Datasets:
Add files using upload-large-folder tool
Browse filesThis view is limited to 50 files because it contains too many changes. See raw diff
- .gitattributes +170 -0
- parse/train/0-uUGPbIjD/0-uUGPbIjD.md +559 -0
- parse/train/0-uUGPbIjD/0-uUGPbIjD_content_list.json +0 -0
- parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi.md +338 -0
- parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_content_list.json +0 -0
- parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_middle.json +0 -0
- parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_model.json +0 -0
- parse/train/5vShUEyjmm/5vShUEyjmm.md +389 -0
- parse/train/5vShUEyjmm/5vShUEyjmm_content_list.json +2026 -0
- parse/train/5vShUEyjmm/5vShUEyjmm_middle.json +0 -0
- parse/train/5vShUEyjmm/5vShUEyjmm_model.json +0 -0
- parse/train/6RB77-6-_oI/6RB77-6-_oI_layout.pdf +3 -0
- parse/train/6RB77-6-_oI/6RB77-6-_oI_origin.pdf +3 -0
- parse/train/6RB77-6-_oI/6RB77-6-_oI_span.pdf +3 -0
- parse/train/79zWncwO2p/79zWncwO2p_layout.pdf +3 -0
- parse/train/79zWncwO2p/79zWncwO2p_origin.pdf +3 -0
- parse/train/79zWncwO2p/79zWncwO2p_span.pdf +3 -0
- parse/train/7_eLEvFjCi3/7_eLEvFjCi3.md +318 -0
- parse/train/7_eLEvFjCi3/7_eLEvFjCi3_content_list.json +1559 -0
- parse/train/7_eLEvFjCi3/7_eLEvFjCi3_middle.json +0 -0
- parse/train/7_eLEvFjCi3/7_eLEvFjCi3_model.json +0 -0
- parse/train/9az9VKjOx00/9az9VKjOx00_layout.pdf +3 -0
- parse/train/9az9VKjOx00/9az9VKjOx00_origin.pdf +3 -0
- parse/train/9az9VKjOx00/9az9VKjOx00_span.pdf +3 -0
- parse/train/B1Kh0SVodY/B1Kh0SVodY.md +347 -0
- parse/train/B1Kh0SVodY/B1Kh0SVodY_content_list.json +1375 -0
- parse/train/B1Kh0SVodY/B1Kh0SVodY_middle.json +0 -0
- parse/train/B1Kh0SVodY/B1Kh0SVodY_model.json +0 -0
- parse/train/B1VZqjAcYX/B1VZqjAcYX_layout.pdf +3 -0
- parse/train/B1VZqjAcYX/B1VZqjAcYX_origin.pdf +3 -0
- parse/train/B1VZqjAcYX/B1VZqjAcYX_span.pdf +3 -0
- parse/train/B1eWbxStPH/B1eWbxStPH_layout.pdf +3 -0
- parse/train/B1eWbxStPH/B1eWbxStPH_origin.pdf +3 -0
- parse/train/B1eWbxStPH/B1eWbxStPH_span.pdf +3 -0
- parse/train/B1mAJI9gl/B1mAJI9gl_layout.pdf +3 -0
- parse/train/B1mAJI9gl/B1mAJI9gl_origin.pdf +3 -0
- parse/train/B1mAJI9gl/B1mAJI9gl_span.pdf +3 -0
- parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w.md +0 -0
- parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_content_list.json +0 -0
- parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_middle.json +0 -0
- parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_model.json +0 -0
- parse/train/BJl6TjRcY7/BJl6TjRcY7_layout.pdf +3 -0
- parse/train/BJl6TjRcY7/BJl6TjRcY7_origin.pdf +3 -0
- parse/train/BJl6TjRcY7/BJl6TjRcY7_span.pdf +3 -0
- parse/train/BJlkgaNKvr/BJlkgaNKvr.md +0 -0
- parse/train/BJlkgaNKvr/BJlkgaNKvr_content_list.json +0 -0
- parse/train/BJlkgaNKvr/BJlkgaNKvr_middle.json +0 -0
- parse/train/BJlkgaNKvr/BJlkgaNKvr_model.json +0 -0
- parse/train/BkV4VS9ll/BkV4VS9ll.md +599 -0
- parse/train/BkV4VS9ll/BkV4VS9ll_content_list.json +0 -0
.gitattributes
CHANGED
|
@@ -7589,3 +7589,173 @@ parse/train/H1f7S3C9YQ/H1f7S3C9YQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
|
| 7589 |
parse/train/H1f7S3C9YQ/H1f7S3C9YQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7590 |
parse/train/H1-nGgWC-/H1-nGgWC-_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7591 |
parse/train/H1-nGgWC-/H1-nGgWC-_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 7589 |
parse/train/H1f7S3C9YQ/H1f7S3C9YQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7590 |
parse/train/H1-nGgWC-/H1-nGgWC-_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7591 |
parse/train/H1-nGgWC-/H1-nGgWC-_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7592 |
+
parse/train/H1-nGgWC-/H1-nGgWC-_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7593 |
+
parse/train/BJl6TjRcY7/BJl6TjRcY7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7594 |
+
parse/train/BJl6TjRcY7/BJl6TjRcY7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7595 |
+
parse/train/BJl6TjRcY7/BJl6TjRcY7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7596 |
+
parse/train/BklEFpEYwS/BklEFpEYwS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7597 |
+
parse/train/BklEFpEYwS/BklEFpEYwS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7598 |
+
parse/train/BklEFpEYwS/BklEFpEYwS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7599 |
+
parse/train/HJxpDiC5tX/HJxpDiC5tX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7600 |
+
parse/train/HJxpDiC5tX/HJxpDiC5tX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7601 |
+
parse/train/HJxpDiC5tX/HJxpDiC5tX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7602 |
+
parse/train/SygWvAVFPr/SygWvAVFPr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7603 |
+
parse/train/SygWvAVFPr/SygWvAVFPr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7604 |
+
parse/train/SygWvAVFPr/SygWvAVFPr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7605 |
+
parse/train/rJNpifWAb/rJNpifWAb_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7606 |
+
parse/train/rJNpifWAb/rJNpifWAb_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7607 |
+
parse/train/rJNpifWAb/rJNpifWAb_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7608 |
+
parse/train/SklGryBtwr/SklGryBtwr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7609 |
+
parse/train/SklGryBtwr/SklGryBtwr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7610 |
+
parse/train/SklGryBtwr/SklGryBtwr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7611 |
+
parse/train/sSjqmfsk95O/sSjqmfsk95O_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7612 |
+
parse/train/sSjqmfsk95O/sSjqmfsk95O_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7613 |
+
parse/train/sSjqmfsk95O/sSjqmfsk95O_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7614 |
+
parse/train/HJxdTxHYvB/HJxdTxHYvB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7615 |
+
parse/train/HJxdTxHYvB/HJxdTxHYvB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7616 |
+
parse/train/HJxdTxHYvB/HJxdTxHYvB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7617 |
+
parse/train/Hke_f0EYPH/Hke_f0EYPH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7618 |
+
parse/train/Hke_f0EYPH/Hke_f0EYPH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7619 |
+
parse/train/Hke_f0EYPH/Hke_f0EYPH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7620 |
+
parse/train/B1mAJI9gl/B1mAJI9gl_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7621 |
+
parse/train/B1mAJI9gl/B1mAJI9gl_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7622 |
+
parse/train/B1mAJI9gl/B1mAJI9gl_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7623 |
+
parse/train/XjYgR6gbCEc/XjYgR6gbCEc_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7624 |
+
parse/train/XjYgR6gbCEc/XjYgR6gbCEc_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7625 |
+
parse/train/XjYgR6gbCEc/XjYgR6gbCEc_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7626 |
+
parse/train/huAdB-Tj4yG/huAdB-Tj4yG_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7627 |
+
parse/train/huAdB-Tj4yG/huAdB-Tj4yG_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7628 |
+
parse/train/huAdB-Tj4yG/huAdB-Tj4yG_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7629 |
+
parse/train/Hyg-JC4FDr/Hyg-JC4FDr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7630 |
+
parse/train/Hyg-JC4FDr/Hyg-JC4FDr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7631 |
+
parse/train/Hyg-JC4FDr/Hyg-JC4FDr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7632 |
+
parse/train/SkhU2fcll/SkhU2fcll_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7633 |
+
parse/train/SkhU2fcll/SkhU2fcll_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7634 |
+
parse/train/SkhU2fcll/SkhU2fcll_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7635 |
+
parse/train/Z7IViZ6Ir1k/Z7IViZ6Ir1k_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7636 |
+
parse/train/Z7IViZ6Ir1k/Z7IViZ6Ir1k_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7637 |
+
parse/train/Z7IViZ6Ir1k/Z7IViZ6Ir1k_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7638 |
+
parse/train/HygsfnR9Ym/HygsfnR9Ym_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7639 |
+
parse/train/HygsfnR9Ym/HygsfnR9Ym_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7640 |
+
parse/train/HygsfnR9Ym/HygsfnR9Ym_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7641 |
+
parse/train/KmykpuSrjcq/KmykpuSrjcq_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7642 |
+
parse/train/KmykpuSrjcq/KmykpuSrjcq_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7643 |
+
parse/train/KmykpuSrjcq/KmykpuSrjcq_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7644 |
+
parse/train/N33d7wjgzde/N33d7wjgzde_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7645 |
+
parse/train/N33d7wjgzde/N33d7wjgzde_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7646 |
+
parse/train/N33d7wjgzde/N33d7wjgzde_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7647 |
+
parse/train/9az9VKjOx00/9az9VKjOx00_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7648 |
+
parse/train/9az9VKjOx00/9az9VKjOx00_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7649 |
+
parse/train/9az9VKjOx00/9az9VKjOx00_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7650 |
+
parse/train/S1E3Ko09F7/S1E3Ko09F7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7651 |
+
parse/train/S1E3Ko09F7/S1E3Ko09F7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7652 |
+
parse/train/S1E3Ko09F7/S1E3Ko09F7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7653 |
+
parse/train/6RB77-6-_oI/6RB77-6-_oI_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7654 |
+
parse/train/6RB77-6-_oI/6RB77-6-_oI_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7655 |
+
parse/train/6RB77-6-_oI/6RB77-6-_oI_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7656 |
+
parse/train/ryfcCo0ctQ/ryfcCo0ctQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7657 |
+
parse/train/ryfcCo0ctQ/ryfcCo0ctQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7658 |
+
parse/train/ryfcCo0ctQ/ryfcCo0ctQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7659 |
+
parse/train/HkwVAXyCW/HkwVAXyCW_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7660 |
+
parse/train/HkwVAXyCW/HkwVAXyCW_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7661 |
+
parse/train/HkwVAXyCW/HkwVAXyCW_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7662 |
+
parse/train/S1l6ITVKPS/S1l6ITVKPS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7663 |
+
parse/train/S1l6ITVKPS/S1l6ITVKPS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7664 |
+
parse/train/S1l6ITVKPS/S1l6ITVKPS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7665 |
+
parse/train/SJJQVZW0b/SJJQVZW0b_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7666 |
+
parse/train/SJJQVZW0b/SJJQVZW0b_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7667 |
+
parse/train/SJJQVZW0b/SJJQVZW0b_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7668 |
+
parse/train/tC6iW2UUbJf/tC6iW2UUbJf_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7669 |
+
parse/train/tC6iW2UUbJf/tC6iW2UUbJf_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7670 |
+
parse/train/tC6iW2UUbJf/tC6iW2UUbJf_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7671 |
+
parse/train/MRYuRHsHzK4/MRYuRHsHzK4_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7672 |
+
parse/train/MRYuRHsHzK4/MRYuRHsHzK4_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7673 |
+
parse/train/MRYuRHsHzK4/MRYuRHsHzK4_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7674 |
+
parse/train/rJWechg0Z/rJWechg0Z_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7675 |
+
parse/train/rJWechg0Z/rJWechg0Z_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7676 |
+
parse/train/rJWechg0Z/rJWechg0Z_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7677 |
+
parse/train/r1xdH3CcKX/r1xdH3CcKX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7678 |
+
parse/train/r1xdH3CcKX/r1xdH3CcKX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7679 |
+
parse/train/r1xdH3CcKX/r1xdH3CcKX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7680 |
+
parse/train/V1ZHVxJ6dSS/V1ZHVxJ6dSS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7681 |
+
parse/train/V1ZHVxJ6dSS/V1ZHVxJ6dSS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7682 |
+
parse/train/V1ZHVxJ6dSS/V1ZHVxJ6dSS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7683 |
+
parse/train/O9bnihsFfXU/O9bnihsFfXU_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7684 |
+
parse/train/O9bnihsFfXU/O9bnihsFfXU_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7685 |
+
parse/train/O9bnihsFfXU/O9bnihsFfXU_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7686 |
+
parse/train/z-X_PpwaroO/z-X_PpwaroO_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7687 |
+
parse/train/z-X_PpwaroO/z-X_PpwaroO_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7688 |
+
parse/train/z-X_PpwaroO/z-X_PpwaroO_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7689 |
+
parse/train/BkmM8Dceg/BkmM8Dceg_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7690 |
+
parse/train/BkmM8Dceg/BkmM8Dceg_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7691 |
+
parse/train/BkmM8Dceg/BkmM8Dceg_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7692 |
+
parse/train/B1VZqjAcYX/B1VZqjAcYX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7693 |
+
parse/train/B1VZqjAcYX/B1VZqjAcYX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7694 |
+
parse/train/B1VZqjAcYX/B1VZqjAcYX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7695 |
+
parse/train/BkVsEMYel/BkVsEMYel_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7696 |
+
parse/train/BkVsEMYel/BkVsEMYel_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7697 |
+
parse/train/BkVsEMYel/BkVsEMYel_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7698 |
+
parse/train/H1gzR2VKDH/H1gzR2VKDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7699 |
+
parse/train/H1gzR2VKDH/H1gzR2VKDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7700 |
+
parse/train/H1gzR2VKDH/H1gzR2VKDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7701 |
+
parse/train/ByME42AqK7/ByME42AqK7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7702 |
+
parse/train/ByME42AqK7/ByME42AqK7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7703 |
+
parse/train/ByME42AqK7/ByME42AqK7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7704 |
+
parse/train/ryCcJaqgl/ryCcJaqgl_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7705 |
+
parse/train/ryCcJaqgl/ryCcJaqgl_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7706 |
+
parse/train/ryCcJaqgl/ryCcJaqgl_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7707 |
+
parse/train/ryggIs0cYQ/ryggIs0cYQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7708 |
+
parse/train/ryggIs0cYQ/ryggIs0cYQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7709 |
+
parse/train/ryggIs0cYQ/ryggIs0cYQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7710 |
+
parse/train/HkgsWxrtPB/HkgsWxrtPB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7711 |
+
parse/train/HkgsWxrtPB/HkgsWxrtPB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7712 |
+
parse/train/HkgsWxrtPB/HkgsWxrtPB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7713 |
+
parse/train/B1eWbxStPH/B1eWbxStPH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7714 |
+
parse/train/B1eWbxStPH/B1eWbxStPH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7715 |
+
parse/train/B1eWbxStPH/B1eWbxStPH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7716 |
+
parse/train/S1fUpoR5FQ/S1fUpoR5FQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7717 |
+
parse/train/S1fUpoR5FQ/S1fUpoR5FQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7718 |
+
parse/train/S1fUpoR5FQ/S1fUpoR5FQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7719 |
+
parse/train/ZsZM-4iMQkH/ZsZM-4iMQkH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7720 |
+
parse/train/ZsZM-4iMQkH/ZsZM-4iMQkH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7721 |
+
parse/train/ZsZM-4iMQkH/ZsZM-4iMQkH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7722 |
+
parse/train/ryE98iR5tm/ryE98iR5tm_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7723 |
+
parse/train/ryE98iR5tm/ryE98iR5tm_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7724 |
+
parse/train/ryE98iR5tm/ryE98iR5tm_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7725 |
+
parse/train/OeWooOxFwDa/OeWooOxFwDa_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7726 |
+
parse/train/OeWooOxFwDa/OeWooOxFwDa_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7727 |
+
parse/train/OeWooOxFwDa/OeWooOxFwDa_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7728 |
+
parse/train/HklY120cYm/HklY120cYm_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7729 |
+
parse/train/HklY120cYm/HklY120cYm_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7730 |
+
parse/train/HklY120cYm/HklY120cYm_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7731 |
+
parse/train/HJgC60EtwB/HJgC60EtwB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7732 |
+
parse/train/HJgC60EtwB/HJgC60EtwB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7733 |
+
parse/train/HJgC60EtwB/HJgC60EtwB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7734 |
+
parse/train/SkxxtgHKPS/SkxxtgHKPS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7735 |
+
parse/train/SkxxtgHKPS/SkxxtgHKPS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7736 |
+
parse/train/SkxxtgHKPS/SkxxtgHKPS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7737 |
+
parse/train/r6I3EvB9eDO/r6I3EvB9eDO_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7738 |
+
parse/train/r6I3EvB9eDO/r6I3EvB9eDO_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7739 |
+
parse/train/r6I3EvB9eDO/r6I3EvB9eDO_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7740 |
+
parse/train/H1ekF2EYDH/H1ekF2EYDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7741 |
+
parse/train/H1ekF2EYDH/H1ekF2EYDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7742 |
+
parse/train/H1ekF2EYDH/H1ekF2EYDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7743 |
+
parse/train/BklfR3EYDH/BklfR3EYDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7744 |
+
parse/train/BklfR3EYDH/BklfR3EYDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7745 |
+
parse/train/BklfR3EYDH/BklfR3EYDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7746 |
+
parse/train/XSLF1XFq5h/XSLF1XFq5h_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7747 |
+
parse/train/XSLF1XFq5h/XSLF1XFq5h_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7748 |
+
parse/train/XSLF1XFq5h/XSLF1XFq5h_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7749 |
+
parse/train/Byg1v1HKDB/Byg1v1HKDB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7750 |
+
parse/train/Byg1v1HKDB/Byg1v1HKDB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7751 |
+
parse/train/Byg1v1HKDB/Byg1v1HKDB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7752 |
+
parse/train/hQGRD1Zael7/hQGRD1Zael7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7753 |
+
parse/train/79zWncwO2p/79zWncwO2p_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7754 |
+
parse/train/79zWncwO2p/79zWncwO2p_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7755 |
+
parse/train/79zWncwO2p/79zWncwO2p_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7756 |
+
parse/train/gDcaUj4Myhn/gDcaUj4Myhn_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7757 |
+
parse/train/gDcaUj4Myhn/gDcaUj4Myhn_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7758 |
+
parse/train/gDcaUj4Myhn/gDcaUj4Myhn_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7759 |
+
parse/train/ByxY8CNtvr/ByxY8CNtvr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7760 |
+
parse/train/ByxY8CNtvr/ByxY8CNtvr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 7761 |
+
parse/train/ByxY8CNtvr/ByxY8CNtvr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
parse/train/0-uUGPbIjD/0-uUGPbIjD.md
ADDED
|
@@ -0,0 +1,559 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# HUMAN-LEVEL PERFORMANCE IN NO-PRESS DIPLOMACY VIA EQUILIBRIUM SEARCH
|
| 2 |
+
|
| 3 |
+
Jonathan Gray∗, Adam Lerer∗, Anton Bakhtin, Noam Brown Facebook AI Research {jsgray,alerer,yolo,noambrown}@fb.com
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Prior AI breakthroughs in complex games have focused on either the purely adversarial or purely cooperative settings. In contrast, Diplomacy is a game of shifting alliances that involves both cooperation and competition. For this reason, Diplomacy has proven to be a formidable research challenge. In this paper we describe an agent for the no-press variant of Diplomacy that combines supervised learning on human data with one-step lookahead search via regret minimization. Regret minimization techniques have been behind previous AI successes in adversarial games, most notably poker, but have not previously been shown to be successful in large-scale games involving cooperation. We show that our agent greatly exceeds the performance of past no-press Diplomacy bots, is unexploitable by expert humans, and ranks in the top $2 \%$ of human players when playing anonymous games on a popular Diplomacy website.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
A primary goal for AI research is to develop agents that can act optimally in real-world multi-agent interactions (i.e., games). In recent years, AI agents have achieved expert-level or even superhuman performance in benchmark games such as backgammon (Tesauro, 1994), chess (Campbell et al., 2002), Go (Silver et al., 2016; 2017; 2018), poker (Moravcík et al., 2017; Brown & Sandholm, ˇ 2017; 2019b), and real-time strategy games (Berner et al., 2019; Vinyals et al., 2019). However, previous large-scale game AI results have focused on either purely competitive or purely cooperative settings. In contrast, real-world games, such as business negotiations, politics, and traffic navigation, involve a far more complex mixture of cooperation and competition. In such settings, the theoretical grounding for the techniques used in previous AI breakthroughs falls apart.
|
| 12 |
+
|
| 13 |
+
In this paper we augment neural policies trained through imitation learning with regret minimization search techniques, and evaluate on the benchmark game of no-press Diplomacy. Diplomacy is a longstanding benchmark for research that features a rich mixture of cooperation and competition. Like previous researchers, we evaluate on the widely played no-press variant of Diplomacy, in which communication can only occur through the actions in the game (i.e., no cheap talk is allowed).
|
| 14 |
+
|
| 15 |
+
Specifically, we begin with a blueprint policy that approximates human play in a dataset of Diplomacy games. We then improve upon the blueprint during play by approximating an equilibrium for the current phase of the game, assuming all players (including our agent) play the blueprint for the remainder of the game. Our agent then plays its part of the computed equilibrium. The equilibrium is computed via regret matching (RM) (Blackwell et al., 1956; Hart & Mas-Colell, 2000).
|
| 16 |
+
|
| 17 |
+
Search via RM has led to remarkable success in poker. However, RM only converges to a Nash equilibrium in two-player zero-sum games and other special cases, and RM was never previously shown to produce strong policies in a mixed cooperative/competitive game as complex as no-press Diplomacy. Nevertheless, we show that our agent exceeds the performance of prior agents and for the first time convincingly achieves human-level performance in no-press Diplomacy. Specifically, we show that our agent soundly defeats previous agents, that our agent is far less exploitable than previous agents, that an expert human cannot exploit our agent even in repeated play, and, most importantly, that our agent ranks in the top $2 \%$ of human players when playing anonymous games on a popular Diplomacy website.
|
| 18 |
+
|
| 19 |
+
# 2 BACKGROUND AND RELATED WORK
|
| 20 |
+
|
| 21 |
+
Search has previously been used in almost every major game AI breakthrough, including backgammon (Tesauro, 1994), chess (Campbell et al., 2002), Go (Silver et al., 2016; 2017; 2018), poker (Moravcík et al., 2017; Brown & Sandholm, 2017; 2019b), and Hanabi (Lerer et al., 2020). ˇ A major exception is real-time strategy games (Vinyals et al., 2019; Berner et al., 2019). Similar to SPARTA as used in Hanabi (Lerer et al., 2020), our agent conducts one-ply lookahead search (i.e., changes the policy just for the current game turn) and thereafter assumes all players play according to the blueprint. Similar to the Pluribus poker agent (Brown & Sandholm, 2019b), our search technique uses regret matching to compute an approximate equilibrium. In a manner similar to the sampled best response algorithm of Anthony et al. (2020), we sample a limited number of actions from the blueprint policy rather than search over all possible actions, which would be intractable.
|
| 22 |
+
|
| 23 |
+
Learning effective policies in games involving cooperation and competition has been studied extensively in the field of multi-agent reinforcement learning (MARL) (Shoham et al., 2003). Nash-Q and CE-Q applied Q learning for general sum games by using Q values derived by computing Nash (or correlated) equilibrium values at the target states (Hu & Wellman, 2003; Greenwald et al., 2003). Friend-or-foe Q learning treats other agents as either cooperative or adversarial, where the Nash Q values are well defined Littman (2001). The recent focus on “deep” MARL has led to learning rules from game theory such as fictitious play and regret minimization being adapted to deep reinforcement learning (Heinrich & Silver, 2016; Brown et al., 2019), as well as work on game-theoretic challenges of mixed cooperative/competitive settings such as social dilemmas and multiple equilibria in the MARL setting (Leibo et al., 2017; Lerer & Peysakhovich, 2017; 2019).
|
| 24 |
+
|
| 25 |
+
Diplomacy in particular has served for decades as a benchmark for multi-agent AI research (Kraus & Lehmann, 1988; Kraus et al., 1994; Kraus & Lehmann, 1995; Johansson & Håård, 2005; Ferreira et al., 2015). Recently, Paquette et al. (2019) applied imitation learning (IL) via deep neural networks on a dataset of more than 150,000 Diplomacy games. This work greatly improved the state of the art for no-press Diplomacy, which was previously a handcrafted agent (van Hal, 2013). Paquette et al. (2019) also tested reinforcement learning (RL) in no-press Diplomacy via Advantage Actor-Critic (A2C) (Mnih et al., 2016). Anthony et al. (2020) introduced sampled best response policy iteration, a self-play technique, which further improved upon the performance of Paquette et al. (2019).
|
| 26 |
+
|
| 27 |
+
# 2.1 DESCRIPTION OF DIPLOMACY
|
| 28 |
+
|
| 29 |
+
The rules of no-press Diplomacy are complex; a full description is provided by Paquette et al. (2019). No-press Diplomacy is a seven-player zero-sum board game in which a map of Europe is divided into 75 provinces. 34 of these provinces contain supply centers (SCs), and the goal of the game is for a player to control a majority (18) of the SCs. Each players begins the game controlling three or four SCs and an equal number of units.
|
| 30 |
+
|
| 31 |
+
The game consists of three types of phases: movement phases in which each player assigns an order to each unit they control, retreat phases in which defeated units retreat to a neighboring province, and adjustment phases in which new units are built or existing units are destroyed.
|
| 32 |
+
|
| 33 |
+
During a movement phase, a player assigns an order to each unit they control. A unit’s order may be to hold (defend its province), move to a neighboring province, convoy a unit over water, or support a neighboring unit’s hold or move order. Support may be provided to units of any player. We refer to a tuple of orders, one order for each of a player’s units, as an action. That is, each player chooses one action each turn. There are an average of 26 valid orders for each unit (Paquette et al., 2019), so the game’s branching factor is massive and on some turns enumerating all actions is intractable.
|
| 34 |
+
|
| 35 |
+
Importantly, all actions occur simultaneously. In live games, players write down their orders and then reveal them at the same time. This makes Diplomacy an imperfect-information game in which an optimal policy may need to be stochastic in order to prevent predictability.
|
| 36 |
+
|
| 37 |
+
Diplomacy is designed in such a way that cooperation with other players is almost essential in order to achieve victory, even though only one player can ultimately win.
|
| 38 |
+
|
| 39 |
+
A game may end in a draw on any turn if all remaining players agree. Draws are a common outcome among experienced players because players will often coordinate to prevent any individual from reaching 18 centers. The two most common scoring systems for draws are draw-size scoring (DSS), in which all surviving players equally split a win, and sum-of-squares scoring (SoS), in which player $i$ receives a score o f C2P ij∈N C2j , where Ci is the number of SCs that player i controls (Fogel, 2020). Throughout this paper we use SoS scoring except in anonymous games against humans where the human host chooses a scoring system.
|
| 40 |
+
|
| 41 |
+
# 2.2 REGRET MATCHING
|
| 42 |
+
|
| 43 |
+
Regret Matching (RM) (Blackwell et al., 1956; Hart & Mas-Colell, 2000) is an iterative algorithm that converges to a Nash equilibrium (NE) (Nash, 1951) in two-player zero-sum games and other special cases, and converges to a coarse correlated equilibrium (CCE) (Hannan, 1957) in general.
|
| 44 |
+
|
| 45 |
+
We consider a game with $\mathcal { N }$ players where each player $i$ chooses an action $a _ { i }$ from a set of actions $\mathbf { \mathcal { A } } _ { i }$ . We denote the joint action as $a = ( a _ { 1 } , a _ { 2 } , \dotsc , a _ { N } )$ , the actions of all players other than $i$ as $a _ { - i }$ , and the set of joint actions as $\mathcal { A }$ . After all players simultaneously choose an action, player $i$ receives a reward of $v _ { i } ( a )$ (which can also be represented as $v _ { i } ( a _ { i } , a _ { - i } ) )$ . Players may also choose a probability distribution over actions, where the probability of action $a _ { i }$ is denoted $\pi _ { i } ( a _ { i } )$ and the vector of probabilities is denoted $\pi _ { i }$ .
|
| 46 |
+
|
| 47 |
+
Normally, each iteration of RM has a computational complexity of $\Pi _ { i \in { \mathcal { N } } } | { \mathcal { A } } _ { i } |$ . In a seven-player game, this is typically intractable. We therefore use a sampled form of RM in which each iteration has a computational complexity of $\textstyle \sum _ { i \in { \mathcal { N } } } | A _ { i } |$ . We now describe this sampled form of RM.
|
| 48 |
+
|
| 49 |
+
Each agent $i$ maintains an external regret value for each action $a _ { i } \in { \mathcal { A } } _ { i }$ , which we refer to simply as regret. The regret on iteration $t$ is denoted $R _ { i } ^ { t } ( a _ { i } )$ . Initially, all regrets are zero. On each iteration $t$ of RM, $\pi _ { i } ^ { t } ( a _ { i } )$ is set according to
|
| 50 |
+
|
| 51 |
+
$$
|
| 52 |
+
\pi _ { i } ^ { t } ( a _ { i } ) = \left\{ \begin{array} { l l } { \frac { \operatorname* { m a x } \{ 0 , R _ { i } ^ { t } ( a _ { i } ) \} } { \sum _ { a _ { i } ^ { \prime } \in A _ { i } } \operatorname* { m a x } \{ 0 , R _ { i } ^ { t } ( a _ { i } ^ { \prime } ) \} } } & { \mathrm { i f } \sum _ { a _ { i } ^ { \prime } \in A _ { i } } \operatorname* { m a x } \{ 0 , R _ { i } ^ { t } ( a _ { i } ^ { \prime } ) \} > 0 } \\ { \frac { 1 } { | A _ { i } | } } & { \mathrm { o t h e r w i s e } } \end{array} \right.
|
| 53 |
+
$$
|
| 54 |
+
|
| 55 |
+
Next, each player samples an action $a _ { i } ^ { * }$ from $\mathbf { \mathcal { A } } _ { i }$ according to $\pi _ { i } ^ { t }$ and all regrets are updated such that
|
| 56 |
+
|
| 57 |
+
$$
|
| 58 |
+
R _ { i } ^ { t + 1 } ( a _ { i } ) = R _ { i } ^ { t } ( a _ { i } ) + v _ { i } ( a _ { i } , a _ { - i } ^ { * } ) - \sum _ { a _ { i } ^ { \prime } \in { \cal A } _ { i } } \pi _ { i } ^ { t } ( a _ { i } ^ { \prime } ) v _ { i } ( a _ { i } ^ { \prime } , a _ { - i } ^ { * } )
|
| 59 |
+
$$
|
| 60 |
+
|
| 61 |
+
This sampled form of RM guarantees that $R _ { i } ^ { t } ( a _ { i } ) \in \mathcal { O } ( \sqrt { t } )$ with high probability (Lanctot et al., 2009). If $R _ { i } ^ { t } ( a _ { i } )$ grows sublinearly for all players’ actions, as is the case in RM, then the average policy over all iterations converges to a NE in two-player zero-sum games and in general the empirical distribution of players’ joint policies converges to a CCE as $t \to \infty$ .
|
| 62 |
+
|
| 63 |
+
In order to improve empirical performance, we use linear RM (Brown & Sandholm, 2019a), which weighs updates on iteration $t$ by $t$ .1 We also use optimism (Syrgkanis et al., 2015), in which the most recent iteration is counted twice when computing regret. Additionally, the action our agent ultimately plays is sampled from the final iteration’s policy, rather than the average policy over all iterations. This reduces the risk of sampling a non-equilibrium action due to insufficient convergence. We explain this modification in more detail in Appendix F.
|
| 64 |
+
|
| 65 |
+
# 3 AGENT DESCRIPTION
|
| 66 |
+
|
| 67 |
+
Our agent is composed of two major components. The first is a blueprint policy and state-value function trained via imitation learning on human data. The second is a search algorithm that utilizes the blueprint. This algorithm is executed on every turn, and approximates an equilibrium policy (for all players, not just the agent) for the current turn via RM, assuming that the blueprint is played by all players for the remaining game beyond the current turn.
|
| 68 |
+
|
| 69 |
+
# 3.1 SUPERVISED LEARNING
|
| 70 |
+
|
| 71 |
+
We construct a blueprint policy via imitation learning on a corpus of 46,148 Diplomacy games collected from online play, building on the methodology and model architecture described by Paquette et al. (2019) and Anthony et al. (2020). A blueprint policy and value function estimated from human play is ideal for performing search in a general-sum game, because it is likely to realistically approximate state values and other players’ actions when playing with humans. Our blueprint supervised model is based on the DipNet agent from Paquette et al. (2019), but we make a number of modifications to the architecture and training. A detailed description of the architecture is provided in Appendix A; in this section we highlight the key differences from prior work.
|
| 72 |
+
|
| 73 |
+
We trained the blueprint policy using only a subset of the data used by Paquette et al. (2019), specifically those games obtained from webdiplomacy.net. For this subset of the data, we obtained metadata about the press variant (full-press vs. no-press) which we add as a feature to the model, and anonymized player IDs for the participants in each game. Using the IDs, we computed ratings $s _ { i }$ for each player $i$ and only trained the policy on actions from players with above-average ratings. Appendix B describes our method for computing these ratings.
|
| 74 |
+
|
| 75 |
+
Our model closely follows the architecture of Paquette et al. (2019), with additional dropout of 0.4 between GNN encoder layers. We model sets of build orders as single tokens because there are a small number of build order combinations and it is tricky to predict sets auto-regressively with teacher forcing. We adopt the encoder changes of Anthony et al. (2020), but do not adopt their relational order decoder because it is more expensive to compute and leads to only marginal accuracy improvements after tuning dropout.
|
| 76 |
+
|
| 77 |
+
We make a small modification to the encoder GNN architecture that improves modeling. In addition to the standard residual that skips the entire GNN layer, we replace the graph convolution2 with the sum of a graph convolution and a linear layer. This allows the model to learn a hierarchy of features for each graph node (through the linear layer) without requiring a concomitant increase in graph smoothing (the GraphConv). The resulting GNN layer computes (modification in red)
|
| 78 |
+
|
| 79 |
+
$$
|
| 80 |
+
x _ { i + 1 } = D r o p o u t ( R e L U ( B N ( G r a p h C o n v ( x _ { i } ) + \mathbf { A x _ { i } } ) ) ) + x _ { i } .
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
where $A$ is a learned linear transformation.
|
| 84 |
+
|
| 85 |
+
Finally, we achieve a substantial improvement in order prediction accuracy using a featurized order decoder. Diplomacy has over 13,000 possible orders, many of which will be observed infrequently in the training data. Therefore, by featurizing the orders by the order type, and encodings of the source, destination, and support locations, we observe improved prediction accuracy.
|
| 86 |
+
|
| 87 |
+
Specifically, in a standard decoder each order $o$ has a learned representation $e _ { o }$ , and for some board encoding $x$ and learned order embedding $e _ { o }$ , $P ( o ) = s o f t m a x ( x \cdot e _ { o } )$ . With order featurization, we use $\tilde { e } _ { o } = e _ { o } + A f _ { o }$ , where $f _ { o }$ are static order features and $A$ is a learned linear transformation. The order featurization we use is the concatenation of the one-hot order type with the board encodings for the source, destination, and support locations. We found that representing order location features by their location encodings works better than one-hot locations, presumably because the model can learn more state-contextual features.3
|
| 88 |
+
|
| 89 |
+
We add an additional value head to the model immediately after the dipnet encoder, that is trained to estimate the final SoS scores given a board situation. We use this value estimate during equilibrium search (Sec. 3.2) to estimate the value of a Monte Carlo rollout after a fixed number of steps. The value head is an MLP with one hidden layer that takes as input the concatenated vector of all board position encodings. A softmax over powers’ SoS scores is applied at the end to enforce that all players’ SoS scores sum to 1.
|
| 90 |
+
|
| 91 |
+
<table><tr><td>Model</td><td>Policy Accuracy</td><td>SoS v. DipNet temp=0.5</td><td>temp=0.1</td></tr><tr><td>DipNet (Paquette et al. (2019))</td><td>60.5%4</td><td>0.143</td><td></td></tr><tr><td>+ combined build orders & encoder dropout + encoder changes from Anthony et al. (2020)</td><td>62.0% 62.4%</td><td>0.150</td><td>0.198</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>switch to webdiplomacy training data only</td><td>61.3%</td><td>0.175</td><td>0.206</td></tr><tr><td>+ output featurization</td><td>62.0%</td><td>0.184 0.183</td><td>0.188 0.205</td></tr><tr><td>+ improved GNN layer</td><td>62.4%</td><td></td><td></td></tr><tr><td>+ merged GNN trunk</td><td>62.9%</td><td>0.199</td><td>0.202</td></tr></table>
|
| 92 |
+
|
| 93 |
+
Table 1: Effect of model and training data changes on supervised model quality. We measure policy accuracy as well as average SoS score achieved by each agent against 6 of the original DipNet model. We measure the SoS scores in two settings: with all 7 agents sampling orders at a temperature of either 0.5 or 0.1.
|
| 94 |
+
|
| 95 |
+
|
| 96 |
+
Figure 1: Left: Score of SearchBot using different numbers of sampled subgame actions, against 6 DipNet agents ((Paquette et al., 2019) at temperature 0.1). A score of $1 4 . 3 \%$ would be a tie. Even when sampling only two actions, SearchBot dramatically outperforms our blueprint, which achieves a score of $2 0 . 2 \%$ . Middle: The effect of the number iterations of sampled regret matching on SearchBot performance. Right: The effect of different rollout lengths on SearchBot performance.
|
| 97 |
+
|
| 98 |
+
# 3.2 EQUILIBRIUM SEARCH
|
| 99 |
+
|
| 100 |
+
The policy that is actually played results from a search algorithm which utilizes the blueprint policy. Let $s$ be the current state of the game. On each turn, the search algorithm computes an equilibrium for a subgame and our agent plays according to its part of the equilibrium solution for its next action.
|
| 101 |
+
|
| 102 |
+
Conceptually, the subgame is a well-defined game that begins at state $s$ . The set of actions available to each player is a subset of the possible actions in state $s$ in the full game, and are referred to as the subgame actions. Each player $i$ chooses a subgame action $a _ { i }$ , resulting in joint subgame action $a$ . After $a$ is taken, the players make no further decisions in the subgame. Instead, the players receive a reward corresponding to the players sampling actions according to the blueprint policy $\pi ^ { b }$ for the remaining game.
|
| 103 |
+
|
| 104 |
+
The subgame actions for player $i$ are the $M k _ { i }$ highest-probability actions according to the blueprint model, where $k _ { i }$ is the number of units controlled by player $i$ and $M$ is a hyperparameter (usually set to 5). The effect of different numbers of subgame actions is plotted in Figure 1 (left).
|
| 105 |
+
|
| 106 |
+
Rolling out $\pi ^ { b }$ to the end of the game is very expensive, so in practice we instead roll out $\pi ^ { b }$ for a small number of turns (usually 2 or 3 movement phases in our experiments) until state $s ^ { \prime }$ is reached, and then use the value for $s ^ { \prime }$ from the blueprint’s value network as the reward vector. Figure 1 (right) shows the performance of our search agent using different rollout lengths. We do not observe improved performance for rolling out farther than 3 or 4 movement phases.
|
| 107 |
+
|
| 108 |
+
We compute a policy for each agent by running the sampled regret matching algorithm described in Section 2.2. The search algorithm used $2 5 6 - 4 { , } 0 9 6$ iterations of RM and typically required between 2 minutes and 20 minutes per turn using a single Volta GPU and 8 CPU cores, depending on the hyperparameters used for the game. Details on the hyperparameters we used are provided in Appendix G.
|
| 109 |
+
|
| 110 |
+
# 4 RESULTS
|
| 111 |
+
|
| 112 |
+
Using the techniques described in Section 3, we developed an agent we call SearchBot. Our experiments focus on two formats. The first evaluates the head-to-head performance of SearchBot playing against the population of human players on a popular Diplomacy website, as well as against prior AI agents. The second measures the exploitability of SearchBot.
|
| 113 |
+
|
| 114 |
+
# 4.1 PERFORMANCE AGAINST A POPULATION OF HUMAN PLAYERS
|
| 115 |
+
|
| 116 |
+
The ultimate test of an AI system is how well it performs in the real world with humans. To measure this, we had SearchBot anonymously play no-press Diplomacy games on the popular Diplomacy website webdiplomacy.net. Since there are 7 players in each game, average human performance is a score of $1 4 . 3 \%$ . In contrast, SearchBot scored $2 6 . 6 \% \pm 3 . 2 \%$ .6 Table 2 shows the agent’s performance and a detailed breakdown is presented in Table 6 in Appendix G. A qualitative analysis of the bot’s play by three-time world Diplomacy champion Andrew Goff is presented in Appendix I.
|
| 117 |
+
|
| 118 |
+
<table><tr><td>Power</td><td>Bot Score</td><td> Human Mean</td><td>Games</td><td>Wins</td><td>Draws</td><td>Losses</td></tr><tr><td>All Games</td><td>26.6% ± 3.2%</td><td>14.3%</td><td>116</td><td>16</td><td>43</td><td>57</td></tr></table>
|
| 119 |
+
|
| 120 |
+
Table 2: Average SoS score of our agent in anonymous games against humans on webdiplomacy.net. Average human performance is $1 4 . 3 \%$ . Score in the case of draws was determined by the rules of the joined game. The $\pm$ shows one standard error.
|
| 121 |
+
|
| 122 |
+
In addition to raw score, we measured SearchBot’s performance using the Ghost-Rating system (Anthony, 2020), which is a Diplomacy rating system inspired by the Elo system that accounts for the relative strength of opponents and that is used to semi-officially rank players on webdiplomacy.net. Among no-press Diplomacy players on the site, our agent ranked 17 out of 901 players with a Ghost-Rating of 183.4.7 Details on the setup for experiments are provided in Appendix G.
|
| 123 |
+
|
| 124 |
+
# 4.2 PERFORMANCE AGAINST OTHER AI AGENTS
|
| 125 |
+
|
| 126 |
+
In addition to playing against humans, we also evaluated SearchBot against prior AI agents from Paquette et al. (2019) (one based on supervised learning, and another based on reinforcement learning), our blueprint agent, and a best-response agent (BRBot). Table 3 shows these results. Following prior work, we compute average scores in ‘1v6’ games containing a single agent of type A and six agents of type B. The average score of an identical agent should therefore be $1 / 7 \stackrel { \cdot } { \approx } 1 4 . 3 \%$ . SearchBot achieves its highest 1v6 score when matched against its own blueprint, since it is most accurately able to approximate the behavior of that agent. It outperforms all three agents by a large margin, and none of the three baselines is able to achieve a score of more than $1 \%$ against our search agent.
|
| 127 |
+
|
| 128 |
+
BRBot is an agent that computes a best response to the blueprint policy in the subgame. Like SearchBot, BRBot considers the $M k _ { i }$ highest-probability actions according to the blueprint. Unlike SearchBot, BRBot searches over only its own actions, and rather than computing an equilibrium, it plays the action that yields the greatest reward assuming other powers play the blueprint policy.
|
| 129 |
+
|
| 130 |
+
Finally, we ran 25 games between one SearchBot and six copies of Albert (van Hal, 2013), the state-of-the-art rule-based bot. SearchBot achieved an average SoS score of $6 9 . 3 \%$ .
|
| 131 |
+
|
| 132 |
+
# 4.3 EXPLOITABILITY
|
| 133 |
+
|
| 134 |
+
While performance of an agent within a population of human players is the most important metric, that metric alone does not capture how the population of players might adapt to the agent’s presence.
|
| 135 |
+
|
| 136 |
+
Table 3: Average SoS scores achieved by each of the agents against each other. DipNet agents from (Paquette et al., 2019) and the Blueprint agent use a temperature of 0.1. BRBot scores higher than SearchBot against the Blueprint, but SearchBot outperforms BRBot in a head-to-head comparison.
|
| 137 |
+
|
| 138 |
+
<table><tr><td>1x↓6x→</td><td>DipNet</td><td>DipNet RL</td><td>Blueprint</td><td>BRBot</td><td>SearchBot</td></tr><tr><td>DipNet</td><td>1</td><td>6.7% ± 0.9%</td><td>11.6% ± 0.1%</td><td>0.1% ± 0.1%</td><td>0.7% ± 0.2%</td></tr><tr><td>DipNet RL</td><td>18.9% ±1.4%</td><td></td><td>10.5% ±1.1%</td><td>0.1% ±0.1%</td><td>0.6% ±0.2%</td></tr><tr><td>Blueprint</td><td>20.2% ±1.3%</td><td>7.5% ± 1.0%</td><td>-</td><td>0.3% ± 0.1%</td><td>0.9% ± 0.2%</td></tr><tr><td>BRBot</td><td>67.3% ± 1.0%</td><td>43.7% ± 1.0%</td><td>69.3% ± 1.7%</td><td></td><td>11.1% ±1.1%</td></tr><tr><td>SearchBot</td><td>51.1% ± 1.9%</td><td>35.2% ± 1.8%</td><td>52.7% ± 1.3%</td><td>17.2% ± 1.3%</td><td>-</td></tr></table>
|
| 139 |
+
|
| 140 |
+
For example, if our agent is extremely strong then over time other players might adopt the bot’s playstyle. As the percentage of players playing like the bot increases, other players might adopt a policy that seeks to exploit this playstyle. Thus, if the bot’s policy is highly exploitable then it might eventually do poorly even if it initially performs well against the population of human players.
|
| 141 |
+
|
| 142 |
+
Motivated by this, we measure the exploitability of our agent. Exploitability of a policy profile $\pi$ (denoted $e ( \pi ) )$ ) measures worst-case performance when all but one agents follows $\pi$ . Formally, the exploitability of $\pi$ is defined as $\begin{array} { r } { e ( \pi ) { \bf \bar { \Phi } } = \sum _ { i \in \mathcal { N } } ( \operatorname* { m a x } _ { \pi _ { i } ^ { * } } v _ { i } ( \pi _ { i } ^ { * } , \pi _ { - i } ) - { \bf \bar { \Phi } } v _ { i } ( \pi ) ) / N } \end{array}$ , where $\pi _ { - i }$ denotes the policies of all players other than $i$ . Agent $i$ ’s best response to $\pi _ { - i }$ is defined as $B R ( \pi _ { - i } ) =$ arg maxπi $v _ { i } ( \pi _ { i } , \pi _ { - i } )$ .
|
| 143 |
+
|
| 144 |
+
We estimate our agent’s full-game exploitability in two ways: by training an RL agent to best respond to the bot, and by having expert humans repeatedly play against six copies of the bot. We also measure the ‘local’ exploitability in the search subgame and show that it converges to an approximate Nash equilibrium.
|
| 145 |
+
|
| 146 |
+
# 4.3.1 PERFORMANCE AGAINST A BEST-RESPONDING AGENT
|
| 147 |
+
|
| 148 |
+
When the policies of all players but one are fixed, the game becomes a Markov Decision Process (MDP) (Howard, 1960) for the non-fixed player because the actions of the fixed players can be viewed as stochastic transitions in the “environment”. Thus, we can estimate the exploitability of $\pi$ by first training a best response policy $B R ( \pi _ { - i } )$ for each agent $i$ using any single-agent RL algorithm, and then computing $\begin{array} { r } { \sum _ { i \in \mathcal { N } } v _ { i } ( B R ( \pi _ { - i } ) , \pi _ { - i } ) / N } \end{array}$ . Since the best response RL policy will not be an exact best response (which is intractable to compute in a game as complex as no-press Diplomacy) this only gives us a lower-bound estimate of the exploitability.
|
| 149 |
+
|
| 150 |
+
Following other work on environments with huge action spaces (Vinyals et al., 2019; Berner et al., 2019), we use a distributed asynchronous actor-critic RL approach to optimize the exploiter policy (Espeholt et al., 2018). We use the same architecture for the exploiter agent as for the fixed model. Moreover, to simplify the training we initialize the exploiter agent from the fixed model.
|
| 151 |
+
|
| 152 |
+
We found that training becomes unstable when the policy entropy gets too low. The standard remedy is to use an entropy regularization term. However, due to the immense action space, an exact computation of the entropy term, $E _ { a } \log p _ { \theta } ( a )$ , is infeasible. Instead, we optimize a surrogate loss that gives an unbiased estimate of the gradient of the entropy loss (see Appendix D). We found this to be critical for the stability of the training.
|
| 153 |
+
|
| 154 |
+
Training an RL agent to exploit SearchBot is prohibitively expensive. Even when choosing hyperparameters that would result in the agent playing as fast as possible, SearchBot typically requires at least a full minute in order to act each turn. Instead, we collect a dataset of self-play games of SearchBot and train a supervised agent on this dataset. The resulting agent, which we refer to as SearchBot-clone, is weaker than SearchBot but requires only a single pass through the neural network in order to act on a turn. By training an agent to exploit SearchBot-clone, we can obtain a (likely) upper bound on what the performance would be if a similar RL agent were trained against SearchBot.
|
| 155 |
+
|
| 156 |
+
We report the reward of the exploiter agents against the blueprint and SearchBot-clone agents in Figure 2. The results show that SearchBot-clone is highly exploitable, with a best responder able to score at least $42 \%$ against SearchBot-clone. Any score above $1 4 . 3 \%$ means the best responder is winning in expectation. However, SearchBot-clone appears to be much less exploitable than the blueprint, which a best responder can beat with a score of at least $54 \%$ . Also, we emphasize again that SearchBot is almost certainly far less exploitable than SearchBot-clone.
|
| 157 |
+
|
| 158 |
+
|
| 159 |
+
Figure 2: Score of the exploiting agent against the blueprint and SearchBot-clone as a function of training time. We report the average of six runs. The shaded area corresponds to three standard errors. We use temperature 0.5 for both agents as it minimizes exploitability for the blueprint. Since SearchBot-clone is trained through imitation learning of SearchBot, the exploitability of SearchBot is almost certainly lower than SearchBot-clone.
|
| 160 |
+
|
| 161 |
+
# 4.3.2 PERFORMANCE AGAINST EXPERT HUMAN EXPLOITERS
|
| 162 |
+
|
| 163 |
+
In addition to training a best-responding agent, we also invited Doug Moore and Marvin Fried, the 1st and 2nd place finishers, respectively, in the 2017 World Diplomacy Convention (widely considered the world championship for full-press Diplomacy) to play games against six copies of our agent. The purpose was to determine whether the human experts could discover exploitable weaknesses in the bot.
|
| 164 |
+
|
| 165 |
+
The humans played games against three types of bots: DipNet (Paquette et al., 2019) (with temperature set to 0.5), our blueprint agent (with temperature set to 0.5), and SearchBot. In total, the participants played 35 games against each bot; each of the seven powers was controlled by a human player five times, while the other six powers were controlled by identical copies of the bot. The performance of the humans is shown in Table 4. While the sample size is relatively small, the results suggest that our agent is less exploitable than prior bots. Our improvements to the supervised learning policy reduced the humans’ score from $3 9 . 1 \%$ to $2 2 . 5 \%$ , which is itself a large improvement. However, the largest gains came from search. Adding search reduced the humans’ score from $2 2 . 5 \%$ to $5 . 7 \%$ . A score below $1 4 . 3 \%$ means the humans are losing on average.
|
| 166 |
+
|
| 167 |
+
<table><tr><td>Power</td><td></td><td>1Human vs. 6 DipNet1Human vs. 6 Blueprint1Human vs.6 SearchBot</td><td></td></tr><tr><td> All Games</td><td>39.1%</td><td>22.5%</td><td>5.7%</td></tr></table>
|
| 168 |
+
|
| 169 |
+
Table 4: Average SoS score of one expert human playing against six bots under repeated play. A score less than $1 4 . 3 \%$ means the human is unable to exploit the bot. Five games were played for each power for each agent, for a total of 35 games per agent. For each power, the human first played all games against DipNet, then the blueprint model described in Section 3.1, and then finally SearchBot.
|
| 170 |
+
|
| 171 |
+
# 4.3.3 EXPLOITABILITY IN LOCAL SUBGAME
|
| 172 |
+
|
| 173 |
+
We also investigate the exploitability of our agent in the local subgame defined by a given board state, sampled actions, and assumed blueprint policy for the rest of the game. We simulate 7 games between a search agent and 6 DipNet agents, and plot the total exploitability of the average strategy of the search procedure as a function of the number of RM iterations, as well as the exploitability of the blueprint policies. Utilities $u _ { i }$ are computed using Monte Carlo rollouts with the same (blueprint)
|
| 174 |
+
|
| 175 |
+
|
| 176 |
+
Figure 3: Left: Distance of the RM average strategy from equilibrium as a function of the RM iteration, computed as the sum of all agents’ exploitability in the matrix game in which RM is employed. RM reduces exploitability, while the blueprint policy has only slightly lower exploitability than the uniform distribution over the 50 sampled actions used in RM (i.e. RM iteration 1). For comparison, our human evaluations used $2 5 6 { - } 2 0 4 8 ~ \mathrm { R M }$ iterations, depending on the time per turn. Right: Comparison of convergence of individual strategies to the average of two independently computed strategies. The similarity of these curves suggests that independent RM computations lead to compatible equilibria. Note: In both figures, exploitability is averaged over all phases in 28 simulated games; per-phase results are provided in Appendix E.
|
| 177 |
+
|
| 178 |
+
rollout policy used during RM, and total exploitability for a joint policy $\pi$ is computed as $e ( \pi ) =$ $\begin{array} { r } { \sum _ { i } \operatorname* { m a x } _ { a _ { i } \in \mathcal { A } _ { i } } u _ { i } \big ( a _ { i } , \pi _ { - i } \big ) - u _ { i } ( \pi ) } \end{array}$ . The exploitability curves aggregated over all phases are shown in Figure 3 (left) and broken down by phase in the Appendix.
|
| 179 |
+
|
| 180 |
+
In Figure 3 (right), we verify that the average of policies from multiple independent executions of RM also converges to an approximate Nash. For example, it is possible that if each agent independently running RM converged to a different incompatible equilibrium and played their part of it, then the joint policy of all the agents would not be an equilibrium. However, we observe that the exploitibility of the average of policies closely matches the exploitability of the individual policies.
|
| 181 |
+
|
| 182 |
+
# 5 CONCLUSIONS
|
| 183 |
+
|
| 184 |
+
No-press Diplomacy is a complex game involving both cooperation and competition that poses major theoretical and practical challenges for past AI techniques. Nevertheless, our AI agent achieves human-level performance in this game with a combination of supervised learning on human data and one-ply search using regret minimization. The massive improvement in performance from conducting search just one action deep matches a larger trend seen in other games, such as chess, Go, poker, and Hanabi, in which search dramatically improves performance. While regret minimization has been behind previous AI breakthroughs in purely competitive games such as poker, it was never previously shown to be successful in a complex game involving cooperation. The success of RM in no-press Diplomacy suggests that its use is not limited to purely adversarial games.
|
| 185 |
+
|
| 186 |
+
Our work points to several avenues for future research. SearchBot conducts search only for the current turn. In principle, this search could extend deeper into the game tree using counterfactual regret minimization (CFR) (Zinkevich et al., 2008). However, the size of the subgame grows exponentially with the depth of the subgame. Developing search techniques that scale more effectively with the depth of the game tree may lead to substantial improvements in performance. Another direction is combining our search technique with reinforcement learning. Combining search with reinforcement learning has led to tremendous success in perfect-information games (Silver et al., 2018) and more recently in two-player zero-sum imperfect-information games as well (Brown et al., 2020). Finally, it remains to be seen whether similar search techniques can be developed for variants of Diplomacy that allow for coordination between agents.
|
| 187 |
+
|
| 188 |
+
# ACKNOWLEDGEMENTS
|
| 189 |
+
|
| 190 |
+
We thank Kestas Kuliukas and the entire webdiplomacy.net team for their cooperation and for providing the dataset used in this research. We also thank Thomas Anthony, Spencer D., and Joshua M. for computing the agent’s Ghost-Rating. We additionally thank Doug Moore and Marvin Fried for participating in the human exploiter experiments in Section 4.3.2, and thank Andrew Goff for providing expert commentary on the bot’s play. Finally, we thank Jakob Foerster for helpful discussions.
|
| 191 |
+
|
| 192 |
+
# REFERENCES
|
| 193 |
+
|
| 194 |
+
Thomas Anthony. Ghost-ratings, 2020. URL https://sites.google.com/view/ webdipinfo/ghost-ratings.
|
| 195 |
+
Thomas Anthony, Tom Eccles, Andrea Tacchetti, János Kramár, Ian Gemp, Thomas C Hudson, Nicolas Porcel, Marc Lanctot, Julien Pérolat, Richard Everett, et al. Learning to play no-press diplomacy with best response policy iteration. arXiv preprint arXiv:2006.04635, 2020.
|
| 196 |
+
Christopher Berner, Greg Brockman, Brooke Chan, Vicki Cheung, Przemysław D˛ebiak, Christy Dennison, David Farhi, Quirin Fischer, Shariq Hashme, Chris Hesse, et al. Dota 2 with large scale deep reinforcement learning. arXiv preprint arXiv:1912.06680, 2019.
|
| 197 |
+
David Blackwell et al. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6(1):1–8, 1956.
|
| 198 |
+
Noam Brown and Tuomas Sandholm. Superhuman AI for heads-up no-limit poker: Libratus beats top professionals. Science, pp. eaao1733, 2017.
|
| 199 |
+
Noam Brown and Tuomas Sandholm. Solving imperfect-information games via discounted regret minimization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 1829–1836, 2019a.
|
| 200 |
+
Noam Brown and Tuomas Sandholm. Superhuman AI for multiplayer poker. Science, pp. eaay2400, 2019b.
|
| 201 |
+
Noam Brown, Adam Lerer, Sam Gross, and Tuomas Sandholm. Deep counterfactual regret minimization. In International Conference on Machine Learning, pp. 793–802, 2019.
|
| 202 |
+
Noam Brown, Anton Bakhtin, Adam Lerer, and Qucheng Gong. Combining deep reinforcement learning and search for imperfect-information games. arXiv preprint arXiv:2007.13544, 2020.
|
| 203 |
+
Murray Campbell, A Joseph Hoane Jr, and Feng-hsiung Hsu. Deep Blue. Artificial intelligence, 134 (1-2):57–83, 2002.
|
| 204 |
+
Arpad E Elo. The rating of chessplayers, past and present. Arco Pub., 1978.
|
| 205 |
+
Lasse Espeholt, Hubert Soyer, Remi Munos, Karen Simonyan, Volodymir Mnih, Tom Ward, Yotam Doron, Vlad Firoiu, Tim Harley, Iain Dunning, et al. Impala: Scalable distributed deep-rl with importance weighted actor-learner architectures. arXiv preprint arXiv:1802.01561, 2018.
|
| 206 |
+
André Ferreira, Henrique Lopes Cardoso, and Luis Paulo Reis. Dipblue: A diplomacy agent with strategic and trust reasoning. In ICAART International Conference on Agents and Artificial Intelligence, Proceedings, 2015.
|
| 207 |
+
Brandon Fogel. To whom tribute is due: The next step in scoring systems, 2020. URL http://windycityweasels.org/wp-content/uploads/2020/04/ 2020-03-To-Whom-Tribute-Is-Due-The-Next-Step-in-Scoring-Systems. pdf.
|
| 208 |
+
Amy Greenwald, Keith Hall, and Roberto Serrano. Correlated q-learning. In ICML, volume 20, pp. 242, 2003.
|
| 209 |
+
James Hannan. Approximation to bayes risk in repeated play. Contributions to the Theory of Games, 3:97–139, 1957.
|
| 210 |
+
|
| 211 |
+
Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5):1127–1150, 2000.
|
| 212 |
+
|
| 213 |
+
Johannes Heinrich and David Silver. Deep reinforcement learning from self-play in imperfectinformation games. arXiv preprint arXiv:1603.01121, 2016.
|
| 214 |
+
|
| 215 |
+
Ralf Herbrich, Tom Minka, and Thore Graepel. TrueskillTM: a bayesian skill rating system. In Advances in neural information processing systems, pp. 569–576, 2007.
|
| 216 |
+
|
| 217 |
+
Ronald A Howard. Dynamic programming and markov processes. 1960.
|
| 218 |
+
|
| 219 |
+
Junling Hu and Michael P Wellman. Nash q-learning for general-sum stochastic games. Journal of machine learning research, 4(Nov):1039–1069, 2003.
|
| 220 |
+
|
| 221 |
+
Stefan J Johansson and Fredrik Håård. Tactical coordination in no-press diplomacy. In International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 423–430, 2005.
|
| 222 |
+
|
| 223 |
+
Sarit Kraus and Daniel Lehmann. Diplomat, an agent in a multi agent environment: An overview. In IEEE International Performance Computing and Communications Conference, pp. 434–435. IEEE Computer Society, 1988.
|
| 224 |
+
|
| 225 |
+
Sarit Kraus and Daniel Lehmann. Designing and building a negotiating automated agent. Computational Intelligence, 11(1):132–171, 1995.
|
| 226 |
+
|
| 227 |
+
Sarit Kraus, Eithan Ephrati, and Daniel Lehmann. Negotiation in a non-cooperative environment. Journal of Experimental & Theoretical Artificial Intelligence, 3(4):255–281, 1994.
|
| 228 |
+
|
| 229 |
+
Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte carlo sampling for regret minimization in extensive games. In Advances in neural information processing systems, pp. 1078–1086, 2009.
|
| 230 |
+
|
| 231 |
+
Joel Z Leibo, Vinicius Zambaldi, Marc Lanctot, Janusz Marecki, and Thore Graepel. Multi-agent reinforcement learning in sequential social dilemmas. arXiv preprint arXiv:1702.03037, 2017.
|
| 232 |
+
|
| 233 |
+
Adam Lerer and Alexander Peysakhovich. Maintaining cooperation in complex social dilemmas using deep reinforcement learning. arXiv preprint arXiv:1707.01068, 2017.
|
| 234 |
+
|
| 235 |
+
Adam Lerer and Alexander Peysakhovich. Learning existing social conventions via observationally augmented self-play. In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pp. 107–114. ACM, 2019.
|
| 236 |
+
|
| 237 |
+
Adam Lerer, Hengyuan Hu, Jakob Foerster, and Noam Brown. Improving policies via search in cooperative partially observable games. In AAAI Conference on Artificial Intelligence, 2020.
|
| 238 |
+
|
| 239 |
+
Michael L Littman. Friend-or-foe q-learning in general-sum games. In ICML, volume 1, pp. 322– 328, 2001.
|
| 240 |
+
|
| 241 |
+
Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International conference on machine learning, pp. 1928–1937, 2016.
|
| 242 |
+
|
| 243 |
+
Matej Moravcík, Martin Schmid, Neil Burch, Viliam Lis ˇ y, Dustin Morrill, Nolan Bard, Trevor \` Davis, Kevin Waugh, Michael Johanson, and Michael Bowling. Deepstack: Expert-level artificial intelligence in heads-up no-limit poker. Science, 356(6337):508–513, 2017.
|
| 244 |
+
|
| 245 |
+
John Nash. Non-cooperative games. Annals of mathematics, pp. 286–295, 1951.
|
| 246 |
+
|
| 247 |
+
Philip Paquette, Yuchen Lu, Seton Steven Bocco, Max Smith, O-G Satya, Jonathan K Kummerfeld, Joelle Pineau, Satinder Singh, and Aaron C Courville. No-press diplomacy: Modeling multi-agent gameplay. In Advances in Neural Information Processing Systems, pp. 4474–4485, 2019.
|
| 248 |
+
|
| 249 |
+
Yoav Shoham, Rob Powers, and Trond Grenager. Multi-agent reinforcement learning: a critical survey. Web manuscript, 2, 2003.
|
| 250 |
+
|
| 251 |
+
David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484, 2016.
|
| 252 |
+
|
| 253 |
+
David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354, 2017.
|
| 254 |
+
|
| 255 |
+
David Silver, Thomas Hubert, Julian Schrittwieser, Ioannis Antonoglou, Matthew Lai, Arthur Guez, Marc Lanctot, Laurent Sifre, Dharshan Kumaran, Thore Graepel, et al. A general reinforcement learning algorithm that masters chess, shogi, and go through self-play. Science, 362(6419):1140– 1144, 2018.
|
| 256 |
+
|
| 257 |
+
Vasilis Syrgkanis, Alekh Agarwal, Haipeng Luo, and Robert E Schapire. Fast convergence of regularized learning in games. In Advances in Neural Information Processing Systems, pp. 2989–2997, 2015.
|
| 258 |
+
|
| 259 |
+
Gerald Tesauro. TD-Gammon, a self-teaching backgammon program, achieves master-level play. Neural computation, 6(2):215–219, 1994.
|
| 260 |
+
|
| 261 |
+
Jason van Hal. Diplomacy AI - Albert, 2013. URL https://sites.google.com/site/ diplomacyai/.
|
| 262 |
+
|
| 263 |
+
Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michaël Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature, 575(7782):350–354, 2019.
|
| 264 |
+
|
| 265 |
+
Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in neural information processing systems, pp. 1729–1736, 2008.
|
| 266 |
+
|
| 267 |
+
In this section, we provide a full description of the model architecture used for the supervised Diplomacy agent.
|
| 268 |
+
|
| 269 |
+
|
| 270 |
+
Figure 4: Architecture of the model used for imitation learning in no-press. Diplomacy.
|
| 271 |
+
|
| 272 |
+
# Board State Encoding
|
| 273 |
+
|
| 274 |
+
1．Unit Type (Army, Fleet, None)
|
| 275 |
+
2．Unit PoWer (AUS,ENG,FRA,ITA,GER,RUS,TUR,None)
|
| 276 |
+
3.Buildable, Removable
|
| 277 |
+
4.Dislodged Unit Type (Army Fleet, None)
|
| 278 |
+
5．Dislodged Unit Power (AUS,., TUR, None)
|
| 279 |
+
6.Area Type (Land,Water, Coast)
|
| 280 |
+
7．Supply Center Owner (AUS,., TUR, None)
|
| 281 |
+
8.Season (Spring,Fall,Winter)
|
| 282 |
+
9.Build Numbers(Integer value for each power)
|
| 283 |
+
10.Press Type (NoPress,Press)
|
| 284 |
+
|
| 285 |
+
|
| 286 |
+
Figure 5: Features used for the board state encoding.
|
| 287 |
+
|
| 288 |
+
|
| 289 |
+
Figure 6: Architecture of an encoder GNN layer.
|
| 290 |
+
|
| 291 |
+
The model architecture is shown in Figure 4 . It consists of a shared encoder and separate policy and value decoder heads.
|
| 292 |
+
|
| 293 |
+
The input to the encoder is a set of features for each of the 81 board locations (75 regions $+ ~ 6$ coasts). The features provided for each location are shown in Figure 5. The first 7 feature types (35 features/location) match those of Paquette et al. (2019), and the last three feature types (in italics) are for global board features that are common across all locations: a 20-d embedding of the current season; the net number of builds for each power during a build phase; and a single feature identifying whether it is a no-press or full-press game.
|
| 294 |
+
|
| 295 |
+
The encoder has two parallel GNN trunks. The Current State Encoder takes the current board state as input, while the Previous State Encoder takes the board state features from the last movement phase, concatenated with a 20-dimensional feature at each location that is a sum of learned embeddings for each order that ocurred at this location during all phases starting at the last movement phase.
|
| 296 |
+
|
| 297 |
+
The current board state and previous board state are each fed to an 8-layer GNN, whose layer architecture is shown in Figure 6. The GraphConv operation consists of a separate learned linear transformation applied at each board location, followed by application of the normalized board adjacency matrix.8 In parallel with the GraphConv is a linear layer that allows the learning of a feature hierarchy without graph smoothing. This is followed by batchnorm, a ReLU non-linearity, and dropout probability 0.4. The $8 1 \times 1 2 0$ output features from each of these GNNs are then concatenated and fed to another 8-layer GNN with dimensional 240. The output of this encoder are 240 features for each board location.
|
| 298 |
+
|
| 299 |
+
The value head takes as input all $8 1 \times 2 4 0$ encoder features concatenated into a single 19440- dimensional vector. An MLP with a single hidden layer followed by a softmax produces predicted sum-of-squares values for each power summing to 1. They are trained to predict the final SoS values with an MSE loss function.
|
| 300 |
+
|
| 301 |
+
The policy head auto-regressively predicts the policy for a particular power one unit at a time with a 2-layer LSTM of width 200. Unit orders are predicted in the global order used by Paquette et al. (2019). The input to the LSTM at each step is the concatenation of (a) the 240-d encoder features of the location whose orders are being predicted; (b) a 60-d embedding of the power whose policy is being computed; (c) an 80-d embedding of the predicted orders for the last unit.
|
| 302 |
+
|
| 303 |
+
A standard decoder matrix can be thought of as a learned vector for each valid order, whose dot product with the LSTM output predicts the logit of the order probability given the state. We featurize these order vectors as shown in Figure 7. The decoder vector for “PAR S RUH - BUR” , for example, would be the sum of a learned vector for this order and learned linear transformations of (a) one-hot features for the order source (RUH), destination (BUR), type (S); and (b) the encoder features of the source (RUH) and destination (BUR) locations.
|
| 304 |
+
|
| 305 |
+
Build orders are different than other order types because the set of source locations for build orders is not known in advance. So the decoder needs to predict an unordered set of build orders during a build phase, which is tricky to compute correctly with an autoregressive decoder and teacher forcing. Luckily, since a power can only build in its 3-4 home supply centers, there are only 170 sets of build orders across all powers, so we just consider each possible set of builds as a single order, rather than predicting them one by one. We do not do featurized order decoding for build orders.
|
| 306 |
+
|
| 307 |
+
# B COMPUTING PLAYER RATINGS IN HUMAN DATA
|
| 308 |
+
|
| 309 |
+
To compute player ratings, we used a regularized logistic outcome model. Specifically, we optimize a vector of ratings s by gradient descent that minimizes the regularized negative log likelihood loss
|
| 310 |
+
|
| 311 |
+
$$
|
| 312 |
+
L ( \mathbf { s } | \mathcal { D } ) = \sum _ { ( i , j ) \in \mathcal { D } } - \log \sigma ( s _ { i } - s _ { j } ) + \lambda | \mathbf { s } | _ { 2 }
|
| 313 |
+
$$
|
| 314 |
+
|
| 315 |
+
over a dataset $\mathcal { D }$ consisting of all pairs of players $( i , j )$ where player $i$ achieved a "better" outcome than player $j$ in a game. We found this approach led to more plausible scores than Elo (Elo, 1978) or TrueSkill (Herbrich et al., 2007) ratings.
|
| 316 |
+
|
| 317 |
+
|
| 318 |
+
Figure 7: Illustration of featurized order decoding.
|
| 319 |
+
|
| 320 |
+
Paquette et al. (2019) took an orthogonal approach to filter poor players from the training data: they only trained on "winning" powers, i.e. those who ended the game with at least 7 SCs. This filtering is sensible for training a policy for play, but is problematic for training policies for search. In a general-sum game, it is crucial for the agent to be able to predict the empirical distribution of actions even for other agents who are destined to lose.
|
| 321 |
+
|
| 322 |
+
# C SEARCH EXAMPLE
|
| 323 |
+
|
| 324 |
+
The listing below shows the policy generated by one run of our search algorithm for the opening move of Diplomacy when running for 512 iterations and considering 8 possible actions per power.
|
| 325 |
+
|
| 326 |
+
For each possible action, the listing below shows probs $:$ The probability of the action in the final strategy. bp_p : The probability of the action in the blueprint strategy. avg_u $:$ The average predicted sum-of-squares utility of this action. orders : The orders for this action
|
| 327 |
+
|
| 328 |
+
AUSTRIA avg_utility=0.15622 probs bp_p avg_u orders 0.53648 0.13268 0.15697 (’A VIE - TRI’, ’F TRI - ALB’, ’A BUD - SER’) 0.46092 0.52008 0.14439 (’A VIE - GAL’, ’F TRI - ALB’, ’A BUD - SER’) 0.00122 0.03470 0.14861 (’A VIE - TRI’, ’F TRI - ALB’, ’A BUD - GAL’) 0.00077 0.03031 0.11967 (’A VIE - BUD’, ’F TRI - ALB’, ’A BUD - SER’) 0.00039 0.05173 0.11655 (’A VIE - GAL’, ’F TRI S A VEN’, ’A BUD - SER’) 0.00015 0.04237 0.12087 (’A VIE - GAL’, ’F TRI H’, ’A BUD - SER’) 0.00007 0.14803 0.09867 (’A VIE - GAL’, ’F TRI - VEN’, ’A BUD - SER’) 0.00000 0.04009 0.03997 (’A VIE H’, ’F TRI H’, ’A BUD H’)
|
| 329 |
+
|
| 330 |
+
ENGLAND avg_utility=0.07112
|
| 331 |
+
|
| 332 |
+
0.41978 0.20069 0.07151 (’F EDI - NTH’, ’F LON - ENG’, ’A LVP - YOR’)
|
| 333 |
+
0.34925 0.29343 0.07161 (’F EDI - NWG’, ’F LON - NTH’, ’A LVP - YOR’)
|
| 334 |
+
0.10536 0.06897 0.07282 (’F EDI - NTH’, ’F LON - ENG’, ’A LVP - WAL’)
|
| 335 |
+
0.07133 0.36475 0.07381 (’F EDI - NWG’, ’F LON - NTH’, ’A LVP - EDI’)
|
| 336 |
+
0.05174 0.01649 0.07202 (’F EDI - NTH’, ’F LON - ENG’, ’A LVP - EDI’)
|
| 337 |
+
0.00249 0.00813 0.06560 (’F EDI - NWG’, ’F LON - NTH’, ’A LVP - WAL’)
|
| 338 |
+
0.00006 0.00820 0.06878 (’F EDI - NWG’, ’F LON - ENG’, ’A LVP - EDI’)
|
| 339 |
+
0.00000 0.03933 0.03118 (’F EDI H’, ’F LON H’, ’A LVP H’)
|
| 340 |
+
|
| 341 |
+
FRANCE avg_utility=0.21569
|
| 342 |
+
|
| 343 |
+
0.92038 0.09075 0.21772 (’F BRE - MAO’, ’A PAR - GAS’, ’A MAR - BUR’)
|
| 344 |
+
0.06968 0.42617 0.18878 (’F BRE - MAO’, ’A PAR - BUR’, ’A MAR S A PAR - BUR’)
|
| 345 |
+
0.00917 0.07987 0.16941 (’F BRE - MAO’, ’A PAR - PIC’, ’A MAR - BUR’)
|
| 346 |
+
0.00049 0.05616 0.16729 (’F BRE - ENG’, ’A PAR - BUR’, ’A MAR - SPA’)
|
| 347 |
+
0.00023 0.17040 0.17665 (’F BRE - MAO’, ’A PAR - BUR’, ’A MAR - SPA’)
|
| 348 |
+
0.00004 0.04265 0.18629 (’F BRE - MAO’, ’A PAR - PIC’, ’A MAR - SPA’)
|
| 349 |
+
0.00001 0.09291 0.15828 (’F BRE - ENG’, ’A PAR - BUR’, ’A MAR S A PAR - BUR’)
|
| 350 |
+
probs bp_p avg_u
|
| 351 |
+
0.39050 0.01382 0.21360
|
| 352 |
+
0.38959 0.02058 0.21381
|
| 353 |
+
0.16608 0.01628 0.21739
|
| 354 |
+
0.04168 0.21879 0.21350
|
| 355 |
+
0.01212 0.47409 0.21287
|
| 356 |
+
0.00003 0.05393 0.14238
|
| 357 |
+
0.00000 0.16896 0.13748
|
| 358 |
+
0.00000 0.03355 0.05917
|
| 359 |
+
|
| 360 |
+
orders (’F KIE - DEN’, ’A MUN - TYR’, ’A BER - KIE’) (’F KIE - DEN’, ’A MUN S A PAR - BUR’, ’A BER - KIE’) (’F KIE - DEN’, ’A MUN H’, ’A BER - KIE’) (’F KIE - DEN’, ’A MUN - BUR’, ’A BER - KIE’) (’F KIE - DEN’, ’A MUN - RUH’, ’A BER - KIE’) (’F KIE - HOL’, ’A MUN - BUR’, ’A BER - KIE’) (’F KIE - HOL’, ’A MUN - RUH’, ’A BER - KIE’) (’F KIE H’, ’A MUN H’, ’A BER H’)
|
| 361 |
+
|
| 362 |
+
ITALY avg_utility=0.13444
|
| 363 |
+
|
| 364 |
+
probs bp_p avg_u orders
|
| 365 |
+
0.41740 0.19181 0.13609 (’F NAP - ION’, ’A ROM - APU’, ’A VEN S F TRI’)
|
| 366 |
+
0.25931 0.07652 0.12465 (’F NAP ION’, ’A ROM - VEN’, ’A VEN - TRI’)
|
| 367 |
+
0.13084 0.29814 0.12831 (’F NAP ION’, ’A ROM - VEN’, ’A VEN - TYR’)
|
| 368 |
+
0.09769 0.03761 0.13193 (’F NAP ION’, ’A ROM - APU’, ’A VEN - TRI’)
|
| 369 |
+
0.09412 0.16622 0.13539 (’F NAP ION’, ’A ROM - APU’, ’A VEN H’)
|
| 370 |
+
0.00034 0.05575 0.11554 (’F NAP - ION’, ’A ROM - APU’, ’A VEN - PIE’)
|
| 371 |
+
0.00028 0.13228 0.10953 (’F NAP - ION’, ’A ROM - VEN’, ’A VEN - PIE’)
|
| 372 |
+
0.00000 0.04167 0.05589 (’F NAP H’, ’A ROM H’, ’A VEN H’)
|
| 373 |
+
|
| 374 |
+
RUSSIA avg_utility=0.06623
|
| 375 |
+
|
| 376 |
+
probs bp_p avg_u orders
|
| 377 |
+
0.64872 0.05988 0.06804 (’F STP/SC - FIN’, ’A MOS - UKR’, ’A WAR - GAL’, ’F SEV - BLA’)
|
| 378 |
+
0.28869 0.07200 0.06801 (’F STP/SC - BOT’, ’A MOS - STP’, ’A WAR - UKR’, ’F SEV - BLA’)
|
| 379 |
+
0.04914 0.67998 0.05929 (’F STP/SC - BOT’, ’A MOS - UKR’, ’A WAR - GAL’, ’F SEV - BLA’)
|
| 380 |
+
0.01133 0.01147 0.05023 (’F STP/SC - BOT’, ’A MOS - SEV’, ’A WAR - UKR’, ’F SEV - RUM’)
|
| 381 |
+
0.00120 0.02509 0.05008 (’F STP/SC - BOT’, ’A MOS - UKR’, ’A WAR - GAL’, ’F SEV - RUM’)
|
| 382 |
+
0.00064 0.09952 0.05883 (’F STP/SC - BOT’, ’A MOS - STP’, ’A WAR - GAL’, ’F SEV - BLA’)
|
| 383 |
+
0.00027 0.01551 0.04404 (’F STP/SC - BOT’, ’A MOS - SEV’, ’A WAR - GAL’, ’F SEV - RUM’)
|
| 384 |
+
0.00000 0.03655 0.02290 (’F STP/SC H’, ’A MOS H’, ’A WAR H’, ’F SEV H’)
|
| 385 |
+
|
| 386 |
+
TURKEY avg_utility=0.13543
|
| 387 |
+
|
| 388 |
+
probs bp_p avg_u orders
|
| 389 |
+
0.82614 0.25313 0.13787 (’F ANK - BLA’, ’A SMY - ARM’, ’A CON - BUL’)
|
| 390 |
+
0.14130 0.00651 0.12942 (’F ANK - BLA’, ’A SMY - ANK’, ’A CON - BUL’)
|
| 391 |
+
0.03080 0.61732 0.12760 (’F ANK - BLA’, ’A SMY - CON’, ’A CON - BUL’)
|
| 392 |
+
0.00074 0.01740 0.11270 (’F ANK - CON’, ’A SMY - ARM’, ’A CON - BUL’)
|
| 393 |
+
0.00069 0.05901 0.12192 (’F ANK - CON’, ’A SMY - ANK’, ’A CON - BUL’)
|
| 394 |
+
0.00030 0.00750 0.11557 (’F ANK - CON’, ’A SMY H’, ’A CON - BUL’)
|
| 395 |
+
0.00001 0.00598 0.10179 (’F ANK S F SEV - BLA’, ’A SMY - CON’, ’A CON - BUL’)
|
| 396 |
+
0.00001 0.03314 0.04464 (’F ANK H’, ’A SMY H’, ’A CON H’)
|
| 397 |
+
|
| 398 |
+
D RL DETAILS
|
| 399 |
+
|
| 400 |
+
Each action $a$ in the MDP is a sequence of orders $\left( o _ { 1 } , \ldots , o _ { t } \right)$ . The probability of the order $a$ under policy $\pi _ { \theta }$ is defined by an LSTM in auto regressive fashion, i.e., $\begin{array} { r } { \pi _ { \theta } ( a ) = \prod _ { i = 1 } ^ { t } \bigl ( \pi _ { \theta } \bigl ( o _ { i } \vert o _ { 1 } \dots o _ { i - 1 } \bigr ) \bigr ) } \end{array}$ . To make training more stable, we would like to prevent entropy $H ( \pi _ { \theta } ) : = - E _ { a \sim \pi _ { \theta } } \log ( \pi _ { \theta } ( a )$ from collapsing to zero. The naive way to optimize the entropy of the joint distribution is to use a sum of entropies for each individual order, i.e., $\begin{array} { r } { \frac { d } { d \theta } H ( \pi _ { \boldsymbol { \theta } } ( \bullet ) ) \approx \sum _ { i = 1 } ^ { t } \frac { \dot { d } } { d \boldsymbol { \theta } } H ( \pi _ { \boldsymbol { \theta } } ( \bullet | o _ { 1 } \ldots o _ { i - 1 } ) ) } \end{array}$ . However, we found that this does not work well for our case probably because there are strong correlations between orders. Instead we use an unbiased estimate of the joint entropy that is agnostic to the size of the action space and requires only to be able to sample from a model and to adjust probabilities of the samples.
|
| 401 |
+
|
| 402 |
+
Statement 1. Let $\pi _ { \boldsymbol { \theta } } ( \bullet )$ be a probability distribution over a discrete set $A$ , such that $\forall a \in A \pi _ { \theta } ( a )$ is a smooth function of a vector of parameters $\theta$ . Then
|
| 403 |
+
|
| 404 |
+
$$
|
| 405 |
+
\frac { d } { d \theta } \left( H ( \pi _ { \theta } ( \bullet ) ) \right) = - E _ { a \sim \pi _ { \theta } } ( 1 + \log \pi _ { \theta } ( a ) ) \frac { d } { d \theta } \log \pi _ { \theta } ( a ) .
|
| 406 |
+
$$
|
| 407 |
+
|
| 408 |
+
Proof. Proof is similar to one for REINFORCE:
|
| 409 |
+
|
| 410 |
+
$$
|
| 411 |
+
\begin{array} { r l } { \frac { d } { d t } ( \mathcal { W } ( \{ \mathbf { x } \} ( \omega ) ( s ) ) ) } & { = \frac { d } { d t } ( - F _ { \mathrm { e x c , \omega , \Psi } } \log \log ( \omega ) ) } \\ { = } & { - \frac { d } { d t } ( \sum _ { \mathbf { x } \in \mathcal { X } _ { \neq } } \omega _ { \mathbf { x } } \log ( s ) \cup \mathbf { x } _ { \mathbf { x } } ) } \\ { = } & { - \frac { d } { \sqrt { d s } } ( \frac { d } { d t } ) \mathrm { e x c } ( \omega ) \mathrm { d } \mathbf { x } \mathrm { g } _ { \mathbf { x } } \mathrm { \cdot } } \\ { = } & { - \sum _ { \mathbf { x } \in \mathcal { X } _ { \neq } } ( \frac { d } { d t } ) \mathrm { e x c } ( \omega ) \mathrm { d } \mathbf { x } \mathrm { g } _ { \mathbf { x } } \mathrm { \cdot } } \\ { = } & { - \sum _ { \mathbf { x } \neq \mathbf { x } } ( \omega ) \frac { d } { d t } \mathrm { e x c } ( \omega ) + \log \mathrm { r c } ( \omega ) \mathrm { d } \frac { d } { d t } \mathrm { e x c } ( \omega ) \mathrm { d } \frac { d } { d t } \mathrm { e x c } ( \omega ) ) } \\ { = } & { - \sum _ { \mathbf { x } \neq \mathbf { x } } ( \omega ) \frac { d } { d t } \mathrm { e x c } ( \omega ) + \log \mathrm { r c } ( \omega ) \mathrm { d } \mathrm { e x c } ( \omega ) \mathrm { d } \frac { d } { d t } \mathrm { e x c } ( \omega ) \mathrm { d } \frac { d } { d t } \mathrm { e x c } ( \omega ) ) } \\ { = } & { - \sum _ { \mathbf { x } \neq \mathbf { x } } ( \omega ) ( ( 1 + \log \mathrm { r c } ( \omega ) ) \frac { d } { d t } \mathrm { e x c } ( \omega ) \mathrm { d } \frac { d } { d t } ) } \\ & { - \sum _ { \mathbf { x } \neq \mathbf { x } } ( \omega ) ( 1 + \log \mathrm { r c } ( \omega ) ) \frac { d } { d t } \mathrm { e x c } ( \omega ) } \\ { = } & - \mathrm { e x c } ( \omega ) \mathrm { d } \frac { d } { d t } \mathrm { e x c } ( \omega ) \mathrm { d } \end{array}
|
| 412 |
+
$$
|
| 413 |
+
|
| 414 |
+
# E SUBGAME EXPLOITABILITY RESULTS
|
| 415 |
+
|
| 416 |
+
Figure 8 plots the total exploitability of joint policies computed by RM in the subgame used for equilibrium search at each phase in 7 simulated Diplomacy games.
|
| 417 |
+
|
| 418 |
+
# F PLAYING ACCORDING TO THE FINAL ITERATION’S POLICY IN RM
|
| 419 |
+
|
| 420 |
+
As described in Section 2.2, in RM it is the average policy over all iterations that converges to an equilibrium, not the final iteration’s policy. Nevertheless, in our experiments we sample an action from the final iteration’s policy. This technique has been used successfully in past poker agents (Brown & Sandholm, 2019b). At first glance this modification may appear to increase exploitability because the final iteration’s policy is not an equilibrium and in practice is often quite pure. However, in this section we provide evidence that sampling from the final iteration’s policy when using a Monte Carlo version of RM may lower exploitability compared to sampling from the average policy, and conjecture an explanation.
|
| 421 |
+
|
| 422 |
+
Critically, in the sampled form of RM the final iteration’s policy depends on the random seed, which is unknown to the opponents (while we assume opponents have access to our code, we assume they do not have access to our run-time random seeds). Therefore, the opponent is unlikely to be able to predict the final iteration’s policy. Instead, from the opponent’s perspective, the policy we play is sampled from an average of final-iteration policies from all possible random seeds. Due to randomness in the behavior of sampled RM, we conjecture that this average has low exploitability.
|
| 423 |
+
|
| 424 |
+
To measure whether this is indeed the case, we ran sampled RM for 256 iterations on two Diplomacy situations (the first turn in the game and the second turn in the game) using 1,024 different random seeds, and averaged together the final iteration of each run. We also ran sampled RM on two random matrix games with 10,000 different random seeds. The results, shown in Table 5, confirm that averaging the final iteration over multiple random seeds leads to low exploitability.
|
| 425 |
+
|
| 426 |
+
We emphasize that while we measured exploitability in these experiments by running sampled regret matching over multiple random seeds, we only need to run it for a single random seed in order to achieve this low exploitability because the opponents do not know which random seed we use. Unfortunately, measuring the exploitability of the final iteration over 1,024 random seeds is about $1 { , } 0 2 4 \mathrm { x }$ as expensive as measuring the exploitability of the average strategy over all iterations (and even 1,024 seeds would provide only an approximate upper bound on exploitability, since computing the true exploitability would require averaging over all possible random seeds). Therefore, in order to keep the computation tractable, we measure the exploitability of the average strategy of single runs in Figure 3.
|
| 427 |
+
|
| 428 |
+
|
| 429 |
+
Figure 8: Exploitability as a function of RM iteration at each phase of 7 simulated games with our search agent (until the game ends or the search agent is eliminated). Aggregate results in Figure 3.
|
| 430 |
+
|
| 431 |
+
Table 5: Exploitability of 256 iterations of RM using either the final iteration’s policy or the average iteration’s policy, and either using the policy from a single run or the policy average over multiple runs. Subgame 1 and Subgame 2 are subgames from Diplomacy S1901 and F1901, respectively. $1 0 \mathrm { x } 1 0$ Random and $1 0 0 \mathrm { x } 1 0 0$ Random are random matrix two-player zero-sum games with entries having value in $[ 0 , 1 )$ sampled uniformly randomly. For Subgame 1 and Subgame 2, “Ave. of Final Policies” and “Ave. of Ave. Policies” are the average of 1,024 runs with different random seeds. For $1 0 \mathrm { x } 1 0$ Random and $1 0 0 \mathrm { x } 1 0 0$ Random, 10,000 seeds were used.
|
| 432 |
+
|
| 433 |
+
<table><tr><td>Method</td><td> Subgame 1</td><td> Subgame 2</td><td>10x10 Random</td><td>100x100 Random</td></tr><tr><td>Ave.of Final Policies</td><td>0.00038</td><td>0.0077</td><td>0.019</td><td>0.063</td></tr><tr><td>Ave. of Ave.Policies</td><td>0.00027</td><td>0.0074</td><td>0.035</td><td>0.092</td></tr><tr><td>Single Ave. Policy</td><td>0.00065</td><td>0.0140</td><td>0.078</td><td>0.225</td></tr><tr><td>Single Final Policy</td><td>0.00880</td><td>0.0580</td><td>0.478</td><td>0.706</td></tr></table>
|
| 434 |
+
|
| 435 |
+
# G DETAILS ON EXPERIMENTAL SETUP
|
| 436 |
+
|
| 437 |
+
Most games on webdiplomacy.net were played with 24-hour turns, though the agent also played in some “live” games with 5-minute turns. Different hyperparameters were used for live games versus non-live games. In non-live games, we typically ran RM for 2,048 iterations with a rollout length of 3 movement phases, and set $M$ (the constant which is multiplied by the number of units to determine the number of subgame actions) equal to 5. This typically required about 20 minutes to compute. In live games (including games in which one human played against six bots) and in games against non-human opponents, we ran RM for 256 iterations with a rollout length of 2 movement phases, and set $M$ equal to 3.5. This typically required about 2 minutes to compute. In all cases, the temperature for the blueprint in rollouts was set to 0.75.
|
| 438 |
+
|
| 439 |
+
Table 6: Average SoS score of our agent in anonymous games against humans on webdiplomacy.net. Average human performance is $1 4 . 3 \%$ . Score in the case of draws was determined by the rules of the joined game. The $\pm$ shows one standard error. Average human performance was calculated based on SoS scoring of historical games on webdiplomacy.net.
|
| 440 |
+
|
| 441 |
+
<table><tr><td>Power</td><td>Score</td><td>Human Mean</td><td>Games</td><td>Wins</td><td>Draws</td><td>Losses</td></tr><tr><td>All Games</td><td>26.6% ± 3.2%</td><td>14.3%</td><td>116</td><td>16</td><td>43</td><td>57</td></tr><tr><td>Normalized By Power</td><td>26.9% ± 3.3%</td><td>14.3%</td><td>116</td><td>16</td><td>43</td><td>57</td></tr><tr><td>Austria</td><td>31.4% ± 9.4%</td><td>11.0%</td><td>17</td><td>3</td><td>6</td><td>8</td></tr><tr><td>England</td><td>38.0% ±10.1%</td><td>12.6%</td><td>16</td><td>4</td><td>6</td><td>6</td></tr><tr><td>France</td><td>19.0% ± 6.2%</td><td>16.9%</td><td>19</td><td>1</td><td>8</td><td>10</td></tr><tr><td>Germany</td><td>36.8% ± 8.0%</td><td>15.0%</td><td>16</td><td>2</td><td>10</td><td>4</td></tr><tr><td>Italy</td><td>31.5% ± 10.5%</td><td>11.6%</td><td>14</td><td>3</td><td>5</td><td>6</td></tr><tr><td>Russia</td><td>17.6% ± 7.6%</td><td>14.8%</td><td>18</td><td>2</td><td>4</td><td>12</td></tr><tr><td>Turkey</td><td>14.6% ± 7.0%</td><td>18.2%</td><td>16</td><td>1</td><td>4</td><td>11</td></tr></table>
|
| 442 |
+
|
| 443 |
+
The experiments on webdiplomacy.net occurred over a three-month timespan, with games commonly taking one to two months to complete (players are typically given 24 hours to act). Freezing research and development over such a period would have been impractical, so our agent was not fixed for the entire time period. Instead, serious bugs were fixed, improvements to the algorithm were made, and the model was updated.
|
| 444 |
+
|
| 445 |
+
In games on webdiplomacy.net, a draw was submitted if SearchBot did not gain a center in two years, or if the agent’s projected sum-of-squares score was less than the score it would achieve by an immediate draw. Since there was no way to submit draws through the webdiplomacy.net API, draws were submitted manually once the above criteria was satisfied. Games against other bots, shown in Table 3, automatically ended in a draw if no player won by 1935.
|
| 446 |
+
|
| 447 |
+
<table><tr><td>Power</td><td>1 Human vs.6DipNet</td><td>1 Human vs. 6 Blueprint</td><td>1Human vs. 6 SearchBot</td></tr><tr><td>All Games</td><td>39.1%</td><td>22.5%</td><td>5.7%</td></tr><tr><td>Austria</td><td>40%</td><td>20%</td><td>0%</td></tr><tr><td>England</td><td>28.4%</td><td>20%</td><td>0%</td></tr><tr><td>France</td><td>20%</td><td>4.3%</td><td>40%</td></tr><tr><td>Germany</td><td>40%</td><td>33%</td><td>0%</td></tr><tr><td>Italy</td><td>60%</td><td>20%</td><td>0%</td></tr><tr><td>Russia</td><td>40%</td><td>20%</td><td>0%</td></tr><tr><td>Turkey</td><td>45.1%</td><td>40%</td><td>0%</td></tr></table>
|
| 448 |
+
|
| 449 |
+
Table 7: Average SoS score of one expert human playing against six bots under repeated play. A score less than $1 4 . 3 \%$ suggests the human is unable to exploit the bot. Five games were played for each power for each agent, for a total of 35 games per agent. For each power, the human first played all games against DipNet, then the blueprint model described in Section 3.1, and then finally SearchBot.
|
| 450 |
+
|
| 451 |
+
# H QUALITATIVE ASSESSMENT OF SEARCHBOT
|
| 452 |
+
|
| 453 |
+
Qualitatively, we observe that SearchBot performs particularly well in the early and mid game. However, we observe that it sometimes struggles with endgame situations. In particular, when it is clear that one power will win unless the others work together to stop it, SearchBot will sometimes continue to attack its would-be allies. There may be multiple contributing factors to this. One important limitation, which we have verified in some situations, is that the sampled subgame actions may not contain any action that could prevent a loss. This is exacerbated by the fact that players typically control far more units in the endgame and the number of possible actions grows exponentially with the number of units, so the sampled subgame actions contain a smaller fraction of all possible actions. Another possible contributing factor is that the state space near the end of the game is far larger, so there is relatively less data for supervised learning in this part of the game. Finally, another possibility is that because the network only has a dependence on the state of the board and the most recent player actions, it is unable to sufficiently model the future behavior of other players in response to being attacked.
|
| 454 |
+
|
| 455 |
+
Although the sample size is small, the results suggest that SearchBot performed particularly well with the central powers of Austria, Germany, and Italy. These powers are considered to be the most difficult to play by humans because they are believed to require an awareness of the interactions between all players in the game.
|
| 456 |
+
|
| 457 |
+
# I ANECDOTAL ANALYSIS OF GAMES BY EXPERT HUMAN
|
| 458 |
+
|
| 459 |
+
Below we present commentary from Andrew Goff, three-time winner of the world championship for Diplomacy. The comments are lightly edited for clarity. This is intended purely as anecdotal commentary from a selection of interesting games.
|
| 460 |
+
|
| 461 |
+
# GAME 1
|
| 462 |
+
|
| 463 |
+
http://webdiplomacy.net/board.php?gameID=319258 (bot as Germany)
|
| 464 |
+
|
| 465 |
+
The bot is doing a lot of things right here, particularly if they are playing for an 18. The opening to Tyrolia is stronger than most players realise, and the return to defend Munich is exactly how to play it. Building 2 fleets is hyper-aggressive but in Gunboat it is a good indication move, particularly after the early fight between England and France. 1902 the board breaks very favourably for Germany, but its play is textbook effective. The bounce in Belgium in Fall is a very computer move. . . 9/10 times it bounces as it does in the game, but the 1/10 times it is a train wreck, as England retreats to Holland and France is angry so it is a risk human players don’t often take.
|
| 466 |
+
|
| 467 |
+
1903 is a bit wilder. The play in the North is solid (Scandinavia is more important than England centres – I agree) but the move to Burgundy seems premature. In gunboat it is probably fine, but why not Ruhr then Belgium? Why not give France greater opportunity to suicidally help you against
|
| 468 |
+
|
| 469 |
+
England and Italy. I suspect this is an artefact of being trained against other AI that would not do that?
|
| 470 |
+
|
| 471 |
+
1904 has a nice convoy (to Yorkshire) and strong moves over the line into Russia. It is all good play, but in fairness the defences put up by the opponents are weak. After that the next few years are good consolidations – a little fidgety in Norway but not unreasonable. 1907 see Turkey dot9 Russia and from here you should expect an $1 8 - { \mathrm { a } }$ terrible move by Turkey. It should be all over now and it is just clean-up. Even as Turkey and Russia patch things up quite quickly, Turkey picks off an Italian centre and now it really is all over. France collapses, Germany cleans up professionally, and score one for AI.
|
| 472 |
+
|
| 473 |
+
Overall, Germany’s play was aggressive early and then clinical late. Some pretty bad errors by Turkey gave Germany the chance, and they took it. 9/10
|
| 474 |
+
|
| 475 |
+
# GAME 2
|
| 476 |
+
|
| 477 |
+
http://webdiplomacy.net/board.php?gameID $=$ 319189 (bot as Italy)
|
| 478 |
+
|
| 479 |
+
Spoiler Alert: this one isn’t so good. The opening is sleepy and the attack on Vienna in 1902 is just wrong against human players. Then the whole rest of the board is playing terribly too – Austria goes for a Hail Mary by attacking their only potential ally. . . Turkey tries a convoy that if it had worked would suggest collusion. . . this is all terrible. Now everyone is fighting everyone and meanwhile Germany and France have cleaned up England. By 1905 at least the AI has given up attacking Austria for the time being – picking the competent over the nonsense – and the day is saved as France and Germany descend into brawling over the spoils.
|
| 480 |
+
|
| 481 |
+
It doesn’t last. Another year another flip/flop. The same attempted trick again (move to Pie then up to Tyrol). Italy requires patience and the AI is showing none at all. Austria gets around to Armenia – they’re playing OK tactically but they have a mad Italian hamstringing them both. Meanwhile France is now officially on top and will present a solo threat soon. The start of 1908 sees Italy finally making some progress against Austria, but France is too big now. . . it is all strategically a mess from 1902 onward and it shows. As a positive though, some of the tactics here are first rate. . . Nap – Alb is tidy, the progress that is being made against Austria is good in isolation.
|
| 482 |
+
|
| 483 |
+
Then follows 8 years of stalemate tactics none of which is interesting. Italy played well, was lucky Turkey didn’t take their centres at the death, and that’s all there is.
|
| 484 |
+
|
| 485 |
+
Overall, I’d say this looked like a human player. And just like most human players: rubbish at Italy. 3/10
|
| 486 |
+
|
| 487 |
+
# GAME 3
|
| 488 |
+
|
| 489 |
+
http://webdiplomacy.net/board.php?gameID $=$ 318752 (bot as Austria)
|
| 490 |
+
|
| 491 |
+
I was about to say 1901 was all boring and then the AI in Austria has built a fleet. That’s wild. With Italy attacking France it isn’t unreasonable. . . but still. Anyhow as the game continues into 1903 he deploys that fleet beautifully. Italy just doesn’t care about it – which is strategically very poor by them. The weird thing here is that Austria seems more intent on attacking Russia than Turkey. If you have the second fleet then surely Turkey is the objective, otherwise you’re splitting forces a lot. In gunboat it is. . . passable. In non-Gunboat this is all-but suicidal. If RT get their heads together they’ll belt Austria, but as we see in this game they don’t so Austria does well from it. It’s similar to game 1 in that this is the most aggressive way to play.
|
| 492 |
+
|
| 493 |
+
One thing really stands out here and that is that the AI is making the moves that are most likely to result in an error by the opponent (See: Turkey, S1904). This is present elsewhere but this turn captures it beautifully. Why did Smyrna go to Con? What a shocker. But without moves that can take advantage of errors like this it doesn’t matter. Italy also makes a key mistake here: Mar holds instead of $ \mathrm { G a s }$ . They’ve lost momentum on the turn Austria gains a lot. In Fall Austria builds 3 armies and Italy gets nothing and the chance is there to solo.
|
| 494 |
+
|
| 495 |
+
The stab comes immediately and appropriately, and Austria has a fleet in the Black Sea and Aegean. Italy is a turn too late into the Northern French centres. There is a subtle problem to the AI’s play here though, not seen in the Germany game. There’s no effort to move over the stalemate positions. At this point in the German game it moved aggressively into Russia. . . here it ignores Germany and aims for the Russian centres again and that shows a lack of awareness of where the 18th centre must come from.
|
| 496 |
+
|
| 497 |
+
The game continues and Austria cleans up the Eastern centres very effectively. I like ignoring the German move to Tyrolia and Vienna – not very human but also very correct (as we will see – there’s nothing there to support it so it just gets yeeted in Spring 1908. However, Spring 1908 has a tactical error (one of the few outright wrong moves I’ve seen!) where:
|
| 498 |
+
|
| 499 |
+
Gre s Adr – Ion; Aeg s Adr – Ion; Adr – Ion.
|
| 500 |
+
|
| 501 |
+
You only need one support and Adriatic is more useful than Greece. . . so better is: Gre – Ion; Aeg s Gre – Ion; Adr – Apu.
|
| 502 |
+
|
| 503 |
+
This allows Nap – Rom; Apu – Nap next turn. A very odd mistake as this is the kind of detail I’d expect AI to do better than human. Ultimately it doesn’t matter so much as Italy is going to concede Tunis to set up a rock-solid defence of Spa/Mar. But still.
|
| 504 |
+
|
| 505 |
+
And now. . . . The ultimate “Not good”. For four turns in a row England fails to defend StP correctly and the AI misses it and therefore the chance to get 18. England is attacking Livonia with 3 and moving only one unit into StP behind it. Germany is not cutting Prussia. So:
|
| 506 |
+
|
| 507 |
+
Pru s Lvn, War s Lvn, Lvn s Mos – StP, Mos – StP
|
| 508 |
+
|
| 509 |
+
And Austria gets an 18. This is a really bad miss by the AI (not to mention England). As in every decent human player or better would have got an 18 here.
|
| 510 |
+
|
| 511 |
+
Overall, Austria played a really good game but didn’t do some of the things I had high hopes for after game 1 – they didn’t move over the stalemate and they didn’t play as tactically precisely. Then the miss at the end. . . makes me feel good about being human. 7/10 until the last move, but in the end 5/10.
|
| 512 |
+
|
| 513 |
+
# GAME 4
|
| 514 |
+
|
| 515 |
+
http://webdiplomacy.net/board.php?gameID=318726 (bot as Russia)
|
| 516 |
+
|
| 517 |
+
StP – Fin. Cute. The whole board just completely opens in the other direction from Russia so they get 7 centres at the end of 1901. I wish that happened to me when I played Russia. Of course, now everyone moves against Russia so we’ll see how the AI defends. Italy has just demolished France, which makes this a strange game strategically too.
