Gradient Coding using the Stochastic Block Model

Transkrypt

1 Gradient Coding using the Stochastic Block Model Zachary Charles (UW-Madison) Joint work with Dimitris Papailiopoulos (UW-Madison)

2 <latexit sha1_base64="xhw+mbjihcmsj4ng5ykmjwtedzk=">aaacaxicbvdlssnafj3uv62vqbvbzwarxjsqikaboelgzux7gcaeywtsdp1mwsxeaepd+ctuxcji1r9w5984bbpq1gmxdufcy733bcmjutn2t1fawl5zxsuvvzy2t7z3zn29lkwyguktjywrnqbjwigntuuvi51uebqhjlsdwfxebz8qiwnc79uwjv6mepxgfcoljd88uiox0m1hvlnzwzgowymjn7tjyzertmvpareju5aqkndwzs83thawe64wq1j2httvxo6eopirccxnjekrhqae6wrkuuykl08/gmnjryqwsoqurubu/t2ro1jkyrzozhipvpz3juj/xjdt0ywxu55mina8wxrldkoetukairuekzbubgfb9a0q95fawonqkjoez/7lrdi6trzbcm7pqvwrio4yoarh4aq44bzuwq1ogcba4be8g1fwzjwzl8a78tfrlrnfzd74a+pzb2h+lzm=</latexit> <latexit sha1_base64="xhw+mbjihcmsj4ng5ykmjwtedzk=">aaacaxicbvdlssnafj3uv62vqbvbzwarxjsqikaboelgzux7gcaeywtsdp1mwsxeaepd+ctuxcji1r9w5984bbpq1gmxdufcy733bcmjutn2t1fawl5zxsuvvzy2t7z3zn29lkwyguktjywrnqbjwigntuuvi51uebqhjlsdwfxebz8qiwnc79uwjv6mepxgfcoljd88uiox0m1hvlnzwzgowymjn7tjyzertmvpareju5aqkndwzs83thawe64wq1j2httvxo6eopirccxnjekrhqae6wrkuuykl08/gmnjryqwsoqurubu/t2ro1jkyrzozhipvpz3juj/xjdt0ywxu55mina8wxrldkoetukairuekzbubgfb9a0q95fawonqkjoez/7lrdi6trzbcm7pqvwrio4yoarh4aq44bzuwq1ogcba4be8g1fwzjwzl8a78tfrlrnfzd74a+pzb2h+lzm=</latexit> <latexit sha1_base64="xhw+mbjihcmsj4ng5ykmjwtedzk=">aaacaxicbvdlssnafj3uv62vqbvbzwarxjsqikaboelgzux7gcaeywtsdp1mwsxeaepd+ctuxcji1r9w5984bbpq1gmxdufcy733bcmjutn2t1fawl5zxsuvvzy2t7z3zn29lkwyguktjywrnqbjwigntuuvi51uebqhjlsdwfxebz8qiwnc79uwjv6mepxgfcoljd88uiox0m1hvlnzwzgowymjn7tjyzertmvpareju5aqkndwzs83thawe64wq1j2httvxo6eopirccxnjekrhqae6wrkuuykl08/gmnjryqwsoqurubu/t2ro1jkyrzozhipvpz3juj/xjdt0ywxu55mina8wxrldkoetukairuekzbubgfb9a0q95fawonqkjoez/7lrdi6trzbcm7pqvwrio4yoarh4aq44bzuwq1ogcba4be8g1fwzjwzl8a78tfrlrnfzd74a+pzb2h+lzm=</latexit> <latexit sha1_base64="xhw+mbjihcmsj4ng5ykmjwtedzk=">aaacaxicbvdlssnafj3uv62vqbvbzwarxjsqikaboelgzux7gcaeywtsdp1mwsxeaepd+ctuxcji1r9w5984bbpq1gmxdufcy733bcmjutn2t1fawl5zxsuvvzy2t7z3zn29lkwyguktjywrnqbjwigntuuvi51uebqhjlsdwfxebz8qiwnc79uwjv6mepxgfcoljd88uiox0m1hvlnzwzgowymjn7tjyzertmvpareju5aqkndwzs83thawe64wq1j2httvxo6eopirccxnjekrhqae6wrkuuykl08/gmnjryqwsoqurubu/t2ro1jkyrzozhipvpz3juj/xjdt0ywxu55mina8wxrldkoetukairuekzbubgfb9a0q95fawonqkjoez/7lrdi6trzbcm7pqvwrio4yoarh4aq44bzuwq1ogcba4be8g1fwzjwzl8a78tfrlrnfzd74a+pzb2h+lzm=</latexit> <latexit sha1_base64="y+oddzjudjltyyfahvx2hs769tk=">aaab8xicbvbns8naej34wetx1aoxybhqjsqikhipepfywx5gg8pmo2mxbjzhd6pu0h/hxymixv033vw3btsctpxbwoo9gwbmbqlnsrvut7w0vlk6tl7ykg5ube/slvb2gypojcu6jxkswwfryjnaumaayyurskkayzmyxk/85gnkxwjxp0cj+hhpcxyysrsr7jvieexx8une6zbkrunoys8slydlyfhrlr46vzimeqpnovgq7bmj9jminamcx8voqjahdej62dzukaivn00vhtvhrunzysxncw1p1d8tgymugkwb6yyihqh5byl+57vthv74grnjqlhq2aiw5bao7cn7do9jpjqpdcfumnortqdeeqpnseutgjf/8ijpndqe63i3z+xqvr5haq7hccrgwtlu4qzquackap7hfd4szb1y79bhrhxjymco4a+szx+afjah</latexit> <latexit sha1_base64="y+oddzjudjltyyfahvx2hs769tk=">aaab8xicbvbns8naej34wetx1aoxybhqjsqikhipepfywx5gg8pmo2mxbjzhd6pu0h/hxymixv033vw3btsctpxbwoo9gwbmbqlnsrvut7w0vlk6tl7ykg5ube/slvb2gypojcu6jxkswwfryjnaumaayyurskkayzmyxk/85gnkxwjxp0cj+hhpcxyysrsr7jvieexx8une6zbkrunoys8slydlyfhrlr46vzimeqpnovgq7bmj9jminamcx8voqjahdej62dzukaivn00vhtvhrunzysxncw1p1d8tgymugkwb6yyihqh5byl+57vthv74grnjqlhq2aiw5bao7cn7do9jpjqpdcfumnortqdeeqpnseutgjf/8ijpndqe63i3z+xqvr5haq7hccrgwtlu4qzquackap7hfd4szb1y79bhrhxjymco4a+szx+afjah</latexit> <latexit sha1_base64="y+oddzjudjltyyfahvx2hs769tk=">aaab8xicbvbns8naej34wetx1aoxybhqjsqikhipepfywx5gg8pmo2mxbjzhd6pu0h/hxymixv033vw3btsctpxbwoo9gwbmbqlnsrvut7w0vlk6tl7ykg5ube/slvb2gypojcu6jxkswwfryjnaumaayyurskkayzmyxk/85gnkxwjxp0cj+hhpcxyysrsr7jvieexx8une6zbkrunoys8slydlyfhrlr46vzimeqpnovgq7bmj9jminamcx8voqjahdej62dzukaivn00vhtvhrunzysxncw1p1d8tgymugkwb6yyihqh5byl+57vthv74grnjqlhq2aiw5bao7cn7do9jpjqpdcfumnortqdeeqpnseutgjf/8ijpndqe63i3z+xqvr5haq7hccrgwtlu4qzquackap7hfd4szb1y79bhrhxjymco4a+szx+afjah</latexit> <latexit sha1_base64="y+oddzjudjltyyfahvx2hs769tk=">aaab8xicbvbns8naej34wetx1aoxybhqjsqikhipepfywx5gg8pmo2mxbjzhd6pu0h/hxymixv033vw3btsctpxbwoo9gwbmbqlnsrvut7w0vlk6tl7ykg5ube/slvb2gypojcu6jxkswwfryjnaumaayyurskkayzmyxk/85gnkxwjxp0cj+hhpcxyysrsr7jvieexx8une6zbkrunoys8slydlyfhrlr46vzimeqpnovgq7bmj9jminamcx8voqjahdej62dzukaivn00vhtvhrunzysxncw1p1d8tgymugkwb6yyihqh5byl+57vthv74grnjqlhq2aiw5bao7cn7do9jpjqpdcfumnortqdeeqpnseutgjf/8ijpndqe63i3z+xqvr5haq7hccrgwtlu4qzquackap7hfd4szb1y79bhrhxjymco4a+szx+afjah</latexit> <latexit sha1_base64="zrt07olzhbci10996rvbrr3i61y=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirfeqo6mjlbfuajobjdnionzmemymlhm7d+ctuxcji1j9w5984bbpq1gmxdufcy733hamjsjvot7wwuls8slpyk65vbg5t2zu7drwnepm6jlkswyfshffb6ppqrlqjjiihjdtd/uxyb94tqwgsbvuwit5hxuejipe2umbdj1mrziorp1iezptchd0j6bhgyopzh4aebxbjqtgtwhni5qqectqc+8vrxdjlrgjmkfjt10m0nygpkwzkvprsrrke+6hl2oykxinys8kni3hola6mymlkadhrf09kics15khp5ej31kw3fv/z2qmozvymiitvrodpoihlumdwhavsuemwzkndejbu3apxd0metqmvaejwz1+ej43jiutu3jutuvuqj6ma9sebkamxniiquay1uacypijn8arercfrxxq3pqatc1y+swf+wpr8at/pmgq=</latexit> <latexit