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 +580 -0
- parse/train/3_MUAtqR0aA/3_MUAtqR0aA.md +217 -0
- parse/train/3_MUAtqR0aA/3_MUAtqR0aA_content_list.json +1142 -0
- parse/train/3_MUAtqR0aA/3_MUAtqR0aA_middle.json +0 -0
- parse/train/3_MUAtqR0aA/3_MUAtqR0aA_model.json +0 -0
- parse/train/HkePNpVKPB/HkePNpVKPB.md +431 -0
- parse/train/HkePNpVKPB/HkePNpVKPB_content_list.json +0 -0
- parse/train/HkePNpVKPB/HkePNpVKPB_middle.json +0 -0
- parse/train/HkePNpVKPB/HkePNpVKPB_model.json +0 -0
- parse/train/HkgxasA5Ym/HkgxasA5Ym.md +259 -0
- parse/train/HkgxasA5Ym/HkgxasA5Ym_content_list.json +1055 -0
- parse/train/HkgxasA5Ym/HkgxasA5Ym_middle.json +0 -0
- parse/train/HkgxasA5Ym/HkgxasA5Ym_model.json +0 -0
- parse/train/h7FqQ6hCK18/h7FqQ6hCK18.md +355 -0
- parse/train/h7FqQ6hCK18/h7FqQ6hCK18_content_list.json +1389 -0
- parse/train/h7FqQ6hCK18/h7FqQ6hCK18_middle.json +0 -0
- parse/train/h7FqQ6hCK18/h7FqQ6hCK18_model.json +0 -0
- parse/train/pvjfA4wogD6/pvjfA4wogD6.md +255 -0
- parse/train/pvjfA4wogD6/pvjfA4wogD6_content_list.json +1197 -0
- parse/train/pvjfA4wogD6/pvjfA4wogD6_middle.json +0 -0
- parse/train/pvjfA4wogD6/pvjfA4wogD6_model.json +0 -0
- pdf/train/--GJkm7nt0.pdf +3 -0
- pdf/train/-_FVMKvxVCQo1.pdf +3 -0
- pdf/train/-oUhJJILWHb.pdf +3 -0
- pdf/train/-qh0M9XWxnv.pdf +3 -0
- pdf/train/03x6x6qNwJ3.pdf +3 -0
- pdf/train/04cII6MumYV.pdf +3 -0
- pdf/train/0NXUSlb6oEu.pdf +3 -0
- pdf/train/0WASBV4xkhy.pdf +3 -0
- pdf/train/0zXJRJecC_.pdf +3 -0
- pdf/train/1FvkSpWosOl.pdf +3 -0
- pdf/train/1toB0Fo9CZy.pdf +3 -0
- pdf/train/2-0wS9-cSL.pdf +3 -0
- pdf/train/2pJZSVcSZz.pdf +3 -0
- pdf/train/33TBJachvOX.pdf +3 -0
- pdf/train/37nvvqkCo5.pdf +3 -0
- pdf/train/3T9iFICe0Y9.pdf +3 -0
- pdf/train/3WbWmdTd8fN.pdf +3 -0
- pdf/train/3hGNqpI4WS.pdf +3 -0
- pdf/train/3qMwV98zLIk.pdf +3 -0
- pdf/train/4G2dEuRZ7eO.pdf +3 -0
- pdf/train/5-GXHFNbq_U.pdf +3 -0
- pdf/train/5CGPY2VeEGb.pdf +3 -0
- pdf/train/5PiSFHhRe2C.pdf +3 -0
- pdf/train/5lhWG3Hj2By.pdf +3 -0
- pdf/train/5nLibPckV2N.pdf +3 -0
- pdf/train/6NFBvWlRXaG.pdf +3 -0
- pdf/train/6UdQLhqJyFD.pdf +3 -0
- pdf/train/6tM849_6RF9.pdf +3 -0
- pdf/train/74RmfBweB60.pdf +3 -0
.gitattributes
CHANGED
|
@@ -11383,3 +11383,583 @@ pdf/train/Hye4WaVYwr.pdf filter=lfs diff=lfs merge=lfs -text
|
|
| 11383 |
pdf/train/SkNSehA9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11384 |
pdf/train/Syx7A3NFvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11385 |
pdf/train/HkCjNI5ex.pdf filter=lfs diff=lfs merge=lfs -text
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 11383 |
pdf/train/SkNSehA9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11384 |
pdf/train/Syx7A3NFvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11385 |
pdf/train/HkCjNI5ex.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11386 |
+
pdf/train/HJgZrsC5t7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11387 |
+
pdf/train/S1jBcueAb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11388 |
+
pdf/train/mjyMGFL8N2.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11389 |
+
pdf/train/ZRcjSOmYraB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11390 |
+
pdf/train/BkV4VS9ll.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11391 |
+
pdf/train/H1WgVz-AZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11392 |
+
pdf/train/B1lgUkBFwr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11393 |
+
pdf/train/SylzhkBtDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11394 |
+
pdf/train/rylNH20qFQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11395 |
+
pdf/train/nzpLWnVAyah.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11396 |
+
pdf/train/SyProzZAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11397 |
+
pdf/train/rJGZq6g0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11398 |
+
pdf/train/r1VPNiA5Fm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11399 |
+
pdf/train/3qMwV98zLIk.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11400 |
+
pdf/train/SkA-IE06W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11401 |
+
pdf/train/SJlJSaEFwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11402 |
+
pdf/train/H11lAfbCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11403 |
+
pdf/train/BJIgi_eCZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11404 |
+
pdf/train/Lpfh1Bpqfk.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11405 |
+
pdf/train/ryuxYmvel.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11406 |
+
pdf/train/zx_uX-BO7CH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11407 |
+
pdf/train/hQGRD1Zael7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11408 |
+
pdf/train/SJeXSo09FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11409 |
+
pdf/train/SkFAWax0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11410 |
+
pdf/train/rndqBJsGoKh.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11411 |
+
pdf/train/HbZTcIuiMAG.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11412 |
+
pdf/train/xgGS6PmzNq6.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11413 |
+
pdf/train/Ec85b0tUwbA.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11414 |
+
pdf/train/Rcmk0xxIQV.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11415 |
+
pdf/train/PS3IMnScugk.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11416 |
+
pdf/train/r1eBeyHFDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11417 |
+
pdf/train/ByIAPUcee.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11418 |
+
pdf/train/BydjJte0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11419 |
+
pdf/train/S1dIzvclg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11420 |
+
pdf/train/rJTutzbA-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11421 |
+
pdf/train/Sy8XvGb0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11422 |
+
pdf/train/O0ow9RQW1nP.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11423 |
+
pdf/train/H1ecDoR5Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11424 |
+
pdf/train/SkFEGHx0Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11425 |
+
pdf/train/r1nmx5l0W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11426 |
+
pdf/train/rkZvSe-RZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11427 |
+
pdf/train/SkeGvaEtPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11428 |
+
pdf/train/HJe6uANtwH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11429 |
+
pdf/train/5PiSFHhRe2C.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11430 |
+
pdf/train/OItvP2-i9j.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11431 |
+
pdf/train/rkFBJv9gg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11432 |
+
pdf/train/HkNEuToge.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11433 |
+
pdf/train/SyhRVm-Rb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11434 |
+
pdf/train/S1E3Ko09F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11435 |
+
pdf/train/xYGNO86OWDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11436 |
+
pdf/train/pLk9yRbRRtF.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11437 |
+
pdf/train/o_V-MjyyGV_.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11438 |
+
pdf/train/SkyQWDcex.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11439 |
+
pdf/train/HkgEQnRqYQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11440 |
+
pdf/train/ByeL1R4FvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11441 |
+
pdf/train/H1gpET4YDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11442 |
+
pdf/train/r154_g-Rb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11443 |
+
pdf/train/HkgSEnA5KQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11444 |
+
pdf/train/LwEQnp6CYev.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11445 |
+
pdf/train/HkejNgBtPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11446 |
+
pdf/train/Sy2ogebAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11447 |
+
pdf/train/SkgsACVKPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11448 |
+
pdf/train/HkgsPhNYPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11449 |
+
pdf/train/HkGzUjR5tQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11450 |
+
pdf/train/qGeqg4_hA2.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11451 |
+
pdf/train/_IY3_4psXuf.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11452 |
+
pdf/train/rylztAEYvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11453 |
+
pdf/train/r1nTpv9eg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11454 |
+
pdf/train/uVPZCMVtsSG.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11455 |
+
pdf/train/MRQJmsNPp8E.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11456 |
+
pdf/train/BkVsWbbAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11457 |
+
pdf/train/HktRlUlAZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11458 |
+
pdf/train/33TBJachvOX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11459 |
+
pdf/train/BkgYPREtPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11460 |
+
pdf/train/Bkeeca4Kvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11461 |
+
pdf/train/tnq_O52RVbR.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11462 |
+
pdf/train/Gl8FHfMVTZu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11463 |
+
pdf/train/KAV7BDCcN6.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11464 |
+
pdf/train/D04TGKz5rfF.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11465 |
+
pdf/train/rkQu4Wb0Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11466 |
+
pdf/train/BJInEZsTb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11467 |
+
pdf/train/MBH29cOeohr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11468 |
+
pdf/train/BJ8c3f-0b.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11469 |
+
pdf/train/SJA7xfb0b.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11470 |
+
pdf/train/_4VxORHq-0g.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11471 |
+
pdf/train/SVsLxTfHa1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11472 |
+
pdf/train/rq_Qr0c1Hyo.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11473 |
+
pdf/train/Bys_NzbC-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11474 |
+
pdf/train/TRDAFiwDq8A.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11475 |
+
pdf/train/SkeRTsAcYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11476 |
+
pdf/train/nVZtXBI6LNn.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11477 |
+
pdf/train/BygNqoR9tm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11478 |
+
pdf/train/lvRTC669EY_.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11479 |
+
pdf/train/DEsIX_D_vR.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11480 |
+
pdf/train/ssohLcmn4-r.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11481 |
+
pdf/train/SkE6PjC9KX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11482 |
+
pdf/train/ALvt7nXa2q.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11483 |
+
pdf/train/rJljdh4KDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11484 |
+
pdf/train/S1ejj64YvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11485 |
+
pdf/train/BJleph4KvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11486 |
+
pdf/train/Sklf1yrYDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11487 |
+
pdf/train/B1gjs6EtDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11488 |
+
pdf/train/BkxfshNYwB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11489 |
+
pdf/train/r1kGbydxg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11490 |
+
pdf/train/WsfXFxqZXRO.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11491 |
+
pdf/train/Syx7WyBtwB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11492 |
+
pdf/train/tqo8sxHyD8.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11493 |
+
pdf/train/S1sqHMZCb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11494 |
+
pdf/train/H1eJxngCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11495 |
+
pdf/train/BJfvknCqFQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11496 |
+
pdf/train/6tM849_6RF9.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11497 |
+
pdf/train/r16Vyf-0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11498 |
+
pdf/train/H1eH4n09KX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11499 |
+
pdf/train/Hy7fDog0b.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11500 |
+
pdf/train/KUDUoRsEphu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11501 |
+
pdf/train/te7PVH1sPxJ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11502 |
+
pdf/train/H1lNPxHKDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11503 |
+
pdf/train/B1excoAqKQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11504 |
+
pdf/train/_0kaDkv3dVf.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11505 |
+
pdf/train/BklfR3EYDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11506 |
+
pdf/train/rJehllrtDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11507 |
+
pdf/train/HkxStoC5F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11508 |
+
pdf/train/r1l-VeSKwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11509 |
+
pdf/train/HyRVBzap-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11510 |
+
pdf/train/St1giarCHLP.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11511 |
+
pdf/train/BJlrF24twB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11512 |
+
pdf/train/BkbY4psgg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11513 |
+
pdf/train/hqDb8d65Vfh.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11514 |
+
pdf/train/pBKOx_dxYAN.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11515 |
+
pdf/train/CRFSrgYtV7m.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11516 |
+
pdf/train/yJqcM36Qvnu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11517 |
+
pdf/train/rkxoNnC5FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11518 |
+
pdf/train/SJlDDnVKwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11519 |
+
pdf/train/G8A_Nl0yim6.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11520 |
+
pdf/train/SJlbGJrtDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11521 |
+
pdf/train/uAX8q61EVRu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11522 |
+
pdf/train/SyGjjsC5tQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11523 |
+
pdf/train/H1e0-30qKm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11524 |
+
pdf/train/SJx7Jrtgl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11525 |
+
pdf/train/MDMV2SxCboX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11526 |
+
pdf/train/ByRWCqvT-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11527 |
+
pdf/train/BJxpIJHKwB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11528 |
+
pdf/train/Q1aiM7sCi1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11529 |
+
pdf/train/r1g1LoAcFm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11530 |
+
pdf/train/rJe2syrtvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11531 |
+
pdf/train/TNkPBBYFkXg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11532 |
+
pdf/train/ajOrOhQOsYx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11533 |
+
pdf/train/Byx91R4twB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11534 |
+
pdf/train/H1e8wsCqYX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11535 |
+
pdf/train/TgSVWXw22FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11536 |
+
pdf/train/SJl5Np4tPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11537 |
+
pdf/train/SJeD3CEFPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11538 |
+
pdf/train/O358nrve1W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11539 |
+
pdf/train/rm0I5y2zkG8.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11540 |
+
pdf/train/EI2KOXKdnP.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11541 |
+
pdf/train/Skgvy64tvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11542 |
+
pdf/train/LwyBkw8Nh6Y1J.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11543 |
+
pdf/train/S1e4Q6EtDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11544 |
+
pdf/train/HJx8HANFDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11545 |
+
pdf/train/SkC_7v5gx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11546 |
+
pdf/train/SyxrxR4KPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11547 |
+
pdf/train/lo7GKwmakFZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11548 |
+
pdf/train/B1xwcyHFDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11549 |
+
pdf/train/JeweO9-QqV-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11550 |
+
pdf/train/BkSDMA36Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11551 |
+
pdf/train/rkgyS0VFvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11552 |
+
pdf/train/BygFVAEKDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11553 |
+
pdf/train/wg_kD_nyAF.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11554 |
+
pdf/train/jLHWRxwc7_f.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11555 |
+
pdf/train/6UdQLhqJyFD.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11556 |
+
pdf/train/rJgBd2NYPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11557 |
+
pdf/train/B1eyO1BFPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11558 |
+
pdf/train/Hklso24Kwr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11559 |
+
pdf/train/HJxDugSFDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11560 |
+
pdf/train/fgd7we_uZa6.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11561 |
+
pdf/train/SkHkeixAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11562 |
+
pdf/train/rJgJDAVKvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11563 |
+
pdf/train/Bkepl7cee.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11564 |
+
pdf/train/ryxB2lBtvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11565 |
+
pdf/train/Q_64PF6XNut.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11566 |
+
pdf/train/d3k38LTDCyO.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11567 |
+
pdf/train/H1gdF34FvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11568 |
+
pdf/train/fmiwLdJCmLS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11569 |
+
pdf/train/DKabt9MFnT.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11570 |
+
pdf/train/Bk9nkMa4G.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11571 |
+
pdf/train/2-0wS9-cSL.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11572 |
+
pdf/train/B1X0mzZCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11573 |
+
pdf/train/BylVcTNtDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11574 |
+
pdf/train/YJL3amnDf2w.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11575 |
+
pdf/train/AAes_3W-2z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11576 |
+
pdf/train/Hygxb2CqKm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11577 |
+
pdf/train/rJeZS3RcYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11578 |
+
pdf/train/1toB0Fo9CZy.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11579 |
+
pdf/train/oh71uL93yay.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11580 |
+
pdf/train/B1l2bp4YwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11581 |
+
pdf/train/SkxW23NtPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11582 |
+
pdf/train/SJg4Y3VFPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11583 |
+
pdf/train/S1lEX04tPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11584 |
+
pdf/train/BylEqnVFDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11585 |
+
pdf/train/B1Gi6LeRZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11586 |
+
pdf/train/oRMRIR4qPC1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11587 |
+
pdf/train/SygvZ209F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11588 |
+
pdf/train/r1gBOxSFwr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11589 |
+
pdf/train/BJgQfkSYDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11590 |
+
pdf/train/EKV158tSfwv.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11591 |
+
pdf/train/w6iVxEdh6bi.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11592 |
+
pdf/train/r1kNDlbCb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11593 |
+
pdf/train/ZD7Ll4pAw7C.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11594 |
+
pdf/train/RHY_9ZVcTa_.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11595 |
+
pdf/train/S1giVsRcYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11596 |
+
pdf/train/BJe-DsC5Fm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11597 |
+
pdf/train/OPyWRrcjVQw.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11598 |
+
pdf/train/HkwZSG-CZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11599 |
+
pdf/train/r1eowANFvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11600 |
+
pdf/train/B1xsqj09Fm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11601 |
+
pdf/train/BJeWUs05KQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11602 |
+
pdf/train/wxobw18IYOxu4.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11603 |
+
pdf/train/rkgO66VKDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11604 |
+
pdf/train/kHSu4ebxFXY.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11605 |
+
pdf/train/Skq89Scxx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11606 |
+
pdf/train/SJxzPsAqFQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11607 |
+
pdf/train/AFiH_CNnVhS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11608 |
+
pdf/train/rJg8yhAqKm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11609 |
+
pdf/train/UcoXdfrORC.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11610 |
+
pdf/train/jSz59N8NvUP.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11611 |
+
pdf/train/ByeMPlHKPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11612 |
+
pdf/train/yILzFBjR0Y.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11613 |
+
pdf/train/S1lTg3RqYQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11614 |
+
pdf/train/yehlf2AvSD_.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11615 |
+
pdf/train/8q_ca26L1fz.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11616 |
+
pdf/train/i7aMbliTkHs.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11617 |
+
pdf/train/HyM7AiA5YX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11618 |
+
pdf/train/HkgTkhRcKQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11619 |
+
pdf/train/9uFiX_HRsIL.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11620 |
+
pdf/train/BJxh2j0qYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11621 |
+
pdf/train/BJgqQ6NYvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11622 |
+
pdf/train/OJLaKwiXSbx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11623 |
+
pdf/train/BJfYvo09Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11624 |
+
pdf/train/ByxdUySKvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11625 |
+
pdf/train/rkdU7tCaZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11626 |
+
pdf/train/LmUJqB1Cz8.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11627 |
+
pdf/train/QIRlze3I6hX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11628 |
+
pdf/train/Hyq4yhile.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11629 |
+
pdf/train/r17lFgZ0Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11630 |
+
pdf/train/BJfOXnActQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11631 |
+
pdf/train/MjNFN44NbZm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11632 |
+
pdf/train/SJxu5iR9KQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11633 |
+
pdf/train/Bkg0u3Etwr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11634 |
+
pdf/train/b4YiFnQH3gN.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11635 |
+
pdf/train/B1QRgziT-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11636 |
+
pdf/train/BklmtJBKDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11637 |
+
pdf/train/rkA1f3NpZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11638 |
+
pdf/train/SyeD0RVtvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11639 |
+
pdf/train/SJfWKsC5K7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11640 |
+
pdf/train/BJe4oxHYPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11641 |
+
pdf/train/r1lUOzWCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11642 |
+
pdf/train/rkxt8oC9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11643 |
+
pdf/train/Hk8N3Sclg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11644 |
+
pdf/train/Hk3mPK5gg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11645 |
+
pdf/train/X3TdREzbZN.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11646 |
+
pdf/train/rklMnyBtPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11647 |
+
pdf/train/rkpdnIqlx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11648 |
+
pdf/train/SJtfOEn6-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11649 |
+
pdf/train/rywHCPkAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11650 |
+
pdf/train/5-GXHFNbq_U.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11651 |
+
pdf/train/rkllGyBFPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11652 |
+
pdf/train/Syg-YfWCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11653 |
+
pdf/train/H1ersoRqtm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11654 |
+
pdf/train/rkgdYhVtvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11655 |
+
pdf/train/EqoXe2zmhrh.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11656 |
+
pdf/train/SyxtJh0qYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11657 |
+
pdf/train/nCY83KxoehA.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11658 |
+
pdf/train/jA5LmT2zLk.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11659 |
+
pdf/train/BkJsCIcgl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11660 |
+
pdf/train/PxMfDdPnTfV.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11661 |
+
pdf/train/fgBxcKcz9ea.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11662 |
+
pdf/train/BJG0voC9YQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11663 |
+
pdf/train/BkluqlSFDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11664 |
+
pdf/train/r1TA9ZbA-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11665 |
+
pdf/train/YNnpaAKeCfx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11666 |
+
pdf/train/r1SuFjkRW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11667 |
+
pdf/train/rkgs0oAqFQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11668 |
+
pdf/train/rJe04p4YDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11669 |
+
pdf/train/Byx83s09Km.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11670 |
+
pdf/train/HkedQp4tPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11671 |
+
pdf/train/hm0i-cunzGW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11672 |
+
pdf/train/IrM64DGB21.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11673 |
+
pdf/train/BJl6AjC5F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11674 |
+
pdf/train/3WbWmdTd8fN.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11675 |
+
pdf/train/BJl_VnR9Km.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11676 |
+
pdf/train/dvyUaK4neD0.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11677 |
+
pdf/train/ByMHvs0cFQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11678 |
+
pdf/train/B1ae1lZRb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11679 |
+
pdf/train/r1ejxnCctX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11680 |
+
pdf/train/S1g2V3Cct7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11681 |
+
pdf/train/Byg9A24tvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11682 |
+
pdf/train/S1lwRjR9YX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11683 |
+
pdf/train/H1uR4GZRZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11684 |
+
pdf/train/BJKYvt5lg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11685 |
+
pdf/train/lxHgXYN4bwl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11686 |
+
pdf/train/LLoe0U9ShkN.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11687 |
+
pdf/train/B1mAJI9gl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11688 |
+
pdf/train/r6cNUjS8cm0.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11689 |
+
pdf/train/SyL9u-WA-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11690 |
+
pdf/train/HkxKH2AcFm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11691 |
+
pdf/train/tkAtoZkcUnm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11692 |
+
pdf/train/SJxzFySKwH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11693 |
+
pdf/train/S1lN69AT-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11694 |
+
pdf/train/uV7hcsjqM-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11695 |
+
pdf/train/rJeKjwvclx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11696 |
+
pdf/train/MudT1U2eIY.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11697 |
+
pdf/train/AAWuCvzaVt.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11698 |
+
pdf/train/tPCrkaLa9Y5ld.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11699 |
+
pdf/train/BkUDvt5gg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11700 |
+
pdf/train/rkzfuiA9F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11701 |
+
pdf/train/HJYoqzbC-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11702 |
+
pdf/train/rJBiunlAW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11703 |
+
pdf/train/D9I3drBz4UC.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11704 |
+
pdf/train/Sk-oDY9ge.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11705 |
+
pdf/train/jfDaBf8PAE.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11706 |
+
pdf/train/6NFBvWlRXaG.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11707 |
+
pdf/train/dwJyEMPZ04I.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11708 |
+
pdf/train/rkQkBnJAb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11709 |
+
pdf/train/SkEYojRqtm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11710 |
+
pdf/train/8PA2nX9v_r2.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11711 |
+
pdf/train/Hkn7CBaTW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11712 |
+
pdf/train/XjIm8hOCxTm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11713 |
+
pdf/train/T4gXBOXoIUr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11714 |
+
pdf/train/BkeaEyBYDB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11715 |
+
pdf/train/r1xjgxBFPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11716 |
+
pdf/train/ByuI-mW0W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11717 |
+
pdf/train/nIAxjsniDzg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11718 |
+
pdf/train/3T9iFICe0Y9.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11719 |
+
pdf/train/r1gelyrtwH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11720 |
+
pdf/train/np96ge7gz0j.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11721 |
+
pdf/train/ToWi1RjuEr8.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11722 |
+
pdf/train/BJC_jUqxe.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11723 |
+
pdf/train/03x6x6qNwJ3.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11724 |
+
pdf/train/SPrVNsXnGd.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11725 |
+
pdf/train/PbEHqvFtcS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11726 |
+
pdf/train/heqv8eIweMY.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11727 |
+
pdf/train/XwATtbX3oCz.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11728 |
+
pdf/train/B1gd0nEFwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11729 |
+
pdf/train/SJeLIgBKPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11730 |
+
pdf/train/S1_pAu9xl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11731 |
+
pdf/train/Hye-LiR5Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11732 |
+
pdf/train/BJuysoFeg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11733 |
+
pdf/train/B1lda1HtvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11734 |
+
pdf/train/BklHpjCqKm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11735 |
+
pdf/train/Db4yerZTYkz.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11736 |
+
pdf/train/SJl2niR9KQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11737 |
+
pdf/train/af_hng9tuNj.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11738 |
+
pdf/train/rJUYGxbCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11739 |
+
pdf/train/ryZ283gAZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11740 |
+
pdf/train/5lhWG3Hj2By.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11741 |
+
pdf/train/UVH3Ucewd-IXZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11742 |
+
pdf/train/Uj7pF-D-YvT.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11743 |
+
pdf/train/37nvvqkCo5.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11744 |
+
pdf/train/_RnHyIeu5Y5.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11745 |
+
pdf/train/HkehD3VtvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11746 |
+
pdf/train/hGmrNwR8qQP.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11747 |
+
pdf/train/8KokDTctkA8e4.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11748 |
+
pdf/train/rJVorjCcKQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11749 |
+
pdf/train/oyZxhRI2RiE.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11750 |
+
pdf/train/Mh1Abj33qI.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11751 |
+
pdf/train/c5QbJ1zob73.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11752 |
+
pdf/train/SklTQCNtvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11753 |
+
pdf/train/j9Rv7qdXjd.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11754 |
+
pdf/train/--GJkm7nt0.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11755 |
+
pdf/train/3hGNqpI4WS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11756 |
+
pdf/train/ByxPYjC5KQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11757 |
+
pdf/train/xyGFYKIPTDJ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11758 |
+
pdf/train/J8_GttYLFgr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11759 |
+
pdf/train/JCodE7xOcc5.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11760 |
+
pdf/train/rkuDV6iex.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11761 |
+
pdf/train/ByxY8CNtvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11762 |
+
pdf/train/r1lIKlSYvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11763 |
+
pdf/train/POWv6hDd9XH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11764 |
+
pdf/train/S1enmaVFvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11765 |
+
pdf/train/HkpYwMZRb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11766 |
+
pdf/train/SJGCiw5gl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11767 |
+
pdf/train/r1gixp4FPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11768 |
+
pdf/train/H1q-TM-AW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11769 |
+
pdf/train/p2XgjS3Qp4X.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11770 |
+
pdf/train/KG2RTUXXU7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11771 |
+
pdf/train/b36m4ZYG1gD.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11772 |
+
pdf/train/HygrdpVKvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11773 |
+
pdf/train/rylqooRqK7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11774 |
+
pdf/train/H1gzR2VKDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11775 |
+
pdf/train/S5S3eTEmouw.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11776 |
+
pdf/train/ryxmrpNtvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11777 |
+
pdf/train/Rt5mjXAqHrY.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11778 |
+
pdf/train/UoVpP8R2Vn.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11779 |
+
pdf/train/ahYIlRBeCFw.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11780 |
+
pdf/train/BJgcwh4FwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11781 |
+
pdf/train/9sF3n8eAco.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11782 |
+
pdf/train/yWkP7JuHX1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11783 |
+
pdf/train/B1eWbxStPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11784 |
+
pdf/train/Wi5KUNlqWty.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11785 |
+
pdf/train/SybqeKgA-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11786 |
+
pdf/train/-_FVMKvxVCQo1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11787 |
+
pdf/train/x8gM-4nFq9b.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11788 |
+
pdf/train/ByxAcjCqt7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11789 |
+
pdf/train/HygnDhEtvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11790 |
+
pdf/train/kHromd7SNA.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11791 |
+
pdf/train/86NHK__yFDl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11792 |
+
pdf/train/ByxGSsR9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11793 |
+
pdf/train/S1VaB4cex.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11794 |
+
pdf/train/n-219jrTht.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11795 |
+
pdf/train/Hkc-TeZ0W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11796 |
+
pdf/train/0WASBV4xkhy.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11797 |
+
pdf/train/Q32U7dzWXpc.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11798 |
+
pdf/train/S1fQSiCcYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11799 |
+
pdf/train/QpU7n-6l0n.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11800 |
+
pdf/train/HyGh4sR9YQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11801 |
+
pdf/train/BkisuzWRW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11802 |
+
pdf/train/rkl03ySYDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11803 |
+
pdf/train/0zXJRJecC_.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11804 |
+
pdf/train/N2dEPuK0vom.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11805 |
+
pdf/train/BkdpaH9ll.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11806 |
+
pdf/train/qbH974jKUVy.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11807 |
+
pdf/train/SyxeqhP9ll.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11808 |
+
pdf/train/rkmu5b0a-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11809 |
+
pdf/train/BkgRe1SFDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11810 |
+
pdf/train/HJlTpCEKvS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11811 |
+
pdf/train/r1iuQjxCZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11812 |
+
pdf/train/B1ckMDqlg.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11813 |
+
pdf/train/SkgzYiRqtX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11814 |
+
pdf/train/zv-typ1gPxA.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11815 |
+
pdf/train/Hkex2a4FPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11816 |
+
pdf/train/rkRwGg-0Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11817 |
+
pdf/train/Pk972F7J8Um.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11818 |
+
pdf/train/BkedwoC5t7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11819 |
+
pdf/train/S1zz2i0cY7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11820 |
+
pdf/train/HkldyTNYwH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11821 |
+
pdf/train/r1HhRfWRZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11822 |
+
pdf/train/wS0UFjsNYjn.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11823 |
+
pdf/train/rkgz2aEKDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11824 |
+
pdf/train/j7u7cJDBo8p.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11825 |
+
pdf/train/_WnAQKse_uK.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11826 |
+
pdf/train/LhY8QdUGSuw.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11827 |
+
pdf/train/SylOlp4FvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11828 |
+
pdf/train/HJlLKjR9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11829 |
+
pdf/train/S1ldO2EFPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11830 |
+
pdf/train/HJMN-xWC-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11831 |
+
pdf/train/il3vPAc9fiH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11832 |
+
pdf/train/1FvkSpWosOl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11833 |
+
pdf/train/Hye4KeSYDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11834 |
+
pdf/train/bqGK5PyI6-N.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11835 |
+
pdf/train/S1esMkHYPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11836 |
+
pdf/train/B1s6xvqlx.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11837 |
+
pdf/train/Cnon5ezMHtu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11838 |
+
pdf/train/4G2dEuRZ7eO.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11839 |
+
pdf/train/ZyugLlWzdO.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11840 |
+
pdf/train/SkeWc2EKPH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11841 |
+
pdf/train/HkgSk2A9Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11842 |
+
pdf/train/SyxnvsAqFm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11843 |
+
pdf/train/Syx4_iCqKQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11844 |
+
pdf/train/LJjC6DmSkgT.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11845 |
+
pdf/train/Nl7VO_Y7K4Q.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11846 |
+
pdf/train/ilVv1LO0Ew.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11847 |
+
pdf/train/TMUR2ovJfjE.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11848 |
+
pdf/train/BJLmN8xRW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11849 |
+
pdf/train/Hk6kPgZA-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11850 |
+
pdf/train/rJlDnoA5Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11851 |
+
pdf/train/CoJibBRjPXQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11852 |
+
pdf/train/Hkz6aNqle.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11853 |
+
pdf/train/HJrDIpiee.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11854 |
+
pdf/train/S1l-C0NtwS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11855 |
+
pdf/train/SylJ1D1C-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11856 |
+
pdf/train/rylnK6VtDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11857 |
+
pdf/train/784_F-WCW46.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11858 |
+
pdf/train/BkewX2C9tX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11859 |
+
pdf/train/J4gRj6d5Qm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11860 |
+
pdf/train/JrsfBJtDFdI.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11861 |
+
pdf/train/HylsTT4FvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11862 |
+
pdf/train/qiydAcw6Re.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11863 |
+
pdf/train/kcI3T5qe1jr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11864 |
+
pdf/train/Skxuk1rFwB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11865 |
+
pdf/train/B1gdkxHFDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11866 |
+
pdf/train/HkGSniC9FQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11867 |
+
pdf/train/BkCV_W-AZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11868 |
+
pdf/train/VzuIzbRDrum.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11869 |
+
pdf/train/jnkE5c5f9m.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11870 |
+
pdf/train/ByBAl2eAZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11871 |
+
pdf/train/SyZI0GWCZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11872 |
+
pdf/train/SJxSOJStPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11873 |
+
pdf/train/rkaqxm-0b.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11874 |
+
pdf/train/HyMTkQZAb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11875 |
+
pdf/train/SylVJTNKDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11876 |
+
pdf/train/ion6Lo5tKtJ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11877 |
+
pdf/train/rkgPnhNFPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11878 |
+
pdf/train/qG4ZVCCyCB0.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11879 |
+
pdf/train/HyydRMZC-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11880 |
+
pdf/train/IpmfpAGoH2KbX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11881 |
+
pdf/train/IBVBtz_sRSm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11882 |
+
pdf/train/ryfcCo0ctQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11883 |
+
pdf/train/SJMeTo09YQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11884 |
+
pdf/train/mEdwVCRJuX4.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11885 |
+
pdf/train/Byl8BnRcYm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11886 |
+
pdf/train/r1gzoaNtvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11887 |
+
pdf/train/P42rXLGZQ07.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11888 |
+
pdf/train/S1eYKlrYvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11889 |
+
pdf/train/HyexAiA5Fm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11890 |
+
pdf/train/Sklv5iRqYX.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11891 |
+
pdf/train/GhMZNcr54zt.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11892 |
+
pdf/train/HJxJ2h4tPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11893 |
+
pdf/train/eedamFFTcK5.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11894 |
+
pdf/train/r1saNM-RW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11895 |
+
pdf/train/f4jw35Vrk6d.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11896 |
+
pdf/train/HJepXaVYDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11897 |
+
pdf/train/N51zJ7F3mw.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11898 |
+
pdf/train/BJe0Gn0cY7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11899 |
+
pdf/train/ByexElSYDr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11900 |
+
pdf/train/BJedHRVtPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11901 |
+
pdf/train/GrB-y0SNkQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11902 |
+
pdf/train/r1xCMyBtPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11903 |
+
pdf/train/Byx93sC9tm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11904 |
+
pdf/train/JOxB9h40A-1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11905 |
+
pdf/train/rkeiQlBFPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11906 |
+
pdf/train/px0-N3_KjA.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11907 |
+
pdf/train/A5VV3UyIQz.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11908 |
+
pdf/train/04cII6MumYV.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11909 |
+
pdf/train/5nLibPckV2N.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11910 |
+
pdf/train/rke3TJrtPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11911 |
+
pdf/train/ryup8-WCW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11912 |
+
pdf/train/EMHoBG0avc1.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11913 |
+
pdf/train/5CGPY2VeEGb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11914 |
+
pdf/train/r1CE9GWR-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11915 |
+
pdf/train/n4IMHNb8_f.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11916 |
+
pdf/train/r1te3Fqel.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11917 |
+
pdf/train/BJlxm30cKm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11918 |
+
pdf/train/S1x1IkHtPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11919 |
+
pdf/train/SyMDXnCcF7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11920 |
+
pdf/train/OGg9XnKxFAH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11921 |
+
pdf/train/bMCfFepJXM.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11922 |
+
pdf/train/Hyg96gBKPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11923 |
+
pdf/train/Hkxvl0EtDH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11924 |
+
pdf/train/BklSv34KvB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11925 |
+
pdf/train/0NXUSlb6oEu.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11926 |
+
pdf/train/NfZ6g2OmXEk.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11927 |
+
pdf/train/h8flNv9x8v-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11928 |
+
pdf/train/cnWSyJNmeCE.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11929 |
+
pdf/train/vQzcqQWIS0q.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11930 |
+
pdf/train/rkgKBhA5Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11931 |
+
pdf/train/2pJZSVcSZz.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11932 |
+
pdf/train/rkeYL1SFvH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11933 |
+
pdf/train/wta_8Hx2KD.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11934 |
+
pdf/train/BkXmYfbAZ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11935 |
+
pdf/train/SJ3dBGZ0Z.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11936 |
+
pdf/train/HyeSin4FPB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11937 |
+
pdf/train/rJliMh09F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11938 |
+
pdf/train/jDdzh5ul-d.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11939 |
+
pdf/train/rJvJXZb0W.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11940 |
+
pdf/train/HJgXsjA5tQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11941 |
+
pdf/train/B1kJ6H9ex.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11942 |
+
pdf/train/BkxthxHYvr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11943 |
+
pdf/train/rk49Mg-CW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11944 |
+
pdf/train/B1l6qiR5F7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11945 |
+
pdf/train/NJS8kp15zzH.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11946 |
+
pdf/train/-qh0M9XWxnv.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11947 |
+
pdf/train/HCSgyPUfeDj.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11948 |
+
pdf/train/OMizHuea_HB.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11949 |
+
pdf/train/BJgkbyHKDS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11950 |
+
pdf/train/rklB76EKPr.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11951 |
+
pdf/train/ByfyHh05tQ.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11952 |
+
pdf/train/r1lnxTEYPS.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11953 |
+
pdf/train/Aeo-xqtb5p.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11954 |
+
pdf/train/SyehMhC9Y7.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11955 |
+
pdf/train/r1lrAiA5Ym.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11956 |
+
pdf/train/SJCPLLpaW.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11957 |
+
pdf/train/ByQPVFull.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11958 |
+
pdf/train/IhiU6AJYpDs.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11959 |
+
pdf/train/crAi7c41xTh.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11960 |
+
pdf/train/HyxAfnA5tm.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11961 |
+
pdf/train/74RmfBweB60.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11962 |
+
pdf/train/-oUhJJILWHb.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11963 |
+
pdf/train/rkGabzZgl.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11964 |
+
pdf/train/rJiaRbk0-.pdf filter=lfs diff=lfs merge=lfs -text
|
| 11965 |
+
pdf/train/Aa5oPXc_1IV.pdf filter=lfs diff=lfs merge=lfs -text
|
parse/train/3_MUAtqR0aA/3_MUAtqR0aA.md
ADDED
|
@@ -0,0 +1,217 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# Packing: Towards 2x NLP BERT Acceleration
|
| 2 |
+
|
| 3 |
+
# Anonymous
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
We find that at sequence length 512 padding tokens represent in excess of $5 0 \%$ of the Wikipedia dataset used for pretraining BERT (Bidirectional Encoder Representations from Transformers). Therefore by removing all padding we achieve a $2 \mathbf { x }$ speed-up in terms of sequences/sec. To exploit this characteristic of the dataset, we develop and contrast two deterministic packing algorithms. Both algorithms rely on the assumption that sequences are interchangeable and therefore packing can be performed on the histogram of sequence lengths, rather than per sample. This transformation of the problem leads to algorithms which are fast and have linear complexity in dataset size. The shortest-pack-first histogram-packing (SPFHP) algorithm determines the packing order for the Wikipedia dataset of over 16M sequences in 0.02 seconds. The non-negative least-squares histogram-packing (NNLSHP) algorithm converges in 28.4 seconds but produces solutions which are more depth efficient, managing to get near optimal packing by combining a maximum of 3 sequences in one sample. Using the dataset with multiple sequences per sample requires additional masking in the attention layer and a modification of the MLM loss function. We demonstrate that both of these changes are straightforward to implement and have relatively little impact on the achievable performance gain on modern hardware. Finally, we pretrain BERT-Large using the packed dataset, demonstrating no loss of convergence and the desired $2 \mathbf { x }$ speed-up.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Since its introduction in 2019, BERT $ { \mathbb { I } }$ has been the backbone driving the most exciting advances in Natural Language Processing (NLP). Pre-training BERT from scratch requires substantial computational resources which may be out of reach for researchers and industry professionals. To some extent this has been addressed by the public release of pre-trained models of different sizes and depths $\left[ \left[ 2 0 \right] \right]$ . Available sizes range from tiny (2 layers with hidden size 128) to large (24 layers with hidden size 1024)[6, 5]. The introduction of ALBERT $\textcircled { 1 1 4 } \textcircled { 1 }$ further improved the accessibility of larger models. However, the dependence on pre-trained models limits the ability of researchers to explore new backbone architectures. Furthermore, it limits the extent to which practitioners in industry can leverage internal datasets and adapt the model to their particular needs. Hence, any approach that speeds up the pre-training process is desirable from an economical as well as environmental perspective.
|
| 12 |
+
|
| 13 |
+
In this paper, we present some methods to enable researchers to accelerate the pre-training of BERT by as much as $2 \mathbf { x }$ . The de-facto pre-training dataset Wikipedia, as well as many other NLP datasets, show a positively skewed distribution of sequence lengths. We show that padding tokens (wasted compute) represent $5 0 \%$ of all tokens of the Wikipedia pre-training dataset at sequence length 512. Overall, the sample lengths range between 5 tokens up to 512 (see Figure 1). Samples of length 512 represent only $2 3 . 5 \%$ of the dataset.
|
| 14 |
+
|
| 15 |
+
While processing the padding tokens wastes compute, it is the most standard approach for leveraging modern massively-parallel compute especially on GPUs. These are most efficient when applying the same operation to each sequence in a batch. By padding all sequences to the same maximum sequence length, they can easily be batched. The most obvious way to reduce the extent of padding in the dataset is to group samples by size before batching, i.e., process the shorter samples together and longer samples together. Typically such an approach would still involve padding but less than if padding all sequences to the same maximum length. For example BERT $\boxed { \pmb { \bigtriangledown } }$ is pre-trained in two phases, where the first phase uses sequence length 128 for 900K steps and the second phase uses sequence length 512 for 100K steps. However even by splitting the training in this way, the wasted compute due to padding is approximately $2 0 \%$ (see Figure $\bigstar \bigstar \bigstar \bigstar$ ). Another example of this approach is Faster Transformer $\overline { { \mathbb { I } \mathbb { 8 } \| } }$ which groups samples of similar size together in one batch and fills up with padding only to the maximum length in this batch.
|
| 16 |
+
|
| 17 |
+
More advanced approaches for reducing the padding overhead rely on custom computational kernels. Loosely these are referred to as “un-padding” approaches. In Effective Transformer $\pmb { \Vert 4 \Vert }$ , the input batch is provided as a padded matrix but padding values are dynamically removed and restored during different calculation stages. While un-padding implementations are highly sophisticated and are able to completely circumvent the processing of padding tokens, they introduce a significant overhead due to the multiple GPU kernel launches (i.e. one kernel per sequence rather than one kernel per batch). Additionally the time to process each batch will fluctuate depending on the sequence lengths in each batch i.e. batches with only shorter sequences will typically be processed faster. When working with more than one accelerator, this variability in throughput results in all devices in the cluster waiting for the device with the most compute intensive batch to finish processing. As such, un-padding approaches are not appropriate for deployment on large clusters.
|
| 18 |
+
|
| 19 |
+
The “packing” based approach introduced in this paper offers significant advantages over un-padding approaches. Firstly, packing is implemented directly at the framework level and requires no additional custom kernel implementations. Secondly, the processing time for each batch is independent of the content of the batch, allowing the packing based approach to maintain the same speed-up whether running on a single device or thousands. Third, each batch now contains a consistent number of real tokens.
|
| 20 |
+
|
| 21 |
+
While we demonstrate the effectiveness of packing specifically on the Wikipedia dataset, packing SQUaD $\mathbb { \lVert \underline { { 9 } } \rVert }$ or GLUE datasets $\mathbb { \lVert 2 2 \rVert 2 3 \rVert }$ for BERT also leads to significant speed-ups (some in excess of 9x) [1] (sections A and B). The effectiveness of packing is a result of both the length distribution of the documents in the source datasets as well as the different text preprocessing steps for BERT [7]. The use of bi-directional self-attention in BERT implies that the input sequences should contain complete sentences. If a sentence is abruptly cut short, the hidden state on other (preceding) tokens in the sequence will be affected. Language models with causal attention (only attending to previous tokens in the input) do not have this issue. For such models, if a sequence is cut short at an arbitrary token, the other tokens (which occur earlier in the sequence) will not be affected. This ability to cut sequences arbitrarily completely trivializes the packing problem. For instance, GPT-3 $\mathbb { \left[ 3 \right] }$ is trained with a maximum sequence length of 2048 where a single sequence may contain multiple segments separated by a special end of segment token. The last segment in each sequence is simply cut to meet the sequence length requirement. In the interest of computational efficiency GPT-3 does not mask the attention between different segments in a sequence. In contrast, the packing approach presented in this paper introduces a mask in the attention layer (see Section $3 . 2 )$ to prevent cross-contamination between examples in a pack. This ensures that the characteristics of the original dataset and model are matched as closely as possible.
|
| 22 |
+
|
| 23 |
+
In summary, the contributions of the paper are as follows. In Section $\bigtriangledown ,$ , we produce histograms of the Wikipedia pre-training dataset showing the high percentage of padding tokens. We present two new deterministic packing algorithms which easily pack datasets with millions of sequences in a matter of seconds (or less). We empirically show that the proposed packing algorithms produce a nearly-optimal packing scheme for Wikipedia pre-training dataset. We show how to compute the per-sequence loss by inexpensively un-packing the loss. We provide code for building an attention mask which prevents attention between tokens of different sequences in the pack. We demonstrate that the convergence of the BERT large model on the packed dataset is equivalent to that on the un-packed dataset. We show that with the packed dataset, we are able to achieve a nearly 2x throughput increase on the Wikipedia sequence length 512 pre-training dataset.
|
| 24 |
+
|
| 25 |
+
# 2 Wikipedia BERT pre-training dataset
|
| 26 |
+
|
| 27 |
+
BERT is pre-trained using masked-language modelling and next-sentence prediction on a large corpus of Wikipedia articles [5]. Each sequence is composed of one ${ \mathrm { < C L S > } }$ token followed by the first part of sentences, followed by a ${ \tt { < S E P > } }$ token, and then finally the second part of sentences. Because parts are created in sentence-level increments there is no token-level control of sequence length. Together with already short parts, empirically, this leads to significant levels of padding, especially for longer maximum sequence lengths (see Figure 1). At sequence length 128 (commonly used in phase 1 of pre-training) the theoretical speed-up is around 1.2, at sequence length 384 this increases to 1.7, and finally at sequence length 512 (commonly used for phase 2 of pre-training) it is 2.0. Despite the widespread use of the Wikipedia dataset for pre-training BERT such histograms have, to the best of our knowledge, not been published previously. This has perhaps lead to the underestimation of the speed-up opportunity available. To put things into perspective, the sequence length 512 dataset contains 8.33 billion tokens, of which 4.17 billion are padding tokens.
|
| 28 |
+
|
| 29 |
+
|
| 30 |
+
Figure 1: Wikipedia BERT pre-training dataset sequence length histograms (token count excluding padding) for different maximum sequence lengths. Based on the Wikipedia article dump from October 1st 2020. The theoretical speed-up relates to not using any padding tokens and not having any overhead from processing the different lengths.
|
| 31 |
+
|
| 32 |
+
# 3 Methods
|
| 33 |
+
|
| 34 |
+
Our approach consists of three distinct components. Firstly, we pack the data efficiently during preprocessing to make full use of the sequence length (Sections $\checkmark$ and $3 . 1 . 2 ,$ see also [1] Section D). Secondly, we adapt the self-attention mask to prevent the model from attending between different sequences in the same pack (Section $\boxed { 3 . 2 }$ . Other components of the model, such as the feed-forward layer $\scriptstyle { \left\| { \overline { { 2 1 } } } \right\| }$ , operate on a per-token basis and do not require any modification. Thirdly, we compute the loss and accuracy on a per-sequence basis to match the canonical BERT implementation (Section $\textcircled { 3 . 3 }$ This is achieved by unpacking the per-pack loss at the framework level, without the use of custom kernels. Additionally, we provide suggestions for hyperparameter adjustment (Section $3 . 4 )$ that lead to analogous convergence behavior between the packed and un-packed BERT implementations.
|
| 35 |
+
|
| 36 |
+
# 3.1 Packing algorithms
|
| 37 |
+
|
| 38 |
+
The problem of optimally concatenating multiple sequences of different length until a maximum combined length is reached can be directly framed as a bin-packing or stock cutting problem. Since an exact solution is strongly NP-complete $\mathbb { \lVert 1 3 \rVert }$ , we propose two new heuristic algorithms that are tailored to this particular instance. A detailed introduction to packing is provided in [1](Section D)
|
| 39 |
+
|
| 40 |
+
# 3.1.1 Shortest-pack-first histogram-packing (SPFHP)
|
| 41 |
+
|
| 42 |
+
Shortest-pack-first histogram-packing (SPFHP) consists of three main components. First, the packing algorithm works on the bins in the sequence length histogram (with bin size 1) rather than the individual samples. Second, we operate on the sorted data from longest to shortest sequences. This comes basically for free due to the use of histograms. Third, we apply the worst-fit algorithm [11, 26] onto this histogram, where the currently observed sample goes to the pack1 that has the most space left to reach maximum packing depth (“shortest-pack-first”). If the sample does not fit, a new pack is created. A variant is to limit the packing depth, in other words the maximum number of sequences that are allowed in a pack. Therefore, we only extend an existing pack if it is not already at maximum packing depth. The detailed code for the algorithm is provided in [1] (listing $\textcircled { 3 }$ .
|
| 43 |
+
|
| 44 |
+
# 3.1.2 Non-negative least squares histogram-packing (NNLSHP)
|
| 45 |
+
|
| 46 |
+
The proposed NNLSHP algorithm is based on re-stating the packing problem as a (weighted) nonnegative least squares problem (NNLS) $\pmb { \mathbb { D } } \|$ of the form $w A x = w b$ where $x \geq 0$ . The vector $b$ is the histogram containing the counts of all the sequence lengths in the dataset. Next, we define the $A$ matrix (the “packing matrix“) by first generating a list of all possible sequence length combinations (“strategies”) that add up exactly to the maximum sequence length. We focus specifically on strategies that consist of at most 3 sequences per pack (independent of $b$ ) and encode each strategy as a column of the sparse matrix $A$ . For example, a strategy consisting of the sequence length 128, 128, and 256 in represented a column vector that has the value 2 at the 128th row, the value 1 at the 256th row, and zero at all other rows. The variable $x$ describes the non-negative repetition count for each strategy. So a 24 in the ith row of $x$ means that the strategy represented by the ith column of $A$ should repeat 24 times. Moreover, in the un-weighted setting, $A x = b$ states that we would like to “mix” the pre-defined strategies (columns of $A$ ) such that the number of samples matches the histogram $b$ , and where each strategy is used $x \geq 0$ times. We use the residual weight $w$ to control the penalization of the $A x - b$ residual on different sequence lengths (different rows of $b$ ). Heuristically, we set the weight of 0.09 for all sequences of length 8 or smaller because they are considered acceptable padding sequences. All other sequence lengths get weight 1. After solving $w A x = w b$ for $x \geq 0$ using an off-the-shelf solver we obtain a floating point solution, which means that the repetition counts are not necessarily integers. Since we cannot use a non-natural number of strategies, we round the solution $\hat { x }$ to the nearest integer. The error introduced by this rounding is found to be negligible. Further details are provided in [1](Section D.4).
|
| 47 |
+
|
| 48 |
+
# 3.2 Attention masking for packed sequences
|
| 49 |
+
|
| 50 |
+
To maintain an implementation that is consistent with the un-packed version, we need to be able to prevent attention between tokens in the pack which belong to different sequences. Other implementations use custom attention kernels which reconstruct padding. Instead, we propose directly masking the attention matrix with a block-diagonal mask to be applied before the attention. This is straightforward to implement in modern frameworks (see Figure 2). Naturally, there is a cost to both the mask construction and applying it to the attention matrix (see Table 1, Section $4 . 1 \dot { } \dot { } )$
|
| 51 |
+
|
| 52 |
+
|
| 53 |
+
Figure 2: Attention mask code sample [left] and example zero-one mask [right].
|
| 54 |
+
|
| 55 |
+
# 3.3 Calculating per-sequence loss and accuracy
|
| 56 |
+
|
| 57 |
+
Canonical implementations of BERT compute the cross-entropy loss for the masked language model on a per-sequence basis. Simply feeding packs of sequences to the same implementation of crossentropy would consequently result in per-pack weighting of the loss. In other words, the overall loss on the micro-batch would sum-up the losses on the individual packs, rather than individual sequences. As a result the packed BERT model would converge to a different optimum. For instance, a pack of a single sequence would contribute to the loss to the same extent as a pack of three sequences. In other words, the long sequence (single per pack) is given the same weight as the three shorter sequences in the pack of three. Empirically, a degradation of masked-language modelling accuracy on shorter sequences is indeed observed when not modifying the loss to account for packing.
|
| 58 |
+
|
| 59 |
+
To recover the per-sequence averaging behavior of the canonical un-packed BERT implementation, it is not sufficient to simply weight the loss (accuracy) on each pack by the number of sequences it contains, because the sequences in the pack have different lengths and therefore should not use the same weight.
|
| 60 |
+
|
| 61 |
+
To implement per-sequence loss, we effectively “unpack” the incoming logits and labels by working with the per-token loss. We compute the loss on all tokens belonging to the first sequence, then all tokens belonging to the second sequence, and so on. However, rather than looping through the sequences index in this way, we compute on all indexes in parallel. This minimizes the latency overhead of un-packing the loss calculation. We use the “masked lm weight” $\pmb { \mathbb { H } }$ input tensor to represent which sequence a given masked token belongs to (0, 1, 2 and so on). This is consistent with the canonical BERT implementation where this input takes a value of either 1 (belonging to the sequence) or 0 (belonging to padding) as detailed in Listing 1. The same methodology can be applied to the next-sentence prediction loss and accuracy.
|
| 62 |
+
|
| 63 |
+
# Listing 1: Loss calculation
|
| 64 |
+
|
| 65 |
+
# The number of sequences in each batch may vary 2 sequences_in_batch $=$ tf. reduce_sum (tf. reduce_max ( masked_lm_weight , -1)) 3 sequences_in_batch $=$ tf. cast ( sequences_in_batch , tf. float32 ) 4 # Create the 0/1 mask that will be used to un - packed sequences 5 masked_lm_weight $=$ tf. reshape ( masked_lm_weight , [B, 1, -1]) 6 sequence_selection $=$ tf. reshape (tf. range (1, max_sequences_per_pack + 1) , [1, -1, 1]) sequence_selection $=$ tf. cast ( masked_lm_weight $= =$ sequence_selection , tf. float32 ) 8 # Apply the mask to un - pack the loss per sequence 9 nll_per_token $=$ tf. reshape ( nll_per_token , [B, 1, -1]) 10 nll_per_sequence $=$ sequence_selection $^ { * }$ nll_per_token 11 # Normalize the per - sequence loss by the number of mlm - tokens in the sequence (as is standard ) 12 attempted $=$ tf. reduce_sum ( sequence_selection , -1, keepdims $=$ True ) 13 attempted $=$ attempted $^ +$ tf. cast ( attempted $= =$ 0, tf. float32 ) # prevent NaNs when dividing by attempted 14 nll_per_sequence $= \texttt { n l 1 }$ _per_sequence / attempted 15 # Average per - batch loss (so contributions from different batches are comparable ) 16 lm_loss $=$ tf. reduce_sum ( nll_per_sequence )/ sequences_in_batch
|
| 66 |
+
|
| 67 |
+
# 3.4 Hyperparameter adjustment
|
| 68 |
+
|
| 69 |
+
In terms of convergence behavior, the primary consequence of packing is an increase in the effective batch size (with respect to sequences and tokens) with some variation over different iterations. For instance, if each pack on average contains two sequences, the batch size (per optimization step) is effectively doubled on average. While one could subsequently reduce the computational batch size by the packing factor (average number of sequences per pack) and keep using the same hyperparameters, this is typically not desirable as it might imply under-utilizing the memory/compute.
|
| 70 |
+
|
| 71 |
+
Instead, we propose an approximate heuristic for updating the decay parameters of the LAMB optimizer $[ [ 2 5 ] ]$ . For a packed dataset with a packing factor $p$ , we update the decay parameters as: $\bar { \beta _ { 1 } } : = \beta _ { 1 } ^ { p }$ , $\beta _ { 2 } : = \beta _ { 2 } ^ { p }$ . For $p = 2$ , this corresponds to the exact parameters for calculating momentum and velocity, when updating with the same gradient twice $\scriptstyle { \mathbb { I I I } } ( { \mathrm { S e c t i o n ~ E } } )$ . A common approach is to scale the learning rate with the batch size. Note however, that we take the mean gradient instead of an accumulated sum and have already a correction by the number of samples in that regard.
|
| 72 |
+
|
| 73 |
+
Since these adjustments are only heuristics the convergence of the model will be comparable but not identical. In particular, it is unlikely that simply adjusting the hyperparameters will fully undo the impact of the increased batch size. However, with these adjustments, researchers should be able to continue to use existing configurations.
|
| 74 |
+
|
| 75 |
+
# 4 Experiments
|
| 76 |
+
|
| 77 |
+
# 4.1 Bin-packing algorithm comparison
|
| 78 |
+
|
| 79 |
+
We evaluate our algorithms using the following metrics: number of packs, number of all tokens, number of padding tokens, solution time of the packing algorithm (after histogram and strategy creation), number of strategies used, packing efficiency (the fraction of non-padding tokens in the packed dataset), the speed-up achieved compared to not packing (depth 1), and the average number of sequences per sample (packing factor). For SPFHP, we analyse different (maximum) packing depth, since packing is less efficient with smaller depth and we want to get a general understanding on how the packing depth influences the processing time. For NNLSHP, we focus on packing depth 3 because it packs the data sufficiently well.
|
| 80 |
+
|
| 81 |
+
For the speed-up analysis, we focus on the intelligence processing unit (IPU) [10] (IPU-M2000, 16 accelerator chips). A GPU dynamically loads the code into the accelerator; in contrast, the IPU works with a static precompiled kernel that gets loaded onto the chip only at the beginning. While other approaches result in excessive padding or continuous changes of the code, our approach can work with the same code for the whole dataset. So in this setting the IPU architecture would especially benefit from our approach since it avoids code changes. Nevertheless, it can be applied to any implementation on GPU or TPU. For determining the speed-up, we take advantage of the precompiled kernel. Since time measurements are quite noisy, we can profile the kernel and how many cycles it takes for processing a batch. That way, we can determine the overhead (in cycles) from processing the additional attention masking and for unpacking the loss. Combining overhead and packing factor, we get the speed-up estimate. No experiment repetitions are required since the algorithms and measurements are deterministic.
|
| 82 |
+
|
| 83 |
+
The main results for the performance metric evaluation are displayed in Table 1. The processing time for SBFHP was around $0 . 0 3 s$ and independent from the packing depth. We see that the overhead slightly increases with packing depth but that the benefits of packing outweigh the cost. The best speed-up is obtained with NNLSHP at depth 3. With a value of 1.913, it is close to the theoretical upper bound of 2.001. The results show that efficiency, packing factor, and speed-up can be viewed inter-changeably. The amount of time needed to process a sample (a pack of sequences) is barely changed relative to the un-packed implementation. The packing factor or the improvement in efficiency effectively provide an accurate estimate of the speed-up.
|
| 84 |
+
|
| 85 |
+
<table><tr><td>pack. depth</td><td>pack. algo.</td><td># packs [M]</td><td>efficiency (%)</td><td>pack. factor</td><td>overhead (%)</td><td>realized speed-up</td></tr><tr><td>1</td><td>none</td><td>16.280</td><td>49.97</td><td>1.000</td><td>0.000</td><td>1.000</td></tr><tr><td>2</td><td>SPFHP</td><td>10.102</td><td>80.52</td><td>1.612</td><td>4.283</td><td>1.544</td></tr><tr><td>3</td><td>SPFHP</td><td>9.095</td><td>89.44</td><td>1.790</td><td>4.287</td><td>1.716</td></tr><tr><td>3</td><td>NNLSHP</td><td>8.155</td><td>99.75</td><td>1.996</td><td>4.287</td><td>1.913</td></tr><tr><td>4</td><td>SPFHP</td><td>8.659</td><td>93.94</td><td>1.880</td><td>4.294</td><td>1.803</td></tr><tr><td>8</td><td>SPFHP</td><td>8.225</td><td>98.90</td><td>1.979</td><td>4.481</td><td>1.895</td></tr><tr><td>16/max</td><td>SPFHP</td><td>8.168</td><td>99.60</td><td>1.993</td><td>4.477</td><td>1.905</td></tr></table>
|
| 86 |
+
|
| 87 |
+
Table 1: Key performance results of proposed packing algorithms (SPFHP and NNLSHP). Packing depth describes the maximum number of packed sequences. Packing depth 1 is the baseline BERT implementation. Setting no limit resulted in a maximum packing depth of 16. The number of packs describes the length of the new packed dataset. Efficiency is the percentage of real tokens in the packed dataset. The packing factor describes the resulting potential speed-up compared to packing depth 1. With overhead, we denote the percentage decrease in throughput due to changes to the model to enable packing (such as the masking scheme introduced in Section $3 . 2 )$ . The realized speed-up is the combination of the speed-up due to packing (the packing factor) and the decrease in throughput due to the overhead. It is used to measure the relative speed-up in throughput and the overhead from masking and loss adjustment.
|
| 88 |
+
|
| 89 |
+
# 4.2 Learning Curves and Hyperparameter Adjustment
|
| 90 |
+
|
| 91 |
+
For depth 1 (classic BERT) and NNLSHP with depth 3, we additionally evaluate on the MLPerf 0.7 BERT pre-training benchmark $\mathbb { \left. \overline { { 1 5 } } \right. }$ . Briefly, this involves training from a standard checkpoint to a masked-language model accuracy of $7 1 . 2 \%$ using 3 million sequences with a maximum length of 512 tokens (refer to $\boxed { 1 6 }$ for details). Following this standardized benchmark supports reproduction of results even on other systems and makes sure that the reproduction effort is moderate and setup rules are clearly documented. We compare the resulting speed-up as well as the respective learning curves by evaluating the data on a held-out validation dataset. The objective of this additional evaluation is to analyse if convergence behavior is changed by the packing strategy and if the theoretical speed-up can be achieved in practice.
|
| 92 |
+
|
| 93 |
+
With packing, we effectively increase the average batch size by the packing factor $( \approx 2 )$ . However, with a different batch size, different hyperparameters are required (see Section $3 . 4 )$ and there is no mapping that will generate exact matching of results but only heuristics. In a first comparison, we use the same hyperparameters when comparing packed and unpacked training except for cutting the accumulation count by half. This way, we make sure that the batch size is constant on average.
|
| 94 |
+
|
| 95 |
+
In the second comparison, we evaluate our heuristics and how they compensate the difference in batch size. This setup is more desirable because it is beneficial to use the hardware to its full potential and cutting the batch size by half usually reduces throughput. In the third comparison, we compare two optimized setups.
|
| 96 |
+
|
| 97 |
+
The learning curves are displayed in Figure $3 .$ In the first setup, we see the curves almost matching perfectly when normalizing by the numbers of samples processed. Differences can be explained by the variation of the number of sequences in the packing batch, and general noise in the training process. Especially after the initial phase, the curves show a near-identical match.
|
| 98 |
+
|
| 99 |
+
The second setup shows bigger differences since changing the batch size and hyperparameters changes the training dynamics. We observe slower convergence early on in training due to the increased batch size. This is expected. The adjustment of the learning rate actually decreases performance probably because we correct for the increased number of sequences already in the modified loss. With the adjustment of the decay parameter of LAMB, we see matching performance at the later training stages. However, it is not feasible to completely recover the early convergence behavior of the smaller batch size by adjusting the hyperparameters. For instance doubling the batch size of unpacked BERT to 3000 and adjusting the LAMB decay parameters leads to more of a slow down in convergence than when running packed BERT with a batch size of 1500 and a packing factor of 2. Overall, in practice we observe a higher acceleration than the estimated 1.913 that goes beyond $2 \mathbf { x }$ . We explain this with slightly better fitting hyperparameters and improved data transfer.
|
| 100 |
+
|
| 101 |
+
|
| 102 |
+
Figure 3: Comparison of learning curves for packed and unpacked processing with [left] same effective batch size (ebs is batch size times packing factor), [middle] different heuristic adjustments of the hyperparameters (batch size 1500 for all runs, such that ebs for packed runs is $1 5 0 0 * 2$ ), and [right] realized time-to-convergence speed-up from packing.
|
| 103 |
+
|
| 104 |
+
# 4.3 Scaling Analysis: Impact of the number of accelerators
|
| 105 |
+
|
| 106 |
+
A further advantage of packing over competing un-padding approaches is the inherent load balancing provided by packing. So called un-padding approaches rely on dynamically launching custom kernels that ignore padding. A stated advantage of such implementations is the ability to avoid computing the complete $( 5 1 2 \mathrm { ~ x ~ } 5 1 2 )$ ) attention matrix. This provides additional computational savings compared to packing, where the attention matrix is computed in its entirety and then masked. Because of these additional savings, un-padding can exceed the theoretical upper bound for speed-up from packing (2.013 on Wikipedia). As a result of the dynamic nature of the approach, the processing time with un-padding is different for each sequence in the batch, and the amount of time required to process a batch of sequences will be determined by the processing time of the longest sequence in the batch (with the sequences being processed in parallel). Furthermore, in the multiple accelerator setting the processing time on each device will vary depending on the sequences in the batch that it receives. Devices which finish early have to wait for the slowest device to finish before exchanging gradients. This load-imbalance between the devices (and inside the batch) leads to a considerable decrease in the speed-up from un-padding as the number of accelerators is increased (see Figure 4).
|
| 107 |
+
|
| 108 |
+
In contrast, packing (our approach) is inherently load-balanced. The processing time on each accelerator is independent of the content inside the batch received by the device. Any number of accelerators can therefore operate in unison without having to wait for the slowest batch to process (all per-device batches are equally fast).
|
| 109 |
+
|
| 110 |
+
To demonstrate the severity of the load-imbalance issue, we consider the scaling of an un-padding approach with a per-device batch size of 32 running on eight devices $\mathbb { \ m }$ . From there, we readily extrapolate the performance to both larger and smaller cluster sizes by fitting a Gumbel distribution to the observed processing times [1] (Section F). On a single device with batch size $3 2 { \mathrm { u n } }$ -padding outperforms packing and exceeds the theoretical upper-bound for packing.
|
| 111 |
+
|
| 112 |
+
As the number of devices increases to two or more, the proposed packing approach outperforms the dynamic un-padding approach. On a cluster with 32 accelerators the speed-up from un-padding drops to $5 0 \%$ and with 2048 devices the speed-up is only $3 0 \%$ . In contrast, the speed-up due to packing is independent of the number of accelerators and stays at 1.913. Switching to a smaller batch size would reduce the load-imbalance issue to some extent, but would also result in under-utilization of the available memory and compute.
|
| 113 |
+
|
| 114 |
+
|
| 115 |
+
Figure 4: Comparison of the theoretical speed-up achievable as the number of accelerators is increased.
|
| 116 |
+
|
| 117 |
+
# 5 Conclusion
|
| 118 |
+
|
| 119 |
+
We showed that packing can be easily implemented without the need for any custom kernels while still providing a 2x speed-up. Additionally, we showed that any additional speed-ups resulting from dynamic un-padding approaches diminish for even moderate batch sizes or when additional accelerators are added. In contrast, packing is load-balanced and maintains the $2 \mathbf { x }$ throughput when scaling to large numbers of accelerators.
|
| 120 |
+
|
| 121 |
+
Furthermore, the computational overhead introduced by the attention mask and the packed persequence loss are small compared to the achieved acceleration. This overhead remains below $5 \%$ for all tested packing depths. The efficient packing algorithms presented in this paper enable us to pack millions of sequences in a matter of seconds. Compared to both the pre-processing time for the Wikipedia dataset and the training runtime, this overhead is negligible. Furthermore, we showed that performing packing as a pre-processing step does not significantly impact the training convergence. Our proposed hyperparameter adjustment scheme additionally helps practitioners easily modify existing validated optimizer settings for use with packed BERT. Further exploration of hyperparameter selection is left to future work.
|
| 122 |
+
|
| 123 |
+
When performing packing as a pre-processing step, the proposed NNLSHP and SPFHP methods achieve near optimal compression efficiency. In this offline setting, we are able to build a histogram of the dataset, and thus achieve linear time complexity with respect to the number of samples. This makes packing modern datasets with millions of sequences possible. In the future, it would be interesting to extend SPFHP to the online setting where a histogram of the entire dataset cannot be built. Another interesting direction is the packing of images of different sizes to help accelerate computer-vision applications. This is especially relevant given the recent advances in the use of transformer-based approaches in the computer vision domain, for example the visual transformer [24]. Masking out the self-attention within transformers is easier to implement than avoiding cross-contamination of convolutions applied to packed images. Finally, packing could potentially eliminate the need for two phase pre-training of BERT. Using short sequences in the first phase to reduce the waste from padding is no longer attractive for packed sequence BERT where the padding is essentially a negligible proportion of the tokens. Furthermore, the argument that the model should first learn short-term dependencies by training on short sequences neglects the fact that these same short-term patterns can be learned from longer sequences. In fact, longer-sequences may contain multiple short patterns, while also maintaining long-range consistency. Future work should explore training packed BERT from scratch and the impact of packing on fine-tuned performance.
|
| 124 |
+
|
| 125 |
+
# Broader Impact
|
| 126 |
+
|
| 127 |
+
We showed that when pre-training BERT on Wikipedia, the computational overhead taken to process padding tokens is roughly $5 0 \%$ . By eliminating this wasted computational time, the approach presented in this paper paves a way to halving the carbon footprint of training BERT-based models.
|
| 128 |
+
|
| 129 |
+
Furthermore, our approach circumvents the need for custom kernels, making the benefits of packing readily accessible to a broader audience of NLP practitioners. As such we are hopeful the research will have a positive impact on the NLP community, and do not see any disadvantage of using this approach.
|
| 130 |
+
|
| 131 |
+
Future work would need to investigate the applicability of packing on text produced by different cultures and in different languages. We have already shown that the speed-up resulting from using our methods does not only occur when pre-training BERT on Wikipedia but also on other datasets such as SQUaD and GLUE. Furthermore, the sentence length distribution of the original English language text shows similar characteristics. Our research leads us to believe that compressible distributions arise naturally in language tasks and beyond, for instance in DNA sequence lengths [9] and protein lengths $\textcircled { 8 }$ . Many such sequence modelling workloads are based on variations of the BERT/transformer architecture and would therefore easily benefit from our acceleration.
|
| 132 |
+
|
| 133 |
+
Failures in NLP can have a big impact on society; many technologies, such as Alexa, Siri, and Google Home, rely on them. Whilst any errors arising from our approach can be avoided, one potential source of error comes from the implementation. Both the attention mask and the per-sequence loss need to be modified to support packing. These changes are significantly smaller than those required by custom kernels, however they may still be time consuming to implement and debug. To help mitigate the risk of any implementation errors, we share our reference implementations of the required changes in the supplemental material [1].
|
| 134 |
+
|
| 135 |
+
# References
|
| 136 |
+
|
| 137 |
+
[1] ANONYMOUS. Supplemental Material for “Packing: Towards 2x NLP BERT Acceleration”, 2021.
|
| 138 |
+
[2] BRO, R., AND DE JONG, S. A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics 11, 5 (sep 1997), 393–401.
|
| 139 |
+
[3] BROWN, T. B., MANN, B., RYDER, N., SUBBIAH, M., KAPLAN, J., DHARIWAL, P., NEELAKANTAN, A., SHYAM, P., SASTRY, G., ASKELL, A., AGARWAL, S., HERBERT-VOSS, A., KRUEGER, G., HENIGHAN, T., CHILD, R., RAMESH, A., ZIEGLER, D. M., WU, J., WINTER, C., HESSE, C., CHEN, M., SIGLER, E., LITWIN, M., GRAY, S., CHESS, B., CLARK, J., BERNER, C., MCCANDLISH, S., RADFORD, A., SUTSKEVER, I., AND AMODEI, D. Language Models are Few-Shot Learners. In Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020) (may 2020).
|
| 140 |
+
[4] BYTEDANCE INC. Effective Transformer. https://github.com/bytedance/effective_ transformer, 2021.
|
| 141 |
+
[5] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. BERT: Pre-training of deep bidirectional transformers for language understanding. NAACL HLT 2019 - 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies - Proceedings of the Conference 1 (oct 2019), 4171–4186.
|
| 142 |
+
[6] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. https://github.com/ google-research/bert, 2019.
|
| 143 |
+
[7] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. Pre-training data creation script for BERT. https://github.com/google-research/bert/blob/master/ create_pretraining_data.py#L243, 2019.
|
| 144 |
+
[8] GUILLÉN, G., DIAZ-CAMINO, C., LOYOLA-TORRES, C., APARICIO-FABRE, R., HERNÁNDEZ-LÓPEZ, A., DÍAZ-SÁNCHEZ, M., AND SANCHEZ, F. Detailed analysis of
|
| 145 |
+
|
| 146 |
+
putative genes encoding small proteins in legume genomes. Frontiers in Plant Science 4 (2013), 208.
|
| 147 |
+
|
| 148 |
+
[9] HANSEN, H. B., DAMGAARD, P. B., MARGARYAN, A., STENDERUP, J., LYNNERUP, N., WILLERSLEV, E., AND ALLENTOFT, M. E. Comparing ancient dna preservation in petrous bone and tooth cementum. PLOS ONE 12, 1 (01 2017), 1–18.
|
| 149 |
+
|
| 150 |
+
[10] JIA, Z., TILLMAN, B., MAGGIONI, M., AND SCARPAZZA, D. P. Dissecting the Graphcore IPU architecture via microbenchmarking. ArXiv abs/1912.03413 (2019).
|
| 151 |
+
|
| 152 |
+
[11] JOHNSON, D. S. Near-optimal bin packing algorithms. PhD thesis, Massachusetts Institute of Technology, 1973.
|
| 153 |
+
|
| 154 |
+
[12] JOHNSON, D. S., AND GAREY, M. R. A 7160 theorem for bin packing. Journal of Complexity 1, 1 (oct 1985), 65–106.
|
| 155 |
+
|
| 156 |
+
[13] KORTE, B., AND VYGEN, J. Combinatorial Optimization, vol. 21 of Algorithms and Combinatorics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012.
|
| 157 |
+
|
| 158 |
+
[14] LAN, Z., CHEN, M., GOODMAN, S., GIMPEL, K., SHARMA, P., AND SORICUT, R. ALBERT: A lite BERT for self-supervised learning of language representations. CoRR abs/1909.11942 (2019).
|
| 159 |
+
|
| 160 |
+
[15] MATTSON, P., REDDI, V. J., CHENG, C., COLEMAN, C., DIAMOS, G., KANTER, D., MICIKEVICIUS, P., PATTERSON, D., SCHMUELLING, G., TANG, H., WEI, G., AND WU, C. MLPerf: An Industry Standard Benchmark Suite for Machine Learning Performance. IEEE Micro 40, 2 (2020), 8–16.
|
| 161 |
+
|
| 162 |
+
[16] MLCOMMONS. v0.7 Results. https://mlcommons.org/en/training-normal-07/, 2020. Result not verified by MLPerf. Throughput/speedup is not the primary metric of MLPerf. MLPerf name and logo are trademarks. See www.mlperf.org for more information.
|
| 163 |
+
|
| 164 |
+
[17] NVIDIA. Reference numbers for BERT un-padding results. https://github.com/ mlcommons/training_results_v0.7/blob/master/NVIDIA/results/dgxa100 ngc20.06_pytorch/bert/result_0.txt, 2020. Throughput/speedup is not the primary metric of MLPerf. MLPerf name and logo are trademarks. See www.mlperf.org for more information.
|
| 165 |
+
|
| 166 |
+
[18] NVIDIA. Faster Transformer. https://github.com/NVIDIA/DeepLearningExamples/ tree/master/FasterTransformer/v1, 2021.
|
| 167 |
+
|
| 168 |
+
[19] RAJPURKAR, P., ZHANG, J., LOPYREV, K., AND LIANG, P. SQuAD: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing (Austin, Texas, Nov. 2016), Association for Computational Linguistics, pp. 2383–2392.
|
| 169 |
+
|
| 170 |
+
[20] TURC, I., CHANG, M.-W., LEE, K., AND TOUTANOVA, K. Well-read students learn better: On the importance of pre-training compact models. arXiv preprint arXiv:1908.08962v2 (2019).
|
| 171 |
+
|
| 172 |
+
[21] VASWANI, A., SHAZEER, N., PARMAR, N., USZKOREIT, J., JONES, L., GOMEZ, A. N., KAISER, U., AND POLOSUKHIN, I. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems (Red Hook, NY, USA, 2017), NIPS’17, Curran Associates Inc., p. 6000–6010.
|
| 173 |
+
|
| 174 |
+
[22] WANG, A., SINGH, A., MICHAEL, J., HILL, F., LEVY, O., AND BOWMAN, S. GLUE: A multi-task benchmark and analysis platform for natural language understanding. In Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP (Brussels, Belgium, Nov. 2018), Association for Computational Linguistics, pp. 353–355.
|
| 175 |
+
|
| 176 |
+
[23] WARSTADT, A., SINGH, A., AND BOWMAN, S. R. Neural network acceptability judgments. arXiv preprint arXiv:1805.12471 (2018).
|
| 177 |
+
|
| 178 |
+
[24] WU, B., XU, C., DAI, X., WAN, A., ZHANG, P., YAN, Z., TOMIZUKA, M., GONZALEZ, J., KEUTZER, K., AND VAJDA, P. Visual transformers: Token-based image representation and processing for computer vision, 2020.
|
| 179 |
+
|
| 180 |
+
[25] YOU, Y., LI, J., REDDI, S., HSEU, J., KUMAR, S., BHOJANAPALLI, S., SONG, X., DEMMEL, J., KEUTZER, K., AND HSIEH, C.-J. Large Batch Optimization for Deep Learning: Training BERT in 76 minutes. arXiv (apr 2019).
|
| 181 |
+
|
| 182 |
+
[26] YUE, M., AND ZHANG, L. A simple proof of the inequality $M F F D ( L ) \leq 7 1 / 6 0 O P T ( L ) +$ $1 , L$ for the MFFD bin-packing algorithm. Acta Mathematicae Applicatae Sinica $^ Ḋ I I Ḍ$ , 3 (jul 1995), 318–330.
|
| 183 |
+
|
| 184 |
+
# Checklist
|
| 185 |
+
|
| 186 |
+
1. For all authors...
|
| 187 |
+
|
| 188 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] Our paper has three main claims. First, in Figure 1[right] we show the distribution of the Wikipedia and the excessive padding that it requires. Second, in Section $^ { 4 . 1 , }$ we show that we can efficiently pack the data which can be easily reproduced with the shared data and code [1]. Third, in Figure 3[right], we clearly show the $2 \mathbf { x }$ performance gain from packing and the related hyperparameter adjustment scheme.
|
| 189 |
+
(b) Did you describe the limitations of your work? [Yes] We see three potential limitations that we discuss in the paper. First, as stated in the broader impact section, our approach is clearly dependent on the sequence length distribution of the dataset. However, we looked into several other datasets beyond Wikipedia and observed even higher potential for acceleration. Second, we explain our focus on the IPU hardware with a static precompiled kernel in Section $\boxed { 4 . 1 }$ Our theoretical analysis in Section $\boxed { 4 . 3 }$ indicates that our approach benefits also other hardware. Third, our changes to the network with a modified attention mask and loss calculation come with some overhead. This is addressed in Table 1 [overhead column] in Section 4.1.
|
| 190 |
+
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We address this point in the “Broader Impact” Section, third paragraph.
|
| 191 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
|
| 192 |
+
|
| 193 |
+
2. If you are including theoretical results...
|
| 194 |
+
|
| 195 |
+
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Detailed algorithm explanations, clarifications of assumptions, and proofs are provided in the supplemental material [1].
|
| 196 |
+
(b) Did you include complete proofs of all theoretical results? [Yes] Section D.5, E, F in the supplemental material [1] provide the necessary derivations on theoretical results.
|
| 197 |
+
|
| 198 |
+
3. If you ran experiments...
|
| 199 |
+
|
| 200 |
+
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] To ensure results are easily reproducible, we follow the MLPerf 0.7 benchmark rules and implementation. Additionally the main packing code is provided in the supplemental material, along with histograms of the datasets, which can be used to confirm the efficiency of the packing algorithms. We are solely relying on open source datasets. The full code related to the changes to BERT will be provided with a future software release, currently anticipated in July. Simplified reference code is provided directly in the paper.
|
| 201 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We follow the MLPerf 0.7 benchmark rules. We document the parameters that we changed and why we change them.
|
| 202 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] The packing algorithms are deterministic and have no error. Other experiments are only once to compare convergence curves.
|
| 203 |
+
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We used 16 Graphcore IPUs for acceleration on an internal cluster.
|
| 204 |
+
|
| 205 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 206 |
+
|
| 207 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] Appropriate references to the BERT authors, all datasets, and the code snippet from the HugginFace inc. are appropriately referenced with citations and links.
|
| 208 |
+
(b) Did you mention the license of the assets? [Yes] For the only taken code snippet, the license is part of the file [Listing 6 in the appendix].
|
| 209 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] New materials like packing code and histograms will be provided under an MIT license and are already listed and linked at the end of the supplemental material.
|
| 210 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We did not curate other people’s data. We only provide a very high level aggregate of the used data.
|
| 211 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We did not curate other people’s data.
|
| 212 |
+
|
| 213 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 214 |
+
|
| 215 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] Our experiments did not include crowdsourcing or human subjects.
|
| 216 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] Our experiments did not include crowdsourcing or human subjects.
|
| 217 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Our experiments did not include crowdsourcing or human subjects.