|
| 518 |
+
|
| 519 |
+
F1903 you could write a whole article on. What it looks like is Russia and Turkey successfully signalling “alliance” and that is impressive for an AI. The tactics are good – ignoring Warsaw to get Budapest, but the bounce in Sweden and drop to Baltic not so much. Why England didn’t support Den – Swe is a mystery and it is probably all over for Russia if they do. But they didn’t so play on.
|
| 520 |
+
|
| 521 |
+
The English play from here on is just great. F1905 the move through Swe – Den is the height of excellence. It obviously helped Russia too, but from an awful position there is now real potential. . . what a great move. Focussing on the AI – it just gets the tactics right. It pushes Germany back and nails the timing to take Vienna. But then it builds a Fleet in Sev. No no no. An Army in Sev is better tactically and it also doesn’t tip off the Turk. Just as it got the signalling right earlier it got it exactly wrong here.
|
| 522 |
+
|
| 523 |
+
In the end the Turk starts blundering under pressure so as the kids say: “Whatever”. The F1906 move of Adr – Alb instead of supporting Tri is egregiously bad and turns a bad position into a terrible one. They proceed to self-immolate and Russia and Italy are cruising now, and England’s play remains first class. The clean-up of Turkish dots is effective, and then the rest is quite boring.
|
| 524 |
+
|
| 525 |
+
The biggest thing that struck me about this game (apart from England’s brilliant recovery) was that Russia didn’t manage unit balance well. Disbanding their only Northern fleet was a long-term error even if it made the most sense at the time; build F Sev instead of A Sev. . . and then ironically being left with too few Southern Fleets anyhow. Russia is hard like that, but the AI played no better than a good human player on that score. To give a practical example – England never had enough Fleets to blow up F Bal, so if you swap A Pru for F Bal then this is a real chance at an 18.
|
| 526 |
+
|
| 527 |
+
Overall, an interesting game characterised by the best and worst of human play (England/Turkey) and a strong but flawed game by the AI. 7/10.
|
| 528 |
+
|
| 529 |
+
# GAME 5
|
| 530 |
+
|
| 531 |
+
http://webdiplomacy.net/board.php?gameID $=$ 318335 (bot as France)
|
| 532 |
+
|
| 533 |
+
Right off the bat I am biased as I love this kind of opening as France. The Fleet in English Channel is not good news though. We get to see defence from the AI, and frankly the next 5 years are boring as we watch a clinical tactics lesson. About F1906/S1907 there needed to be a pivot to help Italy and stop attacking Germany in order to stop the 18, and it just never came. What had been a good display of defence turned into a disaster of AI greed and lack of appreciation for the risk of another player getting an 18. Not only did the AI allow the 18 to happen, but it can fairly reasonably be blamed for it. I think this is something in the reward conditions you might need to look at, as this play makes no sense – it isn’t even worth digging into it further here – it is just flat out strategically wrong.
|
| 534 |
+
|
| 535 |
+
Overall, good tactics but terrible game awareness and strategy. 2/10.
|
| 536 |
+
|
| 537 |
+
GAME 6
|
| 538 |
+
|
| 539 |
+
http://webdiplomacy.net/board.php?gameID $=$ 318329 (bot as England)
|
| 540 |
+
|
| 541 |
+
This is more like it. In fact, I think this is the best game of Gunboat I’ve seen in ages. There’s a pressure and inevitability about it from about 1903 onward. . . Turkey is aggravating everyone near them, and England is moving in second and having people just let it take their centres – it outplays Germany (and punishes F Kie – Hol by bouncing Denmark. . . heartily approved), then Russia and Austria just walk out of their centres for it. This makes the attack against France with Italy easy. The game should draw at this point, but Turkey just can’t help themselves and stabs Italy when England is already on 15 – terrible, inexcusable play. Italy doesn’t even turn around to fight Turkey, but it doesn’t matter as there are just too many disbands and Turkey’s armies are not fast enough to get to the front.
|
| 542 |
+
|
| 543 |
+
It’s an 18 and it is going on 34. Solo victories are usually caused by someone else making terrible play and this is no exception – Turkey was awful. But they also require precise play by the winner. The AI delivered. Even though this is not the style I prefer to play England as, every move makes sense. The builds are precise, the critical moments are executed perfectly (For example: S1908 convoy Lon – Pic, then ignoring Holland to force English Channel. . . that’s a high level play right there) and the sense of not attacking Russia when given the chance a few times is [chef’s kiss]. Then, above all, the presence to stop building fleets and get the armies rolling – this is the inflection point of great England play and it caught Turkey off guard – as they built another fleet, then an army that moved to Armenia, England rushed the middle of the board and Turkey just can’t cover all the gaps alone. This opened the door to “The Mistake” and Turkey walked right through it.
|
| 544 |
+
|
| 545 |
+
This is the easiest game to analyse because there’s just no glaring mistakes by the AI and it takes correct advantage of the situation throughout. This is the only game that made me think the AI could eventually consistently beat human players. As impressive as the German game was it was hyper-win-oriented so it is high risk/high reward. This. . . this is just excellent Gunboat Diplomacy from start to finish.
|
| 546 |
+
|
| 547 |
+
Overall – superb. 10/10.
|
| 548 |
+
|
| 549 |
+
GAME 7
|
| 550 |
+
|
| 551 |
+
http://webdiplomacy.net/board.php?gameID $=$ 319187 (bot as Austria)
|
| 552 |
+
|
| 553 |
+
Sometimes you just get killed.
|
| 554 |
+
|
| 555 |
+
I saw the first four turns and thought “this will be over by $1 9 0 5 '$ but to my delight the AI fought it out bravely and also did its very best to try and get back in the game – rather than accepting being tucked in Greece it made a run for a home centre and then did everything it could to get a second build. All to no avail in the end, but the play is hard to fault.
|
| 556 |
+
|
| 557 |
+
Obviously this is a shorter review as there really isn’t anything to see here. But as a finishing thought it will be this challenge that an AI faces most when it starts playing with natural language as well as just gunboat. In gunboat it is just a crap shoot who gets attacked early and how vicious it is; in non-gunboat this is a factor very much in control of the player. I think the AI has proven tactically savvy enough to cut it once it gets a start, but will it get starts in non-gunboats?
|
| 558 |
+
|
| 559 |
+
Overall – just forget this game ever happened, the same as any good human player would. 5/10.
|
parse/train/0-uUGPbIjD/0-uUGPbIjD_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi.md
ADDED
|
@@ -0,0 +1,338 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# UNCERTAINTY IN GRADIENT BOOSTING VIA ENSEMBLES
|
| 2 |
+
|
| 3 |
+
Andrey Malinin∗
|
| 4 |
+
Yandex; HSE University
|
| 5 |
+
Moscow, Russia
|
| 6 |
+
am969@yandex-team.ru Liudmila Prokhorenkova∗
|
| 7 |
+
Yandex; HSE University;
|
| 8 |
+
Moscow Institute of Physics and Technology Moscow, Russia
|
| 9 |
+
ostroumova-la@yandex-team.ru
|
| 10 |
+
|
| 11 |
+
# Aleksei Ustimenko∗
|
| 12 |
+
|
| 13 |
+
Yandex
|
| 14 |
+
Moscow, Russia
|
| 15 |
+
austimenko@yandex-team.ru
|
| 16 |
+
|
| 17 |
+
# ABSTRACT
|
| 18 |
+
|
| 19 |
+
For many practical, high-risk applications, it is essential to quantify uncertainty in a model’s predictions to avoid costly mistakes. While predictive uncertainty is widely studied for neural networks, the topic seems to be under-explored for models based on gradient boosting. However, gradient boosting often achieves stateof-the-art results on tabular data. This work examines a probabilistic ensemblebased framework for deriving uncertainty estimates in the predictions of gradient boosting classification and regression models. We conducted experiments on a range of synthetic and real datasets and investigated the applicability of ensemble approaches to gradient boosting models that are themselves ensembles of decision trees. Our analysis shows that ensembles of gradient boosting models successfully detect anomalous inputs while having limited ability to improve the predicted total uncertainty. Importantly, we also propose a concept of a virtual ensemble to get the benefits of an ensemble via only one gradient boosting model, which significantly reduces complexity.
|
| 20 |
+
|
| 21 |
+
# 1 INTRODUCTION
|
| 22 |
+
|
| 23 |
+
Gradient boosting (Friedman, 2001) is a widely used machine learning algorithm that achieves stateof-the-art results on tasks containing heterogeneous features, complex dependencies, and noisy data: web search, recommendation systems, weather forecasting, and many others (Burges, 2010; Caruana & Niculescu-Mizil, 2006; Richardson et al., 2007; Roe et al., 2005; Wu et al., 2010; Zhang & Haghani, 2015). Gradient boosting based on decision trees (GBDT) underlies such well-known libraries like XGBoost, LightGBM, and CatBoost. In this paper, we investigate the estimation of predictive uncertainty in GBDT models. Uncertainty estimation is crucial for avoiding costly mistakes in high-risk applications, such as autonomous driving, medical diagnostics, and financial forecasting. For example, in self-driving cars, it is necessary to know when the AI-pilot is confident in its ability to drive and when it is not to avoid a fatal collision. In financial forecasting and medical diagnostics, mistakes on the part of an AI forecasting or diagnostic system could either lead to large financial or reputational loss or to the loss of life. Crucially, both financial and medical data are often represented in heterogeneous tabular form — data on which GBDTs are typically applied, highlighting the relevance of our work on obtaining uncertainty estimates for GBDT models.
|
| 24 |
+
|
| 25 |
+
Approximate Bayesian approaches for uncertainty estimation have been extensively studied for neural network models (Gal, 2016; Malinin, 2019). Bayesian methods for tree-based models (Chipman et al., 2010; Linero, 2017) have also been widely studied in the literature. However, this research did not explicitly focus on studying uncertainty estimation and its applications. Some related work was done by Coulston et al. (2016); Shaker & Hüllermeier (2020), who examined quantifying predictive uncertainty for random forests. However, the area has been otherwise relatively under-explored, especially for GBDT models that are widely used in practice and known to outperform other approaches based on tree ensembles.
|
| 26 |
+
|
| 27 |
+
While for classification problems GDBT models already return a distribution over class labels, for regression tasks they typically yield only point predictions. Recently, this problem was addressed in the NGBoost algorithm (Duan et al., 2020), where a GBDT model is trained to return the mean and variance of a normal distribution over the target variable $y$ for a given feature vector. However, such models only capture data uncertainty (Gal, 2016; Malinin, 2019), also known as aleatoric uncertainty, which arises due to inherent class overlap or noise in the data. However, this does not quantify uncertainty due to the model’s inherent lack of knowledge about inputs from regions either far from the training data or sparsely covered by it, known as knowledge uncertainty, or epistemic uncertainty (Gal, 2016; Malinin, 2019). One class of approaches for capturing knowledge uncertainty are Bayesian ensemble methods, which have recently become popular for estimating predictive uncertainty in neural networks (Depeweg et al., 2017; Gal & Ghahramani, 2016; Kendall et al., 2018; Lakshminarayanan et al., 2017; Maddox et al., 2019; Smith & Gal, 2018). A key feature of ensemble approaches is that they allow overall uncertainty to be decomposed into data uncertainty and knowledge uncertainty within an interpretable probabilistic framework (Depeweg et al., 2017; Gal, 2016; Malinin, 2019). Ensembles are also known to yield improvements in predictive performance.
|
| 28 |
+
|
| 29 |
+
This work examines ensemble-based uncertainty-estimation for GBDT models. The contributions are as follows. First, we consider generating ensembles using both classical Stochastic Gradient Boosting (SGB) as well as the recently proposed Stochastic Gradient Langevin Boosting (SGLB) (Ustimenko & Prokhorenkova, 2020). Importantly, SGLB allows us to guarantee that the models are asymptotically sampled from a true Bayesian posterior. Second, we show that using SGLB we can construct a virtual ensemble using only one gradient boosting model, significantly reducing the computational complexity. Third, to understand the attributes of using ensembles-based uncertainty estimation in GBDT models, we conduct extensive analysis on several synthetic datasets. Finally, we evaluate the proposed approach on a range of real regression and classification datasets. Our results show that this approach successfully enables the detection of anomalous out-of-domain inputs. Importantly, our solution is easy to combine with any implementation of GBDT. Our methods have been implemented within the open-source CatBoost library. The code of our experiments is publicly available at https://github.com/yandex-research/GBDT-uncertainty.
|
| 30 |
+
|
| 31 |
+
# 2 PRELIMINARIES
|
| 32 |
+
|
| 33 |
+
Uncertainty Estimation via Bayesian Ensembles In this work we consider uncertainty estimation within the standard Bayesian ensemble-based framework (Gal, 2016; Malinin, 2019). Here, model parameters $\pmb { \theta }$ are considered random variables and a prior $\mathtt { p } ( \pmb \theta )$ is placed over them to compute a posterior $\mathtt { p } ( \pmb { \theta } | \mathcal { D } )$ via Bayes’ rule:
|
| 34 |
+
|
| 35 |
+
$$
|
| 36 |
+
\mathtt { p } ( \pmb { \theta } | \mathcal { D } ) = \frac { \mathtt { p } ( \mathcal { D } | \pmb { \theta } ) \mathtt { p } ( \pmb { \theta } ) } { \mathtt { p } ( \mathcal { D } ) } .
|
| 37 |
+
$$
|
| 38 |
+
|
| 39 |
+
where $\mathcal { D } = \{ \pmb { x } ^ { ( i ) } , y ^ { ( i ) } \} _ { i = 1 } ^ { N }$ is the training dataset. Each set of parameters can be considered a hypothesis or explanation about how the world works. Samples from the posterior should yield explanations consistent with the observations of the world contained within the training data $\mathcal { D }$ . However, on data far from $\mathcal { D }$ each set of parameters can yield different predictions. Therefore, estimates of knowledge uncertainty can be obtained by examining the diversity of predictions.
|
| 40 |
+
|
| 41 |
+
Consider an ensemble of probabilistic models $\{ \mathsf { P } ( y | \pmb { x } ; \pmb { \theta } ^ { ( m ) } ) \} _ { m = 1 } ^ { M }$ sampled from the posterior $\mathtt { p } ( \pmb { \theta } | \mathcal { D } )$ . Each model $\mathsf { P } ( y | \pmb { x } , \pmb { \theta } ^ { ( m ) } )$ yields a different estimate of data uncertainty, represented by the entropy of its predictive distribution (Malinin, 2019). Uncertainty in predictions due to knowledge uncertainty is expressed as the level of spread, or “disagreement”, of models in the ensemble (Malinin, 2019). Note that exact Bayesian inference is often intractable, and it is common to consider either an explicit or implicit approximation ${ \mathfrak { q } } ( \theta )$ to the true posterior ${ \tt p } ( \pmb \theta | \mathcal { D } )$ . While a range of approximations has been explored for neural network models (Gal & Ghahramani, 2016; Lakshminarayanan et al., 2017; Maddox et al., 2019)1, to the best of our knowledge, limited work has explored Bayesian inference for gradient-boosted trees. Given ${ \tt p } ( \pmb \theta | \mathcal { D } )$ , the predictive posterior of the ensemble is obtained by taking the expectation with respect to the models in the ensemble:
|
| 42 |
+
|
| 43 |
+
$$
|
| 44 |
+
\mathbb { P } \big ( \boldsymbol { y } | \boldsymbol { x } , \mathcal { D } \big ) = \mathbb { E } _ { \mathbb { P } ^ { ( \theta | \mathcal { D } ) } } \big [ \mathbb { P } \big ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } \big ) \big ] \approx \ \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathbb { P } \big ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ^ { ( m ) } \big ) , \ \boldsymbol { \theta } ^ { ( m ) } \sim \mathbb { p } ( \boldsymbol { \theta } | \mathcal { D } ) .
|
| 45 |
+
$$
|
| 46 |
+
|
| 47 |
+
The entropy of the predictive posterior estimates total uncertainty in predictions:
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
\begin{array} { r } { \mathcal { H } \big [ \mathrm { P } ( y | x , \mathcal { D } ) \big ] = \mathbb { E } _ { \mathrm { P } ( y | x , \mathcal { D } ) } \big [ - \ln \mathrm { P } ( y | x , \mathcal { D } ) \big ] . } \end{array}
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
Total uncertainty is due to both data uncertainty and knowledge uncertainty. However, in applications like active learning (Kirsch et al., 2019) and out-of-domain detection it is desirable to estimate knowledge uncertainty separately. The sources of uncertainty can be decomposed by considering the mutual information between the parameters $\pmb \theta$ and the prediction $y$ (Depeweg et al., 2017):
|
| 54 |
+
|
| 55 |
+
$$
|
| 56 |
+
\begin{array} { r l r } { { \underbrace { \underline { { \mathcal { L } \big [ \boldsymbol { y } , \boldsymbol { \theta } | \boldsymbol { x } , \mathcal { D } \big ] } } } _ { \mathrm { K n o w l e d g e ~ U n c e r t a i n t y } } = \underbrace { \mathcal { H } \big [ \mathtt { P } ( \boldsymbol { y } | \boldsymbol { x } , \mathcal { D } ) \big ] } _ { \mathrm { T o t a l ~ U n c e r t a i n t y } } - \underbrace { \mathbb { E } _ { \mathtt { P } ( \boldsymbol { \theta } | \mathcal { D } ) } \big [ \mathcal { H } \big [ \mathtt { P } ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ) \big ] \big ] } _ { \mathrm { E x p e c t e d ~ D a t a ~ U n c e r t a i n t y } } } } \\ & { } & { \approx \mathcal { H } \big [ \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathtt { P } ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ^ { ( m ) } ) \big ] - \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mathcal { H } \big [ \mathtt { P } ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ^ { ( m ) } ) \big ] . } \end{array}
|
| 57 |
+
$$
|
| 58 |
+
|
| 59 |
+
This is expressed as the difference between the entropy of the predictive posterior, a measure of total uncertainty, and the expected entropy of each model in the ensemble, a measure of expected data uncertainty. Their difference is a measure of ensemble diversity and estimates knowledge uncertainty.
|
| 60 |
+
|
| 61 |
+
Unfortunately, when considering ensembles of probabilistic regression models $\{ \mathsf { p } ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ^ { ( m ) } ) \} _ { m = 1 } ^ { M }$ over continuous-valued target $y \in \mathbb { R }$ , it is no longer possible to obtain tractable estimates of the (differential) entropy of the predictive posterior, and, by extension, mutual information. In this cases uncertainty estimates can instead derived via the law of total variation:
|
| 62 |
+
|
| 63 |
+
$$
|
| 64 |
+
\underbrace { \mathbb { V } _ { \mathtt { p } ( y | x , \mathcal { D } ) } [ y ] } _ { \mathrm { T o t a l ~ U n c e r t a i n t y } } = \underbrace { \mathbb { V } _ { \mathtt { p } ( \theta | \mathcal { D } ) } \left[ \mathbb { E } _ { \mathtt { p } ( y | x , \theta ) } [ y ] \right] } _ { \mathrm { K n o w l e d g e ~ U n c e r t a i n t y } } + \underbrace { \mathbb { E } _ { \mathtt { p } ( \theta | \mathcal { D } ) } \left[ \mathbb { V } _ { \mathtt { p } ( y | x , \theta ) } [ y ] \right] } _ { \mathrm { E x p e c t e d ~ D a t a ~ U n c e r t a i n t y } } .
|
| 65 |
+
$$
|
| 66 |
+
|
| 67 |
+
This is conceptually similar to the decomposition (4) obtained via mutual information. For an ensemble of probabilistic regression models which parameterize the normal distribution, and where each models yields a mean and standard-deviation, the total variance can be computed as follows:
|
| 68 |
+
|
| 69 |
+
$$
|
| 70 |
+
\underbrace { \mathbb { V } _ { \mathrm { p } ( y | \mathbf { x } , \mathcal { D } ) } [ y ] } _ { \mathrm { T r a t a l } \Pi _ { \mathrm { m e a r i a n t y } } } \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \Big [ \Big ( \sum _ { m = 1 } ^ { M } \frac { \mu _ { m } } { M } \Big ) - \mu _ { m } \Big ] ^ { 2 } + \quad \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \sigma _ { m } ^ { 2 } \quad , \ \{ \mu _ { m } , \sigma _ { m } \} = f ( \mathbf { x } ; \theta ^ { ( m ) } ) .
|
| 71 |
+
$$
|
| 72 |
+
|
| 73 |
+
However, while these measures are tractable, they are based on only first and second moments, and may therefore miss high-order details in the uncertainty. They are also not scale-invariant, which can cause issues is the scale of prediction on in-domain and out-of-domain data is very different.
|
| 74 |
+
|
| 75 |
+
Gradient boosting is a powerful machine learning technique especially useful on tasks containing heterogeneous features. It iteratively combines weak models, such as decision trees, to obtain more accurate predictions. Formally, given a dataset $\mathcal { D }$ and a loss function $L : \mathbb { R } ^ { 2 } \to \mathbb { R }$ , the gradient boosting algorithm (Friedman, 2001) iteratively constructs a model $F : X \mathbb { R }$ to minimize the empirical risk $\mathcal { L } ( F | \mathcal { D } ) = \mathbb { E } _ { \mathcal { D } } [ L ( F ( \pmb { x } ) , y ) ]$ . At each iteration $t$ the model is updated as:
|
| 76 |
+
|
| 77 |
+
$$
|
| 78 |
+
{ \cal F } ^ { ( t ) } ( { \pmb x } ) = { \cal F } ^ { ( t - 1 ) } ( { \pmb x } ) + \epsilon h ^ { ( t ) } ( { \pmb x } ) ,
|
| 79 |
+
$$
|
| 80 |
+
|
| 81 |
+
where $F ^ { ( t - 1 ) }$ is a model constructed at the previous iteration, $h ^ { ( t ) } ( { \pmb x } ) \in \mathcal { H }$ is a weak learner chosen from some family of functionds $\mathcal { H }$ , and $\epsilon$ is learning rate. The weak learner $h ^ { ( t ) }$ is usually chosen to approximate the negative gradient $\begin{array} { r } { - g ^ { ( t ) } ( \pmb { x } , y ) : = - \frac { \partial L ( y , s ) } { \partial s } \big | _ { s = F ^ { ( t - 1 ) } ( \pmb { x } ) } } \end{array}$ :
|
| 82 |
+
|
| 83 |
+
$$
|
| 84 |
+
\boldsymbol { h } ^ { ( t ) } = \underset { \boldsymbol { h } \in \mathcal { H } } { \arg \operatorname* { m i n } } \mathbb { E } _ { \mathcal { D } } \big [ \big ( - \boldsymbol { g } ^ { ( t ) } ( \boldsymbol { x } , \boldsymbol { y } ) - \boldsymbol { h } ( \boldsymbol { x } ) \big ) ^ { 2 } \big ] .
|
| 85 |
+
$$
|
| 86 |
+
|
| 87 |
+
A weak learner $h ^ { ( t ) }$ is associated with parameters $\phi ^ { ( t ) } \in \mathbb { R } ^ { d }$ . We write $h ^ { ( t ) } ( { \pmb x } , \phi ^ { ( t ) } )$ to reflect this
|
| 88 |
+
dependence. The set of weak learners $\mathcal { H }$ often consists of shallow decision trees, which are models
|
| 89 |
+
that recursively partition the feature space into disjoint regions called leaves. Each leaf $R _ { j }$ of the
|
| 90 |
+
tree iwrite estimated response , so the decision tre $y$ in the correspondings a linear function of n. We c. The fin $\begin{array} { r } { h ( \pmb { x } , \phi ^ { ( t ) } ) = \sum _ { j = 1 } ^ { d } \phi _ { j } ^ { ( t ) } \mathbf { 1 } _ { \{ \pmb { x } \in R _ { j } \} } } \end{array}$ $\phi ^ { ( t ) }$ $F$ $\pmb \theta$
|
| 91 |
+
|
| 92 |
+
For classification tasks, a model yields estimates data uncertainty if it is trained via negative loglikelihood and provides a distribution over class labels. However, classic GBDT regression models yield point predictions, and there has been little research devoted to estimating predictive uncertainty. Recently, this issue was addressed by Duan et al. (2020) via an algorithm called NGBoost (Natural Gradient Boosting), which allows estimating data uncertainty. NGBoost simultaneously estimates the parameters of a conditional distribution $\mathtt { p } ( y | \pmb { x } , \pmb { \theta } )$ over the target $y$ given the features $_ { \textbf { \em x } }$ , by optimizing a proper scoring rule. Typically, a normal distribution over $y$ is assumed and negative log-likelihood is taken as a scoring rule. Formally, given an input $_ { \textbf { \em x } }$ , the model $F$ predicts two parameters of normal distribution - the mean $\mu$ and the logarithm of the standard deviation $\log \sigma$ The loss function is the expected negative log-likelihood:2
|
| 93 |
+
|
| 94 |
+
$$
|
| 95 |
+
\mathrm { p } ( \boldsymbol { y } | \mathbf { \boldsymbol { x } } , \pmb { \theta } ^ { ( t ) } ) = \mathcal { N } ( \boldsymbol { y } | \mu ^ { ( t ) } , \boldsymbol { \sigma } ^ { ( t ) } ) , \quad \{ \mu ^ { ( t ) } , \log \sigma ^ { ( t ) } \} = F ^ { ( t ) } ( \pmb { x } ) .
|
| 96 |
+
$$
|
| 97 |
+
|
| 98 |
+
$$
|
| 99 |
+
\mathcal { L } ( \pmb { \theta } | \mathcal { D } ) = \mathbb { E } _ { \mathcal { D } } [ - \log \mathbb { p } ( \pmb { y } | \pmb { x } , \pmb { \theta } ) ] = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \mathbb { p } ( \pmb { y } ^ { ( i ) } | \pmb { x } ^ { ( i ) } , \pmb { \theta } ) .
|
| 100 |
+
$$
|
| 101 |
+
|
| 102 |
+
Note that $\pmb \theta$ denotes the concatenation of two parameter vectors used to predict $\mu$ and $\log \sigma$ .
|
| 103 |
+
|
| 104 |
+
# 3 GENERATING ENSEMBLES OF GDBT MODELS
|
| 105 |
+
|
| 106 |
+
As discussed in Section 2, knowledge uncertainty can be estimated by considering an ensemble of models $\{ \mathsf { p } ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } ^ { ( m ) } ) \} _ { m = 1 } ^ { M }$ sampled from the posterior ${ \tt p } ( \pmb \theta | \mathcal { D } )$ . The level of diversity or “disagreement” between the models is an estimate of knowledge uncertainty. In this section, we consider three approaches to generating an ensemble of GBDT models. We emphasize that this section discusses ensembles of GBDT models, where a each GBDT model is itself an ensemble of trees.
|
| 107 |
+
|
| 108 |
+
SGB ensembles One way to generate an ensemble is to consider several independent models generated via Stochastic Gradient Boosting (SGB). Stochasticity is added to GBDT models via random subsampling of the data at every iteration (Friedman, 2002). Specifically, at each iteration of (8) we select a subset of training objects $\mathcal { D } ^ { \prime }$ (via bootstrap or uniformly without replacement), which is smaller than the original training dataset $\mathcal { D }$ , and use $\mathcal { D } ^ { \prime }$ to fit the next tree instead of $\mathcal { D }$ . The fraction of chosen objects is called sample rate. This implicitly injects noise into the learning process, effectively inducing a distribution ${ \mathfrak { q } } ( \theta )$ over such models. Thus, an SGB ensemble is an ensemble of independent models $\{ \pmb { \theta } ^ { ( m ) } \} _ { m = 1 } ^ { M }$ built according to SGB with different random seeds for subsampling data. Unfortunately, there are no guarantees on how well the distribution ${ \mathfrak { q } } ( \theta )$ estimates the true posterior $\mathtt { p } ( \pmb { \theta } | \mathcal { D } )$ .
|
| 109 |
+
|
| 110 |
+
SGLB ensembles Remarkably, there is a way to sample GBDT models from the true posterior $\mathtt { p } ( \pmb { \theta } | \mathcal { D } )$ via a recently proposed Stochastic Gradient Langevin Boosting (SGLB) algorithm (Ustimenko $\&$ Prokhorenkova, 2020). SGLB combines gradient boosting with stochastic gradient Langevin dynamics (Raginsky et al., 2017) in order to achieve convergence to the global optimum even for non-convex loss functions. The algorithm has two differences compared with SGB. First, Gaussian noise is explicitly injected into the gradients, so (8) is replaced by:
|
| 111 |
+
|
| 112 |
+
$$
|
| 113 |
+
h ^ { ( t ) } = \underset { h \in \mathcal { H } } { \mathrm { a r g } \mathrm { m i n } } \mathbb { E } _ { \mathcal { D } } \left[ \left( - g ^ { ( t ) } ( \pmb { x } , y ) - h ( \pmb { x } , \phi ) + \nu \right) ^ { 2 } \right] , \nu \sim \mathcal { N } \left( 0 , \frac { 2 } { \beta \epsilon } I _ { | \mathcal { D } | } \right) ,
|
| 114 |
+
$$
|
| 115 |
+
|
| 116 |
+
where $\beta$ is the inverse diffusion temperature and $I _ { | \mathcal { D } | }$ is an identity matrix. This random noise $\nu$ helps to explore the solution space in order to find the global optimum and the diffusion temperature controls the level of exploration. Second, the update (7) is modified as:
|
| 117 |
+
|
| 118 |
+
$$
|
| 119 |
+
F ^ { ( t ) } ( { \pmb x } ) = ( 1 - \gamma \epsilon ) F ^ { ( t - 1 ) } ( { \pmb x } ) + \epsilon h ^ { ( t ) } ( { \pmb x } , { \phi } ^ { ( t ) } ) ,
|
| 120 |
+
$$
|
| 121 |
+
|
| 122 |
+
|
| 123 |
+
Figure 1: Virtual ensemble
|
| 124 |
+
|
| 125 |
+
where $\gamma$ is regularization parameter. If the number of all possible trees is finite (a natural assumption given that the training dataset is finite), then the SGLB parameters ${ \pmb \theta } ^ { ( t ) }$ at each iteration form a Markov chain that weakly converges to the stationary distribution, also called the invariant measure:
|
| 126 |
+
|
| 127 |
+
$$
|
| 128 |
+
p _ { \beta } ^ { * } ( \pmb { \theta } ) \propto \exp ( - \beta \mathcal { L } ( \pmb { \theta } | \mathcal { D } ) - \beta \gamma \| \Gamma \pmb { \theta } \| _ { 2 } ^ { 2 } ) ,
|
| 129 |
+
$$
|
| 130 |
+
|
| 131 |
+
where $\Gamma = \Gamma ^ { T } > 0$ is an implicitly defined regularization matrix which depends on a particular tree construction algorithm (Ustimenko & Prokhorenkova, 2020).
|
| 132 |
+
|
| 133 |
+
While Ustimenko & Prokhorenkova (2020) used the weak convergence to (13) to prove the global convergence, we apply this to enable sampling from the true posterior. For this purpose, we set $\beta = | \mathcal { D } |$ and $\begin{array} { r } { \gamma = \frac { \hat { \mathrm { ~ T ~ } } } { 2 \left| \mathcal { D } \right| } } \end{array}$ . For the negative log-likelihood loss function (10) the invariant measure (13) can be expressed as:
|
| 134 |
+
|
| 135 |
+
$$
|
| 136 |
+
p _ { \beta } ^ { * } ( \pmb { \theta } ) \propto \exp \left( \log \mathbb { p } ( \mathcal { D } | \pmb { \theta } ) - \frac { 1 } { 2 } \| \Gamma \pmb { \theta } \| _ { 2 } ^ { 2 } \right) \propto \mathrm { p } ( \mathcal { D } | \pmb { \theta } ) \mathrm { p } ( \pmb { \theta } ) ,
|
| 137 |
+
$$
|
| 138 |
+
|
| 139 |
+
which is proportional to the true posterior distribution ${ \tt p } ( \pmb \theta | \mathcal { D } )$ under Gaussian prior $\mathsf { p } ( \theta ) = \mathcal { N } ( 0 , \Gamma )$ . Thus, an SGLB ensemble is an ensemble of independent models $\{ \pmb { \theta } ^ { ( m ) } \} _ { m = 1 } ^ { M }$ generated according to the SGLB algorithm using different random seeds. In this case, asymptotically, models are sampled from the true posterior ${ \tt p } ( \pmb \theta | \mathcal { D } )$ .
|
| 140 |
+
|
| 141 |
+
Virtual SGLB ensembles While SGB and SGLB yield ensembles of independent models, their time and space complexity is $M$ times larger than that of a single model, which is a significant overhead. Consequently, generating an ensemble requires either significantly increasing complexity or sacrificing the quality by reducing the number of training iterations. To address this, we introduce the concept of a virtual ensemble that enables generating an ensemble using only one model. This is possible since a GBDT model is itself an ensemble of trees. However, in contrast to random forests formed by independent trees (Shaker & Hüllermeier, 2020), the sequential nature of GBDT models implies that all trees are dependent and individual trees cannot be considered as separate models. Hence, we use “truncated” sub-models of a single GBDT model as elements of an ensemble, as illustrated in Figure 1. Notably, a virtual ensemble can be obtained using any already constructed GBDT model. Below we formally describe this procedure applied to SGLB models since in this case we can guarantee asymptotically sampling from the true posterior ${ \tt p } ( \pmb \theta | \mathcal { D } )$ .
|
| 142 |
+
|
| 143 |
+
Each “truncated” model is described by the vector of parameters $\pmb \theta ^ { ( t ) }$ . As the parameters ${ \pmb \theta } ^ { ( t ) }$ at each iteration of the SGLB algorithm form a Markov chain that weakly convergences to the stationary distribution (14), we can consider using them as an ensemble of models. However, unlike parameters taken from different SGLB trajectories, these will have a high degree of correlation, which adversely affects the ensemble’s quality. This problem can be overcome by retaining only every $K$ -th set of parameters. Formally, fix $K \geq 1$ and consider a set of models $\begin{array} { r } { \Theta _ { T , K } = \{ \pmb { \theta } ^ { ( K t ) } , \left[ \frac { T } { 2 K } \right] \leq t \leq \left[ \frac { T } { K } \right] \} } \end{array}$ i.e., we add to $\Theta _ { T , K }$ every $K$ -th model obtained while constructing one SGLB model using $T$ iterations of gradient boosting. Choosing larger values of $K$ allows us to reduce the correlation between samples from the SGLB Markov chain. Furthermore, we do not include to the ensemble the models $\bar { \pmb { \theta } ( t ) }$ with $t < T / 2$ as (14) holds only asymptotically. The set of $\begin{array} { r } { M \ : = \ : \left[ \frac { T } { 2 K } \right] } \end{array}$ models $\Theta _ { T , K }$ is called a virtual ensemble. Note that virtual ensembles behave similarly to true ensembles in the limit (for large $K$ and $T$ ).
|
| 144 |
+
|
| 145 |
+
Importantly, we can compute the prediction of $\Theta _ { T , K }$ with the same computation time as one ${ \pmb \theta } ^ { ( T ) }$ Indeed, when computing the prediction of one model, we have to sum up the predictions made by individual trees. To get the virtual ensemble, we only have to store the partial sums. For SGLB, we also have to account for regularization (12). Formally, according to (12), for SGLB we have $\begin{array} { r } { \pmb { \theta } ^ { ( T ) } = \sum _ { i = 1 } ^ { T } \epsilon ( 1 - \gamma \epsilon ) ^ { T - i } \pmb { \phi } ^ { ( i ) } } \end{array}$ , where $( 1 - \gamma \epsilon ) ^ { T - i }$ appears due to shrinkage. While computing ${ \pmb \theta } ^ { ( T ) }$ i=1 we store the partial sums $\begin{array} { r } { \pmb { \theta } _ { \leq t } ^ { ( T ) } = \sum _ { i = 1 } ^ { t } \epsilon ( 1 - \gamma \epsilon ) ^ { T - i } \pmb { \phi } ^ { ( i ) } } \end{array}$ . Then, any model $\pmb \theta ^ { ( t ) }$ from $\Theta _ { T , K }$ can easily be obtained from the stored values:
|
| 146 |
+
|
| 147 |
+
|
| 148 |
+
Figure 2: Uncertainty for synthetic regression dataset with two categorical features. Inside the heart (white region on the first figure) there are no training examples.
|
| 149 |
+
|
| 150 |
+
$$
|
| 151 |
+
\pmb \theta ^ { ( t ) } = \sum _ { i = 1 } ^ { t } \epsilon ( 1 - \gamma \epsilon ) ^ { t - i } \pmb \phi ^ { ( i ) } = ( 1 - \gamma \epsilon ) ^ { t - T } \pmb \theta _ { \leq t } ^ { ( T ) } .
|
| 152 |
+
$$
|
| 153 |
+
|
| 154 |
+
# 4 ANALYSIS ON SYNTHETIC DATA
|
| 155 |
+
|
| 156 |
+
In this section, we analyze how ensemble algorithms discussed in Section 3 perform on synthetic data. The aim is to understand the attributes of ensembles of GBDT models for estimating data and knowledge uncertainty in a controllable setting.
|
| 157 |
+
|
| 158 |
+
GBDT models are usually applied to tabular data, where features are often categorical. Hence, we first generate a dataset with each example described by two categorical features $x _ { 1 } , x _ { 2 }$ with 9 values each, resulting in 81 possible combinations. The target depends on the features as $y = a ( x _ { 1 } , x _ { 2 } ) + \varepsilon ( x _ { 1 } , \stackrel { \_ } { x } _ { 2 } )$ , where $\varepsilon ( x _ { 1 } , x _ { 2 } ) \sim \mathcal { N } ( 0 , b ( x _ { 1 } , x _ { 2 } ) )$ and $a ( \overset { \cdot } { x _ { 1 } } , x _ { 2 } ) , b ( \overset { \cdot } { x } _ { 1 } , \overset { \cdot } { x } _ { 2 } )$ are some deterministic functions. The values for $a ( x _ { 1 } , x _ { 2 } )$ are randomly generated according to the uniform distribution over $[ 0 , 1 ]$ . The values for $b ( x _ { 1 } , x _ { 2 } )$ are shown on Figure 2(a). We generate a heartshaped dataset with this distribution: inside the heart (white region on Figure 2(a)) there are no training points, for the other cells we have 1000 examples per cell.
|
| 159 |
+
|
| 160 |
+
We train an ensemble of 10 SGLB models (each model consists of 1000 trees) and observe the following effects. First, Figure 2(b) shows the total uncertainty estimated with SGLB ensemble and we see that the models correctly capture this uncertainty in all cells containing training examples. At the same time, arbitrary values can be predicted inside the heart, as no training data constrain the models’ behavior there. Second, Figure 2(c) shows that estimates of knowledge uncertainty allow us to detect regions that are out-of-domain and are not covered by the training data. Notably, the separation is perfect, as there is no trace of the original heart border.
|
| 161 |
+
|
| 162 |
+
To further analyze ensembles of GBDT models, we apply them to a two-dimensional classification task with continuous-valued features. We consider a 3-class spiral dataset shown on Figure 3(a); and this setting is much harder for gradient boosted trees.3 Figure 3(b) shows the total uncertainty estimated with SGLB ensemble, while Figure 3(c) demonstrates knowledge uncertainty. We observe several effects. First, total uncertainty correctly detects class boundaries and ‘sectors’ of input space outside the training dataset. Second, looking at these ‘sectors’ of high uncertainty, we can better understand how GBDT ensembles work: as decision trees are discriminative functions (Bishop, 2006), if features have values outside the training domain, then the prediction is the same as for the “closest” elements in the dataset. In other words, the models’ behavior on the boundary of the dataset is further extended to the outer regions. Third, estimates of knowledge uncertainty allow discrimination between out-of-domain regions and class boundaries. However, we still can see traces of the class boundaries in Figure 3(c). A possible reason is the fact that for real-valued features, near the class borders, the splitting values may vary across all models in the ensemble, resulting in nonzero estimates of knowledge uncertainty due to decision-boundary ‘jitter’.
|
| 163 |
+
|
| 164 |
+
|
| 165 |
+
Figure 3: Uncertainty for synthetic classification dataset
|
| 166 |
+
|
| 167 |
+
|
| 168 |
+
Figure 4: Comparison of SGLB and vSGLB knowledge uncertainty estimates
|
| 169 |
+
|
| 170 |
+
On both “heart” and “spiral” datasets, we observed that the absolute values of knowledge uncertainty are much smaller than of data uncertainty and therefore contribute very little to total uncertainty. Thus, we expect that while knowledge uncertainty is especially useful for detecting anomalous inputs, the proposed approaches will contribute little to error detection on top of estimates of data uncertainty provided by single models.
|
| 171 |
+
|
| 172 |
+
Finally, on Figure 4, we compare the performance of ‘true’ SGLB ensembles with the virtual SGLB ensembles (vSGBL) on both the “heart” and “spiral” datasets. The virtual ensemble is ten times cheaper to train and infer, but the ensemble members are strongly correlated. We observe that on the “heart” dataset, vSGLB perfectly detects regions not covered by training data. However, the absolute values of knowledge uncertainty are much smaller than for SGLB, which can be explained by the correlations. The “spiral” dataset is more challenging for both SGLB and vSGLB. While having qualitatively similar behavior, virtual ensembles struggle to detect out-of-domain regions and separate them from class boundaries. In all cases, the absolute values of knowledge uncertainty are far lower than for ‘true’ SGLB ensembles. This shows that while vSGLB yields very cheap estimates of knowledge uncertainty by exploiting the ‘ensemble of trees’ structure of GBDT models, the quality of these estimates is inferior to ensembles of independent models.
|
| 173 |
+
|
| 174 |
+
# 5 EXPERIMENTS ON CLASSIFICATION AND REGRESSION DATASETS
|
| 175 |
+
|
| 176 |
+
In this section, we evaluate the performance of ensembles of GBDT models on a range of classification and regression tasks, focusing on their ability to detect errors and out-of-domain inputs.
|
| 177 |
+
|
| 178 |
+
Experimental setup Our implementation of all GBDT models is based on the CatBoost library that is known to achieve state-of-the-art results in a variety of tasks (Prokhorenkova et al., 2018). Classification models yield a probability distribution over binary class labels, while regression models yield the mean and variance of the normal distribution, as discussed in Section 2. All models are trained by optimizing the negative log-likelihood.4 We consider SGB and SGBL single models as the baselines and examine all ensemble methods defined in Section 3. Ensembles of SGB and SGLB models consist of 10 independent (with different seeds) models with 1000 trees each. The virtual ensemble vSGLB is obtained from one model with 1000 trees, where each 50th model from the interval [501, 1000] is added to the ensemble. Thus, vSGLB has the same computational and space complexity as just one SGB or SGLB model. Hyper-parameters are tuned by grid search, for details see Appendix A.2.
|
| 179 |
+
|
| 180 |
+
We compare the algorithms on several classification and regression tasks (Gal & Ghahramani, 2016;
|
| 181 |
+
Prokhorenkova et al., 2018), the description of which is available in Appendix A.3.
|
| 182 |
+
|
| 183 |
+
While not being the focus of the current research, Random Forest (RF) models are naturally suitable for ensemble approaches. Hence, we conduct additional experiments and analyze the performance of ensemble approaches applied to RF models in Appendix C.
|
| 184 |
+
|
| 185 |
+
Detection of errors and anomalous inputs We analyze whether measures of total and knowledge uncertainty can be used to detect errors and out-of-domain inputs. Error detection can be evaluated via the Prediction-Rejection Ratio (PRR) (Malinin, 2019; Malinin et al., 2020), which measures how well uncertainty estimates correlate with errors and rank-order them. The best value is 100, random is 0. Out-of-domain (OOD) detection is assessed via area under the ROC curve (AUCROC) (Hendrycks & Gimpel, 2016). For OOD detection, we need an OOD test-set. However, obtaining ‘real’ OOD examples for the datasets considered in this work is challenging, so we instead create synthetic OOD data as follows. For each dataset, we take its test set as the in-domain examples and sample an OOD dataset of the same size from the Year MSD dataset to get out-of-domain (OOD) data. The only exceptions are KDD datasets (Appetency, Churn, Upselling) and Year MSD, for which we sample OOD data from the Relative location of CT slices on axial axis Data Set (Graf et al., 2011). All numerical features in OOD data are normalized by the per-column mean and variance obtained on the in-domain training data. For categorical features, we sample a random category uniformly at random from the set of all feature’s categories. Total and knowledge uncertainty are estimated via entropy of the predictive posterior (3) and mutual information (4) for classification models and via total variance and variance of the mean (5) for regression ones.
|
| 186 |
+
|
| 187 |
+
Test errors can occur due to both noise and lack of knowledge, so we expect that ranking elements by total uncertainty would give better values of PRR. Table 1 shows that measures of total uncertainty consistently yield better PPR results across all datasets. This is consistent with results obtained for ensembles of neural network models (Lakshminarayanan et al., 2017; Malinin, 2019; Malinin & Gales, 2019; Malinin et al., 2020). However, ensembles do not outperform single models. We believe this occurs for two reasons. First, due to the additive nature of boosting, GDBT models are already ensembles. Second, as we have discussed in Section 4, for GBDT models, estimates of knowledge uncertainty obtained via the approaches considered here contribute little to estimates of total uncertainty.
|
| 188 |
+
|
| 189 |
+
In contrast, Table 1 shows that measures of knowledge uncertainty yield superior OOD detection performance compared to total uncertainty in terms of AUC-ROC, which is consistent with results for non-GBDT models (Malinin, 2019; Malinin & Gales, 2019; Malinin et al., 2020).5 The results also show that SGB and SGLB ensembles performed almost equally well. At the same time, virtual ensembling (vSGLB) performed consistently worse (with one exception) than SGB/SGLB ensembles, which is explained by the presence of strong correlations between the models in a virtual ensemble. However, in classification tasks, estimates of knowledge uncertainty provided by vSGLB nevertheless outperform uncertainty estimates derived from single SGB and SGLB models. This shows that useful measures of knowledge uncertainty can be derived from a single SGLB model by interpreting it as a virtual ensemble at no additional computational or memory cost. For vSGLB, the difference between classification and regression tasks can be explained by the presence or absence of categorical features. In our preliminary experiments on synthetic data, we noticed that categorical features may have a noticeable effect on the diversity of vSGLB models, and our classification datasets contain categorical features.
|
| 190 |
+
|
| 191 |
+
# 6 CONCLUSION
|
| 192 |
+
|
| 193 |
+
This work examined principled, ensemble-based uncertainty-estimation for GBDT models. Two main approaches to generating ensembles of GDBT models, where each model is itself an ensemble of trees, were considered — Stochastic Gradient Boosting (SGB) and Stochastic Gradient Langevin Boosting (SGLB). Based on SGLB, we propose constructing a virtual ensemble (vSGLB) by exploiting the ‘ensemble-of-trees’ nature of GBDT models. Properties of the estimates of total, data, and knowledge uncertainty derived from these ensembles were first analyzed on synthetic data. It was shown that the proposed approach can successfully detect anomalous inputs and is especially successful on tabular data. On continuous data, detecting knowledge uncertainty is still possible, but it becomes harder to differentiate it with data uncertainty due to decision-boundary ‘jitter’. Further experiments on a wide range of classification and regression datasets showed that while ensembles of GDBT models do not offer much advantage in terms of error detection, as each model is already an ensemble of trees, they do yield useful measures of knowledge uncertainty, which enables outof-domain detection in both regression and classification tasks. Notably, measures of knowledge uncertainty, which can only be obtained via ensembles, achieve far better OOD detection performance than measures of total uncertainty. It is also shown that while there is little practical difference between SGB and SGLB ensembles, vSGLB performs noticeably worse. However, for classification tasks containing categorical features, vSGLB still yields useful measures of knowledge uncertainty at the computational time and space complexity of a single SGLB model. Thus, vSGLB allows us to derive the benefits of an ensemble at no additional computational and memory cost.
|
| 194 |
+
|
| 195 |
+
Table 1: Detection of errors and OOD examples for regression and classification tasks
|
| 196 |
+
|
| 197 |
+
<table><tr><td rowspan="2">Dataset</td><td rowspan="2"></td><td colspan="2">Single SGB</td><td colspan="3">Ensemble</td><td colspan="2">Single</td><td colspan="3">Ensemble</td></tr><tr><td></td><td>SGLB</td><td>SGB Classification % PRR (↑)</td><td>SGLB</td><td>vSGLB</td><td>SGB</td><td>SGLB Clasification % AUC-ROC (↑)</td><td>SGB</td><td>SGLB</td><td>vSGLB</td></tr><tr><td>Adult</td><td>TU KU</td><td>72</td><td colspan="3">72 72</td><td>72</td><td colspan="3">53 50</td><td>51</td><td>51</td></tr><tr><td></td><td>TU</td><td>一 71</td><td>一 69</td><td>49 70</td><td>49 68</td><td>38 68</td><td>丨 86</td><td>一 87</td><td>89 86</td><td>89 86</td><td>85 86</td></tr><tr><td>Amazon Click</td><td>KU TU</td><td>丨 43</td><td>一 44</td><td>64 43</td><td>61 44</td><td>40 44</td><td>丨 61</td><td>一 67</td><td>88 64</td><td>74 64</td><td>67 68</td></tr><tr><td>Internet</td><td>KU TU</td><td>丨 76</td><td>一 79</td><td>22 77</td><td>22 79</td><td>11 79</td><td>1 67</td><td>1 68</td><td>91 70</td><td>92 69</td><td>90 68</td></tr><tr><td></td><td>KU TU</td><td>一 68</td><td>1 69</td><td>69 69</td><td>72 69</td><td>61 69</td><td>1 29</td><td>1 48</td><td>87 47</td><td>89 50</td><td>81 52</td></tr><tr><td>KDD-Appetency KDD-Churn</td><td>KU TU</td><td>1 47</td><td>1 45</td><td>64 48</td><td>54 46</td><td>14 46</td><td>1 81</td><td>1 57</td><td>90 82</td><td>91 75</td><td>93 60</td></tr><tr><td>KDD-Upselling</td><td>KU TU</td><td>1 56</td><td>1 56</td><td>33 57</td><td>35 57</td><td>28 56</td><td>1 53</td><td>1 51</td><td>99 62</td><td>98 60</td><td>92 47</td></tr><tr><td>Kick</td><td>KU TU KU</td><td>丨 44</td><td>1 45</td><td>45 44 34</td><td>49 44 34</td><td>33 45 20</td><td>1 45</td><td>1 37</td><td>97 52</td><td>97 58 98</td><td>78 38 89</td></tr><tr><td>Dataset</td><td></td><td colspan="5">Regression % PRR (↑)</td><td colspan="5">98 Regression % AUC-ROC (↑)</td></tr><tr><td>BostonH</td><td>TU KU</td><td>45 一</td><td>45 1</td><td>44 36</td><td>45 37</td><td>46 38</td><td>70 1</td><td>68 1</td><td>71 80</td><td>69 80</td><td>64 49</td></tr><tr><td>Concrete</td><td>TU KU</td><td>45 一</td><td>41 一</td><td>44 27</td><td>42 27</td><td>41 25</td><td>78</td><td>80 1</td><td>79 92</td><td>81 92</td><td>78 56</td></tr><tr><td>Energy</td><td>TU KU</td><td>58 1</td><td>56 1</td><td>58 36</td><td>56 31</td><td>62 54</td><td>67 1</td><td>69 1</td><td>89 100</td><td>89 100</td><td>69 32</td></tr><tr><td>Kin8nm</td><td>TU KU</td><td>59</td><td>59 一</td><td>59 18</td><td>59 19</td><td>58 35</td><td>43 一</td><td>43 1</td><td>43 45</td><td>43 45</td><td>42 45</td></tr><tr><td>Naval-p</td><td>TU KU</td><td>75 一</td><td>76 丨</td><td>82 52</td><td>82 56</td><td>81 69</td><td>99 1</td><td>99 1</td><td>100 100</td><td>100 100</td><td>99 87</td></tr><tr><td>Power-p</td><td>TU KU</td><td>30 丨</td><td>32 1</td><td>31 8</td><td>33 9</td><td>32 13</td><td>48 1</td><td>47 1</td><td>51 72</td><td>49 73</td><td>47 57</td></tr><tr><td>Protein</td><td>TU KU</td><td>49 丨</td><td>48 1</td><td>52 30</td><td>50 29</td><td>48 12</td><td>82 1</td><td>84 1</td><td>92 99</td><td>91 99</td><td>86 94</td></tr><tr><td>Wine-qu</td><td>TU KU</td><td>33 丨</td><td>32 1</td><td>33 25</td><td>32 19</td><td>32 9</td><td>60 1</td><td>56 1</td><td>60 74</td><td>56 72</td><td>56 49</td></tr><tr><td>Yacht</td><td>TU KU</td><td>89 一</td><td>88 1</td><td>88 74</td><td>88 78</td><td>88 66</td><td>57 1</td><td>57 1</td><td>58 62</td><td>58 60</td><td>55 40</td></tr><tr><td>Year</td><td>TU KU</td><td>61 一</td><td>62 1</td><td>62 30</td><td>63 30</td><td>62 25</td><td>59 1</td><td>58 1</td><td>60 67</td><td>60 57</td><td>57 52</td></tr></table>
|
| 198 |
+
|
| 199 |
+
# ACKNOWLEDGMENTS
|
| 200 |
+
|
| 201 |
+
We would like to thank Ekaterina Ermishkina and Stanislav Kirillov for implementing the proposed methods within the CatBoost library.
|
| 202 |
+
|
| 203 |
+
# REFERENCES
|
| 204 |
+
|
| 205 |
+
UCI datasets. https://github.com/yaringal/DropoutUncertaintyExps/tree/ master/UCI_Datasets.
|
| 206 |
+
|
| 207 |
+
Arsenii Ashukha, Alexander Lyzhov, Dmitry Molchanov, and Dmitry Vetrov. Pitfalls of in-domain uncertainty estimation and ensembling in deep learning. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ BJxI5gHKDr.
|
| 208 |
+
|
| 209 |
+
Thierry Bertin-Mahieux, Daniel PW Ellis, Brian Whitman, and Paul Lamere. The million song dataset. In Proceedings of the 12th International Society for Music Information Retrieval Conference (ISMIR 2011), 2011.
|
| 210 |
+
|
| 211 |
+
Christopher M Bishop. Pattern recognition and machine learning. springer, 2006.
|
| 212 |
+
|
| 213 |
+
Christopher JC Burges. From RankNet to LambdaRank to LambdaMART: An overview. Learning, 11(23-581):81, 2010.
|
| 214 |
+
|
| 215 |
+
Rich Caruana and Alexandru Niculescu-Mizil. An empirical comparison of supervised learning algorithms. In Proceedings of the 23rd international conference on Machine learning, pp. 161– 168. ACM, 2006.
|
| 216 |
+
|
| 217 |
+
Hugh A Chipman, Edward I George, Robert E McCulloch, et al. Bart: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266–298, 2010.
|
| 218 |
+
|
| 219 |
+
John W Coulston, Christine E Blinn, Valerie A Thomas, and Randolph H Wynne. Approximating prediction uncertainty for random forest regression models. Photogrammetric Engineering & Remote Sensing, 82(3):189–197, 2016.
|
| 220 |
+
|
| 221 |
+
Stefan Depeweg, José Miguel Hernández-Lobato, Finale Doshi-Velez, and Steffen Udluft. Decomposition of uncertainty for active learning and reliable reinforcement learning in stochastic systems. stat, 1050:11, 2017.
|
| 222 |
+
|
| 223 |
+
Tony Duan, Anand Avati, Daisy Yi Ding, Sanjay Basu, Andrew $\mathrm { ~ Y ~ N ~ g ~ }$ , and Alejandro Schuler. Ngboost: Natural gradient boosting for probabilistic prediction. Proc. 37th International Conference on Machine Learning (ICML), 2020.
|
| 224 |
+
|
| 225 |
+
Jerome H Friedman. Greedy function approximation: a gradient boosting machine. Annals of statistics, pp. 1189–1232, 2001.
|
| 226 |
+
|
| 227 |
+
Jerome H Friedman. Stochastic gradient boosting. Computational Statistics & Data Analysis, 38(4): 367–378, 2002.
|
| 228 |
+
|
| 229 |
+
Yarin Gal. Uncertainty in Deep Learning. PhD thesis, University of Cambridge, 2016.
|
| 230 |
+
|
| 231 |
+
Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. In Proc. 33rd International Conference on Machine Learning (ICML-16), 2016.
|
| 232 |
+
|
| 233 |
+
Franz Graf, Hans-Peter Kriegel, Matthias Schubert, Sebastian Pölsterl, and Alexander Cavallaro. 2d image registration in ct images using radial image descriptors. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 607���614. Springer, 2011.
|
| 234 |
+
|
| 235 |
+
Dan Hendrycks and Kevin Gimpel. A Baseline for Detecting Misclassified and Out-ofDistribution Examples in Neural Networks. http://arxiv.org/abs/1610.02136, 2016. arXiv:1610.02136.
|
| 236 |
+
|
| 237 |
+
Kaggle. "Don’t Get Kicked!". https://www.kaggle.com/c/DontGetKicked, 2011.
|
| 238 |
+
|
| 239 |
+
Kaggle. Amazon dataset. https://www.kaggle.com/bittlingmayer/ amazonreviews, 2017.
|
| 240 |
+
KDD. Kdd cup 2009: Customer relationship prediction. https://www.kdd.org/kdd-cup/ view/kdd-cup-2009/Data, 2009.
|
| 241 |
+
KDD. Kdd cup 2012 (track 2): Predict the click-through rate of ads given the query and user information. https://www.kdd.org/kdd-cup/view/kdd-cup-2012-track-2, 2012.
|
| 242 |
+
Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7482–7491, 2018.
|
| 243 |
+
Andreas Kirsch, Joost van Amersfoort, and Yarin Gal. BatchBALD: Efficient and diverse batch acquisition for deep bayesian active learning. In Advances in Neural Information Processing Systems 32 (NeurIPS 2019), 2019.
|
| 244 |
+
Ron Kohavi. Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, 1996.
|
| 245 |
+
B. Lakshminarayanan, A. Pritzel, and C. Blundell. Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles. In Proc. Conference on Neural Information Processing Systems (NIPS), 2017.
|
| 246 |
+
Antonio R Linero. A review of tree-based bayesian methods. Communications for Statistical Applications and Methods, 24(6), 2017.
|
| 247 |
+
Wesley Maddox, Timur Garipov, Pavel Izmailov, Dmitry Vetrov, and Andrew Gordon Wilson. A simple baseline for bayesian uncertainty in deep learning. arXiv preprint arXiv:1902.02476, 2019.
|
| 248 |
+
Andrey Malinin. Uncertainty Estimation in Deep Learning with application to Spoken Language Assessment. PhD thesis, University of Cambridge, 2019.
|
| 249 |
+
Andrey Malinin and Mark JF Gales. Reverse kl-divergence training of prior networks: Improved uncertainty and adversarial robustness. 2019.
|
| 250 |
+
Andrey Malinin, Bruno Mlodozeniec, and Mark JF Gales. Ensemble distribution distillation. In International Conference on Learning Representations, 2020. URL https://openreview. net/forum?id $=$ BygSP6Vtvr.
|
| 251 |
+
Yaniv Ovadia, Emily Fertig, Jie Ren, Zachary Nado, D Sculley, Sebastian Nowozin, Joshua V Dillon, Balaji Lakshminarayanan, and Jasper Snoek. Can you trust your model’s uncertainty? Evaluating predictive uncertainty under dataset shift. Advances in Neural Information Processing Systems, 2019.
|
| 252 |
+
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine Learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
|
| 253 |
+
Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, and Andrey Gulin. Catboost: unbiased boosting with categorical features. In Proceedings of the 32nd International Conference on Neural Information Processing Systems (NeurIPS), pp. 6638–6648, 2018.
|
| 254 |
+
Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. CoRR, abs/1702.03849, 2017.
|
| 255 |
+
Matthew Richardson, Ewa Dominowska, and Robert Ragno. Predicting clicks: estimating the clickthrough rate for new ads. In Proceedings of the 16th international conference on World Wide Web, pp. 521–530. ACM, 2007.
|
| 256 |
+
Byron P Roe, Hai-Jun Yang, Ji Zhu, Yong Liu, Ion Stancu, and Gordon McGregor. Boosted decision trees as an alternative to artificial neural networks for particle identification. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 543(2):577–584, 2005.
|
| 257 |
+
|
| 258 |
+
Mohammad Hossein Shaker and Eyke Hüllermeier. Aleatoric and epistemic uncertainty with random forests. In International Symposium on Intelligent Data Analysis, pp. 444–456. Springer, 2020.
|
| 259 |
+
|
| 260 |
+
Lewis Smith and Yarin Gal. Understanding Measures of Uncertainty for Adversarial Example Detection. In UAI, 2018.
|
| 261 |
+
|
| 262 |
+
UCI. KDD internet usage data. https://archive.ics.uci.edu/ml/datasets/ Internet $^ +$ Usage+Data, 1997.
|
| 263 |
+
|
| 264 |
+
Aleksei Ustimenko and Liudmila Prokhorenkova. SGLB: Stochastic Gradient Langevin Boosting. arXiv e-prints, art. arXiv:2001.07248, 2020.
|
| 265 |
+
|
| 266 |
+
Qiang Wu, Christopher JC Burges, Krysta M Svore, and Jianfeng Gao. Adapting boosting for information retrieval measures. Information Retrieval, 13(3):254–270, 2010.
|
| 267 |
+
|
| 268 |
+
Yanru Zhang and Ali Haghani. A gradient boosting method to improve travel time prediction. Transportation Research Part C: Emerging Technologies, 58:308–324, 2015.
|
| 269 |
+
|
| 270 |
+
A EXPERIMENTAL SETUP
|
| 271 |
+
|
| 272 |
+
# A.1 OUR IMPLEMENTATION OF DATA UNCERTAINTY
|
| 273 |
+
|
| 274 |
+
As discussed in Section 2.2 of the main text, for regression we simultaneously predict the parameters $\mu$ and $\log \sigma$ of the Normal distribution. Similarly to NGBoost, we use the natural gradients. For our loss and parameterization, the natural gradient is:
|
| 275 |
+
|
| 276 |
+
$$
|
| 277 |
+
g ^ { ( t ) } ( \pmb { x } , y ) = \left( \mu ^ { ( t - 1 ) } - y , \frac { 1 } { 2 } - \frac { 1 } { 2 } \left( \frac { y - \mu ^ { ( t - 1 ) } } { \sigma ^ { ( t - 1 ) } } \right) ^ { 2 } \right) .
|
| 278 |
+
$$
|
| 279 |
+
|
| 280 |
+
At each step of the gradient boosting procedure, we construct one tree predicting both components of $g ^ { ( t ) }$ , similarly to the MultiRMSE regime of CatBoost.6
|
| 281 |
+
|
| 282 |
+
Recall that for classification we optimize the logistic loss.
|
| 283 |
+
|
| 284 |
+
In Table 2, we compare our implementation with NGBoots (Duan et al., 2020) and Deep Ensembles (Lakshminarayanan et al., 2017) on regression datasets. For our implementation, we consider SGB with fixed sample rate (0.5) and perform parameter tuning as described below. The best results are highlighted.
|
| 285 |
+
|
| 286 |
+
Table 2: Comparison of our implementation with existing methods
|
| 287 |
+
|
| 288 |
+
<table><tr><td rowspan="2">Dataset</td><td colspan="3">RMSE</td><td colspan="3">NLL</td></tr><tr><td>Deep. Ens.</td><td>NGBoost</td><td>CatBoost</td><td>Deep. Ens.</td><td>NGBoost</td><td>CatBoost</td></tr><tr><td>Boston</td><td>3.28 ±1.00</td><td>2.94 ± 0.53</td><td>3.06 ± 0.68</td><td>2.41 ± 0.25</td><td>2.43 ± 0.15</td><td>2.47 ± 0.20</td></tr><tr><td>Concrete</td><td>6.03 ± 0.58</td><td>5.06 ± 0.61</td><td>5.21 ± 0.53</td><td>3.06 ± 0.18</td><td>3.04 ± 0.17</td><td>3.06 ± 0.13</td></tr><tr><td>Energy</td><td>2.09 ±0.29</td><td>0.46 ± 0.06</td><td>0.57 ± 0.06</td><td>1.38 ± 0.22</td><td>0.60 ± 0.45</td><td>1.24 ± 1.28</td></tr><tr><td>Kin8nm</td><td>0.09 ± 0.00</td><td>0.16 ± 0.00</td><td>0.14 ± 0.00</td><td>-1.20 ± 0.02</td><td>-0.49 ± 0.02</td><td>- 0.63 ± 0.02</td></tr><tr><td>Naval</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ±0.00</td><td>-5.63 ± 0.05</td><td>-5.34 ± 0.04</td><td>-5.39 ± 0.04</td></tr><tr><td>Power</td><td>4.11 ± 0.17</td><td>3.79 ± 0.18</td><td>3.55 ± 0.27</td><td>2.79 ± 0.04</td><td>2.79 ± 0.11</td><td>2.72 ± 0.12</td></tr><tr><td>Protein</td><td>4.71 ± 0.06</td><td>4.33 ±0.03</td><td>3.92 ± 0.08</td><td>2.83 ± 0.02</td><td>2.81 ± 0.03</td><td>2.73 ± 0.07</td></tr><tr><td>Wine</td><td>0.64 ± 0.04</td><td>0.63 ± 0.04</td><td>0.63 ± 0.04</td><td>0.94 ± 0.12</td><td>0.91 ± 0.06</td><td>0.93 ± 0.08</td></tr><tr><td>Yacht</td><td>1.58 ± 0.48</td><td>0.50 ± 0.20</td><td>0.82 ± 0.40</td><td>1.18 ± 0.21</td><td>0.20 ± 0.26</td><td>0.41 ± 0.39</td></tr><tr><td>Year MSD</td><td>8.89 ± NA</td><td>8.94±NA</td><td>8.99±NA</td><td>3.35± NA</td><td>3.43±NA</td><td>3.43±NA</td></tr></table>
|
| 289 |
+
|
| 290 |
+
# A.2 PARAMETER TUNING
|
| 291 |
+
|
| 292 |
+
For all approaches, we use grid search to tune learning-rate in $\{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 \}$ , tree depth in $\{ 3 , 4 , 5 , \bar { 6 } \bar \}$ . We fix subsample to 0.5 for SGB and to 1 for SGLB. This is done to avoid joint randomization effects of SGB sampling and SGLB noise in gradients. We also set diffusion-temperature $= N$ and model-shrink-rate $\scriptstyle { \dot { \mathbf { \theta } } } = { \frac { \mathbf { \bar { 1 } } } { 2 N } }$ for SGLB.
|
| 293 |
+
|
| 294 |
+
# A.3 DATASETS
|
| 295 |
+
|
| 296 |
+
The datasets are described in Table 3. For regression, we use standard train/validation/test splits (UCI). For classification, we split the datasets into proportion 65/15/20 in train, validation, and test sets. For more details, see our GitHub repository.7
|
| 297 |
+
|
| 298 |
+
# A.4 STATISTICAL SIGNIFICANCE
|
| 299 |
+
|
| 300 |
+
For regression, we perform cross-validation to estimate statistical significance with paired $t$ -test. In the corresponding tables, we highlight the approaches that are insignificantly different from the best one (p-value $> 0 . 0 5$ ).
|
| 301 |
+
|
| 302 |
+
Table 3: Datasets description
|
| 303 |
+
|
| 304 |
+
<table><tr><td>Dataset</td><td># Examples</td><td>#Features</td></tr><tr><td colspan="3">Classification</td></tr><tr><td>Adult (Kohavi, 1996)</td><td>48842</td><td>14</td></tr><tr><td>Amazon (Kaggle,2017)</td><td>32769</td><td>9</td></tr><tr><td>Click (KDD,2012)</td><td>399482</td><td>11</td></tr><tr><td>Internet (UCI, 1997)</td><td>10108</td><td>68</td></tr><tr><td>KDD-Appetency (KDD,2009)</td><td>50000</td><td>419</td></tr><tr><td>KDD-Churn (KDD, 2009)</td><td>50000</td><td>419</td></tr><tr><td>KDD-Upselling (KDD, 2009)</td><td>50000</td><td>419</td></tr><tr><td>Kick (Kaggle,2011)</td><td>72983</td><td>43</td></tr><tr><td colspan="3">Regression</td></tr><tr><td>Boston (UCI)</td><td>506</td><td>13</td></tr><tr><td>Concrete (UCI)</td><td>1030</td><td>8</td></tr><tr><td>Energy (UCI)</td><td>768</td><td>8</td></tr><tr><td>Kin8nm (UCI)</td><td>8192</td><td>8</td></tr><tr><td>Naval (UCI)</td><td>11934</td><td>16</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Power (UCI)</td><td>9568</td><td>4</td></tr><tr><td>Protein (UCI)</td><td>45730</td><td>9</td></tr><tr><td>Wine (UCI)</td><td>1599</td><td>11</td></tr><tr><td>Yacht (UCI)</td><td>308</td><td>6</td></tr><tr><td>Year MSD (Bertin-Mahieux et al.,2011)</td><td>515345</td><td>90</td></tr></table>
|
| 305 |
+
|
| 306 |
+
Table 4: NLL and RMSE/Error rate for regression and classification
|
| 307 |
+
|
| 308 |
+
<table><tr><td rowspan="2">Dataset</td><td colspan="2">Single SGB SGLB</td><td colspan="3">Ensemble SGB SGLB</td><td colspan="2">Single SGB SGLB</td><td colspan="3">Ensemble SGLB vSGLB</td></tr><tr><td></td><td>Classification NLL (↓)</td><td></td><td></td><td>vSGLB</td><td></td><td>SGB Classification % Error (↓)</td><td></td><td></td><td></td></tr><tr><td>Adult</td><td colspan="5">0.273 0.276</td><td colspan="5">12.8 12.7 12.8</td></tr><tr><td>Amazon</td><td>0.276 0.141</td><td>0.142</td><td>0.140</td><td>0.271 0.142</td><td>0.274 0.143</td><td>4.7</td><td>4.6</td><td>4.6</td><td>12.6 4.5</td><td>12.7 4.5</td></tr><tr><td>Click</td><td>0.393</td><td>0.392</td><td>0.392</td><td>0.391</td><td>0.392</td><td>15.6</td><td>15.7</td><td>15.6</td><td>15.6</td><td>15.6</td></tr><tr><td>Internet</td><td>0.224</td><td>0.218</td><td>0.221</td><td>0.217</td><td>0.218</td><td>9.9</td><td>10.0</td><td>9.7</td><td>10.0</td><td>10.0</td></tr><tr><td>Appetency</td><td>0.073</td><td>0.073</td><td>0.073</td><td>0.073</td><td>0.073</td><td>1.8</td><td>1.8</td><td>1.8</td><td>1.8</td><td>1.8</td></tr><tr><td>Churn</td><td>0.235</td><td>0.236</td><td>0.233</td><td>0.234</td><td>0.235</td><td>7.3</td><td>7.2</td><td>7.3</td><td>7.2</td><td>7.2</td></tr><tr><td>Upselling</td><td>0.168</td><td>0.168</td><td>0.168</td><td>0.168</td><td>0.168</td><td>5.0</td><td>4.9</td><td>5.0</td><td>5.0</td><td>4.9</td></tr><tr><td>Kick</td><td>0.287</td><td>0.286</td><td>0.286</td><td>0.285</td><td>0.286</td><td>9.5</td><td>9.6</td><td>9.5</td><td>9.4</td><td>9.6</td></tr><tr><td>Dataset</td><td></td><td colspan="3">Regression NLL (↓)</td><td></td><td colspan="5"></td></tr><tr><td>BostonH</td><td>2.47</td><td>2.52</td><td>2.46</td><td>2.50</td><td>2.50</td><td>3.06</td><td>Regression RMSE (↓) 3.12</td><td>3.04</td><td>3.10</td><td>3.27</td></tr><tr><td>Concrete</td><td>3.06</td><td>3.06</td><td>3.05</td><td>3.05</td><td>3.06</td><td>5.21</td><td>5.11</td><td>5.21</td><td>5.10</td><td>5.37</td></tr><tr><td>Energy</td><td>1.24</td><td>1.70</td><td>1.13</td><td>1.52</td><td>0.70</td><td>0.57</td><td>0.54</td><td>0.57</td><td>0.54</td><td>0.64</td></tr><tr><td>Kin8nm</td><td>-0.63</td><td>-0.65</td><td>-0.63</td><td>-0.65</td><td>-0.60</td><td>0.14</td><td>0.14</td><td>0.14</td><td>0.14</td><td>0.15</td></tr><tr><td>Naval-p</td><td>-5.39</td><td>-5.42</td><td>-5.61</td><td>-5.65</td><td>-5.39</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>Power-p</td><td>2.72</td><td>2.71</td><td>2.66</td><td>2.66</td><td>2.69</td><td>3.55</td><td>3.56</td><td>3.52</td><td>3.54</td><td>3.64</td></tr><tr><td>Protein</td><td>2.73</td><td>2.73</td><td>2.61</td><td>2.64</td><td>2.70</td><td>3.92</td><td>3.96</td><td>3.90</td><td>3.93</td><td>4.02</td></tr><tr><td>Wine-qu</td><td>0.93</td><td>0.99</td><td>0.92</td><td>0.98</td><td>0.96</td><td>0.63</td><td>0.65</td><td>0.63</td><td>0.65</td><td>0.66</td></tr><tr><td>Yacht</td><td>0.41</td><td>0.38</td><td>0.27</td><td>0.32</td><td>0.51</td><td>0.82</td><td>0.84</td><td>0.83</td><td>0.84</td><td>0.97</td></tr><tr><td>Year</td><td>3.43</td><td>3.43</td><td>3.41</td><td>3.40</td><td>3.42</td><td>8.99</td><td>8.96</td><td>8.97</td><td>8.94</td><td>8.98</td></tr></table>
|
| 309 |
+
|
| 310 |
+
For classification (and Year MSD), we measure statistical significance for NLL and error/RMSE on the test set. In the corresponding tables, the approaches that are insignificantly different from the best one are highlighted. For PRR and AUC-ROC (for classification and Year MSD), we highlight the best value.
|
| 311 |
+
|
| 312 |
+
# B ADDITIONAL EXPERIMENTAL RESULTS
|
| 313 |
+
|
| 314 |
+
In Table 4, we compare ensemble approaches with single models in terms of NLL and error rate for classification and in terms of NLL and RMSE for regression tasks. Results for NLL demonstrate an advantage of ensembling approaches compared to single models. However, in some cases the difference is not significant, which can be explained by the additive nature of boosting: averaging several tree ensembles gives another (larger) tree ensemble. Thus, improved NLL can result from the increased complexity of ensemble models. We can make a similar conclusion from the results for RMSE and error rate.
|
| 315 |
+
|
| 316 |
+
# C COMPARISON WITH RANDOM FOREST
|
| 317 |
+
|
| 318 |
+
Our paper specifically focuses on uncertainty estimation in Gradient Boosted Decision Trees (GBDT) models. However, some related work was done for quantifying uncertainty in random forests (Coulston et al., 2016; Shaker & Hüllermeier, 2020), which are also ensembles of decision trees. Thus, for completeness, we also analyze how ensemble approaches perform in combination with random forests.
|
| 319 |
+
|
| 320 |
+
In these experiments, we use the scikit-learn implementation of random forests (Pedregosa et al., 2011). We limit the maximum depth to 10 and keep all other parameters default. For categorical features, we use leave-one-out encoding.
|
| 321 |
+
|
| 322 |
+
Unlike GBDT, where trees are added to correct the previous model’s mistakes, random forests (RF) consist of decision trees that are independently trained on bootstrapped sub-samples of the dataset. Hence, for knowledge uncertainty we can divide RF into several independent parts, each consisting of several trees. Drawing a parallel to virtual SGLB, we call this approach vRF (virtual RF) since it allows estimating knowledge uncertainty using only one trained random forest model. In our experiments with vRF, we divide one RF model into 10 independent parts, each consisting of 100 trees. Similarly, one can also construct an ensemble of several independently trained random forest models, which is expected to be a stronger baseline. However, we expect a small difference between vRF and an ensemble of random forests, as there are, a priori, no correlations between trees both in a single model and across multiple RF models.
|
| 323 |
+
|
| 324 |
+
In Tables 5 and 6, we compare the predictive performance of random forests (both individual models and explicit ensembles of multiple models) to SGLB individual and ensemble models on classification and regression tasks. The results show that generally GBDT models outperform random forest models in terms of classification error rate and NLL. Note that we cannot calculate NLL for RF regression models as they are not naturally probabilistic (do not yield a predicted variance). As a result, they are unable to estimate data uncertainty, and therefore we can only obtain estimates of knowledge uncertainty.
|
| 325 |
+
|
| 326 |
+
Table 7 compares SGLB and RF ensembles in terms of error detection (PRR) and out-of-domain input detection (ROC-AUC). One can see that SGLB usually outperforms RF, especially for OOD detection. Notably, as we expected, for OOD detection vRF and RF give similar results. Thus, we conclude that for random forests, a virtual ensemble is a good and cheap alternative to the true one.
|
| 327 |
+
|
| 328 |
+
Table 5: Comparison with random forest: NLL and error rate for classification
|
| 329 |
+
|
| 330 |
+
<table><tr><td rowspan="3">Dataset</td><td colspan="2">Single</td><td colspan="2">Ensemble</td><td colspan="2">Single</td><td colspan="2">Ensemble</td></tr><tr><td>SGLB</td><td>RF</td><td>SGLB</td><td>RF</td><td>SGLB</td><td>RF</td><td>SGLB</td><td>RF</td></tr><tr><td colspan="4">Classification NLL (↓)</td><td colspan="4">Classification %Error (）</td></tr><tr><td>Adult</td><td>0.273</td><td>0.300</td><td>0.271</td><td>0.300</td><td>12.7</td><td>13.9</td><td>12.6</td><td>13.9</td></tr><tr><td>Amazon</td><td>0.142</td><td>0.183</td><td>0.142</td><td>0.183</td><td>4.6</td><td>5.6</td><td>4.5</td><td>5.6</td></tr><tr><td>Click</td><td>0.392</td><td>0.411</td><td>0.391</td><td>0.411</td><td>15.7</td><td>16.0</td><td>15.6</td><td>16.0</td></tr><tr><td>Internet</td><td>0.218</td><td>0.275</td><td>0.217</td><td>0.274</td><td>10.0</td><td>11.2</td><td>10.0</td><td>11.0</td></tr><tr><td>KDD-Appetency</td><td>0.073</td><td>0.083</td><td>0.073</td><td>0.083</td><td>1.8</td><td>1.8</td><td>1.8</td><td>1.8</td></tr><tr><td>KDD-Churn</td><td>0.236</td><td>0.249</td><td>0.234</td><td>0.249</td><td>7.2</td><td>7.3</td><td>7.2</td><td>7.3</td></tr><tr><td>KDD-Upselling</td><td>0.168</td><td>0.202</td><td>0.168</td><td>0.202</td><td>4.9</td><td>7.4</td><td>5.0</td><td>7.4</td></tr><tr><td>Kick</td><td>0.286</td><td>0.311</td><td>0.285</td><td>0.311</td><td>9.6</td><td>10.4</td><td>9.4</td><td>10.4</td></tr></table>
|
| 331 |
+
|
| 332 |
+
Table 6: Comparison with random forest: RMSE for regression
|
| 333 |
+
|
| 334 |
+
<table><tr><td rowspan="2">Dataset</td><td colspan="2">Single</td><td colspan="2">Ensemble</td></tr><tr><td>SGLB</td><td>RF</td><td>SGLB</td><td>RF</td></tr><tr><td>BostonH</td><td>3.12</td><td>2.98</td><td>3.10</td><td>2.98</td></tr><tr><td>Concrete</td><td>5.11</td><td>4.96</td><td>5.10</td><td>4.95</td></tr><tr><td>Energy</td><td>0.54</td><td>0.50</td><td>0.54</td><td>0.50</td></tr><tr><td>Kin8nm</td><td>0.14</td><td>0.15</td><td>0.14</td><td>0.15</td></tr><tr><td>Naval-p</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>Power-p</td><td>3.56</td><td>3.53</td><td>3.54</td><td>3.53</td></tr><tr><td>Protein</td><td>3.96</td><td>4.19</td><td>3.93</td><td>4.19</td></tr><tr><td>Wine-qu</td><td>0.65</td><td>0.58</td><td>0.65</td><td>0.58</td></tr><tr><td>Yacht</td><td>0.84</td><td>0.84</td><td>0.84</td><td>0.84</td></tr><tr><td>Year</td><td>8.96</td><td>9.43</td><td>8.94</td><td>9.43</td></tr></table>
|
| 335 |
+
|
| 336 |
+
Table 7: Comparison with random forest: detection of errors and OOD examples for regression and classification (virtual and true ensembles)
|
| 337 |
+
|
| 338 |
+
<table><tr><td colspan="2">Dataset</td><td>vSGLB</td><td>vRF</td><td>SGLB</td><td>RF</td><td>vSGLB</td><td>vRF</td><td>SGLB</td><td>RF</td></tr><tr><td colspan="2"></td><td></td><td colspan="3">Classification % PRR (↑)</td><td></td><td>Classfcation % AUC-ROC (1)</td><td></td><td></td></tr><tr><td rowspan="2">Adult</td><td>TU KU</td><td>72</td><td>70</td><td>72</td><td>70</td><td>51</td><td>58</td><td>51</td><td>58</td></tr><tr><td></td><td>38</td><td>20</td><td>49</td><td>22</td><td>85</td><td>86</td><td>89</td><td>87</td></tr><tr><td rowspan="2">Amazon</td><td>TU</td><td>68</td><td>63</td><td>68</td><td>64</td><td>86</td><td>48</td><td>86</td><td>49</td></tr><tr><td>KU</td><td>40</td><td>45</td><td>61</td><td>52</td><td>67</td><td>55</td><td>74</td><td>53</td></tr><tr><td rowspan="2">Click</td><td>TU</td><td>44</td><td>36</td><td>44</td><td>37</td><td>68</td><td>71</td><td>64</td><td>71</td></tr><tr><td>KU</td><td>11</td><td>20</td><td>22</td><td>21</td><td>90</td><td>81</td><td>92</td><td>80</td></tr><tr><td rowspan="2">Internet</td><td>TU</td><td>79</td><td>69</td><td>79</td><td>68</td><td>68</td><td>72</td><td>69</td><td>72</td></tr><tr><td>KU</td><td>61</td><td>38</td><td>72</td><td>36</td><td>81</td><td>79</td><td>89</td><td>79</td></tr><tr><td rowspan="2"> KDD-Appetency</td><td>TU</td><td>69</td><td>56</td><td>69</td><td>56</td><td>52</td><td>82</td><td>50</td><td>81</td></tr><tr><td>KU</td><td>14</td><td>34</td><td>54</td><td>34</td><td>93</td><td>97</td><td>91</td><td>98</td></tr><tr><td rowspan="2">KDD-Churn</td><td>TU</td><td>46</td><td>39</td><td>46</td><td>39</td><td>60</td><td>75</td><td>75</td><td>78</td></tr><tr><td>KU</td><td>28</td><td>13</td><td>35</td><td>9</td><td>92</td><td>93</td><td>98</td><td>94</td></tr><tr><td>KDD-Upselling</td><td>TU KU</td><td>56 33</td><td>66</td><td>57</td><td>66</td><td>47</td><td>73</td><td>60</td><td>72</td></tr><tr><td rowspan="2">Kick</td><td>TU</td><td></td><td>45</td><td>49</td><td>44</td><td>78</td><td>90</td><td>97</td><td>92</td></tr><tr><td>KU</td><td>45 20</td><td>38 27</td><td>44</td><td>38</td><td>38</td><td>64</td><td>58</td><td>63</td></tr><tr><td>Dataset</td><td></td><td>Regression % PRR (1)</td><td></td><td>34</td><td>29</td><td>89</td><td>89</td><td>98</td><td>89</td></tr><tr><td></td><td>TU</td><td></td><td></td><td></td><td></td><td>Regression % AUC-ROC (1)</td><td></td><td></td><td></td></tr><tr><td rowspan="2">BostonH</td><td>KU</td><td>46 38</td><td>1 53</td><td>45</td><td>一</td><td>64</td><td>一</td><td>69</td><td>1</td></tr><tr><td></td><td></td><td></td><td>37</td><td>55</td><td>49</td><td>75</td><td>80</td><td>76</td></tr><tr><td rowspan="2">Concrete</td><td>TU KU</td><td>41</td><td>1</td><td>42</td><td>1</td><td>78</td><td>1</td><td>81</td><td>1</td></tr><tr><td></td><td>25</td><td>43</td><td>27</td><td>42</td><td>56</td><td>80</td><td>92</td><td>80</td></tr><tr><td rowspan="2">Energy</td><td>TU KU</td><td>62</td><td>1</td><td>56</td><td>一</td><td>69</td><td>丨</td><td>89</td><td>丨</td></tr><tr><td></td><td>54</td><td>40</td><td>31</td><td>41</td><td>32</td><td>100</td><td>100</td><td>100</td></tr><tr><td rowspan="2">Kin8nm</td><td>TU KU</td><td>58</td><td>1</td><td>59</td><td>1</td><td>42</td><td>1</td><td>43</td><td>1</td></tr><tr><td></td><td>35</td><td>33</td><td>19</td><td>33</td><td>45</td><td>48</td><td>45</td><td>48</td></tr><tr><td rowspan="2">Power-p</td><td>TU KU</td><td>32</td><td>一</td><td>33</td><td></td><td>47</td><td>一</td><td>49</td><td>一</td></tr><tr><td></td><td>13</td><td>21</td><td>9</td><td>22</td><td>57</td><td>66</td><td>73</td><td>65</td></tr><tr><td rowspan="2">Protein</td><td>TU</td><td>48</td><td>一</td><td>50</td><td></td><td>86</td><td>一</td><td>91</td><td></td></tr><tr><td>KU</td><td>12</td><td>40</td><td>29</td><td>40</td><td>94</td><td>91</td><td>99</td><td>92</td></tr><tr><td rowspan="2">Wine-qu</td><td>TU</td><td>32</td><td></td><td>32</td><td></td><td>56</td><td></td><td>56</td><td></td></tr><tr><td>KU</td><td>9</td><td>35</td><td>19</td><td>32</td><td>49</td><td>74</td><td>72</td><td>74</td></tr><tr><td rowspan="2">Yacht</td><td>TU</td><td>88</td><td>一</td><td>88</td><td>丨</td><td>55</td><td>1</td><td>58</td><td>1</td></tr><tr><td>KU</td><td>66</td><td>79</td><td>78</td><td>81</td><td>40</td><td>52</td><td>60</td><td>52</td></tr><tr><td rowspan="2">Year</td><td>TU</td><td>62</td><td>1</td><td>63</td><td>1</td><td>57</td><td>丨</td><td>60</td><td>1</td></tr><tr><td>KU</td><td>25</td><td>28</td><td>30</td><td>27</td><td>52</td><td>74</td><td>57</td><td>74</td></tr></table>
|
parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/1Jv6b0Zq3qi/1Jv6b0Zq3qi_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/5vShUEyjmm/5vShUEyjmm.md
ADDED
|
@@ -0,0 +1,389 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# DUAL CONTRADISTINCTIVE GENERATIVE AUTOENCODER
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
We present a new generative autoencoder model with dual contradistinctive losses to improve generative autoencoder that performs simultaneous inference (reconstruction) and synthesis (generation). We name our model dual contradistinctive generative autoencoder (DC-VAE) that integrates an instance-level discriminative loss (maintaining the instance-level fidelity for the reconstruction/synthesis) with a set-level adversarial loss (encouraging the set-level fidelity for the reconstruction/synthesis), both being contradistinctive. There also exists a mathematical connection between the instance-based classification and instance-level conditional distribution. DC-VAE achieves competitive results in three tasks, including image synthesis, image reconstruction, and representation learning. DC-VAE is applicable to various tasks in computer vision and machine learning.
|
| 8 |
+
|
| 9 |
+
|
| 10 |
+
(a) DC-VAE (ours) Reconstruction results. Left: $1 2 8 \times 1 2 8$ . Right: $5 1 2 \times 5 1 2$ .
|
| 11 |
+
|
| 12 |
+
|
| 13 |
+
(b) DC-VAE (ours) Sampling results. Left: $1 2 8 \times 1 2 8$ . Right: $5 1 2 \times 5 1 2$
|
| 14 |
+
Figure 1: DC-VAE Reconstruction (top) and Sampling (bottom) on LSUN Bedroom $\mathrm { Y u }$ et al. (2015) at resolution $1 2 8 \times 1 2 8$ (left) and CelebA-HQ (Karras et al., 2018) at resolution $5 1 2 \times 5 1 2$ (right).
|
| 15 |
+
|
| 16 |
+
# 1 INTRODUCTION
|
| 17 |
+
|
| 18 |
+
Tremendous progress has been made in deep learning for the development of various learning frameworks (Krizhevsky et al., 2012; He et al., 2016; Goodfellow et al., 2014; Vaswani et al., 2017). Autoencoder (AE) (LeCun, 1987; Hinton & Zemel, 1994) aims to compactly represent and faithfully reproduce the original input signal by concatenating an encoder and a decoder in an end-to-end learning framework. The goal of AE is to make the encoded representation semantically efficient and sufficient to reproduce the input signal by its decoder. Autoencoder’s generative companion, variational autoencoder (VAE) (Kingma & Welling, 2014), additionally learns a variational model for the latent variables to capture the underlying sample distribution. For the encoder and decoder models separately, tremendous progress has been made in image classification with deep convolutional neural network (CNN) (Krizhevsky et al., 2012; He et al., 2016) (an encoder) and in image generation with generative adversarial network (GAN) (Goodfellow et al., 2014) (a decoder).
|
| 19 |
+
|
| 20 |
+
The key objective for a generative autoencoder is to maintain two types of fidelities: (1) an instancelevel fidelity to make the reconstruction/synthesis faithful to the individual input data sample, and (2) a set-level fidelity to make the reconstruction/synthesis of the decoder faithful to the entire input data set. The VAE/GAN algorithm (Larsen et al., 2016) combines a reconstruction loss with an adversarial loss. However, the result of VAE/GAN is sub-optimal, as shown in Table 1.
|
| 21 |
+
|
| 22 |
+
The pixel-wise reconstruction loss in the standard VAE (Kingma & Welling, 2014) typically results in blurry images with degenerated semantics. A possible solution to resolving the above conflict lies in two aspects: (1) turning the measure in the pixel space into induced feature space that is more semantically meaningful; (2) changing the L2 distance (per-pixel) into a learned instance-level distance function for the entire image (akin to generative adversarial networks which learn set-level distance functions). Taking these two steps allows us to design an instance-level classification loss that is aligned with the adversarial loss in the GAN model enforcing set-level fidelity. Motivated by the above observations, we develop a new generative autoencoder model with dual contradistinctive losses by adopting a discriminative loss performing instance-level classification (enforcing the instance-level fidelity), which is rooted in metric learning (Kulis et al., 2012) and contrastive learning (Hadsell et al., 2006; Wu et al., 2018; van den Oord et al., 2018). Combined with the adversarial losses for the set-level fidelity, both terms are formulated in the induced feature space performing contradistinction: (1) the instance-level contrastive loss considers each input instance (image) itself as a class, and (2) the set-level adversarial loss treats the entire input set as a positive class. We name our method dual contradistinctive generative autoencoder (DC-VAE) and make the following contributions.
|
| 23 |
+
|
| 24 |
+
• We develop a new algorithm, dual contradistinctive generative autoencoder (DC-VAE), by combining instance-level and set-level classification losses in the VAE framework, and systematically show the significance of these two loss terms in DC-VAE.
|
| 25 |
+
• The effectiveness of DC-VAE is illustrated in three tasks altogether, including image synthesis, image reconstruction, and representation learning.
|
| 26 |
+
|
| 27 |
+
# 2 RELATED WORK
|
| 28 |
+
|
| 29 |
+
Related work can be roughly divided into three categories: (1) generative autoencoder, (2) deep generative model, and (3) contrastive learning.
|
| 30 |
+
|
| 31 |
+
Variational autoencoder (VAE) (Kingma & Welling, 2014) points to an exciting direction of generative models by developing an Evidence Lower BOund (ELBO) objective (Higgins et al., 2017; Ding et al., 2020). However, the VAE reconstruction/synthesis is known to be blurry. To improve the image quality, a sequence of VAE based models have been developed (Larsen et al., 2016; Dumoulin et al., 2017; Huang et al., 2018; Brock et al., 2018; Zhang et al., 2019). VAE/GAN (Larsen et al., 2016) adopts an adversarial loss to improve the quality of the image, but its output for both reconstruction and synthesis (new samples) is still unsatisfactory. IntroVAE Huang et al. (2018) adds a loop from the output back to the input and is able to attain image quality that is on par with some modern GANs in some aspects. However, its full illustration for both reconstruction and synthesis is unclear. PGA (Zhang et al., 2019) adds a constraint to the latent variables.
|
| 32 |
+
|
| 33 |
+
Pioneering works of (Tu, 2007; Gutmann & Hyvärinen, 2012) alleviate the difficulty of learning densities by approximating likelihoods via classification (real (positive) samples vs. fake (pseudonegative or adversarial) samples). Generative adversarial network (GAN) (Goodfellow et al., 2014) builds on neural networks and amortized sampling (a decoder network that maps a noise into an image). The subsequent development after GAN (Radford et al., 2016; Arjovsky et al., 2017; Gulrajani et al., 2017; Karras et al., 2018; Gong et al., 2019; Dumoulin et al., 2017; Donahue et al., 2017) has led to a great leap forward in building decoder-based generative models. It has been widely observed that the adversarial loss in GANs contributes significantly to the improved quality of image synthesis. Energy-based generative models (Salakhutdinov & Hinton, 2009; Xie et al., 2016; Jin et al., 2017; Lee et al., 2018) — which aim to directly model data density — are making a steady improvement for a simultaneously generative and discriminative single model.
|
| 34 |
+
|
| 35 |
+
From another angle, contrastive learning (Hadsell et al., 2006; Wu et al., 2018; He et al., 2020; Chen et al., 2020) has lately shown its particular advantage in unsupervised training of CNN features. It overcomes the limitation in unsupervised learning where class label is missing by turning each image instance into one class. Thus, the softmax function in the standard discriminative classification training can be applied. Contrastive learning can be connected to metric learning (Bromley et al., 1993; Chopra et al., 2005; Chechik et al., 2010).
|
| 36 |
+
|
| 37 |
+
In this paper, we aim to improve VAE (Kingma & Welling, 2014) by introducing a contrastive loss (van den Oord et al., 2018) to address instance-level fidelity between the input and the reconstruction in the induced feature space. Unlike in self-supervised representation learning methods (van den Oord et al., 2018; He et al., 2020; Chen et al., 2020), where self-supervision requires generating a transformed input (via data augmentation operations), the reconstruction naturally fits into the contrastive term that encourages the matching between the reconstruction and the input image instance, while pushing the reconstruction away from the rest of the images in the entire training set. Thus, the instance-level and set-level contradistinctive terms collaborate with each to encourage the high fidelity of the reconstruction and synthesis. In Figure 3, we systematically show the significance of with and without the instance-level and the set-level contradistinctive terms. In addition, we explore multi-scale contrastive learning via two schemes in Section 4.1: 1) deep supervision for contrastive learning in different convolution layers, and 2) patch-based contrastive learning for fine-grained data fidelity. In the experiments, we show competitive results for the proposed dual contradistinctive generative autoencoder (DC-VAE) in a number of benchmarks for three tasks, including image synthesis, image reconstruction, and representation learning.
|
| 38 |
+
|
| 39 |
+
# 3 PRELIMINARIES: VAE AND VAE/GAN
|
| 40 |
+
|
| 41 |
+
Variational autoencoder (VAE) Assume a given training set $S = \{ \mathbf { x } _ { i } \} _ { i = 1 } ^ { n }$ where each $\mathbf { x } _ { i } \in \mathbb { R } ^ { m }$ We suppose that each $\mathbf { x } _ { i }$ is sampled from a generative process $p ( \mathbf { x } | \mathbf { z } )$ . In the literature, vector $\mathbf { z }$ refers to latent variables. In practice, latent variables $\mathbf { z }$ and the generative process $p ( \mathbf { x } | \mathbf { z } )$ are unknown. The objectives of a variational autoencoder (VAE) (Kingma & Welling, 2014) is to simultaneously train an inference network $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ and a generator network $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ . In VAE (Kingma $\&$ Welling, 2014), the inference network is a neural network that outputs parameters for Gaussian distribution $q _ { \phi } ( \mathbf { z } | \mathbf { x } ) = \mathcal { N } ( \mu _ { \phi } ( \mathbf { x } ) , \Sigma _ { \phi } ( \mathbf { x } ) )$ . The generator is a deterministic neural network $f _ { \pmb \theta } ( \mathbf z )$ parameterized by $\pmb { \theta }$ . Generative density $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ is assumed to be subject to a Gaussian distribution: $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) = \dot { \mathcal { N } } ( f _ { \pmb { \theta } } ( \mathbf { z } ) , \sigma ^ { 2 } I )$ . These models can be trained by minimizing the negative of evidence lower bound (ELBO) in Eq. (1) below.
|
| 42 |
+
|
| 43 |
+
$$
|
| 44 |
+
\begin{array} { r } { \mathcal { L } _ { \mathrm { E L B O } } ( \pmb { \theta } , \phi ; \mathbf { x } ) = - \mathbb { E } _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \mathrm { l o g } ( p _ { \theta } ( \mathbf { x } | \mathbf { z } ) ) ] + K L [ q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) ] } \end{array}
|
| 45 |
+
$$
|
| 46 |
+
|
| 47 |
+
where $p ( \mathbf { z } )$ is the prior, which is assumed to be $\mathcal { N } ( 0 , I )$ . The first term $- \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log ( p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) ) ]$ reduces to standard pixel-wise reconstruction loss $\mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ | | \mathbf { x } - f _ { \theta } ( \mathbf { z } ) | | _ { 2 } ^ { 2 } ]$ (up to a constant) due to the Gaussian assumption. The second term is the regularization term, which prevents the conditional $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ from deviating from the Gaussian prior $\mathcal { N } ( 0 , I )$ . The inference network and generator network are jointly optimized over training samples by:
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
\operatorname* { m i n } _ { \theta , \phi } \operatorname* { \mathbb { E } } _ { \mathbf { x } \sim p _ { \mathrm { d a t a } } ( \mathbf { x } ) } \mathcal { L } _ { \mathrm { E L B O } } ( \theta , \phi ; \mathbf { x } ) .