sha1_base64="zrt07olzhbci10996rvbrr3i61y=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirfeqo6mjlbfuajobjdnionzmemymlhm7d+ctuxcji1j9w5984bbpq1gmxdufcy733hamjsjvot7wwuls8slpyk65vbg5t2zu7drwnepm6jlkswyfshffb6ppqrlqjjiihjdtd/uxyb94tqwgsbvuwit5hxuejipe2umbdj1mrziorp1iezptchd0j6bhgyopzh4aebxbjqtgtwhni5qqectqc+8vrxdjlrgjmkfjt10m0nygpkwzkvprsrrke+6hl2oykxinys8kni3hola6mymlkadhrf09kics15khp5ej31kw3fv/z2qmozvymiitvrodpoihlumdwhavsuemwzkndejbu3apxd0metqmvaejwz1+ej43jiutu3jutuvuqj6ma9sebkamxniiquay1uacypijn8arercfrxxq3pqatc1y+swf+wpr8at/pmgq=</latexit> <latexit sha1_base64="zrt07olzhbci10996rvbrr3i61y=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirfeqo6mjlbfuajobjdnionzmemymlhm7d+ctuxcji1j9w5984bbpq1gmxdufcy733hamjsjvot7wwuls8slpyk65vbg5t2zu7drwnepm6jlkswyfshffb6ppqrlqjjiihjdtd/uxyb94tqwgsbvuwit5hxuejipe2umbdj1mrziorp1iezptchd0j6bhgyopzh4aebxbjqtgtwhni5qqectqc+8vrxdjlrgjmkfjt10m0nygpkwzkvprsrrke+6hl2oykxinys8kni3hola6mymlkadhrf09kics15khp5ej31kw3fv/z2qmozvymiitvrodpoihlumdwhavsuemwzkndejbu3apxd0metqmvaejwz1+ej43jiutu3jutuvuqj6ma9sebkamxniiquay1uacypijn8arercfrxxq3pqatc1y+swf+wpr8at/pmgq=</latexit> <latexit sha1_base64="zrt07olzhbci10996rvbrr3i61y=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirfeqo6mjlbfuajobjdnionzmemymlhm7d+ctuxcji1j9w5984bbpq1gmxdufcy733hamjsjvot7wwuls8slpyk65vbg5t2zu7drwnepm6jlkswyfshffb6ppqrlqjjiihjdtd/uxyb94tqwgsbvuwit5hxuejipe2umbdj1mrziorp1iezptchd0j6bhgyopzh4aebxbjqtgtwhni5qqectqc+8vrxdjlrgjmkfjt10m0nygpkwzkvprsrrke+6hl2oykxinys8kni3hola6mymlkadhrf09kics15khp5ej31kw3fv/z2qmozvymiitvrodpoihlumdwhavsuemwzkndejbu3apxd0metqmvaejwz1+ej43jiutu3jutuvuqj6ma9sebkamxniiquay1uacypijn8arercfrxxq3pqatc1y+swf+wpr8at/pmgq=</latexit> Machine learning in 30 seconds Training set S = {z 1,...,z n }. Use a loss function `(w; z). nx Goal: min w i=1 `(w; z i ) Solution: (Stochastic) Gradient Descent.

3 <latexit sha1_base64="wxekzj4swb7yfzdh0ck+hormefk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmekf48v7qe0owy2k3bpzhn2n0oj/qlepcji1v/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4zua3h1fphsshm0nqj+hq8pazaqx0/9r3++wkw3xnikvey0kfcjt65a/eigzphniwqbxuem5i/iwqw5naaamxakwog9mhdi2vneltz/ntp+tmkgmsxsqwngsu/p7iakt1japsz0tnsc97m/e/r5ua8mrpuexsg5itfowpicyms7/jgctkrkwsouxxeythi6oomzadkg3bw355lbquqp5b9e4uk/xrpi4inmapnimhnajdltsgcqyg8ayv8oyi58v5dz4wrqunnzmgp3a+fwaiwy2d</latexit> <latexit sha1_base64="wxekzj4swb7yfzdh0ck+hormefk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmekf48v7qe0owy2k3bpzhn2n0oj/qlepcji1v/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4zua3h1fphsshm0nqj+hq8pazaqx0/9r3++wkw3xnikvey0kfcjt65a/eigzphniwqbxuem5i/iwqw5naaamxakwog9mhdi2vneltz/ntp+tmkgmsxsqwngsu/p7iakt1japsz0tnsc97m/e/r5ua8mrpuexsg5itfowpicyms7/jgctkrkwsouxxeythi6oomzadkg3bw355lbquqp5b9e4uk/xrpi4inmapnimhnajdltsgcqyg8ayv8oyi58v5dz4wrqunnzmgp3a+fwaiwy2d</latexit> <latexit sha1_base64="wxekzj4swb7yfzdh0ck+hormefk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmekf48v7qe0owy2k3bpzhn2n0oj/qlepcji1v/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4zua3h1fphsshm0nqj+hq8pazaqx0/9r3++wkw3xnikvey0kfcjt65a/eigzphniwqbxuem5i/iwqw5naaamxakwog9mhdi2vneltz/ntp+tmkgmsxsqwngsu/p7iakt1japsz0tnsc97m/e/r5ua8mrpuexsg5itfowpicyms7/jgctkrkwsouxxeythi6oomzadkg3bw355lbquqp5b9e4uk/xrpi4inmapnimhnajdltsgcqyg8ayv8oyi58v5dz4wrqunnzmgp3a+fwaiwy2d</latexit> <latexit sha1_base64="wxekzj4swb7yfzdh0ck+hormefk=">aaab6nicbvbns8naej3ur1q/qh69lbbbu0leqmekf48v7qe0owy2k3bpzhn2n0oj/qlepcji1v/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4zua3h1fphsshm0nqj+hq8pazaqx0/9r3++wkw3xnikvey0kfcjt65a/eigzphniwqbxuem5i/iwqw5naaamxakwog9mhdi2vneltz/ntp+tmkgmsxsqwngsu/p7iakt1japsz0tnsc97m/e/r5ua8mrpuexsg5itfowpicyms7/jgctkrkwsouxxeythi6oomzadkg3bw355lbquqp5b9e4uk/xrpi4inmapnimhnajdltsgcqyg8ayv8oyi58v5dz4wrqunnzmgp3a+fwaiwy2d</latexit> <latexit sha1_base64="d8ew0nqyjnr3wqndeahyq37u9gc=">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</latexit> <latexit sha1_base64="d8ew0nqyjnr3wqndeahyq37u9gc=">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</latexit> <latexit sha1_base64="d8ew0nqyjnr3wqndeahyq37u9gc=">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</latexit> <latexit sha1_base64="d8ew0nqyjnr3wqndeahyq37u9gc=">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</latexit> B<latexit sha1_base64="izt4ekxtwea32jqpkejfjzb+fki=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpvi8cw7ae0owy2k3btzhn2n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4bua3n1bphsshm0nqj+hq8pazaqzuuo2xk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbvdqhyywqhigv3hzhp0x5935wlqwnhzmgp7a+fwblbsmxq==</latexit> <latexit sha1_base64="izt4ekxtwea32jqpkejfjzb+fki=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpvi8cw7ae0owy2k3btzhn2n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4bua3n1bphsshm0nqj+hq8pazaqzuuo2xk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbvdqhyywqhigv3hzhp0x5935wlqwnhzmgp7a+fwblbsmxq==</latexit> <latexit sha1_base64="izt4ekxtwea32jqpkejfjzb+fki=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpvi8cw7ae0owy2k3btzhn2n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4bua3n1bphsshm0nqj+hq8pazaqzuuo2xk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbvdqhyywqhigv3hzhp0x5935wlqwnhzmgp7a+fwblbsmxq==</latexit> <latexit