|
parse/train/3_MUAtqR0aA/3_MUAtqR0aA_content_list.json
ADDED
|
@@ -0,0 +1,1142 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Packing: Towards 2x NLP BERT Acceleration ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
215,
|
| 8 |
+
122,
|
| 9 |
+
781,
|
| 10 |
+
147
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Anonymous ",
|
| 17 |
+
"text_level": 1,
|
| 18 |
+
"bbox": [
|
| 19 |
+
457,
|
| 20 |
+
202,
|
| 21 |
+
540,
|
| 22 |
+
215
|
| 23 |
+
],
|
| 24 |
+
"page_idx": 0
|
| 25 |
+
},
|
| 26 |
+
{
|
| 27 |
+
"type": "text",
|
| 28 |
+
"text": "Abstract ",
|
| 29 |
+
"text_level": 1,
|
| 30 |
+
"bbox": [
|
| 31 |
+
462,
|
| 32 |
+
251,
|
| 33 |
+
535,
|
| 34 |
+
267
|
| 35 |
+
],
|
| 36 |
+
"page_idx": 0
|
| 37 |
+
},
|
| 38 |
+
{
|
| 39 |
+
"type": "text",
|
| 40 |
+
"text": "We find that at sequence length 512 padding tokens represent in excess of $5 0 \\%$ of the Wikipedia dataset used for pretraining BERT (Bidirectional Encoder Representations from Transformers). Therefore by removing all padding we achieve a $2 \\mathbf { x }$ speed-up in terms of sequences/sec. To exploit this characteristic of the dataset, we develop and contrast two deterministic packing algorithms. Both algorithms rely on the assumption that sequences are interchangeable and therefore packing can be performed on the histogram of sequence lengths, rather than per sample. This transformation of the problem leads to algorithms which are fast and have linear complexity in dataset size. The shortest-pack-first histogram-packing (SPFHP) algorithm determines the packing order for the Wikipedia dataset of over 16M sequences in 0.02 seconds. The non-negative least-squares histogram-packing (NNLSHP) algorithm converges in 28.4 seconds but produces solutions which are more depth efficient, managing to get near optimal packing by combining a maximum of 3 sequences in one sample. Using the dataset with multiple sequences per sample requires additional masking in the attention layer and a modification of the MLM loss function. We demonstrate that both of these changes are straightforward to implement and have relatively little impact on the achievable performance gain on modern hardware. Finally, we pretrain BERT-Large using the packed dataset, demonstrating no loss of convergence and the desired $2 \\mathbf { x }$ speed-up. ",
|
| 41 |
+
"bbox": [
|
| 42 |
+
233,
|
| 43 |
+
281,
|
| 44 |
+
766,
|
| 45 |
+
542
|
| 46 |
+
],
|
| 47 |
+
"page_idx": 0
|
| 48 |
+
},
|
| 49 |
+
{
|
| 50 |
+
"type": "text",
|
| 51 |
+
"text": "1 Introduction ",
|
| 52 |
+
"text_level": 1,
|
| 53 |
+
"bbox": [
|
| 54 |
+
176,
|
| 55 |
+
568,
|
| 56 |
+
310,
|
| 57 |
+
584
|
| 58 |
+
],
|
| 59 |
+
"page_idx": 0
|
| 60 |
+
},
|
| 61 |
+
{
|
| 62 |
+
"type": "text",
|
| 63 |
+
"text": "Since its introduction in 2019, BERT $ { \\mathbb { I } }$ has been the backbone driving the most exciting advances in Natural Language Processing (NLP). Pre-training BERT from scratch requires substantial computational resources which may be out of reach for researchers and industry professionals. To some extent this has been addressed by the public release of pre-trained models of different sizes and depths $\\left[ \\left[ 2 0 \\right] \\right]$ . Available sizes range from tiny (2 layers with hidden size 128) to large (24 layers with hidden size 1024)[6, 5]. The introduction of ALBERT $\\textcircled { 1 1 4 } \\textcircled { 1 }$ further improved the accessibility of larger models. However, the dependence on pre-trained models limits the ability of researchers to explore new backbone architectures. Furthermore, it limits the extent to which practitioners in industry can leverage internal datasets and adapt the model to their particular needs. Hence, any approach that speeds up the pre-training process is desirable from an economical as well as environmental perspective. ",
|
| 64 |
+
"bbox": [
|
| 65 |
+
174,
|
| 66 |
+
598,
|
| 67 |
+
825,
|
| 68 |
+
751
|
| 69 |
+
],
|
| 70 |
+
"page_idx": 0
|
| 71 |
+
},
|
| 72 |
+
{
|
| 73 |
+
"type": "text",
|
| 74 |
+
"text": "In this paper, we present some methods to enable researchers to accelerate the pre-training of BERT by as much as $2 \\mathbf { x }$ . The de-facto pre-training dataset Wikipedia, as well as many other NLP datasets, show a positively skewed distribution of sequence lengths. We show that padding tokens (wasted compute) represent $5 0 \\%$ of all tokens of the Wikipedia pre-training dataset at sequence length 512. Overall, the sample lengths range between 5 tokens up to 512 (see Figure 1). Samples of length 512 represent only $2 3 . 5 \\%$ of the dataset. ",
|
| 75 |
+
"bbox": [
|
| 76 |
+
174,
|
| 77 |
+
757,
|
| 78 |
+
825,
|
| 79 |
+
840
|
| 80 |
+
],
|
| 81 |
+
"page_idx": 0
|
| 82 |
+
},
|
| 83 |
+
{
|
| 84 |
+
"type": "text",
|
| 85 |
+
"text": "While processing the padding tokens wastes compute, it is the most standard approach for leveraging modern massively-parallel compute especially on GPUs. These are most efficient when applying the same operation to each sequence in a batch. By padding all sequences to the same maximum sequence length, they can easily be batched. The most obvious way to reduce the extent of padding in the dataset is to group samples by size before batching, i.e., process the shorter samples together and longer samples together. Typically such an approach would still involve padding but less than if padding all sequences to the same maximum length. For example BERT $\\boxed { \\pmb { \\bigtriangledown } }$ is pre-trained in two phases, where the first phase uses sequence length 128 for 900K steps and the second phase uses sequence length 512 for 100K steps. However even by splitting the training in this way, the wasted compute due to padding is approximately $2 0 \\%$ (see Figure $\\bigstar \\bigstar \\bigstar \\bigstar$ ). Another example of this approach is Faster Transformer $\\overline { { \\mathbb { I } \\mathbb { 8 } \\| } }$ which groups samples of similar size together in one batch and fills up with padding only to the maximum length in this batch. ",
|
| 86 |
+
"bbox": [
|
| 87 |
+
174,
|
| 88 |
+
847,
|
| 89 |
+
825,
|
| 90 |
+
902
|
| 91 |
+
],
|
| 92 |
+
"page_idx": 0
|
| 93 |
+
},
|
| 94 |
+
{
|
| 95 |
+
"type": "text",
|
| 96 |
+
"text": "",
|
| 97 |
+
"bbox": [
|
| 98 |
+
174,
|
| 99 |
+
90,
|
| 100 |
+
825,
|
| 101 |
+
202
|
| 102 |
+
],
|
| 103 |
+
"page_idx": 1
|
| 104 |
+
},
|
| 105 |
+
{
|
| 106 |
+
"type": "text",
|
| 107 |
+
"text": "More advanced approaches for reducing the padding overhead rely on custom computational kernels. Loosely these are referred to as “un-padding” approaches. In Effective Transformer $\\pmb { \\Vert 4 \\Vert }$ , the input batch is provided as a padded matrix but padding values are dynamically removed and restored during different calculation stages. While un-padding implementations are highly sophisticated and are able to completely circumvent the processing of padding tokens, they introduce a significant overhead due to the multiple GPU kernel launches (i.e. one kernel per sequence rather than one kernel per batch). Additionally the time to process each batch will fluctuate depending on the sequence lengths in each batch i.e. batches with only shorter sequences will typically be processed faster. When working with more than one accelerator, this variability in throughput results in all devices in the cluster waiting for the device with the most compute intensive batch to finish processing. As such, un-padding approaches are not appropriate for deployment on large clusters. ",
|
| 108 |
+
"bbox": [
|
| 109 |
+
174,
|
| 110 |
+
208,
|
| 111 |
+
825,
|
| 112 |
+
361
|
| 113 |
+
],
|
| 114 |
+
"page_idx": 1
|
| 115 |
+
},
|
| 116 |
+
{
|
| 117 |
+
"type": "text",
|
| 118 |
+
"text": "The “packing” based approach introduced in this paper offers significant advantages over un-padding approaches. Firstly, packing is implemented directly at the framework level and requires no additional custom kernel implementations. Secondly, the processing time for each batch is independent of the content of the batch, allowing the packing based approach to maintain the same speed-up whether running on a single device or thousands. Third, each batch now contains a consistent number of real tokens. ",
|
| 119 |
+
"bbox": [
|
| 120 |
+
174,
|
| 121 |
+
367,
|
| 122 |
+
825,
|
| 123 |
+
449
|
| 124 |
+
],
|
| 125 |
+
"page_idx": 1
|
| 126 |
+
},
|
| 127 |
+
{
|
| 128 |
+
"type": "text",
|
| 129 |
+
"text": "While we demonstrate the effectiveness of packing specifically on the Wikipedia dataset, packing SQUaD $\\mathbb { \\lVert \\underline { { 9 } } \\rVert }$ or GLUE datasets $\\mathbb { \\lVert 2 2 \\rVert 2 3 \\rVert }$ for BERT also leads to significant speed-ups (some in excess of 9x) [1] (sections A and B). The effectiveness of packing is a result of both the length distribution of the documents in the source datasets as well as the different text preprocessing steps for BERT [7]. The use of bi-directional self-attention in BERT implies that the input sequences should contain complete sentences. If a sentence is abruptly cut short, the hidden state on other (preceding) tokens in the sequence will be affected. Language models with causal attention (only attending to previous tokens in the input) do not have this issue. For such models, if a sequence is cut short at an arbitrary token, the other tokens (which occur earlier in the sequence) will not be affected. This ability to cut sequences arbitrarily completely trivializes the packing problem. For instance, GPT-3 $\\mathbb { \\left[ 3 \\right] }$ is trained with a maximum sequence length of 2048 where a single sequence may contain multiple segments separated by a special end of segment token. The last segment in each sequence is simply cut to meet the sequence length requirement. In the interest of computational efficiency GPT-3 does not mask the attention between different segments in a sequence. In contrast, the packing approach presented in this paper introduces a mask in the attention layer (see Section $3 . 2 )$ to prevent cross-contamination between examples in a pack. This ensures that the characteristics of the original dataset and model are matched as closely as possible. ",
|
| 130 |
+
"bbox": [
|
| 131 |
+
174,
|
| 132 |
+
457,
|
| 133 |
+
825,
|
| 134 |
+
691
|
| 135 |
+
],
|
| 136 |
+
"page_idx": 1
|
| 137 |
+
},
|
| 138 |
+
{
|
| 139 |
+
"type": "text",
|
| 140 |
+
"text": "In summary, the contributions of the paper are as follows. In Section $\\bigtriangledown ,$ , we produce histograms of the Wikipedia pre-training dataset showing the high percentage of padding tokens. We present two new deterministic packing algorithms which easily pack datasets with millions of sequences in a matter of seconds (or less). We empirically show that the proposed packing algorithms produce a nearly-optimal packing scheme for Wikipedia pre-training dataset. We show how to compute the per-sequence loss by inexpensively un-packing the loss. We provide code for building an attention mask which prevents attention between tokens of different sequences in the pack. We demonstrate that the convergence of the BERT large model on the packed dataset is equivalent to that on the un-packed dataset. We show that with the packed dataset, we are able to achieve a nearly 2x throughput increase on the Wikipedia sequence length 512 pre-training dataset. ",
|
| 141 |
+
"bbox": [
|
| 142 |
+
174,
|
| 143 |
+
696,
|
| 144 |
+
825,
|
| 145 |
+
837
|
| 146 |
+
],
|
| 147 |
+
"page_idx": 1
|
| 148 |
+
},
|
| 149 |
+
{
|
| 150 |
+
"type": "text",
|
| 151 |
+
"text": "2 Wikipedia BERT pre-training dataset ",
|
| 152 |
+
"text_level": 1,
|
| 153 |
+
"bbox": [
|
| 154 |
+
174,
|
| 155 |
+
88,
|
| 156 |
+
521,
|
| 157 |
+
107
|
| 158 |
+
],
|
| 159 |
+
"page_idx": 2
|
| 160 |
+
},
|
| 161 |
+
{
|
| 162 |
+
"type": "text",
|
| 163 |
+
"text": "BERT is pre-trained using masked-language modelling and next-sentence prediction on a large corpus of Wikipedia articles [5]. Each sequence is composed of one ${ \\mathrm { < C L S > } }$ token followed by the first part of sentences, followed by a ${ \\tt { < S E P > } }$ token, and then finally the second part of sentences. Because parts are created in sentence-level increments there is no token-level control of sequence length. Together with already short parts, empirically, this leads to significant levels of padding, especially for longer maximum sequence lengths (see Figure 1). At sequence length 128 (commonly used in phase 1 of pre-training) the theoretical speed-up is around 1.2, at sequence length 384 this increases to 1.7, and finally at sequence length 512 (commonly used for phase 2 of pre-training) it is 2.0. Despite the widespread use of the Wikipedia dataset for pre-training BERT such histograms have, to the best of our knowledge, not been published previously. This has perhaps lead to the underestimation of the speed-up opportunity available. To put things into perspective, the sequence length 512 dataset contains 8.33 billion tokens, of which 4.17 billion are padding tokens. ",
|
| 164 |
+
"bbox": [
|
| 165 |
+
173,
|
| 166 |
+
122,
|
| 167 |
+
825,
|
| 168 |
+
287
|
| 169 |
+
],
|
| 170 |
+
"page_idx": 2
|
| 171 |
+
},
|
| 172 |
+
{
|
| 173 |
+
"type": "image",
|
| 174 |
+
"img_path": "images/e7c18d1d548271692d6a27c5121e391cde59a8daf128d6a52e32a07fd3f2934e.jpg",
|
| 175 |
+
"image_caption": [
|
| 176 |
+
"Figure 1: Wikipedia BERT pre-training dataset sequence length histograms (token count excluding padding) for different maximum sequence lengths. Based on the Wikipedia article dump from October 1st 2020. The theoretical speed-up relates to not using any padding tokens and not having any overhead from processing the different lengths. "
|
| 177 |
+
],
|
| 178 |
+
"image_footnote": [],
|
| 179 |
+
"bbox": [
|
| 180 |
+
178,
|
| 181 |
+
304,
|
| 182 |
+
820,
|
| 183 |
+
445
|
| 184 |
+
],
|
| 185 |
+
"page_idx": 2
|
| 186 |
+
},
|
| 187 |
+
{
|
| 188 |
+
"type": "text",
|
| 189 |
+
"text": "3 Methods ",
|
| 190 |
+
"text_level": 1,
|
| 191 |
+
"bbox": [
|
| 192 |
+
174,
|
| 193 |
+
554,
|
| 194 |
+
277,
|
| 195 |
+
570
|
| 196 |
+
],
|
| 197 |
+
"page_idx": 2
|
| 198 |
+
},
|
| 199 |
+
{
|
| 200 |
+
"type": "text",
|
| 201 |
+
"text": "Our approach consists of three distinct components. Firstly, we pack the data efficiently during preprocessing to make full use of the sequence length (Sections $\\checkmark$ and $3 . 1 . 2 ,$ see also [1] Section D). Secondly, we adapt the self-attention mask to prevent the model from attending between different sequences in the same pack (Section $\\boxed { 3 . 2 }$ . Other components of the model, such as the feed-forward layer $\\scriptstyle { \\left\\| { \\overline { { 2 1 } } } \\right\\| }$ , operate on a per-token basis and do not require any modification. Thirdly, we compute the loss and accuracy on a per-sequence basis to match the canonical BERT implementation (Section $\\textcircled { 3 . 3 }$ This is achieved by unpacking the per-pack loss at the framework level, without the use of custom kernels. Additionally, we provide suggestions for hyperparameter adjustment (Section $3 . 4 )$ that lead to analogous convergence behavior between the packed and un-packed BERT implementations. ",
|
| 202 |
+
"bbox": [
|
| 203 |
+
174,
|
| 204 |
+
587,
|
| 205 |
+
826,
|
| 206 |
+
712
|
| 207 |
+
],
|
| 208 |
+
"page_idx": 2
|
| 209 |
+
},
|
| 210 |
+
{
|
| 211 |
+
"type": "text",
|
| 212 |
+
"text": "3.1 Packing algorithms ",
|
| 213 |
+
"text_level": 1,
|
| 214 |
+
"bbox": [
|
| 215 |
+
174,
|
| 216 |
+
731,
|
| 217 |
+
348,
|
| 218 |
+
746
|
| 219 |
+
],
|
| 220 |
+
"page_idx": 2
|
| 221 |
+
},
|
| 222 |
+
{
|
| 223 |
+
"type": "text",
|
| 224 |
+
"text": "The problem of optimally concatenating multiple sequences of different length until a maximum combined length is reached can be directly framed as a bin-packing or stock cutting problem. Since an exact solution is strongly NP-complete $\\mathbb { \\lVert 1 3 \\rVert }$ , we propose two new heuristic algorithms that are tailored to this particular instance. A detailed introduction to packing is provided in [1](Section D) ",
|
| 225 |
+
"bbox": [
|
| 226 |
+
174,
|
| 227 |
+
757,
|
| 228 |
+
825,
|
| 229 |
+
813
|
| 230 |
+
],
|
| 231 |
+
"page_idx": 2
|
| 232 |
+
},
|
| 233 |
+
{
|
| 234 |
+
"type": "text",
|
| 235 |
+
"text": "3.1.1 Shortest-pack-first histogram-packing (SPFHP) ",
|
| 236 |
+
"text_level": 1,
|
| 237 |
+
"bbox": [
|
| 238 |
+
173,
|
| 239 |
+
830,
|
| 240 |
+
558,
|
| 241 |
+
845
|
| 242 |
+
],
|
| 243 |
+
"page_idx": 2
|
| 244 |
+
},
|
| 245 |
+
{
|
| 246 |
+
"type": "text",
|
| 247 |
+
"text": "Shortest-pack-first histogram-packing (SPFHP) consists of three main components. First, the packing algorithm works on the bins in the sequence length histogram (with bin size 1) rather than the individual samples. Second, we operate on the sorted data from longest to shortest sequences. This comes basically for free due to the use of histograms. Third, we apply the worst-fit algorithm [11, 26] onto this histogram, where the currently observed sample goes to the pack1 that has the most space left to reach maximum packing depth (“shortest-pack-first”). If the sample does not fit, a new pack is created. A variant is to limit the packing depth, in other words the maximum number of sequences that are allowed in a pack. Therefore, we only extend an existing pack if it is not already at maximum packing depth. The detailed code for the algorithm is provided in [1] (listing $\\textcircled { 3 }$ . ",
|
| 248 |
+
"bbox": [
|
| 249 |
+
174,
|
| 250 |
+
856,
|
| 251 |
+
825,
|
| 252 |
+
911
|
| 253 |
+
],
|
| 254 |
+
"page_idx": 2
|
| 255 |
+
},
|
| 256 |
+
{
|
| 257 |
+
"type": "text",
|
| 258 |
+
"text": "",
|
| 259 |
+
"bbox": [
|
| 260 |
+
174,
|
| 261 |
+
88,
|
| 262 |
+
825,
|
| 263 |
+
161
|
| 264 |
+
],
|
| 265 |
+
"page_idx": 3
|
| 266 |
+
},
|
| 267 |
+
{
|
| 268 |
+
"type": "text",
|
| 269 |
+
"text": "3.1.2 Non-negative least squares histogram-packing (NNLSHP) ",
|
| 270 |
+
"text_level": 1,
|
| 271 |
+
"bbox": [
|
| 272 |
+
174,
|
| 273 |
+
174,
|
| 274 |
+
625,
|
| 275 |
+
189
|
| 276 |
+
],
|
| 277 |
+
"page_idx": 3
|
| 278 |
+
},
|
| 279 |
+
{
|
| 280 |
+
"type": "text",
|
| 281 |
+
"text": "The proposed NNLSHP algorithm is based on re-stating the packing problem as a (weighted) nonnegative least squares problem (NNLS) $\\pmb { \\mathbb { D } } \\|$ of the form $w A x = w b$ where $x \\geq 0$ . The vector $b$ is the histogram containing the counts of all the sequence lengths in the dataset. Next, we define the $A$ matrix (the “packing matrix“) by first generating a list of all possible sequence length combinations (“strategies”) that add up exactly to the maximum sequence length. We focus specifically on strategies that consist of at most 3 sequences per pack (independent of $b$ ) and encode each strategy as a column of the sparse matrix $A$ . For example, a strategy consisting of the sequence length 128, 128, and 256 in represented a column vector that has the value 2 at the 128th row, the value 1 at the 256th row, and zero at all other rows. The variable $x$ describes the non-negative repetition count for each strategy. So a 24 in the ith row of $x$ means that the strategy represented by the ith column of $A$ should repeat 24 times. Moreover, in the un-weighted setting, $A x = b$ states that we would like to “mix” the pre-defined strategies (columns of $A$ ) such that the number of samples matches the histogram $b$ , and where each strategy is used $x \\geq 0$ times. We use the residual weight $w$ to control the penalization of the $A x - b$ residual on different sequence lengths (different rows of $b$ ). Heuristically, we set the weight of 0.09 for all sequences of length 8 or smaller because they are considered acceptable padding sequences. All other sequence lengths get weight 1. After solving $w A x = w b$ for $x \\geq 0$ using an off-the-shelf solver we obtain a floating point solution, which means that the repetition counts are not necessarily integers. Since we cannot use a non-natural number of strategies, we round the solution $\\hat { x }$ to the nearest integer. The error introduced by this rounding is found to be negligible. Further details are provided in [1](Section D.4). ",
|
| 282 |
+
"bbox": [
|
| 283 |
+
173,
|
| 284 |
+
198,
|
| 285 |
+
825,
|
| 286 |
+
476
|
| 287 |
+
],
|
| 288 |
+
"page_idx": 3
|
| 289 |
+
},
|
| 290 |
+
{
|
| 291 |
+
"type": "text",
|
| 292 |
+
"text": "3.2 Attention masking for packed sequences ",
|
| 293 |
+
"text_level": 1,
|
| 294 |
+
"bbox": [
|
| 295 |
+
174,
|
| 296 |
+
491,
|
| 297 |
+
493,
|
| 298 |
+
505
|
| 299 |
+
],
|
| 300 |
+
"page_idx": 3
|
| 301 |
+
},
|
| 302 |
+
{
|
| 303 |
+
"type": "text",
|
| 304 |
+
"text": "To maintain an implementation that is consistent with the un-packed version, we need to be able to prevent attention between tokens in the pack which belong to different sequences. Other implementations use custom attention kernels which reconstruct padding. Instead, we propose directly masking the attention matrix with a block-diagonal mask to be applied before the attention. This is straightforward to implement in modern frameworks (see Figure 2). Naturally, there is a cost to both the mask construction and applying it to the attention matrix (see Table 1, Section $4 . 1 \\dot { } \\dot { } )$ ",
|
| 305 |
+
"bbox": [
|
| 306 |
+
174,
|
| 307 |
+
515,
|
| 308 |
+
826,
|
| 309 |
+
599
|
| 310 |
+
],
|
| 311 |
+
"page_idx": 3
|
| 312 |
+
},
|
| 313 |
+
{
|
| 314 |
+
"type": "image",
|
| 315 |
+
"img_path": "images/d70f7edb8fbf9c0d3993ae97fbc2e45dca018df57b8a19f0cb36de2895266827.jpg",
|
| 316 |
+
"image_caption": [
|
| 317 |
+
"Figure 2: Attention mask code sample [left] and example zero-one mask [right]. "
|
| 318 |
+
],
|
| 319 |
+
"image_footnote": [],
|
| 320 |
+
"bbox": [
|
| 321 |
+
191,
|
| 322 |
+
614,
|
| 323 |
+
746,
|
| 324 |
+
659
|
| 325 |
+
],
|
| 326 |
+
"page_idx": 3
|
| 327 |
+
},
|
| 328 |
+
{
|
| 329 |
+
"type": "text",
|
| 330 |
+
"text": "3.3 Calculating per-sequence loss and accuracy ",
|
| 331 |
+
"text_level": 1,
|
| 332 |
+
"bbox": [
|
| 333 |
+
173,
|
| 334 |
+
713,
|
| 335 |
+
514,
|
| 336 |
+
728
|
| 337 |
+
],
|
| 338 |
+
"page_idx": 3
|
| 339 |
+
},
|
| 340 |
+
{
|
| 341 |
+
"type": "text",
|
| 342 |
+
"text": "Canonical implementations of BERT compute the cross-entropy loss for the masked language model on a per-sequence basis. Simply feeding packs of sequences to the same implementation of crossentropy would consequently result in per-pack weighting of the loss. In other words, the overall loss on the micro-batch would sum-up the losses on the individual packs, rather than individual sequences. As a result the packed BERT model would converge to a different optimum. For instance, a pack of a single sequence would contribute to the loss to the same extent as a pack of three sequences. In other words, the long sequence (single per pack) is given the same weight as the three shorter sequences in the pack of three. Empirically, a degradation of masked-language modelling accuracy on shorter sequences is indeed observed when not modifying the loss to account for packing. ",
|
| 343 |
+
"bbox": [
|
| 344 |
+
173,
|
| 345 |
+
738,
|
| 346 |
+
825,
|
| 347 |
+
863
|
| 348 |
+
],
|
| 349 |
+
"page_idx": 3
|
| 350 |
+
},
|
| 351 |
+
{
|
| 352 |
+
"type": "text",
|
| 353 |
+
"text": "To recover the per-sequence averaging behavior of the canonical un-packed BERT implementation, it is not sufficient to simply weight the loss (accuracy) on each pack by the number of sequences it contains, because the sequences in the pack have different lengths and therefore should not use the same weight. ",
|
| 354 |
+
"bbox": [
|
| 355 |
+
174,
|
| 356 |
+
92,
|
| 357 |
+
825,
|
| 358 |
+
147
|
| 359 |
+
],
|
| 360 |
+
"page_idx": 4
|
| 361 |
+
},
|
| 362 |
+
{
|
| 363 |
+
"type": "text",
|
| 364 |
+
"text": "To implement per-sequence loss, we effectively “unpack” the incoming logits and labels by working with the per-token loss. We compute the loss on all tokens belonging to the first sequence, then all tokens belonging to the second sequence, and so on. However, rather than looping through the sequences index in this way, we compute on all indexes in parallel. This minimizes the latency overhead of un-packing the loss calculation. We use the “masked lm weight” $\\pmb { \\mathbb { H } }$ input tensor to represent which sequence a given masked token belongs to (0, 1, 2 and so on). This is consistent with the canonical BERT implementation where this input takes a value of either 1 (belonging to the sequence) or 0 (belonging to padding) as detailed in Listing 1. The same methodology can be applied to the next-sentence prediction loss and accuracy. ",
|
| 365 |
+
"bbox": [
|
| 366 |
+
173,
|
| 367 |
+
154,
|
| 368 |
+
825,
|
| 369 |
+
277
|
| 370 |
+
],
|
| 371 |
+
"page_idx": 4
|
| 372 |
+
},
|
| 373 |
+
{
|
| 374 |
+
"type": "text",
|
| 375 |
+
"text": "Listing 1: Loss calculation ",
|
| 376 |
+
"text_level": 1,
|
| 377 |
+
"bbox": [
|
| 378 |
+
410,
|
| 379 |
+
286,
|
| 380 |
+
586,
|
| 381 |
+
299
|
| 382 |
+
],
|
| 383 |
+
"page_idx": 4
|
| 384 |
+
},
|
| 385 |
+
{
|
| 386 |
+
"type": "text",
|
| 387 |
+
"text": "# The number of sequences in each batch may vary 2 sequences_in_batch $=$ tf. reduce_sum (tf. reduce_max ( masked_lm_weight , -1)) 3 sequences_in_batch $=$ tf. cast ( sequences_in_batch , tf. float32 ) 4 # Create the 0/1 mask that will be used to un - packed sequences 5 masked_lm_weight $=$ tf. reshape ( masked_lm_weight , [B, 1, -1]) 6 sequence_selection $=$ tf. reshape (tf. range (1, max_sequences_per_pack + 1) , [1, -1, 1]) sequence_selection $=$ tf. cast ( masked_lm_weight $= =$ sequence_selection , tf. float32 ) 8 # Apply the mask to un - pack the loss per sequence 9 nll_per_token $=$ tf. reshape ( nll_per_token , [B, 1, -1]) 10 nll_per_sequence $=$ sequence_selection $^ { * }$ nll_per_token 11 # Normalize the per - sequence loss by the number of mlm - tokens in the sequence (as is standard ) 12 attempted $=$ tf. reduce_sum ( sequence_selection , -1, keepdims $=$ True ) 13 attempted $=$ attempted $^ +$ tf. cast ( attempted $= =$ 0, tf. float32 ) # prevent NaNs when dividing by attempted 14 nll_per_sequence $= \\texttt { n l 1 }$ _per_sequence / attempted 15 # Average per - batch loss (so contributions from different batches are comparable ) 16 lm_loss $=$ tf. reduce_sum ( nll_per_sequence )/ sequences_in_batch ",
|
| 388 |
+
"bbox": [
|
| 389 |
+
156,
|
| 390 |
+
303,
|
| 391 |
+
820,
|
| 392 |
+
445
|
| 393 |
+
],
|
| 394 |
+
"page_idx": 4
|
| 395 |
+
},
|
| 396 |
+
{
|
| 397 |
+
"type": "text",
|
| 398 |
+
"text": "3.4 Hyperparameter adjustment ",
|
| 399 |
+
"text_level": 1,
|
| 400 |
+
"bbox": [
|
| 401 |
+
174,
|
| 402 |
+
467,
|
| 403 |
+
413,
|
| 404 |
+
481
|
| 405 |
+
],
|
| 406 |
+
"page_idx": 4
|
| 407 |
+
},
|
| 408 |
+
{
|
| 409 |
+
"type": "text",
|
| 410 |
+
"text": "In terms of convergence behavior, the primary consequence of packing is an increase in the effective batch size (with respect to sequences and tokens) with some variation over different iterations. For instance, if each pack on average contains two sequences, the batch size (per optimization step) is effectively doubled on average. While one could subsequently reduce the computational batch size by the packing factor (average number of sequences per pack) and keep using the same hyperparameters, this is typically not desirable as it might imply under-utilizing the memory/compute. ",
|
| 411 |
+
"bbox": [
|
| 412 |
+
173,
|
| 413 |
+
491,
|
| 414 |
+
825,
|
| 415 |
+
575
|
| 416 |
+
],
|
| 417 |
+
"page_idx": 4
|
| 418 |
+
},
|
| 419 |
+
{
|
| 420 |
+
"type": "text",
|
| 421 |
+
"text": "Instead, we propose an approximate heuristic for updating the decay parameters of the LAMB optimizer $[ [ 2 5 ] ]$ . For a packed dataset with a packing factor $p$ , we update the decay parameters as: $\\bar { \\beta _ { 1 } } : = \\beta _ { 1 } ^ { p }$ , $\\beta _ { 2 } : = \\beta _ { 2 } ^ { p }$ . For $p = 2$ , this corresponds to the exact parameters for calculating momentum and velocity, when updating with the same gradient twice $\\scriptstyle { \\mathbb { I I I } } ( { \\mathrm { S e c t i o n ~ E } } )$ . A common approach is to scale the learning rate with the batch size. Note however, that we take the mean gradient instead of an accumulated sum and have already a correction by the number of samples in that regard. ",
|
| 422 |
+
"bbox": [
|
| 423 |
+
174,
|
| 424 |
+
580,
|
| 425 |
+
825,
|
| 426 |
+
665
|
| 427 |
+
],
|
| 428 |
+
"page_idx": 4
|
| 429 |
+
},
|
| 430 |
+
{
|
| 431 |
+
"type": "text",
|
| 432 |
+
"text": "Since these adjustments are only heuristics the convergence of the model will be comparable but not identical. In particular, it is unlikely that simply adjusting the hyperparameters will fully undo the impact of the increased batch size. However, with these adjustments, researchers should be able to continue to use existing configurations. ",
|
| 433 |
+
"bbox": [
|
| 434 |
+
174,
|
| 435 |
+
670,
|
| 436 |
+
825,
|
| 437 |
+
727
|
| 438 |
+
],
|
| 439 |
+
"page_idx": 4
|
| 440 |
+
},
|
| 441 |
+
{
|
| 442 |
+
"type": "text",
|
| 443 |
+
"text": "4 Experiments ",
|
| 444 |
+
"text_level": 1,
|
| 445 |
+
"bbox": [
|
| 446 |
+
174,
|
| 447 |
+
744,
|
| 448 |
+
312,
|
| 449 |
+
762
|
| 450 |
+
],
|
| 451 |
+
"page_idx": 4
|
| 452 |
+
},
|
| 453 |
+
{
|
| 454 |
+
"type": "text",
|
| 455 |
+
"text": "4.1 Bin-packing algorithm comparison ",
|
| 456 |
+
"text_level": 1,
|
| 457 |
+
"bbox": [
|
| 458 |
+
176,
|
| 459 |
+
775,
|
| 460 |
+
455,
|
| 461 |
+
790
|
| 462 |
+
],
|
| 463 |
+
"page_idx": 4
|
| 464 |
+
},
|
| 465 |
+
{
|
| 466 |
+
"type": "text",
|
| 467 |
+
"text": "We evaluate our algorithms using the following metrics: number of packs, number of all tokens, number of padding tokens, solution time of the packing algorithm (after histogram and strategy creation), number of strategies used, packing efficiency (the fraction of non-padding tokens in the packed dataset), the speed-up achieved compared to not packing (depth 1), and the average number of sequences per sample (packing factor). For SPFHP, we analyse different (maximum) packing depth, since packing is less efficient with smaller depth and we want to get a general understanding on how the packing depth influences the processing time. For NNLSHP, we focus on packing depth 3 because it packs the data sufficiently well. ",
|
| 468 |
+
"bbox": [
|
| 469 |
+
173,
|
| 470 |
+
800,
|
| 471 |
+
825,
|
| 472 |
+
911
|
| 473 |
+
],
|
| 474 |
+
"page_idx": 4
|
| 475 |
+
},
|
| 476 |
+
{
|
| 477 |
+
"type": "text",
|
| 478 |
+
"text": "For the speed-up analysis, we focus on the intelligence processing unit (IPU) [10] (IPU-M2000, 16 accelerator chips). A GPU dynamically loads the code into the accelerator; in contrast, the IPU works with a static precompiled kernel that gets loaded onto the chip only at the beginning. While other approaches result in excessive padding or continuous changes of the code, our approach can work with the same code for the whole dataset. So in this setting the IPU architecture would especially benefit from our approach since it avoids code changes. Nevertheless, it can be applied to any implementation on GPU or TPU. For determining the speed-up, we take advantage of the precompiled kernel. Since time measurements are quite noisy, we can profile the kernel and how many cycles it takes for processing a batch. That way, we can determine the overhead (in cycles) from processing the additional attention masking and for unpacking the loss. Combining overhead and packing factor, we get the speed-up estimate. No experiment repetitions are required since the algorithms and measurements are deterministic. ",
|
| 479 |
+
"bbox": [
|
| 480 |
+
174,
|
| 481 |
+
90,
|
| 482 |
+
825,
|
| 483 |
+
257
|
| 484 |
+
],
|
| 485 |
+
"page_idx": 5
|
| 486 |
+
},
|
| 487 |
+
{
|
| 488 |
+
"type": "text",
|
| 489 |
+
"text": "The main results for the performance metric evaluation are displayed in Table 1. The processing time for SBFHP was around $0 . 0 3 s$ and independent from the packing depth. We see that the overhead slightly increases with packing depth but that the benefits of packing outweigh the cost. The best speed-up is obtained with NNLSHP at depth 3. With a value of 1.913, it is close to the theoretical upper bound of 2.001. The results show that efficiency, packing factor, and speed-up can be viewed inter-changeably. The amount of time needed to process a sample (a pack of sequences) is barely changed relative to the un-packed implementation. The packing factor or the improvement in efficiency effectively provide an accurate estimate of the speed-up. ",
|
| 490 |
+
"bbox": [
|
| 491 |
+
174,
|
| 492 |
+
263,
|
| 493 |
+
825,
|
| 494 |
+
375
|
| 495 |
+
],
|
| 496 |
+
"page_idx": 5
|
| 497 |
+
},
|
| 498 |
+
{
|
| 499 |
+
"type": "table",
|
| 500 |
+
"img_path": "images/b14ccfa9c3dd8b8ad6169a5d4c7fb446bed585d6963b476bba2b40f8a9919c80.jpg",
|
| 501 |
+
"table_caption": [],
|
| 502 |
+
"table_footnote": [],
|
| 503 |
+
"table_body": "<table><tr><td>pack. depth</td><td>pack. algo.</td><td># packs [M]</td><td>efficiency (%)</td><td>pack. factor</td><td>overhead (%)</td><td>realized speed-up</td></tr><tr><td>1</td><td>none</td><td>16.280</td><td>49.97</td><td>1.000</td><td>0.000</td><td>1.000</td></tr><tr><td>2</td><td>SPFHP</td><td>10.102</td><td>80.52</td><td>1.612</td><td>4.283</td><td>1.544</td></tr><tr><td>3</td><td>SPFHP</td><td>9.095</td><td>89.44</td><td>1.790</td><td>4.287</td><td>1.716</td></tr><tr><td>3</td><td>NNLSHP</td><td>8.155</td><td>99.75</td><td>1.996</td><td>4.287</td><td>1.913</td></tr><tr><td>4</td><td>SPFHP</td><td>8.659</td><td>93.94</td><td>1.880</td><td>4.294</td><td>1.803</td></tr><tr><td>8</td><td>SPFHP</td><td>8.225</td><td>98.90</td><td>1.979</td><td>4.481</td><td>1.895</td></tr><tr><td>16/max</td><td>SPFHP</td><td>8.168</td><td>99.60</td><td>1.993</td><td>4.477</td><td>1.905</td></tr></table>",
|
| 504 |
+
"bbox": [
|
| 505 |
+
236,
|
| 506 |
+
387,
|
| 507 |
+
761,
|
| 508 |
+
513
|
| 509 |
+
],
|
| 510 |
+
"page_idx": 5
|
| 511 |
+
},
|
| 512 |
+
{
|
| 513 |
+
"type": "text",
|
| 514 |
+
"text": "Table 1: Key performance results of proposed packing algorithms (SPFHP and NNLSHP). Packing depth describes the maximum number of packed sequences. Packing depth 1 is the baseline BERT implementation. Setting no limit resulted in a maximum packing depth of 16. The number of packs describes the length of the new packed dataset. Efficiency is the percentage of real tokens in the packed dataset. The packing factor describes the resulting potential speed-up compared to packing depth 1. With overhead, we denote the percentage decrease in throughput due to changes to the model to enable packing (such as the masking scheme introduced in Section $3 . 2 )$ . The realized speed-up is the combination of the speed-up due to packing (the packing factor) and the decrease in throughput due to the overhead. It is used to measure the relative speed-up in throughput and the overhead from masking and loss adjustment. ",
|
| 515 |
+
"bbox": [
|
| 516 |
+
173,
|
| 517 |
+
513,
|
| 518 |
+
825,
|
| 519 |
+
651
|
| 520 |
+
],
|
| 521 |
+
"page_idx": 5
|
| 522 |
+
},
|
| 523 |
+
{
|
| 524 |
+
"type": "text",
|
| 525 |
+
"text": "4.2 Learning Curves and Hyperparameter Adjustment ",
|
| 526 |
+
"text_level": 1,
|
| 527 |
+
"bbox": [
|
| 528 |
+
174,
|
| 529 |
+
685,
|
| 530 |
+
568,
|
| 531 |
+
700
|
| 532 |
+
],
|
| 533 |
+
"page_idx": 5
|
| 534 |
+
},
|
| 535 |
+
{
|
| 536 |
+
"type": "text",
|
| 537 |
+
"text": "For depth 1 (classic BERT) and NNLSHP with depth 3, we additionally evaluate on the MLPerf 0.7 BERT pre-training benchmark $\\mathbb { \\left. \\overline { { 1 5 } } \\right. }$ . Briefly, this involves training from a standard checkpoint to a masked-language model accuracy of $7 1 . 2 \\%$ using 3 million sequences with a maximum length of 512 tokens (refer to $\\boxed { 1 6 }$ for details). Following this standardized benchmark supports reproduction of results even on other systems and makes sure that the reproduction effort is moderate and setup rules are clearly documented. We compare the resulting speed-up as well as the respective learning curves by evaluating the data on a held-out validation dataset. The objective of this additional evaluation is to analyse if convergence behavior is changed by the packing strategy and if the theoretical speed-up can be achieved in practice. ",
|
| 538 |
+
"bbox": [
|
| 539 |
+
174,
|
| 540 |
+
710,
|
| 541 |
+
825,
|
| 542 |
+
835
|
| 543 |
+
],
|
| 544 |
+
"page_idx": 5
|
| 545 |
+
},
|
| 546 |
+
{
|
| 547 |
+
"type": "text",
|
| 548 |
+
"text": "With packing, we effectively increase the average batch size by the packing factor $( \\approx 2 )$ . However, with a different batch size, different hyperparameters are required (see Section $3 . 4 )$ and there is no mapping that will generate exact matching of results but only heuristics. In a first comparison, we use the same hyperparameters when comparing packed and unpacked training except for cutting the accumulation count by half. This way, we make sure that the batch size is constant on average. ",
|
| 549 |
+
"bbox": [
|
| 550 |
+
174,
|
| 551 |
+
842,
|
| 552 |
+
825,
|
| 553 |
+
911
|
| 554 |
+
],
|
| 555 |
+
"page_idx": 5
|
| 556 |
+
},
|
| 557 |
+
{
|
| 558 |
+
"type": "text",
|
| 559 |
+
"text": "In the second comparison, we evaluate our heuristics and how they compensate the difference in batch size. This setup is more desirable because it is beneficial to use the hardware to its full potential and cutting the batch size by half usually reduces throughput. In the third comparison, we compare two optimized setups. ",
|
| 560 |
+
"bbox": [
|
| 561 |
+
174,
|
| 562 |
+
92,
|
| 563 |
+
825,
|
| 564 |
+
147
|
| 565 |
+
],
|
| 566 |
+
"page_idx": 6
|
| 567 |
+
},
|
| 568 |
+
{
|
| 569 |
+
"type": "text",
|
| 570 |
+
"text": "The learning curves are displayed in Figure $3 .$ In the first setup, we see the curves almost matching perfectly when normalizing by the numbers of samples processed. Differences can be explained by the variation of the number of sequences in the packing batch, and general noise in the training process. Especially after the initial phase, the curves show a near-identical match. ",
|
| 571 |
+
"bbox": [
|
| 572 |
+
174,
|
| 573 |
+
152,
|
| 574 |
+
825,
|
| 575 |
+
208
|
| 576 |
+
],
|
| 577 |
+
"page_idx": 6
|
| 578 |
+
},
|
| 579 |
+
{
|
| 580 |
+
"type": "text",
|
| 581 |
+
"text": "The second setup shows bigger differences since changing the batch size and hyperparameters changes the training dynamics. We observe slower convergence early on in training due to the increased batch size. This is expected. The adjustment of the learning rate actually decreases performance probably because we correct for the increased number of sequences already in the modified loss. With the adjustment of the decay parameter of LAMB, we see matching performance at the later training stages. However, it is not feasible to completely recover the early convergence behavior of the smaller batch size by adjusting the hyperparameters. For instance doubling the batch size of unpacked BERT to 3000 and adjusting the LAMB decay parameters leads to more of a slow down in convergence than when running packed BERT with a batch size of 1500 and a packing factor of 2. Overall, in practice we observe a higher acceleration than the estimated 1.913 that goes beyond $2 \\mathbf { x }$ . We explain this with slightly better fitting hyperparameters and improved data transfer. ",
|
| 582 |
+
"bbox": [
|
| 583 |
+
174,
|
| 584 |
+
215,
|
| 585 |
+
825,
|
| 586 |
+
367
|
| 587 |
+
],
|
| 588 |
+
"page_idx": 6
|
| 589 |
+
},
|
| 590 |
+
{
|
| 591 |
+
"type": "image",
|
| 592 |
+
"img_path": "images/1310340ba5622cee12332ef85d9bbe09a37d18da600b8ab61d8349e186a72c53.jpg",
|
| 593 |
+
"image_caption": [
|
| 594 |
+
"Figure 3: Comparison of learning curves for packed and unpacked processing with [left] same effective batch size (ebs is batch size times packing factor), [middle] different heuristic adjustments of the hyperparameters (batch size 1500 for all runs, such that ebs for packed runs is $1 5 0 0 * 2$ ), and [right] realized time-to-convergence speed-up from packing. "
|
| 595 |
+
],
|
| 596 |
+
"image_footnote": [],
|
| 597 |
+
"bbox": [
|
| 598 |
+
184,
|
| 599 |
+
381,
|
| 600 |
+
812,
|
| 601 |
+
492
|
| 602 |
+
],
|
| 603 |
+
"page_idx": 6
|
| 604 |
+
},
|
| 605 |
+
{
|
| 606 |
+
"type": "text",
|
| 607 |
+
"text": "4.3 Scaling Analysis: Impact of the number of accelerators ",
|
| 608 |
+
"text_level": 1,
|
| 609 |
+
"bbox": [
|
| 610 |
+
174,
|
| 611 |
+
582,
|
| 612 |
+
596,
|
| 613 |
+
597
|
| 614 |
+
],
|
| 615 |
+
"page_idx": 6
|
| 616 |
+
},
|
| 617 |
+
{
|
| 618 |
+
"type": "text",
|
| 619 |
+
"text": "A further advantage of packing over competing un-padding approaches is the inherent load balancing provided by packing. So called un-padding approaches rely on dynamically launching custom kernels that ignore padding. A stated advantage of such implementations is the ability to avoid computing the complete $( 5 1 2 \\mathrm { ~ x ~ } 5 1 2 )$ ) attention matrix. This provides additional computational savings compared to packing, where the attention matrix is computed in its entirety and then masked. Because of these additional savings, un-padding can exceed the theoretical upper bound for speed-up from packing (2.013 on Wikipedia). As a result of the dynamic nature of the approach, the processing time with un-padding is different for each sequence in the batch, and the amount of time required to process a batch of sequences will be determined by the processing time of the longest sequence in the batch (with the sequences being processed in parallel). Furthermore, in the multiple accelerator setting the processing time on each device will vary depending on the sequences in the batch that it receives. Devices which finish early have to wait for the slowest device to finish before exchanging gradients. This load-imbalance between the devices (and inside the batch) leads to a considerable decrease in the speed-up from un-padding as the number of accelerators is increased (see Figure 4). ",
|
| 620 |
+
"bbox": [
|
| 621 |
+
173,
|
| 622 |
+
608,
|
| 623 |
+
825,
|
| 624 |
+
801
|
| 625 |
+
],
|
| 626 |
+
"page_idx": 6
|
| 627 |
+
},
|
| 628 |
+
{
|
| 629 |
+
"type": "text",
|
| 630 |
+
"text": "In contrast, packing (our approach) is inherently load-balanced. The processing time on each accelerator is independent of the content inside the batch received by the device. Any number of accelerators can therefore operate in unison without having to wait for the slowest batch to process (all per-device batches are equally fast). ",
|
| 631 |
+
"bbox": [
|
| 632 |
+
174,
|
| 633 |
+
808,
|
| 634 |
+
825,
|
| 635 |
+
863
|
| 636 |
+
],
|
| 637 |
+
"page_idx": 6
|
| 638 |
+
},
|
| 639 |
+
{
|
| 640 |
+
"type": "text",
|
| 641 |
+
"text": "To demonstrate the severity of the load-imbalance issue, we consider the scaling of an un-padding approach with a per-device batch size of 32 running on eight devices $\\mathbb { \\ m }$ . From there, we readily extrapolate the performance to both larger and smaller cluster sizes by fitting a Gumbel distribution to the observed processing times [1] (Section F). On a single device with batch size $3 2 { \\mathrm { u n } }$ -padding outperforms packing and exceeds the theoretical upper-bound for packing. ",
|
| 642 |
+
"bbox": [
|
| 643 |
+
176,
|
| 644 |
+
869,
|
| 645 |
+
825,
|
| 646 |
+
911
|
| 647 |
+
],
|
| 648 |
+
"page_idx": 6
|
| 649 |
+
},
|
| 650 |
+
{
|
| 651 |
+
"type": "text",
|
| 652 |
+
"text": "",
|
| 653 |
+
"bbox": [
|
| 654 |
+
171,
|
| 655 |
+
90,
|
| 656 |
+
823,
|
| 657 |
+
119
|
| 658 |
+
],
|
| 659 |
+
"page_idx": 7
|
| 660 |
+
},
|
| 661 |
+
{
|
| 662 |
+
"type": "text",
|
| 663 |
+
"text": "As the number of devices increases to two or more, the proposed packing approach outperforms the dynamic un-padding approach. On a cluster with 32 accelerators the speed-up from un-padding drops to $5 0 \\%$ and with 2048 devices the speed-up is only $3 0 \\%$ . In contrast, the speed-up due to packing is independent of the number of accelerators and stays at 1.913. Switching to a smaller batch size would reduce the load-imbalance issue to some extent, but would also result in under-utilization of the available memory and compute. ",
|
| 664 |
+
"bbox": [
|
| 665 |
+
173,
|
| 666 |
+
126,
|
| 667 |
+
825,
|
| 668 |
+
209
|
| 669 |
+
],
|
| 670 |
+
"page_idx": 7
|
| 671 |
+
},
|
| 672 |
+
{
|
| 673 |
+
"type": "image",
|
| 674 |
+
"img_path": "images/11b338cc632273252008ed0bdbeebdc1e9f7ba133a94d971a396c7f4b2bf86d7.jpg",
|
| 675 |
+
"image_caption": [
|
| 676 |
+
"Figure 4: Comparison of the theoretical speed-up achievable as the number of accelerators is increased. "
|
| 677 |
+
],
|
| 678 |
+
"image_footnote": [],
|
| 679 |
+
"bbox": [
|
| 680 |
+
305,
|
| 681 |
+
229,
|
| 682 |
+
691,
|
| 683 |
+
359
|
| 684 |
+
],
|
| 685 |
+
"page_idx": 7
|
| 686 |
+
},
|
| 687 |
+
{
|
| 688 |
+
"type": "text",
|
| 689 |
+
"text": "5 Conclusion ",
|
| 690 |
+
"text_level": 1,
|
| 691 |
+
"bbox": [
|
| 692 |
+
174,
|
| 693 |
+
435,
|
| 694 |
+
299,
|
| 695 |
+
452
|
| 696 |
+
],
|
| 697 |
+
"page_idx": 7
|
| 698 |
+
},
|
| 699 |
+
{
|
| 700 |
+
"type": "text",
|
| 701 |
+
"text": "We showed that packing can be easily implemented without the need for any custom kernels while still providing a 2x speed-up. Additionally, we showed that any additional speed-ups resulting from dynamic un-padding approaches diminish for even moderate batch sizes or when additional accelerators are added. In contrast, packing is load-balanced and maintains the $2 \\mathbf { x }$ throughput when scaling to large numbers of accelerators. ",
|
| 702 |
+
"bbox": [
|
| 703 |
+
174,
|
| 704 |
+
469,
|
| 705 |
+
825,
|
| 706 |
+
539
|
| 707 |
+
],
|
| 708 |
+
"page_idx": 7
|
| 709 |
+
},
|
| 710 |
+
{
|
| 711 |
+
"type": "text",
|
| 712 |
+
"text": "Furthermore, the computational overhead introduced by the attention mask and the packed persequence loss are small compared to the achieved acceleration. This overhead remains below $5 \\%$ for all tested packing depths. The efficient packing algorithms presented in this paper enable us to pack millions of sequences in a matter of seconds. Compared to both the pre-processing time for the Wikipedia dataset and the training runtime, this overhead is negligible. Furthermore, we showed that performing packing as a pre-processing step does not significantly impact the training convergence. Our proposed hyperparameter adjustment scheme additionally helps practitioners easily modify existing validated optimizer settings for use with packed BERT. Further exploration of hyperparameter selection is left to future work. ",
|
| 713 |
+
"bbox": [
|
| 714 |
+
174,
|
| 715 |
+
545,
|
| 716 |
+
825,
|
| 717 |
+
670
|
| 718 |
+
],
|
| 719 |
+
"page_idx": 7
|
| 720 |
+
},
|
| 721 |
+
{
|
| 722 |
+
"type": "text",
|
| 723 |
+
"text": "When performing packing as a pre-processing step, the proposed NNLSHP and SPFHP methods achieve near optimal compression efficiency. In this offline setting, we are able to build a histogram of the dataset, and thus achieve linear time complexity with respect to the number of samples. This makes packing modern datasets with millions of sequences possible. In the future, it would be interesting to extend SPFHP to the online setting where a histogram of the entire dataset cannot be built. Another interesting direction is the packing of images of different sizes to help accelerate computer-vision applications. This is especially relevant given the recent advances in the use of transformer-based approaches in the computer vision domain, for example the visual transformer [24]. Masking out the self-attention within transformers is easier to implement than avoiding cross-contamination of convolutions applied to packed images. Finally, packing could potentially eliminate the need for two phase pre-training of BERT. Using short sequences in the first phase to reduce the waste from padding is no longer attractive for packed sequence BERT where the padding is essentially a negligible proportion of the tokens. Furthermore, the argument that the model should first learn short-term dependencies by training on short sequences neglects the fact that these same short-term patterns can be learned from longer sequences. In fact, longer-sequences may contain multiple short patterns, while also maintaining long-range consistency. Future work should explore training packed BERT from scratch and the impact of packing on fine-tuned performance. ",
|
| 724 |
+
"bbox": [
|
| 725 |
+
174,
|
| 726 |
+
676,
|
| 727 |
+
825,
|
| 728 |
+
911
|
| 729 |
+
],
|
| 730 |
+
"page_idx": 7
|
| 731 |
+
},
|
| 732 |
+
{
|
| 733 |
+
"type": "text",
|
| 734 |
+
"text": "Broader Impact ",
|
| 735 |
+
"text_level": 1,
|
| 736 |
+
"bbox": [
|
| 737 |
+
174,
|
| 738 |
+
89,
|
| 739 |
+
310,
|
| 740 |
+
107
|
| 741 |
+
],
|
| 742 |
+
"page_idx": 8
|
| 743 |
+
},
|
| 744 |
+
{
|
| 745 |
+
"type": "text",
|
| 746 |
+
"text": "We showed that when pre-training BERT on Wikipedia, the computational overhead taken to process padding tokens is roughly $5 0 \\%$ . By eliminating this wasted computational time, the approach presented in this paper paves a way to halving the carbon footprint of training BERT-based models. ",
|
| 747 |
+
"bbox": [
|
| 748 |
+
174,
|
| 749 |
+
121,
|
| 750 |
+
823,
|
| 751 |
+
162
|
| 752 |
+
],
|
| 753 |
+
"page_idx": 8
|
| 754 |
+
},
|
| 755 |
+
{
|
| 756 |
+
"type": "text",
|
| 757 |
+
"text": "Furthermore, our approach circumvents the need for custom kernels, making the benefits of packing readily accessible to a broader audience of NLP practitioners. As such we are hopeful the research will have a positive impact on the NLP community, and do not see any disadvantage of using this approach. ",
|
| 758 |
+
"bbox": [
|
| 759 |
+
174,
|
| 760 |
+
169,
|
| 761 |
+
825,
|
| 762 |
+
224
|
| 763 |
+
],
|
| 764 |
+
"page_idx": 8
|
| 765 |
+
},
|
| 766 |
+
{
|
| 767 |
+
"type": "text",
|
| 768 |
+
"text": "Future work would need to investigate the applicability of packing on text produced by different cultures and in different languages. We have already shown that the speed-up resulting from using our methods does not only occur when pre-training BERT on Wikipedia but also on other datasets such as SQUaD and GLUE. Furthermore, the sentence length distribution of the original English language text shows similar characteristics. Our research leads us to believe that compressible distributions arise naturally in language tasks and beyond, for instance in DNA sequence lengths [9] and protein lengths $\\textcircled { 8 }$ . Many such sequence modelling workloads are based on variations of the BERT/transformer architecture and would therefore easily benefit from our acceleration. ",
|
| 769 |
+
"bbox": [
|
| 770 |
+
174,
|
| 771 |
+
231,
|
| 772 |
+
825,
|
| 773 |
+
342
|
| 774 |
+
],
|
| 775 |
+
"page_idx": 8
|
| 776 |
+
},
|
| 777 |
+
{
|
| 778 |
+
"type": "text",
|
| 779 |
+
"text": "Failures in NLP can have a big impact on society; many technologies, such as Alexa, Siri, and Google Home, rely on them. Whilst any errors arising from our approach can be avoided, one potential source of error comes from the implementation. Both the attention mask and the per-sequence loss need to be modified to support packing. These changes are significantly smaller than those required by custom kernels, however they may still be time consuming to implement and debug. To help mitigate the risk of any implementation errors, we share our reference implementations of the required changes in the supplemental material [1]. ",
|
| 780 |
+
"bbox": [
|
| 781 |
+
174,
|
| 782 |
+
348,
|
| 783 |
+
825,
|
| 784 |
+
445
|
| 785 |
+
],
|
| 786 |
+
"page_idx": 8
|
| 787 |
+
},
|
| 788 |
+
{
|
| 789 |
+
"type": "text",
|
| 790 |
+
"text": "References ",
|
| 791 |
+
"text_level": 1,
|
| 792 |
+
"bbox": [
|
| 793 |
+
174,
|
| 794 |
+
472,
|
| 795 |
+
266,
|
| 796 |
+
488
|
| 797 |
+
],
|
| 798 |
+
"page_idx": 8
|
| 799 |
+
},
|
| 800 |
+
{
|
| 801 |
+
"type": "text",
|
| 802 |
+
"text": "[1] ANONYMOUS. Supplemental Material for “Packing: Towards 2x NLP BERT Acceleration”, 2021. \n[2] BRO, R., AND DE JONG, S. A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics 11, 5 (sep 1997), 393–401. \n[3] BROWN, T. B., MANN, B., RYDER, N., SUBBIAH, M., KAPLAN, J., DHARIWAL, P., NEELAKANTAN, A., SHYAM, P., SASTRY, G., ASKELL, A., AGARWAL, S., HERBERT-VOSS, A., KRUEGER, G., HENIGHAN, T., CHILD, R., RAMESH, A., ZIEGLER, D. M., WU, J., WINTER, C., HESSE, C., CHEN, M., SIGLER, E., LITWIN, M., GRAY, S., CHESS, B., CLARK, J., BERNER, C., MCCANDLISH, S., RADFORD, A., SUTSKEVER, I., AND AMODEI, D. Language Models are Few-Shot Learners. In Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020) (may 2020). \n[4] BYTEDANCE INC. Effective Transformer. https://github.com/bytedance/effective_ transformer, 2021. \n[5] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. BERT: Pre-training of deep bidirectional transformers for language understanding. NAACL HLT 2019 - 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies - Proceedings of the Conference 1 (oct 2019), 4171–4186. \n[6] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. https://github.com/ google-research/bert, 2019. \n[7] DEVLIN, J., CHANG, M. W., LEE, K., AND TOUTANOVA, K. Pre-training data creation script for BERT. https://github.com/google-research/bert/blob/master/ create_pretraining_data.py#L243, 2019. \n[8] GUILLÉN, G., DIAZ-CAMINO, C., LOYOLA-TORRES, C., APARICIO-FABRE, R., HERNÁNDEZ-LÓPEZ, A., DÍAZ-SÁNCHEZ, M., AND SANCHEZ, F. Detailed analysis of ",
|
| 803 |
+
"bbox": [
|
| 804 |
+
176,
|
| 805 |
+
496,
|
| 806 |
+
826,
|
| 807 |
+
912
|
| 808 |
+
],
|
| 809 |
+
"page_idx": 8
|
| 810 |
+
},
|
| 811 |
+
{
|
| 812 |
+
"type": "text",
|
| 813 |
+
"text": "putative genes encoding small proteins in legume genomes. Frontiers in Plant Science 4 (2013), 208. ",
|
| 814 |
+
"bbox": [
|
| 815 |
+
202,
|
| 816 |
+
90,
|
| 817 |
+
825,
|
| 818 |
+
121
|
| 819 |
+
],
|
| 820 |
+
"page_idx": 9
|
| 821 |
+
},
|
| 822 |
+
{
|
| 823 |
+
"type": "text",
|
| 824 |
+
"text": "[9] HANSEN, H. B., DAMGAARD, P. B., MARGARYAN, A., STENDERUP, J., LYNNERUP, N., WILLERSLEV, E., AND ALLENTOFT, M. E. Comparing ancient dna preservation in petrous bone and tooth cementum. PLOS ONE 12, 1 (01 2017), 1–18. ",
|
| 825 |
+
"bbox": [
|
| 826 |
+
178,
|
| 827 |
+
127,
|
| 828 |
+
823,
|
| 829 |
+
171
|
| 830 |
+
],
|
| 831 |
+
"page_idx": 9
|
| 832 |
+
},
|
| 833 |
+
{
|
| 834 |
+
"type": "text",
|
| 835 |
+
"text": "[10] JIA, Z., TILLMAN, B., MAGGIONI, M., AND SCARPAZZA, D. P. Dissecting the Graphcore IPU architecture via microbenchmarking. ArXiv abs/1912.03413 (2019). ",
|
| 836 |
+
"bbox": [
|
| 837 |
+
173,
|
| 838 |
+
178,
|
| 839 |
+
823,
|
| 840 |
+
207
|
| 841 |
+
],
|
| 842 |
+
"page_idx": 9
|
| 843 |
+
},
|
| 844 |
+
{
|
| 845 |
+
"type": "text",
|
| 846 |
+
"text": "[11] JOHNSON, D. S. Near-optimal bin packing algorithms. PhD thesis, Massachusetts Institute of Technology, 1973. ",
|
| 847 |
+
"bbox": [
|
| 848 |
+
174,
|
| 849 |
+
214,
|
| 850 |
+
823,
|
| 851 |
+
243
|
| 852 |
+
],
|
| 853 |
+
"page_idx": 9
|
| 854 |
+
},
|
| 855 |
+
{
|
| 856 |
+
"type": "text",
|
| 857 |
+
"text": "[12] JOHNSON, D. S., AND GAREY, M. R. A 7160 theorem for bin packing. Journal of Complexity 1, 1 (oct 1985), 65–106. ",
|
| 858 |
+
"bbox": [
|
| 859 |
+
176,
|
| 860 |
+
251,
|
| 861 |
+
826,
|
| 862 |
+
280
|
| 863 |
+
],
|
| 864 |
+
"page_idx": 9
|
| 865 |
+
},
|
| 866 |
+
{
|
| 867 |
+
"type": "text",
|
| 868 |
+
"text": "[13] KORTE, B., AND VYGEN, J. Combinatorial Optimization, vol. 21 of Algorithms and Combinatorics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. ",
|
| 869 |
+
"bbox": [
|
| 870 |
+
173,
|
| 871 |
+
286,
|
| 872 |
+
825,
|
| 873 |
+
318
|
| 874 |
+
],
|
| 875 |
+
"page_idx": 9
|
| 876 |
+
},
|
| 877 |
+
{
|
| 878 |
+
"type": "text",
|
| 879 |
+
"text": "[14] LAN, Z., CHEN, M., GOODMAN, S., GIMPEL, K., SHARMA, P., AND SORICUT, R. ALBERT: A lite BERT for self-supervised learning of language representations. CoRR abs/1909.11942 (2019). ",
|
| 880 |
+
"bbox": [
|
| 881 |
+
171,
|
| 882 |
+
324,
|
| 883 |
+
825,
|
| 884 |
+
367
|
| 885 |
+
],
|
| 886 |
+
"page_idx": 9
|
| 887 |
+
},
|
| 888 |
+
{
|
| 889 |
+
"type": "text",
|
| 890 |
+
"text": "[15] MATTSON, P., REDDI, V. J., CHENG, C., COLEMAN, C., DIAMOS, G., KANTER, D., MICIKEVICIUS, P., PATTERSON, D., SCHMUELLING, G., TANG, H., WEI, G., AND WU, C. MLPerf: An Industry Standard Benchmark Suite for Machine Learning Performance. IEEE Micro 40, 2 (2020), 8–16. ",
|
| 891 |
+
"bbox": [
|
| 892 |
+
173,
|
| 893 |
+
375,
|
| 894 |
+
826,
|
| 895 |
+
431
|
| 896 |
+
],
|
| 897 |
+
"page_idx": 9
|
| 898 |
+
},
|
| 899 |
+
{
|
| 900 |
+
"type": "text",
|
| 901 |
+
"text": "[16] MLCOMMONS. v0.7 Results. https://mlcommons.org/en/training-normal-07/, 2020. Result not verified by MLPerf. Throughput/speedup is not the primary metric of MLPerf. MLPerf name and logo are trademarks. See www.mlperf.org for more information. ",
|
| 902 |
+
"bbox": [
|
| 903 |
+
171,
|
| 904 |
+
439,
|
| 905 |
+
826,
|
| 906 |
+
482
|
| 907 |
+
],
|
| 908 |
+
"page_idx": 9
|
| 909 |
+
},
|
| 910 |
+
{
|
| 911 |
+
"type": "text",
|
| 912 |
+
"text": "[17] NVIDIA. Reference numbers for BERT un-padding results. https://github.com/ mlcommons/training_results_v0.7/blob/master/NVIDIA/results/dgxa100 ngc20.06_pytorch/bert/result_0.txt, 2020. Throughput/speedup is not the primary metric of MLPerf. MLPerf name and logo are trademarks. See www.mlperf.org for more information. ",
|
| 913 |
+
"bbox": [
|
| 914 |
+
173,
|
| 915 |
+
488,
|
| 916 |
+
825,
|
| 917 |
+
559
|
| 918 |
+
],
|
| 919 |
+
"page_idx": 9
|
| 920 |
+
},
|
| 921 |
+
{
|
| 922 |
+
"type": "text",
|
| 923 |
+
"text": "[18] NVIDIA. Faster Transformer. https://github.com/NVIDIA/DeepLearningExamples/ tree/master/FasterTransformer/v1, 2021. ",
|
| 924 |
+
"bbox": [
|
| 925 |
+
174,
|
| 926 |
+
566,
|
| 927 |
+
825,
|
| 928 |
+
597
|
| 929 |
+
],
|
| 930 |
+
"page_idx": 9
|
| 931 |
+
},
|
| 932 |
+
{
|
| 933 |
+
"type": "text",
|
| 934 |
+
"text": "[19] RAJPURKAR, P., ZHANG, J., LOPYREV, K., AND LIANG, P. SQuAD: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing (Austin, Texas, Nov. 2016), Association for Computational Linguistics, pp. 2383–2392. ",
|
| 935 |
+
"bbox": [
|
| 936 |
+
174,
|
| 937 |
+
603,
|
| 938 |
+
826,
|
| 939 |
+
660
|
| 940 |
+
],
|
| 941 |
+
"page_idx": 9
|
| 942 |
+
},
|
| 943 |
+
{
|
| 944 |
+
"type": "text",
|
| 945 |
+
"text": "[20] TURC, I., CHANG, M.-W., LEE, K., AND TOUTANOVA, K. Well-read students learn better: On the importance of pre-training compact models. arXiv preprint arXiv:1908.08962v2 (2019). ",
|
| 946 |
+
"bbox": [
|
| 947 |
+
169,
|
| 948 |
+
666,
|
| 949 |
+
826,
|
| 950 |
+
696
|
| 951 |
+
],
|
| 952 |
+
"page_idx": 9
|
| 953 |
+
},
|
| 954 |
+
{
|
| 955 |
+
"type": "text",
|
| 956 |
+
"text": "[21] VASWANI, A., SHAZEER, N., PARMAR, N., USZKOREIT, J., JONES, L., GOMEZ, A. N., KAISER, U., AND POLOSUKHIN, I. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems (Red Hook, NY, USA, 2017), NIPS’17, Curran Associates Inc., p. 6000–6010. ",
|
| 957 |
+
"bbox": [
|
| 958 |
+
173,
|
| 959 |
+
704,
|
| 960 |
+
826,
|
| 961 |
+
761
|
| 962 |
+
],
|
| 963 |
+
"page_idx": 9
|
| 964 |
+
},
|
| 965 |
+
{
|
| 966 |
+
"type": "text",
|
| 967 |
+
"text": "[22] WANG, A., SINGH, A., MICHAEL, J., HILL, F., LEVY, O., AND BOWMAN, S. GLUE: A multi-task benchmark and analysis platform for natural language understanding. In Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP (Brussels, Belgium, Nov. 2018), Association for Computational Linguistics, pp. 353–355. ",
|
| 968 |
+
"bbox": [
|
| 969 |
+
173,
|
| 970 |
+
768,
|
| 971 |
+
826,
|
| 972 |
+
825
|
| 973 |
+
],
|
| 974 |
+
"page_idx": 9
|
| 975 |
+
},
|
| 976 |
+
{
|
| 977 |
+
"type": "text",
|
| 978 |
+
"text": "[23] WARSTADT, A., SINGH, A., AND BOWMAN, S. R. Neural network acceptability judgments. arXiv preprint arXiv:1805.12471 (2018). ",
|
| 979 |
+
"bbox": [
|
| 980 |
+
171,
|
| 981 |
+
832,
|
| 982 |
+
823,
|
| 983 |
+
862
|
| 984 |
+
],
|
| 985 |
+
"page_idx": 9
|
| 986 |
+
},
|
| 987 |
+
{
|
| 988 |
+
"type": "text",
|
| 989 |
+
"text": "[24] WU, B., XU, C., DAI, X., WAN, A., ZHANG, P., YAN, Z., TOMIZUKA, M., GONZALEZ, J., KEUTZER, K., AND VAJDA, P. Visual transformers: Token-based image representation and processing for computer vision, 2020. ",
|
| 990 |
+
"bbox": [
|
| 991 |
+
174,
|
| 992 |
+
869,
|
| 993 |
+
826,
|
| 994 |
+
911
|
| 995 |
+
],
|
| 996 |
+
"page_idx": 9
|
| 997 |
+
},
|
| 998 |
+
{
|
| 999 |
+
"type": "text",
|
| 1000 |
+
"text": "[25] YOU, Y., LI, J., REDDI, S., HSEU, J., KUMAR, S., BHOJANAPALLI, S., SONG, X., DEMMEL, J., KEUTZER, K., AND HSIEH, C.-J. Large Batch Optimization for Deep Learning: Training BERT in 76 minutes. arXiv (apr 2019). ",
|
| 1001 |
+
"bbox": [
|
| 1002 |
+
173,
|
| 1003 |
+
92,
|
| 1004 |
+
823,
|
| 1005 |
+
133
|
| 1006 |
+
],
|
| 1007 |
+
"page_idx": 10
|
| 1008 |
+
},
|
| 1009 |
+
{
|
| 1010 |
+
"type": "text",
|
| 1011 |
+
"text": "[26] YUE, M., AND ZHANG, L. A simple proof of the inequality $M F F D ( L ) \\leq 7 1 / 6 0 O P T ( L ) +$ $1 , L$ for the MFFD bin-packing algorithm. Acta Mathematicae Applicatae Sinica $^ Ḋ I I Ḍ$ , 3 (jul 1995), 318–330. ",
|
| 1012 |
+
"bbox": [
|
| 1013 |
+
173,
|
| 1014 |
+
142,
|
| 1015 |
+
825,
|
| 1016 |
+
184
|
| 1017 |
+
],
|
| 1018 |
+
"page_idx": 10
|
| 1019 |
+
},
|
| 1020 |
+
{
|
| 1021 |
+
"type": "text",
|
| 1022 |
+
"text": "Checklist ",
|
| 1023 |
+
"text_level": 1,
|
| 1024 |
+
"bbox": [
|
| 1025 |
+
174,
|
| 1026 |
+
209,
|
| 1027 |
+
254,
|
| 1028 |
+
226
|
| 1029 |
+
],
|
| 1030 |
+
"page_idx": 10
|
| 1031 |
+
},
|
| 1032 |
+
{
|
| 1033 |
+
"type": "text",
|
| 1034 |
+
"text": "1. For all authors... ",
|
| 1035 |
+
"bbox": [
|
| 1036 |
+
214,
|
| 1037 |
+
236,
|
| 1038 |
+
339,
|
| 1039 |
+
251
|
| 1040 |
+
],
|
| 1041 |
+
"page_idx": 10
|
| 1042 |
+
},
|
| 1043 |
+
{
|
| 1044 |
+
"type": "text",
|
| 1045 |
+
"text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] Our paper has three main claims. First, in Figure 1[right] we show the distribution of the Wikipedia and the excessive padding that it requires. Second, in Section $^ { 4 . 1 , }$ we show that we can efficiently pack the data which can be easily reproduced with the shared data and code [1]. Third, in Figure 3[right], we clearly show the $2 \\mathbf { x }$ performance gain from packing and the related hyperparameter adjustment scheme. \n(b) Did you describe the limitations of your work? [Yes] We see three potential limitations that we discuss in the paper. First, as stated in the broader impact section, our approach is clearly dependent on the sequence length distribution of the dataset. However, we looked into several other datasets beyond Wikipedia and observed even higher potential for acceleration. Second, we explain our focus on the IPU hardware with a static precompiled kernel in Section $\\boxed { 4 . 1 }$ Our theoretical analysis in Section $\\boxed { 4 . 3 }$ indicates that our approach benefits also other hardware. Third, our changes to the network with a modified attention mask and loss calculation come with some overhead. This is addressed in Table 1 [overhead column] in Section 4.1. \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We address this point in the “Broader Impact” Section, third paragraph. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
|
| 1046 |
+
"bbox": [
|
| 1047 |
+
236,
|
| 1048 |
+
255,
|
| 1049 |
+
826,
|
| 1050 |
+
537
|
| 1051 |
+
],
|
| 1052 |
+
"page_idx": 10
|
| 1053 |
+
},
|
| 1054 |
+
{
|
| 1055 |
+
"type": "text",
|
| 1056 |
+
"text": "2. If you are including theoretical results... ",
|
| 1057 |
+
"bbox": [
|
| 1058 |
+
215,
|
| 1059 |
+
542,
|
| 1060 |
+
493,
|
| 1061 |
+
558
|
| 1062 |
+
],
|
| 1063 |
+
"page_idx": 10
|
| 1064 |
+
},
|
| 1065 |
+
{
|
| 1066 |
+
"type": "text",
|
| 1067 |
+
"text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] Detailed algorithm explanations, clarifications of assumptions, and proofs are provided in the supplemental material [1]. \n(b) Did you include complete proofs of all theoretical results? [Yes] Section D.5, E, F in the supplemental material [1] provide the necessary derivations on theoretical results. ",
|
| 1068 |
+
"bbox": [
|
| 1069 |
+
233,
|
| 1070 |
+
560,
|
| 1071 |
+
825,
|
| 1072 |
+
633
|
| 1073 |
+
],
|
| 1074 |
+
"page_idx": 10
|
| 1075 |
+
},
|
| 1076 |
+
{
|
| 1077 |
+
"type": "text",
|
| 1078 |
+
"text": "3. If you ran experiments... ",
|
| 1079 |
+
"bbox": [
|
| 1080 |
+
214,
|
| 1081 |
+
637,
|
| 1082 |
+
393,
|
| 1083 |
+
651
|
| 1084 |
+
],
|
| 1085 |
+
"page_idx": 10
|
| 1086 |
+
},
|
| 1087 |
+
{
|
| 1088 |
+
"type": "text",
|
| 1089 |
+
"text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] To ensure results are easily reproducible, we follow the MLPerf 0.7 benchmark rules and implementation. Additionally the main packing code is provided in the supplemental material, along with histograms of the datasets, which can be used to confirm the efficiency of the packing algorithms. We are solely relying on open source datasets. The full code related to the changes to BERT will be provided with a future software release, currently anticipated in July. Simplified reference code is provided directly in the paper. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We follow the MLPerf 0.7 benchmark rules. We document the parameters that we changed and why we change them. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] The packing algorithms are deterministic and have no error. Other experiments are only once to compare convergence curves. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We used 16 Graphcore IPUs for acceleration on an internal cluster. ",
|
| 1090 |
+
"bbox": [
|
| 1091 |
+
236,
|
| 1092 |
+
656,
|
| 1093 |
+
826,
|
| 1094 |
+
911
|
| 1095 |
+
],
|
| 1096 |
+
"page_idx": 10
|
| 1097 |
+
},
|
| 1098 |
+
{
|
| 1099 |
+
"type": "text",
|
| 1100 |
+
"text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
|
| 1101 |
+
"bbox": [
|
| 1102 |
+
212,
|
| 1103 |
+
90,
|
| 1104 |
+
825,
|
| 1105 |
+
106
|
| 1106 |
+
],
|
| 1107 |
+
"page_idx": 11
|
| 1108 |
+
},
|
| 1109 |
+
{
|
| 1110 |
+
"type": "text",
|
| 1111 |
+
"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] Appropriate references to the BERT authors, all datasets, and the code snippet from the HugginFace inc. are appropriately referenced with citations and links. \n(b) Did you mention the license of the assets? [Yes] For the only taken code snippet, the license is part of the file [Listing 6 in the appendix]. \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] New materials like packing code and histograms will be provided under an MIT license and are already listed and linked at the end of the supplemental material. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We did not curate other people’s data. We only provide a very high level aggregate of the used data. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We did not curate other people’s data. ",
|
| 1112 |
+
"bbox": [
|
| 1113 |
+
238,
|
| 1114 |
+
111,
|
| 1115 |
+
825,
|
| 1116 |
+
299
|
| 1117 |
+
],
|
| 1118 |
+
"page_idx": 11
|
| 1119 |
+
},
|
| 1120 |
+
{
|
| 1121 |
+
"type": "text",
|
| 1122 |
+
"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
|
| 1123 |
+
"bbox": [
|
| 1124 |
+
215,
|
| 1125 |
+
304,
|
| 1126 |
+
707,
|
| 1127 |
+
319
|
| 1128 |
+
],
|
| 1129 |
+
"page_idx": 11
|
| 1130 |
+
},
|
| 1131 |
+
{
|
| 1132 |
+
"type": "text",
|
| 1133 |
+
"text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] Our experiments did not include crowdsourcing or human subjects. \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] Our experiments did not include crowdsourcing or human subjects. \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Our experiments did not include crowdsourcing or human subjects. ",
|
| 1134 |
+
"bbox": [
|
| 1135 |
+
238,
|
| 1136 |
+
323,
|
| 1137 |
+
825,
|
| 1138 |
+
440
|
| 1139 |
+
],
|
| 1140 |
+
"page_idx": 11
|
| 1141 |
+
}
|
| 1142 |
+
]
|
parse/train/3_MUAtqR0aA/3_MUAtqR0aA_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/3_MUAtqR0aA/3_MUAtqR0aA_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/HkePNpVKPB/HkePNpVKPB.md
ADDED
|
@@ -0,0 +1,431 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# COMPOSITIONAL LANGUAGES EMERGE IN A NEURAL ITERATED LEARNING MODEL
|
| 2 |
+
|
| 3 |
+
Yi Ren,1 Shangmin Guo,2 Matthieu Labeau,3 Shay B. Cohen,1 Simon Kirby1
|
| 4 |
+
|
| 5 |
+
1 University of Edinburgh, United Kingdom, 2 University of Cambridge, United Kingdom
|
| 6 |
+
3 LTCI, Tel´ ecom Paris, Institut Polytechnique de Paris, France ´
|
| 7 |
+
1 renyi.joshua@gmail.com, scohen@inf.ed.ac.uk, simon.kirby@ed.ac.uk
|
| 8 |
+
2 sg955@cam.ac.uk, 3 matthieu.labeau@telecom-paris.fr
|
| 9 |
+
|
| 10 |
+
# ABSTRACT
|
| 11 |
+
|
| 12 |
+
The principle of compositionality, which enables natural language to represent complex concepts via a structured combination of simpler ones, allows us to convey an open-ended set of messages using a limited vocabulary. If compositionality is indeed a natural property of language, we may expect it to appear in communication protocols that are created by neural agents in language games. In this paper, we propose an effective neural iterated learning (NIL) algorithm that, when applied to interacting neural agents, facilitates the emergence of a more structured type of language. Indeed, these languages provide learning speed advantages to neural agents during training, which can be incrementally amplified via NIL. We provide a probabilistic model of NIL and an explanation of why the advantage of compositional language exist. Our experiments confirm our analysis, and also demonstrate that the emerged languages largely improve the generalizing power of the neural agent communication.
|
| 13 |
+
|
| 14 |
+
# 1 INTRODUCTION
|
| 15 |
+
|
| 16 |
+
Natural language understanding (NLU), which is exemplified by challenging problems such as machine reading comprehension, question answering and machine translation, plays a crucial role in artificial intelligence systems. So far, most of the existing methods focus on building statistical associations between textual inputs and semantic representations, e.g. using first-order logic (Manning et al., 1999) or other types of representations such as abstract meaning representation (Banarescu et al., 2013). Recently, grounded language learning has gradually attracted attention in various domains, inspired by the hypothesis that early language learning was focused on problemsolving (Kirby & Hurford, 2002). While related to NLU, it focuses on the pragmatics (Clark, 1996) of learning natural language, as it implies learning language from scratch, grounded in experience. This research is often practiced through the development of neural agents which are made to communicate with each other to accomplish specific tasks (for example, playing a game). During this process, the agents build mappings between the concepts they wish to communicate about and the symbols used to represent them. These mappings are usually referred to as ‘emergent language’.
|
| 17 |
+
|
| 18 |
+
So far, an array of recent work (Havrylov & Titov, 2017; Mordatch & Abbeel, 2018; Kottur et al., 2017; Foerster et al., 2016) has shown that in many game settings, the neural agents can use their emergent language to exchange useful coordinating information. While the best way to design games to favour language emergence is still open to debate, there is a consensus on the fact that we should gear these emergent languages towards sharing similarities with natural language. Among the properties of natural language, compositionality is considered to be critical, because it enables representation of complex concepts through the combinination of several simple ones. While work on incorporating compositionality into emergent languages is still in its early stage, several experiments have already demonstrated that by properly choosing the maximum message length and vocabulary size, the agents can be brought together to develop a compositional language that shares similarities with natural language (Li & Bowling, 2019; Lazaridou et al., 2018; Cogswell et al., 2019).
|
| 19 |
+
|
| 20 |
+
In a different body of language research literature, evolutionary linguists have already studied the origins of compositionality for decades (Kirby & Hurford, 2002; Kirby et al., 2014; 2015). They proposed a cultural evolutionary account of the origins of compositionality and designed a framework called iterated learning to simulate the language evolution process, based on the idea that the simulated language must be learned by new speakers at each generation, while also being used for communication. Their experiments show that highly compositional languages may indeed emerge through iterated learning. However, the models they introduced were mainly studied by means of experiments with human participants, in which the compositional languages is favored by the participants because human brain favors structure. Hence, directly applying this framework to ground language learning is not straightforward: we should first verify the existence of the preference of compositional language at the neural agent, and then design an effective training procedure for the neural agent to amplify such an advantage.
|
| 21 |
+
|
| 22 |
+
|
| 23 |
+
Figure 1: Referential communication game and architectures of the agents.
|
| 24 |
+
|
| 25 |
+
In this project, we analyze whether and how the learning speed advantage of the highly compositional languages exists in the context of communication between neural agents playing a game. Then we propose a three-phase neural iterated learning algorithm (NIL) and a probabilistic explanation of it. The experimental results demonstrate that our algorithm can significantly enhance the topological similarity (Brighton & Kirby, 2006) between the emergent language and the original meaning space in a simple referential game (Lewis, 1969). Such highly compositional languages also generalize better, because they perform well on a held-out validation set. We highlight our contribution as:
|
| 26 |
+
|
| 27 |
+
• We discover the learning speed advantages of languages with high topological similarity for neural agents communicating in order to play a referential game. • We propose the NIL based on those advantages, which is quite robust compared to most of the related works. • We propose a probabilistic framework to explain the mechanisms of NIL.
|
| 28 |
+
|
| 29 |
+
# 2 BACKGROUND
|
| 30 |
+
|
| 31 |
+
# 2.1 REFERENTIAL GAME
|
| 32 |
+
|
| 33 |
+
We analyze a typical and straightforward object selection game, in which a speaking agent (Alice, or speaker) and a listening agent (Bob, or listener) must cooperate to accomplish a task. In each round of the game, we show Alice a target object $x$ selected from an object space $\mathcal { X }$ and let her send a discrete-sequence message $\mathbf { m }$ to Bob. We then show Bob $c$ different objects ( $x$ must be one of them) and use $c _ { 1 } , . . . , c _ { c } \in \mathcal { X }$ to represent these candidates. Bob must use the message received from Alice to select the object that Alice refers among the $c$ candidates. If Bob’s selection $\bar { c }$ is correct, both Alice and Bob are rewarded. The objects are shuffled and candidates are randomly selected in each round to avoid the agents recognizing the objects using their order of presentation.
|
| 34 |
+
|
| 35 |
+
In our game, each object in $\mathcal { X }$ has $N _ { a }$ attributes (color and shape are often used in the literature), and each attribute has $N _ { v }$ possible values. To represent objects, similarly to the settings chosen in (Kottur et al., 2017), we encode each attribute as a one-hot vector and concatenate the $N _ { a }$ one-hot vectors to represent one object. The message delivered by Alice is a fixed-length discrete sequence $\mathbf { m } = ( m _ { 1 } , . . . , m _ { { N _ { L } } } )$ , in which each $m _ { i }$ is selected from a fixed size meaningless vocabulary $V$ .
|
| 36 |
+
|
| 37 |
+
# 2.2 NEURAL AGENT STRUCTURES
|
| 38 |
+
|
| 39 |
+
Neural agents usually have separate modules for speaking and listening, which we name Alice and Bob. Their architectures, shown in Figure 1, are similar to those studied in (Havrylov & Titov, 2017) and (Lazaridou et al., 2018). Alice first applies a multi-layer perceptron (MLP) to encode $x$ into an embedding, then feeds it to an encoding LSTM (Hochreiter & Schmidhuber, 1997). Its output will go through a softmax layer, which we use to generate the message $m _ { 1 } , m _ { 2 } , \cdots$ . Bob uses a decoding LSTM to read the message and uses a MLP to encode $c _ { 1 } , . . . , c _ { c }$ into embeddings. Bob then takes the dot product between the hidden states of the decoding LSTM and the embeddings to generate scores $s _ { c }$ for each object. These scores are then used to calculate the cross-entropy loss when training Bob. When Alice and Bob are trained using reinforcement learning, we can use $p _ { A } ( \mathbf { m } | x ; \theta _ { A } )$ and $p _ { B } ( \bar { c } | \mathbf { m } , c _ { 1 } , . . . , c _ { c } ; \theta _ { B } )$ to represent their respective policies, where $\theta _ { A }$ and $\theta _ { B }$ contain the parameters of each of the neural agents. When the agents are trained to play the game together, we use the REINFORCE algorithm (Williams, 1992) to maximize the expected reward under their policies, and add the entropy regularization term to encourage exploration during training, as explained in (Mnih et al., 2016). The gradients of the objective function $J ( \theta _ { A } , \theta _ { B } )$ are:
|
| 40 |
+
|
| 41 |
+
$$
|
| 42 |
+
\begin{array} { r l } & { \nabla _ { \theta _ { A } } J = \mathbb { E } \left[ R ( \bar { c } , x ) \nabla \log p _ { A } ( \mathbf { m } | x ) \right] + \lambda _ { A } \nabla H [ p _ { A } ( \mathbf { m } | x ) ] } \\ & { \nabla _ { \theta _ { B } } J = \mathbb { E } \left[ R ( \bar { c } , x ) \nabla \log p _ { B } ( \bar { c } | \mathbf { m } , c _ { 1 } , . . . , c _ { c } ) \right] + \lambda _ { B } \nabla H [ p _ { B } ( \bar { c } | \mathbf { m } , c _ { 1 } , . . . , c _ { c } ) ] , } \end{array}
|
| 43 |
+
$$
|
| 44 |
+
|
| 45 |
+
where $R ( \bar { c } , x ) = \mathbb { 1 } ( \bar { c } , x )$ is the reward function, $H$ is the standard entropy function, and $\lambda _ { A } , \lambda _ { B } > 0$ are hyperparameters. A formal definition of the agents can be found in Appendix $\textrm { C }$ .
|
| 46 |
+
|
| 47 |
+
# 2.3 MEASURING COMPOSITIONALITY
|
| 48 |
+
|
| 49 |
+
Compositionality is a crucial feature of natural languages, allowing us to use small building blocks (e.g., words, phrases) to generate more complex structures (e.g., sentences), with the meaning of the larger structure being determined by the meaning of its parts (Clark, 1996). However, there is no consensus on how to quantitatively assess it. Besides a subjective human evaluation, topological similarity has been proposed as a possible quantitative measure (Brighton & Kirby, 2006).
|
| 50 |
+
|
| 51 |
+
To define topological similarity, we first define the language studied in this work as $\mathcal { L } ( \cdot ) : \mathcal { X } \mapsto \mathcal { M }$ . Then we need to measure the distances between pairs of objects: $\Delta _ { \chi } ^ { i j } = d _ { \chi } ( x _ { i } , x _ { j } )$ , where $d _ { X } ( \cdot )$ is a distance in $\mathcal { X }$ . Similarly, we compute the corresponding quantity for the associated messages $m _ { i } = \mathcal { L } ( x _ { i } )$ in the message space $\mathcal { M }$ with $\Delta _ { \mathcal { M } } ^ { i j } = \bar { d _ { \mathcal { M } } } \bar { ( m _ { i } , m _ { j } ) }$ , where $d _ { \mathcal { M } } ( \cdot )$ is a distance in $\mathcal { M }$ . Then the topological similarity (i.e., $\rho \mathrm { , }$ M) is defined as the correlation between these quantities across $\mathcal { X }$ . Following the setup of (Lazaridou et al., 2018) and (Li & Bowling, 2019), we use the negative cosine similarity in the object space and Levenshtein distances (Levenshtein, 1966) in the message space. We provide an example in Appendix B to give a better intuition about this metric.
|
| 52 |
+
|
| 53 |
+
# 3 NEURAL ITERATED LEARNING MODEL
|
| 54 |
+
|
| 55 |
+
The idea of iterated learning requires the agent in current generation be partially exposed to the language used in the previous generation. Even this idea is proven to be effective when experimenting with human participants, directly applying it to games played by neural agents is not trivial: for example, we are not sure where to find the preference for high- $\boldsymbol { \rho }$ languages for the neural agents. Besides, we must carefully design an algorithm that can simulate the “partially exposed” procedure, which is essential for the success of iterated learning.
|
| 56 |
+
|
| 57 |
+
# 3.1 LEARNING SPEED ADVANTAGE FOR THE NEURAL AGENTS
|
| 58 |
+
|
| 59 |
+
As mentioned before, the preference of high- $\rho$ language by the learning agents is essential for the success of iterated learning. In language evolution, highly compositional languages are favored because they are structurally simple and hence are easier to learn (Carr et al., 2017). We believe that a similar phenomenon applies to communication between neural agents:
|
| 60 |
+
|
| 61 |
+
Hypothesis 1: High topological similarity improves the learning speed of the speaking neural agent.
|
| 62 |
+
|
| 63 |
+
We speculate that high- $\boldsymbol { \rho }$ languages are easier to emulate for a neural agent than low- $\rho$ languages. Concretely, that means that Alice, when pre-trained with object-message pairs describing a high$\rho$ language at a given generation, will be faster to successfully output the right message for each object. Intuitively, this is because the structured mapping described by a language with high $\rho$ is smoother, and hence has a lower sample complexity, which makes resulting examples easier to learn for the speaker agent (Vapnik, 2013).
|
| 64 |
+
|
| 65 |
+
Hypothesis 2: High topological similarity allows the listening agent to successfully recognize more concepts, using less samples.