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
where $p _ { \mathrm { d a t a } }$ is the distribution induced by the training set $S$ .
|
| 54 |
+
|
| 55 |
+
VAE has an elegant formulation. However, it relies on a pixel-wise reconstruction loss, which is known not ideal to be reflective of perceptual realism (Johnson et al., 2016; Isola et al., 2017), often resulting in blurry images. From another viewpoint, it can be thought of as using a kernel density estimator (with an isotropic Gaussian kernel) in the pixel space. Although allowing efficient training and inference, such a non-parametric approach is overly simplistic for modeling the semantics and perception of natural images.
|
| 56 |
+
|
| 57 |
+
VAE/GAN Generative adversarial networks (GANs) (Goodfellow et al., 2014) and its variants (Radford et al., 2016), on the other hand, are shown to be producing highly realistic images. The success was largely attributed to learning a fidelity function (often referred to as a discriminator) that measures how realistic the generated images are. This can be achieved by learning to contrast (classify) the set of training images with the set of generated images (Tu, 2007; Gutmann & Hyvärinen, 2012; Goodfellow et al., 2014).
|
| 58 |
+
|
| 59 |
+
VAE/GAN (Larsen et al., 2016) augments the ELBO objective (Eq. (2)) with the GAN objective. Specifically, the objective of VAE/GAN consists of two terms, namely the modified ELBO (Eq. (3))
|
| 60 |
+
|
| 61 |
+
and the GAN objective. To make the notations later consistent, we now define the set of given training images as ${ \cal { S } } = \{ { \bf { x } } _ { i } \} _ { i = 1 } ^ { n }$ in which a total number of $n$ unlabeled training images are present. For each input image $\mathbf { x } _ { i }$ , the modified ELBO computes the reconstruction loss in the feature space of the discriminator instead of the pixel space:
|
| 62 |
+
|
| 63 |
+
$$
|
| 64 |
+
\mathcal { L } _ { \mathrm { E L B O } } ( \theta , \phi , D ; \mathbf { x } _ { i } ) = \mathbb { E } _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } [ | | F _ { D } ( \mathbf { x } _ { i } ) - F _ { D } ( f _ { \theta } ( \mathbf { z } ) ) | | _ { 2 } ^ { 2 } ] + K L [ q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) | | p ( \mathbf { z } ) ]
|
| 65 |
+
$$
|
| 66 |
+
|
| 67 |
+
where $F _ { D } ( \cdot )$ denotes the feature embedding from the discriminator $D$ . Feature reconstruction loss (also referred to as perceptual loss), similar to that in style transfer (Johnson et al., 2016). The modified GAN objective considers both reconstructed images (latent code from $q _ { \phi } ( \mathbf { z } | \mathbf { x } ) )$ and sampled images (latent code from the prior $p ( \mathbf { z } ) .$ ) as its fake samples:
|
| 68 |
+
|
| 69 |
+
$$
|
| 70 |
+
\begin{array} { r } { \mathcal { L } _ { \mathrm { G A N } } ( \pmb { \theta } , \phi , D ; \mathbf { x } _ { i } ) = \log D ( \mathbf { x } _ { i } ) + \mathbb { E } _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } \log ( 1 - D ( f _ { \theta } ( \mathbf { z } ) ) + \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } \log ( 1 - D ( f _ { \theta } ( \mathbf { z } ) ) . } \end{array}
|
| 71 |
+
$$
|
| 72 |
+
|
| 73 |
+
The VAE/GAN objective becomes:
|
| 74 |
+
|
| 75 |
+
$$
|
| 76 |
+
\operatorname* { m i n } _ { \theta , \phi } \operatorname* { m a x } _ { D } \sum _ { i = 1 } ^ { n } \left[ \mathcal { L } _ { \mathrm { E L B O } } ( \theta , \phi , D ; \mathbf { x } _ { i } ) + \mathcal { L } _ { \mathrm { G A N } } ( \theta , \phi , D ; \mathbf { x } _ { i } ) \right] .
|
| 77 |
+
$$
|
| 78 |
+
|
| 79 |
+
# 4 DUAL CONTRADISTINCTIVE GENERATIVE AUTOENCODER (DC-VAE)
|
| 80 |
+
|
| 81 |
+
Here we want to address a question: Is the degeneration of the synthesized images by VAE always the case once the decoder is joined with an encoder? Can the problem be remedied by using a more informative loss?
|
| 82 |
+
|
| 83 |
+
Although improving the image qualities of VAE by integrating a set-level contrastive loss (GAN objective of Eq. (4)), VAE/GAN still does not accurately model instance-level fidelity. Inspired by the literature on instance-level classification (Malisiewicz et al., 2011), approximating likelihood by classification (Tu, 2007), and contrastive learning (Hadsell et al., 2006; Wu et al., 2018; He et al., 2020), we propose to model instance-level fidelity by contrastive loss (commonly referred to as InfoNCE loss) (van den Oord et al., 2018). In DC-VAE, we perform the following minimization and loosely call each term a loss.
|
| 84 |
+
|
| 85 |
+
$$
|
| 86 |
+
\mathcal { L } _ { \mathrm { i n s t a n c e } } ( \pmb { \theta } , \phi , D ; i , \{ \mathbf { x } _ { j } \} _ { j = 1 } ^ { n } ) \triangleq - \mathbb { E } _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } \left[ \log \frac { e ^ { h ( \mathbf { x } _ { i } , f _ { \pmb { \theta } } ( \mathbf { z } ) ) } } { \sum _ { j = 1 } ^ { n } e ^ { h ( \mathbf { x } _ { j } , f _ { \pmb { \theta } } ( \mathbf { z } ) ) } } \right] ,
|
| 87 |
+
$$
|
| 88 |
+
|
| 89 |
+
where $i$ is an index for a training sample (instance), $\{ { \bf x } _ { j } \} _ { j = 1 } ^ { n }$ is the union of positive samples and negative samples, $h ( \mathbf x , \mathbf y )$ is the critic function that measures compatibility between $\mathbf { x }$ and $\mathbf { y }$ . Following the popular choice from (He et al., 2020), $h ( \mathbf x , \mathbf y )$ is the cosine similarity between the embeddings of $\mathbf { x }$ and y: $\begin{array} { r } { h ( \mathbf { x } , \mathbf { y } ) = \frac { F _ { D } ( \mathbf { x } ) ^ { \top } F _ { D } ( \mathbf { y } ) } { | | F _ { D } ( \mathbf { x } ) | | _ { 2 } | | F _ { D } ( \mathbf { y } ) | | _ { 2 } } } \end{array}$ . Note that unlike in contrastive self-supervised learning methods (van den Oord et al., 2018; He et al., 2020; Chen et al., 2020) where two views (independent augmentations) of an instance constitutes a positive pair, an input instance $\mathbf { x } _ { i }$ and its reconstruction $f _ { \pmb \theta } ( \mathbf z )$ comprises a positive pair in DC-VAE. Likewise, the reconstruction $f _ { \pmb \theta } ( \mathbf z )$ and any instance that is not $\mathbf { x } _ { i }$ can be a negative pair.
|
| 90 |
+
|
| 91 |
+
To bridge the gap between the instance-level contrastive loss (Eq. (6)) and log-likelihood in ELBO term (Eq. (1)), we show the following observation.
|
| 92 |
+
|
| 93 |
+
Theorem 1 (From (Ma & Collins, 2018; Poole et al., 2019)) The following objective is minimized, i.e., the optimal critic $h$ is achieved, when $h ( f _ { \pmb \theta } ( \mathbf { z } ) , \mathbf { x } ) = \log { p ( \mathbf { x } | \mathbf { z } ) } + \overset { \cdot } { c } ( \mathbf { x } )$ where $c ( \mathbf { x } )$ is any function that does not depend on $\mathbf { z }$ .
|
| 94 |
+
|
| 95 |
+
$$
|
| 96 |
+
I _ { N C E } \triangleq \mathbb { E } _ { { \mathbf { x } } _ { 1 } , \cdots { \mathbf { x } } _ { K } } \mathbb { E } _ { i } \big [ \mathcal { L } _ { i n s t a n c e } ( \pmb { \theta } , \phi , D ; i , \{ { \mathbf { x } } _ { j } \} _ { j = 1 } ^ { n } ) \big ] .
|
| 97 |
+
$$
|
| 98 |
+
|
| 99 |
+
From Theorem 1, we see that the contrastive loss of Eq. (6) implicitly estimates the log-likelihood $\log p _ { \theta } ( \mathbf { x } | \mathbf { z } )$ required for the evidence lower bound (ELBO). Hence, we modify the ELBO objective of Eq. (1) as follows and name it as implicit ELBO (IELBO):
|
| 100 |
+
|
| 101 |
+
$$
|
| 102 |
+
\mathcal { L } _ { \mathrm { I E L B O } } ( \pmb { \theta } , \phi , D ; \mathbf { x } _ { i } ) = \mathcal { L } _ { \mathrm { i n s t a n c e } } ( \pmb { \theta } , \phi , D ; i , \{ \mathbf { x } _ { j } \} _ { j = 1 } ^ { n } ) + K L [ q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) | | p ( \mathbf { z } ) ] .
|
| 103 |
+
$$
|
| 104 |
+
|
| 105 |
+
|
| 106 |
+
Figure 2: Model architecture for the proposed DC-VAE algorithm.
|
| 107 |
+
|
| 108 |
+
Finally, the combined objective for the proposed DC-VAE algorithm becomes:
|
| 109 |
+
|
| 110 |
+
$$
|
| 111 |
+
\operatorname* { m i n } _ { \theta , \phi } \operatorname* { m a x } _ { D } \sum _ { i = 1 } ^ { n } \left[ \mathcal { L } _ { \mathrm { I E L B O } } ( \theta , \phi , D ; \mathbf { x } _ { i } ) + \mathcal { L } _ { \mathrm { G A N } } ( \theta , \phi , D ; \mathbf { x } _ { i } ) \right] .
|
| 112 |
+
$$
|
| 113 |
+
|
| 114 |
+
The definition of $\mathcal { L } _ { \mathrm { G A N } }$ follows Eq. (4). Note here we also consider the term in Eq. (4) as contrasdistinctive since it tries to minimize the difference/discriminative classification between the input (“real”) image set and the reconstructed/generated (“fake”) image set. Below we highlight the significance of the two contradistinctive terms. Figure 2 shows the model architecture.
|
| 115 |
+
|
| 116 |
+
• Instance-level fidelity. The first item in Eq. (8) is an instance-level fidelity term encouraging the reconstruction to be as close as possible to the input image while being different from all the rest of the images. A key advantage of the contrastive loss in Eq. (8) over the standard reconstruction loss in Eq. (3) is its relaxed and background instances aware formulation. In general, the reconstruction in Eq. (3) wants a perfect match between the reconstruction and the input, whereas the contrastive loss in Eq. (8) requests for being the most similar one among the training samples. This way, the contrastive loss becomes more cooperative with less conflict to the GAN loss, compared with the reconstruction loss. The introduction of the contrastive loss results in a significant improvement over VAE and VAE/GAN in which only matching the reconstruction, and the input instance is enforced.
|
| 117 |
+
|
| 118 |
+
• Set-level fidelity. The second item in Eq. (9) is a set-level fidelity term encouraging the entire set of synthesized images to be non distinguishable from the input image set. Having this term (Eq. (4)) is still important since the instance contrastive loss alone (Eq. (9)) will also lead to a degenerated situation: the input image and its reconstruction can be projected to the same point in the new feature space, but without a guarantee that the reconstruction itself lies on the valid “real” image manifold.
|
| 119 |
+
|
| 120 |
+
As shown in Figure 3 and Table 1 for the comparison with and without the individual terms in Eq. (9). We observe evident effectiveness of the proposed DC-VAE combining both the instance-level fidelity term (Eq. (6)) and the set-level fidelity term (Eq. (4)), compared with VAE (using pixel-wise reconstruction loss without the GAN objective), VAE-GAN (using feature reconstruction loss and the GAN objective), and VAE contrastive (using contrastive loss but without the GAN objective).
|
| 121 |
+
|
| 122 |
+
In the experiments, we show that both terms required to achieve faithful reconstruction (captured by InfoNCE loss) with perceptual realism (captured by the GAN loss).
|
| 123 |
+
|
| 124 |
+
# 4.1 MULTI SCALE CONTRASTIVE LEARNING
|
| 125 |
+
|
| 126 |
+
Inspired by (Lee et al., 2015), we utilize information from feature maps at different scales. In addition to contrasting on the last layer of $D$ in Equation 9, we add contrastive objective on $f _ { l } ( \mathbf { z } )$ where $f _ { l }$ is some function on top of an intermediate layer $l$ of D. We do it in two different ways.
|
| 127 |
+
|
| 128 |
+
1. Deep supervision: We use $1 \times 1$ convolution to reduce the dimension channel-wise, and use a linear layer to obtain $f _ { l }$ .
|
| 129 |
+
2. Local patch: We use a random location across channel at layer $l$ (size: $1 \times 1 \times \mathrm { d }$ , where d is the channel depth).
|
| 130 |
+
|
| 131 |
+
The intuition for the second is that in a convolutional neural network, one location at a feature map corresponds to a receptive area (patch) in the original image. Thus, by contrasting locations across channels in the same feature maps, we are encouraging the original image and the reconstruction to image have locally similar content, while encouraging them to have locally dissimilar content in other images. We use deep supervision for initial training, and add local patch after certain iterations.
|
| 132 |
+
|
| 133 |
+
# 5 EXPERIMENTS
|
| 134 |
+
|
| 135 |
+
|
| 136 |
+
Figure 3: Qualitative results of CIFAR-10 (Krizhevsky et al., 2009) images (resolution $3 2 \times 3 2$ ) for experiments in Table 1 (Krizhevsky et al., 2009).
|
| 137 |
+
|
| 138 |
+
# 5.1 IMPLEMENTATION
|
| 139 |
+
|
| 140 |
+
Datasets To validate our method, we train our method on several different datasets — CIFAR-10 (Krizhevsky et al., 2009), STL-10 (Coates et al., 2011), CelebA (Liu et al., 2015), CelebA-HQ (Karras et al., 2018), and LSUN bedroom (Yu et al., 2015). See the appendix for more detailed descriptions.
|
| 141 |
+
|
| 142 |
+
Network architecture For $3 2 \times 3 2$ resolution, we design the encoder and decoder subnetworks of our model in a similar way to the discriminator and generator found through neural architecture search in AutoGAN (Gong et al., 2019). For the higher resolution experiments $1 2 8 \times 1 2 8$ and $5 1 2 \times 5 1 2$ resolution), we use Progressive GAN (Karras et al., 2018) as the backbone. Network architecture diagram is available in the appendix.
|
| 143 |
+
|
| 144 |
+
Training details The number of negative samples for contrastive learning is 8096 for all datasets. The latent dimension for the VAE decoder is 128 for CIFAR-10, STL-10, and 512 for CelebA, CelebA-HQ and LSUN Bedroom. Learning rate is 0.0002 with Adam parameters of $( \beta _ { 1 } , \beta _ { 2 } ) = ( 0 . 0 , 0 . 9 )$ and a batch size of 128 for CIFAR-10 and STL-10. For CelebA, CelebA-HQ, LSUN Bedroom datasets, we use the optimizer parameters given in (Karras et al., 2018). The contrastive embedding dimension used is 16 for each of the experiments.
|
| 145 |
+
|
| 146 |
+
# 5.2 ABLATION STUDY
|
| 147 |
+
|
| 148 |
+
Table 1: Ablation studies on CIFAR-10 for the proposed DC-VAE algorithm. We follow (Johnson et al., 2016) and measure perceptual distance in an relu4_3 layer of a pretrained VGG network. $\downarrow$ means lower is better. ↑ means higher is better.
|
| 149 |
+
|
| 150 |
+
<table><tr><td>Method</td><td>FID↓/IS↑ Sampling</td><td>FID↓/IS↑ Reconstruction</td><td>Pixel↓ Distance</td><td>Perceptual↓ Distance</td></tr><tr><td>VAE</td><td>115.8 /3.8</td><td>108.4/4.3</td><td>21.8</td><td>65.8</td></tr><tr><td>VAE/GAN</td><td>39.8 /7.4</td><td>29.0 /7.6</td><td>62.7</td><td>57.2</td></tr><tr><td>VAE-Contrastive</td><td>240.4 /1.8</td><td>242 /1.9</td><td>53.6</td><td>104.2</td></tr><tr><td>DC-VAE</td><td>17.9 /8.2</td><td>21.4 / 7.9</td><td>45.9</td><td>52.9</td></tr></table>
|
| 151 |
+
|
| 152 |
+
To demonstrate the necessity of the GAN loss (Eq. 4) and contrastive loss (Eq. 8), we conduct four experiments with the same backbone. These experiments are: VAE (No GAN, no Contrastive),
|
| 153 |
+
|
| 154 |
+
VAE/GAN (with GAN, no Contrastive), VAE-Contrastive (No GAN, with Contrastive, and ours (With GAN, with Contrastive). Here, GAN denotes Eq. 4, and Contrastive denotes Eq. 8.
|
| 155 |
+
|
| 156 |
+
Qualitative analysis From Figure 3, we see that without GAN and contrastive, images are blurry; Without GAN, the contrastive head can classify images, but not on the image manifold; Without Contrastive, reconstruction images are on the image manifold because of the discriminator, but they are different from input images. These experiments show that it is necessary to combine both instance-level and set-level fidelity, and in a contradistinctive manner.
|
| 157 |
+
|
| 158 |
+
Quantitative analysis In Table 1 we observe the same trend. VAE generates blurry images; thus the FID/IS (Inception Score) is not ideal. VAE-Contrastive does not generate images on the natural manifold; thus FID/IS is poor. VAE/GAN combines set-level and instance-level information. However the L2 objective is not ideal; thus the FID/IS is sub-optimal. For both reconstruction and sampling tasks, DC-VAE generates high fidelity images and has a favorable FID and Inception score. This illustrates the advantange of having a contradistinctive objective on both set level and instance level. To measure the faithfulness of the reconstructed image we compute the pixelwise L2 distance and the perceptual distance (Johnson et al. (2016)). For the pixel distance, VAE has the lowest value because it directly optimizes this distance during training; our pixel-wise distance is better than VAE/GAN and VAE-Contrastive. For perceptual distance, our method outperforms other three, which confirms that using contrastive learning helps reconstruct images semantically.
|
| 159 |
+
|
| 160 |
+
Table 2: Comparison on CIFAR-10 and STL-10. Average Inception scores (Salimans et al., 2016) and FID scores (Heusel et al., 2017). Results derived from (Gong et al., 2019). Table style derived from (Lee et al., 2019). †Result from Aneja et al. (2020). ∗Result from (Dieng et al., 2019).
|
| 161 |
+
|
| 162 |
+
<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-10</td><td colspan="2">STL-10</td></tr><tr><td>Inception Score↑</td><td>FID↓</td><td>Inception Score↑</td><td>FID↓</td></tr><tr><td>Methods based on GAN:</td><td></td><td></td><td></td><td></td></tr><tr><td>DCGAN (Radford et al.,2016)</td><td>6.6</td><td></td><td></td><td>-</td></tr><tr><td>ProbGAN (He et al.,2019)</td><td>7.8</td><td>24.6</td><td>8.9</td><td>46.7</td></tr><tr><td>WGAN-GP ResNet (Gulrajani et al.,2017)</td><td>7.9</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RaGAN (Jolicoeur-Martineau,2018)</td><td>=</td><td>23.5</td><td>=</td><td>■</td></tr><tr><td>SN-GAN (Miyato et al., 2018)</td><td>8.2</td><td>21.7</td><td>9.1</td><td>40.1</td></tr><tr><td>MGAN (Hoang et al., 2018)</td><td>8.3</td><td>26.7</td><td>-</td><td>-</td></tr><tr><td>Progressive GAN (Karras et al.,2018)</td><td>8.8</td><td>-</td><td>■</td><td>=</td></tr><tr><td>Improving MMD GAN(Wang et al.,2019)</td><td>8.3</td><td>16.2</td><td>9.3</td><td>37.6</td></tr><tr><td>PULSGAN (Guo et al., 2020)</td><td>■</td><td>22.3</td><td>=</td><td>-</td></tr><tr><td>AutoGAN (Gong et al., 2019)</td><td>8.6</td><td>12.4</td><td>9.2</td><td>31.0</td></tr><tr><td>MethodsbasedonVAE:</td><td></td><td></td><td></td><td></td></tr><tr><td>VAE</td><td>3.8</td><td>115.8</td><td></td><td></td></tr><tr><td>VAE/GAN</td><td>7.4</td><td>39.8</td><td></td><td></td></tr><tr><td>VEEGAN* (Srivastava et al.,2017)</td><td>-</td><td>95.2</td><td></td><td></td></tr><tr><td>WAE-GAN (Tolstikhin et al., 2017)</td><td></td><td>93.1</td><td></td><td></td></tr><tr><td>NVAE†(Vahdat & Kautz,202O) Sampling</td><td></td><td>50.8</td><td></td><td></td></tr><tr><td>NVAE† (Vahdat & Kautz,2020) Reconstruction</td><td>=</td><td>2.67</td><td>=</td><td></td></tr><tr><td>DC-VAE Sampling (ours)</td><td>8.2</td><td>17.9</td><td>8.1</td><td>41.9</td></tr><tr><td>DC-VAE Reconstruction (ours)</td><td>7.9</td><td>21.4</td><td>8.4</td><td>43.6</td></tr></table>
|
| 163 |
+
|
| 164 |
+
# 5.3 COMPARISON TO EXISTING GENERATIVE MODELS
|
| 165 |
+
|
| 166 |
+
Table 2 gives a comparison of quantitative measurement for CIFAR-10 and STL-10 dataset. In general, there is a large difference in terms of FID and IS between GAN family and VAE family of models. Our model has state-of-the-art results in VAE family, and is comparable to state-of-the-art GAN models on CIFAR-10. Similarly Table 4 shows that DC-VAE is able to generate images that are comparable to GAN based methods even on higher resolution datasets.
|
| 167 |
+
|
| 168 |
+
# 5.4 LATENT SPACE REPRESENTATION: IMAGE AND STYLE INTERPOLATION
|
| 169 |
+
|
| 170 |
+
We further validate the effectiveness of DC-VAE for representation learning. One benefit of having an AE/VAE framework compared with just a decoder as in GAN Goodfellow et al. (2014) is to be able to directly obtain the latent representation from the input images. The encoder and decoder modules in VAE allows us to readily perform image/style interpolation by mixing the latent variables of different images and reconstruct/synthesize new ones. We demonstrate qualitative results on image interpolation (Fig. 5, Appendix Fig. 9), style interpolation (Appendix Fig. 10) and image editing (Fig. 6). We directly use the trained DC-VAE model without disentanglement learning Karras et al. (2019). Additional latent space analysis and the method used for interpolation and editing is provided in the Appendix. We also quantitatively compare the latent space disentanglement through the perceptual path length (PPL) (Karras et al., 2019) (Table 6). We observe that DC-VAE learns a more disentangled latent space representation than the backbone Progressive GAN (Karras et al., 2018) and StyleALAE (Pidhorskyi et al., 2020) that uses a much more capable StyleGAN (Karras et al., 2019) backbone.
|
| 171 |
+
|
| 172 |
+
|
| 173 |
+
(c) DC-VAE Reconstruction (ours, $5 1 2 \times 5 1 2$ ) Figure 4: Comparison of DC-VAE (resolution $5 1 2 \times 5 1 2$ ) with IntroVAE (Huang et al., 2018) (resolution $1 0 2 4 \times 1 0 2 4 )$ ).
|
| 174 |
+
|
| 175 |
+
Table 3: Quality of image generation (FID) comparison on LSUN Bedrooms. $^ \dag 1 2 8 \times 1 2 8$ resolution. $^ \dag 2 5 6 \times 2 5 6$ resolution. $\downarrow$ means lower is better.
|
| 176 |
+
|
| 177 |
+
<table><tr><td>Method</td><td>FID↓</td></tr><tr><td>Progressive GAN‡ (Karras et al.,2018)</td><td>8.3</td></tr><tr><td>SNGANt (Miyato et al.,2018)(from (Chen et al.,2019))</td><td>16.0</td></tr><tr><td>SSGANt(Chen et al., 2019)</td><td>13.3</td></tr><tr><td>StyleALAE(Pidhorskyi et al.,202O) Reconstruction</td><td>15.92</td></tr><tr><td>StyleALAE(Pidhorskyi et al.,2020) Sampling</td><td>17.13</td></tr><tr><td>DC-VAEt (ours) Reconstruction</td><td>10.57</td></tr><tr><td>DC-VAEt (ours) Sampling</td><td>14.3</td></tr></table>
|
| 178 |
+
|
| 179 |
+
Table 4: FID on CelebA. $^ { * } 6 4 \times 6 4$ resolution.
|
| 180 |
+
|
| 181 |
+
<table><tr><td>Method</td><td>FID↓</td></tr><tr><td>Methods based on GAN:</td><td></td></tr><tr><td>PresGAN*(Dieng et al., 2019)</td><td>29.1</td></tr><tr><td>LSGAN(Mao et al.,2017) (from (Hoshen et al.,2019))</td><td>53.9</td></tr><tr><td>COCO-GAN† (Lin et al., 2019)</td><td>5.7</td></tr><tr><td>PGANt (Karras et al.,2018)(from (Lin et al.,2019))</td><td>7.30</td></tr><tr><td>MethodsbasedonVAE:</td><td></td></tr><tr><td>VEE-GAN† (Srivastava et al.,2017) (from (Dieng et al.,2019))</td><td>46.2</td></tr><tr><td>WAE-GAN*(Tolstikhin et al.,2017)</td><td>42</td></tr><tr><td>DC-VAEt (ours) Reconstruction</td><td>14.3</td></tr><tr><td>DC-VAEt (ours) Sampling</td><td>19.9</td></tr></table>
|
| 182 |
+
|
| 183 |
+
|
| 184 |
+
Figure 5: Interpolation results generated by DC-VAE (ours) on CelebA-HQ (Karras et al., 2018) images $( 5 \bar { 1 } 2 \times 5 1 2$ , left) and LSUN Bedroom Yu et al. (2015) images $1 2 8 \times 1 2 8$ , right). More images can be seen in Appendix Figure 9. (Zoom in for a better visualization.)
|
| 185 |
+
|
| 186 |
+
# 5.5 LATENT SPACE REPRESENTATION: CLSSIFICATION
|
| 187 |
+
|
| 188 |
+
To show that our model learns a good representation, we measure the performance on the downstream MNIST classification task (Ding et al., 2020). The VAE models were trained on MNIST dataset (LeCun, 1998). We feed input images into our VAE encoder and get the latent representation. Then we train a linear classifier on the latent representation to classify the classes of the input images. Results in Table 5 show that our model gives the lowest classification error in most cases. This experiment demonstrates that our model not only gains the ability to do faithful synthesis and reconstruction, but also gains better representation ability on the VAE side.
|
| 189 |
+
|
| 190 |
+
|
| 191 |
+
Figure 6: Image editing on CelebA-HQ (Karras et al., 2018) validation set images (resolution $5 1 2 \times 5 1 2$ ). The method used to generate these is outlined in Appendix A.2.
|
| 192 |
+
|
| 193 |
+
Table 5: Comparison to prior VAE-based representation learning methods. Classification error on MNIST dataset. ↓: lower is better. $9 5 \ \%$ confidence intervals are from 5 trials. Results derived from (Ding et al., 2020).
|
| 194 |
+
Table 6: PPL Comparison of on CelebA-HQ Karras et al. (2018)
|
| 195 |
+
|
| 196 |
+
<table><tr><td>Method</td><td>dz=16↓</td><td>dz=32↓</td><td>dz=64↓</td></tr><tr><td>VAE (Kingma&Welling,2014)</td><td>2.92%±0.12</td><td>3.05%±0.42</td><td>2.98%±0.14</td></tr><tr><td>β-VAE(β=2) (Higgins et al.,2017)</td><td>4.69%±0.18</td><td>5.26%±0.22</td><td>5.40%±0.33</td></tr><tr><td>FactorVAE(y=5) (Kim& Mnih,2018)</td><td>6.07%±0.05</td><td>6.18%±0.20</td><td>6.35%±0.48</td></tr><tr><td>β-TCVAE (α=1,β=5,γ=1) (Chen et al.,2018)</td><td>1.62%±0.07</td><td>1.24%±0.05</td><td>1.32%±0.09</td></tr><tr><td>Guided-VAE (Ding et al.,2020)</td><td>1.85%±0.08</td><td>1.60%±0.08</td><td>1.49%±0.06</td></tr><tr><td>Guided-β-TCVAE (Ding et al.,2020)</td><td>1.47%±0.12</td><td>1.10%±0.03</td><td>1.31%±0.06</td></tr><tr><td>DC-VAE (Ours)</td><td>1.30% ±0.035</td><td>1.27%±0.037</td><td>1.29%±0.034</td></tr><tr><td colspan="4"></td></tr><tr><td>Method</td><td colspan="2">Backbone</td></tr><tr><td>StyleALAE (Pidhorskyi et al., 2020)</td><td colspan="2">StyleGAN</td></tr><tr><td>Progressive GAN (Karras et al., 2018)</td><td colspan="2">Progressive GAN</td></tr><tr><td>DC-VAE (ours)</td><td colspan="2">Progressive GAN</td></tr></table>
|
| 197 |
+
|
| 198 |
+
# 6 CONCLUSION
|
| 199 |
+
|
| 200 |
+
In this paper, we have proposed dual contradistinctive generative autoencoder (DC-VAE), a new framework that integrates an instance-level discriminative loss (InfoNCE) with a set-level adversarial loss (GAN) into a single variational autoencoder framework. Our experiments show competitive results for a single model in several tasks, including image synthesis, image reconstruction, representation learning for image interpolation, and representation learning for classification. DC-VAE points to a encouraging direction that attains high-quality synthesis (decoding) and inference (encoding).
|
| 201 |
+
|
| 202 |
+
# REFERENCES
|
| 203 |
+
|
| 204 |
+
Jyoti Aneja, Alexander Schwing, Jan Kautz, and Arash Vahdat. Ncp-vae: Variational autoencoders with noise contrastive priors, 2020.
|
| 205 |
+
|
| 206 |
+
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In ICML, 2017.
|
| 207 |
+
|
| 208 |
+
Andrew Brock, Jeff Donahue, and Karen Simonyan. Large scale gan training for high fidelity natural image synthesis. arXiv preprint arXiv:1809.11096, 2018.
|
| 209 |
+
|
| 210 |
+
Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Säckinger, and Roopak Shah. Signature verification using a “siamese” time delay neural network. In Neural Information Processing Systems, 1993.
|
| 211 |
+
|
| 212 |
+
Gal Chechik, Varun Sharma, Uri Shalit, and Samy Bengio. Large scale online learning of image similarity through ranking. J. Mach. Learn. Res., 11:1109–1135, March 2010. ISSN 1532-4435.
|
| 213 |
+
|
| 214 |
+
Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, 2018.
|
| 215 |
+
|
| 216 |
+
Ting Chen, Xiaohua Zhai, Marvin Ritter, Mario Lucic, and Neil Houlsby. Self-supervised gans via auxiliary rotation loss. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 12154–12163, 2019.
|
| 217 |
+
|
| 218 |
+
Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv:2002.05709, 2020.
|
| 219 |
+
|
| 220 |
+
Sumit Chopra, Raia Hadsell, and Yann LeCun. Learning a similarity metric discriminatively, with application to face verification. In CVPR, pp. 539–546, 2005. ISBN 0769523722.
|
| 221 |
+
|
| 222 |
+
Adam Coates, Andrew $\mathrm { N g }$ , and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In AISTATS, pp. 215–223, 2011.
|
| 223 |
+
|
| 224 |
+
Adji B Dieng, Francisco JR Ruiz, David M Blei, and Michalis K Titsias. Prescribed generative adversarial networks. arXiv:1910.04302, 2019.
|
| 225 |
+
|
| 226 |
+
Zheng Ding, Yifan Xu, Weijian Xu, Gaurav Parmar, Yang Yang, Max Welling, and Zhuowen Tu. Guided variational autoencoder for disentanglement learning. In CVPR, 2020.
|
| 227 |
+
|
| 228 |
+
Jeff Donahue, Philipp Krähenbühl, and Trevor Darrell. Adversarial feature learning. In ICLR, 2017.
|
| 229 |
+
|
| 230 |
+
Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially learned inference. In ICLR, 2017.
|
| 231 |
+
|
| 232 |
+
Xinyu Gong, Shiyu Chang, Yifan Jiang, and Zhangyang Wang. Autogan: Neural architecture search for generative adversarial networks. In ICCV, 2019.
|
| 233 |
+
|
| 234 |
+
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, 2014.
|
| 235 |
+
|
| 236 |
+
Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of wasserstein gans. In Advances in neural information processing systems, 2017.
|
| 237 |
+
|
| 238 |
+
T. Guo, C. Xu, J. Huang, Y. Wang, B. Shi, C. Xu, and D. Tao. On positive-unlabeled classification in gan. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8382–8390, Los Alamitos, CA, USA, jun 2020. IEEE Computer Society. doi: 10.1109/CVPR42600.2020.00841. URL https://doi.ieeecomputersociety.org/ 10.1109/CVPR42600.2020.00841.
|
| 239 |
+
|
| 240 |
+
Michael U. Gutmann and Aapo Hyvärinen. Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics. J. Mach. Learn. Res., 13(null):307–361, February 2012. ISSN 1532-4435.
|
| 241 |
+
|
| 242 |
+
Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In CVPR, 2006.
|
| 243 |
+
|
| 244 |
+
Hao He, Hao Wang, Guang-He Lee, and Yonglong Tian. Probgan: Towards probabilistic gan with theoretical guarantees. In ICLR, 2019.
|
| 245 |
+
|
| 246 |
+
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016.
|
| 247 |
+
|
| 248 |
+
Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, 2020.
|
| 249 |
+
|
| 250 |
+
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in Neural Information Processing Systems, 2017.
|
| 251 |
+
|
| 252 |
+
Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In ICLR, 2017.
|
| 253 |
+
|
| 254 |
+
Geoffrey E Hinton and Richard S Zemel. Autoencoders, minimum description length and helmholtz free energy. In Advances in neural information processing systems, 1994.
|
| 255 |
+
|
| 256 |
+
Quan Hoang, Tu Dinh Nguyen, Trung Le, and Dinh Phung. Mgan: training generative adversarial nets with multiple generators. In ICLR, 2018.
|
| 257 |
+
|
| 258 |
+
Yedid Hoshen, Ke Li, and Jitendra Malik. Non-adversarial image synthesis with generative latent nearest neighbors. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5811–5819, 2019.
|
| 259 |
+
|
| 260 |
+
Huaibo Huang, Ran He, Zhenan Sun, Tieniu Tan, et al. Introvae: Introspective variational autoencoders for photographic image synthesis. In Advances in Neural Information Processing Systems, pp. 52–63, 2018.
|
| 261 |
+
|
| 262 |
+
Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. CVPR, 2017.
|
| 263 |
+
|
| 264 |
+
Long Jin, Justin Lazarow, and Zhuowen Tu. Introspective classification with convolutional nets. In Advances in Neural Information Processing Systems, 2017.
|
| 265 |
+
|
| 266 |
+
Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In ECCV, 2016.
|
| 267 |
+
|
| 268 |
+
Alexia Jolicoeur-Martineau. The relativistic discriminator: a key element missing from standard gan. arXiv preprint arXiv:1807.00734, 2018.
|
| 269 |
+
|
| 270 |
+
Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. In ICLR, 2018.
|
| 271 |
+
|
| 272 |
+
Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4401–4410, 2019.
|
| 273 |
+
|
| 274 |
+
Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In ICML, 2018.
|
| 275 |
+
|
| 276 |
+
Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In ICLR, 2014.
|
| 277 |
+
|
| 278 |
+
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.
|
| 279 |
+
|
| 280 |
+
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, 2012.
|
| 281 |
+
|
| 282 |
+
Brian Kulis et al. Metric learning: A survey. Foundations and trends in machine learning, 5(4): 287–364, 2012.
|
| 283 |
+
|
| 284 |
+
Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, Hugo Larochelle, and Ole Winther. Autoencoding beyond pixels using a learned similarity metric. In ICML, 2016.
|
| 285 |
+
|
| 286 |
+
Yann LeCun. Modeles connexionnistes de lapprentissage. PhD thesis, PhD thesis, These de Doctorat, Universite Paris 6, 1987.
|
| 287 |
+
|
| 288 |
+
Yann LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/mnist/, 1998.
|
| 289 |
+
|
| 290 |
+
Chen-Yu Lee, Saining Xie, Patrick Gallagher, Zhengyou Zhang, and Zhuowen Tu. Deeply-supervised nets. In Artificial intelligence and statistics, pp. 562–570, 2015.
|
| 291 |
+
|
| 292 |
+
Kwonjoon Lee, Weijian Xu, Fan Fan, and Zhuowen Tu. Wasserstein introspective neural networks. In CVPR, 2018.
|
| 293 |
+
|
| 294 |
+
Kwonjoon Lee, Subhransu Maji, Avinash Ravichandran, and Stefano Soatto. Meta-learning with differentiable convex optimization. In CVPR, 2019.
|
| 295 |
+
|
| 296 |
+
Chieh Hubert Lin, Chia-Che Chang, Yu-Sheng Chen, Da-Cheng Juan, Wei Wei, and Hwann-Tzong Chen. Coco-gan: generation by parts via conditional coordinating. In Proceedings of the IEEE International Conference on Computer Vision, pp. 4512–4521, 2019.
|
| 297 |
+
|
| 298 |
+
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, 2015.
|
| 299 |
+
|
| 300 |
+
Zhuang Ma and Michael Collins. Noise contrastive estimation and negative sampling for conditional models: Consistency and statistical efficiency. In EMNLP, 2018.
|
| 301 |
+
|
| 302 |
+
Tomasz Malisiewicz, Abhinav Gupta, and Alexei A. Efros. Ensemble of exemplar-svms for object detection and beyond. In ICCV, 2011.
|
| 303 |
+
|
| 304 |
+
Xudong Mao, Qing Li, Haoran Xie, Raymond YK Lau, Zhen Wang, and Stephen Paul Smolley. Least squares generative adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2794–2802, 2017.
|
| 305 |
+
|
| 306 |
+
Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. In ICLR, 2018.
|
| 307 |
+
|
| 308 |
+
Stanislav Pidhorskyi, Donald Adjeroh, and Gianfranco Doretto. Adversarial latent autoencoders, 2020.
|
| 309 |
+
|
| 310 |
+
Ben Poole, Sherjil Ozair, Aaron Van Den Oord, Alex Alemi, and George Tucker. On variational bounds of mutual information. In ICML, 2019.
|
| 311 |
+
|
| 312 |
+
Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016.
|
| 313 |
+
|
| 314 |
+
Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In David van Dyk and Max Welling (eds.), AISTATS, volume 5, pp. 448–455, 16–18 Apr 2009.
|
| 315 |
+
|
| 316 |
+
Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, 2016.
|
| 317 |
+
|
| 318 |
+
Akash Srivastava, Lazar Valkov, Chris Russell, Michael U Gutmann, and Charles Sutton. Veegan: Reducing mode collapse in gans using implicit variational learning. In Advances in Neural Information Processing Systems, pp. 3308–3318, 2017.
|
| 319 |
+
|
| 320 |
+
Ilya Tolstikhin, Olivier Bousquet, Sylvain Gelly, and Bernhard Schoelkopf. Wasserstein auto-encoders. arXiv preprint arXiv:1711.01558, 2017.
|
| 321 |
+
|
| 322 |
+
Zhuowen Tu. Learning generative models via discriminative approaches. In CVPR, 2007.
|
| 323 |
+
Arash Vahdat and Jan Kautz. NVAE: A deep hierarchical variational autoencoder. In Neural Information Processing Systems (NeurIPS), 2020.
|
| 324 |
+
Aäron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. CoRR, abs/1807.03748, 2018. URL http://arxiv.org/abs/1807.03748.
|
| 325 |
+
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, 2017.
|
| 326 |
+
Wei Wang, Yuan Sun, and Saman Halgamuge. Improving mmd-gan training with repulsive loss function. In ICLR, 2019.
|
| 327 |
+
Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In CVPR, pp. 3733–3742, 2018.
|
| 328 |
+
Jianwen Xie, Yang Lu, Song-Chun Zhu, and Yingnian Wu. A theory of generative convnet. In ICML, 2016.
|
| 329 |
+
Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. Lsun: Construction of a largescale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015.
|
| 330 |
+
Zijun Zhang, Ruixiang Zhang, Zongpeng Li, Yoshua Bengio, and Liam Paull. Perceptual generative autoencoders. In ICLR, 2019.
|
| 331 |
+
|
| 332 |
+
A APPENDIX
|
| 333 |
+
|
| 334 |
+
# A.1 Additional reconstruction results
|
| 335 |
+
|
| 336 |
+
|
| 337 |
+
Figure 7: Additional CelebA-HQ (Karras et al., 2018) reconstruction images (resolution $5 1 2 \times 5 1 2$ ) generated by DC-VAE (ours)
|
| 338 |
+
|
| 339 |
+
|
| 340 |
+
Figure 8: Additional LSUN Bedroom (Yu et al., 2015) reconstruction images (resolution $1 2 8 \times 1 2 8$
|
| 341 |
+
|
| 342 |
+
# A.2 Analysing the latent space
|
| 343 |
+
|
| 344 |
+
In this section we analyse the smoothness of the latent space learnt by DC-VAE. In Figure 9 we qualitatively show the high resolution $( 5 1 2 \times 5 1 2 )$ CelebA-HQ Karras et al. (2018) images generated by an evenly spaced linear blending between two latent vectors. In Fig. 6 we show that DC-VAE is able to perform meaningful attribute editing on images while retaining the original identity. To perform image editing, we first need to compute the direction vector in the latent space that correspond to a desired attribute (e.g. has glasses, has blonde hair, is a woman, has facial hair). We compute these attribute direction vectors by selecting 20 images that have the attribute and 20 images that do not have the attribute, obtaining the corresponding pairs of 20 latent vectors, and calculating the difference of the mean. The results in Fig. 6 show that these direction vectors can be added to a latent vector to add a diverse combination of desired image attributes while retaining the original identity of the individual.
|
| 345 |
+
|
| 346 |
+
Additionally we corroborate the above qualitative results quantitatively by inspecting the Perceptual Path Length (PPL) Karras et al. (2019) of our learn DC-VAE Decoder (Tab. 6) to measures the disentanglement of the latent space. We note that although ProgressiveGAN (ours base model) has a better FID score, DC-VAE has a lower PPL score which indicated that the latent space learnt is more disentangled.
|
| 347 |
+
|
| 348 |
+
|
| 349 |
+
Figure 9: Additional latent space interpolations on CelebA-HQ (Karras et al., 2018) (resolution $5 1 \bar { 2 } \times 5 1 2 $ )
|
| 350 |
+
|
| 351 |
+
# Source A
|
| 352 |
+
|
| 353 |
+
|
| 354 |
+
Figure 10: Latent Mixing results on CelebA-HQ Karras et al. (2018). Each combined image in the grid is generated by replacing an arbitrary subset of Source A latent with the corresponding Source B latent.
|
| 355 |
+
|
| 356 |
+
# A.3 Effect of negative samples
|
| 357 |
+
|
| 358 |
+
In this section we analyse the effect of varying the number of negative samples used for contrastive learning. The figure 11 shows the reconstruction error on the CIFAR-10 Krizhevsky et al. (2009) test set as the negative samples is varied. We observe that a higher number of negative samples results in better reconstruction. We choose 8096 for all of our experiments because of memory constraints.
|
| 359 |
+
|
| 360 |
+
# A.4 Datasets used
|
| 361 |
+
|
| 362 |
+
CIFAR-10 comprises 50,000 training images and 10,000 test images with a spatial resolution of $3 2 \times 3 2$ . STL-10 is a similar dataset that contains 5,000 training images and 100,000 unlabeled images at $9 6 \times 9 6$ resolution. We follow the procedure in AutoGAN(Gong et al., 2019) and resize the STL-10 images to $3 2 \times 3 2$ . The CelebA dataset has 162,770 training images and 19,962 testing images, CelebA-HQ contains 29,000 training images with 1,000 test images of size $1 0 2 4 \times 1 0 2 4$ and LSUN Bedroom has approximately 3M images. We resize all images progressively in these three datasets from $\left( 4 \times 4 \right)$ to $( 5 1 2 \times 5 1 2 )$ for the progressive training.
|
| 363 |
+
|
| 364 |
+
|
| 365 |
+
Figure 11: Pixel reconstruction error on CIFAR-10 Krizhevsky et al. (2009) test set for varying number of negative samples
|
| 366 |
+
|
| 367 |
+
# A.5 Network architecture diagrams
|
| 368 |
+
|
| 369 |
+
In Figures 15 we show the detailed network architecture of DC-VAE for input resolutions of $3 2 \times 3 2$ Note that the comparison results shown in Figure 3 and Table 1 in the main paper, for VAE, VAE/GAN, VAE w/o GAN, and our proposed DC-VAE are all based on the same network architecture (shown in Figure 15 here), for a fair comparison.
|
| 370 |
+
|
| 371 |
+
The network architectures shown in Figure 15 are adapted closely from the networks discovered by (Gong et al., 2019) through Neural Architecture Search. The DC-VAE developed in our paper is not tied to any particular CNN architecture. We choose the AutoGAN architecture (Gong et al., 2019) to start with a strong baseline. The decoder in Figure 15 matches the generator in (Gong et al., 2019). The encoder is built by modifying the output shape of the final linear layer in the discriminator of AutoGAN (Gong et al., 2019) to match the latent dimension and adding spectral normalization. The discriminator is used both for classifying real/fake images, and contrastive learning. For each layer we choose, we first apply 1x1 convolution and a linear layer, and then use this feature as an input to the contrastive module. For experiments at $3 2 \times 3 2$ , we pick two different positions: the output of second residual conv block (lower level) and the output of the first linear layer (higher level). For experiments on higher resolution datasets we use a Progressive GAN (Karras et al., 2018) Generator and Discriminator as our backbone and apply similar modifications as described above.
|
| 372 |
+
|
| 373 |
+
# A.6 Training infrastructure
|
| 374 |
+
|
| 375 |
+
# .7 Further details about the representation learning experiments
|
| 376 |
+
|
| 377 |
+
As seen in Table 4 in the main paper, we show the representation capability of DC-VAE following the procedure outlined in (Ding et al., 2020). We train our model on the MNIST dataset (LeCun, 1998) and measure the transferability though a classification task on the latent embedding vector. Specifically, we first pretrain the DC-VAE model on the training split of the MNIST dataset. Following that we freeze the DC-VAE model and train a linear classifier that takes latent embedding vector as the input and predicts the class label of the original image.
|
| 378 |
+
|
| 379 |
+
|
| 380 |
+
Figure 12: Visualization of the effect of adding each instance level and set level objectives. Table 1 and Figure 3 contain FID (Heusel et al., 2017) results and qualitative comparisons on the CIFAR-10 (Krizhevsky et al., 2009) that correspond to these settings.
|
| 381 |
+
|
| 382 |
+
|
| 383 |
+
Figure 13: DC-VAE synthesis images on LSUN images (Yu et al., 2015) (resolution $1 2 8 \times 1 2 8$ )
|
| 384 |
+
|
| 385 |
+
|
| 386 |
+
Figure 14: DC-VAE reconstruction (a) and synthesis results (b) on STL10 (Coates et al., 2011) images (resolution $3 2 \times 3 2 ,$ ). In (a) the top two rows are input images and the bottom two rows are the corresponding reconstruction images.
|
| 387 |
+
|
| 388 |
+
|
| 389 |
+
Figure 15: Network architecture of DC-VAE for resolution $3 2 \times 3 2$ for CIFAR-10 (Krizhevsky et al., 2009) and STL-10 (Coates et al., 2011). (a) is the Encoder. (b) is the Decoder. (c) is the Discriminator.
|
parse/train/5vShUEyjmm/5vShUEyjmm_content_list.json
ADDED
|
@@ -0,0 +1,2026 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "DUAL CONTRADISTINCTIVE GENERATIVE AUTOENCODER ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
176,
|
| 8 |
+
99,
|
| 9 |
+
828,
|
| 10 |
+
145
|
| 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 |
+
400,
|
| 21 |
+
199
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
+
234,
|
| 32 |
+
544,
|
| 33 |
+
251
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We present a new generative autoencoder model with dual contradistinctive losses to improve generative autoencoder that performs simultaneous inference (reconstruction) and synthesis (generation). We name our model dual contradistinctive generative autoencoder (DC-VAE) that integrates an instance-level discriminative loss (maintaining the instance-level fidelity for the reconstruction/synthesis) with a set-level adversarial loss (encouraging the set-level fidelity for the reconstruction/synthesis), both being contradistinctive. There also exists a mathematical connection between the instance-based classification and instance-level conditional distribution. DC-VAE achieves competitive results in three tasks, including image synthesis, image reconstruction, and representation learning. DC-VAE is applicable to various tasks in computer vision and machine learning. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
268,
|
| 43 |
+
766,
|
| 44 |
+
421
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "image",
|
| 50 |
+
"img_path": "images/e9f7a014b8b122f06d9ea8d58a8b6a0fe1447fc00a075517d4ddd11c4d3b4d1a.jpg",
|
| 51 |
+
"image_caption": [
|
| 52 |
+
"(a) DC-VAE (ours) Reconstruction results. Left: $1 2 8 \\times 1 2 8$ . Right: $5 1 2 \\times 5 1 2$ . "
|
| 53 |
+
],
|
| 54 |
+
"image_footnote": [],
|
| 55 |
+
"bbox": [
|
| 56 |
+
181,
|
| 57 |
+
439,
|
| 58 |
+
816,
|
| 59 |
+
580
|
| 60 |
+
],
|
| 61 |
+
"page_idx": 0
|
| 62 |
+
},
|
| 63 |
+
{
|
| 64 |
+
"type": "image",
|
| 65 |
+
"img_path": "images/3e30a5806b392d8e8cdd23b95e8ca03d2ba3165503b77b50ae6c325842edc2d8.jpg",
|
| 66 |
+
"image_caption": [
|
| 67 |
+
"(b) DC-VAE (ours) Sampling results. Left: $1 2 8 \\times 1 2 8$ . Right: $5 1 2 \\times 5 1 2$ ",
|
| 68 |
+
"Figure 1: DC-VAE Reconstruction (top) and Sampling (bottom) on LSUN Bedroom $\\mathrm { Y u }$ et al. (2015) at resolution $1 2 8 \\times 1 2 8$ (left) and CelebA-HQ (Karras et al., 2018) at resolution $5 1 2 \\times 5 1 2$ (right). "
|
| 69 |
+
],
|
| 70 |
+
"image_footnote": [],
|
| 71 |
+
"bbox": [
|
| 72 |
+
181,
|
| 73 |
+
604,
|
| 74 |
+
820,
|
| 75 |
+
676
|
| 76 |
+
],
|
| 77 |
+
"page_idx": 0
|
| 78 |
+
},
|
| 79 |
+
{
|
| 80 |
+
"type": "text",
|
| 81 |
+
"text": "1 INTRODUCTION ",
|
| 82 |
+
"text_level": 1,
|
| 83 |
+
"bbox": [
|
| 84 |
+
176,
|
| 85 |
+
744,
|
| 86 |
+
336,
|
| 87 |
+
761
|
| 88 |
+
],
|
| 89 |
+
"page_idx": 0
|
| 90 |
+
},
|
| 91 |
+
{
|
| 92 |
+
"type": "text",
|
| 93 |
+
"text": "Tremendous progress has been made in deep learning for the development of various learning frameworks (Krizhevsky et al., 2012; He et al., 2016; Goodfellow et al., 2014; Vaswani et al., 2017). Autoencoder (AE) (LeCun, 1987; Hinton & Zemel, 1994) aims to compactly represent and faithfully reproduce the original input signal by concatenating an encoder and a decoder in an end-to-end learning framework. The goal of AE is to make the encoded representation semantically efficient and sufficient to reproduce the input signal by its decoder. Autoencoder’s generative companion, variational autoencoder (VAE) (Kingma & Welling, 2014), additionally learns a variational model for the latent variables to capture the underlying sample distribution. For the encoder and decoder models separately, tremendous progress has been made in image classification with deep convolutional neural network (CNN) (Krizhevsky et al., 2012; He et al., 2016) (an encoder) and in image generation with generative adversarial network (GAN) (Goodfellow et al., 2014) (a decoder). ",
|
| 94 |
+
"bbox": [
|
| 95 |
+
174,
|
| 96 |
+
770,
|
| 97 |
+
826,
|
| 98 |
+
924
|
| 99 |
+
],
|
| 100 |
+
"page_idx": 0
|
| 101 |
+
},
|
| 102 |
+
{
|
| 103 |
+
"type": "text",
|
| 104 |
+
"text": "The key objective for a generative autoencoder is to maintain two types of fidelities: (1) an instancelevel fidelity to make the reconstruction/synthesis faithful to the individual input data sample, and (2) a set-level fidelity to make the reconstruction/synthesis of the decoder faithful to the entire input data set. The VAE/GAN algorithm (Larsen et al., 2016) combines a reconstruction loss with an adversarial loss. However, the result of VAE/GAN is sub-optimal, as shown in Table 1. ",
|
| 105 |
+
"bbox": [
|
| 106 |
+
174,
|
| 107 |
+
103,
|
| 108 |
+
825,
|
| 109 |
+
172
|
| 110 |
+
],
|
| 111 |
+
"page_idx": 1
|
| 112 |
+
},
|
| 113 |
+
{
|
| 114 |
+
"type": "text",
|
| 115 |
+
"text": "The pixel-wise reconstruction loss in the standard VAE (Kingma & Welling, 2014) typically results in blurry images with degenerated semantics. A possible solution to resolving the above conflict lies in two aspects: (1) turning the measure in the pixel space into induced feature space that is more semantically meaningful; (2) changing the L2 distance (per-pixel) into a learned instance-level distance function for the entire image (akin to generative adversarial networks which learn set-level distance functions). Taking these two steps allows us to design an instance-level classification loss that is aligned with the adversarial loss in the GAN model enforcing set-level fidelity. Motivated by the above observations, we develop a new generative autoencoder model with dual contradistinctive losses by adopting a discriminative loss performing instance-level classification (enforcing the instance-level fidelity), which is rooted in metric learning (Kulis et al., 2012) and contrastive learning (Hadsell et al., 2006; Wu et al., 2018; van den Oord et al., 2018). Combined with the adversarial losses for the set-level fidelity, both terms are formulated in the induced feature space performing contradistinction: (1) the instance-level contrastive loss considers each input instance (image) itself as a class, and (2) the set-level adversarial loss treats the entire input set as a positive class. We name our method dual contradistinctive generative autoencoder (DC-VAE) and make the following contributions. ",
|
| 116 |
+
"bbox": [
|
| 117 |
+
174,
|
| 118 |
+
181,
|
| 119 |
+
825,
|
| 120 |
+
388
|
| 121 |
+
],
|
| 122 |
+
"page_idx": 1
|
| 123 |
+
},
|
| 124 |
+
{
|
| 125 |
+
"type": "text",
|
| 126 |
+
"text": "• We develop a new algorithm, dual contradistinctive generative autoencoder (DC-VAE), by combining instance-level and set-level classification losses in the VAE framework, and systematically show the significance of these two loss terms in DC-VAE. \n• The effectiveness of DC-VAE is illustrated in three tasks altogether, including image synthesis, image reconstruction, and representation learning. ",
|
| 127 |
+
"bbox": [
|
| 128 |
+
176,
|
| 129 |
+
401,
|
| 130 |
+
826,
|
| 131 |
+
474
|
| 132 |
+
],
|
| 133 |
+
"page_idx": 1
|
| 134 |
+
},
|
| 135 |
+
{
|
| 136 |
+
"type": "text",
|
| 137 |
+
"text": "2 RELATED WORK ",
|
| 138 |
+
"text_level": 1,
|
| 139 |
+
"bbox": [
|
| 140 |
+
176,
|
| 141 |
+
494,
|
| 142 |
+
341,
|
| 143 |
+
511
|
| 144 |
+
],
|
| 145 |
+
"page_idx": 1
|
| 146 |
+
},
|
| 147 |
+
{
|
| 148 |
+
"type": "text",
|
| 149 |
+
"text": "Related work can be roughly divided into three categories: (1) generative autoencoder, (2) deep generative model, and (3) contrastive learning. ",
|
| 150 |
+
"bbox": [
|
| 151 |
+
174,
|
| 152 |
+
526,
|
| 153 |
+
823,
|
| 154 |
+
555
|
| 155 |
+
],
|
| 156 |
+
"page_idx": 1
|
| 157 |
+
},
|
| 158 |
+
{
|
| 159 |
+
"type": "text",
|
| 160 |
+
"text": "Variational autoencoder (VAE) (Kingma & Welling, 2014) points to an exciting direction of generative models by developing an Evidence Lower BOund (ELBO) objective (Higgins et al., 2017; Ding et al., 2020). However, the VAE reconstruction/synthesis is known to be blurry. To improve the image quality, a sequence of VAE based models have been developed (Larsen et al., 2016; Dumoulin et al., 2017; Huang et al., 2018; Brock et al., 2018; Zhang et al., 2019). VAE/GAN (Larsen et al., 2016) adopts an adversarial loss to improve the quality of the image, but its output for both reconstruction and synthesis (new samples) is still unsatisfactory. IntroVAE Huang et al. (2018) adds a loop from the output back to the input and is able to attain image quality that is on par with some modern GANs in some aspects. However, its full illustration for both reconstruction and synthesis is unclear. PGA (Zhang et al., 2019) adds a constraint to the latent variables. ",
|
| 161 |
+
"bbox": [
|
| 162 |
+
174,
|
| 163 |
+
561,
|
| 164 |
+
825,
|
| 165 |
+
700
|
| 166 |
+
],
|
| 167 |
+
"page_idx": 1
|
| 168 |
+
},
|
| 169 |
+
{
|
| 170 |
+
"type": "text",
|
| 171 |
+
"text": "Pioneering works of (Tu, 2007; Gutmann & Hyvärinen, 2012) alleviate the difficulty of learning densities by approximating likelihoods via classification (real (positive) samples vs. fake (pseudonegative or adversarial) samples). Generative adversarial network (GAN) (Goodfellow et al., 2014) builds on neural networks and amortized sampling (a decoder network that maps a noise into an image). The subsequent development after GAN (Radford et al., 2016; Arjovsky et al., 2017; Gulrajani et al., 2017; Karras et al., 2018; Gong et al., 2019; Dumoulin et al., 2017; Donahue et al., 2017) has led to a great leap forward in building decoder-based generative models. It has been widely observed that the adversarial loss in GANs contributes significantly to the improved quality of image synthesis. Energy-based generative models (Salakhutdinov & Hinton, 2009; Xie et al., 2016; Jin et al., 2017; Lee et al., 2018) — which aim to directly model data density — are making a steady improvement for a simultaneously generative and discriminative single model. ",
|
| 172 |
+
"bbox": [
|
| 173 |
+
174,
|
| 174 |
+
708,
|
| 175 |
+
825,
|
| 176 |
+
861
|
| 177 |
+
],
|
| 178 |
+
"page_idx": 1
|
| 179 |
+
},
|
| 180 |
+
{
|
| 181 |
+
"type": "text",
|
| 182 |
+
"text": "From another angle, contrastive learning (Hadsell et al., 2006; Wu et al., 2018; He et al., 2020; Chen et al., 2020) has lately shown its particular advantage in unsupervised training of CNN features. It overcomes the limitation in unsupervised learning where class label is missing by turning each image instance into one class. Thus, the softmax function in the standard discriminative classification training can be applied. Contrastive learning can be connected to metric learning (Bromley et al., 1993; Chopra et al., 2005; Chechik et al., 2010). ",
|
| 183 |
+
"bbox": [
|
| 184 |
+
174,
|
| 185 |
+
867,
|
| 186 |
+
825,
|
| 187 |
+
922
|
| 188 |
+
],
|
| 189 |
+
"page_idx": 1
|
| 190 |
+
},
|
| 191 |
+
{
|
| 192 |
+
"type": "text",
|
| 193 |
+
"text": "",
|
| 194 |
+
"bbox": [
|
| 195 |
+
171,
|
| 196 |
+
103,
|
| 197 |
+
823,
|
| 198 |
+
132
|
| 199 |
+
],
|
| 200 |
+
"page_idx": 2
|
| 201 |
+
},
|
| 202 |
+
{
|
| 203 |
+
"type": "text",
|
| 204 |
+
"text": "In this paper, we aim to improve VAE (Kingma & Welling, 2014) by introducing a contrastive loss (van den Oord et al., 2018) to address instance-level fidelity between the input and the reconstruction in the induced feature space. Unlike in self-supervised representation learning methods (van den Oord et al., 2018; He et al., 2020; Chen et al., 2020), where self-supervision requires generating a transformed input (via data augmentation operations), the reconstruction naturally fits into the contrastive term that encourages the matching between the reconstruction and the input image instance, while pushing the reconstruction away from the rest of the images in the entire training set. Thus, the instance-level and set-level contradistinctive terms collaborate with each to encourage the high fidelity of the reconstruction and synthesis. In Figure 3, we systematically show the significance of with and without the instance-level and the set-level contradistinctive terms. In addition, we explore multi-scale contrastive learning via two schemes in Section 4.1: 1) deep supervision for contrastive learning in different convolution layers, and 2) patch-based contrastive learning for fine-grained data fidelity. In the experiments, we show competitive results for the proposed dual contradistinctive generative autoencoder (DC-VAE) in a number of benchmarks for three tasks, including image synthesis, image reconstruction, and representation learning. ",
|
| 205 |
+
"bbox": [
|
| 206 |
+
173,
|
| 207 |
+
138,
|
| 208 |
+
825,
|
| 209 |
+
347
|
| 210 |
+
],
|
| 211 |
+
"page_idx": 2
|
| 212 |
+
},
|
| 213 |
+
{
|
| 214 |
+
"type": "text",
|
| 215 |
+
"text": "3 PRELIMINARIES: VAE AND VAE/GAN ",
|
| 216 |
+
"text_level": 1,
|
| 217 |
+
"bbox": [
|
| 218 |
+
176,
|
| 219 |
+
367,
|
| 220 |
+
531,
|
| 221 |
+
383
|
| 222 |
+
],
|
| 223 |
+
"page_idx": 2
|
| 224 |
+
},
|
| 225 |
+
{
|
| 226 |
+
"type": "text",
|
| 227 |
+
"text": "Variational autoencoder (VAE) Assume a given training set $S = \\{ \\mathbf { x } _ { i } \\} _ { i = 1 } ^ { n }$ where each $\\mathbf { x } _ { i } \\in \\mathbb { R } ^ { m }$ We suppose that each $\\mathbf { x } _ { i }$ is sampled from a generative process $p ( \\mathbf { x } | \\mathbf { z } )$ . In the literature, vector $\\mathbf { z }$ refers to latent variables. In practice, latent variables $\\mathbf { z }$ and the generative process $p ( \\mathbf { x } | \\mathbf { z } )$ are unknown. The objectives of a variational autoencoder (VAE) (Kingma & Welling, 2014) is to simultaneously train an inference network $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )$ and a generator network $p _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ . In VAE (Kingma $\\&$ Welling, 2014), the inference network is a neural network that outputs parameters for Gaussian distribution $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) = \\mathcal { N } ( \\mu _ { \\phi } ( \\mathbf { x } ) , \\Sigma _ { \\phi } ( \\mathbf { x } ) )$ . The generator is a deterministic neural network $f _ { \\pmb \\theta } ( \\mathbf z )$ parameterized by $\\pmb { \\theta }$ . Generative density $p _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ is assumed to be subject to a Gaussian distribution: $p _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) = \\dot { \\mathcal { N } } ( f _ { \\pmb { \\theta } } ( \\mathbf { z } ) , \\sigma ^ { 2 } I )$ . These models can be trained by minimizing the negative of evidence lower bound (ELBO) in Eq. (1) below. ",
|
| 228 |
+
"bbox": [
|
| 229 |
+
173,
|
| 230 |
+
398,
|
| 231 |
+
825,
|
| 232 |
+
539
|
| 233 |
+
],
|
| 234 |
+
"page_idx": 2
|
| 235 |
+
},
|
| 236 |
+
{
|
| 237 |
+
"type": "equation",
|
| 238 |
+
"img_path": "images/4443ba18266a96f0ff048f953c61a64e1b59fd3cc19b8e4506d21d32103d9576.jpg",
|
| 239 |
+
"text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { E L B O } } ( \\pmb { \\theta } , \\phi ; \\mathbf { x } ) = - \\mathbb { E } _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\mathrm { l o g } ( p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) ) ] + K L [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) | | p ( \\mathbf { z } ) ] } \\end{array}\n$$",
|
| 240 |
+
"text_format": "latex",
|
| 241 |
+
"bbox": [
|
| 242 |
+
274,
|
| 243 |
+
545,
|
| 244 |
+
723,
|
| 245 |
+
564
|
| 246 |
+
],
|
| 247 |
+
"page_idx": 2
|
| 248 |
+
},
|
| 249 |
+
{
|
| 250 |
+
"type": "text",
|
| 251 |
+
"text": "where $p ( \\mathbf { z } )$ is the prior, which is assumed to be $\\mathcal { N } ( 0 , I )$ . The first term $- \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log ( p _ { \\pmb { \\theta } } ( \\mathbf { x } | \\mathbf { z } ) ) ]$ reduces to standard pixel-wise reconstruction loss $\\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ | | \\mathbf { x } - f _ { \\theta } ( \\mathbf { z } ) | | _ { 2 } ^ { 2 } ]$ (up to a constant) due to the Gaussian assumption. The second term is the regularization term, which prevents the conditional $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )$ from deviating from the Gaussian prior $\\mathcal { N } ( 0 , I )$ . The inference network and generator network are jointly optimized over training samples by: ",
|
| 252 |
+
"bbox": [
|
| 253 |
+
173,
|
| 254 |
+
570,
|
| 255 |
+
825,
|
| 256 |
+
643
|
| 257 |
+
],
|
| 258 |
+
"page_idx": 2
|
| 259 |
+
},
|
| 260 |
+
{
|
| 261 |
+
"type": "equation",
|
| 262 |
+
"img_path": "images/56ad4157e632c31398c9797887fdd4a8db748cbae96a32d4cfb37d583c0d750a.jpg",
|
| 263 |
+
"text": "$$\n\\operatorname* { m i n } _ { \\theta , \\phi } \\operatorname* { \\mathbb { E } } _ { \\mathbf { x } \\sim p _ { \\mathrm { d a t a } } ( \\mathbf { x } ) } \\mathcal { L } _ { \\mathrm { E L B O } } ( \\theta , \\phi ; \\mathbf { x } ) .\n$$",
|
| 264 |
+
"text_format": "latex",
|
| 265 |
+
"bbox": [
|
| 266 |
+
400,
|
| 267 |
+
650,
|
| 268 |
+
599,
|
| 269 |
+
678
|
| 270 |
+
],
|
| 271 |
+
"page_idx": 2
|
| 272 |
+
},
|
| 273 |
+
{
|
| 274 |
+
"type": "text",
|
| 275 |
+
"text": "where $p _ { \\mathrm { d a t a } }$ is the distribution induced by the training set $S$ . ",
|
| 276 |
+
"bbox": [
|
| 277 |
+
173,
|
| 278 |
+
684,
|
| 279 |
+
562,
|
| 280 |
+
699
|
| 281 |
+
],
|
| 282 |
+
"page_idx": 2
|
| 283 |
+
},
|
| 284 |
+
{
|
| 285 |
+
"type": "text",
|
| 286 |
+
"text": "VAE has an elegant formulation. However, it relies on a pixel-wise reconstruction loss, which is known not ideal to be reflective of perceptual realism (Johnson et al., 2016; Isola et al., 2017), often resulting in blurry images. From another viewpoint, it can be thought of as using a kernel density estimator (with an isotropic Gaussian kernel) in the pixel space. Although allowing efficient training and inference, such a non-parametric approach is overly simplistic for modeling the semantics and perception of natural images. ",
|
| 287 |
+
"bbox": [
|
| 288 |
+
174,
|
| 289 |
+
704,
|
| 290 |
+
825,
|
| 291 |
+
790
|
| 292 |
+
],
|
| 293 |
+
"page_idx": 2
|
| 294 |
+
},
|
| 295 |
+
{
|
| 296 |
+
"type": "text",
|
| 297 |
+
"text": "VAE/GAN Generative adversarial networks (GANs) (Goodfellow et al., 2014) and its variants (Radford et al., 2016), on the other hand, are shown to be producing highly realistic images. The success was largely attributed to learning a fidelity function (often referred to as a discriminator) that measures how realistic the generated images are. This can be achieved by learning to contrast (classify) the set of training images with the set of generated images (Tu, 2007; Gutmann & Hyvärinen, 2012; Goodfellow et al., 2014). ",
|
| 298 |
+
"bbox": [
|
| 299 |
+
174,
|
| 300 |
+
804,
|
| 301 |
+
825,
|
| 302 |
+
888
|
| 303 |
+
],
|
| 304 |
+
"page_idx": 2
|
| 305 |
+
},
|
| 306 |
+
{
|
| 307 |
+
"type": "text",
|
| 308 |
+
"text": "VAE/GAN (Larsen et al., 2016) augments the ELBO objective (Eq. (2)) with the GAN objective. Specifically, the objective of VAE/GAN consists of two terms, namely the modified ELBO (Eq. (3)) ",
|
| 309 |
+
"bbox": [
|
| 310 |
+
173,
|
| 311 |
+
895,
|
| 312 |
+
823,
|
| 313 |
+
924
|
| 314 |
+
],
|
| 315 |
+
"page_idx": 2
|
| 316 |
+
},
|
| 317 |
+
{
|
| 318 |
+
"type": "text",
|
| 319 |
+
"text": "and the GAN objective. To make the notations later consistent, we now define the set of given training images as ${ \\cal { S } } = \\{ { \\bf { x } } _ { i } \\} _ { i = 1 } ^ { n }$ in which a total number of $n$ unlabeled training images are present. For each input image $\\mathbf { x } _ { i }$ , the modified ELBO computes the reconstruction loss in the feature space of the discriminator instead of the pixel space: ",
|
| 320 |
+
"bbox": [
|
| 321 |
+
173,
|
| 322 |
+
103,
|
| 323 |
+
825,
|
| 324 |
+
160
|
| 325 |
+
],
|
| 326 |
+
"page_idx": 3
|
| 327 |
+
},
|
| 328 |
+
{
|
| 329 |
+
"type": "equation",
|
| 330 |
+
"img_path": "images/37897b85f7f5ecd6120c263aa2a846439593cd38cdeaa9ec3926f79ad3ed9c1b.jpg",
|
| 331 |
+
"text": "$$\n\\mathcal { L } _ { \\mathrm { E L B O } } ( \\theta , \\phi , D ; \\mathbf { x } _ { i } ) = \\mathbb { E } _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } [ | | F _ { D } ( \\mathbf { x } _ { i } ) - F _ { D } ( f _ { \\theta } ( \\mathbf { z } ) ) | | _ { 2 } ^ { 2 } ] + K L [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } ) ]\n$$",
|
| 332 |
+
"text_format": "latex",
|
| 333 |
+
"bbox": [
|
| 334 |
+
218,
|
| 335 |
+
165,
|
| 336 |
+
777,
|
| 337 |
+
185
|
| 338 |
+
],
|
| 339 |
+
"page_idx": 3
|
| 340 |
+
},
|
| 341 |
+
{
|
| 342 |
+
"type": "text",
|
| 343 |
+
"text": "where $F _ { D } ( \\cdot )$ denotes the feature embedding from the discriminator $D$ . Feature reconstruction loss (also referred to as perceptual loss), similar to that in style transfer (Johnson et al., 2016). The modified GAN objective considers both reconstructed images (latent code from $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) )$ and sampled images (latent code from the prior $p ( \\mathbf { z } ) .$ ) as its fake samples: ",
|
| 344 |
+
"bbox": [
|
| 345 |
+
173,
|
| 346 |
+
191,
|
| 347 |
+
825,
|
| 348 |
+
248
|
| 349 |
+
],
|
| 350 |
+
"page_idx": 3
|
| 351 |
+
},
|
| 352 |
+
{
|
| 353 |
+
"type": "equation",
|
| 354 |
+
"img_path": "images/2b9a71252ff1b45fc508a5fa28738a3950356695e4366467f764a97634cf6594.jpg",
|
| 355 |
+
"text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { G A N } } ( \\pmb { \\theta } , \\phi , D ; \\mathbf { x } _ { i } ) = \\log D ( \\mathbf { x } _ { i } ) + \\mathbb { E } _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\log ( 1 - D ( f _ { \\theta } ( \\mathbf { z } ) ) + \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } \\log ( 1 - D ( f _ { \\theta } ( \\mathbf { z } ) ) . } \\end{array}\n$$",
|
| 356 |
+
"text_format": "latex",
|
| 357 |
+
"bbox": [
|
| 358 |
+
179,
|
| 359 |
+
255,
|
| 360 |
+
799,
|
| 361 |
+
273
|
| 362 |
+
],
|
| 363 |
+
"page_idx": 3
|
| 364 |
+
},
|
| 365 |
+
{
|
| 366 |
+
"type": "text",
|
| 367 |
+
"text": "The VAE/GAN objective becomes: ",
|
| 368 |
+
"bbox": [
|
| 369 |
+
174,
|
| 370 |
+
277,
|
| 371 |
+
405,
|
| 372 |
+
292
|
| 373 |
+
],
|
| 374 |
+
"page_idx": 3
|
| 375 |
+
},
|
| 376 |
+
{
|
| 377 |
+
"type": "equation",
|
| 378 |
+
"img_path": "images/4ab3a2ed650879f87aaedb22cfdc783ede43206053fe994bcf286efa25bc5b4e.jpg",
|
| 379 |
+
"text": "$$\n\\operatorname* { m i n } _ { \\theta , \\phi } \\operatorname* { m a x } _ { D } \\sum _ { i = 1 } ^ { n } \\left[ \\mathcal { L } _ { \\mathrm { E L B O } } ( \\theta , \\phi , D ; \\mathbf { x } _ { i } ) + \\mathcal { L } _ { \\mathrm { G A N } } ( \\theta , \\phi , D ; \\mathbf { x } _ { i } ) \\right] .\n$$",
|
| 380 |
+
"text_format": "latex",
|
| 381 |
+
"bbox": [
|
| 382 |
+
310,
|
| 383 |
+
299,
|
| 384 |
+
689,
|
| 385 |
+
340
|
| 386 |
+
],
|
| 387 |
+
"page_idx": 3
|
| 388 |
+
},
|
| 389 |
+
{
|
| 390 |
+
"type": "text",
|
| 391 |
+
"text": "4 DUAL CONTRADISTINCTIVE GENERATIVE AUTOENCODER (DC-VAE) ",
|
| 392 |
+
"text_level": 1,
|
| 393 |
+
"bbox": [
|
| 394 |
+
169,
|
| 395 |
+
357,
|
| 396 |
+
781,
|
| 397 |
+
373
|
| 398 |
+
],
|
| 399 |
+
"page_idx": 3
|
| 400 |
+
},
|
| 401 |
+
{
|
| 402 |
+
"type": "text",
|
| 403 |
+
"text": "Here we want to address a question: Is the degeneration of the synthesized images by VAE always the case once the decoder is joined with an encoder? Can the problem be remedied by using a more informative loss? ",
|
| 404 |
+
"bbox": [
|
| 405 |
+
173,
|
| 406 |
+
387,
|
| 407 |
+
825,
|
| 408 |
+
430
|
| 409 |
+
],
|
| 410 |
+
"page_idx": 3
|
| 411 |
+
},
|
| 412 |
+
{
|
| 413 |
+
"type": "text",
|
| 414 |
+
"text": "Although improving the image qualities of VAE by integrating a set-level contrastive loss (GAN objective of Eq. (4)), VAE/GAN still does not accurately model instance-level fidelity. Inspired by the literature on instance-level classification (Malisiewicz et al., 2011), approximating likelihood by classification (Tu, 2007), and contrastive learning (Hadsell et al., 2006; Wu et al., 2018; He et al., 2020), we propose to model instance-level fidelity by contrastive loss (commonly referred to as InfoNCE loss) (van den Oord et al., 2018). In DC-VAE, we perform the following minimization and loosely call each term a loss. ",
|
| 415 |
+
"bbox": [
|
| 416 |
+
173,
|
| 417 |
+
435,
|
| 418 |
+
826,
|
| 419 |
+
535
|
| 420 |
+
],
|
| 421 |
+
"page_idx": 3
|
| 422 |
+
},
|
| 423 |
+
{
|
| 424 |
+
"type": "equation",
|
| 425 |
+
"img_path": "images/8beffaa4429904a2c4af88de4bb321182a444850b405b331605ee757546a21a9.jpg",
|
| 426 |
+
"text": "$$\n\\mathcal { L } _ { \\mathrm { i n s t a n c e } } ( \\pmb { \\theta } , \\phi , D ; i , \\{ \\mathbf { x } _ { j } \\} _ { j = 1 } ^ { n } ) \\triangleq - \\mathbb { E } _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\left[ \\log \\frac { e ^ { h ( \\mathbf { x } _ { i } , f _ { \\pmb { \\theta } } ( \\mathbf { z } ) ) } } { \\sum _ { j = 1 } ^ { n } e ^ { h ( \\mathbf { x } _ { j } , f _ { \\pmb { \\theta } } ( \\mathbf { z } ) ) } } \\right] ,\n$$",
|
| 427 |
+
"text_format": "latex",
|
| 428 |
+
"bbox": [
|
| 429 |
+
259,
|
| 430 |
+
541,
|
| 431 |
+
736,
|
| 432 |
+
583
|
| 433 |
+
],
|
| 434 |
+
"page_idx": 3
|
| 435 |
+
},
|
| 436 |
+
{
|
| 437 |
+
"type": "text",
|
| 438 |
+
"text": "where $i$ is an index for a training sample (instance), $\\{ { \\bf x } _ { j } \\} _ { j = 1 } ^ { n }$ is the union of positive samples and negative samples, $h ( \\mathbf x , \\mathbf y )$ is the critic function that measures compatibility between $\\mathbf { x }$ and $\\mathbf { y }$ . Following the popular choice from (He et al., 2020), $h ( \\mathbf x , \\mathbf y )$ is the cosine similarity between the embeddings of $\\mathbf { x }$ and y: $\\begin{array} { r } { h ( \\mathbf { x } , \\mathbf { y } ) = \\frac { F _ { D } ( \\mathbf { x } ) ^ { \\top } F _ { D } ( \\mathbf { y } ) } { | | F _ { D } ( \\mathbf { x } ) | | _ { 2 } | | F _ { D } ( \\mathbf { y } ) | | _ { 2 } } } \\end{array}$ . Note that unlike in contrastive self-supervised learning methods (van den Oord et al., 2018; He et al., 2020; Chen et al., 2020) where two views (independent augmentations) of an instance constitutes a positive pair, an input instance $\\mathbf { x } _ { i }$ and its reconstruction $f _ { \\pmb \\theta } ( \\mathbf z )$ comprises a positive pair in DC-VAE. Likewise, the reconstruction $f _ { \\pmb \\theta } ( \\mathbf z )$ and any instance that is not $\\mathbf { x } _ { i }$ can be a negative pair. ",
|
| 439 |
+
"bbox": [
|
| 440 |
+
173,
|
| 441 |
+
588,
|
| 442 |
+
826,
|
| 443 |
+
710
|
| 444 |
+
],
|
| 445 |
+
"page_idx": 3
|
| 446 |
+
},
|
| 447 |
+
{
|
| 448 |
+
"type": "text",
|
| 449 |
+
"text": "To bridge the gap between the instance-level contrastive loss (Eq. (6)) and log-likelihood in ELBO term (Eq. (1)), we show the following observation. ",
|
| 450 |
+
"bbox": [
|
| 451 |
+
173,
|
| 452 |
+
717,
|
| 453 |
+
823,
|
| 454 |
+
746
|
| 455 |
+
],
|
| 456 |
+
"page_idx": 3
|
| 457 |
+
},
|
| 458 |
+
{
|
| 459 |
+
"type": "text",
|
| 460 |
+
"text": "Theorem 1 (From (Ma & Collins, 2018; Poole et al., 2019)) The following objective is minimized, i.e., the optimal critic $h$ is achieved, when $h ( f _ { \\pmb \\theta } ( \\mathbf { z } ) , \\mathbf { x } ) = \\log { p ( \\mathbf { x } | \\mathbf { z } ) } + \\overset { \\cdot } { c } ( \\mathbf { x } )$ where $c ( \\mathbf { x } )$ is any function that does not depend on $\\mathbf { z }$ . ",
|
| 461 |
+
"bbox": [
|
| 462 |
+
173,
|
| 463 |
+
756,
|
| 464 |
+
825,
|
| 465 |
+
800
|
| 466 |
+
],
|
| 467 |
+
"page_idx": 3
|
| 468 |
+
},
|
| 469 |
+
{
|
| 470 |
+
"type": "equation",
|
| 471 |
+
"img_path": "images/e12c8fa70cabfcd1373838fe15025db6581e2e0e5be6cc6e58afbc0bb5d186d6.jpg",
|
| 472 |
+
"text": "$$\nI _ { N C E } \\triangleq \\mathbb { E } _ { { \\mathbf { x } } _ { 1 } , \\cdots { \\mathbf { x } } _ { K } } \\mathbb { E } _ { i } \\big [ \\mathcal { L } _ { i n s t a n c e } ( \\pmb { \\theta } , \\phi , D ; i , \\{ { \\mathbf { x } } _ { j } \\} _ { j = 1 } ^ { n } ) \\big ] .\n$$",
|
| 473 |
+
"text_format": "latex",
|
| 474 |
+
"bbox": [
|
| 475 |
+
331,
|
| 476 |
+
818,
|
| 477 |
+
666,
|
| 478 |
+
838
|
| 479 |
+
],
|
| 480 |
+
"page_idx": 3
|
| 481 |
+
},
|
| 482 |
+
{
|
| 483 |
+
"type": "text",
|
| 484 |
+
"text": "From Theorem 1, we see that the contrastive loss of Eq. (6) implicitly estimates the log-likelihood $\\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } )$ required for the evidence lower bound (ELBO). Hence, we modify the ELBO objective of Eq. (1) as follows and name it as implicit ELBO (IELBO): ",
|
| 485 |
+
"bbox": [
|
| 486 |
+
174,
|
| 487 |
+
845,
|
| 488 |
+
825,
|
| 489 |
+
888
|
| 490 |
+
],
|
| 491 |
+
"page_idx": 3
|
| 492 |
+
},
|
| 493 |
+
{
|
| 494 |
+
"type": "equation",
|
| 495 |
+
"img_path": "images/9a947199439c5d6215ed3a063c17d2dc6562e7e2583690cc7f18fccc32fbeda6.jpg",
|
| 496 |
+
"text": "$$\n\\mathcal { L } _ { \\mathrm { I E L B O } } ( \\pmb { \\theta } , \\phi , D ; \\mathbf { x } _ { i } ) = \\mathcal { L } _ { \\mathrm { i n s t a n c e } } ( \\pmb { \\theta } , \\phi , D ; i , \\{ \\mathbf { x } _ { j } \\} _ { j = 1 } ^ { n } ) + K L [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } ) ] .\n$$",
|
| 497 |
+
"text_format": "latex",
|
| 498 |
+
"bbox": [
|
| 499 |
+
248,
|
| 500 |
+
907,
|
| 501 |
+
748,
|
| 502 |
+
926
|
| 503 |
+
],
|
| 504 |
+
"page_idx": 3
|
| 505 |
+
},
|
| 506 |
+
{
|
| 507 |
+
"type": "image",
|
| 508 |
+
"img_path": "images/99cd070fb7c3f61320d017c470f94dae504adfd88c5f14385c02278c2be0f007.jpg",
|
| 509 |
+
"image_caption": [
|
| 510 |
+
"Figure 2: Model architecture for the proposed DC-VAE algorithm. "
|
| 511 |
+
],
|
| 512 |
+
"image_footnote": [],
|
| 513 |
+
"bbox": [
|
| 514 |
+
210,
|
| 515 |
+
107,
|
| 516 |
+
766,
|
| 517 |
+
306
|
| 518 |
+
],
|
| 519 |
+
"page_idx": 4
|
| 520 |
+
},
|
| 521 |
+
{
|
| 522 |
+
"type": "text",
|
| 523 |
+
"text": "Finally, the combined objective for the proposed DC-VAE algorithm becomes: ",
|
| 524 |
+
"bbox": [
|
| 525 |
+
173,
|
| 526 |
+
344,
|
| 527 |
+
687,
|
| 528 |
+
361
|
| 529 |
+
],
|
| 530 |
+
"page_idx": 4
|
| 531 |
+
},
|
| 532 |
+
{
|
| 533 |
+
"type": "equation",
|
| 534 |
+
"img_path": "images/dde156a271c54af2367645fad4c518c54056bb815723dee40592d305400e5dc5.jpg",
|
| 535 |
+
"text": "$$\n\\operatorname* { m i n } _ { \\theta , \\phi } \\operatorname* { m a x } _ { D } \\sum _ { i = 1 } ^ { n } \\left[ \\mathcal { L } _ { \\mathrm { I E L B O } } ( \\theta , \\phi , D ; \\mathbf { x } _ { i } ) + \\mathcal { L } _ { \\mathrm { G A N } } ( \\theta , \\phi , D ; \\mathbf { x } _ { i } ) \\right] .\n$$",
|
| 536 |
+
"text_format": "latex",
|
| 537 |
+
"bbox": [
|
| 538 |
+
307,
|
| 539 |
+
367,
|
| 540 |
+
691,
|
| 541 |
+
409
|
| 542 |
+
],
|
| 543 |
+
"page_idx": 4
|
| 544 |
+
},
|
| 545 |
+
{
|
| 546 |
+
"type": "text",
|
| 547 |
+
"text": "The definition of $\\mathcal { L } _ { \\mathrm { G A N } }$ follows Eq. (4). Note here we also consider the term in Eq. (4) as contrasdistinctive since it tries to minimize the difference/discriminative classification between the input (“real”) image set and the reconstructed/generated (“fake”) image set. Below we highlight the significance of the two contradistinctive terms. Figure 2 shows the model architecture. ",
|
| 548 |
+
"bbox": [
|
| 549 |
+
174,
|
| 550 |
+
422,
|
| 551 |
+
826,
|
| 552 |
+
479
|
| 553 |
+
],
|
| 554 |
+
"page_idx": 4
|
| 555 |
+
},
|
| 556 |
+
{
|
| 557 |
+
"type": "text",
|
| 558 |
+
"text": "• Instance-level fidelity. The first item in Eq. (8) is an instance-level fidelity term encouraging the reconstruction to be as close as possible to the input image while being different from all the rest of the images. A key advantage of the contrastive loss in Eq. (8) over the standard reconstruction loss in Eq. (3) is its relaxed and background instances aware formulation. In general, the reconstruction in Eq. (3) wants a perfect match between the reconstruction and the input, whereas the contrastive loss in Eq. (8) requests for being the most similar one among the training samples. This way, the contrastive loss becomes more cooperative with less conflict to the GAN loss, compared with the reconstruction loss. The introduction of the contrastive loss results in a significant improvement over VAE and VAE/GAN in which only matching the reconstruction, and the input instance is enforced. ",
|
| 559 |
+
"bbox": [
|
| 560 |
+
176,
|
| 561 |
+
491,
|
| 562 |
+
825,
|
| 563 |
+
631
|
| 564 |
+
],
|
| 565 |
+
"page_idx": 4
|
| 566 |
+
},
|
| 567 |
+
{
|
| 568 |
+
"type": "text",
|
| 569 |
+
"text": "• Set-level fidelity. The second item in Eq. (9) is a set-level fidelity term encouraging the entire set of synthesized images to be non distinguishable from the input image set. Having this term (Eq. (4)) is still important since the instance contrastive loss alone (Eq. (9)) will also lead to a degenerated situation: the input image and its reconstruction can be projected to the same point in the new feature space, but without a guarantee that the reconstruction itself lies on the valid “real” image manifold. ",
|
| 570 |
+
"bbox": [
|
| 571 |
+
176,
|
| 572 |
+
636,
|
| 573 |
+
825,
|
| 574 |
+
719
|
| 575 |
+
],
|
| 576 |
+
"page_idx": 4
|
| 577 |
+
},
|
| 578 |
+
{
|
| 579 |
+
"type": "text",
|
| 580 |
+
"text": "As shown in Figure 3 and Table 1 for the comparison with and without the individual terms in Eq. (9). We observe evident effectiveness of the proposed DC-VAE combining both the instance-level fidelity term (Eq. (6)) and the set-level fidelity term (Eq. (4)), compared with VAE (using pixel-wise reconstruction loss without the GAN objective), VAE-GAN (using feature reconstruction loss and the GAN objective), and VAE contrastive (using contrastive loss but without the GAN objective). ",
|
| 581 |
+
"bbox": [
|
| 582 |
+
174,
|
| 583 |
+
732,
|
| 584 |
+
825,
|
| 585 |
+
801
|
| 586 |
+
],
|
| 587 |
+
"page_idx": 4
|
| 588 |
+
},
|
| 589 |
+
{
|
| 590 |
+
"type": "text",
|
| 591 |
+
"text": "In the experiments, we show that both terms required to achieve faithful reconstruction (captured by InfoNCE loss) with perceptual realism (captured by the GAN loss). ",
|
| 592 |
+
"bbox": [
|
| 593 |
+
174,
|
| 594 |
+
809,
|
| 595 |
+
823,
|
| 596 |
+
837
|
| 597 |
+
],
|
| 598 |
+
"page_idx": 4
|
| 599 |
+
},
|
| 600 |
+
{
|
| 601 |
+
"type": "text",
|
| 602 |
+
"text": "4.1 MULTI SCALE CONTRASTIVE LEARNING ",
|
| 603 |
+
"text_level": 1,
|
| 604 |
+
"bbox": [
|
| 605 |
+
174,
|
| 606 |
+
856,
|
| 607 |
+
488,
|
| 608 |
+
869
|
| 609 |
+
],
|
| 610 |
+
"page_idx": 4
|
| 611 |
+
},
|
| 612 |
+
{
|
| 613 |
+
"type": "text",
|
| 614 |
+
"text": "Inspired by (Lee et al., 2015), we utilize information from feature maps at different scales. In addition to contrasting on the last layer of $D$ in Equation 9, we add contrastive objective on $f _ { l } ( \\mathbf { z } )$ where $f _ { l }$ is some function on top of an intermediate layer $l$ of D. We do it in two different ways. ",
|
| 615 |
+
"bbox": [
|
| 616 |
+
176,
|
| 617 |
+
881,
|
| 618 |
+
823,
|
| 619 |
+
924
|
| 620 |
+
],
|
| 621 |
+
"page_idx": 4
|
| 622 |
+
},
|
| 623 |
+
{
|
| 624 |
+
"type": "text",
|
| 625 |
+
"text": "1. Deep supervision: We use $1 \\times 1$ convolution to reduce the dimension channel-wise, and use a linear layer to obtain $f _ { l }$ . \n2. Local patch: We use a random location across channel at layer $l$ (size: $1 \\times 1 \\times \\mathrm { d }$ , where d is the channel depth). ",
|
| 626 |
+
"bbox": [
|
| 627 |
+
207,
|
| 628 |
+
103,
|
| 629 |
+
825,
|
| 630 |
+
161
|
| 631 |
+
],
|
| 632 |
+
"page_idx": 5
|
| 633 |
+
},
|
| 634 |
+
{
|
| 635 |
+
"type": "text",
|
| 636 |
+
"text": "The intuition for the second is that in a convolutional neural network, one location at a feature map corresponds to a receptive area (patch) in the original image. Thus, by contrasting locations across channels in the same feature maps, we are encouraging the original image and the reconstruction to image have locally similar content, while encouraging them to have locally dissimilar content in other images. We use deep supervision for initial training, and add local patch after certain iterations. ",
|
| 637 |
+
"bbox": [
|
| 638 |
+
174,
|
| 639 |
+
172,
|
| 640 |
+
825,
|
| 641 |
+
242
|
| 642 |
+
],
|
| 643 |
+
"page_idx": 5
|
| 644 |
+
},
|
| 645 |
+
{
|
| 646 |
+
"type": "text",
|
| 647 |
+
"text": "5 EXPERIMENTS ",
|
| 648 |
+
"text_level": 1,
|
| 649 |
+
"bbox": [
|
| 650 |
+
174,
|
| 651 |
+
256,
|
| 652 |
+
326,
|
| 653 |
+
270
|
| 654 |
+
],
|
| 655 |
+
"page_idx": 5
|
| 656 |
+
},
|
| 657 |
+
{
|
| 658 |
+
"type": "image",
|
| 659 |
+
"img_path": "images/e57f118b780d3fccf3c6f98818a1a3b9b6f6a02e5805d2d2957dc516dfe553f7.jpg",
|
| 660 |
+
"image_caption": [
|
| 661 |
+
"Figure 3: Qualitative results of CIFAR-10 (Krizhevsky et al., 2009) images (resolution $3 2 \\times 3 2$ ) for experiments in Table 1 (Krizhevsky et al., 2009). "
|
| 662 |
+
],
|
| 663 |
+
"image_footnote": [],
|
| 664 |
+
"bbox": [
|
| 665 |
+
174,
|
| 666 |
+
273,
|
| 667 |
+
821,
|
| 668 |
+
434
|
| 669 |
+
],
|
| 670 |
+
"page_idx": 5
|
| 671 |
+
},
|
| 672 |
+
{
|
| 673 |
+
"type": "text",
|
| 674 |
+
"text": "5.1 IMPLEMENTATION ",
|
| 675 |
+
"text_level": 1,
|
| 676 |
+
"bbox": [
|
| 677 |
+
174,
|
| 678 |
+
497,
|
| 679 |
+
341,
|
| 680 |
+
512
|
| 681 |
+
],
|
| 682 |
+
"page_idx": 5
|
| 683 |
+
},
|
| 684 |
+
{
|
| 685 |
+
"type": "text",
|
| 686 |
+
"text": "Datasets To validate our method, we train our method on several different datasets — CIFAR-10 (Krizhevsky et al., 2009), STL-10 (Coates et al., 2011), CelebA (Liu et al., 2015), CelebA-HQ (Karras et al., 2018), and LSUN bedroom (Yu et al., 2015). See the appendix for more detailed descriptions. ",
|
| 687 |
+
"bbox": [
|
| 688 |
+
173,
|
| 689 |
+
523,
|
| 690 |
+
826,
|
| 691 |
+
565
|
| 692 |
+
],
|
| 693 |
+
"page_idx": 5
|
| 694 |
+
},
|
| 695 |
+
{
|
| 696 |
+
"type": "text",
|
| 697 |
+
"text": "Network architecture For $3 2 \\times 3 2$ resolution, we design the encoder and decoder subnetworks of our model in a similar way to the discriminator and generator found through neural architecture search in AutoGAN (Gong et al., 2019). For the higher resolution experiments $1 2 8 \\times 1 2 8$ and $5 1 2 \\times 5 1 2$ resolution), we use Progressive GAN (Karras et al., 2018) as the backbone. Network architecture diagram is available in the appendix. ",
|
| 698 |
+
"bbox": [
|
| 699 |
+
173,
|
| 700 |
+
571,
|
| 701 |
+
825,
|
| 702 |
+
642
|
| 703 |
+
],
|
| 704 |
+
"page_idx": 5
|
| 705 |
+
},
|
| 706 |
+
{
|
| 707 |
+
"type": "text",
|
| 708 |
+
"text": "Training details The number of negative samples for contrastive learning is 8096 for all datasets. The latent dimension for the VAE decoder is 128 for CIFAR-10, STL-10, and 512 for CelebA, CelebA-HQ and LSUN Bedroom. Learning rate is 0.0002 with Adam parameters of $( \\beta _ { 1 } , \\beta _ { 2 } ) = ( 0 . 0 , 0 . 9 )$ and a batch size of 128 for CIFAR-10 and STL-10. For CelebA, CelebA-HQ, LSUN Bedroom datasets, we use the optimizer parameters given in (Karras et al., 2018). The contrastive embedding dimension used is 16 for each of the experiments. ",
|
| 709 |
+
"bbox": [
|
| 710 |
+
173,
|
| 711 |
+
648,
|
| 712 |
+
825,
|
| 713 |
+
733
|
| 714 |
+
],
|
| 715 |
+
"page_idx": 5
|
| 716 |
+
},
|
| 717 |
+
{
|
| 718 |
+
"type": "text",
|
| 719 |
+
"text": "5.2 ABLATION STUDY ",
|
| 720 |
+
"text_level": 1,
|
| 721 |
+
"bbox": [
|
| 722 |
+
174,
|
| 723 |
+
742,
|
| 724 |
+
341,
|
| 725 |
+
757
|
| 726 |
+
],
|
| 727 |
+
"page_idx": 5
|
| 728 |
+
},
|
| 729 |
+
{
|
| 730 |
+
"type": "table",
|
| 731 |
+
"img_path": "images/091cb8a9923c44f85da671421afe7b1f159777cdbb96e3e2f94990eac2a81aac.jpg",
|
| 732 |
+
"table_caption": [
|
| 733 |
+
"Table 1: Ablation studies on CIFAR-10 for the proposed DC-VAE algorithm. We follow (Johnson et al., 2016) and measure perceptual distance in an relu4_3 layer of a pretrained VGG network. $\\downarrow$ means lower is better. ↑ means higher is better. "
|
| 734 |
+
],
|
| 735 |
+
"table_footnote": [],
|
| 736 |
+
"table_body": "<table><tr><td>Method</td><td>FID↓/IS↑ Sampling</td><td>FID↓/IS↑ Reconstruction</td><td>Pixel↓ Distance</td><td>Perceptual↓ Distance</td></tr><tr><td>VAE</td><td>115.8 /3.8</td><td>108.4/4.3</td><td>21.8</td><td>65.8</td></tr><tr><td>VAE/GAN</td><td>39.8 /7.4</td><td>29.0 /7.6</td><td>62.7</td><td>57.2</td></tr><tr><td>VAE-Contrastive</td><td>240.4 /1.8</td><td>242 /1.9</td><td>53.6</td><td>104.2</td></tr><tr><td>DC-VAE</td><td>17.9 /8.2</td><td>21.4 / 7.9</td><td>45.9</td><td>52.9</td></tr></table>",
|
| 737 |
+
"bbox": [
|
| 738 |
+
292,
|
| 739 |
+
808,
|
| 740 |
+
704,
|
| 741 |
+
887
|
| 742 |
+
],
|
| 743 |
+
"page_idx": 5
|
| 744 |
+
},
|
| 745 |
+
{
|
| 746 |
+
"type": "text",
|
| 747 |
+
"text": "To demonstrate the necessity of the GAN loss (Eq. 4) and contrastive loss (Eq. 8), we conduct four experiments with the same backbone. These experiments are: VAE (No GAN, no Contrastive), ",
|
| 748 |
+
"bbox": [
|
| 749 |
+
174,
|
| 750 |
+
895,
|
| 751 |
+
820,
|
| 752 |
+
924
|
| 753 |
+
],
|
| 754 |
+
"page_idx": 5
|
| 755 |
+
},
|
| 756 |
+
{
|
| 757 |
+
"type": "text",
|
| 758 |
+
"text": "VAE/GAN (with GAN, no Contrastive), VAE-Contrastive (No GAN, with Contrastive, and ours (With GAN, with Contrastive). Here, GAN denotes Eq. 4, and Contrastive denotes Eq. 8. ",
|
| 759 |
+
"bbox": [
|
| 760 |
+
173,
|
| 761 |
+
103,
|
| 762 |
+
823,
|
| 763 |
+
132
|
| 764 |
+
],
|
| 765 |
+
"page_idx": 6
|
| 766 |
+
},
|
| 767 |
+
{
|
| 768 |
+
"type": "text",
|
| 769 |
+
"text": "Qualitative analysis From Figure 3, we see that without GAN and contrastive, images are blurry; Without GAN, the contrastive head can classify images, but not on the image manifold; Without Contrastive, reconstruction images are on the image manifold because of the discriminator, but they are different from input images. These experiments show that it is necessary to combine both instance-level and set-level fidelity, and in a contradistinctive manner. ",
|
| 770 |
+
"bbox": [
|
| 771 |
+
174,
|
| 772 |
+
138,
|
| 773 |
+
825,
|
| 774 |
+
208
|
| 775 |
+
],
|
| 776 |
+
"page_idx": 6
|
| 777 |
+
},
|
| 778 |
+
{
|
| 779 |
+
"type": "text",
|
| 780 |
+
"text": "Quantitative analysis In Table 1 we observe the same trend. VAE generates blurry images; thus the FID/IS (Inception Score) is not ideal. VAE-Contrastive does not generate images on the natural manifold; thus FID/IS is poor. VAE/GAN combines set-level and instance-level information. However the L2 objective is not ideal; thus the FID/IS is sub-optimal. For both reconstruction and sampling tasks, DC-VAE generates high fidelity images and has a favorable FID and Inception score. This illustrates the advantange of having a contradistinctive objective on both set level and instance level. To measure the faithfulness of the reconstructed image we compute the pixelwise L2 distance and the perceptual distance (Johnson et al. (2016)). For the pixel distance, VAE has the lowest value because it directly optimizes this distance during training; our pixel-wise distance is better than VAE/GAN and VAE-Contrastive. For perceptual distance, our method outperforms other three, which confirms that using contrastive learning helps reconstruct images semantically. ",
|
| 781 |
+
"bbox": [
|
| 782 |
+
174,
|
| 783 |
+
215,
|
| 784 |
+
825,
|
| 785 |
+
368
|
| 786 |
+
],
|
| 787 |
+
"page_idx": 6
|
| 788 |
+
},
|
| 789 |
+
{
|
| 790 |
+
"type": "table",
|
| 791 |
+
"img_path": "images/f7c4e87f62c18bb6f797ca80b8dfcf1a988f46fc36c101c94095929d9307588d.jpg",
|
| 792 |
+
"table_caption": [
|
| 793 |
+
"Table 2: Comparison on CIFAR-10 and STL-10. Average Inception scores (Salimans et al., 2016) and FID scores (Heusel et al., 2017). Results derived from (Gong et al., 2019). Table style derived from (Lee et al., 2019). †Result from Aneja et al. (2020). ∗Result from (Dieng et al., 2019). "
|
| 794 |
+
],
|
| 795 |
+
"table_footnote": [],
|
| 796 |
+
"table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">CIFAR-10</td><td colspan=\"2\">STL-10</td></tr><tr><td>Inception Score↑</td><td>FID↓</td><td>Inception Score↑</td><td>FID↓</td></tr><tr><td>Methods based on GAN:</td><td></td><td></td><td></td><td></td></tr><tr><td>DCGAN (Radford et al.,2016)</td><td>6.6</td><td></td><td></td><td>-</td></tr><tr><td>ProbGAN (He et al.,2019)</td><td>7.8</td><td>24.6</td><td>8.9</td><td>46.7</td></tr><tr><td>WGAN-GP ResNet (Gulrajani et al.,2017)</td><td>7.9</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RaGAN (Jolicoeur-Martineau,2018)</td><td>=</td><td>23.5</td><td>=</td><td>■</td></tr><tr><td>SN-GAN (Miyato et al., 2018)</td><td>8.2</td><td>21.7</td><td>9.1</td><td>40.1</td></tr><tr><td>MGAN (Hoang et al., 2018)</td><td>8.3</td><td>26.7</td><td>-</td><td>-</td></tr><tr><td>Progressive GAN (Karras et al.,2018)</td><td>8.8</td><td>-</td><td>■</td><td>=</td></tr><tr><td>Improving MMD GAN(Wang et al.,2019)</td><td>8.3</td><td>16.2</td><td>9.3</td><td>37.6</td></tr><tr><td>PULSGAN (Guo et al., 2020)</td><td>■</td><td>22.3</td><td>=</td><td>-</td></tr><tr><td>AutoGAN (Gong et al., 2019)</td><td>8.6</td><td>12.4</td><td>9.2</td><td>31.0</td></tr><tr><td>MethodsbasedonVAE:</td><td></td><td></td><td></td><td></td></tr><tr><td>VAE</td><td>3.8</td><td>115.8</td><td></td><td></td></tr><tr><td>VAE/GAN</td><td>7.4</td><td>39.8</td><td></td><td></td></tr><tr><td>VEEGAN* (Srivastava et al.,2017)</td><td>-</td><td>95.2</td><td></td><td></td></tr><tr><td>WAE-GAN (Tolstikhin et al., 2017)</td><td></td><td>93.1</td><td></td><td></td></tr><tr><td>NVAE†(Vahdat & Kautz,202O) Sampling</td><td></td><td>50.8</td><td></td><td></td></tr><tr><td>NVAE† (Vahdat & Kautz,2020) Reconstruction</td><td>=</td><td>2.67</td><td>=</td><td></td></tr><tr><td>DC-VAE Sampling (ours)</td><td>8.2</td><td>17.9</td><td>8.1</td><td>41.9</td></tr><tr><td>DC-VAE Reconstruction (ours)</td><td>7.9</td><td>21.4</td><td>8.4</td><td>43.6</td></tr></table>",
|
| 797 |
+
"bbox": [
|
| 798 |
+
184,
|
| 799 |
+
420,
|
| 800 |
+
785,
|
| 801 |
+
686
|
| 802 |
+
],
|
| 803 |
+
"page_idx": 6
|
| 804 |
+
},
|
| 805 |
+
{
|
| 806 |
+
"type": "text",
|
| 807 |
+
"text": "5.3 COMPARISON TO EXISTING GENERATIVE MODELS ",
|
| 808 |
+
"text_level": 1,
|
| 809 |
+
"bbox": [
|
| 810 |
+
173,
|
| 811 |
+
696,
|
| 812 |
+
558,
|
| 813 |
+
709
|
| 814 |
+
],
|
| 815 |
+
"page_idx": 6
|
| 816 |
+
},
|
| 817 |
+
{
|
| 818 |
+
"type": "text",
|
| 819 |
+
"text": "Table 2 gives a comparison of quantitative measurement for CIFAR-10 and STL-10 dataset. In general, there is a large difference in terms of FID and IS between GAN family and VAE family of models. Our model has state-of-the-art results in VAE family, and is comparable to state-of-the-art GAN models on CIFAR-10. Similarly Table 4 shows that DC-VAE is able to generate images that are comparable to GAN based methods even on higher resolution datasets. ",