sha1_base64="izt4ekxtwea32jqpkejfjzb+fki=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpvi8cw7ae0owy2k3btzhn2n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4bua3n1bphsshm0nqj+hq8pazaqzuuo2xk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbvdqhyywqhigv3hzhp0x5935wlqwnhzmgp7a+fwblbsmxq==</latexit> <latexit sha1_base64="mfcl8vgjvsvovriccm6xvj7iuca=">aaacchicbvdlssnafj34rpuvdencwslutuleubch6mzlbfuajobjdnionzmemylsq5du/bu3lhrx6ye482+ctllo64elh3pu5d57worrpr3n25qbx1hcwi6sfffx1jc27a3thoptiukdxyywrrapwqggdu01i61eesrdrpph/3lkn++ivdqwn3qqej+jrqarxugbkbd3pjxyikpn7vc2dz2bqoy8wlj5/uwhoievwc45fwcmoevcnjrajlpgf3mdgkecci0zuqrtoon2myq1xywmi16qsijwh3vj21cboff+nn5kca+m0ofrle0jdcfq74kmcaugpdsdhomemvzg4n9eo9xrqz9rkasacdxzfkum6hiouoedkgnwbgaiwpkawyhuiymwntkvtqju9muzphfucz2ke31cql7kcrtaltghzecce1afv6ag6gcdr/amxsgb9ws9wo/wx6r1zspndsafwj8/ol6zdw==</latexit> <latexit sha1_base64="mfcl8vgjvsvovriccm6xvj7iuca=">aaacchicbvdlssnafj34rpuvdencwslutuleubch6mzlbfuajobjdnionzmemylsq5du/bu3lhrx6ye482+ctllo64elh3pu5d57worrpr3n25qbx1hcwi6sfffx1jc27a3thoptiukdxyywrrapwqggdu01i61eesrdrpph/3lkn++ivdqwn3qqej+jrqarxugbkbd3pjxyikpn7vc2dz2bqoy8wlj5/uwhoievwc45fwcmoevcnjrajlpgf3mdgkecci0zuqrtoon2myq1xywmi16qsijwh3vj21cboff+nn5kca+m0ofrle0jdcfq74kmcaugpdsdhomemvzg4n9eo9xrqz9rkasacdxzfkum6hiouoedkgnwbgaiwpkawyhuiymwntkvtqju9muzphfucz2ke31cql7kcrtaltghzecce1afv6ag6gcdr/amxsgb9ws9wo/wx6r1zspndsafwj8/ol6zdw==</latexit> <latexit sha1_base64="mfcl8vgjvsvovriccm6xvj7iuca=">aaacchicbvdlssnafj34rpuvdencwslutuleubch6mzlbfuajobjdnionzmemylsq5du/bu3lhrx6ye482+ctllo64elh3pu5d57worrpr3n25qbx1hcwi6sfffx1jc27a3thoptiukdxyywrrapwqggdu01i61eesrdrpph/3lkn++ivdqwn3qqej+jrqarxugbkbd3pjxyikpn7vc2dz2bqoy8wlj5/uwhoievwc45fwcmoevcnjrajlpgf3mdgkecci0zuqrtoon2myq1xywmi16qsijwh3vj21cboff+nn5kca+m0ofrle0jdcfq74kmcaugpdsdhomemvzg4n9eo9xrqz9rkasacdxzfkum6hiouoedkgnwbgaiwpkawyhuiymwntkvtqju9muzphfucz2ke31cql7kcrtaltghzecce1afv6ag6gcdr/amxsgb9ws9wo/wx6r1zspndsafwj8/ol6zdw==</latexit> <latexit sha1_base64="mfcl8vgjvsvovriccm6xvj7iuca=">aaacchicbvdlssnafj34rpuvdencwslutuleubch6mzlbfuajobjdnionzmemylsq5du/bu3lhrx6ye482+ctllo64elh3pu5d57worrpr3n25qbx1hcwi6sfffx1jc27a3thoptiukdxyywrrapwqggdu01i61eesrdrpph/3lkn++ivdqwn3qqej+jrqarxugbkbd3pjxyikpn7vc2dz2bqoy8wlj5/uwhoievwc45fwcmoevcnjrajlpgf3mdgkecci0zuqrtoon2myq1xywmi16qsijwh3vj21cboff+nn5kca+m0ofrle0jdcfq74kmcaugpdsdhomemvzg4n9eo9xrqz9rkasacdxzfkum6hiouoedkgnwbgaiwpkawyhuiymwntkvtqju9muzphfucz2ke31cql7kcrtaltghzecce1afv6ag6gcdr/amxsgb9ws9wo/wx6r1zspndsafwj8/ol6zdw==</latexit> (Stochastic) gradient descent w 0 Initialize. At each iteration: Select a batch. Update: w t+1 = w t B X z2b r`(w; z) To solve ML, we just need kx i=1 r`(w; z i ).

4 <latexit sha1_base64="nlq/z4lxjo/8420ixl6vzdo4aza=">aaacb3icbvdlssnafj34rpuvdsniybhqpiqikihqceoygn1ae8nkommhzkzczespots3/oobf4q49rfc+tdo2yy09ccfwzn3cu89ycko0o7zbc3nlywulrdwiqtr6xub9tz2q8wpxksoyxblvoguyvsquqaakvyiceihi82wfznym3dekhqlgz1iim9rv9ciyqsnfnh7nkp5knfzd3jbh55aiumeyax8f/yq0mpaljkvzww4s9yclecowmb/ez0yp5wijrlsqu06ifyzjdxfjaylxqpignafdunbuie4ux42/mmid4zsgvestqknx+rviqxxpqy8nj0c6z6a9kbif1471dgpn1grpjoipfkupqzqgi5cgr0qcdzsyajckppbie4hiba20rvnco70y7okcvrxnyp7fvyqxurxfmau2adl4iituavxoabqainh8axewzv1zl1y79bhphxoymd2wb9ynz8shjjw</latexit> <latexit sha1_base64="nlq/z4lxjo/8420ixl6vzdo4aza=">aaacb3icbvdlssnafj34rpuvdsniybhqpiqikihqceoygn1ae8nkommhzkzczespots3/oobf4q49rfc+tdo2yy09ccfwzn3cu89ycko0o7zbc3nlywulrdwiqtr6xub9tz2q8wpxksoyxblvoguyvsquqaakvyiceihi82wfznym3dekhqlgz1iim9rv9ciyqsnfnh7nkp5knfzd3jbh55aiumeyax8f/yq0mpaljkvzww4s9yclecowmb/ez0yp5wijrlsqu06ifyzjdxfjaylxqpignafdunbuie4ux42/mmid4zsgvestqknx+rviqxxpqy8nj0c6z6a9kbif1471dgpn1grpjoipfkupqzqgi5cgr0qcdzsyajckppbie4hiba20rvnco70y7okcvrxnyp7fvyqxurxfmau2adl4iituavxoabqainh8axewzv1zl1y79bhphxoymd2wb9ynz8shjjw</latexit> <latexit sha1_base64="nlq/z4lxjo/8420ixl6vzdo4aza=">aaacb3icbvdlssnafj34rpuvdsniybhqpiqikihqceoygn1ae8nkommhzkzczespots3/oobf4q49rfc+tdo2yy09ccfwzn3cu89ycko0o7zbc3nlywulrdwiqtr6xub9tz2q8wpxksoyxblvoguyvsquqaakvyiceihi82wfznym3dekhqlgz1iim9rv9ciyqsnfnh7nkp5knfzd3jbh55aiumeyax8f/yq0mpaljkvzww4s9yclecowmb/ez0yp5wijrlsqu06ifyzjdxfjaylxqpignafdunbuie4ux42/mmid4zsgvestqknx+rviqxxpqy8nj0c6z6a9kbif1471dgpn1grpjoipfkupqzqgi5cgr0qcdzsyajckppbie4hiba20rvnco70y7okcvrxnyp7fvyqxurxfmau2adl4iituavxoabqainh8axewzv1zl1y79bhphxoymd2wb9ynz8shjjw</latexit> <latexit sha1_base64="nlq/z4lxjo/8420ixl6vzdo4aza=">aaacb3icbvdlssnafj34rpuvdsniybhqpiqikihqceoygn1ae8nkommhzkzczespots3/oobf4q49rfc+tdo2yy09ccfwzn3cu89ycko0o7zbc3nlywulrdwiqtr6xub9tz2q8wpxksoyxblvoguyvsquqaakvyiceihi82wfznym3dekhqlgz1iim9rv9ciyqsnfnh7nkp5knfzd3jbh55aiumeyax8f/yq0mpaljkvzww4s9yclecowmb/ez0yp5wijrlsqu06ifyzjdxfjaylxqpignafdunbuie4ux42/mmid4zsgvestqknx+rviqxxpqy8nj0c6z6a9kbif1471dgpn1grpjoipfkupqzqgi5cgr0qcdzsyajckppbie4hiba20rvnco70y7okcvrxnyp7fvyqxurxfmau2adl4iituavxoabqainh8axewzv1zl1y79bhphxoymd2wb9ynz8shjjw</latexit> Distributed learning Computing kx r`(w; z i ) is embarrassingly parallelizable. i=1 Stores the model Gradient Computations

5 Stragglers Some nodes may be much slower than others. g=g 1 +g 2 +g 3 g 1 Total time to compute g 1 g 2 g 3 g 2 g 3 Can we assign tasks in a straggler-resistant way?