|
| 66 |
+
|
| 67 |
+
We speculate that high- $\rho$ languages are easier for a neural agent to recognize. That means that Bob, when pre-trained with message-object pairs corresponding to a high- $\rho$ language, will be faster to successfully choose the right object. Intuitively, the lower topological similarity is, the more difficult it will be to infer unseen object-message pairs from seen examples. The more complex mapping of a low- $\boldsymbol { \rho }$ language implies that more object-message pairs need to be provided to describe it. This translates as an inability for the listening agent to generalize the information it obtained from one object-message associated to a low- $\boldsymbol { \rho }$ language to other examples. Thus, the general performance of Bob on any example will improve much faster when trained with pairs corresponding to a high- $\rho$ language than with a low- $\boldsymbol { \rho }$ language. We provide experimental results in section 4.1 to verify our hypotheses. We also provide a detailed example in Appendix D to illustrate our reasoning.
|
| 68 |
+
|
| 69 |
+
# 3.2 NEURAL ITERATED LEARNING AND PROBABILISTIC ANALYSIS
|
| 70 |
+
|
| 71 |
+
We design the NIL algorithm to exploit these advantages in a robust manner, as detailed in Algorithm 1. The algorithm runs for $I$ generations: at the beginning of each generation $i$ , both the agents are reinitialized. As Alice and Bob have different structures, they are then pre-trained differently (see line 5-7 for Alice and line 8-12 for Bob): this is the learning phase. Alice is pre-trained via categorical cross-entropy, using the data generated at the previous generation, which we denote $D _ { i }$ . Bob is pretrained with REINFORCE, learning from the pre-trained Alice. We note $I _ { a }$ and $I _ { b }$ their respective number of pre-training iterations. With hypothesis 1, the expected $\rho$ of the language spoken by Alice should be higher than that of $D _ { i }$ . Meanwhile, Bob shold be more “familiar with” the language with a higher $\rho$ than $D _ { i }$ , as stated by hypothesis 2. Alice and Bob then play the game together for $I _ { g }$ rounds in the interacting phase, in which both agents are updated via REINFORCE. In this phase, the languages used by them are filtered to be more unambiguous — their language must deliver information accurately to accomplish the task. Finally, in the transmitting phase, we feed all objects to Alice and let it output the corresponding messages to be stored in $D _ { i + 1 }$ for the learning phase of the next generation.
|
| 72 |
+
|
| 73 |
+
To better understand how NIL enhances the expected $\rho$ of the languages generation by generation, we propose a probabilistic model for NIL in Appendix C, as well as a probabilistic analysis of the role played by Alice and Bob in every phase. Intuitively, at the beginning of each generation, the expected $\rho$ of language used by Alice (denoted by $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ] )$ is quite low because of the random initialization. Then during the learning phase, Alice learns from $D _ { i }$ and expected to have the same $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ with $D _ { i }$ if it perfectly learns that data set. However, as the high- $\rho$ language is favored by neural agent during training, the $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ of the weakly pre-trained Alice should be higher than that of $D _ { i }$ . A similar thing may happen when pre-training Bob. Then in the interacting phase, as the game performance has no preference for language with different $\rho$ $, \mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ will not change in this phase.1 Finally, in the transmitting phase, $D _ { i + 1 }$ is sampled based on the language with current $\mathbb { E } _ { \mathcal { L } } [ \bar { \rho } ( \mathcal { L } ) ]$ , which is expected to be higher than that of $D _ { i }$ . In other words, $\mathbb { E } _ { \mathcal { L } } [ \boldsymbol { \rho } ( \bar { \mathcal { L } } ) ]$ would increase generation by generation (the details for derivations are provided in Appendix C):
|
| 74 |
+
|
| 75 |
+
$$
|
| 76 |
+
\begin{array} { r } { \mathbb { E } _ { \mathcal { L } \sim D _ { i + 1 } } [ \rho ( \mathcal { L } ) ] \ge \mathbb { E } _ { \mathcal { L } \sim D _ { i } } [ \rho ( \mathcal { L } ) ] . } \end{array}
|
| 77 |
+
$$
|
| 78 |
+
|
| 79 |
+
# 4 EXPERIMENTS AND DISCUSSIONS
|
| 80 |
+
|
| 81 |
+
In this section, we first verify hypotheses 1 and 2 by directly feeding languages with different $\rho$ to Alice and Bob. Then we examine the behavior and performance of the neural agents, as well as the expected $\rho$ of languages, at each generation. We conduct an ablation study, to examine the effect of pre-training Alice and Bob separately. We then investigate more thoroughly the advantages brought by high- $\rho$ languages, and highlight the ‘interval of advantage’ in pre-training rounds, which could help in selecting reasonable $I _ { a }$ and $I _ { b }$ . Finally, we conduct a series of experiments on a held-out validation set to highlight the positive effect of high- $\boldsymbol { \rho }$ languages on the neural agents generalization ability — which shows the potential of iterated learning for NLU tasks. Details about our experimental setup and our choice of hyper-parameters can be found in Appendix A. More experiments about the robustness of NIL are presented in Appendix E.
|
| 82 |
+
|
| 83 |
+
<table><tr><td>Randomly initialize D1</td></tr><tr><td>for i= 1,2,...,I do</td></tr><tr><td>Re-initialize Alice and Bob, get Alicei and Bobi</td></tr><tr><td>//======= Learning Phase =: for i=1,2,..,Ia do</td></tr><tr><td>Randomly sample an example pair from Di and use it to update Alicei with cross-entropy</td></tr><tr><td>training end for</td></tr><tr><td>for ib = 1,2,...,Ib do</td></tr><tr><td>Alicei generates message based on input objects</td></tr><tr><td>Bobi receives message and selects the target</td></tr><tr><td>Bobi updates its parameters if rewarded</td></tr><tr><td>end for //======= Interacting Phase ===:</td></tr><tr><td>for ig= 1,2,.., Ig do</td></tr><tr><td>Alicei generates message based on input objects</td></tr><tr><td>Bobi receives message and selects the target</td></tr><tr><td>BOTH Alicei and Bobi update parameters if rewarded</td></tr><tr><td>end for</td></tr><tr><td>//======= Transmitting Phase ==:</td></tr><tr><td>for isg = 1,2,...,Is do</td></tr><tr><td>Generate object-message pairs by feeding objects to Alice; and save them to data set D+1</td></tr><tr><td>end for end for</td></tr></table>
|
| 84 |
+
|
| 85 |
+
Algorithm 1: The NIL algorithm. $I _ { a } , I _ { b }$ and $I _ { g }$ are the number of iterations used to pre-train Alice, Bob, and to play the game at each generation.
|
| 86 |
+
|
| 87 |
+
# 4.1 LEARNING SPEED ADVANTAGES
|
| 88 |
+
|
| 89 |
+
We first use the experimental results in Figure 2 to verify hypotheses 1 and 2. In these experiments, we randomly initialize one Alice and feed languages with different expected $\rho$ for it to learn (and repeat the same procedure for Bob). We generate a perfect high- $\rho$ language $( \rho = 1 )$ ) using the method proposed in (Kirby et al., 2015), and randomly permute the messages to generate a low- $\rho$ language with $\rho = 0 . 2 1$ . The other languages are intermediate languages generated during NIL. Note that there is no interacting nor transmitting phase in the experiment in this subsection: we only test the learning behavior of a randomly initialized Alice (or Bob) separately.
|
| 90 |
+
|
| 91 |
+
From the result in Figure 2-(a) and (b), we see that the high- $\rho$ languages indeed has the learning speed advantage at both the speaker and the listener side. One important finding is in Figure 2-(c), which record the expected $\rho _ { \cdot }$ , i.e., $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ , during Alice’s learning. From this figure, we find that when learning a language with low expected $\rho$ , the value of $\mathbb { E } _ { \mathcal { L } } [ \rho ( \bar { \mathcal { L } } ) ]$ will first increase, and finally converge to the $\rho$ of $D$ . This phenomenon, caused by the learning speed advantage, makes the weak pre-train the essential design for the success of NIL: if $I _ { a }$ is correctly chosen, we may expect a higher $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ than that of the data set it learns from.
|
| 92 |
+
|
| 93 |
+
# 4.2 PERFORMANCE OF NIL
|
| 94 |
+
|
| 95 |
+
In this part, we record the game performance (i.e., the rate of successful object selections) and mean $\rho$ of the object-message pairs exchanged by the neural agents every 20 rounds. We run the simulation 10 times, with a different random number seed each time. Although the results are different, they all follow the same trend. In this first series of experiments, we compare the following 4 different methods:
|
| 96 |
+
|
| 97 |
+
|
| 98 |
+
Figure 2: Illustration of the learning speed of Alice and performance improving speed of Bob when pre-training is done with various languages of different topological similarities.
|
| 99 |
+
|
| 100 |
+
|
| 101 |
+
Figure 3: Game performance and average topological similarity for the possible resetting strategies of our proposed iterated learning procedure of 80 generations. In these experiments, $I { = } 8 0$ and $I _ { g } { = } 4 0 0 0$ , with all other hyper-parameters following Table 3.
|
| 102 |
+
|
| 103 |
+
• Iterated learning, with resetting both Alice and Bob at the beginning of each generation.
|
| 104 |
+
• Iterated learning, only resetting Alice at the beginning of each generation;
|
| 105 |
+
• Iterated learning, only resetting Bob at the beginning of each generation;
|
| 106 |
+
• No iterated learning: neither Alice nor Bob are reset during training.
|
| 107 |
+
|
| 108 |
+
From Figure 3-(a), we can see that for the 3 displayed variants of the procedure, neural agents can play the game almost perfectly after a few generations. The curve of the no-reset method will directly converge while the curves of the other two iterated learning procedures will show a loss of accuracy at the beginning of each generation. That is because one or both agents are reset, and are not able to completely re-learn from the data kept from the previous generation during the pre-training phase. However, at the end of each generation, all these algorithms can ensure a perfect game performance.
|
| 109 |
+
|
| 110 |
+
While the use of NIL has little effect on the game performance, given a sufficient number of rounds, these procedures have a clear positive effect on topological similarity. In Figure 3-(b), we can see that the no-reset case has the lowest average $\rho$ while the iterated learning cases all have higher means (and increasing). We provide extra experiments in Appendix E, which demonstrate the robustness of NIL under different scenarios. The discussion on the specific impact of each agent and why the reset-Alice and reset-Bob behave differently is in Section 5.
|
| 111 |
+
|
| 112 |
+
# 4.3 HIGH TOPOLOGICAL SIMILARITY AND INTERVAL OF ADVANTAGE
|
| 113 |
+
|
| 114 |
+
In this section, we explore further the phenomenon caused by the learning speed advantage on NIL. From the discussion in section 3.1 and the experimental results in section 4.1, we know that $I _ { a }$ and $I _ { b }$ play an important role in NIL: they should not be too large nor too small. Intuitively, if $I _ { a }$ is too small, Alice will learn nothing from the previous generation, hence the NIL amounts to playing only one interacting phase. If $I _ { a }$ is too large, from the trend in Figure 2-(c), we may expect that the increase of expected $\rho$ should be small in each generation, because Alice will perfectly learn $D _ { i }$ , and hence have a $\rho$ similar to its predecessor. Hence we speculate that the value of $I _ { a }$ should have a “bottleneck” effect, i.e., a too large one or a too small one will both harm the performance of NIL. A similar argument can also applied in the selection of $I _ { b }$ .
|
| 115 |
+
|
| 116 |
+
<table><tr><td>Ia</td><td>100</td><td>200</td><td>400</td><td>800</td><td>1200</td><td>1500</td><td>2000</td><td>3000</td><td>5000</td><td>8000</td></tr><tr><td>E[r71:80]</td><td>0.293</td><td>0.828</td><td>0.928</td><td>0.951</td><td>0.958</td><td>0.961</td><td>0.952</td><td>0.956</td><td>0.955</td><td>0.949</td></tr><tr><td>E[p1:10]</td><td>0.225</td><td>0.429</td><td>0.452</td><td>0.483</td><td>0.556</td><td>0.575</td><td>0.566</td><td>0.494</td><td>0.481</td><td>0.443</td></tr><tr><td>E[p71:80]</td><td>0.203</td><td>0.706</td><td>0.836</td><td>0.886</td><td>0.899</td><td>0.935</td><td>0.936</td><td>0.929</td><td>0.889</td><td>0.837</td></tr><tr><td>1b</td><td>10</td><td>20</td><td>40</td><td>80</td><td>120</td><td>160</td><td>200</td><td>300</td><td>400</td><td>800</td></tr><tr><td>E[r71:80]</td><td>0.954</td><td>0.946</td><td>0.961</td><td>0.954</td><td>0.962</td><td>0.959</td><td>0.962</td><td>0.957</td><td>0.961</td><td>0.944</td></tr><tr><td>E[p1:10]</td><td>0.415</td><td>0.381</td><td>0.488</td><td>0.496</td><td>0.591</td><td>0.535</td><td>0.557</td><td>0.498</td><td>0.488</td><td>0.448</td></tr><tr><td>E[p71:80]</td><td>0.927</td><td>0.937</td><td>0.929</td><td>0.928</td><td>0.936</td><td>0.891</td><td>0.888</td><td>0.897</td><td>0.891</td><td>0.880</td></tr></table>
|
| 117 |
+
|
| 118 |
+
Table 1: Values of 3 metrics when varying $I _ { a }$ or $I _ { b }$ , highlighting an interval where the topological similarity grows high.
|
| 119 |
+
|
| 120 |
+
To verify our argument, we run NIL with different values of $I _ { a }$ and $I _ { b }$ , examining the behavior of the following 3 different quantitive metrics:
|
| 121 |
+
|
| 122 |
+
• $\mathbb { E } [ r _ { 7 1 : 8 0 } ]$ : The average reward of the last ten generations (game performance);
|
| 123 |
+
• $\mathbb { E } [ \rho _ { 1 : 1 0 } ]$ : The average value of $\rho$ for the first ten generations (converging speed);
|
| 124 |
+
• $\mathbb { E } [ \rho _ { 7 1 : 8 0 } ]$ : The average value of $\rho$ for the last ten generations (converged $\rho$ ).
|
| 125 |
+
|
| 126 |
+
From the results presented in Table 1, we can see the importance of the number of pre-training rounds not being too large nor too small. The suitable $I _ { a }$ and $I _ { b }$ are shown in bold. Furthermore, combining Figure 2 and Table 1, the interval of suitable $I _ { a }$ lies between 1000 to 2000 while it lies between 100 to 200 for $I _ { b }$ , which provides us an effective way to choosing hyper-parameters.
|
| 127 |
+
|
| 128 |
+
# 4.4 TOPOLOGICAL SIMILARITY AND VALIDATION PERFORMANCE
|
| 129 |
+
|
| 130 |
+
In this last series of experiments, we aim to explore the relationship between topological similarity and the generalisation ability of our neural agents, which can also indirectly reflect the expressivity of a language. We measure this ability by looking at their validation game performance: we restrict the training examples to a limited numbers of objects (i.e., the training set), and look at how good are the agents at playing the game on the others (i.e., the validation set). Figure 4-(a) demonstrates the strength of the iterated learning procedure in a validation setting. To illustrate the relationship between $\rho$ and validation performance, we randomly choose ${ { I _ { a } } } \in \left[ 6 0 , 4 0 0 0 \right]$ and $I _ { b } \in [ 5 , 2 0 0 ]$ and conduct a series of experiments. Those for which $I _ { a }$ and $I _ { b }$ are not in their optimal range will yield a worse performance on both validation test and topological similarity. In Figure 4-(b), we record the results from different experimental settings and draw the zero-shot performance given the topological similarity of the emergent language. This shows the linear correlation between these two metrics, and a significance test confirms it: the correlation coefficient is 0.928, and the associated $p$ -value is $3 . 8 * 1 \bar { 0 } ^ { - 1 0 4 }$ . Hence, under various experimental settings, the validation performance and the topological similarity are strongly correlated. Table 2 shows that when the size of validation set increases, using iterated learning can always improve the validation performance: in all the cases, both-reset algorithm always yields the best performance. The fact that the Alice-reset setting performs better than the Bob-reset setting also matches our analysis well.
|
| 131 |
+
|
| 132 |
+
# 5 DISCUSSION: A PARALLEL WITH LANGUAGE EVOLUTION
|
| 133 |
+
|
| 134 |
+
We can observe an interesting phenomenon in Figure 3-(b):2 the topological similarity of the emergent language always increases at first, whether we use iterated learning or not. This is akin to the effect apparent for $\rho$ in Figure 2-(c): continuing training will imply fine-tuning to examples that are not necessarily of good quality. However, through the generational resets and limited number of pre-training examples, iterated learning allows small generational improvements: this is because constraining the agent to learn with smaller amounts of data at each generation — through a ‘bottleneck’ (Kirby & Hurford, 2002) — forces the emergence of a more structured language. This limitation on the amounts of data available corresponds in our algorithm to limiting the number of pre-training rounds of the agents, to a number in what we denoted as the ‘interval of advantage’. In NIL, we use the weak pre-training to simulate this bottleneck, and achieve a good result: the values of $I _ { a }$ and $I _ { b }$ have an effect similar to the bottleneck studied in (Kirby et al., 2015) (more details are provided in Appendix D). Extending this parallel with the evolution of natural language, we can relate the learning speed advantage provided by high- $\cdot \rho$ languages to the speaking agent to the compressibility pressure (Kirby et al., 2015), and the better ability to generalize provided by high- $\rho$ languages to the listening agent to the expressivity pressure (Kirby et al., 2015).
|
| 135 |
+
|
| 136 |
+
Table 2: Validation performance under different validation set sizes.
|
| 137 |
+
|
| 138 |
+
<table><tr><td>Valid set size</td><td colspan="2">0</td><td colspan="2">8</td><td colspan="2">16</td><td colspan="2">32</td></tr><tr><td></td><td>Train</td><td>Valid</td><td>Train</td><td>Valid</td><td>Train</td><td>Valid</td><td>Train</td><td>Valid</td></tr><tr><td>No-reset</td><td>0.985</td><td>-</td><td>0.986</td><td>0.136</td><td>0.990</td><td>0.132</td><td>0.995</td><td>0.102</td></tr><tr><td>Bob-reset</td><td>0.967</td><td>1</td><td>0.943</td><td>0.094</td><td>0.962</td><td>0.152</td><td>0.947</td><td>0.116</td></tr><tr><td>Alice-reset</td><td>0.981</td><td></td><td>0.976</td><td>0.598</td><td>0.979</td><td>0.280</td><td>0.947</td><td>0.210</td></tr><tr><td>Both-reset</td><td>0.988</td><td>1</td><td>0.986</td><td>0.847</td><td>0.984</td><td>0.755</td><td>0.973</td><td>0.558</td></tr></table>
|
| 139 |
+
|
| 140 |
+
|
| 141 |
+
Figure 4: Validation performance and topological similarity with validation size equals eight. NIL leads to the evolution of languages which allow agents to perform well on unseen items.
|
| 142 |
+
|
| 143 |
+
This comparison allows us to address one important difference between our neural iterated learning algorithm and the original version: our speaking and listening agents are not identical. Actually, the speaking module and listening module of human are also not identical, but the works on traditional iterated learning do not pay much attention to such differences. From Figure 3-(b) and Figure 4-(a), it is clear that Alice and Bob are affected differently by the generational resets, and thus do not offer the same contribution to the final performance.3 From this parallel, we retain that iterated learning is also linked to the emergence of a certain form of compositionality when applied to neural agents. Besides, we believe that the correlation between topological similarity and validation performance that we highlight in Section 4.4 is another argument in favor of a relationship between compositionality and generalization, which has recently been explored (Kottur et al., 2017; Choi et al., 2018; Andreas, 2019).
|
| 144 |
+
|
| 145 |
+
# 6 CONCLUSION
|
| 146 |
+
|
| 147 |
+
In this paper, we find and articulate the existence of the learning speed advantages offered by high topological similarity, with which, we propose the NIL algorithm to encourage the dominance of high compositional language in a multi-agent communication game. We show that our procedure, consisting in resetting neural agents playing a referential game and pre-training them on data generated by their predecessors, can incrementally advantage emergent languages with high topological similarity. We demonstrate its interest by obtaining large performance improvements in a validation setting, linking compositionality and ability to generalize to new examples. The robustness of the algorithm is also verified in various experimental settings. Finally, we hope the proposed probabilistic model of NIL could inspire the application of NIL in more complex neural-agents-based systems.
|
| 148 |
+
|
| 149 |
+
# ACKNOWLEDGEMENT
|
| 150 |
+
|
| 151 |
+
We show our sincere gratitude to Kenny Smith, Ivan Titov, Stella Frank and Serhii Havrylov for their helpful discussion and comments that greatly improved the manuscript.
|
| 152 |
+
|
| 153 |
+
We would also like to thank the members from Prof. Jun Zhao’s team at Institute of Automation, Chinese Academy of Sciences, e.g. Dr. Kang Liu, Xiang Zhang and Xinyu Zuo, for sharing computing resources to run some experiments as well as sharing their pearls of wisdom with us during the course of this research, and we thank 3 anonymous reviewers for their insights and comments.
|
| 154 |
+
|
| 155 |
+
# REFERENCES
|
| 156 |
+
|
| 157 |
+
Jacob Andreas. Measuring compositionality in representation learning. arXiv preprint arXiv:1902.07181, 2019.
|
| 158 |
+
|
| 159 |
+
Laura Banarescu, Claire Bonial, Shu Cai, Madalina Georgescu, Kira Griffitt, Ulf Hermjakob, Kevin Knight, Philipp Koehn, Martha Palmer, and Nathan Schneider. Abstract meaning representation for sembanking. In Proceedings of the 7th Linguistic Annotation Workshop and Interoperability with Discourse, pp. 178–186, 2013.
|
| 160 |
+
|
| 161 |
+
Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
|
| 162 |
+
|
| 163 |
+
Henry Brighton. Compositional syntax from cultural transmission. Artificial life, 8(1):25–54, 2002.
|
| 164 |
+
|
| 165 |
+
Henry Brighton and Simon Kirby. Understanding linguistic evolution by visualizing the emergence of topographic mappings. Artificial life, 12(2):229–242, 2006.
|
| 166 |
+
|
| 167 |
+
Jon W Carr, Kenny Smith, Hannah Cornish, and Simon Kirby. The cultural evolution of structured languages in an open-ended, continuous world. Cognitive science, 41(4):892–923, 2017.
|
| 168 |
+
|
| 169 |
+
Edward Choi, Angeliki Lazaridou, and Nando de Freitas. Compositional obverter communication learning from raw visual input. In International Conference on Learning Representations, 2018.
|
| 170 |
+
|
| 171 |
+
Herbert H Clark. Using language. Cambridge university press, 1996.
|
| 172 |
+
|
| 173 |
+
Michael Cogswell, Jiasen Lu, Stefan Lee, Devi Parikh, and Dhruv Batra. Emergence of compositional language with deep generational transmission. arXiv preprint arXiv:1904.09067, 2019.
|
| 174 |
+
|
| 175 |
+
Jakob Foerster, Ioannis Alexandros Assael, Nando de Freitas, and Shimon Whiteson. Learning to communicate with deep multi-agent reinforcement learning. In Advances in Neural Information Processing Systems, pp. 2137–2145, 2016.
|
| 176 |
+
|
| 177 |
+
Serhii Havrylov and Ivan Titov. Emergence of language with multi-agent games: Learning to communicate with sequences of symbols. In Advances in Neural Information Processing Systems, 2017.
|
| 178 |
+
|
| 179 |
+
Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997.
|
| 180 |
+
|
| 181 |
+
Simon Kirby and James R Hurford. The emergence of linguistic structure: An overview of the iterated learning model. In Simulating the evolution of language, pp. 121–147. Springer, 2002.
|
| 182 |
+
|
| 183 |
+
Simon Kirby, Tom Griffiths, and Kenny Smith. Iterated learning and the evolution of language. Current opinion in neurobiology, 28:108–114, 2014.
|
| 184 |
+
|
| 185 |
+
Simon Kirby, Monica Tamariz, Hannah Cornish, and Kenny Smith. Compression and communication in the cultural evolution of linguistic structure. Cognition, 141:87–102, 2015.
|
| 186 |
+
|
| 187 |
+
Satwik Kottur, Jose Moura, Stefan Lee, and Dhruv Batra. Natural language does not emerge natural- ´ lyin multi-agent dialog. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2962–2967, 2017.
|
| 188 |
+
|
| 189 |
+
Angeliki Lazaridou, Karl Moritz Hermann, Karl Tuyls, and Stephen Clark. Emergence of linguistic communication from referential games with symbolic and pixel input. In International Conference on Learning Representations, 2018.
|
| 190 |
+
|
| 191 |
+
Vladimir I Levenshtein. Binary codes capable of correcting deletions, insertions, and reversals. In Soviet physics doklady, volume 10, pp. 707–710, 1966.
|
| 192 |
+
|
| 193 |
+
David Lewis. Convention: A philosophical study. John Wiley & Sons, 1969.
|
| 194 |
+
|
| 195 |
+
Fushan Li and Michael Bowling. Ease-of-teaching and language structure from emergent commu nication. arXiv preprint arXiv:1906.02403, 2019.
|
| 196 |
+
|
| 197 |
+
Christopher D Manning, Christopher D Manning, and Hinrich Schutze. ¨ Foundations of statistical natural language processing. MIT press, 1999.
|
| 198 |
+
|
| 199 |
+
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.
|
| 200 |
+
|
| 201 |
+
Igor Mordatch and Pieter Abbeel. Emergence of grounded compositional language in multi-agent populations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
|
| 202 |
+
|
| 203 |
+
Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 2013.
|
| 204 |
+
|
| 205 |
+
Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
|
| 206 |
+
|
| 207 |
+
# APPENDIX A: PARAMETER SETTINGS
|
| 208 |
+
|
| 209 |
+
Unless specifically stated, the experiments mentioned in this paper use the hyper-parameters given in Table 3. The code is available at https://github.com/Joshua-Ren/Neural_ Iterated_Learning.
|
| 210 |
+
|
| 211 |
+
Table 3: Value of hyper-parameters.
|
| 212 |
+
|
| 213 |
+
<table><tr><td>Notation</td><td>Value</td><td>Description</td></tr><tr><td>Na</td><td>2</td><td>Number of all attributes</td></tr><tr><td>Nu</td><td>8</td><td>Number of possible values for each attribute</td></tr><tr><td>NL</td><td>2,3</td><td>Message length</td></tr><tr><td>|V|</td><td>8+[0,4,8,16,32,64]</td><td>Vocabulary size.</td></tr><tr><td>I</td><td>80,100</td><td>Maximum number of generations</td></tr><tr><td>Ia</td><td>≥ 100,≤8000</td><td>Maximum pre-train rounds for Alice</td></tr><tr><td>Ib</td><td>≥10,≤800</td><td>Maximum pre-train batches for Bob</td></tr><tr><td>1</td><td>≥ 100,≤ 8000</td><td>Maximum interacting rounds</td></tr><tr><td></td><td>10,100,1000</td><td>Maximum rounds for transmitting phase</td></tr><tr><td>Nh</td><td>128</td><td>Hidden layer size</td></tr><tr><td>Nb</td><td>64</td><td>Batch size</td></tr><tr><td>C</td><td>2,5,15,30</td><td>Number of candidates (including the target)</td></tr><tr><td>lr</td><td>≥ 10-5,≤10-3</td><td>Learning rate</td></tr></table>
|
| 214 |
+
|
| 215 |
+
APPENDIX B: DIFFERENT TYPES OF LANGUAGES: A TOY EXAMPLE
|
| 216 |
+
Table 4: Different groups of language and their topological similarity.
|
| 217 |
+
|
| 218 |
+
<table><tr><td>Group</td><td>Compsitional (8)</td><td>Holistic (16)</td><td>Other (232)</td></tr><tr><td>Language Examples</td><td>bluebox=aa red box=ba blue circle = ab</td><td>bluebox=ba red box = aa bluecircle= ab</td><td>bluebox=aa red box=bb blue circle = aa</td></tr><tr><td>p</td><td>red circle = bb 1</td><td>red circle = bb 0.5</td><td>red circle = bb 0.1~ 0.7</td></tr></table>
|
| 219 |
+
|
| 220 |
+
|
| 221 |
+
Figure 5: A simple representation of two languages corresponding to topological similarities of $\rho = 1$ (top) and $\rho = 0 . 5$ (bottom).
|
| 222 |
+
|
| 223 |
+
To better understand how topological similarity can measure the compositionality of one language, and to give some intuitions on how languages having different $\rho$ would like, we provide and illustrate a toy example in this appendix. In this example, the object space is $\scriptscriptstyle \textit { \textbf { X } } =$ {blue box, blue circle, red box, red circle $\}$ and the message space is $\mathcal { M } = \{ \bar { a } a , a b , \bar { b } a , b b \}$ . Any set of mappings from four distinct objects to four messages (not necessarily distinct, i.e. same message could correspond to different objects) forms a language. Hence, there exist $4 ^ { 4 } = 2 5 6$ possible languages in this toy example. Following the principles provided in (Kirby et al., 2015), we define the following concepts for describing a language:
|
| 224 |
+
|
| 225 |
+
• Unambiguous language. A type of language that can unambiguously describe all objects in $\mathcal { X }$ . In other words, the mappings between $\mathcal { X }$ and $\mathcal { M }$ are bijectional. In this example, there exist $4 \times 3 \times 2 \times 1 = 2 4$ such languages. Compositional language. A type of unambiguous language that exhibits systematic compositional structure when forming messages. Such languages can use different symbols to represent different attributes of meaning and combine these symbols in a systematic way to form a message such that the meaning of the whole message is formed from a simple combination of the meaning of its parts. For example, following the rules of $S \to X Y$ , and $X : b l u e b ; X : r e d : b ; Y : b o x a ; X : c i r c l e : b$ , we can derive a compositional language like the example in Table 4. In this example, we have $4 + 4 = 8$ such languages. Holistic language. A type of unambiguous language but does not exhibits full systematic structures. In other words, holistic languages are those unambiguous language who are not compositional languages. In this example, we have $2 4 - 8 = 1 6$ such languages.
|
| 226 |
+
• Degenerate language. A type of ambiguous language that maps all objects to the same message. In this example, we have 4 such languages. Degenerate component. Any ambiguous language having degenerate component, i.e., there may be more than one objects mapping to the same message. The existence of degenerate component makes the language ambiguous.
|
| 227 |
+
|
| 228 |
+
Note that the number of unambiguous languages is usually much smaller than that of ambiguous languages, and the number of compositional languages is usually smaller than that of holistic languages. Using permutation and combination, we can calculate the numbers of all possible languages, unambiguous language, compositional language and holistic language as:
|
| 229 |
+
|
| 230 |
+
$$
|
| 231 |
+
\begin{array} { l } { { \displaystyle \# \mathrm { ~ a l l ~ p o s s i b l e ~ l a n g u a g e s } = \left( | V | ^ { N _ { L } } \right) ^ { ( N _ { v } ^ { N _ { a } } ) } } } \\ { { \displaystyle \# \mathrm { ~ u n a m b i g u o u s ~ l a n g u a g e s } = \frac { \left( | V | ^ { N _ { L } } \right) ! } { \left( | V | ^ { N _ { L } } - N _ { v } ^ { N _ { a } } \right) ! } } } \\ { { \displaystyle \# \mathrm { ~ c o m p o s i t i o n a l ~ l a n g u a g e s } = \frac { N _ { L } ! } { \left( N _ { L } - N _ { a } \right) ! } \cdot \left( \frac { | V | ! } { \left( | V | - N _ { v } \right) ! } \right) ^ { N _ { a } } } } \end{array}
|
| 232 |
+
$$
|
| 233 |
+
|
| 234 |
+
# holistic languages $= \#$ unambiguous languages − # compositional languages
|
| 235 |
+
|
| 236 |
+
From the above equation, it is easy to see that the gap between the number of compositional languages and holistic languages would become larger when $N _ { v }$ ${ _ { v } } , N _ { a } , N _ { L }$ and $| V |$ increase. Further, this means that it becomes even harder to pick a compositional language when randomly sample a language. That could explain why the expected topological similarity of the emergent language may increase when smaller $N _ { L }$ and $| V |$ are applied, as illustrated in (Lazaridou et al., 2018; Cogswell et al., 2019).
|
| 237 |
+
|
| 238 |
+
Besides the numbers of different languages, another key difference among these languages is the topological similarity (i.e., $\rho )$ , as illustrated in section 2.3. As the language studied in this paper is defined as a mapping function from a meaning (i.e., an object) to a message, a compositional language must ensure that the meaning of a symbol is a function of the meaning of its parts. In other words, compositional languages are neighborhood related: nearby meanings tend to be mapped to nearby signals. Or to say, nearby meanings that share similar attributes are likely to share similar message symbols (Brighton & Kirby, 2006). Thus, as the difference between messages are measured by edit distance, the compositional languages will have a higher $\rho$ than the holistic ones. However, the existence of degenerate component also change the value of $\rho$ : the $\rho$ of a degenerate language might be higher than that of a holistic language.
|
| 239 |
+
|
| 240 |
+
From the above discussions, we find that making the highly compositional languages dominate is a challenging task: it occupies a really small portion among all possible languages, and only using topological similarity also cannot tell them apart from those who are highly degenerate. However, the proposed algorithm can solve this problem almost perfectly: it uses the learning speed advantage caused by high topological similarity to increase the posterior probability of high- $\rho$ languages, and uses the interacting phase to rule out the degenerate components. The details of how the probability of languages changes in different phase of our algorithm are illustrated in Appendix C.
|
| 241 |
+
|
| 242 |
+
# APPENDIX C: PROBABILISTIC MODEL OF THE SYSTEM
|
| 243 |
+
|
| 244 |
+
# Probabilistic Model of Emergent Languages:
|
| 245 |
+
|
| 246 |
+
In section 2.3, we define a language as a mapping function from object space $\mathcal { X }$ to the message space $\mathcal { M }$ , i.e., $\mathcal { L } ( \cdot ) : \mathcal { X } \mapsto \mathcal { M }$ . Here we discuss how to describe the probability of a specific language, i.e., $P ( \mathcal { L } )$ .
|
| 247 |
+
|
| 248 |
+
Suppose that we have $N$ possible different objects $( x _ { 1 } , x _ { 2 } , . . . , x _ { N } )$ , where $ { N _ { \mathrm { ~ \scriptsize ~ = ~ } } } N _ { v } ^ { N _ { a } }$ , and the messages are conditionally independent given an object $x _ { n }$ (where $n \in [ 1 , 2 , \ldots , N ] )$ , i.e.:
|
| 249 |
+
|
| 250 |
+
$$
|
| 251 |
+
P ( \mathcal { L } ) = P ( \mathbf { m _ { 1 } } , . . . , \mathbf { m _ { N } } | x _ { 1 } , . . . , x _ { N } ) = \prod _ { n = 1 } ^ { N } P ( \mathbf { m _ { n } } | x _ { 1 } , . . . , x _ { N } ) = \prod _ { n = 1 } ^ { N } P ( \mathbf { m _ { n } } | x _ { n } ) .
|
| 252 |
+
$$
|
| 253 |
+
|
| 254 |
+
Assume that messages are uniformly sampled from $\mathcal { M }$ whose size is $M = | V | ^ { N _ { L } }$ , we could have $\begin{array} { r } { P ( \mathbf { m _ { n } } | x _ { n } ) = \frac { 1 } { M } , \breve { \forall } n \in \{ 1 , 2 , . . . , \bar { N } \} } \end{array}$ . Hence the initial probability (or prior probability) of any possible language is $\left( { \frac { 1 } { M } } \right) ^ { N }$ . We define the posterior distribution of languages as the distribution after our neural iterated learning algorithm (NIL), i.e. $P ( \mathcal { L } | \mathrm { N I L } )$ .
|
| 255 |
+
|
| 256 |
+
Then, our goal is to enhance the posterior probability of the high- $\rho$ languages, which is equivelant to enhance the expectation of $\rho$ , i.e.:
|
| 257 |
+
|
| 258 |
+
$$
|
| 259 |
+
\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ] = \sum _ { i } \rho ( \mathcal { L } _ { i } ) P ( \mathcal { L } _ { i } | \mathrm { N I L } ) .
|
| 260 |
+
$$
|
| 261 |
+
|
| 262 |
+
It is obvious that $\mathbb { E } _ { \mathcal { L } } [ \rho ( \mathcal { L } ) ]$ , the expected topological similarity of languages following the prior probability, is quite low, as the high- $\boldsymbol { \rho }$ languages only occupy an extremely small fraction.
|
| 263 |
+
|
| 264 |
+
# Definition of the Agents:
|
| 265 |
+
|
| 266 |
+
Following the structure provided in Figure 1, we define the speaking agent (Alice) and listening agent (Bob) formally here.
|
| 267 |
+
|
| 268 |
+
Alice is a bunch of neural networks that can map any input object $x$ to a discrete message m. So we define it as $\mathbf { m } = h ( x ) , h : \mathcal { X } \mapsto \mathcal { M }$ . As Alice generate discrete messages with softmax layers, the probabilistic distribution of different words in ${ \bf m } _ { n }$ can be obtained. In the example provided in Figure 1, we can have $P ( m _ { 1 } | x )$ and $P ( m _ { 2 } | x , m _ { 1 } )$ by reading the distribution from softmax layers. In more general cases, we could obtain $P ( m _ { l } | x , m _ { l - 1 } , m _ { l - 2 } , . . . )$ following the same method. Thus, we can directly calculate the probability of specific $\mathbf { m }$ given $x$ for Alice as follow:
|
| 269 |
+
|
| 270 |
+
$$
|
| 271 |
+
P ( \mathbf { m } | x ) = P ( m _ { 1 } | x ) \prod _ { l = 2 } ^ { N _ { L } } P ( m _ { l } | x , m _ { l - 1 } , m _ { l - 2 } , \dots ) .
|
| 272 |
+
$$
|
| 273 |
+
|
| 274 |
+
If we feed all possible $x$ to Alice and calculate the corresponding $P ( { \bf m } | x )$ , we then could calculate the probability distribution of all languages after training Alice, following equation (8) and (9). Then, we can state our goal as to obtain a high $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ by using NIL to update the parameters of the neural network.
|
| 275 |
+
|
| 276 |
+
In our setting, the posterior probability of languages is decided by Alice with its softmax layers. Bob plays a role of assistant to ensure the robustness of NIL, which will be further illustrated in Appendix D and E. From Figure 1, we could see that the inputs of Bob are a discrete message m and $c$ different objects. As Bob will calculate a score $s _ { c }$ for each object $c _ { c }$ , we can denote its function as $s = f ( \mathbf { m } , x ) , \overset { \cdot } { f } : \mathcal { M } \times \mathcal { X } \mapsto \mathbb { R } ^ { 1 }$ .
|
| 277 |
+
|
| 278 |
+
# Probabilistic Description of Language Evolution in NIL:
|
| 279 |
+
|
| 280 |
+
|
| 281 |
+
Figure 6: Probabilitic explanation of different phases in NIL.
|
| 282 |
+
|
| 283 |
+
To avoid confusion, we specify all the probabilities involved in NIL in the left corner of Figure 6. In the figure, the shadow regions with different colors represent the three phases of NIL in ONE generation. Thus, one generation of NIL could be described as:
|
| 284 |
+
|
| 285 |
+
1. Initialization: At the beginning of generation $t$ , the initial probability of Alice[i] is $P _ { 0 } ( \mathcal { L } )$ , which is same as the prior probability of $P ( \mathcal { L } )$ mentioned before, as Alice[i] is always randomly initialized. The initial function of Bob[i] is represented as $f _ { 0 } ( \mathbf { m } , x )$ .
|
| 286 |
+
2. Learning Phase: Then following Algorithm 1, Alice[i] will be pre-trained using the data sampled from the previous generation, i.e. $D _ { i }$ . The pre-trained probability of languages is defined as $P _ { i } ( \mathcal { L } | D _ { i } )$ . Bob[i] will then be pre-trained using the sample generated by $P _ { i } ( \mathcal { L } | D _ { i } )$ , using REINFORCE procedure, after which, its function becomes $\bar { f } _ { i } ( \mathbf { m } , x )$ .
|
| 287 |
+
3. Interacting Phase: The pre-trained Alice[i] and Bob[i] then interact and update their parameters together following the REINFORCE procedure described in section 3.2. In each round of the game, Alice[i] would first use argmax to select m with the highest probability given a randomly selected object $x$ , both agents would then update their parameters if $R = 1$ , i.e. the data pair $\langle \mathbf { m } , x \rangle$ could assist them to accomplish the referential game successfully. We argure that this process has the same effect as the following procedure: we first sample a data set $D _ { * } \sim P _ { i } ( \mathcal { L } | D _ { i } )$ , and then delete the data pairs that cannot unambiguously deliver information to form a refined data set $R _ { i }$ . Then, the interacted probability of Alice[i] can be represented by $P _ { i } ( \mathcal { L } | D _ { i } , R _ { i } )$ . As Bob also update its parameters in this phase, we define its interacted function as $f _ { i * } ( \mathbf { m } , x )$ .
|
| 288 |
+
4. Transmitting Phase: Finally, in the transmitting phase, we sample $D _ { i + 1 } \sim P _ { i } ( \mathcal { L } | D _ { i } , R _ { i } )$ by: i)randomly feeding $x _ { n }$ to Alice[i]; ii) sample a message ${ \bf m } _ { n } \sim P _ { i } ( { \bf m } | x _ { n } , D _ { i } , R _ { i } )$ . Note that Bob[i] is not involved in this phase.
|
| 289 |
+
|
| 290 |
+
From all sections above, we argue that Alice plays an important role in all the phases in NIL while Bob only helps to make the languages effective during interaction phases. As we will discuss the role of Alice and Bob in further details in Appendix $\mathrm { E }$ , we only provide an intuition of how the language changes in NIL in the following paragraphs.
|
| 291 |
+
|
| 292 |
+
Overall, the objective of our NIL design is to ensure the expected topological similarity of emergent languages would increase over generations, as expressed by equation (3). As the languages with higher $\rho$ would be learned faster, which is stated as Hypothesis 1, we can expect those high- $\rho$ languages to have a higher pre-trained probability in $P ( \vec { \mathcal { L } } | \bar { D } _ { i } )$ than in $D _ { i }$ , i.e.:4
|
| 293 |
+
|
| 294 |
+
$$
|
| 295 |
+
\begin{array} { r } { \mathbb E _ { \mathcal L \sim P ( \mathcal L | D _ { i } ) } [ \rho ( \mathcal L ) ] \ge \mathbb E _ { \mathcal L \sim D _ { i } } [ \rho ( \mathcal L ) ] . } \end{array}
|
| 296 |
+
$$
|
| 297 |
+
|
| 298 |
+
Note that this inequality is not a strict corollary, but it is very likely to hold as long as we have an appropriate $I _ { a }$ . In the worst case, we can chose an extremely large $I _ { a }$ to make Alice learn $D _ { i }$ perfectly. However, we could verify it by the experimental results as well as the explanation in Appendix $\mathbf { D }$ that the weak pre-training can indeed help us to achieve a higher expected $\rho$ . Then, in the interacting phase, we may expect:
|
| 299 |
+
|
| 300 |
+
$$
|
| 301 |
+
\begin{array} { r } { \mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | D _ { i } , R _ { i } ) } [ \rho ( \mathcal { L } ) ] = \mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | D _ { i } ) } [ \rho ( \mathcal { L } ) ] , } \end{array}
|
| 302 |
+
$$
|
| 303 |
+
|
| 304 |
+
as the compositional languages and holistic languages are both unambiguous and the game performance cannot tell them apart. Finally, during the transmitting phase, we have $D _ { i + 1 } \sim \bar { P _ { i } } ( \mathcal { L } | \bar { D _ { i } } , R _ { i } )$ . Assuming that we sampled enough $D _ { i + 1 }$ to ensure it has a very similar distribution to $P _ { i } ( \mathcal { L } | D _ { i } , R _ { i } )$ , it is reasonable to have:
|
| 305 |
+
|
| 306 |
+
$$
|
| 307 |
+
\mathbb { E } _ { \mathcal { L } \sim D _ { i + 1 } } [ \rho ( \mathcal { L } ) ] = \mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | D _ { i } , R _ { i } ) } [ \rho ( \mathcal { L } ) ] .
|
| 308 |
+
$$
|
| 309 |
+
|
| 310 |
+
Sum up from the above, equation (3) can be obtained by combining equation (11-13).
|
| 311 |
+
|
| 312 |
+
# APPENDIX D: MORE ON THE LEARNING SPEED ADVANTAGE
|
| 313 |
+
|
| 314 |
+
Amplifying mechanism and learning speed advantage are the two main elements for the success of NIL. The former on is elaborated in section 3.2 and Appendix C, under the assumption that the learning speed advantage of high- $\cdot \rho$ language indeed exist. In this section, we will explain why such an advantage exist by experimental results and a toy example.
|
| 315 |
+
|
| 316 |
+
|
| 317 |
+
Figure 7: Illustration of learning a high- $\rho$ language and low- $\rho$ language.
|
| 318 |
+
|
| 319 |
+
# Example for Supporting Hypothesis 1:
|
| 320 |
+
|
| 321 |
+
This hypothesis claims that a high- $\boldsymbol { \cdot } \rho$ language would be leared faster than a low- $\boldsymbol { \cdot } \rho$ one on the speaker side. As we can directly represent the posterior probability of any language from Alice’s perspective, the assertion of “learned faster” can be converted to “the posterior probability increases faster”. We use a toy example, i.e. two languages in Table 4, to demonstrate how such an advantage emerges and how it works. To make the notation concise, we use “BB, RB, BC, RC” to represent “blue box, red box, blue circle, red circle” respectively. The probability of the compositional language and the holistic language in Table 4 can be represented as:
|
| 322 |
+
|
| 323 |
+
$$
|
| 324 |
+
\begin{array} { r l r } & { } & { P ( \mathcal { L } _ { \mathrm { c m p } } ) = P ( m _ { 1 } = a | B B ) \cdot P ( m _ { 2 } = a | B B , m _ { 1 } = a ) \cdot P ( m _ { 1 } = b | R B ) \cdot P ( m _ { 2 } = a | R B , m _ { 1 } = b ) \cdot C } \\ & { } & { P ( \mathcal { L } _ { \mathrm { b o l } } ) = \underbrace { P ( m _ { 1 } = b | B B ) } _ { \mathrm { ( ~ \overline { { \Omega } } ) } } \cdot \underbrace { P ( m _ { 2 } = a | B B , m _ { 1 } = b ) } _ { \mathrm { ( 2 ) } } \cdot \underbrace { P ( m _ { 1 } = a | R B ) } _ { \mathrm { ( 3 ) } } \cdot \underbrace { P ( m _ { 2 } = a | R B , m _ { 1 } = a ) } _ { \mathrm { ( 4 ) } } \cdot C } \\ & { } & { C = \underbrace { P ( m _ { 1 } = a | B C ) } _ { \mathrm { ( 5 ) } } \cdot \underbrace { P ( m _ { 2 } = b | B C , m _ { 1 } = a ) } _ { \mathrm { ( 6 ) } } \cdot \underbrace { P ( m _ { 1 } = b | R C ) } _ { \mathrm { ( 7 ) } } \cdot \underbrace { P ( m _ { 2 } = b | R C , m _ { 1 } = b ) } _ { \mathrm { ( 8 ) } } } \end{array}
|
| 325 |
+
$$
|
| 326 |
+
|
| 327 |
+
where $C$ is the common part for both languages.
|
| 328 |
+
|
| 329 |
+
As we are using stochastic gradient descent algorithm to update the parameters of Alice, it straightforward to see that the update of gradient from one point will ’pull up’ the neighbourhood region of function $h$ , which is shown in the left panel of Figure 7. Then, we can speculate that if one data sample belonging to both the two language comes, e.g. $\langle a b , B C \rangle$ , the following probabilities would increase at the same time:
|
| 330 |
+
|
| 331 |
+
$$
|
| 332 |
+
P ( m _ { 1 } = a | B C ) ; \quad P ( m _ { 1 } = a | B B ) ; \quad P ( m _ { 1 } = a | R C ) ,
|
| 333 |
+
$$
|
| 334 |
+
|
| 335 |
+
as the input of them are similar with $B C$ (only one attribute changes). As the conditional probabilities must sum to 1, the following probabilities would decrease:
|
| 336 |
+
|
| 337 |
+
$$
|
| 338 |
+
P ( m _ { 1 } = b | B C ) ; \quad P ( m _ { 1 } = b | B B ) ; \quad P ( m _ { 1 } = b | R C ) .
|
| 339 |
+
$$
|
| 340 |
+
|
| 341 |
+
Thus, when Alice learns the data sample $\langle a b , B C \rangle$ , $P ( \mathcal { L } _ { \mathrm { c m p } } )$ may have two terms increased, i.e., terms $\textcircled{5}$ and $\textcircled{1}$ . For $P ( \mathcal { L } _ { \mathrm { h o l } } )$ , however, the decrease of term $\textcircled{1}$ will harm the increase of term $\textcircled{5}$ , hence $P ( \mathcal { L } _ { \mathrm { h o l } } )$ increases slower than $P ( \mathcal { L } _ { \mathrm { c m p } } )$ (The fact that term $\textcircled{7}$ decreases on both sides would not change our deduction).
|
| 342 |
+
|
| 343 |
+
# Example for Supporting Hypothesis 2:
|
| 344 |
+
|
| 345 |
+
We can use a similar explanation for the advantage at Bob. Recall that Bob is defined as a mapping function $f$ from ${ \mathcal { M } } \times { \mathcal { X } }$ to $\mathbb { R } ^ { 1 }$ . Following the principle mentioned above, if Bob learns $\langle a b , B B \rangle$ , a bunch of function values would increase, i.e. $\bar { f } ( a b , \bar { B } B ) , f ( a a , B B ) , f ( b b , B B ) , f ( a b , \bar { B } C ) ,$ and $f ( a b , C B )$ , as they are all close to each other in the input space. Then it is easy to find that two terms in the compositional language in Table 4 are increased while only one term increases in the holistic language. That is, the score of high- $\boldsymbol { \rho }$ language would increases faster.
|
| 346 |
+
|
| 347 |
+
We can also think hypothesis 2 in the following way. With the intuition that a language with higher $\rho$ tends to be smoother and to have fewer inflection points than one with lower $\rho$ , the learning speed advantage given by highly compositional languages can be illustrated by the example provided in Figure 7. In the example, language is considered to be a one-dimensional mapping function, which is represented by the dotted lines in Figure 7. The object-message pairs, which are represented by the cross marks, are the points that satisfy the mapping function. The solid line represents the mapping function of the learning agent. Suppose the target output (i.e. the third cross mark in each figure) is larger than the predicting output (i.e. the circle mark), the optimizer will update the parameters of the neural network following the direction of the gradient, as illustrated by the bold arrows in the figure. Such an update will also pull the neighbouring parts of the function up, as illustrated by the smaller arrows on the solid curve.
|
| 348 |
+
|
| 349 |
+
The smoothness of high- $\boldsymbol { \rho }$ languages implies that the MSE of neighbouring positions will also be reduced by this update, while the MSE of neighbors would be increased in the case of a low- $\rho$ language. Such a trend is represented by the blue arrows and red crossed-arrows in Figure 7: the blue one means a decrease of the MSE at the specific position while the red one means increases of MSE. In other words, for a high- $\cdot \rho$ languages, an update corresponding to one data sample is likely to have a larger positive effect on other data samples, and hence ensure a higher learning speed. Meanwhile, for a low- $\rho$ language, one data sample would have both positive and negative effects on its neighbors and thus lead to a lower learning speed.
|
| 350 |
+
|
| 351 |
+
# APPENDIX E: ROBUSTNESS OF NIL
|
| 352 |
+
|
| 353 |
+
In this section, we will provide experimental results to demonstrate the robustness of the proposed method. The influence of hyperparameters (e.g. vocabulary size, message length) as well as the role played by Alice and Bob are both elaborated.
|
| 354 |
+
|
| 355 |
+
# Robustness for Hyperparameters on Message Space:
|
| 356 |
+
|
| 357 |
+
The message space are decided by the vocabulary size $| V |$ and the message length $N _ { L }$ . Thus, we first make experiments to see the effects of different $| V |$ and $N _ { L }$ on $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ .
|
| 358 |
+
|
| 359 |
+
From the discussion in Appendix B, we know that when $| V |$ and $N _ { L }$ are large, making high- $\rho$ language dominate in the posterior probability is very hard, as the compositional languages only occupy an extremely small portion. Such a trend could also be found in Table 5, as the finally converged expectation of topological similarity becomes lower with larger $| V |$ or $N _ { L }$ .
|
| 360 |
+
|
| 361 |
+
Our algorithm, however, is very robust to different values of $| V |$ and $N _ { L }$ . By comparing different columns in Table 5, $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ decreases very slow with the increasement of $| V |$ and $N _ { L }$ . An extreme example is that, the converged $\rho$ can still be roughly 0.8 with $| V | = 7 2$ . The performance of validation accuracy seems more robust when $| V |$ and $N _ { L }$ changes: the NIL can always obtain more than $80 \%$ accuracy compared to the none reset case (roughly $1 5 \%$ ).
|
| 362 |
+
|
| 363 |
+
Furthermore, compared with $| V | , N _ { L }$ has a stronger impact on the performance in terms of all metrics but the validation performance, as it is shown in Table 5 that the performance with $N _ { L } = 3$ is significantly lower than its counterpart when $N _ { L } ~ = ~ 2$ . One possible explanation is that the increasing of $N _ { L }$ brings an exponential change to the message space. However, no matter how $| V |$ and $N _ { L }$ change, $\bar { \mathbb { E } } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } \big [ \rho ( \mathcal { L } ) \big ]$ is always significantly higher that the compositionality of emergent languages given by baseline model, i.e. 0.3.
|
| 364 |
+
|
| 365 |
+
<table><tr><td></td><td>NL</td><td>V|= 8</td><td>V|=12</td><td>V|=16</td><td>V|=24</td><td>V|=40</td><td>V|=72</td></tr><tr><td rowspan="2">E[p71:80]</td><td>2</td><td>0.986±0.01</td><td>0.937±0.02</td><td>0.933±0.01</td><td>0.854±0.02</td><td>0.830±0.02</td><td>0.793±0.02</td></tr><tr><td>3</td><td>0.712±0.01</td><td>0.833±0.01</td><td>0.798±0.02</td><td>0.777±0.01</td><td>0.793±0.02</td><td>0.780±0.03</td></tr><tr><td rowspan="2">E[p1:10]</td><td>2</td><td>0.767±0.18</td><td>0.690±0.18</td><td>0.684±0.20</td><td>0.630±0.17</td><td>0.668±0.19</td><td>0.572±0.14</td></tr><tr><td>3</td><td>0.528±0.11</td><td>0.647±0.15</td><td>0.640±0.17</td><td>0.664±0.14</td><td>0.637±0.16</td><td>0.628±0.21</td></tr><tr><td rowspan="2">G0.85</td><td>2</td><td>9</td><td>16</td><td>10</td><td>37</td><td>68</td><td>-</td></tr><tr><td>3</td><td>-</td><td>-</td><td>39</td><td>1</td><td>-</td><td>59</td></tr><tr><td rowspan="2">Valid Acc.</td><td>2</td><td>0.868±0.14</td><td>0.914±0.06</td><td>0.833±0.11</td><td>0.866±0.11</td><td>0.801±0.10</td><td>0.828±0.14</td></tr><tr><td>3</td><td>0.804±0.13</td><td>0.677±0.16</td><td>0.773±0.15</td><td>0.858±0.10</td><td>0.867±0.01</td><td>0.900±0.07</td></tr></table>
|
| 366 |
+
|
| 367 |
+
Table 5: Values of 4 metrics when $| V |$ and $N _ { L }$ changes. Metric $G _ { 0 . 8 5 }$ means the first generation that the average $\rho$ of the previous three generations exceed 0.85. The notation “-” means the agents never satisfy the requirement.
|
| 368 |
+
|
| 369 |
+
# Robustness on Degenerate Components:
|
| 370 |
+
|
| 371 |
+
From the discussions in Appendix B, we know that the $\rho$ of a language who has many degenerate components will also be high, and hence can be learned faster by Alice in the learning phase. Thus, it is necessary to check whether our algorithm can avoid the mode collapse caused by the degenerate components. Intuitively, the degenerate components can be filtered out during the interacting phase, as the REINFORCE algorithm ensure that the parameters of the agent will only be updated with respect to $R = 1$ , i.e. the language is effective and thus unambiguous.
|
| 372 |
+
|
| 373 |
+
To verify our hypothesis, we first observe how the number of message types, i.e. the number of different messages used to describe all 64 objects, changes during NIL. It is straightforward to see that a language without any degenerate component would have 64 different message types. As shown in Figure 8, all methods could achieve high numbers if message types, which indicates that the REINFORCE algorithm could always filter out the degenerate components efficiently.
|
| 374 |
+
|
| 375 |
+
Furthermore, we design two challenging tasks for NIL:
|
| 376 |
+
|
| 377 |
+
|
| 378 |
+
Figure 8: Numbers of message types from different settings.
|
| 379 |
+
|
| 380 |
+
1. Degenerate initialized: We let Alice learn from a pure degenerate language at the beginning of each generation, before it learns from $D _ { i }$ .
|
| 381 |
+
|
| 382 |
+
2. Degenerate mixed: We mix the data pair generated by a pure degenerate language to $D _ { i }$ and ensures the proportion of the degenerate pairs is more than $5 0 \%$ , which makes Alice easier to collapse to a degenerate language during learning phase.
|
| 383 |
+
|
| 384 |
+
We then compare the performance, i.e. the expected $\rho$ and validation accuracy, of agents in different tasks. The results shown in Figure 9 demonstrate that our NIL is very robust to the influence of degenerate component, as both $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ and the validation score are much higher than the none reset baseline’s performance.
|
| 385 |
+
|
| 386 |
+
|
| 387 |
+
Figure 9: Two corner case test. NIL with degenerate initialized means Alice is initialized with a degenerate language at the beginning of each generation. NIL with degenerate mixed means Alice is initialized with a degenerate language, AND the $D _ { i }$ is mixed with $I _ { s }$ degenerate language pairs.
|
| 388 |
+
|
| 389 |
+
# The Role of Bob’s Pre-training
|
| 390 |
+
|
| 391 |
+
From the discussions above, it is easy to understand why $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ would gradually increase in NIL and how the REINFORCE applied in interacting phase can filter the degenerate component. However, the role played by Bob, especially in the learning phase where Bob only update its own parameters, is not straightforward. In short, the pre-training of Bob makes the algorithm more robust, especially at the beginning of the interacting phase. We record the value of $\bar { \mathbb { E } } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ every 20 iterations among learning phase and interacting phase, and plot the results of two generations in Figure 10.
|
| 392 |
+
|
| 393 |
+
|
| 394 |
+
Figure 10: The change of $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ in generation 3 and 6.
|
| 395 |
+
|
| 396 |
+
In this figure, the $\mathbf { X }$ -axis is the index of iterations. With $I _ { a } { = } 1 0 0 0$ , $I _ { b } { = } 4 0 0$ , and $I _ { g } { = } 1 6 0 0$ , we split (by dotted lines) each generation to three parts: Alice pre-training, Bob pre-training, and interacting phase. The blue lines are generated by NIL with the pre-training of Bob while the red lines are generated when Bob is not pre-trained (here $I _ { g } { = } 2 0 0 0$ to make a fair comparison). For the blue lines, $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ will not change when Bob is pre-training (begins at the 1000th iteration), because Alice do not update parameters at that time. However, for the red lines, $\mathbb { E } _ { \mathcal { L } \sim P ( \mathcal { L } | \mathrm { N I L } ) } [ \rho ( \mathcal { L } ) ]$ begin to decrease at the $1 0 0 0 \mathrm { { t h } }$ iteration. That is because when Bob is not pre-trained, the language learned by Alice may be impacted by playing with a fresh new Bob at the beginning of interacting phase! That is why the pre-training of Bob can make the NIL more efficient and robust. If Bob are pre-trained by the data generated by Alice in the current generation, Bob would be more “familiar” with Alice’s language, and hence ensures a more stable interacting phase.
|
| 397 |
+
|
| 398 |
+
# Looking at the Emergent Languages
|
| 399 |
+
|
| 400 |
+
From the discussions above, we know that NIL can ensure a high expected $\rho$ of the emergent language, and a high validation performance. Here we show the evolution of the distributions of emergent languages to provide a better intuition on how NIL works.
|
| 401 |
+
|
| 402 |
+
We first provide two examples of converged language (i.e., the language generated by Alice in the last generation) using the none-reset method and the resetting-both method in Table 6 and 7, respectively. In these examples, both languages can almost unambiguously represent all 64 different types of objects in $\mathcal { X }$ , and hence they can help Alice and Bob to play the game successfully. However, the language generated using iterated learning has a clear compositional structure: the first position of the message represents different colors, and the second position represents the shape. Such a structure is quite similar to what humans do, e.g., combine an adjective and a noun to represent a complex concept.
|
| 403 |
+
|
| 404 |
+
To better illustrate the posterior probability of emergent languages as a function of the corresponding value of $\rho$ and the generation, we provide the 3D views of $\bar { P } ( \rho ( \mathcal { L } ) | D _ { i } , R _ { i } )$ in 80 generations in Figure 12 and 13. The heat-map provided in Figure 11 can be considered as the top views of these 3D illustrations. In these two figures, the $\mathbf { X }$ -axis and y-axis represent the index of generation and the topological similarity, and the z-axis represents the probability of languages with a specific value of $\rho$ , in a specific generation. To make the figures easier to read, we smooth the distribution of $\rho$ in each generation using linear interpolation (Boyd & Vandenberghe, 2004).
|
| 405 |
+
|
| 406 |
+
Figure 14-(a) and (b) compare the posterior distributions at some typical generations, which can also be considered as the side views of the 3D illustration from the direction of $\mathbf { X }$ -axis. In these figures, we find that the initial distribution of $\rho$ is not flat. That is because even the prior probability for each language is uniform, the amounts of languages with extremely high $\rho$ and low $\rho$ only occupy a small portion among all possible languages, as stated in (Brighton, 2002). Hence the initial probability of $\rho ( \mathcal { L } )$ is no longer uniform and has a bell shape which is similar to the Gaussian distribution. One new trend provided by these figures is that, in the none-reset case, the width of the curves in different generations do not change much, while in the resetting-both case, the width of the curves will gradually decrease (i.e., becomes more peaky). Such a trends means when iterated learning is applied, language tend to converge to some high- $\cdot \rho$ types.
|
| 407 |
+
|
| 408 |
+
Figure 15-(a) and (b) track the ratio of languages with different values of $\rho$ , which can also be considered as the side views of the 3D illustration from the direction of $_ \textrm { y }$ -axis. In these figures, we divide all possible languages into five groups based on their topological similarity, i.e., languages with $\rho \le 0 . 2 , 0 . 2 < \rho \le 0 . 4 , 0 . 4 < \rho \le 0 . 6 , 0 . 6 < \rho \le 0 . 8$ , and $0 . 8 < \rho$ . We plot the ratio of these five different groups of languages at the end of each generation. From Figure 15-(a), we can see that the high- $\rho$ language, which is represented by the bold curve, always occupy a small portion. The topological similarity of the dominant languages are $\rho < 0 . 4$ . However, in the resetting-both case, as illustrated in Figure 15-(b), the portion of high- $\rho$ language will increase significantly, which further verifies that the iterated learning can gradually make the high- $\boldsymbol { \rho }$ language dominate in posterior.
|
| 409 |
+
|
| 410 |
+
Table 6: Example of the converged language in none-reset case $\rho = 0 . 2 3$
|
| 411 |
+
|
| 412 |
+
<table><tr><td></td><td>blue</td><td>green</td><td>cyan</td><td>brown</td><td>red</td><td>black</td><td>yellow</td><td>white</td></tr><tr><td>box</td><td>aa</td><td>fh</td><td>af</td><td>hh</td><td>cg</td><td>fc</td><td>ha</td><td>hf</td></tr><tr><td>circle</td><td>da</td><td>df</td><td>hb</td><td>db</td><td>fa</td><td>da</td><td>dh</td><td>fb</td></tr><tr><td>triangle</td><td>gc</td><td>ff</td><td>ge</td><td>gf</td><td>gg</td><td>fg</td><td>ge</td><td>he</td></tr><tr><td>square</td><td>ae</td><td>fb</td><td>be</td><td>bb</td><td>bg</td><td>fb</td><td>gb</td><td>ba</td></tr><tr><td>star</td><td>ad</td><td>fd</td><td>de</td><td>db</td><td>dg</td><td>fd</td><td>ce</td><td>hc</td></tr><tr><td>diamond</td><td>ac</td><td>dd</td><td>dc</td><td>db</td><td>dg</td><td>fd</td><td>dc</td><td>dd</td></tr><tr><td>pentagon</td><td>ad</td><td>fe</td><td>ef</td><td>bd</td><td>eg</td><td>fc</td><td>ee</td><td>ed</td></tr><tr><td>capsule</td><td>aa</td><td>dd</td><td>de</td><td>db</td><td>dg</td><td>gd</td><td>de</td><td>fh</td></tr></table>
|
| 413 |
+
|
| 414 |
+
Table 7: Example of the converged language in resetting-both case $\rho = 0 . 9 3$
|
| 415 |
+
|
| 416 |
+
<table><tr><td></td><td>blue</td><td>green</td><td>cyan</td><td>brown</td><td>red</td><td>black</td><td>yellow</td><td>white</td></tr><tr><td>box</td><td>aa</td><td>ea</td><td>ba</td><td>ga</td><td>da</td><td>ca</td><td>ha</td><td>fa</td></tr><tr><td>circle</td><td>ab</td><td>eb</td><td>bb</td><td>gb</td><td>db</td><td>cb</td><td>hb</td><td>fb</td></tr><tr><td>triangle</td><td>ae</td><td>eb</td><td>be</td><td>ge</td><td>de</td><td>ce</td><td>he</td><td>fe</td></tr><tr><td>square</td><td>af</td><td>ef</td><td>bf</td><td>gf</td><td>df</td><td>cf</td><td>hf</td><td>ff</td></tr><tr><td>star</td><td>ac</td><td>ec</td><td>bc</td><td>gc</td><td>dc</td><td>Cc</td><td>dh</td><td>fc</td></tr><tr><td>diamond</td><td>ad</td><td>ed</td><td>bd</td><td>gd</td><td>dd</td><td>cd</td><td>hd</td><td>fd</td></tr><tr><td>pentagon</td><td>ag</td><td>eg</td><td>bg</td><td>gg</td><td>dg</td><td>cg</td><td>hg</td><td>fg</td></tr><tr><td>capsule</td><td>ah</td><td>eh</td><td>bh</td><td>gh</td><td>hc</td><td>ch</td><td>hh</td><td>fh</td></tr></table>
|
| 417 |
+
|
| 418 |
+
|
| 419 |
+
Figure 11: Distribution of $P ( \rho ( \mathcal { L } ) | D _ { i } , R _ { i } )$ through 80 generations. Values of $\rho$ are divided into ten groups. The distribution of $\rho$ in each generation is smoothed using linear interpolation.
|
| 420 |
+
|
| 421 |
+
|
| 422 |
+
Figure 12: Language evolution of none-reset case in a 3D illustration.
|
| 423 |
+
|
| 424 |
+
|
| 425 |
+
Figure 13: Language evolution of resetting-both case in a 3D illustration.
|
| 426 |
+
|
| 427 |
+
|
| 428 |
+
Figure 14: Distribution of $\rho ( \mathcal { L } )$ at different generations.
|
| 429 |
+
|
| 430 |
+
|
| 431 |
+
Figure 15: Evolution of language with different values of $\rho$ .
|
parse/train/HkePNpVKPB/HkePNpVKPB_content_list.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/HkePNpVKPB/HkePNpVKPB_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/HkePNpVKPB/HkePNpVKPB_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/HkgxasA5Ym/HkgxasA5Ym.md
ADDED
|
@@ -0,0 +1,259 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# RELIABLE UNCERTAINTY ESTIMATES IN NEURAL NETWORKS USING NOISE CONTRASTIVE PRIORS
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Obtaining reliable uncertainty estimates of neural network predictions is a long standing challenge. Bayesian neural networks have been proposed as a solution, but it remains open how to specify their prior. In particular, the common practice of a standard normal prior in weight space imposes only weak regularities, causing the function posterior to possibly generalize in unforeseen ways on inputs outside of the training distribution. We propose noise contrastive priors (NCPs) to obtain reliable uncertainty estimates. The key idea is to train the model to output high uncertainty for data points outside of the training distribution. NCPs do so using an input prior, which adds noise to the inputs of the current mini batch, and an output prior, which is a wide distribution given these inputs. NCPs are compatible with any model that can output uncertainty estimates, are easy to scale, and yield reliable uncertainty estimates throughout training. Empirically, we show that NCPs prevent overfitting outside of the training distribution and result in uncertainty estimates that are useful for active learning. We demonstrate the scalability of our method on the flight delays data set, where we significantly improve upon previously published results.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Many successful applications of neural networks (Krizhevsky et al., 2012; Sutskever et al., 2014; van den Oord et al., 2016) are in restricted settings where predictions are only made for inputs similar to the training distribution. In real-world scenarios, neural networks can face truly novel data points during inference, and in these settings it can be valuable to have good estimates of the model’s uncertainty. For example, in healthcare, reliable uncertainty estimates can prevent overconfident decisions for rare or novel patient conditions (Schulam and Saria, 2015). Similarly, autonomous agents that actively explore their environment can use uncertainty estimates to decide what data points will be most informative.
|
| 12 |
+
|
| 13 |
+
Epistemic uncertainty describes the amount of missing knowledge about the data generating function. Uncertainty can in principle be completely reduced by observing more data points at the right locations and training on them. In contrast, the data generating function may also have inherent randomness, which we call aleatoric noise. This noise can be captured by models outputting a distribution rather than a point prediction. Obtaining more data points allows the noise estimate to move closer to the true value, which is usually different from zero. For active learning, it is crucial to separate the two types of randomness: we want to acquire labels in regions of high uncertainty but low noise (MacKay, 1992a).
|
| 14 |
+
|
| 15 |
+
Bayesian analysis provides a principled approach to modeling uncertainty in neural networks (Denker et al., 1987; MacKay, 1992b). Namely, one places a prior over the network’s weights and biases. This effectively places a distribution over the functions that the network represents, capturing uncertainty about which function best fits the data. Specifying this prior remains an open challenge. Common practice is to use a standard normal prior in weight space, which imposes weak shrinkage regularities analogous to weight decay. It is neither informative about the induced function class nor the data (e.g., it is sensitive to parameterization).
|
| 16 |
+
|
| 17 |
+
|
| 18 |
+
Figure 1: Predictive distributions on a low-dimensional active learning task. The predictive distributions are visualized as mean and two standard deviations shaded. They decompose into epistemic uncertainty $\mid$ and aleatoric noise . Data points are only available within two bands, and are selected using the expected information gain . (a) A deterministic network conflates uncertainty as part of the noise and is overconfident outside of the data distribution. (b) A variational Bayesian neural network with standard normal prior represents uncertainty and noise separately but is overconfident outside of the training distribution. (c) On the OOD classifier model, NCP prevents overconfidence. (d) On the Bayesian neural network, NCP produces smooth uncertainty estimates that generalize well to unseen data points. Models trained with NCP also separate uncertainty and noise well. The experimental setup is described in Section 5.1.
|
| 19 |
+
|
| 20 |
+
This can cause the induced function posterior to generalize in unforeseen ways on out-of-distribution (OOD) inputs, which are inputs outside of the distribution that generated the training data.
|
| 21 |
+
|
| 22 |
+
Motivated by these challenges, we introduce noise contrastive priors (NCPs), which encourage uncertainty outside of the training distribution through a loss in data space. NCPs are compatible with any model that represents functional uncertainty as a random variable, are easy to scale, and yield reliable uncertainty estimates that show significantly improved active learning performance.
|
| 23 |
+
|
| 24 |
+
# 2 NOISE CONTRASTIVE PRIORS
|
| 25 |
+
|
| 26 |
+
Specifying priors is intuitive for small probabilistic models, where each variable typically has a clear interpretation (Blei, 2014). It is less intuitive for neural networks, where the parameters serve more as adaptive basis coefficients in a nonparametric function. For example, neural network models are nonidentifiable due to weight symmetries that yield the same function (Müller and Insua, 1998). This makes it difficult to express informative priors on the weights, such as expressing high uncertainty on unfamiliar examples.
|
| 27 |
+
|
| 28 |
+
Data priors Unlike a prior in weight space, a data prior lets one easily express informative assumptions about input-output relationships. Here, we use the example of a prior over a labeled data set $\{ x , y \}$ , although the prior can also be on $x$ and another variable in the model that represents uncertainty and has a clear interpretation. The prior takes the form $p _ { \mathrm { p r i o r } } ( x , y ) = p _ { \mathrm { p r i o r } } ( x ) \ p _ { \mathrm { p r i o r } } ( y \mid x )$ , where $p _ { \mathrm { p r i o r } } ( x )$ denotes the input prior and $p _ { \mathrm { p r i o r } } ( y \mid x )$ denotes the output prior.
|
| 29 |
+
|
| 30 |
+
To prevent overconfident predictions, a good input prior $p _ { \mathrm { p r i o r } } ( x )$ should include OOD examples so that it acts beyond the training distribution. A good output prior $p _ { \mathrm { p r i o r } } ( y \mid x )$ should be a high-entropy distribution, representing high uncertainty about the model output given OOD inputs.
|
| 31 |
+
|
| 32 |
+
|
| 33 |
+
Figure 2: Graphical representations of the two uncertainty-aware models we consider. Circles denote random variables, squares denote deterministic variables, shading denotes observations during training. (a) The Bayesian neural network captures a belief over parameters for the predictive mean, while the predictive variance is a deterministic function of the input. In practice, we only use weight uncertainty for the mean’s output layer and share earlier layers between the mean and variance. (b) The out-of-distribution classifier model uses a binary auxiliary variable $o$ to determine if a given input is out-of-distribution; given its value, the output is drawn from either a neural network prediction or a wide output prior.
|
| 34 |
+
|
| 35 |
+
Generating OOD inputs Exactly generating OOD data is difficult. A priori, we must uniformly represent the input domain. A posteriori, we must represent the complement of the training distribution. Both distributions are typically uniform over infinite support, making them ill-defined. To estimate OOD inputs, we develop an algorithm inspired by noise contrastive estimation (Gutmann and Hyvärinen, 2010a; Mnih and Kavukcuoglu, 2013), where a complement distribution is approximated using random noise.
|
| 36 |
+
|
| 37 |
+
A hypothesis of our work is that in practice it is enough to encourage high uncertainty output near the boundary of the training distribution, and that this effect will propagate to the entire OOD space. This hypothesis is backed up by previous work (Lee et al., 2017) as well as our experiments (see Figure 1). This means we no longer need to sample arbitrary OOD inputs. It is enough to sample OOD points that lie close to the boundary of the training distribution, and to apply our desired prior at those points.
|
| 38 |
+
|
| 39 |
+
Loss function Noise contrastive priors are data priors that are enforced on both training inputs $x$ and inputs $\tilde { x }$ perturbed by noise. For example, in binary and categorical input domains, we approximate OOD inputs by randomly flipping the features to different classes with a certain probability. For continuous valued inputs $x$ we can use additive Gaussian noise to obtain noised up inputs $\tilde { x } = x + \epsilon$ . This expresses the noise contrastive prior where inputs are distributed according to the convolved distribution,
|
| 40 |
+
|
| 41 |
+
$$
|
| 42 |
+
p _ { \mathrm { p r i o r } } ( \tilde { x } ) = \int _ { x } p _ { \mathrm { t r a i n } } ( x ) \mathrm { N o r m a l } ( \tilde { x } - x \mid \mu _ { x } , \sigma _ { x } ^ { 2 } ) d x \qquad p _ { \mathrm { p r i o r } } ( \tilde { y } \mid \tilde { x } ) = \mathrm { N o r m a l } ( \mu _ { y } , \sigma _ { y } ^ { 2 } ) .
|
| 43 |
+
$$
|
| 44 |
+
|
| 45 |
+
The variances $\sigma _ { x } ^ { 2 }$ and $\sigma _ { y } ^ { 2 }$ are hyperparameters that tune how far from the boundary we sample, and how large we want the output uncertainty to be. We choose $\mu _ { x } = 0$ to apply the prior equally in all directions from the data manifold. The output mean $\mu _ { y }$ determines the default prediction of the model outside of the training distribution, for example $\mu _ { y } = 0$ . We set $\mu _ { y } = y$ which corresponds to data augmentation (Matsuoka, 1992; An, 1996), where a model is trained to recover the true labels from perturbed inputs. This way, NCP makes the model uncertain while still trying to generalize to OOD inputs.
|
| 46 |
+
|
| 47 |
+
For training, we minimize the loss function
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
\begin{array} { r l } & { \mathcal { L } ( \theta ) = \mathrm { \mathrm { ~ E } } _ { p _ { \mathrm { t r a i n } } ( x ) } \left[ D _ { \mathrm { K L } } \big [ p _ { \mathrm { t r a i n } } ( y \mid x ) \ \lVert \ p _ { \mathrm { m o d e l } } ( y \mid x , \theta ) \big ] \right] } \\ & { \quad \quad \quad + \gamma \mathrm { \mathrm { E } } _ { p _ { \mathrm { p r i o r } } ( \tilde { x } ) } \left[ D _ { \mathrm { K L } } \big [ p _ { \mathrm { p r i o r } } ( \tilde { y } \mid \tilde { x } ) \ \lVert \ p _ { \mathrm { m o d e l } } ( \tilde { y } \mid \tilde { x } , \theta ) \big ] \right] . } \end{array}
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
The first term represents typical maximum likelihood, in which one minimizes the KL divergence to the empirical training distribution $p _ { \mathrm { t r a i n } } ( y \mid x )$ over training inputs. The second term is added by our method: it represents the analogous term on a data prior. The hyperparameter $\gamma$ sets the relative trade-off between them.
|
| 54 |
+
|
| 55 |
+
Interpretation as function prior The noise contrastive prior can be interpreted as inducing a function prior. This is formalized through the prior predictive distribution,
|
| 56 |
+
|
| 57 |
+
$$
|
| 58 |
+
p ( \boldsymbol { y } \mid \boldsymbol { x } ) = \int p _ { \mathrm { m o d e l } } ( \boldsymbol { y } \mid \boldsymbol { x } , \boldsymbol { \theta } ) p _ { \mathrm { m o d e l } } ( \boldsymbol { \theta } \mid \tilde { \boldsymbol { x } } , \tilde { \boldsymbol { y } } ) p _ { \mathrm { p r i o r } } ( \tilde { \boldsymbol { x } } , \tilde { \boldsymbol { y } } ) d \boldsymbol { \theta } d \tilde { \boldsymbol { x } } d \tilde { \boldsymbol { y } } .
|
| 59 |
+
$$
|
| 60 |
+
|
| 61 |
+
The distribution marginalizes over network parameters $\theta$ as well as data fantasized from the data prior. The distribution $p ( \theta \mid \tilde { x } , \tilde { y } )$ represents the distribution of model parameters after fitting the prior data. That is, the belief over weights is shaped to make $p ( y \mid x )$ highly variable. This parameter belief causes uncertain predictions outside of the training distribution, which we could not specify in weight space directly.
|
| 62 |
+
|
| 63 |
+
Because network weights are constrained to fit the data prior, the prior acts as “pseudo-data.” This is similar to classical work on conjugate priors: a $\mathrm { B e t a } ( \alpha , \beta )$ prior on the probability of a Bernoulli likelihood implies a Beta posterior, and if the posterior mode is chosen as an optimal parameter setting, then the prior translates to $\alpha - 1$ successes and $\beta - 1$ failures. It is also similar to pseudo-data in sparse Gaussian processes (Quiñonero-Candela and Rasmussen, 2005).
|
| 64 |
+
|
| 65 |
+
Data priors encourage learning parameters that not only capture the training data well but also the prior data. In practice, we can combine NCP with other priors, for example the typical standard normal prior in weight space for Bayesian neural networks, although we did not find this necessary in our experiments.
|
| 66 |
+
|
| 67 |
+
# 3 BAYESIAN NEURAL NETWORKS WITH NCP
|
| 68 |
+
|
| 69 |
+
Noise contrastive priors are applicable to any model that represents uncertainty in a random variable. The NCP can then be added to that random variable to make the model uncertain on OOD inputs. In this section, we apply NCP to a Bayesian neural network (BNN) trained via variational inference. Blundell et al. (2015) introduce such a model under the name Bayes by Backprop (BBB) that uses a standard normal prior in weight space. We extend this model with a NCP on the mean predicted by the neural network.
|
| 70 |
+
|
| 71 |
+
Consider a regression task with data $\{ x , y \}$ that we model as $p ( y \mid x , \theta ) = { \mathrm { N o r m a l } } ( \mu ( x ) , \sigma ^ { 2 } ( x ) )$ with mean and variance predicted by a neural network from the inputs. This model is heteroskedastic, meaning that it can predict a different aleatoric noise amount for every point in the input space. We use a weight prior for only the output layer (Lázaro-Gredilla and Figueiras-Vidal, 2010; Calandra et al., 2014) that predicts the mean, resulting in the model
|
| 72 |
+
|
| 73 |
+
$$
|
| 74 |
+
\theta \sim \operatorname { N o r m a l } ( 0 , 0 . 1 ) \qquad y \sim \operatorname { N o r m a l } ( \mu ( x , \theta ) , \sigma ^ { 2 } ( x ) ) .
|
| 75 |
+
$$
|
| 76 |
+
|
| 77 |
+
We do not model uncertainty about the noise estimate, as this is not required for the approximation for the Gaussian expected information gain (MacKay, 1992a) that we use to acquire labels. Therefore, the distribution of the mean induced by the weight prior, $\begin{array} { r } { \dot { q } ( \mu ( x ) ) = \int \mu ( x , \theta ) q _ { \phi } ( \theta ) \dot { d } \theta } \end{array}$ , represents the model’s epistemic uncertainty. Note that this is different from the predictive distribution, which combines both uncertainty and noise. We place an NCP on the distribution of the mean, resulting in the loss function
|
| 78 |
+
|
| 79 |
+
$$
|
| 80 |
+
\mathcal { L } ( \phi ) = - \mathbb { E } _ { q _ { \phi } ( \theta ) } [ \ln p ( y \mid x , \theta ) ] + \beta D _ { \mathrm { K L } } [ q _ { \phi } ( \theta ) \parallel p ( \theta ) ] + \underbrace { \gamma D _ { \mathrm { K L } } [ \mathrm { N o r m a l } ( \mu _ { \mu } , \sigma _ { \mu } ^ { 2 } ) \parallel q ( \mu ( \tilde { x } ) ) ] } _ { \mathrm { N C P l o s s } } .
