolvements of a distribution (top) or an (augmented) sample (bottom) by the forward/backward ODEs pertained to the Poisson field.
|
| 17 |
+
|
| 18 |
+
Our Poisson flow generative process reverses the forward process: we generate a uniform distribution of negative charges on the hemisphere, then track their motion back to the $z = 0$ plane, where they will be distributed as the data distribution. A Poisson flow can be viewed as a type of continuous normalizing flows [4, 10, 33] in the sense that it continuously maps between an arbitrary distribution and an easily sampled one: in the previous works an $N$ -dimensional Gaussian and in PFGM a uniform distribution on an $N$ -dimensional hemisphere. In practice, we implement the Poisson flow by solving a pair of forward/backward ordinary differential equations (ODEs) induced by the electric field (Fig. 1(b)) given by the $N$ -dimensional version of Coulomb’s law (the gradient of the solution to the Poisson’s equation with the data as sources). We will interchangeably refer to this gradient as the Poisson field, since electric fields normally refer to the special case $N = 3$ .
|
| 19 |
+
|
| 20 |
+
The proposed generative model PFGM has a stable training objective and empirically outperforms previously state-of-the-art continuous flow methods [30, 33]. As a different iterative method, PFGM offers two advantages compared to score-based methods [32, 33]. First, the ODE process of PFGM achieves faster sampling speeds than the SDE samplers in [33]. while retaining comparable performance. Second, our backward ODE exhibits better generation performance than the reverse-time ODEs of VE/VP/sub-VP SDEs [33], as well as greater stability on a weaker architecture NSCNv2 [32]. The rationale for robustness is that the time variables in these ODE baselines are strongly correlated with the sample norms during training time, resulting in a less error-tolerant inference. In contrast, the tie between the anchored variable and the sample norm in PFGM is much weaker.
|
| 21 |
+
|
| 22 |
+
Experimentally, we show that PFGM achieves current state-of-the-art performance on CIFAR-10 dataset in the normalizing flow family, with FID/Inception scores of $2 . { \bar { 4 } } 8 / 9 . 6 5$ (w/ $\mathrm { { D D P M + + } }$ [33]) and $2 . 3 5 / 9 . 6 8$ (w/ $\mathrm { { D D P M + + } }$ deep [33]). It performs competitively with current state-of-the-art SDE samplers [33] and provides $1 0 \times$ to $2 0 \times$ speed up across datasets. Notably, the backward ODE in PFGM is the only ODE-based sampler that can produce decent samples on its own on NCSNv2 [32], while other ODE baselines fail without corrections. In addition, PFGM demonstrates the robustness to the step size in the Euler method, with a varying number of function evaluations (NFE) ranging from 10 to 100. We further showcase the utility of the invertible forward/backward ODEs of the Poisson field on likelihood evaluation and image manipulations, and its scalability to higher resolution images on LSUN bedroom $2 5 6 \times 2 5 6$ dataset.
|
| 23 |
+
|
| 24 |
+
# 2 Background and Related works
|
| 25 |
+
|
| 26 |
+
Poisson equation Let $\mathbf { x } \in \mathbb { R } ^ { N }$ and $\rho ( \mathbf { x } ) : \mathbb { R } ^ { N } \mathbb { R }$ be a source function. We assume that the source function has a compact support, $\rho \in \mathcal { C } ^ { 0 }$ and $N \geq 3$ . The Poisson equation is
|
| 27 |
+
|
| 28 |
+
$$
|
| 29 |
+
\nabla ^ { 2 } \varphi ( \mathbf { x } ) = - \rho ( \mathbf { x } ) ,
|
| 30 |
+
$$
|
| 31 |
+
|
| 32 |
+
where $\varphi ( \mathbf { x } ) : \mathbb { R } ^ { N } \mathbb { R }$ is called the potential function, and $\begin{array} { r } { \bigtriangledown ^ { 2 } \equiv \sum _ { i = 1 } ^ { N } \frac { \partial ^ { 2 } } { \partial x _ { i } ^ { 2 } } } \end{array}$ is the Laplacian operator. It is usually helpful to define the gradient field $\mathbf { E } ( \mathbf { x } ) = - \nabla \varphi ( \mathbf { x } )$ and rewrite the Poisson equation as $\nabla \cdot \mathbf { E } = \rho$ , known in physics as Gauss’s law [11]. The Poisson equation is widely used in physics, giving rise to Newton’s gravitational theory [9] and the electrostatic theory [11], when $\rho ( \mathbf { x } )$ is interpreted as mass density or electric charge density, respectively. $\mathbf { E }$ is the $N$ -dimensional analog of the electric field. The Poisson equation Eq. (1) (with zero boundary condition at infinity) admits a unique simple integral solution 2:
|
| 33 |
+
|
| 34 |
+
$$
|
| 35 |
+
\varphi ( \mathbf { x } ) = \int G ( \mathbf { x } , \mathbf { y } ) \rho ( \mathbf { y } ) d \mathbf { y } , \quad G ( \mathbf { x } , \mathbf { y } ) = \frac { 1 } { ( N - 2 ) S _ { N - 1 } ( 1 ) } \frac { 1 } { | | \mathbf { x } - \mathbf { y } | | ^ { N - 2 } } ,
|
| 36 |
+
$$
|
| 37 |
+
|
| 38 |
+
where $S _ { N - 1 } ( 1 )$ is a geometric constant representing the surface area of the unit $( N - 1 )$ -sphere 3, and $G ( \mathbf { x } , \mathbf { y } )$ is the extension of Green’s function in $N$ -dimensional space (details in Appendix A.3). The negative gradient field of $\varphi ( \mathbf x )$ , referred as Poisson field of the source $\rho$ , is
|
| 39 |
+
|
| 40 |
+
$$
|
| 41 |
+
\mathbf { E } ( \mathbf { x } ) = - \nabla \varphi ( \mathbf { x } ) = - \int \nabla _ { \mathbf { x } } G ( \mathbf { x } , \mathbf { y } ) \rho ( \mathbf { y } ) d \mathbf { y } , \quad \nabla _ { \mathbf { x } } G ( \mathbf { x } , \mathbf { y } ) = - \frac { 1 } { S _ { N - 1 } ( 1 ) } \frac { \mathbf { x } - \mathbf { y } } { \left\| \mathbf { x } - \mathbf { y } \right\| ^ { N } } .
|
| 42 |
+
$$
|
| 43 |
+
|
| 44 |
+
Qualitatively, the Poisson field $\mathbf { E } ( \mathbf { x } )$ points away from sources, or equivalently $- \mathbf { E } ( \mathbf { x } )$ points towards sources, as illustrated in Fig. 1. It is straightforward to check that when $\rho ( { \bf x } ) \delta ( { \bf x - y } )$ , we get $\varphi ( \mathbf x ) G ( \mathbf x , \mathbf y )$ and $\mathbf { E } ( \mathbf { x } ) - \nabla _ { \mathbf { x } } G ( \mathbf { x } , \mathbf { y } )$ . This implies that $G ( \mathbf { x } , \mathbf { y } )$ and $- \nabla _ { \mathbf { x } } G ( \mathbf { x } , \mathbf { y } )$ can be interpreted as the potential function and the gradient field generated by a unit point source, e.g., a point charge, located at $\mathbf { y }$ . When $\rho ( \mathbf { x } )$ takes general forms but has bounded support, simple asymptotics exist for $\left\| \mathbf { x } \right\| \gg \left\| \mathbf { y } \right\|$ . To the lowest order, $\mathbf { E ( x ) } = \nabla _ { \mathbf { x } } G ( \mathbf { x } , \mathbf { y } ) | _ { \mathbf { y } = \mathbf { 0 } } \sim \mathbf { x } / \| \mathbf { x } \| ^ { N }$ behaves as if it were generated by a unit point source at $\mathbf y = 0$ . In physics, the power law decay is considered to be long-range (compared to exponential decay) [11].
|
| 45 |
+
|
| 46 |
+
Particle dynamics in a Poisson field The Poisson field immediately defines a flow model, where the probability distribution evolves according to the gradient flow $\partial p _ { t } ( \mathbf { x } ) / \partial t = - \nabla \cdot ( p _ { t } ( \mathbf { x } ) \mathbf { E } ( \mathbf { x } ) )$ . The gradient flow is a special case of the Fokker-Planck equation [28], where the diffusion coefficient is zero. Intuitively we can think of $p _ { t } ( \mathbf { x } )$ as represented by a population of particles. The corresponding (non-diffusion) case of the Ito process is the forward ODE ˆ $\begin{array} { r } { \frac { d \mathbf { x } } { d t } \ = \ \mathbf { E } ( \mathbf { x } ) } \end{array}$ . We can interpret the trajectories of the ODE as particles moving according to the Poisson field $\mathbf { E } ( x )$ , with initial states drawn from $p _ { 0 }$ . The physical picture of the forward ODE is a charged particle under the influence of electric fields in the overdamped limit (details in Appendix F).
|
| 47 |
+
|
| 48 |
+
The dynamics is also rescalable in the sense that the particle trajectory remains the same for $\pm f ( \mathbf { x } ) \mathbf { E } ( \mathbf { x } )$ for $f ( \mathbf { x } ) > 0 , f ( \mathbf { x } ) \in \mathcal { C } ^ { 1 }$ , because the time rescaling $d t \to f ( \mathbf { x } ( t ) ) d t$ recovers $\begin{array} { r } { \frac { d \mathbf { x } } { d t } \ = } \end{array}$ $\begin{array} { r } { \frac { d { \mathbf { x } } } { d t } = } \end{array}$ $\pm \mathbf { E } ( \mathbf { x } )$ . Note that the dynamics is stiff due to the power law factor in the denominator in Eq. (3), posing computational challenges. Luckily the rescalablility allows us to rescale $\mathbf { E } ( \mathbf { x } )$ properly to get new ODEs (formally defined later in Section 3.3) that are more amenable for sampling.
|
| 49 |
+
|
| 50 |
+
Generative Modeling via ODE Generative modeling can be done by transforming a base distribution to a data distribution via mappings defined by ODEs. The ODE-based samplers allow for adaptive sampling, exact likelihood evaluation and modeling of continuous-time dynamics [4, 33]. Previous works broadly fall into two lines. [4, 3] introduce a continuous-time normalizing flow model that can be trained with maximum likelihood by the instantaneous change-of-variables formula [4]. For sampling, they directly integrate the learned invertiable mapping over time. Another work [33] unifies the scored-based model [31, 32] and diffusion model [16] into a general diffusion process, and uses the reverse-time ODE of the diffusion process for sampling. They show that the reverse-time ODE produces high quality samples with improved architecture.
|
| 51 |
+
|
| 52 |
+
# 3 Poisson Flow Generative Models
|
| 53 |
+
|
| 54 |
+
In this section, we start with the properties of the Poisson flow in the augmented space and show how to draw samples from the data distribution by following the backward ODE of the Poisson flow (Section 3.1). We then discuss how to actually learn a normalized Poisson field from data samples through simulations of the forward ODE (Section 3.2) and present an equivalent backward ODE that allows for exponentially decay on $z$ (Section 3.3).
|
| 55 |
+
|
| 56 |
+
|
| 57 |
+
Figure 2: (a) Poisson field (black arrows) and particle trajectories (blue lines) of a 2D uniform disk (red). Left (no augmentation, 2D): all particles collapse to the disk center. Right (augmentation, 3D): particles hit different points on the disk. (b) Proof idea of Theorem 1. By Gauss’s Law, the outflow flux $d \Phi _ { o u t }$ equals the inflow flux $d \Phi _ { i n }$ . The factor of two in $p ( \mathbf { x } ) d A / 2$ is due to the symmetry of Poisson fields in $z < 0$ and $z > 0$ .
|
| 58 |
+
|
| 59 |
+
# 3.1 Augment the data with additional dimension
|
| 60 |
+
|
| 61 |
+
We wish to generate samples $\mathbf { x } \in \mathbb { R } ^ { N }$ from a distribution $p ( \mathbf { x } )$ supported on a bounded region. We may set the source $\rho ( \mathbf { x } ) = p ( \mathbf { x } ) \in \mathcal { C } ^ { 0 \ }$ 4 and compute the resulting gradient field $\mathbf { E } ( \mathbf { x } )$ from Eq. (3). Since $- \mathbf { E } ( \mathbf { x } )$ points towards sources, the backward ODE ${ d { \bf x } } / { d t } = - { \bf E } ( { \bf x } )$ will take samples close to the sources. One may naively hope that the backward ODE is a generative model that recovers $p ( \mathbf { x } )$ . Unfortunately, the backward ODE has the problem of mode collapse. We illustrate this phenomenon with a 2D uniform disk. The reverse Poisson field $- \mathbf { E } ( \mathbf { x } )$ on the 2D $( x , y )$ -plane points towards the center of the disk $O$ (Fig. 2(a) left), so all particle trajectories (blue lines) will eventually hit $O$ . If we instead add an additional dimension $z$ (Fig. 2(a) right), particles can hit different points on the disk and faithfully recover the data distribution.