|
| 820 |
+
"bbox": [
|
| 821 |
+
174,
|
| 822 |
+
714,
|
| 823 |
+
825,
|
| 824 |
+
784
|
| 825 |
+
],
|
| 826 |
+
"page_idx": 6
|
| 827 |
+
},
|
| 828 |
+
{
|
| 829 |
+
"type": "text",
|
| 830 |
+
"text": "5.4 LATENT SPACE REPRESENTATION: IMAGE AND STYLE INTERPOLATION ",
|
| 831 |
+
"text_level": 1,
|
| 832 |
+
"bbox": [
|
| 833 |
+
176,
|
| 834 |
+
794,
|
| 835 |
+
705,
|
| 836 |
+
808
|
| 837 |
+
],
|
| 838 |
+
"page_idx": 6
|
| 839 |
+
},
|
| 840 |
+
{
|
| 841 |
+
"type": "text",
|
| 842 |
+
"text": "We further validate the effectiveness of DC-VAE for representation learning. One benefit of having an AE/VAE framework compared with just a decoder as in GAN Goodfellow et al. (2014) is to be able to directly obtain the latent representation from the input images. The encoder and decoder modules in VAE allows us to readily perform image/style interpolation by mixing the latent variables of different images and reconstruct/synthesize new ones. We demonstrate qualitative results on image interpolation (Fig. 5, Appendix Fig. 9), style interpolation (Appendix Fig. 10) and image editing (Fig. 6). We directly use the trained DC-VAE model without disentanglement learning Karras et al. (2019). Additional latent space analysis and the method used for interpolation and editing is provided in the Appendix. We also quantitatively compare the latent space disentanglement through the perceptual path length (PPL) (Karras et al., 2019) (Table 6). We observe that DC-VAE learns a more disentangled latent space representation than the backbone Progressive GAN (Karras et al., 2018) and StyleALAE (Pidhorskyi et al., 2020) that uses a much more capable StyleGAN (Karras et al., 2019) backbone. ",
|
| 843 |
+
"bbox": [
|
| 844 |
+
174,
|
| 845 |
+
811,
|
| 846 |
+
825,
|
| 847 |
+
924
|
| 848 |
+
],
|
| 849 |
+
"page_idx": 6
|
| 850 |
+
},
|
| 851 |
+
{
|
| 852 |
+
"type": "image",
|
| 853 |
+
"img_path": "images/f1398a564655ab1cc44ff656924a969e29d8cbd65fbc4759bd47b4f57cd9f04b.jpg",
|
| 854 |
+
"image_caption": [
|
| 855 |
+
"(c) DC-VAE Reconstruction (ours, $5 1 2 \\times 5 1 2$ ) Figure 4: Comparison of DC-VAE (resolution $5 1 2 \\times 5 1 2$ ) with IntroVAE (Huang et al., 2018) (resolution $1 0 2 4 \\times 1 0 2 4 )$ ). "
|
| 856 |
+
],
|
| 857 |
+
"image_footnote": [],
|
| 858 |
+
"bbox": [
|
| 859 |
+
271,
|
| 860 |
+
97,
|
| 861 |
+
725,
|
| 862 |
+
308
|
| 863 |
+
],
|
| 864 |
+
"page_idx": 7
|
| 865 |
+
},
|
| 866 |
+
{
|
| 867 |
+
"type": "table",
|
| 868 |
+
"img_path": "images/cd8eff4a945f5b19c667bf5f21e5813c8bf29d8ca563805b92b49d95fe5fca39.jpg",
|
| 869 |
+
"table_caption": [
|
| 870 |
+
"Table 3: Quality of image generation (FID) comparison on LSUN Bedrooms. $^ \\dag 1 2 8 \\times 1 2 8$ resolution. $^ \\dag 2 5 6 \\times 2 5 6$ resolution. $\\downarrow$ means lower is better. "
|
| 871 |
+
],
|
| 872 |
+
"table_footnote": [],
|
| 873 |
+
"table_body": "<table><tr><td>Method</td><td>FID↓</td></tr><tr><td>Progressive GAN‡ (Karras et al.,2018)</td><td>8.3</td></tr><tr><td>SNGANt (Miyato et al.,2018)(from (Chen et al.,2019))</td><td>16.0</td></tr><tr><td>SSGANt(Chen et al., 2019)</td><td>13.3</td></tr><tr><td>StyleALAE(Pidhorskyi et al.,202O) Reconstruction</td><td>15.92</td></tr><tr><td>StyleALAE(Pidhorskyi et al.,2020) Sampling</td><td>17.13</td></tr><tr><td>DC-VAEt (ours) Reconstruction</td><td>10.57</td></tr><tr><td>DC-VAEt (ours) Sampling</td><td>14.3</td></tr></table>",
|
| 874 |
+
"bbox": [
|
| 875 |
+
184,
|
| 876 |
+
405,
|
| 877 |
+
488,
|
| 878 |
+
492
|
| 879 |
+
],
|
| 880 |
+
"page_idx": 7
|
| 881 |
+
},
|
| 882 |
+
{
|
| 883 |
+
"type": "table",
|
| 884 |
+
"img_path": "images/eed3bf761084213200e5594186dc5ad00f40d218f44b846fc1b8ba566596d113.jpg",
|
| 885 |
+
"table_caption": [
|
| 886 |
+
"Table 4: FID on CelebA. $^ { * } 6 4 \\times 6 4$ resolution. "
|
| 887 |
+
],
|
| 888 |
+
"table_footnote": [],
|
| 889 |
+
"table_body": "<table><tr><td>Method</td><td>FID↓</td></tr><tr><td>Methods based on GAN:</td><td></td></tr><tr><td>PresGAN*(Dieng et al., 2019)</td><td>29.1</td></tr><tr><td>LSGAN(Mao et al.,2017) (from (Hoshen et al.,2019))</td><td>53.9</td></tr><tr><td>COCO-GAN† (Lin et al., 2019)</td><td>5.7</td></tr><tr><td>PGANt (Karras et al.,2018)(from (Lin et al.,2019))</td><td>7.30</td></tr><tr><td>MethodsbasedonVAE:</td><td></td></tr><tr><td>VEE-GAN† (Srivastava et al.,2017) (from (Dieng et al.,2019))</td><td>46.2</td></tr><tr><td>WAE-GAN*(Tolstikhin et al.,2017)</td><td>42</td></tr><tr><td>DC-VAEt (ours) Reconstruction</td><td>14.3</td></tr><tr><td>DC-VAEt (ours) Sampling</td><td>19.9</td></tr></table>",
|
| 890 |
+
"bbox": [
|
| 891 |
+
511,
|
| 892 |
+
380,
|
| 893 |
+
807,
|
| 894 |
+
488
|
| 895 |
+
],
|
| 896 |
+
"page_idx": 7
|
| 897 |
+
},
|
| 898 |
+
{
|
| 899 |
+
"type": "text",
|
| 900 |
+
"text": "",
|
| 901 |
+
"bbox": [
|
| 902 |
+
174,
|
| 903 |
+
507,
|
| 904 |
+
826,
|
| 905 |
+
577
|
| 906 |
+
],
|
| 907 |
+
"page_idx": 7
|
| 908 |
+
},
|
| 909 |
+
{
|
| 910 |
+
"type": "image",
|
| 911 |
+
"img_path": "images/84aabba85b39d99a71f1da1de8414c7bd192b096d255b0c02463f04138cf9ba2.jpg",
|
| 912 |
+
"image_caption": [
|
| 913 |
+
"Figure 5: Interpolation results generated by DC-VAE (ours) on CelebA-HQ (Karras et al., 2018) images $( 5 \\bar { 1 } 2 \\times 5 1 2$ , left) and LSUN Bedroom Yu et al. (2015) images $1 2 8 \\times 1 2 8$ , right). More images can be seen in Appendix Figure 9. (Zoom in for a better visualization.) "
|
| 914 |
+
],
|
| 915 |
+
"image_footnote": [],
|
| 916 |
+
"bbox": [
|
| 917 |
+
179,
|
| 918 |
+
595,
|
| 919 |
+
818,
|
| 920 |
+
747
|
| 921 |
+
],
|
| 922 |
+
"page_idx": 7
|
| 923 |
+
},
|
| 924 |
+
{
|
| 925 |
+
"type": "text",
|
| 926 |
+
"text": "5.5 LATENT SPACE REPRESENTATION: CLSSIFICATION ",
|
| 927 |
+
"text_level": 1,
|
| 928 |
+
"bbox": [
|
| 929 |
+
173,
|
| 930 |
+
805,
|
| 931 |
+
566,
|
| 932 |
+
819
|
| 933 |
+
],
|
| 934 |
+
"page_idx": 7
|
| 935 |
+
},
|
| 936 |
+
{
|
| 937 |
+
"type": "text",
|
| 938 |
+
"text": "To show that our model learns a good representation, we measure the performance on the downstream MNIST classification task (Ding et al., 2020). The VAE models were trained on MNIST dataset (LeCun, 1998). We feed input images into our VAE encoder and get the latent representation. Then we train a linear classifier on the latent representation to classify the classes of the input images. Results in Table 5 show that our model gives the lowest classification error in most cases. This experiment demonstrates that our model not only gains the ability to do faithful synthesis and reconstruction, but also gains better representation ability on the VAE side. ",
|
| 939 |
+
"bbox": [
|
| 940 |
+
173,
|
| 941 |
+
827,
|
| 942 |
+
825,
|
| 943 |
+
924
|
| 944 |
+
],
|
| 945 |
+
"page_idx": 7
|
| 946 |
+
},
|
| 947 |
+
{
|
| 948 |
+
"type": "image",
|
| 949 |
+
"img_path": "images/69e115a9134e32a717d3716068609e9514fa03d1262fe8784ee000dbfa557b36.jpg",
|
| 950 |
+
"image_caption": [
|
| 951 |
+
"Figure 6: Image editing on CelebA-HQ (Karras et al., 2018) validation set images (resolution $5 1 2 \\times 5 1 2$ ). The method used to generate these is outlined in Appendix A.2. "
|
| 952 |
+
],
|
| 953 |
+
"image_footnote": [],
|
| 954 |
+
"bbox": [
|
| 955 |
+
186,
|
| 956 |
+
98,
|
| 957 |
+
810,
|
| 958 |
+
476
|
| 959 |
+
],
|
| 960 |
+
"page_idx": 8
|
| 961 |
+
},
|
| 962 |
+
{
|
| 963 |
+
"type": "table",
|
| 964 |
+
"img_path": "images/9aad5b43003101d237f0a9b318831a78ae4c7b6a2e1ffcc0d86eeccc433c5416.jpg",
|
| 965 |
+
"table_caption": [
|
| 966 |
+
"Table 5: Comparison to prior VAE-based representation learning methods. Classification error on MNIST dataset. ↓: lower is better. $9 5 \\ \\%$ confidence intervals are from 5 trials. Results derived from (Ding et al., 2020). ",
|
| 967 |
+
"Table 6: PPL Comparison of on CelebA-HQ Karras et al. (2018) "
|
| 968 |
+
],
|
| 969 |
+
"table_footnote": [],
|
| 970 |
+
"table_body": "<table><tr><td>Method</td><td>dz=16↓</td><td>dz=32↓</td><td>dz=64↓</td></tr><tr><td>VAE (Kingma&Welling,2014)</td><td>2.92%±0.12</td><td>3.05%±0.42</td><td>2.98%±0.14</td></tr><tr><td>β-VAE(β=2) (Higgins et al.,2017)</td><td>4.69%±0.18</td><td>5.26%±0.22</td><td>5.40%±0.33</td></tr><tr><td>FactorVAE(y=5) (Kim& Mnih,2018)</td><td>6.07%±0.05</td><td>6.18%±0.20</td><td>6.35%±0.48</td></tr><tr><td>β-TCVAE (α=1,β=5,γ=1) (Chen et al.,2018)</td><td>1.62%±0.07</td><td>1.24%±0.05</td><td>1.32%±0.09</td></tr><tr><td>Guided-VAE (Ding et al.,2020)</td><td>1.85%±0.08</td><td>1.60%±0.08</td><td>1.49%±0.06</td></tr><tr><td>Guided-β-TCVAE (Ding et al.,2020)</td><td>1.47%±0.12</td><td>1.10%±0.03</td><td>1.31%±0.06</td></tr><tr><td>DC-VAE (Ours)</td><td>1.30% ±0.035</td><td>1.27%±0.037</td><td>1.29%±0.034</td></tr><tr><td colspan=\"4\"></td></tr><tr><td>Method</td><td colspan=\"2\">Backbone</td></tr><tr><td>StyleALAE (Pidhorskyi et al., 2020)</td><td colspan=\"2\">StyleGAN</td></tr><tr><td>Progressive GAN (Karras et al., 2018)</td><td colspan=\"2\">Progressive GAN</td></tr><tr><td>DC-VAE (ours)</td><td colspan=\"2\">Progressive GAN</td></tr></table>",
|
| 971 |
+
"bbox": [
|
| 972 |
+
241,
|
| 973 |
+
555,
|
| 974 |
+
758,
|
| 975 |
+
734
|
| 976 |
+
],
|
| 977 |
+
"page_idx": 8
|
| 978 |
+
},
|
| 979 |
+
{
|
| 980 |
+
"type": "text",
|
| 981 |
+
"text": "6 CONCLUSION ",
|
| 982 |
+
"text_level": 1,
|
| 983 |
+
"bbox": [
|
| 984 |
+
174,
|
| 985 |
+
782,
|
| 986 |
+
320,
|
| 987 |
+
799
|
| 988 |
+
],
|
| 989 |
+
"page_idx": 8
|
| 990 |
+
},
|
| 991 |
+
{
|
| 992 |
+
"type": "text",
|
| 993 |
+
"text": "In this paper, we have proposed dual contradistinctive generative autoencoder (DC-VAE), a new framework that integrates an instance-level discriminative loss (InfoNCE) with a set-level adversarial loss (GAN) into a single variational autoencoder framework. Our experiments show competitive results for a single model in several tasks, including image synthesis, image reconstruction, representation learning for image interpolation, and representation learning for classification. DC-VAE points to a encouraging direction that attains high-quality synthesis (decoding) and inference (encoding). ",
|
| 994 |
+
"bbox": [
|
| 995 |
+
174,
|
| 996 |
+
804,
|
| 997 |
+
826,
|
| 998 |
+
887
|
| 999 |
+
],
|
| 1000 |
+
"page_idx": 8
|
| 1001 |
+
},
|
| 1002 |
+
{
|
| 1003 |
+
"type": "text",
|
| 1004 |
+
"text": "REFERENCES ",
|
| 1005 |
+
"text_level": 1,
|
| 1006 |
+
"bbox": [
|
| 1007 |
+
174,
|
| 1008 |
+
102,
|
| 1009 |
+
287,
|
| 1010 |
+
118
|
| 1011 |
+
],
|
| 1012 |
+
"page_idx": 9
|
| 1013 |
+
},
|
| 1014 |
+
{
|
| 1015 |
+
"type": "text",
|
| 1016 |
+
"text": "Jyoti Aneja, Alexander Schwing, Jan Kautz, and Arash Vahdat. Ncp-vae: Variational autoencoders with noise contrastive priors, 2020. ",
|
| 1017 |
+
"bbox": [
|
| 1018 |
+
171,
|
| 1019 |
+
127,
|
| 1020 |
+
825,
|
| 1021 |
+
155
|
| 1022 |
+
],
|
| 1023 |
+
"page_idx": 9
|
| 1024 |
+
},
|
| 1025 |
+
{
|
| 1026 |
+
"type": "text",
|
| 1027 |
+
"text": "Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In ICML, 2017. ",
|
| 1028 |
+
"bbox": [
|
| 1029 |
+
173,
|
| 1030 |
+
165,
|
| 1031 |
+
825,
|
| 1032 |
+
194
|
| 1033 |
+
],
|
| 1034 |
+
"page_idx": 9
|
| 1035 |
+
},
|
| 1036 |
+
{
|
| 1037 |
+
"type": "text",
|
| 1038 |
+
"text": "Andrew Brock, Jeff Donahue, and Karen Simonyan. Large scale gan training for high fidelity natural image synthesis. arXiv preprint arXiv:1809.11096, 2018. ",
|
| 1039 |
+
"bbox": [
|
| 1040 |
+
171,
|
| 1041 |
+
205,
|
| 1042 |
+
823,
|
| 1043 |
+
234
|
| 1044 |
+
],
|
| 1045 |
+
"page_idx": 9
|
| 1046 |
+
},
|
| 1047 |
+
{
|
| 1048 |
+
"type": "text",
|
| 1049 |
+
"text": "Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Säckinger, and Roopak Shah. Signature verification using a “siamese” time delay neural network. In Neural Information Processing Systems, 1993. ",
|
| 1050 |
+
"bbox": [
|
| 1051 |
+
173,
|
| 1052 |
+
244,
|
| 1053 |
+
826,
|
| 1054 |
+
287
|
| 1055 |
+
],
|
| 1056 |
+
"page_idx": 9
|
| 1057 |
+
},
|
| 1058 |
+
{
|
| 1059 |
+
"type": "text",
|
| 1060 |
+
"text": "Gal Chechik, Varun Sharma, Uri Shalit, and Samy Bengio. Large scale online learning of image similarity through ranking. J. Mach. Learn. Res., 11:1109–1135, March 2010. ISSN 1532-4435. ",
|
| 1061 |
+
"bbox": [
|
| 1062 |
+
173,
|
| 1063 |
+
297,
|
| 1064 |
+
825,
|
| 1065 |
+
327
|
| 1066 |
+
],
|
| 1067 |
+
"page_idx": 9
|
| 1068 |
+
},
|
| 1069 |
+
{
|
| 1070 |
+
"type": "text",
|
| 1071 |
+
"text": "Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, 2018. ",
|
| 1072 |
+
"bbox": [
|
| 1073 |
+
174,
|
| 1074 |
+
337,
|
| 1075 |
+
826,
|
| 1076 |
+
380
|
| 1077 |
+
],
|
| 1078 |
+
"page_idx": 9
|
| 1079 |
+
},
|
| 1080 |
+
{
|
| 1081 |
+
"type": "text",
|
| 1082 |
+
"text": "Ting Chen, Xiaohua Zhai, Marvin Ritter, Mario Lucic, and Neil Houlsby. Self-supervised gans via auxiliary rotation loss. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 12154–12163, 2019. ",
|
| 1083 |
+
"bbox": [
|
| 1084 |
+
176,
|
| 1085 |
+
390,
|
| 1086 |
+
825,
|
| 1087 |
+
434
|
| 1088 |
+
],
|
| 1089 |
+
"page_idx": 9
|
| 1090 |
+
},
|
| 1091 |
+
{
|
| 1092 |
+
"type": "text",
|
| 1093 |
+
"text": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv:2002.05709, 2020. ",
|
| 1094 |
+
"bbox": [
|
| 1095 |
+
174,
|
| 1096 |
+
444,
|
| 1097 |
+
825,
|
| 1098 |
+
473
|
| 1099 |
+
],
|
| 1100 |
+
"page_idx": 9
|
| 1101 |
+
},
|
| 1102 |
+
{
|
| 1103 |
+
"type": "text",
|
| 1104 |
+
"text": "Sumit Chopra, Raia Hadsell, and Yann LeCun. Learning a similarity metric discriminatively, with application to face verification. In CVPR, pp. 539–546, 2005. ISBN 0769523722. ",
|
| 1105 |
+
"bbox": [
|
| 1106 |
+
173,
|
| 1107 |
+
483,
|
| 1108 |
+
821,
|
| 1109 |
+
513
|
| 1110 |
+
],
|
| 1111 |
+
"page_idx": 9
|
| 1112 |
+
},
|
| 1113 |
+
{
|
| 1114 |
+
"type": "text",
|
| 1115 |
+
"text": "Adam Coates, Andrew $\\mathrm { N g }$ , and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In AISTATS, pp. 215–223, 2011. ",
|
| 1116 |
+
"bbox": [
|
| 1117 |
+
173,
|
| 1118 |
+
523,
|
| 1119 |
+
823,
|
| 1120 |
+
553
|
| 1121 |
+
],
|
| 1122 |
+
"page_idx": 9
|
| 1123 |
+
},
|
| 1124 |
+
{
|
| 1125 |
+
"type": "text",
|
| 1126 |
+
"text": "Adji B Dieng, Francisco JR Ruiz, David M Blei, and Michalis K Titsias. Prescribed generative adversarial networks. arXiv:1910.04302, 2019. ",
|
| 1127 |
+
"bbox": [
|
| 1128 |
+
171,
|
| 1129 |
+
563,
|
| 1130 |
+
825,
|
| 1131 |
+
592
|
| 1132 |
+
],
|
| 1133 |
+
"page_idx": 9
|
| 1134 |
+
},
|
| 1135 |
+
{
|
| 1136 |
+
"type": "text",
|
| 1137 |
+
"text": "Zheng Ding, Yifan Xu, Weijian Xu, Gaurav Parmar, Yang Yang, Max Welling, and Zhuowen Tu. Guided variational autoencoder for disentanglement learning. In CVPR, 2020. ",
|
| 1138 |
+
"bbox": [
|
| 1139 |
+
171,
|
| 1140 |
+
602,
|
| 1141 |
+
825,
|
| 1142 |
+
632
|
| 1143 |
+
],
|
| 1144 |
+
"page_idx": 9
|
| 1145 |
+
},
|
| 1146 |
+
{
|
| 1147 |
+
"type": "text",
|
| 1148 |
+
"text": "Jeff Donahue, Philipp Krähenbühl, and Trevor Darrell. Adversarial feature learning. In ICLR, 2017. ",
|
| 1149 |
+
"bbox": [
|
| 1150 |
+
173,
|
| 1151 |
+
641,
|
| 1152 |
+
825,
|
| 1153 |
+
657
|
| 1154 |
+
],
|
| 1155 |
+
"page_idx": 9
|
| 1156 |
+
},
|
| 1157 |
+
{
|
| 1158 |
+
"type": "text",
|
| 1159 |
+
"text": "Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially learned inference. In ICLR, 2017. ",
|
| 1160 |
+
"bbox": [
|
| 1161 |
+
171,
|
| 1162 |
+
667,
|
| 1163 |
+
825,
|
| 1164 |
+
696
|
| 1165 |
+
],
|
| 1166 |
+
"page_idx": 9
|
| 1167 |
+
},
|
| 1168 |
+
{
|
| 1169 |
+
"type": "text",
|
| 1170 |
+
"text": "Xinyu Gong, Shiyu Chang, Yifan Jiang, and Zhangyang Wang. Autogan: Neural architecture search for generative adversarial networks. In ICCV, 2019. ",
|
| 1171 |
+
"bbox": [
|
| 1172 |
+
173,
|
| 1173 |
+
707,
|
| 1174 |
+
825,
|
| 1175 |
+
736
|
| 1176 |
+
],
|
| 1177 |
+
"page_idx": 9
|
| 1178 |
+
},
|
| 1179 |
+
{
|
| 1180 |
+
"type": "text",
|
| 1181 |
+
"text": "Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, 2014. ",
|
| 1182 |
+
"bbox": [
|
| 1183 |
+
174,
|
| 1184 |
+
746,
|
| 1185 |
+
826,
|
| 1186 |
+
790
|
| 1187 |
+
],
|
| 1188 |
+
"page_idx": 9
|
| 1189 |
+
},
|
| 1190 |
+
{
|
| 1191 |
+
"type": "text",
|
| 1192 |
+
"text": "Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of wasserstein gans. In Advances in neural information processing systems, 2017. ",
|
| 1193 |
+
"bbox": [
|
| 1194 |
+
174,
|
| 1195 |
+
800,
|
| 1196 |
+
826,
|
| 1197 |
+
843
|
| 1198 |
+
],
|
| 1199 |
+
"page_idx": 9
|
| 1200 |
+
},
|
| 1201 |
+
{
|
| 1202 |
+
"type": "text",
|
| 1203 |
+
"text": "T. Guo, C. Xu, J. Huang, Y. Wang, B. Shi, C. Xu, and D. Tao. On positive-unlabeled classification in gan. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8382–8390, Los Alamitos, CA, USA, jun 2020. IEEE Computer Society. doi: 10.1109/CVPR42600.2020.00841. URL https://doi.ieeecomputersociety.org/ 10.1109/CVPR42600.2020.00841. ",
|
| 1204 |
+
"bbox": [
|
| 1205 |
+
174,
|
| 1206 |
+
853,
|
| 1207 |
+
825,
|
| 1208 |
+
922
|
| 1209 |
+
],
|
| 1210 |
+
"page_idx": 9
|
| 1211 |
+
},
|
| 1212 |
+
{
|
| 1213 |
+
"type": "text",
|
| 1214 |
+
"text": "Michael U. Gutmann and Aapo Hyvärinen. Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics. J. Mach. Learn. Res., 13(null):307–361, February 2012. ISSN 1532-4435. ",
|
| 1215 |
+
"bbox": [
|
| 1216 |
+
176,
|
| 1217 |
+
103,
|
| 1218 |
+
825,
|
| 1219 |
+
146
|
| 1220 |
+
],
|
| 1221 |
+
"page_idx": 10
|
| 1222 |
+
},
|
| 1223 |
+
{
|
| 1224 |
+
"type": "text",
|
| 1225 |
+
"text": "Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In CVPR, 2006. ",
|
| 1226 |
+
"bbox": [
|
| 1227 |
+
171,
|
| 1228 |
+
155,
|
| 1229 |
+
823,
|
| 1230 |
+
185
|
| 1231 |
+
],
|
| 1232 |
+
"page_idx": 10
|
| 1233 |
+
},
|
| 1234 |
+
{
|
| 1235 |
+
"type": "text",
|
| 1236 |
+
"text": "Hao He, Hao Wang, Guang-He Lee, and Yonglong Tian. Probgan: Towards probabilistic gan with theoretical guarantees. In ICLR, 2019. ",
|
| 1237 |
+
"bbox": [
|
| 1238 |
+
171,
|
| 1239 |
+
194,
|
| 1240 |
+
821,
|
| 1241 |
+
223
|
| 1242 |
+
],
|
| 1243 |
+
"page_idx": 10
|
| 1244 |
+
},
|
| 1245 |
+
{
|
| 1246 |
+
"type": "text",
|
| 1247 |
+
"text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016. ",
|
| 1248 |
+
"bbox": [
|
| 1249 |
+
173,
|
| 1250 |
+
233,
|
| 1251 |
+
823,
|
| 1252 |
+
262
|
| 1253 |
+
],
|
| 1254 |
+
"page_idx": 10
|
| 1255 |
+
},
|
| 1256 |
+
{
|
| 1257 |
+
"type": "text",
|
| 1258 |
+
"text": "Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, 2020. ",
|
| 1259 |
+
"bbox": [
|
| 1260 |
+
173,
|
| 1261 |
+
272,
|
| 1262 |
+
825,
|
| 1263 |
+
301
|
| 1264 |
+
],
|
| 1265 |
+
"page_idx": 10
|
| 1266 |
+
},
|
| 1267 |
+
{
|
| 1268 |
+
"type": "text",
|
| 1269 |
+
"text": "Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in Neural Information Processing Systems, 2017. ",
|
| 1270 |
+
"bbox": [
|
| 1271 |
+
173,
|
| 1272 |
+
310,
|
| 1273 |
+
826,
|
| 1274 |
+
354
|
| 1275 |
+
],
|
| 1276 |
+
"page_idx": 10
|
| 1277 |
+
},
|
| 1278 |
+
{
|
| 1279 |
+
"type": "text",
|
| 1280 |
+
"text": "Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In ICLR, 2017. ",
|
| 1281 |
+
"bbox": [
|
| 1282 |
+
174,
|
| 1283 |
+
363,
|
| 1284 |
+
825,
|
| 1285 |
+
406
|
| 1286 |
+
],
|
| 1287 |
+
"page_idx": 10
|
| 1288 |
+
},
|
| 1289 |
+
{
|
| 1290 |
+
"type": "text",
|
| 1291 |
+
"text": "Geoffrey E Hinton and Richard S Zemel. Autoencoders, minimum description length and helmholtz free energy. In Advances in neural information processing systems, 1994. ",
|
| 1292 |
+
"bbox": [
|
| 1293 |
+
173,
|
| 1294 |
+
415,
|
| 1295 |
+
823,
|
| 1296 |
+
445
|
| 1297 |
+
],
|
| 1298 |
+
"page_idx": 10
|
| 1299 |
+
},
|
| 1300 |
+
{
|
| 1301 |
+
"type": "text",
|
| 1302 |
+
"text": "Quan Hoang, Tu Dinh Nguyen, Trung Le, and Dinh Phung. Mgan: training generative adversarial nets with multiple generators. In ICLR, 2018. ",
|
| 1303 |
+
"bbox": [
|
| 1304 |
+
171,
|
| 1305 |
+
454,
|
| 1306 |
+
823,
|
| 1307 |
+
484
|
| 1308 |
+
],
|
| 1309 |
+
"page_idx": 10
|
| 1310 |
+
},
|
| 1311 |
+
{
|
| 1312 |
+
"type": "text",
|
| 1313 |
+
"text": "Yedid Hoshen, Ke Li, and Jitendra Malik. Non-adversarial image synthesis with generative latent nearest neighbors. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5811–5819, 2019. ",
|
| 1314 |
+
"bbox": [
|
| 1315 |
+
176,
|
| 1316 |
+
493,
|
| 1317 |
+
825,
|
| 1318 |
+
536
|
| 1319 |
+
],
|
| 1320 |
+
"page_idx": 10
|
| 1321 |
+
},
|
| 1322 |
+
{
|
| 1323 |
+
"type": "text",
|
| 1324 |
+
"text": "Huaibo Huang, Ran He, Zhenan Sun, Tieniu Tan, et al. Introvae: Introspective variational autoencoders for photographic image synthesis. In Advances in Neural Information Processing Systems, pp. 52–63, 2018. ",
|
| 1325 |
+
"bbox": [
|
| 1326 |
+
173,
|
| 1327 |
+
545,
|
| 1328 |
+
826,
|
| 1329 |
+
588
|
| 1330 |
+
],
|
| 1331 |
+
"page_idx": 10
|
| 1332 |
+
},
|
| 1333 |
+
{
|
| 1334 |
+
"type": "text",
|
| 1335 |
+
"text": "Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. CVPR, 2017. ",
|
| 1336 |
+
"bbox": [
|
| 1337 |
+
173,
|
| 1338 |
+
598,
|
| 1339 |
+
825,
|
| 1340 |
+
628
|
| 1341 |
+
],
|
| 1342 |
+
"page_idx": 10
|
| 1343 |
+
},
|
| 1344 |
+
{
|
| 1345 |
+
"type": "text",
|
| 1346 |
+
"text": "Long Jin, Justin Lazarow, and Zhuowen Tu. Introspective classification with convolutional nets. In Advances in Neural Information Processing Systems, 2017. ",
|
| 1347 |
+
"bbox": [
|
| 1348 |
+
173,
|
| 1349 |
+
637,
|
| 1350 |
+
825,
|
| 1351 |
+
667
|
| 1352 |
+
],
|
| 1353 |
+
"page_idx": 10
|
| 1354 |
+
},
|
| 1355 |
+
{
|
| 1356 |
+
"type": "text",
|
| 1357 |
+
"text": "Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In ECCV, 2016. ",
|
| 1358 |
+
"bbox": [
|
| 1359 |
+
169,
|
| 1360 |
+
676,
|
| 1361 |
+
825,
|
| 1362 |
+
705
|
| 1363 |
+
],
|
| 1364 |
+
"page_idx": 10
|
| 1365 |
+
},
|
| 1366 |
+
{
|
| 1367 |
+
"type": "text",
|
| 1368 |
+
"text": "Alexia Jolicoeur-Martineau. The relativistic discriminator: a key element missing from standard gan. arXiv preprint arXiv:1807.00734, 2018. ",
|
| 1369 |
+
"bbox": [
|
| 1370 |
+
171,
|
| 1371 |
+
714,
|
| 1372 |
+
825,
|
| 1373 |
+
744
|
| 1374 |
+
],
|
| 1375 |
+
"page_idx": 10
|
| 1376 |
+
},
|
| 1377 |
+
{
|
| 1378 |
+
"type": "text",
|
| 1379 |
+
"text": "Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. In ICLR, 2018. ",
|
| 1380 |
+
"bbox": [
|
| 1381 |
+
169,
|
| 1382 |
+
753,
|
| 1383 |
+
825,
|
| 1384 |
+
784
|
| 1385 |
+
],
|
| 1386 |
+
"page_idx": 10
|
| 1387 |
+
},
|
| 1388 |
+
{
|
| 1389 |
+
"type": "text",
|
| 1390 |
+
"text": "Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4401–4410, 2019. ",
|
| 1391 |
+
"bbox": [
|
| 1392 |
+
176,
|
| 1393 |
+
792,
|
| 1394 |
+
825,
|
| 1395 |
+
835
|
| 1396 |
+
],
|
| 1397 |
+
"page_idx": 10
|
| 1398 |
+
},
|
| 1399 |
+
{
|
| 1400 |
+
"type": "text",
|
| 1401 |
+
"text": "Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In ICML, 2018. ",
|
| 1402 |
+
"bbox": [
|
| 1403 |
+
174,
|
| 1404 |
+
844,
|
| 1405 |
+
687,
|
| 1406 |
+
861
|
| 1407 |
+
],
|
| 1408 |
+
"page_idx": 10
|
| 1409 |
+
},
|
| 1410 |
+
{
|
| 1411 |
+
"type": "text",
|
| 1412 |
+
"text": "Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In ICLR, 2014. ",
|
| 1413 |
+
"bbox": [
|
| 1414 |
+
174,
|
| 1415 |
+
869,
|
| 1416 |
+
748,
|
| 1417 |
+
886
|
| 1418 |
+
],
|
| 1419 |
+
"page_idx": 10
|
| 1420 |
+
},
|
| 1421 |
+
{
|
| 1422 |
+
"type": "text",
|
| 1423 |
+
"text": "Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009. ",
|
| 1424 |
+
"bbox": [
|
| 1425 |
+
176,
|
| 1426 |
+
895,
|
| 1427 |
+
823,
|
| 1428 |
+
924
|
| 1429 |
+
],
|
| 1430 |
+
"page_idx": 10
|
| 1431 |
+
},
|
| 1432 |
+
{
|
| 1433 |
+
"type": "text",
|
| 1434 |
+
"text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, 2012. ",
|
| 1435 |
+
"bbox": [
|
| 1436 |
+
173,
|
| 1437 |
+
103,
|
| 1438 |
+
825,
|
| 1439 |
+
132
|
| 1440 |
+
],
|
| 1441 |
+
"page_idx": 11
|
| 1442 |
+
},
|
| 1443 |
+
{
|
| 1444 |
+
"type": "text",
|
| 1445 |
+
"text": "Brian Kulis et al. Metric learning: A survey. Foundations and trends in machine learning, 5(4): 287–364, 2012. ",
|
| 1446 |
+
"bbox": [
|
| 1447 |
+
173,
|
| 1448 |
+
140,
|
| 1449 |
+
825,
|
| 1450 |
+
170
|
| 1451 |
+
],
|
| 1452 |
+
"page_idx": 11
|
| 1453 |
+
},
|
| 1454 |
+
{
|
| 1455 |
+
"type": "text",
|
| 1456 |
+
"text": "Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, Hugo Larochelle, and Ole Winther. Autoencoding beyond pixels using a learned similarity metric. In ICML, 2016. ",
|
| 1457 |
+
"bbox": [
|
| 1458 |
+
173,
|
| 1459 |
+
178,
|
| 1460 |
+
825,
|
| 1461 |
+
208
|
| 1462 |
+
],
|
| 1463 |
+
"page_idx": 11
|
| 1464 |
+
},
|
| 1465 |
+
{
|
| 1466 |
+
"type": "text",
|
| 1467 |
+
"text": "Yann LeCun. Modeles connexionnistes de lapprentissage. PhD thesis, PhD thesis, These de Doctorat, Universite Paris 6, 1987. ",
|
| 1468 |
+
"bbox": [
|
| 1469 |
+
176,
|
| 1470 |
+
215,
|
| 1471 |
+
825,
|
| 1472 |
+
244
|
| 1473 |
+
],
|
| 1474 |
+
"page_idx": 11
|
| 1475 |
+
},
|
| 1476 |
+
{
|
| 1477 |
+
"type": "text",
|
| 1478 |
+
"text": "Yann LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/mnist/, 1998. ",
|
| 1479 |
+
"bbox": [
|
| 1480 |
+
176,
|
| 1481 |
+
252,
|
| 1482 |
+
821,
|
| 1483 |
+
270
|
| 1484 |
+
],
|
| 1485 |
+
"page_idx": 11
|
| 1486 |
+
},
|
| 1487 |
+
{
|
| 1488 |
+
"type": "text",
|
| 1489 |
+
"text": "Chen-Yu Lee, Saining Xie, Patrick Gallagher, Zhengyou Zhang, and Zhuowen Tu. Deeply-supervised nets. In Artificial intelligence and statistics, pp. 562–570, 2015. ",
|
| 1490 |
+
"bbox": [
|
| 1491 |
+
173,
|
| 1492 |
+
276,
|
| 1493 |
+
823,
|
| 1494 |
+
306
|
| 1495 |
+
],
|
| 1496 |
+
"page_idx": 11
|
| 1497 |
+
},
|
| 1498 |
+
{
|
| 1499 |
+
"type": "text",
|
| 1500 |
+
"text": "Kwonjoon Lee, Weijian Xu, Fan Fan, and Zhuowen Tu. Wasserstein introspective neural networks. In CVPR, 2018. ",
|
| 1501 |
+
"bbox": [
|
| 1502 |
+
173,
|
| 1503 |
+
314,
|
| 1504 |
+
825,
|
| 1505 |
+
343
|
| 1506 |
+
],
|
| 1507 |
+
"page_idx": 11
|
| 1508 |
+
},
|
| 1509 |
+
{
|
| 1510 |
+
"type": "text",
|
| 1511 |
+
"text": "Kwonjoon Lee, Subhransu Maji, Avinash Ravichandran, and Stefano Soatto. Meta-learning with differentiable convex optimization. In CVPR, 2019. ",
|
| 1512 |
+
"bbox": [
|
| 1513 |
+
171,
|
| 1514 |
+
352,
|
| 1515 |
+
823,
|
| 1516 |
+
382
|
| 1517 |
+
],
|
| 1518 |
+
"page_idx": 11
|
| 1519 |
+
},
|
| 1520 |
+
{
|
| 1521 |
+
"type": "text",
|
| 1522 |
+
"text": "Chieh Hubert Lin, Chia-Che Chang, Yu-Sheng Chen, Da-Cheng Juan, Wei Wei, and Hwann-Tzong Chen. Coco-gan: generation by parts via conditional coordinating. In Proceedings of the IEEE International Conference on Computer Vision, pp. 4512–4521, 2019. ",
|
| 1523 |
+
"bbox": [
|
| 1524 |
+
176,
|
| 1525 |
+
388,
|
| 1526 |
+
823,
|
| 1527 |
+
433
|
| 1528 |
+
],
|
| 1529 |
+
"page_idx": 11
|
| 1530 |
+
},
|
| 1531 |
+
{
|
| 1532 |
+
"type": "text",
|
| 1533 |
+
"text": "Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, 2015. ",
|
| 1534 |
+
"bbox": [
|
| 1535 |
+
171,
|
| 1536 |
+
440,
|
| 1537 |
+
825,
|
| 1538 |
+
470
|
| 1539 |
+
],
|
| 1540 |
+
"page_idx": 11
|
| 1541 |
+
},
|
| 1542 |
+
{
|
| 1543 |
+
"type": "text",
|
| 1544 |
+
"text": "Zhuang Ma and Michael Collins. Noise contrastive estimation and negative sampling for conditional models: Consistency and statistical efficiency. In EMNLP, 2018. ",
|
| 1545 |
+
"bbox": [
|
| 1546 |
+
171,
|
| 1547 |
+
478,
|
| 1548 |
+
823,
|
| 1549 |
+
508
|
| 1550 |
+
],
|
| 1551 |
+
"page_idx": 11
|
| 1552 |
+
},
|
| 1553 |
+
{
|
| 1554 |
+
"type": "text",
|
| 1555 |
+
"text": "Tomasz Malisiewicz, Abhinav Gupta, and Alexei A. Efros. Ensemble of exemplar-svms for object detection and beyond. In ICCV, 2011. ",
|
| 1556 |
+
"bbox": [
|
| 1557 |
+
171,
|
| 1558 |
+
515,
|
| 1559 |
+
823,
|
| 1560 |
+
545
|
| 1561 |
+
],
|
| 1562 |
+
"page_idx": 11
|
| 1563 |
+
},
|
| 1564 |
+
{
|
| 1565 |
+
"type": "text",
|
| 1566 |
+
"text": "Xudong Mao, Qing Li, Haoran Xie, Raymond YK Lau, Zhen Wang, and Stephen Paul Smolley. Least squares generative adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2794–2802, 2017. ",
|
| 1567 |
+
"bbox": [
|
| 1568 |
+
176,
|
| 1569 |
+
553,
|
| 1570 |
+
823,
|
| 1571 |
+
597
|
| 1572 |
+
],
|
| 1573 |
+
"page_idx": 11
|
| 1574 |
+
},
|
| 1575 |
+
{
|
| 1576 |
+
"type": "text",
|
| 1577 |
+
"text": "Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. In ICLR, 2018. ",
|
| 1578 |
+
"bbox": [
|
| 1579 |
+
171,
|
| 1580 |
+
604,
|
| 1581 |
+
825,
|
| 1582 |
+
633
|
| 1583 |
+
],
|
| 1584 |
+
"page_idx": 11
|
| 1585 |
+
},
|
| 1586 |
+
{
|
| 1587 |
+
"type": "text",
|
| 1588 |
+
"text": "Stanislav Pidhorskyi, Donald Adjeroh, and Gianfranco Doretto. Adversarial latent autoencoders, 2020. ",
|
| 1589 |
+
"bbox": [
|
| 1590 |
+
171,
|
| 1591 |
+
642,
|
| 1592 |
+
825,
|
| 1593 |
+
671
|
| 1594 |
+
],
|
| 1595 |
+
"page_idx": 11
|
| 1596 |
+
},
|
| 1597 |
+
{
|
| 1598 |
+
"type": "text",
|
| 1599 |
+
"text": "Ben Poole, Sherjil Ozair, Aaron Van Den Oord, Alex Alemi, and George Tucker. On variational bounds of mutual information. In ICML, 2019. ",
|
| 1600 |
+
"bbox": [
|
| 1601 |
+
171,
|
| 1602 |
+
679,
|
| 1603 |
+
825,
|
| 1604 |
+
709
|
| 1605 |
+
],
|
| 1606 |
+
"page_idx": 11
|
| 1607 |
+
},
|
| 1608 |
+
{
|
| 1609 |
+
"type": "text",
|
| 1610 |
+
"text": "Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016. ",
|
| 1611 |
+
"bbox": [
|
| 1612 |
+
171,
|
| 1613 |
+
717,
|
| 1614 |
+
823,
|
| 1615 |
+
746
|
| 1616 |
+
],
|
| 1617 |
+
"page_idx": 11
|
| 1618 |
+
},
|
| 1619 |
+
{
|
| 1620 |
+
"type": "text",
|
| 1621 |
+
"text": "Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In David van Dyk and Max Welling (eds.), AISTATS, volume 5, pp. 448–455, 16–18 Apr 2009. ",
|
| 1622 |
+
"bbox": [
|
| 1623 |
+
171,
|
| 1624 |
+
753,
|
| 1625 |
+
823,
|
| 1626 |
+
785
|
| 1627 |
+
],
|
| 1628 |
+
"page_idx": 11
|
| 1629 |
+
},
|
| 1630 |
+
{
|
| 1631 |
+
"type": "text",
|
| 1632 |
+
"text": "Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, 2016. ",
|
| 1633 |
+
"bbox": [
|
| 1634 |
+
174,
|
| 1635 |
+
791,
|
| 1636 |
+
825,
|
| 1637 |
+
835
|
| 1638 |
+
],
|
| 1639 |
+
"page_idx": 11
|
| 1640 |
+
},
|
| 1641 |
+
{
|
| 1642 |
+
"type": "text",
|
| 1643 |
+
"text": "Akash Srivastava, Lazar Valkov, Chris Russell, Michael U Gutmann, and Charles Sutton. Veegan: Reducing mode collapse in gans using implicit variational learning. In Advances in Neural Information Processing Systems, pp. 3308–3318, 2017. ",
|
| 1644 |
+
"bbox": [
|
| 1645 |
+
173,
|
| 1646 |
+
843,
|
| 1647 |
+
823,
|
| 1648 |
+
887
|
| 1649 |
+
],
|
| 1650 |
+
"page_idx": 11
|
| 1651 |
+
},
|
| 1652 |
+
{
|
| 1653 |
+
"type": "text",
|
| 1654 |
+
"text": "Ilya Tolstikhin, Olivier Bousquet, Sylvain Gelly, and Bernhard Schoelkopf. Wasserstein auto-encoders. arXiv preprint arXiv:1711.01558, 2017. ",
|
| 1655 |
+
"bbox": [
|
| 1656 |
+
174,
|
| 1657 |
+
895,
|
| 1658 |
+
823,
|
| 1659 |
+
924
|
| 1660 |
+
],
|
| 1661 |
+
"page_idx": 11
|
| 1662 |
+
},
|
| 1663 |
+
{
|
| 1664 |
+
"type": "text",
|
| 1665 |
+
"text": "Zhuowen Tu. Learning generative models via discriminative approaches. In CVPR, 2007. \nArash Vahdat and Jan Kautz. NVAE: A deep hierarchical variational autoencoder. In Neural Information Processing Systems (NeurIPS), 2020. \nAäron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. CoRR, abs/1807.03748, 2018. URL http://arxiv.org/abs/1807.03748. \nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, 2017. \nWei Wang, Yuan Sun, and Saman Halgamuge. Improving mmd-gan training with repulsive loss function. In ICLR, 2019. \nZhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In CVPR, pp. 3733–3742, 2018. \nJianwen Xie, Yang Lu, Song-Chun Zhu, and Yingnian Wu. A theory of generative convnet. In ICML, 2016. \nFisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. Lsun: Construction of a largescale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015. \nZijun Zhang, Ruixiang Zhang, Zongpeng Li, Yoshua Bengio, and Liam Paull. Perceptual generative autoencoders. In ICLR, 2019. ",
|
| 1666 |
+
"bbox": [
|
| 1667 |
+
169,
|
| 1668 |
+
101,
|
| 1669 |
+
828,
|
| 1670 |
+
453
|
| 1671 |
+
],
|
| 1672 |
+
"page_idx": 12
|
| 1673 |
+
},
|
| 1674 |
+
{
|
| 1675 |
+
"type": "text",
|
| 1676 |
+
"text": "A APPENDIX ",
|
| 1677 |
+
"bbox": [
|
| 1678 |
+
176,
|
| 1679 |
+
102,
|
| 1680 |
+
299,
|
| 1681 |
+
117
|
| 1682 |
+
],
|
| 1683 |
+
"page_idx": 13
|
| 1684 |
+
},
|
| 1685 |
+
{
|
| 1686 |
+
"type": "text",
|
| 1687 |
+
"text": "A.1 Additional reconstruction results ",
|
| 1688 |
+
"text_level": 1,
|
| 1689 |
+
"bbox": [
|
| 1690 |
+
176,
|
| 1691 |
+
135,
|
| 1692 |
+
449,
|
| 1693 |
+
150
|
| 1694 |
+
],
|
| 1695 |
+
"page_idx": 13
|
| 1696 |
+
},
|
| 1697 |
+
{
|
| 1698 |
+
"type": "image",
|
| 1699 |
+
"img_path": "images/d23b9983778258d4cefbc26c544864a43b704d254867d2e7e3d4798b25f759c0.jpg",
|
| 1700 |
+
"image_caption": [
|
| 1701 |
+
"Figure 7: Additional CelebA-HQ (Karras et al., 2018) reconstruction images (resolution $5 1 2 \\times 5 1 2$ ) generated by DC-VAE (ours) "
|
| 1702 |
+
],
|
| 1703 |
+
"image_footnote": [],
|
| 1704 |
+
"bbox": [
|
| 1705 |
+
214,
|
| 1706 |
+
154,
|
| 1707 |
+
782,
|
| 1708 |
+
872
|
| 1709 |
+
],
|
| 1710 |
+
"page_idx": 13
|
| 1711 |
+
},
|
| 1712 |
+
{
|
| 1713 |
+
"type": "image",
|
| 1714 |
+
"img_path": "images/80075727dd75d44d98158df70e468f65dffede357b4ae2bb9d94d297d9cfdba1.jpg",
|
| 1715 |
+
"image_caption": [
|
| 1716 |
+
"Figure 8: Additional LSUN Bedroom (Yu et al., 2015) reconstruction images (resolution $1 2 8 \\times 1 2 8$ "
|
| 1717 |
+
],
|
| 1718 |
+
"image_footnote": [],
|
| 1719 |
+
"bbox": [
|
| 1720 |
+
191,
|
| 1721 |
+
94,
|
| 1722 |
+
807,
|
| 1723 |
+
734
|
| 1724 |
+
],
|
| 1725 |
+
"page_idx": 14
|
| 1726 |
+
},
|
| 1727 |
+
{
|
| 1728 |
+
"type": "text",
|
| 1729 |
+
"text": "A.2 Analysing the latent space ",
|
| 1730 |
+
"text_level": 1,
|
| 1731 |
+
"bbox": [
|
| 1732 |
+
176,
|
| 1733 |
+
785,
|
| 1734 |
+
400,
|
| 1735 |
+
800
|
| 1736 |
+
],
|
| 1737 |
+
"page_idx": 14
|
| 1738 |
+
},
|
| 1739 |
+
{
|
| 1740 |
+
"type": "text",
|
| 1741 |
+
"text": "In this section we analyse the smoothness of the latent space learnt by DC-VAE. In Figure 9 we qualitatively show the high resolution $( 5 1 2 \\times 5 1 2 )$ CelebA-HQ Karras et al. (2018) images generated by an evenly spaced linear blending between two latent vectors. In Fig. 6 we show that DC-VAE is able to perform meaningful attribute editing on images while retaining the original identity. To perform image editing, we first need to compute the direction vector in the latent space that correspond to a desired attribute (e.g. has glasses, has blonde hair, is a woman, has facial hair). We compute these attribute direction vectors by selecting 20 images that have the attribute and 20 images that do not have the attribute, obtaining the corresponding pairs of 20 latent vectors, and calculating the difference of the mean. The results in Fig. 6 show that these direction vectors can be added to a latent vector to add a diverse combination of desired image attributes while retaining the original identity of the individual. ",
|
| 1742 |
+
"bbox": [
|
| 1743 |
+
174,
|
| 1744 |
+
811,
|
| 1745 |
+
825,
|
| 1746 |
+
924
|
| 1747 |
+
],
|
| 1748 |
+
"page_idx": 14
|
| 1749 |
+
},
|
| 1750 |
+
{
|
| 1751 |
+
"type": "text",
|
| 1752 |
+
"text": "",
|
| 1753 |
+
"bbox": [
|
| 1754 |
+
176,
|
| 1755 |
+
103,
|
| 1756 |
+
823,
|
| 1757 |
+
145
|
| 1758 |
+
],
|
| 1759 |
+
"page_idx": 15
|
| 1760 |
+
},
|
| 1761 |
+
{
|
| 1762 |
+
"type": "text",
|
| 1763 |
+
"text": "Additionally we corroborate the above qualitative results quantitatively by inspecting the Perceptual Path Length (PPL) Karras et al. (2019) of our learn DC-VAE Decoder (Tab. 6) to measures the disentanglement of the latent space. We note that although ProgressiveGAN (ours base model) has a better FID score, DC-VAE has a lower PPL score which indicated that the latent space learnt is more disentangled. ",
|
| 1764 |
+
"bbox": [
|
| 1765 |
+
174,
|
| 1766 |
+
152,
|
| 1767 |
+
825,
|
| 1768 |
+
223
|
| 1769 |
+
],
|
| 1770 |
+
"page_idx": 15
|
| 1771 |
+
},
|
| 1772 |
+
{
|
| 1773 |
+
"type": "image",
|
| 1774 |
+
"img_path": "images/e37eb5cfabccb53487cc405a530b473aadc02fd9f3d473ee9e6f58512be4e24d.jpg",
|
| 1775 |
+
"image_caption": [
|
| 1776 |
+
"Figure 9: Additional latent space interpolations on CelebA-HQ (Karras et al., 2018) (resolution $5 1 \\bar { 2 } \\times 5 1 2 $ ) "
|
| 1777 |
+
],
|
| 1778 |
+
"image_footnote": [],
|
| 1779 |
+
"bbox": [
|
| 1780 |
+
238,
|
| 1781 |
+
98,
|
| 1782 |
+
758,
|
| 1783 |
+
842
|
| 1784 |
+
],
|
| 1785 |
+
"page_idx": 16
|
| 1786 |
+
},
|
| 1787 |
+
{
|
| 1788 |
+
"type": "text",
|
| 1789 |
+
"text": "Source A ",
|
| 1790 |
+
"text_level": 1,
|
| 1791 |
+
"bbox": [
|
| 1792 |
+
514,
|
| 1793 |
+
112,
|
| 1794 |
+
598,
|
| 1795 |
+
130
|
| 1796 |
+
],
|
| 1797 |
+
"page_idx": 17
|
| 1798 |
+
},
|
| 1799 |
+
{
|
| 1800 |
+
"type": "image",
|
| 1801 |
+
"img_path": "images/b104187bb26f4ed10aba9331427859fe1296b02c7adf3efb99f95e7da489e3b2.jpg",
|
| 1802 |
+
"image_caption": [
|
| 1803 |
+
"Figure 10: Latent Mixing results on CelebA-HQ Karras et al. (2018). Each combined image in the grid is generated by replacing an arbitrary subset of Source A latent with the corresponding Source B latent. "
|
| 1804 |
+
],
|
| 1805 |
+
"image_footnote": [],
|
| 1806 |
+
"bbox": [
|
| 1807 |
+
205,
|
| 1808 |
+
133,
|
| 1809 |
+
821,
|
| 1810 |
+
670
|
| 1811 |
+
],
|
| 1812 |
+
"page_idx": 17
|
| 1813 |
+
},
|
| 1814 |
+
{
|
| 1815 |
+
"type": "text",
|
| 1816 |
+
"text": "A.3 Effect of negative samples ",
|
| 1817 |
+
"text_level": 1,
|
| 1818 |
+
"bbox": [
|
| 1819 |
+
176,
|
| 1820 |
+
763,
|
| 1821 |
+
398,
|
| 1822 |
+
779
|
| 1823 |
+
],
|
| 1824 |
+
"page_idx": 17
|
| 1825 |
+
},
|
| 1826 |
+
{
|
| 1827 |
+
"type": "text",
|
| 1828 |
+
"text": "In this section we analyse the effect of varying the number of negative samples used for contrastive learning. The figure 11 shows the reconstruction error on the CIFAR-10 Krizhevsky et al. (2009) test set as the negative samples is varied. We observe that a higher number of negative samples results in better reconstruction. We choose 8096 for all of our experiments because of memory constraints. ",
|
| 1829 |
+
"bbox": [
|
| 1830 |
+
174,
|
| 1831 |
+
791,
|
| 1832 |
+
825,
|
| 1833 |
+
847
|
| 1834 |
+
],
|
| 1835 |
+
"page_idx": 17
|
| 1836 |
+
},
|
| 1837 |
+
{
|
| 1838 |
+
"type": "text",
|
| 1839 |
+
"text": "A.4 Datasets used ",
|
| 1840 |
+
"text_level": 1,
|
| 1841 |
+
"bbox": [
|
| 1842 |
+
174,
|
| 1843 |
+
868,
|
| 1844 |
+
313,
|
| 1845 |
+
882
|
| 1846 |
+
],
|
| 1847 |
+
"page_idx": 17
|
| 1848 |
+
},
|
| 1849 |
+
{
|
| 1850 |
+
"type": "text",
|
| 1851 |
+
"text": "CIFAR-10 comprises 50,000 training images and 10,000 test images with a spatial resolution of $3 2 \\times 3 2$ . STL-10 is a similar dataset that contains 5,000 training images and 100,000 unlabeled images at $9 6 \\times 9 6$ resolution. We follow the procedure in AutoGAN(Gong et al., 2019) and resize the STL-10 images to $3 2 \\times 3 2$ . The CelebA dataset has 162,770 training images and 19,962 testing images, CelebA-HQ contains 29,000 training images with 1,000 test images of size $1 0 2 4 \\times 1 0 2 4$ and LSUN Bedroom has approximately 3M images. We resize all images progressively in these three datasets from $\\left( 4 \\times 4 \\right)$ to $( 5 1 2 \\times 5 1 2 )$ for the progressive training. ",
|
| 1852 |
+
"bbox": [
|
| 1853 |
+
173,
|
| 1854 |
+
895,
|
| 1855 |
+
825,
|
| 1856 |
+
924
|
| 1857 |
+
],
|
| 1858 |
+
"page_idx": 17
|
| 1859 |
+
},
|
| 1860 |
+
{
|
| 1861 |
+
"type": "image",
|
| 1862 |
+
"img_path": "images/ea9b1e55f52c40850b9a7f4890eb3d31c22694689a7b7e4827d598b67ec60185.jpg",
|
| 1863 |
+
"image_caption": [
|
| 1864 |
+
"Figure 11: Pixel reconstruction error on CIFAR-10 Krizhevsky et al. (2009) test set for varying number of negative samples "
|
| 1865 |
+
],
|
| 1866 |
+
"image_footnote": [],
|
| 1867 |
+
"bbox": [
|
| 1868 |
+
241,
|
| 1869 |
+
102,
|
| 1870 |
+
751,
|
| 1871 |
+
378
|
| 1872 |
+
],
|
| 1873 |
+
"page_idx": 18
|
| 1874 |
+
},
|
| 1875 |
+
{
|
| 1876 |
+
"type": "text",
|
| 1877 |
+
"text": "",
|
| 1878 |
+
"bbox": [
|
| 1879 |
+
174,
|
| 1880 |
+
452,
|
| 1881 |
+
825,
|
| 1882 |
+
522
|
| 1883 |
+
],
|
| 1884 |
+
"page_idx": 18
|
| 1885 |
+
},
|
| 1886 |
+
{
|
| 1887 |
+
"type": "text",
|
| 1888 |
+
"text": "A.5 Network architecture diagrams ",
|
| 1889 |
+
"text_level": 1,
|
| 1890 |
+
"bbox": [
|
| 1891 |
+
176,
|
| 1892 |
+
540,
|
| 1893 |
+
436,
|
| 1894 |
+
555
|
| 1895 |
+
],
|
| 1896 |
+
"page_idx": 18
|
| 1897 |
+
},
|
| 1898 |
+
{
|
| 1899 |
+
"type": "text",
|
| 1900 |
+
"text": "In Figures 15 we show the detailed network architecture of DC-VAE for input resolutions of $3 2 \\times 3 2$ Note that the comparison results shown in Figure 3 and Table 1 in the main paper, for VAE, VAE/GAN, VAE w/o GAN, and our proposed DC-VAE are all based on the same network architecture (shown in Figure 15 here), for a fair comparison. ",
|
| 1901 |
+
"bbox": [
|
| 1902 |
+
174,
|
| 1903 |
+
568,
|
| 1904 |
+
825,
|
| 1905 |
+
622
|
| 1906 |
+
],
|
| 1907 |
+
"page_idx": 18
|
| 1908 |
+
},
|
| 1909 |
+
{
|
| 1910 |
+
"type": "text",
|
| 1911 |
+
"text": "The network architectures shown in Figure 15 are adapted closely from the networks discovered by (Gong et al., 2019) through Neural Architecture Search. The DC-VAE developed in our paper is not tied to any particular CNN architecture. We choose the AutoGAN architecture (Gong et al., 2019) to start with a strong baseline. The decoder in Figure 15 matches the generator in (Gong et al., 2019). The encoder is built by modifying the output shape of the final linear layer in the discriminator of AutoGAN (Gong et al., 2019) to match the latent dimension and adding spectral normalization. The discriminator is used both for classifying real/fake images, and contrastive learning. For each layer we choose, we first apply 1x1 convolution and a linear layer, and then use this feature as an input to the contrastive module. For experiments at $3 2 \\times 3 2$ , we pick two different positions: the output of second residual conv block (lower level) and the output of the first linear layer (higher level). For experiments on higher resolution datasets we use a Progressive GAN (Karras et al., 2018) Generator and Discriminator as our backbone and apply similar modifications as described above. ",
|
| 1912 |
+
"bbox": [
|
| 1913 |
+
174,
|
| 1914 |
+
630,
|
| 1915 |
+
825,
|
| 1916 |
+
796
|
| 1917 |
+
],
|
| 1918 |
+
"page_idx": 18
|
| 1919 |
+
},
|
| 1920 |
+
{
|
| 1921 |
+
"type": "text",
|
| 1922 |
+
"text": "A.6 Training infrastructure ",
|
| 1923 |
+
"text_level": 1,
|
| 1924 |
+
"bbox": [
|
| 1925 |
+
176,
|
| 1926 |
+
815,
|
| 1927 |
+
382,
|
| 1928 |
+
829
|
| 1929 |
+
],
|
| 1930 |
+
"page_idx": 18
|
| 1931 |
+
},
|
| 1932 |
+
{
|
| 1933 |
+
"type": "text",
|
| 1934 |
+
"text": ".7 Further details about the representation learning experiments ",
|
| 1935 |
+
"text_level": 1,
|
| 1936 |
+
"bbox": [
|
| 1937 |
+
176,
|
| 1938 |
+
840,
|
| 1939 |
+
648,
|
| 1940 |
+
856
|
| 1941 |
+
],
|
| 1942 |
+
"page_idx": 18
|
| 1943 |
+
},
|
| 1944 |
+
{
|
| 1945 |
+
"type": "text",
|
| 1946 |
+
"text": "As seen in Table 4 in the main paper, we show the representation capability of DC-VAE following the procedure outlined in (Ding et al., 2020). We train our model on the MNIST dataset (LeCun, 1998) and measure the transferability though a classification task on the latent embedding vector. Specifically, we first pretrain the DC-VAE model on the training split of the MNIST dataset. Following that we freeze the DC-VAE model and train a linear classifier that takes latent embedding vector as the input and predicts the class label of the original image. ",
|
| 1947 |
+
"bbox": [
|
| 1948 |
+
176,
|
| 1949 |
+
867,
|
| 1950 |
+
825,
|
| 1951 |
+
924
|
| 1952 |
+
],
|
| 1953 |
+
"page_idx": 18
|
| 1954 |
+
},
|
| 1955 |
+
{
|
| 1956 |
+
"type": "text",
|
| 1957 |
+
"text": "",
|
| 1958 |
+
"bbox": [
|
| 1959 |
+
171,
|
| 1960 |
+
103,
|
| 1961 |
+
825,
|
| 1962 |
+
132
|
| 1963 |
+
],
|
| 1964 |
+
"page_idx": 19
|
| 1965 |
+
},
|
| 1966 |
+
{
|
| 1967 |
+
"type": "image",
|
| 1968 |
+
"img_path": "images/3c5320716f702bad43fe0cf45b0b8ced56928b785c4b34963a6070889c6f11b1.jpg",
|
| 1969 |
+
"image_caption": [
|
| 1970 |
+
"Figure 12: Visualization of the effect of adding each instance level and set level objectives. Table 1 and Figure 3 contain FID (Heusel et al., 2017) results and qualitative comparisons on the CIFAR-10 (Krizhevsky et al., 2009) that correspond to these settings. "
|
| 1971 |
+
],
|
| 1972 |
+
"image_footnote": [],
|
| 1973 |
+
"bbox": [
|
| 1974 |
+
178,
|
| 1975 |
+
146,
|
| 1976 |
+
857,
|
| 1977 |
+
478
|
| 1978 |
+
],
|
| 1979 |
+
"page_idx": 19
|
| 1980 |
+
},
|
| 1981 |
+
{
|
| 1982 |
+
"type": "image",
|
| 1983 |
+
"img_path": "images/f4b28a80df5527e12df8ade4efbd6486e878cf91a31e690082d82738fcabc266.jpg",
|
| 1984 |
+
"image_caption": [
|
| 1985 |
+
"Figure 13: DC-VAE synthesis images on LSUN images (Yu et al., 2015) (resolution $1 2 8 \\times 1 2 8$ ) "
|
| 1986 |
+
],
|
| 1987 |
+
"image_footnote": [],
|
| 1988 |
+
"bbox": [
|
| 1989 |
+
174,
|
| 1990 |
+
122,
|
| 1991 |
+
823,
|
| 1992 |
+
872
|
| 1993 |
+
],
|
| 1994 |
+
"page_idx": 20
|
| 1995 |
+
},
|
| 1996 |
+
{
|
| 1997 |
+
"type": "image",
|
| 1998 |
+
"img_path": "images/7662bdede29d98b93c1830a90a835f97a353ce8185547a07f2d7bd4a6b0d1cb0.jpg",
|
| 1999 |
+
"image_caption": [
|
| 2000 |
+
"Figure 14: DC-VAE reconstruction (a) and synthesis results (b) on STL10 (Coates et al., 2011) images (resolution $3 2 \\times 3 2 ,$ ). In (a) the top two rows are input images and the bottom two rows are the corresponding reconstruction images. "
|
| 2001 |
+
],
|
| 2002 |
+
"image_footnote": [],
|
| 2003 |
+
"bbox": [
|
| 2004 |
+
178,
|
| 2005 |
+
116,
|
| 2006 |
+
820,
|
| 2007 |
+
257
|
| 2008 |
+
],
|
| 2009 |
+
"page_idx": 21
|
| 2010 |
+
},
|
| 2011 |
+
{
|
| 2012 |
+
"type": "image",
|
| 2013 |
+
"img_path": "images/7c04b0b606881dacfe63396278b9d0f260d4b27c18e0964c60792d5765543fc7.jpg",
|
| 2014 |
+
"image_caption": [
|
| 2015 |
+
"Figure 15: Network architecture of DC-VAE for resolution $3 2 \\times 3 2$ for CIFAR-10 (Krizhevsky et al., 2009) and STL-10 (Coates et al., 2011). (a) is the Encoder. (b) is the Decoder. (c) is the Discriminator. "
|
| 2016 |
+
],
|
| 2017 |
+
"image_footnote": [],
|
| 2018 |
+
"bbox": [
|
| 2019 |
+
192,
|
| 2020 |
+
352,
|
| 2021 |
+
830,
|
| 2022 |
+
861
|
| 2023 |
+
],
|
| 2024 |
+
"page_idx": 21
|
| 2025 |
+
}
|
| 2026 |
+
]
|
parse/train/5vShUEyjmm/5vShUEyjmm_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/5vShUEyjmm/5vShUEyjmm_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/6RB77-6-_oI/6RB77-6-_oI_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:a51bfd0cd1a4df28e5fee165c62ab06ffe7dc6a65e7ff998b5acc02496b9cc4c
|
| 3 |
+
size 1312371
|
parse/train/6RB77-6-_oI/6RB77-6-_oI_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:d7ce727a1ee5c3d5a641875f5b55e14ccca45a2abeba79cd7c76b6197a9790fd
|
| 3 |
+
size 1177792
|
parse/train/6RB77-6-_oI/6RB77-6-_oI_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:17eb105e9e655d52bb35f1790fd163702e7c99384ef9c2cb6a34d28aa86e02cc
|
| 3 |
+
size 1323785
|
parse/train/79zWncwO2p/79zWncwO2p_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:a4205a4c3631637d3b829dcb2816bf8c2fddece65ecb84ad7161ce8262a040d4
|
| 3 |
+
size 1339166
|
parse/train/79zWncwO2p/79zWncwO2p_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:1860ead23ddd5220caa7f2ae7aa64860ad8d6cc95368f0835d821684db9f1595
|
| 3 |
+
size 993419
|
parse/train/79zWncwO2p/79zWncwO2p_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:057d9de5d6b7211f5a7337cde901ffa8ab602b3e4ece25228be485fa51101e2c
|
| 3 |
+
size 1347971
|
parse/train/7_eLEvFjCi3/7_eLEvFjCi3.md
ADDED
|
@@ -0,0 +1,318 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# Nested Graph Neural Networks
|
| 2 |
+
|
| 3 |
+
Muhan Zhang1,2,∗ Pan Li3,† 1Institute for Artificial Intelligence, Peking University 2Beijing Institute for General Artificial Intelligence 3Department of Computer Science, Purdue University
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
Graph neural network (GNN)’s success in graph classification is closely related to the Weisfeiler-Lehman (1-WL) algorithm. By iteratively aggregating neighboring node features to a center node, both 1-WL and GNN obtain a node representation that encodes a rooted subtree around the center node. These rooted subtree representations are then pooled into a single representation to represent the whole graph. However, rooted subtrees are of limited expressiveness to represent a nontree graph. To address it, we propose Nested Graph Neural Networks (NGNNs). NGNN represents a graph with rooted subgraphs instead of rooted subtrees, so that two graphs sharing many identical subgraphs (rather than subtrees) tend to have similar representations. The key is to make each node representation encode a subgraph around it more than a subtree. To achieve this, NGNN extracts a local subgraph around each node and applies a base GNN to each subgraph to learn a subgraph representation. The whole-graph representation is then obtained by pooling these subgraph representations. We provide a rigorous theoretical analysis showing that NGNN is strictly more powerful than 1-WL. In particular, we proved that NGNN can discriminate almost all $r$ -regular graphs, where 1-WL always fails. Moreover, unlike other more powerful GNNs, NGNN only introduces a constantfactor higher time complexity than standard GNNs. NGNN is a plug-and-play framework that can be combined with various base GNNs. We test NGNN with different base GNNs on several benchmark datasets. NGNN uniformly improves their performance and shows highly competitive performance on all datasets.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Graph is an important tool to model relational data in the real world. Representation learning over graphs has become a popular topic of machine learning in recent years. While network embedding methods, such as DeepWalk [1], can learn node representations well, they fail to generalize to whole-graph representations, which are crucial for applications such as graph classification, molecule modeling, and drug discovery. On the contrary, although traditional graph kernels [2–7] can be used for graph classification, they define graph similarity often in a heuristic way, which is not parameterized and lacks some flexibility to deal with features.
|
| 12 |
+
|
| 13 |
+
In this context, graph neural networks (GNNs) have regained people’s attention and become the state-of-the-art graph representation learning tool [8–17]. GNNs use message passing to propagate features between connected nodes. By iteratively aggregating neighboring node features to the center node, GNNs learn node representations encoding their local structure and feature information. These node representations can be further pooled into a graph representation, enabling graph-level tasks such as graph classification. In this paper, we will use message passing GNNs to denote this class of GNNs based on repeated neighbor aggregation [18], in order to distinguish them from some high-order GNN variants [19–21] where the effective message passing happens between high-order node tuples instead of nodes.
|
| 14 |
+
|
| 15 |
+
|
| 16 |
+
Figure 1: The two original graphs $G _ { 1 }$ and $G _ { 2 }$ are non-isomorphic. $G _ { 1 }$ is composed of two triangles, while $G _ { 2 }$ is a hexagon. However, both 1-WL and message passing GNNs cannot differentiate them, since all nodes in the two graphs share identical rooted subtrees at any height (see the rooted subtrees around $v _ { 1 }$ and $v _ { 2 }$ in the middle block for example). In comparison, we can discriminate the two graphs by comparing their height-1 rooted subgraphs around any nodes. For example, the height-1 rooted subgraph around $v _ { 1 }$ is a closed triangle, but the height-1 rooted subgraph around $v _ { 2 }$ is an open triangle (see the red boxes in the right block).
|
| 17 |
+
|
| 18 |
+
GNNs’ message passing scheme mimics the 1-dimensional Weisfeiler-Lehman (1-WL) algorithm [22], which iteratively refines a node’s color according to its current color and the multiset of its neighbors’ colors. This procedure essentially encodes a rooted subtree around each node into its final color, where the rooted subtree is constructed by recursively expanding the neighbors of the root node. One critical reason for GNN’s success in graph classification is because two graphs sharing many identical or similar rooted subtrees are more likely classified into the same class, which actually aligns with the inductive bias that two graphs are similar if they have many common substructures [23].
|
| 19 |
+
|
| 20 |
+
Despite this, rooted subtrees are still limited in terms of expressing all possible substructures that can appear in a graph. It is likely that two graphs, despite sharing a lot of identical rooted subtrees, are not similar at all because their other substructure patterns are not similar. Take the two graphs $G _ { 1 }$ and $G _ { 2 }$ in Figure 1 as an example. If we apply 1-WL or a message passing GNN to them, the two graphs will always have the same representation no matter how many iterations/layers we use. This is because all nodes in the two graphs have identical rooted subtrees across all tree heights. However, the two graphs are quite different from a holistic perspective. $G _ { 1 }$ is composed of two triangles, while $G _ { 2 }$ is a hexagon. The intrinsic reason for such a failure is that rooted subtrees have limited expressiveness for representing general graphs, especially those with cycles.
|
| 21 |
+
|
| 22 |
+
To address this issue, we propose Nested Graph Neural Networks (NGNNs). The core idea is, instead of encoding a rooted subtree, we want the final representation of a node to encode a rooted subgraph (local $h$ -hop subgraph) around it. The subgraph is not restricted to be of any particular graph type such as tree, but serves as a general description of the local neighborhood around a node. Rooted subgraphs offer much better representation power than rooted subtrees, e.g., we can easily discriminate the two graphs in Figure 1 by only comparing their height-1 rooted subgraphs.
|
| 23 |
+
|
| 24 |
+
To represent a graph with rooted subgraphs, NGNN uses two levels of GNNs: base (inner) GNNs and an outer GNN. By extracting a local rooted subgraph around each node, NGNN first applies a base GNN to each node’s subgraph independently. Then, a subgraph pooling layer is applied to each subgraph to aggregate the intermediate node representations into a subgraph representation. This subgraph representation is used as the final representation of the root node. Rather than encoding a rooted subtree, this final node representation encodes the local subgraph around it, which contains more information than a subtree. Finally, all the final node representations are further fed into an outer GNN to learn a representation for the entire graph. Figure 2 shows one NGNN implementation using message passing GNNs as the base GNNs and a simple graph pooling layer as the outer GNN.
|
| 25 |
+
|
| 26 |
+
One may wonder that the base GNN seems to still learn only rooted subtrees if it is message-passingbased. Then why is NGNN more powerful than GNN? One key reason lies in the subgraph pooling layer. Take the height-1 rooted subgraphs (marked with red boxes) around $v _ { 1 }$ and $v _ { 2 }$ in Figure 1 as an example. Although $v _ { 1 }$ and $v _ { 2 }$ ’s height-1 rooted subtrees are still the same, their neighbors (labeled by 1 and 2) have different height-1 rooted subtrees. Thus, applying a one-layer message passing GNN plus a subgraph pooling as the base GNN is sufficient to discriminate $G _ { 1 }$ and $G _ { 2 }$ .
|
| 27 |
+
|
| 28 |
+
|
| 29 |
+
Figure 2: A particular implementation of the NGNN framework. It first extracts (copies) a rooted subgraph (height=1) around each node from the original graph, and then applies a base GNN with a subgraph pooling layer to each rooted subgraph independently to learn a subgraph representation. The subgraph representation is used as the root node’s final representation in the original graph. Then, a graph pooling layer is used to summarize the final node representations into a graph representation.
|
| 30 |
+
|
| 31 |
+
The NGNN framework has multiple exclusive advantages. Firstly, it allows freely choosing the base GNN, and can enhance the base GNN’s representation power in a plug-and-play fashion. Theoretically, we proved that NGNN is more powerful than message passing GNNs and 1-WL by being able to discriminate almost all $r$ -regular graphs (where 1-WL always fails). Secondly, by extracting rooted subgraphs, NGNN allows augmenting the initial features of a node with subgraphspecific structural features such as distance encoding [24] to improve the quality of the learned node representations. Thirdly, unlike other more powerful graph neural networks, especially those based on higher-order WL tests [19–21, 25], NGNN still has linear time and space complexity w.r.t. graph size like standard message passing GNNs, thus maintaining good scalability. We demonstrate the effectiveness of the NGNN framework in various synthetic/real-world graph classification/regression datasets. On synthetic datasets, NGNN demonstrates higher-than-1-WL expressive power, matching very well with our theorem. On real-world datasets, NGNN consistently enhances a wide range of base GNNs’ performance, achieving highly competitive results on all datasets.
|
| 32 |
+
|
| 33 |
+
# 2 Preliminaries
|
| 34 |
+
|
| 35 |
+
# 2.1 Notation and problem definition
|
| 36 |
+
|
| 37 |
+
We consider the graph classification/regression problem. Given a graph $G = ( V , E )$ where $V =$ $\{ 1 , 2 , \ldots n \}$ is the node set and $E \subseteq V \times V$ is the edge set, we aim to learn a function mapping $G$ to its class or target value $y$ . The nodes and edges in $G$ can have feature vectors associated with them, denoted by $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ (for node $i$ ) and $e _ { i j }$ (for edge $( i , j ) )$ , respectively.
|
| 38 |
+
|
| 39 |
+
# 2.2 Weisfeiler-Lehman test
|
| 40 |
+
|
| 41 |
+
The Wesfeiler-Lehman (1-WL) test [22] is a popular algorithm for graph isomorphism checking. The classical 1-WL works as follows. At first, all nodes receive a color 1. Each node collects its neighbors’ colors into a multiset. Then, 1-WL will update each node’s color so that two nodes get the same new color if and only if their current colors are the same and they have identical multisets of neighbor colors. Repeat this process until the number of colors does not increase between two iterations. Then, 1-WL will return that two graphs are non-isomorphic if their node colors are different at some iteration, or fail to determine whether they are non-isomorphic. See [7, 26] for more details.
|
| 42 |
+
|
| 43 |
+
1-WL essentially encodes the rooted subtrees around each node at different heights into its color representations. Figure 1 middle shows the rooted subtrees around $v _ { 1 }$ and $v _ { 2 }$ . Two nodes will have the same color at iteration $h$ if and only if their height- $h$ rooted subtrees are the same.
|
| 44 |
+
|
| 45 |
+
# 3 Nested Graph Neural Network
|
| 46 |
+
|
| 47 |
+
In this section, we introduce our Nested Graph Neural Network (NGNN) framework and theoretically demonstrate its higher representation power than message passing GNNs.
|
| 48 |
+
|
| 49 |
+
# 3.1 Limitations of the message passing GNNs
|
| 50 |
+
|
| 51 |
+
Most existing GNNs follow the message passing framework [18]: given a graph $G$ , each node’s hidden state $\bar { h } _ { v } ^ { t + 1 }$ is updated based on its previous state $\boldsymbol { h } _ { v } ^ { t }$ and the messages $m _ { v } ^ { \overline { { t } } + \tilde { 1 } }$ from its neighbors
|
| 52 |
+
|
| 53 |
+
$$
|
| 54 |
+
\pmb { h } _ { v } ^ { t + 1 } = U _ { t } ( \pmb { h } _ { v } ^ { t } , \pmb { m } _ { v } ^ { t + 1 } ) , \mathrm { ~ w h e r e ~ } \pmb { m } _ { v } ^ { t + 1 } = \sum _ { u \in N ( v | G ) } M _ { t } ( \pmb { h } _ { v } ^ { t } , \pmb { h } _ { u } ^ { t } , \pmb { e } _ { v u } ) .
|
| 55 |
+
$$
|
| 56 |
+
|
| 57 |
+
Here $M _ { t } , U _ { t }$ are the message and update functions at time stamp $t$ , $e _ { v u }$ is the feature of edge $( v , u )$ , and $N ( v | G )$ is the set of $v$ ’s neighbors in graph $G$ . The initial hidden states $h _ { v } ^ { 0 }$ are given by the raw node features $\scriptstyle { \mathbf { { \mathit { x } } } } _ { \mathit { v } }$ . After $T$ time stamps (iterations), the final node representations $h _ { v } ^ { T }$ are summarized into a whole-graph representation with a readout (pooling) function $R$ (e.g., mean or sum):
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
\begin{array} { r } { h _ { G } = R ( \{ h _ { v } ^ { T } | v \in G \} ) . } \end{array}
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
Such a message passing (or neighbor aggregation) scheme iteratively aggregates neighbor information into a center node’s hidden state, making it encode a local rooted subtree around the node. The final node representations will contain both the local structure and feature information around nodes, enabling node-level tasks such as node classification. After a pooling layer, these node representations can be further summarized into a graph representation, enabling graph-level tasks. When there is no edge feature and the node features are from a countable space, it is shown that message passing GNNs are at most as powerful as the 1-WL test for discriminating non-isomorphic graphs [27, 19].
|
| 64 |
+
|
| 65 |
+
For an $h$ -layer message passing GNN, it will give two nodes the same final representation if they have identical height- $h$ rooted subtrees (i.e., both the structures and the features on the corresponding nodes/edges are the same). If two graphs have a lot of identical (or similar) rooted subtrees, they will also have similar graph representations after pooling. This insight is crucial for the success of modern GNNs in graph classification, because it aligns with the inductive bias that two graphs are similar if they have many common substructures. Such insight has also been used in designing the WL subtree kernel [7], a state-of-the-art graph classification method before GNNs.
|
| 66 |
+
|
| 67 |
+
However, message passing GNNs have several limitations. Firstly, rooted subtree is only one specific substructure. It is not general enough to represent arbitrary subgraphs, especially those with cycles due to the natural restriction of tree structure. Secondly, using rooted subtree as the elementary substructure results in a discriminating power bounded by the 1-WL test. For example, all $n$ -node $r$ -regular graphs cannot be discriminated by message passing GNNs. Thirdly, standard message passing GNNs do not allow using root-node-specific structural features (such as the distance between a node and the root node) to improve the quality of the learned root node’s representation. We might need to break through such limitations in order to design more powerful GNNs.
|
| 68 |
+
|
| 69 |
+
# 3.2 The NGNN framework
|
| 70 |
+
|
| 71 |
+
To address the above limitations, we propose the Nested Graph Neural Network (NGNN) framework. NGNN no longer aims to encode a rooted subtree around each node. Instead, in NGNN, each node’s final representation encodes the general local subgraph information around it more than a subtree, so that two graphs sharing a lot of identical or similar rooted subgraphs will have similar representations.
|
| 72 |
+
|
| 73 |
+
Definition 1. (Rooted subgraph) Given a graph $G$ and a node v, the height-h rooted subgraph $G _ { v } ^ { h }$ of v is the subgraph induced from $G$ by the nodes within h hops of v (including $h$ -hop nodes).
|
| 74 |
+
|
| 75 |
+
To make a node’s final representation encode a rooted subgraph, we need to compute a subgraph representation. To achieve this, we resort to an arbitrary GNN, which we call the base GNN of NGNN. For example, the base GNN can be simply a message passing GNN, which performs message passing within each rooted subgraph to learn an intermediate representation for every node of the subgraph, and then uses a pooling layer to summarize a subgraph representation from the intermediate node representations. This subgraph representation is used as the final representation of the root node in the original graph. Take root node $w$ as an example. We first perform $T$ rounds of message passing within node $w$ ’s rooted subgraph $G _ { w } ^ { h }$ . Let $v$ be any node appearing in $G _ { w } ^ { h }$ . We have
|
| 76 |
+
|
| 77 |
+
$$
|
| 78 |
+
h _ { v , G _ { w } ^ { h } } ^ { t + 1 } = U _ { t } ( h _ { v , G _ { w } ^ { h } } ^ { t } , m _ { v , G _ { w } ^ { h } } ^ { t + 1 } ) , ~ \mathrm { w h e r e } ~ m _ { v , G _ { w } ^ { h } } ^ { t + 1 } = \sum _ { u \in N ( v | G _ { w } ^ { h } ) } M _ { t } ( h _ { v , G _ { w } ^ { h } } ^ { t } , h _ { u , G _ { w } ^ { h } } ^ { t } , e _ { v u } ) .
|
| 79 |
+
$$
|
| 80 |
+
|
| 81 |
+
Here $M _ { t } , U _ { t }$ are the message and update functions of the base GNN at time stamp $t$ , $N ( v | G _ { w } ^ { h } )$
|
| 82 |
+
denonode s th’s hi set of den st $v$ ’s neighbors within e and message spec $w$ ’s rooted subgraph to rooted subgrap $G _ { w } ^ { h }$ and at ti $h _ { v , G _ { w } ^ { h } } ^ { t + 1 }$ anmp $m _ { v , G _ { w } ^ { h } } ^ { t + 1 }$ denoteote that $v$ $G _ { w } ^ { h }$ $t + 1$ $v$
|
| 83 |
+
different. This is in contrast to standard GNNs where a node’s hidden state and message at time $t$ is
|
| 84 |
+
the same regardless of which root node it contributes to. For example, $\boldsymbol { h } _ { v } ^ { t + 1 }$ and $m _ { v } ^ { t + 1 }$ in Eq. 1 do
|
| 85 |
+
not depend on any particular rooted subgraph.
|
| 86 |
+
|
| 87 |
+
After $T$ rounds of message passing, we apply a subgraph pooling layer to summarize a subgraph representation $h _ { G _ { w } ^ { h } }$ from the intermediate node representations $\{ h _ { v , G _ { w } ^ { h } } ^ { T } | v \in G _ { w } ^ { h } \}$ .
|
| 88 |
+
|
| 89 |
+
$$
|
| 90 |
+
\pmb { h _ { w } } : = \pmb { h } _ { G _ { w } ^ { h } } = R _ { 0 } \big ( \{ \pmb { h } _ { v , G _ { w } ^ { h } } ^ { T } | v \in G _ { w } ^ { h } \} \big ) ,
|
| 91 |
+
$$
|
| 92 |
+
|
| 93 |
+
where $R _ { 0 }$ is the subgraph pooling layer. This subgraph representation $h _ { G _ { w } ^ { h } }$ will be used as root node $w$ ’s final representation $h _ { w }$ in the original graph. Note that the base GNNs are simultaneously applied to all nodes’ rooted subgraphs to return a final node representation for every node in the original graph, and all the base GNNs share the same parameters. With such node representations, NGNN uses an outer GNN to further process and aggregate them into a graph representation of the whole graph. For simplicity, we let the outer GNN be simply a graph pooling layer denoted by $R _ { 1 }$ :
|
| 94 |
+
|
| 95 |
+
$$
|
| 96 |
+
\begin{array} { r } { h _ { G } : = R _ { 1 } ( \{ h _ { w } | w \in G \} ) . } \end{array}
|
| 97 |
+
$$
|
| 98 |
+
|
| 99 |
+
The Nested GNN framework can be understood as a two-level GNN, or a GNN of GNNs—the inner subgraph-level GNNs (base GNNs) are used to learn node representations from their rooted subgraphs, while the outer graph-level GNN is used to return a whole-graph representation from the inner GNNs’ outputs. The inner GNNs all share the same parameters which are trained end-to-end with the outer GNN. Figure 2 depicts the implementation of the NGNN framework described above.
|
| 100 |
+
|
| 101 |
+
Compared to message passing GNNs, NGNN changes the “receptive field” of each node from a rooted subtree to a rooted subgraph, in order to capture better local substructure information. The rooted subgraph is read by a base GNN to learn a subgraph representation. Finally, the outer GNN reads the subgraph representations output by the base GNNs to return a graph representation.
|
| 102 |
+
|
| 103 |
+
Note that, when we apply the base GNN to a rooted subgraph, this rooted subgraph is extracted (copied) out of the original graph and treated as a completely independent graph from the other rooted subgraphs and the original graph. This allows the same node to have different representations within different rooted subgraphs. For example, in Figure 2, the same node $B$ appears in four different rooted subgraphs. Sometimes it is the root node, while other times it is a 1-hop neighbor of the root node. NGNN enables learning different representations for the same node when it appears in different rooted subgraphs, in contrast to standard GNNs where a node only has one single representation at one time stamp (Eq. 1). Similarly, NGNN also enables using different initial features for the same node when it appears in different rooted subgraphs. This allows us to customize a node’s initial features based on its structural role within a rooted subgraph, as opposed to using the same initial features for a node across all rooted subgraphs. For example, we can optionally augment node $B$ ’s initial features with the distance between node $B$ and the root—when node $B$ is the root node, we give it an additional feature 0; and when $B$ is a $k$ -hop neighbor of the root, we give it an additional feature $k$ . Such feature augmentation may help better capture a node’s structural role within a rooted subgraph. It is an exclusive advantage of NGNN and is not possible in standard GNNs.
|
| 104 |
+
|
| 105 |
+
# 3.3 The representation power of NGNN
|
| 106 |
+
|
| 107 |
+
We theoretically characterize the additional expressive power of NGNN (using message passing GNNs as base GNNs) as opposed to standard message passing GNNs. We focus on the ability to discriminate regular graphs because they form an important category of graphs which standard GNNs cannot represent well. Using 1-WL or message passing GNNs, any two $n$ -sized $r$ -regular graphs will have the same representation, unless discriminative node features are available. In contrast, we prove that NGNN can distinguish almost all pairs of $n$ -sized $r$ -regular graphs regardless of node features.
|
| 108 |
+
|
| 109 |
+
Definition 2. If the message passing (Eq. 3) and the two-level graph pooling (Eqs. 4,5) are all injective given input from a countable space, then the NGNN is called proper.
|
| 110 |
+
|
| 111 |
+
A proper NGNN always exists due to the representation power of fully-connected neural networks used for message passing and Deep Set for graph pooling [28]. For all pairs of graphs that 1-WL can discriminate, there always exists a proper NGNN that can also discriminate them, because two graphs discriminated by 1-WL means they must have different multisets of rooted subtrees at some height $h$ , while a rooted subtree is always included in a rooted subgraph with the same height.
|
| 112 |
+
|
| 113 |
+
Now we present our main theorem.
|
| 114 |
+
|
| 115 |
+
Theorem 1. Consider all pairs of $n$ -sized $r$ -regular graphs, where $3 \leq r < ( 2 \log n ) ^ { 1 / 2 }$ . For any small constant $\epsilon > 0$ , there exists a proper NGNN using at most $\begin{array} { r } { \lceil ( \frac { 1 } { 2 } + \epsilon ) \frac { \log n } { \log ( r - 1 - \epsilon ) } \rceil } \end{array}$ -height rooted subgraphs and $\left\lceil \epsilon \frac { \log n } { \log ( r - 1 - \epsilon ) } \right\rceil$ -layer message passing, which distinguishes almost all $( 1 - o ( 1 ) ,$ ) such pairs of graphs.
|
| 116 |
+
|
| 117 |
+
We include the proof in Appendix A. Theorem 1 has three implications. Firstly, since NGNN can discriminate almost all $r$ -regular graphs where 1-WL always fails, it is strictly more powerful than 1-WL and message passing GNNs. Secondly, it implies that NGNN does not need to extract subgraphs with a too large height (abou t 12 log nlog (r−1) ) to be more powerful. Moreover, NGNN is already powerful with very few layers, i.e., an arbitrarily small constant $\epsilon$ times $\frac { \log n } { \log ( r - 1 ) }$ (as few as 1 layer). This benefit comes from the subgraph pooling (Eq. 4), freeing us from using deep base GNNs. We further conduct a simulation experiment in Appendix D to verify Theorem 1 by testing how well NGNN discriminates $r$ -regular graphs in practice. The results match almost perfectly with our theory.
|
| 118 |
+
|
| 119 |
+
Although NGNN is strictly more powerful than 1-WL and 2-WL (1-WL and 2-WL have the same discriminating power [20]), it is unclear whether NGNN is more powerful than 3-WL. Our early-stage analysis shows both NGNN and 3-WL cannot discriminate strongly regular graphs with the same parameters [29]. We leave the exact comparison between NGNN and 3-WL to future work.
|
| 120 |
+
|
| 121 |
+
# 3.4 Discussion
|
| 122 |
+
|
| 123 |
+
Base GNN. NGNN is a general plug-and-play framework to increase the power of a base GNN. For the base GNN, we are not restricted to message passing GNNs as described in Section 3.2. For example, we can also use GNNs approximating the power of higher-dimensional WL tests, such as 1-2-3-GNN [19] and PPGN/Ring-GNN [20, 21], as the base GNN. In fact, one limitation of these high-order GNNs is their $\mathcal { O } ( n ^ { 3 } )$ complexity. Using the NGNN framework we can greatly alleviate this by applying the higher-order GNN to multiple small rooted subgraphs instead of the whole graph. Suppose a rooted subgraph has at most $c$ nodes, then by applying a high-order GNN to all $n$ rooted subgraphs, we can reduce the time complexity from $\dot { \mathcal { O } ( n ^ { 3 } ) }$ to $\bar { \mathcal { O } } \bar { ( n c ^ { 3 } ) }$ .
|
| 124 |
+
|
| 125 |
+
Complexity. We compare the time complexity of NGNN (using message passing GNNs as base GNNs) with a standard message passing GNN. Suppose the graph has $n$ nodes with a maximum degree $d$ , and the maximum number of nodes in a rooted subgraph is $c$ . Each message passing iteration in a standard message passing GNN takes $\mathcal { O } ( n d )$ operations. In NGNN, we need to perform message passing over all $n$ nodes’ rooted subgraphs, which takes $O ( n \cdot c d )$ . We will keep $c$ small (which can be achieved by using a small $h$ ) to improve NGNN’s scalability. Additionally, a small $c$ enables the base GNN to focus on learning local subgraph patterns.
|
| 126 |
+
|
| 127 |
+
In Appendix B, we discuss some other design choices of NGNN.
|
| 128 |
+
|
| 129 |
+
# 4 Related work
|
| 130 |
+
|
| 131 |
+
Understanding GNN’s representation power is a fundamental problem in GNN research. Xu et al. [27] and Morris et al. [19] first proved that the discriminating power of message passing GNNs is bounded by the 1-WL test, namely they cannot discriminate two non-isomorphic graphs that 1-WL fails to discriminate (such as $r$ -regular graphs). Since then, there is increasing effort in enhancing GNN’s discriminating power beyond 1-WL [19, 21, 20, 30, 24, 31–33, 25]. Many GNNs have been proposed to mimic higher-dimensional WL tests, such as 1-2-3-GNN [19], Ring-GNN [21] and PPGN [20]. However, these models generally require learning the representations of all node tuples of certain cardinality (e.g., node pairs, node triples and so on), thus cannot leverage the sparsity of graph structure and are difficult to scale to large graphs. Some works study the universality of GNNs for approximating any invariant or equivariant functions over graphs [34, 21, 35–37]. However, reaching universality would require polynomial $( n )$ -order tensors, which hold more theoretical value than practical applicability. Dasoulas et al. [38] propose to augment nodes of identical attributes with different colors, which requires exhausting all the coloring choices to reach universality. Similarly, Relational Pooling (RP) [30] uses the ensemble of permutation-aware functions over graphs to reach universality, which requires exhausting all $n !$ permutations to achieve its theoretical power. Its local version Local Relational Pooling (LRP) [39] applies RP over subgraphs around nodes, which is similar to our work yet still requires exhausting node permutations in local subgraphs and even more loses RP’s theoretical power. In contrast, NGNN maintains a controllable cost by only applying a message passing GNN to local subgraphs, and is guaranteed to be more powerful than 1-WL.
|
| 132 |
+
|
| 133 |
+
Because of the high cost of mimicking high-dimensional WL tests, several works have been proposed to increase GNN’s representation power within the message passing framework. Observing that different neighbors are indistinguishable during neighbor aggregation, some works propose to add one-hot node index features or random features to GNNs [40, 41]. These methods work well when nodes naturally have distinct identities irrespective of the graph structure. However, although making GNNs more discriminative, they also lose some of GNNs’ generalization ability by not being able to guarantee nodes with identical neighborhoods to have the same embedding; the resulting models are also no longer permutation invariant. Repeating random initialization helps with avoiding such an issue but gets much slower convergence [42]. An exception is structural message-passing (SMP) [43], which propagates one-hot node index features to learn a global $n \times d$ feature matrix for each node. The feature matrix is further pooled to learn a permutation-invariant node representation.
|
| 134 |
+
|
| 135 |
+
On the contrary, some works propose to use structural features to augment GNNs without hurting the generalization ability of GNNs. SEAL [44, 45], IGMC [46] and DE [24] use distance-based features, where a distance vector w.r.t. the target node set to predict is calculated for each node as its additional features. Our NGNN framework is naturally compatible with such distance-based features due to its independent rooted subgraph processing. GSN [31] uses the count of certain substructures to augment node/edge features, which also surpasses 1-WL theoretically. However, GSN needs a properly defined substructure set to incorporate domain-specific inductive biases, while NGNN aims to learn arbitrary substructures around nodes without the need to predefine a substructure set.
|
| 136 |
+
|
| 137 |
+
Concurrent to our work, You et al. [32] propose Identity-aware GNN (ID-GNN). ID-GNN uses different weight parameters between each root node and its context nodes during message passing. It also extracts a rooted subgraph around each node, and thus can be viewed as a special case of NGNN with: 1) the number of message passing layers equivalent to the subgraph height, 2) directly using the root node’s intermediate representation as its final representation without subgraph pooling, and 3) augmenting initial node features with 0/1 “identity”. However, the extra power of ID-GNN only comes from the “identity” feature, while the power of NGNN comes from the subgraph pooling— without using any node features, NGNN is still provably more discriminative than 1-WL. Another similar work to ours is natural graph network (NGN) [47]. NGN argues that graph convolution weights need not be shared among all nodes but only (locally) isomorphic nodes. If we view our distance-based node features as refining the graph convolution weights so that nodes within a center node’s neighborhood are no longer treated symmetrically, then our NGNN reduces to an NGN.
|
| 138 |
+
|
| 139 |
+
The idea of independently performing message passing within $k$ -hop neighborhood is also explored in $k$ -hop GNN [48] and MixHop [49]. However, MixHop directly concatenates the aggregation results of neighbors at different hops as the root representation, which ignores the connections between other nodes in the rooted subgraph. $k$ -hop GNN sequentially performs message passing for $k$ -hop, $k - 1$ -hop, ..., and 0-hop node (the update of $( i - 1 )$ -hop nodes depend on the updated states of $i$ -hop nodes), while NGNN simultaneously performs message passing for all nodes in the subgraph thus is more parallelizable. Both MixHop and $k$ -hop GNN directly use the root node’s representation as its final node representation. In contrast, NGNN uses a subgraph pooling to summarize all node representations within the subgraph as the final root representation, which distinguishes NGNN from other $k$ -hop models. As Theorem 1 shows, the subgraph pooling enables using a much smaller number of message passing layers $l$ (as small as 1) than the depth $k$ of the subgraph, while MixHop and $k$ -hop GNN always require $l \geq k$ . MixHop and $k$ -hop GNN also do not have the strong theoretical power of NGNN to discriminate $r$ -regular graphs. Like SEAL and $k$ -hop GNN, G-Meta [50] is another work extracting subgraphs around nodes/links. It focuses specifically on a meta-learning setting.
|
| 140 |
+
|
| 141 |
+
Table 1: Statistics and evaluation metrics of the QM9 and OGB datasets.
|
| 142 |
+
|
| 143 |
+
<table><tr><td>Dataset</td><td>#Graphs</td><td>Avg. #nodes</td><td>Avg.#edges</td><td>Split ratio</td><td>#Tasks</td><td>Task type</td><td>Metric</td></tr><tr><td>QM9</td><td>129,433</td><td>18.0</td><td>18.6</td><td>80/10/10</td><td>12</td><td>Regression</td><td>MAE</td></tr><tr><td>ogbl-molhiv</td><td>41,127</td><td>25.5</td><td>27.5</td><td>80/10/10</td><td>1</td><td>Classification</td><td>ROC-AUC</td></tr><tr><td>ogbl-molpcba</td><td>437,929</td><td>26.0</td><td>28.1</td><td>80/10/10</td><td>128</td><td>Classification</td><td>AP</td></tr></table>
|
| 144 |
+
|
| 145 |
+
# 5 Experiments
|
| 146 |
+
|
| 147 |
+
In this section, we study the effectiveness of the NGNN framework for graph classification and regression tasks. In particular, we want to answer the following questions:
|
| 148 |
+
|
| 149 |
+
Q1 Can NGNN reach its theoretical power to discriminate 1-WL-indistinguishable graphs? Q2 How often and how much does NGNN improve the performance of a base GNN? Q3 How does NGNN perform in comparison to state-of-the-art GNN methods in open benchmarks? Q4 How much extra computation time does NGNN incur?
|
| 150 |
+
|
| 151 |
+
We implement the NGNN framework based on the PyTorch Geometric library [51]. Our code is available at https://github.com/muhanzhang/NestedGNN.
|
| 152 |
+
|
| 153 |
+
# 5.1 Datasets
|
| 154 |
+
|
| 155 |
+
To answer Q1, we use a simulation dataset of $r$ -regular graphs and the EXP dataset [42] containing 600 pairs of 1-WL-indistinguishable but non-isomorphic graphs. To answer Q2, we use the QM9 dataset [52, 53] and the TU datasets [54]. QM9 contains $1 3 0 \mathrm { K }$ small molecules. The task here is to perform regression on twelve targets representing energetic, electronic, geometric, and thermodynamic properties, based on the graph structure and node/edge features. TU contains five graph classification datasets including D&D [55], MUTAG [56], PROTEINS [55], PTC_MR [57], and ENZYMES [58]. We used the datasets provided by PyTorch Geometric [51], where for QM9 we performed unit conversions to match the units used by [19]. The evaluation metric is Mean Absolute Error (MAE) for QM9 and Accuracy $( \% )$ for TU. To answer Q3, we use two Open Graph Benchmark (OGB) datasets [59], ogbg-molhiv and ogbg-molpcba. The ogbg-molhiv dataset contains 41K small molecules, the task of which is to classify whether a molecule inhibits HIV virus or not. ROC-AUC is used for evaluation. The ogbg-molpcba dataset contains 438K molecules with 128 classification tasks. The evaluation metric is Average Precision (AP) averaged over all the tasks. We include the statistics for QM9 and OGB datasets in Table 1.
|
| 156 |
+
|
| 157 |
+
# 5.2 Models
|
| 158 |
+
|
| 159 |
+
QM9. We use 1-GNN, 1-2-GNN, 1-3-GNN, and 1-2-3-GNN from [19] as both the baselines and the base GNNs of NGNN. Among them, 1-GNN is a standard message passing GNN with 1-WL power. 1-2-GNN is a GNN mimicking 2-WL, where message passing happens among 2-tuples of nodes. 1-3-GNN and 1-2-3-GNN mimic 3-WL, where message passing happens among 3-tuples of nodes. 1-2-GNN and 1-3-GNN use features computed by 1-GNN as initial node features, and 1-2-3-GNN uses the concatenated features from 1-2-GNN and 1-3-GNN. We additionally include numbers provided by [53] and Deep LRP [39] as baselines. Note that we omit more recent methods [60–62] using advanced physical representations calculated from angles, atom coordinates, and quantum mechanics, which may obscure the comparison of models’ pure graph representation power. For NGNN, we uniformly use height-3 rooted subgraphs. For a fair comparison, the base GNNs in NGNN use exactly the same hyperparameters as when they are used alone, except for 1-GNN where we increase the number of message passing layers from 3 to 5 to make the number of layers larger than the subgraph height, similar to [63]. For subgraph pooling and graph pooling layers, we uniformly use mean pooling. All other settings follow [19].
|
| 160 |
+
|
| 161 |
+
TU. We use four widely adopted GNNs as the baselines and the base GNNs of NGNN: GCN [12], GraphSAGE [64], GIN [27], and GAT [15]. Since TU datasets suffer from inconsistent evaluation standards [65], we uniformly use the 10-fold cross validation framework provided by PyTorch Geomtric [66] for all the models to ensure a fair comparison. For GNNs, we search the number of message passing layers in $\{ 2 , 3 , 4 , 5 \}$ . For NGNNs, we similarly search the subgraph height $h$ in $\{ 2 , 3 , 4 , 5 \}$ , so that both NGNNs and GNNs can have equal-depth local receptive fields. For NGNNs, we always use $h + 1$ message passing layers instead of searching it together with $h$ , because that will make NGNNs have more hyperparameters to tune. All models have 32 hidden dimensions, and are trained for 100 epochs with a batch size of 128. For each fold, we record the test accuracy with the hyperparameters chosen based on the best validation performance of this fold. Finally, we report the average test accuracy across all the 10 folds.
|
| 162 |
+
|
| 163 |
+
OGB. We use GNNs achieving top places on the OGB graph classification leaderboard3 (at the time of submission) as the baselines, including GCN [12], GIN [27], DeeperGCN [67], Deep LRP [39], PNA [68], DGN [33], GINE [69], and PHC-GNN [70]. Note that those high-order GNNs [19–21, 25] are not included here, because despite being theoretically more discriminative, these GNNs are not among the GNNs with the best empirical performance on modern large-scale graph benchmarks, and their $\bar { \mathcal { O } } ( n ^ { 3 } )$ complexity also raises a scalability issue. For NGNN, we use GIN as the base GNN (although GIN is not among the strongest baselines here). Some baselines additionally use the virtual node technique [18, 11, 71], which are marked by $\therefore$ . For NGNN, we search the subgraph height $h$ in $\{ 3 , 4 , 5 \}$ , and the number of layers in $\{ 4 , 5 , 6 \}$ . We train the NGNN models for 100 and 150 epochs for ogbg-molhiv and ogbg-molpcba, respectively, and report the validation and test scores at the best validation epoch. We also find that our models are subject to high performance variance across epochs, likely due to the increased expressiveness. Thus, we save a model checkpoint every 10 epochs, and additionally report the ensemble performance by averaging the predictions from all checkpoints. The final hyperparameter choices and more details about the experimental settings are included in Appendix C. All results are averaged over 10 independent runs.
|
| 164 |
+
|
| 165 |
+
In the following, we uniformly use “Nested GNN” to denote an NGNN model using “GNN” as the base GNN. For example, Nested GIN denotes an NGNN model using GIN [27] as the base GNN. For the NGNN models in QM9, TU and OGB datasets, we augment the initial features of a node with Distance Encoding (DE) [24], which uses the (generalized) distance between a node and the root as its additional feature, due to DE’s successful applications in link-level tasks [44, 46]. Note that such feature augmentation is not applicable to the baseline models as discussed in Section 3.2. An ablation study on the effects of the DE features is included in Appendix E.
|
| 166 |
+
|
| 167 |
+
# 5.3 Results and discussion
|
| 168 |
+
|
| 169 |
+
To answer Q1, we first run a simulation to test NGNN’s power for discriminating $r$ -regular graphs. The results are presented in Appendix D. They match almost perfectly with Theorem 1, demonstrating that a practical NGNN can fulfil its theoretical power for discriminating $r$ -regular graphs.
|
| 170 |
+
|
| 171 |
+
Table 2: Results $( \% )$ on EXP.
|
| 172 |
+
|
| 173 |
+
<table><tr><td>Method</td><td>Test Accuracy</td></tr><tr><td>GCN-RNI [42]</td><td>98.0±1.85</td></tr><tr><td>PPGN [20]</td><td>50.0±0.00</td></tr><tr><td>1-2-3-GNN [19]</td><td>50.0±0.00</td></tr><tr><td>3-GCN[42]</td><td>99.7±0.004</td></tr><tr><td>Nested GIN</td><td>99.9±0.26</td></tr></table>
|
| 174 |
+
|
| 175 |
+
We also test NGNN’s expressive power using the EXP dataset provided by [42], which contains 600 carefully constructed 1-WL indistinguishable but non-isomorphic graph pairs. Each pair of graphs have different labels, thus a standard message passing GNN cannot predict them both correctly, resulting in an expected classification accuracy of only $50 \%$ . We exactly follow the experimental settings and copy the baseline results in [42]. In Table 2, our Nested GIN model achieves a $9 9 . 9 \%$ classification accuracy, which outperforms all the baselines and distinguishes almost all the 1-WL indistinguishable graph pairs. These results verified that NGNN’s expressive power is indeed beyond 1-WL and message passing GNNs.