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
Here, $\tilde { x }$ are the perturbed inputs and $q _ { \phi } ( \theta )$ forms an approximate posterior over weights.1 Because we only use the weight belief for the linear output layer, we can compute the KL-divergence of the NCP loss analytically. In other models, it could be estimated using samples.
|
| 84 |
+
|
| 85 |
+
The loss function applies weight regularization in order for network weights to regress to a standard normal prior; like other regularization techniques, this assists in improving the network’s generalization in-distribution.
|
| 86 |
+
|
| 87 |
+
The NCP loss encourages the network’s generalization OOD by matching the mean distribution to the output prior. Minimizing the KL divergence to a wide output prior results in high uncertainty on OOD inputs, so the model will explore these data points during active learning.
|
| 88 |
+
|
| 89 |
+
In practice, we find that NCP is sufficient as a prior for the BNN and set $\beta = 0$ . The appendix (Appendix B includes an alternative interpretation explaining why NCP might be sufficient, which represents the weight space KL-divergence in data space after a change of variables.
|
| 90 |
+
|
| 91 |
+
# 4 RELATED WORK
|
| 92 |
+
|
| 93 |
+
Priors for neural networks Classic work has investigated entropic priors (Buntine and Weigend, 1991) and hierarchical priors (MacKay, 1992b; Neal, 2012; Lampinen and Vehtari, 2001). More recently, Depeweg et al. (2018) introduce networks with latent variables in order to disentangle forms of uncertainty, and FlamShepherd et al. (2017) propose general-purpose weight priors based on approximating Gaussian processes. Other works have analyzed priors for compression and model selection (Ghosh and Doshi-Velez, 2017; Louizos et al., 2017). Instead of a prior in weight space (or latent inputs as in Depeweg et al. (2018)), NCPs take the functional view by imposing explicit regularities in terms of the network’s inputs and outputs. Malinin and Gales (2018) propose prior networks to avoid an explicit belief over parameters for classification tasks.
|
| 94 |
+
|
| 95 |
+
Input and output regularization There is classic work on adding noise to inputs for improved generalization (Matsuoka, 1992; An, 1996; Bishop, 1995). For example, denoising autoencoders (Vincent et al., 2008) encourage reconstructions given noisy encodings. Output regularization is also a classic idea from the maximum entropy principle (Jaynes, 1957), where it has motivated label smoothing (Szegedy et al., 2016) and entropy penalties (Pereyra et al., 2017). Also related is virtual adversarial training (Miyato et al., 2015), which includes examples that are close to the current input but cause a maximal change in the model output, and mixup (Zhang et al., 2018), which includes examples under the vicinity of training data. These methods are orthogonal to NCPs: they aim to improve generalization from finite data within the training distribution (interpolation), while we aim to improve uncertainty estimates outside of the training distribution (extrapolation).
|
| 96 |
+
|
| 97 |
+
Classifying out-of-distribution inputs A simple approach for neural network uncertainty is to classify whether data points belong to the data distribution, or are OOD (Hendrycks and Gimpel, 2017). This is core to noise contrastive estimation (Gutmann and Hyvärinen, 2010b), a training method for intractable probabilistic models. More recently, Lee et al. (2017) introduce a GAN to generate OOD samples, and Liang et al. (2018) add perturbations to the input, applying an “OOD detector” to improve softmax scores on OOD samples by scaling the temperature. Extending these directions of research, we connect to Bayesian principles and focus on uncertainty estimates that are useful for active data acquisition.
|
| 98 |
+
|
| 99 |
+
# 5 EXPERIMENTS
|
| 100 |
+
|
| 101 |
+
To demonstrate their usefulness, we evaluate NCPs on various tasks where uncertainty estimates are desired. Our focus is on active learning for regression tasks, where only few targets are visible in the beginning, and additional targets are selected regularly based on an acquisition function. We use two data sets: a toy example and a large flights data set. We also evaluate how sensitive our method is to the choice of input noise. Finally, we show that NCP scales to large data sets by training on the full flights data set in a passive learning setting. Our implementation uses TensorFlow Probability (Dillon et al., 2017; Tran et al., 2016) and is open-sourced at https://<hidden-for-review>.
|
| 102 |
+
|
| 103 |
+
We compare four neural network models, all using leaky ReLU activations (Maas et al., 2013) and trained using Adam (Kingma and Ba, 2014). The four models are:
|
| 104 |
+
|
| 105 |
+
|
| 106 |
+
Figure 3: Active learning on the 1-dimensional regression problem, mean and standard deviation over 20 seeds. The test root mean squared error (RMSE) and negative log predictive density (NLPD) of the models trained with NCP decreases during the active learning run, while the baseline models select less informative data and overfit. The deterministic network is barely visible in the plots as it overfits quickly. Figure 1 shows the predictive distributions of the models.
|
| 107 |
+
|
| 108 |
+
• Deterministic neural network (Det) A neural network that predicts the mean and variance of a normal distribution. The name stands for deterministic, as there is no weight uncertainty.
|
| 109 |
+
• Bayes by Backprop (BBB) A Bayesian neural network trained via gradient-based variational inference with a standard normal prior in weight space (Blundell et al., 2015; Kucukelbir et al., 2017). We use the same model as in Section 3 but without the NCP loss term.
|
| 110 |
+
• Bayes by Backprop with noise contrastive prior $\mathbf { ( B B B + N C P ) }$ ) Bayes by Backprop with NCP on the predicted mean distribution as described in Section 3.
|
| 111 |
+
• Out-of-distribution classifier with noise contrastive prior $\mathbf { ( O C D + N C P ) }$ ) An uncertainty classifier model described in Appendix A. It is a deterministic neural network combined with NCP which we use as a baseline alternative to Bayes by Backprop with NCP.
|
| 112 |
+
|
| 113 |
+
For active learning, we select new data points $\{ x , y \}$ for which $x$ maximizes the expected information gain $\operatorname { E } _ { q ( y | x ) } [ D _ { \mathrm { K L } } [ q ( \theta \mid x , y ) \parallel _ { . } q ( \theta ) ] ]$ under the model ${ \\begin{array} { r } { { \dot { q } } ( y \mid x ) = \int p ( y \mid x , \theta ) q ( \theta ) d \theta } \end{array} }$ . Intuitively, this objective function is higher where the model has high epistemic uncertainty and predicts low aleatoric noise.
|
| 114 |
+
|
| 115 |
+
We use an approximation from MacKay (1992a) for Gaussian posterior predictive distributions. Moreover, we place a softmax distribution on the information gain for all available data points and acquire labels by sampling with a temperature of $\tau = 0 . 5$ to get diversity when selecting batches of labels,
|
| 116 |
+
|
| 117 |
+
$$
|
| 118 |
+
\{ x _ { \mathrm { n e w } } \} \sim p _ { \mathrm { n e w } } ( x ) = \frac { 1 } { Z } \exp \Big ( \frac { 1 } { 2 \tau } \ln \big ( 1 + \mathrm { V a r } [ q ( \mu ( x ) ) ] / \sigma ^ { 2 } ( x ) \big ) \Big ) = \frac { 1 } { Z } \big ( 1 + \mathrm { V a r } [ q ( \mu ( x ) ) ] / \sigma ^ { 2 } ( x ) \big ) ,
|
| 119 |
+
$$
|
| 120 |
+
|
| 121 |
+
where $\sigma ^ { 2 } ( x )$ is the estimated aleatoric noise and $q ( \mu ( x ) )$ is the epistemic uncertainty projected into output space. Since our Bayesian neural networks only use a weight belief for the output layer, $\operatorname { V a r } [ q ( \mu ( x ) ) ]$ is Gaussian and can be computed in closed form. In general, it the epistemic part of the predictive variance would be estimated by sampling. In the classifier model, we use the OOD probability $p ( o = 1 | x )$ for this. For the deterministic neural network, we use $\mathrm { V a r } [ p ( y \mid x ) ]$ as proxy since it does not output an estimate of epistemic uncertainty.
|
| 122 |
+
|
| 123 |
+
# 5.1 LOW-DIMENSIONAL ACTIVE LEARNING
|
| 124 |
+
|
| 125 |
+
For visualization purposes, we start with experiments on a 1-dimensional regression task that consists of a sine function with a small slope and increasing variance for higher inputs. Training data can be acquired within two bands, and the model is evaluated on all data points that are not visible to the model. This structured split between training and testing data causes a distributional shift at test time, requiring successful models to have reliable uncertainty estimates to avoid mispredictions for OOD inputs.
|
| 126 |
+
|
| 127 |
+
|
| 128 |
+
Figure 4: Active learning on the flights data set. The models trained with NCP achieve significantly lower negative log predictive density (NLPD) on the test set, and Bayes by Backprop with NCP achieves the lowest root mean squared error (RMSE). The test NLPD for the baseline models diverges as they overfit to the visible data points. Plots show mean and std over 10 runs.
|
| 129 |
+
|
| 130 |
+
For this experiment, we use two layers of 200 hidden units, a batch size of 10, and a learning rate of $3 \times 1 0 ^ { - 4 }$ for all models. NCP models use noise $\epsilon \sim \mathrm { N o r m a l } ( 0 , 0 . 5 )$ . We start with 10 randomly selected initial targets, and select 1 additional target every 1000 epochs. Figure 3 shows the root mean squared error (RMSE) and negative log predictive density (NLPD) throughout learning. The two baseline models severely overfit to the training distribution early on when only few data points are visible. Models with NCP outperform BBB, which in turn outperforms Det. Figure 1 visualizes the models’ predictive distributions at the end of training, showing that NCP prevents overconfident generalization.
|
| 131 |
+
|
| 132 |
+
# 5.2 ACTIVE LEARNING ON FLIGHT DELAYS
|
| 133 |
+
|
| 134 |
+
We consider the flight delay data set (Hensman et al., 2013; Deisenroth and Ng, 2015; Lakshminarayanan et al., 2016), a large scale regression benchmark with several published results. The data set has 8 input variables describing a flight, and the target is the delay of the flight in minutes. There are 700K training examples and 100K test examples. The test set has a subtle distributional shift, since the 100K data points temporally follow after the training data.
|
| 135 |
+
|
| 136 |
+
We use two layers with 50 units each, a batch size of 10, and a learning rate of $1 0 ^ { - 4 }$ . For NCP models, $\epsilon \sim \mathrm { N o r m a l } ( 0 , 0 . 1 )$ . Starting from 10 labels, the models select a batch of 10 additional labels every 50 epochs. The 700K data points of the training data set are available for acquisition, and we evaluate performance on the typical test split. Figure 4 shows the performance for the visible data points and the test set respectively. We note that BBB and $\mathrm { B B B + N C P }$ show similar NLPD on the visible data points, but the NCP models generalize better to unseen data. Moreover, the Bayesian neural network with NCP achieves lower RMSE than the one without and the classifier based model achieves lower RMSE than the deterministic neural network. All uncertainty-based models outperform the deterministic neural network.
|
| 137 |
+
|
| 138 |
+
# 5.3 ROBUSTNESS TO NOISE PATTERNS
|
| 139 |
+
|
| 140 |
+
The choice of input noise might seem like a critical hyper parameter for NCP. In this experiment, we find that our method is robust to the choice of input noise. The experimental setup is the same as for the active learning experiment described in Section 5.2, but with uniform or normal input noise with different variance $( \sigma _ { x } ^ { 2 } \in \bar { \{ 0 . 1 , 0 . 2 , \cdot \cdot \cdot , 1 . 0 \} } )$ . For uniform input noise, this means noise is drawn from the interval $[ - 2 \sigma _ { x } , 2 \sigma _ { x } ]$ We observe that $\mathbf { B B B + N C P }$ is robust to the size of the input noise. NCP consistently improves RMSE for the tested noise sizes and yields the best NLPD for all noise sizes below 0.6. For our ODC baseline, we observe an intuitive trade-off: smaller input noise increases the regularization strength, leading to better NLPD but reduced RMSE. Robustness to the choice of input noise is further supported by the analogous experiment on toy data set, where above a small threshold $( \mathbf { \bar { B } B B + N C P } \ \sigma _ { x } ^ { 2 } \geq 0 . 3 $ and ${ \mathrm { O D C + N C P } }$ $\sigma _ { x } ^ { 2 } \ge \bar { 0 . 1 } \bar { , }$ ), NCP consistently performs well (Figure 6).
|
| 141 |
+
|
| 142 |
+
|
| 143 |
+
Figure 5: Robustness to different noise patterns. Plots show the final test performance on the flights active learning task (mean and stddev over 5 seeds). Lower is better. NCP is robust to the choice of input noise and improves over the baselines in all settings (compare Figure 4).
|
| 144 |
+
|
| 145 |
+
# 5.4 LARGE SCALE REGRESSION OF FLIGHT DELAYS
|
| 146 |
+
|
| 147 |
+
In addition to the active learning experiments, we perform a passive learning run on all 700K data points of the flights data set to explore the scalability of NCP. We use networks of 3 layers with 1000 units and a learning rate of $1 0 ^ { - 4 }$ . Table 1 compares the performance of our models to previously published results. We significantly improve state of the art performance on this data set.
|
| 148 |
+
|
| 149 |
+
# 6 DISCUSSION
|
| 150 |
+
|
| 151 |
+
We develop noise contrastive priors (NCPs), a prior for neural networks in data space. NCPs encourage network weights that not only explain the training data but also capture high uncertainty on OOD inputs. We show that NCPs offer strong improvements over baselines and scale to large regression tasks.
|
| 152 |
+
|
| 153 |
+
We focused on active learning for regression tasks, where uncertainty is crucial for determining which data points to select next. In future work it would be interesting to apply NCPs to alternative settings where uncertainty is important, such as image classification and learning with sparse or missing data. In addition, NCPs are only one form of a data prior, designed to encourage uncertainty on OOD inputs. Priors in data space can easily capture other properties such as periodicity or spatial invariance, and they may provide a scalable alternative to Gaussian process priors.
|
| 154 |
+
|
| 155 |
+
Table 1: Performance on all 700K data points of the flights data set. While uncertainty estimates are not necessary when a large data set that is similar to the test data set is available, it shows that our method scales easily to large data sets.
|
| 156 |
+
|
| 157 |
+
<table><tr><td>Model</td><td>NLPD</td><td>RMSE</td></tr><tr><td>gPoE (Deisenroth & Ng 2015)</td><td>8.1</td><td></td></tr><tr><td>SAVIGP (Bonilla et al. 2016)</td><td>5.02</td><td></td></tr><tr><td>SVI GP (Hensman et al. 2013)</td><td>一</td><td>32.60</td></tr><tr><td>HGP (Ng & Deisenroth 2014)</td><td>一</td><td>27.45</td></tr><tr><td>MF F (Lakshminarayanan et al. 2016)</td><td>4.89</td><td>26.57</td></tr><tr><td>BBB</td><td>4.38</td><td>24.59</td></tr><tr><td>BBB+NCP</td><td>4.38</td><td>24.71</td></tr><tr><td>ODC+NCP</td><td>4.38</td><td>24.68</td></tr></table>
|
| 158 |
+
|
| 159 |
+
# REFERENCES
|
| 160 |
+
|
| 161 |
+
G. An. The effects of adding noise during backpropagation training on a generalization performance. Neural computation, 8(3):643–674, 1996.
|
| 162 |
+
C. M. Bishop. Training with noise is equivalent to tikhonov regularization. Neural computation, 7(1):108–116, 1995.
|
| 163 |
+
D. M. Blei. Build, compute, critique, repeat: Data analysis with latent variable models. Annual Review of Statistics and Its Application, 1:203–232, 2014.
|
| 164 |
+
C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra. Weight uncertainty in neural networks. arXiv preprint arXiv:1505.05424, 2015.
|
| 165 |
+
W. L. Buntine and A. S. Weigend. Bayesian back-propagation. Complex Systems, 5(6):603–643, 1991.
|
| 166 |
+
R. Calandra, J. Peters, C. E. Rasmussen, and M. P. Deisenroth. Manifold gaussian processes for regression. arXiv preprint arXiv:1402.5876, 2014.
|
| 167 |
+
S. Dasgupta. Analysis of a greedy active learning strategy. In Neural Information Processing Systems (NIPS), 2004.
|
| 168 |
+
M. P. Deisenroth and J. W. Ng. Distributed gaussian processes. arXiv preprint arXiv:1502.02843, 2015.
|
| 169 |
+
J. Denker, D. Schwartz, B. Wittner, S. Solla, R. Howard, L. Jackel, and J. Hopfield. Large automatic learning, rule extraction, and generalization. Complex Systems, 1(5):877–922, 1987.
|
| 170 |
+
S. Depeweg, J. M. Hernández-Lobato, F. Doshi-Velez, and S. Udluft. Uncertainty decomposition in bayesian neural networks with latent variables. In International Conference on Machine Learning, 2018.
|
| 171 |
+
J. V. Dillon, I. Langmore, D. Tran, E. Brevdo, S. Vasudevan, D. Moore, B. Patton, A. Alemi, M. Hoffman, and R. A. Saurous. Tensorflow distributions. arXiv preprint arXiv:1711.10604, 2017.
|
| 172 |
+
D. Flam-Shepherd, J. Requeima, and D. Duvenaud. Mapping gaussian process priors to bayesian neural networks. In NIPS Workshop, 2017.
|
| 173 |
+
Y. Freund, H. S. Seung, E. Shamir, and N. Tishby. Selective sampling using the query by committee algorithm. Machine learning, 28(2-3):133–168, 1997.
|
| 174 |
+
Y. Gal, R. Islam, and Z. Ghahramani. Deep bayesian active learning with image data. arXiv preprint arXiv:1703.02910, 2017.
|
| 175 |
+
S. Ghosh and F. Doshi-Velez. Model selection in bayesian neural networks via horseshoe priors. arXiv preprint arXiv:1705.10388, 2017.
|
| 176 |
+
M. Gutmann and A. Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 297–304, 2010a.
|
| 177 |
+
M. Gutmann and A. Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 297–304, 2010b.
|
| 178 |
+
D. Hendrycks and K. Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. In International Conference on Learning Representations, 2017.
|
| 179 |
+
J. Hensman, N. Fusi, and N. D. Lawrence. Gaussian processes for big data. arXiv preprint arXiv:1309.6835, 2013.
|
| 180 |
+
I. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. International Conference on Learning Representations, 2016.
|
| 181 |
+
N. Houlsby, F. Huszár, Z. Ghahramani, and M. Lengyel. Bayesian active learning for classification and preference learning. arXiv preprint arXiv:1112.5745, 2011.
|
| 182 |
+
E. T. Jaynes. Information theory and statistical mechanics. Physical review, 106(4):620, 1957.
|
| 183 |
+
A. J. Joshi, F. Porikli, and N. Papanikolopoulos. Multi-class active learning for image classification. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 2372–2379. IEEE, 2009.
|
| 184 |
+
A. Kapoor, K. Grauman, R. Urtasun, and T. Darrell. Active learning with gaussian processes for object categorization. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, pages 1–8. IEEE, 2007.
|
| 185 |
+
D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
|
| 186 |
+
A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Neural Information Processing Systems, pages 1097–1105, 2012.
|
| 187 |
+
A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei. Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1):430–474, 2017.
|
| 188 |
+
B. Lakshminarayanan, D. M. Roy, and Y. W. Teh. Mondrian forests for large-scale regression when uncertainty matters. In Artificial Intelligence and Statistics, pages 1478–1487, 2016.
|
| 189 |
+
J. Lampinen and A. Vehtari. Bayesian approach for neural networks—review and case studies. Neural networks, 14(3):257–274, 2001.
|
| 190 |
+
M. Lázaro-Gredilla and A. R. Figueiras-Vidal. Marginalized neural network mixtures for large-scale regression. IEEE transactions on neural networks, 21(8):1345–1351, 2010.
|
| 191 |
+
K. Lee, H. Lee, K. Lee, and J. Shin. Training confidence-calibrated classifiers for detecting out-of-distribution samples. arXiv preprint arXiv:1711.09325, 2017.
|
| 192 |
+
D. D. Lewis and W. A. Gale. A sequential algorithm for training text classifiers. In Proceedings of the 17th annual international ACM SIGIR conference on Research and development in information retrieval, pages 3–12. Springer-Verlag New York, Inc., 1994.
|
| 193 |
+
X. Li and Y. Guo. Adaptive active learning for image classification. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pages 859–866. IEEE, 2013.
|
| 194 |
+
S. Liang, Y. Li, and R. Srikant. Principled detection of out-of-distribution examples in neural networks. In International Conference on Learning Representations, 2018.
|
| 195 |
+
C. Louizos, K. Ullrich, and M. Welling. Bayesian compression for deep learning. In Neural Information Processing Systems, pages 3290–3300, 2017.
|
| 196 |
+
A. L. Maas, A. Y. Hannun, and A. Y. Ng. Rectifier nonlinearities improve neural network acoustic models. In International Conference on Machine Learning (ICML), 2013.
|
| 197 |
+
D. J. MacKay. Information-based objective functions for active data selection. Neural computation, 4(4): 590–604, 1992a.
|
| 198 |
+
D. J. MacKay. A practical bayesian framework for backpropagation networks. Neural computation, 4(3): 448–472, 1992b.
|
| 199 |
+
A. Malinin and M. Gales. Predictive uncertainty estimation via prior networks. arXiv preprint arXiv:1802.10501, 2018.
|
| 200 |
+
K. Matsuoka. Noise injection into inputs in back-propagation learning. IEEE Transactions on Systems, Man, and Cybernetics, 22(3):436–440, 1992.
|
| 201 |
+
A. K. McCallumzy and K. Nigamy. Employing em and pool-based active learning for text classification. In Proc. International Conference on Machine Learning (ICML), pages 359–367. Citeseer, 1998.
|
| 202 |
+
T. Miyato, S. Maeda, M. Koyama, K. Nakae, and S. Ishii. Distributional smoothing with virtual adversarial training. arXiv preprint arXiv:1507.00677, 2015.
|
| 203 |
+
A. Mnih and K. Kavukcuoglu. Learning word embeddings efficiently with noise-contrastive estimation. In Advances in neural information processing systems, pages 2265–2273, 2013.
|
| 204 |
+
P. Müller and D. R. Insua. Issues in bayesian analysis of neural network models. Neural Computation, 10(3): 749–770, 1998.
|
| 205 |
+
R. M. Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012.
|
| 206 |
+
G. Pereyra, G. Tucker, J. Chorowski, Ł. Kaiser, and G. Hinton. Regularizing neural networks by penalizing confident output distributions. arXiv preprint arXiv:1701.06548, 2017.
|
| 207 |
+
J. Quiñonero-Candela and C. E. Rasmussen. A unifying view of sparse approximate gaussian process regression. Journal of Machine Learning Research, 6(Dec):1939–1959, 2005.
|
| 208 |
+
H. A. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IIEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 1998.
|
| 209 |
+
N. Roy and A. McCallum. Toward optimal active learning through monte carlo estimation of error reduction. ICML, Williamstown, pages 441–448, 2001.
|
| 210 |
+
P. Schulam and S. Saria. A framework for individualizing predictions of disease trajectories by exploiting multi-resolution structure. In Advances in Neural Information Processing Systems, pages 748–756, 2015.
|
| 211 |
+
B. Settles. Active learning literature survey. Computer Sciences Technical Report 1648, University of Wisconsin–Madison, 2009.
|
| 212 |
+
H. S. Seung, M. Opper, and H. Sompolinsky. Query by committee. In Proceedings of the fifth annual workshop on Computational learning theory, pages 287–294. ACM, 1992.
|
| 213 |
+
K.-K. Sung. Learning and example selection for object and pattern detection. MIT A.I. Memo No. 1521, 1994.
|
| 214 |
+
I. Sutskever, O. Vinyals, and Q. V. Le. Sequence to sequence learning with neural networks. In Neural Information Processing Systems, 2014.
|
| 215 |
+
C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2818–2826, 2016.
|
| 216 |
+
S. Tong and D. Koller. Support vector machine active learning with applications to text classification. Journal of machine learning research, 2(Nov):45–66, 2001.
|
| 217 |
+
D. Tran, A. Kucukelbir, A. B. Dieng, M. Rudolph, D. Liang, and D. M. Blei. Edward: A library for probabilistic modeling, inference, and criticism. arXiv preprint arXiv:1610.09787, 2016.
|
| 218 |
+
A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.
|
| 219 |
+
P. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pages 1096–1103. ACM, 2008.
|
| 220 |
+
K. Wang, D. Zhang, Y. Li, R. Zhang, and L. Lin. Cost-effective active learning for deep image classification. IEEE Transactions on Circuits and Systems for Video Technology, 27(12):2591–2600, 2017.
|
| 221 |
+
H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018.
|
| 222 |
+
|
| 223 |
+
We showed how to apply NCP to a Bayesian neural network model that captures function uncertainty in a belief over parameters. An alternative approach to capture uncertainty is to make explicit predictions about whether an input is OOD. There is no belief over weights in this model. Figure 2b shows such a mixture model via a binary variable $o$ ,
|
| 224 |
+
|
| 225 |
+
$$
|
| 226 |
+
\begin{array} { r l } & { o \sim \mathrm { B e r n o u l l i } ( \pi ( x , \theta ) ) } \\ & { y \sim \left\{ \begin{array} { l l } { \mathrm { N o r m a l } ( \mu ( x , \theta ) , \sigma ^ { 2 } ( x , \theta ) ) } & { \mathrm { i f } o = 0 } \\ { \mathrm { N o r m a l } ( \mu _ { y } , \sigma _ { y } ^ { 2 } ) } & { \mathrm { i f } o = 1 , } \end{array} \right. } \end{array}
|
| 227 |
+
$$
|
| 228 |
+
|
| 229 |
+
where $p ( o = 1 \mid x )$ is the OOD probability of $x$ . If $o = 0$ (“in distribution”), the model outputs the neural network prediction. Otherwise, if $o = 1$ (“out of distribution”), the model uses a fixed output prior. The neural network weights $\theta$ are estimated using a point estimate, so we do not maintain a belief distribution over them.
|
| 230 |
+
|
| 231 |
+
The classifier prediction $p ( o \mid x , \theta )$ captures uncertainty in this model. We apply the NCP $p ( o \mid \tilde { x } , \theta ) =$ $\delta ( o = 1 | \tilde { x } , \theta )$ to this variable, which assumes noised-up inputs to be OOD. During training on the data set, $\{ x , y \}$ and $o = 0$ are observed, as training data are in-distribution by definition. Following Equation 2, the loss function is
|
| 232 |
+
|
| 233 |
+
$$
|
| 234 |
+
\begin{array} { r l } & { \mathcal { L } ( \theta ) = D _ { \mathrm { K L } } [ p _ { \mathrm { f r a i n } } ( y \mid x ) \mid | p _ { \mathrm { m o d e l } } ( y \mid x , o = 0 , \theta ) ] + D _ { \mathrm { K L } } [ p _ { \mathrm { p r i o r } } ( \tilde { o } \mid \tilde { x } ) \mid | p _ { \mathrm { m o d e l } } ( \tilde { o } \mid \tilde { x } , \theta ) ] } \\ & { \quad \quad = - \ln p ( y , o = 0 \mid x , \theta ) - \ln p ( y , o = 1 \mid \tilde { x } , \theta ) } \\ & { \quad \quad = - \ln \mathrm { N o r m a l } ( y \mid \mu ( x , \theta ) , \sigma ^ { 2 } ( x , \theta ) ) - \ln \mathrm { B e r n o u l i } ( 0 \mid \pi ( x , \theta ) ) \frac { - \ln \mathrm { B e r n o u l i } ( 1 \mid \pi ( \tilde { x } , \theta ) ) } { \mathrm { N C P l o s s } } . } \end{array}
|
| 235 |
+
$$
|
| 236 |
+
|
| 237 |
+
Analogously to the Bayesian neural network model in Section 3, we can either set $\mu _ { y } , \sigma _ { y } ^ { 2 }$ manually or use the neural network prediction for potentially improved generalization. In our experiments, we implement the OOD classifier model using a single neural network with two output layers that parameterize the Gaussian distribution and the binary distribution.
|
| 238 |
+
|
| 239 |
+
# B BNN WITH NCP USING REVERSE KL
|
| 240 |
+
|
| 241 |
+
In Section 3, we derived the Bayes by Backprop model with NCP by adding a forward KL-divergence from the mean prior to the model mean to the loss. An alternative derivation uses the fact that the KL-divergence is invariant to parameterization to replace the reverse KL-divergence in weight space by a KL-divergence in output space,
|
| 242 |
+
|
| 243 |
+
$$
|
| 244 |
+
\begin{array} { r l } & { \mathrm { E } _ { p ( x , y ) } \big [ \ln p ( y \mid x ) \big ] = \mathrm { E } _ { p ( x , y ) } \Big [ \ln \displaystyle \int p ( y \mid x , \theta ) p ( \theta ) \frac { q ( \theta ) } { q ( \theta ) } d \theta \Big ] } \\ & { \qquad \quad \ge \mathrm { E } _ { p ( x , y ) } \Big [ \int q ( \theta ) \ln p ( y \mid x , \theta ) \frac { p ( \theta ) } { q ( \theta ) } d \theta \Big ] } \\ & { \qquad = \mathrm { E } _ { p ( x , y ) } \big [ \mathrm { E } _ { q ( \theta ) } [ \ln p ( y \mid x , \theta ) ] - D _ { \mathrm { K L } } [ q ( \theta ) \mid p ( \theta ) ] \big ] } \\ & { \qquad = \mathrm { E } _ { p ( x , y ) } \big [ \mathrm { E } _ { q ( \theta ) } [ \ln p ( y \mid x , \theta ) ] - \mathrm { E } _ { p ( \tilde { x } \mid x ) } [ D _ { \mathrm { K L } } [ q ( \theta ) \mid \mid p ( \theta ) ] ] \big ] } \\ & { \qquad \approx \mathrm { E } _ { p ( x , y ) } \big [ \mathrm { E } _ { q ( \theta ) } [ \ln p ( y \mid x , \theta ) ] - \mathrm { E } _ { p ( \tilde { x } \mid x ) } [ D _ { \mathrm { K L } } [ q ( \mu ( \tilde { x } ) ) \mid \mid p ( \mu ( \tilde { x } ) \mid x ) ] ] \big ] , } \end{array}
|
| 245 |
+
$$
|
| 246 |
+
|
| 247 |
+
where $\begin{array} { r } { p ( \mu ( \tilde { x } ) ) = \int \mu ( \tilde { x } , \theta ) p ( \theta ) d \theta } \end{array}$ and $\begin{array} { r } { q ( \mu ( \tilde { x } ) ) = \int \mu ( \tilde { x } , \theta ) q ( \theta ) d \theta } \end{array}$ are the distributions of the predicted mean induces by the weight beliefs. As a result, instead of specifying a prior in weight space, we can specify a prior in output space.
|
| 248 |
+
|
| 249 |
+
Above, we reparameteterized the $\mathrm { K L }$ in weight space as a KL in output space; by the change of variables, this is equivalent if the mapping $\mu ( \cdot , \theta )$ is continuous and 1-1 with respect to $\theta$ . This assumption does not hold for neural nets as multiple parameter vectors can lead to the same predictive distribution, thus the approximation above. A compact reparameterization of the neural network (equivalence class of parameteters) would make this an equality.
|
| 250 |
+
|
| 251 |
+
|
| 252 |
+
C ROBUSTNESS EXPERIMENT ON TOY DATASET
|
| 253 |
+
Figure 6: Robustness to different noise patterns. Plots show the final test performance on the low-dimensional active learning task (mean and stddev over 5 seeds). Lower is better. The baseline performances are RMSE: BBB $( 0 . 7 5 \pm 0 . 3 1 )$ , Det $( 1 . 4 6 \pm 0 . 6 4 )$ and NLPD: BBB $( 1 0 . 2 9 \pm 8 . 0 5 )$ , Det $( 1 . 3 \times 1 0 ^ { 8 } \pm 1 . 7 \times 1 0 ^ { 8 } )$ ). NCP works with both Gaussian and uniform input noise $\epsilon$ and is robust to $\sigma _ { x } ^ { 2 }$ .
|
| 254 |
+
|
| 255 |
+
# D RELATED ACTIVE LEARNING WORK
|
| 256 |
+
|
| 257 |
+
Active learning is often employed in domains where data is cheap but labeling is expensive, and is motivated by the idea that not all data points are equally valuable when it comes to learning (Settles, 2009; Dasgupta, 2004). Active learning techniques can be coarsely grouped into three categories. Ensemble methods (Seung et al., 1992; McCallumzy and Nigamy, 1998; Freund et al., 1997) generate queries that have the greatest disagreement between a set of classifiers. Error reduction approaches incorporate the select data based on the predicted reduction in classifier error based on information (MacKay, 1992a), Monte Carlo estimation (Roy and McCallum, 2001), or hard-negative example mining (Sung, 1994; Rowley et al., 1998).
|
| 258 |
+
|
| 259 |
+
Uncertainty-based techniques select samples for which the classifier is most uncertain. Approaches include maximum entropy (Joshi et al., 2009), distance from the decision boundary (Tong and Koller, 2001), pseudo labelling high confidence examples (Wang et al., 2017), and mixtures of information density and uncertainty measures (Li and Guo, 2013). Within this category, the area most related to our work are Bayesian methods. Kapoor et al. (2007) estimate expected improvement using a Gaussian process. Other approaches use classifier confidence (Lewis and Gale, 1994), predicted expected error (Roy and McCallum, 2001), or model disagreement (Houlsby et al., 2011). Recently, Gal et al. (2017) applied a convolutional neural network with dropout uncertainty to images.
|
parse/train/HkgxasA5Ym/HkgxasA5Ym_content_list.json
ADDED
|
@@ -0,0 +1,1055 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "RELIABLE UNCERTAINTY ESTIMATES IN NEURAL NETWORKS USING NOISE CONTRASTIVE PRIORS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
148,
|
| 8 |
+
114,
|
| 9 |
+
743,
|
| 10 |
+
162
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
156,
|
| 19 |
+
185,
|
| 20 |
+
372,
|
| 21 |
+
215
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
+
253,
|
| 32 |
+
544,
|
| 33 |
+
268
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "Obtaining reliable uncertainty estimates of neural network predictions is a long standing challenge. Bayesian neural networks have been proposed as a solution, but it remains open how to specify their prior. In particular, the common practice of a standard normal prior in weight space imposes only weak regularities, causing the function posterior to possibly generalize in unforeseen ways on inputs outside of the training distribution. We propose noise contrastive priors (NCPs) to obtain reliable uncertainty estimates. The key idea is to train the model to output high uncertainty for data points outside of the training distribution. NCPs do so using an input prior, which adds noise to the inputs of the current mini batch, and an output prior, which is a wide distribution given these inputs. NCPs are compatible with any model that can output uncertainty estimates, are easy to scale, and yield reliable uncertainty estimates throughout training. Empirically, we show that NCPs prevent overfitting outside of the training distribution and result in uncertainty estimates that are useful for active learning. We demonstrate the scalability of our method on the flight delays data set, where we significantly improve upon previously published results. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
207,
|
| 42 |
+
284,
|
| 43 |
+
792,
|
| 44 |
+
478
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
150,
|
| 54 |
+
502,
|
| 55 |
+
310,
|
| 56 |
+
518
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Many successful applications of neural networks (Krizhevsky et al., 2012; Sutskever et al., 2014; van den Oord et al., 2016) are in restricted settings where predictions are only made for inputs similar to the training distribution. In real-world scenarios, neural networks can face truly novel data points during inference, and in these settings it can be valuable to have good estimates of the model’s uncertainty. For example, in healthcare, reliable uncertainty estimates can prevent overconfident decisions for rare or novel patient conditions (Schulam and Saria, 2015). Similarly, autonomous agents that actively explore their environment can use uncertainty estimates to decide what data points will be most informative. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
148,
|
| 65 |
+
532,
|
| 66 |
+
851,
|
| 67 |
+
630
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "Epistemic uncertainty describes the amount of missing knowledge about the data generating function. Uncertainty can in principle be completely reduced by observing more data points at the right locations and training on them. In contrast, the data generating function may also have inherent randomness, which we call aleatoric noise. This noise can be captured by models outputting a distribution rather than a point prediction. Obtaining more data points allows the noise estimate to move closer to the true value, which is usually different from zero. For active learning, it is crucial to separate the two types of randomness: we want to acquire labels in regions of high uncertainty but low noise (MacKay, 1992a). ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
148,
|
| 76 |
+
636,
|
| 77 |
+
851,
|
| 78 |
+
733
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "Bayesian analysis provides a principled approach to modeling uncertainty in neural networks (Denker et al., 1987; MacKay, 1992b). Namely, one places a prior over the network’s weights and biases. This effectively places a distribution over the functions that the network represents, capturing uncertainty about which function best fits the data. Specifying this prior remains an open challenge. Common practice is to use a standard normal prior in weight space, which imposes weak shrinkage regularities analogous to weight decay. It is neither informative about the induced function class nor the data (e.g., it is sensitive to parameterization). ",
|
| 85 |
+
"bbox": [
|
| 86 |
+
148,
|
| 87 |
+
738,
|
| 88 |
+
851,
|
| 89 |
+
821
|
| 90 |
+
],
|
| 91 |
+
"page_idx": 0
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "image",
|
| 95 |
+
"img_path": "images/767f706f6bc8f210b4fd373d7486b4ea5d44b60217b42db25c4bfed3c0c2e015.jpg",
|
| 96 |
+
"image_caption": [
|
| 97 |
+
"Figure 1: Predictive distributions on a low-dimensional active learning task. The predictive distributions are visualized as mean and two standard deviations shaded. They decompose into epistemic uncertainty $\\mid$ and aleatoric noise \u0004. Data points are only available within two bands, and are selected using the expected information gain \u0004. (a) A deterministic network conflates uncertainty as part of the noise and is overconfident outside of the data distribution. (b) A variational Bayesian neural network with standard normal prior represents uncertainty and noise separately but is overconfident outside of the training distribution. (c) On the OOD classifier model, NCP prevents overconfidence. (d) On the Bayesian neural network, NCP produces smooth uncertainty estimates that generalize well to unseen data points. Models trained with NCP also separate uncertainty and noise well. The experimental setup is described in Section 5.1. "
|
| 98 |
+
],
|
| 99 |
+
"image_footnote": [],
|
| 100 |
+
"bbox": [
|
| 101 |
+
145,
|
| 102 |
+
117,
|
| 103 |
+
852,
|
| 104 |
+
319
|
| 105 |
+
],
|
| 106 |
+
"page_idx": 1
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "This can cause the induced function posterior to generalize in unforeseen ways on out-of-distribution (OOD) inputs, which are inputs outside of the distribution that generated the training data. ",
|
| 111 |
+
"bbox": [
|
| 112 |
+
147,
|
| 113 |
+
469,
|
| 114 |
+
848,
|
| 115 |
+
498
|
| 116 |
+
],
|
| 117 |
+
"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "Motivated by these challenges, we introduce noise contrastive priors (NCPs), which encourage uncertainty outside of the training distribution through a loss in data space. NCPs are compatible with any model that represents functional uncertainty as a random variable, are easy to scale, and yield reliable uncertainty estimates that show significantly improved active learning performance. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
148,
|
| 124 |
+
503,
|
| 125 |
+
849,
|
| 126 |
+
559
|
| 127 |
+
],
|
| 128 |
+
"page_idx": 1
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "2 NOISE CONTRASTIVE PRIORS ",
|
| 133 |
+
"text_level": 1,
|
| 134 |
+
"bbox": [
|
| 135 |
+
148,
|
| 136 |
+
577,
|
| 137 |
+
428,
|
| 138 |
+
594
|
| 139 |
+
],
|
| 140 |
+
"page_idx": 1
|
| 141 |
+
},
|
| 142 |
+
{
|
| 143 |
+
"type": "text",
|
| 144 |
+
"text": "Specifying priors is intuitive for small probabilistic models, where each variable typically has a clear interpretation (Blei, 2014). It is less intuitive for neural networks, where the parameters serve more as adaptive basis coefficients in a nonparametric function. For example, neural network models are nonidentifiable due to weight symmetries that yield the same function (Müller and Insua, 1998). This makes it difficult to express informative priors on the weights, such as expressing high uncertainty on unfamiliar examples. ",
|
| 145 |
+
"bbox": [
|
| 146 |
+
148,
|
| 147 |
+
607,
|
| 148 |
+
851,
|
| 149 |
+
691
|
| 150 |
+
],
|
| 151 |
+
"page_idx": 1
|
| 152 |
+
},
|
| 153 |
+
{
|
| 154 |
+
"type": "text",
|
| 155 |
+
"text": "Data priors Unlike a prior in weight space, a data prior lets one easily express informative assumptions about input-output relationships. Here, we use the example of a prior over a labeled data set $\\{ x , y \\}$ , although the prior can also be on $x$ and another variable in the model that represents uncertainty and has a clear interpretation. The prior takes the form $p _ { \\mathrm { p r i o r } } ( x , y ) = p _ { \\mathrm { p r i o r } } ( x ) \\ p _ { \\mathrm { p r i o r } } ( y \\mid x )$ , where $p _ { \\mathrm { p r i o r } } ( x )$ denotes the input prior and $p _ { \\mathrm { p r i o r } } ( y \\mid x )$ denotes the output prior. ",
|
| 156 |
+
"bbox": [
|
| 157 |
+
148,
|
| 158 |
+
704,
|
| 159 |
+
851,
|
| 160 |
+
775
|
| 161 |
+
],
|
| 162 |
+
"page_idx": 1
|
| 163 |
+
},
|
| 164 |
+
{
|
| 165 |
+
"type": "text",
|
| 166 |
+
"text": "To prevent overconfident predictions, a good input prior $p _ { \\mathrm { p r i o r } } ( x )$ should include OOD examples so that it acts beyond the training distribution. A good output prior $p _ { \\mathrm { p r i o r } } ( y \\mid x )$ should be a high-entropy distribution, representing high uncertainty about the model output given OOD inputs. ",
|
| 167 |
+
"bbox": [
|
| 168 |
+
148,
|
| 169 |
+
780,
|
| 170 |
+
851,
|
| 171 |
+
823
|
| 172 |
+
],
|
| 173 |
+
"page_idx": 1
|
| 174 |
+
},
|
| 175 |
+
{
|
| 176 |
+
"type": "image",
|
| 177 |
+
"img_path": "images/2e73abd3bb62c6ac1cc6c59de31a5ef1fe62b9161e35ec9ecf56ffb1850ec3f4.jpg",
|
| 178 |
+
"image_caption": [
|
| 179 |
+
"Figure 2: Graphical representations of the two uncertainty-aware models we consider. Circles denote random variables, squares denote deterministic variables, shading denotes observations during training. (a) The Bayesian neural network captures a belief over parameters for the predictive mean, while the predictive variance is a deterministic function of the input. In practice, we only use weight uncertainty for the mean’s output layer and share earlier layers between the mean and variance. (b) The out-of-distribution classifier model uses a binary auxiliary variable $o$ to determine if a given input is out-of-distribution; given its value, the output is drawn from either a neural network prediction or a wide output prior. "
|
| 180 |
+
],
|
| 181 |
+
"image_footnote": [],
|
| 182 |
+
"bbox": [
|
| 183 |
+
256,
|
| 184 |
+
117,
|
| 185 |
+
751,
|
| 186 |
+
247
|
| 187 |
+
],
|
| 188 |
+
"page_idx": 2
|
| 189 |
+
},
|
| 190 |
+
{
|
| 191 |
+
"type": "text",
|
| 192 |
+
"text": "Generating OOD inputs Exactly generating OOD data is difficult. A priori, we must uniformly represent the input domain. A posteriori, we must represent the complement of the training distribution. Both distributions are typically uniform over infinite support, making them ill-defined. To estimate OOD inputs, we develop an algorithm inspired by noise contrastive estimation (Gutmann and Hyvärinen, 2010a; Mnih and Kavukcuoglu, 2013), where a complement distribution is approximated using random noise. ",
|
| 193 |
+
"bbox": [
|
| 194 |
+
147,
|
| 195 |
+
367,
|
| 196 |
+
852,
|
| 197 |
+
438
|
| 198 |
+
],
|
| 199 |
+
"page_idx": 2
|
| 200 |
+
},
|
| 201 |
+
{
|
| 202 |
+
"type": "text",
|
| 203 |
+
"text": "A hypothesis of our work is that in practice it is enough to encourage high uncertainty output near the boundary of the training distribution, and that this effect will propagate to the entire OOD space. This hypothesis is backed up by previous work (Lee et al., 2017) as well as our experiments (see Figure 1). This means we no longer need to sample arbitrary OOD inputs. It is enough to sample OOD points that lie close to the boundary of the training distribution, and to apply our desired prior at those points. ",
|
| 204 |
+
"bbox": [
|
| 205 |
+
147,
|
| 206 |
+
443,
|
| 207 |
+
851,
|
| 208 |
+
513
|
| 209 |
+
],
|
| 210 |
+
"page_idx": 2
|
| 211 |
+
},
|
| 212 |
+
{
|
| 213 |
+
"type": "text",
|
| 214 |
+
"text": "Loss function Noise contrastive priors are data priors that are enforced on both training inputs $x$ and inputs $\\tilde { x }$ perturbed by noise. For example, in binary and categorical input domains, we approximate OOD inputs by randomly flipping the features to different classes with a certain probability. For continuous valued inputs $x$ we can use additive Gaussian noise to obtain noised up inputs $\\tilde { x } = x + \\epsilon$ . This expresses the noise contrastive prior where inputs are distributed according to the convolved distribution, ",
|
| 215 |
+
"bbox": [
|
| 216 |
+
147,
|
| 217 |
+
525,
|
| 218 |
+
851,
|
| 219 |
+
595
|
| 220 |
+
],
|
| 221 |
+
"page_idx": 2
|
| 222 |
+
},
|
| 223 |
+
{
|
| 224 |
+
"type": "equation",
|
| 225 |
+
"img_path": "images/b9a300e715aa1e363cc7de3f59126eca803819b0c48b2cffd7bf47e1e3356123.jpg",
|
| 226 |
+
"text": "$$\np _ { \\mathrm { p r i o r } } ( \\tilde { x } ) = \\int _ { x } p _ { \\mathrm { t r a i n } } ( x ) \\mathrm { N o r m a l } ( \\tilde { x } - x \\mid \\mu _ { x } , \\sigma _ { x } ^ { 2 } ) d x \\qquad p _ { \\mathrm { p r i o r } } ( \\tilde { y } \\mid \\tilde { x } ) = \\mathrm { N o r m a l } ( \\mu _ { y } , \\sigma _ { y } ^ { 2 } ) .\n$$",
|
| 227 |
+
"text_format": "latex",
|
| 228 |
+
"bbox": [
|
| 229 |
+
200,
|
| 230 |
+
598,
|
| 231 |
+
795,
|
| 232 |
+
630
|
| 233 |
+
],
|
| 234 |
+
"page_idx": 2
|
| 235 |
+
},
|
| 236 |
+
{
|
| 237 |
+
"type": "text",
|
| 238 |
+
"text": "The variances $\\sigma _ { x } ^ { 2 }$ and $\\sigma _ { y } ^ { 2 }$ are hyperparameters that tune how far from the boundary we sample, and how large we want the output uncertainty to be. We choose $\\mu _ { x } = 0$ to apply the prior equally in all directions from the data manifold. The output mean $\\mu _ { y }$ determines the default prediction of the model outside of the training distribution, for example $\\mu _ { y } = 0$ . We set $\\mu _ { y } = y$ which corresponds to data augmentation (Matsuoka, 1992; An, 1996), where a model is trained to recover the true labels from perturbed inputs. This way, NCP makes the model uncertain while still trying to generalize to OOD inputs. ",
|
| 239 |
+
"bbox": [
|
| 240 |
+
148,
|
| 241 |
+
632,
|
| 242 |
+
852,
|
| 243 |
+
718
|
| 244 |
+
],
|
| 245 |
+
"page_idx": 2
|
| 246 |
+
},
|
| 247 |
+
{
|
| 248 |
+
"type": "text",
|
| 249 |
+
"text": "For training, we minimize the loss function ",
|
| 250 |
+
"bbox": [
|
| 251 |
+
148,
|
| 252 |
+
723,
|
| 253 |
+
433,
|
| 254 |
+
737
|
| 255 |
+
],
|
| 256 |
+
"page_idx": 2
|
| 257 |
+
},
|
| 258 |
+
{
|
| 259 |
+
"type": "equation",
|
| 260 |
+
"img_path": "images/87245183910b71d74f51cccde61598ad15d4c08b0295afba7c72dd45db3d5430.jpg",
|
| 261 |
+
"text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( \\theta ) = \\mathrm { \\mathrm { ~ E } } _ { p _ { \\mathrm { t r a i n } } ( x ) } \\left[ D _ { \\mathrm { K L } } \\big [ p _ { \\mathrm { t r a i n } } ( y \\mid x ) \\ \\lVert \\ p _ { \\mathrm { m o d e l } } ( y \\mid x , \\theta ) \\big ] \\right] } \\\\ & { \\quad \\quad \\quad + \\gamma \\mathrm { \\mathrm { E } } _ { p _ { \\mathrm { p r i o r } } ( \\tilde { x } ) } \\left[ D _ { \\mathrm { K L } } \\big [ p _ { \\mathrm { p r i o r } } ( \\tilde { y } \\mid \\tilde { x } ) \\ \\lVert \\ p _ { \\mathrm { m o d e l } } ( \\tilde { y } \\mid \\tilde { x } , \\theta ) \\big ] \\right] . } \\end{array}\n$$",
|
| 262 |
+
"text_format": "latex",
|
| 263 |
+
"bbox": [
|
| 264 |
+
305,
|
| 265 |
+
738,
|
| 266 |
+
691,
|
| 267 |
+
780
|
| 268 |
+
],
|
| 269 |
+
"page_idx": 2
|
| 270 |
+
},
|
| 271 |
+
{
|
| 272 |
+
"type": "text",
|
| 273 |
+
"text": "The first term represents typical maximum likelihood, in which one minimizes the KL divergence to the empirical training distribution $p _ { \\mathrm { t r a i n } } ( y \\mid x )$ over training inputs. The second term is added by our method: it represents the analogous term on a data prior. The hyperparameter $\\gamma$ sets the relative trade-off between them. ",
|
| 274 |
+
"bbox": [
|
| 275 |
+
148,
|
| 276 |
+
780,
|
| 277 |
+
851,
|
| 278 |
+
823
|
| 279 |
+
],
|
| 280 |
+
"page_idx": 2
|
| 281 |
+
},
|
| 282 |
+
{
|
| 283 |
+
"type": "text",
|
| 284 |
+
"text": "Interpretation as function prior The noise contrastive prior can be interpreted as inducing a function prior. This is formalized through the prior predictive distribution, ",
|
| 285 |
+
"bbox": [
|
| 286 |
+
142,
|
| 287 |
+
119,
|
| 288 |
+
851,
|
| 289 |
+
148
|
| 290 |
+
],
|
| 291 |
+
"page_idx": 3
|
| 292 |
+
},
|
| 293 |
+
{
|
| 294 |
+
"type": "equation",
|
| 295 |
+
"img_path": "images/546418b8dac28fd9e4f9f39ba4c7b1a854afc87be812c7ce67d94a17c05497ac.jpg",
|
| 296 |
+
"text": "$$\np ( \\boldsymbol { y } \\mid \\boldsymbol { x } ) = \\int p _ { \\mathrm { m o d e l } } ( \\boldsymbol { y } \\mid \\boldsymbol { x } , \\boldsymbol { \\theta } ) p _ { \\mathrm { m o d e l } } ( \\boldsymbol { \\theta } \\mid \\tilde { \\boldsymbol { x } } , \\tilde { \\boldsymbol { y } } ) p _ { \\mathrm { p r i o r } } ( \\tilde { \\boldsymbol { x } } , \\tilde { \\boldsymbol { y } } ) d \\boldsymbol { \\theta } d \\tilde { \\boldsymbol { x } } d \\tilde { \\boldsymbol { y } } .\n$$",
|
| 297 |
+
"text_format": "latex",
|
| 298 |
+
"bbox": [
|
| 299 |
+
274,
|
| 300 |
+
155,
|
| 301 |
+
723,
|
| 302 |
+
188
|
| 303 |
+
],
|
| 304 |
+
"page_idx": 3
|
| 305 |
+
},
|
| 306 |
+
{
|
| 307 |
+
"type": "text",
|
| 308 |
+
"text": "The distribution marginalizes over network parameters $\\theta$ as well as data fantasized from the data prior. The distribution $p ( \\theta \\mid \\tilde { x } , \\tilde { y } )$ represents the distribution of model parameters after fitting the prior data. That is, the belief over weights is shaped to make $p ( y \\mid x )$ highly variable. This parameter belief causes uncertain predictions outside of the training distribution, which we could not specify in weight space directly. ",
|
| 309 |
+
"bbox": [
|
| 310 |
+
147,
|
| 311 |
+
193,
|
| 312 |
+
852,
|
| 313 |
+
250
|
| 314 |
+
],
|
| 315 |
+
"page_idx": 3
|
| 316 |
+
},
|
| 317 |
+
{
|
| 318 |
+
"type": "text",
|
| 319 |
+
"text": "Because network weights are constrained to fit the data prior, the prior acts as “pseudo-data.” This is similar to classical work on conjugate priors: a $\\mathrm { B e t a } ( \\alpha , \\beta )$ prior on the probability of a Bernoulli likelihood implies a Beta posterior, and if the posterior mode is chosen as an optimal parameter setting, then the prior translates to $\\alpha - 1$ successes and $\\beta - 1$ failures. It is also similar to pseudo-data in sparse Gaussian processes (Quiñonero-Candela and Rasmussen, 2005). ",
|
| 320 |
+
"bbox": [
|
| 321 |
+
148,
|
| 322 |
+
253,
|
| 323 |
+
851,
|
| 324 |
+
324
|
| 325 |
+
],
|
| 326 |
+
"page_idx": 3
|
| 327 |
+
},
|
| 328 |
+
{
|
| 329 |
+
"type": "text",
|
| 330 |
+
"text": "Data priors encourage learning parameters that not only capture the training data well but also the prior data. In practice, we can combine NCP with other priors, for example the typical standard normal prior in weight space for Bayesian neural networks, although we did not find this necessary in our experiments. ",
|
| 331 |
+
"bbox": [
|
| 332 |
+
148,
|
| 333 |
+
329,
|
| 334 |
+
851,
|
| 335 |
+
372
|
| 336 |
+
],
|
| 337 |
+
"page_idx": 3
|
| 338 |
+
},
|
| 339 |
+
{
|
| 340 |
+
"type": "text",
|
| 341 |
+
"text": "3 BAYESIAN NEURAL NETWORKS WITH NCP ",
|
| 342 |
+
"text_level": 1,
|
| 343 |
+
"bbox": [
|
| 344 |
+
148,
|
| 345 |
+
390,
|
| 346 |
+
542,
|
| 347 |
+
406
|
| 348 |
+
],
|
| 349 |
+
"page_idx": 3
|
| 350 |
+
},
|
| 351 |
+
{
|
| 352 |
+
"type": "text",
|
| 353 |
+
"text": "Noise contrastive priors are applicable to any model that represents uncertainty in a random variable. The NCP can then be added to that random variable to make the model uncertain on OOD inputs. In this section, we apply NCP to a Bayesian neural network (BNN) trained via variational inference. Blundell et al. (2015) introduce such a model under the name Bayes by Backprop (BBB) that uses a standard normal prior in weight space. We extend this model with a NCP on the mean predicted by the neural network. ",
|
| 354 |
+
"bbox": [
|
| 355 |
+
147,
|
| 356 |
+
420,
|
| 357 |
+
852,
|
| 358 |
+
491
|
| 359 |
+
],
|
| 360 |
+
"page_idx": 3
|
| 361 |
+
},
|
| 362 |
+
{
|
| 363 |
+
"type": "text",
|
| 364 |
+
"text": "Consider a regression task with data $\\{ x , y \\}$ that we model as $p ( y \\mid x , \\theta ) = { \\mathrm { N o r m a l } } ( \\mu ( x ) , \\sigma ^ { 2 } ( x ) )$ with mean and variance predicted by a neural network from the inputs. This model is heteroskedastic, meaning that it can predict a different aleatoric noise amount for every point in the input space. We use a weight prior for only the output layer (Lázaro-Gredilla and Figueiras-Vidal, 2010; Calandra et al., 2014) that predicts the mean, resulting in the model ",
|
| 365 |
+
"bbox": [
|
| 366 |
+
147,
|
| 367 |
+
494,
|
| 368 |
+
851,
|
| 369 |
+
565
|
| 370 |
+
],
|
| 371 |
+
"page_idx": 3
|
| 372 |
+
},
|
| 373 |
+
{
|
| 374 |
+
"type": "equation",
|
| 375 |
+
"img_path": "images/3b86c6ccf944501c17d46d2ce5d21402cfd0a57aae3dece6e3b13f3193224af2.jpg",
|
| 376 |
+
"text": "$$\n\\theta \\sim \\operatorname { N o r m a l } ( 0 , 0 . 1 ) \\qquad y \\sim \\operatorname { N o r m a l } ( \\mu ( x , \\theta ) , \\sigma ^ { 2 } ( x ) ) .\n$$",
|
| 377 |
+
"text_format": "latex",
|
| 378 |
+
"bbox": [
|
| 379 |
+
316,
|
| 380 |
+
571,
|
| 381 |
+
679,
|
| 382 |
+
589
|
| 383 |
+
],
|
| 384 |
+
"page_idx": 3
|
| 385 |
+
},
|
| 386 |
+
{
|
| 387 |
+
"type": "text",
|
| 388 |
+
"text": "We do not model uncertainty about the noise estimate, as this is not required for the approximation for the Gaussian expected information gain (MacKay, 1992a) that we use to acquire labels. Therefore, the distribution of the mean induced by the weight prior, $\\begin{array} { r } { \\dot { q } ( \\mu ( x ) ) = \\int \\mu ( x , \\theta ) q _ { \\phi } ( \\theta ) \\dot { d } \\theta } \\end{array}$ , represents the model’s epistemic uncertainty. Note that this is different from the predictive distribution, which combines both uncertainty and noise. We place an NCP on the distribution of the mean, resulting in the loss function ",
|
| 389 |
+
"bbox": [
|
| 390 |
+
147,
|
| 391 |
+
594,
|
| 392 |
+
851,
|
| 393 |
+
665
|
| 394 |
+
],
|
| 395 |
+
"page_idx": 3
|
| 396 |
+
},
|
| 397 |
+
{
|
| 398 |
+
"type": "equation",
|
| 399 |
+
"img_path": "images/da5b634a3d4518fbeaed572417e3b6e2f900cdc524a6e6cead3a447998218022.jpg",
|
| 400 |
+
"text": "$$\n\\mathcal { L } ( \\phi ) = - \\mathbb { E } _ { q _ { \\phi } ( \\theta ) } [ \\ln p ( y \\mid x , \\theta ) ] + \\beta D _ { \\mathrm { K L } } [ q _ { \\phi } ( \\theta ) \\parallel p ( \\theta ) ] + \\underbrace { \\gamma D _ { \\mathrm { K L } } [ \\mathrm { N o r m a l } ( \\mu _ { \\mu } , \\sigma _ { \\mu } ^ { 2 } ) \\parallel q ( \\mu ( \\tilde { x } ) ) ] } _ { \\mathrm { N C P l o s s } } .\n$$",
|
| 401 |
+
"text_format": "latex",
|
| 402 |
+
"bbox": [
|
| 403 |
+
189,
|
| 404 |
+
670,
|
| 405 |
+
807,
|
| 406 |
+
702
|
| 407 |
+
],
|
| 408 |
+
"page_idx": 3
|
| 409 |
+
},
|
| 410 |
+
{
|
| 411 |
+
"type": "text",
|
| 412 |
+
"text": "Here, $\\tilde { x }$ are the perturbed inputs and $q _ { \\phi } ( \\theta )$ forms an approximate posterior over weights.1 Because we only use the weight belief for the linear output layer, we can compute the KL-divergence of the NCP loss analytically. In other models, it could be estimated using samples. ",
|
| 413 |
+
"bbox": [
|
| 414 |
+
147,
|
| 415 |
+
709,
|
| 416 |
+
852,
|
| 417 |
+
752
|
| 418 |
+
],
|
| 419 |
+
"page_idx": 3
|
| 420 |
+
},
|
| 421 |
+
{
|
| 422 |
+
"type": "text",
|
| 423 |
+
"text": "The loss function applies weight regularization in order for network weights to regress to a standard normal prior; like other regularization techniques, this assists in improving the network’s generalization in-distribution. ",
|
| 424 |
+
"bbox": [
|
| 425 |
+
147,
|
| 426 |
+
757,
|
| 427 |
+
849,
|
| 428 |
+
786
|
| 429 |
+
],
|
| 430 |
+
"page_idx": 3
|
| 431 |
+
},
|
| 432 |
+
{
|
| 433 |
+
"type": "text",
|
| 434 |
+
"text": "The NCP loss encourages the network’s generalization OOD by matching the mean distribution to the output prior. Minimizing the KL divergence to a wide output prior results in high uncertainty on OOD inputs, so the model will explore these data points during active learning. ",
|
| 435 |
+
"bbox": [
|
| 436 |
+
148,
|
| 437 |
+
121,
|
| 438 |
+
851,
|
| 439 |
+
162
|
| 440 |
+
],
|
| 441 |
+
"page_idx": 4
|
| 442 |
+
},
|
| 443 |
+
{
|
| 444 |
+
"type": "text",
|
| 445 |
+
"text": "In practice, we find that NCP is sufficient as a prior for the BNN and set $\\beta = 0$ . The appendix (Appendix B includes an alternative interpretation explaining why NCP might be sufficient, which represents the weight space KL-divergence in data space after a change of variables. ",
|
| 446 |
+
"bbox": [
|
| 447 |
+
148,
|
| 448 |
+
167,
|
| 449 |
+
851,
|
| 450 |
+
210
|
| 451 |
+
],
|
| 452 |
+
"page_idx": 4
|
| 453 |
+
},
|
| 454 |
+
{
|
| 455 |
+
"type": "text",
|
| 456 |
+
"text": "4 RELATED WORK ",
|
| 457 |
+
"text_level": 1,
|
| 458 |
+
"bbox": [
|
| 459 |
+
150,
|
| 460 |
+
229,
|
| 461 |
+
318,
|
| 462 |
+
246
|
| 463 |
+
],
|
| 464 |
+
"page_idx": 4
|
| 465 |
+
},
|
| 466 |
+
{
|
| 467 |
+
"type": "text",
|
| 468 |
+
"text": "Priors for neural networks Classic work has investigated entropic priors (Buntine and Weigend, 1991) and hierarchical priors (MacKay, 1992b; Neal, 2012; Lampinen and Vehtari, 2001). More recently, Depeweg et al. (2018) introduce networks with latent variables in order to disentangle forms of uncertainty, and FlamShepherd et al. (2017) propose general-purpose weight priors based on approximating Gaussian processes. Other works have analyzed priors for compression and model selection (Ghosh and Doshi-Velez, 2017; Louizos et al., 2017). Instead of a prior in weight space (or latent inputs as in Depeweg et al. (2018)), NCPs take the functional view by imposing explicit regularities in terms of the network’s inputs and outputs. Malinin and Gales (2018) propose prior networks to avoid an explicit belief over parameters for classification tasks. ",
|
| 469 |
+
"bbox": [
|
| 470 |
+
148,
|
| 471 |
+
261,
|
| 472 |
+
852,
|
| 473 |
+
372
|
| 474 |
+
],
|
| 475 |
+
"page_idx": 4
|
| 476 |
+
},
|
| 477 |
+
{
|
| 478 |
+
"type": "text",
|
| 479 |
+
"text": "Input and output regularization There is classic work on adding noise to inputs for improved generalization (Matsuoka, 1992; An, 1996; Bishop, 1995). For example, denoising autoencoders (Vincent et al., 2008) encourage reconstructions given noisy encodings. Output regularization is also a classic idea from the maximum entropy principle (Jaynes, 1957), where it has motivated label smoothing (Szegedy et al., 2016) and entropy penalties (Pereyra et al., 2017). Also related is virtual adversarial training (Miyato et al., 2015), which includes examples that are close to the current input but cause a maximal change in the model output, and mixup (Zhang et al., 2018), which includes examples under the vicinity of training data. These methods are orthogonal to NCPs: they aim to improve generalization from finite data within the training distribution (interpolation), while we aim to improve uncertainty estimates outside of the training distribution (extrapolation). ",
|
| 480 |
+
"bbox": [
|
| 481 |
+
148,
|
| 482 |
+
387,
|
| 483 |
+
852,
|
| 484 |
+
526
|
| 485 |
+
],
|
| 486 |
+
"page_idx": 4
|
| 487 |
+
},
|
| 488 |
+
{
|
| 489 |
+
"type": "text",
|
| 490 |
+
"text": "Classifying out-of-distribution inputs A simple approach for neural network uncertainty is to classify whether data points belong to the data distribution, or are OOD (Hendrycks and Gimpel, 2017). This is core to noise contrastive estimation (Gutmann and Hyvärinen, 2010b), a training method for intractable probabilistic models. More recently, Lee et al. (2017) introduce a GAN to generate OOD samples, and Liang et al. (2018) add perturbations to the input, applying an “OOD detector” to improve softmax scores on OOD samples by scaling the temperature. Extending these directions of research, we connect to Bayesian principles and focus on uncertainty estimates that are useful for active data acquisition. ",
|
| 491 |
+
"bbox": [
|
| 492 |
+
148,
|
| 493 |
+
542,
|
| 494 |
+
851,
|
| 495 |
+
640
|
| 496 |
+
],
|
| 497 |
+
"page_idx": 4
|
| 498 |
+
},
|
| 499 |
+
{
|
| 500 |
+
"type": "text",
|
| 501 |
+
"text": "5 EXPERIMENTS ",
|
| 502 |
+
"text_level": 1,
|
| 503 |
+
"bbox": [
|
| 504 |
+
148,
|
| 505 |
+
660,
|
| 506 |
+
300,
|
| 507 |
+
676
|
| 508 |
+
],
|
| 509 |
+
"page_idx": 4
|
| 510 |
+
},
|
| 511 |
+
{
|
| 512 |
+
"type": "text",
|
| 513 |
+
"text": "To demonstrate their usefulness, we evaluate NCPs on various tasks where uncertainty estimates are desired. Our focus is on active learning for regression tasks, where only few targets are visible in the beginning, and additional targets are selected regularly based on an acquisition function. We use two data sets: a toy example and a large flights data set. We also evaluate how sensitive our method is to the choice of input noise. Finally, we show that NCP scales to large data sets by training on the full flights data set in a passive learning setting. Our implementation uses TensorFlow Probability (Dillon et al., 2017; Tran et al., 2016) and is open-sourced at https://<hidden-for-review>. ",
|
| 514 |
+
"bbox": [
|
| 515 |
+
148,
|
| 516 |
+
690,
|
| 517 |
+
852,
|
| 518 |
+
789
|
| 519 |
+
],
|
| 520 |
+
"page_idx": 4
|
| 521 |
+
},
|
| 522 |
+
{
|
| 523 |
+
"type": "text",
|
| 524 |
+
"text": "We compare four neural network models, all using leaky ReLU activations (Maas et al., 2013) and trained using Adam (Kingma and Ba, 2014). The four models are: ",
|
| 525 |
+
"bbox": [
|
| 526 |
+
145,
|
| 527 |
+
794,
|
| 528 |
+
849,
|
| 529 |
+
821
|
| 530 |
+
],
|
| 531 |
+
"page_idx": 4
|
| 532 |
+
},
|
| 533 |
+
{
|
| 534 |
+
"type": "image",
|
| 535 |
+
"img_path": "images/aed734cb24c8e8c6ee03698202b9c4bfd0733711982169668b9b7cccfffa6f40.jpg",
|
| 536 |
+
"image_caption": [
|
| 537 |
+
"Figure 3: Active learning on the 1-dimensional regression problem, mean and standard deviation over 20 seeds. The test root mean squared error (RMSE) and negative log predictive density (NLPD) of the models trained with NCP decreases during the active learning run, while the baseline models select less informative data and overfit. The deterministic network is barely visible in the plots as it overfits quickly. Figure 1 shows the predictive distributions of the models. "
|
| 538 |
+
],
|
| 539 |
+
"image_footnote": [],
|
| 540 |
+
"bbox": [
|
| 541 |
+
143,
|
| 542 |
+
114,
|
| 543 |
+
851,
|
| 544 |
+
255
|
| 545 |
+
],
|
| 546 |
+
"page_idx": 5
|
| 547 |
+
},
|
| 548 |
+
{
|
| 549 |
+
"type": "text",
|
| 550 |
+
"text": "• Deterministic neural network (Det) A neural network that predicts the mean and variance of a normal distribution. The name stands for deterministic, as there is no weight uncertainty. \n• Bayes by Backprop (BBB) A Bayesian neural network trained via gradient-based variational inference with a standard normal prior in weight space (Blundell et al., 2015; Kucukelbir et al., 2017). We use the same model as in Section 3 but without the NCP loss term. \n• Bayes by Backprop with noise contrastive prior $\\mathbf { ( B B B + N C P ) }$ ) Bayes by Backprop with NCP on the predicted mean distribution as described in Section 3. \n• Out-of-distribution classifier with noise contrastive prior $\\mathbf { ( O C D + N C P ) }$ ) An uncertainty classifier model described in Appendix A. It is a deterministic neural network combined with NCP which we use as a baseline alternative to Bayes by Backprop with NCP. ",
|
| 551 |
+
"bbox": [
|
| 552 |
+
176,
|
| 553 |
+
349,
|
| 554 |
+
854,
|
| 555 |
+
508
|
| 556 |
+
],
|
| 557 |
+
"page_idx": 5
|
| 558 |
+
},
|
| 559 |
+
{
|
| 560 |
+
"type": "text",
|
| 561 |
+
"text": "For active learning, we select new data points $\\{ x , y \\}$ for which $x$ maximizes the expected information gain $\\operatorname { E } _ { q ( y | x ) } [ D _ { \\mathrm { K L } } [ q ( \\theta \\mid x , y ) \\parallel _ { . } q ( \\theta ) ] ]$ under the model ${ \\\\begin{array} { r } { { \\dot { q } } ( y \\mid x ) = \\int p ( y \\mid x , \\theta ) q ( \\theta ) d \\theta } \\end{array} }$ . Intuitively, this objective function is higher where the model has high epistemic uncertainty and predicts low aleatoric noise. ",
|
| 562 |
+
"bbox": [
|
| 563 |
+
148,
|
| 564 |
+
518,
|
| 565 |
+
852,
|
| 566 |
+
561
|
| 567 |
+
],
|
| 568 |
+
"page_idx": 5
|
| 569 |
+
},
|
| 570 |
+
{
|
| 571 |
+
"type": "text",
|
| 572 |
+
"text": "We use an approximation from MacKay (1992a) for Gaussian posterior predictive distributions. Moreover, we place a softmax distribution on the information gain for all available data points and acquire labels by sampling with a temperature of $\\tau = 0 . 5$ to get diversity when selecting batches of labels, ",
|
| 573 |
+
"bbox": [
|
| 574 |
+
147,
|
| 575 |
+
565,
|
| 576 |
+
856,
|
| 577 |
+
609
|
| 578 |
+
],
|
| 579 |
+
"page_idx": 5
|
| 580 |
+
},
|
| 581 |
+
{
|
| 582 |
+
"type": "equation",
|
| 583 |
+
"img_path": "images/c370bbeabf682c72c76e47e63e3bc469f580ab093ff181b83562e56c00b8bee2.jpg",
|
| 584 |
+
"text": "$$\n\\{ x _ { \\mathrm { n e w } } \\} \\sim p _ { \\mathrm { n e w } } ( x ) = \\frac { 1 } { Z } \\exp \\Big ( \\frac { 1 } { 2 \\tau } \\ln \\big ( 1 + \\mathrm { V a r } [ q ( \\mu ( x ) ) ] / \\sigma ^ { 2 } ( x ) \\big ) \\Big ) = \\frac { 1 } { Z } \\big ( 1 + \\mathrm { V a r } [ q ( \\mu ( x ) ) ] / \\sigma ^ { 2 } ( x ) \\big ) ,\n$$",
|
| 585 |
+
"text_format": "latex",
|
| 586 |
+
"bbox": [
|
| 587 |
+
161,
|
| 588 |
+
616,
|
| 589 |
+
813,
|
| 590 |
+
647
|
| 591 |
+
],
|
| 592 |
+
"page_idx": 5
|
| 593 |
+
},
|
| 594 |
+
{
|
| 595 |
+
"type": "text",
|
| 596 |
+
"text": "where $\\sigma ^ { 2 } ( x )$ is the estimated aleatoric noise and $q ( \\mu ( x ) )$ is the epistemic uncertainty projected into output space. Since our Bayesian neural networks only use a weight belief for the output layer, $\\operatorname { V a r } [ q ( \\mu ( x ) ) ]$ is Gaussian and can be computed in closed form. In general, it the epistemic part of the predictive variance would be estimated by sampling. In the classifier model, we use the OOD probability $p ( o = 1 | x )$ for this. For the deterministic neural network, we use $\\mathrm { V a r } [ p ( y \\mid x ) ]$ as proxy since it does not output an estimate of epistemic uncertainty. ",
|
| 597 |
+
"bbox": [
|
| 598 |
+
148,
|
| 599 |
+
654,
|
| 600 |
+
852,
|
| 601 |
+
739
|
| 602 |
+
],
|
| 603 |
+
"page_idx": 5
|
| 604 |
+
},
|
| 605 |
+
{
|
| 606 |
+
"type": "text",
|
| 607 |
+
"text": "5.1 LOW-DIMENSIONAL ACTIVE LEARNING ",
|
| 608 |
+
"text_level": 1,
|
| 609 |
+
"bbox": [
|
| 610 |
+
150,
|
| 611 |
+
755,
|
| 612 |
+
459,
|
| 613 |
+
768
|
| 614 |
+
],
|
| 615 |
+
"page_idx": 5
|
| 616 |
+
},
|
| 617 |
+
{
|
| 618 |
+
"type": "text",
|
| 619 |
+
"text": "For visualization purposes, we start with experiments on a 1-dimensional regression task that consists of a sine function with a small slope and increasing variance for higher inputs. Training data can be acquired within two bands, and the model is evaluated on all data points that are not visible to the model. This structured split between training and testing data causes a distributional shift at test time, requiring successful models to have reliable uncertainty estimates to avoid mispredictions for OOD inputs. ",
|
| 620 |
+
"bbox": [
|
| 621 |
+
148,
|
| 622 |
+
780,
|
| 623 |
+
851,
|
| 624 |
+
823
|
| 625 |
+
],
|
| 626 |
+
"page_idx": 5
|
| 627 |
+
},
|
| 628 |
+
{
|
| 629 |
+
"type": "image",
|
| 630 |
+
"img_path": "images/105bdfd66b8a5dcf9cd8c426a1ea28c174da90a784da759877af6ae3a1cc842e.jpg",
|
| 631 |
+
"image_caption": [
|
| 632 |
+
"Figure 4: Active learning on the flights data set. The models trained with NCP achieve significantly lower negative log predictive density (NLPD) on the test set, and Bayes by Backprop with NCP achieves the lowest root mean squared error (RMSE). The test NLPD for the baseline models diverges as they overfit to the visible data points. Plots show mean and std over 10 runs. "
|
| 633 |
+
],
|
| 634 |
+
"image_footnote": [],
|
| 635 |
+
"bbox": [
|
| 636 |
+
145,
|
| 637 |
+
114,
|
| 638 |
+
851,
|
| 639 |
+
255
|
| 640 |
+
],
|
| 641 |
+
"page_idx": 6
|
| 642 |
+
},
|
| 643 |
+
{
|
| 644 |
+
"type": "text",
|
| 645 |
+
"text": "",
|
| 646 |
+
"bbox": [
|
| 647 |
+
147,
|
| 648 |
+
335,
|
| 649 |
+
849,
|
| 650 |
+
363
|
| 651 |
+
],
|
| 652 |
+
"page_idx": 6
|
| 653 |
+
},
|
| 654 |
+
{
|
| 655 |
+
"type": "text",
|
| 656 |
+
"text": "For this experiment, we use two layers of 200 hidden units, a batch size of 10, and a learning rate of $3 \\times 1 0 ^ { - 4 }$ for all models. NCP models use noise $\\epsilon \\sim \\mathrm { N o r m a l } ( 0 , 0 . 5 )$ . We start with 10 randomly selected initial targets, and select 1 additional target every 1000 epochs. Figure 3 shows the root mean squared error (RMSE) and negative log predictive density (NLPD) throughout learning. The two baseline models severely overfit to the training distribution early on when only few data points are visible. Models with NCP outperform BBB, which in turn outperforms Det. Figure 1 visualizes the models’ predictive distributions at the end of training, showing that NCP prevents overconfident generalization. ",
|
| 657 |
+
"bbox": [
|
| 658 |
+
148,
|
| 659 |
+
369,
|
| 660 |
+
852,
|
| 661 |
+
467
|
| 662 |
+
],
|
| 663 |
+
"page_idx": 6
|
| 664 |
+
},
|
| 665 |
+
{
|
| 666 |
+
"type": "text",
|
| 667 |
+
"text": "5.2 ACTIVE LEARNING ON FLIGHT DELAYS",
|
| 668 |
+
"text_level": 1,
|
| 669 |
+
"bbox": [
|
| 670 |
+
150,
|
| 671 |
+
482,
|
| 672 |
+
457,
|
| 673 |
+
496
|
| 674 |
+
],
|
| 675 |
+
"page_idx": 6
|
| 676 |
+
},
|
| 677 |
+
{
|
| 678 |
+
"type": "text",
|
| 679 |
+
"text": "We consider the flight delay data set (Hensman et al., 2013; Deisenroth and Ng, 2015; Lakshminarayanan et al., 2016), a large scale regression benchmark with several published results. The data set has 8 input variables describing a flight, and the target is the delay of the flight in minutes. There are 700K training examples and 100K test examples. The test set has a subtle distributional shift, since the 100K data points temporally follow after the training data. ",
|
| 680 |
+
"bbox": [
|
| 681 |
+
148,
|
| 682 |
+
506,
|
| 683 |
+
851,
|
| 684 |
+
577
|
| 685 |
+
],
|
| 686 |
+
"page_idx": 6
|
| 687 |
+
},
|
| 688 |
+
{
|
| 689 |
+
"type": "text",
|
| 690 |
+
"text": "We use two layers with 50 units each, a batch size of 10, and a learning rate of $1 0 ^ { - 4 }$ . For NCP models, $\\epsilon \\sim \\mathrm { N o r m a l } ( 0 , 0 . 1 )$ . Starting from 10 labels, the models select a batch of 10 additional labels every 50 epochs. The 700K data points of the training data set are available for acquisition, and we evaluate performance on the typical test split. Figure 4 shows the performance for the visible data points and the test set respectively. We note that BBB and $\\mathrm { B B B + N C P }$ show similar NLPD on the visible data points, but the NCP models generalize better to unseen data. Moreover, the Bayesian neural network with NCP achieves lower RMSE than the one without and the classifier based model achieves lower RMSE than the deterministic neural network. All uncertainty-based models outperform the deterministic neural network. ",
|
| 691 |
+
"bbox": [
|
| 692 |
+
148,
|
| 693 |
+
582,
|
| 694 |
+
851,
|
| 695 |
+
693
|
| 696 |
+
],
|
| 697 |
+
"page_idx": 6
|
| 698 |
+
},
|
| 699 |
+
{
|
| 700 |
+
"type": "text",
|
| 701 |
+
"text": "5.3 ROBUSTNESS TO NOISE PATTERNS ",
|
| 702 |
+
"text_level": 1,
|
| 703 |
+
"bbox": [
|
| 704 |
+
151,
|
| 705 |
+
708,
|
| 706 |
+
423,
|
| 707 |
+
722
|
| 708 |
+
],
|
| 709 |
+
"page_idx": 6
|
| 710 |
+
},
|
| 711 |
+
{
|
| 712 |
+
"type": "text",
|
| 713 |
+
"text": "The choice of input noise might seem like a critical hyper parameter for NCP. In this experiment, we find that our method is robust to the choice of input noise. The experimental setup is the same as for the active learning experiment described in Section 5.2, but with uniform or normal input noise with different variance $( \\sigma _ { x } ^ { 2 } \\in \\bar { \\{ 0 . 1 , 0 . 2 , \\cdot \\cdot \\cdot , 1 . 0 \\} } )$ . For uniform input noise, this means noise is drawn from the interval $[ - 2 \\sigma _ { x } , 2 \\sigma _ { x } ]$ We observe that $\\mathbf { B B B + N C P }$ is robust to the size of the input noise. NCP consistently improves RMSE for the tested noise sizes and yields the best NLPD for all noise sizes below 0.6. For our ODC baseline, we observe an intuitive trade-off: smaller input noise increases the regularization strength, leading to better NLPD but reduced RMSE. Robustness to the choice of input noise is further supported by the analogous experiment on toy data set, where above a small threshold $( \\mathbf { \\bar { B } B B + N C P } \\ \\sigma _ { x } ^ { 2 } \\geq 0 . 3 $ and ${ \\mathrm { O D C + N C P } }$ $\\sigma _ { x } ^ { 2 } \\ge \\bar { 0 . 1 } \\bar { , }$ ), NCP consistently performs well (Figure 6). ",
|
| 714 |
+
"bbox": [
|
| 715 |
+
145,
|
| 716 |
+
732,
|
| 717 |
+
851,
|
| 718 |
+
821
|
| 719 |
+
],
|
| 720 |
+
"page_idx": 6
|
| 721 |
+
},
|
| 722 |
+
{
|
| 723 |
+
"type": "image",
|
| 724 |
+
"img_path": "images/a661f6b87ed31f78adf003c889209c95d7cab4df4a62bf2724fc76daca9d045b.jpg",
|
| 725 |
+
"image_caption": [
|
| 726 |
+
"Figure 5: Robustness to different noise patterns. Plots show the final test performance on the flights active learning task (mean and stddev over 5 seeds). Lower is better. NCP is robust to the choice of input noise and improves over the baselines in all settings (compare Figure 4). "
|
| 727 |
+
],
|
| 728 |
+
"image_footnote": [],
|
| 729 |
+
"bbox": [
|
| 730 |
+
143,
|
| 731 |
+
114,
|
| 732 |
+
849,
|
| 733 |
+
255
|
| 734 |
+
],
|
| 735 |
+
"page_idx": 7
|
| 736 |
+
},
|
| 737 |
+
{
|
| 738 |
+
"type": "text",
|
| 739 |
+
"text": "",
|
| 740 |
+
"bbox": [
|
| 741 |
+
148,
|
| 742 |
+
321,
|
| 743 |
+
851,
|
| 744 |
+
378
|
| 745 |
+
],
|
| 746 |
+
"page_idx": 7
|
| 747 |
+
},
|
| 748 |
+
{
|
| 749 |
+
"type": "text",
|
| 750 |
+
"text": "5.4 LARGE SCALE REGRESSION OF FLIGHT DELAYS",
|
| 751 |
+
"text_level": 1,
|
| 752 |
+
"bbox": [
|
| 753 |
+
148,
|
| 754 |
+
393,
|
| 755 |
+
516,
|
| 756 |
+
407
|
| 757 |
+
],
|
| 758 |
+
"page_idx": 7
|
| 759 |
+
},
|
| 760 |
+
{
|
| 761 |
+
"type": "text",
|
| 762 |
+
"text": "In addition to the active learning experiments, we perform a passive learning run on all 700K data points of the flights data set to explore the scalability of NCP. We use networks of 3 layers with 1000 units and a learning rate of $1 0 ^ { - 4 }$ . Table 1 compares the performance of our models to previously published results. We significantly improve state of the art performance on this data set. ",
|
| 763 |
+
"bbox": [
|
| 764 |
+
148,
|
| 765 |
+
417,
|
| 766 |
+
851,
|
| 767 |
+
474
|
| 768 |
+
],
|
| 769 |
+
"page_idx": 7
|
| 770 |
+
},
|
| 771 |
+
{
|
| 772 |
+
"type": "text",
|
| 773 |
+
"text": "6 DISCUSSION ",
|
| 774 |
+
"text_level": 1,
|
| 775 |
+
"bbox": [
|
| 776 |
+
148,
|
| 777 |
+
492,
|
| 778 |
+
284,
|
| 779 |
+
508
|
| 780 |
+
],
|
| 781 |
+
"page_idx": 7
|
| 782 |
+
},
|
| 783 |
+
{
|
| 784 |
+
"type": "text",
|
| 785 |
+
"text": "We develop noise contrastive priors (NCPs), a prior for neural networks in data space. NCPs encourage network weights that not only explain the training data but also capture high uncertainty on OOD inputs. We show that NCPs offer strong improvements over baselines and scale to large regression tasks. ",
|
| 786 |
+
"bbox": [
|
| 787 |
+
148,
|
| 788 |
+
523,
|
| 789 |
+
508,
|
| 790 |
+
607
|
| 791 |
+
],
|
| 792 |
+
"page_idx": 7
|
| 793 |
+
},
|
| 794 |
+
{
|
| 795 |
+
"type": "text",
|
| 796 |
+
"text": "We focused on active learning for regression tasks, where uncertainty is crucial for determining which data points to select next. In future work it would be interesting to apply NCPs to alternative settings where uncertainty is important, such as image classification and learning with sparse or missing data. In addition, NCPs are only one form of a data prior, designed to encourage uncertainty on OOD inputs. Priors in data space can easily capture other properties such as periodicity or spatial invariance, and they may provide a scalable alternative to Gaussian process priors. ",
|
| 797 |
+
"bbox": [
|
| 798 |
+
148,
|
| 799 |
+
613,
|
| 800 |
+
509,
|
| 801 |
+
765
|
| 802 |
+
],
|
| 803 |
+
"page_idx": 7
|
| 804 |
+
},
|
| 805 |
+
{
|
| 806 |
+
"type": "table",
|
| 807 |
+
"img_path": "images/73d319b3bda629d47574bc7d9555e21e18f67b66ed923dd89a8982f993165806.jpg",
|
| 808 |
+
"table_caption": [
|
| 809 |
+
"Table 1: Performance on all 700K data points of the flights data set. While uncertainty estimates are not necessary when a large data set that is similar to the test data set is available, it shows that our method scales easily to large data sets. "
|
| 810 |
+
],
|
| 811 |
+
"table_footnote": [],
|
| 812 |
+
"table_body": "<table><tr><td>Model</td><td>NLPD</td><td>RMSE</td></tr><tr><td>gPoE (Deisenroth & Ng 2015)</td><td>8.1</td><td></td></tr><tr><td>SAVIGP (Bonilla et al. 2016)</td><td>5.02</td><td></td></tr><tr><td>SVI GP (Hensman et al. 2013)</td><td>一</td><td>32.60</td></tr><tr><td>HGP (Ng & Deisenroth 2014)</td><td>一</td><td>27.45</td></tr><tr><td>MF F (Lakshminarayanan et al. 2016)</td><td>4.89</td><td>26.57</td></tr><tr><td>BBB</td><td>4.38</td><td>24.59</td></tr><tr><td>BBB+NCP</td><td>4.38</td><td>24.71</td></tr><tr><td>ODC+NCP</td><td>4.38</td><td>24.68</td></tr></table>",
|
| 813 |
+
"bbox": [
|
| 814 |
+
519,
|
| 815 |
+
606,
|
| 816 |
+
859,
|
| 817 |
+
763
|
| 818 |
+
],
|
| 819 |
+
"page_idx": 7
|
| 820 |
+
},
|
| 821 |
+
{
|
| 822 |
+
"type": "text",
|
| 823 |
+
"text": "REFERENCES ",
|
| 824 |
+
"text_level": 1,
|
| 825 |
+
"bbox": [
|
| 826 |
+
148,
|
| 827 |
+
119,
|
| 828 |
+
261,
|
| 829 |
+
135
|
| 830 |
+
],
|
| 831 |
+
"page_idx": 8
|
| 832 |
+
},
|
| 833 |
+
{
|
| 834 |
+
"type": "text",
|
| 835 |
+
"text": "G. An. The effects of adding noise during backpropagation training on a generalization performance. Neural computation, 8(3):643–674, 1996. \nC. M. Bishop. Training with noise is equivalent to tikhonov regularization. Neural computation, 7(1):108–116, 1995. \nD. M. Blei. Build, compute, critique, repeat: Data analysis with latent variable models. Annual Review of Statistics and Its Application, 1:203–232, 2014. \nC. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra. Weight uncertainty in neural networks. arXiv preprint arXiv:1505.05424, 2015. \nW. L. Buntine and A. S. Weigend. Bayesian back-propagation. Complex Systems, 5(6):603–643, 1991. \nR. Calandra, J. Peters, C. E. Rasmussen, and M. P. Deisenroth. Manifold gaussian processes for regression. arXiv preprint arXiv:1402.5876, 2014. \nS. Dasgupta. Analysis of a greedy active learning strategy. In Neural Information Processing Systems (NIPS), 2004. \nM. P. Deisenroth and J. W. Ng. Distributed gaussian processes. arXiv preprint arXiv:1502.02843, 2015. \nJ. Denker, D. Schwartz, B. Wittner, S. Solla, R. Howard, L. Jackel, and J. Hopfield. Large automatic learning, rule extraction, and generalization. Complex Systems, 1(5):877–922, 1987. \nS. Depeweg, J. M. Hernández-Lobato, F. Doshi-Velez, and S. Udluft. Uncertainty decomposition in bayesian neural networks with latent variables. In International Conference on Machine Learning, 2018. \nJ. V. Dillon, I. Langmore, D. Tran, E. Brevdo, S. Vasudevan, D. Moore, B. Patton, A. Alemi, M. Hoffman, and R. A. Saurous. Tensorflow distributions. arXiv preprint arXiv:1711.10604, 2017. \nD. Flam-Shepherd, J. Requeima, and D. Duvenaud. Mapping gaussian process priors to bayesian neural networks. In NIPS Workshop, 2017. \nY. Freund, H. S. Seung, E. Shamir, and N. Tishby. Selective sampling using the query by committee algorithm. Machine learning, 28(2-3):133–168, 1997. \nY. Gal, R. Islam, and Z. Ghahramani. Deep bayesian active learning with image data. arXiv preprint arXiv:1703.02910, 2017. \nS. Ghosh and F. Doshi-Velez. Model selection in bayesian neural networks via horseshoe priors. arXiv preprint arXiv:1705.10388, 2017. \nM. Gutmann and A. Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 297–304, 2010a. \nM. Gutmann and A. Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 297–304, 2010b. \nD. Hendrycks and K. Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. In International Conference on Learning Representations, 2017. \nJ. Hensman, N. Fusi, and N. D. Lawrence. Gaussian processes for big data. arXiv preprint arXiv:1309.6835, 2013. \nI. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. International Conference on Learning Representations, 2016. \nN. Houlsby, F. Huszár, Z. Ghahramani, and M. Lengyel. Bayesian active learning for classification and preference learning. arXiv preprint arXiv:1112.5745, 2011. \nE. T. Jaynes. Information theory and statistical mechanics. Physical review, 106(4):620, 1957. \nA. J. Joshi, F. Porikli, and N. Papanikolopoulos. Multi-class active learning for image classification. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 2372–2379. IEEE, 2009. \nA. Kapoor, K. Grauman, R. Urtasun, and T. Darrell. Active learning with gaussian processes for object categorization. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, pages 1–8. IEEE, 2007. \nD. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Neural Information Processing Systems, pages 1097–1105, 2012. \nA. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei. Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1):430–474, 2017. \nB. Lakshminarayanan, D. M. Roy, and Y. W. Teh. Mondrian forests for large-scale regression when uncertainty matters. In Artificial Intelligence and Statistics, pages 1478–1487, 2016. \nJ. Lampinen and A. Vehtari. Bayesian approach for neural networks—review and case studies. Neural networks, 14(3):257–274, 2001. \nM. Lázaro-Gredilla and A. R. Figueiras-Vidal. Marginalized neural network mixtures for large-scale regression. IEEE transactions on neural networks, 21(8):1345–1351, 2010. \nK. Lee, H. Lee, K. Lee, and J. Shin. Training confidence-calibrated classifiers for detecting out-of-distribution samples. arXiv preprint arXiv:1711.09325, 2017. \nD. D. Lewis and W. A. Gale. A sequential algorithm for training text classifiers. In Proceedings of the 17th annual international ACM SIGIR conference on Research and development in information retrieval, pages 3–12. Springer-Verlag New York, Inc., 1994. \nX. Li and Y. Guo. Adaptive active learning for image classification. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pages 859–866. IEEE, 2013. \nS. Liang, Y. Li, and R. Srikant. Principled detection of out-of-distribution examples in neural networks. In International Conference on Learning Representations, 2018. \nC. Louizos, K. Ullrich, and M. Welling. Bayesian compression for deep learning. In Neural Information Processing Systems, pages 3290–3300, 2017. \nA. L. Maas, A. Y. Hannun, and A. Y. Ng. Rectifier nonlinearities improve neural network acoustic models. In International Conference on Machine Learning (ICML), 2013. \nD. J. MacKay. Information-based objective functions for active data selection. Neural computation, 4(4): 590–604, 1992a. \nD. J. MacKay. A practical bayesian framework for backpropagation networks. Neural computation, 4(3): 448–472, 1992b. \nA. Malinin and M. Gales. Predictive uncertainty estimation via prior networks. arXiv preprint arXiv:1802.10501, 2018. \nK. Matsuoka. Noise injection into inputs in back-propagation learning. IEEE Transactions on Systems, Man, and Cybernetics, 22(3):436–440, 1992. \nA. K. McCallumzy and K. Nigamy. Employing em and pool-based active learning for text classification. In Proc. International Conference on Machine Learning (ICML), pages 359–367. Citeseer, 1998. \nT. Miyato, S. Maeda, M. Koyama, K. Nakae, and S. Ishii. Distributional smoothing with virtual adversarial training. arXiv preprint arXiv:1507.00677, 2015. \nA. Mnih and K. Kavukcuoglu. Learning word embeddings efficiently with noise-contrastive estimation. In Advances in neural information processing systems, pages 2265–2273, 2013. \nP. Müller and D. R. Insua. Issues in bayesian analysis of neural network models. Neural Computation, 10(3): 749–770, 1998. \nR. M. Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012. \nG. Pereyra, G. Tucker, J. Chorowski, Ł. Kaiser, and G. Hinton. Regularizing neural networks by penalizing confident output distributions. arXiv preprint arXiv:1701.06548, 2017. \nJ. Quiñonero-Candela and C. E. Rasmussen. A unifying view of sparse approximate gaussian process regression. Journal of Machine Learning Research, 6(Dec):1939–1959, 2005. \nH. A. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IIEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 1998. \nN. Roy and A. McCallum. Toward optimal active learning through monte carlo estimation of error reduction. ICML, Williamstown, pages 441–448, 2001. \nP. Schulam and S. Saria. A framework for individualizing predictions of disease trajectories by exploiting multi-resolution structure. In Advances in Neural Information Processing Systems, pages 748–756, 2015. \nB. Settles. Active learning literature survey. Computer Sciences Technical Report 1648, University of Wisconsin–Madison, 2009. \nH. S. Seung, M. Opper, and H. Sompolinsky. Query by committee. In Proceedings of the fifth annual workshop on Computational learning theory, pages 287–294. ACM, 1992. \nK.-K. Sung. Learning and example selection for object and pattern detection. MIT A.I. Memo No. 1521, 1994. \nI. Sutskever, O. Vinyals, and Q. V. Le. Sequence to sequence learning with neural networks. In Neural Information Processing Systems, 2014. \nC. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2818–2826, 2016. \nS. Tong and D. Koller. Support vector machine active learning with applications to text classification. Journal of machine learning research, 2(Nov):45–66, 2001. \nD. Tran, A. Kucukelbir, A. B. Dieng, M. Rudolph, D. Liang, and D. M. Blei. Edward: A library for probabilistic modeling, inference, and criticism. arXiv preprint arXiv:1610.09787, 2016. \nA. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016. \nP. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pages 1096–1103. ACM, 2008. \nK. Wang, D. Zhang, Y. Li, R. Zhang, and L. Lin. Cost-effective active learning for deep image classification. IEEE Transactions on Circuits and Systems for Video Technology, 27(12):2591–2600, 2017. \nH. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018. ",
|
| 836 |
+
"bbox": [
|
| 837 |
+
145,
|
| 838 |
+
138,
|
| 839 |
+
854,
|
| 840 |
+
828
|
| 841 |
+
],
|
| 842 |
+
"page_idx": 8
|
| 843 |
+
},
|
| 844 |
+
{
|
| 845 |
+
"type": "text",
|
| 846 |
+
"text": "",
|
| 847 |
+
"bbox": [
|
| 848 |
+
142,
|
| 849 |
+
118,
|
| 850 |
+
856,
|
| 851 |
+
825
|
| 852 |
+
],
|
| 853 |
+
"page_idx": 9
|
| 854 |
+
},
|
| 855 |
+
{
|
| 856 |
+
"type": "text",
|
| 857 |
+
"text": "",
|
| 858 |
+
"bbox": [
|
| 859 |
+
142,
|
| 860 |
+
108,
|
| 861 |
+
856,
|
| 862 |
+
829
|
| 863 |
+
],
|
| 864 |
+
"page_idx": 10
|
| 865 |
+
},
|
| 866 |
+
{
|
| 867 |
+
"type": "text",
|
| 868 |
+
"text": "",
|
| 869 |
+
"bbox": [
|
| 870 |
+
143,
|
| 871 |
+
118,
|
| 872 |
+
854,
|
| 873 |
+
352
|
| 874 |
+
],
|
| 875 |
+
"page_idx": 11
|
| 876 |
+
},
|
| 877 |
+
{
|
| 878 |
+
"type": "text",
|
| 879 |
+
"text": "We showed how to apply NCP to a Bayesian neural network model that captures function uncertainty in a belief over parameters. An alternative approach to capture uncertainty is to make explicit predictions about whether an input is OOD. There is no belief over weights in this model. Figure 2b shows such a mixture model via a binary variable $o$ , ",
|
| 880 |
+
"bbox": [
|
| 881 |
+
147,
|
| 882 |
+
148,
|
| 883 |
+
852,
|
| 884 |
+
205
|
| 885 |
+
],
|
| 886 |
+
"page_idx": 12
|
| 887 |
+
},
|
| 888 |
+
{
|
| 889 |
+
"type": "equation",
|
| 890 |
+
"img_path": "images/2b1e89167d04bca8797f415bc926d580c43e4f946176c957f4ff77a0bdcc371d.jpg",
|
| 891 |
+
"text": "$$\n\\begin{array} { r l } & { o \\sim \\mathrm { B e r n o u l l i } ( \\pi ( x , \\theta ) ) } \\\\ & { y \\sim \\left\\{ \\begin{array} { l l } { \\mathrm { N o r m a l } ( \\mu ( x , \\theta ) , \\sigma ^ { 2 } ( x , \\theta ) ) } & { \\mathrm { i f } o = 0 } \\\\ { \\mathrm { N o r m a l } ( \\mu _ { y } , \\sigma _ { y } ^ { 2 } ) } & { \\mathrm { i f } o = 1 , } \\end{array} \\right. } \\end{array}\n$$",
|
| 892 |
+
"text_format": "latex",
|
| 893 |
+
"bbox": [
|
| 894 |
+
351,
|
| 895 |
+
210,
|
| 896 |
+
645,
|
| 897 |
+
270
|
| 898 |
+
],
|
| 899 |
+
"page_idx": 12
|
| 900 |
+
},
|
| 901 |
+
{
|
| 902 |
+
"type": "text",
|
| 903 |
+
"text": "where $p ( o = 1 \\mid x )$ is the OOD probability of $x$ . If $o = 0$ (“in distribution”), the model outputs the neural network prediction. Otherwise, if $o = 1$ (“out of distribution”), the model uses a fixed output prior. The neural network weights $\\theta$ are estimated using a point estimate, so we do not maintain a belief distribution over them. ",
|
| 904 |
+
"bbox": [
|
| 905 |
+
147,
|
| 906 |
+
275,
|
| 907 |
+
851,
|
| 908 |
+
332
|
| 909 |
+
],
|
| 910 |
+
"page_idx": 12
|
| 911 |
+
},
|
| 912 |
+
{
|
| 913 |
+
"type": "text",
|
| 914 |
+
"text": "The classifier prediction $p ( o \\mid x , \\theta )$ captures uncertainty in this model. We apply the NCP $p ( o \\mid \\tilde { x } , \\theta ) =$ $\\delta ( o = 1 | \\tilde { x } , \\theta )$ to this variable, which assumes noised-up inputs to be OOD. During training on the data set, $\\{ x , y \\}$ and $o = 0$ are observed, as training data are in-distribution by definition. Following Equation 2, the loss function is ",
|
| 915 |
+
"bbox": [
|
| 916 |
+
148,
|
| 917 |
+
337,
|
| 918 |
+
852,
|
| 919 |
+
393
|
| 920 |
+
],
|
| 921 |
+
"page_idx": 12
|
| 922 |
+
},
|
| 923 |
+
{
|
| 924 |
+
"type": "equation",
|
| 925 |
+
"img_path": "images/516882d706937319b059bea56cd05f9a5685f7eed7661b3ffb7068f8253a2036.jpg",
|
| 926 |
+
"text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( \\theta ) = D _ { \\mathrm { K L } } [ p _ { \\mathrm { f r a i n } } ( y \\mid x ) \\mid | p _ { \\mathrm { m o d e l } } ( y \\mid x , o = 0 , \\theta ) ] + D _ { \\mathrm { K L } } [ p _ { \\mathrm { p r i o r } } ( \\tilde { o } \\mid \\tilde { x } ) \\mid | p _ { \\mathrm { m o d e l } } ( \\tilde { o } \\mid \\tilde { x } , \\theta ) ] } \\\\ & { \\quad \\quad = - \\ln p ( y , o = 0 \\mid x , \\theta ) - \\ln p ( y , o = 1 \\mid \\tilde { x } , \\theta ) } \\\\ & { \\quad \\quad = - \\ln \\mathrm { N o r m a l } ( y \\mid \\mu ( x , \\theta ) , \\sigma ^ { 2 } ( x , \\theta ) ) - \\ln \\mathrm { B e r n o u l i } ( 0 \\mid \\pi ( x , \\theta ) ) \\frac { - \\ln \\mathrm { B e r n o u l i } ( 1 \\mid \\pi ( \\tilde { x } , \\theta ) ) } { \\mathrm { N C P l o s s } } . } \\end{array}\n$$",
|
| 927 |
+
"text_format": "latex",
|
| 928 |
+
"bbox": [
|
| 929 |
+
160,
|
| 930 |
+
396,
|
| 931 |
+
816,
|
| 932 |
+
463
|
| 933 |
+
],
|
| 934 |
+
"page_idx": 12
|
| 935 |
+
},
|
| 936 |
+
{
|
| 937 |
+
"type": "text",
|
| 938 |
+
"text": "Analogously to the Bayesian neural network model in Section 3, we can either set $\\mu _ { y } , \\sigma _ { y } ^ { 2 }$ manually or use the neural network prediction for potentially improved generalization. In our experiments, we implement the OOD classifier model using a single neural network with two output layers that parameterize the Gaussian distribution and the binary distribution. ",
|
| 939 |
+
"bbox": [
|
| 940 |
+
148,
|
| 941 |
+
472,
|
| 942 |
+
852,
|
| 943 |
+
530
|
| 944 |
+
],
|
| 945 |
+
"page_idx": 12
|
| 946 |
+
},
|
| 947 |
+
{
|
| 948 |
+
"type": "text",
|
| 949 |
+
"text": "B BNN WITH NCP USING REVERSE KL ",
|
| 950 |
+
"text_level": 1,
|
| 951 |
+
"bbox": [
|
| 952 |
+
148,
|
| 953 |
+
547,
|
| 954 |
+
496,
|
| 955 |
+
565
|
| 956 |
+
],
|
| 957 |
+
"page_idx": 12
|
| 958 |
+
},
|
| 959 |
+
{
|
| 960 |
+
"type": "text",
|
| 961 |
+
"text": "In Section 3, we derived the Bayes by Backprop model with NCP by adding a forward KL-divergence from the mean prior to the model mean to the loss. An alternative derivation uses the fact that the KL-divergence is invariant to parameterization to replace the reverse KL-divergence in weight space by a KL-divergence in output space, ",
|
| 962 |
+
"bbox": [
|
| 963 |
+
147,
|
| 964 |
+
578,
|
| 965 |
+
851,
|
| 966 |
+
636
|
| 967 |
+
],
|
| 968 |
+
"page_idx": 12
|
| 969 |
+
},
|
| 970 |
+
{
|
| 971 |
+
"type": "equation",
|
| 972 |
+
"img_path": "images/4ef571e3f9c8dcdc862b3ebde728c945331f851bd201952a3e882c112ba71595.jpg",
|
| 973 |
+
"text": "$$\n\\begin{array} { r l } & { \\mathrm { E } _ { p ( x , y ) } \\big [ \\ln p ( y \\mid x ) \\big ] = \\mathrm { E } _ { p ( x , y ) } \\Big [ \\ln \\displaystyle \\int p ( y \\mid x , \\theta ) p ( \\theta ) \\frac { q ( \\theta ) } { q ( \\theta ) } d \\theta \\Big ] } \\\\ & { \\qquad \\quad \\ge \\mathrm { E } _ { p ( x , y ) } \\Big [ \\int q ( \\theta ) \\ln p ( y \\mid x , \\theta ) \\frac { p ( \\theta ) } { q ( \\theta ) } d \\theta \\Big ] } \\\\ & { \\qquad = \\mathrm { E } _ { p ( x , y ) } \\big [ \\mathrm { E } _ { q ( \\theta ) } [ \\ln p ( y \\mid x , \\theta ) ] - D _ { \\mathrm { K L } } [ q ( \\theta ) \\mid p ( \\theta ) ] \\big ] } \\\\ & { \\qquad = \\mathrm { E } _ { p ( x , y ) } \\big [ \\mathrm { E } _ { q ( \\theta ) } [ \\ln p ( y \\mid x , \\theta ) ] - \\mathrm { E } _ { p ( \\tilde { x } \\mid x ) } [ D _ { \\mathrm { K L } } [ q ( \\theta ) \\mid \\mid p ( \\theta ) ] ] \\big ] } \\\\ & { \\qquad \\approx \\mathrm { E } _ { p ( x , y ) } \\big [ \\mathrm { E } _ { q ( \\theta ) } [ \\ln p ( y \\mid x , \\theta ) ] - \\mathrm { E } _ { p ( \\tilde { x } \\mid x ) } [ D _ { \\mathrm { K L } } [ q ( \\mu ( \\tilde { x } ) ) \\mid \\mid p ( \\mu ( \\tilde { x } ) \\mid x ) ] ] \\big ] , } \\end{array}\n$$",
|
| 974 |
+
"text_format": "latex",
|
| 975 |
+
"bbox": [
|
| 976 |
+
191,
|
| 977 |
+
642,
|
| 978 |
+
807,
|
| 979 |
+
773
|
| 980 |
+
],
|
| 981 |
+
"page_idx": 12
|
| 982 |
+
},
|
| 983 |
+
{
|
| 984 |
+
"type": "text",
|
| 985 |
+
"text": "where $\\begin{array} { r } { p ( \\mu ( \\tilde { x } ) ) = \\int \\mu ( \\tilde { x } , \\theta ) p ( \\theta ) d \\theta } \\end{array}$ and $\\begin{array} { r } { q ( \\mu ( \\tilde { x } ) ) = \\int \\mu ( \\tilde { x } , \\theta ) q ( \\theta ) d \\theta } \\end{array}$ are the distributions of the predicted mean induces by the weight beliefs. As a result, instead of specifying a prior in weight space, we can specify a prior in output space. ",
|
| 986 |
+
"bbox": [
|
| 987 |
+
147,
|
| 988 |
+
779,
|
| 989 |
+
849,
|
| 990 |
+
823
|
| 991 |
+
],
|
| 992 |
+
"page_idx": 12
|
| 993 |
+
},
|
| 994 |
+
{
|
| 995 |
+
"type": "text",
|
| 996 |
+
"text": "Above, we reparameteterized the $\\mathrm { K L }$ in weight space as a KL in output space; by the change of variables, this is equivalent if the mapping $\\mu ( \\cdot , \\theta )$ is continuous and 1-1 with respect to $\\theta$ . This assumption does not hold for neural nets as multiple parameter vectors can lead to the same predictive distribution, thus the approximation above. A compact reparameterization of the neural network (equivalence class of parameteters) would make this an equality. ",
|
| 997 |
+
"bbox": [
|
| 998 |
+
147,
|
| 999 |
+
119,
|
| 1000 |
+
851,
|
| 1001 |
+
190
|
| 1002 |
+
],
|
| 1003 |
+
"page_idx": 13
|
| 1004 |
+
},
|
| 1005 |
+
{
|
| 1006 |
+
"type": "image",
|
| 1007 |
+
"img_path": "images/993aa791d17cff9990aa55e866fe978c95a36d2466bb36590c90a258714536a2.jpg",
|
| 1008 |
+
"image_caption": [
|
| 1009 |
+
"C ROBUSTNESS EXPERIMENT ON TOY DATASET ",
|
| 1010 |
+
"Figure 6: Robustness to different noise patterns. Plots show the final test performance on the low-dimensional active learning task (mean and stddev over 5 seeds). Lower is better. The baseline performances are RMSE: BBB $( 0 . 7 5 \\pm 0 . 3 1 )$ , Det $( 1 . 4 6 \\pm 0 . 6 4 )$ and NLPD: BBB $( 1 0 . 2 9 \\pm 8 . 0 5 )$ , Det $( 1 . 3 \\times 1 0 ^ { 8 } \\pm 1 . 7 \\times 1 0 ^ { 8 } )$ ). NCP works with both Gaussian and uniform input noise $\\epsilon$ and is robust to $\\sigma _ { x } ^ { 2 }$ . "
|
| 1011 |
+
],
|
| 1012 |
+
"image_footnote": [],
|
| 1013 |
+
"bbox": [
|
| 1014 |
+
145,
|
| 1015 |
+
239,
|
| 1016 |
+
851,
|
| 1017 |
+
369
|
| 1018 |
+
],
|
| 1019 |
+
"page_idx": 13
|
| 1020 |
+
},
|
| 1021 |
+
{
|
| 1022 |
+
"type": "text",
|
| 1023 |
+
"text": "D RELATED ACTIVE LEARNING WORK ",
|
| 1024 |
+
"text_level": 1,
|
| 1025 |
+
"bbox": [
|
| 1026 |
+
148,
|
| 1027 |
+
465,
|
| 1028 |
+
488,
|
| 1029 |
+
482
|
| 1030 |
+
],
|
| 1031 |
+
"page_idx": 13
|
| 1032 |
+
},
|
| 1033 |
+
{
|
| 1034 |
+
"type": "text",
|
| 1035 |
+
"text": "Active learning is often employed in domains where data is cheap but labeling is expensive, and is motivated by the idea that not all data points are equally valuable when it comes to learning (Settles, 2009; Dasgupta, 2004). Active learning techniques can be coarsely grouped into three categories. Ensemble methods (Seung et al., 1992; McCallumzy and Nigamy, 1998; Freund et al., 1997) generate queries that have the greatest disagreement between a set of classifiers. Error reduction approaches incorporate the select data based on the predicted reduction in classifier error based on information (MacKay, 1992a), Monte Carlo estimation (Roy and McCallum, 2001), or hard-negative example mining (Sung, 1994; Rowley et al., 1998). ",
|
| 1036 |
+
"bbox": [
|
| 1037 |
+
147,
|
| 1038 |
+
494,
|
| 1039 |
+
852,
|
| 1040 |
+
593
|
| 1041 |
+
],
|
| 1042 |
+
"page_idx": 13
|
| 1043 |
+
},
|
| 1044 |
+
{
|
| 1045 |
+
"type": "text",
|
| 1046 |
+
"text": "Uncertainty-based techniques select samples for which the classifier is most uncertain. Approaches include maximum entropy (Joshi et al., 2009), distance from the decision boundary (Tong and Koller, 2001), pseudo labelling high confidence examples (Wang et al., 2017), and mixtures of information density and uncertainty measures (Li and Guo, 2013). Within this category, the area most related to our work are Bayesian methods. Kapoor et al. (2007) estimate expected improvement using a Gaussian process. Other approaches use classifier confidence (Lewis and Gale, 1994), predicted expected error (Roy and McCallum, 2001), or model disagreement (Houlsby et al., 2011). Recently, Gal et al. (2017) applied a convolutional neural network with dropout uncertainty to images. ",
|
| 1047 |
+
"bbox": [
|
| 1048 |
+
148,
|
| 1049 |
+
598,
|
| 1050 |
+
852,
|
| 1051 |
+
710
|
| 1052 |
+
],
|
| 1053 |
+
"page_idx": 13
|
| 1054 |
+
}
|
| 1055 |
+
]
|
parse/train/HkgxasA5Ym/HkgxasA5Ym_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/HkgxasA5Ym/HkgxasA5Ym_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/h7FqQ6hCK18/h7FqQ6hCK18.md
ADDED
|
@@ -0,0 +1,355 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients
|
| 2 |
+
|
| 3 |
+
Aritra Mitra Rayana Jaafar George J. Pappas Hamed Hassani Department of Electrical and Systems Engineering {amitra20,rayanaj,pappasg,hassani} $@$ seas.upenn.edu
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
We consider a standard federated learning (FL) setup where a group of clients periodically coordinate with a central server to train a statistical model. We develop a general algorithmic framework called FedLin to tackle some of the key challenges intrinsic to FL, namely objective heterogeneity, systems heterogeneity, and infrequent and imprecise communication. Our framework is motivated by the observation that under these challenges, various existing FL algorithms suffer from a fundamental speed-accuracy conflict: they either guarantee linear convergence but to an incorrect point, or convergence to the global minimum but at a sub-linear rate, i.e., fast convergence comes at the expense of accuracy. In contrast, when the clients’ local loss functions are smooth and strongly convex, we show that FedLin guarantees linear convergence to the global minimum, despite arbitrary objective and systems heterogeneity. We then establish matching upper and lower bounds on the convergence rate of FedLin that highlight the effects of infrequent, periodic communication. Finally, we show that FedLin preserves linear convergence rates under aggressive gradient sparsification, and quantify the effect of the compression level on the convergence rate. Notably, our work is the first to provide tight linear convergence rate guarantees, and constitutes the first comprehensive analysis of gradient sparsification in FL.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
In a canonical federated learning (FL) architecture, a set $s$ of clients periodically communicate with a central server to find a global statistical model that solves the following problem [1–5]:
|
| 12 |
+
|
| 13 |
+
$$
|
| 14 |
+
\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) , { \mathrm { ~ w h e r e ~ } } f ( x ) = { \frac { 1 } { m } } \sum _ { i = 1 } ^ { m } f _ { i } ( x ) .