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+
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| 63 |
+
Consequently, instead of solving the Poisson equation $\nabla ^ { 2 } \varphi ( \mathbf { x } ) = - p ( \mathbf { x } )$ in the original data space, we solve the Poisson equation in an augmented space $\tilde { \mathbf { x } } = ( \mathbf { x } , z ) \in \mathbb { R } ^ { N + 1 }$ with an additional variable $z \in \mathbb { R }$ . We augment the training data $\tilde { \mathbf { x } }$ in the new space by setting $z = 0$ such that $\tilde { \mathbf { x } } = ( \mathbf { x } , 0 )$ . As a consequence, the data distribution in the augmented space is $\tilde { p } ( \tilde { \mathbf { x } } ) = p ( \mathbf { x } ) \delta ( z )$ , where $\delta$ is the Dirac delta function. By Eq. (3), the Poisson field by solving the new Poisson equation $\nabla ^ { 2 } \varphi ( \tilde { \mathbf { x } } ) = - \tilde { p } ( \tilde { \mathbf { x } } )$ has an analytical form:
|
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| 65 |
+
$$
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| 66 |
+
\forall \tilde { { \mathbf { x } } } \in \mathbb { R } ^ { N + 1 } , \mathbf { E } ( \tilde { { \mathbf { x } } } ) = - \nabla \varphi ( \tilde { { \mathbf { x } } } ) = \frac { 1 } { S _ { N } ( 1 ) } \int \frac { \tilde { { \mathbf { x } } } - \tilde { { \mathbf { y } } } } { \left\| \tilde { { \mathbf { x } } } - \tilde { { \mathbf { y } } } \right\| ^ { N + 1 } } \tilde { p } ( \tilde { { \mathbf { y } } } ) d \tilde { { \mathbf { y } } }
|
| 67 |
+
$$
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+
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The associated forward/backward ODEs of the Poisson field are $d \tilde { \mathbf { x } } / d t = \mathbf { E } ( \tilde { \mathbf { x } } ) , d \tilde { \mathbf { x } } / d t = - \mathbf { E } ( \tilde { \mathbf { x } } )$ . Intuitively, theses ODEs uniquely define trajectories of particles between the $z = 0$ hyperplane and an enclosing hemisphere (cf. Fig. 1(a)). In the following theorem, we show that the backward ODE defines a transformation between the uniform distribution on an infinite hemisphere and the data distribution $\tilde { p } ( \tilde { { \mathbf x } } )$ in the $z = 0$ plane. We present the formal proof to Appendix A, illustrated by Fig. 2(b). The proof is based on the idea that when the radius of hemisphere $r \infty$ , the data distribution $\tilde { p } ( \tilde { \mathbf { x } } )$ can be effectively viewed as a delta distribution at origin. Consequently, the Poisson field points in the radial direction at $r \infty$ , perpendicular to $S _ { N } ^ { + } ( r )$ (Green arrows in Fig. 2(b)).
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Theorem 1. Suppose particles are sampled from a uniform distribution on the upper $( z > 0 ,$ half of the sphere of radius r and evolved by the backward ODE $\begin{array} { r } { \frac { d \tilde { \mathbf { x } } } { d t } = - \mathbf { E } \big ( \tilde { \mathbf { x } } \big ) } \end{array}$ until they reach the $z = 0$ hyperplane, where the Poisson field $\mathbf { E } ( \tilde { \mathbf { x } } )$ is generated by the source $\tilde { p } ( \tilde { { \mathbf x } } )$ . In the $r \infty$ limit, under some mild conditions detailed in Appendix $\cdot$ , this process generates a particle distribution $\tilde { p } ( \tilde { { \mathbf x } } )$ , i.e., a distribution $p ( \mathbf { x } )$ in the $z = 0$ hyperplane.
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Proof sketch. Suppose the flux of the backward ODE connects a solid angle $d \Omega$ (on $S _ { N } ^ { + } ( r ) )$ with an area $d A$ (on $\operatorname { s u p p } ( \tilde { p } ( \tilde { \mathbf { x } } ) )$ . According to Gauss’s law, the outflow flux $d \Phi _ { o u t } = d \Omega / S _ { N } \bar { ( 1 ) }$ on the hemisphere (Green arrows in Fig. 2(b)) equals the inflow flux $d \Phi _ { i n } = p ( { \bf x } ) d A / 2$ on $\operatorname { s u p p } ( \tilde { p } ( \tilde { \mathbf { x } } ) )$ (Red arrows in Fig. 2(b)). $d \Phi _ { i n } = d \Phi _ { o u t }$ gives $d \Omega / d A = p ( \mathbf { x } ) S _ { N } ( 1 ) / 2 \propto \tilde { p } ( \mathbf { \tilde { x } } )$ . Together, by change-ofvariable, we conclude that the final distribution in the $z = 0$ hyperplane is $p ( \mathbf { x } )$ . □
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The theorem states that starting from an infinite hemisphere, one can recover the data distribution $\tilde { p }$ by following the inverse Poisson field $- \mathbf { E } ( \tilde { \mathbf { x } } )$ . We defer the formal proof and technical assumptions of the theorem to Appendix A. The property allows generative modeling by following the Poisson flow of $\nabla ^ { 2 } \varphi ( \tilde { \mathbf { x } } ) = - \tilde { p } ( \tilde { \mathbf { x } } )$ .
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# 3.2 Learning the normalized Poisson Field
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Given a set of training data $\mathcal { D } = \{ \mathbf { x } _ { i } \} _ { i = 1 } ^ { n }$ i.i.d sampled from the data distribution $p ( \mathbf { x } )$ , we define the =empirical version of the Poisson field (Eq. (4)) as follows:
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$$
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\hat { \bf E } ( \tilde { \bf x } ) = c ( \tilde { \bf x } ) \sum _ { i = 1 } ^ { n } \frac { \tilde { \bf x } - \tilde { \bf x } _ { i } } { \| \tilde { \bf x } - \tilde { \bf x } _ { i } \| ^ { N + 1 } }
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+
$$
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where the gradient field is calculated on $n$ augmented datapoints $\{ \tilde { \mathbf { x } } _ { i } = ( \mathbf { x } _ { i } , 0 ) \} _ { i = 1 } ^ { n }$ , and $c ( \tilde { \mathbf { x } } ) =$ $\textstyle 1 / \sum _ { i = 1 } ^ { n } { \frac { 1 } { \left\| \tilde { \mathbf { x } } - \tilde { \mathbf { x } } _ { i } \right\| ^ { N + 1 } } }$ is the multiplier for numerical stability. We further normalize the field to resolve the variations in the magnitude of the norm $\Vert \hat { \textbf { E } } ( \tilde { \textbf { x } } ) \ \Vert _ { 2 }$ , and fit the neural network to the more amenable negative normalized field $\mathbf { v } ( \tilde { \mathbf { x } } ) = - \sqrt { N } \hat { \mathbf { E } } ( \tilde { \mathbf { x } } ) / \| \hat { \mathbf { E } } ( \tilde { \mathbf { x } } ) \| _ { 2 }$ . The Poisson field is rescalable (cf. Section 2) and thus trajectories of its forward/backward ODEs are invariant under normalization. We denote the empirical field calculated on batch data $\boldsymbol { B }$ by $\hat { \mathbf { E } } _ { B }$ and the negative normalized field as $\mathbf { v } _ { B } ( \tilde { \mathbf { x } } ) = - \sqrt { N } \hat { \mathbf { E } } _ { B } ( \tilde { \mathbf { x } } ) / \| \hat { \mathbf { E } } _ { B } ( \tilde { \mathbf { x } } ) \| _ { 2 }$ .
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Similar to the scored-based models, we sample points inside the hemisphere by perturbing the augmented training data. Given a training point $\textbf { x } \in \mathcal { D }$ , we add noise to its augmented version $\{ \tilde { \mathbf { x } } _ { i } ^ { - } = ( \mathbf { x } _ { i } , 0 ) \} _ { i = 1 } ^ { n }$ to construct the perturbed point $( \mathbf { y } , z )$ :
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$$
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\mathbf { y } = \mathbf { x } + \parallel \epsilon _ { \mathbf { x } } \parallel ( 1 + \tau ) ^ { m } \mathbf { u } , \quad z = | \epsilon _ { z } | ( 1 + \tau ) ^ { m }
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+
$$
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where ${ \epsilon } = \left( { \epsilon } _ { \mathbf { x } } , { \epsilon } _ { z } \right) \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { N + 1 \times N + 1 } )$ , $\mathbf { u } \sim \mathcal { U } ( S _ { N - 1 } ( 1 ) )$ and $m \sim \mathcal { U } [ 0 , M ]$ . The upper limit $M$ , standard deviation $\sigma$ and $\tau$ are hyper-parameters. With fixed $\epsilon$ and $\mathbf { u }$ , the added noise increases exponentially with $m$ . The rationale behind the design is that points farther away from the data support play a less important role in generative modeling, sharing a similar spirit with the choice of noisy scales in score-based models [32, 33].
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In practice, we sample the p ts urbing a mini-batch data $B = \{ \mathbf { x } _ { i } \} _ { i = 1 } ^ { | B | }$ each iteration. We $m$ $[ 0 , M ]$ =for each datapoint. We select a large $M$ 300) to ensure the perturbed points can reach a large enough hemisphere. We use a larger batch $\boldsymbol { B } _ { L }$ for the estimation of normalized field since the empirical normalized field is biased, which empirically gives better results. Denoting the set of perturbed points as $\{ \tilde { \mathbf { y } } _ { i } \} _ { i = 1 } ^ { | B | }$ , we train the neural network $f _ { \theta }$ =on these points to estimate the negative normalized field by minimizing the following loss:
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$$
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\mathcal { L } ( \boldsymbol { \theta } ) = \frac { 1 } { | \mathcal { B } | } \sum _ { i = 1 } ^ { | \mathcal { B } | } \parallel f _ { \boldsymbol { \theta } } \big ( \tilde { \mathbf { y } } _ { i } \big ) - \mathbf { v } _ { \mathcal { B } _ { L } } \big ( \tilde { \mathbf { y } } _ { i } \big ) \parallel _ { 2 } ^ { 2 }
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$$
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We summarize the training process in Algorithm 1. In practice, we add a small constant $\gamma$ to the denominator of the normalized field to overcome the numerical issue when $\exists i , \left\| \tilde { \mathbf { x } } - \tilde { \mathbf { x } } _ { i } \right\| \approx 0$ .
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# 3.3 Backward ODE anchored by the additional dimension
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After estimating the normalized field $\mathbf { v }$ , we can sample from the data distribution by the backward ODE $d { \tilde { \mathbf { x } } } = \mathbf { v } ( { \tilde { \mathbf { x } } } ) d t$ . Nevertheless, the boundary condition of the above ODE is unclear: the starting and terminal time $t$ of the ODE are both unknown. To remedy the issue, we propose an equivalent backward ODE in which $\mathbf { x }$ evolves with the augmented variable $z$ :
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+
$$
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+
d ( \mathbf { x } , z ) = ( \frac { d \mathbf { x } } { d t } \frac { d t } { d z } d z , d z ) = ( \mathbf { v } ( \tilde { \mathbf { x } } ) _ { \mathbf { x } } \mathbf { v } ( \tilde { \mathbf { x } } ) _ { z } ^ { - 1 } , 1 ) d z
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$$
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Algorithm 1 Learning the normalized Poisson Field
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<table><tr><td>Input: Training iteration T,Initial model fe,dataset D,constant y,learning rate n. fort=1...Tdo</td></tr><tr><td>from BL |B|</td></tr><tr><td>Simulate the ODE: {yi = perturb(xi) Ji=1 Calculate the normalized field by BL: VB (yi)=-√NEB,(yi)/(ll EB (yi) Il2 +γ), ∀i</td></tr><tr><td>|l f(yi)-vBL(yi)l² i1</td></tr><tr><td>Update the model parameter: 0 = 0 - nVL(0)</td></tr><tr><td>end for</td></tr><tr><td>return fe</td></tr></table>
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# Algorithm 2 perturb $\cdot$
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<table><tr><td>Sample the power m ~U[0,M] Sample the initial noise (∈x,∈z)~N(O,σ²I(N+1)x(N+1))</td></tr><tr><td>Uniformly sample the vector from the unit ball u ~U(SN(1))</td></tr><tr><td>Construct training point y = x+ | x I (1 + 𝑇)mu, z = |∈zl(1 + T)m</td></tr><tr><td>return y = (y, z)</td></tr></table>
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where $\mathbf { v } ( \tilde { \mathbf { x } } ) _ { \mathbf { x } } , \mathbf { v } ( \tilde { \mathbf { x } } ) _ { z }$ are the corresponding components of $\mathbf x , z$ in vector $\mathbf { v } ( \tilde { \mathbf { x } } )$ . In the new ODE, we replace the time variable $t$ with the physically meaningful variable $z$ , permitting explicit starting and terminal conditions: when $z = 0$ , we arrive at the data distribution and we can freely choose a large $z _ { \mathrm { m a x } }$ as the starting point in the backward ODE. The backward ODE is compatible with general-purpose ODE solvers, e.g., RK45 method [23] and forward Euler method. The popular black-box ODE solvers, such as the one in Scipy library [37], typically use a common starting time for the same batch of samples. Since the distribution on the $z = z _ { \mathrm { m a x } }$ hyperplane is no longer uniform, we derive the prior distribution by radially projecting uniform distribution on the hemisphere with radius $r = z _ { \mathrm { m a x } }$ to the $z = z _ { \mathrm { m a x } }$ hyperplane:
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+
$$
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+
p _ { \mathrm { p r i o r } } ( \mathbf { x } ) = { \frac { 2 z _ { \mathrm { m a x } } ^ { N + 1 } } { S _ { N } { \big ( } z _ { \mathrm { m a x } } { \big ) } { \big ( } \| \mathbf { x } \| _ { 2 } ^ { 2 } + z _ { \mathrm { m a x } } ^ { 2 } { \big ) } ^ { \frac { N + 1 } { 2 } } } } = { \frac { 2 z _ { \mathrm { m a x } } } { S _ { N } { \big ( } 1 { \big ) } { \big ( } \| \mathbf { x } \| _ { 2 } ^ { 2 } + z _ { \mathrm { m a x } } ^ { 2 } { \big ) } ^ { \frac { N + 1 } { 2 } } } }
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$$
|
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+
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where $S _ { N } ( r )$ is the surface area of $N$ -sphere with radius $r$ . The reason behind the radial projection is that the Poisson field points in the radial direction at $r \infty$ . The new backward ODE also defines a bijective transformation between $p _ { \mathrm { p r i o r } } ( \mathbf { x } )$ on the infinite hyperplane $z _ { \mathrm { m a x } } \infty$ ) and the data distribution the norm (r $\tilde { p } ( \tilde { { \mathbf x } } )$ , analogous to Theorem ) from the distribution: $p _ { \mathrm { p r i o r } } ( \mathbf { x } )$ to sampleand then $p _ { \mathrm { r a d i u s } } ( \| \textbf { x } \| _ { 2 } ) \propto \| \textbf { x } \| _ { 2 } ^ { N - 1 } / ( \| \textbf { x } \| _ { 2 } ^ { 2 } + z _ { \operatorname* { m a x } } ^ { 2 } ) ^ { \frac { N + 1 } { 2 } }$ uniformly sample its angle. We provide detailed derivations and practical sampling procedure in Appendix A.4. We further achieve exponential decay on the $z$ dimension by introducing a new variable $t ^ { \prime }$ :
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+
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+
$$
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+
\begin{array} { r l } { [ \mathrm { B a c k w a r d ~ O D E } ] } & { { } d ( \mathbf { x } , z ) = ( \mathbf { v } ( \tilde { \mathbf { x } } ) _ { \mathbf { x } } \mathbf { v } ( \tilde { \mathbf { x } } ) _ { z } ^ { - 1 } z , z ) d t ^ { \prime } } \end{array}
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+
$$
|
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+
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+
The $z$ component in the backward ODE, i.e., $d z = z d t ^ { \prime }$ , can be solved by $z = e ^ { t ^ { \prime } }$ . Since $z$ reaches zero as $t ^ { \prime } \to - \infty$ , we instead choose a tiny positive number $z _ { \mathrm { m i n } }$ as the terminal condition. The corresponding starting/terminal time of the variable $t ^ { \prime }$ are $\log z _ { \operatorname* { m a x } } / \log z _ { \operatorname* { m i n } }$ respectively. Empirically, this simple change of variable leads to $2 \times$ faster sampling with almost no harm to the sample quality. In addition, we substitue the predicted $\mathbf { v } ( \tilde { \mathbf { x } } ) _ { z }$ with a more accurate one when $z$ is small (Appendix B.2.3). We defer more details of the simulation of backward ODE to Appendix B.2.