|
| 176 |
+
|
| 177 |
+
To answer Q2, we adopt the QM9 and TU datasets. We show the QM9 results in Table 3. If the Nested version of a base GNN achieves a better result than the base GNN itself, we color that cell with light green. As we can see, NGNN brings performance gains to all base GNNs on most targets, sometimes by large margins. We also show the results on TU in Table 4. NGNNs also show improvement over their base GNNs in most cases. These results indicate that NGNN is a general framework for improving a GNN’s power. We further compute the maximum reduction of MAE for QM9 and maximum improvement of accuracy for TU before and after applying NGNN. NGNN reduces the MAE by up to 7.9 times for QM9, and increases the accuracy by up to $1 4 . 3 \%$ for TU. These results answer Q2, indicating that NGNN can bring steady and significant improvement to base GNNs.
|
| 178 |
+
|
| 179 |
+
To answer Q3, we compare Nested GIN with leading methods on the OGB leaderboard. The results are shown in Table 5. Nested GIN achieves highly competitive performance with these leading GNN models, albeit using a relatively weak base GNN (GIN). Compared to GIN alone, Nested GIN shows clear performance gains. It achieves test scores up to 79.86 and 30.07 on ogbg-molhiv and ogbg-molpcba, respectively, which outperform all the baselines. In particular, for the challenging ogbg-molpcba, our Nested GIN can achieve 30.07 and 28.32 test AP with and without ensemble, respectively, outperforming the plain GIN model (with 27.03 test AP) significantly. These results demonstrate the great empirical performance and potential of NGNN even compared to heavily tuned open leaderboard models, despite using only GIN as the base GNN.
|
| 180 |
+
|
| 181 |
+
Table 3: MAE results on QM9 (smaller the better). A colored cell means NGNN is better than the base GNN.
|
| 182 |
+
|
| 183 |
+
<table><tr><td rowspan="2">Target</td><td colspan="10">Method (Ne. forNested)</td></tr><tr><td>DTNN</td><td>MPNN</td><td>Deep LRP</td><td>1-GNN1-2-GNN</td><td></td><td>1-3-GNN</td><td></td><td></td><td></td><td></td><td></td><td>1-2-3-GNN|Ne.1-GNN Ne.1-2-GNN Ne.1-3-GNN Ne.1-2-3-GNN|Max.reduction</td></tr><tr><td>μ</td><td>0.244</td><td>0.358</td><td>0.364</td><td>0.493</td><td>0.493</td><td>0.473</td><td>0.476</td><td>0.428</td><td>0.437</td><td>0.436</td><td>0.433</td><td>1.2×</td></tr><tr><td>α</td><td>0.95</td><td>0.89</td><td>0.298</td><td>0.78</td><td>0.27</td><td>0.46</td><td>0.27</td><td>0.29</td><td>0.278</td><td>0.261</td><td>0.265</td><td>2.7×</td></tr><tr><td>εHOMO</td><td>0.00388</td><td>0.00541</td><td>0.00254</td><td>0.00321</td><td>0.00331</td><td>0.00328</td><td>0.00337</td><td>0.00265</td><td>0.00275</td><td>0.00265</td><td>0.00279</td><td>1.2×</td></tr><tr><td>εLUMO</td><td>0.00512</td><td>0.00623</td><td>0.00277</td><td>0.00355</td><td>0.00350</td><td>0.00354</td><td>0.00351</td><td>0.00297</td><td>0.00271</td><td>0.00269</td><td>0.00276</td><td>1.3×</td></tr><tr><td>△</td><td>0.0112</td><td>0.0066</td><td>0.00353</td><td>0.0049</td><td>0.0047</td><td>0.0046</td><td>0.0048</td><td>0.0038</td><td>0.0039</td><td>0.0039</td><td>0.0039</td><td>1.8×</td></tr><tr><td>(R²）</td><td>17.0</td><td>28.5</td><td>19.3</td><td>34.1</td><td>21.5</td><td>25.8</td><td>22.9</td><td>20.5</td><td>20.4</td><td>20.2</td><td>20.1</td><td>1.7×</td></tr><tr><td>ZPVE</td><td>0.00172</td><td>0.00216</td><td>0.00055</td><td>0.00124</td><td>0.00018</td><td>0.00064</td><td>0.00019</td><td>0.00020</td><td>0.00017</td><td>0.00017</td><td>0.00015</td><td>6.2×</td></tr><tr><td>Uo</td><td>2.43</td><td>2.05</td><td>0.413</td><td>2.32</td><td>0.0357</td><td>0.6855</td><td>0.0427</td><td>0.295</td><td>0.252</td><td>0.291</td><td>0.205</td><td>7.9×</td></tr><tr><td>U</td><td>2.43</td><td>2.00</td><td>0.413</td><td>2.08</td><td>0.107</td><td>0.686</td><td>0.111</td><td>0.361</td><td>0.265</td><td>0.278</td><td>0.200</td><td>5.8×</td></tr><tr><td>H</td><td>2.43</td><td>2.02</td><td>0.413</td><td>2.23</td><td>0.070</td><td>0.794</td><td>0.0419</td><td>0.305</td><td>0.241</td><td>0.267</td><td>0.249</td><td>7.3×</td></tr><tr><td>G</td><td>2.43</td><td>2.02</td><td>0.413</td><td>1.94</td><td>0.140</td><td>0.587</td><td>0.0469</td><td>0.489</td><td>0.272</td><td>0.287</td><td>0.253</td><td>4.0×</td></tr><tr><td>C</td><td>0.27</td><td>0.42</td><td>0.129</td><td>0.27</td><td>0.0989</td><td>0.158</td><td>0.0944</td><td>0.174</td><td>0.0891</td><td>0.0879</td><td>0.0811</td><td>1.8×</td></tr></table>
|
| 184 |
+
|
| 185 |
+
Table 4: Accuracy results $( \% )$ on TU datasets.
|
| 186 |
+
|
| 187 |
+
<table><tr><td>#Graphs</td><td>D&D 1178</td><td>MUTAG 188</td><td>PROTEINS 1113</td><td>PTC_MR 344</td><td>ENZYMES 600</td></tr><tr><td>Avg. #nodes</td><td>284.32</td><td>17.93</td><td>39.06</td><td>14.29</td><td>32.63</td></tr><tr><td>GCN GraphSAGE</td><td>71.6±2.8 71.6±3.0</td><td>73.4±10.8</td><td>71.7±4.7</td><td>56.4±7.1 57.0±5.5</td><td>27.3±5.5 30.7±6.3</td></tr><tr><td></td><td>70.5±3.9</td><td>74.0±8.8</td><td>71.2±5.2</td><td></td><td></td></tr><tr><td>GIN GAT</td><td>71.0±4.4</td><td>84.5±8.9 73.9±10.7</td><td>70.6±4.3 72.0±3.3</td><td>51.2±9.2 57.0±7.3</td><td>38.3±6.4 30.2±4.2</td></tr><tr><td>Nested GCN</td><td>76.3±3.8</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>77.4±4.2</td><td>82.9±11.1</td><td>73.3±4.0</td><td>57.3±7.7</td><td>31.2±6.7</td></tr><tr><td>Nested GraphSAGE Nested GIN</td><td>77.8±3.9</td><td>83.9±10.7 87.9±8.2</td><td>74.2±3.7 73.9±5.1</td><td>57.0±5.9 54.1±7.7</td><td>30.7±6.3 29.0±8.0</td></tr><tr><td>Nested GAT</td><td>76.0±4.4</td><td>81.9±10.2</td><td>73.7±4.8</td><td>56.7±8.1</td><td>29.5±5.7</td></tr><tr><td>Max. improvement</td><td>10.4%</td><td>13.4%</td><td>4.7%</td><td>5.7%</td><td>14.3%</td></tr></table>
|
| 188 |
+
|
| 189 |
+
Table 5: Results $( \% )$ on OGB datasets (\* virtual node).
|
| 190 |
+
|
| 191 |
+
<table><tr><td></td><td colspan="2">ogbg-molhiv (AUC)</td><td colspan="2">ogbg-molpcba (AP)</td></tr><tr><td>Method</td><td>Validation</td><td>Test</td><td>Validation</td><td>Test</td></tr><tr><td>CCN*</td><td>83.84±0.91</td><td>75.99±1.19</td><td>24.95±0.42</td><td>24.24±0.34</td></tr><tr><td>GIN*</td><td>84.79±0.68</td><td>77.07±1.49</td><td>27.98±0.25</td><td>27.03±0.23</td></tr><tr><td>Deep LRP</td><td>82.09±1.16</td><td>77.19±1.40</td><td></td><td></td></tr><tr><td>DeeperGCN*</td><td></td><td></td><td>29.20±0.25</td><td>27.81±0.38</td></tr><tr><td>HIMP</td><td></td><td>78.80±0.82</td><td></td><td></td></tr><tr><td>PNA</td><td>85.19±0.99</td><td>79.05±1.32</td><td></td><td></td></tr><tr><td>DGN</td><td>84.70±0.47</td><td>79.70±0.97-</td><td></td><td></td></tr><tr><td>GINE*</td><td></td><td></td><td>30.65±0.30</td><td>29.17±0.15</td></tr><tr><td>PHC-GNN</td><td>82.17±0.89</td><td>79.34±1.16</td><td>30.68±0.25</td><td>29.47±0.26</td></tr><tr><td>Nested GIN*</td><td>83.17±1.99</td><td>78.34±1.86</td><td>29.15±0.35</td><td>28.32±0.41</td></tr><tr><td>NestedGIN* (ens)</td><td>80.80±2.78</td><td>79.86±1.05</td><td>30.59±0.56</td><td>30.07±0.37</td></tr></table>
|
| 192 |
+
|
| 193 |
+
To answer Q4, we report the training time per epoch for GIN and Nested GIN on OGB datasets. On ogbg-molhiv, GIN takes 54s per epoch, while Nested GIN takes 183s. On ogbg-molpcba, GIN takes 10min per epoch, while Nested GIN takes $2 0 \mathrm { { m i n } }$ . This verifies that NGNN has comparable time complexity with message passing GNNs. The extra complexity comes from independently learning better node representations from rooted subgraphs, which is a trade-off for the higher expressivity.
|
| 194 |
+
|
| 195 |
+
In summary, our experiments have firmly shown that NGNN is a theoretically sound method which brings consistent gains to its base GNNs in a plug-and-play way. Furthermore, NGNN still maintains a controllable time complexity compared to other more powerful GNNs.
|
| 196 |
+
|
| 197 |
+
Finally, we point out one memory limitation of the current NGNN implementation. Currently, NGNN does not scale to graph datasets with a large average node number (such as REDDIT-BINARY) or datasets with a large average node degree (such as ogbg-ppa) due to copying a rooted subgraph for each node to the GPU memory. Reducing batch size or subgraph height helps, but at the same time leads to performance degradation. One may wonder why materializing all the subgraphs into GPU memory is necessary. The reason is that we want to batch-process all the subgraphs simultaneously. Otherwise, we have to sequentially extract subgraphs on the fly, which results in a much higher latency. We leave the exploration of memory efficient NGNN to the future work.
|
| 198 |
+
|
| 199 |
+
# 6 Conclusions
|
| 200 |
+
|
| 201 |
+
We have proposed Nested Graph Neural Network (NGNN), a general framework for improving GNN’s representation power. NGNN learns node representations encoding rooted subgraphs instead of rooted subtrees. Theoretically, we prove NGNN can discriminate almost all $r$ -regular graphs where 1-WL always fails. Empirically, NGNN consistently improves the performance of various base GNNs across different datasets without incurring the $\mathcal { O } ( \bar { n } ^ { 3 } )$ complexity like other more powerful GNNs.
|
| 202 |
+
|
| 203 |
+
# Acknowledge
|
| 204 |
+
|
| 205 |
+
The authors greatly thank the actionable suggestions from the reviewers to improve the manuscript. Li is partly supported by the 2021 JP Morgan Faculty Award and the National Science Foundation (NSF) award HDR-2117997.
|
| 206 |
+
|
| 207 |
+
References
|
| 208 |
+
[1] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 701–710. ACM, 2014.
|
| 209 |
+
[2] David Haussler. Convolution kernels on discrete structures. Technical report, Citeseer, 1999.
|
| 210 |
+
[3] Nino Shervashidze, SVN Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten M Borgwardt. Efficient graphlet kernels for large graph comparison. In AISTATS, volume 5, pages 488–495, 2009.
|
| 211 |
+
[4] Risi Kondor, Nino Shervashidze, and Karsten M Borgwardt. The graphlet spectrum. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 529–536. ACM, 2009.
|
| 212 |
+
[5] Karsten M Borgwardt and Hans-Peter Kriegel. Shortest-path kernels on graphs. In 5th IEEE International Conference on Data Mining, pages 8–pp. IEEE, 2005.
|
| 213 |
+
[6] Marion Neumann, Roman Garnett, Christian Bauckhage, and Kristian Kersting. Propagation kernels: efficient graph kernels from propagated information. Machine Learning, 102(2): 209–245, 2016.
|
| 214 |
+
[7] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research, 12(Sep): 2539–2561, 2011.
|
| 215 |
+
[8] 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, 2009.
|
| 216 |
+
[9] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013.
|
| 217 |
+
[10] David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in neural information processing systems, pages 2224– 2232, 2015.
|
| 218 |
+
[11] Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.
|
| 219 |
+
[12] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.
|
| 220 |
+
[13] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3837–3845, 2016.
|
| 221 |
+
[14] Hanjun Dai, Bo Dai, and Le Song. Discriminative embeddings of latent variable models for structured data. In Proceedings of The 33rd International Conference on Machine Learning, pages 2702–2711, 2016.
|
| 222 |
+
[15] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903, 2017.
|
| 223 |
+
[16] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In AAAI, pages 4438–4445, 2018.
|
| 224 |
+
[17] Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems, pages 4800–4810, 2018.
|
| 225 |
+
[18] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1263–1272. JMLR. org, 2017.
|
| 226 |
+
|
| 227 |
+
[19] Christopher Morris, Martin Ritzert, Matthias Fey, William L Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 4602–4609, 2019.
|
| 228 |
+
|
| 229 |
+
[20] Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, and Yaron Lipman. Provably powerful graph networks. In Advances in Neural Information Processing Systems, pages 2156–2167, 2019.
|
| 230 |
+
|
| 231 |
+
[21] Zhengdao Chen, Soledad Villar, Lei Chen, and Joan Bruna. On the equivalence between graph isomorphism testing and function approximation with gnns. In Advances in Neural Information Processing Systems, pages 15894–15902, 2019.
|
| 232 |
+
|
| 233 |
+
[22] Boris Weisfeiler and AA Lehman. A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, 2(9):12–16, 1968.
|
| 234 |
+
|
| 235 |
+
[23] S Vichy N Vishwanathan, Nicol N Schraudolph, Risi Kondor, and Karsten M Borgwardt. Graph kernels. Journal of Machine Learning Research, 11(Apr):1201–1242, 2010.
|
| 236 |
+
|
| 237 |
+
[24] Pan Li, Yanbang Wang, Hongwei Wang, and Jure Leskovec. Distance encoding–design provably more powerful gnns for structural representation learning. arXiv preprint arXiv:2009.00142, 2020.
|
| 238 |
+
|
| 239 |
+
[25] Christopher Morris, Gaurav Rattan, and Petra Mutzel. Weisfeiler and leman go sparse: Towards scalable higher-order graph embeddings. 2020.
|
| 240 |
+
|
| 241 |
+
[26] Muhan Zhang and Yixin Chen. Weisfeiler-lehman neural machine for link prediction. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 575–583. ACM, 2017.
|
| 242 |
+
|
| 243 |
+
[27] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018.
|
| 244 |
+
|
| 245 |
+
[28] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in Neural Information Processing Systems, pages 3391–3401, 2017.
|
| 246 |
+
|
| 247 |
+
[29] Andries E Brouwer and Willem H Haemers. Strongly regular graphs. In Spectra of Graphs, pages 115–149. Springer, 2012.
|
| 248 |
+
|
| 249 |
+
[30] Ryan Murphy, Balasubramaniam Srinivasan, Vinayak Rao, and Bruno Ribeiro. Relational pooling for graph representations. In International Conference on Machine Learning, pages 4663–4673. PMLR, 2019.
|
| 250 |
+
|
| 251 |
+
[31] Giorgos Bouritsas, Fabrizio Frasca, Stefanos Zafeiriou, and Michael M Bronstein. Improving graph neural network expressivity via subgraph isomorphism counting. arXiv preprint arXiv:2006.09252, 2020.
|
| 252 |
+
|
| 253 |
+
[32] Jiaxuan You, Jonathan Gomes-Selman, Rex Ying, and Jure Leskovec. Identity-aware graph neural networks. arXiv preprint arXiv:2101.10320, 2021.
|
| 254 |
+
|
| 255 |
+
[33] Dominique Beaini, Saro Passaro, Vincent Létourneau, William L Hamilton, Gabriele Corso, and Pietro Liò. Directional graph networks. arXiv preprint arXiv:2010.02863, 2020.
|
| 256 |
+
|
| 257 |
+
[34] Haggai Maron, Heli Ben-Hamu, Nadav Shamir, and Yaron Lipman. Invariant and equivariant graph networks. arXiv preprint arXiv:1812.09902, 2018.
|
| 258 |
+
|
| 259 |
+
[35] Haggai Maron, Ethan Fetaya, Nimrod Segol, and Yaron Lipman. On the universality of invariant networks. In International conference on machine learning, pages 4363–4371. PMLR, 2019.
|
| 260 |
+
|
| 261 |
+
[36] Nicolas Keriven and Gabriel Peyré. Universal invariant and equivariant graph neural networks. arXiv preprint arXiv:1905.04943, 2019.
|
| 262 |
+
|
| 263 |
+
[37] Waïss Azizian and Marc Lelarge. Characterizing the expressive power of invariant and equivariant graph neural networks. arXiv preprint arXiv:2006.15646, 2020.
|
| 264 |
+
|
| 265 |
+
[38] George Dasoulas, Ludovic Dos Santos, Kevin Scaman, and Aladin Virmaux. Coloring graph neural networks for node disambiguation. arXiv preprint arXiv:1912.06058, 2019.
|
| 266 |
+
|
| 267 |
+
[39] Zhengdao Chen, Lei Chen, Soledad Villar, and Joan Bruna. Can graph neural networks count substructures? Advances in neural information processing systems, 2020.
|
| 268 |
+
|
| 269 |
+
[40] Andreas Loukas. What graph neural networks cannot learn: depth vs width. arXiv preprint arXiv:1907.03199, 2019.
|
| 270 |
+
|
| 271 |
+
[41] Ryoma Sato, Makoto Yamada, and Hisashi Kashima. Random features strengthen graph neural networks. arXiv preprint arXiv:2002.03155, 2020.
|
| 272 |
+
|
| 273 |
+
[42] Ralph Abboud, ˙Ismail ˙Ilkan Ceylan, Martin Grohe, and Thomas Lukasiewicz. The surprising power of graph neural networks with random node initialization. arXiv preprint arXiv:2010.01179, 2020.
|
| 274 |
+
|
| 275 |
+
[43] Clément Vignac, Andreas Loukas, and Pascal Frossard. Building powerful and equivariant graph neural networks with structural message-passing. arXiv e-prints, pages arXiv–2006, 2020.
|
| 276 |
+
|
| 277 |
+
[44] Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pages 5165–5175, 2018.
|
| 278 |
+
|
| 279 |
+
[45] Muhan Zhang, Pan Li, Yinglong Xia, Kai Wang, and Long Jin. Revisiting graph neural networks for link prediction. arXiv preprint arXiv:2010.16103, 2020.
|
| 280 |
+
|
| 281 |
+
[46] Muhan Zhang and Yixin Chen. Inductive matrix completion based on graph neural networks. In International Conference on Learning Representations, 2020. URL https://openreview. net/forum?id $\equiv$ ByxxgCEYDS.
|
| 282 |
+
|
| 283 |
+
[47] Pim de Haan, Taco Cohen, and Max Welling. Natural graph networks. arXiv preprint arXiv:2007.08349, 2020.
|
| 284 |
+
|
| 285 |
+
[48] Giannis Nikolentzos, George Dasoulas, and Michalis Vazirgiannis. k-hop graph neural networks. Neural Networks, 130:195–205, 2020.
|
| 286 |
+
|
| 287 |
+
[49] Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pages 21–29. PMLR, 2019.
|
| 288 |
+
|
| 289 |
+
[50] Kexin Huang and Marinka Zitnik. Graph meta learning via local subgraphs. Advances in Neural Information Processing Systems, 33, 2020.
|
| 290 |
+
|
| 291 |
+
[51] Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric. arXiv preprint arXiv:1903.02428, 2019.
|
| 292 |
+
|
| 293 |
+
[52] Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific data, 1(1):1–7, 2014.
|
| 294 |
+
|
| 295 |
+
[53] Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl Leswing, and Vijay Pande. Moleculenet: a benchmark for molecular machine learning. Chemical science, 9(2):513–530, 2018.
|
| 296 |
+
|
| 297 |
+
[54] Kristian Kersting, Nils M. Kriege, Christopher Morris, Petra Mutzel, and Marion Neumann. Benchmark data sets for graph kernels, 2016. URL http://graphkernels.cs. tu-dortmund.de.
|
| 298 |
+
|
| 299 |
+
[55] Paul D Dobson and Andrew J Doig. Distinguishing enzyme structures from non-enzymes without alignments. Journal of molecular biology, 330(4):771–783, 2003.
|
| 300 |
+
|
| 301 |
+
[56] Asim Kumar Debnath, de Compadre RL Lopez, Gargi Debnath, Alan J Shusterman, and Corwin Hansch. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity. Journal of medicinal chemistry, 34(2):786–797, 1991.
|
| 302 |
+
[57] Hannu Toivonen, Ashwin Srinivasan, Ross D King, Stefan Kramer, and Christoph Helma. Statistical evaluation of the predictive toxicology challenge 2000–2001. Bioinformatics, 19(10): 1183–1193, 2003.
|
| 303 |
+
[58] Ida Schomburg, Antje Chang, Christian Ebeling, Marion Gremse, Christian Heldt, Gregor Huhn, and Dietmar Schomburg. Brenda, the enzyme database: updates and major new developments. Nucleic acids research, 32(suppl_1):D431–D433, 2004.
|
| 304 |
+
[59] 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. arXiv preprint arXiv:2005.00687, 2020.
|
| 305 |
+
[60] Brandon Anderson, Truong-Son Hy, and Risi Kondor. Cormorant: Covariant molecular neural networks. arXiv preprint arXiv:1906.04015, 2019.
|
| 306 |
+
[61] Johannes Klicpera, Janek Groß, and Stephan Günnemann. Directional message passing for molecular graphs. arXiv preprint arXiv:2003.03123, 2020.
|
| 307 |
+
[62] Zhuoran Qiao, Matthew Welborn, Animashree Anandkumar, Frederick R Manby, and Thomas F Miller III. Orbnet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features. The Journal of Chemical Physics, 153(12):124111, 2020.
|
| 308 |
+
[63] Hanqing Zeng, Muhan Zhang, Yinglong Xia, Ajitesh Srivastava, Andrey Malevich, Rajgopal Kannan, Viktor Prasanna, Long Jin, and Ren Chen. Deep graph neural networks with shallow subgraph samplers. arXiv preprint arXiv:2012.01380, 2020.
|
| 309 |
+
[64] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pages 1025–1035, 2017.
|
| 310 |
+
[65] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. arXiv preprint arXiv:1912.09893, 2019.
|
| 311 |
+
[66] Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019.
|
| 312 |
+
[67] Guohao Li, Chenxin Xiong, Ali Thabet, and Bernard Ghanem. Deepergcn: All you need to train deeper gcns. arXiv preprint arXiv:2006.07739, 2020.
|
| 313 |
+
[68] Gabriele Corso, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Velickovi ˇ c. Principal ´ neighbourhood aggregation for graph nets. arXiv preprint arXiv:2004.05718, 2020.
|
| 314 |
+
[69] Rémy Brossard, Oriel Frigo, and David Dehaene. Graph convolutions that can finally model local structure. arXiv preprint arXiv:2011.15069, 2020.
|
| 315 |
+
[70] Tuan Le, Marco Bertolini, Frank Noé, and Djork-Arné Clevert. Parameterized hypercomplex graph neural networks for graph classification. arXiv preprint arXiv:2103.16584, 2021.
|
| 316 |
+
[71] Katsuhiko Ishiguro, Shin-ichi Maeda, and Masanori Koyama. Graph warp module: an auxiliary module for boosting the power of graph neural networks. arXiv preprint arXiv:1902.01020, 2019.
|
| 317 |
+
[72] Hongyang Gao and Shuiwang Ji. Graph u-nets. arXiv preprint arXiv:1905.05178, 2019.
|
| 318 |
+
[73] Douglas J Klein and Milan Randic. Resistance distance. ´ Journal of Mathematical Chemistry, 12(1):81–95, 1993.
|
parse/train/7_eLEvFjCi3/7_eLEvFjCi3_content_list.json
ADDED
|
@@ -0,0 +1,1559 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Nested Graph Neural Networks ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
305,
|
| 8 |
+
122,
|
| 9 |
+
691,
|
| 10 |
+
147
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Muhan Zhang1,2,∗ Pan Li3,† 1Institute for Artificial Intelligence, Peking University 2Beijing Institute for General Artificial Intelligence 3Department of Computer Science, Purdue University ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
320,
|
| 19 |
+
199,
|
| 20 |
+
678,
|
| 21 |
+
258
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Abstract ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
462,
|
| 31 |
+
294,
|
| 32 |
+
535,
|
| 33 |
+
310
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "Graph neural network (GNN)’s success in graph classification is closely related to the Weisfeiler-Lehman (1-WL) algorithm. By iteratively aggregating neighboring node features to a center node, both 1-WL and GNN obtain a node representation that encodes a rooted subtree around the center node. These rooted subtree representations are then pooled into a single representation to represent the whole graph. However, rooted subtrees are of limited expressiveness to represent a nontree graph. To address it, we propose Nested Graph Neural Networks (NGNNs). NGNN represents a graph with rooted subgraphs instead of rooted subtrees, so that two graphs sharing many identical subgraphs (rather than subtrees) tend to have similar representations. The key is to make each node representation encode a subgraph around it more than a subtree. To achieve this, NGNN extracts a local subgraph around each node and applies a base GNN to each subgraph to learn a subgraph representation. The whole-graph representation is then obtained by pooling these subgraph representations. We provide a rigorous theoretical analysis showing that NGNN is strictly more powerful than 1-WL. In particular, we proved that NGNN can discriminate almost all $r$ -regular graphs, where 1-WL always fails. Moreover, unlike other more powerful GNNs, NGNN only introduces a constantfactor higher time complexity than standard GNNs. NGNN is a plug-and-play framework that can be combined with various base GNNs. We test NGNN with different base GNNs on several benchmark datasets. NGNN uniformly improves their performance and shows highly competitive performance on all datasets. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
325,
|
| 43 |
+
766,
|
| 44 |
+
616
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
176,
|
| 54 |
+
642,
|
| 55 |
+
310,
|
| 56 |
+
660
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Graph is an important tool to model relational data in the real world. Representation learning over graphs has become a popular topic of machine learning in recent years. While network embedding methods, such as DeepWalk [1], can learn node representations well, they fail to generalize to whole-graph representations, which are crucial for applications such as graph classification, molecule modeling, and drug discovery. On the contrary, although traditional graph kernels [2–7] can be used for graph classification, they define graph similarity often in a heuristic way, which is not parameterized and lacks some flexibility to deal with features. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
675,
|
| 66 |
+
825,
|
| 67 |
+
771
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "In this context, graph neural networks (GNNs) have regained people’s attention and become the state-of-the-art graph representation learning tool [8–17]. GNNs use message passing to propagate features between connected nodes. By iteratively aggregating neighboring node features to the center node, GNNs learn node representations encoding their local structure and feature information. These node representations can be further pooled into a graph representation, enabling graph-level tasks such as graph classification. In this paper, we will use message passing GNNs to denote this class of GNNs based on repeated neighbor aggregation [18], in order to distinguish them from some high-order GNN variants [19–21] where the effective message passing happens between high-order node tuples instead of nodes. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
779,
|
| 77 |
+
823,
|
| 78 |
+
862
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "image",
|
| 84 |
+
"img_path": "images/def539561826dd11f9391009109ccd60e1cc43c574755925f1e343b3b82d5a5f.jpg",
|
| 85 |
+
"image_caption": [
|
| 86 |
+
"Figure 1: The two original graphs $G _ { 1 }$ and $G _ { 2 }$ are non-isomorphic. $G _ { 1 }$ is composed of two triangles, while $G _ { 2 }$ is a hexagon. However, both 1-WL and message passing GNNs cannot differentiate them, since all nodes in the two graphs share identical rooted subtrees at any height (see the rooted subtrees around $v _ { 1 }$ and $v _ { 2 }$ in the middle block for example). In comparison, we can discriminate the two graphs by comparing their height-1 rooted subgraphs around any nodes. For example, the height-1 rooted subgraph around $v _ { 1 }$ is a closed triangle, but the height-1 rooted subgraph around $v _ { 2 }$ is an open triangle (see the red boxes in the right block). "
|
| 87 |
+
],
|
| 88 |
+
"image_footnote": [],
|
| 89 |
+
"bbox": [
|
| 90 |
+
196,
|
| 91 |
+
88,
|
| 92 |
+
802,
|
| 93 |
+
260
|
| 94 |
+
],
|
| 95 |
+
"page_idx": 1
|
| 96 |
+
},
|
| 97 |
+
{
|
| 98 |
+
"type": "text",
|
| 99 |
+
"text": "",
|
| 100 |
+
"bbox": [
|
| 101 |
+
176,
|
| 102 |
+
380,
|
| 103 |
+
823,
|
| 104 |
+
422
|
| 105 |
+
],
|
| 106 |
+
"page_idx": 1
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "GNNs’ message passing scheme mimics the 1-dimensional Weisfeiler-Lehman (1-WL) algorithm [22], which iteratively refines a node’s color according to its current color and the multiset of its neighbors’ colors. This procedure essentially encodes a rooted subtree around each node into its final color, where the rooted subtree is constructed by recursively expanding the neighbors of the root node. One critical reason for GNN’s success in graph classification is because two graphs sharing many identical or similar rooted subtrees are more likely classified into the same class, which actually aligns with the inductive bias that two graphs are similar if they have many common substructures [23]. ",
|
| 111 |
+
"bbox": [
|
| 112 |
+
174,
|
| 113 |
+
428,
|
| 114 |
+
825,
|
| 115 |
+
525
|
| 116 |
+
],
|
| 117 |
+
"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "Despite this, rooted subtrees are still limited in terms of expressing all possible substructures that can appear in a graph. It is likely that two graphs, despite sharing a lot of identical rooted subtrees, are not similar at all because their other substructure patterns are not similar. Take the two graphs $G _ { 1 }$ and $G _ { 2 }$ in Figure 1 as an example. If we apply 1-WL or a message passing GNN to them, the two graphs will always have the same representation no matter how many iterations/layers we use. This is because all nodes in the two graphs have identical rooted subtrees across all tree heights. However, the two graphs are quite different from a holistic perspective. $G _ { 1 }$ is composed of two triangles, while $G _ { 2 }$ is a hexagon. The intrinsic reason for such a failure is that rooted subtrees have limited expressiveness for representing general graphs, especially those with cycles. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
174,
|
| 124 |
+
531,
|
| 125 |
+
825,
|
| 126 |
+
656
|
| 127 |
+
],
|
| 128 |
+
"page_idx": 1
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "To address this issue, we propose Nested Graph Neural Networks (NGNNs). The core idea is, instead of encoding a rooted subtree, we want the final representation of a node to encode a rooted subgraph (local $h$ -hop subgraph) around it. The subgraph is not restricted to be of any particular graph type such as tree, but serves as a general description of the local neighborhood around a node. Rooted subgraphs offer much better representation power than rooted subtrees, e.g., we can easily discriminate the two graphs in Figure 1 by only comparing their height-1 rooted subgraphs. ",
|
| 133 |
+
"bbox": [
|
| 134 |
+
174,
|
| 135 |
+
662,
|
| 136 |
+
825,
|
| 137 |
+
746
|
| 138 |
+
],
|
| 139 |
+
"page_idx": 1
|
| 140 |
+
},
|
| 141 |
+
{
|
| 142 |
+
"type": "text",
|
| 143 |
+
"text": "To represent a graph with rooted subgraphs, NGNN uses two levels of GNNs: base (inner) GNNs and an outer GNN. By extracting a local rooted subgraph around each node, NGNN first applies a base GNN to each node’s subgraph independently. Then, a subgraph pooling layer is applied to each subgraph to aggregate the intermediate node representations into a subgraph representation. This subgraph representation is used as the final representation of the root node. Rather than encoding a rooted subtree, this final node representation encodes the local subgraph around it, which contains more information than a subtree. Finally, all the final node representations are further fed into an outer GNN to learn a representation for the entire graph. Figure 2 shows one NGNN implementation using message passing GNNs as the base GNNs and a simple graph pooling layer as the outer GNN. ",
|
| 144 |
+
"bbox": [
|
| 145 |
+
174,
|
| 146 |
+
752,
|
| 147 |
+
825,
|
| 148 |
+
877
|
| 149 |
+
],
|
| 150 |
+
"page_idx": 1
|
| 151 |
+
},
|
| 152 |
+
{
|
| 153 |
+
"type": "text",
|
| 154 |
+
"text": "One may wonder that the base GNN seems to still learn only rooted subtrees if it is message-passingbased. Then why is NGNN more powerful than GNN? One key reason lies in the subgraph pooling layer. Take the height-1 rooted subgraphs (marked with red boxes) around $v _ { 1 }$ and $v _ { 2 }$ in Figure 1 as an example. Although $v _ { 1 }$ and $v _ { 2 }$ ’s height-1 rooted subtrees are still the same, their neighbors (labeled by 1 and 2) have different height-1 rooted subtrees. Thus, applying a one-layer message passing GNN plus a subgraph pooling as the base GNN is sufficient to discriminate $G _ { 1 }$ and $G _ { 2 }$ . ",
|
| 155 |
+
"bbox": [
|
| 156 |
+
174,
|
| 157 |
+
883,
|
| 158 |
+
823,
|
| 159 |
+
911
|
| 160 |
+
],
|
| 161 |
+
"page_idx": 1
|
| 162 |
+
},
|
| 163 |
+
{
|
| 164 |
+
"type": "image",
|
| 165 |
+
"img_path": "images/980bc2b24f862ccec031254479260a4ea420149715060948ad40aeec60dfa428.jpg",
|
| 166 |
+
"image_caption": [
|
| 167 |
+
"Figure 2: A particular implementation of the NGNN framework. It first extracts (copies) a rooted subgraph (height=1) around each node from the original graph, and then applies a base GNN with a subgraph pooling layer to each rooted subgraph independently to learn a subgraph representation. The subgraph representation is used as the root node’s final representation in the original graph. Then, a graph pooling layer is used to summarize the final node representations into a graph representation. "
|
| 168 |
+
],
|
| 169 |
+
"image_footnote": [],
|
| 170 |
+
"bbox": [
|
| 171 |
+
178,
|
| 172 |
+
88,
|
| 173 |
+
818,
|
| 174 |
+
248
|
| 175 |
+
],
|
| 176 |
+
"page_idx": 2
|
| 177 |
+
},
|
| 178 |
+
{
|
| 179 |
+
"type": "text",
|
| 180 |
+
"text": "",
|
| 181 |
+
"bbox": [
|
| 182 |
+
174,
|
| 183 |
+
347,
|
| 184 |
+
825,
|
| 185 |
+
402
|
| 186 |
+
],
|
| 187 |
+
"page_idx": 2
|
| 188 |
+
},
|
| 189 |
+
{
|
| 190 |
+
"type": "text",
|
| 191 |
+
"text": "The NGNN framework has multiple exclusive advantages. Firstly, it allows freely choosing the base GNN, and can enhance the base GNN’s representation power in a plug-and-play fashion. Theoretically, we proved that NGNN is more powerful than message passing GNNs and 1-WL by being able to discriminate almost all $r$ -regular graphs (where 1-WL always fails). Secondly, by extracting rooted subgraphs, NGNN allows augmenting the initial features of a node with subgraphspecific structural features such as distance encoding [24] to improve the quality of the learned node representations. Thirdly, unlike other more powerful graph neural networks, especially those based on higher-order WL tests [19–21, 25], NGNN still has linear time and space complexity w.r.t. graph size like standard message passing GNNs, thus maintaining good scalability. We demonstrate the effectiveness of the NGNN framework in various synthetic/real-world graph classification/regression datasets. On synthetic datasets, NGNN demonstrates higher-than-1-WL expressive power, matching very well with our theorem. On real-world datasets, NGNN consistently enhances a wide range of base GNNs’ performance, achieving highly competitive results on all datasets. ",
|
| 192 |
+
"bbox": [
|
| 193 |
+
174,
|
| 194 |
+
409,
|
| 195 |
+
825,
|
| 196 |
+
589
|
| 197 |
+
],
|
| 198 |
+
"page_idx": 2
|
| 199 |
+
},
|
| 200 |
+
{
|
| 201 |
+
"type": "text",
|
| 202 |
+
"text": "2 Preliminaries ",
|
| 203 |
+
"text_level": 1,
|
| 204 |
+
"bbox": [
|
| 205 |
+
174,
|
| 206 |
+
609,
|
| 207 |
+
318,
|
| 208 |
+
626
|
| 209 |
+
],
|
| 210 |
+
"page_idx": 2
|
| 211 |
+
},
|
| 212 |
+
{
|
| 213 |
+
"type": "text",
|
| 214 |
+
"text": "2.1 Notation and problem definition ",
|
| 215 |
+
"text_level": 1,
|
| 216 |
+
"bbox": [
|
| 217 |
+
176,
|
| 218 |
+
641,
|
| 219 |
+
436,
|
| 220 |
+
656
|
| 221 |
+
],
|
| 222 |
+
"page_idx": 2
|
| 223 |
+
},
|
| 224 |
+
{
|
| 225 |
+
"type": "text",
|
| 226 |
+
"text": "We consider the graph classification/regression problem. Given a graph $G = ( V , E )$ where $V =$ $\\{ 1 , 2 , \\ldots n \\}$ is the node set and $E \\subseteq V \\times V$ is the edge set, we aim to learn a function mapping $G$ to its class or target value $y$ . The nodes and edges in $G$ can have feature vectors associated with them, denoted by $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ (for node $i$ ) and $e _ { i j }$ (for edge $( i , j ) )$ , respectively. ",
|
| 227 |
+
"bbox": [
|
| 228 |
+
174,
|
| 229 |
+
666,
|
| 230 |
+
825,
|
| 231 |
+
723
|
| 232 |
+
],
|
| 233 |
+
"page_idx": 2
|
| 234 |
+
},
|
| 235 |
+
{
|
| 236 |
+
"type": "text",
|
| 237 |
+
"text": "2.2 Weisfeiler-Lehman test ",
|
| 238 |
+
"text_level": 1,
|
| 239 |
+
"bbox": [
|
| 240 |
+
174,
|
| 241 |
+
739,
|
| 242 |
+
372,
|
| 243 |
+
755
|
| 244 |
+
],
|
| 245 |
+
"page_idx": 2
|
| 246 |
+
},
|
| 247 |
+
{
|
| 248 |
+
"type": "text",
|
| 249 |
+
"text": "The Wesfeiler-Lehman (1-WL) test [22] is a popular algorithm for graph isomorphism checking. The classical 1-WL works as follows. At first, all nodes receive a color 1. Each node collects its neighbors’ colors into a multiset. Then, 1-WL will update each node’s color so that two nodes get the same new color if and only if their current colors are the same and they have identical multisets of neighbor colors. Repeat this process until the number of colors does not increase between two iterations. Then, 1-WL will return that two graphs are non-isomorphic if their node colors are different at some iteration, or fail to determine whether they are non-isomorphic. See [7, 26] for more details. ",
|
| 250 |
+
"bbox": [
|
| 251 |
+
174,
|
| 252 |
+
766,
|
| 253 |
+
825,
|
| 254 |
+
863
|
| 255 |
+
],
|
| 256 |
+
"page_idx": 2
|
| 257 |
+
},
|
| 258 |
+
{
|
| 259 |
+
"type": "text",
|
| 260 |
+
"text": "1-WL essentially encodes the rooted subtrees around each node at different heights into its color representations. Figure 1 middle shows the rooted subtrees around $v _ { 1 }$ and $v _ { 2 }$ . Two nodes will have the same color at iteration $h$ if and only if their height- $h$ rooted subtrees are the same. ",
|
| 261 |
+
"bbox": [
|
| 262 |
+
176,
|
| 263 |
+
869,
|
| 264 |
+
823,
|
| 265 |
+
911
|
| 266 |
+
],
|
| 267 |
+
"page_idx": 2
|
| 268 |
+
},
|
| 269 |
+
{
|
| 270 |
+
"type": "text",
|
| 271 |
+
"text": "3 Nested Graph Neural Network ",
|
| 272 |
+
"text_level": 1,
|
| 273 |
+
"bbox": [
|
| 274 |
+
174,
|
| 275 |
+
88,
|
| 276 |
+
462,
|
| 277 |
+
107
|
| 278 |
+
],
|
| 279 |
+
"page_idx": 3
|
| 280 |
+
},
|
| 281 |
+
{
|
| 282 |
+
"type": "text",
|
| 283 |
+
"text": "In this section, we introduce our Nested Graph Neural Network (NGNN) framework and theoretically demonstrate its higher representation power than message passing GNNs. ",
|
| 284 |
+
"bbox": [
|
| 285 |
+
173,
|
| 286 |
+
121,
|
| 287 |
+
823,
|
| 288 |
+
150
|
| 289 |
+
],
|
| 290 |
+
"page_idx": 3
|
| 291 |
+
},
|
| 292 |
+
{
|
| 293 |
+
"type": "text",
|
| 294 |
+
"text": "3.1 Limitations of the message passing GNNs ",
|
| 295 |
+
"text_level": 1,
|
| 296 |
+
"bbox": [
|
| 297 |
+
174,
|
| 298 |
+
166,
|
| 299 |
+
501,
|
| 300 |
+
181
|
| 301 |
+
],
|
| 302 |
+
"page_idx": 3
|
| 303 |
+
},
|
| 304 |
+
{
|
| 305 |
+
"type": "text",
|
| 306 |
+
"text": "Most existing GNNs follow the message passing framework [18]: given a graph $G$ , each node’s hidden state $\\bar { h } _ { v } ^ { t + 1 }$ is updated based on its previous state $\\boldsymbol { h } _ { v } ^ { t }$ and the messages $m _ { v } ^ { \\overline { { t } } + \\tilde { 1 } }$ from its neighbors ",
|
| 307 |
+
"bbox": [
|
| 308 |
+
171,
|
| 309 |
+
191,
|
| 310 |
+
823,
|
| 311 |
+
220
|
| 312 |
+
],
|
| 313 |
+
"page_idx": 3
|
| 314 |
+
},
|
| 315 |
+
{
|
| 316 |
+
"type": "equation",
|
| 317 |
+
"img_path": "images/c16e21203998c13dbb2549d80933ff72d41d032f8b2e100abb589b45ae97cfd2.jpg",
|
| 318 |
+
"text": "$$\n\\pmb { h } _ { v } ^ { t + 1 } = U _ { t } ( \\pmb { h } _ { v } ^ { t } , \\pmb { m } _ { v } ^ { t + 1 } ) , \\mathrm { ~ w h e r e ~ } \\pmb { m } _ { v } ^ { t + 1 } = \\sum _ { u \\in N ( v | G ) } M _ { t } ( \\pmb { h } _ { v } ^ { t } , \\pmb { h } _ { u } ^ { t } , \\pmb { e } _ { v u } ) .\n$$",
|
| 319 |
+
"text_format": "latex",
|
| 320 |
+
"bbox": [
|
| 321 |
+
271,
|
| 322 |
+
227,
|
| 323 |
+
727,
|
| 324 |
+
263
|
| 325 |
+
],
|
| 326 |
+
"page_idx": 3
|
| 327 |
+
},
|
| 328 |
+
{
|
| 329 |
+
"type": "text",
|
| 330 |
+
"text": "Here $M _ { t } , U _ { t }$ are the message and update functions at time stamp $t$ , $e _ { v u }$ is the feature of edge $( v , u )$ , and $N ( v | G )$ is the set of $v$ ’s neighbors in graph $G$ . The initial hidden states $h _ { v } ^ { 0 }$ are given by the raw node features $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { \\mathit { v } }$ . After $T$ time stamps (iterations), the final node representations $h _ { v } ^ { T }$ are summarized into a whole-graph representation with a readout (pooling) function $R$ (e.g., mean or sum): ",
|
| 331 |
+
"bbox": [
|
| 332 |
+
174,
|
| 333 |
+
271,
|
| 334 |
+
825,
|
| 335 |
+
328
|
| 336 |
+
],
|
| 337 |
+
"page_idx": 3
|
| 338 |
+
},
|
| 339 |
+
{
|
| 340 |
+
"type": "equation",
|
| 341 |
+
"img_path": "images/b438f2d7ae6cab6b1e605b0891f962f643cc598074e0cfc23387d94ffde0bc5a.jpg",
|
| 342 |
+
"text": "$$\n\\begin{array} { r } { h _ { G } = R ( \\{ h _ { v } ^ { T } | v \\in G \\} ) . } \\end{array}\n$$",
|
| 343 |
+
"text_format": "latex",
|
| 344 |
+
"bbox": [
|
| 345 |
+
418,
|
| 346 |
+
334,
|
| 347 |
+
578,
|
| 348 |
+
353
|
| 349 |
+
],
|
| 350 |
+
"page_idx": 3
|
| 351 |
+
},
|
| 352 |
+
{
|
| 353 |
+
"type": "text",
|
| 354 |
+
"text": "Such a message passing (or neighbor aggregation) scheme iteratively aggregates neighbor information into a center node’s hidden state, making it encode a local rooted subtree around the node. The final node representations will contain both the local structure and feature information around nodes, enabling node-level tasks such as node classification. After a pooling layer, these node representations can be further summarized into a graph representation, enabling graph-level tasks. When there is no edge feature and the node features are from a countable space, it is shown that message passing GNNs are at most as powerful as the 1-WL test for discriminating non-isomorphic graphs [27, 19]. ",
|
| 355 |
+
"bbox": [
|
| 356 |
+
173,
|
| 357 |
+
367,
|
| 358 |
+
825,
|
| 359 |
+
465
|
| 360 |
+
],
|
| 361 |
+
"page_idx": 3
|
| 362 |
+
},
|
| 363 |
+
{
|
| 364 |
+
"type": "text",
|
| 365 |
+
"text": "For an $h$ -layer message passing GNN, it will give two nodes the same final representation if they have identical height- $h$ rooted subtrees (i.e., both the structures and the features on the corresponding nodes/edges are the same). If two graphs have a lot of identical (or similar) rooted subtrees, they will also have similar graph representations after pooling. This insight is crucial for the success of modern GNNs in graph classification, because it aligns with the inductive bias that two graphs are similar if they have many common substructures. Such insight has also been used in designing the WL subtree kernel [7], a state-of-the-art graph classification method before GNNs. ",
|
| 366 |
+
"bbox": [
|
| 367 |
+
173,
|
| 368 |
+
469,
|
| 369 |
+
825,
|
| 370 |
+
569
|
| 371 |
+
],
|
| 372 |
+
"page_idx": 3
|
| 373 |
+
},
|
| 374 |
+
{
|
| 375 |
+
"type": "text",
|
| 376 |
+
"text": "However, message passing GNNs have several limitations. Firstly, rooted subtree is only one specific substructure. It is not general enough to represent arbitrary subgraphs, especially those with cycles due to the natural restriction of tree structure. Secondly, using rooted subtree as the elementary substructure results in a discriminating power bounded by the 1-WL test. For example, all $n$ -node $r$ -regular graphs cannot be discriminated by message passing GNNs. Thirdly, standard message passing GNNs do not allow using root-node-specific structural features (such as the distance between a node and the root node) to improve the quality of the learned root node’s representation. We might need to break through such limitations in order to design more powerful GNNs. ",
|
| 377 |
+
"bbox": [
|
| 378 |
+
174,
|
| 379 |
+
573,
|
| 380 |
+
825,
|
| 381 |
+
685
|
| 382 |
+
],
|
| 383 |
+
"page_idx": 3
|
| 384 |
+
},
|
| 385 |
+
{
|
| 386 |
+
"type": "text",
|
| 387 |
+
"text": "3.2 The NGNN framework ",
|
| 388 |
+
"text_level": 1,
|
| 389 |
+
"bbox": [
|
| 390 |
+
174,
|
| 391 |
+
702,
|
| 392 |
+
374,
|
| 393 |
+
717
|
| 394 |
+
],
|
| 395 |
+
"page_idx": 3
|
| 396 |
+
},
|
| 397 |
+
{
|
| 398 |
+
"type": "text",
|
| 399 |
+
"text": "To address the above limitations, we propose the Nested Graph Neural Network (NGNN) framework. NGNN no longer aims to encode a rooted subtree around each node. Instead, in NGNN, each node’s final representation encodes the general local subgraph information around it more than a subtree, so that two graphs sharing a lot of identical or similar rooted subgraphs will have similar representations. ",
|
| 400 |
+
"bbox": [
|
| 401 |
+
173,
|
| 402 |
+
727,
|
| 403 |
+
825,
|
| 404 |
+
784
|
| 405 |
+
],
|
| 406 |
+
"page_idx": 3
|
| 407 |
+
},
|
| 408 |
+
{
|
| 409 |
+
"type": "text",
|
| 410 |
+
"text": "Definition 1. (Rooted subgraph) Given a graph $G$ and a node v, the height-h rooted subgraph $G _ { v } ^ { h }$ of v is the subgraph induced from $G$ by the nodes within h hops of v (including $h$ -hop nodes). ",
|
| 411 |
+
"bbox": [
|
| 412 |
+
173,
|
| 413 |
+
787,
|
| 414 |
+
823,
|
| 415 |
+
816
|
| 416 |
+
],
|
| 417 |
+
"page_idx": 3
|
| 418 |
+
},
|
| 419 |
+
{
|
| 420 |
+
"type": "text",
|
| 421 |
+
"text": "To make a node’s final representation encode a rooted subgraph, we need to compute a subgraph representation. To achieve this, we resort to an arbitrary GNN, which we call the base GNN of NGNN. For example, the base GNN can be simply a message passing GNN, which performs message passing within each rooted subgraph to learn an intermediate representation for every node of the subgraph, and then uses a pooling layer to summarize a subgraph representation from the intermediate node representations. This subgraph representation is used as the final representation of the root node in the original graph. Take root node $w$ as an example. We first perform $T$ rounds of message passing within node $w$ ’s rooted subgraph $G _ { w } ^ { h }$ . Let $v$ be any node appearing in $G _ { w } ^ { h }$ . We have ",
|
| 422 |
+
"bbox": [
|
| 423 |
+
174,
|
| 424 |
+
827,
|
| 425 |
+
825,
|
| 426 |
+
911
|
| 427 |
+
],
|
| 428 |
+
"page_idx": 3
|
| 429 |
+
},
|
| 430 |
+
{
|
| 431 |
+
"type": "text",
|
| 432 |
+
"text": "",
|
| 433 |
+
"bbox": [
|
| 434 |
+
171,
|
| 435 |
+
90,
|
| 436 |
+
825,
|
| 437 |
+
119
|
| 438 |
+
],
|
| 439 |
+
"page_idx": 4
|
| 440 |
+
},
|
| 441 |
+
{
|
| 442 |
+
"type": "equation",
|
| 443 |
+
"img_path": "images/49bd5cdeeb1cb959927081dccc7da5f789b7c04c6c8c1e49c62332dc2a3a1a95.jpg",
|
| 444 |
+
"text": "$$\nh _ { v , G _ { w } ^ { h } } ^ { t + 1 } = U _ { t } ( h _ { v , G _ { w } ^ { h } } ^ { t } , m _ { v , G _ { w } ^ { h } } ^ { t + 1 } ) , ~ \\mathrm { w h e r e } ~ m _ { v , G _ { w } ^ { h } } ^ { t + 1 } = \\sum _ { u \\in N ( v | G _ { w } ^ { h } ) } M _ { t } ( h _ { v , G _ { w } ^ { h } } ^ { t } , h _ { u , G _ { w } ^ { h } } ^ { t } , e _ { v u } ) .\n$$",
|
| 445 |
+
"text_format": "latex",
|
| 446 |
+
"bbox": [
|
| 447 |
+
217,
|
| 448 |
+
122,
|
| 449 |
+
779,
|
| 450 |
+
159
|
| 451 |
+
],
|
| 452 |
+
"page_idx": 4
|
| 453 |
+
},
|
| 454 |
+
{
|
| 455 |
+
"type": "text",
|
| 456 |
+
"text": "Here $M _ { t } , U _ { t }$ are the message and update functions of the base GNN at time stamp $t$ , $N ( v | G _ { w } ^ { h } )$ \ndenonode s th’s hi set of den st $v$ ’s neighbors within e and message spec $w$ ’s rooted subgraph to rooted subgrap $G _ { w } ^ { h }$ and at ti $h _ { v , G _ { w } ^ { h } } ^ { t + 1 }$ anmp $m _ { v , G _ { w } ^ { h } } ^ { t + 1 }$ denoteote that $v$ $G _ { w } ^ { h }$ $t + 1$ $v$ \ndifferent. This is in contrast to standard GNNs where a node’s hidden state and message at time $t$ is \nthe same regardless of which root node it contributes to. For example, $\\boldsymbol { h } _ { v } ^ { t + 1 }$ and $m _ { v } ^ { t + 1 }$ in Eq. 1 do \nnot depend on any particular rooted subgraph. ",
|
| 457 |
+
"bbox": [
|
| 458 |
+
173,
|
| 459 |
+
161,
|
| 460 |
+
825,
|
| 461 |
+
267
|
| 462 |
+
],
|
| 463 |
+
"page_idx": 4
|
| 464 |
+
},
|
| 465 |
+
{
|
| 466 |
+
"type": "text",
|
| 467 |
+
"text": "After $T$ rounds of message passing, we apply a subgraph pooling layer to summarize a subgraph representation $h _ { G _ { w } ^ { h } }$ from the intermediate node representations $\\{ h _ { v , G _ { w } ^ { h } } ^ { T } | v \\in G _ { w } ^ { h } \\}$ . ",
|
| 468 |
+
"bbox": [
|
| 469 |
+
173,
|
| 470 |
+
272,
|
| 471 |
+
820,
|
| 472 |
+
303
|
| 473 |
+
],
|
| 474 |
+
"page_idx": 4
|
| 475 |
+
},
|
| 476 |
+
{
|
| 477 |
+
"type": "equation",
|
| 478 |
+
"img_path": "images/e22e0eaee50333f733f5ae6f3824ec6d716e40d8f76b94df46abd153c7e35ccf.jpg",
|
| 479 |
+
"text": "$$\n\\pmb { h _ { w } } : = \\pmb { h } _ { G _ { w } ^ { h } } = R _ { 0 } \\big ( \\{ \\pmb { h } _ { v , G _ { w } ^ { h } } ^ { T } | v \\in G _ { w } ^ { h } \\} \\big ) ,\n$$",
|
| 480 |
+
"text_format": "latex",
|
| 481 |
+
"bbox": [
|
| 482 |
+
369,
|
| 483 |
+
306,
|
| 484 |
+
627,
|
| 485 |
+
328
|
| 486 |
+
],
|
| 487 |
+
"page_idx": 4
|
| 488 |
+
},
|
| 489 |
+
{
|
| 490 |
+
"type": "text",
|
| 491 |
+
"text": "where $R _ { 0 }$ is the subgraph pooling layer. This subgraph representation $h _ { G _ { w } ^ { h } }$ will be used as root node $w$ ’s final representation $h _ { w }$ in the original graph. Note that the base GNNs are simultaneously applied to all nodes’ rooted subgraphs to return a final node representation for every node in the original graph, and all the base GNNs share the same parameters. With such node representations, NGNN uses an outer GNN to further process and aggregate them into a graph representation of the whole graph. For simplicity, we let the outer GNN be simply a graph pooling layer denoted by $R _ { 1 }$ : ",
|
| 492 |
+
"bbox": [
|
| 493 |
+
174,
|
| 494 |
+
329,
|
| 495 |
+
825,
|
| 496 |
+
415
|
| 497 |
+
],
|
| 498 |
+
"page_idx": 4
|
| 499 |
+
},
|
| 500 |
+
{
|
| 501 |
+
"type": "equation",
|
| 502 |
+
"img_path": "images/52897b1345726c977981e90bee5b417406153e182506617e6f30a3becbcf5a35.jpg",
|
| 503 |
+
"text": "$$\n\\begin{array} { r } { h _ { G } : = R _ { 1 } ( \\{ h _ { w } | w \\in G \\} ) . } \\end{array}\n$$",
|
| 504 |
+
"text_format": "latex",
|
| 505 |
+
"bbox": [
|
| 506 |
+
410,
|
| 507 |
+
416,
|
| 508 |
+
586,
|
| 509 |
+
434
|
| 510 |
+
],
|
| 511 |
+
"page_idx": 4
|
| 512 |
+
},
|
| 513 |
+
{
|
| 514 |
+
"type": "text",
|
| 515 |
+
"text": "The Nested GNN framework can be understood as a two-level GNN, or a GNN of GNNs—the inner subgraph-level GNNs (base GNNs) are used to learn node representations from their rooted subgraphs, while the outer graph-level GNN is used to return a whole-graph representation from the inner GNNs’ outputs. The inner GNNs all share the same parameters which are trained end-to-end with the outer GNN. Figure 2 depicts the implementation of the NGNN framework described above. ",
|
| 516 |
+
"bbox": [
|
| 517 |
+
173,
|
| 518 |
+
441,
|
| 519 |
+
825,
|
| 520 |
+
512
|
| 521 |
+
],
|
| 522 |
+
"page_idx": 4
|
| 523 |
+
},
|
| 524 |
+
{
|
| 525 |
+
"type": "text",
|
| 526 |
+
"text": "Compared to message passing GNNs, NGNN changes the “receptive field” of each node from a rooted subtree to a rooted subgraph, in order to capture better local substructure information. The rooted subgraph is read by a base GNN to learn a subgraph representation. Finally, the outer GNN reads the subgraph representations output by the base GNNs to return a graph representation. ",
|
| 527 |
+
"bbox": [
|
| 528 |
+
173,
|
| 529 |
+
517,
|
| 530 |
+
825,
|
| 531 |
+
574
|
| 532 |
+
],
|
| 533 |
+
"page_idx": 4
|
| 534 |
+
},
|
| 535 |
+
{
|
| 536 |
+
"type": "text",
|
| 537 |
+
"text": "Note that, when we apply the base GNN to a rooted subgraph, this rooted subgraph is extracted (copied) out of the original graph and treated as a completely independent graph from the other rooted subgraphs and the original graph. This allows the same node to have different representations within different rooted subgraphs. For example, in Figure 2, the same node $B$ appears in four different rooted subgraphs. Sometimes it is the root node, while other times it is a 1-hop neighbor of the root node. NGNN enables learning different representations for the same node when it appears in different rooted subgraphs, in contrast to standard GNNs where a node only has one single representation at one time stamp (Eq. 1). Similarly, NGNN also enables using different initial features for the same node when it appears in different rooted subgraphs. This allows us to customize a node’s initial features based on its structural role within a rooted subgraph, as opposed to using the same initial features for a node across all rooted subgraphs. For example, we can optionally augment node $B$ ’s initial features with the distance between node $B$ and the root—when node $B$ is the root node, we give it an additional feature 0; and when $B$ is a $k$ -hop neighbor of the root, we give it an additional feature $k$ . Such feature augmentation may help better capture a node’s structural role within a rooted subgraph. It is an exclusive advantage of NGNN and is not possible in standard GNNs. ",
|
| 538 |
+
"bbox": [
|
| 539 |
+
173,
|
| 540 |
+
580,
|
| 541 |
+
825,
|
| 542 |
+
787
|
| 543 |
+
],
|
| 544 |
+
"page_idx": 4
|
| 545 |
+
},
|
| 546 |
+
{
|
| 547 |
+
"type": "text",
|
| 548 |
+
"text": "3.3 The representation power of NGNN ",
|
| 549 |
+
"text_level": 1,
|
| 550 |
+
"bbox": [
|
| 551 |
+
176,
|
| 552 |
+
803,
|
| 553 |
+
460,
|
| 554 |
+
818
|
| 555 |
+
],
|
| 556 |
+
"page_idx": 4
|
| 557 |
+
},
|
| 558 |
+
{
|
| 559 |
+
"type": "text",
|
| 560 |
+
"text": "We theoretically characterize the additional expressive power of NGNN (using message passing GNNs as base GNNs) as opposed to standard message passing GNNs. We focus on the ability to discriminate regular graphs because they form an important category of graphs which standard GNNs cannot represent well. Using 1-WL or message passing GNNs, any two $n$ -sized $r$ -regular graphs will have the same representation, unless discriminative node features are available. In contrast, we prove that NGNN can distinguish almost all pairs of $n$ -sized $r$ -regular graphs regardless of node features. ",
|
| 561 |
+
"bbox": [
|
| 562 |
+
174,
|
| 563 |
+
827,
|
| 564 |
+
825,
|
| 565 |
+
911
|
| 566 |
+
],
|
| 567 |
+
"page_idx": 4
|
| 568 |
+
},
|
| 569 |
+
{
|
| 570 |
+
"type": "text",
|
| 571 |
+
"text": "Definition 2. If the message passing (Eq. 3) and the two-level graph pooling (Eqs. 4,5) are all injective given input from a countable space, then the NGNN is called proper. ",
|
| 572 |
+
"bbox": [
|
| 573 |
+
173,
|
| 574 |
+
90,
|
| 575 |
+
823,
|
| 576 |
+
119
|
| 577 |
+
],
|
| 578 |
+
"page_idx": 5
|
| 579 |
+
},
|
| 580 |
+
{
|
| 581 |
+
"type": "text",
|
| 582 |
+
"text": "A proper NGNN always exists due to the representation power of fully-connected neural networks used for message passing and Deep Set for graph pooling [28]. For all pairs of graphs that 1-WL can discriminate, there always exists a proper NGNN that can also discriminate them, because two graphs discriminated by 1-WL means they must have different multisets of rooted subtrees at some height $h$ , while a rooted subtree is always included in a rooted subgraph with the same height. ",
|
| 583 |
+
"bbox": [
|
| 584 |
+
174,
|
| 585 |
+
130,
|
| 586 |
+
825,
|
| 587 |
+
200
|
| 588 |
+
],
|
| 589 |
+
"page_idx": 5
|
| 590 |
+
},
|
| 591 |
+
{
|
| 592 |
+
"type": "text",
|
| 593 |
+
"text": "Now we present our main theorem. ",
|
| 594 |
+
"bbox": [
|
| 595 |
+
176,
|
| 596 |
+
207,
|
| 597 |
+
403,
|
| 598 |
+
220
|
| 599 |
+
],
|
| 600 |
+
"page_idx": 5
|
| 601 |
+
},
|
| 602 |
+
{
|
| 603 |
+
"type": "text",
|
| 604 |
+
"text": "Theorem 1. Consider all pairs of $n$ -sized $r$ -regular graphs, where $3 \\leq r < ( 2 \\log n ) ^ { 1 / 2 }$ . For any small constant $\\epsilon > 0$ , there exists a proper NGNN using at most $\\begin{array} { r } { \\lceil ( \\frac { 1 } { 2 } + \\epsilon ) \\frac { \\log n } { \\log ( r - 1 - \\epsilon ) } \\rceil } \\end{array}$ -height rooted subgraphs and $\\left\\lceil \\epsilon \\frac { \\log n } { \\log ( r - 1 - \\epsilon ) } \\right\\rceil$ -layer message passing, which distinguishes almost all $( 1 - o ( 1 ) ,$ ) such pairs of graphs. ",
|
| 605 |
+
"bbox": [
|
| 606 |
+
174,
|
| 607 |
+
226,
|
| 608 |
+
825,
|
| 609 |
+
294
|
| 610 |
+
],
|
| 611 |
+
"page_idx": 5
|
| 612 |
+
},
|
| 613 |
+
{
|
| 614 |
+
"type": "text",
|
| 615 |
+
"text": "We include the proof in Appendix A. Theorem 1 has three implications. Firstly, since NGNN can discriminate almost all $r$ -regular graphs where 1-WL always fails, it is strictly more powerful than 1-WL and message passing GNNs. Secondly, it implies that NGNN does not need to extract subgraphs with a too large height (abou t 12 log nlog (r−1) ) to be more powerful. Moreover, NGNN is already powerful with very few layers, i.e., an arbitrarily small constant $\\epsilon$ times $\\frac { \\log n } { \\log ( r - 1 ) }$ (as few as 1 layer). This benefit comes from the subgraph pooling (Eq. 4), freeing us from using deep base GNNs. We further conduct a simulation experiment in Appendix D to verify Theorem 1 by testing how well NGNN discriminates $r$ -regular graphs in practice. The results match almost perfectly with our theory. ",
|
| 616 |
+
"bbox": [
|
| 617 |
+
174,
|
| 618 |
+
303,
|
| 619 |
+
825,
|
| 620 |
+
426
|
| 621 |
+
],
|
| 622 |
+
"page_idx": 5
|
| 623 |
+
},
|
| 624 |
+
{
|
| 625 |
+
"type": "text",
|
| 626 |
+
"text": "Although NGNN is strictly more powerful than 1-WL and 2-WL (1-WL and 2-WL have the same discriminating power [20]), it is unclear whether NGNN is more powerful than 3-WL. Our early-stage analysis shows both NGNN and 3-WL cannot discriminate strongly regular graphs with the same parameters [29]. We leave the exact comparison between NGNN and 3-WL to future work. ",
|
| 627 |
+
"bbox": [
|
| 628 |
+
174,
|
| 629 |
+
431,
|
| 630 |
+
825,
|
| 631 |
+
487
|
| 632 |
+
],
|
| 633 |
+
"page_idx": 5
|
| 634 |
+
},
|
| 635 |
+
{
|
| 636 |
+
"type": "text",
|
| 637 |
+
"text": "3.4 Discussion ",
|
| 638 |
+
"text_level": 1,
|
| 639 |
+
"bbox": [
|
| 640 |
+
174,
|
| 641 |
+
503,
|
| 642 |
+
285,
|
| 643 |
+
517
|
| 644 |
+
],
|
| 645 |
+
"page_idx": 5
|
| 646 |
+
},
|
| 647 |
+
{
|
| 648 |
+
"type": "text",
|
| 649 |
+
"text": "Base GNN. NGNN is a general plug-and-play framework to increase the power of a base GNN. For the base GNN, we are not restricted to message passing GNNs as described in Section 3.2. For example, we can also use GNNs approximating the power of higher-dimensional WL tests, such as 1-2-3-GNN [19] and PPGN/Ring-GNN [20, 21], as the base GNN. In fact, one limitation of these high-order GNNs is their $\\mathcal { O } ( n ^ { 3 } )$ complexity. Using the NGNN framework we can greatly alleviate this by applying the higher-order GNN to multiple small rooted subgraphs instead of the whole graph. Suppose a rooted subgraph has at most $c$ nodes, then by applying a high-order GNN to all $n$ rooted subgraphs, we can reduce the time complexity from $\\dot { \\mathcal { O } ( n ^ { 3 } ) }$ to $\\bar { \\mathcal { O } } \\bar { ( n c ^ { 3 } ) }$ . ",
|
| 650 |
+
"bbox": [
|
| 651 |
+
173,
|
| 652 |
+
529,
|
| 653 |
+
825,
|
| 654 |
+
640
|
| 655 |
+
],
|
| 656 |
+
"page_idx": 5
|
| 657 |
+
},
|
| 658 |
+
{
|
| 659 |
+
"type": "text",
|
| 660 |
+
"text": "Complexity. We compare the time complexity of NGNN (using message passing GNNs as base GNNs) with a standard message passing GNN. Suppose the graph has $n$ nodes with a maximum degree $d$ , and the maximum number of nodes in a rooted subgraph is $c$ . Each message passing iteration in a standard message passing GNN takes $\\mathcal { O } ( n d )$ operations. In NGNN, we need to perform message passing over all $n$ nodes’ rooted subgraphs, which takes $O ( n \\cdot c d )$ . We will keep $c$ small (which can be achieved by using a small $h$ ) to improve NGNN’s scalability. Additionally, a small $c$ enables the base GNN to focus on learning local subgraph patterns. ",
|
| 661 |
+
"bbox": [
|
| 662 |
+
174,
|
| 663 |
+
645,
|
| 664 |
+
825,
|
| 665 |
+
743
|
| 666 |
+
],
|
| 667 |
+
"page_idx": 5
|
| 668 |
+
},
|
| 669 |
+
{
|
| 670 |
+
"type": "text",
|
| 671 |
+
"text": "In Appendix B, we discuss some other design choices of NGNN. ",
|
| 672 |
+
"bbox": [
|
| 673 |
+
174,
|
| 674 |
+
748,
|
| 675 |
+
598,
|
| 676 |
+
763
|
| 677 |
+
],
|
| 678 |
+
"page_idx": 5
|
| 679 |
+
},
|
| 680 |
+
{
|
| 681 |
+
"type": "text",
|
| 682 |
+
"text": "4 Related work ",
|
| 683 |
+
"text_level": 1,
|
| 684 |
+
"bbox": [
|
| 685 |
+
174,
|
| 686 |
+
782,
|
| 687 |
+
316,
|
| 688 |
+
799
|
| 689 |
+
],
|
| 690 |
+
"page_idx": 5
|
| 691 |
+
},
|
| 692 |
+
{
|
| 693 |
+
"type": "text",
|
| 694 |
+
"text": "Understanding GNN’s representation power is a fundamental problem in GNN research. Xu et al. [27] and Morris et al. [19] first proved that the discriminating power of message passing GNNs is bounded by the 1-WL test, namely they cannot discriminate two non-isomorphic graphs that 1-WL fails to discriminate (such as $r$ -regular graphs). Since then, there is increasing effort in enhancing GNN’s discriminating power beyond 1-WL [19, 21, 20, 30, 24, 31–33, 25]. Many GNNs have been proposed to mimic higher-dimensional WL tests, such as 1-2-3-GNN [19], Ring-GNN [21] and PPGN [20]. However, these models generally require learning the representations of all node tuples of certain cardinality (e.g., node pairs, node triples and so on), thus cannot leverage the sparsity of graph structure and are difficult to scale to large graphs. Some works study the universality of GNNs for approximating any invariant or equivariant functions over graphs [34, 21, 35–37]. However, reaching universality would require polynomial $( n )$ -order tensors, which hold more theoretical value than practical applicability. Dasoulas et al. [38] propose to augment nodes of identical attributes with different colors, which requires exhausting all the coloring choices to reach universality. Similarly, Relational Pooling (RP) [30] uses the ensemble of permutation-aware functions over graphs to reach universality, which requires exhausting all $n !$ permutations to achieve its theoretical power. Its local version Local Relational Pooling (LRP) [39] applies RP over subgraphs around nodes, which is similar to our work yet still requires exhausting node permutations in local subgraphs and even more loses RP’s theoretical power. In contrast, NGNN maintains a controllable cost by only applying a message passing GNN to local subgraphs, and is guaranteed to be more powerful than 1-WL. ",
|
| 695 |
+
"bbox": [
|
| 696 |
+
174,
|
| 697 |
+
814,
|
| 698 |
+
825,
|
| 699 |
+
911
|
| 700 |
+
],
|
| 701 |
+
"page_idx": 5
|
| 702 |
+
},
|
| 703 |
+
{
|
| 704 |
+
"type": "text",
|
| 705 |
+
"text": "",
|
| 706 |
+
"bbox": [
|
| 707 |
+
174,
|
| 708 |
+
90,
|
| 709 |
+
825,
|
| 710 |
+
257
|
| 711 |
+
],
|
| 712 |
+
"page_idx": 6
|
| 713 |
+
},
|
| 714 |
+
{
|
| 715 |
+
"type": "text",
|
| 716 |
+
"text": "Because of the high cost of mimicking high-dimensional WL tests, several works have been proposed to increase GNN’s representation power within the message passing framework. Observing that different neighbors are indistinguishable during neighbor aggregation, some works propose to add one-hot node index features or random features to GNNs [40, 41]. These methods work well when nodes naturally have distinct identities irrespective of the graph structure. However, although making GNNs more discriminative, they also lose some of GNNs’ generalization ability by not being able to guarantee nodes with identical neighborhoods to have the same embedding; the resulting models are also no longer permutation invariant. Repeating random initialization helps with avoiding such an issue but gets much slower convergence [42]. An exception is structural message-passing (SMP) [43], which propagates one-hot node index features to learn a global $n \\times d$ feature matrix for each node. The feature matrix is further pooled to learn a permutation-invariant node representation. ",
|
| 717 |
+
"bbox": [
|
| 718 |
+
174,
|
| 719 |
+
263,
|
| 720 |
+
825,
|
| 721 |
+
415
|
| 722 |
+
],
|
| 723 |
+
"page_idx": 6
|
| 724 |
+
},
|
| 725 |
+
{
|
| 726 |
+
"type": "text",
|
| 727 |
+
"text": "On the contrary, some works propose to use structural features to augment GNNs without hurting the generalization ability of GNNs. SEAL [44, 45], IGMC [46] and DE [24] use distance-based features, where a distance vector w.r.t. the target node set to predict is calculated for each node as its additional features. Our NGNN framework is naturally compatible with such distance-based features due to its independent rooted subgraph processing. GSN [31] uses the count of certain substructures to augment node/edge features, which also surpasses 1-WL theoretically. However, GSN needs a properly defined substructure set to incorporate domain-specific inductive biases, while NGNN aims to learn arbitrary substructures around nodes without the need to predefine a substructure set. ",
|
| 728 |
+
"bbox": [
|
| 729 |
+
174,
|
| 730 |
+
421,
|
| 731 |
+
825,
|
| 732 |
+
532
|
| 733 |
+
],
|
| 734 |
+
"page_idx": 6
|
| 735 |
+
},
|
| 736 |
+
{
|
| 737 |
+
"type": "text",
|
| 738 |
+
"text": "Concurrent to our work, You et al. [32] propose Identity-aware GNN (ID-GNN). ID-GNN uses different weight parameters between each root node and its context nodes during message passing. It also extracts a rooted subgraph around each node, and thus can be viewed as a special case of NGNN with: 1) the number of message passing layers equivalent to the subgraph height, 2) directly using the root node’s intermediate representation as its final representation without subgraph pooling, and 3) augmenting initial node features with 0/1 “identity”. However, the extra power of ID-GNN only comes from the “identity” feature, while the power of NGNN comes from the subgraph pooling— without using any node features, NGNN is still provably more discriminative than 1-WL. Another similar work to ours is natural graph network (NGN) [47]. NGN argues that graph convolution weights need not be shared among all nodes but only (locally) isomorphic nodes. If we view our distance-based node features as refining the graph convolution weights so that nodes within a center node’s neighborhood are no longer treated symmetrically, then our NGNN reduces to an NGN. ",
|
| 739 |
+
"bbox": [
|
| 740 |
+
174,
|
| 741 |
+
539,
|
| 742 |
+
825,
|
| 743 |
+
704
|
| 744 |
+
],
|
| 745 |
+
"page_idx": 6
|
| 746 |
+
},
|
| 747 |
+
{
|
| 748 |
+
"type": "text",
|
| 749 |
+
"text": "The idea of independently performing message passing within $k$ -hop neighborhood is also explored in $k$ -hop GNN [48] and MixHop [49]. However, MixHop directly concatenates the aggregation results of neighbors at different hops as the root representation, which ignores the connections between other nodes in the rooted subgraph. $k$ -hop GNN sequentially performs message passing for $k$ -hop, $k - 1$ -hop, ..., and 0-hop node (the update of $( i - 1 )$ -hop nodes depend on the updated states of $i$ -hop nodes), while NGNN simultaneously performs message passing for all nodes in the subgraph thus is more parallelizable. Both MixHop and $k$ -hop GNN directly use the root node’s representation as its final node representation. In contrast, NGNN uses a subgraph pooling to summarize all node representations within the subgraph as the final root representation, which distinguishes NGNN from other $k$ -hop models. As Theorem 1 shows, the subgraph pooling enables using a much smaller number of message passing layers $l$ (as small as 1) than the depth $k$ of the subgraph, while MixHop and $k$ -hop GNN always require $l \\geq k$ . MixHop and $k$ -hop GNN also do not have the strong theoretical power of NGNN to discriminate $r$ -regular graphs. Like SEAL and $k$ -hop GNN, G-Meta [50] is another work extracting subgraphs around nodes/links. It focuses specifically on a meta-learning setting. ",
|
| 750 |
+
"bbox": [
|
| 751 |
+
174,
|
| 752 |
+
712,
|
| 753 |
+
825,
|
| 754 |
+
905
|
| 755 |
+
],
|
| 756 |
+
"page_idx": 6
|
| 757 |
+
},
|
| 758 |
+
{
|
| 759 |
+
"type": "table",
|
| 760 |
+
"img_path": "images/b1bea4fa2c113eed1db8ae3974b9e5005c8284bfee926c1e61e8969d735decab.jpg",
|
| 761 |
+
"table_caption": [
|
| 762 |
+
"Table 1: Statistics and evaluation metrics of the QM9 and OGB datasets. "
|
| 763 |
+
],
|
| 764 |
+
"table_footnote": [],
|
| 765 |
+
"table_body": "<table><tr><td>Dataset</td><td>#Graphs</td><td>Avg. #nodes</td><td>Avg.#edges</td><td>Split ratio</td><td>#Tasks</td><td>Task type</td><td>Metric</td></tr><tr><td>QM9</td><td>129,433</td><td>18.0</td><td>18.6</td><td>80/10/10</td><td>12</td><td>Regression</td><td>MAE</td></tr><tr><td>ogbl-molhiv</td><td>41,127</td><td>25.5</td><td>27.5</td><td>80/10/10</td><td>1</td><td>Classification</td><td>ROC-AUC</td></tr><tr><td>ogbl-molpcba</td><td>437,929</td><td>26.0</td><td>28.1</td><td>80/10/10</td><td>128</td><td>Classification</td><td>AP</td></tr></table>",
|
| 766 |
+
"bbox": [
|
| 767 |
+
209,
|
| 768 |
+
107,
|
| 769 |
+
779,
|
| 770 |
+
162
|
| 771 |
+
],
|
| 772 |
+
"page_idx": 7
|
| 773 |
+
},
|
| 774 |
+
{
|
| 775 |
+
"type": "text",
|
| 776 |
+
"text": "5 Experiments ",
|
| 777 |
+
"text_level": 1,
|
| 778 |
+
"bbox": [
|
| 779 |
+
173,
|
| 780 |
+
179,
|
| 781 |
+
312,
|
| 782 |
+
196
|
| 783 |
+
],
|
| 784 |
+
"page_idx": 7
|
| 785 |
+
},
|
| 786 |
+
{
|
| 787 |
+
"type": "text",
|
| 788 |
+
"text": "In this section, we study the effectiveness of the NGNN framework for graph classification and regression tasks. In particular, we want to answer the following questions: ",
|
| 789 |
+
"bbox": [
|
| 790 |
+
176,
|
| 791 |
+
212,
|
| 792 |
+
821,
|
| 793 |
+
239
|
| 794 |
+
],
|
| 795 |
+
"page_idx": 7
|
| 796 |
+
},
|
| 797 |
+
{
|
| 798 |
+
"type": "text",
|
| 799 |
+
"text": "Q1 Can NGNN reach its theoretical power to discriminate 1-WL-indistinguishable graphs? Q2 How often and how much does NGNN improve the performance of a base GNN? Q3 How does NGNN perform in comparison to state-of-the-art GNN methods in open benchmarks? Q4 How much extra computation time does NGNN incur? ",
|
| 800 |
+
"bbox": [
|
| 801 |
+
174,
|
| 802 |
+
246,
|
| 803 |
+
825,
|
| 804 |
+
301
|
| 805 |
+
],
|
| 806 |
+
"page_idx": 7
|
| 807 |
+
},
|
| 808 |
+
{
|
| 809 |
+
"type": "text",
|
| 810 |
+
"text": "We implement the NGNN framework based on the PyTorch Geometric library [51]. Our code is available at https://github.com/muhanzhang/NestedGNN. ",
|
| 811 |
+
"bbox": [
|
| 812 |
+
174,
|
| 813 |
+
308,
|
| 814 |
+
821,
|
| 815 |
+
337
|
| 816 |
+
],
|
| 817 |
+
"page_idx": 7
|
| 818 |
+
},
|
| 819 |
+
{
|
| 820 |
+
"type": "text",
|
| 821 |
+
"text": "5.1 Datasets ",
|
| 822 |
+
"text_level": 1,
|
| 823 |
+
"bbox": [
|
| 824 |
+
173,
|
| 825 |
+
353,
|
| 826 |
+
271,
|
| 827 |
+
367
|
| 828 |
+
],
|
| 829 |
+
"page_idx": 7
|
| 830 |
+
},
|
| 831 |
+
{
|
| 832 |
+
"type": "text",
|
| 833 |
+
"text": "To answer Q1, we use a simulation dataset of $r$ -regular graphs and the EXP dataset [42] containing 600 pairs of 1-WL-indistinguishable but non-isomorphic graphs. To answer Q2, we use the QM9 dataset [52, 53] and the TU datasets [54]. QM9 contains $1 3 0 \\mathrm { K }$ small molecules. The task here is to perform regression on twelve targets representing energetic, electronic, geometric, and thermodynamic properties, based on the graph structure and node/edge features. TU contains five graph classification datasets including D&D [55], MUTAG [56], PROTEINS [55], PTC_MR [57], and ENZYMES [58]. We used the datasets provided by PyTorch Geometric [51], where for QM9 we performed unit conversions to match the units used by [19]. The evaluation metric is Mean Absolute Error (MAE) for QM9 and Accuracy $( \\% )$ for TU. To answer Q3, we use two Open Graph Benchmark (OGB) datasets [59], ogbg-molhiv and ogbg-molpcba. The ogbg-molhiv dataset contains 41K small molecules, the task of which is to classify whether a molecule inhibits HIV virus or not. ROC-AUC is used for evaluation. The ogbg-molpcba dataset contains 438K molecules with 128 classification tasks. The evaluation metric is Average Precision (AP) averaged over all the tasks. We include the statistics for QM9 and OGB datasets in Table 1. ",
|
| 834 |
+
"bbox": [
|
| 835 |
+
174,
|
| 836 |
+
378,
|
| 837 |
+
825,
|
| 838 |
+
571
|
| 839 |
+
],
|
| 840 |
+
"page_idx": 7
|
| 841 |
+
},
|
| 842 |
+
{
|
| 843 |
+
"type": "text",
|
| 844 |
+
"text": "5.2 Models ",
|
| 845 |
+
"text_level": 1,
|
| 846 |
+
"bbox": [
|
| 847 |
+
174,
|
| 848 |
+
588,
|
| 849 |
+
263,
|
| 850 |
+
603
|
| 851 |
+
],
|
| 852 |
+
"page_idx": 7
|
| 853 |
+
},
|
| 854 |
+
{
|
| 855 |
+
"type": "text",
|
| 856 |
+
"text": "QM9. We use 1-GNN, 1-2-GNN, 1-3-GNN, and 1-2-3-GNN from [19] as both the baselines and the base GNNs of NGNN. Among them, 1-GNN is a standard message passing GNN with 1-WL power. 1-2-GNN is a GNN mimicking 2-WL, where message passing happens among 2-tuples of nodes. 1-3-GNN and 1-2-3-GNN mimic 3-WL, where message passing happens among 3-tuples of nodes. 1-2-GNN and 1-3-GNN use features computed by 1-GNN as initial node features, and 1-2-3-GNN uses the concatenated features from 1-2-GNN and 1-3-GNN. We additionally include numbers provided by [53] and Deep LRP [39] as baselines. Note that we omit more recent methods [60–62] using advanced physical representations calculated from angles, atom coordinates, and quantum mechanics, which may obscure the comparison of models’ pure graph representation power. For NGNN, we uniformly use height-3 rooted subgraphs. For a fair comparison, the base GNNs in NGNN use exactly the same hyperparameters as when they are used alone, except for 1-GNN where we increase the number of message passing layers from 3 to 5 to make the number of layers larger than the subgraph height, similar to [63]. For subgraph pooling and graph pooling layers, we uniformly use mean pooling. All other settings follow [19]. ",
|
| 857 |
+
"bbox": [
|
| 858 |
+
174,
|
| 859 |
+
614,
|
| 860 |
+
825,
|
| 861 |
+
808
|
| 862 |
+
],
|
| 863 |
+
"page_idx": 7
|
| 864 |
+
},
|
| 865 |
+
{
|
| 866 |
+
"type": "text",
|
| 867 |
+
"text": "TU. We use four widely adopted GNNs as the baselines and the base GNNs of NGNN: GCN [12], GraphSAGE [64], GIN [27], and GAT [15]. Since TU datasets suffer from inconsistent evaluation standards [65], we uniformly use the 10-fold cross validation framework provided by PyTorch Geomtric [66] for all the models to ensure a fair comparison. For GNNs, we search the number of message passing layers in $\\{ 2 , 3 , 4 , 5 \\}$ . For NGNNs, we similarly search the subgraph height $h$ in $\\{ 2 , 3 , 4 , 5 \\}$ , so that both NGNNs and GNNs can have equal-depth local receptive fields. For NGNNs, we always use $h + 1$ message passing layers instead of searching it together with $h$ , because that will make NGNNs have more hyperparameters to tune. All models have 32 hidden dimensions, and are trained for 100 epochs with a batch size of 128. For each fold, we record the test accuracy with the hyperparameters chosen based on the best validation performance of this fold. Finally, we report the average test accuracy across all the 10 folds. ",
|
| 868 |
+
"bbox": [
|
| 869 |
+
174,
|
| 870 |
+
814,
|
| 871 |
+
825,
|
| 872 |
+
911
|
| 873 |
+
],
|
| 874 |
+
"page_idx": 7
|
| 875 |
+
},
|
| 876 |
+
{
|
| 877 |
+
"type": "text",
|
| 878 |
+
"text": "",
|
| 879 |
+
"bbox": [
|
| 880 |
+
174,
|
| 881 |
+
92,
|
| 882 |
+
825,
|
| 883 |
+
146
|
| 884 |
+
],
|
| 885 |
+
"page_idx": 8
|
| 886 |
+
},
|
| 887 |
+
{
|
| 888 |
+
"type": "text",
|
| 889 |
+
"text": "OGB. We use GNNs achieving top places on the OGB graph classification leaderboard3 (at the time of submission) as the baselines, including GCN [12], GIN [27], DeeperGCN [67], Deep LRP [39], PNA [68], DGN [33], GINE [69], and PHC-GNN [70]. Note that those high-order GNNs [19–21, 25] are not included here, because despite being theoretically more discriminative, these GNNs are not among the GNNs with the best empirical performance on modern large-scale graph benchmarks, and their $\\bar { \\mathcal { O } } ( n ^ { 3 } )$ complexity also raises a scalability issue. For NGNN, we use GIN as the base GNN (although GIN is not among the strongest baselines here). Some baselines additionally use the virtual node technique [18, 11, 71], which are marked by $\\therefore$ . For NGNN, we search the subgraph height $h$ in $\\{ 3 , 4 , 5 \\}$ , and the number of layers in $\\{ 4 , 5 , 6 \\}$ . We train the NGNN models for 100 and 150 epochs for ogbg-molhiv and ogbg-molpcba, respectively, and report the validation and test scores at the best validation epoch. We also find that our models are subject to high performance variance across epochs, likely due to the increased expressiveness. Thus, we save a model checkpoint every 10 epochs, and additionally report the ensemble performance by averaging the predictions from all checkpoints. The final hyperparameter choices and more details about the experimental settings are included in Appendix C. All results are averaged over 10 independent runs. ",
|
| 890 |
+
"bbox": [
|
| 891 |
+
173,
|
| 892 |
+
154,
|
| 893 |
+
825,
|
| 894 |
+
359
|
| 895 |
+
],
|
| 896 |
+
"page_idx": 8
|
| 897 |
+
},
|
| 898 |
+
{
|
| 899 |
+
"type": "text",
|
| 900 |
+
"text": "In the following, we uniformly use “Nested GNN” to denote an NGNN model using “GNN” as the base GNN. For example, Nested GIN denotes an NGNN model using GIN [27] as the base GNN. For the NGNN models in QM9, TU and OGB datasets, we augment the initial features of a node with Distance Encoding (DE) [24], which uses the (generalized) distance between a node and the root as its additional feature, due to DE’s successful applications in link-level tasks [44, 46]. Note that such feature augmentation is not applicable to the baseline models as discussed in Section 3.2. An ablation study on the effects of the DE features is included in Appendix E. ",
|
| 901 |
+
"bbox": [
|
| 902 |
+
174,
|
| 903 |
+
367,
|
| 904 |
+
825,
|
| 905 |
+
464
|
| 906 |
+
],
|
| 907 |
+
"page_idx": 8
|
| 908 |
+
},
|
| 909 |
+
{
|
| 910 |
+
"type": "text",
|
| 911 |
+
"text": "5.3 Results and discussion ",
|
| 912 |
+
"text_level": 1,
|
| 913 |
+
"bbox": [
|
| 914 |
+
174,
|
| 915 |
+
481,
|
| 916 |
+
367,
|
| 917 |
+
494
|
| 918 |
+
],
|
| 919 |
+
"page_idx": 8
|
| 920 |
+
},
|
| 921 |
+
{
|
| 922 |
+
"type": "text",
|
| 923 |
+
"text": "To answer Q1, we first run a simulation to test NGNN’s power for discriminating $r$ -regular graphs. The results are presented in Appendix D. They match almost perfectly with Theorem 1, demonstrating that a practical NGNN can fulfil its theoretical power for discriminating $r$ -regular graphs. ",
|
| 924 |
+
"bbox": [
|
| 925 |
+
176,
|
| 926 |
+
506,
|
| 927 |
+
826,
|
| 928 |
+
549
|
| 929 |
+
],
|
| 930 |
+
"page_idx": 8
|
| 931 |
+
},
|
| 932 |
+
{
|
| 933 |
+
"type": "table",
|
| 934 |
+
"img_path": "images/e0c2e2996886613b3885eb9bad587b13fe2f2c9203041c0c21dbc0f862b34d92.jpg",
|
| 935 |
+
"table_caption": [
|
| 936 |
+
"Table 2: Results $( \\% )$ on EXP. "
|
| 937 |
+
],
|
| 938 |
+
"table_footnote": [],
|
| 939 |
+
"table_body": "<table><tr><td>Method</td><td>Test Accuracy</td></tr><tr><td>GCN-RNI [42]</td><td>98.0±1.85</td></tr><tr><td>PPGN [20]</td><td>50.0±0.00</td></tr><tr><td>1-2-3-GNN [19]</td><td>50.0±0.00</td></tr><tr><td>3-GCN[42]</td><td>99.7±0.004</td></tr><tr><td>Nested GIN</td><td>99.9±0.26</td></tr></table>",
|
| 940 |
+
"bbox": [
|
| 941 |
+
176,
|
| 942 |
+
575,
|
| 943 |
+
377,
|
| 944 |
+
655
|
| 945 |
+
],
|
| 946 |
+
"page_idx": 8
|
| 947 |
+
},
|
| 948 |
+
{
|
| 949 |
+
"type": "text",
|
| 950 |
+
"text": "We also test NGNN’s expressive power using the EXP dataset provided by [42], which contains 600 carefully constructed 1-WL indistinguishable but non-isomorphic graph pairs. Each pair of graphs have different labels, thus a standard message passing GNN cannot predict them both correctly, resulting in an expected classification accuracy of only $50 \\%$ . We exactly follow the experimental settings and copy the baseline results in [42]. In Table 2, our Nested GIN model achieves a $9 9 . 9 \\%$ classification accuracy, which outperforms all the baselines and distinguishes almost all the 1-WL indistinguishable graph pairs. These results verified that NGNN’s expressive power is indeed beyond 1-WL and message passing GNNs. ",
|
| 951 |
+
"bbox": [
|
| 952 |
+
385,
|
| 953 |
+
555,
|
| 954 |
+
825,
|
| 955 |
+
665
|
| 956 |
+
],
|
| 957 |
+
"page_idx": 8
|
| 958 |
+
},
|
| 959 |
+
{
|
| 960 |
+
"type": "text",
|
| 961 |
+
"text": "",
|
| 962 |
+
"bbox": [
|
| 963 |
+
171,
|
| 964 |
+
665,
|
| 965 |
+
823,
|
| 966 |
+
693
|
| 967 |
+
],
|
| 968 |
+
"page_idx": 8
|
| 969 |
+
},
|
| 970 |
+
{
|
| 971 |
+
"type": "text",
|
| 972 |
+
"text": "To answer Q2, we adopt the QM9 and TU datasets. We show the QM9 results in Table 3. If the Nested version of a base GNN achieves a better result than the base GNN itself, we color that cell with light green. As we can see, NGNN brings performance gains to all base GNNs on most targets, sometimes by large margins. We also show the results on TU in Table 4. NGNNs also show improvement over their base GNNs in most cases. These results indicate that NGNN is a general framework for improving a GNN’s power. We further compute the maximum reduction of MAE for QM9 and maximum improvement of accuracy for TU before and after applying NGNN. NGNN reduces the MAE by up to 7.9 times for QM9, and increases the accuracy by up to $1 4 . 3 \\%$ for TU. These results answer Q2, indicating that NGNN can bring steady and significant improvement to base GNNs. ",
|
| 973 |
+
"bbox": [
|
| 974 |
+
174,
|
| 975 |
+
699,
|
| 976 |
+
825,
|
| 977 |
+
824
|
| 978 |
+
],
|
| 979 |
+
"page_idx": 8
|
| 980 |
+
},
|
| 981 |
+
{
|
| 982 |
+
"type": "text",
|
| 983 |
+
"text": "To answer Q3, we compare Nested GIN with leading methods on the OGB leaderboard. The results are shown in Table 5. Nested GIN achieves highly competitive performance with these leading GNN models, albeit using a relatively weak base GNN (GIN). Compared to GIN alone, Nested GIN shows clear performance gains. It achieves test scores up to 79.86 and 30.07 on ogbg-molhiv and ogbg-molpcba, respectively, which outperform all the baselines. In particular, for the challenging ogbg-molpcba, our Nested GIN can achieve 30.07 and 28.32 test AP with and without ensemble, respectively, outperforming the plain GIN model (with 27.03 test AP) significantly. These results demonstrate the great empirical performance and potential of NGNN even compared to heavily tuned open leaderboard models, despite using only GIN as the base GNN. ",
|
| 984 |
+
"bbox": [
|
| 985 |
+
176,
|
| 986 |
+
830,
|
| 987 |
+
825,
|
| 988 |
+
886
|
| 989 |
+
],
|
| 990 |
+
"page_idx": 8
|
| 991 |
+
},
|
| 992 |
+
{
|
| 993 |
+
"type": "table",
|
| 994 |
+
"img_path": "images/1644df3cea685f3dde4348717f0af14e887311ee4c1aebcfd74aba560f625458.jpg",
|
| 995 |
+
"table_caption": [
|
| 996 |
+
"Table 3: MAE results on QM9 (smaller the better). A colored cell means NGNN is better than the base GNN. "
|
| 997 |
+
],
|
| 998 |
+
"table_footnote": [],
|
| 999 |
+
"table_body": "<table><tr><td rowspan=\"2\">Target</td><td colspan=\"10\">Method (Ne. forNested)</td></tr><tr><td>DTNN</td><td>MPNN</td><td>Deep LRP</td><td>1-GNN1-2-GNN</td><td></td><td>1-3-GNN</td><td></td><td></td><td></td><td></td><td></td><td>1-2-3-GNN|Ne.1-GNN Ne.1-2-GNN Ne.1-3-GNN Ne.1-2-3-GNN|Max.reduction</td></tr><tr><td>μ</td><td>0.244</td><td>0.358</td><td>0.364</td><td>0.493</td><td>0.493</td><td>0.473</td><td>0.476</td><td>0.428</td><td>0.437</td><td>0.436</td><td>0.433</td><td>1.2×</td></tr><tr><td>α</td><td>0.95</td><td>0.89</td><td>0.298</td><td>0.78</td><td>0.27</td><td>0.46</td><td>0.27</td><td>0.29</td><td>0.278</td><td>0.261</td><td>0.265</td><td>2.7×</td></tr><tr><td>εHOMO</td><td>0.00388</td><td>0.00541</td><td>0.00254</td><td>0.00321</td><td>0.00331</td><td>0.00328</td><td>0.00337</td><td>0.00265</td><td>0.00275</td><td>0.00265</td><td>0.00279</td><td>1.2×</td></tr><tr><td>εLUMO</td><td>0.00512</td><td>0.00623</td><td>0.00277</td><td>0.00355</td><td>0.00350</td><td>0.00354</td><td>0.00351</td><td>0.00297</td><td>0.00271</td><td>0.00269</td><td>0.00276</td><td>1.3×</td></tr><tr><td>△</td><td>0.0112</td><td>0.0066</td><td>0.00353</td><td>0.0049</td><td>0.0047</td><td>0.0046</td><td>0.0048</td><td>0.0038</td><td>0.0039</td><td>0.0039</td><td>0.0039</td><td>1.8×</td></tr><tr><td>(R²）</td><td>17.0</td><td>28.5</td><td>19.3</td><td>34.1</td><td>21.5</td><td>25.8</td><td>22.9</td><td>20.5</td><td>20.4</td><td>20.2</td><td>20.1</td><td>1.7×</td></tr><tr><td>ZPVE</td><td>0.00172</td><td>0.00216</td><td>0.00055</td><td>0.00124</td><td>0.00018</td><td>0.00064</td><td>0.00019</td><td>0.00020</td><td>0.00017</td><td>0.00017</td><td>0.00015</td><td>6.2×</td></tr><tr><td>Uo</td><td>2.43</td><td>2.05</td><td>0.413</td><td>2.32</td><td>0.0357</td><td>0.6855</td><td>0.0427</td><td>0.295</td><td>0.252</td><td>0.291</td><td>0.205</td><td>7.9×</td></tr><tr><td>U</td><td>2.43</td><td>2.00</td><td>0.413</td><td>2.08</td><td>0.107</td><td>0.686</td><td>0.111</td><td>0.361</td><td>0.265</td><td>0.278</td><td>0.200</td><td>5.8×</td></tr><tr><td>H</td><td>2.43</td><td>2.02</td><td>0.413</td><td>2.23</td><td>0.070</td><td>0.794</td><td>0.0419</td><td>0.305</td><td>0.241</td><td>0.267</td><td>0.249</td><td>7.3×</td></tr><tr><td>G</td><td>2.43</td><td>2.02</td><td>0.413</td><td>1.94</td><td>0.140</td><td>0.587</td><td>0.0469</td><td>0.489</td><td>0.272</td><td>0.287</td><td>0.253</td><td>4.0×</td></tr><tr><td>C</td><td>0.27</td><td>0.42</td><td>0.129</td><td>0.27</td><td>0.0989</td><td>0.158</td><td>0.0944</td><td>0.174</td><td>0.0891</td><td>0.0879</td><td>0.0811</td><td>1.8×</td></tr></table>",
|
| 1000 |
+
"bbox": [
|
| 1001 |
+
176,
|
| 1002 |
+
103,
|
| 1003 |
+
820,
|
| 1004 |
+
228
|
| 1005 |
+
],
|
| 1006 |
+
"page_idx": 9
|
| 1007 |
+
},
|
| 1008 |
+
{
|
| 1009 |
+
"type": "table",
|
| 1010 |
+
"img_path": "images/ab71ddc002796d60889fc8096882ebbe44f1bac28f20b4342fbb6dd37ee20759.jpg",
|
| 1011 |
+
"table_caption": [
|
| 1012 |
+
"Table 4: Accuracy results $( \\% )$ on TU datasets. "
|
| 1013 |
+
],
|
| 1014 |
+
"table_footnote": [],
|
| 1015 |
+
"table_body": "<table><tr><td>#Graphs</td><td>D&D 1178</td><td>MUTAG 188</td><td>PROTEINS 1113</td><td>PTC_MR 344</td><td>ENZYMES 600</td></tr><tr><td>Avg. #nodes</td><td>284.32</td><td>17.93</td><td>39.06</td><td>14.29</td><td>32.63</td></tr><tr><td>GCN GraphSAGE</td><td>71.6±2.8 71.6±3.0</td><td>73.4±10.8</td><td>71.7±4.7</td><td>56.4±7.1 57.0±5.5</td><td>27.3±5.5 30.7±6.3</td></tr><tr><td></td><td>70.5±3.9</td><td>74.0±8.8</td><td>71.2±5.2</td><td></td><td></td></tr><tr><td>GIN GAT</td><td>71.0±4.4</td><td>84.5±8.9 73.9±10.7</td><td>70.6±4.3 72.0±3.3</td><td>51.2±9.2 57.0±7.3</td><td>38.3±6.4 30.2±4.2</td></tr><tr><td>Nested GCN</td><td>76.3±3.8</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>77.4±4.2</td><td>82.9±11.1</td><td>73.3±4.0</td><td>57.3±7.7</td><td>31.2±6.7</td></tr><tr><td>Nested GraphSAGE Nested GIN</td><td>77.8±3.9</td><td>83.9±10.7 87.9±8.2</td><td>74.2±3.7 73.9±5.1</td><td>57.0±5.9 54.1±7.7</td><td>30.7±6.3 29.0±8.0</td></tr><tr><td>Nested GAT</td><td>76.0±4.4</td><td>81.9±10.2</td><td>73.7±4.8</td><td>56.7±8.1</td><td>29.5±5.7</td></tr><tr><td>Max. improvement</td><td>10.4%</td><td>13.4%</td><td>4.7%</td><td>5.7%</td><td>14.3%</td></tr></table>",
|
| 1016 |
+
"bbox": [
|
| 1017 |
+
171,
|
| 1018 |
+
251,
|
| 1019 |
+
521,
|
| 1020 |
+
364
|
| 1021 |
+
],
|
| 1022 |
+
"page_idx": 9
|
| 1023 |
+
},
|
| 1024 |
+
{
|
| 1025 |
+
"type": "table",
|
| 1026 |
+
"img_path": "images/64875b6dc6426ea3feca6813017ceab4ff03572a7ac2e8a502735f0662b8322d.jpg",
|
| 1027 |
+
"table_caption": [
|
| 1028 |
+
"Table 5: Results $( \\% )$ on OGB datasets (\\* virtual node). "
|
| 1029 |
+
],
|
| 1030 |
+
"table_footnote": [],
|
| 1031 |
+
"table_body": "<table><tr><td></td><td colspan=\"2\">ogbg-molhiv (AUC)</td><td colspan=\"2\">ogbg-molpcba (AP)</td></tr><tr><td>Method</td><td>Validation</td><td>Test</td><td>Validation</td><td>Test</td></tr><tr><td>CCN*</td><td>83.84±0.91</td><td>75.99±1.19</td><td>24.95±0.42</td><td>24.24±0.34</td></tr><tr><td>GIN*</td><td>84.79±0.68</td><td>77.07±1.49</td><td>27.98±0.25</td><td>27.03±0.23</td></tr><tr><td>Deep LRP</td><td>82.09±1.16</td><td>77.19±1.40</td><td></td><td></td></tr><tr><td>DeeperGCN*</td><td></td><td></td><td>29.20±0.25</td><td>27.81±0.38</td></tr><tr><td>HIMP</td><td></td><td>78.80±0.82</td><td></td><td></td></tr><tr><td>PNA</td><td>85.19±0.99</td><td>79.05±1.32</td><td></td><td></td></tr><tr><td>DGN</td><td>84.70±0.47</td><td>79.70±0.97-</td><td></td><td></td></tr><tr><td>GINE*</td><td></td><td></td><td>30.65±0.30</td><td>29.17±0.15</td></tr><tr><td>PHC-GNN</td><td>82.17±0.89</td><td>79.34±1.16</td><td>30.68±0.25</td><td>29.47±0.26</td></tr><tr><td>Nested GIN*</td><td>83.17±1.99</td><td>78.34±1.86</td><td>29.15±0.35</td><td>28.32±0.41</td></tr><tr><td>NestedGIN* (ens)</td><td>80.80±2.78</td><td>79.86±1.05</td><td>30.59±0.56</td><td>30.07±0.37</td></tr></table>",
|
| 1032 |
+
"bbox": [
|
| 1033 |
+
531,
|
| 1034 |
+
251,
|
| 1035 |
+
823,
|
| 1036 |
+
364
|
| 1037 |
+
],
|
| 1038 |
+
"page_idx": 9
|
| 1039 |
+
},
|
| 1040 |
+
{
|
| 1041 |
+
"type": "text",
|
| 1042 |
+
"text": "",
|
| 1043 |
+
"bbox": [
|
| 1044 |
+
174,
|
| 1045 |
+
390,
|
| 1046 |
+
825,
|
| 1047 |
+
459
|
| 1048 |
+
],
|
| 1049 |
+
"page_idx": 9
|
| 1050 |
+
},
|
| 1051 |
+
{
|
| 1052 |
+
"type": "text",
|
| 1053 |
+
"text": "To answer Q4, we report the training time per epoch for GIN and Nested GIN on OGB datasets. On ogbg-molhiv, GIN takes 54s per epoch, while Nested GIN takes 183s. On ogbg-molpcba, GIN takes 10min per epoch, while Nested GIN takes $2 0 \\mathrm { { m i n } }$ . This verifies that NGNN has comparable time complexity with message passing GNNs. The extra complexity comes from independently learning better node representations from rooted subgraphs, which is a trade-off for the higher expressivity. ",