6 Gradient coding With redundancy, we may not need every node to finish. g=g 1 +g 2 +g 3 +g 4 g 1 +g 2 g 1 +g 2 g 3 +g 4 g 3 +g 4 A gradient code that can tolerate one straggler. Theorem [Tandon et al., 2016]: If each worker computes gradients, we can tolerate at most s 1 stragglers. s

7 <latexit sha1_base64="l4akujsf4v/9rekbtitqh5wnfzo=">aaacthicbvdlssnafj1urbw+qi7ddbbfvule0i1qcahlcvybtsmt6u07ddijmxoxhh6ggxfu/ao3lhqrnkrbtpxcgxppfcyd40wckw3bl1zhaxmlufpak69vbg5tv3z2wyqmjyumdxkoox5rwjmapmaaqyesqakpq9sbx6x19j1ixujxpycr9aiyfmxnlggj9ss0ct0fx0/xbxy9gdkrrahrkj1mhxyeu9gpxlc8n9s5z/sfhmtgz1o/urvrdhz4ktg5qai8gv3kszsiarya0jqtpbqoheleqqrmlmo07mykiklhzahdqwujqpwsziwppjtkapuhnbaaz+rviyqesk0cz3sa+0zqvpak/9w6sfbpewktuaxb0nldfsyxdnhqlb4wcvtzisgesmzuxxrejkha+f82jjjzx14krzoay9ec29nq/tk3o4t20qe6rg46q3v0gxqoish6rk/ohx1yt9ab9wl9zvolvj6zh/5eofgn0rmojq==</latexit> <latexit sha1_base64="l4akujsf4v/9rekbtitqh5wnfzo=">aaacthicbvdlssnafj1urbw+qi7ddbbfvule0i1qcahlcvybtsmt6u07ddijmxoxhh6ggxfu/ao3lhqrnkrbtpxcgxppfcyd40wckw3bl1zhaxmlufpak69vbg5tv3z2wyqmjyumdxkoox5rwjmapmaaqyesqakpq9sbx6x19j1ixujxpycr9aiyfmxnlggj9ss0ct0fx0/xbxy9gdkrrahrkj1mhxyeu9gpxlc8n9s5z/sfhmtgz1o/urvrdhz4ktg5qai8gv3kszsiarya0jqtpbqoheleqqrmlmo07mykiklhzahdqwujqpwsziwppjtkapuhnbaaz+rviyqesk0cz3sa+0zqvpak/9w6sfbpewktuaxb0nldfsyxdnhqlb4wcvtzisgesmzuxxrejkha+f82jjjzx14krzoay9ec29nq/tk3o4t20qe6rg46q3v0gxqoish6rk/ohx1yt9ab9wl9zvolvj6zh/5eofgn0rmojq==</latexit> <latexit sha1_base64="l4akujsf4v/9rekbtitqh5wnfzo=">aaacthicbvdlssnafj1urbw+qi7ddbbfvule0i1qcahlcvybtsmt6u07ddijmxoxhh6ggxfu/ao3lhqrnkrbtpxcgxppfcyd40wckw3bl1zhaxmlufpak69vbg5tv3z2wyqmjyumdxkoox5rwjmapmaaqyesqakpq9sbx6x19j1ixujxpycr9aiyfmxnlggj9ss0ct0fx0/xbxy9gdkrrahrkj1mhxyeu9gpxlc8n9s5z/sfhmtgz1o/urvrdhz4ktg5qai8gv3kszsiarya0jqtpbqoheleqqrmlmo07mykiklhzahdqwujqpwsziwppjtkapuhnbaaz+rviyqesk0cz3sa+0zqvpak/9w6sfbpewktuaxb0nldfsyxdnhqlb4wcvtzisgesmzuxxrejkha+f82jjjzx14krzoay9ec29nq/tk3o4t20qe6rg46q3v0gxqoish6rk/ohx1yt9ab9wl9zvolvj6zh/5eofgn0rmojq==</latexit> <latexit sha1_base64="l4akujsf4v/9rekbtitqh5wnfzo=">aaacthicbvdlssnafj1urbw+qi7ddbbfvule0i1qcahlcvybtsmt6u07ddijmxoxhh6ggxfu/ao3lhqrnkrbtpxcgxppfcyd40wckw3bl1zhaxmlufpak69vbg5tv3z2wyqmjyumdxkoox5rwjmapmaaqyesqakpq9sbx6x19j1ixujxpycr9aiyfmxnlggj9ss0ct0fx0/xbxy9gdkrrahrkj1mhxyeu9gpxlc8n9s5z/sfhmtgz1o/urvrdhz4ktg5qai8gv3kszsiarya0jqtpbqoheleqqrmlmo07mykiklhzahdqwujqpwsziwppjtkapuhnbaaz+rviyqesk0cz3sa+0zqvpak/9w6sfbpewktuaxb0nldfsyxdnhqlb4wcvtzisgesmzuxxrejkha+f82jjjzx14krzoay9ec29nq/tk3o4t20qe6rg46q3v0gxqoish6rk/ohx1yt9ab9wl9zvolvj6zh/5eofgn0rmojq==</latexit> Function assignment matrix Each node is assigned a linear combination of gradients. g=g 1 +g 2 +g 3 +g 4 G = g 1 +g 2 g 1 +g 2 g 3 +g 4 g 3 +g 4 k B A compute nodes k gradients The server takes a linear combination of non-straggler columns. Recovering the sum of gradients The all ones vector is in the column span

8 Approximate gradient coding Approximations may be acceptable. Ex: Stochastic gradients. Allows us to tolerate more stragglers with less work. Theorem [Tandon et al., 2016]: If each worker computes s gradients, we can tolerate at most s 1 stragglers if we want exact recovery.

9 <latexit sha1_base64="qaemc0ynf1b1y85fsc52yy5i4yo=">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</latexit> <latexit sha1_base64="qaemc0ynf1b1y85fsc52yy5i4yo=">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</latexit> <latexit sha1_base64="qaemc0ynf1b1y85fsc52yy5i4yo=">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</latexit> <latexit sha1_base64="qaemc0ynf1b1y85fsc52yy5i4yo=">aaacmnicbvbnswmxem36wdevqkcvwaj4swxu0iugefbvfawwukvjprm1njtdkqxylv1nxvwlggc9kolvh2fai6h14jgxnzpmzattwbxxvcdnbhxicmq6nopozs0vljaxli91kikgdzairdvcqkfwcxxdjybgqodgoycrshvcz1/dgni8kreml0iq047kewfuwklvpsv9mminbt7afggdlvm0pkbx22ibb2bi4fm+69l3xwit+xmiflveb9n+rm+vk17vgwqejwrikmgytvb5ww8nlitbgiao1k3ipsbiqtkccshcp9oqutalhwhakmkmosghjxd43spthcxkqho8uh925dtwuhehttlud63/5vrif7lmzql9iocyzqxi9juoygq2ce77h9tcatoizwllittdmbumijjjxxatcetvyapkcrtkvco5360chg3tkkfvtiy2euf76bcdohqqi4bu0cn6qa/ovfpsvdnvx6vjzrbnbf0k5+mtvz6kya==</latexit> v<latexit sha1_base64="rrecnss8lo3mmiev2lvneuodblw=">aaab7nicbvbns8naej3ur1q/oh69lbbbu0le0gpfi8ck9gpaudbbtbt0swm7k0ij/rfepcji1d/jzx/jts1bwx8mpn6bywzemeph0po+ndlg5tb2tnm3srd/chjkhp+0tjjpxpsskynuhnrwkrrvokdjo6nmna4lb4fj+7nfnnbtrkkecjryikzdjslbkfqpnffciexmfbfq1bwfydrxc1kfao2++9ubjcyluuimqtfd30sxyklgwssfvxqz4sllyzrkxusvjbkj8sw5m3jhlqgjem1livmovydyghszjupbgvmcmvvvlv7ndtomboncqdrdrthyuzrjggmz/04gqnogcmojzvrywwkbuu0z2oqqngr/9ev10rqq+v7nf7yu1u+kompwbudwct7cqb0eoafnydcgz3ifnyd1xpx352pzwnkkmvp4a+fzbyuzj3e=</latexit> <latexit sha1_base64="rrecnss8lo3mmiev2lvneuodblw=">aaab7nicbvbns8naej3ur1q/oh69lbbbu0le0gpfi8ck9gpaudbbtbt0swm7k0ij/rfepcji1d/jzx/jts1bwx8mpn6bywzemeph0po+ndlg5tb2tnm3srd/chjkhp+0tjjpxpsskynuhnrwkrrvokdjo6nmna4lb4fj+7nfnnbtrkkecjryikzdjslbkfqpnffciexmfbfq1bwfydrxc1kfao2++9ubjcyluuimqtfd30sxyklgwssfvxqz4sllyzrkxusvjbkj8sw5m3jhlqgjem1livmovydyghszjupbgvmcmvvvlv7ndtomboncqdrdrthyuzrjggmz/04gqnogcmojzvrywwkbuu0z2oqqngr/9ev10rqq+v7nf7yu1u+kompwbudwct7cqb0eoafnydcgz3ifnyd1xpx352pzwnkkmvp4a+fzbyuzj3e=</latexit> <latexit