|
| 15 |
+
$$
|
| 16 |
+
|
| 17 |
+
Here, $m$ is the number of clients, $f _ { i } : \mathbb { R } ^ { d } \mathbb { R }$ is the local objective (loss) function of client $i$ , and $f ( x )$ is the global objective function. Some of the core distinguishing tenets of the FL paradigm are as follows [1–5]. First, due to privacy considerations, clients cannot directly share their local training data with the server. Second, differences in the clients’ data-sets may cause the clients to have nonidentical loss functions with different minima - this is known as statistical or objective heterogeneity. Third, due to variability in hardware (CPU, memory) and power (battery level), i.e., due to systems or device heterogeneity, the client devices may have different computation speeds; in particular, this may lead to slow and straggling devices that affect convergence guarantees. Finally, communicationefficiency is a major concern, dictating the need to reduce the number of communication rounds, and also the size of the messages transmitted in each round. The above considerations pose unique technical challenges that we aim to address in this paper.
|
| 18 |
+
|
| 19 |
+
In a typical FL setting, to reduce the number of communication rounds, clients perform multiple local training steps in isolation before communicating with the server. Due to such local steps, the popular
|
| 20 |
+
|
| 21 |
+
FedAvg algorithm suffers from a “client-drift phenomenon" under objective heterogeneity [6–11]: the local iterates of each client drift-off towards the minimum of their own local loss function, leading to slow convergence rates. For analysis on FedAvg, we refer the reader to [6, 8, 12–21]. Recently, several new algorithms such as FedProx [22], SCAFFOLD [11], FedSplit [10], and FedNova [23] have been proposed as improvements to FedAvg. Despite these advances, there remain gaps in our understanding of the extent to which these algorithms match the guarantees of a centralized baseline.1
|
| 22 |
+
|
| 23 |
+
For instance, even for simple, deterministic settings, FedProx [22] and FedNova [23] exhibit a fundamental speed-accuracy conflict under objective heterogeneity; see [8, 9] and Section 2. Specifically, with constant step-sizes, these algorithms converge linearly, but potentially to an incorrect point. Thus, convergence to the minimum of the global loss function necessitates diminishing step-sizes, which, in turn, leads to sub-linear convergence. Thus, fast convergence comes at the expense of accuracy. Although SCAFFOLD [11] and FedSplit [10] employ variance-reduction and operatorsplitting techniques, respectively, to tackle objective heterogeneity, it is not known whether the rates in these papers are tight. More importantly, neither SCAFFOLD nor FedSplit account for the effects of systems heterogeneity or compression, both of which are key challenges in FL. Indeed, due to systems heterogeneity, the number of local steps may vary across clients, causing some clients to make much less progress than others in each round [23]. Moreover, while empirical studies [24, 25] have revealed significant benefits of biased sparsification, theoretical guarantees for such methods in a federated setting have remained elusive. In this context, our contributions are as follows.
|
| 24 |
+
|
| 25 |
+
• A New Algorithm: Motivated by the above concerns, we develop a general algorithmic framework called FedLin that simultaneously accounts for objective heterogeneity, systems heterogeneity, and gradient sparsification. The key components of FedLin include a gradient correction term in the local update rule that exploits memory; the use of client-specific learning rates; and error-feedback mechanisms at the clients and the server.
|
| 26 |
+
|
| 27 |
+
• Matching Centralized Rates: For smooth and strongly convex losses, we show that FedLin converges to the global minimum linearly in the deterministic setting, and with a $O ( 1 / T )$ rate for a general stochastic oracle model, thereby matching centralized rates (up to constants). We then present matching rates for smooth, convex and non-convex settings as well. Importantly, our results hold under arbitrary objective and systems heterogeneity. In contrast, the only other work in FL (as far as we are aware) that investigates both objective and systems heterogeneity [23] provides results only for the non-convex setting, under a bounded dissimilarity assumption. Moreover, the FedNova algorithm in [23] suffers from the speed-accuracy conflict, while FedLin does not.
|
| 28 |
+
|
| 29 |
+
• Quantifying the Price of Multiple Local Steps: We establish a lower bound for FedLin that matches the upper-bound we obtain for smooth, strongly convex losses. In doing so, we provide the first (as far as we are aware) tight linear convergence rate analysis. Our lower bound highlights the price paid for performing multiple local steps, i.e., the effect of infrequent communication on the convergence rate. In particular, our analysis reveals, perhaps surprisingly, that there exist simple instances (involving quadratic losses) for which performing multiple local steps does not improve the rate of convergence, indicating that even mild statistical heterogeneity can hurt. Our analysis also provides valuable insights into the limitations of gradient-tracking/variance-reduction techniques.
|
| 30 |
+
|
| 31 |
+
• Analyzing the Impacts of Gradient Sparsification at Server and at Clients: While several works explore the effect of unbiased random quantization in distributed settings [26–31], there are only a handful of papers [15, 32] that also consider the effect of local steps in FL. Different from all these works, we explore the impacts of sparsifying gradients using a biased TOP-k operator, both at the server side and at the clients. Our results in this context (i) constitute the first formal study of gradient sparsification in a federated setting; (ii) reveal key differences between up-link and down-link compression; and (iii) quantify the effect of the compression level on the convergence rate. Notably, FedLin preserves linear convergence rates despite aggressive gradient sparsification.
|
| 32 |
+
|
| 33 |
+
Basic Notation and Terminology: Referring to (1), let $\begin{array} { r } { x ^ { * } \ \in \ \operatorname { a r g m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) } \end{array}$ , and $x _ { i } ^ { * } \in$ $\mathrm { a r g m i n } _ { x \in \mathbb { R } ^ { d } }$ $f _ { i } ( x )$ . Every FL algorithm mentioned in this paper operates in rounds $t \in \{ 1 , \ldots , T \}$ In each round $t$ , every client performs a certain number of local steps in isolation, starting from a common global model $\bar { x } _ { t }$ . We will denote by $x _ { i , \ell } ^ { ( t ) }$ client $i$ ’s estimate of the model at the $\ell \cdot$ -th local step of round $t$ . In particular, $x _ { i , 0 } ^ { ( t ) } = \bar { x } _ { t } , \forall i \in \mathcal { S }$ .
|
| 34 |
+
|
| 35 |
+
<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Linear Convergencetox*</td><td rowspan=1 colspan=1>Lower Bounds</td><td rowspan=1 colspan=1>Variable ClientSpeeds</td><td rowspan=1 colspan=1>Sparsification/Compression</td></tr><tr><td rowspan=1 colspan=1>FedAvg[2]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Thm. I in [11]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedProx[22]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedNova[23]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedSplit[10]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>SCAFFOLD[11]</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedLin (Sec.3)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>Thm. 5</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td></tr></table>
|
| 36 |
+
|
| 37 |
+
Table 1: Comparison of our proposed algorithm FedLin with popular FL algorithms. We indicate whether or not each algorithm (i) guarantees linear convergence to $x ^ { * }$ for smooth, strongly convex losses in a deterministic setting under objective heterogeneity; (ii) comes with lower bounds; (iii) accounts for variable local steps across clients (systems heterogeneity); and (iv) performs compression.
|
| 38 |
+
|
| 39 |
+
# 2 Motivation: Speed-Accuracy Trade-Off
|
| 40 |
+
|
| 41 |
+
To motivate our work, we first show how some recently proposed FL algorithms, namely FedProx [22] and FedNova [23], exhibit a fundamental speed-accuracy trade-off even in simple, deterministic settings. Specifically, we show that these schemes do not, in general, guarantee convergence to the global minimum with constant step-sizes. This, in turn, necessitates diminishing step-sizes, leading to sub-linear convergence rates. Our analysis here is inspired by that in [8] for FedAvg. We consider a deterministic quadratic model where the local loss function of client $i$ is given by
|
| 42 |
+
|
| 43 |
+
|
| 44 |
+
Figure 1: Simulations comparing FedProx, FedNova, and FedLin for two clients with $f _ { 1 } ( { x } ) ~ = ~ ( 1 / 2 ) ( x - 3 ) ^ { 2 }$ and $f _ { 2 } ( x ) \ = \ ( x - 5 0 ) ^ { 2 }$ . Left: Clients perform the same number of local steps, $H = 5 0$ . For FedProx, we set $\beta = 5$ . Right: Clients 1 and 2 perform 50 and 30 local steps, respectively.
|
| 45 |
+
|
| 46 |
+
$f _ { i } ( x ) = 1 / 2 \| A _ { i } ^ { 1 / 2 } ( x - c _ { i } ) \| ^ { 2 }$ , where $A _ { i }$ is a symmetric positive-definite matrix. We begin by assuming that all clients perform the same number of local steps $H$ . The following is the FedProx update rule where a proximal term is added to mitigate client-drift:
|
| 47 |
+
|
| 48 |
+
$$
|
| 49 |
+
x _ { i , \ell + 1 } ^ { ( t ) } = x _ { i , \ell } ^ { ( t ) } - \eta \bigg ( \nabla f _ { i } ( x _ { i , \ell } ^ { ( t ) } ) + \beta ( x _ { i , \ell } ^ { ( t ) } - \bar { x } _ { t } ) \bigg ) , \ell = 0 , \dots , H - 1 ; \bar { x } _ { t + 1 } = \frac { 1 } { m } \sum _ { i \in S } x _ { i , H } ^ { ( t ) } .
|
| 50 |
+
$$
|
| 51 |
+
|
| 52 |
+
Proposition 1. For any step-size $\eta > 0$ , $T$ rounds of FedProx amount to performing $T$ rounds of parallel $G D$ on the surrogate optimization problem given by
|
| 53 |
+
|
| 54 |
+
$$
|
| 55 |
+
\operatorname* { m i n } _ { x } \frac { 1 } { m } \sum _ { i \in \mathcal { S } } \frac { 1 } { 2 } \bigg \| \bigg ( \sum _ { \ell = 0 } ^ { H - 1 } [ I - \eta ( A _ { i } + \beta I ) ] ^ { \ell } A _ { i } \bigg ) ^ { 1 / 2 } ( x - c _ { i } ) \bigg \| ^ { 2 } .
|
| 56 |
+
$$
|
| 57 |
+
|
| 58 |
+
Proposition 1 shows that even when clients perform the same number of local updates, FedProx minimizes a surrogate objective function (3) whose minimum may not, in general, coincide with the minimum of the original problem. When $\beta = 0$ , FedProx reduces to FedAvg, and our observations continue to hold. To capture systems heterogeneity as in [23], suppose now that client $i$ performs $\tau _ { i }$ local steps. Define $\tau _ { e f f } \triangleq 1 / m \sum _ { i \in \mathcal { S } } \tau _ { i }$ and $\alpha _ { i } \triangleq \tau _ { e f f } / \tau _ { i }$ , $\forall i \in S$ . The update rule of FedNova relies on normalized aggregation of cumulative local gradients, and is given by
|
| 59 |
+
|
| 60 |
+
$$
|
| 61 |
+
\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) ; ~ \bar { \boldsymbol { x } } _ { t + 1 } = \bar { \boldsymbol { x } } _ { t } - \frac { \eta } { m } \sum _ { i \in S } \alpha _ { i } \sum _ { \ell = 0 } ^ { \tau _ { i } - 1 } \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) ,
|
| 62 |
+
$$
|
| 63 |
+
|
| 64 |
+
where $\ell = 0 , \dots , \tau _ { i } - 1 , \ i \in \mathcal { S }$ . Although FedNova can accommodate any local solver whose accumulated gradients are expressible as a linear combination of local gradients, we choose gradient descent, a simple solver, to isolate the impact of normalized aggregation - the essence of FedNova.
|
| 65 |
+
|
| 66 |
+
# Algorithm 1 FedLin
|
| 67 |
+
|
| 68 |
+
1: Input: Client step-sizes $\eta _ { i } , i ~ \in ~ S$ , compression levels $\delta _ { c }$ and $\delta _ { s }$ , initial iterate $\overline { { \bar { x } _ { 1 } ~ \in ~ \mathbb { R } ^ { d } } }$
|
| 69 |
+
$\bar { g _ { 1 } = \nabla f ( \bar { x } _ { 1 } ) }$ , initial compression errors $\rho _ { i , 1 } = 0 , \forall i \in \mathcal { S }$ and $e _ { 1 } = 0$
|
| 70 |
+
2: for $t = 1 , \dots , T$ do
|
| 71 |
+
3: for $i = 1 , \ldots , m$ do
|
| 72 |
+
4: for $\ell = 0 , \ldots , \tau _ { i } - 1$ do
|
| 73 |
+
5: $x _ { i , \ell + 1 } ^ { ( t ) } \gets x _ { i , \ell } ^ { ( t ) } - \eta _ { i } ( \nabla f _ { i } ( x _ { i , \ell } ^ { ( t ) } ) - \nabla f _ { i } ( \bar { x } _ { t } ) + g _ { t } ) ; x _ { i , 0 } ^ { ( t ) } = \bar { x } _ { t }$
|
| 74 |
+
6: end for
|
| 75 |
+
7: Transmit x(t)i,τi to server
|
| 76 |
+
8: end for
|
| 77 |
+
9: Server transmits x¯t+1 = 1/m Pi∈S x(t)i,τi
|
| 78 |
+
10: for $i = 1 , \ldots , m$ do
|
| 79 |
+
11: Transmit $h _ { i , t + 1 } = \mathcal { C } _ { \delta _ { c } } ( \rho _ { i , t } + \nabla f _ { i } ( \bar { x } _ { t + 1 } ) )$ to server
|
| 80 |
+
12: $\rho _ { i , t + 1 } \rho _ { i , t } + \nabla f _ { i } ( \bar { x } _ { t + 1 } ) - h _ { i , t + 1 }$
|
| 81 |
+
13: end for
|
| 82 |
+
14: Server transmits $\begin{array} { r } { g _ { t + 1 } = \mathcal { C } _ { \delta _ { s } } ( e _ { t } + 1 / m \sum _ { i \in S } h _ { i , t + 1 } ) } \end{array}$
|
| 83 |
+
15: $e _ { t + 1 } e _ { t } + 1 / m \sum _ { i \in S } h _ { i , t + 1 } - g _ { t + 1 }$
|
| 84 |
+
16: end for
|
| 85 |
+
|
| 86 |
+
Proposition 2. For any step-size $\eta > 0$ , $T$ rounds of FedNova amount to performing $T$ rounds of parallel GD on the surrogate optimization problem given by
|
| 87 |
+
|
| 88 |
+
$$
|
| 89 |
+
\operatorname* { m i n } _ { x } \frac { 1 } { m } \sum _ { i \in S } \frac { 1 } { 2 } \bigg \| \bigg ( \sum _ { \ell = 0 } ^ { \tau _ { i } - 1 } [ I - \eta A _ { i } ] ^ { \ell } \alpha _ { i } A _ { i } \bigg ) ^ { 1 / 2 } ( x - c _ { i } ) \bigg \| ^ { 2 } .
|
| 90 |
+
$$
|
| 91 |
+
|
| 92 |
+
For the proofs of Propositions 1 and 2, see Appendix B. Proposition 2 shows that in the presence of both objective and systems heterogeneity, FedNova minimizes a surrogate loss function whose minimum may not coincide with $x ^ { * }$ .2 Observe from (3) and (5) that using a larger learning rate $\eta$ introduces more distortion to the original problem. In Figure 1, we see how FedProx and FedNova both converge to incorrect minimizers, even for simple instances with two clients and deterministic, quadratic losses. In contrast, FedLin, our proposed approach that we develop in the next section, guarantees linear convergence to the global minimum.
|
| 93 |
+
|
| 94 |
+
Main Takeaway: The main message we want to convey here is that even for deterministic settings, there are non-trivial challenges posed by objective and systems heterogeneity that only get amplified when one additionally considers biased compression. For such scenarios, it is not at all apparent whether (and to what extent) one can match even the basic centralized benchmark of achieving linear convergence for smooth, strongly convex loss functions. To focus on the above unresolved issues, we will primarily consider a deterministic model in this paper. Nonetheless, the general approach we develop applies to the stochastic setting as well, as aptly demonstrated by Theorem 4 in Section 4.
|
| 95 |
+
|
| 96 |
+
# 3 Proposed Algorithm: FedLin
|
| 97 |
+
|
| 98 |
+
In this section, we develop our proposed algorithm FedLin, formally described in Algorithm 1. FedLin is initialized from a common global iterate $\bar { x } _ { 1 } ~ \in ~ \mathbb { R } ^ { d }$ . For simplicity, we assume that $g _ { 1 } = \nabla f ( { \bar { x } } _ { 1 } )$ , i.e., every client has access to the true gradient of $f ( \cdot )$ initially; we can allow $g _ { 1 }$ to be arbitrary as well without affecting the convergence guarantees. FedLin proceeds in rounds: in each round $t$ , starting from a common global model $\bar { x } _ { t }$ , each client $i$ performs $\tau _ { i }$ local training steps in parallel, as per line 5 of Algorithm 1. The key features of our local update rule are as follows: exploiting past gradients to account for objective heterogeneity, using client-specific step-sizes to tackle systems heterogeneity, and employing error-feedback to account for gradient sparsification. We now discuss each of these features in detail.
|
| 99 |
+
|
| 100 |
+
To gain intuition regarding the local step in line 5, note that the ideal local update at client $i$ is $\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta _ { i } \nabla f ( \boldsymbol { x } _ { i , \ell } ^ { ( \bar { t } ) } )$ . However, this requires client $i$ to have access to the gradients of all other clients - which it does not, since clients do not communicate between rounds. To get around this, client $i$ exploits memory, and uses the gradient of the global function $\nabla f ( { \bar { x } } _ { t } )$ from the beginning of round $t$ (when the clients last communicated) as a guiding direction in its update rule. However, since $\nabla f ( { \bar { x } } _ { t } )$ is evaluated at a stale point $x _ { i , 0 } ^ { ( t ) } = \bar { x } _ { t }$ , client $i$ subtracts off $\nabla f _ { i } ( \bar { x } _ { t } )$ from $\nabla f ( { \bar { x } } _ { t } )$ , and adds in the most recently evaluated gradient $\nabla f _ { i } ( x _ { i , \ell } ^ { ( t ) } )$ . This results in the update rule: $\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta _ { i } ( \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) - \nabla f _ { i } ( \bar { \boldsymbol { x } } _ { t } ) + \nabla f ( \bar { \boldsymbol { x } } _ { t } ) )$ . Our local update rule in line 5 is precisely of the above form, where $g _ { t }$ is an inexact version of $\nabla f ( { \bar { x } } _ { t } )$ to account for gradient sparsification.
|
| 101 |
+
|
| 102 |
+
When each client $i$ performs $\tau _ { i }$ local-steps, our analysis reveals that the bound on the drift-term $\| x _ { i , \ell } - \bar { x } _ { t } \|$ scales linearly in $\tau _ { i }$ (see Lemma 9 in Appendix F). Accordingly, to compensate for such drift at client $i$ , the step-size $\eta _ { i }$ needs to be chosen to vary inversely with the number of local steps $\tau _ { i }$ . In fact, the requirement that $\eta _ { i } \propto 1 / \tau _ { i }$ also turns out to be necessary (see Theorem 5), providing further motivation for the choice of client-specific learning rates in FedLin.
|
| 103 |
+
|
| 104 |
+
To explain the gradient sparsification module, let us denote by $\mathcal { C } _ { \delta } : \mathbb { R } ^ { d } \mathbb { R } ^ { d }$ the TOP- $\mathtt { . k }$ operator, where $\delta = d / k$ , and $k \in \bar { \{ 1 , \ldots , d \} }$ . Given any $x \in \mathbb { R } ^ { d }$ , let ${ \mathcal { E } } _ { \delta } ( x )$ be a set containing the indices of the $k$ largest-magnitude components of $x$ . Then, the TOP- ${ \bf \nabla } \cdot { \bf k }$ operator we consider is given by $( \mathcal { C } _ { \delta } ( x ) ) _ { j } = \mathsf { \bar { \Psi } } ( x ) _ { j }$ if $j \in \mathcal { E } _ { \delta } ( x )$ , and $\left( \boldsymbol { \mathcal { C } } _ { \delta } ( \boldsymbol { x } ) \right) _ { j } = 0$ otherwise. Here, we use $( x ) _ { j }$ to denote the $j$ -th component of a vector $x$ . Clearly, a larger $\delta$ implies more aggressive compression. We employ a standard error-feedback mechanism [33–35] at both the server and the clients to account for gradient sparsification. At client $i$ , $\rho _ { i , t }$ represents the accumulated error due to gradient sparsification. At the end of round $t$ , instead of just compressing $\nabla f _ { i } ( \bar { x } _ { t + 1 } )$ , client $i$ instead compresses $\nabla f _ { i } ( \bar { x } _ { t + 1 } ) + \rho _ { i , t }$ , to account for gradient coordinates not transmitted in the past. It then updates the aggregate error via line 12. An analogous description applies to the error-feedback scheme at the server, where $e _ { t }$ is the aggregate error at the beginning of round $t$ . The parameters of FedLin are the client step-sizes $\{ \eta _ { i } \} _ { i \in { \cal S } }$ , and the compression levels $\delta _ { c }$ and $\delta _ { s }$ at the clients and at the server, respectively. We now comment on some related algorithmic ideas.
|
| 105 |
+
|
| 106 |
+
Related Algorithmic Approaches: In the related but different setting of distributed optimization, we note that the idea of exploiting past gradients has been used to design gradient-tracking algorithms [36–40]. In the context of FL, this idea is also related to the variance-reduction technique employed in SCAFFOLD [11]. A major difference of FedLin with the above works is that none of them consider the effect of systems heterogeneity or biased compression. In particular, accounting for the inexact gradient term $g _ { t }$ in our update rule introduces new technical challenges that we address in this paper.
|
| 107 |
+
|
| 108 |
+
There are some additional basic differences between FedLin and SCAFFOLD. To see this, consider the update rule of FedLin without sparsification: $\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta _ { i } ( \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) - \nabla f _ { i } ( \bar { \boldsymbol { x } } _ { t } ) + \nabla f ( \bar { \boldsymbol { x } } _ { t } ) )$ Now suppose the global model $\hat { x } _ { t }$ at the beginning of round $t$ has already converged to $x ^ { * }$ . Since $x _ { i , 0 } ^ { ( t ) } = \bar { x } _ { t } , \forall i \in \mathcal { S }$ , and $\nabla f ( x ^ { * } ) = 0$ , it is easy to see that the iterates of the clients do not evolve any further, as one would ideally want. Thus, the global optimum $x ^ { * }$ can be viewed as a fixed-point of the FedLin update rule. Adapting to our notation, and considering the case when there is no noise in the gradients, the update rule of SCAFFOLD takes the form $\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta ( \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) - \boldsymbol { c } _ { i } + \boldsymbol { c } )$ , where $c _ { i }$ is a ‘control-variate’ maintained by client $i$ , and $c$ is the average of the $c _ { i }$ ’s. Importantly, the control variates $\{ c _ { i } \} _ { i \in { \mathcal { S } } }$ used in round $t$ of SCAFFOLD contain stale terms from round $t - 1$ . As a result, even if $\bar { x } _ { t } = x ^ { * }$ , it may very well be that $( \nabla f _ { i } ( { \bar { x } } _ { t } ) - c _ { i } + c ) \neq 0$ , causing the iterates of the clients to move away from $x ^ { * }$ , and requiring further rounds of communication to average out the imbalance. Thus, the fixed-point property we discussed for FedLin does not hold in general for SCAFFOLD. Our simulations in Section 7 reveal that FedLin converges much faster relative to SCAFFOLD on a simple linear regression model; we conjecture it is precisely due to the reason described above.
|
| 109 |
+
|
| 110 |
+
Keeping aside the differences due to systems heterogeneity and compression, the FedSVRG algorithm in [1] includes a similar gradient correction term as in FedLin, but makes use of certain additional diagonal scaling and pre-conditioning matrices. Although promising empirical results are reported for FedSVRG in [1], these results come with no supporting theoretical guarantees of convergence. In contrast, we will develop rigorous complexity guarantees for FedLin in the following sections. Specifically, we will show that FedLin guarantees linear convergence rates despite the challenges of objective heterogeneity, systems heterogeneity, and aggressive gradient sparsification.
|
| 111 |
+
|
| 112 |
+
# 4 Matching Centralized Rates under Objective and Systems Heterogeneity
|
| 113 |
+
|
| 114 |
+
In this section, we will analyze the performance of FedLin in the face of both objective and systems heterogeneity. To focus solely on the effects of client heterogeneity, we will assume throughout this section that there is no gradient sparsification, i.e., $\delta _ { c } = \delta _ { s } = 1$ . Accordingly, observe that $\rho _ { i , t } = 0 , e _ { t } = 0 , \forall i \in \mathcal { S } , \forall t \in \{ 1 , . . . , \bar { T } \}$ . Thus, the local update rule for FedLin simplifies to
|
| 115 |
+
|
| 116 |
+
$$
|
| 117 |
+
\begin{array} { r } { \boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta _ { i } ( \nabla f _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) - \nabla f _ { i } ( \bar { \boldsymbol { x } } _ { t } ) + \nabla f ( \bar { \boldsymbol { x } } _ { t } ) ) . } \end{array}
|
| 118 |
+
$$
|
| 119 |
+
|
| 120 |
+
Let us denote by $\kappa = L / \mu$ the condition number of an $L$ -smooth and $\mu$ -strongly convex function. Also, let $\eta _ { i } = \bar { \eta } / \tau _ { i } , \forall i \in \mathcal { S }$ , where $\bar { \eta } \in ( 0 , 1 )$ is a flexible parameter that we will specify based on context. We are now ready to state the main results of this section.
|
| 121 |
+
|
| 122 |
+
Theorem 1. (Strongly convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\mu$ -strongly convex. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ , and $\delta _ { c } = \delta _ { s } = 1$ . Then, with $\begin{array} { r } { \eta _ { i } = \frac { 1 } { 6 L \tau _ { i } } , \forall i \dot { \in } \mathcal { S } } \end{array}$ , FedL $_ { i n }$ guarantees:
|
| 123 |
+
|
| 124 |
+
$$
|
| 125 |
+
f ( \bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \leq \left( 1 - \frac { 1 } { 6 \kappa } \right) ^ { T } ( f ( \bar { x } _ { 1 } ) - f ( x ^ { * } ) ) .
|
| 126 |
+
$$
|
| 127 |
+
|
| 128 |
+
Theorem 2. (Convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth and convex. Moreover, suppose $\tau _ { i } \geq$ $1 , \forall i \in S$ , and $\delta _ { c } = \delta _ { s } = 1$ . Then, with $\begin{array} { r } { \eta _ { i } \stackrel { \cdot \cdot } { = } \frac { \mathrm { ~ i ~ } } { 1 0 L \tau _ { i } } , \forall i \in \mathcal { S } , } \end{array}$ , FedLin guarantees:
|
| 129 |
+
|
| 130 |
+
$$
|
| 131 |
+
f \left( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \bar { x } _ { t } \right) - f ( x ^ { * } ) \leq \frac { 1 0 L } { T } \left( \Vert \bar { x } _ { 1 } - x ^ { * } \Vert ^ { 2 } - \Vert \bar { x } _ { T + 1 } - x ^ { * } \Vert ^ { 2 } \right) .
|
| 132 |
+
$$
|
| 133 |
+
|
| 134 |
+
Theorem 3. (Non-convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ and $\delta _ { c } = \delta _ { s } = 1$ . Then, with $\begin{array} { r } { \eta _ { i } = \frac { 1 } { 2 6 L \tau _ { i } } , \forall i \in \mathcal { S } } \end{array}$ , FedLin guarantees:
|
| 135 |
+
|
| 136 |
+
$$
|
| 137 |
+
\operatorname* { m i n } _ { t \in \left[ T \right] } \left\| \nabla f ( \bar { x } _ { t } ) \right\| ^ { 2 } \leq \frac { 5 2 L } { T } ( f ( \bar { x } _ { 1 } ) - f ( \bar { x } _ { T + 1 } ) ) .
|
| 138 |
+
$$
|
| 139 |
+
|
| 140 |
+
Noisy Case Analysis: We now analyze the performance of FedLin under a general stochastic oracle model. For each $i \in S$ and $x \in \mathbb { R } ^ { d }$ , let $q _ { i } ( \bar { x } )$ be an unbiased estimate of the gradient $\nabla f _ { i } ( x )$ with variance bounded above by σ2. We consider the update rule: x(t)i,\`+1 = x(t)i,\` $\boldsymbol { x } _ { i , \ell + 1 } ^ { ( t ) } = \boldsymbol { x } _ { i , \ell } ^ { ( t ) } - \eta _ { i } \big ( q _ { i } ( \boldsymbol { x } _ { i , \ell } ^ { ( t ) } ) - q _ { i } ( \bar { \boldsymbol { x } } _ { t } ) +$ $q ( \bar { x } _ { t } ) )$ , where $\begin{array} { r } { q ( x ) \triangleq 1 / m \sum _ { i \in S } q _ { i } ( x ) , \forall x \in \mathbb { R } ^ { d } } \end{array}$ . We then have the following result.
|
| 141 |
+
|
| 142 |
+
Theorem 4. (Strongly convex case with noise) Consider the above stochastic oracle model. Suppose each $f _ { i } ( x )$ is $L$ -smooth and -strongly convex. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ , and $\delta _ { c } = \delta _ { s } = 1$ . For each $i \in S$ , let $\begin{array} { r } { \eta _ { i } = \frac { \bar { \eta } } { \tau _ { i } } } \end{array}$ , where $\bar { \eta } \in ( 0 , 1 )$ satisfies $\begin{array} { r } { \bar { \eta } < \frac { 1 } { 6 L } } \end{array}$ . Then, $\forall t \in [ T ]$ , FedLin guarantees:
|
| 143 |
+
|
| 144 |
+
$$
|
| 145 |
+
\mathbb { E } [ \| \bar { x } _ { t + 1 } - x ^ { * } \| ^ { 2 } ] \leq \left( 1 - \frac { \bar { \eta } \mu } { 2 } \right) \mathbb { E } [ \| \bar { x } _ { t } - x ^ { * } \| ^ { 2 } ] + 2 5 \bar { \eta } ^ { 2 } \sigma ^ { 2 } .
|
| 146 |
+
$$
|
| 147 |
+
|
| 148 |
+
The proofs of Theorems 1, 2, 3, and 4 are provided in Appendix F.
|
| 149 |
+
|
| 150 |
+
Main Takeaways: From Theorems 1, 2, and 3, we note that FedLin matches the convergence guarantees of centralized gradient descent (up to constants) for smooth, strongly convex, convex, and non-convex settings, respectively. As far as we are aware, this is the first work to provide such comprehensive guarantees under arbitrary objective and systems heterogeneity. In fact, all our results continue to hold even when the operating speeds of the client machines vary across rounds, i.e., $\tau _ { i }$ is allowed to be a function of $t$ . Each client $i$ can simply adjust its learning rate $\eta _ { i } \propto 1 / \tau _ { i } ( t )$ locally to account for such variations. The bound for the noisy case in Theorem 4 resembles that of centralized SGD [41]: with a time-varying parameter $\bar { \eta _ { t } } = \dot { O } ( 1 / t )$ , we get the standard $O ( 1 / T )$ rate after $T$ rounds (using the exact same arguments as in [41]). The key thing to note here is that despite arbitrary heterogeneity, the assumptions we make on the stochastic gradients are the same as those made in the analysis of centralized SGD: unbiased gradients with bounded variance, nothing more.
|
| 151 |
+
|
| 152 |
+
Comparison with Related Work: In the recent paper [10], the authors propose FedSplit, and analyze it in a deterministic setting. For strongly-convex and smooth loss functions, FedSplit guarantees linear convergence, but only to a non-vanishing neighborhood of $x ^ { * }$ . Thus, like FedAvg [2], FedProx [22], and FedNova [23], FedSplit fails to guarantee exact linear convergence to $x ^ { * }$ . Empirically, we observe that FedSplit diverges on certain instances; see Appendix J. Compared to these algorithms, we see from Theorem 1 that FedLin guarantees linear convergence to $x ^ { * }$ . Notably, the linear convergence rate we obtain in Theorem 1 under both objective and systems heterogeneity is the best rate we know of in $F L$ , and matches that of SCAFFOLD [11] where only objective heterogeneity is considered.3 The model of systems heterogeneity we study is taken from [23], where the authors provide guarantees only for the non-convex case under a bounded dissimilarity assumption. In contrast, our results cover all the three standard settings - strongly-convex, convex, and non-convex - without requiring any bounded dissimilarity assumption. For further related work on straggler-robust distributed learning algorithms (without objective heterogeneity or local steps), see [43–48].
|
| 153 |
+
|
| 154 |
+
# 4.1 The Price of Infrequent Communication
|
| 155 |
+
|
| 156 |
+
In this section, we take a closer look at the effect of performing multiple local steps on the convergence rate. To do so, we assume that all clients perform the same number of local steps $H$ , i.e., there is no communication for $H$ consecutive time-steps between two communication rounds. Now consider a centralized baseline where each client can communicate with every other client at all times (i.e., even between rounds). In this case, since each client can always access $\nabla f ( x )$ , gradient descent yields
|
| 157 |
+
|
| 158 |
+
$$
|
| 159 |
+
f ( \bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \leq \exp ( - \frac { 1 } { \kappa } T H ) ( f ( \bar { x } _ { 1 } ) - f ( x ^ { * } ) )
|
| 160 |
+
$$
|
| 161 |
+
|
| 162 |
+
after $T$ rounds, with $H$ synchronized local iterations within each round. Based on Theorem 1, observe that we lose out by a factor of $H$ in the exponent relative to the centralized baseline. Notably, both in the centralized case, and in FedLin, each client queries the gradient of its local objective $H$ times in each round, thereby making $T H$ gradient queries over $T$ rounds. Thus, relative to a centralized baseline, FedLin incurs the same computational cost in terms of gradient queries, and reduces communication by a factor of $H$ , at the expense of a convergence rate that is slower by a factor of $H$ . We emphasize here that just as with FedLin, $H$ does not show up in the convergence rate (exponent) of algorithms like FedSplit [10] and SCAFFOLD [11] either.
|
| 163 |
+
|
| 164 |
+
The primary reason for the slower convergence rate (relative to a centralized baseline) stems from the need to set $\eta \propto 1 / H$ to mitigate client-drift under objective heterogeneity. At this stage, one may conjecture that the above requirement is simply an artifact of a conservative analysis of Algorithm 1, and that a more refined analysis will reveal the utility of performing more local steps even in the heterogeneous setting. Our next result suggests otherwise; for a proof, see Appendix $\mathrm { E }$ .
|
| 165 |
+
|
| 166 |
+
Theorem 5. (Lower bound for FedLin) Suppose $\delta _ { c } = \delta _ { s } = 1$ , and $\tau _ { i } = H , \eta _ { i } = \eta , \forall i \in S$ . Then, given any $L \ge 1 4$ and $H \geq 2$ , there exists an instance involving 2 clients where each $f _ { i } ( x ) , i \in \{ 1 , 2 \}$ , is 1-strongly convex and $L$ -smooth, and an initial condition $\scriptstyle { \bar { x } } _ { 1 }$ , such that FedLin initialized from $\scriptstyle { \bar { x } } _ { 1 }$ generates a sequence of iterates $\{ \bar { x } _ { t } \}$ satisfying the following for any $T \geq 1$ :
|
| 167 |
+
|
| 168 |
+
$$
|
| 169 |
+
\begin{array} { r } { \| \bar { x } _ { T + 1 } - x ^ { * } \| ^ { 2 } \geq \exp \left( - 4 T \right) \| \bar { x } _ { 1 } - x ^ { * } \| ^ { 2 } ; f ( \bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \geq \exp ( - 4 T ) ( f ( \bar { x } _ { 1 } ) - f ( x ^ { * } ) ) . } \end{array}
|
| 170 |
+
$$
|
| 171 |
+
|
| 172 |
+
Main Takeaways: There are several key implications of Theorem 5. First, it complements Theorem 1 by providing a matching lower bound. We believe ours is the first work to provide a tight linear convergence rate analysis: [11] and [10] only provide upper-bounds for SCAFFOLD and FedSplit, respectively. Second, our analysis of Theorem 5 in Appendix E indicates that there are problem instances where setting $\eta \propto 1 / H$ is in fact necessary to guarantee convergence to $x ^ { * }$ . As a result, for such problem instances, no matter how many local steps $H$ each client performs, the error at the end of $T$ rounds remains bounded below by an $H$ -independent quantity, as is apparent from (10). Perhaps surprisingly, we show in Appendix E that the lower bound in Theorem 5 even applies to simple instances with non-identical quadratic losses (across clients) where every $f _ { i } ( x )$ has the same minimum! This is particularly insightful since it highlights the limitations of exploiting stale gradient terms in the local update rule (as is done in both FedLin and SCAFFOLD), and suggests the need for more informed updating schemes that explicitly take into account the level of statistical heterogeneity.
|
| 173 |
+
|
| 174 |
+
Proof Idea for Theorem 5: To establish Theorem 5, we set up an instance involving two clients with quadratic loss functions. Our main idea is to relate the convergence of FedLin to the Schur stability of an appropriate discrete-time linear time-invariant (LTI) system. Based on this connection, we show that guaranteeing stability necessitates setting $\eta \propto 1 / H$ , which immediately leads to the lower bound. We believe that the same technique can be used to establish a similar lower bound for SCAFFOLD.
|
| 175 |
+
|
| 176 |
+
# 5 Gradient Sparsification at Server
|
| 177 |
+
|
| 178 |
+
In this section, our focus will be on addressing the following question: For strongly convex and smooth deterministic functions, and in the presence of both objective and systems heterogeneity, can we still hope for linear convergence to $x ^ { * }$ when gradients are sparsified at the server? Interestingly, we will show that not only is it possible to converge linearly to $x ^ { * }$ , it is possible to do so without any error-feedback. Moreover, this claim holds regardless of how aggressive the server is in its sparsification scheme: it may even transmit just a single component of the aggregated gradient vector.
|
| 179 |
+
|
| 180 |
+
To isolate the impact of server-level sparsification, we will assume throughout this section that gradients are not sparsified at the clients, i.e., $\delta _ { c } = 1$ . Consequently, $h _ { i , t + 1 } = \nabla f _ { i } ( \bar { x } _ { t + 1 } ) , \forall i \in$ $\mathcal { S } , \forall t \in \{ 1 , \ldots , T \}$ . We begin by considering a simpler variant of FedLin with no error-feedback at the server side, i.e., line 15 is skipped, and $g _ { t + 1 }$ in line 14 of Algo. 1 is instead updated as follows
|
| 181 |
+
|
| 182 |
+
$$
|
| 183 |
+
g _ { t + 1 } = { \mathcal C } _ { \delta _ { s } } \left( \frac { 1 } { m } \sum _ { i \in \mathcal { S } } \nabla f _ { i } ( \bar { x } _ { t + 1 } ) \right) = { \mathcal C } _ { \delta _ { s } } \left( \nabla f ( \bar { x } _ { t + 1 } ) \right) .
|
| 184 |
+
$$
|
| 185 |
+
|
| 186 |
+
Theorem 6. (Sparsification at server with no error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\mu$ -strongly convex. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ , and $\delta _ { c } = 1$ . Consider a variant of FedLin, where line $^ { I 4 }$ is replaced by equation (11), and line 15 is skipped, i.e., there is no error-feedback. Then, with ηi = 12(2+√δs)Lτi , , this variant of FedLin guarantees
|
| 187 |
+
|
| 188 |
+
$$
|
| 189 |
+
f ( \bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \leq \bigg ( 1 - \frac { 1 } { 2 \delta _ { s } \left( 2 + \sqrt { \delta _ { s } } \right) \kappa } \bigg ) ^ { T } ( f ( \bar { x } _ { 1 } ) - f ( x ^ { * } ) ) .
|
| 190 |
+
$$
|
| 191 |
+
|
| 192 |
+
Main Takeaways: From Theorem 6, we see that even without error-feedback, it is possible to linearly converge to $x ^ { * }$ ; the rate of convergence, however, is inversely proportional to $\delta _ { s } ^ { \frac { 3 } { 2 } }$ . Thus, Theorem 6 quantifies the trade-off between the level of sparsification at the server, and the rate of convergence. When there is no gradient compression, i.e., when $\delta _ { s } = 1$ , we exactly recover Theorem 1.
|
| 193 |
+
|
| 194 |
+
One may ask: Is there any potential benefit to employing error-feedback when gradients are sparsified at the server? Our next result answers this question in the affirmative.
|
| 195 |
+
|
| 196 |
+
Theorem 7. (Sparsification at server with error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\mu$ -strongly convex. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ , and $\delta _ { c } = 1$ . Let the step-size for client i be chosen as $\begin{array} { r } { \eta _ { i } = \frac { 1 } { 7 2 L \delta _ { s } \tau _ { i } } } \end{array}$ . Then, FedLin guarantees:
|
| 197 |
+
|
| 198 |
+
$$
|
| 199 |
+
f ( \bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \leq 2 \kappa \bigg ( 1 - \frac { 1 } { 9 6 \delta _ { s } \kappa } \bigg ) ^ { T } \left( f ( \bar { x } _ { 1 } ) - f ( x ^ { * } ) \right) .
|
| 200 |
+
$$
|
| 201 |
+
|
| 202 |
+
For proofs of Theorems 6 and 7, see Appendix G and I.
|
| 203 |
+
|
| 204 |
+
Main Takeaways: Comparing the guarantee of Theorem 6 with that of Theorem 7, we note that the convergence rate is inversely proportional to $\delta _ { s } ^ { \frac { 3 } { 2 } }$ in the former, and inversely proportional to $\delta _ { s }$ in the latter. Thus, the main message here is that employing error-feedback leads to a faster convergence rate by improving the dependence of the rate on $\delta _ { s }$ .
|
| 205 |
+
|
| 206 |
+
# 6 Gradient Sparsification at Clients
|
| 207 |
+
|
| 208 |
+
In this section, we will turn our attention to the case when gradients are sparsified at the clients prior to being transmitted to the server. Throughout this section, we will assume that gradients are not compressed any further at the server side, i.e., $\delta _ { s } = 1$ . To proceed, we will need to make the following bounded gradient dissimilarity assumption.
|
| 209 |
+
|
| 210 |
+
Assumption 1. There exist constants $C \geq 1$ and $D \geq 0$ such that the following holds $\forall x \in \mathbb { R } ^ { d }$
|
| 211 |
+
|
| 212 |
+
$$
|
| 213 |
+
{ \frac { 1 } { m } } \sum _ { i = 1 } ^ { m } \left\| \nabla f _ { i } ( x ) \right\| ^ { 2 } \leq C \left\| \nabla f ( x ) \right\| ^ { 2 } + D .
|
| 214 |
+
$$
|
| 215 |
+
|
| 216 |
+
The following is the main result of this section; for a proof, see Appendix H.
|
| 217 |
+
|
| 218 |
+
Theorem 8. (Sparsification at clients with error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and -strongly convex, and suppose Assumption 1 holds. Moreover, suppose $\tau _ { i } \geq 1 , \forall i \in S$ , and $\delta _ { s } = 1$ . Let the step-size for client i be chosen as $\begin{array} { r } { \eta _ { i } = \frac { \bar { \eta } } { \tau _ { i } } } \end{array}$ , where $\bar { \eta } \in ( 0 , 1 )$ satisfies $\begin{array} { r } { \bar { \eta } \le \frac { 1 } { 7 2 L \delta _ { c } C } } \end{array}$ . Then, FedLin guarantees:
|
| 219 |
+
|
| 220 |
+
$$
|
| 221 |
+
\left\| { \bar { x } } _ { T + 1 } - x ^ { * } \right\| ^ { 2 } \leq 2 { \left( 1 - \frac { 3 } { 4 } { \bar { \eta } } \mu \right) } ^ { T } \left\| { \bar { x } } _ { 1 } - x ^ { * } \right\| ^ { 2 } + \frac { 1 6 } { 3 } { \bar { \eta } } \left( \frac { 6 } { \delta _ { c } C } + \delta _ { c } \right) \frac { D } { \mu } .
|
| 222 |
+
$$
|
| 223 |
+
|
| 224 |
+
Main Takeaways: Intuitively, one would expect that sparsifying gradients at each client prior to aggregation at the server would inject more errors than when gradients are first accurately aggregated at the server, and then the aggregated gradient vector is sparsified: Theorems 6 and 8 support this intuition. For the former, we neither required error-feedback nor Assumption 1 to guarantee linear convergence to the global minimum $x ^ { * }$ ; for the latter, even with error-feedback and the bounded gradient dissimilarity assumption, we can establish linear convergence to only a neighborhood of $x ^ { * }$ , in general. From (13), we note that the size of this neighborhood scales linearly with $D$ - a measure of objective heterogeneity. In particular, when $D = 0$ , the iterates $\bar { x } _ { t }$ converge exactly to $x ^ { * }$ .
|
| 225 |
+
|
| 226 |
+
Remark 1. To the best of our knowledge, our results in Sections 5 and 6 constitute the first formal analysis of biased gradient sparsification in FL. In particular, we significantly generalize the recent results in [49] for a single worker to a multi-client $F L$ setting with both objective and systems heterogeneity. To arrive at these results, we develop a new potential-function based proof technique in Appendix H. For more related work on compression in distributed learning, see Appendix A.
|
| 227 |
+
|
| 228 |
+
Extensions: We studied the effect of compressing information at the server and at the clients separately, with the goal of identifying the key differences between each of these mechanisms. The analysis techniques we developed in the process pave the way for studying various natural extensions: (i) combined sparsification at both the clients and the server; (ii) gradient sparsification in tandem with model parameter compression; and (iii) stochastic counterparts of Theorems 6, 7, and 8.
|
| 229 |
+
|
| 230 |
+
# 7 Experimental Results
|
| 231 |
+
|
| 232 |
+
In this section, we provide numerical results for FedLin on a least squares problem to validate our theory. In Appendix K, we also provide additional numerical results on a logistic regression problem.
|
| 233 |
+
|
| 234 |
+
For now, we consider the following least squares regression problem:
|
| 235 |
+
|
| 236 |
+
$$
|
| 237 |
+
\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) = \operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \frac { 1 } { 2 } \| A _ { i } x - b _ { i } \| ^ { 2 } ,
|
| 238 |
+
$$
|
| 239 |
+
|
| 240 |
+
where $A _ { i } \in \mathbb { R } ^ { 5 0 0 \times 1 0 0 }$ is a design matrix and $b _ { i } \in \mathbb { R } ^ { 5 0 0 }$ is a response vector. The client objective functions, $f _ { i } ( x )$ are strongly convex. Assuming that all design matrices are full column rank, problem (14) admits a unique minimizer. To generate synthetic data, for each client $i \in \mathcal { S } = \{ 1 , . . . , \bar { 2 0 } \}$ , we generate $A _ { i }$ and $b _ { i }$ according to the model $b _ { i } = A _ { i } x _ { i } + \varepsilon _ { i }$ , where $x _ { i }$ is a weight vector and $\varepsilon _ { i } \in \mathrm { \mathbb { R } } ^ { 5 0 0 }$ is a disturbance. In particular, we generate $[ A _ { i } ] _ { j k } \stackrel { i . i . d . } { \sim } { \cal N } ( 0 , 1 )$ , and $\varepsilon _ { i } \sim \mathcal { N } ( 0 , 0 . 5 I _ { 5 0 0 } )$ , $\forall i \in S$ To capture statistical heterogeneity, the entries of the local true parameter of client $i$ are modeled as $[ x _ { i } ] _ { k } \overset { \vartriangle } { \sim } \mathcal { N } ( u _ { i } , 1 )$ , $k \in \{ 1 , \ldots , 1 0 0 \}$ , where $u _ { i } \sim \mathcal { N } ( 0 , \alpha )$ and $\alpha \geq 0$ . Hence, $\alpha$ controls the level of statistical heterogeneity. To model the effect of systems heterogeneity, for each client $i \in S$ , the number of local steps is drawn uniformly and independently from [2, 100]. We will primarily focus on a deterministic setting here for our experiments; in Appendix $\mathrm { L }$ , we evaluate FedLin on a standard stochastic oracle model. Our experiments in Appendix $\mathrm { L }$ reveal that under a noisy oracle, FedLin guarantees linear convergence to a ball around the true minimum, exactly as suggested by Thm. 4.
|
| 241 |
+
|
| 242 |
+
Gradient Sparsification at Server. We first consider a variant of FedLin where gradient sparsification is implemented only at the server side and without any error-feedback. In particular, we consider the cases where $\delta _ { s } \in \{ 2 , 4 \}$ , which correspond to the implementation of a TOP-50 and a TOP-25 operator, respectively. For comparison, we also plot the resulting performance when no gradient sparsification is implemented at the server. To examine the effect of statistical heterogeneity on the performance of FedLin, we generate two synthetic datasets corresponding to two different levels of heterogeneity in the clients’ local objectives, namely $\alpha = 1 0$ and $\alpha = 5 0$ . As illustrated in Fig. 2, irrespective of the level of gradient sparsification on the server side, FedLin achieves linear convergence to the true minimum in the presence of both objective and systems heterogeneity, confirming Theorem 6. Also, both the convergence speed and accuracy of FedLin remain unaffected as the level of heterogeneity in the clients’ objective functions increases.
|
| 243 |
+
|
| 244 |
+
|
| 245 |
+
|
| 246 |
+
|
| 247 |
+
Figure 2: Server-side sparsification results for FedLin. The constant $\bar { \eta }$ is fixed at $1 \dot { 0 } ^ { - 2 }$ . Left: $\alpha = 1 0$ . Right: $\alpha = 5 0$ .
|
| 248 |
+
|
| 249 |
+
Gradient Sparsification at Clients. Next, we implement gradient sparsification only at the clients’ side, i.e. $\delta _ { s } ~ = ~ 1$ . In particular, we consider the cases where $\delta _ { c } \in \{ 4 / 3 , 2 \}$ , which correspond to the implementation of a TOP-75 and a TOP-50 operator, respectively. Once again, we generate two synthetic datasets with different levels of objective heterogeneity, namely $\alpha = 1$ and $\alpha = 1 0$ . As illustrated in Fig. 3, unlike the server case, FedLin with sparsification at the clients’ side converges linearly, but with a non-vanishing error that increases as the value of $\delta _ { c }$ increases. This aligns with the conclusions of Theorem 8. Furthermore, the level of objective heterogeneity has a direct impact on the convergence error. In particular, for the same level of gradient sparsification, higher levels of objective heterogeneity result in larger values of the convergence error.
|
| 250 |
+
|
| 251 |
+
|
| 252 |
+
Figure 3: Client-side sparsification results for FedLin. The constant $\bar { \eta }$ is fixed at $5 \times 1 0 ^ { - 4 }$ . Left: $\alpha = 1$ . Right: $\alpha = 1 0$ .
|
| 253 |
+
Figure 4: Comparison of FedLin with SCAFFOLD. (Left) Deterministic setting. (Right) General stochastic oracle model: unbiased gradients with variance $\sigma = 1 0 ^ { - 1 }$ .
|
| 254 |
+
|
| 255 |
+
Comparison with SCAFFOLD. We now compare FedLin with SCAFFOLD on the least squares regression setup described above. To make a fair comparison, we assume that there is no systems heterogeneity or gradient compression. For implementing SCAFFOLD, we use Option II in their paper [11] for updating the control variates. We set the number of local steps $H = 2 0$ , the statistical heterogeneity parameter $\alpha = 1 0$ , and use a step-size of $1 0 ^ { - 3 }$ for both algorithms (the step-size was tuned to get best results). For the deterministic setting, we note from Fig. 4 that FedLin con
|
| 256 |
+
|
| 257 |
+
verges much faster compared to SCAFFOLD. This trend persists when we perturb the gradients with zero-mean Gaussian noise with variance $\sigma = 1 0 ^ { - 1 }$ . We conjecture that the faster convergence of FedLin stems from the fact that it uses less stale gradient correction terms relative to the control variates of SCAFFOLD; see the discussion about the fixed point property of FedLin in Sec. 3.
|
| 258 |
+
|
| 259 |
+
# 8 Conclusion
|
| 260 |
+
|
| 261 |
+
We developed a novel algorithmic framework called FedLin to tackle some of the key challenges in FL, namely objective heterogeneity, systems heterogeneity, and imprecise communication. We showed that FedLin enjoys strong theoretical guarantees: (i) FedLin matches centralized rates, and, in particular, guarantees linear convergence to the global minimum under arbitrary objective and systems heterogeneity; and (ii) preserves linear convergence rates despite aggressive gradient sparsification. We also established a tight lower-bound for FedLin, highlighting that even mild statistical heterogeneity can end up hurting convergence rates - this is the first such result in FL. Our current approach requires two passes of communication between the clients and the server in each round. Moreover, our analysis does not account for partial client participation. As future work, we plan to address these limitations. We also plan to investigate other federated learning formulations (beyond supervised learning) where statistical heterogeneity can potentially help.
|
| 262 |
+
|
| 263 |
+
Acknowledgement: This work was supported by NSF Award 1837253, NSF CAREER award CIF 1943064, and the Air Force Office of Scientific Research Young Investigator Program (AFOSR-YIP) under award FA9550-20-1-0111.
|
| 264 |
+
|
| 265 |
+
References
|
| 266 |
+
[1] Jakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richtárik. Federated optimiza-\` tion: Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016.
|
| 267 |
+
[2] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282. PMLR, 2017.
|
| 268 |
+
[3] Keith Bonawitz, Hubert Eichner, Wolfgang Grieskamp, Dzmitry Huba, Alex Ingerman, Vladimir Ivanov, Chloe Kiddon, Jakub Konecnˇ y, Stefano Mazzocchi, H Brendan McMahan, et al. \` Towards federated learning at scale: System design. arXiv preprint arXiv:1902.01046, 2019.
|
| 269 |
+
[4] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.
|
| 270 |
+
[5] Tian Li, Anit Kumar Sahu, Ameet Talwalkar, and Virginia Smith. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020.
|
| 271 |
+
[6] Xiang Li, Kaixuan Huang, Wenhao Yang, Shusen Wang, and Zhihua Zhang. On the convergence of fedavg on non-iid data. arXiv preprint arXiv:1907.02189, 2019.
|
| 272 |
+
[7] Grigory Malinovskiy, Dmitry Kovalev, Elnur Gasanov, Laurent Condat, and Peter Richtarik. From local sgd to local fixed-point methods for federated learning. In International Conference on Machine Learning, pages 6692–6701. PMLR, 2020.
|
| 273 |
+
[8] Zachary Charles and Jakub Konecnˇ y. On the outsized importance of learning rates in local \` update methods. arXiv preprint arXiv:2007.00878, 2020.
|
| 274 |
+
[9] Zachary Charles and Jakub Konecnˇ y. Convergence and accuracy trade-offs in federated learning \` and meta-learning. In International Conference on Artificial Intelligence and Statistics, pages 2575–2583. PMLR, 2021.
|
| 275 |
+
[10] Reese Pathak and Martin J Wainwright. Fedsplit: An algorithmic framework for fast federated optimization. arXiv preprint arXiv:2005.05238, 2020.
|
| 276 |
+
[11] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.
|
| 277 |
+
[12] Sebastian U Stich. Local sgd converges fast and communicates little. arXiv preprint arXiv:1805.09767, 2018.
|
| 278 |
+
[13] Jianyu Wang and Gauri Joshi. Cooperative sgd: A unified framework for the design and analysis of communication-efficient sgd algorithms. arXiv preprint arXiv:1808.07576, 2018.
|
| 279 |
+
[14] Artin Spiridonoff, Alex Olshevsky, and Ioannis Ch Paschalidis. Local sgd with a communication overhead depending only on the number of workers. arXiv preprint arXiv:2006.02582, 2020.
|
| 280 |
+
[15] Amirhossein Reisizadeh, Aryan Mokhtari, Hamed Hassani, Ali Jadbabaie, and Ramtin Pedarsani. Fedpaq: A communication-efficient federated learning method with periodic averaging and quantization. In International Conference on Artificial Intelligence and Statistics, pages 2021– 2031. PMLR, 2020.
|
| 281 |
+
[16] Farzin Haddadpour, Mohammad Mahdi Kamani, Mehrdad Mahdavi, and Viveck Cadambe. Local sgd with periodic averaging: Tighter analysis and adaptive synchronization. In Advances in Neural Information Processing Systems, pages 11082–11094, 2019.
|
| 282 |
+
[17] Blake Woodworth, Kumar Kshitij Patel, Sebastian U Stich, Zhen Dai, Brian Bullins, H Brendan McMahan, Ohad Shamir, and Nathan Srebro. Is local sgd better than minibatch sgd? arXiv preprint arXiv:2002.07839, 2020.
|
| 283 |
+
[18] Ahmed Khaled, Konstantin Mishchenko, and Peter Richtárik. First analysis of local gd on heterogeneous data. arXiv preprint arXiv:1909.04715, 2019.
|
| 284 |
+
[19] Ahmed Khaled, Konstantin Mishchenko, and Peter Richtárik. Tighter theory for local sgd on identical and heterogeneous data. In International Conference on Artificial Intelligence and Statistics, pages 4519–4529. PMLR, 2020.
|
| 285 |
+
[20] Farzin Haddadpour and Mehrdad Mahdavi. On the convergence of local descent methods in federated learning. arXiv preprint arXiv:1910.14425, 2019.
|
| 286 |
+
[21] Anastasia Koloskova, Nicolas Loizou, Sadra Boreiri, Martin Jaggi, and Sebastian U Stich. A unified theory of decentralized sgd with changing topology and local updates. arXiv preprint arXiv:2003.10422, 2020.
|
| 287 |
+
[22] Anit Kumar Sahu, Tian Li, Maziar Sanjabi, Manzil Zaheer, Ameet Talwalkar, and Virginia Smith. On the convergence of federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 3, 2018.
|
| 288 |
+
[23] Jianyu Wang, Qinghua Liu, Hao Liang, Gauri Joshi, and H Vincent Poor. Tackling the objective inconsistency problem in heterogeneous federated optimization. Advances in Neural Information Processing Systems, 33, 2020.
|
| 289 |
+
[24] Alham Fikri Aji and Kenneth Heafield. Sparse communication for distributed gradient descent. arXiv preprint arXiv:1704.05021, 2017.
|
| 290 |
+
[25] Yujun Lin, Song Han, Huizi Mao, Yu Wang, and William J Dally. Deep gradient compression: Reducing the communication bandwidth for distributed training. arXiv preprint arXiv:1712.01887, 2017.
|
| 291 |
+
[26] Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Qsgd: Communication-efficient sgd via gradient quantization and encoding. Advances in Neural Information Processing Systems, 30:1709–1720, 2017.
|
| 292 |
+
[27] Wei Wen, Cong Xu, Feng Yan, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Terngrad: Ternary gradients to reduce communication in distributed deep learning. In Advances in neural information processing systems, pages 1509–1519, 2017.
|
| 293 |
+
[28] Sarit Khirirat, Hamid Reza Feyzmahdavian, and Mikael Johansson. Distributed learning with compressed gradients. arXiv preprint arXiv:1806.06573, 2018.
|
| 294 |
+
[29] Konstantin Mishchenko, Eduard Gorbunov, Martin Takác, and Peter Richtárik. Distributed ˇ learning with compressed gradient differences. arXiv preprint arXiv:1901.09269, 2019.
|
| 295 |
+
[30] Zhize Li, Dmitry Kovalev, Xun Qian, and Peter Richtárik. Acceleration for compressed gradient descent in distributed and federated optimization. arXiv preprint arXiv:2002.11364, 2020.
|
| 296 |
+
[31] Samuel Horváth, Dmitry Kovalev, Konstantin Mishchenko, Sebastian Stich, and Peter Richtárik. Stochastic distributed learning with gradient quantization and variance reduction. arXiv preprint arXiv:1904.05115, 2019.
|
| 297 |
+
[32] Farzin Haddadpour, Mohammad Mahdi Kamani, Aryan Mokhtari, and Mehrdad Mahdavi. Federated learning with compression: Unified analysis and sharp guarantees. arXiv preprint arXiv:2007.01154, 2020.
|
| 298 |
+
[33] Frank Seide, Hao Fu, Jasha Droppo, Gang Li, and Dong Yu. 1-bit stochastic gradient descent and its application to data-parallel distributed training of speech dnns. In Fifteenth Annual Conference of the International Speech Communication Association, 2014.
|
| 299 |
+
[34] Sebastian U Stich, Jean-Baptiste Cordonnier, and Martin Jaggi. Sparsified sgd with memory. In Advances in Neural Information Processing Systems, pages 4447–4458, 2018.
|
| 300 |
+
[35] Dan Alistarh, Torsten Hoefler, Mikael Johansson, Nikola Konstantinov, Sarit Khirirat, and Cédric Renggli. The convergence of sparsified gradient methods. In Advances in Neural Information Processing Systems, pages 5973–5983, 2018.
|
| 301 |
+
[36] Guannan Qu and Na Li. Harnessing smoothness to accelerate distributed optimization. IEEE Transactions on Control of Network Systems, 5(3):1245–1260, 2017.
|
| 302 |
+
[37] Angelia Nedic, Alex Olshevsky, and Wei Shi. Achieving geometric convergence for distributed optimization over time-varying graphs. SIAM Journal on Optimization, 27(4):2597–2633, 2017.
|
| 303 |
+
[38] Chenguang Xi, Ran Xin, and Usman A Khan. Add-opt: Accelerated distributed directed optimization. IEEE Transactions on Automatic Control, 63(5):1329–1339, 2017.
|
| 304 |
+
[39] Shi Pu and Angelia Nedic. Distributed stochastic gradient tracking methods. ´ Mathematical Programming, pages 1–49, 2020.
|
| 305 |
+
[40] Ran Xin, Anit Kumar Sahu, Usman A Khan, and Soummya Kar. Distributed stochastic optimization with gradient tracking over strongly-connected networks. In Proc. of the 58th IEEE Conference on Decision and Control (CDC), pages 8353–8358, 2019.
|
| 306 |
+
[41] Arkadi Nemirovski, Anatoli Juditsky, Guanghui Lan, and Alexander Shapiro. Robust stochastic approximation approach to stochastic programming. SIAM Journal on optimization, 19(4):1574– 1609, 2009.
|
| 307 |
+
[42] Eduard Gorbunov, Filip Hanzely, and Peter Richtárik. Local sgd: Unified theory and new efficient methods. In International Conference on Artificial Intelligence and Statistics, pages 3556–3564. PMLR, 2021.
|
| 308 |
+
[43] Amirhossein Reisizadeh, Isidoros Tziotis, Hamed Hassani, Aryan Mokhtari, and Ramtin Pedarsani. Straggler-resilient federated learning: Leveraging the interplay between statistical accuracy and system heterogeneity. arXiv preprint arXiv:2012.14453, 2020.
|
| 309 |
+
[44] Sanghamitra Dutta, Jianyu Wang, and Gauri Joshi. Slow and stale gradients can win the race. arXiv preprint arXiv:2003.10579, 2020.
|
| 310 |
+
[45] Jianyu Wang and Gauri Joshi. Adaptive communication strategies to achieve the best errorruntime trade-off in local-update sgd. arXiv preprint arXiv:1810.08313, 2018.
|
| 311 |
+
[46] Rawad Bitar, Mary Wootters, and Salim El Rouayheb. Stochastic gradient coding for straggler mitigation in distributed learning. IEEE Journal on Selected Areas in Information Theory, 2020.
|
| 312 |
+
[47] Nuwan Ferdinand and Stark C Draper. Anytime stochastic gradient descent: A time to hear from all the workers. In 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 552–559. IEEE, 2018.
|
| 313 |
+
[48] Amirhossein Reisizadeh, Hossein Taheri, Aryan Mokhtari, Hamed Hassani, and Ramtin Pedarsani. Robust and communication-efficient collaborative learning. In Advances in Neural Information Processing Systems, pages 8388–8399, 2019.
|
| 314 |
+
[49] Aleksandr Beznosikov, Samuel Horváth, Peter Richtárik, and Mher Safaryan. On biased compression for distributed learning. arXiv preprint arXiv:2002.12410, 2020.