|
| 1054 |
+
"bbox": [
|
| 1055 |
+
174,
|
| 1056 |
+
465,
|
| 1057 |
+
823,
|
| 1058 |
+
535
|
| 1059 |
+
],
|
| 1060 |
+
"page_idx": 9
|
| 1061 |
+
},
|
| 1062 |
+
{
|
| 1063 |
+
"type": "text",
|
| 1064 |
+
"text": "In summary, our experiments have firmly shown that NGNN is a theoretically sound method which brings consistent gains to its base GNNs in a plug-and-play way. Furthermore, NGNN still maintains a controllable time complexity compared to other more powerful GNNs. ",
|
| 1065 |
+
"bbox": [
|
| 1066 |
+
176,
|
| 1067 |
+
540,
|
| 1068 |
+
821,
|
| 1069 |
+
583
|
| 1070 |
+
],
|
| 1071 |
+
"page_idx": 9
|
| 1072 |
+
},
|
| 1073 |
+
{
|
| 1074 |
+
"type": "text",
|
| 1075 |
+
"text": "Finally, we point out one memory limitation of the current NGNN implementation. Currently, NGNN does not scale to graph datasets with a large average node number (such as REDDIT-BINARY) or datasets with a large average node degree (such as ogbg-ppa) due to copying a rooted subgraph for each node to the GPU memory. Reducing batch size or subgraph height helps, but at the same time leads to performance degradation. One may wonder why materializing all the subgraphs into GPU memory is necessary. The reason is that we want to batch-process all the subgraphs simultaneously. Otherwise, we have to sequentially extract subgraphs on the fly, which results in a much higher latency. We leave the exploration of memory efficient NGNN to the future work. ",
|
| 1076 |
+
"bbox": [
|
| 1077 |
+
174,
|
| 1078 |
+
588,
|
| 1079 |
+
825,
|
| 1080 |
+
700
|
| 1081 |
+
],
|
| 1082 |
+
"page_idx": 9
|
| 1083 |
+
},
|
| 1084 |
+
{
|
| 1085 |
+
"type": "text",
|
| 1086 |
+
"text": "6 Conclusions ",
|
| 1087 |
+
"text_level": 1,
|
| 1088 |
+
"bbox": [
|
| 1089 |
+
174,
|
| 1090 |
+
719,
|
| 1091 |
+
307,
|
| 1092 |
+
736
|
| 1093 |
+
],
|
| 1094 |
+
"page_idx": 9
|
| 1095 |
+
},
|
| 1096 |
+
{
|
| 1097 |
+
"type": "text",
|
| 1098 |
+
"text": "We have proposed Nested Graph Neural Network (NGNN), a general framework for improving GNN’s representation power. NGNN learns node representations encoding rooted subgraphs instead of rooted subtrees. Theoretically, we prove NGNN can discriminate almost all $r$ -regular graphs where 1-WL always fails. Empirically, NGNN consistently improves the performance of various base GNNs across different datasets without incurring the $\\mathcal { O } ( \\bar { n } ^ { 3 } )$ complexity like other more powerful GNNs. ",
|
| 1099 |
+
"bbox": [
|
| 1100 |
+
174,
|
| 1101 |
+
750,
|
| 1102 |
+
825,
|
| 1103 |
+
819
|
| 1104 |
+
],
|
| 1105 |
+
"page_idx": 9
|
| 1106 |
+
},
|
| 1107 |
+
{
|
| 1108 |
+
"type": "text",
|
| 1109 |
+
"text": "Acknowledge ",
|
| 1110 |
+
"text_level": 1,
|
| 1111 |
+
"bbox": [
|
| 1112 |
+
174,
|
| 1113 |
+
838,
|
| 1114 |
+
287,
|
| 1115 |
+
856
|
| 1116 |
+
],
|
| 1117 |
+
"page_idx": 9
|
| 1118 |
+
},
|
| 1119 |
+
{
|
| 1120 |
+
"type": "text",
|
| 1121 |
+
"text": "The authors greatly thank the actionable suggestions from the reviewers to improve the manuscript. Li is partly supported by the 2021 JP Morgan Faculty Award and the National Science Foundation (NSF) award HDR-2117997. ",
|
| 1122 |
+
"bbox": [
|
| 1123 |
+
176,
|
| 1124 |
+
869,
|
| 1125 |
+
825,
|
| 1126 |
+
910
|
| 1127 |
+
],
|
| 1128 |
+
"page_idx": 9
|
| 1129 |
+
},
|
| 1130 |
+
{
|
| 1131 |
+
"type": "text",
|
| 1132 |
+
"text": "References \n[1] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 701–710. ACM, 2014. \n[2] David Haussler. Convolution kernels on discrete structures. Technical report, Citeseer, 1999. \n[3] Nino Shervashidze, SVN Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten M Borgwardt. Efficient graphlet kernels for large graph comparison. In AISTATS, volume 5, pages 488–495, 2009. \n[4] Risi Kondor, Nino Shervashidze, and Karsten M Borgwardt. The graphlet spectrum. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 529–536. ACM, 2009. \n[5] Karsten M Borgwardt and Hans-Peter Kriegel. Shortest-path kernels on graphs. In 5th IEEE International Conference on Data Mining, pages 8–pp. IEEE, 2005. \n[6] Marion Neumann, Roman Garnett, Christian Bauckhage, and Kristian Kersting. Propagation kernels: efficient graph kernels from propagated information. Machine Learning, 102(2): 209–245, 2016. \n[7] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research, 12(Sep): 2539–2561, 2011. \n[8] 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, 2009. \n[9] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013. \n[10] David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in neural information processing systems, pages 2224– 2232, 2015. \n[11] Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015. \n[12] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016. \n[13] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3837–3845, 2016. \n[14] Hanjun Dai, Bo Dai, and Le Song. Discriminative embeddings of latent variable models for structured data. In Proceedings of The 33rd International Conference on Machine Learning, pages 2702–2711, 2016. \n[15] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903, 2017. \n[16] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In AAAI, pages 4438–4445, 2018. \n[17] Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems, pages 4800–4810, 2018. \n[18] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1263–1272. JMLR. org, 2017. ",
|
| 1133 |
+
"bbox": [
|
| 1134 |
+
169,
|
| 1135 |
+
77,
|
| 1136 |
+
828,
|
| 1137 |
+
916
|
| 1138 |
+
],
|
| 1139 |
+
"page_idx": 10
|
| 1140 |
+
},
|
| 1141 |
+
{
|
| 1142 |
+
"type": "text",
|
| 1143 |
+
"text": "[19] Christopher Morris, Martin Ritzert, Matthias Fey, William L Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 4602–4609, 2019. ",
|
| 1144 |
+
"bbox": [
|
| 1145 |
+
173,
|
| 1146 |
+
90,
|
| 1147 |
+
826,
|
| 1148 |
+
147
|
| 1149 |
+
],
|
| 1150 |
+
"page_idx": 11
|
| 1151 |
+
},
|
| 1152 |
+
{
|
| 1153 |
+
"type": "text",
|
| 1154 |
+
"text": "[20] Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, and Yaron Lipman. Provably powerful graph networks. In Advances in Neural Information Processing Systems, pages 2156–2167, 2019. ",
|
| 1155 |
+
"bbox": [
|
| 1156 |
+
174,
|
| 1157 |
+
155,
|
| 1158 |
+
823,
|
| 1159 |
+
198
|
| 1160 |
+
],
|
| 1161 |
+
"page_idx": 11
|
| 1162 |
+
},
|
| 1163 |
+
{
|
| 1164 |
+
"type": "text",
|
| 1165 |
+
"text": "[21] Zhengdao Chen, Soledad Villar, Lei Chen, and Joan Bruna. On the equivalence between graph isomorphism testing and function approximation with gnns. In Advances in Neural Information Processing Systems, pages 15894–15902, 2019. ",
|
| 1166 |
+
"bbox": [
|
| 1167 |
+
174,
|
| 1168 |
+
207,
|
| 1169 |
+
823,
|
| 1170 |
+
250
|
| 1171 |
+
],
|
| 1172 |
+
"page_idx": 11
|
| 1173 |
+
},
|
| 1174 |
+
{
|
| 1175 |
+
"type": "text",
|
| 1176 |
+
"text": "[22] Boris Weisfeiler and AA Lehman. A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, 2(9):12–16, 1968. ",
|
| 1177 |
+
"bbox": [
|
| 1178 |
+
173,
|
| 1179 |
+
257,
|
| 1180 |
+
821,
|
| 1181 |
+
286
|
| 1182 |
+
],
|
| 1183 |
+
"page_idx": 11
|
| 1184 |
+
},
|
| 1185 |
+
{
|
| 1186 |
+
"type": "text",
|
| 1187 |
+
"text": "[23] S Vichy N Vishwanathan, Nicol N Schraudolph, Risi Kondor, and Karsten M Borgwardt. Graph kernels. Journal of Machine Learning Research, 11(Apr):1201–1242, 2010. ",
|
| 1188 |
+
"bbox": [
|
| 1189 |
+
173,
|
| 1190 |
+
294,
|
| 1191 |
+
821,
|
| 1192 |
+
324
|
| 1193 |
+
],
|
| 1194 |
+
"page_idx": 11
|
| 1195 |
+
},
|
| 1196 |
+
{
|
| 1197 |
+
"type": "text",
|
| 1198 |
+
"text": "[24] Pan Li, Yanbang Wang, Hongwei Wang, and Jure Leskovec. Distance encoding–design provably more powerful gnns for structural representation learning. arXiv preprint arXiv:2009.00142, 2020. ",
|
| 1199 |
+
"bbox": [
|
| 1200 |
+
176,
|
| 1201 |
+
332,
|
| 1202 |
+
823,
|
| 1203 |
+
373
|
| 1204 |
+
],
|
| 1205 |
+
"page_idx": 11
|
| 1206 |
+
},
|
| 1207 |
+
{
|
| 1208 |
+
"type": "text",
|
| 1209 |
+
"text": "[25] Christopher Morris, Gaurav Rattan, and Petra Mutzel. Weisfeiler and leman go sparse: Towards scalable higher-order graph embeddings. 2020. ",
|
| 1210 |
+
"bbox": [
|
| 1211 |
+
173,
|
| 1212 |
+
382,
|
| 1213 |
+
823,
|
| 1214 |
+
411
|
| 1215 |
+
],
|
| 1216 |
+
"page_idx": 11
|
| 1217 |
+
},
|
| 1218 |
+
{
|
| 1219 |
+
"type": "text",
|
| 1220 |
+
"text": "[26] Muhan Zhang and Yixin Chen. Weisfeiler-lehman neural machine for link prediction. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 575–583. ACM, 2017. ",
|
| 1221 |
+
"bbox": [
|
| 1222 |
+
174,
|
| 1223 |
+
419,
|
| 1224 |
+
823,
|
| 1225 |
+
462
|
| 1226 |
+
],
|
| 1227 |
+
"page_idx": 11
|
| 1228 |
+
},
|
| 1229 |
+
{
|
| 1230 |
+
"type": "text",
|
| 1231 |
+
"text": "[27] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018. ",
|
| 1232 |
+
"bbox": [
|
| 1233 |
+
173,
|
| 1234 |
+
469,
|
| 1235 |
+
823,
|
| 1236 |
+
500
|
| 1237 |
+
],
|
| 1238 |
+
"page_idx": 11
|
| 1239 |
+
},
|
| 1240 |
+
{
|
| 1241 |
+
"type": "text",
|
| 1242 |
+
"text": "[28] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in Neural Information Processing Systems, pages 3391–3401, 2017. ",
|
| 1243 |
+
"bbox": [
|
| 1244 |
+
176,
|
| 1245 |
+
507,
|
| 1246 |
+
825,
|
| 1247 |
+
550
|
| 1248 |
+
],
|
| 1249 |
+
"page_idx": 11
|
| 1250 |
+
},
|
| 1251 |
+
{
|
| 1252 |
+
"type": "text",
|
| 1253 |
+
"text": "[29] Andries E Brouwer and Willem H Haemers. Strongly regular graphs. In Spectra of Graphs, pages 115–149. Springer, 2012. ",
|
| 1254 |
+
"bbox": [
|
| 1255 |
+
174,
|
| 1256 |
+
558,
|
| 1257 |
+
823,
|
| 1258 |
+
588
|
| 1259 |
+
],
|
| 1260 |
+
"page_idx": 11
|
| 1261 |
+
},
|
| 1262 |
+
{
|
| 1263 |
+
"type": "text",
|
| 1264 |
+
"text": "[30] Ryan Murphy, Balasubramaniam Srinivasan, Vinayak Rao, and Bruno Ribeiro. Relational pooling for graph representations. In International Conference on Machine Learning, pages 4663–4673. PMLR, 2019. ",
|
| 1265 |
+
"bbox": [
|
| 1266 |
+
173,
|
| 1267 |
+
594,
|
| 1268 |
+
823,
|
| 1269 |
+
638
|
| 1270 |
+
],
|
| 1271 |
+
"page_idx": 11
|
| 1272 |
+
},
|
| 1273 |
+
{
|
| 1274 |
+
"type": "text",
|
| 1275 |
+
"text": "[31] Giorgos Bouritsas, Fabrizio Frasca, Stefanos Zafeiriou, and Michael M Bronstein. Improving graph neural network expressivity via subgraph isomorphism counting. arXiv preprint arXiv:2006.09252, 2020. ",
|
| 1276 |
+
"bbox": [
|
| 1277 |
+
174,
|
| 1278 |
+
646,
|
| 1279 |
+
823,
|
| 1280 |
+
689
|
| 1281 |
+
],
|
| 1282 |
+
"page_idx": 11
|
| 1283 |
+
},
|
| 1284 |
+
{
|
| 1285 |
+
"type": "text",
|
| 1286 |
+
"text": "[32] Jiaxuan You, Jonathan Gomes-Selman, Rex Ying, and Jure Leskovec. Identity-aware graph neural networks. arXiv preprint arXiv:2101.10320, 2021. ",
|
| 1287 |
+
"bbox": [
|
| 1288 |
+
173,
|
| 1289 |
+
696,
|
| 1290 |
+
823,
|
| 1291 |
+
727
|
| 1292 |
+
],
|
| 1293 |
+
"page_idx": 11
|
| 1294 |
+
},
|
| 1295 |
+
{
|
| 1296 |
+
"type": "text",
|
| 1297 |
+
"text": "[33] Dominique Beaini, Saro Passaro, Vincent Létourneau, William L Hamilton, Gabriele Corso, and Pietro Liò. Directional graph networks. arXiv preprint arXiv:2010.02863, 2020. ",
|
| 1298 |
+
"bbox": [
|
| 1299 |
+
174,
|
| 1300 |
+
733,
|
| 1301 |
+
823,
|
| 1302 |
+
763
|
| 1303 |
+
],
|
| 1304 |
+
"page_idx": 11
|
| 1305 |
+
},
|
| 1306 |
+
{
|
| 1307 |
+
"type": "text",
|
| 1308 |
+
"text": "[34] Haggai Maron, Heli Ben-Hamu, Nadav Shamir, and Yaron Lipman. Invariant and equivariant graph networks. arXiv preprint arXiv:1812.09902, 2018. ",
|
| 1309 |
+
"bbox": [
|
| 1310 |
+
173,
|
| 1311 |
+
771,
|
| 1312 |
+
823,
|
| 1313 |
+
801
|
| 1314 |
+
],
|
| 1315 |
+
"page_idx": 11
|
| 1316 |
+
},
|
| 1317 |
+
{
|
| 1318 |
+
"type": "text",
|
| 1319 |
+
"text": "[35] Haggai Maron, Ethan Fetaya, Nimrod Segol, and Yaron Lipman. On the universality of invariant networks. In International conference on machine learning, pages 4363–4371. PMLR, 2019. ",
|
| 1320 |
+
"bbox": [
|
| 1321 |
+
174,
|
| 1322 |
+
808,
|
| 1323 |
+
823,
|
| 1324 |
+
838
|
| 1325 |
+
],
|
| 1326 |
+
"page_idx": 11
|
| 1327 |
+
},
|
| 1328 |
+
{
|
| 1329 |
+
"type": "text",
|
| 1330 |
+
"text": "[36] Nicolas Keriven and Gabriel Peyré. Universal invariant and equivariant graph neural networks. arXiv preprint arXiv:1905.04943, 2019. ",
|
| 1331 |
+
"bbox": [
|
| 1332 |
+
174,
|
| 1333 |
+
845,
|
| 1334 |
+
823,
|
| 1335 |
+
875
|
| 1336 |
+
],
|
| 1337 |
+
"page_idx": 11
|
| 1338 |
+
},
|
| 1339 |
+
{
|
| 1340 |
+
"type": "text",
|
| 1341 |
+
"text": "[37] Waïss Azizian and Marc Lelarge. Characterizing the expressive power of invariant and equivariant graph neural networks. arXiv preprint arXiv:2006.15646, 2020. ",
|
| 1342 |
+
"bbox": [
|
| 1343 |
+
174,
|
| 1344 |
+
883,
|
| 1345 |
+
823,
|
| 1346 |
+
911
|
| 1347 |
+
],
|
| 1348 |
+
"page_idx": 11
|
| 1349 |
+
},
|
| 1350 |
+
{
|
| 1351 |
+
"type": "text",
|
| 1352 |
+
"text": "[38] George Dasoulas, Ludovic Dos Santos, Kevin Scaman, and Aladin Virmaux. Coloring graph neural networks for node disambiguation. arXiv preprint arXiv:1912.06058, 2019. ",
|
| 1353 |
+
"bbox": [
|
| 1354 |
+
173,
|
| 1355 |
+
90,
|
| 1356 |
+
825,
|
| 1357 |
+
121
|
| 1358 |
+
],
|
| 1359 |
+
"page_idx": 12
|
| 1360 |
+
},
|
| 1361 |
+
{
|
| 1362 |
+
"type": "text",
|
| 1363 |
+
"text": "[39] Zhengdao Chen, Lei Chen, Soledad Villar, and Joan Bruna. Can graph neural networks count substructures? Advances in neural information processing systems, 2020. ",
|
| 1364 |
+
"bbox": [
|
| 1365 |
+
173,
|
| 1366 |
+
131,
|
| 1367 |
+
823,
|
| 1368 |
+
160
|
| 1369 |
+
],
|
| 1370 |
+
"page_idx": 12
|
| 1371 |
+
},
|
| 1372 |
+
{
|
| 1373 |
+
"type": "text",
|
| 1374 |
+
"text": "[40] Andreas Loukas. What graph neural networks cannot learn: depth vs width. arXiv preprint arXiv:1907.03199, 2019. ",
|
| 1375 |
+
"bbox": [
|
| 1376 |
+
174,
|
| 1377 |
+
170,
|
| 1378 |
+
821,
|
| 1379 |
+
200
|
| 1380 |
+
],
|
| 1381 |
+
"page_idx": 12
|
| 1382 |
+
},
|
| 1383 |
+
{
|
| 1384 |
+
"type": "text",
|
| 1385 |
+
"text": "[41] Ryoma Sato, Makoto Yamada, and Hisashi Kashima. Random features strengthen graph neural networks. arXiv preprint arXiv:2002.03155, 2020. ",
|
| 1386 |
+
"bbox": [
|
| 1387 |
+
173,
|
| 1388 |
+
210,
|
| 1389 |
+
823,
|
| 1390 |
+
241
|
| 1391 |
+
],
|
| 1392 |
+
"page_idx": 12
|
| 1393 |
+
},
|
| 1394 |
+
{
|
| 1395 |
+
"type": "text",
|
| 1396 |
+
"text": "[42] Ralph Abboud, ˙Ismail ˙Ilkan Ceylan, Martin Grohe, and Thomas Lukasiewicz. The surprising power of graph neural networks with random node initialization. arXiv preprint arXiv:2010.01179, 2020. ",
|
| 1397 |
+
"bbox": [
|
| 1398 |
+
174,
|
| 1399 |
+
251,
|
| 1400 |
+
823,
|
| 1401 |
+
294
|
| 1402 |
+
],
|
| 1403 |
+
"page_idx": 12
|
| 1404 |
+
},
|
| 1405 |
+
{
|
| 1406 |
+
"type": "text",
|
| 1407 |
+
"text": "[43] Clément Vignac, Andreas Loukas, and Pascal Frossard. Building powerful and equivariant graph neural networks with structural message-passing. arXiv e-prints, pages arXiv–2006, 2020. ",
|
| 1408 |
+
"bbox": [
|
| 1409 |
+
174,
|
| 1410 |
+
304,
|
| 1411 |
+
825,
|
| 1412 |
+
348
|
| 1413 |
+
],
|
| 1414 |
+
"page_idx": 12
|
| 1415 |
+
},
|
| 1416 |
+
{
|
| 1417 |
+
"type": "text",
|
| 1418 |
+
"text": "[44] Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pages 5165–5175, 2018. ",
|
| 1419 |
+
"bbox": [
|
| 1420 |
+
173,
|
| 1421 |
+
358,
|
| 1422 |
+
823,
|
| 1423 |
+
388
|
| 1424 |
+
],
|
| 1425 |
+
"page_idx": 12
|
| 1426 |
+
},
|
| 1427 |
+
{
|
| 1428 |
+
"type": "text",
|
| 1429 |
+
"text": "[45] Muhan Zhang, Pan Li, Yinglong Xia, Kai Wang, and Long Jin. Revisiting graph neural networks for link prediction. arXiv preprint arXiv:2010.16103, 2020. ",
|
| 1430 |
+
"bbox": [
|
| 1431 |
+
173,
|
| 1432 |
+
398,
|
| 1433 |
+
823,
|
| 1434 |
+
429
|
| 1435 |
+
],
|
| 1436 |
+
"page_idx": 12
|
| 1437 |
+
},
|
| 1438 |
+
{
|
| 1439 |
+
"type": "text",
|
| 1440 |
+
"text": "[46] Muhan Zhang and Yixin Chen. Inductive matrix completion based on graph neural networks. In International Conference on Learning Representations, 2020. URL https://openreview. net/forum?id $\\equiv$ ByxxgCEYDS. ",
|
| 1441 |
+
"bbox": [
|
| 1442 |
+
174,
|
| 1443 |
+
439,
|
| 1444 |
+
825,
|
| 1445 |
+
482
|
| 1446 |
+
],
|
| 1447 |
+
"page_idx": 12
|
| 1448 |
+
},
|
| 1449 |
+
{
|
| 1450 |
+
"type": "text",
|
| 1451 |
+
"text": "[47] Pim de Haan, Taco Cohen, and Max Welling. Natural graph networks. arXiv preprint arXiv:2007.08349, 2020. ",
|
| 1452 |
+
"bbox": [
|
| 1453 |
+
169,
|
| 1454 |
+
492,
|
| 1455 |
+
825,
|
| 1456 |
+
522
|
| 1457 |
+
],
|
| 1458 |
+
"page_idx": 12
|
| 1459 |
+
},
|
| 1460 |
+
{
|
| 1461 |
+
"type": "text",
|
| 1462 |
+
"text": "[48] Giannis Nikolentzos, George Dasoulas, and Michalis Vazirgiannis. k-hop graph neural networks. Neural Networks, 130:195–205, 2020. ",
|
| 1463 |
+
"bbox": [
|
| 1464 |
+
173,
|
| 1465 |
+
532,
|
| 1466 |
+
823,
|
| 1467 |
+
563
|
| 1468 |
+
],
|
| 1469 |
+
"page_idx": 12
|
| 1470 |
+
},
|
| 1471 |
+
{
|
| 1472 |
+
"type": "text",
|
| 1473 |
+
"text": "[49] Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pages 21–29. PMLR, 2019. ",
|
| 1474 |
+
"bbox": [
|
| 1475 |
+
173,
|
| 1476 |
+
573,
|
| 1477 |
+
826,
|
| 1478 |
+
631
|
| 1479 |
+
],
|
| 1480 |
+
"page_idx": 12
|
| 1481 |
+
},
|
| 1482 |
+
{
|
| 1483 |
+
"type": "text",
|
| 1484 |
+
"text": "[50] Kexin Huang and Marinka Zitnik. Graph meta learning via local subgraphs. Advances in Neural Information Processing Systems, 33, 2020. ",
|
| 1485 |
+
"bbox": [
|
| 1486 |
+
171,
|
| 1487 |
+
640,
|
| 1488 |
+
825,
|
| 1489 |
+
670
|
| 1490 |
+
],
|
| 1491 |
+
"page_idx": 12
|
| 1492 |
+
},
|
| 1493 |
+
{
|
| 1494 |
+
"type": "text",
|
| 1495 |
+
"text": "[51] Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric. arXiv preprint arXiv:1903.02428, 2019. ",
|
| 1496 |
+
"bbox": [
|
| 1497 |
+
171,
|
| 1498 |
+
680,
|
| 1499 |
+
825,
|
| 1500 |
+
710
|
| 1501 |
+
],
|
| 1502 |
+
"page_idx": 12
|
| 1503 |
+
},
|
| 1504 |
+
{
|
| 1505 |
+
"type": "text",
|
| 1506 |
+
"text": "[52] Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific data, 1(1):1–7, 2014. ",
|
| 1507 |
+
"bbox": [
|
| 1508 |
+
173,
|
| 1509 |
+
720,
|
| 1510 |
+
828,
|
| 1511 |
+
763
|
| 1512 |
+
],
|
| 1513 |
+
"page_idx": 12
|
| 1514 |
+
},
|
| 1515 |
+
{
|
| 1516 |
+
"type": "text",
|
| 1517 |
+
"text": "[53] Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl Leswing, and Vijay Pande. Moleculenet: a benchmark for molecular machine learning. Chemical science, 9(2):513–530, 2018. ",
|
| 1518 |
+
"bbox": [
|
| 1519 |
+
173,
|
| 1520 |
+
775,
|
| 1521 |
+
823,
|
| 1522 |
+
818
|
| 1523 |
+
],
|
| 1524 |
+
"page_idx": 12
|
| 1525 |
+
},
|
| 1526 |
+
{
|
| 1527 |
+
"type": "text",
|
| 1528 |
+
"text": "[54] Kristian Kersting, Nils M. Kriege, Christopher Morris, Petra Mutzel, and Marion Neumann. Benchmark data sets for graph kernels, 2016. URL http://graphkernels.cs. tu-dortmund.de. ",
|
| 1529 |
+
"bbox": [
|
| 1530 |
+
173,
|
| 1531 |
+
829,
|
| 1532 |
+
823,
|
| 1533 |
+
871
|
| 1534 |
+
],
|
| 1535 |
+
"page_idx": 12
|
| 1536 |
+
},
|
| 1537 |
+
{
|
| 1538 |
+
"type": "text",
|
| 1539 |
+
"text": "[55] Paul D Dobson and Andrew J Doig. Distinguishing enzyme structures from non-enzymes without alignments. Journal of molecular biology, 330(4):771–783, 2003. ",
|
| 1540 |
+
"bbox": [
|
| 1541 |
+
171,
|
| 1542 |
+
882,
|
| 1543 |
+
821,
|
| 1544 |
+
912
|
| 1545 |
+
],
|
| 1546 |
+
"page_idx": 12
|
| 1547 |
+
},
|
| 1548 |
+
{
|
| 1549 |
+
"type": "text",
|
| 1550 |
+
"text": "[56] Asim Kumar Debnath, de Compadre RL Lopez, Gargi Debnath, Alan J Shusterman, and Corwin Hansch. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity. Journal of medicinal chemistry, 34(2):786–797, 1991. \n[57] Hannu Toivonen, Ashwin Srinivasan, Ross D King, Stefan Kramer, and Christoph Helma. Statistical evaluation of the predictive toxicology challenge 2000–2001. Bioinformatics, 19(10): 1183–1193, 2003. \n[58] Ida Schomburg, Antje Chang, Christian Ebeling, Marion Gremse, Christian Heldt, Gregor Huhn, and Dietmar Schomburg. Brenda, the enzyme database: updates and major new developments. Nucleic acids research, 32(suppl_1):D431–D433, 2004. \n[59] 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. arXiv preprint arXiv:2005.00687, 2020. \n[60] Brandon Anderson, Truong-Son Hy, and Risi Kondor. Cormorant: Covariant molecular neural networks. arXiv preprint arXiv:1906.04015, 2019. \n[61] Johannes Klicpera, Janek Groß, and Stephan Günnemann. Directional message passing for molecular graphs. arXiv preprint arXiv:2003.03123, 2020. \n[62] Zhuoran Qiao, Matthew Welborn, Animashree Anandkumar, Frederick R Manby, and Thomas F Miller III. Orbnet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features. The Journal of Chemical Physics, 153(12):124111, 2020. \n[63] Hanqing Zeng, Muhan Zhang, Yinglong Xia, Ajitesh Srivastava, Andrey Malevich, Rajgopal Kannan, Viktor Prasanna, Long Jin, and Ren Chen. Deep graph neural networks with shallow subgraph samplers. arXiv preprint arXiv:2012.01380, 2020. \n[64] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pages 1025–1035, 2017. \n[65] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. arXiv preprint arXiv:1912.09893, 2019. \n[66] Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019. \n[67] Guohao Li, Chenxin Xiong, Ali Thabet, and Bernard Ghanem. Deepergcn: All you need to train deeper gcns. arXiv preprint arXiv:2006.07739, 2020. \n[68] Gabriele Corso, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Velickovi ˇ c. Principal ´ neighbourhood aggregation for graph nets. arXiv preprint arXiv:2004.05718, 2020. \n[69] Rémy Brossard, Oriel Frigo, and David Dehaene. Graph convolutions that can finally model local structure. arXiv preprint arXiv:2011.15069, 2020. \n[70] Tuan Le, Marco Bertolini, Frank Noé, and Djork-Arné Clevert. Parameterized hypercomplex graph neural networks for graph classification. arXiv preprint arXiv:2103.16584, 2021. \n[71] Katsuhiko Ishiguro, Shin-ichi Maeda, and Masanori Koyama. Graph warp module: an auxiliary module for boosting the power of graph neural networks. arXiv preprint arXiv:1902.01020, 2019. \n[72] Hongyang Gao and Shuiwang Ji. Graph u-nets. arXiv preprint arXiv:1905.05178, 2019. \n[73] Douglas J Klein and Milan Randic. Resistance distance. ´ Journal of Mathematical Chemistry, 12(1):81–95, 1993. ",
|
| 1551 |
+
"bbox": [
|
| 1552 |
+
171,
|
| 1553 |
+
53,
|
| 1554 |
+
828,
|
| 1555 |
+
910
|
| 1556 |
+
],
|
| 1557 |
+
"page_idx": 13
|
| 1558 |
+
}
|
| 1559 |
+
]
|
parse/train/7_eLEvFjCi3/7_eLEvFjCi3_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/7_eLEvFjCi3/7_eLEvFjCi3_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/9az9VKjOx00/9az9VKjOx00_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:7beffa3bb1f07200f1e8c446b828b03dfb9013b09927428db377abd496cb6c9a
|
| 3 |
+
size 911951
|
parse/train/9az9VKjOx00/9az9VKjOx00_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:8ebac7d8b71d1450879748775445058fa3f854a9a46d5c6f3bb4eb0348e0b53b
|
| 3 |
+
size 728554
|
parse/train/9az9VKjOx00/9az9VKjOx00_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:a3fc09c7fe3f160723307668abb20bfa1aba04b68133bd04e4be3e3d42c67606
|
| 3 |
+
size 916004
|
parse/train/B1Kh0SVodY/B1Kh0SVodY.md
ADDED
|
@@ -0,0 +1,347 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# Learning curves of generic features maps for realistic datasets with a teacher-student model
|
| 2 |
+
|
| 3 |
+
Bruno Loureiro IdePHICS, EPFL, Lausanne
|
| 4 |
+
|
| 5 |
+
Cédric Gerbelot Lab. de Physique de l’École Normale Supérieure, Paris
|
| 6 |
+
|
| 7 |
+
Hugo Cui SPOC, EPFL, Lausanne
|
| 8 |
+
|
| 9 |
+
Sebastian Goldt SISSA, Trieste
|
| 10 |
+
|
| 11 |
+
Florent Krzakala IdePHICS, EPFL, Lausanne
|
| 12 |
+
|
| 13 |
+
Marc Mézard École Normale Supérieure, Paris
|
| 14 |
+
|
| 15 |
+
Lenka Zdeborová SPOC, EPFL, Lausanne
|
| 16 |
+
|
| 17 |
+
# Abstract
|
| 18 |
+
|
| 19 |
+
Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: first, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones - such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.
|
| 20 |
+
|
| 21 |
+
# 1 Introduction
|
| 22 |
+
|
| 23 |
+
Teacher-student models are a popular framework to study the high-dimensional asymptotic performance of learning problems with synthetic data, and have been the subject of intense investigations spanning three decades [1–7]. In the wake of understanding the limitations of classical statistical learning approaches [8–10], this direction is witnessing a renewal of interest [10–15]. However, this framework is often assuming the input data to be Gaussian i.i.d., which is arguably too simplistic to be able to capture properties of realistic data. In this paper, we redeem this line of work by defining a Gaussian covariate model where the teacher and student act on different Gaussian correlated spaces with arbitrary covariance. We derive a rigorous asymptotic solution of this model generalizing the formulas found in the above mentioned classical works.
|
| 24 |
+
|
| 25 |
+
We then put forward a theory, supported by universality arguments and numerical experiments, that this model captures learning curves, i.e. the dependence of the training and test errors on the number of samples, for a generic class of feature maps applied to realistic datasets. These maps can be deterministic, random, or even learnt from the data. This analysis thus gives a unified framework to describe the learning curves of, for example, kernel regression and classification, the analysis of feature maps – random projections [16], neural tangent kernels [17], scattering transforms [18] – as well as the analysis of transfer learning performance on data generated by generative adversarial networks [19]. We also discuss limits of applicability of our results, by showing concrete situations where the learning curves of the Gaussian covariate model differ from the actual ones.
|
| 26 |
+
|
| 27 |
+
Model definition — The Gaussian covariate teacher-student model is defined via two vectors $\mathbf { \boldsymbol { \mathscr { u } } } \in \mathbb { R } ^ { p }$ and $\pmb { v } \in \mathbb { R } ^ { d }$ , with correlation matrices $\Psi \in \mathbb { R } ^ { p \times p } , \Omega \in \mathbb { R } ^ { d \times d }$ and $\Phi \in \mathbb { R } ^ { p \times d }$ , from which we draw $n$ independent samples:
|
| 28 |
+
|
| 29 |
+
$$
|
| 30 |
+
\left[ { \pmb u } ^ { \mu } \right] \in \mathbb { R } ^ { p + d } \underset { \mathrm { i . i . d . } } { \sim } \mathcal { N } \left( 0 , \left[ \frac { \Psi } { \Phi ^ { \top } } \quad \frac { \Phi } { \Omega } \right] \right) , \qquad \ \mu = 1 , \cdots , n .
|
| 31 |
+
$$
|
| 32 |
+
|
| 33 |
+
The labels $y ^ { \mu }$ are generated by a teacher function that is only using the vectors $\pmb { u } ^ { \mu }$ :
|
| 34 |
+
|
| 35 |
+
$$
|
| 36 |
+
y ^ { \mu } = f _ { 0 } \left( \frac { 1 } { \sqrt { p } } \pmb { \theta } _ { 0 } ^ { \top } { \pmb u } ^ { \mu } \right) ,
|
| 37 |
+
$$
|
| 38 |
+
|
| 39 |
+
where $f _ { 0 } : \mathbb { R } \to \mathbb { R }$ is a function that may include randomness such as, for instance, an additive Gaussian noise, and $\pmb \theta _ { 0 } \in \mathbb { R } ^ { p }$ is a vector of teacher-weights with finite norm which can be either random or deterministic. Learning is performed by the student with weights $\pmb { w }$ via empirical risk minimization that has access only to the features $v ^ { \mu }$ :
|
| 40 |
+
|
| 41 |
+
$$
|
| 42 |
+
\pmb { \hat { w } } = \underset { \pmb { w } \in \mathbb { R } ^ { d } } { \arg \operatorname* { m i n } } \left[ \sum _ { \mu = 1 } ^ { n } g \left( \frac { \pmb { w } ^ { \top } \pmb { v } ^ { \mu } } { \sqrt { d } } , y ^ { \mu } \right) + r ( \pmb { w } ) \right] ,
|
| 43 |
+
$$
|
| 44 |
+
|
| 45 |
+
where $r$ and $g$ are proper, convex, lower-semicontinuous functions of $\pmb { w } \in \mathbb { R } ^ { d }$ (e.g. $g$ can be a logistic or a square loss and $r$ a $\ell _ { p }$ $( p = 1 , 2 )$ regularization). The key quantities we want to compute in this model are the averaged training and generalisation errors for the estimator $\pmb { w }$ ,
|
| 46 |
+
|
| 47 |
+
$$
|
| 48 |
+
\begin{array} { r } { \mathrm { ~ \xi ~ } _ { \mathrm { t r a i n . } } ( w ) \equiv \displaystyle \frac { 1 } { n } \sum _ { \mu = 1 } ^ { n } g \left( \frac { w ^ { \top } v ^ { \mu } } { \sqrt { d } } , y ^ { \mu } \right) \quad \mathrm { a n d } \quad \mathcal { E } _ { \mathrm { g e n . } } ( w ) \equiv \mathbb { E } \left[ \hat { g } \left( \hat { f } \left( \frac { v _ { \mathrm { n e w } } ^ { \top } w } { \sqrt { d } } \right) , f _ { 0 } \left( \frac { u _ { \mathrm { n e w } } ^ { \top } \theta _ { 0 } } { \sqrt { p } } \right) \right) \right] . } \end{array}
|
| 49 |
+
$$
|
| 50 |
+
|
| 51 |
+
where $g$ is the loss function in eq. (3), $\hat { f }$ is a prediction function (e.g. $\hat { f } = \mathrm { s i g n }$ for a classification task), $\hat { g }$ is a performance measure (e.g. $\hat { \boldsymbol g } ( \hat { y } , y ) = ( \hat { y } - y ) ^ { 2 }$ for regression or $\hat { g } ( \hat { y } , y ) = \mathbb { P } ( \hat { y } \neq y )$ for classification) and $( \boldsymbol { \mathbf { \mathit { u } } } _ { \mathrm { n e w } } , \boldsymbol { \mathbf { \mathit { v } } } _ { \mathrm { n e w } } )$ is a fresh sample from the joint distribution of $\textbf { \em u }$ and $\textbf { { v } }$ .
|
| 52 |
+
|
| 53 |
+
# Our two main technical contributions are:
|
| 54 |
+
|
| 55 |
+
(C1) In Theorems 1 & 2, we give a rigorous closed-form characterisation of the properties of the estimator $\hat { \pmb { w } }$ for the Gaussian covariate model (1), and the corresponding training and generalisation errors in the high-dimensional limit. We prove our result using Gaussian comparison inequalities [20]; (C2) We show how the same expression can be obtained using the replica method from statistical physics [21]. This is of additional interest given the wide range of applications of the replica approach in machine learning and computer science [22]. In particular, this allows to put on a rigorous basis many results previously derived with the replica method.
|
| 56 |
+
|
| 57 |
+
Towards realistic data — In the second part of our paper, we argue that the above Gaussian covariate model (1) is generic enough to capture the learning behaviour of a broad range of realistic data. Let $\{ { \pmb x } ^ { \mu } \} _ { \mu = 1 } ^ { n }$ denote a data set with $n$ independent samples on $\boldsymbol { \mathcal { X } } \subset \mathbb { R } ^ { D }$ . Based on this input, the features $_ { u , v }$ are given by (potentially) elaborated transformations of $_ { \textbf { \em x } }$ , i.e.
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
\pmb { u } = \varphi _ { t } ( \pmb { x } ) \in \mathbb { R } ^ { p } \quad \mathrm { a n d } \quad \pmb { v } = \varphi _ { s } ( \pmb { x } ) \in \mathbb { R } ^ { d }
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
for given centred feature maps $\varphi _ { t } : \mathcal { X } \to \mathbb { R } ^ { p }$ and $\varphi _ { s } : \mathcal { X } \mathbb { R } ^ { d }$ , see Fig. 1. Uncentered features can be taken into account by shifting the covariances, but we focus on the centred case to lighten notation.
|
| 64 |
+
|
| 65 |
+
The Gaussian covariate model (1) is exact in the case where $_ { \textbf { \em x } }$ are Gaussian variables and the feature maps $( \varphi _ { s } , \varphi _ { s } )$ preserve the Gaussianity, for example linear features. In particular, this is the case for $\pmb { u } = \pmb { v } = \pmb { x }$ , which is the widely-studied vanilla teacher-student model [24]. The interest of the model (1) is that it also captures a range of cases in which the feature maps $\varphi _ { t }$ and $\varphi _ { s }$ are deterministic, or even learnt from the data. The covariance matrices $\Psi , \Phi$ , and $\Omega$ then represent different aspects of the data-generative process and learning model. The student (3) then corresponds to the last layer of the learning model. These observation can be distilled into the following conjecture:
|
| 66 |
+
|
| 67 |
+
|
| 68 |
+
Figure 1: Left: Given a data set $\{ { \pmb x } ^ { \mu } \} _ { \mu = 1 } ^ { n }$ , teacher $\pmb { u } = \varphi _ { t } ( \pmb { x } )$ and student maps $\pmb { v } = \varphi _ { t } ( \pmb { x } )$ , we assume $[ { \pmb u } , { \pmb v } ]$ to be jointly Gaussian random variables and apply the results of the Gaussian covariate model (1). Right: Illustration on real data, here ridge regression on even vs odd MNIST digits, with regularisation $\lambda { = } 1 0 ^ { - 2 }$ . Full line is theory, points are simulations. We show the performance with no feature map (blue), random feature map with $\sigma = \operatorname { e r f } \delta$ & Gaussian projection (orange), the scattering transform with parameters $J = 3 , L = 8$ [18] (green), and of the limiting kernel of the random map [23] (red). The covariance $\Omega$ is empirically estimated from the full data set, while the other quantities appearing in the Theorem 1 are expressed directly as a function of the labels, see Section 3.4. Simulations are averaged over 10 independent runs.
|
| 69 |
+
|
| 70 |
+
Conjecture 1. (Gaussian equivalent model) For a wide class of data distributions $\{ { \pmb x } ^ { \mu } \} _ { \mu = 1 } ^ { n }$ , and features maps ${ \pmb u } = \varphi _ { t } ( { \pmb x } ) , { \pmb v } = \varphi _ { s } ( { \pmb x } )$ , the generalisation and training errors of estimator (3) are asymptotically captured by the equivalent Gaussian model $( I ) _ { : }$ , where $[ { \pmb u } , { \pmb v } ]$ are jointly Gaussian variables, and thus by the closed-form expressions of Theorem 1.
|
| 71 |
+
|
| 72 |
+
# The second part of our main contributions are:
|
| 73 |
+
|
| 74 |
+
(C3) In Sec. 3.3 we show that the theoretical predictions from (C1) captures the learning curves in non-trivial cases, e.g. when input data are generated using a trained generative adversarial network, while extracting both the feature maps from a neural network trained on real data.
|
| 75 |
+
|
| 76 |
+
(C4) In Sec. 3.4, we show empirically that for ridge regression the asymptotic formula of Theorem 1 can be applied directly to real data sets, even though the Gaussian hypothesis is not satisfied. This universality-like property is a consequence of Theorem 3 and is illustrated in Fig. 1 (right) where the real learning curve of several features maps learning the odd-versus-even digit task on MNIST is compared to the theoretical prediction.
|
| 77 |
+
|
| 78 |
+
Related work — Rigorous results for teacher-student models: The Gaussian covariate model (1) contains the vanilla teacher-student model as a special case where one takes $\textbf { \em u }$ and $\textbf { { v } }$ identical, with unique covariance matrix $\Omega$ . This special case has been extensively studied in the statistical physics community using the heuristic replica method [1–3, 24, 25]. Many recent rigorous results for such models can be rederived as a special case of our formula, e.g. refs. [10–15, 26–29]. Numerous of these results are based on the same proof technique as we employed here: the Gordon’s Gaussian min-max inequalities [20, 30, 31]. The asymptotic analysis of kernel ridge regression [32], of margin-based classification [33] also follow from our theorem. See also Appendix A.6 for the details on these connections. Other examples include models of the double descent phenomenon [34]. Closer to our work is the recent work of [35] on the random feature model. For ridge regression, there are also precise predictions thanks to random matrix theory [12, 36–41]. A related set of results was obtained in [42] for orthogonal random matrix models. The main technical novelty of our proof is the handling of a generic loss and regularisation, not only ridge, representing convex empirical risk minimization, for both classification and regression, with the generic correlation structure of the model (1).
|
| 79 |
+
|
| 80 |
+
Gaussian equivalence: A similar Gaussian conjecture has been discussed in a series of recent works, and some authors proved partial results in this direction [11, 12, 28, 35, 43–46]. Ref. [45] analyses a special case of the Gaussian model (corresponding to $\varphi _ { t } ~ = \mathrm { i d }$ here), and proves a Gaussian equivalence theorem (GET) for feature maps $\varphi _ { s }$ given by single-layer neural networks with fixed weights. They also show that for Gaussian data $\pmb { x } \sim \mathcal { N } ( \mathbf { 0 } , \mathrm { I } _ { D } )$ , feature maps of the form $\pmb { v } = \sigma ( \mathbf { W } \pmb { x } )$ (with some technical restriction on the weights) led to the jointly-Gaussian property for the two scalars $( \pmb { v } \cdot \pmb { w } , \pmb { u } \cdot \pmb { \theta } _ { 0 } )$ for almost any vector $\textbf { \em w }$ . However, their stringent assumptions on random teacher weights limited the scope of applications to unrealistic label models. A related line of work discussed similar universality through the lens of random matrix theory [47–49]. In particular, Seddik et al. [50] showed that, in our notations, vectors $[ u , v ]$ obtained from Gaussian inputs $\pmb { x } \sim \mathcal { N } ( \mathbf { 0 } , \mathrm { I } _ { D } )$ with Lipschitz feature maps satisfy a concentration property. In this case, again, one can expect the two scalars $\begin{array} { r } { ( \pmb { v } \cdot \pmb { w } , \pmb { u } \cdot \pmb { \theta } _ { 0 } ) } \end{array}$ to be jointly Gaussian with high-probability on $\textbf { \em w }$ . Remarkably, in the case of random feature maps, [46] could go beyond this central-limit-like behavior and established the universality of the Gaussian covariate model (1) for the actual learned weights $\hat { \pmb { w } }$ .
|
| 81 |
+
|
| 82 |
+
# 2 Main technical results
|
| 83 |
+
|
| 84 |
+
Our main technical result is a closed-form expression for the asymptotic training and generalisation errors (4) of the Gaussian covariate model introduced above. We start by presenting our result in the most relevant setting for the applications of interest in Section 3, which is the case of the $\ell _ { 2 }$ regularization. Next, we briefly present our result in larger generality, which includes non-asymptotic results for non-separable losses and regularizations.
|
| 85 |
+
|
| 86 |
+
We start by defining key quantities that we will use to characterize the estimator $\hat { \pmb { w } }$ . Let $\Omega =$ $\mathbf { S } ^ { \top } \mathrm { d i a g } ( \omega _ { i } ) \mathbf { \bar { S } }$ be the spectral decomposition of $\Omega$ . Let:
|
| 87 |
+
|
| 88 |
+
$$
|
| 89 |
+
\rho \equiv \frac { 1 } { d } \pmb { \theta } _ { 0 } ^ { \top } \Psi \pmb { \theta } _ { 0 } \in \mathbb { R } , \qquad \quad \bar { \pmb { \theta } } \equiv \frac { { \bf S } \Phi ^ { \top } \pmb { \theta } _ { 0 } } { \sqrt { \rho } } \in \mathbb { R } ^ { d }
|
| 90 |
+
$$
|
| 91 |
+
|
| 92 |
+
and define the joint empirical density $\hat { \mu } _ { d }$ between $( \omega _ { i } , \bar { \theta } _ { i } )$ :
|
| 93 |
+
|
| 94 |
+
$$
|
| 95 |
+
\hat { \mu } _ { d } ( \omega , \bar { \theta } ) \equiv { \frac { 1 } { d } } \sum _ { i = 1 } ^ { d } \delta ( \omega - \omega _ { i } ) \delta ( \bar { \theta } - \bar { \theta } _ { i } ) .
|
| 96 |
+
$$
|
| 97 |
+
|
| 98 |
+
Note that $\Phi ^ { \top } \pmb \theta _ { 0 }$ is the projection of the teacher weights on the student space, and therefore $\bar { \pmb { \theta } }$ is the rotated projection on the basis of the student covariance, rescaled by the teacher variance. Together with the student eigenvalues $\omega _ { i }$ , these are relevant statistics of the model, encoded here in the joint distribution $\hat { \mu } _ { d }$ .
|
| 99 |
+
|
| 100 |
+
Assumptions — Consider the high-dimensional limit in which the number of samples $n$ and the dimensions $p , d \mathrm { g o }$ to infinity with fixed ratios:
|
| 101 |
+
|
| 102 |
+
$$
|
| 103 |
+
\alpha \equiv { \frac { n } { d } } , \mathrm { a n d } \gamma \equiv { \frac { p } { d } } .
|
| 104 |
+
$$
|
| 105 |
+
|
| 106 |
+
Assume that the covariance matrices $\Psi , \Omega$ are positive-definite and that the Schur complement of the block covariance in equation (1) is positive semi-definite. Additionally, the spectral distributions of the matrices $\Phi , \Psi$ and $\Omega$ converge to distributions such that the limiting joint distribution $\mu$ is well-defined, and their maximum singular values are bounded with high probability as $n , p , d \to \infty$ . Finally, regularity assumptions are made on the loss and regularization functions mainly to ensure feasibility of the minimization problem. We assume that the cost function $r + g$ is coercive, i.e. $\begin{array} { r } { \operatorname* { l i m } _ { \| \pmb { w } \| _ { 2 } + \infty } ( r + g ) ( \pmb { w } ) = + \infty } \end{array}$ and that the following scaling condition holds : for all $n , d \in$ $\mathbb { N } , z \in \mathbb { R } ^ { n }$ and any constant $c > 0$ , there exist a finite, positive constant $C$ , such that, for any standard normal random vectors $\pmb { h } \in \mathbb { R } ^ { d }$ and $\pmb { g } \in \mathbb { R } ^ { n }$ :
|
| 107 |
+
|
| 108 |
+
$$
|
| 109 |
+
\| z \| _ { 2 } \leqslant c \sqrt { n } \implies \operatorname* { s u p } _ { x \in \partial g ( z ) } \| x \| _ { 2 } \leqslant C \sqrt { n } , \qquad \frac { 1 } { d } \mathbb { E } \left[ r ( h ) \right] < + \infty , \qquad \frac { 1 } { n } \mathbb { E } \left[ g ( g ) \right] < + \infty
|
| 110 |
+
$$
|
| 111 |
+
|
| 112 |
+
The relevance of these assumptions in a supervised machine learning context is discussed in Appendix B.1. We are now in a position to state our result.
|
| 113 |
+
|
| 114 |
+
Theorem 1. (Closed-form asymptotics for $\ell _ { 2 }$ regularization) In the asymptotic limit defined above, the training and generalisation errors (4) of the estimator $\hat { \textbf { \textit { w } } } \in \mathbb { R } ^ { \dot { d } }$ solving the empirical risk minimisation problem in eq. (3) with $\ell _ { 2 }$ regularization $\begin{array} { r } { r ( \pmb { w } ) = \frac { \lambda } { 2 } | | \pmb { w } | | _ { 2 } ^ { 2 } } \end{array}$ verify:
|
| 115 |
+
|
| 116 |
+
$$
|
| 117 |
+
\begin{array} { r l } & { \mathcal { E } _ { \mathrm { t r a i n . } } ( \hat { w } ) \xrightarrow [ d \infty ] { P } \mathbb { E } _ { s , h \sim \mathcal { N } ( 0 , 1 ) } [ g ( p r o x _ { V ^ { \star } g ( \cdot , f _ { 0 } ( \sqrt { \rho } s ) ) } ( \frac { m ^ { \star } } { \sqrt { \rho } } s + \sqrt { q ^ { \star } - \frac { m ^ { \star ^ { 2 } } } { \rho } } h ) , f _ { 0 } ( \sqrt { \rho } s ) ) ] } \\ & { \mathcal { E } _ { \mathrm { g e n . } } ( \hat { w } ) \xrightarrow [ d \infty ] { P } \mathbb { E } _ { ( \nu , \lambda ) } [ \hat { g } ( \hat { f } ( \lambda ) , f _ { 0 } ( \nu ) ) ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( 1 6 ^ { d } ) } \end{array}
|
| 118 |
+
$$
|
| 119 |
+
|
| 120 |
+
where prox stands for the proximal operator defined as
|
| 121 |
+
|
| 122 |
+
$$
|
| 123 |
+
p r o x _ { V g ( . , y ) } ( x ) = \arg \operatorname* { m i n } _ { z } \{ g ( z , y ) + \frac { 1 } { 2 V } ( x - z ) ^ { 2 } \}
|
| 124 |
+
$$
|
| 125 |
+
|
| 126 |
+
and where $( \nu , \lambda )$ are jointly Gaussian scalar variables:
|
| 127 |
+
|
| 128 |
+
$$
|
| 129 |
+
( \nu , \lambda ) \sim \mathcal { N } \left( 0 , \left[ \rho \begin{array} { c c } { \rho } & { m ^ { \star } } \end{array} \right] \right) ,
|
| 130 |
+
$$
|
| 131 |
+
|
| 132 |
+
and the overlap parameters $( V ^ { \star } , q ^ { \star } , m ^ { \star } )$ are prescribed by the unique fixed point of the following set of self-consistent equations:
|
| 133 |
+
|
| 134 |
+
$$
|
| 135 |
+
\left\{ \begin{array} { l } { V = \mathbb { E } _ { ( \omega , \bar { \theta } ) \sim \mu } \left[ \frac { \omega } { \lambda + \hat { V } \omega } \right] } \\ { m = \frac { \hat { m } } { \sqrt { \gamma } } \mathbb { E } _ { ( \omega , \bar { \theta } ) \sim \mu } \left[ \frac { \theta } { \lambda + \hat { V } \omega } \right] } \\ { q = \mathbb { E } _ { ( \omega , \bar { \theta } ) \sim \mu } \left[ \frac { \hat { m } ^ { 2 } \bar { \theta } ^ { 2 } \omega + \hat { q } \omega ^ { 2 } } { \left( \lambda + \hat { V } \omega \right) ^ { 2 } } \right] } \end{array} \right. , \left\{ \begin{array} { l } { \hat { V } = \frac { \alpha } { V } ( 1 - \mathbb { E } _ { s , h \sim \mathcal { N } ( 0 , 1 ) } [ f _ { g } ^ { \prime } ( V , m , q ) ] ) } \\ { \hat { m } = \frac { 1 } { \sqrt { \rho \gamma } } \frac { \alpha } { V } \mathbb { E } _ { s , h \sim \mathcal { N } ( 0 , 1 ) } \left[ s f _ { g } ( V , m , q ) - \frac { m } { \sqrt { \rho } } f _ { g } ^ { \prime } ( V , m , q ) \right] } \\ { \hat { q } = \frac { \alpha } { V ^ { 2 } } \mathbb { E } _ { s , h \sim \mathcal { N } ( 0 , 1 ) } \left[ \left( \frac { m } { \sqrt { \rho } } s + \sqrt { q - \frac { m ^ { 2 } } { \rho } } h - f _ { g } ( V , m , q ) \right) ^ { 2 } \right] } \end{array} \right.
|
| 136 |
+
$$
|
| 137 |
+
|
| 138 |
+
where we defined the scalar random functions $f _ { g } ( V , m , q ) ~ = ~ p r o x _ { V g ( . , f _ { 0 } ( \sqrt { \rho } s ) ) } ( \rho ^ { - 1 / 2 } m s ~ + ~$ $\sqrt { q - \rho ^ { - 1 } m ^ { 2 } } h )$ and $f _ { g } ^ { \prime } ( V , m , h ) = p r o x _ { V g ( . , f _ { 0 } ( \sqrt { \rho } s ) ) } ^ { \prime } ( \rho ^ { - 1 / 2 } m s + \sqrt { q - \rho ^ { - 1 } m ^ { 2 } } h )$ as the first derivative of the proximal operator.
|
| 139 |
+
|
| 140 |
+
Proof : This result is a consequence of Theorem 2, whose proof can be found in appendix B.
|
| 141 |
+
|
| 142 |
+
The parameters of the model $( \pmb \theta _ { 0 } , \Omega , \Phi , \Psi )$ only appear trough $\rho$ , eq. (6), and the asymptotic limit $\mu$ of the joint distribution eq. (7) and $( f _ { 0 } , \hat { f } , g , \lambda )$ . One can easily iterate the above equations to find their fixed point, and extract $( q ^ { * } , m ^ { * } )$ which appear in the expressions for the training and generalisation errors $( \mathcal { E } _ { \mathrm { t r a i n } } ^ { \star } , \mathcal { E } _ { \mathrm { g e n } } ^ { \star } )$ , see eq. (4). Note that $( q ^ { \star } , m ^ { \star } )$ have an intuitive interpretation in terms of the estimator $\hat { \pmb w } \in \mathbb { R } ^ { d }$ :
|
| 143 |
+
|
| 144 |
+
$$
|
| 145 |
+
\boldsymbol { q } ^ { \star } \equiv \frac { 1 } { d } \hat { \boldsymbol { w } } ^ { \top } \boldsymbol { \Omega } \hat { \boldsymbol { w } } , m ^ { \star } \equiv \frac { 1 } { \sqrt { d p } } \boldsymbol { \theta } _ { 0 } ^ { \top } \Phi \hat { \boldsymbol { w } }
|
| 146 |
+
$$
|
| 147 |
+
|
| 148 |
+
Or in words: $m ^ { \star }$ is the correlation between the estimator projected in the teacher space, while $q ^ { \star }$ is the reweighted norm of the estimator by the covariance $\Omega$ . The parameter $V ^ { * }$ also has a concrete interpretation $:$ it parametrizes the deformation that must be applied to a Gaussian field specified by the solution of the fixed point equations to obtain the asymptotic behaviour of $\hat { \mathbf { z } }$ . It prescribes the degree of non-linearity given to the linear output by the chosen loss function. This is coherent with the robust regression viewpoint, where one introduces non-square losses to deal with the potential non-linearity of the generative model. $\hat { V } ^ { * }$ plays a similar role for the estimator wˆ through the proximal operator of the regularisation, see Theorem 4 and 5 in the Appendix. Two cases are of particular relevance for the experiments that follow. The first is the case of ridge regression, in which $f _ { 0 } ( x ) = { \hat { f } } ( x )$ and both the loss $g$ and the performance measure $\hat { g }$ are taken to be the meansquared error $\mathrm { m s e } ( y , \hat { y } ) = \textstyle { \frac { 1 } { 2 } } ( y - \hat { y } ) ^ { 2 }$ , and the asymptotic errors are given by the simple closed-form expression:
|
| 149 |
+
|
| 150 |
+
$$
|
| 151 |
+
\mathcal { E } _ { \mathrm { g e n } } ^ { \star } = \rho + q ^ { \star } - 2 m ^ { \star } , \qquad \mathcal { E } _ { \mathrm { t r a i n } } ^ { \star } = \frac { \mathcal { E } _ { \mathrm { g e n } } ^ { \star } } { ( 1 + V ^ { \star } ) ^ { 2 } } ,
|
| 152 |
+
$$
|
| 153 |
+
|
| 154 |
+
The second case of interest is the one of a binary classification task, for which $f _ { 0 } ( x ) = { \hat { f } } ( x ) =$ $\mathrm { s i g n } ( x )$ , and we choose the performance measure to be the classification error $\hat { g } ( y , \hat { y } ) = \mathbb { P } ( y \neq \hat { y } )$ . In the same notation as before, the asymptotic generalisation error in this case reads:
|
| 155 |
+
|
| 156 |
+
$$
|
| 157 |
+
\mathcal { E } _ { \mathrm { g e n } } ^ { \star } = \frac { 1 } { \pi } \cos ^ { - 1 } \left( \frac { m ^ { \star } } { \sqrt { \rho q ^ { \star } } } \right) ,
|
| 158 |
+
$$
|
| 159 |
+
|
| 160 |
+
$\mathcal { E } _ { \mathrm { t r a i n } } ^ { \star }$ depends on the choice of ll of the binary classificat $g$ - which we will take to be the logistic lossn experiments. $g ( y , x ) = \log { ( 1 + e ^ { - x y } ) }$
|
| 161 |
+
|
| 162 |
+
As mentioned above, this paper includes stronger technical results including finite size corrections and precise characterization of the distribution of the estimator $\hat { \pmb { w } }$ , for generic, non-separable loss and regularization $g$ and $r$ . This type of distributional statement is encountered for special cases of the model in related works such as [28, 29, 51]. Define $\boldsymbol { \mathcal { V } } \in \mathbb { R } ^ { n \times d }$ as the matrix of concatenated samples used by the student. Informally, in high-dimension, the estimator $\hat { \pmb w }$ and $\begin{array} { r } { \hat { z } = \frac { 1 } { \sqrt { d } } \mathcal { V } \hat { w } } \end{array}$ roughly behave as non-linear transforms of Gaussian random variables centered around the teacher vector $\theta _ { 0 }$ (or its projection on the covariance spaces) as follows:
|
| 163 |
+
|
| 164 |
+
$$
|
| 165 |
+
{ \boldsymbol w } ^ { * } = \Omega ^ { - 1 / 2 } \frac { \mathrm { p r o x } } { \bar { \nu } ^ { * } r ( \Omega ^ { - 1 / 2 } . ) } \left( \frac { 1 } { \hat { V } ^ { * } } ( \hat { m } ^ { * } t + \sqrt { \hat { q } ^ { * } } g ) \right) , \boldsymbol z ^ { * } = \operatorname * { p r o x } _ { V ^ { * } g ( . , \boldsymbol z ) } \left( \frac { m ^ { * } } { \sqrt { \rho } } \boldsymbol s + \sqrt { \boldsymbol q ^ { * } - \frac { ( m ^ { * } ) ^ { 2 } } { \rho } } \boldsymbol h \right) .
|
| 166 |
+
$$
|
| 167 |
+
|
| 168 |
+
where $s , h \sim \mathcal { N } ( 0 , \mathrm { I } _ { n } )$ and $\mathbf { \mathscr { g } } \sim \mathcal { N } ( 0 , \mathrm { I } _ { d } )$ are random vectors independent of the other quantities, $t =$ $\Omega ^ { - 1 / 2 } \Phi ^ { \top } \pmb { \theta } _ { 0 }$ , $\pmb { y } = \pmb { f } _ { 0 } \left( \sqrt { \rho } \pmb { s } \right)$ , and $( V ^ { * } , \hat { V } ^ { * } , q ^ { * } , \hat { q } ^ { * } , m ^ { * } , \hat { m } ^ { * } )$ is the unique solution to the fixed point equations presented in Lemma 12 of appendix B. Those fixed point equations are the generalization of (13) to generic, non-separable loss function and regularization. The formal concentration of measure result can then be stated in the following way:
|
| 169 |
+
|
| 170 |
+
Theorem 2. (Non-asymptotic version, generic loss and regularization) Under Assumption $_ { ( B . I ) }$ , consider any optimal solution $\hat { \pmb { w } }$ to 3. Then, there exist constants $C , c , c ^ { \prime } > 0$ such that, for any Lipschitz function $\phi _ { 1 } : \mathbb { R } ^ { d } \mathbb { R }$ , and separable, pseudo-Lipschitz function $\phi _ { 2 } : \mathbb { R } ^ { n } \to \mathbb { R }$ and any $0 < \epsilon < c ^ { \prime }$ :
|
| 171 |
+
|
| 172 |
+
$$
|
| 173 |
+
\mathbb { P } \left( \left| \phi _ { 1 } \left( \frac { \hat { w } } { \sqrt { d } } \right) - \mathbb { E } \phi _ { 1 } \left( \frac { w ^ { * } } { \sqrt { d } } \right) \right| \geqslant \epsilon \right) \leqslant \frac { C } { \epsilon ^ { 2 } } e ^ { - c n \epsilon ^ { 4 } } , \mathbb { P } \left( \left| \phi _ { 2 } \left( \frac { \hat { z } } { \sqrt { n } } \right) - \mathbb { E } \phi _ { 2 } \left( \frac { z ^ { * } } { \sqrt { n } } \right) \right| \geqslant \epsilon \right) \leqslant \frac { C } { \epsilon ^ { 2 } } e ^ { - c n \epsilon ^ { 4 } } .
|
| 174 |
+
$$
|
| 175 |
+
|
| 176 |
+
Note that in this form, the dimensions $n , p , d$ still appear explicitly, as we are characterizing the convergence of the estimator’s distribution for large but finite dimension. The clearer, one-dimensional statements are recovered by taking the $n , p , d \infty$ limit with separable functions and an $\ell _ { 2 }$ regularization. Other simplified formulas can also be obtained from our general result in the case of an $\ell _ { 1 }$ penalty, but since this breaks rotational invariance, they do look more involved than the $\ell _ { 2 }$ case. From Theorem 2, one can deduce the expressions of a number of observables, represented by the test functions $\phi _ { 1 } , \phi _ { 2 }$ , characterizing the performance of $\hat { \pmb { w } }$ , for instance the training and generalization error. A more detailed statement, along with the proof, is given in appendix B.
|
| 177 |
+
|
| 178 |
+
# 3 Applications of the Gaussian model
|
| 179 |
+
|
| 180 |
+
We now discuss how the theorems above are applied to characterise the learning curves for a range of concrete cases. We present a number of cases – some rather surprising – for which Conjecture 1 seems valid, and point out some where it is not. An out-of-the-box iterator for all the cases studied hereafter is provided in the GitHub repository for this manuscript at https: //github.com/IdePHICS/GCMProject.
|
| 181 |
+
|
| 182 |
+
# 3.1 Random kitchen sink with Gaussian data
|
| 183 |
+
|
| 184 |
+
If we choose random feature maps $\varphi _ { s } ( \pmb { x } ) = \sigma \left( \mathrm { F } \pmb { x } \right)$ for a random matrix $\mathrm { F }$ and a chosen scalar function $\sigma$ acting component-wise, we obtain the random kitchen sink model [16]. This model has seen a surge of interest recently, and a sharp asymptotic analysis was provided in the particular case of uncorrelated Gaussian data $\pmb { x } \sim \mathcal { N } ( \mathbf { 0 } , \tilde { \mathrm { I } } _ { D } )$ and $\varphi _ { t } ( \pmb { x } ) = \pmb { x }$ in [11, 12] for ridge regression and generalised by [43, 46] for generic convex losses. Both results can be framed as a Gaussian covariate model with:
|
| 185 |
+
|
| 186 |
+
$$
|
| 187 |
+
\Psi = { \bf I } _ { p } , \Phi = \kappa _ { 1 } \mathrm { \bf F } ^ { \top } , \Omega = \kappa _ { 0 } ^ { 2 } { \bf 1 } _ { d } { \bf 1 } _ { d } ^ { \top } + \kappa _ { 1 } ^ { 2 } \frac { \mathrm { \bf F } \mathrm { \bf F } ^ { \top } } { d } + \kappa _ { \star } ^ { 2 } { \bf I } _ { d } ,
|
| 188 |
+
$$
|
| 189 |
+
|
| 190 |
+
where $\mathbf { 1 } _ { d } \in \mathbb { R } ^ { d }$ is the all-one vector and the constants $( \kappa _ { 0 } , \kappa _ { 1 } , \kappa _ { \star } )$ are related to the non-linearity $\sigma$
|
| 191 |
+
|
| 192 |
+
$$
|
| 193 |
+
\begin{array} { r } { \kappa _ { 0 } = \mathbb { E } _ { z \sim \mathcal { N } ( 0 , 1 ) } \left[ \sigma ( z ) \right] , \quad \kappa _ { 1 } = \mathbb { E } _ { z \sim \mathcal { N } ( 0 , 1 ) } \left[ z \sigma ( z ) \right] , \quad \kappa _ { \star } = \sqrt { \mathbb { E } _ { z \sim \mathcal { N } ( 0 , 1 ) } \left[ \sigma ( z ) ^ { 2 } \right] - \kappa _ { 0 } ^ { 2 } - \kappa _ { 1 } ^ { 2 } } . } \end{array}
|
| 194 |
+
$$
|
| 195 |
+
|
| 196 |
+
In this case, the averages over $\mu$ in eq. (13) can be directly expressed in terms of the Stieltjes transform associated with the spectral density of $\mathrm { F F } ^ { \top }$ . Note, however, that our present framework can accommodate more involved random sinks models, such as when the teacher features are also a random feature model or multi-layer random architectures.
|
| 197 |
+
|
| 198 |
+
# 3.2 Kernel methods with Gaussian data
|
| 199 |
+
|
| 200 |
+
|
| 201 |
+
Figure 2: Learning in kernel space: Teacher and student live in the same (Hilbert) feature space $\pmb { v } = \pmb { u } \in \mathbb { R } ^ { d }$ with $d \gg n$ , and the performance only depends on the relative decay between the student spectrum $\omega _ { i } = d i ^ { - 2 }$ (the capacity) and the teacher weights in feature space $\theta _ { 0 i } ^ { 2 } \bar { \omega _ { i } } = \dot { d } i ^ { - a }$ (the source). Top: a task with sign teacher (in kernel space), fitted with a max-margin support vector machine (logistic regression with vanishing regularisation [52]). Bottom: a task with linear teacher (in kernel space) fitted via kernel ridge regression with vanishing regularisation. Points are simulation that matches the theory (lines). Simulations are averaged over 10 independent runs.
|
| 202 |
+
|
| 203 |
+
Another direct application of our formalism is to kernel methods. Kernel methods admit a dual representation in terms of optimization over feature space [53]. The connection is given by Mercer’s theorem, which provides an eigen-decomposition of the kernel and of the target function in the feature basis, effectively mapping kernel regression to a teacherstudent problem on feature space. The classical way of studying the performance of kernel methods [54, 55] is then to directly analyse the performance of convex learning in this space. In our notation, the teacher and student feature maps are equal, and we thus set $p = d , \Psi = \Phi = \Omega = \mathrm { d i a g } ( \omega _ { \mathrm { i } } )$ where $\omega _ { i }$ are the eigenvalues of the kernel and we take the teacher weights $\pmb { \theta } _ { 0 }$ to be the decomposition of the target function in the kernel feature basis.
|
| 204 |
+
|
| 205 |
+
There are many results in classical learning theory on this problem for the case of ridge regression (where the teacher is usually called "the source" and the eigenvalues of the kernel matrix the "capacity", see e.g. [54, 56]). However, these are worst case approaches, where no assumption is made on the true distribution of the data. In contrast, here we follow a typical case analysis, assuming Gaussianity in feature space. Through Theorem 1, this allows us to go beyond the restriction of the ridge loss. An example for logistic loss is in Fig. 2.
|
| 206 |
+
|
| 207 |
+
For the particular case of kernel ridge regression, Th. 1 provides a rigorous proof of the formula conjectured in [32]. App. A.6 presents an explicit mapping to their results. Hard-margin Support Vector Machines (SVMs) have also been studied using the heuristic replica method from statistical physics in [57, 58]. In our framework, this corresponds to the hinge loss $g ( x , y ) = \operatorname* { m a x } ( 0 , 1 - y x )$ when $\lambda 0 ^ { + }$ . Our theorem thus puts also these works on rigorous grounds, and extends them to more general losses and regularization.
|
| 208 |
+
|
| 209 |
+
# 3.3 GAN-generated data and learned teachers
|
| 210 |
+
|
| 211 |
+
To approach more realistic data sets, we now consider the case in which the input data $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ is given by a generative neural network $\begin{array} { r } { \pmb { x } = \pmb { \mathcal { G } } ( \pmb { z } ) } \end{array}$ , where $_ z$ is a Gaussian i.i.d. latent vector. Therefore, the covariates $[ { \pmb u } , { \pmb v } ]$ are the result of the following Markov chain:
|
| 212 |
+
|
| 213 |
+
$$
|
| 214 |
+
z \underset { \mathcal { G } } { \mapsto } x \in \mathcal { X } \underset { \varphi _ { t } } { \mapsto } u \in \mathbb { R } ^ { p } , \qquad z \underset { \mathcal { G } } { \mapsto } x \in \mathcal { X } \underset { \varphi _ { s } } { \mapsto } v \in \mathbb { R } ^ { d } .
|
| 215 |
+
$$
|
| 216 |
+
|
| 217 |
+
With a model for the covariates, the missing ingredient is the teacher weights $\pmb \theta _ { 0 } \in \mathbb { R } ^ { p }$ , which determine the label assignment: $\boldsymbol { y } = f _ { 0 } ( \boldsymbol { \mathbf { u } } ^ { \intercal } \bar { \boldsymbol { \theta } } _ { 0 } )$ . In the experiments that follow, we fit the teacher weights from the original data set in which the generative model $\mathcal { G }$ was trained. Different choices for the fitting yield different teacher weights, and the quality of label assignment can be accessed by the performance of the fit on the test set. The set $\left( \varphi _ { t } , \varphi _ { s } , \mathcal { G } , \theta _ { 0 } \right)$ defines the data generative process. For predicting the learning curves from the iterative eqs. (13) we need to sample from the spectral measure $\mu$ , which amounts to estimating the population covariances $( \Psi , \Phi , \Omega )$ . This is done from the generative process in eq. (19) with a Monte Carlo sampling algorithm. This pipeline is explained in detail in Appendix D. An open source implementation of the algorithms used in the experiments is available online at https://github.com/IdePHICS/GCMProject.
|
| 218 |
+
|
| 219 |
+
Fig. 3 shows an example of the learning curves resulting from the pipeline discussed above in a logistic regression task on data generated by a GAN trained on CIFAR10 images. More concretely, we used a pre-trained five-layer deep convolutional GAN (dcGAN) from [59], which maps 100 dimensional i.i.d. Gaussian noise into $k = 3 2 \times 3 2 \times 3$ realistic looking CIFAR10-like images: $\mathcal { G } : \boldsymbol { z } \in \mathbb { R } ^ { 1 0 0 } \mapsto \boldsymbol { x } \in \mathbb { R } ^ { 3 2 \times 3 2 \times 3 }$ . To generate labels, we trained a simple fully-connected four-layer neural network on the real CIFAR10 data set, on a odd $( y = + 1 )$ ) vs. even $( y = - 1$ ) task, achieving $\sim 7 5 \%$ classification accuracy on the test set. The teacher weights $\pmb \theta _ { 0 } \in \mathbb { R } ^ { p }$ were taken from the last layer of the network, and the teacher feature map $\varphi _ { t }$ from the three previous layers. For the student model, we trained a completely independent fully connected 3-layer neural network on the dcGAN-generated CIFAR10-like images and took snapshots of the feature maps $\varphi _ { s } ^ { i }$ induced by the 2-first layers during the first $i \in \{ 0 , 5 , 5 0 , 2 0 0 \}$ epochs of training. Finally, once $\left( \mathcal { G } , \varphi _ { t } , \varphi _ { s } ^ { i } , \theta _ { 0 } \right)$ have been fixed, we estimated the covariances $( \Psi , \Phi , \Omega )$ with a Monte Carlo algorithm. Details of the architectures used and of the training procedure can be found in Appendix. D.1.
|
| 220 |
+
|
| 221 |
+
|
| 222 |
+
Figure 3: Left: generalisation classification error (top) and (unregularised) training loss (bottom) vs the sample complexity $\alpha = n / d$ for logistic regression on a learned feature map trained on dcGAN-generated CIFAR10-like images labelled by a teacher fully-connected neural network (see Appendix D.1 for architecture details), with vanishing $\ell _ { 2 }$ regularisation. The different curves compare featured maps at different epochs of training. The theoretical predictions based on the Gaussian covariate model (full lines) are in very good agreement with the actual performance (points). Right: Test classification error (top) and (unregularised) training loss, (bottom) for logistic regression as a function of the number of samples $n$ for an animal vs not-animal binary classification task with $\ell _ { 2 }$ regularization $\lambda = 1 0 ^ { - 2 }$ , comparing real CIFAR10 grey-scale images (blue) with dcGAN-generated CIFAR10-like gray-scale images (red). The real-data learning curve was estimated, just as in Figs. 4 from the population covariances on the full data set, and it is not in agreement with the theory in this case. On the very right we depict the histograms of the variable $\begin{array} { r } { \frac { 1 } { \sqrt { d } } \pmb { v } ^ { \top } \hat { \pmb { w } } } \end{array}$ for a fixed number of samples $n = 2 d = 2 0 4 8$ and the respective theoretical predictions (solid line). Simulations are averaged over 10 independent runs.
|
| 223 |
+
|
| 224 |
+
Fig. 3 depicts the resulting learning curves obtained by training the last layer of the student. Interestingly, the performance of the feature map at epoch 0 (random initialisation) beats the performance of the learned features during early phases of training in this experiment. Another interesting behaviour is given by the separability threshold of the learned features, i.e. the number of samples for which the training loss becomes larger than 0 in logistic regression. At epoch 50 the learned features are separable at lower sample complexity $\alpha = n / d$ than at epoch 200 - even though in the later the training and generalisation performances are better.
|
| 225 |
+
|
| 226 |
+
# 3.4 Learning from real data sets
|
| 227 |
+
|
| 228 |
+
Applying teacher/students to a real data set — Given that the learning curves of realistic-looking inputs can be captured by the Gaussian covariate model, it is fair to ask whether the same might be true for real data sets. To test this idea, we first need to cast the real data set into the teacher-student formalism, and then compute the covariance matrices $\Omega , \Psi , \Phi$ and teacher vector $\pmb { \theta } _ { 0 }$ required by model (1).
|
| 229 |
+
|
| 230 |
+
Let $\{ x ^ { \mu } , y ^ { \mu } \} _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } }$ denote a real data set, e.g. MNIST or Fashion-MNIST for concreteness, where $n _ { \mathrm { t o t } } = 7 \times 1 0 ^ { 4 }$ , $\pmb { x } ^ { \mu } \in \mathbb { R } ^ { D }$ with $D = 7 8 4$ . Without loss of generality, we can assume the data is centred. To generate the teacher, let $\pmb { u } ^ { \mu } = \varphi _ { t } ( \pmb { x } ^ { \mu } ) \in \mathbb { R } ^ { p }$ be a feature map such that data is invertible in feature space, i.e. that $y ^ { \mu } = \pmb \theta _ { 0 } ^ { \top } { \pmb u } ^ { \mu }$ for some teacher weights $\pmb \theta _ { 0 } \in \mathbb { R } ^ { p }$ , which should be computed from the samples. Similarly, let $\mathbf { \bar { \boldsymbol { v } } } ^ { \mu } = \varphi _ { s } ( \mathbf { \boldsymbol { x } } ^ { \mu } ) \in \mathbb { R } ^ { d }$ be a feature map we are interested in studying. Then, we can estimate the population covariances $( \Psi , \Phi , \Omega )$ empirically from the entire data set as:
|
| 231 |
+
|
| 232 |
+
$$
|
| 233 |
+
\Psi = \sum _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } } { \frac { u ^ { \mu } { \pmb u } ^ { \mu } ^ { \top } } { n _ { \mathrm { t o t } } } } , \qquad \Phi = \sum _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } } { \frac { u ^ { \mu } \pmb v ^ { \mu } ^ { \top } } { n _ { \mathrm { t o t } } } } , \qquad \Omega = \sum _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } } { \frac { v ^ { \mu } \pmb v ^ { \mu } ^ { \top } } { n _ { \mathrm { t o t } } } } .
|
| 234 |
+
$$
|
| 235 |
+
|
| 236 |
+
At this point, we have all we need to run the self-consistent equations (13). The issue with this approach is that there is not a unique teacher map $\varphi _ { t }$ and teacher vector $\pmb { \theta } _ { 0 }$ that fit the true labels. However, we can show that all interpolating linear teachers are equivalent:
|
| 237 |
+
|
| 238 |
+
Theorem 3. (Universality of linear teachers) For any teacher feature map $\varphi _ { t }$ , and for any $\pmb { \theta } _ { 0 }$ that interpolates the data so that $\mathbf { \bar { \boldsymbol { y } } } ^ { \mu } = \pmb { \theta } _ { 0 } ^ { \top } \pmb { u } ^ { \mu } \forall \mu$ , the asymptotic predictions of model $( l )$ are equivalent. Proof. It follows from the fact that the teacher weights and covariances only appear in eq. (13) through $\begin{array} { r } { \rho = \frac { 1 } { p } \pmb { \theta } _ { 0 } ^ { \top } \Psi \pmb { \theta } _ { 0 } } \end{array}$ and the projection $\Phi ^ { \top } \pmb \theta _ { 0 }$ . Using the estimation (20) and the assumption that it exists $y ^ { \mu } = \pmb { \theta } _ { 0 } ^ { \top } \pmb { u } ^ { \mu }$ , one can write these quantities directly from the labels $y ^ { \mu }$ :
|
| 239 |
+
|
| 240 |
+
$$
|
| 241 |
+
\rho = \frac { 1 } { n _ { \mathrm { t o t } } } \sum _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } } \left( y ^ { \mu } \right) ^ { 2 } , \qquad \Phi ^ { \top } \pmb { \theta } _ { 0 } = \frac { 1 } { n _ { \mathrm { t o t } } } \sum _ { \mu = 1 } ^ { n _ { \mathrm { t o t } } } y ^ { \mu } \pmb { v } ^ { \mu } .
|
| 242 |
+
$$
|
| 243 |
+
|
| 244 |
+
For linear interpolating teachers, results are thus independent of the choice of the teacher.
|
| 245 |
+
|
| 246 |
+
Although this result might seen surprising at first sight, it is quite intuitive. Indeed, the information about the teacher model only enters the Gaussian covariate model (1) through the statistics of $\pmb { u } ^ { \top } \pmb { \theta } _ { 0 }$ . For a linear teacher $f _ { 0 } ( x ) = x$ , this is precisely given by the labels.
|
| 247 |
+
|
| 248 |
+
|
| 249 |
+
Figure 4: Test and training mean-squared errors eqs. (15) as a function of the number of samples $n$ for ridge regression. The Fashion-MNIST data set, with vanishing regularisation $\lambda = 1 0 ^ { - 5 }$ . In this plot, the student feature map $\varphi _ { s }$ is a 3-layer fully-connected neural network with $d \ : = \ : 2 3 5 2$ hidden neurons trained on the full data set with the square loss. Different curves correspond to the feature map obtained at different stages of training. Simulations are averaged over 10 independent runs. Further details on the simulations are described in Appendix D.1
|
| 250 |
+
|
| 251 |
+
# Ridge Regression with linear teachers —
|
| 252 |
+
|
| 253 |
+
We now test the prediction of model (1) on real data sets, and show that it is surprisingly effective in predicting the learning curves, at least for the ridge regression task. We have trained a 3-layer fully connected neural network with ReLU activations on the full Fashion-MNIST data set to distinguish clothing used above vs. below the waist [60]. The student feature map $\varphi _ { s } : \mathbb { R } ^ { 7 8 4 } \to \mathbb { R } ^ { \bar { d } }$ is obtained by removing the last layer, see Appendix D.1 for a detailed description. In Fig. 4 we show the test and training errors of the ridge estimator on a sub-sample of $n \ < \ n _ { \mathrm { t o t } }$ on the Fashion-MNIST images. We observe remarkable agreement between the learning curve obtained from simulations and the theoretical prediction by the matching Gaussian covariate model. Note that for the square loss and for $\lambda \ll 1$ , the worst performance peak is located at the point in which the linear system becomes invertible. Curiously, Fig. 4 shows that the fully-connected network progressively learns a low-rank representation of the data as training proceeds. This can be directly verified by counting the number of zero eigenvalues of $\Omega$ , which go from a full-rank matrix to a matrix of rank 380 after 200 epochs of training.
|
| 254 |
+
|
| 255 |
+
Fig. 1 (right) shows a similar experiment on the MNIST data set, but for different out-of-the-box feature maps, such as random features and the scattering transform [61], and we chose the number of random features $d = 1 9 5 3$ to match the number of features from the scattering transform. Note the characteristic double-descent behaviour [9, 25, 62], and the accurate prediction of the peak where the interpolation transition occurs. We note in Appendix D.1 that for both Figs. 4 and 1, for a number of samples $n$ closer to $n _ { \mathrm { t o t } }$ we start to see deviations between the real learning curve and the theory. This is to be expected since in the teacher-student framework the student can, in principle, express the same function as the teacher if it recovers its weights exactly. Recovering the teacher weights becomes possible with a large training set. In that case, its test error will be zero. However, in our setup the test error on real data remains finite even if more training data is added, leading to the discrepancy between teacher-student learning curve and real data, see Appendix D.1 for further discussion.
|
| 256 |
+
|
| 257 |
+
Why is the Gaussian model so effective for describing learning with data that are not Gaussian? The point is that ridge regression is sensitive only to second order statistics, and not to the full distribution of the data. It is a classical property (see Appendix E) that the training and generalisation errors are only a function of the spectrum of the empirical and population covariances, and of their products. Random matrix theory teaches us that such quantities are very robust, and their asymptotic behaviour is universal for a broad class of distributions of $[ u , v ]$ [49, 63–65]. The asymptotic behavior of kernel matrices has indeed been the subject of intense scrutiny [11, 47, 48, 50, 66, 67]. Indeed, a universality result akin to Theorem 3 was noted in [41] in the specific case of kernel methods. We thus expect the validity of model (1) for ridge regression, with a linear teacher, to go way beyond the Gaussian assumption.
|
| 258 |
+
|
| 259 |
+
Beyond ridge regression — The same strategy fails beyond ridge regression and mean-squared test error. This suggests a limit in the application of model (1) to real (non-Gaussian) data to the universal linear teacher. To illustrate this, consider the setting of Figs. 4, and compare the model predictions for the binary classification error instead of the $\ell _ { 2 }$ one. There is a clear mismatch between the simulated performance and prediction given by the theory (see Appendix D.1) due to the fact that the classification error does not depends only on the first two moments.
|
| 260 |
+
|
| 261 |
+
We present an additional experiment in Fig. 3. We compare the learning curves of logistic regression on a classification task on the real CIFAR10 images with the real labels versus the one on dcGANgenerated CIFAR10-like images and teacher generated labels from Sec. 3.3. While the Gaussian theory captures well the behaviour of the later, it fails on the former. A histogram of the distribution of the product $\mathbf { \Delta } u ^ { \top } \hat { \mathbf { \Gamma } } w$ for a fixed number of samples illustrates well the deviation from the prediction of the theory with the real case, in particular on the tails of the distribution. The difference between GAN generated data (that fits the Gaussian theory) and real data is clear. Given that for classification problems there exists a number of choices of "sign" teachers and feature maps that give the exact same labels as in the data set, an interesting open question is: is there a teacher that allows to reproduce the learning curves more accurately? This question is left for future works.
|
| 262 |
+
|
| 263 |
+
# Acknowledgements
|
| 264 |
+
|
| 265 |
+
We thank Romain Couillet, Cosme Louart, Loucas Pillaud-Vivien, Matthieu Wyart, Federica Gerace, Luca Saglietti and Yue Lu for discussions. We are grateful to Kabir Aladin Chandrasekher, Ashwin Pananjady and Christos Thrampoulidis for pointing out discrepancies in the finite size rates and insightful related discussions. We acknowledge funding from the ERC under the European Union’s Horizon 2020 Research and Innovation Programme Grant Agreement 714608-SMiLe, and from the French National Research Agency grants ANR-17-CE23-0023-01 PAIL.
|
| 266 |
+
|
| 267 |
+
References
|
| 268 |
+
[1] Hyunjune Sebastian Seung, Haim Sompolinsky, and Naftali Tishby. Statistical mechanics of learning from examples. Physical review A, 45(8):6056, 1992.
|
| 269 |
+
[2] Timothy LH Watkin, Albrecht Rau, and Michael Biehl. The statistical mechanics of learning a rule. Reviews of Modern Physics, 65(2):499, 1993.
|
| 270 |
+
[3] Andreas Engel and Christian Van den Broeck. Statistical mechanics of learning. Cambridge University Press, 2001.
|
| 271 |
+
[4] David L Donoho, Arian Maleki, and Andrea Montanari. Message-passing algorithms for compressed sensing. Proceedings of the National Academy of Sciences, 106(45):18914–18919, 2009.
|
| 272 |
+
[5] Noureddine El Karoui, Derek Bean, Peter J Bickel, Chinghway Lim, and Bin Yu. On robust regression with high-dimensional predictors. Proceedings of the National Academy of Sciences, 110(36):14557–14562, 2013.
|
| 273 |
+
[6] Lenka Zdeborová and Florent Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453–552, 2016.
|
| 274 |
+
[7] David Donoho and Andrea Montanari. High dimensional robust m-estimation: Asymptotic variance via approximate message passing. Probability Theory and Related Fields, 166(3- 4):935–969, 2016.
|
| 275 |
+
[8] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. Understanding deep learning requires rethinking generalization. In ICLR, 2017.
|
| 276 |
+
[9] Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machinelearning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116(32):15849–15854, 2019.
|
| 277 |
+
[10] Mikhail Belkin, Daniel Hsu, and Ji Xu. Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2(4):1167–1180, 2020.
|
| 278 |
+
[11] Song Mei and Andrea Montanari. The generalization error of random features regression: Precise asymptotics and double descent curve. arXiv preprint arXiv:1908.05355, 2019.
|
| 279 |
+
[12] Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J Tibshirani. Surprises in highdimensional ridgeless least squares interpolation. arXiv preprint arXiv:1903.08560, 2019.
|
| 280 |
+
[13] Emmanuel J Candès, Pragya Sur, et al. The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression. The Annals of Statistics, 48(1):27– 42, 2020.
|
| 281 |
+
[14] Benjamin Aubin, Florent Krzakala, Yue M Lu, and Lenka Zdeborová. Generalization error in high-dimensional perceptrons: Approaching bayes error with convex optimization. In Advances in Neural Information Processing Systems, volume 33, 2020.
|
| 282 |
+
[15] Fariborz Salehi, Ehsan Abbasi, and Babak Hassibi. The performance analysis of generalized margin maximizers on separable data. In International Conference on Machine Learning, pages 8417–8426. PMLR, 2020.
|
| 283 |
+
[16] Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. In Advances in neural information processing systems, pages 1177–1184, 2008.
|
| 284 |
+
[17] Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In Advances in neural information processing systems, pages 8571–8580, 2018.
|
| 285 |
+
[18] Mathieu Andreux, Tomás Angles, Georgios Exarchakis, Roberto Leonarduzzi, Gaspar Rochette, Louis Thiry, John Zarka, Stéphane Mallat, Joakim Andén, Eugene Belilovsky, et al. Kymatio: Scattering transforms in python. Journal of Machine Learning Research, 21(60):1–6, 2020.
|
| 286 |
+
[19] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks, 2014.
|
| 287 |
+
[20] Yehoram Gordon. Some inequalities for gaussian processes and applications. Israel Journal of Mathematics, 50(4):265–289, 1985.
|
| 288 |
+
[21] Marc Mézard, Giorgio Parisi, and Miguel Virasoro. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, volume 9. World Scientific Publishing Company, 1987.
|
| 289 |
+
[22] Marc Mézard and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009.
|
| 290 |
+
[23] Christopher K. I. Williams. Computing with infinite networks. In Proceedings of the 9th International Conference on Neural Information Processing Systems, NIPS’96, page 295–301, Cambridge, MA, USA, 1996. MIT Press.
|
| 291 |
+
[24] Elizabeth Gardner and Bernard Derrida. Three unfinished works on the optimal storage capacity of networks. Journal of Physics A: Mathematical and General, 22(12):1983, 1989.
|
| 292 |
+
[25] Manfred Opper and Wolfgang Kinzel. Statistical mechanics of generalization. In Models of neural networks III, pages 151–209. Springer, 1996.
|
| 293 |
+
[26] Behrooz Ghorbani, Song Mei, Theodor Misiakiewicz, and Andrea Montanari. When do neural networks outperform kernel methods? In Advances in Neural Information Processing Systems, volume 33, 2020.
|
| 294 |
+
[27] Christos Thrampoulidis, Ehsan Abbasi, and Babak Hassibi. Precise error analysis of regularized $m$ -estimators in high dimensions. IEEE Transactions on Information Theory, 64(8):5592–5628, 2018.
|
| 295 |
+
[28] Andrea Montanari, Feng Ruan, Youngtak Sohn, and Jun Yan. The generalization error of max-margin linear classifiers: High-dimensional asymptotics in the overparametrized regime. arXiv preprint arXiv:1911.01544, 2019.
|
| 296 |
+
[29] Michael Celentano, Andrea Montanari, and Yuting Wei. The lasso with general gaussian designs with applications to hypothesis testing. arXiv preprint arXiv:2007.13716, 2020.
|
| 297 |
+
[30] Mihailo Stojnic. A framework to characterize performance of lasso algorithms. arXiv preprint arXiv:1303.7291, 2013.
|
| 298 |
+
[31] Samet Oymak, Christos Thrampoulidis, and Babak Hassibi. The squared-error of generalized lasso: A precise analysis. In 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1002–1009. IEEE, 2013.
|
| 299 |
+
[32] Blake Bordelon, Abdulkadir Canatar, and Cengiz Pehlevan. Spectrum dependent learning curves in kernel regression and wide neural networks. In International Conference on Machine Learning, pages 1024–1034. PMLR, 2020.
|
| 300 |
+
[33] Hanwen Huang and Qinglong Yang. Large scale analysis of generalization error in learning using margin based classification methods. Journal of Statistical Mechanics: Theory and Experiment, 2020(10):103407, 2020.
|
| 301 |
+
[34] Partha P Mitra. Understanding overfitting peaks in generalization error: Analytical risk curves for $l _ { - 2 }$ and $l _ { - } 1$ penalized interpolation. arXiv preprint arXiv:1906.03667, 2019.
|
| 302 |
+
[35] Oussama Dhifallah and Yue M Lu. A precise performance analysis of learning with random features. arXiv preprint arXiv:2008.11904, 2020.
|
| 303 |
+
[36] Edgar Dobriban, Stefan Wager, et al. High-dimensional asymptotics of prediction: Ridge regression and classification. The Annals of Statistics, 46(1):247–279, 2018.
|
| 304 |
+
[37] Denny Wu and Ji Xu. On the optimal weighted $\ell _ { 2 }$ regularization in overparameterized linear regression. In Advances in Neural Information Processing Systems, volume 33, 2020.
|
| 305 |
+
[38] Zhenyu Liao, Romain Couillet, and Michael W Mahoney. A random matrix analysis of random fourier features: beyond the gaussian kernel, a precise phase transition, and the corresponding double descent. In Advances in Neural Information Processing Systems, volume 33, 2020.
|
| 306 |
+
[39] Fanghui Liu, Zhenyu Liao, and Johan AK Suykens. Kernel regression in high dimension: Refined analysis beyond double descent. arXiv preprint arXiv:2010.02681, 2020.
|
| 307 |
+
[40] Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117(48):30063–30070, 2020.
|
| 308 |
+
[41] Arthur Jacot, Berfin ¸Sim¸sek, Francesco Spadaro, Clément Hongler, and Franck Gabriel. Kernel alignment risk estimator: Risk prediction from training data. arXiv preprint arXiv:2006.09796, 2020.
|
| 309 |
+
[42] Cédric Gerbelot, Alia Abbara, and Florent Krzakala. Asymptotic errors for high-dimensional convex penalized linear regression beyond gaussian matrices. In Conference on Learning Theory, pages 1682–1713. PMLR, 2020.
|
| 310 |
+
[43] F. Gerace, B. Loureiro, F. Krzakala, M. Mézard, and L. Zdeborová. Generalisation error in learning with random features and the hidden manifold model. In 37th International Conference on Machine Learning, 2020.
|
| 311 |
+
[44] S. Goldt, M. Mézard, F. Krzakala, and L. Zdeborová. Modeling the influence of data structure on learning in neural networks: The hidden manifold model. Phys. Rev. X, 10(4):041044, 2020.
|
| 312 |
+
[45] Sebastian Goldt, Bruno Loureiro, Galen Reeves, Marc Mézard, Florent Krzakala, and Lenka Zdeborová. The gaussian equivalence of generative models for learning with two-layer neural networks. In Mathematical and Scientific Machine Learning, 2021.
|
| 313 |
+
[46] Hong Hu and Yue M Lu. Universality laws for high-dimensional learning with random features. arXiv preprint arXiv:2009.07669, 2020.
|
| 314 |
+
[47] Noureddine El Karoui et al. The spectrum of kernel random matrices. Annals of statistics, 38(1):1–50, 2010.
|
| 315 |
+
[48] Jeffrey Pennington and Pratik Worah. Nonlinear random matrix theory for deep learning. In Advances in Neural Information Processing Systems, volume 30, pages 2637–2646, 2017.
|
| 316 |
+
[49] Cosme Louart and Romain Couillet. Concentration of measure and large random matrices with an application to sample covariance matrices. arXiv preprint arXiv:1805.08295, 2018.
|
| 317 |
+
[50] Mohamed El Amine Seddik, Cosme Louart, Mohamed Tamaazousti, and Romain Couillet. Random matrix theory proves that deep learning representations of gan-data behave as gaussian mixtures. In International Conference on Machine Learning, pages 8573–8582. PMLR, 2020.
|
| 318 |
+
[51] Léo Miolane and Andrea Montanari. The distribution of the lasso: Uniform control over sparse balls and adaptive parameter tuning. arXiv preprint arXiv:1811.01212, 2018.
|
| 319 |
+
[52] Saharon Rosset, Ji Zhu, and Trevor Hastie. Margin maximizing loss functions. In NIPS, pages 1237–1244, 2003.
|
| 320 |
+
[53] B. Scholkopf and A.J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. Adaptive Computation and Machine Learning. MIT Press, 2018.
|
| 321 |
+
[54] Ingo Steinwart, Don R Hush, Clint Scovel, et al. Optimal rates for regularized least squares regression. In COLT, pages 79–93, 2009.
|
| 322 |
+
[55] Andrea Caponnetto and Ernesto De Vito. Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, 7(3):331–368, 2007.
|
| 323 |
+
[56] Loucas Pillaud-Vivien, Alessandro Rudi, and Francis Bach. Statistical optimality of stochastic gradient descent on hard learning problems through multiple passes. In Advances in Neural Information Processing Systems, volume 31, pages 8114–8124, 2018.
|
| 324 |
+
[57] Rainer Dietrich, Manfred Opper, and Haim Sompolinsky. Statistical mechanics of support vector networks. Phys. Rev. Lett., 82:2975–2978, Apr 1999.
|
| 325 |
+
[58] M. Opper and R. Urbanczik. Universal learning curves of support vector machines. Phys. Rev. Lett., 86:4410–4413, May 2001.
|
| 326 |
+
[59] A. Radford, L. Metz, and S. Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016.
|
| 327 |
+
[60] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017.
|
| 328 |
+
[61] J. Bruna and S. Mallat. Invariant scattering convolution networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1872–1886, 2013.
|
| 329 |
+
[62] Stefano Spigler, Mario Geiger, Stéphane d’Ascoli, Levent Sagun, Giulio Biroli, and Matthieu Wyart. A jamming transition from under-to over-parametrization affects generalization in deep learning. Journal of Physics A: Mathematical and Theoretical, 52(47):474001, 2019.
|
| 330 |
+
[63] Zhidong Bai and Wang Zhou. Large sample covariance matrices without independence structures in columns. Statistica Sinica, pages 425–442, 2008.
|
| 331 |
+
[64] Olivier Ledoit and Sandrine Péché. Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 151(1):233–264, 2011.
|
| 332 |
+
[65] Noureddine El Karoui et al. Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond. Annals of Applied Probability, 19(6):2362–2405, 2009.
|
| 333 |
+
[66] Xiuyuan Cheng and Amit Singer. The spectrum of random inner-product kernel matrices. Random Matrices: Theory and Applications, 2(04):1350010, 2013.
|
| 334 |
+
[67] Zhou Fan and Andrea Montanari. The spectral norm of random inner-product kernel matrices. Probability Theory and Related Fields, 173(1):27–85, 2019.
|
| 335 |
+
[68] Heinz H Bauschke, Patrick L Combettes, et al. Convex analysis and monotone operator theory in Hilbert spaces, volume 408. Springer, 2011.
|
| 336 |
+
[69] Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration inequalities: A nonasymptotic theory of independence. Oxford university press, 2013.
|
| 337 |
+
[70] Yanting Ma, Cynthia Rush, and Dror Baron. Analysis of approximate message passing with a class of non-separable denoisers. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 231–235. IEEE, 2017.
|
| 338 |
+
[71] Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Y. Eldar and G. Kutyniok., editors, Compressed Sensing, Theory and Applications. Cambridge University Press, 2012.
|
| 339 |
+
[72] Rick Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019.
|
| 340 |
+
[73] Per Kragh Andersen and Richard D Gill. Cox’s regression model for counting processes: a large sample study. The annals of statistics, pages 1100–1120, 1982.
|
| 341 |
+
[74] Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018.
|
| 342 |
+
[75] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. 2015 IEEE International Conference on Computer Vision (ICCV), pages 1026–1034, 2015.
|
| 343 |
+
[76] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference for Learning Representations, volume 3, 2015.
|
| 344 |
+
[77] Greg W Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices. Number 118. Cambridge university press, 2010.
|
| 345 |
+
[78] Vladimir Alexandrovich Marchenko and Leonid Andreevich Pastur. Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik, 114(4):507–536, 1967.
|
| 346 |
+
[79] Djalil Chafaï and Konstantin Tikhomirov. On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence. Probability Theory and Related Fields, 170(3):847–889, 2018.
|
| 347 |
+
[80] Walid Hachem, Philippe Loubaton, Jamal Najim, et al. Deterministic equivalents for certain functionals of large random matrices. The Annals of Applied Probability, 17(3):875–930, 2007.