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# 4 Generative Modeling via the Backward ODE
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In this section, we demonstrate the effectiveness of the backward ODE associated with PFGM on image generation tasks. In Section 4.1, we show that PFGM achieves currently best in class performance in the normalizing flow family. In comparison to the existing state-of-the-art SDE or MCMC approaches, PFGM exhibits $1 0 \times$ or $2 0 \times$ acceleration while maintaining competitive or higher generation quality. Meanwhile, unlike existing ODE baselines that heavily rely on corrector to generate decent samples on weaker architectures, PFGM exhibits greater stability against error (Section 4.2). Finally, we show that PFGM is robust to the step size in the Euler method (Section 4.3), and its associated ODE allows for likelihood evaluation and image manipulation by editing the latent space (Section 4.4).
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Table 1: CIFAR-10 sample quality (FID, Inception) and number of function evaluation (NFE).
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+
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<table><tr><td></td><td>Invertible?</td><td>Inception ↑</td><td>FID↓</td><td>NFE↓</td></tr><tr><td>PixelCNN[36]</td><td></td><td>4.60</td><td>65.9</td><td>1024</td></tr><tr><td>IGEBM[8]</td><td></td><td>6.02</td><td>40.6</td><td>60</td></tr><tr><td>ViTGAN [24]</td><td></td><td>9.30</td><td>6.66</td><td>1</td></tr><tr><td>StyleGAN2-ADA [17]</td><td></td><td>9.83</td><td>2.92</td><td>1</td></tr><tr><td>StyleGAN2-ADA (cond.) [17]</td><td>xxxxxxxxxx</td><td>10.14</td><td>2.42</td><td>1</td></tr><tr><td>NCSN[31]</td><td></td><td>8.87</td><td>25.32</td><td>1001</td></tr><tr><td>NCSNv2 [32]</td><td></td><td>8.40</td><td>10.87</td><td>1161</td></tr><tr><td>DDPM[16]</td><td></td><td>9.46</td><td>3.17</td><td>1000</td></tr><tr><td>NCSN++ VE-SDE[33]</td><td></td><td>9.83</td><td>2.38</td><td>2000</td></tr><tr><td>NCSN++ deep VE-SDE [33]</td><td></td><td>9.89</td><td>2.20</td><td>2000</td></tr><tr><td>Glow [19]</td><td></td><td>3.92</td><td>48.9</td><td>1</td></tr><tr><td>DDIM,T=50 [30]</td><td></td><td>-</td><td>4.67</td><td>50</td></tr><tr><td>DDIM, T=100 [30]</td><td></td><td>1</td><td>4.16</td><td>100</td></tr><tr><td>NCSN++ VE-ODE [33]</td><td></td><td>9.34</td><td>5.29</td><td>194</td></tr><tr><td>NCSN++ deep VE-ODE[33]</td><td></td><td>9.17</td><td>7.66</td><td>194</td></tr><tr><td colspan="5">DDPM++backbone</td></tr><tr><td>VP-SDE[33]</td><td></td><td>9.58</td><td>2.55</td><td>1000</td></tr><tr><td>sub-VP-SDE[33]</td><td>xx-</td><td>9.56</td><td>2.61</td><td>1000</td></tr><tr><td>VP-ODE [33]</td><td></td><td>9.46</td><td>2.97</td><td>134</td></tr><tr><td>sub-VP-ODE [33]</td><td></td><td>9.30</td><td>3.16</td><td>146</td></tr><tr><td>PFGM (ours)</td><td></td><td>9.65</td><td>2.48</td><td>104</td></tr><tr><td colspan="5">DDPM++ deep backbone</td></tr><tr><td>VP-SDE [33]</td><td></td><td>9.68</td><td>2.41</td><td>1000</td></tr><tr><td>sub-VP-SDE[33]</td><td>xx-</td><td>9.57</td><td>2.41</td><td>1000</td></tr><tr><td>VP-ODE [33]</td><td></td><td>9.47</td><td>2.86</td><td>134</td></tr><tr><td>sub-VP-ODE [33]</td><td></td><td>9.40</td><td>3.05</td><td>146</td></tr><tr><td>PFGM (ours)</td><td></td><td>9.68</td><td>2.35</td><td>110</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>
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+
# 4.1 Efficient image generation by PFGM
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Setup For image generation tasks, we consider the CIFAR-10 [22], CelebA $6 4 \times 6 4$ [38] and LSUN bedroom $2 5 6 \times 2 5 6$ [39]. Following [32], we first center-crop the CelebA images and then resize them to $6 4 \times 6 4$ . We choose $M \ : = \ : 2 9 1$ CIFAR-10 and CelebA 356 LSUN bedroom , $\sigma ~ = ~ 0 . 0 1$ and $\tau { \it \Delta \phi } = 0 . 0 3$ for the perturbation Algorithm 2, and $z _ { \mathrm { m i n } } ~ = ~ 1 e \mathrm { ~ - ~ } 3$ , $\begin{array} { r l } { z _ { \operatorname* { m a x } } } & { { } = } \end{array}$ 40 CIFAR-10 60 CelebA $6 4 ^ { 2 }$ 100 LSUN bedroom for the backward ODE. We further clip the norms of initial samples into $( 0 , 3 0 0 0 )$ for CIFAR-10, $( 0 , 6 0 0 0 )$ for CelebA $6 4 ^ { 2 }$ and $( 0 , 3 0 0 0 0 )$ for LSUN bedroom. We adopt the $\mathrm { { D D P M + + } }$ and $\mathrm { { D D P M + + } }$ deep architectures [33] as our backbones. We add the scalar $z$ (resp. predicted direction on $z$ ) as input (resp. output) to accommodate the additional dimension. We take the same set of hyper-parameters, such as batch size, learning rate and training iterations from [33]. We provide more training details in Appendix B.1, and discuss how to set these hyper-parameters for general datasets in B.1.1 and B.2.1.
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Baselines We compare PFGM to modern autoregressive model [36], GAN [17, 24], normalizing flow [19] and EBM [8]. We also compare with variants of score-based models such as DDIM [30] and current state-of-the-art SDE/ODE methods [33]. We denote the methods that use forward-time SDEs in [33] such as Variance Exploding (VE) SDE/Variance Preserving (VP) SDE/ sub-Variance Preserving (sub-VP), and the corresponding backward SDE/ODE, as A-B, where $\mathbf { A } \in \{ \mathrm { V E }$ , VP, sub- $\mathrm { V P } \}$ and $\mathbf { B } \in \{ \mathrm { S D E } , \mathrm { O D E } \}$ . We follow the model selection protocol in [33], which selects the checkpoint with the smallest FID score over the course of training every 50k iterations.
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Figure 3: Uncurated samples on datasets of increasing resolution. From left to right: CIFAR-10 $3 2 \times 3 2$ , CelebA $6 4 \times 6 4$ and LSUN bedroom $2 5 6 \times 2 5 6$ .
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Numerical Solvers The backward ODE (Eq. (6)) is compatible with any general purpose ODE solver. In our experiments, the default solver of ODEs is the black box solver in the Scipy library [37] with the RK45 [7] method (RK45), unless otherwise specified. For VE/VP/subVP-SDEs, we use the predictor-corrector (PC) sampler introduced in [33]. For VP/sub-VP-SDEs, we apply the predictor-only sampler, because its performance is on par with the PC sampler while requiring half computation.
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Results For quantitative evaluation on CIFAR-10, we report the Inception [29] (higher is better) and FID [13] scores (lower is better) in Table 1. We also include our preliminary experimental results on a weaker architecture NCSNv2 [32] in Appendix D.2. We measure the inference speed by the average NFE (number of function evaluation). We also explicitly indicate which methods belong to the invertible flow family.
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Our main findings are: (1) PFGM achieves the best Inception scores and FID scores among the normalizing flow models. Specifically, PFGM obtains an Inception score of 9.68 and a FID score of 2.48 using the $\mathrm { D D P M + + }$ deep architecture. To our best knowledge, these are the highest FID and Inception scores by flow models on CIFAR-10. (2) PFGM achieves a $1 0 \times \sim 2 0 \times$ faster inference speed than the SDE methods using the similar architectures, while retaining comparable sample quality. As shown in Table 1, PFGM requires NFEs of 110 whereas the SDE methods typically use $1 0 0 0 \sim 2 0 0 0$ inference steps. PFGM outperforms all the baselines on $\mathrm { { D D P M + + } }$ in all metrics. In addition, PFGM generally samples faster than other ODE baselines with the same RK45 solver. (3) The backward ODE in PFGM is compatible with architectures with varying capacities. PFGM consistently outperforms other ODE baselines on $\mathrm { D D P M + + }$ (Table 1) or NCSNv2 (Appendix D.2) backbones. (4) PFGM shows scalability to higher resolution datasets. In Appendix D.1, we show that PFGM are capable of scale-up to LSUN bedroom $2 5 6 \times 2 5 6$ . In particular, PFGM has comparable performance with VE-SDE with $1 5 \times$ fewer NFE.
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In Fig. 3, we visualize the uncurated samples from PFGM on CIFAR-10, CelebA $6 4 \times 6 4$ and LSUN bedroom $2 5 6 \times 2 5 6$ . We provides more samples in Appendix E.
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# 4.2 Failure of VE/VP-ODEs on NCSNv2 architecture
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In our preliminary experiments on NCSNv2 architectures, we empirically observe that the VE/VP-ODEs have FID scores greater than 90 on CIFAR-10. In particular, VE/VP-ODEs can only generate decent samples when applying the Langevin dynamics corrector, and even then, their performances are still inferior to PFGM (Table 9, Table 10). The poor performance on NCSNv2 stands in striking contrast to their high sample quality on $\mathrm { N C S N + + / D D P M + + }$ in [33]. It indicates that the VE/VP-ODEs are
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Figure 4: Sample norm distributions with varying time variables $\sigma$ for VE-ODE and $z$ for PFGM)
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Figure 5: (a) Norm- $\cdot \sigma ( t )$ relation during the backward sampling of VE-ODE (Euler). (b) Norm- $z ( t ^ { \prime } )$ relation during the backward sampling of PFGM (Euler). The shaded areas mean the standard deviation of norms. (c) Number of steps versus FID score.
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more susceptible to estimation errors than PFGM. We hypothesize that the strong norm- $\sigma$ correla tion seen during the training of score-based models causes the problem.
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For score-based models, the $l _ { 2 }$ norms of perturbed training samples and the standard deviations $\sigma ( t )$ of Gaussian noises have strong correlation, e. $g . , l _ { 2 } \ \mathrm { n o r m } \approx \sigma ( t ) \sqrt { N }$ for large $\sigma ( t )$ in VE [33]. In contrast, as shown in Fig. 4, PFGM allocates high mass across a wide spectrum of the training sample norms. During sampling, VE/VP-ODEs could break down when the trajectories of backward ODEs deviate from the norm- $\cdot \sigma ( t )$ relation to which most training samples pertain. The weaker NCSNv2 backbone incurs larger errors and thus leads to their failure. The PFGM is more resistant to estimate errors because of the greater range of training sample norms.
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To further verify the hypothesis above, we split a batch of VE-ODE samples into cleaner and noisier samples according to visual quality (Fig. 8(a)). In Fig. 5(a), we investigate the relation for cleaner and noisier samples during the forward Euler simulation of VE-ODE when $\sigma ( t ) < 1 5$ . We can see that the trajectory of cleaner samples stays close to the norm- $\sigma ( t )$ relation (the red dash line), whereas that of the noisier samples diverges from the relation. The Langevin dynamics corrector changes the trajectory of noisier samples to align with the relation. Fig. 5(b) further shows that the anchored variable $z ( t ^ { \prime } )$ and the norms in the backward ODE of PFGM are not strongly correlated, giving rise to the robustness against the imprecise estimation on NCSNv2. We defer more details to Appendix C.
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# 4.3 Effects of step size in the forward Euler method
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In order to accelerate the inference speed of ODEs, we can increase the step size (decrease the NFEs) in numerical solvers such as the forward Euler method. It also enables the trade-off between sample quality and computational efficiency in real-world deployment. We study the effects of increasing step size on PFGM, VP-ODE and DDIM [30] using the forward Euler method, with a varying NFE ranging from 10 to 100.
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In Fig. 5(c), we report the sample quality measured by FID scores on CIFAR-10. As expected, all the methods have higher FID scores when decreasing the NFE. We observe that the sample quality of PFGM degrades gracefully as we decrease the NFE. Our method shows significantly better robustness to step sizes than the VP-ODE, especially when only taking a few Euler steps. In addition, PFGM obtains better FID scores than DDIM on most NFEs except for 10 where PFGM is marginally worse. This suggests that the PFGM is a promising method for accommodating instantaneous resource availability, as high-quality samples can be generated in limited steps.