|
| 315 |
+
[50] Nikko Strom. Scalable distributed dnn training using commodity gpu cloud computing. In Sixteenth Annual Conference of the International Speech Communication Association, 2015.
|
| 316 |
+
[51] Sai Praneeth Karimireddy, Quentin Rebjock, Sebastian U Stich, and Martin Jaggi. Error feedback fixes signsgd and other gradient compression schemes. arXiv preprint arXiv:1901.09847, 2019.
|
| 317 |
+
[52] Sebastian U Stich and Sai Praneeth Karimireddy. The error-feedback framework: Better rates for sgd with delayed gradients and compressed communication. arXiv preprint arXiv:1909.05350, 2019.
|
| 318 |
+
[53] Eduard Gorbunov, Dmitry Kovalev, Dmitry Makarenko, and Peter Richtárik. Linearly converging error compensated sgd. Advances in Neural Information Processing Systems, 33, 2020.
|
| 319 |
+
[54] Yurii Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.
|
| 320 |
+
[55] Sébastien Bubeck. Convex optimization: Algorithms and complexity. arXiv preprint arXiv:1405.4980, 2014.
|
| 321 |
+
[56] Horia Mania, Xinghao Pan, Dimitris Papailiopoulos, Benjamin Recht, Kannan Ramchandran, and Michael I Jordan. Perturbed iterate analysis for asynchronous stochastic optimization. SIAM Journal on Optimization, 27(4):2202–2229, 2017.
|
| 322 |
+
|
| 323 |
+
# Checklist
|
| 324 |
+
|
| 325 |
+
1. For all authors...
|
| 326 |
+
|
| 327 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
|
| 328 |
+
(b) Did you describe the limitations of your work? [Yes] We provide a lower bound for our algorithm in Theorem 5 of Section 4.1 that suggests the need for more informed local updating schemes.
|
| 329 |
+
(c) Did you discuss any potential negative societal impacts of your work? [No] We could not think of any potential negative societal impacts.
|
| 330 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
|
| 331 |
+
|
| 332 |
+
2. If you are including theoretical results...
|
| 333 |
+
|
| 334 |
+
(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] We provide complete proofs of all our results in the supplemental material.
|
| 335 |
+
|
| 336 |
+
3. If you ran experiments...
|
| 337 |
+
|
| 338 |
+
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No]
|
| 339 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
|
| 340 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]
|
| 341 |
+
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No]
|
| 342 |
+
|
| 343 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 344 |
+
|
| 345 |
+
(a) If your work uses existing assets, did you cite the creators? [N/A]
|
| 346 |
+
(b) Did you mention the license of the assets? [N/A]
|
| 347 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
|
| 348 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
|
| 349 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
|
| 350 |
+
|
| 351 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 352 |
+
|
| 353 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
|
| 354 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
|
| 355 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
|
parse/train/h7FqQ6hCK18/h7FqQ6hCK18_content_list.json
ADDED
|
@@ -0,0 +1,1389 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
184,
|
| 8 |
+
122,
|
| 9 |
+
813,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Aritra Mitra Rayana Jaafar George J. Pappas Hamed Hassani Department of Electrical and Systems Engineering {amitra20,rayanaj,pappasg,hassani} $@$ seas.upenn.edu ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
232,
|
| 19 |
+
224,
|
| 20 |
+
767,
|
| 21 |
+
268
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Abstract ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
462,
|
| 31 |
+
303,
|
| 32 |
+
535,
|
| 33 |
+
320
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We consider a standard federated learning (FL) setup where a group of clients periodically coordinate with a central server to train a statistical model. We develop a general algorithmic framework called FedLin to tackle some of the key challenges intrinsic to FL, namely objective heterogeneity, systems heterogeneity, and infrequent and imprecise communication. Our framework is motivated by the observation that under these challenges, various existing FL algorithms suffer from a fundamental speed-accuracy conflict: they either guarantee linear convergence but to an incorrect point, or convergence to the global minimum but at a sub-linear rate, i.e., fast convergence comes at the expense of accuracy. In contrast, when the clients’ local loss functions are smooth and strongly convex, we show that FedLin guarantees linear convergence to the global minimum, despite arbitrary objective and systems heterogeneity. We then establish matching upper and lower bounds on the convergence rate of FedLin that highlight the effects of infrequent, periodic communication. Finally, we show that FedLin preserves linear convergence rates under aggressive gradient sparsification, and quantify the effect of the compression level on the convergence rate. Notably, our work is the first to provide tight linear convergence rate guarantees, and constitutes the first comprehensive analysis of gradient sparsification in FL. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
232,
|
| 42 |
+
334,
|
| 43 |
+
766,
|
| 44 |
+
582
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
174,
|
| 54 |
+
606,
|
| 55 |
+
312,
|
| 56 |
+
623
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "In a canonical federated learning (FL) architecture, a set $s$ of clients periodically communicate with a central server to find a global statistical model that solves the following problem [1–5]: ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
637,
|
| 66 |
+
823,
|
| 67 |
+
665
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "equation",
|
| 73 |
+
"img_path": "images/abcf07216834b093f75a8b6ff23d92323852ab579bf4a57fa469f0b8d1dcee76.jpg",
|
| 74 |
+
"text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x ) , { \\mathrm { ~ w h e r e ~ } } f ( x ) = { \\frac { 1 } { m } } \\sum _ { i = 1 } ^ { m } f _ { i } ( x ) .\n$$",
|
| 75 |
+
"text_format": "latex",
|
| 76 |
+
"bbox": [
|
| 77 |
+
362,
|
| 78 |
+
670,
|
| 79 |
+
633,
|
| 80 |
+
710
|
| 81 |
+
],
|
| 82 |
+
"page_idx": 0
|
| 83 |
+
},
|
| 84 |
+
{
|
| 85 |
+
"type": "text",
|
| 86 |
+
"text": "Here, $m$ is the number of clients, $f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }$ is the local objective (loss) function of client $i$ , and $f ( x )$ is the global objective function. Some of the core distinguishing tenets of the FL paradigm are as follows [1–5]. First, due to privacy considerations, clients cannot directly share their local training data with the server. Second, differences in the clients’ data-sets may cause the clients to have nonidentical loss functions with different minima - this is known as statistical or objective heterogeneity. Third, due to variability in hardware (CPU, memory) and power (battery level), i.e., due to systems or device heterogeneity, the client devices may have different computation speeds; in particular, this may lead to slow and straggling devices that affect convergence guarantees. Finally, communicationefficiency is a major concern, dictating the need to reduce the number of communication rounds, and also the size of the messages transmitted in each round. The above considerations pose unique technical challenges that we aim to address in this paper. ",
|
| 87 |
+
"bbox": [
|
| 88 |
+
173,
|
| 89 |
+
715,
|
| 90 |
+
825,
|
| 91 |
+
869
|
| 92 |
+
],
|
| 93 |
+
"page_idx": 0
|
| 94 |
+
},
|
| 95 |
+
{
|
| 96 |
+
"type": "text",
|
| 97 |
+
"text": "In a typical FL setting, to reduce the number of communication rounds, clients perform multiple local training steps in isolation before communicating with the server. Due to such local steps, the popular ",
|
| 98 |
+
"bbox": [
|
| 99 |
+
174,
|
| 100 |
+
875,
|
| 101 |
+
823,
|
| 102 |
+
904
|
| 103 |
+
],
|
| 104 |
+
"page_idx": 0
|
| 105 |
+
},
|
| 106 |
+
{
|
| 107 |
+
"type": "text",
|
| 108 |
+
"text": "FedAvg algorithm suffers from a “client-drift phenomenon\" under objective heterogeneity [6–11]: the local iterates of each client drift-off towards the minimum of their own local loss function, leading to slow convergence rates. For analysis on FedAvg, we refer the reader to [6, 8, 12–21]. Recently, several new algorithms such as FedProx [22], SCAFFOLD [11], FedSplit [10], and FedNova [23] have been proposed as improvements to FedAvg. Despite these advances, there remain gaps in our understanding of the extent to which these algorithms match the guarantees of a centralized baseline.1 ",
|
| 109 |
+
"bbox": [
|
| 110 |
+
174,
|
| 111 |
+
90,
|
| 112 |
+
825,
|
| 113 |
+
174
|
| 114 |
+
],
|
| 115 |
+
"page_idx": 1
|
| 116 |
+
},
|
| 117 |
+
{
|
| 118 |
+
"type": "text",
|
| 119 |
+
"text": "For instance, even for simple, deterministic settings, FedProx [22] and FedNova [23] exhibit a fundamental speed-accuracy conflict under objective heterogeneity; see [8, 9] and Section 2. Specifically, with constant step-sizes, these algorithms converge linearly, but potentially to an incorrect point. Thus, convergence to the minimum of the global loss function necessitates diminishing step-sizes, which, in turn, leads to sub-linear convergence. Thus, fast convergence comes at the expense of accuracy. Although SCAFFOLD [11] and FedSplit [10] employ variance-reduction and operatorsplitting techniques, respectively, to tackle objective heterogeneity, it is not known whether the rates in these papers are tight. More importantly, neither SCAFFOLD nor FedSplit account for the effects of systems heterogeneity or compression, both of which are key challenges in FL. Indeed, due to systems heterogeneity, the number of local steps may vary across clients, causing some clients to make much less progress than others in each round [23]. Moreover, while empirical studies [24, 25] have revealed significant benefits of biased sparsification, theoretical guarantees for such methods in a federated setting have remained elusive. In this context, our contributions are as follows. ",
|
| 120 |
+
"bbox": [
|
| 121 |
+
174,
|
| 122 |
+
180,
|
| 123 |
+
825,
|
| 124 |
+
359
|
| 125 |
+
],
|
| 126 |
+
"page_idx": 1
|
| 127 |
+
},
|
| 128 |
+
{
|
| 129 |
+
"type": "text",
|
| 130 |
+
"text": "• A New Algorithm: Motivated by the above concerns, we develop a general algorithmic framework called FedLin that simultaneously accounts for objective heterogeneity, systems heterogeneity, and gradient sparsification. The key components of FedLin include a gradient correction term in the local update rule that exploits memory; the use of client-specific learning rates; and error-feedback mechanisms at the clients and the server. ",
|
| 131 |
+
"bbox": [
|
| 132 |
+
174,
|
| 133 |
+
367,
|
| 134 |
+
825,
|
| 135 |
+
435
|
| 136 |
+
],
|
| 137 |
+
"page_idx": 1
|
| 138 |
+
},
|
| 139 |
+
{
|
| 140 |
+
"type": "text",
|
| 141 |
+
"text": "• Matching Centralized Rates: For smooth and strongly convex losses, we show that FedLin converges to the global minimum linearly in the deterministic setting, and with a $O ( 1 / T )$ rate for a general stochastic oracle model, thereby matching centralized rates (up to constants). We then present matching rates for smooth, convex and non-convex settings as well. Importantly, our results hold under arbitrary objective and systems heterogeneity. In contrast, the only other work in FL (as far as we are aware) that investigates both objective and systems heterogeneity [23] provides results only for the non-convex setting, under a bounded dissimilarity assumption. Moreover, the FedNova algorithm in [23] suffers from the speed-accuracy conflict, while FedLin does not. ",
|
| 142 |
+
"bbox": [
|
| 143 |
+
173,
|
| 144 |
+
443,
|
| 145 |
+
825,
|
| 146 |
+
554
|
| 147 |
+
],
|
| 148 |
+
"page_idx": 1
|
| 149 |
+
},
|
| 150 |
+
{
|
| 151 |
+
"type": "text",
|
| 152 |
+
"text": "• Quantifying the Price of Multiple Local Steps: We establish a lower bound for FedLin that matches the upper-bound we obtain for smooth, strongly convex losses. In doing so, we provide the first (as far as we are aware) tight linear convergence rate analysis. Our lower bound highlights the price paid for performing multiple local steps, i.e., the effect of infrequent communication on the convergence rate. In particular, our analysis reveals, perhaps surprisingly, that there exist simple instances (involving quadratic losses) for which performing multiple local steps does not improve the rate of convergence, indicating that even mild statistical heterogeneity can hurt. Our analysis also provides valuable insights into the limitations of gradient-tracking/variance-reduction techniques. ",
|
| 153 |
+
"bbox": [
|
| 154 |
+
173,
|
| 155 |
+
559,
|
| 156 |
+
825,
|
| 157 |
+
670
|
| 158 |
+
],
|
| 159 |
+
"page_idx": 1
|
| 160 |
+
},
|
| 161 |
+
{
|
| 162 |
+
"type": "text",
|
| 163 |
+
"text": "• Analyzing the Impacts of Gradient Sparsification at Server and at Clients: While several works explore the effect of unbiased random quantization in distributed settings [26–31], there are only a handful of papers [15, 32] that also consider the effect of local steps in FL. Different from all these works, we explore the impacts of sparsifying gradients using a biased TOP-k operator, both at the server side and at the clients. Our results in this context (i) constitute the first formal study of gradient sparsification in a federated setting; (ii) reveal key differences between up-link and down-link compression; and (iii) quantify the effect of the compression level on the convergence rate. Notably, FedLin preserves linear convergence rates despite aggressive gradient sparsification. ",
|
| 164 |
+
"bbox": [
|
| 165 |
+
173,
|
| 166 |
+
676,
|
| 167 |
+
825,
|
| 168 |
+
787
|
| 169 |
+
],
|
| 170 |
+
"page_idx": 1
|
| 171 |
+
},
|
| 172 |
+
{
|
| 173 |
+
"type": "text",
|
| 174 |
+
"text": "Basic Notation and Terminology: Referring to (1), let $\\begin{array} { r } { x ^ { * } \\ \\in \\ \\operatorname { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x ) } \\end{array}$ , and $x _ { i } ^ { * } \\in$ $\\mathrm { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } }$ $f _ { i } ( x )$ . Every FL algorithm mentioned in this paper operates in rounds $t \\in \\{ 1 , \\ldots , T \\}$ In each round $t$ , every client performs a certain number of local steps in isolation, starting from a common global model $\\bar { x } _ { t }$ . We will denote by $x _ { i , \\ell } ^ { ( t ) }$ client $i$ ’s estimate of the model at the $\\ell \\cdot$ -th local step of round $t$ . In particular, $x _ { i , 0 } ^ { ( t ) } = \\bar { x } _ { t } , \\forall i \\in \\mathcal { S }$ . ",
|
| 175 |
+
"bbox": [
|
| 176 |
+
173,
|
| 177 |
+
794,
|
| 178 |
+
825,
|
| 179 |
+
875
|
| 180 |
+
],
|
| 181 |
+
"page_idx": 1
|
| 182 |
+
},
|
| 183 |
+
{
|
| 184 |
+
"type": "table",
|
| 185 |
+
"img_path": "images/9b79b22049de5abf515a2896375a8aa79d42c6fc159cbdae536695172c40af75.jpg",
|
| 186 |
+
"table_caption": [],
|
| 187 |
+
"table_footnote": [],
|
| 188 |
+
"table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Linear Convergencetox*</td><td rowspan=1 colspan=1>Lower Bounds</td><td rowspan=1 colspan=1>Variable ClientSpeeds</td><td rowspan=1 colspan=1>Sparsification/Compression</td></tr><tr><td rowspan=1 colspan=1>FedAvg[2]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Thm. I in [11]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedProx[22]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedNova[23]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedSplit[10]</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>SCAFFOLD[11]</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td></tr><tr><td rowspan=1 colspan=1>FedLin (Sec.3)</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>Thm. 5</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td></tr></table>",
|
| 189 |
+
"bbox": [
|
| 190 |
+
178,
|
| 191 |
+
88,
|
| 192 |
+
815,
|
| 193 |
+
210
|
| 194 |
+
],
|
| 195 |
+
"page_idx": 2
|
| 196 |
+
},
|
| 197 |
+
{
|
| 198 |
+
"type": "text",
|
| 199 |
+
"text": "Table 1: Comparison of our proposed algorithm FedLin with popular FL algorithms. We indicate whether or not each algorithm (i) guarantees linear convergence to $x ^ { * }$ for smooth, strongly convex losses in a deterministic setting under objective heterogeneity; (ii) comes with lower bounds; (iii) accounts for variable local steps across clients (systems heterogeneity); and (iv) performs compression. ",
|
| 200 |
+
"bbox": [
|
| 201 |
+
173,
|
| 202 |
+
220,
|
| 203 |
+
826,
|
| 204 |
+
277
|
| 205 |
+
],
|
| 206 |
+
"page_idx": 2
|
| 207 |
+
},
|
| 208 |
+
{
|
| 209 |
+
"type": "text",
|
| 210 |
+
"text": "2 Motivation: Speed-Accuracy Trade-Off ",
|
| 211 |
+
"text_level": 1,
|
| 212 |
+
"bbox": [
|
| 213 |
+
174,
|
| 214 |
+
304,
|
| 215 |
+
537,
|
| 216 |
+
321
|
| 217 |
+
],
|
| 218 |
+
"page_idx": 2
|
| 219 |
+
},
|
| 220 |
+
{
|
| 221 |
+
"type": "text",
|
| 222 |
+
"text": "To motivate our work, we first show how some recently proposed FL algorithms, namely FedProx [22] and FedNova [23], exhibit a fundamental speed-accuracy trade-off even in simple, deterministic settings. Specifically, we show that these schemes do not, in general, guarantee convergence to the global minimum with constant step-sizes. This, in turn, necessitates diminishing step-sizes, leading to sub-linear convergence rates. Our analysis here is inspired by that in [8] for FedAvg. We consider a deterministic quadratic model where the local loss function of client $i$ is given by ",
|
| 223 |
+
"bbox": [
|
| 224 |
+
173,
|
| 225 |
+
334,
|
| 226 |
+
405,
|
| 227 |
+
570
|
| 228 |
+
],
|
| 229 |
+
"page_idx": 2
|
| 230 |
+
},
|
| 231 |
+
{
|
| 232 |
+
"type": "image",
|
| 233 |
+
"img_path": "images/12026a3b1c700db0ec32d7f9f5748052971b27798d8fae0292a601023096c6e5.jpg",
|
| 234 |
+
"image_caption": [
|
| 235 |
+
"Figure 1: Simulations comparing FedProx, FedNova, and FedLin for two clients with $f _ { 1 } ( { x } ) ~ = ~ ( 1 / 2 ) ( x - 3 ) ^ { 2 }$ and $f _ { 2 } ( x ) \\ = \\ ( x - 5 0 ) ^ { 2 }$ . Left: Clients perform the same number of local steps, $H = 5 0$ . For FedProx, we set $\\beta = 5$ . Right: Clients 1 and 2 perform 50 and 30 local steps, respectively. "
|
| 236 |
+
],
|
| 237 |
+
"image_footnote": [],
|
| 238 |
+
"bbox": [
|
| 239 |
+
416,
|
| 240 |
+
332,
|
| 241 |
+
818,
|
| 242 |
+
476
|
| 243 |
+
],
|
| 244 |
+
"page_idx": 2
|
| 245 |
+
},
|
| 246 |
+
{
|
| 247 |
+
"type": "text",
|
| 248 |
+
"text": "$f _ { i } ( x ) = 1 / 2 \\| A _ { i } ^ { 1 / 2 } ( x - c _ { i } ) \\| ^ { 2 }$ , where $A _ { i }$ is a symmetric positive-definite matrix. We begin by assuming that all clients perform the same number of local steps $H$ . The following is the FedProx update rule where a proximal term is added to mitigate client-drift: ",
|
| 249 |
+
"bbox": [
|
| 250 |
+
178,
|
| 251 |
+
573,
|
| 252 |
+
826,
|
| 253 |
+
614
|
| 254 |
+
],
|
| 255 |
+
"page_idx": 2
|
| 256 |
+
},
|
| 257 |
+
{
|
| 258 |
+
"type": "equation",
|
| 259 |
+
"img_path": "images/8ec256fd923185ede44ea4accdc9a0161f84f243031987cb7d10c5a4320a77c5.jpg",
|
| 260 |
+
"text": "$$\nx _ { i , \\ell + 1 } ^ { ( t ) } = x _ { i , \\ell } ^ { ( t ) } - \\eta \\bigg ( \\nabla f _ { i } ( x _ { i , \\ell } ^ { ( t ) } ) + \\beta ( x _ { i , \\ell } ^ { ( t ) } - \\bar { x } _ { t } ) \\bigg ) , \\ell = 0 , \\dots , H - 1 ; \\bar { x } _ { t + 1 } = \\frac { 1 } { m } \\sum _ { i \\in S } x _ { i , H } ^ { ( t ) } .\n$$",
|
| 261 |
+
"text_format": "latex",
|
| 262 |
+
"bbox": [
|
| 263 |
+
199,
|
| 264 |
+
616,
|
| 265 |
+
781,
|
| 266 |
+
652
|
| 267 |
+
],
|
| 268 |
+
"page_idx": 2
|
| 269 |
+
},
|
| 270 |
+
{
|
| 271 |
+
"type": "text",
|
| 272 |
+
"text": "Proposition 1. For any step-size $\\eta > 0$ , $T$ rounds of FedProx amount to performing $T$ rounds of parallel $G D$ on the surrogate optimization problem given by ",
|
| 273 |
+
"bbox": [
|
| 274 |
+
171,
|
| 275 |
+
656,
|
| 276 |
+
825,
|
| 277 |
+
685
|
| 278 |
+
],
|
| 279 |
+
"page_idx": 2
|
| 280 |
+
},
|
| 281 |
+
{
|
| 282 |
+
"type": "equation",
|
| 283 |
+
"img_path": "images/2bd3b94b6691e7e60215871deb5669de35a897f51d983830ca8d95cf99e7a0d9.jpg",
|
| 284 |
+
"text": "$$\n\\operatorname* { m i n } _ { x } \\frac { 1 } { m } \\sum _ { i \\in \\mathcal { S } } \\frac { 1 } { 2 } \\bigg \\| \\bigg ( \\sum _ { \\ell = 0 } ^ { H - 1 } [ I - \\eta ( A _ { i } + \\beta I ) ] ^ { \\ell } A _ { i } \\bigg ) ^ { 1 / 2 } ( x - c _ { i } ) \\bigg \\| ^ { 2 } .\n$$",
|
| 285 |
+
"text_format": "latex",
|
| 286 |
+
"bbox": [
|
| 287 |
+
302,
|
| 288 |
+
686,
|
| 289 |
+
696,
|
| 290 |
+
731
|
| 291 |
+
],
|
| 292 |
+
"page_idx": 2
|
| 293 |
+
},
|
| 294 |
+
{
|
| 295 |
+
"type": "text",
|
| 296 |
+
"text": "Proposition 1 shows that even when clients perform the same number of local updates, FedProx minimizes a surrogate objective function (3) whose minimum may not, in general, coincide with the minimum of the original problem. When $\\beta = 0$ , FedProx reduces to FedAvg, and our observations continue to hold. To capture systems heterogeneity as in [23], suppose now that client $i$ performs $\\tau _ { i }$ local steps. Define $\\tau _ { e f f } \\triangleq 1 / m \\sum _ { i \\in \\mathcal { S } } \\tau _ { i }$ and $\\alpha _ { i } \\triangleq \\tau _ { e f f } / \\tau _ { i }$ , $\\forall i \\in S$ . The update rule of FedNova relies on normalized aggregation of cumulative local gradients, and is given by ",
|
| 297 |
+
"bbox": [
|
| 298 |
+
173,
|
| 299 |
+
737,
|
| 300 |
+
826,
|
| 301 |
+
824
|
| 302 |
+
],
|
| 303 |
+
"page_idx": 2
|
| 304 |
+
},
|
| 305 |
+
{
|
| 306 |
+
"type": "equation",
|
| 307 |
+
"img_path": "images/400c9157968106c6dcca41cc4e015bf8b82053dafd79117ec975812eeed6566f.jpg",
|
| 308 |
+
"text": "$$\n\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) ; ~ \\bar { \\boldsymbol { x } } _ { t + 1 } = \\bar { \\boldsymbol { x } } _ { t } - \\frac { \\eta } { m } \\sum _ { i \\in S } \\alpha _ { i } \\sum _ { \\ell = 0 } ^ { \\tau _ { i } - 1 } \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) ,\n$$",
|
| 309 |
+
"text_format": "latex",
|
| 310 |
+
"bbox": [
|
| 311 |
+
272,
|
| 312 |
+
824,
|
| 313 |
+
723,
|
| 314 |
+
868
|
| 315 |
+
],
|
| 316 |
+
"page_idx": 2
|
| 317 |
+
},
|
| 318 |
+
{
|
| 319 |
+
"type": "text",
|
| 320 |
+
"text": "where $\\ell = 0 , \\dots , \\tau _ { i } - 1 , \\ i \\in \\mathcal { S }$ . Although FedNova can accommodate any local solver whose accumulated gradients are expressible as a linear combination of local gradients, we choose gradient descent, a simple solver, to isolate the impact of normalized aggregation - the essence of FedNova. ",
|
| 321 |
+
"bbox": [
|
| 322 |
+
174,
|
| 323 |
+
869,
|
| 324 |
+
825,
|
| 325 |
+
912
|
| 326 |
+
],
|
| 327 |
+
"page_idx": 2
|
| 328 |
+
},
|
| 329 |
+
{
|
| 330 |
+
"type": "text",
|
| 331 |
+
"text": "Algorithm 1 FedLin ",
|
| 332 |
+
"text_level": 1,
|
| 333 |
+
"bbox": [
|
| 334 |
+
174,
|
| 335 |
+
92,
|
| 336 |
+
316,
|
| 337 |
+
106
|
| 338 |
+
],
|
| 339 |
+
"page_idx": 3
|
| 340 |
+
},
|
| 341 |
+
{
|
| 342 |
+
"type": "text",
|
| 343 |
+
"text": "1: Input: Client step-sizes $\\eta _ { i } , i ~ \\in ~ S$ , compression levels $\\delta _ { c }$ and $\\delta _ { s }$ , initial iterate $\\overline { { \\bar { x } _ { 1 } ~ \\in ~ \\mathbb { R } ^ { d } } }$ \n$\\bar { g _ { 1 } = \\nabla f ( \\bar { x } _ { 1 } ) }$ , initial compression errors $\\rho _ { i , 1 } = 0 , \\forall i \\in \\mathcal { S }$ and $e _ { 1 } = 0$ \n2: for $t = 1 , \\dots , T$ do \n3: for $i = 1 , \\ldots , m$ do \n4: for $\\ell = 0 , \\ldots , \\tau _ { i } - 1$ do \n5: $x _ { i , \\ell + 1 } ^ { ( t ) } \\gets x _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } ( \\nabla f _ { i } ( x _ { i , \\ell } ^ { ( t ) } ) - \\nabla f _ { i } ( \\bar { x } _ { t } ) + g _ { t } ) ; x _ { i , 0 } ^ { ( t ) } = \\bar { x } _ { t }$ \n6: end for \n7: Transmit x(t)i,τi to server \n8: end for \n9: Server transmits x¯t+1 = 1/m Pi∈S x(t)i,τi \n10: for $i = 1 , \\ldots , m$ do \n11: Transmit $h _ { i , t + 1 } = \\mathcal { C } _ { \\delta _ { c } } ( \\rho _ { i , t } + \\nabla f _ { i } ( \\bar { x } _ { t + 1 } ) )$ to server \n12: $\\rho _ { i , t + 1 } \\rho _ { i , t } + \\nabla f _ { i } ( \\bar { x } _ { t + 1 } ) - h _ { i , t + 1 }$ \n13: end for \n14: Server transmits $\\begin{array} { r } { g _ { t + 1 } = \\mathcal { C } _ { \\delta _ { s } } ( e _ { t } + 1 / m \\sum _ { i \\in S } h _ { i , t + 1 } ) } \\end{array}$ \n15: $e _ { t + 1 } e _ { t } + 1 / m \\sum _ { i \\in S } h _ { i , t + 1 } - g _ { t + 1 }$ \n16: end for ",
|
| 344 |
+
"bbox": [
|
| 345 |
+
176,
|
| 346 |
+
109,
|
| 347 |
+
823,
|
| 348 |
+
353
|
| 349 |
+
],
|
| 350 |
+
"page_idx": 3
|
| 351 |
+
},
|
| 352 |
+
{
|
| 353 |
+
"type": "text",
|
| 354 |
+
"text": "Proposition 2. For any step-size $\\eta > 0$ , $T$ rounds of FedNova amount to performing $T$ rounds of parallel GD on the surrogate optimization problem given by ",
|
| 355 |
+
"bbox": [
|
| 356 |
+
173,
|
| 357 |
+
385,
|
| 358 |
+
823,
|
| 359 |
+
414
|
| 360 |
+
],
|
| 361 |
+
"page_idx": 3
|
| 362 |
+
},
|
| 363 |
+
{
|
| 364 |
+
"type": "equation",
|
| 365 |
+
"img_path": "images/1f20cc43c467740b444c57e7ee7bda4fe5608c3a54c8f024e4da3468894e2820.jpg",
|
| 366 |
+
"text": "$$\n\\operatorname* { m i n } _ { x } \\frac { 1 } { m } \\sum _ { i \\in S } \\frac { 1 } { 2 } \\bigg \\| \\bigg ( \\sum _ { \\ell = 0 } ^ { \\tau _ { i } - 1 } [ I - \\eta A _ { i } ] ^ { \\ell } \\alpha _ { i } A _ { i } \\bigg ) ^ { 1 / 2 } ( x - c _ { i } ) \\bigg \\| ^ { 2 } .\n$$",
|
| 367 |
+
"text_format": "latex",
|
| 368 |
+
"bbox": [
|
| 369 |
+
320,
|
| 370 |
+
424,
|
| 371 |
+
678,
|
| 372 |
+
468
|
| 373 |
+
],
|
| 374 |
+
"page_idx": 3
|
| 375 |
+
},
|
| 376 |
+
{
|
| 377 |
+
"type": "text",
|
| 378 |
+
"text": "For the proofs of Propositions 1 and 2, see Appendix B. Proposition 2 shows that in the presence of both objective and systems heterogeneity, FedNova minimizes a surrogate loss function whose minimum may not coincide with $x ^ { * }$ .2 Observe from (3) and (5) that using a larger learning rate $\\eta$ introduces more distortion to the original problem. In Figure 1, we see how FedProx and FedNova both converge to incorrect minimizers, even for simple instances with two clients and deterministic, quadratic losses. In contrast, FedLin, our proposed approach that we develop in the next section, guarantees linear convergence to the global minimum. ",
|
| 379 |
+
"bbox": [
|
| 380 |
+
173,
|
| 381 |
+
483,
|
| 382 |
+
825,
|
| 383 |
+
582
|
| 384 |
+
],
|
| 385 |
+
"page_idx": 3
|
| 386 |
+
},
|
| 387 |
+
{
|
| 388 |
+
"type": "text",
|
| 389 |
+
"text": "Main Takeaway: The main message we want to convey here is that even for deterministic settings, there are non-trivial challenges posed by objective and systems heterogeneity that only get amplified when one additionally considers biased compression. For such scenarios, it is not at all apparent whether (and to what extent) one can match even the basic centralized benchmark of achieving linear convergence for smooth, strongly convex loss functions. To focus on the above unresolved issues, we will primarily consider a deterministic model in this paper. Nonetheless, the general approach we develop applies to the stochastic setting as well, as aptly demonstrated by Theorem 4 in Section 4. ",
|
| 390 |
+
"bbox": [
|
| 391 |
+
173,
|
| 392 |
+
587,
|
| 393 |
+
825,
|
| 394 |
+
685
|
| 395 |
+
],
|
| 396 |
+
"page_idx": 3
|
| 397 |
+
},
|
| 398 |
+
{
|
| 399 |
+
"type": "text",
|
| 400 |
+
"text": "3 Proposed Algorithm: FedLin ",
|
| 401 |
+
"text_level": 1,
|
| 402 |
+
"bbox": [
|
| 403 |
+
176,
|
| 404 |
+
699,
|
| 405 |
+
447,
|
| 406 |
+
718
|
| 407 |
+
],
|
| 408 |
+
"page_idx": 3
|
| 409 |
+
},
|
| 410 |
+
{
|
| 411 |
+
"type": "text",
|
| 412 |
+
"text": "In this section, we develop our proposed algorithm FedLin, formally described in Algorithm 1. FedLin is initialized from a common global iterate $\\bar { x } _ { 1 } ~ \\in ~ \\mathbb { R } ^ { d }$ . For simplicity, we assume that $g _ { 1 } = \\nabla f ( { \\bar { x } } _ { 1 } )$ , i.e., every client has access to the true gradient of $f ( \\cdot )$ initially; we can allow $g _ { 1 }$ to be arbitrary as well without affecting the convergence guarantees. FedLin proceeds in rounds: in each round $t$ , starting from a common global model $\\bar { x } _ { t }$ , each client $i$ performs $\\tau _ { i }$ local training steps in parallel, as per line 5 of Algorithm 1. The key features of our local update rule are as follows: exploiting past gradients to account for objective heterogeneity, using client-specific step-sizes to tackle systems heterogeneity, and employing error-feedback to account for gradient sparsification. We now discuss each of these features in detail. ",
|
| 413 |
+
"bbox": [
|
| 414 |
+
173,
|
| 415 |
+
733,
|
| 416 |
+
826,
|
| 417 |
+
858
|
| 418 |
+
],
|
| 419 |
+
"page_idx": 3
|
| 420 |
+
},
|
| 421 |
+
{
|
| 422 |
+
"type": "text",
|
| 423 |
+
"text": "To gain intuition regarding the local step in line 5, note that the ideal local update at client $i$ is $\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } \\nabla f ( \\boldsymbol { x } _ { i , \\ell } ^ { ( \\bar { t } ) } )$ . However, this requires client $i$ to have access to the gradients of all other clients - which it does not, since clients do not communicate between rounds. To get around this, client $i$ exploits memory, and uses the gradient of the global function $\\nabla f ( { \\bar { x } } _ { t } )$ from the beginning of round $t$ (when the clients last communicated) as a guiding direction in its update rule. However, since $\\nabla f ( { \\bar { x } } _ { t } )$ is evaluated at a stale point $x _ { i , 0 } ^ { ( t ) } = \\bar { x } _ { t }$ , client $i$ subtracts off $\\nabla f _ { i } ( \\bar { x } _ { t } )$ from $\\nabla f ( { \\bar { x } } _ { t } )$ , and adds in the most recently evaluated gradient $\\nabla f _ { i } ( x _ { i , \\ell } ^ { ( t ) } )$ . This results in the update rule: $\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } ( \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) - \\nabla f _ { i } ( \\bar { \\boldsymbol { x } } _ { t } ) + \\nabla f ( \\bar { \\boldsymbol { x } } _ { t } ) )$ . Our local update rule in line 5 is precisely of the above form, where $g _ { t }$ is an inexact version of $\\nabla f ( { \\bar { x } } _ { t } )$ to account for gradient sparsification. ",
|
| 424 |
+
"bbox": [
|
| 425 |
+
173,
|
| 426 |
+
90,
|
| 427 |
+
826,
|
| 428 |
+
239
|
| 429 |
+
],
|
| 430 |
+
"page_idx": 4
|
| 431 |
+
},
|
| 432 |
+
{
|
| 433 |
+
"type": "text",
|
| 434 |
+
"text": "When each client $i$ performs $\\tau _ { i }$ local-steps, our analysis reveals that the bound on the drift-term $\\| x _ { i , \\ell } - \\bar { x } _ { t } \\|$ scales linearly in $\\tau _ { i }$ (see Lemma 9 in Appendix F). Accordingly, to compensate for such drift at client $i$ , the step-size $\\eta _ { i }$ needs to be chosen to vary inversely with the number of local steps $\\tau _ { i }$ . In fact, the requirement that $\\eta _ { i } \\propto 1 / \\tau _ { i }$ also turns out to be necessary (see Theorem 5), providing further motivation for the choice of client-specific learning rates in FedLin. ",
|
| 435 |
+
"bbox": [
|
| 436 |
+
173,
|
| 437 |
+
244,
|
| 438 |
+
825,
|
| 439 |
+
314
|
| 440 |
+
],
|
| 441 |
+
"page_idx": 4
|
| 442 |
+
},
|
| 443 |
+
{
|
| 444 |
+
"type": "text",
|
| 445 |
+
"text": "To explain the gradient sparsification module, let us denote by $\\mathcal { C } _ { \\delta } : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ the TOP- $\\mathtt { . k }$ operator, where $\\delta = d / k$ , and $k \\in \\bar { \\{ 1 , \\ldots , d \\} }$ . Given any $x \\in \\mathbb { R } ^ { d }$ , let ${ \\mathcal { E } } _ { \\delta } ( x )$ be a set containing the indices of the $k$ largest-magnitude components of $x$ . Then, the TOP- ${ \\bf \\nabla } \\cdot { \\bf k }$ operator we consider is given by $( \\mathcal { C } _ { \\delta } ( x ) ) _ { j } = \\mathsf { \\bar { \\Psi } } ( x ) _ { j }$ if $j \\in \\mathcal { E } _ { \\delta } ( x )$ , and $\\left( \\boldsymbol { \\mathcal { C } } _ { \\delta } ( \\boldsymbol { x } ) \\right) _ { j } = 0$ otherwise. Here, we use $( x ) _ { j }$ to denote the $j$ -th component of a vector $x$ . Clearly, a larger $\\delta$ implies more aggressive compression. We employ a standard error-feedback mechanism [33–35] at both the server and the clients to account for gradient sparsification. At client $i$ , $\\rho _ { i , t }$ represents the accumulated error due to gradient sparsification. At the end of round $t$ , instead of just compressing $\\nabla f _ { i } ( \\bar { x } _ { t + 1 } )$ , client $i$ instead compresses $\\nabla f _ { i } ( \\bar { x } _ { t + 1 } ) + \\rho _ { i , t }$ , to account for gradient coordinates not transmitted in the past. It then updates the aggregate error via line 12. An analogous description applies to the error-feedback scheme at the server, where $e _ { t }$ is the aggregate error at the beginning of round $t$ . The parameters of FedLin are the client step-sizes $\\{ \\eta _ { i } \\} _ { i \\in { \\cal S } }$ , and the compression levels $\\delta _ { c }$ and $\\delta _ { s }$ at the clients and at the server, respectively. We now comment on some related algorithmic ideas. ",
|
| 446 |
+
"bbox": [
|
| 447 |
+
173,
|
| 448 |
+
320,
|
| 449 |
+
826,
|
| 450 |
+
501
|
| 451 |
+
],
|
| 452 |
+
"page_idx": 4
|
| 453 |
+
},
|
| 454 |
+
{
|
| 455 |
+
"type": "text",
|
| 456 |
+
"text": "Related Algorithmic Approaches: In the related but different setting of distributed optimization, we note that the idea of exploiting past gradients has been used to design gradient-tracking algorithms [36–40]. In the context of FL, this idea is also related to the variance-reduction technique employed in SCAFFOLD [11]. A major difference of FedLin with the above works is that none of them consider the effect of systems heterogeneity or biased compression. In particular, accounting for the inexact gradient term $g _ { t }$ in our update rule introduces new technical challenges that we address in this paper. ",
|
| 457 |
+
"bbox": [
|
| 458 |
+
174,
|
| 459 |
+
508,
|
| 460 |
+
825,
|
| 461 |
+
592
|
| 462 |
+
],
|
| 463 |
+
"page_idx": 4
|
| 464 |
+
},
|
| 465 |
+
{
|
| 466 |
+
"type": "text",
|
| 467 |
+
"text": "There are some additional basic differences between FedLin and SCAFFOLD. To see this, consider the update rule of FedLin without sparsification: $\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } ( \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) - \\nabla f _ { i } ( \\bar { \\boldsymbol { x } } _ { t } ) + \\nabla f ( \\bar { \\boldsymbol { x } } _ { t } ) )$ Now suppose the global model $\\hat { x } _ { t }$ at the beginning of round $t$ has already converged to $x ^ { * }$ . Since $x _ { i , 0 } ^ { ( t ) } = \\bar { x } _ { t } , \\forall i \\in \\mathcal { S }$ , and $\\nabla f ( x ^ { * } ) = 0$ , it is easy to see that the iterates of the clients do not evolve any further, as one would ideally want. Thus, the global optimum $x ^ { * }$ can be viewed as a fixed-point of the FedLin update rule. Adapting to our notation, and considering the case when there is no noise in the gradients, the update rule of SCAFFOLD takes the form $\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta ( \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) - \\boldsymbol { c } _ { i } + \\boldsymbol { c } )$ , where $c _ { i }$ is a ‘control-variate’ maintained by client $i$ , and $c$ is the average of the $c _ { i }$ ’s. Importantly, the control variates $\\{ c _ { i } \\} _ { i \\in { \\mathcal { S } } }$ used in round $t$ of SCAFFOLD contain stale terms from round $t - 1$ . As a result, even if $\\bar { x } _ { t } = x ^ { * }$ , it may very well be that $( \\nabla f _ { i } ( { \\bar { x } } _ { t } ) - c _ { i } + c ) \\neq 0$ , causing the iterates of the clients to move away from $x ^ { * }$ , and requiring further rounds of communication to average out the imbalance. Thus, the fixed-point property we discussed for FedLin does not hold in general for SCAFFOLD. Our simulations in Section 7 reveal that FedLin converges much faster relative to SCAFFOLD on a simple linear regression model; we conjecture it is precisely due to the reason described above. ",
|
| 468 |
+
"bbox": [
|
| 469 |
+
173,
|
| 470 |
+
597,
|
| 471 |
+
825,
|
| 472 |
+
805
|
| 473 |
+
],
|
| 474 |
+
"page_idx": 4
|
| 475 |
+
},
|
| 476 |
+
{
|
| 477 |
+
"type": "text",
|
| 478 |
+
"text": "Keeping aside the differences due to systems heterogeneity and compression, the FedSVRG algorithm in [1] includes a similar gradient correction term as in FedLin, but makes use of certain additional diagonal scaling and pre-conditioning matrices. Although promising empirical results are reported for FedSVRG in [1], these results come with no supporting theoretical guarantees of convergence. In contrast, we will develop rigorous complexity guarantees for FedLin in the following sections. Specifically, we will show that FedLin guarantees linear convergence rates despite the challenges of objective heterogeneity, systems heterogeneity, and aggressive gradient sparsification. ",
|
| 479 |
+
"bbox": [
|
| 480 |
+
174,
|
| 481 |
+
811,
|
| 482 |
+
825,
|
| 483 |
+
909
|
| 484 |
+
],
|
| 485 |
+
"page_idx": 4
|
| 486 |
+
},
|
| 487 |
+
{
|
| 488 |
+
"type": "text",
|
| 489 |
+
"text": "4 Matching Centralized Rates under Objective and Systems Heterogeneity ",
|
| 490 |
+
"text_level": 1,
|
| 491 |
+
"bbox": [
|
| 492 |
+
169,
|
| 493 |
+
88,
|
| 494 |
+
810,
|
| 495 |
+
107
|
| 496 |
+
],
|
| 497 |
+
"page_idx": 5
|
| 498 |
+
},
|
| 499 |
+
{
|
| 500 |
+
"type": "text",
|
| 501 |
+
"text": "In this section, we will analyze the performance of FedLin in the face of both objective and systems heterogeneity. To focus solely on the effects of client heterogeneity, we will assume throughout this section that there is no gradient sparsification, i.e., $\\delta _ { c } = \\delta _ { s } = 1$ . Accordingly, observe that $\\rho _ { i , t } = 0 , e _ { t } = 0 , \\forall i \\in \\mathcal { S } , \\forall t \\in \\{ 1 , . . . , \\bar { T } \\}$ . Thus, the local update rule for FedLin simplifies to ",
|
| 502 |
+
"bbox": [
|
| 503 |
+
173,
|
| 504 |
+
119,
|
| 505 |
+
825,
|
| 506 |
+
176
|
| 507 |
+
],
|
| 508 |
+
"page_idx": 5
|
| 509 |
+
},
|
| 510 |
+
{
|
| 511 |
+
"type": "equation",
|
| 512 |
+
"img_path": "images/efe5aeadd1ee6952fa03e328561fcf700820a9545942accc5eb9b4d08adeed2f.jpg",
|
| 513 |
+
"text": "$$\n\\begin{array} { r } { \\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } ( \\nabla f _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) - \\nabla f _ { i } ( \\bar { \\boldsymbol { x } } _ { t } ) + \\nabla f ( \\bar { \\boldsymbol { x } } _ { t } ) ) . } \\end{array}\n$$",
|
| 514 |
+
"text_format": "latex",
|
| 515 |
+
"bbox": [
|
| 516 |
+
321,
|
| 517 |
+
183,
|
| 518 |
+
676,
|
| 519 |
+
205
|
| 520 |
+
],
|
| 521 |
+
"page_idx": 5
|
| 522 |
+
},
|
| 523 |
+
{
|
| 524 |
+
"type": "text",
|
| 525 |
+
"text": "Let us denote by $\\kappa = L / \\mu$ the condition number of an $L$ -smooth and $\\mu$ -strongly convex function. Also, let $\\eta _ { i } = \\bar { \\eta } / \\tau _ { i } , \\forall i \\in \\mathcal { S }$ , where $\\bar { \\eta } \\in ( 0 , 1 )$ is a flexible parameter that we will specify based on context. We are now ready to state the main results of this section. ",
|
| 526 |
+
"bbox": [
|
| 527 |
+
173,
|
| 528 |
+
210,
|
| 529 |
+
826,
|
| 530 |
+
253
|
| 531 |
+
],
|
| 532 |
+
"page_idx": 5
|
| 533 |
+
},
|
| 534 |
+
{
|
| 535 |
+
"type": "text",
|
| 536 |
+
"text": "Theorem 1. (Strongly convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\\mu$ -strongly convex. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ , and $\\delta _ { c } = \\delta _ { s } = 1$ . Then, with $\\begin{array} { r } { \\eta _ { i } = \\frac { 1 } { 6 L \\tau _ { i } } , \\forall i \\dot { \\in } \\mathcal { S } } \\end{array}$ , FedL $_ { i n }$ guarantees: ",
|
| 537 |
+
"bbox": [
|
| 538 |
+
173,
|
| 539 |
+
257,
|
| 540 |
+
823,
|
| 541 |
+
287
|
| 542 |
+
],
|
| 543 |
+
"page_idx": 5
|
| 544 |
+
},
|
| 545 |
+
{
|
| 546 |
+
"type": "equation",
|
| 547 |
+
"img_path": "images/279400b94f37c515d2db692590a11f9b90c9bf80ab6a79d70fec3f377c093fb0.jpg",
|
| 548 |
+
"text": "$$\nf ( \\bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \\leq \\left( 1 - \\frac { 1 } { 6 \\kappa } \\right) ^ { T } ( f ( \\bar { x } _ { 1 } ) - f ( x ^ { * } ) ) .\n$$",
|
| 549 |
+
"text_format": "latex",
|
| 550 |
+
"bbox": [
|
| 551 |
+
326,
|
| 552 |
+
295,
|
| 553 |
+
669,
|
| 554 |
+
332
|
| 555 |
+
],
|
| 556 |
+
"page_idx": 5
|
| 557 |
+
},
|
| 558 |
+
{
|
| 559 |
+
"type": "text",
|
| 560 |
+
"text": "Theorem 2. (Convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth and convex. Moreover, suppose $\\tau _ { i } \\geq$ $1 , \\forall i \\in S$ , and $\\delta _ { c } = \\delta _ { s } = 1$ . Then, with $\\begin{array} { r } { \\eta _ { i } \\stackrel { \\cdot \\cdot } { = } \\frac { \\mathrm { ~ i ~ } } { 1 0 L \\tau _ { i } } , \\forall i \\in \\mathcal { S } , } \\end{array}$ , FedLin guarantees: ",
|
| 561 |
+
"bbox": [
|
| 562 |
+
171,
|
| 563 |
+
333,
|
| 564 |
+
821,
|
| 565 |
+
366
|
| 566 |
+
],
|
| 567 |
+
"page_idx": 5
|
| 568 |
+
},
|
| 569 |
+
{
|
| 570 |
+
"type": "equation",
|
| 571 |
+
"img_path": "images/cbb3c87f1b8b827f8345c4cbafa94335774b9c0bfb0f59e8690210f4a3aa0c86.jpg",
|
| 572 |
+
"text": "$$\nf \\left( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\bar { x } _ { t } \\right) - f ( x ^ { * } ) \\leq \\frac { 1 0 L } { T } \\left( \\Vert \\bar { x } _ { 1 } - x ^ { * } \\Vert ^ { 2 } - \\Vert \\bar { x } _ { T + 1 } - x ^ { * } \\Vert ^ { 2 } \\right) .\n$$",
|
| 573 |
+
"text_format": "latex",
|
| 574 |
+
"bbox": [
|
| 575 |
+
281,
|
| 576 |
+
371,
|
| 577 |
+
717,
|
| 578 |
+
415
|
| 579 |
+
],
|
| 580 |
+
"page_idx": 5
|
| 581 |
+
},
|
| 582 |
+
{
|
| 583 |
+
"type": "text",
|
| 584 |
+
"text": "Theorem 3. (Non-convex case) Suppose each $f _ { i } ( x )$ is $L$ -smooth. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ and $\\delta _ { c } = \\delta _ { s } = 1$ . Then, with $\\begin{array} { r } { \\eta _ { i } = \\frac { 1 } { 2 6 L \\tau _ { i } } , \\forall i \\in \\mathcal { S } } \\end{array}$ , FedLin guarantees: ",
|
| 585 |
+
"bbox": [
|
| 586 |
+
173,
|
| 587 |
+
420,
|
| 588 |
+
823,
|
| 589 |
+
450
|
| 590 |
+
],
|
| 591 |
+
"page_idx": 5
|
| 592 |
+
},
|
| 593 |
+
{
|
| 594 |
+
"type": "equation",
|
| 595 |
+
"img_path": "images/bac2f6b970fb497e0652efc9bf7c3af0ab18ad59236ac69047aa4e7ec732353f.jpg",
|
| 596 |
+
"text": "$$\n\\operatorname* { m i n } _ { t \\in \\left[ T \\right] } \\left\\| \\nabla f ( \\bar { x } _ { t } ) \\right\\| ^ { 2 } \\leq \\frac { 5 2 L } { T } ( f ( \\bar { x } _ { 1 } ) - f ( \\bar { x } _ { T + 1 } ) ) .\n$$",
|
| 597 |
+
"text_format": "latex",
|
| 598 |
+
"bbox": [
|
| 599 |
+
348,
|
| 600 |
+
457,
|
| 601 |
+
648,
|
| 602 |
+
489
|
| 603 |
+
],
|
| 604 |
+
"page_idx": 5
|
| 605 |
+
},
|
| 606 |
+
{
|
| 607 |
+
"type": "text",
|
| 608 |
+
"text": "Noisy Case Analysis: We now analyze the performance of FedLin under a general stochastic oracle model. For each $i \\in S$ and $x \\in \\mathbb { R } ^ { d }$ , let $q _ { i } ( \\bar { x } )$ be an unbiased estimate of the gradient $\\nabla f _ { i } ( x )$ with variance bounded above by σ2. We consider the update rule: x(t)i,\\`+1 = x(t)i,\\` $\\boldsymbol { x } _ { i , \\ell + 1 } ^ { ( t ) } = \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } - \\eta _ { i } \\big ( q _ { i } ( \\boldsymbol { x } _ { i , \\ell } ^ { ( t ) } ) - q _ { i } ( \\bar { \\boldsymbol { x } } _ { t } ) +$ $q ( \\bar { x } _ { t } ) )$ , where $\\begin{array} { r } { q ( x ) \\triangleq 1 / m \\sum _ { i \\in S } q _ { i } ( x ) , \\forall x \\in \\mathbb { R } ^ { d } } \\end{array}$ . We then have the following result. ",
|
| 609 |
+
"bbox": [
|
| 610 |
+
173,
|
| 611 |
+
501,
|
| 612 |
+
826,
|
| 613 |
+
569
|
| 614 |
+
],
|
| 615 |
+
"page_idx": 5
|
| 616 |
+
},
|
| 617 |
+
{
|
| 618 |
+
"type": "text",
|
| 619 |
+
"text": "Theorem 4. (Strongly convex case with noise) Consider the above stochastic oracle model. Suppose each $f _ { i } ( x )$ is $L$ -smooth and -strongly convex. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ , and $\\delta _ { c } = \\delta _ { s } = 1$ . For each $i \\in S$ , let $\\begin{array} { r } { \\eta _ { i } = \\frac { \\bar { \\eta } } { \\tau _ { i } } } \\end{array}$ , where $\\bar { \\eta } \\in ( 0 , 1 )$ satisfies $\\begin{array} { r } { \\bar { \\eta } < \\frac { 1 } { 6 L } } \\end{array}$ . Then, $\\forall t \\in [ T ]$ , FedLin guarantees: ",
|
| 620 |
+
"bbox": [
|
| 621 |
+
173,
|
| 622 |
+
570,
|
| 623 |
+
826,
|
| 624 |
+
616
|
| 625 |
+
],
|
| 626 |
+
"page_idx": 5
|
| 627 |
+
},
|
| 628 |
+
{
|
| 629 |
+
"type": "equation",
|
| 630 |
+
"img_path": "images/e13ac5d119eda5a4b78bd9e7700dfc45c7635820e276303bfb73cc6426b606d5.jpg",
|
| 631 |
+
"text": "$$\n\\mathbb { E } [ \\| \\bar { x } _ { t + 1 } - x ^ { * } \\| ^ { 2 } ] \\leq \\left( 1 - \\frac { \\bar { \\eta } \\mu } { 2 } \\right) \\mathbb { E } [ \\| \\bar { x } _ { t } - x ^ { * } \\| ^ { 2 } ] + 2 5 \\bar { \\eta } ^ { 2 } \\sigma ^ { 2 } .\n$$",
|
| 632 |
+
"text_format": "latex",
|
| 633 |
+
"bbox": [
|
| 634 |
+
308,
|
| 635 |
+
622,
|
| 636 |
+
687,
|
| 637 |
+
651
|
| 638 |
+
],
|
| 639 |
+
"page_idx": 5
|
| 640 |
+
},
|
| 641 |
+
{
|
| 642 |
+
"type": "text",
|
| 643 |
+
"text": "The proofs of Theorems 1, 2, 3, and 4 are provided in Appendix F. ",
|
| 644 |
+
"bbox": [
|
| 645 |
+
174,
|
| 646 |
+
662,
|
| 647 |
+
609,
|
| 648 |
+
678
|
| 649 |
+
],
|
| 650 |
+
"page_idx": 5
|
| 651 |
+
},
|
| 652 |
+
{
|
| 653 |
+
"type": "text",
|
| 654 |
+
"text": "Main Takeaways: From Theorems 1, 2, and 3, we note that FedLin matches the convergence guarantees of centralized gradient descent (up to constants) for smooth, strongly convex, convex, and non-convex settings, respectively. As far as we are aware, this is the first work to provide such comprehensive guarantees under arbitrary objective and systems heterogeneity. In fact, all our results continue to hold even when the operating speeds of the client machines vary across rounds, i.e., $\\tau _ { i }$ is allowed to be a function of $t$ . Each client $i$ can simply adjust its learning rate $\\eta _ { i } \\propto 1 / \\tau _ { i } ( t )$ locally to account for such variations. The bound for the noisy case in Theorem 4 resembles that of centralized SGD [41]: with a time-varying parameter $\\bar { \\eta _ { t } } = \\dot { O } ( 1 / t )$ , we get the standard $O ( 1 / T )$ rate after $T$ rounds (using the exact same arguments as in [41]). The key thing to note here is that despite arbitrary heterogeneity, the assumptions we make on the stochastic gradients are the same as those made in the analysis of centralized SGD: unbiased gradients with bounded variance, nothing more. ",
|
| 655 |
+
"bbox": [
|
| 656 |
+
173,
|
| 657 |
+
683,
|
| 658 |
+
825,
|
| 659 |
+
835
|
| 660 |
+
],
|
| 661 |
+
"page_idx": 5
|
| 662 |
+
},
|
| 663 |
+
{
|
| 664 |
+
"type": "text",
|
| 665 |
+
"text": "Comparison with Related Work: In the recent paper [10], the authors propose FedSplit, and analyze it in a deterministic setting. For strongly-convex and smooth loss functions, FedSplit guarantees linear convergence, but only to a non-vanishing neighborhood of $x ^ { * }$ . Thus, like FedAvg [2], FedProx [22], and FedNova [23], FedSplit fails to guarantee exact linear convergence to $x ^ { * }$ . Empirically, we observe that FedSplit diverges on certain instances; see Appendix J. Compared to these algorithms, we see from Theorem 1 that FedLin guarantees linear convergence to $x ^ { * }$ . Notably, the linear convergence rate we obtain in Theorem 1 under both objective and systems heterogeneity is the best rate we know of in $F L$ , and matches that of SCAFFOLD [11] where only objective heterogeneity is considered.3 The model of systems heterogeneity we study is taken from [23], where the authors provide guarantees only for the non-convex case under a bounded dissimilarity assumption. In contrast, our results cover all the three standard settings - strongly-convex, convex, and non-convex - without requiring any bounded dissimilarity assumption. For further related work on straggler-robust distributed learning algorithms (without objective heterogeneity or local steps), see [43–48]. ",
|
| 666 |
+
"bbox": [
|
| 667 |
+
174,
|
| 668 |
+
842,
|
| 669 |
+
825,
|
| 670 |
+
911
|
| 671 |
+
],
|
| 672 |
+
"page_idx": 5
|
| 673 |
+
},
|
| 674 |
+
{
|
| 675 |
+
"type": "text",
|
| 676 |
+
"text": "",
|
| 677 |
+
"bbox": [
|
| 678 |
+
173,
|
| 679 |
+
90,
|
| 680 |
+
826,
|
| 681 |
+
203
|
| 682 |
+
],
|
| 683 |
+
"page_idx": 6
|
| 684 |
+
},
|
| 685 |
+
{
|
| 686 |
+
"type": "text",
|
| 687 |
+
"text": "4.1 The Price of Infrequent Communication ",
|
| 688 |
+
"text_level": 1,
|
| 689 |
+
"bbox": [
|
| 690 |
+
173,
|
| 691 |
+
217,
|
| 692 |
+
493,
|
| 693 |
+
232
|
| 694 |
+
],
|
| 695 |
+
"page_idx": 6
|
| 696 |
+
},
|
| 697 |
+
{
|
| 698 |
+
"type": "text",
|
| 699 |
+
"text": "In this section, we take a closer look at the effect of performing multiple local steps on the convergence rate. To do so, we assume that all clients perform the same number of local steps $H$ , i.e., there is no communication for $H$ consecutive time-steps between two communication rounds. Now consider a centralized baseline where each client can communicate with every other client at all times (i.e., even between rounds). In this case, since each client can always access $\\nabla f ( x )$ , gradient descent yields ",
|
| 700 |
+
"bbox": [
|
| 701 |
+
173,
|
| 702 |
+
242,
|
| 703 |
+
825,
|
| 704 |
+
313
|
| 705 |
+
],
|
| 706 |
+
"page_idx": 6
|
| 707 |
+
},
|
| 708 |
+
{
|
| 709 |
+
"type": "equation",
|
| 710 |
+
"img_path": "images/019e10795ecc009657fb98c8a585892f63f2ec5932ef9e35b615b76bba9c2350.jpg",
|
| 711 |
+
"text": "$$\nf ( \\bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \\leq \\exp ( - \\frac { 1 } { \\kappa } T H ) ( f ( \\bar { x } _ { 1 } ) - f ( x ^ { * } ) )\n$$",
|
| 712 |
+
"text_format": "latex",
|
| 713 |
+
"bbox": [
|
| 714 |
+
326,
|
| 715 |
+
315,
|
| 716 |
+
671,
|
| 717 |
+
344
|
| 718 |
+
],
|
| 719 |
+
"page_idx": 6
|
| 720 |
+
},
|
| 721 |
+
{
|
| 722 |
+
"type": "text",
|
| 723 |
+
"text": "after $T$ rounds, with $H$ synchronized local iterations within each round. Based on Theorem 1, observe that we lose out by a factor of $H$ in the exponent relative to the centralized baseline. Notably, both in the centralized case, and in FedLin, each client queries the gradient of its local objective $H$ times in each round, thereby making $T H$ gradient queries over $T$ rounds. Thus, relative to a centralized baseline, FedLin incurs the same computational cost in terms of gradient queries, and reduces communication by a factor of $H$ , at the expense of a convergence rate that is slower by a factor of $H$ . We emphasize here that just as with FedLin, $H$ does not show up in the convergence rate (exponent) of algorithms like FedSplit [10] and SCAFFOLD [11] either. ",
|
| 724 |
+
"bbox": [
|
| 725 |
+
173,
|
| 726 |
+
345,
|
| 727 |
+
825,
|
| 728 |
+
458
|
| 729 |
+
],
|
| 730 |
+
"page_idx": 6
|
| 731 |
+
},
|
| 732 |
+
{
|
| 733 |
+
"type": "text",
|
| 734 |
+
"text": "The primary reason for the slower convergence rate (relative to a centralized baseline) stems from the need to set $\\eta \\propto 1 / H$ to mitigate client-drift under objective heterogeneity. At this stage, one may conjecture that the above requirement is simply an artifact of a conservative analysis of Algorithm 1, and that a more refined analysis will reveal the utility of performing more local steps even in the heterogeneous setting. Our next result suggests otherwise; for a proof, see Appendix $\\mathrm { E }$ . ",
|
| 735 |
+
"bbox": [
|
| 736 |
+
174,
|
| 737 |
+
463,
|
| 738 |
+
825,
|
| 739 |
+
534
|
| 740 |
+
],
|
| 741 |
+
"page_idx": 6
|
| 742 |
+
},
|
| 743 |
+
{
|
| 744 |
+
"type": "text",
|
| 745 |
+
"text": "Theorem 5. (Lower bound for FedLin) Suppose $\\delta _ { c } = \\delta _ { s } = 1$ , and $\\tau _ { i } = H , \\eta _ { i } = \\eta , \\forall i \\in S$ . Then, given any $L \\ge 1 4$ and $H \\geq 2$ , there exists an instance involving 2 clients where each $f _ { i } ( x ) , i \\in \\{ 1 , 2 \\}$ , is 1-strongly convex and $L$ -smooth, and an initial condition $\\scriptstyle { \\bar { x } } _ { 1 }$ , such that FedLin initialized from $\\scriptstyle { \\bar { x } } _ { 1 }$ generates a sequence of iterates $\\{ \\bar { x } _ { t } \\}$ satisfying the following for any $T \\geq 1$ : ",
|
| 746 |
+
"bbox": [
|
| 747 |
+
174,
|
| 748 |
+
535,
|
| 749 |
+
825,
|
| 750 |
+
592
|
| 751 |
+
],
|
| 752 |
+
"page_idx": 6
|
| 753 |
+
},
|
| 754 |
+
{
|
| 755 |
+
"type": "equation",
|
| 756 |
+
"img_path": "images/005f32c9850827547b8afbca1d53d94f8c49a6030bfed1faf9dbc4b7ed001ae8.jpg",
|
| 757 |
+
"text": "$$\n\\begin{array} { r } { \\| \\bar { x } _ { T + 1 } - x ^ { * } \\| ^ { 2 } \\geq \\exp \\left( - 4 T \\right) \\| \\bar { x } _ { 1 } - x ^ { * } \\| ^ { 2 } ; f ( \\bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \\geq \\exp ( - 4 T ) ( f ( \\bar { x } _ { 1 } ) - f ( x ^ { * } ) ) . } \\end{array}\n$$",
|
| 758 |
+
"text_format": "latex",
|
| 759 |
+
"bbox": [
|
| 760 |
+
183,
|
| 761 |
+
593,
|
| 762 |
+
787,
|
| 763 |
+
613
|
| 764 |
+
],
|
| 765 |
+
"page_idx": 6
|
| 766 |
+
},
|
| 767 |
+
{
|
| 768 |
+
"type": "text",
|
| 769 |
+
"text": "Main Takeaways: There are several key implications of Theorem 5. First, it complements Theorem 1 by providing a matching lower bound. We believe ours is the first work to provide a tight linear convergence rate analysis: [11] and [10] only provide upper-bounds for SCAFFOLD and FedSplit, respectively. Second, our analysis of Theorem 5 in Appendix E indicates that there are problem instances where setting $\\eta \\propto 1 / H$ is in fact necessary to guarantee convergence to $x ^ { * }$ . As a result, for such problem instances, no matter how many local steps $H$ each client performs, the error at the end of $T$ rounds remains bounded below by an $H$ -independent quantity, as is apparent from (10). Perhaps surprisingly, we show in Appendix E that the lower bound in Theorem 5 even applies to simple instances with non-identical quadratic losses (across clients) where every $f _ { i } ( x )$ has the same minimum! This is particularly insightful since it highlights the limitations of exploiting stale gradient terms in the local update rule (as is done in both FedLin and SCAFFOLD), and suggests the need for more informed updating schemes that explicitly take into account the level of statistical heterogeneity. ",
|
| 770 |
+
"bbox": [
|
| 771 |
+
173,
|
| 772 |
+
621,
|
| 773 |
+
826,
|
| 774 |
+
787
|
| 775 |
+
],
|
| 776 |
+
"page_idx": 6
|
| 777 |
+
},
|
| 778 |
+
{
|
| 779 |
+
"type": "text",
|
| 780 |
+
"text": "Proof Idea for Theorem 5: To establish Theorem 5, we set up an instance involving two clients with quadratic loss functions. Our main idea is to relate the convergence of FedLin to the Schur stability of an appropriate discrete-time linear time-invariant (LTI) system. Based on this connection, we show that guaranteeing stability necessitates setting $\\eta \\propto 1 / H$ , which immediately leads to the lower bound. We believe that the same technique can be used to establish a similar lower bound for SCAFFOLD. ",
|
| 781 |
+
"bbox": [
|
| 782 |
+
174,
|
| 783 |
+
795,
|
| 784 |
+
825,
|
| 785 |
+
864
|
| 786 |
+
],
|
| 787 |
+
"page_idx": 6
|
| 788 |
+
},
|
| 789 |
+
{
|
| 790 |
+
"type": "text",
|
| 791 |
+
"text": "5 Gradient Sparsification at Server ",
|
| 792 |
+
"text_level": 1,
|
| 793 |
+
"bbox": [
|
| 794 |
+
173,
|
| 795 |
+
88,
|
| 796 |
+
482,
|
| 797 |
+
107
|
| 798 |
+
],
|
| 799 |
+
"page_idx": 7
|
| 800 |
+
},
|
| 801 |
+
{
|
| 802 |
+
"type": "text",
|
| 803 |
+
"text": "In this section, our focus will be on addressing the following question: For strongly convex and smooth deterministic functions, and in the presence of both objective and systems heterogeneity, can we still hope for linear convergence to $x ^ { * }$ when gradients are sparsified at the server? Interestingly, we will show that not only is it possible to converge linearly to $x ^ { * }$ , it is possible to do so without any error-feedback. Moreover, this claim holds regardless of how aggressive the server is in its sparsification scheme: it may even transmit just a single component of the aggregated gradient vector. ",
|
| 804 |
+
"bbox": [
|
| 805 |
+
173,
|
| 806 |
+
119,
|
| 807 |
+
826,
|
| 808 |
+
204
|
| 809 |
+
],
|
| 810 |
+
"page_idx": 7
|
| 811 |
+
},
|
| 812 |
+
{
|
| 813 |
+
"type": "text",
|
| 814 |
+
"text": "To isolate the impact of server-level sparsification, we will assume throughout this section that gradients are not sparsified at the clients, i.e., $\\delta _ { c } = 1$ . Consequently, $h _ { i , t + 1 } = \\nabla f _ { i } ( \\bar { x } _ { t + 1 } ) , \\forall i \\in$ $\\mathcal { S } , \\forall t \\in \\{ 1 , \\ldots , T \\}$ . We begin by considering a simpler variant of FedLin with no error-feedback at the server side, i.e., line 15 is skipped, and $g _ { t + 1 }$ in line 14 of Algo. 1 is instead updated as follows ",
|
| 815 |
+
"bbox": [
|
| 816 |
+
174,
|
| 817 |
+
209,
|
| 818 |
+
825,
|
| 819 |
+
266
|
| 820 |
+
],
|
| 821 |
+
"page_idx": 7
|
| 822 |
+
},
|
| 823 |
+
{
|
| 824 |
+
"type": "equation",
|
| 825 |
+
"img_path": "images/39c7bdff6b7c3d3887b86eda5bb923e711eaf4093cbf8ee02fe5c52cc2236353.jpg",
|
| 826 |
+
"text": "$$\ng _ { t + 1 } = { \\mathcal C } _ { \\delta _ { s } } \\left( \\frac { 1 } { m } \\sum _ { i \\in \\mathcal { S } } \\nabla f _ { i } ( \\bar { x } _ { t + 1 } ) \\right) = { \\mathcal C } _ { \\delta _ { s } } \\left( \\nabla f ( \\bar { x } _ { t + 1 } ) \\right) .\n$$",
|
| 827 |
+
"text_format": "latex",
|
| 828 |
+
"bbox": [
|
| 829 |
+
320,
|
| 830 |
+
270,
|
| 831 |
+
678,
|
| 832 |
+
313
|
| 833 |
+
],
|
| 834 |
+
"page_idx": 7
|
| 835 |
+
},
|
| 836 |
+
{
|
| 837 |
+
"type": "text",
|
| 838 |
+
"text": "Theorem 6. (Sparsification at server with no error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\\mu$ -strongly convex. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ , and $\\delta _ { c } = 1$ . Consider a variant of FedLin, where line $^ { I 4 }$ is replaced by equation (11), and line 15 is skipped, i.e., there is no error-feedback. Then, with ηi = 12(2+√δs)Lτi , , this variant of FedLin guarantees ",
|
| 839 |
+
"bbox": [
|
| 840 |
+
173,
|
| 841 |
+
318,
|
| 842 |
+
826,
|
| 843 |
+
381
|
| 844 |
+
],
|
| 845 |
+
"page_idx": 7
|
| 846 |
+
},
|
| 847 |
+
{
|
| 848 |
+
"type": "equation",
|
| 849 |
+
"img_path": "images/3d9b57316c1a7f6cb2863a7337a2d88054299ee9195772b73cfec0377bf82abd.jpg",
|
| 850 |
+
"text": "$$\nf ( \\bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \\leq \\bigg ( 1 - \\frac { 1 } { 2 \\delta _ { s } \\left( 2 + \\sqrt { \\delta _ { s } } \\right) \\kappa } \\bigg ) ^ { T } ( f ( \\bar { x } _ { 1 } ) - f ( x ^ { * } ) ) .\n$$",
|
| 851 |
+
"text_format": "latex",
|
| 852 |
+
"bbox": [
|
| 853 |
+
281,
|
| 854 |
+
385,
|
| 855 |
+
717,
|
| 856 |
+
430
|
| 857 |
+
],
|
| 858 |
+
"page_idx": 7
|
| 859 |
+
},
|
| 860 |
+
{
|
| 861 |
+
"type": "text",
|
| 862 |
+
"text": "Main Takeaways: From Theorem 6, we see that even without error-feedback, it is possible to linearly converge to $x ^ { * }$ ; the rate of convergence, however, is inversely proportional to $\\delta _ { s } ^ { \\frac { 3 } { 2 } }$ . Thus, Theorem 6 quantifies the trade-off between the level of sparsification at the server, and the rate of convergence. When there is no gradient compression, i.e., when $\\delta _ { s } = 1$ , we exactly recover Theorem 1. ",
|
| 863 |
+
"bbox": [
|
| 864 |
+
173,
|
| 865 |
+
439,
|
| 866 |
+
826,
|
| 867 |
+
502
|
| 868 |
+
],
|
| 869 |
+
"page_idx": 7
|
| 870 |
+
},
|
| 871 |
+
{
|
| 872 |
+
"type": "text",
|
| 873 |
+
"text": "One may ask: Is there any potential benefit to employing error-feedback when gradients are sparsified at the server? Our next result answers this question in the affirmative. ",
|
| 874 |
+
"bbox": [
|
| 875 |
+
174,
|
| 876 |
+
507,
|
| 877 |
+
823,
|
| 878 |
+
536
|
| 879 |
+
],
|
| 880 |
+
"page_idx": 7
|
| 881 |
+
},
|
| 882 |
+
{
|
| 883 |
+
"type": "text",
|
| 884 |
+
"text": "Theorem 7. (Sparsification at server with error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and $\\mu$ -strongly convex. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ , and $\\delta _ { c } = 1$ . Let the step-size for client i be chosen as $\\begin{array} { r } { \\eta _ { i } = \\frac { 1 } { 7 2 L \\delta _ { s } \\tau _ { i } } } \\end{array}$ . Then, FedLin guarantees: ",
|
| 885 |
+
"bbox": [
|
| 886 |
+
176,
|
| 887 |
+
539,
|
| 888 |
+
821,
|
| 889 |
+
583
|
| 890 |
+
],
|
| 891 |
+
"page_idx": 7
|
| 892 |
+
},
|
| 893 |
+
{
|
| 894 |
+
"type": "equation",
|
| 895 |
+
"img_path": "images/c7a297fb8c0aefdff46a386e4be7700572192b92a64da773bf8ce0d83ba89ee6.jpg",
|
| 896 |
+
"text": "$$\nf ( \\bar { x } _ { T + 1 } ) - f ( x ^ { * } ) \\leq 2 \\kappa \\bigg ( 1 - \\frac { 1 } { 9 6 \\delta _ { s } \\kappa } \\bigg ) ^ { T } \\left( f ( \\bar { x } _ { 1 } ) - f ( x ^ { * } ) \\right) .\n$$",
|
| 897 |
+
"text_format": "latex",
|
| 898 |
+
"bbox": [
|
| 899 |
+
303,
|
| 900 |
+
587,
|
| 901 |
+
692,
|
| 902 |
+
626
|
| 903 |
+
],
|
| 904 |
+
"page_idx": 7
|
| 905 |
+
},
|
| 906 |
+
{
|
| 907 |
+
"type": "text",
|
| 908 |
+
"text": "For proofs of Theorems 6 and 7, see Appendix G and I. ",
|
| 909 |
+
"bbox": [
|
| 910 |
+
174,
|
| 911 |
+
636,
|
| 912 |
+
537,
|
| 913 |
+
651
|
| 914 |
+
],
|
| 915 |
+
"page_idx": 7
|
| 916 |
+
},
|
| 917 |
+
{
|
| 918 |
+
"type": "text",
|
| 919 |
+
"text": "Main Takeaways: Comparing the guarantee of Theorem 6 with that of Theorem 7, we note that the convergence rate is inversely proportional to $\\delta _ { s } ^ { \\frac { 3 } { 2 } }$ in the former, and inversely proportional to $\\delta _ { s }$ in the latter. Thus, the main message here is that employing error-feedback leads to a faster convergence rate by improving the dependence of the rate on $\\delta _ { s }$ . ",
|
| 920 |
+
"bbox": [
|
| 921 |
+
174,
|
| 922 |
+
656,
|
| 923 |
+
826,
|
| 924 |
+
718
|
| 925 |
+
],
|
| 926 |
+
"page_idx": 7
|
| 927 |
+
},
|
| 928 |
+
{
|
| 929 |
+
"type": "text",
|
| 930 |
+
"text": "6 Gradient Sparsification at Clients ",
|
| 931 |
+
"text_level": 1,
|
| 932 |
+
"bbox": [
|
| 933 |
+
173,
|
| 934 |
+
736,
|
| 935 |
+
488,
|
| 936 |
+
753
|
| 937 |
+
],
|
| 938 |
+
"page_idx": 7
|
| 939 |
+
},
|
| 940 |
+
{
|
| 941 |
+
"type": "text",
|
| 942 |
+
"text": "In this section, we will turn our attention to the case when gradients are sparsified at the clients prior to being transmitted to the server. Throughout this section, we will assume that gradients are not compressed any further at the server side, i.e., $\\delta _ { s } = 1$ . To proceed, we will need to make the following bounded gradient dissimilarity assumption. ",
|
| 943 |
+
"bbox": [
|
| 944 |
+
173,
|
| 945 |
+
767,
|
| 946 |
+
825,
|
| 947 |
+
824
|
| 948 |
+
],
|
| 949 |
+
"page_idx": 7
|
| 950 |
+
},
|
| 951 |
+
{
|
| 952 |
+
"type": "text",
|
| 953 |
+
"text": "Assumption 1. There exist constants $C \\geq 1$ and $D \\geq 0$ such that the following holds $\\forall x \\in \\mathbb { R } ^ { d }$ ",
|
| 954 |
+
"bbox": [
|
| 955 |
+
174,
|
| 956 |
+
825,
|
| 957 |
+
800,
|
| 958 |
+
842
|
| 959 |
+
],
|
| 960 |
+
"page_idx": 7
|
| 961 |
+
},
|
| 962 |
+
{
|
| 963 |
+
"type": "equation",
|
| 964 |
+
"img_path": "images/6b5e3df6d32862dfd030b00e40467430624c20ffd029fbb3f68ead23613a14cd.jpg",
|
| 965 |
+
"text": "$$\n{ \\frac { 1 } { m } } \\sum _ { i = 1 } ^ { m } \\left\\| \\nabla f _ { i } ( x ) \\right\\| ^ { 2 } \\leq C \\left\\| \\nabla f ( x ) \\right\\| ^ { 2 } + D .\n$$",
|
| 966 |
+
"text_format": "latex",
|
| 967 |
+
"bbox": [
|
| 968 |
+
367,
|
| 969 |
+
845,
|
| 970 |
+
632,
|
| 971 |
+
886
|
| 972 |
+
],
|
| 973 |
+
"page_idx": 7
|
| 974 |
+
},
|
| 975 |
+
{
|
| 976 |
+
"type": "text",
|
| 977 |
+
"text": "The following is the main result of this section; for a proof, see Appendix H. ",
|
| 978 |
+
"bbox": [
|
| 979 |
+
171,
|
| 980 |
+
896,
|
| 981 |
+
674,
|
| 982 |
+
912
|
| 983 |
+
],
|
| 984 |
+
"page_idx": 7
|
| 985 |
+
},
|
| 986 |
+
{
|
| 987 |
+
"type": "text",
|
| 988 |
+
"text": "Theorem 8. (Sparsification at clients with error-feedback) Suppose each $f _ { i } ( x )$ is $L$ -smooth and -strongly convex, and suppose Assumption 1 holds. Moreover, suppose $\\tau _ { i } \\geq 1 , \\forall i \\in S$ , and $\\delta _ { s } = 1$ . Let the step-size for client i be chosen as $\\begin{array} { r } { \\eta _ { i } = \\frac { \\bar { \\eta } } { \\tau _ { i } } } \\end{array}$ , where $\\bar { \\eta } \\in ( 0 , 1 )$ satisfies $\\begin{array} { r } { \\bar { \\eta } \\le \\frac { 1 } { 7 2 L \\delta _ { c } C } } \\end{array}$ . Then, FedLin guarantees: ",
|
| 989 |
+
"bbox": [
|
| 990 |
+
174,
|
| 991 |
+
90,
|
| 992 |
+
826,
|
| 993 |
+
147
|
| 994 |
+
],
|
| 995 |
+
"page_idx": 8
|
| 996 |
+
},
|
| 997 |
+
{
|
| 998 |
+
"type": "equation",
|
| 999 |
+
"img_path": "images/ccc8165bcf68d27885294bbf7754d6c702b920297dc840974a72b8b5b9e3e41a.jpg",
|
| 1000 |
+
"text": "$$\n\\left\\| { \\bar { x } } _ { T + 1 } - x ^ { * } \\right\\| ^ { 2 } \\leq 2 { \\left( 1 - \\frac { 3 } { 4 } { \\bar { \\eta } } \\mu \\right) } ^ { T } \\left\\| { \\bar { x } } _ { 1 } - x ^ { * } \\right\\| ^ { 2 } + \\frac { 1 6 } { 3 } { \\bar { \\eta } } \\left( \\frac { 6 } { \\delta _ { c } C } + \\delta _ { c } \\right) \\frac { D } { \\mu } .\n$$",
|
| 1001 |
+
"text_format": "latex",
|
| 1002 |
+
"bbox": [
|
| 1003 |
+
266,
|
| 1004 |
+
147,
|
| 1005 |
+
732,
|
| 1006 |
+
185
|
| 1007 |
+
],
|
| 1008 |
+
"page_idx": 8
|
| 1009 |
+
},
|
| 1010 |
+
{
|
| 1011 |
+
"type": "text",
|
| 1012 |
+
"text": "Main Takeaways: Intuitively, one would expect that sparsifying gradients at each client prior to aggregation at the server would inject more errors than when gradients are first accurately aggregated at the server, and then the aggregated gradient vector is sparsified: Theorems 6 and 8 support this intuition. For the former, we neither required error-feedback nor Assumption 1 to guarantee linear convergence to the global minimum $x ^ { * }$ ; for the latter, even with error-feedback and the bounded gradient dissimilarity assumption, we can establish linear convergence to only a neighborhood of $x ^ { * }$ , in general. From (13), we note that the size of this neighborhood scales linearly with $D$ - a measure of objective heterogeneity. In particular, when $D = 0$ , the iterates $\\bar { x } _ { t }$ converge exactly to $x ^ { * }$ . ",
|
| 1013 |
+
"bbox": [
|
| 1014 |
+
173,
|
| 1015 |
+
191,
|
| 1016 |
+
825,
|
| 1017 |
+
304
|
| 1018 |
+
],
|
| 1019 |
+
"page_idx": 8
|
| 1020 |
+
},
|
| 1021 |
+
{
|
| 1022 |
+
"type": "text",
|
| 1023 |
+
"text": "Remark 1. To the best of our knowledge, our results in Sections 5 and 6 constitute the first formal analysis of biased gradient sparsification in FL. In particular, we significantly generalize the recent results in [49] for a single worker to a multi-client $F L$ setting with both objective and systems heterogeneity. To arrive at these results, we develop a new potential-function based proof technique in Appendix H. For more related work on compression in distributed learning, see Appendix A. ",
|
| 1024 |
+
"bbox": [
|
| 1025 |
+
174,
|
| 1026 |
+
305,
|
| 1027 |
+
825,
|
| 1028 |
+
375
|
| 1029 |
+
],
|
| 1030 |
+
"page_idx": 8
|
| 1031 |
+
},
|
| 1032 |
+
{
|
| 1033 |
+
"type": "text",
|
| 1034 |
+
"text": "Extensions: We studied the effect of compressing information at the server and at the clients separately, with the goal of identifying the key differences between each of these mechanisms. The analysis techniques we developed in the process pave the way for studying various natural extensions: (i) combined sparsification at both the clients and the server; (ii) gradient sparsification in tandem with model parameter compression; and (iii) stochastic counterparts of Theorems 6, 7, and 8. ",
|
| 1035 |
+
"bbox": [
|
| 1036 |
+
173,
|
| 1037 |
+
382,
|
| 1038 |
+
825,
|
| 1039 |
+
453
|
| 1040 |
+
],
|
| 1041 |
+
"page_idx": 8
|
| 1042 |
+
},
|
| 1043 |
+
{
|
| 1044 |
+
"type": "text",
|
| 1045 |
+
"text": "7 Experimental Results ",
|
| 1046 |
+
"text_level": 1,
|
| 1047 |
+
"bbox": [
|
| 1048 |
+
174,
|
| 1049 |
+
463,
|
| 1050 |
+
387,
|
| 1051 |
+
481
|
| 1052 |
+
],
|
| 1053 |
+
"page_idx": 8
|
| 1054 |
+
},
|
| 1055 |
+
{
|
| 1056 |
+
"type": "text",
|
| 1057 |
+
"text": "In this section, we provide numerical results for FedLin on a least squares problem to validate our theory. In Appendix K, we also provide additional numerical results on a logistic regression problem. ",
|
| 1058 |
+
"bbox": [
|
| 1059 |
+
173,
|
| 1060 |
+
487,
|
| 1061 |
+
825,
|
| 1062 |
+
516
|
| 1063 |
+
],
|
| 1064 |
+
"page_idx": 8
|
| 1065 |
+
},
|
| 1066 |
+
{
|
| 1067 |
+
"type": "text",
|
| 1068 |
+
"text": "For now, we consider the following least squares regression problem: ",
|
| 1069 |
+
"bbox": [
|
| 1070 |
+
174,
|
| 1071 |
+
521,
|
| 1072 |
+
627,
|
| 1073 |
+
536
|
| 1074 |
+
],
|
| 1075 |
+
"page_idx": 8
|
| 1076 |
+
},
|
| 1077 |
+
{
|
| 1078 |
+
"type": "equation",
|
| 1079 |
+
"img_path": "images/bd503ea818e0e04580ed1e5a9441d2de19b88bf2539749f63359dccd7e81e24b.jpg",
|
| 1080 |
+
"text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x ) = \\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\frac { 1 } { 2 } \\| A _ { i } x - b _ { i } \\| ^ { 2 } ,\n$$",
|
| 1081 |
+
"text_format": "latex",
|
| 1082 |
+
"bbox": [
|
| 1083 |
+
362,
|
| 1084 |
+
536,
|
| 1085 |
+
633,
|
| 1086 |
+
577
|
| 1087 |
+
],
|
| 1088 |
+
"page_idx": 8
|
| 1089 |
+
},
|
| 1090 |
+
{
|
| 1091 |
+
"type": "text",
|
| 1092 |
+
"text": "where $A _ { i } \\in \\mathbb { R } ^ { 5 0 0 \\times 1 0 0 }$ is a design matrix and $b _ { i } \\in \\mathbb { R } ^ { 5 0 0 }$ is a response vector. The client objective functions, $f _ { i } ( x )$ are strongly convex. Assuming that all design matrices are full column rank, problem (14) admits a unique minimizer. To generate synthetic data, for each client $i \\in \\mathcal { S } = \\{ 1 , . . . , \\bar { 2 0 } \\}$ , we generate $A _ { i }$ and $b _ { i }$ according to the model $b _ { i } = A _ { i } x _ { i } + \\varepsilon _ { i }$ , where $x _ { i }$ is a weight vector and $\\varepsilon _ { i } \\in \\mathrm { \\mathbb { R } } ^ { 5 0 0 }$ is a disturbance. In particular, we generate $[ A _ { i } ] _ { j k } \\stackrel { i . i . d . } { \\sim } { \\cal N } ( 0 , 1 )$ , and $\\varepsilon _ { i } \\sim \\mathcal { N } ( 0 , 0 . 5 I _ { 5 0 0 } )$ , $\\forall i \\in S$ To capture statistical heterogeneity, the entries of the local true parameter of client $i$ are modeled as $[ x _ { i } ] _ { k } \\overset { \\vartriangle } { \\sim } \\mathcal { N } ( u _ { i } , 1 )$ , $k \\in \\{ 1 , \\ldots , 1 0 0 \\}$ , where $u _ { i } \\sim \\mathcal { N } ( 0 , \\alpha )$ and $\\alpha \\geq 0$ . Hence, $\\alpha$ controls the level of statistical heterogeneity. To model the effect of systems heterogeneity, for each client $i \\in S$ , the number of local steps is drawn uniformly and independently from [2, 100]. We will primarily focus on a deterministic setting here for our experiments; in Appendix $\\mathrm { L }$ , we evaluate FedLin on a standard stochastic oracle model. Our experiments in Appendix $\\mathrm { L }$ reveal that under a noisy oracle, FedLin guarantees linear convergence to a ball around the true minimum, exactly as suggested by Thm. 4. ",
|
| 1093 |
+
"bbox": [
|
| 1094 |
+
173,
|
| 1095 |
+
578,
|
| 1096 |
+
825,
|
| 1097 |
+
751
|