|
parse/train/B1Kh0SVodY/B1Kh0SVodY_content_list.json
ADDED
|
@@ -0,0 +1,1375 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Learning curves of generic features maps for realistic datasets with a teacher-student model ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
222,
|
| 8 |
+
122,
|
| 9 |
+
779,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Bruno Loureiro IdePHICS, EPFL, Lausanne ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
205,
|
| 19 |
+
222,
|
| 20 |
+
392,
|
| 21 |
+
250
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Cédric Gerbelot Lab. de Physique de l’École Normale Supérieure, Paris ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
428,
|
| 30 |
+
222,
|
| 31 |
+
794,
|
| 32 |
+
252
|
| 33 |
+
],
|
| 34 |
+
"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Hugo Cui SPOC, EPFL, Lausanne ",
|
| 39 |
+
"bbox": [
|
| 40 |
+
218,
|
| 41 |
+
272,
|
| 42 |
+
377,
|
| 43 |
+
301
|
| 44 |
+
],
|
| 45 |
+
"page_idx": 0
|
| 46 |
+
},
|
| 47 |
+
{
|
| 48 |
+
"type": "text",
|
| 49 |
+
"text": "Sebastian Goldt SISSA, Trieste ",
|
| 50 |
+
"bbox": [
|
| 51 |
+
428,
|
| 52 |
+
272,
|
| 53 |
+
542,
|
| 54 |
+
301
|
| 55 |
+
],
|
| 56 |
+
"page_idx": 0
|
| 57 |
+
},
|
| 58 |
+
{
|
| 59 |
+
"type": "text",
|
| 60 |
+
"text": "Florent Krzakala IdePHICS, EPFL, Lausanne ",
|
| 61 |
+
"bbox": [
|
| 62 |
+
593,
|
| 63 |
+
272,
|
| 64 |
+
779,
|
| 65 |
+
301
|
| 66 |
+
],
|
| 67 |
+
"page_idx": 0
|
| 68 |
+
},
|
| 69 |
+
{
|
| 70 |
+
"type": "text",
|
| 71 |
+
"text": "Marc Mézard École Normale Supérieure, Paris ",
|
| 72 |
+
"bbox": [
|
| 73 |
+
263,
|
| 74 |
+
321,
|
| 75 |
+
478,
|
| 76 |
+
352
|
| 77 |
+
],
|
| 78 |
+
"page_idx": 0
|
| 79 |
+
},
|
| 80 |
+
{
|
| 81 |
+
"type": "text",
|
| 82 |
+
"text": "Lenka Zdeborová SPOC, EPFL, Lausanne ",
|
| 83 |
+
"bbox": [
|
| 84 |
+
575,
|
| 85 |
+
321,
|
| 86 |
+
735,
|
| 87 |
+
351
|
| 88 |
+
],
|
| 89 |
+
"page_idx": 0
|
| 90 |
+
},
|
| 91 |
+
{
|
| 92 |
+
"type": "text",
|
| 93 |
+
"text": "Abstract ",
|
| 94 |
+
"text_level": 1,
|
| 95 |
+
"bbox": [
|
| 96 |
+
462,
|
| 97 |
+
387,
|
| 98 |
+
535,
|
| 99 |
+
404
|
| 100 |
+
],
|
| 101 |
+
"page_idx": 0
|
| 102 |
+
},
|
| 103 |
+
{
|
| 104 |
+
"type": "text",
|
| 105 |
+
"text": "Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: first, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones - such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework. ",
|
| 106 |
+
"bbox": [
|
| 107 |
+
233,
|
| 108 |
+
420,
|
| 109 |
+
766,
|
| 110 |
+
641
|
| 111 |
+
],
|
| 112 |
+
"page_idx": 0
|
| 113 |
+
},
|
| 114 |
+
{
|
| 115 |
+
"type": "text",
|
| 116 |
+
"text": "1 Introduction ",
|
| 117 |
+
"text_level": 1,
|
| 118 |
+
"bbox": [
|
| 119 |
+
174,
|
| 120 |
+
667,
|
| 121 |
+
310,
|
| 122 |
+
684
|
| 123 |
+
],
|
| 124 |
+
"page_idx": 0
|
| 125 |
+
},
|
| 126 |
+
{
|
| 127 |
+
"type": "text",
|
| 128 |
+
"text": "Teacher-student models are a popular framework to study the high-dimensional asymptotic performance of learning problems with synthetic data, and have been the subject of intense investigations spanning three decades [1–7]. In the wake of understanding the limitations of classical statistical learning approaches [8–10], this direction is witnessing a renewal of interest [10–15]. However, this framework is often assuming the input data to be Gaussian i.i.d., which is arguably too simplistic to be able to capture properties of realistic data. In this paper, we redeem this line of work by defining a Gaussian covariate model where the teacher and student act on different Gaussian correlated spaces with arbitrary covariance. We derive a rigorous asymptotic solution of this model generalizing the formulas found in the above mentioned classical works. ",
|
| 129 |
+
"bbox": [
|
| 130 |
+
174,
|
| 131 |
+
699,
|
| 132 |
+
825,
|
| 133 |
+
824
|
| 134 |
+
],
|
| 135 |
+
"page_idx": 0
|
| 136 |
+
},
|
| 137 |
+
{
|
| 138 |
+
"type": "text",
|
| 139 |
+
"text": "We then put forward a theory, supported by universality arguments and numerical experiments, that this model captures learning curves, i.e. the dependence of the training and test errors on the number of samples, for a generic class of feature maps applied to realistic datasets. These maps can be deterministic, random, or even learnt from the data. This analysis thus gives a unified framework to describe the learning curves of, for example, kernel regression and classification, the analysis of feature maps – random projections [16], neural tangent kernels [17], scattering transforms [18] – as well as the analysis of transfer learning performance on data generated by generative adversarial networks [19]. We also discuss limits of applicability of our results, by showing concrete situations where the learning curves of the Gaussian covariate model differ from the actual ones. ",
|
| 140 |
+
"bbox": [
|
| 141 |
+
174,
|
| 142 |
+
830,
|
| 143 |
+
825,
|
| 144 |
+
900
|
| 145 |
+
],
|
| 146 |
+
"page_idx": 0
|
| 147 |
+
},
|
| 148 |
+
{
|
| 149 |
+
"type": "text",
|
| 150 |
+
"text": "",
|
| 151 |
+
"bbox": [
|
| 152 |
+
174,
|
| 153 |
+
90,
|
| 154 |
+
825,
|
| 155 |
+
147
|
| 156 |
+
],
|
| 157 |
+
"page_idx": 1
|
| 158 |
+
},
|
| 159 |
+
{
|
| 160 |
+
"type": "text",
|
| 161 |
+
"text": "Model definition — The Gaussian covariate teacher-student model is defined via two vectors $\\mathbf { \\boldsymbol { \\mathscr { u } } } \\in \\mathbb { R } ^ { p }$ and $\\pmb { v } \\in \\mathbb { R } ^ { d }$ , with correlation matrices $\\Psi \\in \\mathbb { R } ^ { p \\times p } , \\Omega \\in \\mathbb { R } ^ { d \\times d }$ and $\\Phi \\in \\mathbb { R } ^ { p \\times d }$ , from which we draw $n$ independent samples: ",
|
| 162 |
+
"bbox": [
|
| 163 |
+
173,
|
| 164 |
+
161,
|
| 165 |
+
825,
|
| 166 |
+
204
|
| 167 |
+
],
|
| 168 |
+
"page_idx": 1
|
| 169 |
+
},
|
| 170 |
+
{
|
| 171 |
+
"type": "equation",
|
| 172 |
+
"img_path": "images/a436876ad7d00c399dcf7ae17ba3f7669e3d7478cd2fcf3e14e3a202d924fc2e.jpg",
|
| 173 |
+
"text": "$$\n\\left[ { \\pmb u } ^ { \\mu } \\right] \\in \\mathbb { R } ^ { p + d } \\underset { \\mathrm { i . i . d . } } { \\sim } \\mathcal { N } \\left( 0 , \\left[ \\frac { \\Psi } { \\Phi ^ { \\top } } \\quad \\frac { \\Phi } { \\Omega } \\right] \\right) , \\qquad \\ \\mu = 1 , \\cdots , n .\n$$",
|
| 174 |
+
"text_format": "latex",
|
| 175 |
+
"bbox": [
|
| 176 |
+
271,
|
| 177 |
+
207,
|
| 178 |
+
725,
|
| 179 |
+
243
|
| 180 |
+
],
|
| 181 |
+
"page_idx": 1
|
| 182 |
+
},
|
| 183 |
+
{
|
| 184 |
+
"type": "text",
|
| 185 |
+
"text": "The labels $y ^ { \\mu }$ are generated by a teacher function that is only using the vectors $\\pmb { u } ^ { \\mu }$ : ",
|
| 186 |
+
"bbox": [
|
| 187 |
+
173,
|
| 188 |
+
246,
|
| 189 |
+
722,
|
| 190 |
+
261
|
| 191 |
+
],
|
| 192 |
+
"page_idx": 1
|
| 193 |
+
},
|
| 194 |
+
{
|
| 195 |
+
"type": "equation",
|
| 196 |
+
"img_path": "images/22fb70f92d2870befdf4e171564a385ac288ed674ff58e10e2bbfb65ead72494.jpg",
|
| 197 |
+
"text": "$$\ny ^ { \\mu } = f _ { 0 } \\left( \\frac { 1 } { \\sqrt { p } } \\pmb { \\theta } _ { 0 } ^ { \\top } { \\pmb u } ^ { \\mu } \\right) ,\n$$",
|
| 198 |
+
"text_format": "latex",
|
| 199 |
+
"bbox": [
|
| 200 |
+
418,
|
| 201 |
+
265,
|
| 202 |
+
578,
|
| 203 |
+
301
|
| 204 |
+
],
|
| 205 |
+
"page_idx": 1
|
| 206 |
+
},
|
| 207 |
+
{
|
| 208 |
+
"type": "text",
|
| 209 |
+
"text": "where $f _ { 0 } : \\mathbb { R } \\to \\mathbb { R }$ is a function that may include randomness such as, for instance, an additive Gaussian noise, and $\\pmb \\theta _ { 0 } \\in \\mathbb { R } ^ { p }$ is a vector of teacher-weights with finite norm which can be either random or deterministic. Learning is performed by the student with weights $\\pmb { w }$ via empirical risk minimization that has access only to the features $v ^ { \\mu }$ : ",
|
| 210 |
+
"bbox": [
|
| 211 |
+
173,
|
| 212 |
+
305,
|
| 213 |
+
825,
|
| 214 |
+
361
|
| 215 |
+
],
|
| 216 |
+
"page_idx": 1
|
| 217 |
+
},
|
| 218 |
+
{
|
| 219 |
+
"type": "equation",
|
| 220 |
+
"img_path": "images/e27c520ae58331875b92707c786e7ebb9d0972c5e5019688a1f34f39bcf83d2e.jpg",
|
| 221 |
+
"text": "$$\n\\pmb { \\hat { w } } = \\underset { \\pmb { w } \\in \\mathbb { R } ^ { d } } { \\arg \\operatorname* { m i n } } \\left[ \\sum _ { \\mu = 1 } ^ { n } g \\left( \\frac { \\pmb { w } ^ { \\top } \\pmb { v } ^ { \\mu } } { \\sqrt { d } } , y ^ { \\mu } \\right) + r ( \\pmb { w } ) \\right] ,\n$$",
|
| 222 |
+
"text_format": "latex",
|
| 223 |
+
"bbox": [
|
| 224 |
+
341,
|
| 225 |
+
366,
|
| 226 |
+
655,
|
| 227 |
+
409
|
| 228 |
+
],
|
| 229 |
+
"page_idx": 1
|
| 230 |
+
},
|
| 231 |
+
{
|
| 232 |
+
"type": "text",
|
| 233 |
+
"text": "where $r$ and $g$ are proper, convex, lower-semicontinuous functions of $\\pmb { w } \\in \\mathbb { R } ^ { d }$ (e.g. $g$ can be a logistic or a square loss and $r$ a $\\ell _ { p }$ $( p = 1 , 2 )$ regularization). The key quantities we want to compute in this model are the averaged training and generalisation errors for the estimator $\\pmb { w }$ , ",
|
| 234 |
+
"bbox": [
|
| 235 |
+
174,
|
| 236 |
+
415,
|
| 237 |
+
825,
|
| 238 |
+
458
|
| 239 |
+
],
|
| 240 |
+
"page_idx": 1
|
| 241 |
+
},
|
| 242 |
+
{
|
| 243 |
+
"type": "equation",
|
| 244 |
+
"img_path": "images/4f15fd883d3d510342d932a53a559956b244a937b57fd090d4f7f027633b977a.jpg",
|
| 245 |
+
"text": "$$\n\\begin{array} { r } { \\mathrm { ~ \\xi ~ } _ { \\mathrm { t r a i n . } } ( w ) \\equiv \\displaystyle \\frac { 1 } { n } \\sum _ { \\mu = 1 } ^ { n } g \\left( \\frac { w ^ { \\top } v ^ { \\mu } } { \\sqrt { d } } , y ^ { \\mu } \\right) \\quad \\mathrm { a n d } \\quad \\mathcal { E } _ { \\mathrm { g e n . } } ( w ) \\equiv \\mathbb { E } \\left[ \\hat { g } \\left( \\hat { f } \\left( \\frac { v _ { \\mathrm { n e w } } ^ { \\top } w } { \\sqrt { d } } \\right) , f _ { 0 } \\left( \\frac { u _ { \\mathrm { n e w } } ^ { \\top } \\theta _ { 0 } } { \\sqrt { p } } \\right) \\right) \\right] . } \\end{array}\n$$",
|
| 246 |
+
"text_format": "latex",
|
| 247 |
+
"bbox": [
|
| 248 |
+
181,
|
| 249 |
+
462,
|
| 250 |
+
825,
|
| 251 |
+
506
|
| 252 |
+
],
|
| 253 |
+
"page_idx": 1
|
| 254 |
+
},
|
| 255 |
+
{
|
| 256 |
+
"type": "text",
|
| 257 |
+
"text": "where $g$ is the loss function in eq. (3), $\\hat { f }$ is a prediction function (e.g. $\\hat { f } = \\mathrm { s i g n }$ for a classification task), $\\hat { g }$ is a performance measure (e.g. $\\hat { \\boldsymbol g } ( \\hat { y } , y ) = ( \\hat { y } - y ) ^ { 2 }$ for regression or $\\hat { g } ( \\hat { y } , y ) = \\mathbb { P } ( \\hat { y } \\neq y )$ for classification) and $( \\boldsymbol { \\mathbf { \\mathit { u } } } _ { \\mathrm { n e w } } , \\boldsymbol { \\mathbf { \\mathit { v } } } _ { \\mathrm { n e w } } )$ is a fresh sample from the joint distribution of $\\textbf { \\em u }$ and $\\textbf { { v } }$ . ",
|
| 258 |
+
"bbox": [
|
| 259 |
+
173,
|
| 260 |
+
525,
|
| 261 |
+
825,
|
| 262 |
+
570
|
| 263 |
+
],
|
| 264 |
+
"page_idx": 1
|
| 265 |
+
},
|
| 266 |
+
{
|
| 267 |
+
"type": "text",
|
| 268 |
+
"text": "Our two main technical contributions are: ",
|
| 269 |
+
"text_level": 1,
|
| 270 |
+
"bbox": [
|
| 271 |
+
176,
|
| 272 |
+
575,
|
| 273 |
+
460,
|
| 274 |
+
590
|
| 275 |
+
],
|
| 276 |
+
"page_idx": 1
|
| 277 |
+
},
|
| 278 |
+
{
|
| 279 |
+
"type": "text",
|
| 280 |
+
"text": "(C1) In Theorems 1 & 2, we give a rigorous closed-form characterisation of the properties of the estimator $\\hat { \\pmb { w } }$ for the Gaussian covariate model (1), and the corresponding training and generalisation errors in the high-dimensional limit. We prove our result using Gaussian comparison inequalities [20]; (C2) We show how the same expression can be obtained using the replica method from statistical physics [21]. This is of additional interest given the wide range of applications of the replica approach in machine learning and computer science [22]. In particular, this allows to put on a rigorous basis many results previously derived with the replica method. ",
|
| 281 |
+
"bbox": [
|
| 282 |
+
173,
|
| 283 |
+
593,
|
| 284 |
+
825,
|
| 285 |
+
690
|
| 286 |
+
],
|
| 287 |
+
"page_idx": 1
|
| 288 |
+
},
|
| 289 |
+
{
|
| 290 |
+
"type": "text",
|
| 291 |
+
"text": "Towards realistic data — In the second part of our paper, we argue that the above Gaussian covariate model (1) is generic enough to capture the learning behaviour of a broad range of realistic data. Let $\\{ { \\pmb x } ^ { \\mu } \\} _ { \\mu = 1 } ^ { n }$ denote a data set with $n$ independent samples on $\\boldsymbol { \\mathcal { X } } \\subset \\mathbb { R } ^ { D }$ . Based on this input, the features $_ { u , v }$ are given by (potentially) elaborated transformations of $_ { \\textbf { \\em x } }$ , i.e. ",
|
| 292 |
+
"bbox": [
|
| 293 |
+
173,
|
| 294 |
+
694,
|
| 295 |
+
825,
|
| 296 |
+
751
|
| 297 |
+
],
|
| 298 |
+
"page_idx": 1
|
| 299 |
+
},
|
| 300 |
+
{
|
| 301 |
+
"type": "equation",
|
| 302 |
+
"img_path": "images/fdd06a8045e83e2a82d39c31c78314b07dde097c8370000fa79bd998f17bf31e.jpg",
|
| 303 |
+
"text": "$$\n\\pmb { u } = \\varphi _ { t } ( \\pmb { x } ) \\in \\mathbb { R } ^ { p } \\quad \\mathrm { a n d } \\quad \\pmb { v } = \\varphi _ { s } ( \\pmb { x } ) \\in \\mathbb { R } ^ { d }\n$$",
|
| 304 |
+
"text_format": "latex",
|
| 305 |
+
"bbox": [
|
| 306 |
+
354,
|
| 307 |
+
755,
|
| 308 |
+
643,
|
| 309 |
+
773
|
| 310 |
+
],
|
| 311 |
+
"page_idx": 1
|
| 312 |
+
},
|
| 313 |
+
{
|
| 314 |
+
"type": "text",
|
| 315 |
+
"text": "for given centred feature maps $\\varphi _ { t } : \\mathcal { X } \\to \\mathbb { R } ^ { p }$ and $\\varphi _ { s } : \\mathcal { X } \\mathbb { R } ^ { d }$ , see Fig. 1. Uncentered features can be taken into account by shifting the covariances, but we focus on the centred case to lighten notation. ",
|
| 316 |
+
"bbox": [
|
| 317 |
+
171,
|
| 318 |
+
779,
|
| 319 |
+
825,
|
| 320 |
+
809
|
| 321 |
+
],
|
| 322 |
+
"page_idx": 1
|
| 323 |
+
},
|
| 324 |
+
{
|
| 325 |
+
"type": "text",
|
| 326 |
+
"text": "The Gaussian covariate model (1) is exact in the case where $_ { \\textbf { \\em x } }$ are Gaussian variables and the feature maps $( \\varphi _ { s } , \\varphi _ { s } )$ preserve the Gaussianity, for example linear features. In particular, this is the case for $\\pmb { u } = \\pmb { v } = \\pmb { x }$ , which is the widely-studied vanilla teacher-student model [24]. The interest of the model (1) is that it also captures a range of cases in which the feature maps $\\varphi _ { t }$ and $\\varphi _ { s }$ are deterministic, or even learnt from the data. The covariance matrices $\\Psi , \\Phi$ , and $\\Omega$ then represent different aspects of the data-generative process and learning model. The student (3) then corresponds to the last layer of the learning model. These observation can be distilled into the following conjecture: ",
|
| 327 |
+
"bbox": [
|
| 328 |
+
173,
|
| 329 |
+
814,
|
| 330 |
+
825,
|
| 331 |
+
912
|
| 332 |
+
],
|
| 333 |
+
"page_idx": 1
|
| 334 |
+
},
|
| 335 |
+
{
|
| 336 |
+
"type": "image",
|
| 337 |
+
"img_path": "images/3bd0bd1af9521b9c4b1e543e2b4cc154c52c16fb9df7f6367a872b05ea3bab26.jpg",
|
| 338 |
+
"image_caption": [
|
| 339 |
+
"Figure 1: Left: Given a data set $\\{ { \\pmb x } ^ { \\mu } \\} _ { \\mu = 1 } ^ { n }$ , teacher $\\pmb { u } = \\varphi _ { t } ( \\pmb { x } )$ and student maps $\\pmb { v } = \\varphi _ { t } ( \\pmb { x } )$ , we assume $[ { \\pmb u } , { \\pmb v } ]$ to be jointly Gaussian random variables and apply the results of the Gaussian covariate model (1). Right: Illustration on real data, here ridge regression on even vs odd MNIST digits, with regularisation $\\lambda { = } 1 0 ^ { - 2 }$ . Full line is theory, points are simulations. We show the performance with no feature map (blue), random feature map with $\\sigma = \\operatorname { e r f } \\delta$ & Gaussian projection (orange), the scattering transform with parameters $J = 3 , L = 8$ [18] (green), and of the limiting kernel of the random map [23] (red). The covariance $\\Omega$ is empirically estimated from the full data set, while the other quantities appearing in the Theorem 1 are expressed directly as a function of the labels, see Section 3.4. Simulations are averaged over 10 independent runs. "
|
| 340 |
+
],
|
| 341 |
+
"image_footnote": [],
|
| 342 |
+
"bbox": [
|
| 343 |
+
202,
|
| 344 |
+
88,
|
| 345 |
+
761,
|
| 346 |
+
258
|
| 347 |
+
],
|
| 348 |
+
"page_idx": 2
|
| 349 |
+
},
|
| 350 |
+
{
|
| 351 |
+
"type": "text",
|
| 352 |
+
"text": "Conjecture 1. (Gaussian equivalent model) For a wide class of data distributions $\\{ { \\pmb x } ^ { \\mu } \\} _ { \\mu = 1 } ^ { n }$ , and features maps ${ \\pmb u } = \\varphi _ { t } ( { \\pmb x } ) , { \\pmb v } = \\varphi _ { s } ( { \\pmb x } )$ , the generalisation and training errors of estimator (3) are asymptotically captured by the equivalent Gaussian model $( I ) _ { : }$ , where $[ { \\pmb u } , { \\pmb v } ]$ are jointly Gaussian variables, and thus by the closed-form expressions of Theorem 1. ",
|
| 353 |
+
"bbox": [
|
| 354 |
+
173,
|
| 355 |
+
393,
|
| 356 |
+
825,
|
| 357 |
+
452
|
| 358 |
+
],
|
| 359 |
+
"page_idx": 2
|
| 360 |
+
},
|
| 361 |
+
{
|
| 362 |
+
"type": "text",
|
| 363 |
+
"text": "The second part of our main contributions are: ",
|
| 364 |
+
"text_level": 1,
|
| 365 |
+
"bbox": [
|
| 366 |
+
174,
|
| 367 |
+
463,
|
| 368 |
+
486,
|
| 369 |
+
478
|
| 370 |
+
],
|
| 371 |
+
"page_idx": 2
|
| 372 |
+
},
|
| 373 |
+
{
|
| 374 |
+
"type": "text",
|
| 375 |
+
"text": "(C3) In Sec. 3.3 we show that the theoretical predictions from (C1) captures the learning curves in non-trivial cases, e.g. when input data are generated using a trained generative adversarial network, while extracting both the feature maps from a neural network trained on real data. ",
|
| 376 |
+
"bbox": [
|
| 377 |
+
176,
|
| 378 |
+
483,
|
| 379 |
+
823,
|
| 380 |
+
523
|
| 381 |
+
],
|
| 382 |
+
"page_idx": 2
|
| 383 |
+
},
|
| 384 |
+
{
|
| 385 |
+
"type": "text",
|
| 386 |
+
"text": "(C4) In Sec. 3.4, we show empirically that for ridge regression the asymptotic formula of Theorem 1 can be applied directly to real data sets, even though the Gaussian hypothesis is not satisfied. This universality-like property is a consequence of Theorem 3 and is illustrated in Fig. 1 (right) where the real learning curve of several features maps learning the odd-versus-even digit task on MNIST is compared to the theoretical prediction. ",
|
| 387 |
+
"bbox": [
|
| 388 |
+
174,
|
| 389 |
+
525,
|
| 390 |
+
825,
|
| 391 |
+
593
|
| 392 |
+
],
|
| 393 |
+
"page_idx": 2
|
| 394 |
+
},
|
| 395 |
+
{
|
| 396 |
+
"type": "text",
|
| 397 |
+
"text": "Related work — Rigorous results for teacher-student models: The Gaussian covariate model (1) contains the vanilla teacher-student model as a special case where one takes $\\textbf { \\em u }$ and $\\textbf { { v } }$ identical, with unique covariance matrix $\\Omega$ . This special case has been extensively studied in the statistical physics community using the heuristic replica method [1–3, 24, 25]. Many recent rigorous results for such models can be rederived as a special case of our formula, e.g. refs. [10–15, 26–29]. Numerous of these results are based on the same proof technique as we employed here: the Gordon’s Gaussian min-max inequalities [20, 30, 31]. The asymptotic analysis of kernel ridge regression [32], of margin-based classification [33] also follow from our theorem. See also Appendix A.6 for the details on these connections. Other examples include models of the double descent phenomenon [34]. Closer to our work is the recent work of [35] on the random feature model. For ridge regression, there are also precise predictions thanks to random matrix theory [12, 36–41]. A related set of results was obtained in [42] for orthogonal random matrix models. The main technical novelty of our proof is the handling of a generic loss and regularisation, not only ridge, representing convex empirical risk minimization, for both classification and regression, with the generic correlation structure of the model (1). ",
|
| 398 |
+
"bbox": [
|
| 399 |
+
173,
|
| 400 |
+
601,
|
| 401 |
+
825,
|
| 402 |
+
794
|
| 403 |
+
],
|
| 404 |
+
"page_idx": 2
|
| 405 |
+
},
|
| 406 |
+
{
|
| 407 |
+
"type": "text",
|
| 408 |
+
"text": "Gaussian equivalence: A similar Gaussian conjecture has been discussed in a series of recent works, and some authors proved partial results in this direction [11, 12, 28, 35, 43–46]. Ref. [45] analyses a special case of the Gaussian model (corresponding to $\\varphi _ { t } ~ = \\mathrm { i d }$ here), and proves a Gaussian equivalence theorem (GET) for feature maps $\\varphi _ { s }$ given by single-layer neural networks with fixed weights. They also show that for Gaussian data $\\pmb { x } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathrm { I } _ { D } )$ , feature maps of the form $\\pmb { v } = \\sigma ( \\mathbf { W } \\pmb { x } )$ (with some technical restriction on the weights) led to the jointly-Gaussian property for the two scalars $( \\pmb { v } \\cdot \\pmb { w } , \\pmb { u } \\cdot \\pmb { \\theta } _ { 0 } )$ for almost any vector $\\textbf { \\em w }$ . However, their stringent assumptions on random teacher weights limited the scope of applications to unrealistic label models. A related line of work discussed similar universality through the lens of random matrix theory [47–49]. In particular, Seddik et al. [50] showed that, in our notations, vectors $[ u , v ]$ obtained from Gaussian inputs $\\pmb { x } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathrm { I } _ { D } )$ with Lipschitz feature maps satisfy a concentration property. In this case, again, one can expect the two scalars $\\begin{array} { r } { ( \\pmb { v } \\cdot \\pmb { w } , \\pmb { u } \\cdot \\pmb { \\theta } _ { 0 } ) } \\end{array}$ to be jointly Gaussian with high-probability on $\\textbf { \\em w }$ . Remarkably, in the case of random feature maps, [46] could go beyond this central-limit-like behavior and established the universality of the Gaussian covariate model (1) for the actual learned weights $\\hat { \\pmb { w } }$ . ",
|
| 409 |
+
"bbox": [
|
| 410 |
+
174,
|
| 411 |
+
800,
|
| 412 |
+
825,
|
| 413 |
+
911
|
| 414 |
+
],
|
| 415 |
+
"page_idx": 2
|
| 416 |
+
},
|
| 417 |
+
{
|
| 418 |
+
"type": "text",
|
| 419 |
+
"text": "",
|
| 420 |
+
"bbox": [
|
| 421 |
+
173,
|
| 422 |
+
90,
|
| 423 |
+
826,
|
| 424 |
+
175
|
| 425 |
+
],
|
| 426 |
+
"page_idx": 3
|
| 427 |
+
},
|
| 428 |
+
{
|
| 429 |
+
"type": "text",
|
| 430 |
+
"text": "2 Main technical results ",
|
| 431 |
+
"text_level": 1,
|
| 432 |
+
"bbox": [
|
| 433 |
+
174,
|
| 434 |
+
193,
|
| 435 |
+
392,
|
| 436 |
+
210
|
| 437 |
+
],
|
| 438 |
+
"page_idx": 3
|
| 439 |
+
},
|
| 440 |
+
{
|
| 441 |
+
"type": "text",
|
| 442 |
+
"text": "Our main technical result is a closed-form expression for the asymptotic training and generalisation errors (4) of the Gaussian covariate model introduced above. We start by presenting our result in the most relevant setting for the applications of interest in Section 3, which is the case of the $\\ell _ { 2 }$ regularization. Next, we briefly present our result in larger generality, which includes non-asymptotic results for non-separable losses and regularizations. ",
|
| 443 |
+
"bbox": [
|
| 444 |
+
173,
|
| 445 |
+
224,
|
| 446 |
+
825,
|
| 447 |
+
295
|
| 448 |
+
],
|
| 449 |
+
"page_idx": 3
|
| 450 |
+
},
|
| 451 |
+
{
|
| 452 |
+
"type": "text",
|
| 453 |
+
"text": "We start by defining key quantities that we will use to characterize the estimator $\\hat { \\pmb { w } }$ . Let $\\Omega =$ $\\mathbf { S } ^ { \\top } \\mathrm { d i a g } ( \\omega _ { i } ) \\mathbf { \\bar { S } }$ be the spectral decomposition of $\\Omega$ . Let: ",
|
| 454 |
+
"bbox": [
|
| 455 |
+
173,
|
| 456 |
+
300,
|
| 457 |
+
821,
|
| 458 |
+
330
|
| 459 |
+
],
|
| 460 |
+
"page_idx": 3
|
| 461 |
+
},
|
| 462 |
+
{
|
| 463 |
+
"type": "equation",
|
| 464 |
+
"img_path": "images/2b02e9fd1d756ffe3de383e15e7638ef4545a1befd3653c350f11dac22014ed5.jpg",
|
| 465 |
+
"text": "$$\n\\rho \\equiv \\frac { 1 } { d } \\pmb { \\theta } _ { 0 } ^ { \\top } \\Psi \\pmb { \\theta } _ { 0 } \\in \\mathbb { R } , \\qquad \\quad \\bar { \\pmb { \\theta } } \\equiv \\frac { { \\bf S } \\Phi ^ { \\top } \\pmb { \\theta } _ { 0 } } { \\sqrt { \\rho } } \\in \\mathbb { R } ^ { d }\n$$",
|
| 466 |
+
"text_format": "latex",
|
| 467 |
+
"bbox": [
|
| 468 |
+
303,
|
| 469 |
+
334,
|
| 470 |
+
694,
|
| 471 |
+
372
|
| 472 |
+
],
|
| 473 |
+
"page_idx": 3
|
| 474 |
+
},
|
| 475 |
+
{
|
| 476 |
+
"type": "text",
|
| 477 |
+
"text": "and define the joint empirical density $\\hat { \\mu } _ { d }$ between $( \\omega _ { i } , \\bar { \\theta } _ { i } )$ : ",
|
| 478 |
+
"bbox": [
|
| 479 |
+
173,
|
| 480 |
+
377,
|
| 481 |
+
555,
|
| 482 |
+
393
|
| 483 |
+
],
|
| 484 |
+
"page_idx": 3
|
| 485 |
+
},
|
| 486 |
+
{
|
| 487 |
+
"type": "equation",
|
| 488 |
+
"img_path": "images/5c1bb19865603127057d9785e32ffd08740f65a3b1755fe20c292e787097b471.jpg",
|
| 489 |
+
"text": "$$\n\\hat { \\mu } _ { d } ( \\omega , \\bar { \\theta } ) \\equiv { \\frac { 1 } { d } } \\sum _ { i = 1 } ^ { d } \\delta ( \\omega - \\omega _ { i } ) \\delta ( \\bar { \\theta } - \\bar { \\theta } _ { i } ) .\n$$",
|
| 490 |
+
"text_format": "latex",
|
| 491 |
+
"bbox": [
|
| 492 |
+
370,
|
| 493 |
+
397,
|
| 494 |
+
627,
|
| 495 |
+
441
|
| 496 |
+
],
|
| 497 |
+
"page_idx": 3
|
| 498 |
+
},
|
| 499 |
+
{
|
| 500 |
+
"type": "text",
|
| 501 |
+
"text": "Note that $\\Phi ^ { \\top } \\pmb \\theta _ { 0 }$ is the projection of the teacher weights on the student space, and therefore $\\bar { \\pmb { \\theta } }$ is the rotated projection on the basis of the student covariance, rescaled by the teacher variance. Together with the student eigenvalues $\\omega _ { i }$ , these are relevant statistics of the model, encoded here in the joint distribution $\\hat { \\mu } _ { d }$ . ",
|
| 502 |
+
"bbox": [
|
| 503 |
+
173,
|
| 504 |
+
446,
|
| 505 |
+
825,
|
| 506 |
+
503
|
| 507 |
+
],
|
| 508 |
+
"page_idx": 3
|
| 509 |
+
},
|
| 510 |
+
{
|
| 511 |
+
"type": "text",
|
| 512 |
+
"text": "Assumptions — Consider the high-dimensional limit in which the number of samples $n$ and the dimensions $p , d \\mathrm { g o }$ to infinity with fixed ratios: ",
|
| 513 |
+
"bbox": [
|
| 514 |
+
173,
|
| 515 |
+
517,
|
| 516 |
+
823,
|
| 517 |
+
546
|
| 518 |
+
],
|
| 519 |
+
"page_idx": 3
|
| 520 |
+
},
|
| 521 |
+
{
|
| 522 |
+
"type": "equation",
|
| 523 |
+
"img_path": "images/d61ae675129e30cfa121fbccf9ee58a389cf9a6e0ea44eb6e5b7198a2f3ef477.jpg",
|
| 524 |
+
"text": "$$\n\\alpha \\equiv { \\frac { n } { d } } , \\mathrm { a n d } \\gamma \\equiv { \\frac { p } { d } } .\n$$",
|
| 525 |
+
"text_format": "latex",
|
| 526 |
+
"bbox": [
|
| 527 |
+
423,
|
| 528 |
+
550,
|
| 529 |
+
575,
|
| 530 |
+
578
|
| 531 |
+
],
|
| 532 |
+
"page_idx": 3
|
| 533 |
+
},
|
| 534 |
+
{
|
| 535 |
+
"type": "text",
|
| 536 |
+
"text": "Assume that the covariance matrices $\\Psi , \\Omega$ are positive-definite and that the Schur complement of the block covariance in equation (1) is positive semi-definite. Additionally, the spectral distributions of the matrices $\\Phi , \\Psi$ and $\\Omega$ converge to distributions such that the limiting joint distribution $\\mu$ is well-defined, and their maximum singular values are bounded with high probability as $n , p , d \\to \\infty$ . Finally, regularity assumptions are made on the loss and regularization functions mainly to ensure feasibility of the minimization problem. We assume that the cost function $r + g$ is coercive, i.e. $\\begin{array} { r } { \\operatorname* { l i m } _ { \\| \\pmb { w } \\| _ { 2 } + \\infty } ( r + g ) ( \\pmb { w } ) = + \\infty } \\end{array}$ and that the following scaling condition holds : for all $n , d \\in$ $\\mathbb { N } , z \\in \\mathbb { R } ^ { n }$ and any constant $c > 0$ , there exist a finite, positive constant $C$ , such that, for any standard normal random vectors $\\pmb { h } \\in \\mathbb { R } ^ { d }$ and $\\pmb { g } \\in \\mathbb { R } ^ { n }$ : ",
|
| 537 |
+
"bbox": [
|
| 538 |
+
173,
|
| 539 |
+
580,
|
| 540 |
+
826,
|
| 541 |
+
707
|
| 542 |
+
],
|
| 543 |
+
"page_idx": 3
|
| 544 |
+
},
|
| 545 |
+
{
|
| 546 |
+
"type": "equation",
|
| 547 |
+
"img_path": "images/d91ccee4c93aefab5fa9a06f0c93ce81c8cc38a910bf6cc2eb80d92241918b1c.jpg",
|
| 548 |
+
"text": "$$\n\\| z \\| _ { 2 } \\leqslant c \\sqrt { n } \\implies \\operatorname* { s u p } _ { x \\in \\partial g ( z ) } \\| x \\| _ { 2 } \\leqslant C \\sqrt { n } , \\qquad \\frac { 1 } { d } \\mathbb { E } \\left[ r ( h ) \\right] < + \\infty , \\qquad \\frac { 1 } { n } \\mathbb { E } \\left[ g ( g ) \\right] < + \\infty\n$$",
|
| 549 |
+
"text_format": "latex",
|
| 550 |
+
"bbox": [
|
| 551 |
+
200,
|
| 552 |
+
710,
|
| 553 |
+
779,
|
| 554 |
+
746
|
| 555 |
+
],
|
| 556 |
+
"page_idx": 3
|
| 557 |
+
},
|
| 558 |
+
{
|
| 559 |
+
"type": "text",
|
| 560 |
+
"text": "The relevance of these assumptions in a supervised machine learning context is discussed in Appendix B.1. We are now in a position to state our result. ",
|
| 561 |
+
"bbox": [
|
| 562 |
+
173,
|
| 563 |
+
750,
|
| 564 |
+
826,
|
| 565 |
+
779
|
| 566 |
+
],
|
| 567 |
+
"page_idx": 3
|
| 568 |
+
},
|
| 569 |
+
{
|
| 570 |
+
"type": "text",
|
| 571 |
+
"text": "Theorem 1. (Closed-form asymptotics for $\\ell _ { 2 }$ regularization) In the asymptotic limit defined above, the training and generalisation errors (4) of the estimator $\\hat { \\textbf { \\textit { w } } } \\in \\mathbb { R } ^ { \\dot { d } }$ solving the empirical risk minimisation problem in eq. (3) with $\\ell _ { 2 }$ regularization $\\begin{array} { r } { r ( \\pmb { w } ) = \\frac { \\lambda } { 2 } | | \\pmb { w } | | _ { 2 } ^ { 2 } } \\end{array}$ verify: ",
|
| 572 |
+
"bbox": [
|
| 573 |
+
173,
|
| 574 |
+
781,
|
| 575 |
+
825,
|
| 576 |
+
825
|
| 577 |
+
],
|
| 578 |
+
"page_idx": 3
|
| 579 |
+
},
|
| 580 |
+
{
|
| 581 |
+
"type": "equation",
|
| 582 |
+
"img_path": "images/366ad29212d5e8bccf0062cde00a6a384690a53eb455d919f5d8e876f0073aaa.jpg",
|
| 583 |
+
"text": "$$\n\\begin{array} { r l } & { \\mathcal { E } _ { \\mathrm { t r a i n . } } ( \\hat { w } ) \\xrightarrow [ d \\infty ] { P } \\mathbb { E } _ { s , h \\sim \\mathcal { N } ( 0 , 1 ) } [ g ( p r o x _ { V ^ { \\star } g ( \\cdot , f _ { 0 } ( \\sqrt { \\rho } s ) ) } ( \\frac { m ^ { \\star } } { \\sqrt { \\rho } } s + \\sqrt { q ^ { \\star } - \\frac { m ^ { \\star ^ { 2 } } } { \\rho } } h ) , f _ { 0 } ( \\sqrt { \\rho } s ) ) ] } \\\\ & { \\mathcal { E } _ { \\mathrm { g e n . } } ( \\hat { w } ) \\xrightarrow [ d \\infty ] { P } \\mathbb { E } _ { ( \\nu , \\lambda ) } [ \\hat { g } ( \\hat { f } ( \\lambda ) , f _ { 0 } ( \\nu ) ) ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( 1 6 ^ { d } ) } \\end{array}\n$$",
|
| 584 |
+
"text_format": "latex",
|
| 585 |
+
"bbox": [
|
| 586 |
+
181,
|
| 587 |
+
830,
|
| 588 |
+
812,
|
| 589 |
+
911
|
| 590 |
+
],
|
| 591 |
+
"page_idx": 3
|
| 592 |
+
},
|
| 593 |
+
{
|
| 594 |
+
"type": "text",
|
| 595 |
+
"text": "where prox stands for the proximal operator defined as ",
|
| 596 |
+
"bbox": [
|
| 597 |
+
176,
|
| 598 |
+
92,
|
| 599 |
+
534,
|
| 600 |
+
106
|
| 601 |
+
],
|
| 602 |
+
"page_idx": 4
|
| 603 |
+
},
|
| 604 |
+
{
|
| 605 |
+
"type": "equation",
|
| 606 |
+
"img_path": "images/8e66047afc976ba370ceca079999be4e47540b7356f09a6f6403487ad20d51ef.jpg",
|
| 607 |
+
"text": "$$\np r o x _ { V g ( . , y ) } ( x ) = \\arg \\operatorname* { m i n } _ { z } \\{ g ( z , y ) + \\frac { 1 } { 2 V } ( x - z ) ^ { 2 } \\}\n$$",
|
| 608 |
+
"text_format": "latex",
|
| 609 |
+
"bbox": [
|
| 610 |
+
328,
|
| 611 |
+
111,
|
| 612 |
+
666,
|
| 613 |
+
142
|
| 614 |
+
],
|
| 615 |
+
"page_idx": 4
|
| 616 |
+
},
|
| 617 |
+
{
|
| 618 |
+
"type": "text",
|
| 619 |
+
"text": "and where $( \\nu , \\lambda )$ are jointly Gaussian scalar variables: ",
|
| 620 |
+
"bbox": [
|
| 621 |
+
173,
|
| 622 |
+
148,
|
| 623 |
+
537,
|
| 624 |
+
164
|
| 625 |
+
],
|
| 626 |
+
"page_idx": 4
|
| 627 |
+
},
|
| 628 |
+
{
|
| 629 |
+
"type": "equation",
|
| 630 |
+
"img_path": "images/e10314f58c38a151ac13eb27a864b85a544c0e15015536a2aff0de4054a46669.jpg",
|
| 631 |
+
"text": "$$\n( \\nu , \\lambda ) \\sim \\mathcal { N } \\left( 0 , \\left[ \\rho \\begin{array} { c c } { \\rho } & { m ^ { \\star } } \\end{array} \\right] \\right) ,\n$$",
|
| 632 |
+
"text_format": "latex",
|
| 633 |
+
"bbox": [
|
| 634 |
+
397,
|
| 635 |
+
169,
|
| 636 |
+
601,
|
| 637 |
+
204
|
| 638 |
+
],
|
| 639 |
+
"page_idx": 4
|
| 640 |
+
},
|
| 641 |
+
{
|
| 642 |
+
"type": "text",
|
| 643 |
+
"text": "and the overlap parameters $( V ^ { \\star } , q ^ { \\star } , m ^ { \\star } )$ are prescribed by the unique fixed point of the following set of self-consistent equations: ",
|
| 644 |
+
"bbox": [
|
| 645 |
+
173,
|
| 646 |
+
209,
|
| 647 |
+
825,
|
| 648 |
+
238
|
| 649 |
+
],
|
| 650 |
+
"page_idx": 4
|
| 651 |
+
},
|
| 652 |
+
{
|
| 653 |
+
"type": "equation",
|
| 654 |
+
"img_path": "images/90328d4037f4f577a7eccfc05572ebeb2e0d64cbcbd751a0bc3126e1a0332d3a.jpg",
|
| 655 |
+
"text": "$$\n\\left\\{ \\begin{array} { l } { V = \\mathbb { E } _ { ( \\omega , \\bar { \\theta } ) \\sim \\mu } \\left[ \\frac { \\omega } { \\lambda + \\hat { V } \\omega } \\right] } \\\\ { m = \\frac { \\hat { m } } { \\sqrt { \\gamma } } \\mathbb { E } _ { ( \\omega , \\bar { \\theta } ) \\sim \\mu } \\left[ \\frac { \\theta } { \\lambda + \\hat { V } \\omega } \\right] } \\\\ { q = \\mathbb { E } _ { ( \\omega , \\bar { \\theta } ) \\sim \\mu } \\left[ \\frac { \\hat { m } ^ { 2 } \\bar { \\theta } ^ { 2 } \\omega + \\hat { q } \\omega ^ { 2 } } { \\left( \\lambda + \\hat { V } \\omega \\right) ^ { 2 } } \\right] } \\end{array} \\right. , \\left\\{ \\begin{array} { l } { \\hat { V } = \\frac { \\alpha } { V } ( 1 - \\mathbb { E } _ { s , h \\sim \\mathcal { N } ( 0 , 1 ) } [ f _ { g } ^ { \\prime } ( V , m , q ) ] ) } \\\\ { \\hat { m } = \\frac { 1 } { \\sqrt { \\rho \\gamma } } \\frac { \\alpha } { V } \\mathbb { E } _ { s , h \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ s f _ { g } ( V , m , q ) - \\frac { m } { \\sqrt { \\rho } } f _ { g } ^ { \\prime } ( V , m , q ) \\right] } \\\\ { \\hat { q } = \\frac { \\alpha } { V ^ { 2 } } \\mathbb { E } _ { s , h \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ \\left( \\frac { m } { \\sqrt { \\rho } } s + \\sqrt { q - \\frac { m ^ { 2 } } { \\rho } } h - f _ { g } ( V , m , q ) \\right) ^ { 2 } \\right] } \\end{array} \\right.\n$$",
|
| 656 |
+
"text_format": "latex",
|
| 657 |
+
"bbox": [
|
| 658 |
+
179,
|
| 659 |
+
242,
|
| 660 |
+
785,
|
| 661 |
+
323
|
| 662 |
+
],
|
| 663 |
+
"page_idx": 4
|
| 664 |
+
},
|
| 665 |
+
{
|
| 666 |
+
"type": "text",
|
| 667 |
+
"text": "where we defined the scalar random functions $f _ { g } ( V , m , q ) ~ = ~ p r o x _ { V g ( . , f _ { 0 } ( \\sqrt { \\rho } s ) ) } ( \\rho ^ { - 1 / 2 } m s ~ + ~$ $\\sqrt { q - \\rho ^ { - 1 } m ^ { 2 } } h )$ and $f _ { g } ^ { \\prime } ( V , m , h ) = p r o x _ { V g ( . , f _ { 0 } ( \\sqrt { \\rho } s ) ) } ^ { \\prime } ( \\rho ^ { - 1 / 2 } m s + \\sqrt { q - \\rho ^ { - 1 } m ^ { 2 } } h )$ as the first derivative of the proximal operator. ",
|
| 668 |
+
"bbox": [
|
| 669 |
+
173,
|
| 670 |
+
344,
|
| 671 |
+
828,
|
| 672 |
+
396
|
| 673 |
+
],
|
| 674 |
+
"page_idx": 4
|
| 675 |
+
},
|
| 676 |
+
{
|
| 677 |
+
"type": "text",
|
| 678 |
+
"text": "Proof : This result is a consequence of Theorem 2, whose proof can be found in appendix B. ",
|
| 679 |
+
"bbox": [
|
| 680 |
+
171,
|
| 681 |
+
405,
|
| 682 |
+
777,
|
| 683 |
+
421
|
| 684 |
+
],
|
| 685 |
+
"page_idx": 4
|
| 686 |
+
},
|
| 687 |
+
{
|
| 688 |
+
"type": "text",
|
| 689 |
+
"text": "The parameters of the model $( \\pmb \\theta _ { 0 } , \\Omega , \\Phi , \\Psi )$ only appear trough $\\rho$ , eq. (6), and the asymptotic limit $\\mu$ of the joint distribution eq. (7) and $( f _ { 0 } , \\hat { f } , g , \\lambda )$ . One can easily iterate the above equations to find their fixed point, and extract $( q ^ { * } , m ^ { * } )$ which appear in the expressions for the training and generalisation errors $( \\mathcal { E } _ { \\mathrm { t r a i n } } ^ { \\star } , \\mathcal { E } _ { \\mathrm { g e n } } ^ { \\star } )$ , see eq. (4). Note that $( q ^ { \\star } , m ^ { \\star } )$ have an intuitive interpretation in terms of the estimator $\\hat { \\pmb w } \\in \\mathbb { R } ^ { d }$ : ",
|
| 690 |
+
"bbox": [
|
| 691 |
+
173,
|
| 692 |
+
426,
|
| 693 |
+
826,
|
| 694 |
+
502
|
| 695 |
+
],
|
| 696 |
+
"page_idx": 4
|
| 697 |
+
},
|
| 698 |
+
{
|
| 699 |
+
"type": "equation",
|
| 700 |
+
"img_path": "images/4451419d6149d1ff503cefd8b3ae0ce14325307145a2245d973bf98f54bc10e5.jpg",
|
| 701 |
+
"text": "$$\n\\boldsymbol { q } ^ { \\star } \\equiv \\frac { 1 } { d } \\hat { \\boldsymbol { w } } ^ { \\top } \\boldsymbol { \\Omega } \\hat { \\boldsymbol { w } } , m ^ { \\star } \\equiv \\frac { 1 } { \\sqrt { d p } } \\boldsymbol { \\theta } _ { 0 } ^ { \\top } \\Phi \\hat { \\boldsymbol { w } }\n$$",
|
| 702 |
+
"text_format": "latex",
|
| 703 |
+
"bbox": [
|
| 704 |
+
312,
|
| 705 |
+
505,
|
| 706 |
+
686,
|
| 707 |
+
539
|
| 708 |
+
],
|
| 709 |
+
"page_idx": 4
|
| 710 |
+
},
|
| 711 |
+
{
|
| 712 |
+
"type": "text",
|
| 713 |
+
"text": "Or in words: $m ^ { \\star }$ is the correlation between the estimator projected in the teacher space, while $q ^ { \\star }$ is the reweighted norm of the estimator by the covariance $\\Omega$ . The parameter $V ^ { * }$ also has a concrete interpretation $:$ it parametrizes the deformation that must be applied to a Gaussian field specified by the solution of the fixed point equations to obtain the asymptotic behaviour of $\\hat { \\mathbf { z } }$ . It prescribes the degree of non-linearity given to the linear output by the chosen loss function. This is coherent with the robust regression viewpoint, where one introduces non-square losses to deal with the potential non-linearity of the generative model. $\\hat { V } ^ { * }$ plays a similar role for the estimator wˆ through the proximal operator of the regularisation, see Theorem 4 and 5 in the Appendix. Two cases are of particular relevance for the experiments that follow. The first is the case of ridge regression, in which $f _ { 0 } ( x ) = { \\hat { f } } ( x )$ and both the loss $g$ and the performance measure $\\hat { g }$ are taken to be the meansquared error $\\mathrm { m s e } ( y , \\hat { y } ) = \\textstyle { \\frac { 1 } { 2 } } ( y - \\hat { y } ) ^ { 2 }$ , and the asymptotic errors are given by the simple closed-form expression: ",
|
| 714 |
+
"bbox": [
|
| 715 |
+
173,
|
| 716 |
+
542,
|
| 717 |
+
826,
|
| 718 |
+
714
|
| 719 |
+
],
|
| 720 |
+
"page_idx": 4
|
| 721 |
+
},
|
| 722 |
+
{
|
| 723 |
+
"type": "equation",
|
| 724 |
+
"img_path": "images/4434f3a83862d80a7ecc0394c6598969a4ba8f64d5f659c12a633fa79878219a.jpg",
|
| 725 |
+
"text": "$$\n\\mathcal { E } _ { \\mathrm { g e n } } ^ { \\star } = \\rho + q ^ { \\star } - 2 m ^ { \\star } , \\qquad \\mathcal { E } _ { \\mathrm { t r a i n } } ^ { \\star } = \\frac { \\mathcal { E } _ { \\mathrm { g e n } } ^ { \\star } } { ( 1 + V ^ { \\star } ) ^ { 2 } } ,\n$$",
|
| 726 |
+
"text_format": "latex",
|
| 727 |
+
"bbox": [
|
| 728 |
+
295,
|
| 729 |
+
718,
|
| 730 |
+
700,
|
| 731 |
+
753
|
| 732 |
+
],
|
| 733 |
+
"page_idx": 4
|
| 734 |
+
},
|
| 735 |
+
{
|
| 736 |
+
"type": "text",
|
| 737 |
+
"text": "The second case of interest is the one of a binary classification task, for which $f _ { 0 } ( x ) = { \\hat { f } } ( x ) =$ $\\mathrm { s i g n } ( x )$ , and we choose the performance measure to be the classification error $\\hat { g } ( y , \\hat { y } ) = \\mathbb { P } ( y \\neq \\hat { y } )$ . In the same notation as before, the asymptotic generalisation error in this case reads: ",
|
| 738 |
+
"bbox": [
|
| 739 |
+
173,
|
| 740 |
+
761,
|
| 741 |
+
825,
|
| 742 |
+
804
|
| 743 |
+
],
|
| 744 |
+
"page_idx": 4
|
| 745 |
+
},
|
| 746 |
+
{
|
| 747 |
+
"type": "equation",
|
| 748 |
+
"img_path": "images/82ec30892366c57adcaaf69cac6f80977aec0ede4f8af2a28d65808ee4cc1f1e.jpg",
|
| 749 |
+
"text": "$$\n\\mathcal { E } _ { \\mathrm { g e n } } ^ { \\star } = \\frac { 1 } { \\pi } \\cos ^ { - 1 } \\left( \\frac { m ^ { \\star } } { \\sqrt { \\rho q ^ { \\star } } } \\right) ,\n$$",
|
| 750 |
+
"text_format": "latex",
|
| 751 |
+
"bbox": [
|
| 752 |
+
406,
|
| 753 |
+
809,
|
| 754 |
+
591,
|
| 755 |
+
844
|
| 756 |
+
],
|
| 757 |
+
"page_idx": 4
|
| 758 |
+
},
|
| 759 |
+
{
|
| 760 |
+
"type": "text",
|
| 761 |
+
"text": "$\\mathcal { E } _ { \\mathrm { t r a i n } } ^ { \\star }$ depends on the choice of ll of the binary classificat $g$ - which we will take to be the logistic lossn experiments. $g ( y , x ) = \\log { ( 1 + e ^ { - x y } ) }$ ",
|
| 762 |
+
"bbox": [
|
| 763 |
+
174,
|
| 764 |
+
848,
|
| 765 |
+
823,
|
| 766 |
+
877
|
| 767 |
+
],
|
| 768 |
+
"page_idx": 4
|
| 769 |
+
},
|
| 770 |
+
{
|
| 771 |
+
"type": "text",
|
| 772 |
+
"text": "As mentioned above, this paper includes stronger technical results including finite size corrections and precise characterization of the distribution of the estimator $\\hat { \\pmb { w } }$ , for generic, non-separable loss and regularization $g$ and $r$ . This type of distributional statement is encountered for special cases of the model in related works such as [28, 29, 51]. Define $\\boldsymbol { \\mathcal { V } } \\in \\mathbb { R } ^ { n \\times d }$ as the matrix of concatenated samples used by the student. Informally, in high-dimension, the estimator $\\hat { \\pmb w }$ and $\\begin{array} { r } { \\hat { z } = \\frac { 1 } { \\sqrt { d } } \\mathcal { V } \\hat { w } } \\end{array}$ roughly behave as non-linear transforms of Gaussian random variables centered around the teacher vector $\\theta _ { 0 }$ (or its projection on the covariance spaces) as follows: ",
|
| 773 |
+
"bbox": [
|
| 774 |
+
173,
|
| 775 |
+
882,
|
| 776 |
+
821,
|
| 777 |
+
912
|
| 778 |
+
],
|
| 779 |
+
"page_idx": 4
|
| 780 |
+
},
|
| 781 |
+
{
|
| 782 |
+
"type": "text",
|
| 783 |
+
"text": "",
|
| 784 |
+
"bbox": [
|
| 785 |
+
174,
|
| 786 |
+
90,
|
| 787 |
+
825,
|
| 788 |
+
165
|
| 789 |
+
],
|
| 790 |
+
"page_idx": 5
|
| 791 |
+
},
|
| 792 |
+
{
|
| 793 |
+
"type": "equation",
|
| 794 |
+
"img_path": "images/235bc2cca111450ff663d4cf8c733523585ef192af3887addb777f90ddf71e19.jpg",
|
| 795 |
+
"text": "$$\n{ \\boldsymbol w } ^ { * } = \\Omega ^ { - 1 / 2 } \\frac { \\mathrm { p r o x } } { \\bar { \\nu } ^ { * } r ( \\Omega ^ { - 1 / 2 } . ) } \\left( \\frac { 1 } { \\hat { V } ^ { * } } ( \\hat { m } ^ { * } t + \\sqrt { \\hat { q } ^ { * } } g ) \\right) , \\boldsymbol z ^ { * } = \\operatorname * { p r o x } _ { V ^ { * } g ( . , \\boldsymbol z ) } \\left( \\frac { m ^ { * } } { \\sqrt { \\rho } } \\boldsymbol s + \\sqrt { \\boldsymbol q ^ { * } - \\frac { ( m ^ { * } ) ^ { 2 } } { \\rho } } \\boldsymbol h \\right) .\n$$",
|
| 796 |
+
"text_format": "latex",
|
| 797 |
+
"bbox": [
|
| 798 |
+
187,
|
| 799 |
+
166,
|
| 800 |
+
807,
|
| 801 |
+
210
|
| 802 |
+
],
|
| 803 |
+
"page_idx": 5
|
| 804 |
+
},
|
| 805 |
+
{
|
| 806 |
+
"type": "text",
|
| 807 |
+
"text": "where $s , h \\sim \\mathcal { N } ( 0 , \\mathrm { I } _ { n } )$ and $\\mathbf { \\mathscr { g } } \\sim \\mathcal { N } ( 0 , \\mathrm { I } _ { d } )$ are random vectors independent of the other quantities, $t =$ $\\Omega ^ { - 1 / 2 } \\Phi ^ { \\top } \\pmb { \\theta } _ { 0 }$ , $\\pmb { y } = \\pmb { f } _ { 0 } \\left( \\sqrt { \\rho } \\pmb { s } \\right)$ , and $( V ^ { * } , \\hat { V } ^ { * } , q ^ { * } , \\hat { q } ^ { * } , m ^ { * } , \\hat { m } ^ { * } )$ is the unique solution to the fixed point equations presented in Lemma 12 of appendix B. Those fixed point equations are the generalization of (13) to generic, non-separable loss function and regularization. The formal concentration of measure result can then be stated in the following way: ",
|
| 808 |
+
"bbox": [
|
| 809 |
+
173,
|
| 810 |
+
210,
|
| 811 |
+
825,
|
| 812 |
+
285
|
| 813 |
+
],
|
| 814 |
+
"page_idx": 5
|
| 815 |
+
},
|
| 816 |
+
{
|
| 817 |
+
"type": "text",
|
| 818 |
+
"text": "Theorem 2. (Non-asymptotic version, generic loss and regularization) Under Assumption $_ { ( B . I ) }$ , consider any optimal solution $\\hat { \\pmb { w } }$ to 3. Then, there exist constants $C , c , c ^ { \\prime } > 0$ such that, for any Lipschitz function $\\phi _ { 1 } : \\mathbb { R } ^ { d } \\mathbb { R }$ , and separable, pseudo-Lipschitz function $\\phi _ { 2 } : \\mathbb { R } ^ { n } \\to \\mathbb { R }$ and any $0 < \\epsilon < c ^ { \\prime }$ : ",
|
| 819 |
+
"bbox": [
|
| 820 |
+
173,
|
| 821 |
+
286,
|
| 822 |
+
825,
|
| 823 |
+
342
|
| 824 |
+
],
|
| 825 |
+
"page_idx": 5
|
| 826 |
+
},
|
| 827 |
+
{
|
| 828 |
+
"type": "equation",
|
| 829 |
+
"img_path": "images/27ce51ccbd86e6c0f512f69532a01a72dac1b4bae852c65377e6f2fc78addc65.jpg",
|
| 830 |
+
"text": "$$\n\\mathbb { P } \\left( \\left| \\phi _ { 1 } \\left( \\frac { \\hat { w } } { \\sqrt { d } } \\right) - \\mathbb { E } \\phi _ { 1 } \\left( \\frac { w ^ { * } } { \\sqrt { d } } \\right) \\right| \\geqslant \\epsilon \\right) \\leqslant \\frac { C } { \\epsilon ^ { 2 } } e ^ { - c n \\epsilon ^ { 4 } } , \\mathbb { P } \\left( \\left| \\phi _ { 2 } \\left( \\frac { \\hat { z } } { \\sqrt { n } } \\right) - \\mathbb { E } \\phi _ { 2 } \\left( \\frac { z ^ { * } } { \\sqrt { n } } \\right) \\right| \\geqslant \\epsilon \\right) \\leqslant \\frac { C } { \\epsilon ^ { 2 } } e ^ { - c n \\epsilon ^ { 4 } } .\n$$",
|
| 831 |
+
"text_format": "latex",
|
| 832 |
+
"bbox": [
|
| 833 |
+
181,
|
| 834 |
+
342,
|
| 835 |
+
830,
|
| 836 |
+
377
|
| 837 |
+
],
|
| 838 |
+
"page_idx": 5
|
| 839 |
+
},
|
| 840 |
+
{
|
| 841 |
+
"type": "text",
|
| 842 |
+
"text": "Note that in this form, the dimensions $n , p , d$ still appear explicitly, as we are characterizing the convergence of the estimator’s distribution for large but finite dimension. The clearer, one-dimensional statements are recovered by taking the $n , p , d \\infty$ limit with separable functions and an $\\ell _ { 2 }$ regularization. Other simplified formulas can also be obtained from our general result in the case of an $\\ell _ { 1 }$ penalty, but since this breaks rotational invariance, they do look more involved than the $\\ell _ { 2 }$ case. From Theorem 2, one can deduce the expressions of a number of observables, represented by the test functions $\\phi _ { 1 } , \\phi _ { 2 }$ , characterizing the performance of $\\hat { \\pmb { w } }$ , for instance the training and generalization error. A more detailed statement, along with the proof, is given in appendix B. ",
|
| 843 |
+
"bbox": [
|
| 844 |
+
173,
|
| 845 |
+
383,
|
| 846 |
+
826,
|
| 847 |
+
497
|
| 848 |
+
],
|
| 849 |
+
"page_idx": 5
|
| 850 |
+
},
|
| 851 |
+
{
|
| 852 |
+
"type": "text",
|
| 853 |
+
"text": "3 Applications of the Gaussian model ",
|
| 854 |
+
"text_level": 1,
|
| 855 |
+
"bbox": [
|
| 856 |
+
173,
|
| 857 |
+
513,
|
| 858 |
+
501,
|
| 859 |
+
531
|
| 860 |
+
],
|
| 861 |
+
"page_idx": 5
|
| 862 |
+
},
|
| 863 |
+
{
|
| 864 |
+
"type": "text",
|
| 865 |
+
"text": "We now discuss how the theorems above are applied to characterise the learning curves for a range of concrete cases. We present a number of cases – some rather surprising – for which Conjecture 1 seems valid, and point out some where it is not. An out-of-the-box iterator for all the cases studied hereafter is provided in the GitHub repository for this manuscript at https: //github.com/IdePHICS/GCMProject. ",
|
| 866 |
+
"bbox": [
|
| 867 |
+
173,
|
| 868 |
+
544,
|
| 869 |
+
826,
|
| 870 |
+
614
|
| 871 |
+
],
|
| 872 |
+
"page_idx": 5
|
| 873 |
+
},
|
| 874 |
+
{
|
| 875 |
+
"type": "text",
|
| 876 |
+
"text": "3.1 Random kitchen sink with Gaussian data ",
|
| 877 |
+
"text_level": 1,
|
| 878 |
+
"bbox": [
|
| 879 |
+
174,
|
| 880 |
+
630,
|
| 881 |
+
500,
|
| 882 |
+
645
|
| 883 |
+
],
|
| 884 |
+
"page_idx": 5
|
| 885 |
+
},
|
| 886 |
+
{
|
| 887 |
+
"type": "text",
|
| 888 |
+
"text": "If we choose random feature maps $\\varphi _ { s } ( \\pmb { x } ) = \\sigma \\left( \\mathrm { F } \\pmb { x } \\right)$ for a random matrix $\\mathrm { F }$ and a chosen scalar function $\\sigma$ acting component-wise, we obtain the random kitchen sink model [16]. This model has seen a surge of interest recently, and a sharp asymptotic analysis was provided in the particular case of uncorrelated Gaussian data $\\pmb { x } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\mathrm { I } } _ { D } )$ and $\\varphi _ { t } ( \\pmb { x } ) = \\pmb { x }$ in [11, 12] for ridge regression and generalised by [43, 46] for generic convex losses. Both results can be framed as a Gaussian covariate model with: ",
|
| 889 |
+
"bbox": [
|
| 890 |
+
173,
|
| 891 |
+
654,
|
| 892 |
+
825,
|
| 893 |
+
738
|
| 894 |
+
],
|
| 895 |
+
"page_idx": 5
|
| 896 |
+
},
|
| 897 |
+
{
|
| 898 |
+
"type": "equation",
|
| 899 |
+
"img_path": "images/f930377a3855f81ffbc31d91549af556a3f761a80b7253d7ae840ea10dac685c.jpg",
|
| 900 |
+
"text": "$$\n\\Psi = { \\bf I } _ { p } , \\Phi = \\kappa _ { 1 } \\mathrm { \\bf F } ^ { \\top } , \\Omega = \\kappa _ { 0 } ^ { 2 } { \\bf 1 } _ { d } { \\bf 1 } _ { d } ^ { \\top } + \\kappa _ { 1 } ^ { 2 } \\frac { \\mathrm { \\bf F } \\mathrm { \\bf F } ^ { \\top } } { d } + \\kappa _ { \\star } ^ { 2 } { \\bf I } _ { d } ,\n$$",
|
| 901 |
+
"text_format": "latex",
|
| 902 |
+
"bbox": [
|
| 903 |
+
250,
|
| 904 |
+
737,
|
| 905 |
+
746,
|
| 906 |
+
770
|
| 907 |
+
],
|
| 908 |
+
"page_idx": 5
|
| 909 |
+
},
|
| 910 |
+
{
|
| 911 |
+
"type": "text",
|
| 912 |
+
"text": "where $\\mathbf { 1 } _ { d } \\in \\mathbb { R } ^ { d }$ is the all-one vector and the constants $( \\kappa _ { 0 } , \\kappa _ { 1 } , \\kappa _ { \\star } )$ are related to the non-linearity $\\sigma$ ",
|
| 913 |
+
"bbox": [
|
| 914 |
+
174,
|
| 915 |
+
773,
|
| 916 |
+
825,
|
| 917 |
+
789
|
| 918 |
+
],
|
| 919 |
+
"page_idx": 5
|
| 920 |
+
},
|
| 921 |
+
{
|
| 922 |
+
"type": "equation",
|
| 923 |
+
"img_path": "images/00682252ac67909e95a69519e38a5b2827ed5ca9ff0e2f2e6c72381e2e6ad2c3.jpg",
|
| 924 |
+
"text": "$$\n\\begin{array} { r } { \\kappa _ { 0 } = \\mathbb { E } _ { z \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ \\sigma ( z ) \\right] , \\quad \\kappa _ { 1 } = \\mathbb { E } _ { z \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ z \\sigma ( z ) \\right] , \\quad \\kappa _ { \\star } = \\sqrt { \\mathbb { E } _ { z \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ \\sigma ( z ) ^ { 2 } \\right] - \\kappa _ { 0 } ^ { 2 } - \\kappa _ { 1 } ^ { 2 } } . } \\end{array}\n$$",
|
| 925 |
+
"text_format": "latex",
|
| 926 |
+
"bbox": [
|
| 927 |
+
184,
|
| 928 |
+
791,
|
| 929 |
+
787,
|
| 930 |
+
818
|
| 931 |
+
],
|
| 932 |
+
"page_idx": 5
|
| 933 |
+
},
|
| 934 |
+
{
|
| 935 |
+
"type": "text",
|
| 936 |
+
"text": "In this case, the averages over $\\mu$ in eq. (13) can be directly expressed in terms of the Stieltjes transform associated with the spectral density of $\\mathrm { F F } ^ { \\top }$ . Note, however, that our present framework can accommodate more involved random sinks models, such as when the teacher features are also a random feature model or multi-layer random architectures. ",
|
| 937 |
+
"bbox": [
|
| 938 |
+
174,
|
| 939 |
+
819,
|
| 940 |
+
825,
|
| 941 |
+
877
|
| 942 |
+
],
|
| 943 |
+
"page_idx": 5
|
| 944 |
+
},
|
| 945 |
+
{
|
| 946 |
+
"type": "text",
|
| 947 |
+
"text": "3.2 Kernel methods with Gaussian data ",
|
| 948 |
+
"text_level": 1,
|
| 949 |
+
"bbox": [
|
| 950 |
+
174,
|
| 951 |
+
892,
|
| 952 |
+
460,
|
| 953 |
+
907
|
| 954 |
+
],
|
| 955 |
+
"page_idx": 5
|
| 956 |
+
},
|
| 957 |
+
{
|
| 958 |
+
"type": "image",
|
| 959 |
+
"img_path": "images/fda2f723c2bfabdd64bd29334132dc590cd64755980f0db719efb7f75659d7e9.jpg",
|
| 960 |
+
"image_caption": [
|
| 961 |
+
"Figure 2: Learning in kernel space: Teacher and student live in the same (Hilbert) feature space $\\pmb { v } = \\pmb { u } \\in \\mathbb { R } ^ { d }$ with $d \\gg n$ , and the performance only depends on the relative decay between the student spectrum $\\omega _ { i } = d i ^ { - 2 }$ (the capacity) and the teacher weights in feature space $\\theta _ { 0 i } ^ { 2 } \\bar { \\omega _ { i } } = \\dot { d } i ^ { - a }$ (the source). Top: a task with sign teacher (in kernel space), fitted with a max-margin support vector machine (logistic regression with vanishing regularisation [52]). Bottom: a task with linear teacher (in kernel space) fitted via kernel ridge regression with vanishing regularisation. Points are simulation that matches the theory (lines). Simulations are averaged over 10 independent runs. "
|
| 962 |
+
],
|
| 963 |
+
"image_footnote": [],
|
| 964 |
+
"bbox": [
|
| 965 |
+
202,
|
| 966 |
+
95,
|
| 967 |
+
486,
|
| 968 |
+
315
|
| 969 |
+
],
|
| 970 |
+
"page_idx": 6
|
| 971 |
+
},
|
| 972 |
+
{
|
| 973 |
+
"type": "text",
|
| 974 |
+
"text": "Another direct application of our formalism is to kernel methods. Kernel methods admit a dual representation in terms of optimization over feature space [53]. The connection is given by Mercer’s theorem, which provides an eigen-decomposition of the kernel and of the target function in the feature basis, effectively mapping kernel regression to a teacherstudent problem on feature space. The classical way of studying the performance of kernel methods [54, 55] is then to directly analyse the performance of convex learning in this space. In our notation, the teacher and student feature maps are equal, and we thus set $p = d , \\Psi = \\Phi = \\Omega = \\mathrm { d i a g } ( \\omega _ { \\mathrm { i } } )$ where $\\omega _ { i }$ are the eigenvalues of the kernel and we take the teacher weights $\\pmb { \\theta } _ { 0 }$ to be the decomposition of the target function in the kernel feature basis. ",
|
| 975 |
+
"bbox": [
|
| 976 |
+
527,
|
| 977 |
+
90,
|
| 978 |
+
825,
|
| 979 |
+
340
|
| 980 |
+
],
|
| 981 |
+
"page_idx": 6
|
| 982 |
+
},
|
| 983 |
+
{
|
| 984 |
+
"type": "text",
|
| 985 |
+
"text": "There are many results in classical learning theory on this problem for the case of ridge regression (where the teacher is usually called \"the source\" and the eigenvalues of the kernel matrix the \"capacity\", see e.g. [54, 56]). However, these are worst case approaches, where no assumption is made on the true distribution of the data. In contrast, here we follow a typical case analysis, assuming Gaussianity in feature space. Through Theorem 1, this allows us to go beyond the restriction of the ridge loss. An example for logistic loss is in Fig. 2. ",
|
| 986 |
+
"bbox": [
|
| 987 |
+
527,
|
| 988 |
+
347,
|
| 989 |
+
825,
|
| 990 |
+
525
|
| 991 |
+
],
|
| 992 |
+
"page_idx": 6
|
| 993 |
+
},
|
| 994 |
+
{
|
| 995 |
+
"type": "text",
|
| 996 |
+
"text": "For the particular case of kernel ridge regression, Th. 1 provides a rigorous proof of the formula conjectured in [32]. App. A.6 presents an explicit mapping to their results. Hard-margin Support Vector Machines (SVMs) have also been studied using the heuristic replica method from statistical physics in [57, 58]. In our framework, this corresponds to the hinge loss $g ( x , y ) = \\operatorname* { m a x } ( 0 , 1 - y x )$ when $\\lambda 0 ^ { + }$ . Our theorem thus puts also these works on rigorous grounds, and extends them to more general losses and regularization. ",
|
| 997 |
+
"bbox": [
|
| 998 |
+
174,
|
| 999 |
+
531,
|
| 1000 |
+
825,
|
| 1001 |
+
616
|
| 1002 |
+
],
|
| 1003 |
+
"page_idx": 6
|
| 1004 |
+
},
|
| 1005 |
+
{
|
| 1006 |
+
"type": "text",
|
| 1007 |
+
"text": "3.3 GAN-generated data and learned teachers ",
|
| 1008 |
+
"text_level": 1,
|
| 1009 |
+
"bbox": [
|
| 1010 |
+
174,
|
| 1011 |
+
631,
|
| 1012 |
+
508,
|
| 1013 |
+
647
|
| 1014 |
+
],
|
| 1015 |
+
"page_idx": 6
|
| 1016 |
+
},
|
| 1017 |
+
{
|
| 1018 |
+
"type": "text",
|
| 1019 |
+
"text": "To approach more realistic data sets, we now consider the case in which the input data $\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }$ is given by a generative neural network $\\begin{array} { r } { \\pmb { x } = \\pmb { \\mathcal { G } } ( \\pmb { z } ) } \\end{array}$ , where $_ z$ is a Gaussian i.i.d. latent vector. Therefore, the covariates $[ { \\pmb u } , { \\pmb v } ]$ are the result of the following Markov chain: ",
|
| 1020 |
+
"bbox": [
|
| 1021 |
+
173,
|
| 1022 |
+
656,
|
| 1023 |
+
826,
|
| 1024 |
+
699
|
| 1025 |
+
],
|
| 1026 |
+
"page_idx": 6
|
| 1027 |
+
},
|
| 1028 |
+
{
|
| 1029 |
+
"type": "equation",
|
| 1030 |
+
"img_path": "images/f14a89eba4c7cb07edaa3c2a72b4179ec3b4ec45505832a79e5e7989ce1e769e.jpg",
|
| 1031 |
+
"text": "$$\nz \\underset { \\mathcal { G } } { \\mapsto } x \\in \\mathcal { X } \\underset { \\varphi _ { t } } { \\mapsto } u \\in \\mathbb { R } ^ { p } , \\qquad z \\underset { \\mathcal { G } } { \\mapsto } x \\in \\mathcal { X } \\underset { \\varphi _ { s } } { \\mapsto } v \\in \\mathbb { R } ^ { d } .\n$$",
|
| 1032 |
+
"text_format": "latex",
|
| 1033 |
+
"bbox": [
|
| 1034 |
+
281,
|
| 1035 |
+
705,
|
| 1036 |
+
715,
|
| 1037 |
+
732
|
| 1038 |
+
],
|
| 1039 |
+
"page_idx": 6
|
| 1040 |
+
},
|
| 1041 |
+
{
|
| 1042 |
+
"type": "text",
|
| 1043 |
+
"text": "With a model for the covariates, the missing ingredient is the teacher weights $\\pmb \\theta _ { 0 } \\in \\mathbb { R } ^ { p }$ , which determine the label assignment: $\\boldsymbol { y } = f _ { 0 } ( \\boldsymbol { \\mathbf { u } } ^ { \\intercal } \\bar { \\boldsymbol { \\theta } } _ { 0 } )$ . In the experiments that follow, we fit the teacher weights from the original data set in which the generative model $\\mathcal { G }$ was trained. Different choices for the fitting yield different teacher weights, and the quality of label assignment can be accessed by the performance of the fit on the test set. The set $\\left( \\varphi _ { t } , \\varphi _ { s } , \\mathcal { G } , \\theta _ { 0 } \\right)$ defines the data generative process. For predicting the learning curves from the iterative eqs. (13) we need to sample from the spectral measure $\\mu$ , which amounts to estimating the population covariances $( \\Psi , \\Phi , \\Omega )$ . This is done from the generative process in eq. (19) with a Monte Carlo sampling algorithm. This pipeline is explained in detail in Appendix D. An open source implementation of the algorithms used in the experiments is available online at https://github.com/IdePHICS/GCMProject. ",
|
| 1044 |
+
"bbox": [
|
| 1045 |
+
173,
|
| 1046 |
+
737,
|
| 1047 |
+
825,
|
| 1048 |
+
877
|
| 1049 |
+
],
|
| 1050 |
+
"page_idx": 6
|
| 1051 |
+
},
|
| 1052 |
+
{
|
| 1053 |
+
"type": "text",
|
| 1054 |
+
"text": "Fig. 3 shows an example of the learning curves resulting from the pipeline discussed above in a logistic regression task on data generated by a GAN trained on CIFAR10 images. More concretely, we used a pre-trained five-layer deep convolutional GAN (dcGAN) from [59], which maps 100 dimensional i.i.d. Gaussian noise into $k = 3 2 \\times 3 2 \\times 3$ realistic looking CIFAR10-like images: $\\mathcal { G } : \\boldsymbol { z } \\in \\mathbb { R } ^ { 1 0 0 } \\mapsto \\boldsymbol { x } \\in \\mathbb { R } ^ { 3 2 \\times 3 2 \\times 3 }$ . To generate labels, we trained a simple fully-connected four-layer neural network on the real CIFAR10 data set, on a odd $( y = + 1 )$ ) vs. even $( y = - 1$ ) task, achieving $\\sim 7 5 \\%$ classification accuracy on the test set. The teacher weights $\\pmb \\theta _ { 0 } \\in \\mathbb { R } ^ { p }$ were taken from the last layer of the network, and the teacher feature map $\\varphi _ { t }$ from the three previous layers. For the student model, we trained a completely independent fully connected 3-layer neural network on the dcGAN-generated CIFAR10-like images and took snapshots of the feature maps $\\varphi _ { s } ^ { i }$ induced by the 2-first layers during the first $i \\in \\{ 0 , 5 , 5 0 , 2 0 0 \\}$ epochs of training. Finally, once $\\left( \\mathcal { G } , \\varphi _ { t } , \\varphi _ { s } ^ { i } , \\theta _ { 0 } \\right)$ have been fixed, we estimated the covariances $( \\Psi , \\Phi , \\Omega )$ with a Monte Carlo algorithm. Details of the architectures used and of the training procedure can be found in Appendix. D.1. ",
|
| 1055 |
+
"bbox": [
|
| 1056 |
+
174,
|
| 1057 |
+
883,
|
| 1058 |
+
823,
|
| 1059 |
+
911
|
| 1060 |
+
],
|
| 1061 |
+
"page_idx": 6
|
| 1062 |
+
},
|
| 1063 |
+
{
|
| 1064 |
+
"type": "image",
|
| 1065 |
+
"img_path": "images/edd9c243ed0e77099d9f480cf66562b3905600f50f057fa33195362b146bffea.jpg",
|
| 1066 |
+
"image_caption": [
|
| 1067 |
+
"Figure 3: Left: generalisation classification error (top) and (unregularised) training loss (bottom) vs the sample complexity $\\alpha = n / d$ for logistic regression on a learned feature map trained on dcGAN-generated CIFAR10-like images labelled by a teacher fully-connected neural network (see Appendix D.1 for architecture details), with vanishing $\\ell _ { 2 }$ regularisation. The different curves compare featured maps at different epochs of training. The theoretical predictions based on the Gaussian covariate model (full lines) are in very good agreement with the actual performance (points). Right: Test classification error (top) and (unregularised) training loss, (bottom) for logistic regression as a function of the number of samples $n$ for an animal vs not-animal binary classification task with $\\ell _ { 2 }$ regularization $\\lambda = 1 0 ^ { - 2 }$ , comparing real CIFAR10 grey-scale images (blue) with dcGAN-generated CIFAR10-like gray-scale images (red). The real-data learning curve was estimated, just as in Figs. 4 from the population covariances on the full data set, and it is not in agreement with the theory in this case. On the very right we depict the histograms of the variable $\\begin{array} { r } { \\frac { 1 } { \\sqrt { d } } \\pmb { v } ^ { \\top } \\hat { \\pmb { w } } } \\end{array}$ for a fixed number of samples $n = 2 d = 2 0 4 8$ and the respective theoretical predictions (solid line). Simulations are averaged over 10 independent runs. "
|
| 1068 |
+
],
|
| 1069 |
+
"image_footnote": [],
|
| 1070 |
+
"bbox": [
|
| 1071 |
+
187,
|
| 1072 |
+
93,
|
| 1073 |
+
810,
|
| 1074 |
+
272
|
| 1075 |
+
],
|
| 1076 |
+
"page_idx": 7
|
| 1077 |
+
},
|
| 1078 |
+
{
|
| 1079 |
+
"type": "text",
|
| 1080 |
+
"text": "",
|
| 1081 |
+
"bbox": [
|
| 1082 |
+
173,
|
| 1083 |
+
522,
|
| 1084 |
+
825,
|
| 1085 |
+
679
|
| 1086 |
+
],
|
| 1087 |
+
"page_idx": 7
|
| 1088 |
+
},
|
| 1089 |
+
{
|
| 1090 |
+
"type": "text",
|
| 1091 |
+
"text": "Fig. 3 depicts the resulting learning curves obtained by training the last layer of the student. Interestingly, the performance of the feature map at epoch 0 (random initialisation) beats the performance of the learned features during early phases of training in this experiment. Another interesting behaviour is given by the separability threshold of the learned features, i.e. the number of samples for which the training loss becomes larger than 0 in logistic regression. At epoch 50 the learned features are separable at lower sample complexity $\\alpha = n / d$ than at epoch 200 - even though in the later the training and generalisation performances are better. ",
|
| 1092 |
+
"bbox": [
|
| 1093 |
+
174,
|
| 1094 |
+
685,
|
| 1095 |
+
825,
|
| 1096 |
+
781
|
| 1097 |
+
],
|
| 1098 |
+
"page_idx": 7
|
| 1099 |
+
},
|
| 1100 |
+
{
|
| 1101 |
+
"type": "text",
|
| 1102 |
+
"text": "3.4 Learning from real data sets ",
|
| 1103 |
+
"text_level": 1,
|
| 1104 |
+
"bbox": [
|
| 1105 |
+
176,
|
| 1106 |
+
811,
|
| 1107 |
+
410,
|
| 1108 |
+
825
|
| 1109 |
+
],
|
| 1110 |
+
"page_idx": 7
|
| 1111 |
+
},
|
| 1112 |
+
{
|
| 1113 |
+
"type": "text",
|
| 1114 |
+
"text": "Applying teacher/students to a real data set — Given that the learning curves of realistic-looking inputs can be captured by the Gaussian covariate model, it is fair to ask whether the same might be true for real data sets. To test this idea, we first need to cast the real data set into the teacher-student formalism, and then compute the covariance matrices $\\Omega , \\Psi , \\Phi$ and teacher vector $\\pmb { \\theta } _ { 0 }$ required by model (1). ",
|
| 1115 |
+
"bbox": [
|
| 1116 |
+
174,
|
| 1117 |
+
842,
|
| 1118 |
+
825,
|
| 1119 |
+
911
|
| 1120 |
+
],
|
| 1121 |
+
"page_idx": 7
|
| 1122 |
+
},
|
| 1123 |
+
{
|
| 1124 |
+
"type": "text",
|
| 1125 |
+
"text": "Let $\\{ x ^ { \\mu } , y ^ { \\mu } \\} _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } }$ denote a real data set, e.g. MNIST or Fashion-MNIST for concreteness, where $n _ { \\mathrm { t o t } } = 7 \\times 1 0 ^ { 4 }$ , $\\pmb { x } ^ { \\mu } \\in \\mathbb { R } ^ { D }$ with $D = 7 8 4$ . Without loss of generality, we can assume the data is centred. To generate the teacher, let $\\pmb { u } ^ { \\mu } = \\varphi _ { t } ( \\pmb { x } ^ { \\mu } ) \\in \\mathbb { R } ^ { p }$ be a feature map such that data is invertible in feature space, i.e. that $y ^ { \\mu } = \\pmb \\theta _ { 0 } ^ { \\top } { \\pmb u } ^ { \\mu }$ for some teacher weights $\\pmb \\theta _ { 0 } \\in \\mathbb { R } ^ { p }$ , which should be computed from the samples. Similarly, let $\\mathbf { \\bar { \\boldsymbol { v } } } ^ { \\mu } = \\varphi _ { s } ( \\mathbf { \\boldsymbol { x } } ^ { \\mu } ) \\in \\mathbb { R } ^ { d }$ be a feature map we are interested in studying. Then, we can estimate the population covariances $( \\Psi , \\Phi , \\Omega )$ empirically from the entire data set as: ",
|
| 1126 |
+
"bbox": [
|
| 1127 |
+
173,
|
| 1128 |
+
90,
|
| 1129 |
+
826,
|
| 1130 |
+
181
|
| 1131 |
+
],
|
| 1132 |
+
"page_idx": 8
|
| 1133 |
+
},
|
| 1134 |
+
{
|
| 1135 |
+
"type": "equation",
|
| 1136 |
+
"img_path": "images/93716dffe232d62851fc16c3463b0c29f10a9349a82b8247ef44836d75c03e7a.jpg",
|
| 1137 |
+
"text": "$$\n\\Psi = \\sum _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } } { \\frac { u ^ { \\mu } { \\pmb u } ^ { \\mu } ^ { \\top } } { n _ { \\mathrm { t o t } } } } , \\qquad \\Phi = \\sum _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } } { \\frac { u ^ { \\mu } \\pmb v ^ { \\mu } ^ { \\top } } { n _ { \\mathrm { t o t } } } } , \\qquad \\Omega = \\sum _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } } { \\frac { v ^ { \\mu } \\pmb v ^ { \\mu } ^ { \\top } } { n _ { \\mathrm { t o t } } } } .\n$$",
|
| 1138 |
+
"text_format": "latex",
|
| 1139 |
+
"bbox": [
|
| 1140 |
+
246,
|
| 1141 |
+
185,
|
| 1142 |
+
751,
|
| 1143 |
+
228
|
| 1144 |
+
],
|
| 1145 |
+
"page_idx": 8
|
| 1146 |
+
},
|
| 1147 |
+
{
|
| 1148 |
+
"type": "text",
|
| 1149 |
+
"text": "At this point, we have all we need to run the self-consistent equations (13). The issue with this approach is that there is not a unique teacher map $\\varphi _ { t }$ and teacher vector $\\pmb { \\theta } _ { 0 }$ that fit the true labels. However, we can show that all interpolating linear teachers are equivalent: ",
|
| 1150 |
+
"bbox": [
|
| 1151 |
+
173,
|
| 1152 |
+
233,
|
| 1153 |
+
825,
|
| 1154 |
+
275
|
| 1155 |
+
],
|
| 1156 |
+
"page_idx": 8
|
| 1157 |
+
},
|
| 1158 |
+
{
|
| 1159 |
+
"type": "text",
|
| 1160 |
+
"text": "Theorem 3. (Universality of linear teachers) For any teacher feature map $\\varphi _ { t }$ , and for any $\\pmb { \\theta } _ { 0 }$ that interpolates the data so that $\\mathbf { \\bar { \\boldsymbol { y } } } ^ { \\mu } = \\pmb { \\theta } _ { 0 } ^ { \\top } \\pmb { u } ^ { \\mu } \\forall \\mu$ , the asymptotic predictions of model $( l )$ are equivalent. Proof. It follows from the fact that the teacher weights and covariances only appear in eq. (13) through $\\begin{array} { r } { \\rho = \\frac { 1 } { p } \\pmb { \\theta } _ { 0 } ^ { \\top } \\Psi \\pmb { \\theta } _ { 0 } } \\end{array}$ and the projection $\\Phi ^ { \\top } \\pmb \\theta _ { 0 }$ . Using the estimation (20) and the assumption that it exists $y ^ { \\mu } = \\pmb { \\theta } _ { 0 } ^ { \\top } \\pmb { u } ^ { \\mu }$ , one can write these quantities directly from the labels $y ^ { \\mu }$ : ",
|
| 1161 |
+
"bbox": [
|
| 1162 |
+
173,
|
| 1163 |
+
279,
|
| 1164 |
+
825,
|
| 1165 |
+
353
|
| 1166 |
+
],
|
| 1167 |
+
"page_idx": 8
|
| 1168 |
+
},
|
| 1169 |
+
{
|
| 1170 |
+
"type": "equation",
|
| 1171 |
+
"img_path": "images/0961fe501a4a899372ad700d192ac4ec3ab7482c4da93c697d46cd13ef4f4359.jpg",
|
| 1172 |
+
"text": "$$\n\\rho = \\frac { 1 } { n _ { \\mathrm { t o t } } } \\sum _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } } \\left( y ^ { \\mu } \\right) ^ { 2 } , \\qquad \\Phi ^ { \\top } \\pmb { \\theta } _ { 0 } = \\frac { 1 } { n _ { \\mathrm { t o t } } } \\sum _ { \\mu = 1 } ^ { n _ { \\mathrm { t o t } } } y ^ { \\mu } \\pmb { v } ^ { \\mu } .\n$$",
|
| 1173 |
+
"text_format": "latex",
|
| 1174 |
+
"bbox": [
|
| 1175 |
+
290,
|
| 1176 |
+
356,
|
| 1177 |
+
707,
|
| 1178 |
+
401
|
| 1179 |
+
],
|
| 1180 |
+
"page_idx": 8
|
| 1181 |
+
},
|
| 1182 |
+
{
|
| 1183 |
+
"type": "text",
|
| 1184 |
+
"text": "For linear interpolating teachers, results are thus independent of the choice of the teacher. ",
|
| 1185 |
+
"bbox": [
|
| 1186 |
+
171,
|
| 1187 |
+
405,
|
| 1188 |
+
758,
|
| 1189 |
+
420
|
| 1190 |
+
],
|
| 1191 |
+
"page_idx": 8
|
| 1192 |
+
},
|
| 1193 |
+
{
|
| 1194 |
+
"type": "text",
|
| 1195 |
+
"text": "Although this result might seen surprising at first sight, it is quite intuitive. Indeed, the information about the teacher model only enters the Gaussian covariate model (1) through the statistics of $\\pmb { u } ^ { \\top } \\pmb { \\theta } _ { 0 }$ . For a linear teacher $f _ { 0 } ( x ) = x$ , this is precisely given by the labels. ",
|
| 1196 |
+
"bbox": [
|
| 1197 |
+
174,
|
| 1198 |
+
434,
|
| 1199 |
+
826,
|
| 1200 |
+
477
|
| 1201 |
+
],
|
| 1202 |
+
"page_idx": 8
|
| 1203 |
+
},
|
| 1204 |
+
{
|
| 1205 |
+
"type": "image",
|
| 1206 |
+
"img_path": "images/99b7a281af657064e0cd4cb2ea6e0425a214ca066260d5d3f39f34785295ce74.jpg",
|
| 1207 |
+
"image_caption": [
|
| 1208 |
+
"Figure 4: Test and training mean-squared errors eqs. (15) as a function of the number of samples $n$ for ridge regression. The Fashion-MNIST data set, with vanishing regularisation $\\lambda = 1 0 ^ { - 5 }$ . In this plot, the student feature map $\\varphi _ { s }$ is a 3-layer fully-connected neural network with $d \\ : = \\ : 2 3 5 2$ hidden neurons trained on the full data set with the square loss. Different curves correspond to the feature map obtained at different stages of training. Simulations are averaged over 10 independent runs. Further details on the simulations are described in Appendix D.1 "
|
| 1209 |
+
],
|
| 1210 |
+
"image_footnote": [],
|
| 1211 |
+
"bbox": [
|
| 1212 |
+
178,
|
| 1213 |
+
510,
|
| 1214 |
+
465,
|
| 1215 |
+
664
|
| 1216 |
+
],
|
| 1217 |
+
"page_idx": 8
|
| 1218 |
+
},
|
| 1219 |
+
{
|
| 1220 |
+
"type": "text",
|
| 1221 |
+
"text": "Ridge Regression with linear teachers — ",
|
| 1222 |
+
"text_level": 1,
|
| 1223 |
+
"bbox": [
|
| 1224 |
+
514,
|
| 1225 |
+
491,
|
| 1226 |
+
825,
|
| 1227 |
+
505
|
| 1228 |
+
],
|
| 1229 |
+
"page_idx": 8
|
| 1230 |
+
},
|
| 1231 |
+
{
|
| 1232 |
+
"type": "text",
|
| 1233 |
+
"text": "We now test the prediction of model (1) on real data sets, and show that it is surprisingly effective in predicting the learning curves, at least for the ridge regression task. We have trained a 3-layer fully connected neural network with ReLU activations on the full Fashion-MNIST data set to distinguish clothing used above vs. below the waist [60]. The student feature map $\\varphi _ { s } : \\mathbb { R } ^ { 7 8 4 } \\to \\mathbb { R } ^ { \\bar { d } }$ is obtained by removing the last layer, see Appendix D.1 for a detailed description. In Fig. 4 we show the test and training errors of the ridge estimator on a sub-sample of $n \\ < \\ n _ { \\mathrm { t o t } }$ on the Fashion-MNIST images. We observe remarkable agreement between the learning curve obtained from simulations and the theoretical prediction by the matching Gaussian covariate model. Note that for the square loss and for $\\lambda \\ll 1$ , the worst performance peak is located at the point in which the linear system becomes invertible. Curiously, Fig. 4 shows that the fully-connected network progressively learns a low-rank representation of the data as training proceeds. This can be directly verified by counting the number of zero eigenvalues of $\\Omega$ , which go from a full-rank matrix to a matrix of rank 380 after 200 epochs of training. ",
|
| 1234 |
+
"bbox": [
|
| 1235 |
+
514,
|
| 1236 |
+
503,
|
| 1237 |
+
825,
|
| 1238 |
+
862
|
| 1239 |
+
],
|
| 1240 |
+
"page_idx": 8
|
| 1241 |
+
},
|
| 1242 |
+
{
|
| 1243 |
+
"type": "text",
|
| 1244 |
+
"text": "Fig. 1 (right) shows a similar experiment on the MNIST data set, but for different out-of-the-box feature maps, such as random features and the scattering transform [61], and we chose the number of random features $d = 1 9 5 3$ to match the number of features from the scattering transform. Note the characteristic double-descent behaviour [9, 25, 62], and the accurate prediction of the peak where the interpolation transition occurs. We note in Appendix D.1 that for both Figs. 4 and 1, for a number of samples $n$ closer to $n _ { \\mathrm { t o t } }$ we start to see deviations between the real learning curve and the theory. This is to be expected since in the teacher-student framework the student can, in principle, express the same function as the teacher if it recovers its weights exactly. Recovering the teacher weights becomes possible with a large training set. In that case, its test error will be zero. However, in our setup the test error on real data remains finite even if more training data is added, leading to the discrepancy between teacher-student learning curve and real data, see Appendix D.1 for further discussion. ",
|
| 1245 |
+
"bbox": [
|
| 1246 |
+
174,
|
| 1247 |
+
869,
|
| 1248 |
+
825,
|
| 1249 |
+
911
|
| 1250 |
+
],
|
| 1251 |
+
"page_idx": 8
|
| 1252 |
+
},
|
| 1253 |
+
{
|
| 1254 |
+
"type": "text",
|
| 1255 |
+
"text": "",
|
| 1256 |
+
"bbox": [
|
| 1257 |
+
174,
|
| 1258 |
+
90,
|
| 1259 |
+
825,
|
| 1260 |
+
215
|
| 1261 |
+
],
|
| 1262 |
+
"page_idx": 9
|
| 1263 |
+
},
|
| 1264 |
+
{
|
| 1265 |
+
"type": "text",
|
| 1266 |
+
"text": "Why is the Gaussian model so effective for describing learning with data that are not Gaussian? The point is that ridge regression is sensitive only to second order statistics, and not to the full distribution of the data. It is a classical property (see Appendix E) that the training and generalisation errors are only a function of the spectrum of the empirical and population covariances, and of their products. Random matrix theory teaches us that such quantities are very robust, and their asymptotic behaviour is universal for a broad class of distributions of $[ u , v ]$ [49, 63–65]. The asymptotic behavior of kernel matrices has indeed been the subject of intense scrutiny [11, 47, 48, 50, 66, 67]. Indeed, a universality result akin to Theorem 3 was noted in [41] in the specific case of kernel methods. We thus expect the validity of model (1) for ridge regression, with a linear teacher, to go way beyond the Gaussian assumption. ",
|
| 1267 |
+
"bbox": [
|
| 1268 |
+
174,
|
| 1269 |
+
222,
|
| 1270 |
+
825,
|
| 1271 |
+
361
|
| 1272 |
+
],
|
| 1273 |
+
"page_idx": 9
|
| 1274 |
+
},
|
| 1275 |
+
{
|
| 1276 |
+
"type": "text",
|
| 1277 |
+
"text": "Beyond ridge regression — The same strategy fails beyond ridge regression and mean-squared test error. This suggests a limit in the application of model (1) to real (non-Gaussian) data to the universal linear teacher. To illustrate this, consider the setting of Figs. 4, and compare the model predictions for the binary classification error instead of the $\\ell _ { 2 }$ one. There is a clear mismatch between the simulated performance and prediction given by the theory (see Appendix D.1) due to the fact that the classification error does not depends only on the first two moments. ",
|
| 1278 |
+
"bbox": [
|
| 1279 |
+
174,
|
| 1280 |
+
368,
|
| 1281 |
+
825,
|
| 1282 |
+
450
|
| 1283 |
+
],
|
| 1284 |
+
"page_idx": 9
|
| 1285 |
+
},
|
| 1286 |
+
{
|
| 1287 |
+
"type": "text",
|
| 1288 |
+
"text": "We present an additional experiment in Fig. 3. We compare the learning curves of logistic regression on a classification task on the real CIFAR10 images with the real labels versus the one on dcGANgenerated CIFAR10-like images and teacher generated labels from Sec. 3.3. While the Gaussian theory captures well the behaviour of the later, it fails on the former. A histogram of the distribution of the product $\\mathbf { \\Delta } u ^ { \\top } \\hat { \\mathbf { \\Gamma } } w$ for a fixed number of samples illustrates well the deviation from the prediction of the theory with the real case, in particular on the tails of the distribution. The difference between GAN generated data (that fits the Gaussian theory) and real data is clear. Given that for classification problems there exists a number of choices of \"sign\" teachers and feature maps that give the exact same labels as in the data set, an interesting open question is: is there a teacher that allows to reproduce the learning curves more accurately? This question is left for future works. ",
|
| 1289 |
+
"bbox": [
|
| 1290 |
+
174,
|
| 1291 |
+
458,
|
| 1292 |
+
825,
|
| 1293 |
+
597
|
| 1294 |
+
],
|
| 1295 |
+
"page_idx": 9
|
| 1296 |
+
},
|
| 1297 |
+
{
|
| 1298 |
+
"type": "text",
|
| 1299 |
+
"text": "Acknowledgements ",
|
| 1300 |
+
"text_level": 1,
|
| 1301 |
+
"bbox": [
|
| 1302 |
+
176,
|
| 1303 |
+
608,
|
| 1304 |
+
338,
|
| 1305 |
+
626
|
| 1306 |
+
],
|
| 1307 |
+
"page_idx": 9
|
| 1308 |
+
},
|
| 1309 |
+
{
|
| 1310 |
+
"type": "text",
|
| 1311 |
+
"text": "We thank Romain Couillet, Cosme Louart, Loucas Pillaud-Vivien, Matthieu Wyart, Federica Gerace, Luca Saglietti and Yue Lu for discussions. We are grateful to Kabir Aladin Chandrasekher, Ashwin Pananjady and Christos Thrampoulidis for pointing out discrepancies in the finite size rates and insightful related discussions. We acknowledge funding from the ERC under the European Union’s Horizon 2020 Research and Innovation Programme Grant Agreement 714608-SMiLe, and from the French National Research Agency grants ANR-17-CE23-0023-01 PAIL. ",
|
| 1312 |
+
"bbox": [
|
| 1313 |
+
174,
|
| 1314 |
+
640,
|
| 1315 |
+
825,
|
| 1316 |
+
722
|
| 1317 |
+
],
|
| 1318 |
+
"page_idx": 9
|
| 1319 |
+
},
|
| 1320 |
+
{
|
| 1321 |
+
"type": "text",
|
| 1322 |
+
"text": "References \n[1] Hyunjune Sebastian Seung, Haim Sompolinsky, and Naftali Tishby. Statistical mechanics of learning from examples. Physical review A, 45(8):6056, 1992. \n[2] Timothy LH Watkin, Albrecht Rau, and Michael Biehl. The statistical mechanics of learning a rule. Reviews of Modern Physics, 65(2):499, 1993. \n[3] Andreas Engel and Christian Van den Broeck. Statistical mechanics of learning. Cambridge University Press, 2001. \n[4] David L Donoho, Arian Maleki, and Andrea Montanari. Message-passing algorithms for compressed sensing. Proceedings of the National Academy of Sciences, 106(45):18914–18919, 2009. \n[5] Noureddine El Karoui, Derek Bean, Peter J Bickel, Chinghway Lim, and Bin Yu. On robust regression with high-dimensional predictors. Proceedings of the National Academy of Sciences, 110(36):14557–14562, 2013. \n[6] Lenka Zdeborová and Florent Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453–552, 2016. \n[7] David Donoho and Andrea Montanari. High dimensional robust m-estimation: Asymptotic variance via approximate message passing. Probability Theory and Related Fields, 166(3- 4):935–969, 2016. \n[8] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. Understanding deep learning requires rethinking generalization. In ICLR, 2017. \n[9] Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machinelearning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116(32):15849–15854, 2019. \n[10] Mikhail Belkin, Daniel Hsu, and Ji Xu. Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2(4):1167–1180, 2020. \n[11] Song Mei and Andrea Montanari. The generalization error of random features regression: Precise asymptotics and double descent curve. arXiv preprint arXiv:1908.05355, 2019. \n[12] Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J Tibshirani. Surprises in highdimensional ridgeless least squares interpolation. arXiv preprint arXiv:1903.08560, 2019. \n[13] Emmanuel J Candès, Pragya Sur, et al. The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression. The Annals of Statistics, 48(1):27– 42, 2020. \n[14] Benjamin Aubin, Florent Krzakala, Yue M Lu, and Lenka Zdeborová. Generalization error in high-dimensional perceptrons: Approaching bayes error with convex optimization. In Advances in Neural Information Processing Systems, volume 33, 2020. \n[15] Fariborz Salehi, Ehsan Abbasi, and Babak Hassibi. The performance analysis of generalized margin maximizers on separable data. In International Conference on Machine Learning, pages 8417–8426. PMLR, 2020. \n[16] Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. In Advances in neural information processing systems, pages 1177–1184, 2008. \n[17] Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In Advances in neural information processing systems, pages 8571–8580, 2018. \n[18] Mathieu Andreux, Tomás Angles, Georgios Exarchakis, Roberto Leonarduzzi, Gaspar Rochette, Louis Thiry, John Zarka, Stéphane Mallat, Joakim Andén, Eugene Belilovsky, et al. Kymatio: Scattering transforms in python. Journal of Machine Learning Research, 21(60):1–6, 2020. \n[19] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks, 2014. \n[20] Yehoram Gordon. Some inequalities for gaussian processes and applications. Israel Journal of Mathematics, 50(4):265–289, 1985. \n[21] Marc Mézard, Giorgio Parisi, and Miguel Virasoro. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, volume 9. World Scientific Publishing Company, 1987. \n[22] Marc Mézard and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009. \n[23] Christopher K. I. Williams. Computing with infinite networks. In Proceedings of the 9th International Conference on Neural Information Processing Systems, NIPS’96, page 295–301, Cambridge, MA, USA, 1996. MIT Press. \n[24] Elizabeth Gardner and Bernard Derrida. Three unfinished works on the optimal storage capacity of networks. Journal of Physics A: Mathematical and General, 22(12):1983, 1989. \n[25] Manfred Opper and Wolfgang Kinzel. Statistical mechanics of generalization. In Models of neural networks III, pages 151–209. Springer, 1996. \n[26] Behrooz Ghorbani, Song Mei, Theodor Misiakiewicz, and Andrea Montanari. When do neural networks outperform kernel methods? In Advances in Neural Information Processing Systems, volume 33, 2020. \n[27] Christos Thrampoulidis, Ehsan Abbasi, and Babak Hassibi. Precise error analysis of regularized $m$ -estimators in high dimensions. IEEE Transactions on Information Theory, 64(8):5592–5628, 2018. \n[28] Andrea Montanari, Feng Ruan, Youngtak Sohn, and Jun Yan. The generalization error of max-margin linear classifiers: High-dimensional asymptotics in the overparametrized regime. arXiv preprint arXiv:1911.01544, 2019. \n[29] Michael Celentano, Andrea Montanari, and Yuting Wei. The lasso with general gaussian designs with applications to hypothesis testing. arXiv preprint arXiv:2007.13716, 2020. \n[30] Mihailo Stojnic. A framework to characterize performance of lasso algorithms. arXiv preprint arXiv:1303.7291, 2013. \n[31] Samet Oymak, Christos Thrampoulidis, and Babak Hassibi. The squared-error of generalized lasso: A precise analysis. In 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1002–1009. IEEE, 2013. \n[32] Blake Bordelon, Abdulkadir Canatar, and Cengiz Pehlevan. Spectrum dependent learning curves in kernel regression and wide neural networks. In International Conference on Machine Learning, pages 1024–1034. PMLR, 2020. \n[33] Hanwen Huang and Qinglong Yang. Large scale analysis of generalization error in learning using margin based classification methods. Journal of Statistical Mechanics: Theory and Experiment, 2020(10):103407, 2020. \n[34] Partha P Mitra. Understanding overfitting peaks in generalization error: Analytical risk curves for $l _ { - 2 }$ and $l _ { - } 1$ penalized interpolation. arXiv preprint arXiv:1906.03667, 2019. \n[35] Oussama Dhifallah and Yue M Lu. A precise performance analysis of learning with random features. arXiv preprint arXiv:2008.11904, 2020. \n[36] Edgar Dobriban, Stefan Wager, et al. High-dimensional asymptotics of prediction: Ridge regression and classification. The Annals of Statistics, 46(1):247–279, 2018. \n[37] Denny Wu and Ji Xu. On the optimal weighted $\\ell _ { 2 }$ regularization in overparameterized linear regression. In Advances in Neural Information Processing Systems, volume 33, 2020. \n[38] Zhenyu Liao, Romain Couillet, and Michael W Mahoney. A random matrix analysis of random fourier features: beyond the gaussian kernel, a precise phase transition, and the corresponding double descent. In Advances in Neural Information Processing Systems, volume 33, 2020. \n[39] Fanghui Liu, Zhenyu Liao, and Johan AK Suykens. Kernel regression in high dimension: Refined analysis beyond double descent. arXiv preprint arXiv:2010.02681, 2020. \n[40] Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117(48):30063–30070, 2020. \n[41] Arthur Jacot, Berfin ¸Sim¸sek, Francesco Spadaro, Clément Hongler, and Franck Gabriel. Kernel alignment risk estimator: Risk prediction from training data. arXiv preprint arXiv:2006.09796, 2020. \n[42] Cédric Gerbelot, Alia Abbara, and Florent Krzakala. Asymptotic errors for high-dimensional convex penalized linear regression beyond gaussian matrices. In Conference on Learning Theory, pages 1682–1713. PMLR, 2020. \n[43] F. Gerace, B. Loureiro, F. Krzakala, M. Mézard, and L. Zdeborová. Generalisation error in learning with random features and the hidden manifold model. In 37th International Conference on Machine Learning, 2020. \n[44] S. Goldt, M. Mézard, F. Krzakala, and L. Zdeborová. Modeling the influence of data structure on learning in neural networks: The hidden manifold model. Phys. Rev. X, 10(4):041044, 2020. \n[45] Sebastian Goldt, Bruno Loureiro, Galen Reeves, Marc Mézard, Florent Krzakala, and Lenka Zdeborová. The gaussian equivalence of generative models for learning with two-layer neural networks. In Mathematical and Scientific Machine Learning, 2021. \n[46] Hong Hu and Yue M Lu. Universality laws for high-dimensional learning with random features. arXiv preprint arXiv:2009.07669, 2020. \n[47] Noureddine El Karoui et al. The spectrum of kernel random matrices. Annals of statistics, 38(1):1–50, 2010. \n[48] Jeffrey Pennington and Pratik Worah. Nonlinear random matrix theory for deep learning. In Advances in Neural Information Processing Systems, volume 30, pages 2637–2646, 2017. \n[49] Cosme Louart and Romain Couillet. Concentration of measure and large random matrices with an application to sample covariance matrices. arXiv preprint arXiv:1805.08295, 2018. \n[50] Mohamed El Amine Seddik, Cosme Louart, Mohamed Tamaazousti, and Romain Couillet. Random matrix theory proves that deep learning representations of gan-data behave as gaussian mixtures. In International Conference on Machine Learning, pages 8573–8582. PMLR, 2020. \n[51] Léo Miolane and Andrea Montanari. The distribution of the lasso: Uniform control over sparse balls and adaptive parameter tuning. arXiv preprint arXiv:1811.01212, 2018. \n[52] Saharon Rosset, Ji Zhu, and Trevor Hastie. Margin maximizing loss functions. In NIPS, pages 1237–1244, 2003. \n[53] B. Scholkopf and A.J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. Adaptive Computation and Machine Learning. MIT Press, 2018. \n[54] Ingo Steinwart, Don R Hush, Clint Scovel, et al. Optimal rates for regularized least squares regression. In COLT, pages 79–93, 2009. \n[55] Andrea Caponnetto and Ernesto De Vito. Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, 7(3):331–368, 2007. \n[56] Loucas Pillaud-Vivien, Alessandro Rudi, and Francis Bach. Statistical optimality of stochastic gradient descent on hard learning problems through multiple passes. In Advances in Neural Information Processing Systems, volume 31, pages 8114–8124, 2018. \n[57] Rainer Dietrich, Manfred Opper, and Haim Sompolinsky. Statistical mechanics of support vector networks. Phys. Rev. Lett., 82:2975–2978, Apr 1999. \n[58] M. Opper and R. Urbanczik. Universal learning curves of support vector machines. Phys. Rev. Lett., 86:4410–4413, May 2001. \n[59] A. Radford, L. Metz, and S. Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016. \n[60] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017. \n[61] J. Bruna and S. Mallat. Invariant scattering convolution networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1872–1886, 2013. \n[62] Stefano Spigler, Mario Geiger, Stéphane d’Ascoli, Levent Sagun, Giulio Biroli, and Matthieu Wyart. A jamming transition from under-to over-parametrization affects generalization in deep learning. Journal of Physics A: Mathematical and Theoretical, 52(47):474001, 2019. \n[63] Zhidong Bai and Wang Zhou. Large sample covariance matrices without independence structures in columns. Statistica Sinica, pages 425–442, 2008. \n[64] Olivier Ledoit and Sandrine Péché. Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 151(1):233–264, 2011. \n[65] Noureddine El Karoui et al. Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond. Annals of Applied Probability, 19(6):2362–2405, 2009. \n[66] Xiuyuan Cheng and Amit Singer. The spectrum of random inner-product kernel matrices. Random Matrices: Theory and Applications, 2(04):1350010, 2013. \n[67] Zhou Fan and Andrea Montanari. The spectral norm of random inner-product kernel matrices. Probability Theory and Related Fields, 173(1):27–85, 2019. \n[68] Heinz H Bauschke, Patrick L Combettes, et al. Convex analysis and monotone operator theory in Hilbert spaces, volume 408. Springer, 2011. \n[69] Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration inequalities: A nonasymptotic theory of independence. Oxford university press, 2013. \n[70] Yanting Ma, Cynthia Rush, and Dror Baron. Analysis of approximate message passing with a class of non-separable denoisers. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 231–235. IEEE, 2017. \n[71] Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Y. Eldar and G. Kutyniok., editors, Compressed Sensing, Theory and Applications. Cambridge University Press, 2012. \n[72] Rick Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019. \n[73] Per Kragh Andersen and Richard D Gill. Cox’s regression model for counting processes: a large sample study. The annals of statistics, pages 1100–1120, 1982. \n[74] Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018. \n[75] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. 2015 IEEE International Conference on Computer Vision (ICCV), pages 1026–1034, 2015. \n[76] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference for Learning Representations, volume 3, 2015. \n[77] Greg W Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices. Number 118. Cambridge university press, 2010. \n[78] Vladimir Alexandrovich Marchenko and Leonid Andreevich Pastur. Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik, 114(4):507–536, 1967. \n[79] Djalil Chafaï and Konstantin Tikhomirov. On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence. Probability Theory and Related Fields, 170(3):847–889, 2018. \n[80] Walid Hachem, Philippe Loubaton, Jamal Najim, et al. Deterministic equivalents for certain functionals of large random matrices. The Annals of Applied Probability, 17(3):875–930, 2007. ",
|
| 1323 |
+
"bbox": [
|
| 1324 |
+
171,
|
| 1325 |
+
65,
|
| 1326 |
+
828,
|
| 1327 |
+
912
|
| 1328 |
+
],
|
| 1329 |
+
"page_idx": 10
|
| 1330 |
+
},
|
| 1331 |
+
{
|
| 1332 |
+
"type": "text",
|
| 1333 |
+
"text": "",
|
| 1334 |
+
"bbox": [
|
| 1335 |
+
171,
|
| 1336 |
+
54,
|
| 1337 |
+
828,
|
| 1338 |
+
916
|
| 1339 |
+
],
|
| 1340 |
+
"page_idx": 11
|
| 1341 |
+
},
|
| 1342 |
+
{
|
| 1343 |
+
"type": "text",
|
| 1344 |
+
"text": "",
|
| 1345 |
+
"bbox": [
|
| 1346 |
+
171,
|
| 1347 |
+
34,
|
| 1348 |
+
828,
|
| 1349 |
+
916
|
| 1350 |
+
],
|
| 1351 |
+
"page_idx": 12
|
| 1352 |
+
},
|
| 1353 |
+
{
|
| 1354 |
+
"type": "text",
|
| 1355 |
+
"text": "",
|
| 1356 |
+
"bbox": [
|
| 1357 |
+
171,
|
| 1358 |
+
60,
|
| 1359 |
+
828,
|
| 1360 |
+
916
|
| 1361 |
+
],
|
| 1362 |
+
"page_idx": 13
|
| 1363 |
+
},
|
| 1364 |
+
{
|
| 1365 |
+
"type": "text",
|
| 1366 |
+
"text": "",
|
| 1367 |
+
"bbox": [
|
| 1368 |
+
171,
|
| 1369 |
+
90,
|
| 1370 |
+
828,
|
| 1371 |
+
247
|
| 1372 |
+
],
|
| 1373 |
+
"page_idx": 14
|
| 1374 |
+
}
|
| 1375 |
+
]
|
parse/train/B1Kh0SVodY/B1Kh0SVodY_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/B1Kh0SVodY/B1Kh0SVodY_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/B1VZqjAcYX/B1VZqjAcYX_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:8077dea64eeca5fe2d72d5ac3018f4938eb95ff16e1aeb12a32cf2903ae17a30
|
| 3 |
+
size 1896738
|
parse/train/B1VZqjAcYX/B1VZqjAcYX_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:56ddfd5ffa505904f3f8cf0357645b165d424c90115e5f789b8de8642293980a
|
| 3 |
+
size 1546949
|
parse/train/B1VZqjAcYX/B1VZqjAcYX_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:9cea02b862a2424a1ccfeb3b1c2fc12fbaf331c5b93faf544ceb55e74a15111e
|
| 3 |
+
size 1899860
|
parse/train/B1eWbxStPH/B1eWbxStPH_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:17145782499cd676b5c422c60a9a9674261beb50b6f5d0f78128b14b2e224b9e
|
| 3 |
+
size 1646134
|
parse/train/B1eWbxStPH/B1eWbxStPH_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:1a155741580805ea2d12edd30b01c09c158102a76076598c075e3a5205f58657
|
| 3 |
+
size 1395722
|
parse/train/B1eWbxStPH/B1eWbxStPH_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:7f994d025fa94f78b19924442aae519f67ba1aa2ba3e387d675058d81dc04937
|
| 3 |
+
size 1656978
|
parse/train/B1mAJI9gl/B1mAJI9gl_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:b66624b42c8e457326ff207855ad530cddfa06285cf02973800ec559819b443f
|
| 3 |
+
size 1969781
|
parse/train/B1mAJI9gl/B1mAJI9gl_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:c9d11fdac457a2868bad8892d45c33a3d250da3321a7786325c3a768f53cd9c7
|
| 3 |
+
size 1695734
|
parse/train/B1mAJI9gl/B1mAJI9gl_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:81d2d5a2fed2d80cffbe6b3688814d7349cd0b159e063cf7cae7e27522b6a102
|
| 3 |
+
size 1991486
|
parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w.md
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/B7v4QMR6Z9w/B7v4QMR6Z9w_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BJl6TjRcY7/BJl6TjRcY7_layout.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:818d55a3b296061a5a3b14f3a82136858ab9a363b5739bdbe88ef737c0930560
|
| 3 |
+
size 6109195
|
parse/train/BJl6TjRcY7/BJl6TjRcY7_origin.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:f8bb84d14a09e4460be087a5bb600f3e9fa854ebb12249988416e964e8bf8255
|
| 3 |
+
size 5969410
|
parse/train/BJl6TjRcY7/BJl6TjRcY7_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:f7e6ae65dc5668448e9d4515eeddaa31591f12e9c0f17b3c258cef7bf8ab7c22
|
| 3 |
+
size 6110823
|
parse/train/BJlkgaNKvr/BJlkgaNKvr.md
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BJlkgaNKvr/BJlkgaNKvr_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BJlkgaNKvr/BJlkgaNKvr_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BJlkgaNKvr/BJlkgaNKvr_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BkV4VS9ll/BkV4VS9ll.md
ADDED
|
@@ -0,0 +1,599 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# THE INCREDIBLE SHRINKING NEURAL NETWORK: NEW PERSPECTIVES ON LEARNING REPRESENTATIONS THROUGH THE LENS OF PRUNING
|
| 2 |
+
|
| 3 |
+
Nikolas Wolfe, Aditya Sharma & Bhiksha Raj
|
| 4 |
+
School of Computer Science
|
| 5 |
+
Carnegie Mellon University
|
| 6 |
+
Pittsburgh, PA 15213, USA
|
| 7 |
+
{nwolfe, bhiksha}@cs.cmu.edu, adityasharma@cmu.edu
|
| 8 |
+
|
| 9 |
+
Lukas Drude Universitat Paderborn drude@nt.upb.de
|
| 10 |
+
|
| 11 |
+
# ABSTRACT
|
| 12 |
+
|
| 13 |
+
How much can pruning algorithms teach us about the fundamentals of learning representations in neural networks? A lot, it turns out. Neural network model compression has become a topic of great interest in recent years, and many different techniques have been proposed to address this problem. In general, this is motivated by the idea that smaller models typically lead to better generalization. At the same time, the decision of what to prune and when to prune necessarily forces us to confront our assumptions about how neural networks actually learn to represent patterns in data. In this work we set out to test several long-held hypotheses about neural network learning representations and numerical approaches to pruning. To accomplish this we first reviewed the historical literature and derived a novel algorithm to prune whole neurons (as opposed to the traditional method of pruning weights) from optimally trained networks using a second-order Taylor method. We then set about testing the performance of our algorithm and analyzing the quality of the decisions it made. As a baseline for comparison we used a first-order Taylor method based on the Skeletonization algorithm and an exhaustive brute-force serial pruning algorithm. Our proposed algorithm worked well compared to a first-order method, but not nearly as well as the brute-force method. Our error analysis led us to question the validity of many widely-held assumptions behind pruning algorithms in general and the trade-offs we often make in the interest of reducing computational complexity. We discovered that there is a straightforward way, however expensive, to serially prune $40 \%$ of the neurons in a trained network with minimal effect on the learning representation and without any re-training.