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# 4.4 Utilities of ODE: likelihood evaluation and latent representation
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Similar to the family of discrete normalizing flows [6, 19, 14] and continuous probability flow [33], the forward ODE in PFGM defines an invertible mapping between the data space and latent space with a known prior. Formally, we define the invertible forward $\mathcal { M }$ mapping by integrating the corresponding forward ODE $\begin{array} { r } { \dot { d } ( \mathbf { x } , z ) = ( \mathbf { v } ( \tilde { \mathbf { x } } ) _ { \mathbf { x } } \mathbf { v } ( \tilde { \mathbf { x } } ) _ { z } ^ { - 1 } z , z ) d t ^ { \prime } } \end{array}$ of Eq. (6):
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$$
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\mathbf { x } ( \log z _ { \operatorname* { m a x } } ) = \mathcal { M } ( \mathbf { x } ( \log z _ { \operatorname* { m i n } } ) ) \equiv \mathbf { x } ( \log z _ { \operatorname* { m i n } } ) + \int _ { \log z _ { \operatorname* { m i n } } } ^ { \log z _ { \operatorname* { m a x } } } \mathbf { v } ( \mathbf { x } ( t ^ { \prime } ) ) _ { \mathbf { x } } \mathbf { v } ( \tilde { \mathbf { x } } ( t ^ { \prime } ) ) _ { z } ^ { - 1 } e ^ { t ^ { \prime } } d t ^ { \prime }
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$$
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where $\log z _ { \mathrm { m i n } } / \log z _ { \mathrm { m a x } }$ are the starting/terminal time in the forward ODE. The forward mapping transfers the data distribution to the prior distribution $p _ { \mathrm { p r i o r } }$ on the $z = z _ { \mathrm { m a x } }$ hyperplane (cf. Section 3.3): $p _ { \mathrm { p r i o r } } \big ( \mathbf { x } \big ( \log z _ { \mathrm { m a x } } \big ) \big ) = \mathcal { M } \big ( p ( \mathbf { x } ( \log \bar { z } _ { \mathrm { m i n } } ) ) \big )$ . The invertibility enables likelihood evaluation and creates a meaningful latent space on the $z = z _ { \mathrm { m a x } }$ hyperplane. In addition, we can adapt to the computational constraints by adjusting the step size or the precision in numerical ODE solvers.
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Likelihood evaluation We evaluate the data likelihood by the instantaneous change-of-variable formula [4, 33]. In Table 2, we report the bits/dim on the uniformly dequantized CIFAR-10 test set and compare with existing baselines that use the same setup. We observe that PFGM achieves better likelihoods than discrete normalizing flow models, even without maximum likelihood training. Among the continuous flow models, sub-VP-ODE shows the lowest bits/dim, although its sample quality is worse than VP-ODE and PFGM (Table 1). The exploration of the seeming trade-off between likelihood and sample quality is left for future works.
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Table 2: Bits/dim on CIFAR-10
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<table><tr><td></td><td>bits/dim ↓</td></tr><tr><td>RealNVP [6]</td><td>3.49</td></tr><tr><td>Glow [19] Residual Flow [3]</td><td>3.35 3.28</td></tr><tr><td>Flow++ [14]</td><td>3.29</td></tr><tr><td>DDPM(L)[16]</td><td>≤3.70*</td></tr><tr><td>DDPM++backbone</td><td></td></tr><tr><td>VP-ODE [33]</td><td>3.20</td></tr><tr><td>sub-VP-ODE[33]</td><td>3.02</td></tr><tr><td>PFGM (ours)</td><td>3.19</td></tr></table>
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Latent representation Since the samples are uniquely identifiable by their latents via the invertible mapping $\mathcal { M }$ , PFGM further supports image manipulation using its latent representation on the $z =$ $z _ { \mathrm { m a x } }$ hyperplane. We include the results of image interpolation and the temperature scaling [6, 19, 33] to Appendix D.4 and Appendix D.5. For interpolation, it shows that we can travel along the latent space to obtain perceptually consistent interpolations between CelebA images.
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# 5 Conclusion
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We present a new deep generative model by solving the Poisson equation whose source term is the data distribution. We estimate the normalized gradient field of the solution in an augmented space with an additional dimension. For sampling, we devise a backward ODE that exponential decays on the physically meaningful additional dimension. Empirically, our approach has currently best performance over other normalizing flow baselines, and achieving $1 0 \times$ to $2 0 \times$ acceleration over the stochastic methods. Our backward ODE shows greater stability against errors than popular ODE-based methods, and enables efficient adaptive sampling. We further demonstrate the utilities of the forward ODE on likelihood evaluation and image interpolation. Future directions include improving the normalization of Poisson fields. More principled approaches can be used to get around the divergent near-field behavior. For example, we may exploit renormalization, a useful tool in physics, to make the Poisson field well-behaved in near fields.
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# Acknowledgements
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We are grateful to Shangyuan Tong, Timur Garipov and Yang Song for helpful discussion. We would like to thank Octavian Ganea and Wengong Jin for reviewing an early draft of this paper. YX and TJ acknowledge support from MIT-DSTA Singapore collaboration, from NSF Expeditions grant (award 1918839) ”Understanding the World Through Code”, and from MIT-IBM Grand Challenge project. ZL and MT would like to thank the Center for Brains, Minds, and Machines (CBMM) for hospitality. ZL and MT are supported by The Casey and Family Foundation, the Foundational Questions Institute, the Rothberg Family Fund for Cognitive Science and IAIFI through NSF grant PHY-2019786.
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# Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes] See Appendix G.
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(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix H.
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix A. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.
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(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] See the abstract.
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix B.1.
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]
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(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] All the experiments are run on a single NVIDIA A100 GPU.
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [N/A] The assets are public/open-source datasets and codes.
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(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
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(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(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": "Poisson Flow Generative Models ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
300,
|
| 8 |
+
122,
|
| 9 |
+
696,
|
| 10 |
+
147
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Yilun Xu∗ , Ziming Liu∗ , Max Tegmark, Tommi Jaakkola Massachusetts Institute of Technology ylxu, zmliu, tegmark @mit.edu; tommi@csail.mit.edu ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
279,
|
| 19 |
+
195,
|
| 20 |
+
720,
|
| 21 |
+
238
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Abstract ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
462,
|
| 31 |
+
275,
|
| 32 |
+
535,
|
| 33 |
+
291
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We propose a new “Poisson flow” generative model (PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the $z = 0$ hyperplane in a space augmented with an additional dimension $z$ , generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the $z \\ = \\ 0$ plane transforms into a distribution on the hemisphere of radius $r$ that becomes uniform in the $r \\infty$ limit. To learn the bijective transformation, we estimate the normalized field in the augmented space. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the (unaugmented) data manifold when the $z$ reaches zero. Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of 9.68 and a FID score of 2.35. It also performs on par with the state-of-the-art SDE approaches while offering $1 0 \\times$ to $2 0 \\times$ acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method. The code is available at https: //github.com/Newbeeer/poisson_flow. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
306,
|
| 43 |
+
766,
|
| 44 |
+
553
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
174,
|
| 54 |
+
578,
|
| 55 |
+
310,
|
| 56 |
+
595
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Deep generative models are a prominent approach for data generation, and have been used to produce high quality samples in image [1], text [2] and audio [35], as well as improve semi-supervised learning [20], domain generalization [25] and imitation learning [15]. However, current deep generative models also have limitations, such as unstable training objectives (GANs [1, 12, 17]) and low sample quality (VAEs [21], normalizing flows [6]). New techniques [12, 24] are introduced to stablize the training of CNN-based or ViT-based GAN models. Although recent advances on diffusion [16] and scored-based models [33] achieve comparable sample quality to GAN’s without adversarial training, these models have a slow stochastic sampling process. [33] proposes backward ODE samplers (normalizing flow) that speed up the sampling process but these methods have not yet performed on par with the SDE counterparts. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
609,
|
| 66 |
+
825,
|
| 67 |
+
747
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "We present a new “Poisson flow” generative model (PFGM), exploiting a remarkable physics fact that generalizes to $N$ dimensions. As illustrated in Fig. 1(a), motion in a viscous fluid transforms any planar charge distribution into a uniform angular distribution. Specifically, we interpret $N$ - dimensional data points $\\mathbf { x }$ (images, say) as positive electric charges in the $z \\ = \\ 0$ plane of an $N + 1$ -dimensional space (see Fig. 1(a)) filled with a viscous liquid (say honey). A positive charge with $z > 0$ will be repelled by the other charges and move in the direction of their repulsive force, eventually crossing an imaginary hemisphere of radius $r$ . We show that, remarkably, if the the original charge distribution is let loose just above $z = 0$ , this law of motion will cause a uniform distribution for their hemisphere crossings in the $r \\infty$ limit. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
753,
|
| 77 |
+
825,
|
| 78 |
+
878
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "image",
|
| 84 |
+
"img_path": "images/54726ebf34e46087bef020e3ddcf1a0c1fce7fffed017e15273765115d92e791.jpg",
|
| 85 |
+
"image_caption": [
|
| 86 |
+
"Figure 1: (a) 3D Poisson field trajectories for a heart-shaped distribution (b) The evolvements of a distribution (top) or an (augmented) sample (bottom) by the forward/backward ODEs pertained to the Poisson field. "
|
| 87 |
+
],
|
| 88 |
+
"image_footnote": [],
|
| 89 |
+
"bbox": [
|
| 90 |
+
171,
|
| 91 |
+
84,
|
| 92 |
+
802,
|
| 93 |
+
270
|
| 94 |
+
],
|
| 95 |
+
"page_idx": 1
|
| 96 |
+
},
|
| 97 |
+
{
|
| 98 |
+
"type": "text",
|
| 99 |
+
"text": "Our Poisson flow generative process reverses the forward process: we generate a uniform distribution of negative charges on the hemisphere, then track their motion back to the $z = 0$ plane, where they will be distributed as the data distribution. A Poisson flow can be viewed as a type of continuous normalizing flows [4, 10, 33] in the sense that it continuously maps between an arbitrary distribution and an easily sampled one: in the previous works an $N$ -dimensional Gaussian and in PFGM a uniform distribution on an $N$ -dimensional hemisphere. In practice, we implement the Poisson flow by solving a pair of forward/backward ordinary differential equations (ODEs) induced by the electric field (Fig. 1(b)) given by the $N$ -dimensional version of Coulomb’s law (the gradient of the solution to the Poisson’s equation with the data as sources). We will interchangeably refer to this gradient as the Poisson field, since electric fields normally refer to the special case $N = 3$ . ",
|
| 100 |
+
"bbox": [
|
| 101 |
+
173,
|
| 102 |
+
351,
|
| 103 |
+
825,
|
| 104 |
+
489
|
| 105 |
+
],
|
| 106 |
+
"page_idx": 1
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "The proposed generative model PFGM has a stable training objective and empirically outperforms previously state-of-the-art continuous flow methods [30, 33]. As a different iterative method, PFGM offers two advantages compared to score-based methods [32, 33]. First, the ODE process of PFGM achieves faster sampling speeds than the SDE samplers in [33]. while retaining comparable performance. Second, our backward ODE exhibits better generation performance than the reverse-time ODEs of VE/VP/sub-VP SDEs [33], as well as greater stability on a weaker architecture NSCNv2 [32]. The rationale for robustness is that the time variables in these ODE baselines are strongly correlated with the sample norms during training time, resulting in a less error-tolerant inference. In contrast, the tie between the anchored variable and the sample norm in PFGM is much weaker. ",
|
| 111 |
+
"bbox": [
|
| 112 |
+
173,
|
| 113 |
+
496,
|
| 114 |
+
825,
|
| 115 |
+
621
|
| 116 |
+
],
|
| 117 |
+
"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "Experimentally, we show that PFGM achieves current state-of-the-art performance on CIFAR-10 dataset in the normalizing flow family, with FID/Inception scores of $2 . { \\bar { 4 } } 8 / 9 . 6 5$ (w/ $\\mathrm { { D D P M + + } }$ [33]) and $2 . 3 5 / 9 . 6 8$ (w/ $\\mathrm { { D D P M + + } }$ deep [33]). It performs competitively with current state-of-the-art SDE samplers [33] and provides $1 0 \\times$ to $2 0 \\times$ speed up across datasets. Notably, the backward ODE in PFGM is the only ODE-based sampler that can produce decent samples on its own on NCSNv2 [32], while other ODE baselines fail without corrections. In addition, PFGM demonstrates the robustness to the step size in the Euler method, with a varying number of function evaluations (NFE) ranging from 10 to 100. We further showcase the utility of the invertible forward/backward ODEs of the Poisson field on likelihood evaluation and image manipulations, and its scalability to higher resolution images on LSUN bedroom $2 5 6 \\times 2 5 6$ dataset. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
173,
|
| 124 |
+
626,
|
| 125 |
+
825,
|
| 126 |
+
765
|
| 127 |
+
],
|
| 128 |
+
"page_idx": 1
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "2 Background and Related works ",
|
| 133 |
+
"text_level": 1,
|
| 134 |
+
"bbox": [
|
| 135 |
+
174,
|
| 136 |
+
784,
|
| 137 |
+
470,
|
| 138 |
+
801
|
| 139 |
+
],
|
| 140 |
+
"page_idx": 1
|
| 141 |
+
},
|
| 142 |
+
{
|
| 143 |
+
"type": "text",
|
| 144 |
+
"text": "Poisson equation Let $\\mathbf { x } \\in \\mathbb { R } ^ { N }$ and $\\rho ( \\mathbf { x } ) : \\mathbb { R } ^ { N } \\mathbb { R }$ be a source function. We assume that the source function has a compact support, $\\rho \\in \\mathcal { C } ^ { 0 }$ and $N \\geq 3$ . The Poisson equation is ",
|
| 145 |
+
"bbox": [
|
| 146 |
+
169,
|
| 147 |
+
815,
|
| 148 |
+
823,
|
| 149 |
+
845
|
| 150 |
+
],
|
| 151 |
+
"page_idx": 1
|
| 152 |
+
},
|
| 153 |
+
{
|
| 154 |
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"type": "equation",
|
| 155 |
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"img_path": "images/5c3b5ccc1ca4ab2e1f96ee83cf1cfbe378716b7b8c4e5fe11bd0803b24e03f54.jpg",
|
| 156 |
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"text": "$$\n\\nabla ^ { 2 } \\varphi ( \\mathbf { x } ) = - \\rho ( \\mathbf { x } ) ,\n$$",
