Gradient Coding using the Stochastic Block Model Zachary Charles (UW-Madison) Joint work with Dimitris Papailiopoulos (UW-Madison)
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Use a loss function `(w; z). nx Goal: min w i=1 `(w; z i ) Solution: (Stochastic) Gradient Descent.
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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 ).
<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
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?
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
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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.
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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. 0 1 0 1 2 1 0 Straggler removal 2 0 G = @ 0 3 1 A A = @ 0 1 A 1 0 1 1 1 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.
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]
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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 = 0 1 1 s s 0 s s... 0 s s 0 s s 1 s s... 0 s s B....... @... 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].
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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) B @..... 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.
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sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">aaacphicbva9swnbfnyl3/erammzgaqfcxciacoiwlgmaltixbc3etgre3vn7jsxhpfdbpwrdly2forywrujktrmyggyey99m0eshuhxfxzyi6nj4xotu/npmdm5+clc4pmjes2hyimz6yuagzbcqrufsriinbawkhae3bx0/fm70eze6hq7mdrddqves3cgvmoutsobtej9oagfmmwhqxqcxabgrxgcosbz/xy5kyqn64zuuh/hhtn90nlavg6dyjdndr1rkloltwf6n3h9uir9lbufj78z8sqehvwyy2qeg2m9zrofl5dl/crazpgnu4kapyrz2+ppl3xgv63spk1i26eq9ttfgykljemegz3sjjsdxlcc5tusbo3uu6hibehxn49aiaqy0w6ttck0cjqdsxjxwt5keztpxth2nbcleior/5ozzzlnlrzkvnhvsf/hjfkmk2snegsb7jejuizvwskdesfv5n15df6dd+fzzztn9hewyb84x983kqzr</latexit> <latexit sha1_base64="duc4gkqmqzbcpfbuci0mm5neada=">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</latexit> Stochastic block code G = 0 1 P Q... Q Q P... Q B....... @... 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