| 1098 |
+
],
|
| 1099 |
+
"page_idx": 8
|
| 1100 |
+
},
|
| 1101 |
+
{
|
| 1102 |
+
"type": "text",
|
| 1103 |
+
"text": "Gradient Sparsification at Server. We first consider a variant of FedLin where gradient sparsification is implemented only at the server side and without any error-feedback. In particular, we consider the cases where $\\delta _ { s } \\in \\{ 2 , 4 \\}$ , which correspond to the implementation of a TOP-50 and a TOP-25 operator, respectively. For comparison, we also plot the resulting performance when no gradient sparsification is implemented at the server. To examine the effect of statistical heterogeneity on the performance of FedLin, we generate two synthetic datasets corresponding to two different levels of heterogeneity in the clients’ local objectives, namely $\\alpha = 1 0$ and $\\alpha = 5 0$ . As illustrated in Fig. 2, irrespective of the level of gradient sparsification on the server side, FedLin achieves linear convergence to the true minimum in the presence of both objective and systems heterogeneity, confirming Theorem 6. Also, both the convergence speed and accuracy of FedLin remain unaffected as the level of heterogeneity in the clients’ objective functions increases. ",
|
| 1104 |
+
"bbox": [
|
| 1105 |
+
173,
|
| 1106 |
+
756,
|
| 1107 |
+
825,
|
| 1108 |
+
909
|
| 1109 |
+
],
|
| 1110 |
+
"page_idx": 8
|
| 1111 |
+
},
|
| 1112 |
+
{
|
| 1113 |
+
"type": "image",
|
| 1114 |
+
"img_path": "images/b9946e0016d1f3222cc7ea00abb207e4eede4c856c80fec12e713362b741c193.jpg",
|
| 1115 |
+
"image_caption": [],
|
| 1116 |
+
"image_footnote": [],
|
| 1117 |
+
"bbox": [
|
| 1118 |
+
178,
|
| 1119 |
+
90,
|
| 1120 |
+
558,
|
| 1121 |
+
227
|
| 1122 |
+
],
|
| 1123 |
+
"page_idx": 9
|
| 1124 |
+
},
|
| 1125 |
+
{
|
| 1126 |
+
"type": "image",
|
| 1127 |
+
"img_path": "images/f407b5b418f6c30639d8a94a97f87a7309d80f646c9ff33d1cff0da62d1fd645.jpg",
|
| 1128 |
+
"image_caption": [
|
| 1129 |
+
"Figure 2: Server-side sparsification results for FedLin. The constant $\\bar { \\eta }$ is fixed at $1 \\dot { 0 } ^ { - 2 }$ . Left: $\\alpha = 1 0$ . Right: $\\alpha = 5 0$ . "
|
| 1130 |
+
],
|
| 1131 |
+
"image_footnote": [],
|
| 1132 |
+
"bbox": [
|
| 1133 |
+
176,
|
| 1134 |
+
276,
|
| 1135 |
+
557,
|
| 1136 |
+
415
|
| 1137 |
+
],
|
| 1138 |
+
"page_idx": 9
|
| 1139 |
+
},
|
| 1140 |
+
{
|
| 1141 |
+
"type": "text",
|
| 1142 |
+
"text": "Gradient Sparsification at Clients. Next, we implement gradient sparsification only at the clients’ side, i.e. $\\delta _ { s } ~ = ~ 1$ . In particular, we consider the cases where $\\delta _ { c } \\in \\{ 4 / 3 , 2 \\}$ , which correspond to the implementation of a TOP-75 and a TOP-50 operator, respectively. Once again, we generate two synthetic datasets with different levels of objective heterogeneity, namely $\\alpha = 1$ and $\\alpha = 1 0$ . As illustrated in Fig. 3, unlike the server case, FedLin with sparsification at the clients’ side converges linearly, but with a non-vanishing error that increases as the value of $\\delta _ { c }$ increases. This aligns with the conclusions of Theorem 8. Furthermore, the level of objective heterogeneity has a direct impact on the convergence error. In particular, for the same level of gradient sparsification, higher levels of objective heterogeneity result in larger values of the convergence error. ",
|
| 1143 |
+
"bbox": [
|
| 1144 |
+
578,
|
| 1145 |
+
90,
|
| 1146 |
+
825,
|
| 1147 |
+
421
|
| 1148 |
+
],
|
| 1149 |
+
"page_idx": 9
|
| 1150 |
+
},
|
| 1151 |
+
{
|
| 1152 |
+
"type": "image",
|
| 1153 |
+
"img_path": "images/156a4dd073fbf246f89c9c33a7fb3226e915f347ca07f6e341188a94f998d89e.jpg",
|
| 1154 |
+
"image_caption": [
|
| 1155 |
+
"Figure 3: Client-side sparsification results for FedLin. The constant $\\bar { \\eta }$ is fixed at $5 \\times 1 0 ^ { - 4 }$ . Left: $\\alpha = 1$ . Right: $\\alpha = 1 0$ . ",
|
| 1156 |
+
"Figure 4: Comparison of FedLin with SCAFFOLD. (Left) Deterministic setting. (Right) General stochastic oracle model: unbiased gradients with variance $\\sigma = 1 0 ^ { - 1 }$ . "
|
| 1157 |
+
],
|
| 1158 |
+
"image_footnote": [],
|
| 1159 |
+
"bbox": [
|
| 1160 |
+
183,
|
| 1161 |
+
473,
|
| 1162 |
+
563,
|
| 1163 |
+
584
|
| 1164 |
+
],
|
| 1165 |
+
"page_idx": 9
|
| 1166 |
+
},
|
| 1167 |
+
{
|
| 1168 |
+
"type": "text",
|
| 1169 |
+
"text": "Comparison with SCAFFOLD. We now compare FedLin with SCAFFOLD on the least squares regression setup described above. To make a fair comparison, we assume that there is no systems heterogeneity or gradient compression. For implementing SCAFFOLD, we use Option II in their paper [11] for updating the control variates. We set the number of local steps $H = 2 0$ , the statistical heterogeneity parameter $\\alpha = 1 0$ , and use a step-size of $1 0 ^ { - 3 }$ for both algorithms (the step-size was tuned to get best results). For the deterministic setting, we note from Fig. 4 that FedLin con",
|
| 1170 |
+
"bbox": [
|
| 1171 |
+
578,
|
| 1172 |
+
430,
|
| 1173 |
+
823,
|
| 1174 |
+
648
|
| 1175 |
+
],
|
| 1176 |
+
"page_idx": 9
|
| 1177 |
+
},
|
| 1178 |
+
{
|
| 1179 |
+
"type": "text",
|
| 1180 |
+
"text": "verges much faster compared to SCAFFOLD. This trend persists when we perturb the gradients with zero-mean Gaussian noise with variance $\\sigma = 1 0 ^ { - 1 }$ . We conjecture that the faster convergence of FedLin stems from the fact that it uses less stale gradient correction terms relative to the control variates of SCAFFOLD; see the discussion about the fixed point property of FedLin in Sec. 3. ",
|
| 1181 |
+
"bbox": [
|
| 1182 |
+
174,
|
| 1183 |
+
650,
|
| 1184 |
+
825,
|
| 1185 |
+
705
|
| 1186 |
+
],
|
| 1187 |
+
"page_idx": 9
|
| 1188 |
+
},
|
| 1189 |
+
{
|
| 1190 |
+
"type": "text",
|
| 1191 |
+
"text": "8 Conclusion ",
|
| 1192 |
+
"text_level": 1,
|
| 1193 |
+
"bbox": [
|
| 1194 |
+
174,
|
| 1195 |
+
724,
|
| 1196 |
+
299,
|
| 1197 |
+
741
|
| 1198 |
+
],
|
| 1199 |
+
"page_idx": 9
|
| 1200 |
+
},
|
| 1201 |
+
{
|
| 1202 |
+
"type": "text",
|
| 1203 |
+
"text": "We developed a novel algorithmic framework called FedLin to tackle some of the key challenges in FL, namely objective heterogeneity, systems heterogeneity, and imprecise communication. We showed that FedLin enjoys strong theoretical guarantees: (i) FedLin matches centralized rates, and, in particular, guarantees linear convergence to the global minimum under arbitrary objective and systems heterogeneity; and (ii) preserves linear convergence rates despite aggressive gradient sparsification. We also established a tight lower-bound for FedLin, highlighting that even mild statistical heterogeneity can end up hurting convergence rates - this is the first such result in FL. Our current approach requires two passes of communication between the clients and the server in each round. Moreover, our analysis does not account for partial client participation. As future work, we plan to address these limitations. We also plan to investigate other federated learning formulations (beyond supervised learning) where statistical heterogeneity can potentially help. ",
|
| 1204 |
+
"bbox": [
|
| 1205 |
+
174,
|
| 1206 |
+
756,
|
| 1207 |
+
825,
|
| 1208 |
+
907
|
| 1209 |
+
],
|
| 1210 |
+
"page_idx": 9
|
| 1211 |
+
},
|
| 1212 |
+
{
|
| 1213 |
+
"type": "text",
|
| 1214 |
+
"text": "Acknowledgement: This work was supported by NSF Award 1837253, NSF CAREER award CIF 1943064, and the Air Force Office of Scientific Research Young Investigator Program (AFOSR-YIP) under award FA9550-20-1-0111. ",
|
| 1215 |
+
"bbox": [
|
| 1216 |
+
173,
|
| 1217 |
+
92,
|
| 1218 |
+
823,
|
| 1219 |
+
133
|
| 1220 |
+
],
|
| 1221 |
+
"page_idx": 10
|
| 1222 |
+
},
|
| 1223 |
+
{
|
| 1224 |
+
"type": "text",
|
| 1225 |
+
"text": "References \n[1] Jakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richtárik. Federated optimiza-\\` tion: Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016. \n[2] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282. PMLR, 2017. \n[3] Keith Bonawitz, Hubert Eichner, Wolfgang Grieskamp, Dzmitry Huba, Alex Ingerman, Vladimir Ivanov, Chloe Kiddon, Jakub Konecnˇ y, Stefano Mazzocchi, H Brendan McMahan, et al. \\` Towards federated learning at scale: System design. arXiv preprint arXiv:1902.01046, 2019. \n[4] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019. \n[5] Tian Li, Anit Kumar Sahu, Ameet Talwalkar, and Virginia Smith. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020. \n[6] Xiang Li, Kaixuan Huang, Wenhao Yang, Shusen Wang, and Zhihua Zhang. On the convergence of fedavg on non-iid data. arXiv preprint arXiv:1907.02189, 2019. \n[7] Grigory Malinovskiy, Dmitry Kovalev, Elnur Gasanov, Laurent Condat, and Peter Richtarik. From local sgd to local fixed-point methods for federated learning. In International Conference on Machine Learning, pages 6692–6701. PMLR, 2020. \n[8] Zachary Charles and Jakub Konecnˇ y. On the outsized importance of learning rates in local \\` update methods. arXiv preprint arXiv:2007.00878, 2020. \n[9] Zachary Charles and Jakub Konecnˇ y. Convergence and accuracy trade-offs in federated learning \\` and meta-learning. In International Conference on Artificial Intelligence and Statistics, pages 2575–2583. PMLR, 2021. \n[10] Reese Pathak and Martin J Wainwright. Fedsplit: An algorithmic framework for fast federated optimization. arXiv preprint arXiv:2005.05238, 2020. \n[11] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020. \n[12] Sebastian U Stich. Local sgd converges fast and communicates little. arXiv preprint arXiv:1805.09767, 2018. \n[13] Jianyu Wang and Gauri Joshi. Cooperative sgd: A unified framework for the design and analysis of communication-efficient sgd algorithms. arXiv preprint arXiv:1808.07576, 2018. \n[14] Artin Spiridonoff, Alex Olshevsky, and Ioannis Ch Paschalidis. Local sgd with a communication overhead depending only on the number of workers. arXiv preprint arXiv:2006.02582, 2020. \n[15] Amirhossein Reisizadeh, Aryan Mokhtari, Hamed Hassani, Ali Jadbabaie, and Ramtin Pedarsani. Fedpaq: A communication-efficient federated learning method with periodic averaging and quantization. In International Conference on Artificial Intelligence and Statistics, pages 2021– 2031. PMLR, 2020. \n[16] Farzin Haddadpour, Mohammad Mahdi Kamani, Mehrdad Mahdavi, and Viveck Cadambe. Local sgd with periodic averaging: Tighter analysis and adaptive synchronization. In Advances in Neural Information Processing Systems, pages 11082–11094, 2019. \n[17] Blake Woodworth, Kumar Kshitij Patel, Sebastian U Stich, Zhen Dai, Brian Bullins, H Brendan McMahan, Ohad Shamir, and Nathan Srebro. Is local sgd better than minibatch sgd? arXiv preprint arXiv:2002.07839, 2020. \n[18] Ahmed Khaled, Konstantin Mishchenko, and Peter Richtárik. First analysis of local gd on heterogeneous data. arXiv preprint arXiv:1909.04715, 2019. \n[19] Ahmed Khaled, Konstantin Mishchenko, and Peter Richtárik. Tighter theory for local sgd on identical and heterogeneous data. In International Conference on Artificial Intelligence and Statistics, pages 4519–4529. PMLR, 2020. \n[20] Farzin Haddadpour and Mehrdad Mahdavi. On the convergence of local descent methods in federated learning. arXiv preprint arXiv:1910.14425, 2019. \n[21] Anastasia Koloskova, Nicolas Loizou, Sadra Boreiri, Martin Jaggi, and Sebastian U Stich. A unified theory of decentralized sgd with changing topology and local updates. arXiv preprint arXiv:2003.10422, 2020. \n[22] Anit Kumar Sahu, Tian Li, Maziar Sanjabi, Manzil Zaheer, Ameet Talwalkar, and Virginia Smith. On the convergence of federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 3, 2018. \n[23] Jianyu Wang, Qinghua Liu, Hao Liang, Gauri Joshi, and H Vincent Poor. Tackling the objective inconsistency problem in heterogeneous federated optimization. Advances in Neural Information Processing Systems, 33, 2020. \n[24] Alham Fikri Aji and Kenneth Heafield. Sparse communication for distributed gradient descent. arXiv preprint arXiv:1704.05021, 2017. \n[25] Yujun Lin, Song Han, Huizi Mao, Yu Wang, and William J Dally. Deep gradient compression: Reducing the communication bandwidth for distributed training. arXiv preprint arXiv:1712.01887, 2017. \n[26] Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Qsgd: Communication-efficient sgd via gradient quantization and encoding. Advances in Neural Information Processing Systems, 30:1709–1720, 2017. \n[27] Wei Wen, Cong Xu, Feng Yan, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Terngrad: Ternary gradients to reduce communication in distributed deep learning. In Advances in neural information processing systems, pages 1509–1519, 2017. \n[28] Sarit Khirirat, Hamid Reza Feyzmahdavian, and Mikael Johansson. Distributed learning with compressed gradients. arXiv preprint arXiv:1806.06573, 2018. \n[29] Konstantin Mishchenko, Eduard Gorbunov, Martin Takác, and Peter Richtárik. Distributed ˇ learning with compressed gradient differences. arXiv preprint arXiv:1901.09269, 2019. \n[30] Zhize Li, Dmitry Kovalev, Xun Qian, and Peter Richtárik. Acceleration for compressed gradient descent in distributed and federated optimization. arXiv preprint arXiv:2002.11364, 2020. \n[31] Samuel Horváth, Dmitry Kovalev, Konstantin Mishchenko, Sebastian Stich, and Peter Richtárik. Stochastic distributed learning with gradient quantization and variance reduction. arXiv preprint arXiv:1904.05115, 2019. \n[32] Farzin Haddadpour, Mohammad Mahdi Kamani, Aryan Mokhtari, and Mehrdad Mahdavi. Federated learning with compression: Unified analysis and sharp guarantees. arXiv preprint arXiv:2007.01154, 2020. \n[33] Frank Seide, Hao Fu, Jasha Droppo, Gang Li, and Dong Yu. 1-bit stochastic gradient descent and its application to data-parallel distributed training of speech dnns. In Fifteenth Annual Conference of the International Speech Communication Association, 2014. \n[34] Sebastian U Stich, Jean-Baptiste Cordonnier, and Martin Jaggi. Sparsified sgd with memory. In Advances in Neural Information Processing Systems, pages 4447–4458, 2018. \n[35] Dan Alistarh, Torsten Hoefler, Mikael Johansson, Nikola Konstantinov, Sarit Khirirat, and Cédric Renggli. The convergence of sparsified gradient methods. In Advances in Neural Information Processing Systems, pages 5973–5983, 2018. \n[36] Guannan Qu and Na Li. Harnessing smoothness to accelerate distributed optimization. IEEE Transactions on Control of Network Systems, 5(3):1245–1260, 2017. \n[37] Angelia Nedic, Alex Olshevsky, and Wei Shi. Achieving geometric convergence for distributed optimization over time-varying graphs. SIAM Journal on Optimization, 27(4):2597–2633, 2017. \n[38] Chenguang Xi, Ran Xin, and Usman A Khan. Add-opt: Accelerated distributed directed optimization. IEEE Transactions on Automatic Control, 63(5):1329–1339, 2017. \n[39] Shi Pu and Angelia Nedic. Distributed stochastic gradient tracking methods. ´ Mathematical Programming, pages 1–49, 2020. \n[40] Ran Xin, Anit Kumar Sahu, Usman A Khan, and Soummya Kar. Distributed stochastic optimization with gradient tracking over strongly-connected networks. In Proc. of the 58th IEEE Conference on Decision and Control (CDC), pages 8353–8358, 2019. \n[41] Arkadi Nemirovski, Anatoli Juditsky, Guanghui Lan, and Alexander Shapiro. Robust stochastic approximation approach to stochastic programming. SIAM Journal on optimization, 19(4):1574– 1609, 2009. \n[42] Eduard Gorbunov, Filip Hanzely, and Peter Richtárik. Local sgd: Unified theory and new efficient methods. In International Conference on Artificial Intelligence and Statistics, pages 3556–3564. PMLR, 2021. \n[43] Amirhossein Reisizadeh, Isidoros Tziotis, Hamed Hassani, Aryan Mokhtari, and Ramtin Pedarsani. Straggler-resilient federated learning: Leveraging the interplay between statistical accuracy and system heterogeneity. arXiv preprint arXiv:2012.14453, 2020. \n[44] Sanghamitra Dutta, Jianyu Wang, and Gauri Joshi. Slow and stale gradients can win the race. arXiv preprint arXiv:2003.10579, 2020. \n[45] Jianyu Wang and Gauri Joshi. Adaptive communication strategies to achieve the best errorruntime trade-off in local-update sgd. arXiv preprint arXiv:1810.08313, 2018. \n[46] Rawad Bitar, Mary Wootters, and Salim El Rouayheb. Stochastic gradient coding for straggler mitigation in distributed learning. IEEE Journal on Selected Areas in Information Theory, 2020. \n[47] Nuwan Ferdinand and Stark C Draper. Anytime stochastic gradient descent: A time to hear from all the workers. In 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 552–559. IEEE, 2018. \n[48] Amirhossein Reisizadeh, Hossein Taheri, Aryan Mokhtari, Hamed Hassani, and Ramtin Pedarsani. Robust and communication-efficient collaborative learning. In Advances in Neural Information Processing Systems, pages 8388–8399, 2019. \n[49] Aleksandr Beznosikov, Samuel Horváth, Peter Richtárik, and Mher Safaryan. On biased compression for distributed learning. arXiv preprint arXiv:2002.12410, 2020. \n[50] Nikko Strom. Scalable distributed dnn training using commodity gpu cloud computing. In Sixteenth Annual Conference of the International Speech Communication Association, 2015. \n[51] Sai Praneeth Karimireddy, Quentin Rebjock, Sebastian U Stich, and Martin Jaggi. Error feedback fixes signsgd and other gradient compression schemes. arXiv preprint arXiv:1901.09847, 2019. \n[52] Sebastian U Stich and Sai Praneeth Karimireddy. The error-feedback framework: Better rates for sgd with delayed gradients and compressed communication. arXiv preprint arXiv:1909.05350, 2019. \n[53] Eduard Gorbunov, Dmitry Kovalev, Dmitry Makarenko, and Peter Richtárik. Linearly converging error compensated sgd. Advances in Neural Information Processing Systems, 33, 2020. \n[54] Yurii Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013. \n[55] Sébastien Bubeck. Convex optimization: Algorithms and complexity. arXiv preprint arXiv:1405.4980, 2014. \n[56] Horia Mania, Xinghao Pan, Dimitris Papailiopoulos, Benjamin Recht, Kannan Ramchandran, and Michael I Jordan. Perturbed iterate analysis for asynchronous stochastic optimization. SIAM Journal on Optimization, 27(4):2202–2229, 2017. ",
|
| 1226 |
+
"bbox": [
|
| 1227 |
+
171,
|
| 1228 |
+
133,
|
| 1229 |
+
826,
|
| 1230 |
+
920
|
| 1231 |
+
],
|
| 1232 |
+
"page_idx": 10
|
| 1233 |
+
},
|
| 1234 |
+
{
|
| 1235 |
+
"type": "text",
|
| 1236 |
+
"text": "",
|
| 1237 |
+
"bbox": [
|
| 1238 |
+
171,
|
| 1239 |
+
51,
|
| 1240 |
+
828,
|
| 1241 |
+
919
|
| 1242 |
+
],
|
| 1243 |
+
"page_idx": 11
|
| 1244 |
+
},
|
| 1245 |
+
{
|
| 1246 |
+
"type": "text",
|
| 1247 |
+
"text": "",
|
| 1248 |
+
"bbox": [
|
| 1249 |
+
171,
|
| 1250 |
+
69,
|
| 1251 |
+
828,
|
| 1252 |
+
919
|
| 1253 |
+
],
|
| 1254 |
+
"page_idx": 12
|
| 1255 |
+
},
|
| 1256 |
+
{
|
| 1257 |
+
"type": "text",
|
| 1258 |
+
"text": "",
|
| 1259 |
+
"bbox": [
|
| 1260 |
+
173,
|
| 1261 |
+
92,
|
| 1262 |
+
828,
|
| 1263 |
+
256
|
| 1264 |
+
],
|
| 1265 |
+
"page_idx": 13
|
| 1266 |
+
},
|
| 1267 |
+
{
|
| 1268 |
+
"type": "text",
|
| 1269 |
+
"text": "Checklist ",
|
| 1270 |
+
"text_level": 1,
|
| 1271 |
+
"bbox": [
|
| 1272 |
+
174,
|
| 1273 |
+
280,
|
| 1274 |
+
254,
|
| 1275 |
+
296
|
| 1276 |
+
],
|
| 1277 |
+
"page_idx": 13
|
| 1278 |
+
},
|
| 1279 |
+
{
|
| 1280 |
+
"type": "text",
|
| 1281 |
+
"text": "1. For all authors... ",
|
| 1282 |
+
"bbox": [
|
| 1283 |
+
214,
|
| 1284 |
+
305,
|
| 1285 |
+
339,
|
| 1286 |
+
319
|
| 1287 |
+
],
|
| 1288 |
+
"page_idx": 13
|
| 1289 |
+
},
|
| 1290 |
+
{
|
| 1291 |
+
"type": "text",
|
| 1292 |
+
"text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] We provide a lower bound for our algorithm in Theorem 5 of Section 4.1 that suggests the need for more informed local updating schemes. \n(c) Did you discuss any potential negative societal impacts of your work? [No] We could not think of any potential negative societal impacts. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
|
| 1293 |
+
"bbox": [
|
| 1294 |
+
238,
|
| 1295 |
+
323,
|
| 1296 |
+
825,
|
| 1297 |
+
454
|
| 1298 |
+
],
|
| 1299 |
+
"page_idx": 13
|
| 1300 |
+
},
|
| 1301 |
+
{
|
| 1302 |
+
"type": "text",
|
| 1303 |
+
"text": "2. If you are including theoretical results... ",
|
| 1304 |
+
"bbox": [
|
| 1305 |
+
214,
|
| 1306 |
+
458,
|
| 1307 |
+
493,
|
| 1308 |
+
472
|
| 1309 |
+
],
|
| 1310 |
+
"page_idx": 13
|
| 1311 |
+
},
|
| 1312 |
+
{
|
| 1313 |
+
"type": "text",
|
| 1314 |
+
"text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] We provide complete proofs of all our results in the supplemental material. ",
|
| 1315 |
+
"bbox": [
|
| 1316 |
+
236,
|
| 1317 |
+
476,
|
| 1318 |
+
825,
|
| 1319 |
+
520
|
| 1320 |
+
],
|
| 1321 |
+
"page_idx": 13
|
| 1322 |
+
},
|
| 1323 |
+
{
|
| 1324 |
+
"type": "text",
|
| 1325 |
+
"text": "3. If you ran experiments... ",
|
| 1326 |
+
"bbox": [
|
| 1327 |
+
212,
|
| 1328 |
+
523,
|
| 1329 |
+
393,
|
| 1330 |
+
537
|
| 1331 |
+
],
|
| 1332 |
+
"page_idx": 13
|
| 1333 |
+
},
|
| 1334 |
+
{
|
| 1335 |
+
"type": "text",
|
| 1336 |
+
"text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] ",
|
| 1337 |
+
"bbox": [
|
| 1338 |
+
238,
|
| 1339 |
+
541,
|
| 1340 |
+
825,
|
| 1341 |
+
659
|
| 1342 |
+
],
|
| 1343 |
+
"page_idx": 13
|
| 1344 |
+
},
|
| 1345 |
+
{
|
| 1346 |
+
"type": "text",
|
| 1347 |
+
"text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
|
| 1348 |
+
"bbox": [
|
| 1349 |
+
212,
|
| 1350 |
+
661,
|
| 1351 |
+
823,
|
| 1352 |
+
676
|
| 1353 |
+
],
|
| 1354 |
+
"page_idx": 13
|
| 1355 |
+
},
|
| 1356 |
+
{
|
| 1357 |
+
"type": "text",
|
| 1358 |
+
"text": "(a) If your work uses existing assets, did you cite the creators? [N/A] \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
|
| 1359 |
+
"bbox": [
|
| 1360 |
+
238,
|
| 1361 |
+
680,
|
| 1362 |
+
825,
|
| 1363 |
+
797
|
| 1364 |
+
],
|
| 1365 |
+
"page_idx": 13
|
| 1366 |
+
},
|
| 1367 |
+
{
|
| 1368 |
+
"type": "text",
|
| 1369 |
+
"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
|
| 1370 |
+
"bbox": [
|
| 1371 |
+
214,
|
| 1372 |
+
801,
|
| 1373 |
+
705,
|
| 1374 |
+
815
|
| 1375 |
+
],
|
| 1376 |
+
"page_idx": 13
|
| 1377 |
+
},
|
| 1378 |
+
{
|
| 1379 |
+
"type": "text",
|
| 1380 |
+
"text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
|
| 1381 |
+
"bbox": [
|
| 1382 |
+
238,
|
| 1383 |
+
820,
|
| 1384 |
+
825,
|
| 1385 |
+
907
|
| 1386 |
+
],
|
| 1387 |
+
"page_idx": 13
|
| 1388 |
+
}
|
| 1389 |
+
]
|
parse/train/h7FqQ6hCK18/h7FqQ6hCK18_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/h7FqQ6hCK18/h7FqQ6hCK18_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/pvjfA4wogD6/pvjfA4wogD6.md
ADDED
|
@@ -0,0 +1,255 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
# Video Instance Segmentation using Inter-Frame Communication Transformers
|
| 2 |
+
|
| 3 |
+
Sukjun Hwang1 Miran Heo1 Seoung Wug Oh2 Seon Joo Kim1 1Yonsei University 2Adobe Research {sj.hwang, miran, seonjookim}@yonsei.ac.kr seoh@adobe.com
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
We propose a novel end-to-end solution for video instance segmentation (VIS) based on transformers. Recently, the per-clip pipeline shows superior performance over per-frame methods leveraging richer information from multiple frames. However, previous per-clip models require heavy computation and memory usage to achieve frame-to-frame communications, limiting practicality. In this work, we propose Inter-frame Communication Transformers (IFC), which significantly reduces the overhead for information-passing between frames by efficiently encoding the context within the input clip. Specifically, we propose to utilize concise memory tokens as a means of conveying information as well as summarizing each frame scene. The features of each frame are enriched and correlated with other frames through exchange of information between the precisely encoded memory tokens. We validate our method on the latest benchmark sets and achieved state-of-the-art performance (AP 42.6 on YouTube-VIS 2019 val set using the offline inference) while having a considerably fast runtime (89.4 FPS). Our method can also be applied to near-online inference for processing a video in real-time with only a small delay. The code is available at https://github.com/sukjunhwang/IFC.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
With the growing interest toward the video domain in computer vision, the task of video instance segmentation (VIS) is emerging [1]. Most of the current approaches [1, 2, 3, 4] extend image instance segmentation models [5, 6, 7, 8] and take frame-wise inputs. These per-frame methods extend the concept of temporal tracking by matching frame-wise predictions of high similarities. The models can be easily customized to real-world applications as they run in an online [9] fashion, but they show limitations in dealing with occlusions and motion blur that are common in videos.
|
| 12 |
+
|
| 13 |
+
On the contrary, per-clip models are designed to overcome such challenges by incorporating multiple frames while sacrificing the efficiency. Previous per-clip approaches [10, 11, 12] aggregate information within a clip to generate instance-specific features. As the features are generated per instance, the number of instances in addition to the number of frames has a significant impact on the overall computation. Recently proposed VisTR [11] adapted DETR [13] to the VIS task and reduced the inference time by inserting the entire video, not a clip, to its offline end-to-end network. However, its full self-attention transformers [14] over the space-time inputs involve explosive computations and memories. In this work, we raise the following question: can a per-clip method be efficient while attaining great accuracy?
|
| 14 |
+
|
| 15 |
+
To achieve our goal, we introduce Inter-frame Communication Transformers (IFC) to greatly reduce the computations of the full space-time transformers. Similar to recent works [15, 16, 17] that alleviate the explosive computational growth inherent in attention-based models [14, 18], IFC takes a decomposition strategy utilizing two transformers. The first transformer (Encode-Receive, $\mathcal { E }$ ) encodes each frame independently. To exchange the information between frames, the second transformer (Gather-Communicate, $\mathcal { G }$ ) executes attention between a small number of memory tokens that hold concise information of the clip. The memory tokens are utilized to store the overall context of the clip, for example “a hand over a lizard” in Fig. 1. The concise information assists detecting the lizard that is largely occluded by the hand in the first frame, without employing an expensive pixel-level attention over space and time. The memory tokens are only in charge of the communications between frames, and the features of each frame are enriched and correlated through the memory tokens.
|
| 16 |
+
|
| 17 |
+
We further reduce overheads while taking advantage of per-clip pipelines by concisely representing each instance with a unique convolutional weight [7]. Despite the changes of appearances at different frames, the instances of the same identity share commonalities because the frames originated from the same source video. Therefore, we can effectively capture instance-specific characteristics in a clip with dynamically generated convolutional weights. In companion with the segmentation, we track instances by uniformly applying the weights to all frames in a clip. Moreover, all executions of our spatial decoder are instance-agnostic except for the final layer which applies instance-specific weights. Accordingly, our model is highly efficient and also suitable for scenes with numerous instances.
|
| 18 |
+
|
| 19 |
+
In addition to the efficient modeling, we provide optimizations and an instance tracking algorithm that are designed to be VIS-centric. By the definition of $\mathsf { A P } ^ { \mathtt { V I S } }$ , the VIS task [1] aims to maximize the objective similarity: space-time mask IoU. Inspired by previous works [13, 19, 20], our model is optimized to maximize the similarity between bipartitely matched pairs of ground truth masks and predicted masks. Furthermore, we again adopt the similarity maximization for tracking instances of same identities, which effectively links predicted space-time masks using bipartite matching. As both of our training and inference algorithms are fundamentally designed to address the key challenge of VIS task, our method attains an outstanding accuracy.
|
| 20 |
+
|
| 21 |
+
From these improvements, IFC sets the new state-of-the-art: $4 2 . 6 \%$ AP and more surprisingly, in 89.4 fps. Furthermore, our model also shows great speed-accuracy balance under near-online settings, which leads to a huge practicality. We believe that our model can be a powerful baseline for video instance segmentation approaches that follow the per-clip execution.
|
| 22 |
+
|
| 23 |
+
# 2 Related Work
|
| 24 |
+
|
| 25 |
+
Video instance segmentation The VIS task [1] extends the concept of tracking to the image instance segmentation task. The early solutions [1, 2] follow the per-frame pipeline, which utilize additional tracking head to the models that are mainly designed to solve image instance segmentation. More advanced algorithms that are recently proposed [3, 4] take video characteristics into consideration, which result in improved performance.
|
| 26 |
+
|
| 27 |
+
Per-clip models [10, 11, 12] dedicate computations to extract information from multiple frames for higher accuracy. By exploiting multiple frames, per-clip models can effectively handle typical challenges in video, i.e., motion blurs and occlusions. Our model is designed to be highly efficient while following the per-clip pipeline, which leads to fast and accurate predictions.
|
| 28 |
+
|
| 29 |
+
Transformers Recently, transformers [14] are greatly impacting many tasks in computer vision. After the huge success of DETR [13], which has brought a new paradigm to the object detection task, numerous vision tasks are incorporating transformers [21, 22] in place of CNNs. For classification tasks in both NLP and computer vision, many adopt an extra classification token to the input of transformers [21, 23]. All the input tokens affect each other as the encoders are mainly composed of the self-attention, thus the classification token can be used to determine the class of the overall input. Similarly, DeiT [24] inserts an additional distillation token to transformers, and the novel usage leads to a higher data efficiency. MaX-DeepLab [20] adopted the concept of memory and proposed a novel dual-path transformer for the panoptic segmentation task [25]. By making use of numerous memory tokens to convey information, MaX-DeepLab integrates the transformer and the CNN by making both feedback itself and the other.
|
| 30 |
+
|
| 31 |
+
We further utilize the concept of the memory tokens to the videos. Using Inter-frame Communication Transformers, each frame runs independently while sharing their information with interim communications. The communications lead to higher accuracy while the execution independence between frames accelerates the inference.
|
| 32 |
+
|
| 33 |
+
|
| 34 |
+
Figure 1: Overview of IFC framework. Our transformer encoder block has two components: 1) Encode-Receive $( \mathcal { E } )$ simultaneously encodes frame tokens and memory tokens. 2) Only memory tokens pass Gather-Communicate $( { \mathcal { G } } )$ to perform communications between frames. The output from the stack of $N _ { E }$ encoder blocks goes into two modules, spatial decoder and transformer decoder, to generate segmentation masks.
|
| 35 |
+
|
| 36 |
+
# 3 Method
|
| 37 |
+
|
| 38 |
+
The proposed method follows a per-clip pipeline which takes a video clip as input and outputs clip-level results. We also introduce Inter-frame Communication Transformers, which can effectively share frame-wise information within a clip with a high efficiency.
|
| 39 |
+
|
| 40 |
+
# 3.1 Model architecture
|
| 41 |
+
|
| 42 |
+
Inspired by DETR [13], our network consists of a CNN backbone and transformer encoder-decoder layers (Fig. 1). The input clip is first independently embedded into a feature map through the backbone. Then, the embedded clip passes through our inter-frame communication encoder blocks that enrich the feature map by allowing information exchange between frames. Next, a set of transformer decoder layers that take the encoder outputs and object queries as inputs predict unique convolutional weights for each instance in the clip. Finally, the masks for each instance across the clip are computed in one shot by convolving the encoded feature map with the unique convolutional weight.
|
| 43 |
+
|
| 44 |
+
Backbone Given an input clip $\{ x _ { i } \} _ { i = 1 } ^ { T } \ \in \ \mathbb { R } ^ { T \times H _ { 0 } \times W _ { 0 } \times 3 }$ , composed of $T$ frames with 3 color channels, the CNN backbone processes the input clip frame-by-frame. As the result, the clip is encoded into a set of low-resolution features, $\bar { \{ f _ { i } ^ { 0 } \} _ { i = 1 } ^ { T } } \in \mathbb { R } ^ { T \times H \times W \times C }$ , where $C$ is the number of channels and $\begin{array} { r } { H , W = \frac { H _ { 0 } } { 3 2 } , \frac { W _ { 0 } } { 3 2 } } \end{array}$ .
|
| 45 |
+
|
| 46 |
+
Inter-Frame Communication Encoder Given an image, humans can effortlessly summarize the scene with only a few words. Also, frames from a same video share a lot of commonalities, the difference between them is sufficiently summarized and communicated even with a small bandwidth. Based on this hypothesis, we propose an inter-frame communication encoder to make the computation to be mostly frame-wise independent with some communications between frames. Specifically, we adopt memory tokens for both summarizing per-frame scenes and the means of communications.
|
| 47 |
+
|
| 48 |
+
Our encoder blocks are composed of two phases of separate transformers: Encode-Receive $( \mathcal { E } )$ and Gather-Communicate $( { \mathcal { G } } )$ . Both Encode-Receive and Gather-Communicate follow the typical transformer encoder architecture [14], which consists of an addition of fixed positional encoding, a multi-head self-attention module, and a feed forward network.
|
| 49 |
+
|
| 50 |
+
Encode-Receive operates in a per-frame manner, taking a frame-level feature map and corresponding memory tokens. Passing through Encode-Receive, we expect two functionalities: (1) image features encode per-frame information to the memory tokens, and (2) image features receive information of different frames that are gathered in the memory tokens. Gather-Communicate operates across frames to form a clip-level knowledge. It takes the memory tokens from each frame as inputs and performs communications between frames. Alternating two phases through multiple layers, the encoder can efficiently learn consensus representations across frames.
|
| 51 |
+
|
| 52 |
+
Table 1: Complexity comparison. Various transformer encoders for space-time input. As the overall FLOPs can vary by the number of detected instances, listed values are measured only at the encoders.
|
| 53 |
+
|
| 54 |
+
<table><tr><td rowspan="3">Communication Type</td><td rowspan="3">Complexity per Layer</td><td colspan="4">FLOPs (G)1</td></tr><tr><td></td><td>360 × 640</td><td></td><td>720×1280</td></tr><tr><td>T=5</td><td>T=36</td><td>T=5</td><td>T=36</td></tr><tr><td>No Comm</td><td>O(C²THW + CT(HW)2)</td><td>5.17</td><td>37.23</td><td>24.62</td><td>177.29</td></tr><tr><td>Full THW</td><td>O(C²THW + C(THW)2)</td><td>6.94</td><td>148.70</td><td>50.63</td><td>1815.38</td></tr><tr><td>Decompose T-HW</td><td>O(C²THW + CT(HW)² + CT²HW)</td><td>8.33</td><td>60.24</td><td>36.73</td><td>265.50</td></tr><tr><td>IFC (M = 8)</td><td>O(C²THW +CT(HW)2)</td><td>5.52</td><td>39.73</td><td>25.05</td><td>180.39</td></tr></table>
|
| 55 |
+
|
| 56 |
+
In more detail, given the frame embedding $\{ f _ { i } ^ { 0 } \} _ { i = 1 } ^ { T }$ , we spatially flatten each feature $\mathbb { R } ^ { H \times W \times C } $ $\mathbb { R } ^ { H W \times C }$ . The initial memory tokens $m ^ { 0 }$ of size $M$ are copied per frame and concatenated to each frame feature as follows:
|
| 57 |
+
|
| 58 |
+
$$
|
| 59 |
+
[ f _ { t } ^ { 0 } , m _ { t } ^ { 0 } ] \in \mathbb { R } ^ { ( H W + M ) \times C } , \qquad t \in \{ 1 , 2 , \cdots , T \} ,
|
| 60 |
+
$$
|
| 61 |
+
|
| 62 |
+
where $[ \cdot , \cdot ]$ indicates a concatenation of two feature vectors. Note that the initial memory tokens $m ^ { 0 }$ are trainable parameters learnt during training.
|
| 63 |
+
|
| 64 |
+
The first phase of IFC is Encode-Receive, which processes frames individually as follows:
|
| 65 |
+
|
| 66 |
+
$$
|
| 67 |
+
[ f _ { t } ^ { l } , \widehat { m } _ { t } ^ { l } ] = \mathcal { E } ^ { l } ( [ f _ { t } ^ { l - 1 } , m _ { t } ^ { l - 1 } ] ) ,
|
| 68 |
+
$$
|
| 69 |
+
|
| 70 |
+
where ${ \mathcal { E } } ^ { l }$ denotes the $l$ -th Encode-Receive layer. With a self-attention computed over the frame pixel locations and the memory tokens, the information of each frame can be passed to the memory tokens and vise-versa.
|
| 71 |
+
|
| 72 |
+
The outputs of Encode-Receive are grouped by memory indices and formulate the inputs for GatherCommunicate layer. The grouping can be understood as a decomposition of memory tokens, and becomes computationally beneficial when the total size of gathered memory tokens increases.
|
| 73 |
+
|
| 74 |
+
$$
|
| 75 |
+
\begin{array} { r l } & { [ m _ { 1 } ^ { l } ( i ) , m _ { 2 } ^ { l } ( i ) , \cdots , m _ { T } ^ { l } ( i ) ] = \mathcal { G } ^ { l } ( [ \widehat { m } _ { 1 } ^ { l } ( i ) , \widehat { m } _ { 2 } ^ { l } ( i ) , \cdots , \widehat { m } _ { T } ^ { l } ( i ) ] ) , \qquad i \in \{ 1 , 2 , \cdots , M \} , } \end{array}
|
| 76 |
+
$$
|
| 77 |
+
|
| 78 |
+
where $\mathcal { G } ^ { l }$ denotes the $l$ -th Gather-Communicate layer. The processed outputs are redistributed to the originated frame and get concatenated as $m _ { t } \overset { \cdot } { = } \left[ m _ { t } ( 1 ) , \overset { \cdot } { m } _ { t } ( 2 ) , \cdot \cdot \cdot , \overset { \cdot } { m } _ { t } ( M ) \right]$ . Unlike EncodeReceive, Gather-Communicate utilizes the attention mechanism to convey the information from different frames over the input clip.
|
| 79 |
+
|
| 80 |
+
Defining the $l$ -th inter-frame encoder block $( \mathrm { I F C } ^ { l } )$ as ${ \mathcal { E } } ^ { l }$ followed by $\mathcal { G } ^ { l }$ , the stack of $N _ { E }$ encoder blocks can be inductively formulated as:
|
| 81 |
+
|
| 82 |
+
$$
|
| 83 |
+
[ f _ { t } ^ { l } , m _ { t } ^ { l } ] = \mathrm { I F C } ^ { l } ( [ f _ { t } ^ { l - 1 } , m _ { t } ^ { l - 1 } ] ) , \qquad 1 \leq l \leq N _ { E } ,
|
| 84 |
+
$$
|
| 85 |
+
|
| 86 |
+
where $\left[ f _ { t } ^ { N _ { E } } , m _ { t } ^ { N _ { E } } \right]$ is the final result. The stacking of multiple encoder layers brings communications between frames, thus each frame can have coincidence to the other, specifying the identities of instances in a given clip.
|
| 87 |
+
|
| 88 |
+
Complexity comparison In Table 1, we analyze the computational complexity of transformer encoder variants applied for video input in terms of the Big-O complexity and FLOPs. The complexity of the original transformer encoder layer [14] is $\mathcal { O } ( C ^ { 2 } N ^ { ' } { + } C N ^ { 2 } )$ , where $N$ is the number of inputs. Without any communication between frames, No Comm, it shows the smallest amount of computation $( { \mathcal O } ( C ^ { 2 } T \dot { H W } + C T ( H W ) ^ { 2 } ) )$ ). As indicated as Full THW in Table 1, the complexity of VisTR [11] that performs a full space-time self-attention is $\mathcal { O } ( C ^ { 2 } ( T H W ) + C ( T H W ) ^ { 2 } )$ thus either a higher resolution or an increase of number of input frames leads to a massive increase in computations. VisTR bypasses the problem by highly reducing the input resolution and utilizing GPUs with tremendous memory capacity. However, as such solutions cannot resolve the fundamental issues, it is impractical to real-world videos. Moreover, VisTR remains as a complete offline strategy because it takes the entire video as an input.
|
| 89 |
+
|
| 90 |
+
An intriguing improvement for the naïve full self-attention would be the decomposition of the attention into space and time axis [16, 17, 26]. In Decompose T-HW, we decompose attention computation into spatial and temporal attention. The complexity of the separation of space-time leads to the sum of the two transformer encoder: $\mathcal { O } ( T ( C ^ { 2 } ( H \bar { W } ) + \bar { C } ( H W ) ^ { 2 } ) )$ and $\mathcal { O } ( H \bar { W } ( C ^ { 2 } T + C T ^ { 2 } ) )$ . In comparison to the full self-attention, the decomposition lowers the computational growth relative to the number of frames.
|
| 91 |
+
|
| 92 |
+
Our encoder, IFC, that communicates between frames using the memory tokens leads to a huge benefit to the total computations adding only a small amount of computation over No Comm while providing sufficient channels for communication. The complexity of each phase in our proposed encoder is: $\mathcal { O } ( C ^ { 2 } T ( H W + M ) + C T ( H W + M ) ^ { 2 } )$ for Encode-Receive and $\mathcal { O } ( C ^ { 2 } T M + \bar { C } \bar { T } ^ { 2 } M )$ for Gather-Communicate respectively. Assuming that $M$ is kept small (e.g., 8), the computation needed for Gather-Communicate can be neglected, while the complexity of Encode-Receive can be approximated to $\mathcal { O } ( C ^ { 2 } T H W + C T ( H W ) ^ { 2 } )$ as shown in Table 1. Finally, with respect to the number of frames of the input, we can expect approximate linear increase rather than the high increase of computation occurred in VisTR.
|
| 93 |
+
|
| 94 |
+
Decoders and output heads As depicted in Fig. 1, the transformer decoder of our model is stacked with $N _ { D }$ layers [14]. Contrary to VisTR, where the number of object queries increases proportionally to the number of frames, our model receives learnt encodings of fixed size $N _ { q }$ for object queries. Also, by utilizing these encodings throughout the entire frames, our model can effectively deal with clips of various lengths. A set of projection matrices are applied to $\{ f _ { t } ^ { N _ { E } } , m _ { t } ^ { N _ { E } } \} _ { t = 1 } ^ { T }$ for the generation of keys and values. The object queries turn into output embeddings by the transformer decoder, and the embeddings are eventually used as an input to the output heads.
|
| 95 |
+
|
| 96 |
+
There are two output heads on top of the transformer decoder, a class head and a segmentation head, each composed of two fully-connected layers. The output embeddings from the transformer decoder are independently inserted to the heads, resulting in $N _ { q }$ predictions per a clip. The class head outputs a class probability distribution of instances $\hat { p } ( c ) \in \mathbb { R } ^ { N _ { q } \times | \mathbb { C } | }$ . Note that the possible classes $\mathbb { C } \ni c$ include no object $\mathcal { D }$ class in addition to the given classes of a dataset.
|
| 97 |
+
|
| 98 |
+
The segmentation head generates $N _ { q }$ conditional convolutional weights $w \in \mathbb { R } ^ { N _ { q } \times C }$ in a manner similar to [7, 20]. For the conditional convolution, the output feature of the encoder reused by undoing the flatten operation. For the upsampling, the encoder feature pa $\{ f _ { t } ^ { N _ { E } } \} _ { t = 1 } ^ { T }$ isgh fpn-style [27] spatial decoder without temporal connections resulting in $T$ feature maps that are $1 / 8$ of the input resolution. Finally, the resulting feature maps $f ^ { \prime }$ are convolved with each convolutional weight to generate a segmentation mask as follows:
|
| 99 |
+
|
| 100 |
+
$$
|
| 101 |
+
\hat { s } _ { i } = \{ f _ { t } ^ { \prime } \circ w _ { i } \} _ { t = 1 } ^ { T } ,
|
| 102 |
+
$$
|
| 103 |
+
|
| 104 |
+
where $w _ { i }$ is $i$ -th convolutional weight, $\circ$ indicate $1 \times 1$ spatial convolution operation, and the result $\hat { s _ { i } }$ is a spatial-temporal object mask in shape of $\mathbb { R } ^ { T \times H ^ { \prime } \times W ^ { \prime } }$ where $\begin{array} { r } { H ^ { \prime } = \frac { H _ { 0 } } { 8 } } \end{array}$ , $\begin{array} { r } { W ^ { \prime } = \frac { W _ { 0 } } { 8 } } \\ { . } \end{array}$ . Note that, for an instance, a common weight is applied throughout the video clip. Our spatial decoder is an instanceagnostic design, which is much more efficient than instance-specific decoders [10, 11, 12, 13] as the number of detected instances increases. Meanwhile, thanks to our segmentation head which specifies and captures the characteristics of an instance, IFC can conduct both segmentation and tracking at once within a clip.
|
| 105 |
+
|
| 106 |
+
# 3.2 Instance matching and loss
|
| 107 |
+
|
| 108 |
+
To train our network, we first assign the ground truth for each instance estimation and then a set of loss function between each the ground truth and prediction pair. For a given input clip, our model generate a fixed-size set of class-labeled masks $\{ \hat { y } _ { i } \} _ { i = 1 } ^ { N _ { q } } = \{ ( \hat { p } _ { i } ( \boldsymbol { c } ) , \hat { s } _ { i } ) \} _ { i = 1 } ^ { N _ { q } }$ . The ground truth set of the clip can be represented as $y _ { i } = ( c _ { i } , s _ { i } )$ ; $c _ { i }$ is the target class label including $\mathcal { D }$ , and $s _ { i }$ is the target mask which is down-sampled to the size of the prediction masks for efficient similarity calculation. One-to-one bipartite matching between the prediction set $\{ \hat { y } _ { i } \} _ { i = 1 } ^ { N _ { q } }$ and the ground truth set $\{ y _ { i } \} _ { i = 1 } ^ { K }$ is performed to find the best assignment of a prediction to a ground truth. The objective can be formally
|
| 109 |
+
|
| 110 |
+
described as:
|
| 111 |
+
|
| 112 |
+
$$
|
| 113 |
+
\hat { \sigma } = \underset { \sigma \in \mathfrak { S } _ { N _ { q } } } { \arg \operatorname* { m a x } } \sum _ { i = 1 } ^ { K } \sin ( y _ { i } , \hat { y } _ { \sigma ( i ) } ) ,
|
| 114 |
+
$$
|
| 115 |
+
|
| 116 |
+
where $\sin ( y _ { i } , \hat { y } _ { \sigma ( i ) } )$ refers a pair-wise similarity over a permutation of $\underset { \_ w } { \sigma } \in \mathfrak { S } _ { N _ { g } }$ . Following prior work [13, 20, 28], the bipartite matching is efficiently computed using Hungarian algorithm [19]. We find that box-based similarity measurement as used in DETR [13] shows weaknesses in matching instances in video clip due to the case of occlusion and disappear-and-reappear. Therefore, we define $\sin ( y _ { i } , \hat { y } _ { \sigma ( i ) } )$ to be mask-based term as $\mathbb { 1 } _ { \{ c _ { i } \neq \infty \} } [ \hat { p } _ { \sigma ( i ) } ( c _ { i } ) + \bar { \lambda } _ { 0 } \mathrm { D I C E } ( s _ { i } , \bar { \hat { s } } _ { \sigma ( i ) } ^ { - } ) ]$ , where DICE denotes dice coefficients [29].
|
| 117 |
+
|
| 118 |
+
Given the optimal assignment $\hat { \sigma }$ , we refer to the $K$ matched predictions and $( N _ { q } - K )$ non-matched predictions as positive and negative pairs respectively. The positive pairs aim to predict the ground truth masks and classes while the negative pairs are optimized to predict the $\mathcal { D }$ class. The final loss is a sum of the losses from positive pairs and negative pairs where each can be computed as follows:
|
| 119 |
+
|
| 120 |
+
$$
|
| 121 |
+
\begin{array} { r l } & { \mathcal { L } _ { p o s } = \displaystyle \sum _ { i = 1 } ^ { K } [ \underbrace { - \log \hat { p } _ { \hat { \sigma } ( i ) } ( c _ { i } ) } _ { \mathrm { C r o s s - e n t r o p y ~ l o s s } } + \lambda _ { 1 } ( \underbrace { 1 - \operatorname { D I C E } \bigl ( s _ { i } , \hat { s } _ { \hat { \sigma } ( i ) } \bigr ) } _ { \mathrm { D i c e ~ l o s s ~ } [ 2 9 ] } ) + \lambda _ { 2 } \underbrace { \operatorname { F O C A L } \bigl ( s _ { i } , \hat { s } _ { \hat { \sigma } ( i ) } \bigr ) } _ { \mathrm { S i g m o i d - f o c a l ~ l o s s ~ } [ 3 0 ] } ] , } \\ & { \quad \quad \quad \quad \quad \quad \mathcal { L } _ { n e g } = \displaystyle \sum _ { i = k + 1 } ^ { N _ { q } } [ - \log \hat { p } _ { \hat { \sigma } ( i ) } ( \emptyset ) ] . } \end{array}
|
| 122 |
+
$$
|
| 123 |
+
|
| 124 |
+
As $( N _ { q } - K )$ is likely to be much greater than $K$ , we down-weight $\mathcal { L } _ { n e g }$ by a factor of 10 to resolve the imbalance, following prior work [13]. The goal of video instance segmentation [1] is to maximize the space-time IoU between a prediction and a ground truth mask. Therefore, our mask-related losses (Dice loss and Sigmoid-focal loss) are spatio-temporally calculated over an entire clip, rather than averaging the losses that are accumulated frame-by-frame.
|
| 125 |
+
|
| 126 |
+
# 3.3 Clip-level instance tracking
|
| 127 |
+
|
| 128 |
+
To infer a video input that is longer than the clip length, we match instances using the predicted masks of overlapping frames. Let $\mathcal { V } _ { I }$ and ${ \mathcal { V } } _ { A }$ be the result sets of clip $I$ and $A$ excluding the $\mathcal { D }$ class. The goal is to perform matching of same identities between pre-collected instance set $\mathcal { V } _ { I }$ and $\mathcal { V } _ { A }$ . We first calculate the matching scores which are space-time soft IoU at intersecting frames between $\mathcal { V } _ { I }$ and $\mathcal { V } _ { A }$ . Then, we find optimal paired indices $\hat { \sigma } _ { S }$ using Hungarian algorithm [19] to the gathered matching score $\mathcal { S } \in [ 0 , \bar { 1 ] } ^ { | \mathcal { N } _ { I } | \times | \bar { \mathcal { V } } _ { A } | }$ . We update $\mathscr { D } _ { I } ( i )$ by concatenating $\mathcal { V } _ { A } ( \hat { \sigma } _ { S } ( i ) )$ if $\bar { \cal S } ( i , \bar { \sigma } s ( i ) )$ is above a certain threshold, and add non-matched prediction sets to $\mathcal { V } _ { I }$ as new instances. Note that a previous per-clip model (MaskProp [10]) also utilizes soft IoU for tracking instances, but the matching scores are computed per-frame and averaged for intersecting frames. Different from MaskProp, using space-time soft IoU leads to an accurate tracking as it can better represent the definition of mask similarities between clips which brings at most $2 \%$ AP increase. The overall tracking pipeline can be effectively implemented in a GPU-friendly manner.
|
| 129 |
+
|
| 130 |
+
# 4 Experiments
|
| 131 |
+
|
| 132 |
+
In this section, we evaluate the proposed method using YouTube-VIS 2019 and 2021 [1]. For every listed score, we report the mean of five runs as the results may vary by each run due to the insufficient number of training and testing set of YouTube-VIS dataset. We demonstrate the effectiveness of our model regarding both accuracy and speed. We further examine how different settings affect the overall performance and efficiency of IFC encoder. Unless specified, all models for measurements used $\bar { N _ { E } } = 3 , \bar { N _ { D } } = 3$ , stride of 1, and ResNet-50.
|
| 133 |
+
|
| 134 |
+
# 4.1 Implementation Details
|
| 135 |
+
|
| 136 |
+
We used detectron2 [33] for our code basis, and hyper-parameters mostly follow the settings of DETR [13] unless specified. We used AdamW [34] optimizer with initial learning rate of $1 0 ^ { - 4 }$ for transformers, and $1 \bar { 0 } ^ { - 5 }$ for backbone. We first pre-train the model for image instance segmentation on COCO [35] by setting our model to $T = 1$ . The pre-train procedure follows the shortened training schedule of DETR [13], which runs 300 epochs with a decay of the learning rate by a factor of 10 at 200 epochs. Using the pre-trained weights, the models are trained on a targeted dataset using the batch size of 16, each clip composed of $T = 5$ frames downscaled to either $3 6 0 \mathrm { p }$ or $4 8 0 \mathrm { p }$ . For the sampling of each clip, a reference frame index $t$ is randomly chosen. The remaining $T - 1$ frame indices are then sampled within an interval of 20. The models are trained for 8 epochs, and decays the learning rate by 10 at 5th epoch.
|
| 137 |
+
|
| 138 |
+
Table 2: Evaluations on various settings.
|
| 139 |
+
(a) AP and FPS on YouTube-VIS 2019 val set. For fairness, FPS is measured on a same machine, using a single RTX 2080Ti GPU. We used the official codes and checkpoints provided by the authors for the measurements. We report the clip settings of [10, 11]. T : window size.
|
| 140 |
+
(b) Accuracy on YTVIS 2021 val set
|
| 141 |
+
(d) Effect of strides
|
| 142 |
+
|
| 143 |
+
<table><tr><td colspan="3">Method (Settings)</td><td>Backbone [31]</td><td>FPS²</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td rowspan="9">prij.ite</td><td colspan="2">MaskTrack R-CNN[1]</td><td>ResNet-50</td><td>26.1</td><td>30.3</td><td>51.1</td><td>32.6</td><td>31.0</td><td>35.5</td></tr><tr><td colspan="2">MaskTrack R-CNN[1]</td><td>ResNet-101</td><td>=</td><td>31.8</td><td>53.0</td><td>33.6</td><td>33.2</td><td>37.6</td></tr><tr><td colspan="2">SipMask [2]</td><td>ResNet-50</td><td>35.5</td><td>33.7</td><td>54.1</td><td>35.8</td><td>35.4</td><td>40.1</td></tr><tr><td colspan="2">SG-Net [4]</td><td>ResNet-50</td><td>1</td><td>34.8</td><td>56.1</td><td>36.8</td><td>35.8</td><td>40.8</td></tr><tr><td colspan="2">SG-Net [4]</td><td>ResNet-101</td><td>1</td><td>36.3</td><td>57.1</td><td>39.6</td><td>35.9</td><td>43.0</td></tr><tr><td colspan="2">Cross VIS [3]</td><td>ResNet-50</td><td>=</td><td>36.3</td><td>56.8</td><td>38.9</td><td>35.6</td><td>40.7</td></tr><tr><td colspan="2">Cross VIS [3]</td><td>ResNet-101</td><td>1</td><td>36.6</td><td>57.3</td><td>39.7</td><td>36.0</td><td>42.0</td></tr><tr><td colspan="2">STEm-Seg [32]</td><td>ResNet-101</td><td>3.0</td><td>34.6</td><td>55.8</td><td>37.9</td><td>34.4</td><td>41.6</td></tr><tr><td rowspan="7">VisTR[11]</td><td>VisTR[11]</td><td>(T=36)</td><td>ResNet-50</td><td>51.1</td><td>35.6</td><td>56.8</td><td>37.0</td><td>35.2</td><td>40.2</td></tr><tr><td></td><td>(T=36)</td><td>ResNet-101</td><td>43.5</td><td>38.6</td><td>61.3</td><td>42.3</td><td>37.6</td><td>44.2</td></tr><tr><td>MaskProp[10]</td><td>(T=13)</td><td>ResNet-50</td><td>1</td><td>40.0</td><td>1</td><td>42.9</td><td>1</td><td>-</td></tr><tr><td>MaskProp [10]</td><td>(T=13)</td><td>ResNet-101</td><td>1</td><td>42.5</td><td>1</td><td>45.6</td><td>1</td><td>1</td></tr><tr><td>OurSnear-online</td><td>(T=5)</td><td>ResNet-50</td><td>46.5</td><td>39.0</td><td>60.4</td><td>42.7</td><td>41.7</td><td>51.6</td></tr><tr><td>OurSoffline</td><td>(T=36)</td><td>ResNet-50</td><td>107.1</td><td>41.2</td><td>65.1</td><td>44.6</td><td>42.3</td><td>49.6</td></tr><tr><td>OurSoffline</td><td>(T=36)</td><td>ResNet-101</td><td>89.4</td><td>42.6</td><td>66.6</td><td>46.3</td><td>43.5</td><td>51.4</td></tr></table>
|
| 144 |
+
|
| 145 |
+
(c) Bipartite matching
|
| 146 |
+
|
| 147 |
+
<table><tr><td></td><td>AP</td></tr><tr><td>Box-based</td><td>37.5</td></tr><tr><td>Mask-based</td><td>39.6</td></tr></table>
|
| 148 |
+
|
| 149 |
+
<table><tr><td></td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>MaskTrack-RCNN</td><td>28.6</td><td>48.9</td><td>29.6</td></tr><tr><td>SipMask</td><td>31.7</td><td>52.5</td><td>34.0</td></tr><tr><td>CrossVIS</td><td>34.2</td><td>54.4</td><td>37.9</td></tr><tr><td>Ours</td><td>35.2</td><td>57.2</td><td>37.5</td></tr></table>
|
| 150 |
+
|
| 151 |
+
<table><tr><td></td><td>AP</td><td>AP75</td><td>FPS</td></tr><tr><td>T=5</td><td>S=3 S=5</td><td>38.7 42.1</td><td>72.7</td></tr><tr><td>T=10</td><td>39.5</td><td>42.8</td><td>83.0</td></tr><tr><td>T=15 S=8</td><td>39.7</td><td>43.0</td><td>92.5</td></tr><tr><td>T=20 S=10</td><td>40.4</td><td>43.3</td><td>95.7</td></tr></table>
|
| 152 |
+
|
| 153 |
+
During inference, our model takes inputs as follows. Let an input video has $V$ frames, $T$ is the number of frames per clip and $S$ is the stride of clips. We start from inserting a clip of frame indices $[ 1 , T ]$ and sequentially insert clips of $[ 1 + S , T + \bar { S } ] , [ 1 + 2 S , T + 2 S ] , \therefore , [ 1 + n S , T + n S ]$ . It repeats until the end frame index $T + n S$ is equal to or greater than $V$ . If the end frame index of the last clip $T + n S$ is greater than $V$ , we change the frame indices of the last clip to $[ V - T + 1 , V ]$ . The resolution of input videos are downscaled to $3 6 0 \mathrm { p }$ , which follows MaskTrack R-CNN [1].
|
| 154 |
+
|
| 155 |
+
# 4.2 Main Results
|
| 156 |
+
|
| 157 |
+
YouTube-VIS 2019 evaluation results We compare our proposed IFC to the state-of-the-art models in the video instance segmentation task on YouTube-VIS $2 0 1 9 \ \mathtt { v a l }$ in Table 2 (a). We measure the accuracy by AP and our model sets the highest score among all online, near-online, and offline models while presenting the fastest runtime. As mentioned earlier, IFC is highly efficient during the inference thanks to three advantages: (1) memory token-based decomposition for transformer encoder (2) instance-agnostic spatial decoder (3) GPU-friendly instance matching. Moreover, our model does not make use of any heavy modules such as deformable convolutions [36] or cascading networks [37]. Thanks to these advantages, IFC achieves an outstanding runtime, which is faster speed than online models [1, 2].
|
| 158 |
+
|
| 159 |
+
During the inference, our method is able to freely adjust the length of the clip $( T )$ as needed. If the input clip length is set to contain entire video frames, our method becomes an offline method (like
|
| 160 |
+
|
| 161 |
+
|
| 162 |
+
Figure 2: Visualization of predictions from VisTR and our model. Instances with the same identity are displayed in the same color.
|
| 163 |
+
|
| 164 |
+
VisTR [11]) that processes the entire video in one shot. As the offline inference can skip matching between clips and maximize the GPU utilization, our method represents surprisingly fast runtime (107.1 FPS). On the other hand, if the application requires instant outputs given a video stream, we can reduce the clip length to make our method near-online. In the near-online scenario with $T = 5$ our system is still able to process a video in real-time (46.5 FPS) with only a small delay.
|
| 165 |
+
|
| 166 |
+
YouTube-VIS 2021 evaluation results The recently introduced dataset YouTube-VIS 2021 is an improved version of YouTube-VIS 2019. The newly added videos in the dataset include higher number of instances and frames. For the new dataset, we use 32 memory tokens. In Table 2 (b), we refer the results reported in [3], which evaluated [1, 2] using official implementations. Again, our model achieves the best performance.
|
| 167 |
+
|
| 168 |
+
Qualitative result comparison We compare some qualitative results predicted by our model and VisTR [11] in Fig. 2. In terms of both tracking accuracy and segmentation quality, IFC yields better results than VisTR.
|
| 169 |
+
|
| 170 |
+
# 4.3 Ablation Study
|
| 171 |
+
|
| 172 |
+
In this section, we provide ablation studies and discuss how different settings impact the overall performance. The experiments are conducted using YouTube-VIS 2019 val set.
|
| 173 |
+
|
| 174 |
+
Box-based and mask-based bipartite matching We observe how the different policies for bipartite matching affect the performance. As our model does is a box-free method, we adjust our model to predict bounding boxes similar to VisTR [11] and conduct bipartite matching [13, 19] using the predicted boxes. The change of optimization from mask-based to box-based brings a noticeable performance drop as shown in Table 2 (c). With the VIS-centric design, the mask-based optimization shows more robustness than box-based optimizations under typical video circumstances such as instances with heavy overlaps and partial occlusions.
|
| 175 |
+
|
| 176 |
+
Differing window strides In addition to the clip length $T$ , we further optimize our runtime placing a stride $S$ between clips, as shown in Table 2 (d). IFC can be used in a near-online manner, which takes clips that are consecutively extracted from a video. The placement of a larger stride reduces temporal intersections, which lessens computational overheads but also causes difficulty in matching instances. By enlarging the stride from $S = 1$ to $S = 3$ , IFC accomplishes approximately $150 \%$ speed improvement with only $0 . 1 \%$ AP drop. The tendency of high speed gain and low accuracy drop persists under various conditions. Therefore, our model can be applied to conditions where the enlargement of strides is necessary, i.e., using devices that are not powerful enough but has to maintain high inference speed.
|
| 177 |
+
|
| 178 |
+
Table 3: Encoder variations. We show how different encoders affect the overall performance.
|
| 179 |
+
(a) Various encoders taking clips of different lengths (see Table 1)
|
| 180 |
+
|
| 181 |
+
<table><tr><td></td><td colspan="3">T=5</td><td colspan="3">T=10</td><td colspan="3">T=15</td><td colspan="3">T=20</td></tr><tr><td></td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td></tr><tr><td>No Comm</td><td>37.4</td><td>39.9</td><td>38.1</td><td>38.8</td><td>41.6</td><td>40.8</td><td>39.3</td><td>41.7</td><td>46.7</td><td>39.6</td><td>41.9</td><td>52.9</td></tr><tr><td>Full THW</td><td>37.2</td><td>40.0</td><td>37.6</td><td>38.8</td><td>41.2</td><td>35.5</td><td>39.8</td><td>42.6</td><td>32.9</td><td>39.7</td><td>42.8</td><td>34.8</td></tr><tr><td>Decomp T-HW</td><td>37.2</td><td>39.8</td><td>35.7</td><td>38.3</td><td>40.9</td><td>37.9</td><td>38.5</td><td>41.5</td><td>42.6</td><td>39.0</td><td>41.9</td><td>49.4</td></tr><tr><td>IFC</td><td>39.0</td><td>42.7</td><td>36.3</td><td>39.6</td><td>43.0</td><td>38.9</td><td>39.8</td><td>43.0</td><td>43.7</td><td>40.4</td><td>43.4</td><td>50.2</td></tr></table>
|
| 182 |
+
|
| 183 |
+
(b) Image instance segmentation on COCO val set
|
| 184 |
+
(c) Number of memory tokens (AP)
|
| 185 |
+
(d) Index-wise memory decomposition
|
| 186 |
+
|
| 187 |
+
<table><tr><td></td><td>T=5</td><td>T=10</td><td>T=15</td><td>T=20</td></tr><tr><td>M=1</td><td>37.6</td><td>39.2</td><td>39.4</td><td>39.4</td></tr><tr><td>M=2</td><td>37.9</td><td>39.2</td><td>39.6</td><td>39.8</td></tr><tr><td>M=4</td><td>38.0</td><td>39.5</td><td>39.7</td><td>39.9</td></tr><tr><td>M=8</td><td>39.0</td><td>39.6</td><td>39.8</td><td>40.4</td></tr><tr><td>M=16</td><td>38.1</td><td>39.1</td><td>39.7</td><td>39.9</td></tr></table>
|
| 188 |
+
|
| 189 |
+
<table><tr><td></td><td>APCOcO</td><td>APOC</td></tr><tr><td>w/o mem</td><td>35.0</td><td>56.6</td></tr><tr><td>w/mem</td><td>35.1</td><td>56.5</td></tr></table>
|
| 190 |
+
|
| 191 |
+
<table><tr><td></td><td>T=5</td><td>T=10</td><td>T=15</td><td>T=20</td></tr><tr><td>Unified</td><td>38.1</td><td>38.9</td><td>39.7</td><td>39.9</td></tr><tr><td>Decomp</td><td>39.0</td><td>39.6</td><td>39.8</td><td>40.4</td></tr></table>
|
| 192 |
+
|
| 193 |
+
Various decomposition strategies of encoders In Table 1, we observed the computational gaps derived from the decomposition of the encoder layers. Extending Table 1, we now investigate the how the decomposition strategies affect the accuracy in Table 3.
|
| 194 |
+
|
| 195 |
+
The models are evaluated with variety of window sizes $( T = 5 , 1 0 , 1 5 , 2 0 )$ as an increase of window size $T$ has pros and cons. When matching predictions from different clips, greater $T$ is advantageous due to an enlargement of temporal intersections between clips. On the contrary, frames in longer clips are likely to be composed of diverse appearances, which disrupt tracking and segmenting instances within a clip. Therefore, the key to the performance enhancement is to cope with the appearance changes by precisely encoding and correlating space-time inputs.
|
| 196 |
+
|
| 197 |
+
As shown in Table 3 (a), the full self-attention [11] surpasses the encoder without communications as the length of clips increase. However, the enlargement of the window size highly slows down the inference speed, and the improvements are marginal that the tremendous computation and memory usage cannot be compensated. The decomposition of space-time maintains comparable speed even if the window is large, but fails to achieve high accuracy.
|
| 198 |
+
|
| 199 |
+
Our model shows fast inference as the only additional computations of IFC are from utilizing a small number of memory tokens. Furthermore, by effectively encoding the space-time inputs with the communications between frames, IFC can take advantages of enlarging the window size, and surpasses other encoders.
|
| 200 |
+
|
| 201 |
+
Memory tokens We also study the effects of utilizing memory tokens. As mentioned, the motivation of using the memory tokens is to build communications between frames. Different from the video instance segmentation task, the image segmentation task is consisted of a single frame. Therefore, the use of the memory tokens does not lead to improvements to the image instance segmentation task as mutual communications cannot be solely made (see Table 3 (b)). Meanwhile, the utilization of the memory tokens achieves great improvements by effectively passing the information between frames. Results in Table 3 (a, c) demonstrate that the use of memory tokens achieves higher accuracy than the encoder without any communications (No comm), which emphasizes the importance of the communications. We evaluate how the size of the memory tokens affect the overall accuracy in Table 3 (c) and set the default size of the tokens $M$ to be 8.
|
| 202 |
+
|
| 203 |
+
In Section 3.1, we demonstrated the formulation of the inputs for Gather-Communicate layer, which groups the outputs of Encode-Receive by memory indices. As aforementioned, the formulation can be considered as a decomposition of memory tokens: insertion to the Gather-Communicate layer by separate $M$ groups each consisting of $T$ tokens. In Table 3 (d), we investigate the impact of inserting the unified $M T$ tokens as a whole. Compared to the unified insertion, the decomposition brings better accuracy as the memories of same indices have more correspondences, which ease the encoders to build attentions in between.
|
| 204 |
+
|
| 205 |
+
|
| 206 |
+
Figure 3: Visualizations of results and attention maps of memory tokens.
|
| 207 |
+
|
| 208 |
+
We choose a memory index attending foreground instances and visualize the attention map in Fig. 3. As shown in the results of the upper clip, we find that the memory token has more interests to instances that are relatively difficult to detect; it more attends the heavily occluded car at the rear. The clip at the bottom is composed of frames with huge motion blurs and appearance changes. With the communications of memory tokens, IFC successfully tracks and segments the rabbit.
|
| 209 |
+
|
| 210 |
+
# 5 Conclusion
|
| 211 |
+
|
| 212 |
+
In this paper, we have proposed a novel video instance segmentation network using Inter-frame Communication Transformers (IFC), which alleviates full space-time attention and successfully builds communications between frames. Finally, our network presents a rapid inference and sets the new state-of-the-art on the YouTube-VIS dataset. For the future work, we plan to integrate temporal information, which indeed would take a step further to the human video understanding.
|
| 213 |
+
|
| 214 |
+
# Acknowledgments
|
| 215 |
+
|
| 216 |
+
This research was grant funded by the Artificial Intelligence Graduate School Program of Yonsei University, under Grant 2020-0-01361, Korea Evaluation Institute of Industrial Technology (KEIT) funded by the Ministry of Trade, Industry and Energy (10073129), and also supported by the Advanced Robotics Laboratory, part of the Future Technology Center at LG Electronics.
|
| 217 |
+
|
| 218 |
+
# References
|
| 219 |
+
|
| 220 |
+
[2] Cao, J., R. M. Anwer, H. Cholakkal, et al. Sipmask: Spatial information preservation for fast image and video instance segmentation. In ECCV. 2020.
|
| 221 |
+
[3] Yang, S., Y. Fang, X. Wang, et al. Crossover learning for fast online video instance segmentation. In ICCV. 2021.
|
| 222 |
+
[4] Liu, D., Y. Cui, W. Tan, et al. Sg-net: Spatial granularity network for one-stage video instance segmentation. In CVPR. 2021.
|
| 223 |
+
[5] He, K., G. Gkioxari, P. Dollar, et al. Mask r-cnn. In ICCV. 2017.
|
| 224 |
+
[6] Bolya, D., C. Zhou, F. Xiao, et al. Yolact: Real-time instance segmentation. In ICCV. 2019.
|
| 225 |
+
[7] Tian, Z., C. Shen, H. Chen. Conditional convolutions for instance segmentation. In ECCV. 2020.
|
| 226 |
+
[8] Chen, H., K. Sun, Z. Tian, et al. Blendmask: Top-down meets bottom-up for instance segmentation. In CVPR. 2020.
|
| 227 |
+
[9] Luo, W., J. Xing, A. Milan, et al. Multiple object tracking: A literature review. Artificial Intelligence, 2020.
|
| 228 |
+
[10] Bertasius, G., L. Torresani. Classifying, segmenting, and tracking object instances in video with mask propagation. In CVPR. 2020.
|
| 229 |
+
[11] Wang, Y., Z. Xu, X. Wang, et al. End-to-end video instance segmentation with transformers. In CVPR. 2020.
|
| 230 |
+
[12] Lin, H., R. Wu, S. Liu, et al. Video instance segmentation with a propose-reduce paradigm. In ICCV. 2021.
|
| 231 |
+
[13] Carion, N., F. Massa, G. Synnaeve, et al. End-to-end object detection with transformers. In ECCV. 2020.
|
| 232 |
+
[14] Vaswani, A., N. Shazeer, N. Parmar, et al. Attention is all you need. In NeurIPS. 2017.
|
| 233 |
+
[15] Wang, H., Y. Zhu, B. Green, et al. Axial-deeplab: Stand-alone axial-attention for panoptic segmentation. In ECCV. 2020.
|
| 234 |
+
[16] Bertasius, G., H. Wang, L. Torresani. Is space-time attention all you need for video understanding? In ICML. 2021.
|
| 235 |
+
[17] Arnab, A., M. Dehghani, G. Heigold, et al. Vivit: A video vision transformer. arXiv preprint arXiv:2103.15691, 2021.
|
| 236 |
+
[18] Wang, X., R. Girshick, A. Gupta, et al. Non-local neural networks. In CVPR. 2018.
|
| 237 |
+
[19] Kuhn, H. W. The hungarian method for the assignment problem. In Naval research logistics quarterly. 1955.
|
| 238 |
+
[20] Wang, H., Y. Zhu, H. Adam, et al. Max-deeplab: End-to-end panoptic segmentation with mask transformers. In CVPR. 2021.
|
| 239 |
+
[21] Dosovitskiy, A., L. Beyer, A. Kolesnikov, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR. 2021.
|
| 240 |
+
[22] Ranftl, R., A. Bochkovskiy, V. Koltun. Vision transformers for dense prediction. In ICCV. 2021.
|
| 241 |
+
[23] Devlin, J., M.-W. Chang, K. Lee, et al. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL. 2019.
|
| 242 |
+
[24] Touvron, H., M. Cord, M. Douze, et al. Training data-efficient image transformers & distillation through attention. In ICML. 2021.
|
| 243 |
+
[25] Kirillov, A., K. He, R. Girshick, et al. Panoptic segmentation. In CVPR. 2019.
|
| 244 |
+
[26] Tran, D., H. Wang, L. Torresani, et al. A closer look at spatiotemporal convolutions for action recognition. In CVPR. 2018.
|
| 245 |
+
[27] Lin, T.-Y., P. Dollar, R. Girshick, et al. Feature pyramid networks for object detection. In CVPR. 2017.
|
| 246 |
+
[28] Stewart, R., M. Andriluka, A. Y. Ng. End-to-end people detection in crowded scenes. In CVPR. 2016.
|
| 247 |
+
[29] Milletari, F., N. Navab, S.-A. Ahmadi. V-net: Fully convolutional neural networks for volumetric medical image segmentation. In 3DV. 2016.
|
| 248 |
+
[30] Lin, T.-Y., P. Goyal, R. Girshick, et al. Focal loss for dense object detection. In ICCV. 2017.
|
| 249 |
+
[31] He, K., X. Zhang, S. Ren, et al. Deep residual learning for image recognition. In CVPR. 2016.
|
| 250 |
+
[32] Athar, A., S. Mahadevan, A. Ošep, et al. Stem-seg: Spatio-temporal embeddings for instance segmentation in videos. In ECCV. 2020.
|
| 251 |
+
[33] Wu, Y., A. Kirillov, F. Massa, et al. Detectron2. https://github.com/facebookresearch/ detectron2, 2019.
|
| 252 |
+
[34] Loshchilov, I., F. Hutter. Decoupled weight decay regularization. In ICLR. 2019.
|
| 253 |
+
[35] Lin, T.-Y., M. Maire, S. Belongie, et al. Microsoft coco: Common objects in context. In ECCV. 2014.
|
| 254 |
+
[36] Dai, J., H. Qi, Y. Xiong, et al. Deformable convolutional networks. In ICCV. 2017.
|
| 255 |
+
[37] Cai, Z., N. Vasconcelos. Cascade r-cnn: Delving into high quality object detection. In CVPR. 2018.