|
| 157 |
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"text_format": "latex",
|
| 158 |
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"bbox": [
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"text": "where $\\varphi ( \\mathbf { x } ) : \\mathbb { R } ^ { N } \\mathbb { R }$ is called the potential function, and $\\begin{array} { r } { \\bigtriangledown ^ { 2 } \\equiv \\sum _ { i = 1 } ^ { N } \\frac { \\partial ^ { 2 } } { \\partial x _ { i } ^ { 2 } } } \\end{array}$ is the Laplacian operator. It is usually helpful to define the gradient field $\\mathbf { E } ( \\mathbf { x } ) = - \\nabla \\varphi ( \\mathbf { x } )$ and rewrite the Poisson equation as $\\nabla \\cdot \\mathbf { E } = \\rho$ , known in physics as Gauss’s law [11]. The Poisson equation is widely used in physics, giving rise to Newton’s gravitational theory [9] and the electrostatic theory [11], when $\\rho ( \\mathbf { x } )$ is interpreted as mass density or electric charge density, respectively. $\\mathbf { E }$ is the $N$ -dimensional analog of the electric field. The Poisson equation Eq. (1) (with zero boundary condition at infinity) admits a unique simple integral solution 2: ",
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"text": "$$\n\\varphi ( \\mathbf { x } ) = \\int G ( \\mathbf { x } , \\mathbf { y } ) \\rho ( \\mathbf { y } ) d \\mathbf { y } , \\quad G ( \\mathbf { x } , \\mathbf { y } ) = \\frac { 1 } { ( N - 2 ) S _ { N - 1 } ( 1 ) } \\frac { 1 } { | | \\mathbf { x } - \\mathbf { y } | | ^ { N - 2 } } ,\n$$",
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"text_format": "latex",
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"text": "where $S _ { N - 1 } ( 1 )$ is a geometric constant representing the surface area of the unit $( N - 1 )$ -sphere 3, and $G ( \\mathbf { x } , \\mathbf { y } )$ is the extension of Green’s function in $N$ -dimensional space (details in Appendix A.3). The negative gradient field of $\\varphi ( \\mathbf x )$ , referred as Poisson field of the source $\\rho$ , is ",
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"text": "$$\n\\mathbf { E } ( \\mathbf { x } ) = - \\nabla \\varphi ( \\mathbf { x } ) = - \\int \\nabla _ { \\mathbf { x } } G ( \\mathbf { x } , \\mathbf { y } ) \\rho ( \\mathbf { y } ) d \\mathbf { y } , \\quad \\nabla _ { \\mathbf { x } } G ( \\mathbf { x } , \\mathbf { y } ) = - \\frac { 1 } { S _ { N - 1 } ( 1 ) } \\frac { \\mathbf { x } - \\mathbf { y } } { \\left\\| \\mathbf { x } - \\mathbf { y } \\right\\| ^ { N } } .\n$$",
|
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"text_format": "latex",
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"text": "Qualitatively, the Poisson field $\\mathbf { E } ( \\mathbf { x } )$ points away from sources, or equivalently $- \\mathbf { E } ( \\mathbf { x } )$ points towards sources, as illustrated in Fig. 1. It is straightforward to check that when $\\rho ( { \\bf x } ) \\delta ( { \\bf x - y } )$ , we get $\\varphi ( \\mathbf x ) G ( \\mathbf x , \\mathbf y )$ and $\\mathbf { E } ( \\mathbf { x } ) - \\nabla _ { \\mathbf { x } } G ( \\mathbf { x } , \\mathbf { y } )$ . This implies that $G ( \\mathbf { x } , \\mathbf { y } )$ and $- \\nabla _ { \\mathbf { x } } G ( \\mathbf { x } , \\mathbf { y } )$ can be interpreted as the potential function and the gradient field generated by a unit point source, e.g., a point charge, located at $\\mathbf { y }$ . When $\\rho ( \\mathbf { x } )$ takes general forms but has bounded support, simple asymptotics exist for $\\left\\| \\mathbf { x } \\right\\| \\gg \\left\\| \\mathbf { y } \\right\\|$ . To the lowest order, $\\mathbf { E ( x ) } = \\nabla _ { \\mathbf { x } } G ( \\mathbf { x } , \\mathbf { y } ) | _ { \\mathbf { y } = \\mathbf { 0 } } \\sim \\mathbf { x } / \\| \\mathbf { x } \\| ^ { N }$ behaves as if it were generated by a unit point source at $\\mathbf y = 0$ . In physics, the power law decay is considered to be long-range (compared to exponential decay) [11]. ",
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"text": "Particle dynamics in a Poisson field The Poisson field immediately defines a flow model, where the probability distribution evolves according to the gradient flow $\\partial p _ { t } ( \\mathbf { x } ) / \\partial t = - \\nabla \\cdot ( p _ { t } ( \\mathbf { x } ) \\mathbf { E } ( \\mathbf { x } ) )$ . The gradient flow is a special case of the Fokker-Planck equation [28], where the diffusion coefficient is zero. Intuitively we can think of $p _ { t } ( \\mathbf { x } )$ as represented by a population of particles. The corresponding (non-diffusion) case of the Ito process is the forward ODE ˆ $\\begin{array} { r } { \\frac { d \\mathbf { x } } { d t } \\ = \\ \\mathbf { E } ( \\mathbf { x } ) } \\end{array}$ . We can interpret the trajectories of the ODE as particles moving according to the Poisson field $\\mathbf { E } ( x )$ , with initial states drawn from $p _ { 0 }$ . The physical picture of the forward ODE is a charged particle under the influence of electric fields in the overdamped limit (details in Appendix F). ",
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"text": "The dynamics is also rescalable in the sense that the particle trajectory remains the same for $\\pm f ( \\mathbf { x } ) \\mathbf { E } ( \\mathbf { x } )$ for $f ( \\mathbf { x } ) > 0 , f ( \\mathbf { x } ) \\in \\mathcal { C } ^ { 1 }$ , because the time rescaling $d t \\to f ( \\mathbf { x } ( t ) ) d t$ recovers $\\begin{array} { r } { \\frac { d \\mathbf { x } } { d t } \\ = } \\end{array}$ $\\begin{array} { r } { \\frac { d { \\mathbf { x } } } { d t } = } \\end{array}$ $\\pm \\mathbf { E } ( \\mathbf { x } )$ . Note that the dynamics is stiff due to the power law factor in the denominator in Eq. (3), posing computational challenges. Luckily the rescalablility allows us to rescale $\\mathbf { E } ( \\mathbf { x } )$ properly to get new ODEs (formally defined later in Section 3.3) that are more amenable for sampling. ",
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"text": "Generative Modeling via ODE Generative modeling can be done by transforming a base distribution to a data distribution via mappings defined by ODEs. The ODE-based samplers allow for adaptive sampling, exact likelihood evaluation and modeling of continuous-time dynamics [4, 33]. Previous works broadly fall into two lines. [4, 3] introduce a continuous-time normalizing flow model that can be trained with maximum likelihood by the instantaneous change-of-variables formula [4]. For sampling, they directly integrate the learned invertiable mapping over time. Another work [33] unifies the scored-based model [31, 32] and diffusion model [16] into a general diffusion process, and uses the reverse-time ODE of the diffusion process for sampling. They show that the reverse-time ODE produces high quality samples with improved architecture. ",
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"type": "text",
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"text": "3 Poisson Flow Generative Models ",
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"text": "In this section, we start with the properties of the Poisson flow in the augmented space and show how to draw samples from the data distribution by following the backward ODE of the Poisson flow (Section 3.1). We then discuss how to actually learn a normalized Poisson field from data samples through simulations of the forward ODE (Section 3.2) and present an equivalent backward ODE that allows for exponentially decay on $z$ (Section 3.3). ",
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"type": "image",
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"img_path": "images/e2f896cb7811162de5ce22a937be497e9cd000c7801fea03b15b727a4f369139.jpg",
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"image_caption": [
|
| 296 |
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"Figure 2: (a) Poisson field (black arrows) and particle trajectories (blue lines) of a 2D uniform disk (red). Left (no augmentation, 2D): all particles collapse to the disk center. Right (augmentation, 3D): particles hit different points on the disk. (b) Proof idea of Theorem 1. By Gauss’s Law, the outflow flux $d \\Phi _ { o u t }$ equals the inflow flux $d \\Phi _ { i n }$ . The factor of two in $p ( \\mathbf { x } ) d A / 2$ is due to the symmetry of Poisson fields in $z < 0$ and $z > 0$ . "
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"type": "text",
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"text": "3.1 Augment the data with additional dimension ",
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"text": "We wish to generate samples $\\mathbf { x } \\in \\mathbb { R } ^ { N }$ from a distribution $p ( \\mathbf { x } )$ supported on a bounded region. We may set the source $\\rho ( \\mathbf { x } ) = p ( \\mathbf { x } ) \\in \\mathcal { C } ^ { 0 \\ }$ 4 and compute the resulting gradient field $\\mathbf { E } ( \\mathbf { x } )$ from Eq. (3). Since $- \\mathbf { E } ( \\mathbf { x } )$ points towards sources, the backward ODE ${ d { \\bf x } } / { d t } = - { \\bf E } ( { \\bf x } )$ will take samples close to the sources. One may naively hope that the backward ODE is a generative model that recovers $p ( \\mathbf { x } )$ . Unfortunately, the backward ODE has the problem of mode collapse. We illustrate this phenomenon with a 2D uniform disk. The reverse Poisson field $- \\mathbf { E } ( \\mathbf { x } )$ on the 2D $( x , y )$ -plane points towards the center of the disk $O$ (Fig. 2(a) left), so all particle trajectories (blue lines) will eventually hit $O$ . If we instead add an additional dimension $z$ (Fig. 2(a) right), particles can hit different points on the disk and faithfully recover the data distribution. ",
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"text": "Consequently, instead of solving the Poisson equation $\\nabla ^ { 2 } \\varphi ( \\mathbf { x } ) = - p ( \\mathbf { x } )$ in the original data space, we solve the Poisson equation in an augmented space $\\tilde { \\mathbf { x } } = ( \\mathbf { x } , z ) \\in \\mathbb { R } ^ { N + 1 }$ with an additional variable $z \\in \\mathbb { R }$ . We augment the training data $\\tilde { \\mathbf { x } }$ in the new space by setting $z = 0$ such that $\\tilde { \\mathbf { x } } = ( \\mathbf { x } , 0 )$ . As a consequence, the data distribution in the augmented space is $\\tilde { p } ( \\tilde { \\mathbf { x } } ) = p ( \\mathbf { x } ) \\delta ( z )$ , where $\\delta$ is the Dirac delta function. By Eq. (3), the Poisson field by solving the new Poisson equation $\\nabla ^ { 2 } \\varphi ( \\tilde { \\mathbf { x } } ) = - \\tilde { p } ( \\tilde { \\mathbf { x } } )$ has an analytical form: ",
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| 342 |
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"type": "equation",
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| 343 |
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"img_path": "images/f56f7ed322ed6d07de15ab2cec692248f6cf98d936a9d732fd2c626117ea7697.jpg",
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| 344 |
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"text": "$$\n\\forall \\tilde { { \\mathbf { x } } } \\in \\mathbb { R } ^ { N + 1 } , \\mathbf { E } ( \\tilde { { \\mathbf { x } } } ) = - \\nabla \\varphi ( \\tilde { { \\mathbf { x } } } ) = \\frac { 1 } { S _ { N } ( 1 ) } \\int \\frac { \\tilde { { \\mathbf { x } } } - \\tilde { { \\mathbf { y } } } } { \\left\\| \\tilde { { \\mathbf { x } } } - \\tilde { { \\mathbf { y } } } \\right\\| ^ { N + 1 } } \\tilde { p } ( \\tilde { { \\mathbf { y } } } ) d \\tilde { { \\mathbf { y } } }\n$$",
|
| 345 |
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| 346 |
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"text": "The associated forward/backward ODEs of the Poisson field are $d \\tilde { \\mathbf { x } } / d t = \\mathbf { E } ( \\tilde { \\mathbf { x } } ) , d \\tilde { \\mathbf { x } } / d t = - \\mathbf { E } ( \\tilde { \\mathbf { x } } )$ . Intuitively, theses ODEs uniquely define trajectories of particles between the $z = 0$ hyperplane and an enclosing hemisphere (cf. Fig. 1(a)). In the following theorem, we show that the backward ODE defines a transformation between the uniform distribution on an infinite hemisphere and the data distribution $\\tilde { p } ( \\tilde { { \\mathbf x } } )$ in the $z = 0$ plane. We present the formal proof to Appendix A, illustrated by Fig. 2(b). The proof is based on the idea that when the radius of hemisphere $r \\infty$ , the data distribution $\\tilde { p } ( \\tilde { \\mathbf { x } } )$ can be effectively viewed as a delta distribution at origin. Consequently, the Poisson field points in the radial direction at $r \\infty$ , perpendicular to $S _ { N } ^ { + } ( r )$ (Green arrows in Fig. 2(b)). ",
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| 357 |
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"type": "text",
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"text": "Theorem 1. Suppose particles are sampled from a uniform distribution on the upper $( z > 0 ,$ half of the sphere of radius r and evolved by the backward ODE $\\begin{array} { r } { \\frac { d \\tilde { \\mathbf { x } } } { d t } = - \\mathbf { E } \\big ( \\tilde { \\mathbf { x } } \\big ) } \\end{array}$ until they reach the $z = 0$ hyperplane, where the Poisson field $\\mathbf { E } ( \\tilde { \\mathbf { x } } )$ is generated by the source $\\tilde { p } ( \\tilde { { \\mathbf x } } )$ . In the $r \\infty$ limit, under some mild conditions detailed in Appendix $\\cdot$ , this process generates a particle distribution $\\tilde { p } ( \\tilde { { \\mathbf x } } )$ , i.e., a distribution $p ( \\mathbf { x } )$ in the $z = 0$ hyperplane. ",
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| 368 |
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| 377 |
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"type": "text",
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| 378 |
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"text": "Proof sketch. Suppose the flux of the backward ODE connects a solid angle $d \\Omega$ (on $S _ { N } ^ { + } ( r ) )$ with an area $d A$ (on $\\operatorname { s u p p } ( \\tilde { p } ( \\tilde { \\mathbf { x } } ) )$ . According to Gauss’s law, the outflow flux $d \\Phi _ { o u t } = d \\Omega / S _ { N } \\bar { ( 1 ) }$ on the hemisphere (Green arrows in Fig. 2(b)) equals the inflow flux $d \\Phi _ { i n } = p ( { \\bf x } ) d A / 2$ on $\\operatorname { s u p p } ( \\tilde { p } ( \\tilde { \\mathbf { x } } ) )$ (Red arrows in Fig. 2(b)). $d \\Phi _ { i n } = d \\Phi _ { o u t }$ gives $d \\Omega / d A = p ( \\mathbf { x } ) S _ { N } ( 1 ) / 2 \\propto \\tilde { p } ( \\mathbf { \\tilde { x } } )$ . Together, by change-ofvariable, we conclude that the final distribution in the $z = 0$ hyperplane is $p ( \\mathbf { x } )$ . □ ",
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| 388 |
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"type": "text",
|
| 389 |
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"text": "",
|
| 390 |
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"text": "The theorem states that starting from an infinite hemisphere, one can recover the data distribution $\\tilde { p }$ by following the inverse Poisson field $- \\mathbf { E } ( \\tilde { \\mathbf { x } } )$ . We defer the formal proof and technical assumptions of the theorem to Appendix A. The property allows generative modeling by following the Poisson flow of $\\nabla ^ { 2 } \\varphi ( \\tilde { \\mathbf { x } } ) = - \\tilde { p } ( \\tilde { \\mathbf { x } } )$ . ",
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},
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| 410 |
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"type": "text",
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| 411 |