sha1_base64="rrecnss8lo3mmiev2lvneuodblw=">aaab7nicbvbns8naej3ur1q/oh69lbbbu0le0gpfi8ck9gpaudbbtbt0swm7k0ij/rfepcji1d/jzx/jts1bwx8mpn6bywzemeph0po+ndlg5tb2tnm3srd/chjkhp+0tjjpxpsskynuhnrwkrrvokdjo6nmna4lb4fj+7nfnnbtrkkecjryikzdjslbkfqpnffciexmfbfq1bwfydrxc1kfao2++9ubjcyluuimqtfd30sxyklgwssfvxqz4sllyzrkxusvjbkj8sw5m3jhlqgjem1livmovydyghszjupbgvmcmvvvlv7ndtomboncqdrdrthyuzrjggmz/04gqnogcmojzvrywwkbuu0z2oqqngr/9ev10rqq+v7nf7yu1u+kompwbudwct7cqb0eoafnydcgz3ifnyd1xpx352pzwnkkmvp4a+fzbyuzj3e=</latexit> <latexit sha1_base64="xlfewbo2mml+99iejipfksusovu=">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</latexit> <latexit sha1_base64="xlfewbo2mml+99iejipfksusovu=">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</latexit> <latexit sha1_base64="xlfewbo2mml+99iejipfksusovu=">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</latexit> <latexit sha1_base64="xlfewbo2mml+99iejipfksusovu=">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</latexit> <latexit sha1_base64="en/cfenuoudgya1xzjerob3o7v4=">aaacjnicbzblswmxemezpmt9vt16crbfi2vtbl0uwrx4rgaf0c0lm862odnskmtfsvttepgrepfqefhmrzf9ino6epljpzpmzn+pbdfgdt+dldw19y3nzfz2e2d3bz93cfjxuaiy1fgkitx0qqbbjdqmnwkasqia+gia/ubmkm88gni8kvdmgem7pd3ja86osvinv0o9p8cves5hz4cel2kcuqp446iiz7drevnx/phyibyuiaey+1ptyexdgjsnvaxkdnk0j2onn/a6eutckiyjqnwlulfpp1qzzgsmsl6iiazsqhvqsihpclqdts8c4vordheqkfukwvp1d0dkq62how8r7x59vzibip/lwokjrtspl3fiqllzocar2er44hnucgxmikefyhs3u2lwp4oyy53nwhpi4snluc8wifsgd5f5cmvurwydoxn0jgi6qmv0i6qohhh6qi9ojn6cz+fvexc+zqurzrzncp0j5+sbg7wiha==</latexit> <latexit sha1_base64="en/cfenuoudgya1xzjerob3o7v4=">aaacjnicbzblswmxemezpmt9vt16crbfi2vtbl0uwrx4rgaf0c0lm862odnskmtfsvttepgrepfqefhmrzf9ino6epljpzpmzn+pbdfgdt+dldw19y3nzfz2e2d3bz93cfjxuaiy1fgkitx0qqbbjdqmnwkasqia+gia/ubmkm88gni8kvdmgem7pd3ja86osvinv0o9p8cves5hz4cel2kcuqp446iiz7drevnx/phyibyuiaey+1ptyexdgjsnvaxkdnk0j2onn/a6eutckiyjqnwlulfpp1qzzgsmsl6iiazsqhvqsihpclqdts8c4vordheqkfukwvp1d0dkq62how8r7x59vzibip/lwokjrtspl3fiqllzocar2er44hnucgxmikefyhs3u2lwp4oyy53nwhpi4snluc8wifsgd5f5cmvurwydoxn0jgi6qmv0i6qohhh6qi9ojn6cz+fvexc+zqurzrzncp0j5+sbg7wiha==</latexit> <latexit sha1_base64="en/cfenuoudgya1xzjerob3o7v4=">aaacjnicbzblswmxemezpmt9vt16crbfi2vtbl0uwrx4rgaf0c0lm862odnskmtfsvttepgrepfqefhmrzf9ino6epljpzpmzn+pbdfgdt+dldw19y3nzfz2e2d3bz93cfjxuaiy1fgkitx0qqbbjdqmnwkasqia+gia/ubmkm88gni8kvdmgem7pd3ja86osvinv0o9p8cves5hz4cel2kcuqp446iiz7drevnx/phyibyuiaey+1ptyexdgjsnvaxkdnk0j2onn/a6eutckiyjqnwlulfpp1qzzgsmsl6iiazsqhvqsihpclqdts8c4vordheqkfukwvp1d0dkq62how8r7x59vzibip/lwokjrtspl3fiqllzocar2er44hnucgxmikefyhs3u2lwp4oyy53nwhpi4snluc8wifsgd5f5cmvurwydoxn0jgi6qmv0i6qohhh6qi9ojn6cz+fvexc+zqurzrzncp0j5+sbg7wiha==</latexit> <latexit sha1_base64="en/cfenuoudgya1xzjerob3o7v4=">aaacjnicbzblswmxemezpmt9vt16crbfi2vtbl0uwrx4rgaf0c0lm862odnskmtfsvttepgrepfqefhmrzf9ino6epljpzpmzn+pbdfgdt+dldw19y3nzfz2e2d3bz93cfjxuaiy1fgkitx0qqbbjdqmnwkasqia+gia/ubmkm88gni8kvdmgem7pd3ja86osvinv0o9p8cves5hz4cel2kcuqp446iiz7drevnx/phyibyuiaey+1ptyexdgjsnvaxkdnk0j2onn/a6eutckiyjqnwlulfpp1qzzgsmsl6iiazsqhvqsihpclqdts8c4vordheqkfukwvp1d0dkq62how8r7x59vzibip/lwokjrtspl3fiqllzocar2er44hnucgxmikefyhs3u2lwp4oyy53nwhpi4snluc8wifsgd5f5cmvurwydoxn0jgi6qmv0i6qohhh6qi9ojn6cz+fvexc+zqurzrzncp0j5+sbg7wiha==</latexit> Approximate gradient coding Approximations may be acceptable. Ex: Stochastic gradients. Allows us to tolerate more stragglers with less work. Goal: Assign tasks so that stragglers incur small error Straggler removal 2 0 G A A 0 1 A Decoding: Find s.t. err(v) := 1 <latexit sha1_base64="rrecnss8lo3mmiev2lvneuodblw=">aaab7nicbvbns8naej3ur1q/oh69lbbbu0le0gpfi8ck9gpaudbbtbt0swm7k0ij/rfepcji1d/jzx/jts1bwx8mpn6bywzemeph0po+ndlg5tb2tnm3srd/chjkhp+0tjjpxpsskynuhnrwkrrvokdjo6nmna4lb4fj+7nfnnbtrkkecjryikzdjslbkfqpnffciexmfbfq1bwfydrxc1kfao2++9ubjcyluuimqtfd30sxyklgwssfvxqz4sllyzrkxusvjbkj8sw5m3jhlqgjem1livmovydyghszjupbgvmcmvvvlv7ndtomboncqdrdrthyuzrjggmz/04gqnogcmojzvrywwkbuu0z2oqqngr/9ev10rqq+v7nf7yu1u+kompwbudwct7cqb0eoafnydcgz3ifnyd1xpx352pzwnkkmvp4a+fzbyuzj3e=</latexit> is small. k kav 1 kk 2 2 Linear time decoding. Optimal decoding.

10 Prior art Coded computation: Linear functions [Lee, Lam, Pedarsani, Papailiopoulos, Kannan, 2015] MapReduce [Li, Maddah-Ali, Avestimehr, 2015] Large linear transforms [Dutta, Cadambe, Grover, 2016] Caching [Ghasemi, Ramamoorthy, 2017] Heterogeneous clusters [Reisizadeh, Prakash, Pedarsani, Avestimehr, 2017] High-dimensional matrix multiplication [Yu, Maddah-Ali, Avestimehr, 2017] Gradient codes: Fractional repetition code [Tandon, Lei, Dimakis, Karmatziakis, 2016] Cyclic repetition code [Tandon, Lei, Dimakis, Karmatziakis, 2016] MDS code [Raviv, Tamo, Tandon, Dimakis, 2017] Data encoding [Karakus, Sun, Diggavi, Yin, 2017] Batched coupon collector [Li, Kalan, Avestimehr, Soltanolkotabi, 2017] Polynomial codes [Ye, Abbe, 2018] Approximate gradient codes: Expander graphs [Raviv, Tamo, Tandon, Dimakis, 2017] Fractional repetition code [C., Papailiopoulos, 2017] Bernoulli gradient code [C., Papailiopoulos, 2017]