|
| 14 |
+
|
| 15 |
+
# 1 INTRODUCTION
|
| 16 |
+
|
| 17 |
+
In this work we propose and evaluate a novel algorithm for pruning whole neurons from a trained neural network without any re-training and examine its performance compared to two simpler methods. We then analyze the kinds of errors made by our algorithm and use this as a stepping off point to launch an investigation into the fundamental nature of learning representations in neural networks. Our results corroborate an insightful though largely forgotten observation by Mozer & Smolensky (1989a) concerning the nature of neural network learning. This observation is best summarized in a quotation from Segee & Carter (1991) on the notion of fault-tolerance in multilayer perceptron networks:
|
| 18 |
+
|
| 19 |
+
Contrary to the belief widely held, multilayer networks are not inherently fault tolerant. In fact, the loss of a single weight is frequently sufficient to completely disrupt a learned function approximation. Furthermore, having a large number of weights does not seem to improve fault tolerance. [Emphasis added]
|
| 20 |
+
|
| 21 |
+
Essentially, Mozer & Smolensky (1989b) observed that during training neural networks do not distribute the learning representation evenly or equitably across hidden units. What actually happens is that a few, elite neurons learn an approximation of the input-output function, and the remaining units must learn a complex interdependence function which cancels out their respective influence on the network output. Furthermore, assuming enough units exist to learn the function in question, increasing the number of parameters does not increase the richness or robustness of the learned approximation, but rather simply increases the likelihood of overfitting and the number of noisy parameters to be canceled during training. This is evinced by the fact that in many cases, multiple neurons can be removed from a network with no re-training and with negligible impact on the quality of the output approximation. In other words, there are few bipartisan units in a trained network. A unit is typically either part of the (possibly overfit) input-output function approximation, or it is part of an elaborate noise cancellation task force. Assuming this is the case, most of the compute-time spent training a neural network is likely occupied by this arguably wasteful procedure of silencing superfluous parameters, and pruning can be viewed as a necessary procedure to “trim the fat.”
|
| 22 |
+
|
| 23 |
+
We observed copious evidence of this phenomenon in our experiments, and this is the motivation behind our decision to evaluate the pruning algorithms in this study on the simple criteria of their ability to trim neurons without any re-training. If we were to employ re-training as part of our evaluation criteria, we would arguably not be evaluating the quality of our algorithm’s pruning decisions per se but rather the ability of back-propagation trained networks to recover from faults caused by non-ideal pruning decisions, as suggested by the conclusions of Segee & Carter (1991) and Mozer & Smolensky (1989a). Moreover, as Fahlman & Lebiere (1989) discuss, due to the “herd effect” and “moving target” phenomena in back-propagation learning, the remaining units in a network will simply shift course to account for whatever error signal is re-introduced as a result of a bad pruning decision or network fault. So long as there are enough critical parameters to learn the function in question, a network can typically recover faults with additional training. This limits the conclusions we can draw about the quality of our pruning criteria when we employ re-training.
|
| 24 |
+
|
| 25 |
+
In terms of removing units without re-training, what we discovered is that predicting the behavior of a network when a unit is to be pruned is very difficult, and most of the approximation techniques put forth in existing pruning algorithms do not fare well at all when compared to a brute-force search. To begin our discussion of how we arrived at our algorithm and set up our experiments, we review of the existing literature.
|
| 26 |
+
|
| 27 |
+
# 2 LITERATURE REVIEW
|
| 28 |
+
|
| 29 |
+
Pruning algorithms, as comprehensively surveyed by Reed (1993), are a useful set of heuristics designed to identify and remove elements from a neural network which are either redundant or do not significantly contribute to the output of the network. This is motivated by the observed tendency of neural networks to overfit to the idiosyncrasies of their training data given too many trainable parameters or too few input patterns from which to generalize, as stated by Chauvin (1990).
|
| 30 |
+
|
| 31 |
+
Network architecture design and hyperparameter selection are inherently difficult tasks typically approached using a few well-known rules of thumb, e.g. various weight initialization procedures, choosing the width and number of layers, different activation functions, learning rates, momentum, etc. Some of this “black art” appears unavoidable. For problems which cannot be solved using linear threshold units alone, Baum & Haussler (1989) demonstrate that there is no way to precisely determine the appropriate size of a neural network a priori given any random set of training instances. Using too few neurons seems to inhibit learning, and so in practice it is common to attempt to overparameterize networks initially using a large number of hidden units and weights, and then prune or compress them afterwards if necessary. Of course, as the old saying goes, there’s more than one way to skin a neural network.
|
| 32 |
+
|
| 33 |
+
# 2.1 NON-PRUNING BASED GENERALIZATION & COMPRESSION TECHNIQUES
|
| 34 |
+
|
| 35 |
+
The generalization behavior of neural networks has been well studied, and apart from pruning algorithms many heuristics have been used to avoid overfitting, such as dropout (Srivastava et al.
|
| 36 |
+
|
| 37 |
+
(2014)), maxout (Goodfellow et al. (2013)), and cascade correlation (Fahlman & Lebiere (1989)), among others. Of course, while cascade correlation specifically tries to construct of minimal networks, many techniques to improve network generalization do not explicitly attempt to reduce the total number of parameters or the memory footprint of a trained network per se.
|
| 38 |
+
|
| 39 |
+
Model compression often has benefits with respect to generalization performance and the portability of neural networks to operate in memory-constrained or embedded environments. Without explicitly removing parameters from the network, weight quantization allows for a reduction in the number of bytes used to represent each weight parameter, as investigated by Balzer et al. (1991), Dundar & Rose (1994), and Hoehfeld & Fahlman (1992).
|
| 40 |
+
|
| 41 |
+
A recently proposed method for compressing recurrent neural networks (Prabhavalkar et al. (2016)) uses the singular values of a trained weight matrix as basis vectors from which to derive a compressed hidden layer. Øland & Raj (2015) successfully implemented network compression through weight quantization with an encoding step while others such as Han et al. (2016) have tried to expand on this by adding weight-pruning as a preceding step to quantization and encoding.
|
| 42 |
+
|
| 43 |
+
In summary, we can say that there are many different ways to improve network generalization by altering the training procedure, the objective error function, or by using compressed representations of the network parameters. But these are not, strictly speaking, examples of techniques to reduce the number of parameters in a network. For this we must employ some form of pruning criteria.
|
| 44 |
+
|
| 45 |
+
# 2.2 PRUNING TECHNIQUES
|
| 46 |
+
|
| 47 |
+
If we wanted to continually shrink a neural network down to minimum size, the most straightforward brute-force way to do it is to individually switch each element off and measure the increase in total error on the training set. We then pick the element which has the least impact on the total error, and remove it. Rinse and repeat. This is extremely computationally expensive, given a reasonably large neural network and training set. Alternatively, we might accomplish this using any number of much faster off-the-shelf pruning algorithms, such as Skeletonization (Mozer & Smolensky (1989a)), Optimal Brain Damage (LeCun et al. (1989)), or later variants such as Optimal Brain Surgeon (Hassibi & Stork (1993)). In fact, we borrow much of our inspiration from these algorithms, with one major variation: Instead of pruning individual weights, we prune entire neurons, thereby eliminating all of their incoming and outgoing weight parameters in one go, resulting in more memory saved, faster.
|
| 48 |
+
|
| 49 |
+
The algorithm developed for this paper is targeted at reducing the total number of neurons in a trained network, which is one way of reducing its computational memory footprint. This is often a desirable criteria to minimize in the case of resource-constrained or embedded devices, and also allows us to probe the limitations of pruning down to the very last essential network elements. In terms of generalization as well, we can measure the error of the network on the test set as each element is sequentially removed from the network. With an oracle pruning algorithm, what we expect to observe is that the output of the network remains stable as the first few superfluous neurons are removed, and as we start to bite into the more crucial members of the function approximation, the error should start to rise dramatically. In this paper, the brute-force approach described at the beginning of this section serves as a proxy for an oracle pruning algorithm.
|
| 50 |
+
|
| 51 |
+
One reason to choose to rank and prune individual neurons as opposed to weights is that there are far fewer elements to consider. Furthermore, the removal of a single weight from a large network is a drop in the bucket in terms of reducing a network’s core memory footprint. If we want to reduce the size of a network as efficiently as possible, we argue that pruning neurons instead of weights is more efficient computationally as well as practically in terms of quickly reaching a hypothetical target reduction in memory consumption. This approach also offers downstream applications a realistic expectation of the minimal increase in error resulting from the removal of a specified percentage of neurons. Such trade-offs are unavoidable, but performance impacts can be limited if a principled approach is used to find the best candidate neurons for removal.
|
| 52 |
+
|
| 53 |
+
It is well known that too many free parameters in a neural network can lead to overfitting. Regardless of the number of weights used in a given network, as Segee & Carter (1991) assert, the representation of a learned function approximation is almost never evenly distributed over the hidden units, and thus the removal of any single hidden unit at random can actually result in a network fault. Mozer & Smolensky (1989b) argue that only a subset of the hidden units in a neural network actually latch on to the invariant or generalizing properties of the training inputs, and the rest learn to either mutually cancel each other’s influence or begin overfitting to the noise in the data. We leverage this idea in the current work to rank all neurons in pre-trained networks based on their effective contributions to the overall performance. We then remove the unnecessary neurons to reduce the network’s footprint. Through our experiments we not only concretely validate the theory put forth by Mozer & Smolensky (1989b) but we also successfully build on it to prune networks to 40 to 60 $\%$ of their original size without any major loss in performance.
|
| 54 |
+
|
| 55 |
+
# 3 PRUNING NEURONS TO SHRINK NEURAL NETWORKS
|
| 56 |
+
|
| 57 |
+
As discussed in Section 1 our aim is to leverage the highly non-uniform distribution of the learning representation in pre-trained neural networks to eliminate redundant neurons, without focusing on individual weight parameters. Taking this approach enables us to remove all the weights (incoming and outgoing) associated with a non-contributing neuron at once. We would like to note here that in an ideal scenario, based on the neuron interdependency theory put forward by Mozer & Smolensky (1989a), one would evaluate all possible combinations of neurons to remove (one at a time, two at a time, three at a time and so forth) to find the optimal subset of neurons to keep. This is computationally unacceptable, and so we will only focus on removing one neuron at a time and explore more “greedy” algorithms to do this in a more efficient manner.
|
| 58 |
+
|
| 59 |
+
The general approach taken to prune an optimally trained neural network here is to create a ranked list of all the neurons in the network based off of one of the 3 proposed ranking criteria: a brute force approximation, a linear approximation and a quadratic approximation of the neuron’s impact on the output of the network. We then test the effects of removing neurons on the accuracy and error of the network. All the algorithms and methods presented here are easily parallelizable as well.
|
| 60 |
+
|
| 61 |
+
One last thing to note here before moving forward is that the methods discussed in this section involve some non-trivial derivations which are beyond the scope of this paper. We are more focused on analyzing the implications of these methods on our understanding of neural network learning representations. However, a complete step-by-step derivation and proof of all the results presented is provided in the Supplementary Material as an Appendix.
|
| 62 |
+
|
| 63 |
+
# 3.1 BRUTE FORCE REMOVAL APPROACH
|
| 64 |
+
|
| 65 |
+
This is perhaps the most naive yet the most accurate method for pruning the network. It is also the slowest and hence possibly unusable on large-scale neural networks with thousands of neurons. This method explicitly evaluates each neuron in the network. The idea is to manually check the effect of every single neuron on the output. This is done by running a forward propagation on the validation set $K$ times (where $K$ is the total number of neurons in the network), turning off exactly one neuron each time (keeping all other neurons active) and noting down the change in error. Turning a neuron off can be achieved by simply setting its output to 0. This results in all the outgoing weights from that neuron being turned off. This change in error is then used to generate the ranked list.
|
| 66 |
+
|
| 67 |
+
# 3.2 TAYLOR SERIES REPRESENTATION OF ERROR
|
| 68 |
+
|
| 69 |
+
Let us denote the total error from the optimally trained neural network for any given validation dataset by $E$ . $E$ can be seen as a function of $O$ , where $O$ is the output of any general neuron in the network. This error can be approximated at a particular neuron’s output (say $O _ { k }$ ) by using the 2nd order Taylor Series as,
|
| 70 |
+
|
| 71 |
+
$$
|
| 72 |
+
\hat { E } ( \boldsymbol { O } ) \approx E ( \boldsymbol { O } _ { k } ) + ( \boldsymbol { O } - \boldsymbol { O } _ { k } ) \cdot \left. \frac { \partial E } { \partial \boldsymbol { O } } \right| _ { \boldsymbol { O } _ { k } } + 0 . 5 \cdot ( \boldsymbol { O } - \boldsymbol { O } _ { k } ) ^ { 2 } \cdot \left. \frac { \partial ^ { 2 } E } { \partial \boldsymbol { O } ^ { 2 } } \right| _ { \boldsymbol { O } _ { k } } ,
|
| 73 |
+
$$
|
| 74 |
+
|
| 75 |
+
When a neuron is pruned, its output $O$ becomes 0.
|
| 76 |
+
|
| 77 |
+
Replacing $O$ by $O _ { k }$ in equation 1 shows us that the error is approximated perfectly by equation 1 at $O _ { k }$ . So:
|
| 78 |
+
|
| 79 |
+
$$
|
| 80 |
+
\Delta E _ { k } = \hat { E } ( 0 ) - \hat { E } ( O _ { k } ) = - O _ { k } \cdot \left. \frac { \partial E } { \partial O } \right| _ { O _ { k } } + 0 . 5 \cdot O _ { k } ^ { 2 } \cdot \left. \frac { \partial ^ { 2 } E } { \partial O ^ { 2 } } \right| _ { O _ { k } } ,
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
where $\Delta E _ { k }$ is the change in the total error of the network when exactly one neuron $( k )$ is turned off. Most of the terms in this equation are fairly easy to compute, as we have $O _ { k }$ already from the activations of the hidden units and we already compute $\frac { \partial E } { \partial O } \big | _ { O _ { k } } ^ { \star }$ for each training instance during backpropagation. The ∂2E∂O2 |Ok terms are a little more difficult to compute. This is derived in the appendix and summarized in the sections below.
|
| 84 |
+
|
| 85 |
+
# 3.2.1 LINEAR APPROXIMATION APPROACH
|
| 86 |
+
|
| 87 |
+
We can use equation 2 to get the linear error approximation of the change in error due to the $k$ th neuron being turned off and represent it as $\Delta E _ { k } ^ { 1 }$ as follows:
|
| 88 |
+
|
| 89 |
+
$$
|
| 90 |
+
\Delta E _ { k } ^ { 1 } = - O _ { k } \cdot \left. \frac { \partial E } { \partial O } \right| _ { O _ { k } }
|
| 91 |
+
$$
|
| 92 |
+
|
| 93 |
+
The derivative term above is the first-order gradient which represents the change in error with respect to the output a given neuron. This term can be collected during back-propagation. As we shall see further in this section, linear approximations are not reliable indicators of change in error but they provide us with an interesting basis for comparison with the other methods discussed in this paper.
|
| 94 |
+
|
| 95 |
+
# 3.2.2 QUADRATIC APPROXIMATION APPROACH
|
| 96 |
+
|
| 97 |
+
As above, we can use equation 2 to get the quadratic error approximation of the change in error due to the $k$ th neuron being turned off and represent it as $\Delta E _ { k } ^ { 2 }$ as follows:
|
| 98 |
+
|
| 99 |
+
$$
|
| 100 |
+
\Delta E _ { k } ^ { 2 } = - O _ { k } \cdot \left. \frac { \partial E } { \partial O } \right| _ { O _ { k } } + 0 . 5 \cdot O _ { k } ^ { 2 } \cdot \left. \frac { \partial ^ { 2 } E } { \partial O ^ { 2 } } \right| _ { O _ { k } }
|
| 101 |
+
$$
|
| 102 |
+
|
| 103 |
+
The additional second-order gradient term appearing above represents the quadratic change in error with respect to the output of a given neuron. This term can be generated by performing backpropagation using second order derivatives. Collecting these quadratic gradients involves some non-trivial mathematics, the entire step-by-step derivation procedure of which is provided in the Supplementary Material as an Appendix.
|
| 104 |
+
|
| 105 |
+
# 3.3 PROPOSED PRUNING ALGORITHM
|
| 106 |
+
|
| 107 |
+
Figure 1 shows a random error function plotted against the output of any given neuron. Note that this figure is for illustration purposes only. The error function is minimized at a particular value of the neuron output as can be seen in the figure. The process of training a neural network is essentially the process of finding these minimizing output values for all the neurons in the network. Pruning this particular neuron (which translates to getting a zero output from it will result in a change in the total overall error. This change in error is represented by distance between the original minimum error (shown by the dashed line) and the top red arrow. This neuron is clearly a bad candidate for removal since removing it will result in a huge error increase.
|
| 108 |
+
|
| 109 |
+
The straight red line in the figure represents the first-order approximation of the error using Taylor Series as described before while the parabola represents a second-order approximation. It can be clearly seen that the second-order approximation is a much better estimate of the change in error.
|
| 110 |
+
|
| 111 |
+
One thing to note here is that it is possible in some cases that there is some thresholding required when trying to approximate the error using the 2nd order Taylor Series expansion. These cases might arise when the parabolic approximation undergoes a steep slope change. To take into account such cases, mean and median thresholding were employed, where any change above a certain threshold was assigned a mean or median value respectively.
|
| 112 |
+
|
| 113 |
+
|
| 114 |
+
Figure 1: The intuition behind 1st & 2nd order neuron pruning decisions
|
| 115 |
+
|
| 116 |
+
Two pruning algorithms are proposed here. They are different in the way the neurons are ranked but both of them use $\Delta E _ { k }$ , the approximation of the change in error as the basis for the ranking. $\Delta E _ { k }$ can be calculated using the Brute Force method, or one of the two Taylor Series approximations discussed previously.
|
| 117 |
+
|
| 118 |
+
The first step in both the algorithms is to decide a stopping criterion. This can vary depending on the application but some intuitive stopping criteria can be: maximum number of neurons to remove, percentage scaling needed, maximum allowable accuracy drop etc.
|
| 119 |
+
|
| 120 |
+
# 3.3.1 ALGORITHM I: SINGLE OVERALL RANKING
|
| 121 |
+
|
| 122 |
+
The complete algorithm is shown in Algorithm 1. The idea here is to generate a single ranked list based on the values of $\Delta E _ { k }$ . This involves a single pass of second-order back-propagation (without weight updates) to collect the gradients for each neuron. The neurons from this rank-list (with the lowest values of $\Delta E _ { k }$ ) are then pruned according to the stopping criterion decided. We note here that this algorithm is intentionally naive and is used for comparison only.
|
| 123 |
+
|
| 124 |
+
Data: optimally trained network, training set
|
| 125 |
+
Result: A pruned network
|
| 126 |
+
initialize and define stopping criterion ;
|
| 127 |
+
perform forward propagation over the training set ;
|
| 128 |
+
perform second-order back-propagation without updating weights and collect linear and quadratic gradients ;
|
| 129 |
+
rank the remaining neurons based on $\Delta E _ { k }$ ;
|
| 130 |
+
while stopping criterion is not met do
|
| 131 |
+
remove the last ranked neuron ;
|
| 132 |
+
end
|
| 133 |
+
|
| 134 |
+
Algorithm 1: Single Overall Ranking
|
| 135 |
+
|
| 136 |
+
# 3.3.2 ALGORITHM II: ITERATIVE RE-RANKING
|
| 137 |
+
|
| 138 |
+
In this greedy variation of the algorithm (Algorithm 2), after each neuron removal, the remaining network undergoes a single forward and backward pass of second-order back-propagation (without weight updates) and the rank list is formed again. Hence, each removal involves a new pass through the network. This method is computationally more expensive but takes into account the dependencies the neurons might have on one another which would lead to a change in error contribution every time a dependent neuron is removed.
|
| 139 |
+
|
| 140 |
+
Data: optimally trained network, training set
|
| 141 |
+
Result: A pruned network
|
| 142 |
+
initialize and define stopping criterion ;
|
| 143 |
+
while stopping criterion is not met do perform forward propagation over the training set ; perform second-order back-propagation without updating weights and collect linear and quadratic gradients ; rank the remaining neurons based on $\Delta E _ { k }$ ; remove the worst neuron based on the ranking ;
|
| 144 |
+
end
|
| 145 |
+
|
| 146 |
+
Algorithm 2: Iterative Re-Ranking
|
| 147 |
+
|
| 148 |
+
# 4 EXPERIMENTAL RESULTS
|
| 149 |
+
|
| 150 |
+
# 4.1 EXAMPLE REGRESSION PROBLEM
|
| 151 |
+
|
| 152 |
+
This problem serves as a quick example to demonstrate many of the phenomena described in previous sections. We trained two networks to learn the cosine function, with one input and one output. This is a task which requires no more than 11 sigmoid neurons to solve entirely, and in this case we don’t care about overfitting because the cosine function has a precise definition. Furthermore, the cosine function is a good toy example because it is a smooth continuous function and, as demonstrated by Nielsen (2015), if we were to tinker directly with the weights and bias parameters of the network, we could allocate individual units within the network to be responsible for constrained ranges of inputs, similar to a basis spline function with many control points. This would distribute the learned function approximation evenly across all hidden units, and thus we have presented the network with a problem in which it could productively use as many hidden units as we give it. In this case, a pruning algorithm would observe a fairly consistent increase in error after the removal of each successive unit. In practice however, regardless of the number of experimental trials, this is not what happens. The network will always use 10-11 hidden units and leave the rest to cancel each other’s influence.
|
| 153 |
+
|
| 154 |
+
|
| 155 |
+
Figure 2: Degradation in squared error after pruning a two-layer network trained to compute the cosine function (Left Network: 2 layers, 10 neurons each, 1 output, logistic sigmoid activation, starting test accuracy: 0.9999993, Right Network: 2 layers, 50 neurons each, 1 output, logistic sigmoid activation, starting test accuracy: 0.9999996)
|
| 156 |
+
|
| 157 |
+
Figure 2 shows two graphs. Both graphs demonstrate the use of the iterative re-ranking algorithm and the comparative performance of the brute-force pruning method (in blue), the first order method (in green), and the second order method (in red). The graph on the left shows the performance of these algorithms starting from a network with two layers of 10 neurons (20 total), and the graph on the right shows a network with two layers of 50 neurons (100 total).
|
| 158 |
+
|
| 159 |
+
In the left graph, we see that the brute-force method shows a graceful degradation, and the error only begins to rise sharply after $50 \%$ of the total neurons have been removed. The error is basically constant up to that point. In the first and second order methods, we see evidence of poor decision making in the sense that both made mistakes early on, which disrupted the output function approximation. The first order method made a large error early on, though we see after a few more neurons were removed this error was corrected somewhat (though it only got worse from there). This is direct evidence of the lack of fault tolerance in a trained neural network. This phenomenon is even more starkly demonstrated in the second order method. After making a few poor neuron removal decisions in a row, the error signal rose sharply, and then went back to zero after the 6th neuron was removed. This is due to the fact that the neurons it chose to remove were trained to cancel each others’ influence within a localized part of the network. After the entire group was eliminated, the approximation returned to normal. This can only happen if the output function approximation is not evenly distributed over the hidden units in a trained network.
|
| 160 |
+
|
| 161 |
+
This phenomenon is even more starkly demonstrated in the graph on the right. Here we see the first order method got “lucky” in the beginning and made decent decisions up to about the 40th removed neuron. The second order method had a small error in the beginning which it recovered from gracefully and proceeded to pass the 50 neuron point before finally beginning to unravel. The brute force method, in sharp contrast, shows little to no increase in error at all until $90 \%$ of the neurons in the network have been obliterated. Clearly first and second order methods have some value in that they do not make completely arbitrary choices, but the brute force method is far better at this task.
|
| 162 |
+
|
| 163 |
+
This also demonstrates the sharp dualism in neuron roles within a trained network. These networks were trained to near-perfect precision and each pruning method was applied without any re-training of any kind. Clearly, in the case of the brute force or oracle method, up to $90 \%$ of the network can be completely extirpated before the output approximation even begins to show any signs of degradation. This would be impossible if the learning representation were evenly or equitably distributed. Note, for example, that the degradation point in both cases is approximately the same. This example is not a real-world application of course, but it brings into very clear focus the kind of phenomena we will discuss in the following sections.
|
| 164 |
+
|
| 165 |
+
# 4.2 RESULTS ON MNIST DATASET
|
| 166 |
+
|
| 167 |
+
For all the results presented in this section, the MNIST database of Handwritten Digits by LeCun & Cortes (2010) was used. It is worth noting that due to the time taken by the brute force algorithm we rather used a 5000 image subset of the MNIST database in which we have normalized the pixel values between 0 and 1.0, and compressed the image sizes to $2 0 \mathrm { x } 2 0$ images rather than $2 8 \mathbf { x } 2 8$ , so the starting test accuracy reported here appears higher than those reported by LeCun et al. We do not believe that this affects the interpretation of the presented results because the basic learning problem does not change with a larger dataset or input dimension.
|
| 168 |
+
|
| 169 |
+
# 4.3 PRUNING A 1-LAYER NETWORK
|
| 170 |
+
|
| 171 |
+
The network architecture in this case consisted of 1 layer, 100 neurons, 10 outputs, logistic sigmoid activations, and a starting test accuracy of 0.998.
|
| 172 |
+
|
| 173 |
+
# 4.3.1 SINGLE OVERALL RANKING ALGORITHM
|
| 174 |
+
|
| 175 |
+
We first present the results for a single-layer neural network in Figure 3, using the Single Overall algorithm (Algorithm 1) as proposed in Section 3. (We again note that this algorithm is intentionally naive and is used for comparison only. Its performance should be expected to be poor.) After training, each neuron is assigned its permanent ranking based on the three criteria discussed previously: A brute force “ground truth” ranking, and two approximations of this ranking using first and second order Taylor estimations of the change in network output error resulting from the removal of each neuron.
|
| 176 |
+
|
| 177 |
+
An interesting observation here is that with only a single layer, no criteria for ranking the neurons in the network (brute force or the two Taylor Series variants) using Algorithm 1 emerges superior, indicating that the 1st and 2nd order Taylor Series methods are actually reasonable approximations of the brute force method under certain conditions. Of course, this method is still quite bad in terms of the rate of degradation of the classification accuracy and in practice we would likely follow Algorithm 2 which is takes into account Mozer & Smolensky (1989a)’s observations stated in the Related Work section. The purpose of the present investigation, however, is to demonstrate how much of a trained network can be theoretically removed without altering the network’s learned parameters in any way.
|
| 178 |
+
|
| 179 |
+
|
| 180 |
+
Figure 3: Degradation in squared error (left) and classification accuracy (right) after pruning a single-layer network using The Single Overall Ranking algorithm (Network: 1 layer, 100 neurons, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.998)
|
| 181 |
+
|
| 182 |
+
|
| 183 |
+
4.3.2 ITERATIVE RE-RANKING ALGORITHM
|
| 184 |
+
Figure 4: Degradation in squared error (left) and classification accuracy (right) after pruning a single-layer network the iterative re-ranking algorithm (Network: 1 layer, 100 neurons, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.998)
|
| 185 |
+
|
| 186 |
+
In Figure 4 we present our results using Algorithm 2 (The iterative re-ranking Algorithm) in which all remaining neurons are re-ranked after each successive neuron is switched off. We compute the same brute force rankings and Taylor series approximations of error deltas over the remaining active neurons in the network after each pruning decision. This is intended to account for the effects of cancelling interactions between neurons.
|
| 187 |
+
|
| 188 |
+
There are 2 key observations here. Using the brute force ranking criteria, almost $60 \%$ of the neurons in the network can be pruned away without any major loss in performance. The other noteworthy observation here is that the 2nd order Taylor Series approximation of the error performs consistently better than its 1st order version, in most situations, though Figure 21 is a poignant counter-example.
|
| 189 |
+
|
| 190 |
+
# 4.3.3 VISUALIZATION OF ERROR SURFACE & PRUNING DECISIONS
|
| 191 |
+
|
| 192 |
+
As explained in Section 3, these graphs are a visualization of the error surface of the network output with respect to the neurons chosen for removal using each of the 3 ranking criteria, represented in intervals of 10 neurons. In each graph, the error surface of the network output is displayed in log space (left) and in real space (right) with respect to each candidate neuron chosen for removal. We create these plots during the pruning exercise by picking a neuron to switch off, and then multiplying its output by a scalar gain value $\alpha$ which is adjusted from 0.0 to 10.0 with a step size of 0.001. When the value of $\alpha$ is 1.0, this represents the unperturbed neuron output learned during training. Between 0.0 and 1.0, we are graphing the literal effect of turning the neuron off $( \alpha = 0$ ), and when $\alpha > 1 . 0$ we are simulating a boosting of the neuron’s influence in the network, i.e. inflating the value of its outgoing weight parameters.
|
| 193 |
+
|
| 194 |
+
We graph the effect of boosting the neuron’s output to demonstrate that for certain neurons in the network, even doubling, tripling, or quadrupling the scalar output of the neuron has no effect on the overall error of the network, indicating the remarkable degree to which the network has learned to ignore the value of certain parameters. In other cases, we can get a sense of the sensitivity of the network’s output to the value of a given neuron when the curve rises steeply after the red 1.0 line. This indicates that the learned value of the parameters emanating from a given neuron are relatively important, and this is why we should ideally see sharper upticks in the curves for the later-removed neurons in the network, that is, when the neurons crucial to the learning representation start to be picked off. Some very interesting observations can be made in each of these graphs.
|
| 195 |
+
|
| 196 |
+
Remember that lower is better in terms of the height of the curve and minimal (or negative) horizontal change between the vertical red line at 1.0 (neuron on, $\alpha = 1 . 0$ ) and 0.0 (neuron off, $\alpha = 0 . 0$ ) is indicative of a good candidate neuron to prune, i.e. there will be minimal effect on the network output when the neuron is removed.
|
| 197 |
+
|
| 198 |
+
|
| 199 |
+
4.3.4 VISUALIZATION OF BRUTE FORCE PRUNING DECISIONS
|
| 200 |
+
Figure 5: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the brute force criterion; (Network: 1 layer, 100 neurons, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.998)
|
| 201 |
+
|
| 202 |
+
In Figure ??, we notice how low to the floor and flat most of the curves are. It’s only until the 90th removed neuron that we see a higher curve with a more convex shape (clearly a more sensitive, influential piece of the network).
|
| 203 |
+
|
| 204 |
+
# 4.3.5 VISUALIZATION OF 1ST ORDER APPROXIMATION PRUNING DECISIONS
|
| 205 |
+
|
| 206 |
+
It can be seen in Figure 6 that most choices seem to have flat or negatively sloped curves, indicating that the first order approximation seems to be pretty good, but examining the brute force choices shows they could be better.
|
| 207 |
+
|
| 208 |
+
# 4.3.6 VISUALIZATION OF 2ND ORDER APPROXIMATION PRUNING DECISIONS
|
| 209 |
+
|
| 210 |
+
The method in Figure 7 looks similar to the brute force method choices, though clearly not as good (they’re more spread out). Notice the difference in convexity between the 2nd and 1st order method
|
| 211 |
+
|
| 212 |
+
|
| 213 |
+
Figure 6: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the 1st order Taylor Series error approximation criterion; (Network: 1 layer, 100 neurons, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.998)
|
| 214 |
+
|
| 215 |
+
|
| 216 |
+
Figure 7: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the 2nd order Taylor Series error approximation criterion; (Network: 1 layer, 100 neurons, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.998)
|
| 217 |
+
|
| 218 |
+
choices. It’s clear that the first order method is fitting a line and the 2nd order method is fitting a parabola in their approximation.
|
| 219 |
+
|
| 220 |
+
# 4.4 PRUNING A 2-LAYER NETWORK
|
| 221 |
+
|
| 222 |
+
The network architecture in this case consisted of 2 layers, 50 neurons per layer, 10 outputs, logistic sigmoid activations, and a starting test accuracy of 1.000.
|
| 223 |
+
|
| 224 |
+
# 4.4.1 SINGLE OVERALL RANKING ALGORITHM
|
| 225 |
+
|
| 226 |
+
Figure 8 shows the pruning results for Algorithm 1 on a 2-layer network. The ranking procedure is identical to the one used to generate Figure 3. (We again note that this algorithm is intentionally naive and is used for comparison only. Its performance should be expected to be poor.)
|
| 227 |
+
|
| 228 |
+
Unsurprisingly, a 2-layer network is harder to prune because a single overall ranking will never capture the interdependencies between neurons in different layers. It makes sense that this is worse
|
| 229 |
+
|
| 230 |
+
|
| 231 |
+
Figure 8: Degradation in squared error (left) and classification accuracy (right) after pruning a 2- layer network using the Single Overall Ranking algorithm; (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 1.000)
|
| 232 |
+
|
| 233 |
+
than the performance on the 1-layer network, even if this method is already known to be bad, and we’d likely never use it in practice.
|
| 234 |
+
|
| 235 |
+
# 4.4.2 ITERATIVE RE-RANKING ALGORITHM
|
| 236 |
+
|
| 237 |
+
|
| 238 |
+
Figure 9: Degradation in squared error (left) and classification accuracy (right) after pruning a 2- layer network using the iterative re-ranking algorithm; (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 1.000)
|
| 239 |
+
|
| 240 |
+
Figure 9 shows the results from using Algorithm 2 on a 2-layer network. We compute the same brute force rankings and Taylor series approximations of error deltas over the remaining active neurons in the network after each pruning decision used to generate Figure 4. Again, this is intended to account for the effects of cancelling interactions between neurons.
|
| 241 |
+
|
| 242 |
+
It is clear that it becomes harder to remove neurons 1-by-1 with a deeper network (which makes sense because the neurons have more interdependencies in a deeper network), but we see an overall better performance with 2nd order method vs. 1st order, except for the first $20 \%$ of the neurons (but this doesn’t seem to make much difference for classification accuracy.)
|
| 243 |
+
|
| 244 |
+
Perhaps a more important observation here is that even with a more complex network, it is possible to remove up to $40 \%$ of the neurons with no major loss in performance which is clearly illustrated by the brute force curve. This shows the clear potential of an ideal pruning technique and also shows how inconsistent 1st and 2nd order Taylor Series approximations of the error can be as ranking criteria.
|
| 245 |
+
|
| 246 |
+
# 4.4.3 VISUALIZATION OF ERROR SURFACE & PRUNING DECISIONS
|
| 247 |
+
|
| 248 |
+
As seen in the case of a single layered network, these graphs are a visualization the error surface of the network output with respect to the neurons chosen for removal using each algorithm, represented in intervals of 10 neurons.
|
| 249 |
+
|
| 250 |
+
|
| 251 |
+
4.4.4 VISUALIZATION OF BRUTE FORCE PRUNING DECISIONS
|
| 252 |
+
Figure 10: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the brute force criterion; (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 1.000)
|
| 253 |
+
|
| 254 |
+
In Figure 10, it is clear why these neurons got chosen, their graphs clearly show little change when neuron is removed, are mostly near the floor, and show convex behaviour of error surface, which argues for the rationalization of using 2nd order methods to estimate difference in error when they are turned off.
|
| 255 |
+
|
| 256 |
+
|
| 257 |
+
4.4.5 VISUALIZATION OF 1ST ORDER APPROXIMATION PRUNING DECISIONS
|
| 258 |
+
Figure 11: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the 1st order Taylor Series error approximation criterion; (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 1.000)
|
| 259 |
+
|
| 260 |
+
Drawing a flat line at the point of each neurons intersection with the red vertical line (no change in gain) shows that the 1st derivative method is actually accurate for estimation of change in error in these cases, but still ultimately leads to poor decisions.
|
| 261 |
+
|
| 262 |
+
|
| 263 |
+
Figure 12: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the 2nd order Taylor Series error approximation criterion; (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 1.000)
|
| 264 |
+
|
| 265 |
+
Clearly these neurons are not overtly poor candidates for removal (error doesn’t change much between 1.0 & zero-crossing left-hand-side), but could be better (as described above in the brute force Criterion discussion).
|
| 266 |
+
|
| 267 |
+
# 4.5 INVESTIGATION OF PRUNING PERFORMANCE WITH IMPERFECT STARTING CONDITIONS
|
| 268 |
+
|
| 269 |
+
In our experiments thus far we have tacitly assumed that we start with a network which has learned an “optimal” representation of the training objective, i.e. it has been trained to the point where we accept its performance on the test set. Here we explore what happens when we prune with a sub-optimal starting network.
|
| 270 |
+
|
| 271 |
+
If the assumptions of this paper regarding the nature of neural network learning are correct, we expect that two processes are essentially at work during back-propagation training. First, we expect that the neurons which directly participate in the fundamental learning representation (even if redundantly) work together to reduce error on the training data. Second, we expect that neurons which do not directly participate in the learning representation work to cancel each other’s negative influence. Furthermore, we expect that these two groups are essentially distinct, as evinced by the fact that multiple neurons can often be removed as a group with little to no effect on the network output. Some non-trivial portion of the training time, then, is spent doing work which has nothing intrinsically to do with the learning representation and essentially functions as noise cancellation.
|
| 272 |
+
|
| 273 |
+
If this is the case, when we attempt to prune a network which has not fully canceled the noisy influence of extraneous or redundant units, we might expect to see the error actually improve after removing a few bad apples. This is in fact what we observe, as demonstrated in the following experiments.
|
| 274 |
+
|
| 275 |
+
For each experiment in this section we trained with the full MNIST training set (LeCun & Cortes (2010)), uncompressed and without any data normalization. We trained three different networks to learn to distinguish a single handwritten digit from the rest of the data. The network architectures were each composed of 784 inputs, 1 hidden layer with 100 neurons, and 2 soft-max outputs; one to say yes, and the other to say no. These networks were trained to distinguish the digits 0, 1, and 2, and their respective starting accuracies were a sub-optimal 0.9757, 0.9881, and 0.9513. Finally, we only consider the iterative re-ranking algorithm, as the single overall ranking algorithm is clearly nonviable.
|
| 276 |
+
|
| 277 |
+
# 4.5.1 MNIST SINGLE DIGIT CLASSIFICATION: DIGIT 0
|
| 278 |
+
|
| 279 |
+
Figure 13 shows the degradation in squared error after removing neurons from a network trained to distinguish the digit 0. What we observe is that the first and second order methods both fail in different ways, though clearly the second order method makes better decisions overall. The first order method explodes spectacularly in the first few iterations. The brute force method, in stark contrast, actually improves in the first few iterations, and remains essentially flat until around the $60 \%$ mark, at which point it begins to gradually increase and meet the other curves.
|
| 280 |
+
|
| 281 |
+
|
| 282 |
+
Figure 13: Degradation in squared error after pruning a single-layer network trained to do a oneversus-all classification of the digit 0 using the iterative re-ranking algorithm
|
| 283 |
+
|
| 284 |
+
The behavior of the brute force method here demonstrates that the network was essentially working to cancel the effect of a few bad neurons when the training convergence criteria were met, i.e. the network was no longer able to make progress on the training set. After removing these neurons during pruning, the output improved. We can investigate this by looking at the error surface with respect to the neurons chosen for removal by each method in turn. Below in Figure 14 is the graph of the brute force method.
|
| 285 |
+
|
| 286 |
+
|
| 287 |
+
Figure 14: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the brute force iterative re-ranking removal criterion
|
| 288 |
+
|
| 289 |
+
Figure 14 shows an interesting phenomenon, which we will see in later experiments as well. The high blue curve corresponding to neuron 0 is negatively sloped in the beginning and clearly after removing this neuron, the output will improve. The rest of the curves, in correspondence with the squared error degradation curve above, are mostly flat and tightly layered together, indicating that they are good neurons to remove.
|
| 290 |
+
|
| 291 |
+
In Figure 15 below, we observe a stark contrast to this. The curves corresponding to neurons 0 and 10 are mostly flat, and fairly lower than the rest, though clearly a mistake was made early on and the rest of the curves are clearly bad choices. In all of these cases however, we see that the curves are easily approximated with a straight line and so the first order method may have been fairly accurate in its predictions, even though it still made poor decisions.
|
| 292 |
+
|
| 293 |
+
|
| 294 |
+
Figure 15: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the first-order iterative re-ranking removal criterion
|
| 295 |
+
|
| 296 |
+
Figure 15 is an example of how things can go south once a few bad mistakes are made at the outset. Figure 16 shows a much better set of choices made by the second order method, though clearly not as good as the brute force method. The log-space plots make it a bit easier to see the difference between the brute force and second order methods in Figures 14 and 16, respectively.
|
| 297 |
+
|
| 298 |
+
|
| 299 |
+
Figure 16: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the second-order iterative re-ranking removal criterion
|
| 300 |
+
|
| 301 |
+
# 4.5.2 MNIST SINGLE DIGIT CLASSIFICATION: DIGIT 1
|
| 302 |
+
|
| 303 |
+
Examining Figure 17, we see a much starker example of the previous phenomenon, in which the brute force method continues to improve the performance of the network after removing $80 \%$ of the neurons in the network. The first and second order methods fail early and proceed in fits and starts (clearly demonstrating evidence of interrelated groups of noise-canceling neurons), and never fully recover. It should be noted that it would be impossible to see curves like this if neural networks evenly distributed the learning representation evenly or equitably over their hidden units.
|
| 304 |
+
|
| 305 |
+
One of the most striking things about the blue curve in Figure 17 is the fact that the network never drops below its starting error until it crosses the $80 \%$ mark, indicating that only $20 \%$ of the neurons in this network are actually essential to the learning the training objective. In this sense, we can only wonder how much of the training time was spent winnowing the error out of the remaining $80 \%$ of the network.
|
| 306 |
+
|
| 307 |
+
|
| 308 |
+
Figure 17: Degradation in squared error after pruning a single-layer network trained to do a oneversus-all classification of the digit 1 using the iterative re-ranking algorithm
|
| 309 |
+
|
| 310 |
+
|
| 311 |
+
Figure 18: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the brute force iterative re-ranking removal criterion
|
| 312 |
+
|
| 313 |
+
In Figures 18, 19 and 20 we can examine the choices made by the respective methods. The brute force method serves as our example of a near-optimal pruning regimen, and the rest are first and second order approximations of this. Small differences, clearly, can lead to large effects on the network output as shown in Figure 17.
|
| 314 |
+
|
| 315 |
+
# 4.5.3 MNIST SINGLE DIGIT CLASSIFICATION: DIGIT 2
|
| 316 |
+
|
| 317 |
+
Figure 21 is an interesting case because it shatters our confidence in the reliability of the second order method to make good pruning decisions, and further demonstrates the phenomenon of how much the error can improve if the right neurons are removed after training gets stuck. In this case, though still a poor performance overall, the first order method vastly outperforms the second order method.
|
| 318 |
+
|
| 319 |
+
Figure 22 shows us a clear example of the first element to remove having a negative error slope, and improving the output as a result. The rest of the pruning decisions are reasonable. Comparing with the blue curve in Figure 21, we see the correspondence between the first pruning decision improving the output, and the remaining pruning decisions keeping the output fairly flat. Clearly, however, there isn’t much room to get worse given our starting point with a sub-optimal network, and we see that the ending sum of squared errors is not much higher than the starting point. At the same time, we can still see the contrast in performance if we make optimal pruning decisions, and most of the neurons in this network were clearly doing nothing.
|
| 320 |
+
|
| 321 |
+
|
| 322 |
+
Figure 19: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the first-order iterative re-ranking removal criterion
|
| 323 |
+
|
| 324 |
+
|
| 325 |
+
Figure 20: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the second-order iterative re-ranking removal criterion
|
| 326 |
+
|
| 327 |
+
|
| 328 |
+
Figure 21: Degradation in squared error after pruning a single-layer network trained to do a oneversus-all classification of the digit 2 using the iterative re-ranking algorithm
|
| 329 |
+
|
| 330 |
+
In Figure 23, we see a mixed bag in which the decisions are clearly sub-optimal, though much better than Figure 24, in which we can observe how a bad first decision essentially ruined the network for good. The jagged edges of the red curve in Figure 21 correspond with the positive and negative slopes of the cluster of bad pruning decisions in 24. Once again, these are not necessarily bad decisions, but the starting point is already bad and this cannot be recovered without re-training the network.
|
| 331 |
+
|
| 332 |
+
|
| 333 |
+
Figure 22: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the brute force iterative re-ranking removal criterion
|
| 334 |
+
|
| 335 |
+
|
| 336 |
+
Figure 23: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the first-order iterative re-ranking removal criterion
|
| 337 |
+
|
| 338 |
+
# 4.5.4 ASIDE: IMPLICATIONS OF THIS EXPERIMENT
|
| 339 |
+
|
| 340 |
+
From the three examples above, we see that in each case, starting from a sub-optimal network, a brute force removal technique consistently improves performance for the first few pruning iterations, and the sum of squared errors does not degrade beyond the starting point until around $60 \%$ of the neurons have been removed. This is only possible if we have an essentially strict dichotomy between the roles of different neurons during training. If the network needs only $20 \%$ of the neurons it began with, the training process is essentially dominated by the task of canceling the residual noise of redundant neurons. Furthermore, the network can get stuck in training with redundant units and distort the final output. This is strong evidence of our thesis that the learning representation is neither equitably nor evenly distributed and that most of the neurons which do not directly participate in the learning representation can be removed without any retraining.
|
| 341 |
+
|
| 342 |
+
# 4.6 EXPERIMENTS ON TOY DATASETS
|
| 343 |
+
|
| 344 |
+
As can be seen from the experiments on MNIST, even though the 2nd-order approximation criterion is consistently better than 1st-order, its performance is not nearly as good as brute force based ranking, especially beyond the first layer. What is interesting to note is that from some other experiments conducted on toy datasets (predicting whether a given point would lie inside a given shape on the Cartesian plane), the performance of the 2nd-order method was found to be exceptionally good and produced results very close to the brute force method. The 1st-order method, as expected, performed poorly here as well. Some of these results are illustrated in Figure 25.
|
| 345 |
+
|
| 346 |
+
|
| 347 |
+
Figure 24: Error surface of the network output in log space (left) and real space (right) with respect to each candidate neuron chosen for removal using the second-order iterative re-ranking removal criterion
|
| 348 |
+
|
| 349 |
+
|
| 350 |
+
Figure 25: Degradation in squared error (left) and classification accuracy (right) after pruning a 2- layer network using the iterative re-ranking algorithm on a toy “diamond” shape dataset (top) and a toy “random shape” dataset (below); (Network: 2 layers, 50 neurons/layer, 10 outputs, logistic sigmoid activation, starting test accuracy: 0.992(diamond); 0.986(random shape)
|
| 351 |
+
|
| 352 |
+
# 5 CONCLUSIONS & FUTURE WORK
|
| 353 |
+
|
| 354 |
+
In conclusion, we must first re-assert that we do not present this work as a bench-marking study of the algorithm we derived and tested. We have merely used this algorithm as a jumping off point to investigate the nature of learning representations in neural networks. What we discovered is that first and second order methods do not make particularly good pruning decisions, and can get hopelessly lost after making a bad pruning decision resulting in a network fault. Furthermore, the brute-force algorithm does surprisingly well, despite being computationally expensive. This method does so well in fact, we argue that further investigation is warranted to make this algorithm computationally tractable, though we do not speculate on how that should be done here.
|
| 355 |
+
|
| 356 |
+
We also observed strong evidence for the hypotheses of Mozer & Smolensky (1989a) regarding the “dualist” nature of hidden units, i.e. that learning representations are divided between units which either participate in the output approximation or learn to cancel each others influence. This suggests that neural networks may in fact learn a minimal network implicitly, though we cannot say for sure that this is the case without further investigation. A necessary experiment to this end would be to compare the size of network constructed using cascade correlation (Fahlman & Lebiere (1989)) and compare it to the results described herein.
|
| 357 |
+
|
| 358 |
+
We have presented a novel algorithm for pruning whole neurons from a trained neural network using a second-order Taylor series approximation of the change in error resulting from the removal a given neuron as a pruning criteria. We compared this method to a first order method and a bruteforce serial removal method which exhaustively found the next best single neuron to remove at each stage. Our algorithm relies on a combination of assumptions similar to the ones made by Mozer & Smolensky (1989a) and LeCun et al. (1989) in the formulation of the Skeletonization and Optimal Brain Damage algorithms.
|
| 359 |
+
|
| 360 |
+
First, we assumed that the error function with respect to each individual neuron can be approximated with a straight line or more precisely with a parabola. Second, for second derivative terms we consider only the diagonal elements of the Hessian matrix, i.e. we assume that each neuron-weight connection can be treated independently of the other elements in the network. Third, we assumed that pruning could be done in a serial fashion in which we find the single least productive element in the network, remove it, and move on. We found that all of these assumptions are deeply flawed in the sense that the true relevance of a neuron can only be partially approximated by a first or second order method, and only at certain stages of the pruning process.
|
| 361 |
+
|
| 362 |
+
For most problems, these methods can usually remove between $10 { - } 3 0 \%$ of the neurons in a trained network, but beyond this point their reliability breaks down. For certain problems, none of the described methods seem to perform very well, though for obvious reasons the brute-force method always exhibits the best results. The reason for this is that the error function with respect to each hidden unit is more complex than a simple second-order Taylor series can approximate. Furthermore, we have not directly taken into account the interdependence of elements within a network, though the work of Hassibi & Stork (1993) could provide some guidance in this regard. This is another critical issue to investigate in the future.
|
| 363 |
+
|
| 364 |
+
Re-training may help in this regard. We freely admit that our algorithm does not use re-training to recover from errors made in pruning decisions. We argue that evaluating a network pruning algorithm using re-training does not allow us to make fair comparisons between the kinds of decisions made by these algorithms. Neural networks are very good at recovering from the removal of individual elements with re-training and so this compensates for sub-optimal pruning criteria.
|
| 365 |
+
|
| 366 |
+
We have observed that pruning whole neurons from an optimally trained network without major loss in performance is not only possible but also enables compressing networks to $40 \%$ of their original size, which is of great importance in constrained memory environments like embedded devices. We cite the results of our experiments using the brute force criterion as evidence of this conclusion. However expensive, it would be extremely easy to parallelize this method, or potentially approximate it using a subset of the training data to decide which neurons to prune. This avoids the problem of trying to approximate the importance of a unit and potentially making a mistake.
|
| 367 |
+
|
| 368 |
+
It would also be interesting to see how these methods perform on deeper networks and on some other popular and real world datasets. In our case, on the MNIST dataset, we observed that it was more difficult to prune neurons from a deeper network than from one with a single layer. We should expect this trend to continue as networks get deeper and deeper, which also calls into further question the reliability of the described first and second order methods. We did investigate the order in which neurons were plucked from each layer of the networks and we found that the brute force method primarily removes neurons from the deepest layer of the network first, but there was no obvious pattern in layer preference for the other two methods.
|
| 369 |
+
|
| 370 |
+
Our experiments using the visualization of error surfaces and pruning decisions concretely establish the fact that not all neurons in a network contribute to its performance in the same way, and the observed complexity of these functions demonstrates limitations of the approximations we used.
|
| 371 |
+
|
| 372 |
+
Finally, we encourage the readers of this work to take these results into consideration when making decisions as to which methods to use to improve network generalization or compress their models. It should be remembered that various heuristics may perform well in practice for reasons which are in fact orthogonal to the accepted justifications given by their proponents.
|
| 373 |
+
|
| 374 |
+
# REFERENCES
|
| 375 |
+
|
| 376 |
+
Wolfgang Balzer, Masanobu Takahashi, Jun Ohta, and Kazuo Kyuma. Weight quantization in boltzmann machines. Neural Networks, 4(3):405–409, 1991.
|
| 377 |
+
|
| 378 |
+
Eric B Baum and David Haussler. What size net gives valid generalization? Neural computation, 1 (1):151–160, 1989.
|
| 379 |
+
|
| 380 |
+
Yves Chauvin. Generalization performance of overtrained back-propagation networks. In Neural Networks, pp. 45–55. Springer, 1990.
|
| 381 |
+
|
| 382 |
+
Gunhan Dundar and Kenneth Rose. The effects of quantization on multilayer neural networks. IEEE transactions on neural networks/a publication of the IEEE Neural Networks Council, 6(6): 1446–1451, 1994.
|
| 383 |
+
|
| 384 |
+
Scott E Fahlman and Christian Lebiere. The cascade-correlation learning architecture. 1989.
|
| 385 |
+
|
| 386 |
+
Ian J Goodfellow, David Warde-Farley, Mehdi Mirza, Aaron Courville, and Yoshua Bengio. Maxout networks. arXiv preprint arXiv:1302.4389, 2013.
|
| 387 |
+
|
| 388 |
+
Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149v5, 2016.
|
| 389 |
+
|
| 390 |
+
Babak Hassibi and David G Stork. Second order derivatives for network pruning: Optimal brain surgeon. Morgan Kaufmann, 1993.
|
| 391 |
+
|
| 392 |
+
Markus Hoehfeld and Scott E Fahlman. Learning with limited numerical precision using the cascade-correlation algorithm. IEEE Transactions on Neural Networks, 3(4):602–611, 1992.
|
| 393 |
+
|
| 394 |
+
Yann LeCun and Corinna Cortes. MNIST handwritten digit database. 2010.
|
| 395 |
+
|
| 396 |
+
Yann LeCun, John S Denker, Sara A Solla, Richard E Howard, and Lawrence D Jackel. Optimal brain damage. In NIPs, volume 89, 1989.
|
| 397 |
+
|
| 398 |
+
Michael C Mozer and Paul Smolensky. Skeletonization: A technique for trimming the fat from a network via relevance assessment. In Advances in neural information processing systems, pp. 107–115, 1989a.
|
| 399 |
+
|
| 400 |
+
Michael C Mozer and Paul Smolensky. Using relevance to reduce network size automatically. Connection Science, 1(1):3–16, 1989b.
|
| 401 |
+
|
| 402 |
+
Michael A Nielsen. Neural networks and deep learning. http://neuralnetworksanddeeplearning.com/chap4.html.(visited: 01.11. 2014), 2015.
|
| 403 |
+
|
| 404 |
+
URL:
|
| 405 |
+
|
| 406 |
+
Anders Øland and Bhiksha Raj. Reducing communication overhead in distributed learning by an order of magnitude (almost). In IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 2219–2223, 2015.
|
| 407 |
+
|
| 408 |
+
Rohit Prabhavalkar, Ouais Alsharif, Antoine Bruguier, and Lan McGraw. On the compression of recurrent neural networks with an application to lvcsr acoustic modeling for embedded speech recognition. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5970–5974. IEEE, 2016.
|
| 409 |
+
|
| 410 |
+
Russell Reed. Pruning algorithms-a survey. Neural Networks, IEEE Transactions on, 4(5):740–747, 1993.
|
| 411 |
+
|
| 412 |
+
Bruce E Segee and Michael J Carter. Fault tolerance of pruned multilayer networks. In Neural Networks, 1991., IJCNN-91-Seattle International Joint Conference on, volume 2, pp. 447–452. IEEE, 1991.
|
| 413 |
+
|
| 414 |
+
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014.
|
| 415 |
+
|
| 416 |
+
# APPENDIX
|
| 417 |
+
|
| 418 |
+
# A SECOND DERIVATIVE BACK-PROPAGATION
|
| 419 |
+
|
| 420 |
+
|
| 421 |
+
Figure 26: A computational graph of a simple feed-forward network illustrating the naming of different variables, where $\sigma ( \cdot )$ is the nonlinearity, MSE is the mean-squared error cost function and $E$ is the overall loss.
|
| 422 |
+
|
| 423 |
+
Name and network definitions:
|
| 424 |
+
|
| 425 |
+
$$
|
| 426 |
+
E = { \frac { 1 } { 2 } } \sum _ { i } ( o _ { i } ^ { ( 0 ) } - t _ { i } ) ^ { 2 } \quad o _ { i } ^ { ( m ) } = \sigma ( x _ { i } ^ { ( m ) } ) \quad x _ { i } ^ { ( m ) } = \sum _ { j } w _ { j i } ^ { ( m ) } o _ { j } ^ { ( m + 1 ) } \quad c _ { j i } ^ { ( m ) } = w _ { j i } ^ { ( m ) } o _ { j } ^ { ( m + 1 ) }
|
| 427 |
+
$$
|
| 428 |
+
|
| 429 |
+
Superscripts represent the index of the layer of the network in question, with 0 representing the output layer. $E$ is the squared-error network cost function. $o _ { i } ^ { ( m ) }$ is the $i$ th output in layer $m$ generated by the activation function $\sigma$ , which in this paper is is the standard logistic sigmoid . x(m)i i s the weighted sumneuron in the puts to the ith neuron in the layer to the input of the ith $m$ th layer, and ron in the m $c _ { j i } ^ { ( m ) }$ is the contribution of the er. $j$ th $m + 1$
|
| 430 |
+
|
| 431 |
+
# A.1 FIRST AND SECOND DERIVATIVES
|
| 432 |
+
|
| 433 |
+
The first and second derivatives of the cost function with respect to the outputs:
|
| 434 |
+
|
| 435 |
+
$$
|
| 436 |
+
\frac { \partial E } { \partial o _ { i } ^ { ( 0 ) } } = o _ { i } ^ { ( 0 ) } - t _ { i }
|
| 437 |
+
$$
|
| 438 |
+
|
| 439 |
+
$$
|
| 440 |
+
\frac { \partial ^ { 2 } E } { \partial o _ { i } ^ { ( 0 ) ^ { 2 } } } = 1
|
| 441 |
+
$$
|
| 442 |
+
|
| 443 |
+
The first and second derivatives of the sigmoid function in forms depending only on the output:
|
| 444 |
+
|
| 445 |
+
$$
|
| 446 |
+
\begin{array} { l } { { \sigma ^ { \prime } ( x ) = \sigma ( x ) \left( 1 - \sigma ( x ) \right) } } \\ { { \sigma ^ { \prime \prime } ( x ) = \sigma ^ { \prime } ( x ) \left( 1 - 2 \sigma ( x ) \right) } } \end{array}
|
| 447 |
+
$$
|
| 448 |
+
|
| 449 |
+
The second derivative of the sigmoid is easily derived from the first derivative:
|
| 450 |
+
|
| 451 |
+
$$
|
| 452 |
+
{ \begin{array} { l } { \displaystyle \sigma ^ { \prime } ( x ) = \sigma ( x ) \left( 1 - \sigma ( x ) \right) } \\ { \displaystyle \sigma ^ { \prime \prime } ( x ) = { \frac { \mathrm { d } } { \mathrm { d } x } } { \frac { \sigma ( x ) } { f ( x ) } } { \frac { \left( 1 - \sigma ( x ) \right) } { g ( x ) } } } \\ { \displaystyle \sigma ^ { \prime \prime } ( x ) = f ^ { \prime } ( x ) g ( x ) + f ( x ) g ^ { \prime } ( x ) } \\ { \displaystyle \sigma ^ { \prime \prime } ( x ) = \sigma ^ { \prime } ( x ) \left( 1 - \sigma ( x ) \right) - \sigma ( x ) \sigma ^ { \prime } ( x ) } \\ { \displaystyle \sigma ^ { \prime \prime } ( x ) = \sigma ^ { \prime } ( x ) - 2 \sigma ( x ) \sigma ^ { \prime } ( x ) } \\ { \displaystyle \sigma ^ { \prime \prime } ( x ) = \sigma ^ { \prime } ( x ) \left( 1 - 2 \sigma ( x ) \right) } \end{array} }
|
| 453 |
+
$$
|
| 454 |
+
|
| 455 |
+
And for future convenience:
|
| 456 |
+
|
| 457 |
+
$$
|
| 458 |
+
\begin{array} { l } { { \displaystyle { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \frac { \mathrm { d } } { \mathrm { d } x _ { i } ^ { ( m ) } } \left( \sigma _ { i } ^ { ( m ) } = \sigma ( x _ { i } ^ { ( m ) } ) \right) } } } \\ { { \displaystyle { \phantom { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \left( \sigma _ { i } ^ { ( m ) } \right) \left( 1 - \sigma _ { i } ^ { ( m ) } \right) } } } } \\ { { \displaystyle { \phantom { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \sigma ^ { \prime } \left( x _ { i } ^ { ( m ) } \right) } \left( \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \left( \sigma _ { i } ^ { ( m ) } \right) \left( 1 - \sigma _ { i } ^ { ( m ) } \right) \right) } } } \\ { { \displaystyle { \phantom { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \frac { \mathrm { d } } { \mathrm { d } x _ { i } ^ { ( m ) } } \left( \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \left( \sigma _ { i } ^ { ( m ) } \right) \left( 1 - 2 \sigma _ { i } ^ { ( m ) } \right) \right) } } } } \\ { \displaystyle { \phantom { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \left( \sigma _ { i } ^ { ( m ) } \left( 1 - \sigma _ { i } ^ { ( m ) } \right) \right) \left( 1 - 2 \sigma _ { i } ^ { ( m ) } \right) } } } \\ { \displaystyle { \phantom { \frac { \mathrm { d } { \partial _ { i } ^ { ( m ) } } } { \mathrm { d } x _ { i } ^ { ( m ) } } = \sigma ^ { \prime } \left( x _ { i } ^ { ( m ) } \right) } } } \end{array}
|
| 459 |
+
$$
|
| 460 |
+
|
| 461 |
+
Derivative of the error with respect to the ith neuron’s input $x _ { i } ^ { ( 0 ) }$ in the output layer:
|
| 462 |
+
|
| 463 |
+
$$
|
| 464 |
+
\begin{array} { r l } & { \frac { \partial E } { \partial x _ { i } ^ { ( 0 ) } } = \frac { \partial E } { \partial \sigma _ { i } ^ { ( 0 ) } } \frac { \partial \sigma _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } } \\ & { \qquad = \underbrace { \left( \sigma _ { i } ^ { ( 0 ) } - t _ { i } \right) } _ { \mathrm { f r o m ~ ( 6 ) } } \underbrace { \sigma \left( x _ { i } ^ { ( 0 ) } \right) \left( 1 - \sigma \left( x _ { i } ^ { ( 0 ) } \right) \right) } _ { \mathrm { f r o m ~ ( 8 ) } } } \\ & { \qquad = \left( \sigma _ { i } ^ { ( 0 ) } - t _ { i } \right) \left( \sigma _ { i } ^ { ( 0 ) } \left( 1 - \sigma _ { i } ^ { ( 0 ) } \right) \right) } \\ & { \qquad = \left( \sigma _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) } \end{array}
|
| 465 |
+
$$
|
| 466 |
+
|
| 467 |
+
Second derivative of the error with respect to the ith neuron’s input $x _ { i } ^ { ( 0 ) }$ in the output layer:
|
| 468 |
+
|
| 469 |
+
$$
|
| 470 |
+
\begin{array} { r l r } { { \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( 0 ) ^ { 2 } } } = \frac { \partial } { \partial x _ { i } ^ { ( 0 ) } } ( \frac { \partial E } { \partial \sigma _ { i } ^ { ( 0 ) } } \frac { \partial \sigma _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } ) } } \\ & { } & { = \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( 0 ) } \partial \sigma _ { i } ^ { ( 0 ) } } \frac { \partial \sigma _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } + \frac { \partial E } { \partial \sigma _ { i } ^ { ( 0 ) } } \frac { \partial ^ { 2 } \sigma _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) ^ { 2 } } } } \\ & { } & { = \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( 0 ) } \partial \sigma _ { i } ^ { ( 0 ) } } \underbrace { ( \frac { \sigma _ { i } ^ { ( 0 ) } } { i } ( 1 - \sigma _ { i } ^ { ( 0 ) } ) ) } _ { \mathrm { f r o m ~ ( 8 ) } } + \underbrace { ( \sigma _ { i } ^ { ( 0 ) } - t _ { i } ) } _ { \mathrm { f r o m ~ ( 6 ) } } \underbrace { ( \sigma _ { i } ^ { ( 0 ) } ( 1 - \sigma _ { i } ^ { ( 0 ) } ) ) ( 1 - 2 \sigma _ { i } ^ { ( 0 ) } ) } _ { \mathrm { f r o m ~ ( 9 ) } } } \end{array}
|
| 471 |
+
$$
|
| 472 |
+
|
| 473 |
+
$$
|
| 474 |
+
\left( \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( 0 ) } \partial \sigma _ { i } ^ { ( 0 ) } } \right) = \frac { \partial } { \partial x _ { i } ^ { ( 0 ) } } \frac { \partial E } { \partial \sigma _ { i } ^ { ( 0 ) } } = \frac { \partial } { \partial x _ { i } ^ { ( 0 ) } } \underbrace { \left( \sigma _ { i } ^ { ( 0 ) } - t _ { i } \right) } _ { \left( \partial x _ { i } ^ { ( 0 ) } \right) } = \frac { \partial \sigma _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } = \underbrace { \left( \sigma _ { i } ^ { ( 0 ) } \left( 1 - \sigma _ { i } ^ { ( 0 ) } \right) \right) } _ { \left( \partial x _ { i } ^ { ( 0 ) } \right) }
|
| 475 |
+
$$
|
| 476 |
+
|
| 477 |
+
$$
|
| 478 |
+
\begin{array} { c } { { \displaystyle { \frac { \partial ^ { 2 } E } { \partial { x _ { i } ^ { ( 0 ) } } ^ { 2 } } = \left( o _ { i } ^ { ( 0 ) } \left( 1 - o _ { i } ^ { ( 0 ) } \right) \right) ^ { 2 } + \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \left( o _ { i } ^ { ( 0 ) } \left( 1 - o _ { i } ^ { ( 0 ) } \right) \right) \left( 1 - 2 o _ { i } ^ { ( 0 ) } \right) } } } \\ { { \mathrm { } } } \\ { { \displaystyle { = \left( \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) \right) ^ { 2 } + \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime \prime } \left( x _ { i } ^ { ( 0 ) } \right) } } } \end{array}
|
| 479 |
+
$$
|
| 480 |
+
|
| 481 |
+
First derivative of the error with respect to a single input contribution $c _ { j i } ^ { ( 0 ) }$ from neuron $j$ to neuron i with weight w(0)ji in the output layer:
|
| 482 |
+
|
| 483 |
+
$$
|
| 484 |
+
\begin{array} { r l r } { { \frac { \partial E } { \partial c _ { j i } ^ { ( 0 ) } } = \frac { \partial E } { \partial o _ { i } ^ { ( 0 ) } } \frac { \partial o _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } \frac { \partial x _ { i } ^ { ( 0 ) } } { \partial c _ { j i } ^ { ( 0 ) } } } } \\ & { } & { = \underbrace { ( o _ { i } ^ { ( 0 ) } - t _ { i } ) } _ { \mathrm { f r o m ~ ( 6 ) } } \underbrace { ( o _ { i } ^ { ( 0 ) } ( 1 - o _ { i } ^ { ( 0 ) } ) ) } _ { \mathrm { f r o m ~ ( 8 ) } } \frac { \partial x _ { i } ^ { ( 0 ) } } { \partial c _ { j i } ^ { ( 0 ) } } } \end{array}
|
| 485 |
+
$$
|
| 486 |
+
|
| 487 |
+
$$
|
| 488 |
+
\begin{array} { r l } & { \left( \displaystyle \frac { \partial x _ { i } ^ { ( m ) } } { \partial c _ { j i } ^ { ( m ) } } \right) = \displaystyle \frac { \partial } { \partial c _ { j i } ^ { ( m ) } } \left( x _ { i } ^ { ( m ) } = \sum _ { j } w _ { j i } ^ { ( m ) } o _ { j } ^ { ( m + 1 ) } \right) = \displaystyle \frac { \partial } { \partial c _ { j i } ^ { ( m ) } } \left( c _ { j i } ^ { ( m ) } + k \right) = 1 } \\ & { \qquad \displaystyle \frac { \partial E } { \partial c _ { j i } ^ { ( 0 ) } } = \left( o _ { i } ^ { ( 0 ) } - l _ { i } \right) \left( o _ { i } ^ { ( 0 ) } \left( 1 - o _ { i } ^ { ( 0 ) } \right) \right) } \\ & { \qquad \displaystyle = \underbrace { \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) } _ { \mathrm { t r o m ~ ( 2 5 ) } } } \\ & { \qquad \displaystyle \frac { \partial E } { \partial c _ { j i } ^ { ( 0 ) } } = \frac { \partial E } { \partial x _ { i } ^ { ( 0 ) } } } \end{array}
|
| 489 |
+
$$
|
| 490 |
+
|
| 491 |
+
Second derivative of the error with respect to a single input contribution $c _ { j i } ^ { ( 0 ) }$ :
|
| 492 |
+
|
| 493 |
+
$$
|
| 494 |
+
\begin{array} { r l } { \frac { \partial ^ { 2 } E } { \partial t _ { 3 } ^ { ( 0 ) } } ^ { 2 } = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \frac { \partial E } { \partial x _ { 3 } ^ { ( 0 ) } } ^ { 2 } = \underbrace { ( \alpha ^ { ( 0 ) } - i \delta ) \sigma ^ { \prime } ( x _ { 3 } ^ { ( 0 ) } ) } _ { i \mathrm { m i n } ( \infty ( s ) ) } ) } & { } \\ { = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( ( \varepsilon ( \frac { i \delta } { s } ) ^ { 2 } ) - i _ { z } ) \sigma ^ { \prime } ( \overline { { \alpha } } _ { i } ^ { ( 0 ) } ) } & { } \\ { = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \tau ( \sum _ { s } w _ { i } ^ { ( 0 ) } , \sigma _ { i j } ^ { ( 0 ) } + 1 ) - \kappa _ { k } ) \sigma ^ { \prime } ( \sum _ { s } w _ { i } ^ { ( 0 ) } , \sigma _ { j i } ^ { ( 0 + 1 ) } ) } & { } \\ { = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \tau ( \sum _ { s } w _ { i } ^ { ( 0 ) } ) - i _ { z } ) \sigma ^ { \prime } ( \sum _ { s } w _ { i } ^ { ( 0 ) } ) } & { } \\ { = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \frac { \partial \sigma ( x _ { i j } ^ { ( 0 ) } + k ) } { \partial t _ { 3 } ^ { ( 0 ) } } - 1 ) \sigma ^ { \prime } ( \sum _ { s } w _ { i } ^ { ( 0 ) } ) } & { } \\ = \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } \frac { \partial } { \partial t _ { 3 } ^ { ( 0 ) } } ( \frac { \partial \sigma ( x _ { i j } ^ { ( 0 ) } + k ) } { \partial t _ { 3 } ^ { ( 0 ) } ( \varepsilon ( x _ { i j } ^ { ( 0 ) } ) } - 1 ) \sigma ^ { \prime } ( \sum _ { s } w _ { i } ^ \end{array}
|
| 495 |
+
$$
|
| 496 |
+
|
| 497 |
+
We now make use of the abbreviations $f$ and $g$ :
|
| 498 |
+
|
| 499 |
+
$$
|
| 500 |
+
\begin{array} { r l } & { \quad = \bar { f } \left( \mathcal { L } _ { \beta } ^ { ( 0 ) } \right) g \left( c _ { \beta \gamma } ^ { ( 0 ) } \right) + \bar { f } \left( c _ { \beta \gamma } ^ { ( 0 ) } \right) g ^ { \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } \right) } \\ & { \quad = \sigma ^ { \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) \sigma ^ { \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) + \left( \sigma \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) - l _ { \downarrow } \right) \sigma ^ { \prime \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) } \\ & { \quad = \sigma ^ { \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) ^ { 2 } + \left( \sigma _ { \downarrow } ^ { ( 0 ) } - t _ { \downarrow } \right) \sigma ^ { \prime \prime } \left( c _ { \beta \gamma } ^ { ( 0 ) } + k \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \frac { \partial ^ { 2 } F } { \partial c _ { \beta \gamma } ^ { ( 0 ) } } } \\ & { \quad \quad \quad \quad \quad \frac { \partial ^ { 2 } F } { \partial c _ { \beta \gamma } ^ { ( 0 ) } } ^ { 2 } = \underbrace { \left( \sigma ^ { \prime } \left( x _ { \langle } ^ { ( 0 ) } \right) \right) ^ { 2 } + \left( \sigma _ { \downarrow } ^ { ( 0 ) } - t _ { \downarrow } \right) \sigma ^ { \prime \prime } \left( c _ { \downarrow } ^ { ( 0 ) } \right) } _ { \mathrm { e m ~ \ c a l ~ \lambda \mit t } } } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array}
|
| 501 |
+
$$
|
| 502 |
+
|
| 503 |
+
# A.1.1 SUMMARY OF OUTPUT LAYER DERIVATIVES
|
| 504 |
+
|
| 505 |
+
$$
|
| 506 |
+
\frac { \partial E } { \partial o _ { i } ^ { ( 0 ) } } = o _ { i } ^ { ( 0 ) } - t _ { i } \quad \quad \quad \quad \quad \frac { \partial ^ { 2 } E } { { \partial o _ { i } ^ { ( 0 ) } } ^ { 2 } } = 1
|
| 507 |
+
$$
|
| 508 |
+
|
| 509 |
+
$$
|
| 510 |
+
\frac { \partial E } { \partial x _ { i } ^ { ( 0 ) } } = \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) \quad \quad \frac { \partial ^ { 2 } E } { \partial { x _ { i } ^ { ( 0 ) } } ^ { 2 } } = \left( \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) \right) ^ { 2 } + \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime \prime } \left( x _ { i } ^ { ( 0 ) } \right)
|
| 511 |
+
$$
|
| 512 |
+
|
| 513 |
+
$$
|
| 514 |
+
{ \frac { \partial E } { \partial c _ { j i } ^ { ( 0 ) } } } = { \frac { \partial E } { \partial x _ { i } ^ { ( 0 ) } } } \qquad { \frac { \partial ^ { 2 } E } { \partial c _ { j i } ^ { ( 0 ) ^ { 2 } } } } = { \frac { \partial ^ { 2 } E } { \partial { x _ { i } ^ { ( 0 ) } } ^ { 2 } } }
|
| 515 |
+
$$
|
| 516 |
+
|
| 517 |
+
# A.1.2 HIDDEN LAYER DERIVATIVES
|
| 518 |
+
|
| 519 |
+
The first derivative of the error with respect to a neuron with output $o _ { j } ^ { ( 1 ) }$ in the first hidden layer, summing over all partial derivative contributions from the output layer:
|
| 520 |
+
|
| 521 |
+
$$
|
| 522 |
+
\frac { \partial E } { \partial o _ { j } ^ { ( 1 ) } } = \sum _ { i } \frac { \partial E } { \partial o _ { i } ^ { ( 0 ) } } \frac { \partial o _ { i } ^ { ( 0 ) } } { \partial x _ { i } ^ { ( 0 ) } } \frac { \partial x _ { i } ^ { ( 0 ) } } { \partial c _ { j i } ^ { ( 0 ) } } \frac { \partial c _ { j i } ^ { ( 0 ) } } { \partial o _ { j } ^ { ( 1 ) } } = \sum _ { i } \underbrace { \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime } \left( x _ { i } ^ { ( 0 ) } \right) } _ { \textit { i } } w _ { j i } ^ { ( 0 ) }
|
| 523 |
+
$$
|
| 524 |
+
|
| 525 |
+
$$
|
| 526 |
+
\begin{array} { c } { { \displaystyle { \frac { \partial c _ { j i } ^ { ( m ) } } { \partial o _ { j } ^ { ( m + 1 ) } } = \frac { \partial } { \partial o _ { j } ^ { ( m + 1 ) } } \left( c _ { j i } ^ { ( m ) } = w _ { j i } ^ { ( m ) } o _ { j } ^ { ( m + 1 ) } \right) = w _ { j i } ^ { ( m ) } } } } \\ { { \displaystyle { \frac { \partial E } { \partial o _ { j } ^ { ( 1 ) } } = \sum _ { i } \frac { \partial E } { \partial x _ { i } ^ { ( 0 ) } } w _ { j i } ^ { ( 0 ) } } } } \end{array}
|
| 527 |
+
$$
|
| 528 |
+
|
| 529 |
+
Note that this equation does not depend on the specific form o f ∂E∂x(0) , whether it involves a sigmoid or any other activation function. We can therefore replace the specific indexes with general ones, and use this equation in the future.
|
| 530 |
+
|
| 531 |
+
$$
|
| 532 |
+
\frac { \partial E } { \partial o _ { j } ^ { ( m + 1 ) } } = \sum _ { i } \frac { \partial E } { \partial x _ { i } ^ { ( m ) } } w _ { j i } ^ { ( m ) }
|
| 533 |
+
$$
|
| 534 |
+
|
| 535 |
+
The second derivative of the error with respect to a neuron with output $o _ { j } ^ { ( 1 ) }$ in the first hidden layer:
|
| 536 |
+
|
| 537 |
+
$$
|
| 538 |
+
\begin{array} { l } { { \displaystyle { \frac { \partial ^ { 2 } E } { \partial { o _ { j } ^ { ( 1 ) } } ^ { 2 } } = \frac { \partial } { \partial { o _ { j } ^ { ( 1 ) } } } \frac { \partial E } { \partial { o _ { j } ^ { ( 1 ) } } } } } } \\ { { \displaystyle { \ = \frac { \partial } { \partial { o _ { j } ^ { ( 1 ) } } } \sum _ { i } \frac { \partial E } { \partial { x _ { i } ^ { ( 0 ) } } } w _ { j i } ^ { ( 0 ) } } } } \\ { { \displaystyle { \ = \frac { \partial } { \partial { o _ { j } ^ { ( 1 ) } } } \sum _ { i } \left( o _ { i } ^ { ( 0 ) } - t _ { i } \right) \sigma ^ { \prime } \left( { x _ { i } ^ { ( 0 ) } } \right) w _ { j i } ^ { ( 0 ) } } } } \end{array}
|
| 539 |
+
$$
|
| 540 |
+
|
| 541 |
+
If we now make use of the fact, that $\begin{array} { r } { o _ { i } ^ { ( 0 ) } = \sigma \left( x _ { i } ^ { ( 0 ) } \right) = \sigma \left( \sum _ { j } \left( w _ { j i } ^ { ( 0 ) } o _ { j } ^ { ( 1 ) } \right) \right) } \end{array}$ , we can evaluate the expression further.
|
| 542 |
+
|
| 543 |
+
$$
|
| 544 |
+
\begin{array} { r l } { \frac { \partial \hat { \mathcal { F } } ( R ) } { \partial \theta _ { 0 } ^ { ( 1 ) } } = \frac { \partial } { \partial \theta _ { 0 } ^ { ( 1 ) } } \sum _ { t } ( \underbrace { \sigma ( \sum _ { s } \alpha _ { 0 } ^ { ( 0 , 0 , 1 ) } \alpha _ { 0 } ^ { ( 1 ) } ) } _ { t \in \mathcal { N } _ { 1 } ^ { ( 0 , 1 ) } } - \nu _ { s } ) \cdot ( \sum _ { s } \gamma _ { s } \alpha _ { 0 } ^ { ( 0 , 0 , 0 ) } ) \frac { \alpha _ { 0 } ^ { ( 1 ) } } { s ! } } & { = } \\ & { \quad - \sum _ { s } ( \mathcal { C } ( \frac { \partial ^ { ( 0 , 0 ) } } { \partial s } ) \alpha _ { 0 } ^ { ( 0 , 0 ) } ) - f ( \alpha _ { 0 } ^ { ( 1 ) } - 1 ) \cdot \sqrt { ( \alpha _ { 0 } ^ { ( 0 , 1 ) } ) } } & { = } \\ & { \quad - \sum _ { s ^ { \prime } } ( ( \sum _ { s } \alpha _ { 0 } ^ { ( 0 , 0 ) } \beta _ { 0 } ^ { ( 1 ) } ) \alpha _ { 0 } ^ { ( 0 , 0 ) } - f ( \sum _ { s } \alpha _ { 0 } ^ { ( 1 ) } ) ) \cdot ( \alpha _ { 0 } ^ { ( 0 , 1 ) } ) } \\ & { \quad \sum _ { s ^ { \prime } } ( \varepsilon ( \sum _ { s } \gamma _ { s } ^ { ( 0 , 0 , 1 ) } ) \alpha _ { 0 } ^ { ( 1 ) } - \varepsilon ( \sum _ { s } \alpha _ { 0 } ^ { ( 0 , 0 ) } \beta _ { 0 } ^ { ( 1 ) } ) ) \frac { \alpha _ { 0 } ^ { ( 1 ) } } { s ^ { 2 } } + \dots } \\ & \quad \sum _ { s ^ { \prime } } ( \varepsilon ( \sum _ { s } \gamma _ { s } ^ { ( 0 , 0 , 1 ) } ) - ( \sum _ { s } \gamma _ { s } ^ { ( 0 , 0 , 0 ) } \varepsilon ) \frac { \alpha _ { 0 } ^ { ( 1 ) } } { s ^ { 2 } } ) \cdot ( \sum _ { s } \alpha _ { 0 } ^ { ( 1 ) } \varepsilon ) \frac { \partial ^ { ( 0 , 0 ) } } { \partial s ^ { \prime } } ) ( \varepsilon ( \frac { \partial ^ { ( 0 , 0 ) } } \end{array}
|
| 545 |
+
$$
|
| 546 |
+
|
| 547 |
+
Summing up, we obtain the more general expression:
|
| 548 |
+
|
| 549 |
+
$$
|
| 550 |
+
\frac { \partial ^ { 2 } E } { { \partial o _ { j } ^ { ( 1 ) } } ^ { 2 } } = \sum _ { i } \frac { \partial ^ { 2 } E } { { \partial x _ { i } ^ { ( 0 ) } } ^ { 2 } } \left( w _ { j i } ^ { ( 0 ) } \right) ^ { 2 }
|
| 551 |
+
$$
|
| 552 |
+
|
| 553 |
+
Note that the equation in (65) does not depend on the form o f ∂2E∂x(0)x 2 , which means we can replace the specific indexes with general ones:
|
| 554 |
+
|
| 555 |
+
$$
|
| 556 |
+
\frac { \partial ^ { 2 } E } { \partial o _ { j } ^ { ( m + 1 ) ^ { 2 } } } = \sum _ { i } \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( m ) ^ { 2 } } } \left( w _ { j i } ^ { ( m ) } \right) ^ { 2 }
|
| 557 |
+
$$
|
| 558 |
+
|
| 559 |
+
At this point we are beginning to see the recursion in the form of the 2nd derivative terms which can be thought of analogously to the first derivative recursion which is central to the back-propagation algorithm. The formulation above which makes specific reference to layer indexes also works in the general case.
|
| 560 |
+
|
| 561 |
+
Consider the $i$ th neuron in any layer $m$ with output $o _ { i } ^ { ( m ) }$ and input $x _ { i } ^ { ( m ) }$ . The first and second derivatives of the error $E$ with respect to this neuron’s input are:
|
| 562 |
+
|
| 563 |
+
$$
|
| 564 |
+
\frac { \partial E } { \partial x _ { i } ^ { ( m ) } } = \frac { \partial E } { \partial o _ { i } ^ { ( m ) } } \frac { \partial o _ { i } ^ { ( m ) } } { \partial x _ { i } ^ { ( m ) } }
|
| 565 |
+
$$
|
| 566 |
+
|
| 567 |
+
$$
|
| 568 |
+
\begin{array} { r l } & { \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = \frac { \partial } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } \frac { \partial E } { \partial x _ { 1 } ^ { ( \mathrm { i n k } ) } } } \\ & { \phantom { \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = } - \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } \frac { \partial E } { \partial x _ { 1 } ^ { ( \mathrm { i n k } ) } } } \\ & { \phantom { \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = } - \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } ( \frac { \partial ^ { 2 } E } { \partial z _ { 1 } ^ { ( \mathrm { i n k } ) } } \frac { \partial ^ { 2 } E ^ { ( \mathrm { i n k } ) } } { \partial z _ { 1 } ^ { ( \mathrm { i n k } ) } } ) } \\ & { \phantom { \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = } - \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } \frac { \partial ^ { 2 } E } { \partial z _ { 1 } ^ { ( \mathrm { i n k } ) } } \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } } \\ & { \phantom { \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = } \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } \partial z _ { 1 } ^ { ( \mathrm { i n k } ) } } \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } } \\ & \phantom \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } = \frac { \partial ^ { 2 } E } { \partial z _ { 2 } ^ { ( \mathrm { i n k } ) } } ( \frac { \partial ^ { 2 } E } { \partial z _ { 1 } ^ { ( \mathrm { i n k } ) } } - \frac { \partial ^ { 2 } E } \partial z _ { 1 } ^ ( \ \end{array}
|
| 569 |
+
$$
|
| 570 |
+
|
| 571 |
+
Note the form of this equation is the general form of what was derived for the output layer in (31). Both of the above first and second terms are easily computable and can be stored as we propagate back from the output of the network to the input. With respect to the output layer, the first and second derivative terms have already been derived above. In the case of the $m + 1$ hidden layer during back propagation, there is a summation of terms calculated in the mth layer. For the first derivative, we have this from (55).
|
| 572 |
+
|
| 573 |
+
$$
|
| 574 |
+
\frac { \partial E } { \partial o _ { j } ^ { ( m + 1 ) } } = \sum _ { i } \frac { \partial E } { \partial x _ { i } ^ { ( m ) } } w _ { j i } ^ { ( m ) }
|
| 575 |
+
$$
|
| 576 |
+
|
| 577 |
+
And the second derivative for the $j$ th neuron in the $m + 1$ layer:
|
| 578 |
+
|
| 579 |
+
$$
|
| 580 |
+
\frac { \partial ^ { 2 } E } { \partial x _ { j } ^ { ( m + 1 ) ^ { 2 } } } = \frac { \partial ^ { 2 } E } { \partial o _ { j } ^ { ( m + 1 ) ^ { 2 } } } \left( \sigma ^ { \prime } \left( x _ { j } ^ { ( m + 1 ) } \right) \right) ^ { 2 } + \frac { \partial E } { \partial o _ { j } ^ { ( m + 1 ) } } \sigma ^ { \prime \prime } \left( x _ { j } ^ { ( m + 1 ) } \right)
|
| 581 |
+
$$
|
| 582 |
+
|
| 583 |
+
We can replace both derivative terms with the forms which depend on the previous layer:
|
| 584 |
+
|
| 585 |
+
$$
|
| 586 |
+
\frac { \partial ^ { 2 } E } { \partial x _ { j } ^ { ( m + 1 ) ^ { 2 } } } = \underbrace { \sum _ { i } \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( 0 ) ^ { 2 } } } \left( w _ { j i } ^ { ( 0 ) } \right) ^ { 2 } } _ { \mathrm { f r o m ~ ( 6 6 ) } } \left( \sigma ^ { \prime } \left( x _ { j } ^ { ( m + 1 ) } \right) \right) ^ { 2 } + \underbrace { \sum _ { i } \frac { \partial E } { \partial x _ { i } ^ { ( m ) } } w _ { j i } ^ { ( m ) } } _ { \mathrm { f r o m ~ ( 5 5 ) } } \sigma ^ { \prime \prime } \left( x _ { j } ^ { ( m + 1 ) } \right)
|
| 587 |
+
$$
|
| 588 |
+
|
| 589 |
+
And this horrible mouthful of an equation gives you a general form for any neuron in the $j$ th position of the $m + 1$ layer. Taking very careful note of the indexes, this can be more or less translated painlessly to code. You are welcome, world.
|
| 590 |
+
|
| 591 |
+
# A.1.3 SUMMARY OF HIDDEN LAYER DERIVATIVES
|
| 592 |
+
|
| 593 |
+
$$
|
| 594 |
+
\frac { \partial E } { \partial \sigma _ { j } ^ { ( m + 1 ) } } = \sum _ { i } \frac { \partial E } { \partial x _ { i } ^ { ( m ) } } w _ { j i } ^ { ( m ) } \qquad \frac { \partial ^ { 2 } E } { \partial \sigma _ { j } ^ { ( m + 1 ) ^ { 2 } } } = \sum _ { i } \frac { \partial ^ { 2 } E } { \partial x _ { i } ^ { ( m ) ^ { 2 } } } \left( w _ { j i } ^ { ( m ) } \right) ^ { 2 }
|
| 595 |
+
$$
|
| 596 |
+
|
| 597 |
+
$$
|
| 598 |
+
\begin{array} { r l r } { { \frac { \partial E } { \partial x _ { i } ^ { ( m ) } } = \frac { \partial E } { \partial o _ { i } ^ { ( m ) } } \frac { \partial o _ { i } ^ { ( m ) } } { \partial x _ { i } ^ { ( m ) } } } } \\ & { \frac { \partial ^ { 2 } E } { \partial x _ { j } ^ { ( m + 1 ) ^ { 2 } } } = \frac { \partial ^ { 2 } E } { \partial o _ { j } ^ { ( m + 1 ) ^ { 2 } } } ( \sigma ^ { \prime } ( x _ { j } ^ { ( m + 1 ) } ) ) ^ { 2 } + \frac { \partial E } { \partial o _ { j } ^ { ( m + 1 ) } } \sigma ^ { \prime \prime } ( x _ { j } ^ { ( m + 1 ) } ) } \end{array}
|
| 599 |
+
$$
|
parse/train/BkV4VS9ll/BkV4VS9ll_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|