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"text": "3.2 Learning the normalized Poisson Field ",
|
| 412 |
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|
| 421 |
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{
|
| 422 |
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"type": "text",
|
| 423 |
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"text": "Given a set of training data $\\mathcal { D } = \\{ \\mathbf { x } _ { i } \\} _ { i = 1 } ^ { n }$ i.i.d sampled from the data distribution $p ( \\mathbf { x } )$ , we define the =empirical version of the Poisson field (Eq. (4)) as follows: ",
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| 424 |
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| 433 |
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"type": "equation",
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"img_path": "images/8de60ed6b47f961bff7ed279b4d1c104d1dd296b9f1066034aae261490653d35.jpg",
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| 435 |
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"text": "$$\n\\hat { \\bf E } ( \\tilde { \\bf x } ) = c ( \\tilde { \\bf x } ) \\sum _ { i = 1 } ^ { n } \\frac { \\tilde { \\bf x } - \\tilde { \\bf x } _ { i } } { \\| \\tilde { \\bf x } - \\tilde { \\bf x } _ { i } \\| ^ { N + 1 } }\n$$",
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"text_format": "latex",
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"bbox": [
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| 446 |
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"type": "text",
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| 447 |
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"text": "where the gradient field is calculated on $n$ augmented datapoints $\\{ \\tilde { \\mathbf { x } } _ { i } = ( \\mathbf { x } _ { i } , 0 ) \\} _ { i = 1 } ^ { n }$ , and $c ( \\tilde { \\mathbf { x } } ) =$ $\\textstyle 1 / \\sum _ { i = 1 } ^ { n } { \\frac { 1 } { \\left\\| \\tilde { \\mathbf { x } } - \\tilde { \\mathbf { x } } _ { i } \\right\\| ^ { N + 1 } } }$ is the multiplier for numerical stability. We further normalize the field to resolve the variations in the magnitude of the norm $\\Vert \\hat { \\textbf { E } } ( \\tilde { \\textbf { x } } ) \\ \\Vert _ { 2 }$ , and fit the neural network to the more amenable negative normalized field $\\mathbf { v } ( \\tilde { \\mathbf { x } } ) = - \\sqrt { N } \\hat { \\mathbf { E } } ( \\tilde { \\mathbf { x } } ) / \\| \\hat { \\mathbf { E } } ( \\tilde { \\mathbf { x } } ) \\| _ { 2 }$ . The Poisson field is rescalable (cf. Section 2) and thus trajectories of its forward/backward ODEs are invariant under normalization. We denote the empirical field calculated on batch data $\\boldsymbol { B }$ by $\\hat { \\mathbf { E } } _ { B }$ and the negative normalized field as $\\mathbf { v } _ { B } ( \\tilde { \\mathbf { x } } ) = - \\sqrt { N } \\hat { \\mathbf { E } } _ { B } ( \\tilde { \\mathbf { x } } ) / \\| \\hat { \\mathbf { E } } _ { B } ( \\tilde { \\mathbf { x } } ) \\| _ { 2 }$ . ",
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"bbox": [
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| 457 |
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"type": "text",
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| 458 |
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"text": "Similar to the scored-based models, we sample points inside the hemisphere by perturbing the augmented training data. Given a training point $\\textbf { x } \\in \\mathcal { D }$ , we add noise to its augmented version $\\{ \\tilde { \\mathbf { x } } _ { i } ^ { - } = ( \\mathbf { x } _ { i } , 0 ) \\} _ { i = 1 } ^ { n }$ to construct the perturbed point $( \\mathbf { y } , z )$ : ",
|
| 459 |
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"bbox": [
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"type": "equation",
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"img_path": "images/e7a81e3d8227333b0b956c9d6fdaa92c8336caa2824b8c93179516f33f32bf04.jpg",
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"text": "$$\n\\mathbf { y } = \\mathbf { x } + \\parallel \\epsilon _ { \\mathbf { x } } \\parallel ( 1 + \\tau ) ^ { m } \\mathbf { u } , \\quad z = | \\epsilon _ { z } | ( 1 + \\tau ) ^ { m }\n$$",
|
| 471 |
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"text_format": "latex",
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| 472 |
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"bbox": [
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|
| 480 |
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{
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| 481 |
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"type": "text",
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| 482 |
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"text": "where ${ \\epsilon } = \\left( { \\epsilon } _ { \\mathbf { x } } , { \\epsilon } _ { z } \\right) \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { N + 1 \\times N + 1 } )$ , $\\mathbf { u } \\sim \\mathcal { U } ( S _ { N - 1 } ( 1 ) )$ and $m \\sim \\mathcal { U } [ 0 , M ]$ . The upper limit $M$ , standard deviation $\\sigma$ and $\\tau$ are hyper-parameters. With fixed $\\epsilon$ and $\\mathbf { u }$ , the added noise increases exponentially with $m$ . The rationale behind the design is that points farther away from the data support play a less important role in generative modeling, sharing a similar spirit with the choice of noisy scales in score-based models [32, 33]. ",
|
| 483 |
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"bbox": [
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"type": "text",
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| 493 |
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"text": "In practice, we sample the p ts urbing a mini-batch data $B = \\{ \\mathbf { x } _ { i } \\} _ { i = 1 } ^ { | B | }$ each iteration. We $m$ $[ 0 , M ]$ =for each datapoint. We select a large $M$ 300) to ensure the perturbed points can reach a large enough hemisphere. We use a larger batch $\\boldsymbol { B } _ { L }$ for the estimation of normalized field since the empirical normalized field is biased, which empirically gives better results. Denoting the set of perturbed points as $\\{ \\tilde { \\mathbf { y } } _ { i } \\} _ { i = 1 } ^ { | B | }$ , we train the neural network $f _ { \\theta }$ =on these points to estimate the negative normalized field by minimizing the following loss: ",
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"type": "equation",
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"img_path": "images/267bd119595567558b0a74a4a1831aacb48d9aad01458f9169bbd40b572ce3f8.jpg",
|
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"text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } ) = \\frac { 1 } { | \\mathcal { B } | } \\sum _ { i = 1 } ^ { | \\mathcal { B } | } \\parallel f _ { \\boldsymbol { \\theta } } \\big ( \\tilde { \\mathbf { y } } _ { i } \\big ) - \\mathbf { v } _ { \\mathcal { B } _ { L } } \\big ( \\tilde { \\mathbf { y } } _ { i } \\big ) \\parallel _ { 2 } ^ { 2 }\n$$",
|
| 506 |
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"text_format": "latex",
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| 507 |
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"bbox": [
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| 516 |
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"type": "text",
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| 517 |
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"text": "We summarize the training process in Algorithm 1. In practice, we add a small constant $\\gamma$ to the denominator of the normalized field to overcome the numerical issue when $\\exists i , \\left\\| \\tilde { \\mathbf { x } } - \\tilde { \\mathbf { x } } _ { i } \\right\\| \\approx 0$ . ",
|
| 518 |
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"type": "text",
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"text": "3.3 Backward ODE anchored by the additional dimension ",
|
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"type": "text",
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"text": "After estimating the normalized field $\\mathbf { v }$ , we can sample from the data distribution by the backward ODE $d { \\tilde { \\mathbf { x } } } = \\mathbf { v } ( { \\tilde { \\mathbf { x } } } ) d t$ . Nevertheless, the boundary condition of the above ODE is unclear: the starting and terminal time $t$ of the ODE are both unknown. To remedy the issue, we propose an equivalent backward ODE in which $\\mathbf { x }$ evolves with the augmented variable $z$ : ",
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"type": "equation",
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"img_path": "images/6a077a3c370ba6f955394864da8dd52fa6d1d64d41dacb58fe358b5d5b9cba4a.jpg",
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"text": "$$\nd ( \\mathbf { x } , z ) = ( \\frac { d \\mathbf { x } } { d t } \\frac { d t } { d z } d z , d z ) = ( \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { \\mathbf { x } } \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { z } ^ { - 1 } , 1 ) d z\n$$",
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|
| 563 |
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"type": "text",
|
| 564 |
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"text": "Algorithm 1 Learning the normalized Poisson Field ",
|
| 565 |
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"type": "table",
|
| 575 |
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"img_path": "images/7fb9bff56e4d77cca1acec77cbc61c1cdab6b3654cb5169569ef79914e39aaec.jpg",
|
| 576 |
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"table_caption": [],
|
| 577 |
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"table_footnote": [],
|
| 578 |
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"table_body": "<table><tr><td>Input: Training iteration T,Initial model fe,dataset D,constant y,learning rate n. fort=1...Tdo</td></tr><tr><td>from BL |B|</td></tr><tr><td>Simulate the ODE: {yi = perturb(xi) Ji=1 Calculate the normalized field by BL: VB (yi)=-√NEB,(yi)/(ll EB (yi) Il2 +γ), ∀i</td></tr><tr><td>|l f(yi)-vBL(yi)l² i1</td></tr><tr><td>Update the model parameter: 0 = 0 - nVL(0)</td></tr><tr><td>end for</td></tr><tr><td>return fe</td></tr></table>",
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| 579 |
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| 588 |
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"type": "text",
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| 589 |
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"text": "Algorithm 2 perturb $\\cdot$ ",
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"text_level": 1,
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{
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| 600 |
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"type": "table",
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| 601 |
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"img_path": "images/74417a23e4e9403cebdbaa2f88f9cfd5ae1d39e86836ea35c9978940c4283dea.jpg",
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"table_caption": [],
|
| 603 |
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"table_footnote": [],
|
| 604 |
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"table_body": "<table><tr><td>Sample the power m ~U[0,M] Sample the initial noise (∈x,∈z)~N(O,σ²I(N+1)x(N+1))</td></tr><tr><td>Uniformly sample the vector from the unit ball u ~U(SN(1))</td></tr><tr><td>Construct training point y = x+ | x I (1 + 𝑇)mu, z = |∈zl(1 + T)m</td></tr><tr><td>return y = (y, z)</td></tr></table>",
|
| 605 |
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"bbox": [
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{
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| 614 |
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"type": "text",
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| 615 |
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"text": "where $\\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { \\mathbf { x } } , \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { z }$ are the corresponding components of $\\mathbf x , z$ in vector $\\mathbf { v } ( \\tilde { \\mathbf { x } } )$ . In the new ODE, we replace the time variable $t$ with the physically meaningful variable $z$ , permitting explicit starting and terminal conditions: when $z = 0$ , we arrive at the data distribution and we can freely choose a large $z _ { \\mathrm { m a x } }$ as the starting point in the backward ODE. The backward ODE is compatible with general-purpose ODE solvers, e.g., RK45 method [23] and forward Euler method. The popular black-box ODE solvers, such as the one in Scipy library [37], typically use a common starting time for the same batch of samples. Since the distribution on the $z = z _ { \\mathrm { m a x } }$ hyperplane is no longer uniform, we derive the prior distribution by radially projecting uniform distribution on the hemisphere with radius $r = z _ { \\mathrm { m a x } }$ to the $z = z _ { \\mathrm { m a x } }$ hyperplane: ",
|
| 616 |
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| 625 |
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"type": "equation",
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"img_path": "images/03be1377e92c170e073289fe6be3fd20a032dbd69bc6fb33c5bc2307c712fdc3.jpg",
|
| 627 |
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"text": "$$\np _ { \\mathrm { p r i o r } } ( \\mathbf { x } ) = { \\frac { 2 z _ { \\mathrm { m a x } } ^ { N + 1 } } { S _ { N } { \\big ( } z _ { \\mathrm { m a x } } { \\big ) } { \\big ( } \\| \\mathbf { x } \\| _ { 2 } ^ { 2 } + z _ { \\mathrm { m a x } } ^ { 2 } { \\big ) } ^ { \\frac { N + 1 } { 2 } } } } = { \\frac { 2 z _ { \\mathrm { m a x } } } { S _ { N } { \\big ( } 1 { \\big ) } { \\big ( } \\| \\mathbf { x } \\| _ { 2 } ^ { 2 } + z _ { \\mathrm { m a x } } ^ { 2 } { \\big ) } ^ { \\frac { N + 1 } { 2 } } } }\n$$",
|
| 628 |
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"text_format": "latex",
|
| 629 |
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"bbox": [
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| 634 |
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| 636 |
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| 637 |
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{
|
| 638 |
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"type": "text",
|
| 639 |
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"text": "where $S _ { N } ( r )$ is the surface area of $N$ -sphere with radius $r$ . The reason behind the radial projection is that the Poisson field points in the radial direction at $r \\infty$ . The new backward ODE also defines a bijective transformation between $p _ { \\mathrm { p r i o r } } ( \\mathbf { x } )$ on the infinite hyperplane $z _ { \\mathrm { m a x } } \\infty$ ) and the data distribution the norm (r $\\tilde { p } ( \\tilde { { \\mathbf x } } )$ , analogous to Theorem ) from the distribution: $p _ { \\mathrm { p r i o r } } ( \\mathbf { x } )$ to sampleand then $p _ { \\mathrm { r a d i u s } } ( \\| \\textbf { x } \\| _ { 2 } ) \\propto \\| \\textbf { x } \\| _ { 2 } ^ { N - 1 } / ( \\| \\textbf { x } \\| _ { 2 } ^ { 2 } + z _ { \\operatorname* { m a x } } ^ { 2 } ) ^ { \\frac { N + 1 } { 2 } }$ uniformly sample its angle. We provide detailed derivations and practical sampling procedure in Appendix A.4. We further achieve exponential decay on the $z$ dimension by introducing a new variable $t ^ { \\prime }$ : ",
|
| 640 |
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| 647 |
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| 648 |
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|
| 649 |
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"type": "equation",
|
| 650 |