|
parse/train/pvjfA4wogD6/pvjfA4wogD6_content_list.json
ADDED
|
@@ -0,0 +1,1197 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Video Instance Segmentation using Inter-Frame Communication Transformers ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
235,
|
| 8 |
+
122,
|
| 9 |
+
758,
|
| 10 |
+
172
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Sukjun Hwang1 Miran Heo1 Seoung Wug Oh2 Seon Joo Kim1 1Yonsei University 2Adobe Research {sj.hwang, miran, seonjookim}@yonsei.ac.kr seoh@adobe.com ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
233,
|
| 19 |
+
220,
|
| 20 |
+
761,
|
| 21 |
+
265
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Abstract ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
462,
|
| 31 |
+
310,
|
| 32 |
+
535,
|
| 33 |
+
327
|
| 34 |
+
],
|
| 35 |
+
"page_idx": 0
|
| 36 |
+
},
|
| 37 |
+
{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We propose a novel end-to-end solution for video instance segmentation (VIS) based on transformers. Recently, the per-clip pipeline shows superior performance over per-frame methods leveraging richer information from multiple frames. However, previous per-clip models require heavy computation and memory usage to achieve frame-to-frame communications, limiting practicality. In this work, we propose Inter-frame Communication Transformers (IFC), which significantly reduces the overhead for information-passing between frames by efficiently encoding the context within the input clip. Specifically, we propose to utilize concise memory tokens as a means of conveying information as well as summarizing each frame scene. The features of each frame are enriched and correlated with other frames through exchange of information between the precisely encoded memory tokens. We validate our method on the latest benchmark sets and achieved state-of-the-art performance (AP 42.6 on YouTube-VIS 2019 val set using the offline inference) while having a considerably fast runtime (89.4 FPS). Our method can also be applied to near-online inference for processing a video in real-time with only a small delay. The code is available at https://github.com/sukjunhwang/IFC. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
343,
|
| 43 |
+
766,
|
| 44 |
+
565
|
| 45 |
+
],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
174,
|
| 54 |
+
592,
|
| 55 |
+
310,
|
| 56 |
+
608
|
| 57 |
+
],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "With the growing interest toward the video domain in computer vision, the task of video instance segmentation (VIS) is emerging [1]. Most of the current approaches [1, 2, 3, 4] extend image instance segmentation models [5, 6, 7, 8] and take frame-wise inputs. These per-frame methods extend the concept of temporal tracking by matching frame-wise predictions of high similarities. The models can be easily customized to real-world applications as they run in an online [9] fashion, but they show limitations in dealing with occlusions and motion blur that are common in videos. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
623,
|
| 66 |
+
825,
|
| 67 |
+
707
|
| 68 |
+
],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "On the contrary, per-clip models are designed to overcome such challenges by incorporating multiple frames while sacrificing the efficiency. Previous per-clip approaches [10, 11, 12] aggregate information within a clip to generate instance-specific features. As the features are generated per instance, the number of instances in addition to the number of frames has a significant impact on the overall computation. Recently proposed VisTR [11] adapted DETR [13] to the VIS task and reduced the inference time by inserting the entire video, not a clip, to its offline end-to-end network. However, its full self-attention transformers [14] over the space-time inputs involve explosive computations and memories. In this work, we raise the following question: can a per-clip method be efficient while attaining great accuracy? ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
713,
|
| 77 |
+
825,
|
| 78 |
+
838
|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "To achieve our goal, we introduce Inter-frame Communication Transformers (IFC) to greatly reduce the computations of the full space-time transformers. Similar to recent works [15, 16, 17] that alleviate the explosive computational growth inherent in attention-based models [14, 18], IFC takes a decomposition strategy utilizing two transformers. The first transformer (Encode-Receive, $\\mathcal { E }$ ) encodes each frame independently. To exchange the information between frames, the second transformer (Gather-Communicate, $\\mathcal { G }$ ) executes attention between a small number of memory tokens that hold concise information of the clip. The memory tokens are utilized to store the overall context of the clip, for example “a hand over a lizard�� in Fig. 1. The concise information assists detecting the lizard that is largely occluded by the hand in the first frame, without employing an expensive pixel-level attention over space and time. The memory tokens are only in charge of the communications between frames, and the features of each frame are enriched and correlated through the memory tokens. ",
|
| 85 |
+
"bbox": [
|
| 86 |
+
174,
|
| 87 |
+
844,
|
| 88 |
+
823,
|
| 89 |
+
900
|
| 90 |
+
],
|
| 91 |
+
"page_idx": 0
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "text",
|
| 95 |
+
"text": "",
|
| 96 |
+
"bbox": [
|
| 97 |
+
174,
|
| 98 |
+
90,
|
| 99 |
+
825,
|
| 100 |
+
188
|
| 101 |
+
],
|
| 102 |
+
"page_idx": 1
|
| 103 |
+
},
|
| 104 |
+
{
|
| 105 |
+
"type": "text",
|
| 106 |
+
"text": "We further reduce overheads while taking advantage of per-clip pipelines by concisely representing each instance with a unique convolutional weight [7]. Despite the changes of appearances at different frames, the instances of the same identity share commonalities because the frames originated from the same source video. Therefore, we can effectively capture instance-specific characteristics in a clip with dynamically generated convolutional weights. In companion with the segmentation, we track instances by uniformly applying the weights to all frames in a clip. Moreover, all executions of our spatial decoder are instance-agnostic except for the final layer which applies instance-specific weights. Accordingly, our model is highly efficient and also suitable for scenes with numerous instances. ",
|
| 107 |
+
"bbox": [
|
| 108 |
+
174,
|
| 109 |
+
194,
|
| 110 |
+
825,
|
| 111 |
+
305
|
| 112 |
+
],
|
| 113 |
+
"page_idx": 1
|
| 114 |
+
},
|
| 115 |
+
{
|
| 116 |
+
"type": "text",
|
| 117 |
+
"text": "In addition to the efficient modeling, we provide optimizations and an instance tracking algorithm that are designed to be VIS-centric. By the definition of $\\mathsf { A P } ^ { \\mathtt { V I S } }$ , the VIS task [1] aims to maximize the objective similarity: space-time mask IoU. Inspired by previous works [13, 19, 20], our model is optimized to maximize the similarity between bipartitely matched pairs of ground truth masks and predicted masks. Furthermore, we again adopt the similarity maximization for tracking instances of same identities, which effectively links predicted space-time masks using bipartite matching. As both of our training and inference algorithms are fundamentally designed to address the key challenge of VIS task, our method attains an outstanding accuracy. ",
|
| 118 |
+
"bbox": [
|
| 119 |
+
174,
|
| 120 |
+
311,
|
| 121 |
+
825,
|
| 122 |
+
422
|
| 123 |
+
],
|
| 124 |
+
"page_idx": 1
|
| 125 |
+
},
|
| 126 |
+
{
|
| 127 |
+
"type": "text",
|
| 128 |
+
"text": "From these improvements, IFC sets the new state-of-the-art: $4 2 . 6 \\%$ AP and more surprisingly, in 89.4 fps. Furthermore, our model also shows great speed-accuracy balance under near-online settings, which leads to a huge practicality. We believe that our model can be a powerful baseline for video instance segmentation approaches that follow the per-clip execution. ",
|
| 129 |
+
"bbox": [
|
| 130 |
+
174,
|
| 131 |
+
429,
|
| 132 |
+
825,
|
| 133 |
+
484
|
| 134 |
+
],
|
| 135 |
+
"page_idx": 1
|
| 136 |
+
},
|
| 137 |
+
{
|
| 138 |
+
"type": "text",
|
| 139 |
+
"text": "2 Related Work ",
|
| 140 |
+
"text_level": 1,
|
| 141 |
+
"bbox": [
|
| 142 |
+
174,
|
| 143 |
+
508,
|
| 144 |
+
321,
|
| 145 |
+
526
|
| 146 |
+
],
|
| 147 |
+
"page_idx": 1
|
| 148 |
+
},
|
| 149 |
+
{
|
| 150 |
+
"type": "text",
|
| 151 |
+
"text": "Video instance segmentation The VIS task [1] extends the concept of tracking to the image instance segmentation task. The early solutions [1, 2] follow the per-frame pipeline, which utilize additional tracking head to the models that are mainly designed to solve image instance segmentation. More advanced algorithms that are recently proposed [3, 4] take video characteristics into consideration, which result in improved performance. ",
|
| 152 |
+
"bbox": [
|
| 153 |
+
174,
|
| 154 |
+
545,
|
| 155 |
+
825,
|
| 156 |
+
614
|
| 157 |
+
],
|
| 158 |
+
"page_idx": 1
|
| 159 |
+
},
|
| 160 |
+
{
|
| 161 |
+
"type": "text",
|
| 162 |
+
"text": "Per-clip models [10, 11, 12] dedicate computations to extract information from multiple frames for higher accuracy. By exploiting multiple frames, per-clip models can effectively handle typical challenges in video, i.e., motion blurs and occlusions. Our model is designed to be highly efficient while following the per-clip pipeline, which leads to fast and accurate predictions. ",
|
| 163 |
+
"bbox": [
|
| 164 |
+
174,
|
| 165 |
+
621,
|
| 166 |
+
825,
|
| 167 |
+
676
|
| 168 |
+
],
|
| 169 |
+
"page_idx": 1
|
| 170 |
+
},
|
| 171 |
+
{
|
| 172 |
+
"type": "text",
|
| 173 |
+
"text": "Transformers Recently, transformers [14] are greatly impacting many tasks in computer vision. After the huge success of DETR [13], which has brought a new paradigm to the object detection task, numerous vision tasks are incorporating transformers [21, 22] in place of CNNs. For classification tasks in both NLP and computer vision, many adopt an extra classification token to the input of transformers [21, 23]. All the input tokens affect each other as the encoders are mainly composed of the self-attention, thus the classification token can be used to determine the class of the overall input. Similarly, DeiT [24] inserts an additional distillation token to transformers, and the novel usage leads to a higher data efficiency. MaX-DeepLab [20] adopted the concept of memory and proposed a novel dual-path transformer for the panoptic segmentation task [25]. By making use of numerous memory tokens to convey information, MaX-DeepLab integrates the transformer and the CNN by making both feedback itself and the other. ",
|
| 174 |
+
"bbox": [
|
| 175 |
+
174,
|
| 176 |
+
696,
|
| 177 |
+
826,
|
| 178 |
+
849
|
| 179 |
+
],
|
| 180 |
+
"page_idx": 1
|
| 181 |
+
},
|
| 182 |
+
{
|
| 183 |
+
"type": "text",
|
| 184 |
+
"text": "We further utilize the concept of the memory tokens to the videos. Using Inter-frame Communication Transformers, each frame runs independently while sharing their information with interim communications. The communications lead to higher accuracy while the execution independence between frames accelerates the inference. ",
|
| 185 |
+
"bbox": [
|
| 186 |
+
176,
|
| 187 |
+
856,
|
| 188 |
+
825,
|
| 189 |
+
911
|
| 190 |
+
],
|
| 191 |
+
"page_idx": 1
|
| 192 |
+
},
|
| 193 |
+
{
|
| 194 |
+
"type": "image",
|
| 195 |
+
"img_path": "images/38190ff2e29f8cfb09f1e13cb0bf969f482baa8226d8a40c6907883122649d66.jpg",
|
| 196 |
+
"image_caption": [
|
| 197 |
+
"Figure 1: Overview of IFC framework. Our transformer encoder block has two components: 1) Encode-Receive $( \\mathcal { E } )$ simultaneously encodes frame tokens and memory tokens. 2) Only memory tokens pass Gather-Communicate $( { \\mathcal { G } } )$ to perform communications between frames. The output from the stack of $N _ { E }$ encoder blocks goes into two modules, spatial decoder and transformer decoder, to generate segmentation masks. "
|
| 198 |
+
],
|
| 199 |
+
"image_footnote": [],
|
| 200 |
+
"bbox": [
|
| 201 |
+
173,
|
| 202 |
+
87,
|
| 203 |
+
825,
|
| 204 |
+
306
|
| 205 |
+
],
|
| 206 |
+
"page_idx": 2
|
| 207 |
+
},
|
| 208 |
+
{
|
| 209 |
+
"type": "text",
|
| 210 |
+
"text": "3 Method ",
|
| 211 |
+
"text_level": 1,
|
| 212 |
+
"bbox": [
|
| 213 |
+
174,
|
| 214 |
+
406,
|
| 215 |
+
271,
|
| 216 |
+
422
|
| 217 |
+
],
|
| 218 |
+
"page_idx": 2
|
| 219 |
+
},
|
| 220 |
+
{
|
| 221 |
+
"type": "text",
|
| 222 |
+
"text": "The proposed method follows a per-clip pipeline which takes a video clip as input and outputs clip-level results. We also introduce Inter-frame Communication Transformers, which can effectively share frame-wise information within a clip with a high efficiency. ",
|
| 223 |
+
"bbox": [
|
| 224 |
+
174,
|
| 225 |
+
436,
|
| 226 |
+
825,
|
| 227 |
+
479
|
| 228 |
+
],
|
| 229 |
+
"page_idx": 2
|
| 230 |
+
},
|
| 231 |
+
{
|
| 232 |
+
"type": "text",
|
| 233 |
+
"text": "3.1 Model architecture ",
|
| 234 |
+
"text_level": 1,
|
| 235 |
+
"bbox": [
|
| 236 |
+
174,
|
| 237 |
+
494,
|
| 238 |
+
346,
|
| 239 |
+
510
|
| 240 |
+
],
|
| 241 |
+
"page_idx": 2
|
| 242 |
+
},
|
| 243 |
+
{
|
| 244 |
+
"type": "text",
|
| 245 |
+
"text": "Inspired by DETR [13], our network consists of a CNN backbone and transformer encoder-decoder layers (Fig. 1). The input clip is first independently embedded into a feature map through the backbone. Then, the embedded clip passes through our inter-frame communication encoder blocks that enrich the feature map by allowing information exchange between frames. Next, a set of transformer decoder layers that take the encoder outputs and object queries as inputs predict unique convolutional weights for each instance in the clip. Finally, the masks for each instance across the clip are computed in one shot by convolving the encoded feature map with the unique convolutional weight. ",
|
| 246 |
+
"bbox": [
|
| 247 |
+
174,
|
| 248 |
+
520,
|
| 249 |
+
825,
|
| 250 |
+
617
|
| 251 |
+
],
|
| 252 |
+
"page_idx": 2
|
| 253 |
+
},
|
| 254 |
+
{
|
| 255 |
+
"type": "text",
|
| 256 |
+
"text": "Backbone Given an input clip $\\{ x _ { i } \\} _ { i = 1 } ^ { T } \\ \\in \\ \\mathbb { R } ^ { T \\times H _ { 0 } \\times W _ { 0 } \\times 3 }$ , composed of $T$ frames with 3 color channels, the CNN backbone processes the input clip frame-by-frame. As the result, the clip is encoded into a set of low-resolution features, $\\bar { \\{ f _ { i } ^ { 0 } \\} _ { i = 1 } ^ { T } } \\in \\mathbb { R } ^ { T \\times H \\times W \\times C }$ , where $C$ is the number of channels and $\\begin{array} { r } { H , W = \\frac { H _ { 0 } } { 3 2 } , \\frac { W _ { 0 } } { 3 2 } } \\end{array}$ . ",
|
| 257 |
+
"bbox": [
|
| 258 |
+
174,
|
| 259 |
+
630,
|
| 260 |
+
825,
|
| 261 |
+
693
|
| 262 |
+
],
|
| 263 |
+
"page_idx": 2
|
| 264 |
+
},
|
| 265 |
+
{
|
| 266 |
+
"type": "text",
|
| 267 |
+
"text": "Inter-Frame Communication Encoder Given an image, humans can effortlessly summarize the scene with only a few words. Also, frames from a same video share a lot of commonalities, the difference between them is sufficiently summarized and communicated even with a small bandwidth. Based on this hypothesis, we propose an inter-frame communication encoder to make the computation to be mostly frame-wise independent with some communications between frames. Specifically, we adopt memory tokens for both summarizing per-frame scenes and the means of communications. ",
|
| 268 |
+
"bbox": [
|
| 269 |
+
174,
|
| 270 |
+
704,
|
| 271 |
+
825,
|
| 272 |
+
787
|
| 273 |
+
],
|
| 274 |
+
"page_idx": 2
|
| 275 |
+
},
|
| 276 |
+
{
|
| 277 |
+
"type": "text",
|
| 278 |
+
"text": "Our encoder blocks are composed of two phases of separate transformers: Encode-Receive $( \\mathcal { E } )$ and Gather-Communicate $( { \\mathcal { G } } )$ . Both Encode-Receive and Gather-Communicate follow the typical transformer encoder architecture [14], which consists of an addition of fixed positional encoding, a multi-head self-attention module, and a feed forward network. ",
|
| 279 |
+
"bbox": [
|
| 280 |
+
174,
|
| 281 |
+
794,
|
| 282 |
+
825,
|
| 283 |
+
849
|
| 284 |
+
],
|
| 285 |
+
"page_idx": 2
|
| 286 |
+
},
|
| 287 |
+
{
|
| 288 |
+
"type": "text",
|
| 289 |
+
"text": "Encode-Receive operates in a per-frame manner, taking a frame-level feature map and corresponding memory tokens. Passing through Encode-Receive, we expect two functionalities: (1) image features encode per-frame information to the memory tokens, and (2) image features receive information of different frames that are gathered in the memory tokens. Gather-Communicate operates across frames to form a clip-level knowledge. It takes the memory tokens from each frame as inputs and performs communications between frames. Alternating two phases through multiple layers, the encoder can efficiently learn consensus representations across frames. ",
|
| 290 |
+
"bbox": [
|
| 291 |
+
176,
|
| 292 |
+
856,
|
| 293 |
+
825,
|
| 294 |
+
911
|
| 295 |
+
],
|
| 296 |
+
"page_idx": 2
|
| 297 |
+
},
|
| 298 |
+
{
|
| 299 |
+
"type": "table",
|
| 300 |
+
"img_path": "images/2778f3271558aa37551fff29a82d9708bb645857c02dc5d266ffa8b91f63e862.jpg",
|
| 301 |
+
"table_caption": [
|
| 302 |
+
"Table 1: Complexity comparison. Various transformer encoders for space-time input. As the overall FLOPs can vary by the number of detected instances, listed values are measured only at the encoders. "
|
| 303 |
+
],
|
| 304 |
+
"table_footnote": [],
|
| 305 |
+
"table_body": "<table><tr><td rowspan=\"3\">Communication Type</td><td rowspan=\"3\">Complexity per Layer</td><td colspan=\"4\">FLOPs (G)1</td></tr><tr><td></td><td>360 × 640</td><td></td><td>720×1280</td></tr><tr><td>T=5</td><td>T=36</td><td>T=5</td><td>T=36</td></tr><tr><td>No Comm</td><td>O(C²THW + CT(HW)2)</td><td>5.17</td><td>37.23</td><td>24.62</td><td>177.29</td></tr><tr><td>Full THW</td><td>O(C²THW + C(THW)2)</td><td>6.94</td><td>148.70</td><td>50.63</td><td>1815.38</td></tr><tr><td>Decompose T-HW</td><td>O(C²THW + CT(HW)² + CT²HW)</td><td>8.33</td><td>60.24</td><td>36.73</td><td>265.50</td></tr><tr><td>IFC (M = 8)</td><td>O(C²THW +CT(HW)2)</td><td>5.52</td><td>39.73</td><td>25.05</td><td>180.39</td></tr></table>",
|
| 306 |
+
"bbox": [
|
| 307 |
+
173,
|
| 308 |
+
126,
|
| 309 |
+
821,
|
| 310 |
+
234
|
| 311 |
+
],
|
| 312 |
+
"page_idx": 3
|
| 313 |
+
},
|
| 314 |
+
{
|
| 315 |
+
"type": "text",
|
| 316 |
+
"text": "",
|
| 317 |
+
"bbox": [
|
| 318 |
+
173,
|
| 319 |
+
261,
|
| 320 |
+
825,
|
| 321 |
+
304
|
| 322 |
+
],
|
| 323 |
+
"page_idx": 3
|
| 324 |
+
},
|
| 325 |
+
{
|
| 326 |
+
"type": "text",
|
| 327 |
+
"text": "In more detail, given the frame embedding $\\{ f _ { i } ^ { 0 } \\} _ { i = 1 } ^ { T }$ , we spatially flatten each feature $\\mathbb { R } ^ { H \\times W \\times C } $ $\\mathbb { R } ^ { H W \\times C }$ . The initial memory tokens $m ^ { 0 }$ of size $M$ are copied per frame and concatenated to each frame feature as follows: ",
|
| 328 |
+
"bbox": [
|
| 329 |
+
173,
|
| 330 |
+
308,
|
| 331 |
+
825,
|
| 332 |
+
353
|
| 333 |
+
],
|
| 334 |
+
"page_idx": 3
|
| 335 |
+
},
|
| 336 |
+
{
|
| 337 |
+
"type": "equation",
|
| 338 |
+
"img_path": "images/e4b554f7c3b51d417e4fa32104b4e1afaf99cea19f48f18c3f542f26fc18db65.jpg",
|
| 339 |
+
"text": "$$\n[ f _ { t } ^ { 0 } , m _ { t } ^ { 0 } ] \\in \\mathbb { R } ^ { ( H W + M ) \\times C } , \\qquad t \\in \\{ 1 , 2 , \\cdots , T \\} ,\n$$",
|
| 340 |
+
"text_format": "latex",
|
| 341 |
+
"bbox": [
|
| 342 |
+
334,
|
| 343 |
+
358,
|
| 344 |
+
663,
|
| 345 |
+
377
|
| 346 |
+
],
|
| 347 |
+
"page_idx": 3
|
| 348 |
+
},
|
| 349 |
+
{
|
| 350 |
+
"type": "text",
|
| 351 |
+
"text": "where $[ \\cdot , \\cdot ]$ indicates a concatenation of two feature vectors. Note that the initial memory tokens $m ^ { 0 }$ are trainable parameters learnt during training. ",
|
| 352 |
+
"bbox": [
|
| 353 |
+
173,
|
| 354 |
+
383,
|
| 355 |
+
823,
|
| 356 |
+
414
|
| 357 |
+
],
|
| 358 |
+
"page_idx": 3
|
| 359 |
+
},
|
| 360 |
+
{
|
| 361 |
+
"type": "text",
|
| 362 |
+
"text": "The first phase of IFC is Encode-Receive, which processes frames individually as follows: ",
|
| 363 |
+
"bbox": [
|
| 364 |
+
171,
|
| 365 |
+
417,
|
| 366 |
+
764,
|
| 367 |
+
434
|
| 368 |
+
],
|
| 369 |
+
"page_idx": 3
|
| 370 |
+
},
|
| 371 |
+
{
|
| 372 |
+
"type": "equation",
|
| 373 |
+
"img_path": "images/83fbfa4a19e414b3cff0aa02cd7f01046ec0f97e1c0c8e04bed22a54a7f98810.jpg",
|
| 374 |
+
"text": "$$\n[ f _ { t } ^ { l } , \\widehat { m } _ { t } ^ { l } ] = \\mathcal { E } ^ { l } ( [ f _ { t } ^ { l - 1 } , m _ { t } ^ { l - 1 } ] ) ,\n$$",
|
| 375 |
+
"text_format": "latex",
|
| 376 |
+
"bbox": [
|
| 377 |
+
401,
|
| 378 |
+
440,
|
| 379 |
+
594,
|
| 380 |
+
459
|
| 381 |
+
],
|
| 382 |
+
"page_idx": 3
|
| 383 |
+
},
|
| 384 |
+
{
|
| 385 |
+
"type": "text",
|
| 386 |
+
"text": "where ${ \\mathcal { E } } ^ { l }$ denotes the $l$ -th Encode-Receive layer. With a self-attention computed over the frame pixel locations and the memory tokens, the information of each frame can be passed to the memory tokens and vise-versa. ",
|
| 387 |
+
"bbox": [
|
| 388 |
+
176,
|
| 389 |
+
467,
|
| 390 |
+
826,
|
| 391 |
+
510
|
| 392 |
+
],
|
| 393 |
+
"page_idx": 3
|
| 394 |
+
},
|
| 395 |
+
{
|
| 396 |
+
"type": "text",
|
| 397 |
+
"text": "The outputs of Encode-Receive are grouped by memory indices and formulate the inputs for GatherCommunicate layer. The grouping can be understood as a decomposition of memory tokens, and becomes computationally beneficial when the total size of gathered memory tokens increases. ",
|
| 398 |
+
"bbox": [
|
| 399 |
+
174,
|
| 400 |
+
515,
|
| 401 |
+
825,
|
| 402 |
+
558
|
| 403 |
+
],
|
| 404 |
+
"page_idx": 3
|
| 405 |
+
},
|
| 406 |
+
{
|
| 407 |
+
"type": "equation",
|
| 408 |
+
"img_path": "images/826462796f0e0ff15e14208c5e355a04a4b54bcc9c8e69822d8f63c911cb9cb9.jpg",
|
| 409 |
+
"text": "$$\n\\begin{array} { r l } & { [ m _ { 1 } ^ { l } ( i ) , m _ { 2 } ^ { l } ( i ) , \\cdots , m _ { T } ^ { l } ( i ) ] = \\mathcal { G } ^ { l } ( [ \\widehat { m } _ { 1 } ^ { l } ( i ) , \\widehat { m } _ { 2 } ^ { l } ( i ) , \\cdots , \\widehat { m } _ { T } ^ { l } ( i ) ] ) , \\qquad i \\in \\{ 1 , 2 , \\cdots , M \\} , } \\end{array}\n$$",
|
| 410 |
+
"text_format": "latex",
|
| 411 |
+
"bbox": [
|
| 412 |
+
215,
|
| 413 |
+
564,
|
| 414 |
+
781,
|
| 415 |
+
584
|
| 416 |
+
],
|
| 417 |
+
"page_idx": 3
|
| 418 |
+
},
|
| 419 |
+
{
|
| 420 |
+
"type": "text",
|
| 421 |
+
"text": "where $\\mathcal { G } ^ { l }$ denotes the $l$ -th Gather-Communicate layer. The processed outputs are redistributed to the originated frame and get concatenated as $m _ { t } \\overset { \\cdot } { = } \\left[ m _ { t } ( 1 ) , \\overset { \\cdot } { m } _ { t } ( 2 ) , \\cdot \\cdot \\cdot , \\overset { \\cdot } { m } _ { t } ( M ) \\right]$ . Unlike EncodeReceive, Gather-Communicate utilizes the attention mechanism to convey the information from different frames over the input clip. ",
|
| 422 |
+
"bbox": [
|
| 423 |
+
173,
|
| 424 |
+
590,
|
| 425 |
+
826,
|
| 426 |
+
648
|
| 427 |
+
],
|
| 428 |
+
"page_idx": 3
|
| 429 |
+
},
|
| 430 |
+
{
|
| 431 |
+
"type": "text",
|
| 432 |
+
"text": "Defining the $l$ -th inter-frame encoder block $( \\mathrm { I F C } ^ { l } )$ as ${ \\mathcal { E } } ^ { l }$ followed by $\\mathcal { G } ^ { l }$ , the stack of $N _ { E }$ encoder blocks can be inductively formulated as: ",
|
| 433 |
+
"bbox": [
|
| 434 |
+
174,
|
| 435 |
+
654,
|
| 436 |
+
823,
|
| 437 |
+
684
|
| 438 |
+
],
|
| 439 |
+
"page_idx": 3
|
| 440 |
+
},
|
| 441 |
+
{
|
| 442 |
+
"type": "equation",
|
| 443 |
+
"img_path": "images/12201cbca6981cd6046b05a17f902beadf6a56e673115e912cbb891df68857c4.jpg",
|
| 444 |
+
"text": "$$\n[ f _ { t } ^ { l } , m _ { t } ^ { l } ] = \\mathrm { I F C } ^ { l } ( [ f _ { t } ^ { l - 1 } , m _ { t } ^ { l - 1 } ] ) , \\qquad 1 \\leq l \\leq N _ { E } ,\n$$",
|
| 445 |
+
"text_format": "latex",
|
| 446 |
+
"bbox": [
|
| 447 |
+
333,
|
| 448 |
+
690,
|
| 449 |
+
665,
|
| 450 |
+
710
|
| 451 |
+
],
|
| 452 |
+
"page_idx": 3
|
| 453 |
+
},
|
| 454 |
+
{
|
| 455 |
+
"type": "text",
|
| 456 |
+
"text": "where $\\left[ f _ { t } ^ { N _ { E } } , m _ { t } ^ { N _ { E } } \\right]$ is the final result. The stacking of multiple encoder layers brings communications between frames, thus each frame can have coincidence to the other, specifying the identities of instances in a given clip. ",
|
| 457 |
+
"bbox": [
|
| 458 |
+
174,
|
| 459 |
+
718,
|
| 460 |
+
825,
|
| 461 |
+
761
|
| 462 |
+
],
|
| 463 |
+
"page_idx": 3
|
| 464 |
+
},
|
| 465 |
+
{
|
| 466 |
+
"type": "text",
|
| 467 |
+
"text": "Complexity comparison In Table 1, we analyze the computational complexity of transformer encoder variants applied for video input in terms of the Big-O complexity and FLOPs. The complexity of the original transformer encoder layer [14] is $\\mathcal { O } ( C ^ { 2 } N ^ { ' } { + } C N ^ { 2 } )$ , where $N$ is the number of inputs. Without any communication between frames, No Comm, it shows the smallest amount of computation $( { \\mathcal O } ( C ^ { 2 } T \\dot { H W } + C T ( H W ) ^ { 2 } ) )$ ). As indicated as Full THW in Table 1, the complexity of VisTR [11] that performs a full space-time self-attention is $\\mathcal { O } ( C ^ { 2 } ( T H W ) + C ( T H W ) ^ { 2 } )$ thus either a higher resolution or an increase of number of input frames leads to a massive increase in computations. VisTR bypasses the problem by highly reducing the input resolution and utilizing GPUs with tremendous memory capacity. However, as such solutions cannot resolve the fundamental issues, it is impractical to real-world videos. Moreover, VisTR remains as a complete offline strategy because it takes the entire video as an input. ",
|
| 468 |
+
"bbox": [
|
| 469 |
+
173,
|
| 470 |
+
775,
|
| 471 |
+
826,
|
| 472 |
+
887
|
| 473 |
+
],
|
| 474 |
+
"page_idx": 3
|
| 475 |
+
},
|
| 476 |
+
{
|
| 477 |
+
"type": "text",
|
| 478 |
+
"text": "",
|
| 479 |
+
"bbox": [
|
| 480 |
+
176,
|
| 481 |
+
90,
|
| 482 |
+
823,
|
| 483 |
+
133
|
| 484 |
+
],
|
| 485 |
+
"page_idx": 4
|
| 486 |
+
},
|
| 487 |
+
{
|
| 488 |
+
"type": "text",
|
| 489 |
+
"text": "An intriguing improvement for the naïve full self-attention would be the decomposition of the attention into space and time axis [16, 17, 26]. In Decompose T-HW, we decompose attention computation into spatial and temporal attention. The complexity of the separation of space-time leads to the sum of the two transformer encoder: $\\mathcal { O } ( T ( C ^ { 2 } ( H \\bar { W } ) + \\bar { C } ( H W ) ^ { 2 } ) )$ and $\\mathcal { O } ( H \\bar { W } ( C ^ { 2 } T + C T ^ { 2 } ) )$ . In comparison to the full self-attention, the decomposition lowers the computational growth relative to the number of frames. ",
|
| 490 |
+
"bbox": [
|
| 491 |
+
173,
|
| 492 |
+
138,
|
| 493 |
+
825,
|
| 494 |
+
222
|
| 495 |
+
],
|
| 496 |
+
"page_idx": 4
|
| 497 |
+
},
|
| 498 |
+
{
|
| 499 |
+
"type": "text",
|
| 500 |
+
"text": "Our encoder, IFC, that communicates between frames using the memory tokens leads to a huge benefit to the total computations adding only a small amount of computation over No Comm while providing sufficient channels for communication. The complexity of each phase in our proposed encoder is: $\\mathcal { O } ( C ^ { 2 } T ( H W + M ) + C T ( H W + M ) ^ { 2 } )$ for Encode-Receive and $\\mathcal { O } ( C ^ { 2 } T M + \\bar { C } \\bar { T } ^ { 2 } M )$ for Gather-Communicate respectively. Assuming that $M$ is kept small (e.g., 8), the computation needed for Gather-Communicate can be neglected, while the complexity of Encode-Receive can be approximated to $\\mathcal { O } ( C ^ { 2 } T H W + C T ( H W ) ^ { 2 } )$ as shown in Table 1. Finally, with respect to the number of frames of the input, we can expect approximate linear increase rather than the high increase of computation occurred in VisTR. ",
|
| 501 |
+
"bbox": [
|
| 502 |
+
173,
|
| 503 |
+
228,
|
| 504 |
+
825,
|
| 505 |
+
353
|
| 506 |
+
],
|
| 507 |
+
"page_idx": 4
|
| 508 |
+
},
|
| 509 |
+
{
|
| 510 |
+
"type": "text",
|
| 511 |
+
"text": "Decoders and output heads As depicted in Fig. 1, the transformer decoder of our model is stacked with $N _ { D }$ layers [14]. Contrary to VisTR, where the number of object queries increases proportionally to the number of frames, our model receives learnt encodings of fixed size $N _ { q }$ for object queries. Also, by utilizing these encodings throughout the entire frames, our model can effectively deal with clips of various lengths. A set of projection matrices are applied to $\\{ f _ { t } ^ { N _ { E } } , m _ { t } ^ { N _ { E } } \\} _ { t = 1 } ^ { T }$ for the generation of keys and values. The object queries turn into output embeddings by the transformer decoder, and the embeddings are eventually used as an input to the output heads. ",
|
| 512 |
+
"bbox": [
|
| 513 |
+
173,
|
| 514 |
+
367,
|
| 515 |
+
825,
|
| 516 |
+
468
|
| 517 |
+
],
|
| 518 |
+
"page_idx": 4
|
| 519 |
+
},
|
| 520 |
+
{
|
| 521 |
+
"type": "text",
|
| 522 |
+
"text": "There are two output heads on top of the transformer decoder, a class head and a segmentation head, each composed of two fully-connected layers. The output embeddings from the transformer decoder are independently inserted to the heads, resulting in $N _ { q }$ predictions per a clip. The class head outputs a class probability distribution of instances $\\hat { p } ( c ) \\in \\mathbb { R } ^ { N _ { q } \\times | \\mathbb { C } | }$ . Note that the possible classes $\\mathbb { C } \\ni c$ include no object $\\mathcal { D }$ class in addition to the given classes of a dataset. ",
|
| 523 |
+
"bbox": [
|
| 524 |
+
173,
|
| 525 |
+
473,
|
| 526 |
+
825,
|
| 527 |
+
545
|
| 528 |
+
],
|
| 529 |
+
"page_idx": 4
|
| 530 |
+
},
|
| 531 |
+
{
|
| 532 |
+
"type": "text",
|
| 533 |
+
"text": "The segmentation head generates $N _ { q }$ conditional convolutional weights $w \\in \\mathbb { R } ^ { N _ { q } \\times C }$ in a manner similar to [7, 20]. For the conditional convolution, the output feature of the encoder reused by undoing the flatten operation. For the upsampling, the encoder feature pa $\\{ f _ { t } ^ { N _ { E } } \\} _ { t = 1 } ^ { T }$ isgh fpn-style [27] spatial decoder without temporal connections resulting in $T$ feature maps that are $1 / 8$ of the input resolution. Finally, the resulting feature maps $f ^ { \\prime }$ are convolved with each convolutional weight to generate a segmentation mask as follows: ",
|
| 534 |
+
"bbox": [
|
| 535 |
+
173,
|
| 536 |
+
550,
|
| 537 |
+
825,
|
| 538 |
+
637
|
| 539 |
+
],
|
| 540 |
+
"page_idx": 4
|
| 541 |
+
},
|
| 542 |
+
{
|
| 543 |
+
"type": "equation",
|
| 544 |
+
"img_path": "images/f866c1f8d7c32e015f24637c2e527d6695ac75ee24e332d9eb40e370b70203ec.jpg",
|
| 545 |
+
"text": "$$\n\\hat { s } _ { i } = \\{ f _ { t } ^ { \\prime } \\circ w _ { i } \\} _ { t = 1 } ^ { T } ,\n$$",
|
| 546 |
+
"text_format": "latex",
|
| 547 |
+
"bbox": [
|
| 548 |
+
434,
|
| 549 |
+
643,
|
| 550 |
+
562,
|
| 551 |
+
662
|
| 552 |
+
],
|
| 553 |
+
"page_idx": 4
|
| 554 |
+
},
|
| 555 |
+
{
|
| 556 |
+
"type": "text",
|
| 557 |
+
"text": "where $w _ { i }$ is $i$ -th convolutional weight, $\\circ$ indicate $1 \\times 1$ spatial convolution operation, and the result $\\hat { s _ { i } }$ is a spatial-temporal object mask in shape of $\\mathbb { R } ^ { T \\times H ^ { \\prime } \\times W ^ { \\prime } }$ where $\\begin{array} { r } { H ^ { \\prime } = \\frac { H _ { 0 } } { 8 } } \\end{array}$ , $\\begin{array} { r } { W ^ { \\prime } = \\frac { W _ { 0 } } { 8 } } \\\\ { . } \\end{array}$ . Note that, for an instance, a common weight is applied throughout the video clip. Our spatial decoder is an instanceagnostic design, which is much more efficient than instance-specific decoders [10, 11, 12, 13] as the number of detected instances increases. Meanwhile, thanks to our segmentation head which specifies and captures the characteristics of an instance, IFC can conduct both segmentation and tracking at once within a clip. ",
|
| 558 |
+
"bbox": [
|
| 559 |
+
173,
|
| 560 |
+
666,
|
| 561 |
+
825,
|
| 562 |
+
767
|
| 563 |
+
],
|
| 564 |
+
"page_idx": 4
|
| 565 |
+
},
|
| 566 |
+
{
|
| 567 |
+
"type": "text",
|
| 568 |
+
"text": "3.2 Instance matching and loss ",
|
| 569 |
+
"text_level": 1,
|
| 570 |
+
"bbox": [
|
| 571 |
+
174,
|
| 572 |
+
782,
|
| 573 |
+
400,
|
| 574 |
+
797
|
| 575 |
+
],
|
| 576 |
+
"page_idx": 4
|
| 577 |
+
},
|
| 578 |
+
{
|
| 579 |
+
"type": "text",
|
| 580 |
+
"text": "To train our network, we first assign the ground truth for each instance estimation and then a set of loss function between each the ground truth and prediction pair. For a given input clip, our model generate a fixed-size set of class-labeled masks $\\{ \\hat { y } _ { i } \\} _ { i = 1 } ^ { N _ { q } } = \\{ ( \\hat { p } _ { i } ( \\boldsymbol { c } ) , \\hat { s } _ { i } ) \\} _ { i = 1 } ^ { N _ { q } }$ . The ground truth set of the clip can be represented as $y _ { i } = ( c _ { i } , s _ { i } )$ ; $c _ { i }$ is the target class label including $\\mathcal { D }$ , and $s _ { i }$ is the target mask which is down-sampled to the size of the prediction masks for efficient similarity calculation. One-to-one bipartite matching between the prediction set $\\{ \\hat { y } _ { i } \\} _ { i = 1 } ^ { N _ { q } }$ and the ground truth set $\\{ y _ { i } \\} _ { i = 1 } ^ { K }$ is performed to find the best assignment of a prediction to a ground truth. The objective can be formally ",
|
| 581 |
+
"bbox": [
|
| 582 |
+
173,
|
| 583 |
+
808,
|
| 584 |
+
826,
|
| 585 |
+
912
|
| 586 |
+
],
|
| 587 |
+
"page_idx": 4
|
| 588 |
+
},
|
| 589 |
+
{
|
| 590 |
+
"type": "text",
|
| 591 |
+
"text": "described as: ",
|
| 592 |
+
"bbox": [
|
| 593 |
+
173,
|
| 594 |
+
92,
|
| 595 |
+
259,
|
| 596 |
+
106
|
| 597 |
+
],
|
| 598 |
+
"page_idx": 5
|
| 599 |
+
},
|
| 600 |
+
{
|
| 601 |
+
"type": "equation",
|
| 602 |
+
"img_path": "images/fb019df8ddd4dd097b5a31d29542133f0c5bd974273ae59f3e201ecf228fb3e4.jpg",
|
| 603 |
+
"text": "$$\n\\hat { \\sigma } = \\underset { \\sigma \\in \\mathfrak { S } _ { N _ { q } } } { \\arg \\operatorname* { m a x } } \\sum _ { i = 1 } ^ { K } \\sin ( y _ { i } , \\hat { y } _ { \\sigma ( i ) } ) ,\n$$",
|
| 604 |
+
"text_format": "latex",
|
| 605 |
+
"bbox": [
|
| 606 |
+
392,
|
| 607 |
+
102,
|
| 608 |
+
604,
|
| 609 |
+
145
|
| 610 |
+
],
|
| 611 |
+
"page_idx": 5
|
| 612 |
+
},
|
| 613 |
+
{
|
| 614 |
+
"type": "text",
|
| 615 |
+
"text": "where $\\sin ( y _ { i } , \\hat { y } _ { \\sigma ( i ) } )$ refers a pair-wise similarity over a permutation of $\\underset { \\_ w } { \\sigma } \\in \\mathfrak { S } _ { N _ { g } }$ . Following prior work [13, 20, 28], the bipartite matching is efficiently computed using Hungarian algorithm [19]. We find that box-based similarity measurement as used in DETR [13] shows weaknesses in matching instances in video clip due to the case of occlusion and disappear-and-reappear. Therefore, we define $\\sin ( y _ { i } , \\hat { y } _ { \\sigma ( i ) } )$ to be mask-based term as $\\mathbb { 1 } _ { \\{ c _ { i } \\neq \\infty \\} } [ \\hat { p } _ { \\sigma ( i ) } ( c _ { i } ) + \\bar { \\lambda } _ { 0 } \\mathrm { D I C E } ( s _ { i } , \\bar { \\hat { s } } _ { \\sigma ( i ) } ^ { - } ) ]$ , where DICE denotes dice coefficients [29]. ",
|
| 616 |
+
"bbox": [
|
| 617 |
+
173,
|
| 618 |
+
146,
|
| 619 |
+
825,
|
| 620 |
+
231
|
| 621 |
+
],
|
| 622 |
+
"page_idx": 5
|
| 623 |
+
},
|
| 624 |
+
{
|
| 625 |
+
"type": "text",
|
| 626 |
+
"text": "Given the optimal assignment $\\hat { \\sigma }$ , we refer to the $K$ matched predictions and $( N _ { q } - K )$ non-matched predictions as positive and negative pairs respectively. The positive pairs aim to predict the ground truth masks and classes while the negative pairs are optimized to predict the $\\mathcal { D }$ class. The final loss is a sum of the losses from positive pairs and negative pairs where each can be computed as follows: ",
|
| 627 |
+
"bbox": [
|
| 628 |
+
176,
|
| 629 |
+
236,
|
| 630 |
+
825,
|
| 631 |
+
292
|
| 632 |
+
],
|
| 633 |
+
"page_idx": 5
|
| 634 |
+
},
|
| 635 |
+
{
|
| 636 |
+
"type": "equation",
|
| 637 |
+
"img_path": "images/9b23e415d7a2efc89846c697f04f297d9fc9948c5400dcc957d4d7992db68d04.jpg",
|
| 638 |
+
"text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { p o s } = \\displaystyle \\sum _ { i = 1 } ^ { K } [ \\underbrace { - \\log \\hat { p } _ { \\hat { \\sigma } ( i ) } ( c _ { i } ) } _ { \\mathrm { C r o s s - e n t r o p y ~ l o s s } } + \\lambda _ { 1 } ( \\underbrace { 1 - \\operatorname { D I C E } \\bigl ( s _ { i } , \\hat { s } _ { \\hat { \\sigma } ( i ) } \\bigr ) } _ { \\mathrm { D i c e ~ l o s s ~ } [ 2 9 ] } ) + \\lambda _ { 2 } \\underbrace { \\operatorname { F O C A L } \\bigl ( s _ { i } , \\hat { s } _ { \\hat { \\sigma } ( i ) } \\bigr ) } _ { \\mathrm { S i g m o i d - f o c a l ~ l o s s ~ } [ 3 0 ] } ] , } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\mathcal { L } _ { n e g } = \\displaystyle \\sum _ { i = k + 1 } ^ { N _ { q } } [ - \\log \\hat { p } _ { \\hat { \\sigma } ( i ) } ( \\emptyset ) ] . } \\end{array}\n$$",
|
| 639 |
+
"text_format": "latex",
|
| 640 |
+
"bbox": [
|
| 641 |
+
240,
|
| 642 |
+
297,
|
| 643 |
+
754,
|
| 644 |
+
397
|
| 645 |
+
],
|
| 646 |
+
"page_idx": 5
|
| 647 |
+
},
|
| 648 |
+
{
|
| 649 |
+
"type": "text",
|
| 650 |
+
"text": "As $( N _ { q } - K )$ is likely to be much greater than $K$ , we down-weight $\\mathcal { L } _ { n e g }$ by a factor of 10 to resolve the imbalance, following prior work [13]. The goal of video instance segmentation [1] is to maximize the space-time IoU between a prediction and a ground truth mask. Therefore, our mask-related losses (Dice loss and Sigmoid-focal loss) are spatio-temporally calculated over an entire clip, rather than averaging the losses that are accumulated frame-by-frame. ",
|
| 651 |
+
"bbox": [
|
| 652 |
+
173,
|
| 653 |
+
401,
|
| 654 |
+
825,
|
| 655 |
+
472
|
| 656 |
+
],
|
| 657 |
+
"page_idx": 5
|
| 658 |
+
},
|
| 659 |
+
{
|
| 660 |
+
"type": "text",
|
| 661 |
+
"text": "3.3 Clip-level instance tracking ",
|
| 662 |
+
"text_level": 1,
|
| 663 |
+
"bbox": [
|
| 664 |
+
176,
|
| 665 |
+
488,
|
| 666 |
+
403,
|
| 667 |
+
503
|
| 668 |
+
],
|
| 669 |
+
"page_idx": 5
|
| 670 |
+
},
|
| 671 |
+
{
|
| 672 |
+
"type": "text",
|
| 673 |
+
"text": "To infer a video input that is longer than the clip length, we match instances using the predicted masks of overlapping frames. Let $\\mathcal { V } _ { I }$ and ${ \\mathcal { V } } _ { A }$ be the result sets of clip $I$ and $A$ excluding the $\\mathcal { D }$ class. The goal is to perform matching of same identities between pre-collected instance set $\\mathcal { V } _ { I }$ and $\\mathcal { V } _ { A }$ . We first calculate the matching scores which are space-time soft IoU at intersecting frames between $\\mathcal { V } _ { I }$ and $\\mathcal { V } _ { A }$ . Then, we find optimal paired indices $\\hat { \\sigma } _ { S }$ using Hungarian algorithm [19] to the gathered matching score $\\mathcal { S } \\in [ 0 , \\bar { 1 ] } ^ { | \\mathcal { N } _ { I } | \\times | \\bar { \\mathcal { V } } _ { A } | }$ . We update $\\mathscr { D } _ { I } ( i )$ by concatenating $\\mathcal { V } _ { A } ( \\hat { \\sigma } _ { S } ( i ) )$ if $\\bar { \\cal S } ( i , \\bar { \\sigma } s ( i ) )$ is above a certain threshold, and add non-matched prediction sets to $\\mathcal { V } _ { I }$ as new instances. Note that a previous per-clip model (MaskProp [10]) also utilizes soft IoU for tracking instances, but the matching scores are computed per-frame and averaged for intersecting frames. Different from MaskProp, using space-time soft IoU leads to an accurate tracking as it can better represent the definition of mask similarities between clips which brings at most $2 \\%$ AP increase. The overall tracking pipeline can be effectively implemented in a GPU-friendly manner. ",
|
| 674 |
+
"bbox": [
|
| 675 |
+
173,
|
| 676 |
+
512,
|
| 677 |
+
826,
|
| 678 |
+
680
|
| 679 |
+
],
|
| 680 |
+
"page_idx": 5
|
| 681 |
+
},
|
| 682 |
+
{
|
| 683 |
+
"type": "text",
|
| 684 |
+
"text": "4 Experiments ",
|
| 685 |
+
"text_level": 1,
|
| 686 |
+
"bbox": [
|
| 687 |
+
174,
|
| 688 |
+
699,
|
| 689 |
+
312,
|
| 690 |
+
717
|
| 691 |
+
],
|
| 692 |
+
"page_idx": 5
|
| 693 |
+
},
|
| 694 |
+
{
|
| 695 |
+
"type": "text",
|
| 696 |
+
"text": "In this section, we evaluate the proposed method using YouTube-VIS 2019 and 2021 [1]. For every listed score, we report the mean of five runs as the results may vary by each run due to the insufficient number of training and testing set of YouTube-VIS dataset. We demonstrate the effectiveness of our model regarding both accuracy and speed. We further examine how different settings affect the overall performance and efficiency of IFC encoder. Unless specified, all models for measurements used $\\bar { N _ { E } } = 3 , \\bar { N _ { D } } = 3$ , stride of 1, and ResNet-50. ",
|
| 697 |
+
"bbox": [
|
| 698 |
+
174,
|
| 699 |
+
729,
|
| 700 |
+
825,
|
| 701 |
+
814
|
| 702 |
+
],
|
| 703 |
+
"page_idx": 5
|
| 704 |
+
},
|
| 705 |
+
{
|
| 706 |
+
"type": "text",
|
| 707 |
+
"text": "4.1 Implementation Details ",
|
| 708 |
+
"text_level": 1,
|
| 709 |
+
"bbox": [
|
| 710 |
+
174,
|
| 711 |
+
830,
|
| 712 |
+
377,
|
| 713 |
+
844
|
| 714 |
+
],
|
| 715 |
+
"page_idx": 5
|
| 716 |
+
},
|
| 717 |
+
{
|
| 718 |
+
"type": "text",
|
| 719 |
+
"text": "We used detectron2 [33] for our code basis, and hyper-parameters mostly follow the settings of DETR [13] unless specified. We used AdamW [34] optimizer with initial learning rate of $1 0 ^ { - 4 }$ for transformers, and $1 \\bar { 0 } ^ { - 5 }$ for backbone. We first pre-train the model for image instance segmentation on COCO [35] by setting our model to $T = 1$ . The pre-train procedure follows the shortened training schedule of DETR [13], which runs 300 epochs with a decay of the learning rate by a factor of 10 at 200 epochs. Using the pre-trained weights, the models are trained on a targeted dataset using the batch size of 16, each clip composed of $T = 5$ frames downscaled to either $3 6 0 \\mathrm { p }$ or $4 8 0 \\mathrm { p }$ . For the sampling of each clip, a reference frame index $t$ is randomly chosen. The remaining $T - 1$ frame indices are then sampled within an interval of 20. The models are trained for 8 epochs, and decays the learning rate by 10 at 5th epoch. ",
|
| 720 |
+
"bbox": [
|
| 721 |
+
174,
|
| 722 |
+
856,
|
| 723 |
+
825,
|
| 724 |
+
911
|
| 725 |
+
],
|
| 726 |
+
"page_idx": 5
|
| 727 |
+
},
|
| 728 |
+
{
|
| 729 |
+
"type": "table",
|
| 730 |
+
"img_path": "images/8117fe1192cb18961b36b519cc6594c8dcf53d69eaf7891be356bd1f19b81efb.jpg",
|
| 731 |
+
"table_caption": [
|
| 732 |
+
"Table 2: Evaluations on various settings. ",
|
| 733 |
+
"(a) AP and FPS on YouTube-VIS 2019 val set. For fairness, FPS is measured on a same machine, using a single RTX 2080Ti GPU. We used the official codes and checkpoints provided by the authors for the measurements. We report the clip settings of [10, 11]. T : window size. ",
|
| 734 |
+
"(b) Accuracy on YTVIS 2021 val set ",
|
| 735 |
+
"(d) Effect of strides "
|
| 736 |
+
],
|
| 737 |
+
"table_footnote": [],
|
| 738 |
+
"table_body": "<table><tr><td colspan=\"3\">Method (Settings)</td><td>Backbone [31]</td><td>FPS²</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td rowspan=\"9\">prij.ite</td><td colspan=\"2\">MaskTrack R-CNN[1]</td><td>ResNet-50</td><td>26.1</td><td>30.3</td><td>51.1</td><td>32.6</td><td>31.0</td><td>35.5</td></tr><tr><td colspan=\"2\">MaskTrack R-CNN[1]</td><td>ResNet-101</td><td>=</td><td>31.8</td><td>53.0</td><td>33.6</td><td>33.2</td><td>37.6</td></tr><tr><td colspan=\"2\">SipMask [2]</td><td>ResNet-50</td><td>35.5</td><td>33.7</td><td>54.1</td><td>35.8</td><td>35.4</td><td>40.1</td></tr><tr><td colspan=\"2\">SG-Net [4]</td><td>ResNet-50</td><td>1</td><td>34.8</td><td>56.1</td><td>36.8</td><td>35.8</td><td>40.8</td></tr><tr><td colspan=\"2\">SG-Net [4]</td><td>ResNet-101</td><td>1</td><td>36.3</td><td>57.1</td><td>39.6</td><td>35.9</td><td>43.0</td></tr><tr><td colspan=\"2\">Cross VIS [3]</td><td>ResNet-50</td><td>=</td><td>36.3</td><td>56.8</td><td>38.9</td><td>35.6</td><td>40.7</td></tr><tr><td colspan=\"2\">Cross VIS [3]</td><td>ResNet-101</td><td>1</td><td>36.6</td><td>57.3</td><td>39.7</td><td>36.0</td><td>42.0</td></tr><tr><td colspan=\"2\">STEm-Seg [32]</td><td>ResNet-101</td><td>3.0</td><td>34.6</td><td>55.8</td><td>37.9</td><td>34.4</td><td>41.6</td></tr><tr><td rowspan=\"7\">VisTR[11]</td><td>VisTR[11]</td><td>(T=36)</td><td>ResNet-50</td><td>51.1</td><td>35.6</td><td>56.8</td><td>37.0</td><td>35.2</td><td>40.2</td></tr><tr><td></td><td>(T=36)</td><td>ResNet-101</td><td>43.5</td><td>38.6</td><td>61.3</td><td>42.3</td><td>37.6</td><td>44.2</td></tr><tr><td>MaskProp[10]</td><td>(T=13)</td><td>ResNet-50</td><td>1</td><td>40.0</td><td>1</td><td>42.9</td><td>1</td><td>-</td></tr><tr><td>MaskProp [10]</td><td>(T=13)</td><td>ResNet-101</td><td>1</td><td>42.5</td><td>1</td><td>45.6</td><td>1</td><td>1</td></tr><tr><td>OurSnear-online</td><td>(T=5)</td><td>ResNet-50</td><td>46.5</td><td>39.0</td><td>60.4</td><td>42.7</td><td>41.7</td><td>51.6</td></tr><tr><td>OurSoffline</td><td>(T=36)</td><td>ResNet-50</td><td>107.1</td><td>41.2</td><td>65.1</td><td>44.6</td><td>42.3</td><td>49.6</td></tr><tr><td>OurSoffline</td><td>(T=36)</td><td>ResNet-101</td><td>89.4</td><td>42.6</td><td>66.6</td><td>46.3</td><td>43.5</td><td>51.4</td></tr></table>",
|
| 739 |
+
"bbox": [
|
| 740 |
+
191,
|
| 741 |
+
164,
|
| 742 |
+
805,
|
| 743 |
+
397
|
| 744 |
+
],
|
| 745 |
+
"page_idx": 6
|
| 746 |
+
},
|
| 747 |
+
{
|
| 748 |
+
"type": "table",
|
| 749 |
+
"img_path": "images/5ab3ca7c1737bd925d9ae4b548b92d8d15eff27f8f43313d185e2057fbaaf5f4.jpg",
|
| 750 |
+
"table_caption": [
|
| 751 |
+
"(c) Bipartite matching "
|
| 752 |
+
],
|
| 753 |
+
"table_footnote": [],
|
| 754 |
+
"table_body": "<table><tr><td></td><td>AP</td></tr><tr><td>Box-based</td><td>37.5</td></tr><tr><td>Mask-based</td><td>39.6</td></tr></table>",
|
| 755 |
+
"bbox": [
|
| 756 |
+
434,
|
| 757 |
+
435,
|
| 758 |
+
555,
|
| 759 |
+
483
|
| 760 |
+
],
|
| 761 |
+
"page_idx": 6
|
| 762 |
+
},
|
| 763 |
+
{
|
| 764 |
+
"type": "table",
|
| 765 |
+
"img_path": "images/02d3486867af5ac7a6130285042039e6ba267181f9277cc547809b1ae852f288.jpg",
|
| 766 |
+
"table_caption": [],
|
| 767 |
+
"table_footnote": [],
|
| 768 |
+
"table_body": "<table><tr><td></td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>MaskTrack-RCNN</td><td>28.6</td><td>48.9</td><td>29.6</td></tr><tr><td>SipMask</td><td>31.7</td><td>52.5</td><td>34.0</td></tr><tr><td>CrossVIS</td><td>34.2</td><td>54.4</td><td>37.9</td></tr><tr><td>Ours</td><td>35.2</td><td>57.2</td><td>37.5</td></tr></table>",
|
| 769 |
+
"bbox": [
|
| 770 |
+
174,
|
| 771 |
+
426,
|
| 772 |
+
416,
|
| 773 |
+
492
|
| 774 |
+
],
|
| 775 |
+
"page_idx": 6
|
| 776 |
+
},
|
| 777 |
+
{
|
| 778 |
+
"type": "table",
|
| 779 |
+
"img_path": "images/ac98e7dbd4939eaaab0aeaf8797477a7278c98fd33b3f68dcef33538aa21f86f.jpg",
|
| 780 |
+
"table_caption": [],
|
| 781 |
+
"table_footnote": [],
|
| 782 |
+
"table_body": "<table><tr><td></td><td>AP</td><td>AP75</td><td>FPS</td></tr><tr><td>T=5</td><td>S=3 S=5</td><td>38.7 42.1</td><td>72.7</td></tr><tr><td>T=10</td><td>39.5</td><td>42.8</td><td>83.0</td></tr><tr><td>T=15 S=8</td><td>39.7</td><td>43.0</td><td>92.5</td></tr><tr><td>T=20 S=10</td><td>40.4</td><td>43.3</td><td>95.7</td></tr></table>",
|
| 783 |
+
"bbox": [
|
| 784 |
+
584,
|
| 785 |
+
426,
|
| 786 |
+
813,
|
| 787 |
+
492
|
| 788 |
+
],
|
| 789 |
+
"page_idx": 6
|
| 790 |
+
},
|
| 791 |
+
{
|
| 792 |
+
"type": "text",
|
| 793 |
+
"text": "",
|
| 794 |
+
"bbox": [
|
| 795 |
+
173,
|
| 796 |
+
512,
|
| 797 |
+
825,
|
| 798 |
+
595
|
| 799 |
+
],
|
| 800 |
+
"page_idx": 6
|
| 801 |
+
},
|
| 802 |
+
{
|
| 803 |
+
"type": "text",
|
| 804 |
+
"text": "During inference, our model takes inputs as follows. Let an input video has $V$ frames, $T$ is the number of frames per clip and $S$ is the stride of clips. We start from inserting a clip of frame indices $[ 1 , T ]$ and sequentially insert clips of $[ 1 + S , T + \\bar { S } ] , [ 1 + 2 S , T + 2 S ] , \\therefore , [ 1 + n S , T + n S ]$ . It repeats until the end frame index $T + n S$ is equal to or greater than $V$ . If the end frame index of the last clip $T + n S$ is greater than $V$ , we change the frame indices of the last clip to $[ V - T + 1 , V ]$ . The resolution of input videos are downscaled to $3 6 0 \\mathrm { p }$ , which follows MaskTrack R-CNN [1]. ",
|
| 805 |
+
"bbox": [
|
| 806 |
+
174,
|
| 807 |
+
602,
|
| 808 |
+
825,
|
| 809 |
+
685
|
| 810 |
+
],
|
| 811 |
+
"page_idx": 6
|
| 812 |
+
},
|
| 813 |
+
{
|
| 814 |
+
"type": "text",
|
| 815 |
+
"text": "4.2 Main Results ",
|
| 816 |
+
"text_level": 1,
|
| 817 |
+
"bbox": [
|
| 818 |
+
174,
|
| 819 |
+
702,
|
| 820 |
+
305,
|
| 821 |
+
717
|
| 822 |
+
],
|
| 823 |
+
"page_idx": 6
|
| 824 |
+
},
|
| 825 |
+
{
|
| 826 |
+
"type": "text",
|
| 827 |
+
"text": "YouTube-VIS 2019 evaluation results We compare our proposed IFC to the state-of-the-art models in the video instance segmentation task on YouTube-VIS $2 0 1 9 \\ \\mathtt { v a l }$ in Table 2 (a). We measure the accuracy by AP and our model sets the highest score among all online, near-online, and offline models while presenting the fastest runtime. As mentioned earlier, IFC is highly efficient during the inference thanks to three advantages: (1) memory token-based decomposition for transformer encoder (2) instance-agnostic spatial decoder (3) GPU-friendly instance matching. Moreover, our model does not make use of any heavy modules such as deformable convolutions [36] or cascading networks [37]. Thanks to these advantages, IFC achieves an outstanding runtime, which is faster speed than online models [1, 2]. ",
|
| 828 |
+
"bbox": [
|
| 829 |
+
173,
|
| 830 |
+
727,
|
| 831 |
+
825,
|
| 832 |
+
853
|
| 833 |
+
],
|
| 834 |
+
"page_idx": 6
|
| 835 |
+
},
|
| 836 |
+
{
|
| 837 |
+
"type": "text",
|
| 838 |
+
"text": "During the inference, our method is able to freely adjust the length of the clip $( T )$ as needed. If the input clip length is set to contain entire video frames, our method becomes an offline method (like ",
|
| 839 |
+
"bbox": [
|
| 840 |
+
176,
|
| 841 |
+
858,
|
| 842 |
+
821,
|
| 843 |
+
887
|
| 844 |
+
],
|
| 845 |
+
"page_idx": 6
|
| 846 |
+
},
|
| 847 |
+
{
|
| 848 |
+
"type": "image",
|
| 849 |
+
"img_path": "images/670bbd204f67e3d539074c5cfb25f24385665e804acb9763ac21417ab8d1c742.jpg",
|
| 850 |
+
"image_caption": [
|
| 851 |
+
"Figure 2: Visualization of predictions from VisTR and our model. Instances with the same identity are displayed in the same color. "
|
| 852 |
+
],
|
| 853 |
+
"image_footnote": [],
|
| 854 |
+
"bbox": [
|
| 855 |
+
178,
|
| 856 |
+
88,
|
| 857 |
+
823,
|
| 858 |
+
396
|
| 859 |
+
],
|
| 860 |
+
"page_idx": 7
|
| 861 |
+
},
|
| 862 |
+
{
|
| 863 |
+
"type": "text",
|
| 864 |
+
"text": "VisTR [11]) that processes the entire video in one shot. As the offline inference can skip matching between clips and maximize the GPU utilization, our method represents surprisingly fast runtime (107.1 FPS). On the other hand, if the application requires instant outputs given a video stream, we can reduce the clip length to make our method near-online. In the near-online scenario with $T = 5$ our system is still able to process a video in real-time (46.5 FPS) with only a small delay. ",
|
| 865 |
+
"bbox": [
|
| 866 |
+
174,
|
| 867 |
+
449,
|
| 868 |
+
825,
|
| 869 |
+
517
|
| 870 |
+
],
|
| 871 |
+
"page_idx": 7
|
| 872 |
+
},
|
| 873 |
+
{
|
| 874 |
+
"type": "text",
|
| 875 |
+
"text": "YouTube-VIS 2021 evaluation results The recently introduced dataset YouTube-VIS 2021 is an improved version of YouTube-VIS 2019. The newly added videos in the dataset include higher number of instances and frames. For the new dataset, we use 32 memory tokens. In Table 2 (b), we refer the results reported in [3], which evaluated [1, 2] using official implementations. Again, our model achieves the best performance. ",
|
| 876 |
+
"bbox": [
|
| 877 |
+
174,
|
| 878 |
+
532,
|
| 879 |
+
823,
|
| 880 |
+
603
|
| 881 |
+
],
|
| 882 |
+
"page_idx": 7
|
| 883 |
+
},
|
| 884 |
+
{
|
| 885 |
+
"type": "text",
|
| 886 |
+
"text": "Qualitative result comparison We compare some qualitative results predicted by our model and VisTR [11] in Fig. 2. In terms of both tracking accuracy and segmentation quality, IFC yields better results than VisTR. ",
|
| 887 |
+
"bbox": [
|
| 888 |
+
174,
|
| 889 |
+
617,
|
| 890 |
+
823,
|
| 891 |
+
659
|
| 892 |
+
],
|
| 893 |
+
"page_idx": 7
|
| 894 |
+
},
|
| 895 |
+
{
|
| 896 |
+
"type": "text",
|
| 897 |
+
"text": "4.3 Ablation Study ",
|
| 898 |
+
"text_level": 1,
|
| 899 |
+
"bbox": [
|
| 900 |
+
174,
|
| 901 |
+
675,
|
| 902 |
+
316,
|
| 903 |
+
690
|
| 904 |
+
],
|
| 905 |
+
"page_idx": 7
|
| 906 |
+
},
|
| 907 |
+
{
|
| 908 |
+
"type": "text",
|
| 909 |
+
"text": "In this section, we provide ablation studies and discuss how different settings impact the overall performance. The experiments are conducted using YouTube-VIS 2019 val set. ",
|
| 910 |
+
"bbox": [
|
| 911 |
+
174,
|
| 912 |
+
700,
|
| 913 |
+
823,
|
| 914 |
+
728
|
| 915 |
+
],
|
| 916 |
+
"page_idx": 7
|
| 917 |
+
},
|
| 918 |
+
{
|
| 919 |
+
"type": "text",
|
| 920 |
+
"text": "Box-based and mask-based bipartite matching We observe how the different policies for bipartite matching affect the performance. As our model does is a box-free method, we adjust our model to predict bounding boxes similar to VisTR [11] and conduct bipartite matching [13, 19] using the predicted boxes. The change of optimization from mask-based to box-based brings a noticeable performance drop as shown in Table 2 (c). With the VIS-centric design, the mask-based optimization shows more robustness than box-based optimizations under typical video circumstances such as instances with heavy overlaps and partial occlusions. ",
|
| 921 |
+
"bbox": [
|
| 922 |
+
174,
|
| 923 |
+
743,
|
| 924 |
+
825,
|
| 925 |
+
840
|
| 926 |
+
],
|
| 927 |
+
"page_idx": 7
|
| 928 |
+
},
|
| 929 |
+
{
|
| 930 |
+
"type": "text",
|
| 931 |
+
"text": "Differing window strides In addition to the clip length $T$ , we further optimize our runtime placing a stride $S$ between clips, as shown in Table 2 (d). IFC can be used in a near-online manner, which takes clips that are consecutively extracted from a video. The placement of a larger stride reduces temporal intersections, which lessens computational overheads but also causes difficulty in matching instances. By enlarging the stride from $S = 1$ to $S = 3$ , IFC accomplishes approximately $150 \\%$ speed improvement with only $0 . 1 \\%$ AP drop. The tendency of high speed gain and low accuracy drop persists under various conditions. Therefore, our model can be applied to conditions where the enlargement of strides is necessary, i.e., using devices that are not powerful enough but has to maintain high inference speed. ",
|
| 932 |
+
"bbox": [
|
| 933 |
+
176,
|
| 934 |
+
856,
|
| 935 |
+
823,
|
| 936 |
+
911
|
| 937 |
+
],
|
| 938 |
+
"page_idx": 7
|
| 939 |
+
},
|
| 940 |
+
{
|
| 941 |
+
"type": "table",
|
| 942 |
+
"img_path": "images/88c13316f2d0a9844c2c8e721ac46405845d71124807c943ba03f119e9f6101d.jpg",
|
| 943 |
+
"table_caption": [
|
| 944 |
+
"Table 3: Encoder variations. We show how different encoders affect the overall performance. ",
|
| 945 |
+
"(a) Various encoders taking clips of different lengths (see Table 1) "
|
| 946 |
+
],
|
| 947 |
+
"table_footnote": [],
|
| 948 |
+
"table_body": "<table><tr><td></td><td colspan=\"3\">T=5</td><td colspan=\"3\">T=10</td><td colspan=\"3\">T=15</td><td colspan=\"3\">T=20</td></tr><tr><td></td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td><td>AP</td><td>AP75</td><td>FPS</td></tr><tr><td>No Comm</td><td>37.4</td><td>39.9</td><td>38.1</td><td>38.8</td><td>41.6</td><td>40.8</td><td>39.3</td><td>41.7</td><td>46.7</td><td>39.6</td><td>41.9</td><td>52.9</td></tr><tr><td>Full THW</td><td>37.2</td><td>40.0</td><td>37.6</td><td>38.8</td><td>41.2</td><td>35.5</td><td>39.8</td><td>42.6</td><td>32.9</td><td>39.7</td><td>42.8</td><td>34.8</td></tr><tr><td>Decomp T-HW</td><td>37.2</td><td>39.8</td><td>35.7</td><td>38.3</td><td>40.9</td><td>37.9</td><td>38.5</td><td>41.5</td><td>42.6</td><td>39.0</td><td>41.9</td><td>49.4</td></tr><tr><td>IFC</td><td>39.0</td><td>42.7</td><td>36.3</td><td>39.6</td><td>43.0</td><td>38.9</td><td>39.8</td><td>43.0</td><td>43.7</td><td>40.4</td><td>43.4</td><td>50.2</td></tr></table>",
|
| 949 |
+
"bbox": [
|
| 950 |
+
200,
|
| 951 |
+
140,
|
| 952 |
+
795,
|
| 953 |
+
223
|
| 954 |
+
],
|
| 955 |
+
"page_idx": 8
|
| 956 |
+
},
|
| 957 |
+
{
|
| 958 |
+
"type": "table",
|
| 959 |
+
"img_path": "images/debc584ae906fc54321b5664cf7d19a77a9a1664305f2a25e5ab8dde23604c8c.jpg",
|
| 960 |
+
"table_caption": [
|
| 961 |
+
"(b) Image instance segmentation on COCO val set ",
|
| 962 |
+
"(c) Number of memory tokens (AP) ",
|
| 963 |
+
"(d) Index-wise memory decomposition "
|
| 964 |
+
],
|
| 965 |
+
"table_footnote": [],
|
| 966 |
+
"table_body": "<table><tr><td></td><td>T=5</td><td>T=10</td><td>T=15</td><td>T=20</td></tr><tr><td>M=1</td><td>37.6</td><td>39.2</td><td>39.4</td><td>39.4</td></tr><tr><td>M=2</td><td>37.9</td><td>39.2</td><td>39.6</td><td>39.8</td></tr><tr><td>M=4</td><td>38.0</td><td>39.5</td><td>39.7</td><td>39.9</td></tr><tr><td>M=8</td><td>39.0</td><td>39.6</td><td>39.8</td><td>40.4</td></tr><tr><td>M=16</td><td>38.1</td><td>39.1</td><td>39.7</td><td>39.9</td></tr></table>",
|
| 967 |
+
"bbox": [
|
| 968 |
+
356,
|
| 969 |
+
252,
|
| 970 |
+
576,
|
| 971 |
+
328
|
| 972 |
+
],
|
| 973 |
+
"page_idx": 8
|
| 974 |
+
},
|
| 975 |
+
{
|
| 976 |
+
"type": "table",
|
| 977 |
+
"img_path": "images/90345da31da3a96282ed6643f7f1dd5778c6427b3195e3ddd5654470ce88a6f0.jpg",
|
| 978 |
+
"table_caption": [],
|
| 979 |
+
"table_footnote": [],
|
| 980 |
+
"table_body": "<table><tr><td></td><td>APCOcO</td><td>APOC</td></tr><tr><td>w/o mem</td><td>35.0</td><td>56.6</td></tr><tr><td>w/mem</td><td>35.1</td><td>56.5</td></tr></table>",
|
| 981 |
+
"bbox": [
|
| 982 |
+
176,
|
| 983 |
+
273,
|
| 984 |
+
346,
|
| 985 |
+
320
|
| 986 |
+
],
|
| 987 |
+
"page_idx": 8
|
| 988 |
+
},
|
| 989 |
+
{
|
| 990 |
+
"type": "table",
|
| 991 |
+
"img_path": "images/1234d522c58cf56d442e84da24a4460290fd027ec461ccc29f23126f2b2fdeb1.jpg",
|
| 992 |
+
"table_caption": [],
|
| 993 |
+
"table_footnote": [],
|
| 994 |
+
"table_body": "<table><tr><td></td><td>T=5</td><td>T=10</td><td>T=15</td><td>T=20</td></tr><tr><td>Unified</td><td>38.1</td><td>38.9</td><td>39.7</td><td>39.9</td></tr><tr><td>Decomp</td><td>39.0</td><td>39.6</td><td>39.8</td><td>40.4</td></tr></table>",
|
| 995 |
+
"bbox": [
|
| 996 |
+
583,
|
| 997 |
+
267,
|
| 998 |
+
818,
|
| 999 |
+
314
|
| 1000 |
+
],
|
| 1001 |
+
"page_idx": 8
|
| 1002 |
+
},
|
| 1003 |
+
{
|
| 1004 |
+
"type": "text",
|
| 1005 |
+
"text": "",
|
| 1006 |
+
"bbox": [
|
| 1007 |
+
174,
|
| 1008 |
+
342,
|
| 1009 |
+
825,
|
| 1010 |
+
412
|
| 1011 |
+
],
|
| 1012 |
+
"page_idx": 8
|
| 1013 |
+
},
|
| 1014 |
+
{
|
| 1015 |
+
"type": "text",
|
| 1016 |
+
"text": "Various decomposition strategies of encoders In Table 1, we observed the computational gaps derived from the decomposition of the encoder layers. Extending Table 1, we now investigate the how the decomposition strategies affect the accuracy in Table 3. ",
|
| 1017 |
+
"bbox": [
|
| 1018 |
+
174,
|
| 1019 |
+
426,
|
| 1020 |
+
820,
|
| 1021 |
+
469
|
| 1022 |
+
],
|
| 1023 |
+
"page_idx": 8
|
| 1024 |
+
},
|
| 1025 |
+
{
|
| 1026 |
+
"type": "text",
|
| 1027 |
+
"text": "The models are evaluated with variety of window sizes $( T = 5 , 1 0 , 1 5 , 2 0 )$ as an increase of window size $T$ has pros and cons. When matching predictions from different clips, greater $T$ is advantageous due to an enlargement of temporal intersections between clips. On the contrary, frames in longer clips are likely to be composed of diverse appearances, which disrupt tracking and segmenting instances within a clip. Therefore, the key to the performance enhancement is to cope with the appearance changes by precisely encoding and correlating space-time inputs. ",
|
| 1028 |
+
"bbox": [
|
| 1029 |
+
173,
|
| 1030 |
+
474,
|
| 1031 |
+
825,
|
| 1032 |
+
559
|
| 1033 |
+
],
|
| 1034 |
+
"page_idx": 8
|
| 1035 |
+
},
|
| 1036 |
+
{
|
| 1037 |
+
"type": "text",
|
| 1038 |
+
"text": "As shown in Table 3 (a), the full self-attention [11] surpasses the encoder without communications as the length of clips increase. However, the enlargement of the window size highly slows down the inference speed, and the improvements are marginal that the tremendous computation and memory usage cannot be compensated. The decomposition of space-time maintains comparable speed even if the window is large, but fails to achieve high accuracy. ",
|
| 1039 |
+
"bbox": [
|
| 1040 |
+
174,
|
| 1041 |
+
564,
|
| 1042 |
+
825,
|
| 1043 |
+
635
|
| 1044 |
+
],
|
| 1045 |
+
"page_idx": 8
|
| 1046 |
+
},
|
| 1047 |
+
{
|
| 1048 |
+
"type": "text",
|
| 1049 |
+
"text": "Our model shows fast inference as the only additional computations of IFC are from utilizing a small number of memory tokens. Furthermore, by effectively encoding the space-time inputs with the communications between frames, IFC can take advantages of enlarging the window size, and surpasses other encoders. ",
|
| 1050 |
+
"bbox": [
|
| 1051 |
+
173,
|
| 1052 |
+
640,
|
| 1053 |
+
825,
|
| 1054 |
+
695
|
| 1055 |
+
],
|
| 1056 |
+
"page_idx": 8
|
| 1057 |
+
},
|
| 1058 |
+
{
|
| 1059 |
+
"type": "text",
|
| 1060 |
+
"text": "Memory tokens We also study the effects of utilizing memory tokens. As mentioned, the motivation of using the memory tokens is to build communications between frames. Different from the video instance segmentation task, the image segmentation task is consisted of a single frame. Therefore, the use of the memory tokens does not lead to improvements to the image instance segmentation task as mutual communications cannot be solely made (see Table 3 (b)). Meanwhile, the utilization of the memory tokens achieves great improvements by effectively passing the information between frames. Results in Table 3 (a, c) demonstrate that the use of memory tokens achieves higher accuracy than the encoder without any communications (No comm), which emphasizes the importance of the communications. We evaluate how the size of the memory tokens affect the overall accuracy in Table 3 (c) and set the default size of the tokens $M$ to be 8. ",
|
| 1061 |
+
"bbox": [
|
| 1062 |
+
174,
|
| 1063 |
+
710,
|
| 1064 |
+
825,
|
| 1065 |
+
849
|
| 1066 |
+
],
|
| 1067 |
+
"page_idx": 8
|
| 1068 |
+
},
|
| 1069 |
+
{
|
| 1070 |
+
"type": "text",
|
| 1071 |
+
"text": "In Section 3.1, we demonstrated the formulation of the inputs for Gather-Communicate layer, which groups the outputs of Encode-Receive by memory indices. As aforementioned, the formulation can be considered as a decomposition of memory tokens: insertion to the Gather-Communicate layer by separate $M$ groups each consisting of $T$ tokens. In Table 3 (d), we investigate the impact of inserting the unified $M T$ tokens as a whole. Compared to the unified insertion, the decomposition brings better accuracy as the memories of same indices have more correspondences, which ease the encoders to build attentions in between. ",
|
| 1072 |
+
"bbox": [
|
| 1073 |
+
174,
|
| 1074 |
+
856,
|
| 1075 |
+
823,
|
| 1076 |
+
911
|
| 1077 |
+
],
|
| 1078 |
+
"page_idx": 8
|
| 1079 |
+
},
|
| 1080 |
+
{
|
| 1081 |
+
"type": "image",
|
| 1082 |
+
"img_path": "images/4b931f85da0d48797b1b4c756cba7dd5676ea7a6fc6286067da15b0c72e1db60.jpg",
|
| 1083 |
+
"image_caption": [
|
| 1084 |
+
"Figure 3: Visualizations of results and attention maps of memory tokens. "
|
| 1085 |
+
],
|
| 1086 |
+
"image_footnote": [],
|
| 1087 |
+
"bbox": [
|
| 1088 |
+
174,
|
| 1089 |
+
88,
|
| 1090 |
+
823,
|
| 1091 |
+
314
|
| 1092 |
+
],
|
| 1093 |
+
"page_idx": 9
|
| 1094 |
+
},
|
| 1095 |
+
{
|
| 1096 |
+
"type": "text",
|
| 1097 |
+
"text": "",
|
| 1098 |
+
"bbox": [
|
| 1099 |
+
176,
|
| 1100 |
+
353,
|
| 1101 |
+
823,
|
| 1102 |
+
395
|
| 1103 |
+
],
|
| 1104 |
+
"page_idx": 9
|
| 1105 |
+
},
|
| 1106 |
+
{
|
| 1107 |
+
"type": "text",
|
| 1108 |
+
"text": "We choose a memory index attending foreground instances and visualize the attention map in Fig. 3. As shown in the results of the upper clip, we find that the memory token has more interests to instances that are relatively difficult to detect; it more attends the heavily occluded car at the rear. The clip at the bottom is composed of frames with huge motion blurs and appearance changes. With the communications of memory tokens, IFC successfully tracks and segments the rabbit. ",
|
| 1109 |
+
"bbox": [
|
| 1110 |
+
174,
|
| 1111 |
+
401,
|
| 1112 |
+
825,
|
| 1113 |
+
470
|
| 1114 |
+
],
|
| 1115 |
+
"page_idx": 9
|
| 1116 |
+
},
|
| 1117 |
+
{
|
| 1118 |
+
"type": "text",
|
| 1119 |
+
"text": "5 Conclusion ",
|
| 1120 |
+
"text_level": 1,
|
| 1121 |
+
"bbox": [
|
| 1122 |
+
174,
|
| 1123 |
+
489,
|
| 1124 |
+
299,
|
| 1125 |
+
507
|
| 1126 |
+
],
|
| 1127 |
+
"page_idx": 9
|
| 1128 |
+
},
|
| 1129 |
+
{
|
| 1130 |
+
"type": "text",
|
| 1131 |
+
"text": "In this paper, we have proposed a novel video instance segmentation network using Inter-frame Communication Transformers (IFC), which alleviates full space-time attention and successfully builds communications between frames. Finally, our network presents a rapid inference and sets the new state-of-the-art on the YouTube-VIS dataset. For the future work, we plan to integrate temporal information, which indeed would take a step further to the human video understanding. ",
|
| 1132 |
+
"bbox": [
|
| 1133 |
+
174,
|
| 1134 |
+
522,
|
| 1135 |
+
825,
|
| 1136 |
+
590
|
| 1137 |
+
],
|
| 1138 |
+
"page_idx": 9
|
| 1139 |
+
},
|
| 1140 |
+
{
|
| 1141 |
+
"type": "text",
|
| 1142 |
+
"text": "Acknowledgments ",
|
| 1143 |
+
"text_level": 1,
|
| 1144 |
+
"bbox": [
|
| 1145 |
+
176,
|
| 1146 |
+
611,
|
| 1147 |
+
328,
|
| 1148 |
+
627
|
| 1149 |
+
],
|
| 1150 |
+
"page_idx": 9
|
| 1151 |
+
},
|
| 1152 |
+
{
|
| 1153 |
+
"type": "text",
|
| 1154 |
+
"text": "This research was grant funded by the Artificial Intelligence Graduate School Program of Yonsei University, under Grant 2020-0-01361, Korea Evaluation Institute of Industrial Technology (KEIT) funded by the Ministry of Trade, Industry and Energy (10073129), and also supported by the Advanced Robotics Laboratory, part of the Future Technology Center at LG Electronics. ",
|
| 1155 |
+
"bbox": [
|
| 1156 |
+
174,
|
| 1157 |
+
642,
|
| 1158 |
+
825,
|
| 1159 |
+
698
|
| 1160 |
+
],
|
| 1161 |
+
"page_idx": 9
|
| 1162 |
+
},
|
| 1163 |
+
{
|
| 1164 |
+
"type": "text",
|
| 1165 |
+
"text": "References ",
|
| 1166 |
+
"text_level": 1,
|
| 1167 |
+
"bbox": [
|
| 1168 |
+
174,
|
| 1169 |
+
89,
|
| 1170 |
+
267,
|
| 1171 |
+
106
|
| 1172 |
+
],
|
| 1173 |
+
"page_idx": 10
|
| 1174 |
+
},
|
| 1175 |
+
{
|
| 1176 |
+
"type": "text",
|
| 1177 |
+
"text": "[2] Cao, J., R. M. Anwer, H. Cholakkal, et al. Sipmask: Spatial information preservation for fast image and video instance segmentation. In ECCV. 2020. \n[3] Yang, S., Y. Fang, X. Wang, et al. Crossover learning for fast online video instance segmentation. In ICCV. 2021. \n[4] Liu, D., Y. Cui, W. Tan, et al. Sg-net: Spatial granularity network for one-stage video instance segmentation. In CVPR. 2021. \n[5] He, K., G. Gkioxari, P. Dollar, et al. Mask r-cnn. In ICCV. 2017. \n[6] Bolya, D., C. Zhou, F. Xiao, et al. Yolact: Real-time instance segmentation. In ICCV. 2019. \n[7] Tian, Z., C. Shen, H. Chen. Conditional convolutions for instance segmentation. In ECCV. 2020. \n[8] Chen, H., K. Sun, Z. Tian, et al. Blendmask: Top-down meets bottom-up for instance segmentation. In CVPR. 2020. \n[9] Luo, W., J. Xing, A. Milan, et al. Multiple object tracking: A literature review. Artificial Intelligence, 2020. \n[10] Bertasius, G., L. Torresani. Classifying, segmenting, and tracking object instances in video with mask propagation. In CVPR. 2020. \n[11] Wang, Y., Z. Xu, X. Wang, et al. End-to-end video instance segmentation with transformers. In CVPR. 2020. \n[12] Lin, H., R. Wu, S. Liu, et al. Video instance segmentation with a propose-reduce paradigm. In ICCV. 2021. \n[13] Carion, N., F. Massa, G. Synnaeve, et al. End-to-end object detection with transformers. In ECCV. 2020. \n[14] Vaswani, A., N. Shazeer, N. Parmar, et al. Attention is all you need. In NeurIPS. 2017. \n[15] Wang, H., Y. Zhu, B. Green, et al. Axial-deeplab: Stand-alone axial-attention for panoptic segmentation. In ECCV. 2020. \n[16] Bertasius, G., H. Wang, L. Torresani. Is space-time attention all you need for video understanding? In ICML. 2021. \n[17] Arnab, A., M. Dehghani, G. Heigold, et al. Vivit: A video vision transformer. arXiv preprint arXiv:2103.15691, 2021. \n[18] Wang, X., R. Girshick, A. Gupta, et al. Non-local neural networks. In CVPR. 2018. \n[19] Kuhn, H. W. The hungarian method for the assignment problem. In Naval research logistics quarterly. 1955. \n[20] Wang, H., Y. Zhu, H. Adam, et al. Max-deeplab: End-to-end panoptic segmentation with mask transformers. In CVPR. 2021. \n[21] Dosovitskiy, A., L. Beyer, A. Kolesnikov, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR. 2021. \n[22] Ranftl, R., A. Bochkovskiy, V. Koltun. Vision transformers for dense prediction. In ICCV. 2021. \n[23] Devlin, J., M.-W. Chang, K. Lee, et al. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL. 2019. \n[24] Touvron, H., M. Cord, M. Douze, et al. Training data-efficient image transformers & distillation through attention. In ICML. 2021. \n[25] Kirillov, A., K. He, R. Girshick, et al. Panoptic segmentation. In CVPR. 2019. \n[26] Tran, D., H. Wang, L. Torresani, et al. A closer look at spatiotemporal convolutions for action recognition. In CVPR. 2018. \n[27] Lin, T.-Y., P. Dollar, R. Girshick, et al. Feature pyramid networks for object detection. In CVPR. 2017. \n[28] Stewart, R., M. Andriluka, A. Y. Ng. End-to-end people detection in crowded scenes. In CVPR. 2016. \n[29] Milletari, F., N. Navab, S.-A. Ahmadi. V-net: Fully convolutional neural networks for volumetric medical image segmentation. In 3DV. 2016. \n[30] Lin, T.-Y., P. Goyal, R. Girshick, et al. Focal loss for dense object detection. In ICCV. 2017. \n[31] He, K., X. Zhang, S. Ren, et al. Deep residual learning for image recognition. In CVPR. 2016. \n[32] Athar, A., S. Mahadevan, A. Ošep, et al. Stem-seg: Spatio-temporal embeddings for instance segmentation in videos. In ECCV. 2020. \n[33] Wu, Y., A. Kirillov, F. Massa, et al. Detectron2. https://github.com/facebookresearch/ detectron2, 2019. \n[34] Loshchilov, I., F. Hutter. Decoupled weight decay regularization. In ICLR. 2019. \n[35] Lin, T.-Y., M. Maire, S. Belongie, et al. Microsoft coco: Common objects in context. In ECCV. 2014. \n[36] Dai, J., H. Qi, Y. Xiong, et al. Deformable convolutional networks. In ICCV. 2017. \n[37] Cai, Z., N. Vasconcelos. Cascade r-cnn: Delving into high quality object detection. In CVPR. 2018. ",
|
| 1178 |
+
"bbox": [
|
| 1179 |
+
171,
|
| 1180 |
+
136,
|
| 1181 |
+
828,
|
| 1182 |
+
910
|
| 1183 |
+
],
|
| 1184 |
+
"page_idx": 10
|
| 1185 |
+
},
|
| 1186 |
+
{
|
| 1187 |
+
"type": "text",
|
| 1188 |
+
"text": "",
|
| 1189 |
+
"bbox": [
|
| 1190 |
+
171,
|
| 1191 |
+
89,
|
| 1192 |
+
830,
|
| 1193 |
+
359
|
| 1194 |
+
],
|
| 1195 |
+
"page_idx": 11
|
| 1196 |
+
}
|
| 1197 |
+
]
|
parse/train/pvjfA4wogD6/pvjfA4wogD6_middle.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/pvjfA4wogD6/pvjfA4wogD6_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
pdf/train/--GJkm7nt0.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:0406394e01d7fd12d705bf2d9ae241f9d3e1ca42d2146c04776175192bdf4037
|
| 3 |
+
size 10544207
|
pdf/train/-_FVMKvxVCQo1.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:15c70da44dd370b55486c0a167f01a5c69007713fff294e0f92b114610102815
|
| 3 |
+
size 562682
|
pdf/train/-oUhJJILWHb.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:fcf99272625c2e1f336c0e21faa684248490e746da4e8f778dde5c4a23d9a21c
|
| 3 |
+
size 2938508
|
pdf/train/-qh0M9XWxnv.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:33b670e621b2af828b18b4f9e1a60a3a26c4abeab7ae7a064d74d5017075c020
|
| 3 |
+
size 3296975
|
pdf/train/03x6x6qNwJ3.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:084cad957adc2577bd6d2ccda75d67b7309242fe992026278ec6a8ae1db4c0ed
|
| 3 |
+
size 3027635
|
pdf/train/04cII6MumYV.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:fee66249b42bf28e77f5c67d6257595a64f9426ed5346824be3e299421e20b2d
|
| 3 |
+
size 15057207
|
pdf/train/0NXUSlb6oEu.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:12a321b96bf1f9626b307cff5084253c01f9596d42a8246aa0f618f09b129b1c
|
| 3 |
+
size 940554
|
pdf/train/0WASBV4xkhy.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:3f48617cf81fa5f6af6774557f161fee5adf23673df517b400632ee245af0b73
|
| 3 |
+
size 569550
|
pdf/train/0zXJRJecC_.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:39ec44b1c50cd4a2e3501bebdfa814d6b977d14bee1dbf9f77e4c0cc8c4cee91
|
| 3 |
+
size 1097573
|
pdf/train/1FvkSpWosOl.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:f36633ab19c852b0d73f32605a852434e5a94b820580f095853a937f75a0b851
|
| 3 |
+
size 960329
|
pdf/train/1toB0Fo9CZy.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:cc57aebd2f196ffa35c84fd367b468b1a0335bf6f8b83689463eccf5164e7917
|
| 3 |
+
size 1645915
|
pdf/train/2-0wS9-cSL.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:053630fbec30f199f185c598ed5fdd0656a7079e31ce46251a66de8e1ba23aac
|
| 3 |
+
size 1274100
|
pdf/train/2pJZSVcSZz.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:ac09fb23420b775bb5141b48ce25001f66add91738768186619a34f30aa1afaf
|
| 3 |
+
size 1460315
|
pdf/train/33TBJachvOX.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:1e8ddcae1a9f2cc9a0d020723415270d632d2874d0ab59100222dc68977ad9b1
|
| 3 |
+
size 631544
|
pdf/train/37nvvqkCo5.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:dfab00ed83ac0ff562e50fc3a2615645e7b37e3649aac132d52719f2cbd3ffcb
|
| 3 |
+
size 896229
|
pdf/train/3T9iFICe0Y9.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:dbacff8144d4bf8c42bbfcaa021acdf4726549233b1dbf730de3ea7561342781
|
| 3 |
+
size 1202734
|
pdf/train/3WbWmdTd8fN.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:0a4cf2b50f74c5e7bd02bf9fc345b59fd99e423ebf944409b8c803f6610a9014
|
| 3 |
+
size 557620
|
pdf/train/3hGNqpI4WS.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:4eb46a5961dd96a8e6ac6626c3dba56b52801af0ec518ae9cea2fd644cee3d79
|
| 3 |
+
size 4391699
|
pdf/train/3qMwV98zLIk.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:ce82db8d76331e3ffc8443036f2faef18969c3ae4bccbf767e73d4a1766b03ed
|
| 3 |
+
size 578800
|
pdf/train/4G2dEuRZ7eO.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:f62c3b2bd248c374b38ffb1533f69e6af433737a26785a813946f8ab3c8bfd2a
|
| 3 |
+
size 1692428
|
pdf/train/5-GXHFNbq_U.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:39262a26947c11e0892ac98a115142aabb39998e2702e61bb16f6e00cd7e8199
|
| 3 |
+
size 26411543
|
pdf/train/5CGPY2VeEGb.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:80aa9e2e6db6bbf46b1345570fa9b643de57b108572eec099032bc504d4aef7f
|
| 3 |
+
size 896507
|
pdf/train/5PiSFHhRe2C.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:3d839f03b81324dfbbc39ef57491558bc8dbc45defc6b92776e65c7a20fb7031
|
| 3 |
+
size 485822
|
pdf/train/5lhWG3Hj2By.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:130025b8ed457090b70e6c9612e9f9f144cdfdccb3b651dbe4a41da0fc9e15f1
|
| 3 |
+
size 586917
|
pdf/train/5nLibPckV2N.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:b6fe99eb230761a7be49424b60eb1371d5308f4c53fa606d0459cbd5c1b53c89
|
| 3 |
+
size 893080
|
pdf/train/6NFBvWlRXaG.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:4d1a45c65c111fece7979e21424f07dbc63ecb3875f60c0b03c96a28655d5795
|
| 3 |
+
size 1451217
|
pdf/train/6UdQLhqJyFD.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:b019bd4333051920819ed2881e50c45f9a0d0b6e8d6142cf5b6750dca9e52d55
|
| 3 |
+
size 601122
|
pdf/train/6tM849_6RF9.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:1f907b8039a520e0ea7844c54d2c86f661b55c0fef431f2a98f41945ac1edc0e
|
| 3 |
+
size 793625
|
pdf/train/74RmfBweB60.pdf
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:33f586d906ef6c567b858ea5e6a24e778630c2ef9f8946190d4331a6c3c80da3
|
| 3 |
+
size 594111
|