11 <latexit sha1_base64="b4yjobkgnyyfgx9yckhvhsnafim=">aaacdhicbvc7sgnbfj31gemramkzgiskibsiacneboymyb6qdwf2cjczmvtg5m4wlpkag3/fxkirwz/azr9x8ig08cda4zxzuxopf0uh0ba/raxlldw19cxgdnnre2c3t7df01giofr5jcpv8jggkukookajjvgbczwjda9/pfbra1baroe9dmnobawbcl9whkzq5/iuwgn6fgpkjqqp6/l0mcrss3pbcgxupf2tftgk7ji9av0kzozkyqyvdu7l7uq8csbelpnwtceoszuyhyjlggxdrepmej91owloyalqrxryzigeg6vd/uizfykdql8nuhzopqw8kwwy9vs8nxb/85oj+hetvirxghdy6si/krqjom6gdoqcjnjoconkml9s3mokctt9zu0jzvzji6r2wnlsknn3li9fzerikenyrareieektg5ihvqjj4/kmbysn+vjerhery9pdmmazryqp7a+fwc5+jov</latexit> <latexit sha1_base64="b4yjobkgnyyfgx9yckhvhsnafim=">aaacdhicbvc7sgnbfj31gemramkzgiskibsiacneboymyb6qdwf2cjczmvtg5m4wlpkag3/fxkirwz/azr9x8ig08cda4zxzuxopf0uh0ba/raxlldw19cxgdnnre2c3t7df01giofr5jcpv8jggkukookajjvgbczwjda9/pfbra1baroe9dmnobawbcl9whkzq5/iuwgn6fgpkjqqp6/l0mcrss3pbcgxupf2tftgk7ji9av0kzozkyqyvdu7l7uq8csbelpnwtceoszuyhyjlggxdrepmej91owloyalqrxryzigeg6vd/uizfykdql8nuhzopqw8kwwy9vs8nxb/85oj+hetvirxghdy6si/krqjom6gdoqcjnjoconkml9s3mokctt9zu0jzvzji6r2wnlsknn3li9fzerikenyrareieektg5ihvqjj4/kmbysn+vjerhery9pdmmazryqp7a+fwc5+jov</latexit> <latexit sha1_base64="b4yjobkgnyyfgx9yckhvhsnafim=">aaacdhicbvc7sgnbfj31gemramkzgiskibsiacneboymyb6qdwf2cjczmvtg5m4wlpkag3/fxkirwz/azr9x8ig08cda4zxzuxopf0uh0ba/raxlldw19cxgdnnre2c3t7df01giofr5jcpv8jggkukookajjvgbczwjda9/pfbra1baroe9dmnobawbcl9whkzq5/iuwgn6fgpkjqqp6/l0mcrss3pbcgxupf2tftgk7ji9av0kzozkyqyvdu7l7uq8csbelpnwtceoszuyhyjlggxdrepmej91owloyalqrxryzigeg6vd/uizfykdql8nuhzopqw8kwwy9vs8nxb/85oj+hetvirxghdy6si/krqjom6gdoqcjnjoconkml9s3mokctt9zu0jzvzji6r2wnlsknn3li9fzerikenyrareieektg5ihvqjj4/kmbysn+vjerhery9pdmmazryqp7a+fwc5+jov</latexit> <latexit sha1_base64="b4yjobkgnyyfgx9yckhvhsnafim=">aaacdhicbvc7sgnbfj31gemramkzgiskibsiacneboymyb6qdwf2cjczmvtg5m4wlpkag3/fxkirwz/azr9x8ig08cda4zxzuxopf0uh0ba/raxlldw19cxgdnnre2c3t7df01giofr5jcpv8jggkukookajjvgbczwjda9/pfbra1baroe9dmnobawbcl9whkzq5/iuwgn6fgpkjqqp6/l0mcrss3pbcgxupf2tftgk7ji9av0kzozkyqyvdu7l7uq8csbelpnwtceoszuyhyjlggxdrepmej91owloyalqrxryzigeg6vd/uizfykdql8nuhzopqw8kwwy9vs8nxb/85oj+hetvirxghdy6si/krqjom6gdoqcjnjoconkml9s3mokctt9zu0jzvzji6r2wnlsknn3li9fzerikenyrareieektg5ihvqjj4/kmbysn+vjerhery9pdmmazryqp7a+fwc5+jov</latexit> <latexit sha1_base64="5wwtfepymqwgxhnjpe5eunz5t6o=">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</latexit> <latexit sha1_base64="5wwtfepymqwgxhnjpe5eunz5t6o=">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</latexit> <latexit sha1_base64="5wwtfepymqwgxhnjpe5eunz5t6o=">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</latexit> <latexit sha1_base64="5wwtfepymqwgxhnjpe5eunz5t6o=">aaadfxichvjbaxnbfj7dwtvgqkk99jiylauh7bzbl0lbgj1wmg0he8ls7ntk6ozsmvm2ncz7j7z4v3rxobsvgjf/jzptom0azyobj+97b743b16uk2kxch55/tqd9ycbm1utr9upnzxtd3zobvyyax2rqcycr9yckhr6kfhbew6ap5gcs+ji3vw/m4kxmtmfczbdmovjlrmpodpq1pfelixk6pukvqusgrhuzz5ynpkyqowwgpwwmpqpwgorukdropjtt1mm4gztcpmx1vkifzisuitng/kvio8w//db0fpcowx0/gcio3y36av10psgbecxnheyav9kcsakfdqkxa0dhegow5iblejb1wkfhzylcz6ggyoao9nhwf9qrv84jqzjztzrsgv2dkxju2tnaeqyxx8tu6jnywxaomdkzbcuoi8qtlgxsgpfmapzfagxncbqzrzgwkjxkxutbrhat0gtn4rw8cn3welblwx64ydx3cojzhybzjc8j/skjk/jitkmj6rphpfju/k+et/8z/4x/9r/fppqe03nm3in/b+/axrl9/e=</latexit> <latexit sha1_base64="imqhhuiiat2thnlswzjnkmpt4ce=">aaab+3icbvdlssnafj34rpuv69lnybhqpiqi6eaouhfnbfuajptjdjionueymygl9ffcufdert/izr9x2mahrqcuhm65l3vvivjgtfg8b2dldw19y7o0vd7e2d3bdw8qbs0zhuklsyzvn0kamcpiy1ddsddvbpgiku40upn6nueinjxiwyxtenkucbptjiyv+m5fwysy3hgsofrazajhp3236tw9geay8qtsbqwaffcrgeiccsimzkjrnu+ljsyrmhqzmikhmsypwioukj6langiw3x2+wsewguay6lscqnn6u+jhhgtxzyynryzov70puj/xi8z8wwyu5fmhgg8xxrndbojp0haavuegza2bgff7a0qd5fc2ni4yjyef/hlzdi+q/te3b8/rzauizhk4agcgxrwwqvogfvqbc2awrn4bq/gzzk4l8678zfvxxgkmupwb87nd+njkw0=</latexit> <latexit sha1_base64="imqhhuiiat2thnlswzjnkmpt4ce=">aaab+3icbvdlssnafj34rpuv69lnybhqpiqi6eaouhfnbfuajptjdjionueymygl9ffcufdert/izr9x2mahrqcuhm65l3vvivjgtfg8b2dldw19y7o0vd7e2d3bdw8qbs0zhuklsyzvn0kamcpiy1ddsddvbpgiku40upn6nueinjxiwyxtenkucbptjiyv+m5fwysy3hgsofrazajhp3236tw9geay8qtsbqwaffcrgeiccsimzkjrnu+ljsyrmhqzmikhmsypwioukj6langiw3x2+wsewguay6lscqnn6u+jhhgtxzyynryzov70puj/xi8z8wwyu5fmhgg8xxrndbojp0haavuegza2bgff7a0qd5fc2ni4yjyef/hlzdi+q/te3b8/rzauizhk4agcgxrwwqvogfvqbc2awrn4bq/gzzk4l8678zfvxxgkmupwb87nd+njkw0=</latexit> <latexit sha1_base64="imqhhuiiat2thnlswzjnkmpt4ce=">aaab+3icbvdlssnafj34rpuv69lnybhqpiqi6eaouhfnbfuajptjdjionueymygl9ffcufdert/izr9x2mahrqcuhm65l3vvivjgtfg8b2dldw19y7o0vd7e2d3bdw8qbs0zhuklsyzvn0kamcpiy1ddsddvbpgiku40upn6nueinjxiwyxtenkucbptjiyv+m5fwysy3hgsofrazajhp3236tw9geay8qtsbqwaffcrgeiccsimzkjrnu+ljsyrmhqzmikhmsypwioukj6langiw3x2+wsewguay6lscqnn6u+jhhgtxzyynryzov70puj/xi8z8wwyu5fmhgg8xxrndbojp0haavuegza2bgff7a0qd5fc2ni4yjyef/hlzdi+q/te3b8/rzauizhk4agcgxrwwqvogfvqbc2awrn4bq/gzzk4l8678zfvxxgkmupwb87nd+njkw0=</latexit> <latexit sha1_base64="imqhhuiiat2thnlswzjnkmpt4ce=">aaab+3icbvdlssnafj34rpuv69lnybhqpiqi6eaouhfnbfuajptjdjionueymygl9ffcufdert/izr9x2mahrqcuhm65l3vvivjgtfg8b2dldw19y7o0vd7e2d3bdw8qbs0zhuklsyzvn0kamcpiy1ddsddvbpgiku40upn6nueinjxiwyxtenkucbptjiyv+m5fwysy3hgsofrazajhp3236tw9geay8qtsbqwaffcrgeiccsimzkjrnu+ljsyrmhqzmikhmsypwioukj6langiw3x2+wsewguay6lscqnn6u+jhhgtxzyynryzov70puj/xi8z8wwyu5fmhgg8xxrndbojp0haavuegza2bgff7a0qd5fc2ni4yjyef/hlzdi+q/te3b8/rzauizhk4agcgxrwwqvogfvqbc2awrn4bq/gzzk4l8678zfvxxgkmupwb87nd+njkw0=</latexit> Fractional repetition code G = s s 0 s s... 0 s s 0 s s 1 s s... 0 s s C.. A 0 s s 0 s s... 1 s s { Adversarial selection Theorem: If the stragglers are random and then w.h.p. err(v) = O(log k/k). s = (log k) Holds even for a constant fraction of stragglers. Issue: Adversarial stragglers. Permuting does not help. Adversarial straggler section is NP-Hard [C., Papailiopoulos, 2017].