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"img_path": "images/646814215122bed857b46f4e0d543f1b8287dd5a0d4e674bbbfce9f8fe8215a8.jpg",
|
| 651 |
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"text": "$$\n\\begin{array} { r l } { [ \\mathrm { B a c k w a r d ~ O D E } ] } & { { } d ( \\mathbf { x } , z ) = ( \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { \\mathbf { x } } \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { z } ^ { - 1 } z , z ) d t ^ { \\prime } } \\end{array}\n$$",
|
| 652 |
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"text_format": "latex",
|
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| 660 |
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|
| 661 |
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{
|
| 662 |
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"type": "text",
|
| 663 |
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"text": "The $z$ component in the backward ODE, i.e., $d z = z d t ^ { \\prime }$ , can be solved by $z = e ^ { t ^ { \\prime } }$ . Since $z$ reaches zero as $t ^ { \\prime } \\to - \\infty$ , we instead choose a tiny positive number $z _ { \\mathrm { m i n } }$ as the terminal condition. The corresponding starting/terminal time of the variable $t ^ { \\prime }$ are $\\log z _ { \\operatorname* { m a x } } / \\log z _ { \\operatorname* { m i n } }$ respectively. Empirically, this simple change of variable leads to $2 \\times$ faster sampling with almost no harm to the sample quality. In addition, we substitue the predicted $\\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { z }$ with a more accurate one when $z$ is small (Appendix B.2.3). We defer more details of the simulation of backward ODE to Appendix B.2. ",
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|
| 673 |
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"type": "text",
|
| 674 |
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"text": "4 Generative Modeling via the Backward ODE ",
|
| 675 |
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| 686 |
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"text": "In this section, we demonstrate the effectiveness of the backward ODE associated with PFGM on image generation tasks. In Section 4.1, we show that PFGM achieves currently best in class performance in the normalizing flow family. In comparison to the existing state-of-the-art SDE or MCMC approaches, PFGM exhibits $1 0 \\times$ or $2 0 \\times$ acceleration while maintaining competitive or higher generation quality. Meanwhile, unlike existing ODE baselines that heavily rely on corrector to generate decent samples on weaker architectures, PFGM exhibits greater stability against error (Section 4.2). Finally, we show that PFGM is robust to the step size in the Euler method (Section 4.3), and its associated ODE allows for likelihood evaluation and image manipulation by editing the latent space (Section 4.4). ",
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{
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"type": "table",
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"img_path": "images/7c3d937df96eeba6b761d69681617432cc07eabcf1315e03c91e0d3578953cd3.jpg",
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"table_caption": [
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"Table 1: CIFAR-10 sample quality (FID, Inception) and number of function evaluation (NFE). "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td></td><td>Invertible?</td><td>Inception ↑</td><td>FID↓</td><td>NFE↓</td></tr><tr><td>PixelCNN[36]</td><td></td><td>4.60</td><td>65.9</td><td>1024</td></tr><tr><td>IGEBM[8]</td><td></td><td>6.02</td><td>40.6</td><td>60</td></tr><tr><td>ViTGAN [24]</td><td></td><td>9.30</td><td>6.66</td><td>1</td></tr><tr><td>StyleGAN2-ADA [17]</td><td></td><td>9.83</td><td>2.92</td><td>1</td></tr><tr><td>StyleGAN2-ADA (cond.) [17]</td><td>xxxxxxxxxx</td><td>10.14</td><td>2.42</td><td>1</td></tr><tr><td>NCSN[31]</td><td></td><td>8.87</td><td>25.32</td><td>1001</td></tr><tr><td>NCSNv2 [32]</td><td></td><td>8.40</td><td>10.87</td><td>1161</td></tr><tr><td>DDPM[16]</td><td></td><td>9.46</td><td>3.17</td><td>1000</td></tr><tr><td>NCSN++ VE-SDE[33]</td><td></td><td>9.83</td><td>2.38</td><td>2000</td></tr><tr><td>NCSN++ deep VE-SDE [33]</td><td></td><td>9.89</td><td>2.20</td><td>2000</td></tr><tr><td>Glow [19]</td><td></td><td>3.92</td><td>48.9</td><td>1</td></tr><tr><td>DDIM,T=50 [30]</td><td></td><td>-</td><td>4.67</td><td>50</td></tr><tr><td>DDIM, T=100 [30]</td><td></td><td>1</td><td>4.16</td><td>100</td></tr><tr><td>NCSN++ VE-ODE [33]</td><td></td><td>9.34</td><td>5.29</td><td>194</td></tr><tr><td>NCSN++ deep VE-ODE[33]</td><td></td><td>9.17</td><td>7.66</td><td>194</td></tr><tr><td colspan=\"5\">DDPM++backbone</td></tr><tr><td>VP-SDE[33]</td><td></td><td>9.58</td><td>2.55</td><td>1000</td></tr><tr><td>sub-VP-SDE[33]</td><td>xx-</td><td>9.56</td><td>2.61</td><td>1000</td></tr><tr><td>VP-ODE [33]</td><td></td><td>9.46</td><td>2.97</td><td>134</td></tr><tr><td>sub-VP-ODE [33]</td><td></td><td>9.30</td><td>3.16</td><td>146</td></tr><tr><td>PFGM (ours)</td><td></td><td>9.65</td><td>2.48</td><td>104</td></tr><tr><td colspan=\"5\">DDPM++ deep backbone</td></tr><tr><td>VP-SDE [33]</td><td></td><td>9.68</td><td>2.41</td><td>1000</td></tr><tr><td>sub-VP-SDE[33]</td><td>xx-</td><td>9.57</td><td>2.41</td><td>1000</td></tr><tr><td>VP-ODE [33]</td><td></td><td>9.47</td><td>2.86</td><td>134</td></tr><tr><td>sub-VP-ODE [33]</td><td></td><td>9.40</td><td>3.05</td><td>146</td></tr><tr><td>PFGM (ours)</td><td></td><td>9.68</td><td>2.35</td><td>110</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr></table>",
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"type": "text",
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"text": "4.1 Efficient image generation by PFGM ",
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"text": "Setup For image generation tasks, we consider the CIFAR-10 [22], CelebA $6 4 \\times 6 4$ [38] and LSUN bedroom $2 5 6 \\times 2 5 6$ [39]. Following [32], we first center-crop the CelebA images and then resize them to $6 4 \\times 6 4$ . We choose $M \\ : = \\ : 2 9 1$ CIFAR-10 and CelebA 356 LSUN bedroom , $\\sigma ~ = ~ 0 . 0 1$ and $\\tau { \\it \\Delta \\phi } = 0 . 0 3$ for the perturbation Algorithm 2, and $z _ { \\mathrm { m i n } } ~ = ~ 1 e \\mathrm { ~ - ~ } 3$ , $\\begin{array} { r l } { z _ { \\operatorname* { m a x } } } & { { } = } \\end{array}$ 40 CIFAR-10 60 CelebA $6 4 ^ { 2 }$ 100 LSUN bedroom for the backward ODE. We further clip the norms of initial samples into $( 0 , 3 0 0 0 )$ for CIFAR-10, $( 0 , 6 0 0 0 )$ for CelebA $6 4 ^ { 2 }$ and $( 0 , 3 0 0 0 0 )$ for LSUN bedroom. We adopt the $\\mathrm { { D D P M + + } }$ and $\\mathrm { { D D P M + + } }$ deep architectures [33] as our backbones. We add the scalar $z$ (resp. predicted direction on $z$ ) as input (resp. output) to accommodate the additional dimension. We take the same set of hyper-parameters, such as batch size, learning rate and training iterations from [33]. We provide more training details in Appendix B.1, and discuss how to set these hyper-parameters for general datasets in B.1.1 and B.2.1. ",
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"type": "text",
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"text": "Baselines We compare PFGM to modern autoregressive model [36], GAN [17, 24], normalizing flow [19] and EBM [8]. We also compare with variants of score-based models such as DDIM [30] and current state-of-the-art SDE/ODE methods [33]. We denote the methods that use forward-time SDEs in [33] such as Variance Exploding (VE) SDE/Variance Preserving (VP) SDE/ sub-Variance Preserving (sub-VP), and the corresponding backward SDE/ODE, as A-B, where $\\mathbf { A } \\in \\{ \\mathrm { V E }$ , VP, sub- $\\mathrm { V P } \\}$ and $\\mathbf { B } \\in \\{ \\mathrm { S D E } , \\mathrm { O D E } \\}$ . We follow the model selection protocol in [33], which selects the checkpoint with the smallest FID score over the course of training every 50k iterations. ",
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{
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"type": "image",
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"img_path": "images/765d37140b6f169aca43c335a32e1ca5fc5bf14d8e90ab0f9dcca04e90401859.jpg",
|
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"image_caption": [
|
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"Figure 3: Uncurated samples on datasets of increasing resolution. From left to right: CIFAR-10 $3 2 \\times 3 2$ , CelebA $6 4 \\times 6 4$ and LSUN bedroom $2 5 6 \\times 2 5 6$ . "
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"image_footnote": [],
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"type": "text",
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"text": "Numerical Solvers The backward ODE (Eq. (6)) is compatible with any general purpose ODE solver. In our experiments, the default solver of ODEs is the black box solver in the Scipy library [37] with the RK45 [7] method (RK45), unless otherwise specified. For VE/VP/subVP-SDEs, we use the predictor-corrector (PC) sampler introduced in [33]. For VP/sub-VP-SDEs, we apply the predictor-only sampler, because its performance is on par with the PC sampler while requiring half computation. ",
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"type": "text",
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"text": "Results For quantitative evaluation on CIFAR-10, we report the Inception [29] (higher is better) and FID [13] scores (lower is better) in Table 1. We also include our preliminary experimental results on a weaker architecture NCSNv2 [32] in Appendix D.2. We measure the inference speed by the average NFE (number of function evaluation). We also explicitly indicate which methods belong to the invertible flow family. ",
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"type": "text",
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"text": "Our main findings are: (1) PFGM achieves the best Inception scores and FID scores among the normalizing flow models. Specifically, PFGM obtains an Inception score of 9.68 and a FID score of 2.48 using the $\\mathrm { D D P M + + }$ deep architecture. To our best knowledge, these are the highest FID and Inception scores by flow models on CIFAR-10. (2) PFGM achieves a $1 0 \\times \\sim 2 0 \\times$ faster inference speed than the SDE methods using the similar architectures, while retaining comparable sample quality. As shown in Table 1, PFGM requires NFEs of 110 whereas the SDE methods typically use $1 0 0 0 \\sim 2 0 0 0$ inference steps. PFGM outperforms all the baselines on $\\mathrm { { D D P M + + } }$ in all metrics. In addition, PFGM generally samples faster than other ODE baselines with the same RK45 solver. (3) The backward ODE in PFGM is compatible with architectures with varying capacities. PFGM consistently outperforms other ODE baselines on $\\mathrm { D D P M + + }$ (Table 1) or NCSNv2 (Appendix D.2) backbones. (4) PFGM shows scalability to higher resolution datasets. In Appendix D.1, we show that PFGM are capable of scale-up to LSUN bedroom $2 5 6 \\times 2 5 6$ . In particular, PFGM has comparable performance with VE-SDE with $1 5 \\times$ fewer NFE. ",
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"type": "text",
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"text": "In Fig. 3, we visualize the uncurated samples from PFGM on CIFAR-10, CelebA $6 4 \\times 6 4$ and LSUN bedroom $2 5 6 \\times 2 5 6$ . We provides more samples in Appendix E. ",
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"type": "text",
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"text": "4.2 Failure of VE/VP-ODEs on NCSNv2 architecture ",
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"text_level": 1,
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"type": "text",
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"text": "In our preliminary experiments on NCSNv2 architectures, we empirically observe that the VE/VP-ODEs have FID scores greater than 90 on CIFAR-10. In particular, VE/VP-ODEs can only generate decent samples when applying the Langevin dynamics corrector, and even then, their performances are still inferior to PFGM (Table 9, Table 10). The poor performance on NCSNv2 stands in striking contrast to their high sample quality on $\\mathrm { N C S N + + / D D P M + + }$ in [33]. It indicates that the VE/VP-ODEs are ",
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{
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"type": "image",
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"img_path": "images/b8860bcda32dab2d49f72ab93a489b264c43c232817b6c997c0142ac14f89d96.jpg",
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"image_caption": [
|
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"Figure 4: Sample norm distributions with varying time variables $\\sigma$ for VE-ODE and $z$ for PFGM) "
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],
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"image_footnote": [],
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{
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"img_path": "images/8f1d6bfd2fc3172f6a771fae4fc3892a6d968815e5d64788e0ff673cb1dd4cc0.jpg",
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"image_caption": [
|
| 857 |
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"Figure 5: (a) Norm- $\\cdot \\sigma ( t )$ relation during the backward sampling of VE-ODE (Euler). (b) Norm- $z ( t ^ { \\prime } )$ relation during the backward sampling of PFGM (Euler). The shaded areas mean the standard deviation of norms. (c) Number of steps versus FID score. "
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|
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"image_footnote": [],
|
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{
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"type": "text",
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"text": "more susceptible to estimation errors than PFGM. We hypothesize that the strong norm- $\\sigma$ correla tion seen during the training of score-based models causes the problem. ",
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"page_idx": 8
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"type": "text",
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"text": "For score-based models, the $l _ { 2 }$ norms of perturbed training samples and the standard deviations $\\sigma ( t )$ of Gaussian noises have strong correlation, e. $g . , l _ { 2 } \\ \\mathrm { n o r m } \\approx \\sigma ( t ) \\sqrt { N }$ for large $\\sigma ( t )$ in VE [33]. In contrast, as shown in Fig. 4, PFGM allocates high mass across a wide spectrum of the training sample norms. During sampling, VE/VP-ODEs could break down when the trajectories of backward ODEs deviate from the norm- $\\cdot \\sigma ( t )$ relation to which most training samples pertain. The weaker NCSNv2 backbone incurs larger errors and thus leads to their failure. The PFGM is more resistant to estimate errors because of the greater range of training sample norms. ",
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},
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{
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"type": "text",
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"text": "To further verify the hypothesis above, we split a batch of VE-ODE samples into cleaner and noisier samples according to visual quality (Fig. 8(a)). In Fig. 5(a), we investigate the relation for cleaner and noisier samples during the forward Euler simulation of VE-ODE when $\\sigma ( t ) < 1 5$ . We can see that the trajectory of cleaner samples stays close to the norm- $\\sigma ( t )$ relation (the red dash line), whereas that of the noisier samples diverges from the relation. The Langevin dynamics corrector changes the trajectory of noisier samples to align with the relation. Fig. 5(b) further shows that the anchored variable $z ( t ^ { \\prime } )$ and the norms in the backward ODE of PFGM are not strongly correlated, giving rise to the robustness against the imprecise estimation on NCSNv2. We defer more details to Appendix C. ",