12 <latexit sha1_base64="pcljcms8pmfdjwrfwhdywvtuhju=">aaac23icnvjlt9taef6blkdaaibjl6tgvhcjbfqjlkgikobipszeiq1ovr6hfeu1ttuoicxz4qcqeuwpcenv9bewersvhbmjrftt983mzmojxaqdnvfkuasfpi4ula/upn1exftsx99om6zqhfo8k5nurmyafapakfbcj9fa0kjcrxr1pnivbqcnynqvhoyqpqyvrci4q0v16n/lieroauupabbbx6gytxlqcv0fcndyhogutvmd+p3o3wwcozkvgqd2/sdbp+efxk+y9ychfb801qs3vky3njop/clokkmd9+qpqzzxigwfxdjjur6xy1gyjyjlqgpbysbn/ir1owuhyimysbzvpqjblolpkml7fnix+39eyvjjhmlkpw19l2zwg5fvad0ck/2wfcovebsfpjqukmjgr4umsddauq4tyfwlwyvll0wzjvy71owq/nmw50f7t+l7tf/nj8bhyxqcy+qr+ua2iu/2yce5i+ekrbjtcw6co+fedd1b97f7z+lqotoytflk3idnsgfhvq==</latexit> <latexit sha1_base64="pcljcms8pmfdjwrfwhdywvtuhju=">aaac23icnvjlt9taef6blkdaaibjl6tgvhcjbfqjlkgikobipszeiq1ovr6hfeu1ttuoicxz4qcqeuwpcenv9bewersvhbmjrftt983mzmojxaqdnvfkuasfpi4ula/upn1exftsx99om6zqhfo8k5nurmyafapakfbcj9fa0kjcrxr1pnivbqcnynqvhoyqpqyvrci4q0v16n/lieroauupabbbx6gytxlqcv0fcndyhogutvmd+p3o3wwcozkvgqd2/sdbp+efxk+y9ychfb801qs3vky3njop/clokkmd9+qpqzzxigwfxdjjur6xy1gyjyjlqgpbysbn/ir1owuhyimysbzvpqjblolpkml7fnix+39eyvjjhmlkpw19l2zwg5fvad0ck/2wfcovebsfpjqukmjgr4umsddauq4tyfwlwyvll0wzjvy71owq/nmw50f7t+l7tf/nj8bhyxqcy+qr+ua2iu/2yce5i+ekrbjtcw6co+fedd1b97f7z+lqotoytflk3idnsgfhvq==</latexit> <latexit sha1_base64="pcljcms8pmfdjwrfwhdywvtuhju=">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</latexit> <latexit sha1_base64="pcljcms8pmfdjwrfwhdywvtuhju=">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</latexit> <latexit sha1_base64="7kntz11+08uybl1isewtonjnw/0=">aaab7nicbvbnswmxej3ur1q/qh69bivgqe6kobeh4svjbwsl7vkyabynm82gjcuupt/ciwdfvpp7vplvtns9aoudgcd7m8zmc5xgxnrenyqtrk6tb5q3k1vbo7t71f2dr5nmmriwtuwqoyexthdjwpzbwtpkm5kegrxd+hbqt5+ynjyvd3aswjcqoeqrp8q6qa3wntzncb9a8+redhiz+awpqyfmv/rvg6q0s5i0vbbjur6nbjatbtkvbflpzyypqmmyzf1hjumycflzurn84pqbjlltslo8u39p5cqxzpyerjmhdmqwvan4n9fnbhqv5fyqzdjj54uitgcb4unvema1o1amhsfuc3crpioicbuuoyolwv98ezk8ntd9r+7fx9qan0uczticyzgfhy6haxfqhbzqioezxuenkfsc3thhvlweipld+ap0+qmst465</latexit> <latexit sha1_base64="7kntz11+08uybl1isewtonjnw/0=">aaab7nicbvbnswmxej3ur1q/qh69bivgqe6kobeh4svjbwsl7vkyabynm82gjcuupt/ciwdfvpp7vplvtns9aoudgcd7m8zmc5xgxnrenyqtrk6tb5q3k1vbo7t71f2dr5nmmriwtuwqoyexthdjwpzbwtpkm5kegrxd+hbqt5+ynjyvd3aswjcqoeqrp8q6qa3wntzncb9a8+redhiz+awpqyfmv/rvg6q0s5i0vbbjur6nbjatbtkvbflpzyypqmmyzf1hjumycflzurn84pqbjlltslo8u39p5cqxzpyerjmhdmqwvan4n9fnbhqv5fyqzdjj54uitgcb4unvema1o1amhsfuc3crpioicbuuoyolwv98ezk8ntd9r+7fx9qan0uczticyzgfhy6haxfqhbzqioezxuenkfsc3thhvlweipld+ap0+qmst465</latexit> <latexit sha1_base64="7kntz11+08uybl1isewtonjnw/0=">aaab7nicbvbnswmxej3ur1q/qh69bivgqe6kobeh4svjbwsl7vkyabynm82gjcuupt/ciwdfvpp7vplvtns9aoudgcd7m8zmc5xgxnrenyqtrk6tb5q3k1vbo7t71f2dr5nmmriwtuwqoyexthdjwpzbwtpkm5kegrxd+hbqt5+ynjyvd3aswjcqoeqrp8q6qa3wntzncb9a8+redhiz+awpqyfmv/rvg6q0s5i0vbbjur6nbjatbtkvbflpzyypqmmyzf1hjumycflzurn84pqbjlltslo8u39p5cqxzpyerjmhdmqwvan4n9fnbhqv5fyqzdjj54uitgcb4unvema1o1amhsfuc3crpioicbuuoyolwv98ezk8ntd9r+7fx9qan0uczticyzgfhy6haxfqhbzqioezxuenkfsc3thhvlweipld+ap0+qmst465</latexit> <latexit sha1_base64="7kntz11+08uybl1isewtonjnw/0=">aaab7nicbvbnswmxej3ur1q/qh69bivgqe6kobeh4svjbwsl7vkyabynm82gjcuupt/ciwdfvpp7vplvtns9aoudgcd7m8zmc5xgxnrenyqtrk6tb5q3k1vbo7t71f2dr5nmmriwtuwqoyexthdjwpzbwtpkm5kegrxd+hbqt5+ynjyvd3aswjcqoeqrp8q6qa3wntzncb9a8+redhiz+awpqyfmv/rvg6q0s5i0vbbjur6nbjatbtkvbflpzyypqmmyzf1hjumycflzurn84pqbjlltslo8u39p5cqxzpyerjmhdmqwvan4n9fnbhqv5fyqzdjj54uitgcb4unvema1o1amhsfuc3crpioicbuuoyolwv98ezk8ntd9r+7fx9qan0uczticyzgfhy6haxfqhbzqioezxuenkfsc3thhvlweipld+ap0+qmst465</latexit> <latexit sha1_base64="56hm4ubafcidedtrcp1ipapbnw0=">aaacb3icbvdjsgnbeo2jw4xb1kmgjufilnfgbl0ies/ejgawselo6dqktxowumucycjni7/ixymixv0fb/6nnewgiq8khu9vuvxpjatqanvfvmppewv1lb2e2djc2t7j7u5vdrgrdhueyldvxazbigaqkfbcpvlaffdcze1fj/3aajqwyxcpwwhapusgwhocozha2cmmwgo6xgjkjfjj0/xoyfsgl/q275zoqjubs4v2bhsrodosizou29mvzifksq8bcsm0bjh2hk2ekrrcwijtjdvejpdzfxqgbswh3uomf4zosve61auvqqdprp09ktbf66hvmk6fyu/pe2pxp68ro3frskqqxqgbny7yykkxponqaeco4cihhjcuhlmv8h5tjkojlmnccozfxitv06jjf527s1zpahzhmhyqi5indjknjxjdyqrcohkkz+svvflp1ov1bn1mw1pwbgaf/ih1+qpwe5fz</latexit> <latexit sha1_base64="56hm4ubafcidedtrcp1ipapbnw0=">aaacb3icbvdjsgnbeo2jw4xb1kmgjufilnfgbl0ies/ejgawselo6dqktxowumucycjni7/ixymixv0fb/6nnewgiq8khu9vuvxpjatqanvfvmppewv1lb2e2djc2t7j7u5vdrgrdhueyldvxazbigaqkfbcpvlaffdcze1fj/3aajqwyxcpwwhapusgwhocozha2cmmwgo6xgjkjfjj0/xoyfsgl/q275zoqjubs4v2bhsrodosizou29mvzifksq8bcsm0bjh2hk2ekrrcwijtjdvejpdzfxqgbswh3uomf4zosve61auvqqdprp09ktbf66hvmk6fyu/pe2pxp68ro3frskqqxqgbny7yykkxponqaeco4cihhjcuhlmv8h5tjkojlmnccozfxitv06jjf527s1zpahzhmhyqi5indjknjxjdyqrcohkkz+svvflp1ov1bn1mw1pwbgaf/ih1+qpwe5fz</latexit> <latexit sha1_base64="56hm4ubafcidedtrcp1ipapbnw0=">aaacb3icbvdjsgnbeo2jw4xb1kmgjufilnfgbl0ies/ejgawselo6dqktxowumucycjni7/ixymixv0fb/6nnewgiq8khu9vuvxpjatqanvfvmppewv1lb2e2djc2t7j7u5vdrgrdhueyldvxazbigaqkfbcpvlaffdcze1fj/3aajqwyxcpwwhapusgwhocozha2cmmwgo6xgjkjfjj0/xoyfsgl/q275zoqjubs4v2bhsrodosizou29mvzifksq8bcsm0bjh2hk2ekrrcwijtjdvejpdzfxqgbswh3uomf4zosve61auvqqdprp09ktbf66hvmk6fyu/pe2pxp68ro3frskqqxqgbny7yykkxponqaeco4cihhjcuhlmv8h5tjkojlmnccozfxitv06jjf527s1zpahzhmhyqi5indjknjxjdyqrcohkkz+svvflp1ov1bn1mw1pwbgaf/ih1+qpwe5fz</latexit> <latexit sha1_base64="56hm4ubafcidedtrcp1ipapbnw0=">aaacb3icbvdjsgnbeo2jw4xb1kmgjufilnfgbl0ies/ejgawselo6dqktxowumucycjni7/ixymixv0fb/6nnewgiq8khu9vuvxpjatqanvfvmppewv1lb2e2djc2t7j7u5vdrgrdhueyldvxazbigaqkfbcpvlaffdcze1fj/3aajqwyxcpwwhapusgwhocozha2cmmwgo6xgjkjfjj0/xoyfsgl/q275zoqjubs4v2bhsrodosizou29mvzifksq8bcsm0bjh2hk2ekrrcwijtjdvejpdzfxqgbswh3uomf4zosve61auvqqdprp09ktbf66hvmk6fyu/pe2pxp68ro3frskqqxqgbny7yykkxponqaeco4cihhjcuhlmv8h5tjkojlmnccozfxitv06jjf527s1zpahzhmhyqi5indjknjxjdyqrcohkkz+svvflp1ov1bn1mw1pwbgaf/ih1+qpwe5fz</latexit> Bernoulli gradient code G = 0 1 Ber(p) Ber(p)... Ber(p) Ber(p) Ber(p)... Ber(p) C. A Ber(p) Ber(p)... Ber(p) Decoding: Sum up non-stragglers and scale appropriately. Robust to random stragglers but with worse error. Theorem: If p = s/k, then w.h.p. err(v) =O(1/s). Holds even for a constant fraction of stragglers. Not as easily attacked by adversaries.