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"text": "4.3 Effects of step size in the forward Euler method ",
|
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"text_level": 1,
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"text": "In order to accelerate the inference speed of ODEs, we can increase the step size (decrease the NFEs) in numerical solvers such as the forward Euler method. It also enables the trade-off between sample quality and computational efficiency in real-world deployment. We study the effects of increasing step size on PFGM, VP-ODE and DDIM [30] using the forward Euler method, with a varying NFE ranging from 10 to 100. ",
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"text": "In Fig. 5(c), we report the sample quality measured by FID scores on CIFAR-10. As expected, all the methods have higher FID scores when decreasing the NFE. We observe that the sample quality of PFGM degrades gracefully as we decrease the NFE. Our method shows significantly better robustness to step sizes than the VP-ODE, especially when only taking a few Euler steps. In addition, PFGM obtains better FID scores than DDIM on most NFEs except for 10 where PFGM is marginally worse. This suggests that the PFGM is a promising method for accommodating instantaneous resource availability, as high-quality samples can be generated in limited steps. ",
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"type": "text",
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"text": "4.4 Utilities of ODE: likelihood evaluation and latent representation ",
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"text_level": 1,
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},
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"type": "text",
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"text": "Similar to the family of discrete normalizing flows [6, 19, 14] and continuous probability flow [33], the forward ODE in PFGM defines an invertible mapping between the data space and latent space with a known prior. Formally, we define the invertible forward $\\mathcal { M }$ mapping by integrating the corresponding forward ODE $\\begin{array} { r } { \\dot { d } ( \\mathbf { x } , z ) = ( \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { \\mathbf { x } } \\mathbf { v } ( \\tilde { \\mathbf { x } } ) _ { z } ^ { - 1 } z , z ) d t ^ { \\prime } } \\end{array}$ of Eq. (6): ",
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"type": "equation",
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"img_path": "images/de973ce12148d28895bba204e84e1d40f37ef0fa8b8705e90efc4d27d0ee8a19.jpg",
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"text": "$$\n\\mathbf { x } ( \\log z _ { \\operatorname* { m a x } } ) = \\mathcal { M } ( \\mathbf { x } ( \\log z _ { \\operatorname* { m i n } } ) ) \\equiv \\mathbf { x } ( \\log z _ { \\operatorname* { m i n } } ) + \\int _ { \\log z _ { \\operatorname* { m i n } } } ^ { \\log z _ { \\operatorname* { m a x } } } \\mathbf { v } ( \\mathbf { x } ( t ^ { \\prime } ) ) _ { \\mathbf { x } } \\mathbf { v } ( \\tilde { \\mathbf { x } } ( t ^ { \\prime } ) ) _ { z } ^ { - 1 } e ^ { t ^ { \\prime } } d t ^ { \\prime }\n$$",
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"text_format": "latex",
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"bbox": [
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"type": "text",
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"text": "where $\\log z _ { \\mathrm { m i n } } / \\log z _ { \\mathrm { m a x } }$ are the starting/terminal time in the forward ODE. The forward mapping transfers the data distribution to the prior distribution $p _ { \\mathrm { p r i o r } }$ on the $z = z _ { \\mathrm { m a x } }$ hyperplane (cf. Section 3.3): $p _ { \\mathrm { p r i o r } } \\big ( \\mathbf { x } \\big ( \\log z _ { \\mathrm { m a x } } \\big ) \\big ) = \\mathcal { M } \\big ( p ( \\mathbf { x } ( \\log \\bar { z } _ { \\mathrm { m i n } } ) ) \\big )$ . The invertibility enables likelihood evaluation and creates a meaningful latent space on the $z = z _ { \\mathrm { m a x } }$ hyperplane. In addition, we can adapt to the computational constraints by adjusting the step size or the precision in numerical ODE solvers. ",
|
| 974 |
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"text": "Likelihood evaluation We evaluate the data likelihood by the instantaneous change-of-variable formula [4, 33]. In Table 2, we report the bits/dim on the uniformly dequantized CIFAR-10 test set and compare with existing baselines that use the same setup. We observe that PFGM achieves better likelihoods than discrete normalizing flow models, even without maximum likelihood training. Among the continuous flow models, sub-VP-ODE shows the lowest bits/dim, although its sample quality is worse than VP-ODE and PFGM (Table 1). The exploration of the seeming trade-off between likelihood and sample quality is left for future works. ",
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"type": "table",
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"img_path": "images/c452efcad2fd82bbc0d70344e1698d29d6fa874798cc81029b380d62e987b32d.jpg",
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"table_caption": [
|
| 997 |
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"Table 2: Bits/dim on CIFAR-10 "
|
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],
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| 999 |
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"table_footnote": [],
|
| 1000 |
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"table_body": "<table><tr><td></td><td>bits/dim ↓</td></tr><tr><td>RealNVP [6]</td><td>3.49</td></tr><tr><td>Glow [19] Residual Flow [3]</td><td>3.35 3.28</td></tr><tr><td>Flow++ [14]</td><td>3.29</td></tr><tr><td>DDPM(L)[16]</td><td>≤3.70*</td></tr><tr><td>DDPM++backbone</td><td></td></tr><tr><td>VP-ODE [33]</td><td>3.20</td></tr><tr><td>sub-VP-ODE[33]</td><td>3.02</td></tr><tr><td>PFGM (ours)</td><td>3.19</td></tr></table>",
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"text": "Latent representation Since the samples are uniquely identifiable by their latents via the invertible mapping $\\mathcal { M }$ , PFGM further supports image manipulation using its latent representation on the $z =$ $z _ { \\mathrm { m a x } }$ hyperplane. We include the results of image interpolation and the temperature scaling [6, 19, 33] to Appendix D.4 and Appendix D.5. For interpolation, it shows that we can travel along the latent space to obtain perceptually consistent interpolations between CelebA images. ",
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"type": "text",
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"text": "5 Conclusion ",
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"text_level": 1,
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"text": "We present a new deep generative model by solving the Poisson equation whose source term is the data distribution. We estimate the normalized gradient field of the solution in an augmented space with an additional dimension. For sampling, we devise a backward ODE that exponential decays on the physically meaningful additional dimension. Empirically, our approach has currently best performance over other normalizing flow baselines, and achieving $1 0 \\times$ to $2 0 \\times$ acceleration over the stochastic methods. Our backward ODE shows greater stability against errors than popular ODE-based methods, and enables efficient adaptive sampling. We further demonstrate the utilities of the forward ODE on likelihood evaluation and image interpolation. Future directions include improving the normalization of Poisson fields. More principled approaches can be used to get around the divergent near-field behavior. For example, we may exploit renormalization, a useful tool in physics, to make the Poisson field well-behaved in near fields. ",
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"type": "text",
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"text": "Acknowledgements ",
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"text_level": 1,
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"bbox": [
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"text": "We are grateful to Shangyuan Tong, Timur Garipov and Yang Song for helpful discussion. We would like to thank Octavian Ganea and Wengong Jin for reviewing an early draft of this paper. YX and TJ acknowledge support from MIT-DSTA Singapore collaboration, from NSF Expeditions grant (award 1918839) ”Understanding the World Through Code”, and from MIT-IBM Grand Challenge project. ZL and MT would like to thank the Center for Brains, Minds, and Machines (CBMM) for hospitality. ZL and MT are supported by The Casey and Family Foundation, the Foundational Questions Institute, the Rothberg Family Fund for Cognitive Science and IAIFI through NSF grant PHY-2019786. ",
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| 1251 |
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| 1252 |
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|
| 1253 |
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|
| 1254 |
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| 1255 |
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"text": "[36] Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Koray Kavukcuoglu, Oriol Vinyals, ¨ and Alex Graves. Conditional image generation with pixelcnn decoders. In NIPS, 2016. ",
|
| 1256 |
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| 1264 |
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"text": "[37] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stefan J. ´ van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, Ilhan Polat, Yu Feng, Eric W. Moore, J. Vanderplas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Daniel Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antonio H. Ribeiro, Fabian Pedregosa, ˆ Paul van Mulbregt, Aditya Alessandro Pietro Alex Andreas Andreas Anthony Ant Vijaykumar Bardelli Rothberg Hilboll Kloeckner Sco, Aditya Vijaykumar, Alessandro Pietro Bardelli, Alex Rothberg, Andreas Hilboll, Andre Kloeckner, Anthony M. Scopatz, Antony Lee, Ariel S. Rokem, C. Nathan Woods, Chad Fulton, Charles Masson, Christian Haggstr ¨ om, Clark Fitzgerald, ¨ ",
|
| 1267 |
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| 1274 |
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},
|
| 1275 |
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|
| 1276 |
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| 1277 |
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"text": "David A. Nicholson, David R. Hagen, Dmitrii V. Pasechnik, Emanuele Olivetti, Eric Martin, Eric Wieser, Fabrice Silva, Felix Lenders, Florian Wilhelm, Gert Young, Gavin A. Price, GertLudwig Ingold, Gregory E. Allen, Gregory R. Lee, Herve Audren, Irvin Probst, Jorg P. Dietrich, ´ Jacob Silterra, James T. Webber, Janko Slavi, Joel Nothman, Johannes Buchner, Johannes Kulick, Johannes L. Schonberger, Jos ¨ e Vin ´ ´ıcius de Miranda Cardoso, Joscha Reimer, Joseph E. Harrington, Juan Luis Cano Rodr´ıguez, Juan Nunez-Iglesias, Justin Kuczynski, Kevin Lee Tritz, Martin Dr Thoma, Matt Newville, Matthias Kummerer, Maximilian Bolingbroke, Michael ¨ Tartre, Mikhail Pak, Nathaniel J. Smith, Nikolai Nowaczyk, Nikolay Shebanov, Oleksandr Pavlyk, Per Andreas Brodtkorb, Perry Lee, Robert T. McGibbon, Roman Feldbauer, Sam Lewis, Sam Tygier, Scott Sievert, Sebastiano Vigna, Stefan Peterson, Surhud More, Tadeusz Pudlik, Taku Oshima, Thomas J. Pingel, Thomas P. Robitaille, Thomas Spura, Thouis Raymond Jones, Tim Cera, Tim Leslie, Tiziano Zito, Tom Krauss, U. Upadhyay, Yaroslav O. Halchenko, and Y. Vazquez-Baeza. Scipy 1.0: fundamental algorithms for scientific computing in python. ´ Nature Methods, 17:261 – 272, 2020. \n[38] Shuo Yang, Ping Luo, Chen Change Loy, and Xiaoou Tang. From facial parts responses to face detection: A deep learning approach. 2015 IEEE International Conference on Computer Vision (ICCV), pages 3676–3684, 2015. \n[39] Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. ArXiv, abs/1506.03365, 2015. ",
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| 1285 |
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|
| 1286 |
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{
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| 1287 |
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| 1288 |
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"text": "Checklist ",
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| 1289 |
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| 1290 |
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| 1291 |
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| 1297 |
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| 1298 |
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|
| 1299 |
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"type": "text",
|
| 1300 |
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"text": "1. For all authors... ",
|
| 1301 |
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|
| 1302 |
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| 1310 |
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| 1311 |
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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? [Yes] See Appendix G. \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix H. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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|
| 1320 |
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| 1321 |
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"text": "2. If you are including theoretical results... ",
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| 1323 |
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| 1332 |
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"text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix A. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A. ",
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| 1334 |
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| 1344 |
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"text": "3. If you ran experiments... ",
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| 1345 |
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| 1353 |
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| 1354 |
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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] See the abstract. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix B.1. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] \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] All the experiments are run on a single NVIDIA A100 GPU. ",
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|
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| 1375 |
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|
| 1376 |
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| 1377 |
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"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [N/A] The assets are public/open-source datasets and codes. \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
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| 1386 |
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| 1387 |
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| 1388 |
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"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
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| 1389 |
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| 1399 |
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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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