13 <latexit sha1_base64="s6vsmmkmrkkfk1k/zb7fudddrtg=">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</latexit> <latexit sha1_base64="s6vsmmkmrkkfk1k/zb7fudddrtg=">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</latexit> <latexit sha1_base64="s6vsmmkmrkkfk1k/zb7fudddrtg=">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</latexit> <latexit sha1_base64="s6vsmmkmrkkfk1k/zb7fudddrtg=">aaacb3icbvhlsgmxfm2m7/qqdefckwbrdfnmrncnufdq5rssck0pmcydgprjdelgleo3fqa7/8gnf2babx/vcxfopfezkzatxbvpe3xcmdm5+yxfpcryyuraenwjdqvtxdfos1sk6j6kggsx0dbccljpfnakfhaxdi7g+bthujqn8symm+gmtc95zbk1lupvnwssxvhqhm8xcahpzzel1cj+narwaw5zjyjkjr4hheyiyiqskmcy+abrl8bwtkanbqrk9lwsv617dw9i+c/ws1bhpqw96gujupynia0tvouo72wmw1blobmwqpbcq0bzgpahy6gkcehumdfrhpcte+e4vdalwrp2z0dbe62hswgr7x0pejo3jv/ldxitn3ullrpcggsfi+jcyjpisfg44gqyeumlkfpc3orza1wugftffsucp/3kv+d2uof7db91um9elnisom20hw6rj05re12jalurq29ozdl2dpx3d8vddffnqeuupzvol7lhh6y8txw=</latexit> <latexit sha1_base64="gzvjlqrtvsusht4gwwr2dkl5px0=">aaab7xicbvdlsgnbeoynrxhfuy85obget8uul3gril48rjapsjywo5lnxszordozqlgcfoixd4p49x/05n84erw0sachqoqmuytmonpg876d3mrq2vpgfrowtb2zu1fcp2homspc60ryqvoh1pqzqeuggu5biai4djlthsorid98oeozkw7nkkfbjpucryxgy6vggi7qvdstlj3xmwite39oytxs0dcjans6xc9ot5i0psiqjrvu+15iggwrwwin40in1ttbzij7tg2pwdhvqta9doxornjdkvs2heft9fdehmotr3foo2nsbnrrm4j/ee3urodbxkssgiribfgucmqkmryoekxryvjiekwus7cimsake2mdktgq/mwxl0njzpu917+xavzcdhkowtgcgg8vqmi11kaobo7gcv7g1zhos/pmvm9ac8585hd+wpn4abhbkao=</latexit> <latexit sha1_base64="v49lkebjzmwtwaooejegnqsij90=">aaab7xicbvdlsgnbeoynrxhfuy9bhaycp2xxi14ces8ei5ghjeuyncwmy2zn1plziszxg7x4umsr/6mn/8bj46drgoaiqpvurjdhtbvp+3jys8srq2v59clg5tb2tnf3r6flqgite8mlaovyu84errtmog0liui45lqzdi8nfvoeks2kudgjhayx7gswmyknlrojqqa7t1sse643bfpl/dkpv0uhn/khcldrfj86punsmapdona67xujctksdcocjgudvnmekyhu07alasdub9n02je6tkoprvlzegzn1z8tgy61hswh7yyxgehfbyl+57vte50hgrnjaqggs0vrypgrapi66jffieejszbrzn6kyaartiwnqgbd8bdf/ksap67vuf61temczshdcy7gbhw4gypcqq3qqoawhuezxhzppdmvztusnefmz/bhf5z3b7vpkne=</latexit> <latexit sha1_base64="v49lkebjzmwtwaooejegnqsij90=">aaab7xicbvdlsgnbeoynrxhfuy9bhaycp2xxi14ces8ei5ghjeuyncwmy2zn1plziszxg7x4umsr/6mn/8bj46drgoaiqpvurjdhtbvp+3jys8srq2v59clg5tb2tnf3r6flqgite8mlaovyu84errtmog0liui45lqzdi8nfvoeks2kudgjhayx7gswmyknlrojqqa7t1sse643bfpl/dkpv0uhn/khcldrfj86punsmapdona67xujctksdcocjgudvnmekyhu07alasdub9n02je6tkoprvlzegzn1z8tgy61hswh7yyxgehfbyl+57vte50hgrnjaqggs0vrypgrapi66jffieejszbrzn6kyaartiwnqgbd8bdf/ksap67vuf61temczshdcy7gbhw4gypcqq3qqoawhuezxhzppdmvztusnefmz/bhf5z3b7vpkne=</latexit> <latexit sha1_base64="ffi2d5ohvb+xupe8bxhyxk4zzgw=">aaab7xicbvbnswmxej31s9avqkcvwsj4wrje9cjuvhisyd+gxuo2zbax2wrnskjz+h+8efdeq//hm//gtn2dtj4yelw3w8y8kbxcwiy/vzxvtfwnzdjwextnd2+/cndyncrtldwoekq3i2ky4ji1llectvpnsbij1opgn1o/9cs04ure23hkwoqmji85jdzjzrrdoue/v6lih8+alklqkcouqpcqx92+olncpkwcgnmjcgrdngjlqwctcjczlcv0raas46gkctnhprt2gk6d0kex0q6krtp190roempgseq6e2khztgbiv95nczgl2hozzpzjul8uzwjzbwavo76xdnqxdgrqjv3tyi6jjpq6wiquxccxzexsfpcd7af3ofq7bqiowthcajnemaf1oaw6taacg/wdk/w5invxxv3puatk14xcwr/4h3+aepjjj0=</latexit> <latexit sha1_base64="vbj2g3fgp2p7qcbz8+uyg612ani=">aaab9hicbvdlsgmxfl2pr1pfvzdugkvwicomclorkm5cvrapaiesstntacyzttkfmvq73lhqxk0f486/mw1noa0hlvdwzr3k5gsj4nq47jcqrk1vbg4vt0s7u3v7b+xdo4aou0vzncyivq2aaca4zhxdjwctrdesbyi1g+h9zg+omdi8lk9mkja/in3jq06jszkf4fvsxecrba7tlvdcx50drxivjxxiueuwvzq9mkyrk4ykonxbcxpjz0qztgwbljqpzgmhq9jnbuslizj2s/nru3xmlr4oy2vlgjxxf29kjnj6egv2mijmoje9mfif105neonnxcapyziuhgptgu2mzwnghlemgjgxhfdf7a2ydogi1nicsjyeb/nlq6rx6xiu4z1evap3erxfoiftoacprqekd1cdolaywto8whsaoxf0jj4wowwu7xzdh6dphxjsj7e=</latexit> <latexit sha1_base64="vbj2g3fgp2p7qcbz8+uyg612ani=">aaab9hicbvdlsgmxfl2pr1pfvzdugkvwicomclorkm5cvrapaiesstntacyzttkfmvq73lhqxk0f486/mw1noa0hlvdwzr3k5gsj4nq47jcqrk1vbg4vt0s7u3v7b+xdo4aou0vzncyivq2aaca4zhxdjwctrdesbyi1g+h9zg+omdi8lk9mkja/in3jq06jszkf4fvsxecrba7tlvdcx50drxivjxxiueuwvzq9mkyrk4ykonxbcxpjz0qztgwbljqpzgmhq9jnbuslizj2s/nru3xmlr4oy2vlgjxxf29kjnj6egv2mijmoje9mfif105neonnxcapyziuhgptgu2mzwnghlemgjgxhfdf7a2ydogi1nicsjyeb/nlq6rx6xiu4z1evap3erxfoiftoacprqekd1cdolaywto8whsaoxf0jj4wowwu7xzdh6dphxjsj7e=</latexit> <latexit sha1_base64="vbj2g3fgp2p7qcbz8+uyg612ani=">aaab9hicbvdlsgmxfl2pr1pfvzdugkvwicomclorkm5cvrapaiesstntacyzttkfmvq73lhqxk0f486/mw1noa0hlvdwzr3k5gsj4nq47jcqrk1vbg4vt0s7u3v7b+xdo4aou0vzncyivq2aaca4zhxdjwctrdesbyi1g+h9zg+omdi8lk9mkja/in3jq06jszkf4fvsxecrba7tlvdcx50drxivjxxiueuwvzq9mkyrk4ykonxbcxpjz0qztgwbljqpzgmhq9jnbuslizj2s/nru3xmlr4oy2vlgjxxf29kjnj6egv2mijmoje9mfif105neonnxcapyziuhgptgu2mzwnghlemgjgxhfdf7a2ydogi1nicsjyeb/nlq6rx6xiu4z1evap3erxfoiftoacprqekd1cdolaywto8whsaoxf0jj4wowwu7xzdh6dphxjsj7e=</latexit> <latexit sha1_base64="vbj2g3fgp2p7qcbz8+uyg612ani=">aaab9hicbvdlsgmxfl2pr1pfvzdugkvwicomclorkm5cvrapaiesstntacyzttkfmvq73lhqxk0f486/mw1noa0hlvdwzr3k5gsj4nq47jcqrk1vbg4vt0s7u3v7b+xdo4aou0vzncyivq2aaca4zhxdjwctrdesbyi1g+h9zg+omdi8lk9mkja/in3jq06jszkf4fvsxecrba7tlvdcx50drxivjxxiueuwvzq9mkyrk4ykonxbcxpjz0qztgwbljqpzgmhq9jnbuslizj2s/nru3xmlr4oy2vlgjxxf29kjnj6egv2mijmoje9mfif105neonnxcapyziuhgptgu2mzwnghlemgjgxhfdf7a2ydogi1nicsjyeb/nlq6rx6xiu4z1evap3erxfoiftoacprqekd1cdolaywto8whsaoxf0jj4wowwu7xzdh6dphxjsj7e=</latexit> <latexit sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">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</latexit> <latexit sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">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</latexit> <latexit sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">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</latexit> <latexit sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">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</latexit> Stochastic block code G = 0 1 P Q... Q Q P... Q C.. A Q Q... P P, Q 2 R s s P i,j =Ber(p) Q i,j =Ber(q) Fractional repetition code: p =1,q =0. Bernoulli gradient code: p = q. Function assignment matrix Stochastic block model