Closing the convergence gap of SGD without replacement
Shashank Rajput, Anant Gupta, Dimitris Papailiopoulos
Closing the convergence gap of SGD without replacement Shashank - - PowerPoint PPT Presentation
Closing the convergence gap of SGD without replacement Shashank Rajput, Anant Gupta, Dimitris Papailiopoulos <latexit
Shashank Rajput, Anant Gupta, Dimitris Papailiopoulos
1. 2. 3. Repeat for T iterations
Sampling with replacement
xt+1 := xt αrfi(xt), α is the step size
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n
i=1 fi(x)
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1. Sample fi uniformly at random from I 2. Remove fi from I 3.
Epoch
With v/s Without replacement [2]
xt+1 = xt αrfi(xt)
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sha1_base64="wh7qNt7IHaW/WEg1mkEojUZq/MQ=">ACIHicbZDLSgMxFIYzXmu9V26CbZCizrMDELdFApuXLioaC/Q1pJM21o5kJyRlpKH8WNr+LGhSK606cxvSy09UDg5/vPSXJ+NxJcgWV9GUvLK6tr64mN5ObW9s5uam+/osJYUlamoQhlzSWKCR6wMnAQrBZJRnxXsKrbuxz71QcmFQ+DOxhErOmTsA9Tglo1ErlrxngjNeys/1cIds/sXP3TuZ0TJwpOZsQE98CkYCJbu4XrIzZSqUt05oUXhT2TKTRrEqt1GejHdLYZwFQZSq21YEzaG+lFPBRslGrFhEaI90WF3LgPhMNYeTBUf4WJM29kKpTwB4Qn9PDImv1MB3dadPoKvmvTH8z6vH4F0hzyIYmABnT7kxQJDiMdp4TaXjIYaEGo5PqvmHaJBR0pkdgj2/8qKoOKZtmfaNky6ez+JIoEN0hLIRnlURFeohMqIokf0jF7Rm/FkvBjvxse0dcmYzRygP2V8/wC1KJ0T</latexit><latexit 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sha1_base64="wh7qNt7IHaW/WEg1mkEojUZq/MQ=">ACIHicbZDLSgMxFIYzXmu9V26CbZCizrMDELdFApuXLioaC/Q1pJM21o5kJyRlpKH8WNr+LGhSK606cxvSy09UDg5/vPSXJ+NxJcgWV9GUvLK6tr64mN5ObW9s5uam+/osJYUlamoQhlzSWKCR6wMnAQrBZJRnxXsKrbuxz71QcmFQ+DOxhErOmTsA9Tglo1ErlrxngjNeys/1cIds/sXP3TuZ0TJwpOZsQE98CkYCJbu4XrIzZSqUt05oUXhT2TKTRrEqt1GejHdLYZwFQZSq21YEzaG+lFPBRslGrFhEaI90WF3LgPhMNYeTBUf4WJM29kKpTwB4Qn9PDImv1MB3dadPoKvmvTH8z6vH4F0hzyIYmABnT7kxQJDiMdp4TaXjIYaEGo5PqvmHaJBR0pkdgj2/8qKoOKZtmfaNky6ez+JIoEN0hLIRnlURFeohMqIokf0jF7Rm/FkvBjvxse0dcmYzRygP2V8/wC1KJ0T</latexit><latexit sha1_base64="wh7qNt7IHaW/WEg1mkEojUZq/MQ=">ACIHicbZDLSgMxFIYzXmu9V26CbZCizrMDELdFApuXLioaC/Q1pJM21o5kJyRlpKH8WNr+LGhSK606cxvSy09UDg5/vPSXJ+NxJcgWV9GUvLK6tr64mN5ObW9s5uam+/osJYUlamoQhlzSWKCR6wMnAQrBZJRnxXsKrbuxz71QcmFQ+DOxhErOmTsA9Tglo1ErlrxngjNeys/1cIds/sXP3TuZ0TJwpOZsQE98CkYCJbu4XrIzZSqUt05oUXhT2TKTRrEqt1GejHdLYZwFQZSq21YEzaG+lFPBRslGrFhEaI90WF3LgPhMNYeTBUf4WJM29kKpTwB4Qn9PDImv1MB3dadPoKvmvTH8z6vH4F0hzyIYmABnT7kxQJDiMdp4TaXjIYaEGo5PqvmHaJBR0pkdgj2/8qKoOKZtmfaNky6ez+JIoEN0hLIRnlURFeohMqIokf0jF7Rm/FkvBjvxse0dcmYzRygP2V8/wC1KJ0T</latexit>SGD: Easy because E[rfi(xt)] = rF(xt)
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[3]: Jeffery Z HaoChen and Suvrit Sra. Random shuffling beats sgd after finite epochs. 2018 [4]: Prateek Jain, Dheeraj Nagaraj, and Praneeth Netrapalli. Sgd without replacement: Sharper rates for general smooth convex functions. 2019 [5]: Itay Safran and Ohad Shamir. How good is sgd with random shuffling?
n = # functions K = # epochs T = nK SGD error = O (1/T )
Upper bound [3,4] 𝑃
! "! + #" "" , 𝑃 # "!
Lower bound [5] Ω
! "! + #! ""
𝑃
! "! + #! ""
F is strongly convex quadratic
O ⇣
1 T 2 + n3 T 3
⌘ , O n
T 2
Ω ⇣
1 T 2 + n2 T 3
⌘
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1 T 2 + n2 T 3
⌘
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Our upper bound
[4]: Prateek Jain, Dheeraj Nagaraj, and Praneeth Netrapalli. Sgd without replacement: Sharper rates for general smooth convex functions. 2019 [5]: Itay Safran and Ohad Shamir. How good is sgd with random shuffling?
n = # functions K = # epochs T = nK SGD error = O (1/T )
Upper bound [4] 𝑃
# "!
Lower bound [5] Ω
! "! + #! ""
Ω
# "!
F is strongly convex smooth function
O n
T 2
Ω ⇣
1 T 2 + n2 T 3
⌘
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T 2
Our lower bound
Surprisingly, lower bound is different for non-quadratics!
#
$(#) 𝑦# ≈ % #
$(#) 𝑦" = 𝛽𝑜∇𝐺 𝑦"
n steps of gradient descent!
[3]: Jeffery Z HaoChen and Suvrit Sra. Random shuffling beats sgd after finite epochs. 2018 [4]: Prateek Jain, Dheeraj Nagaraj, and Praneeth Netrapalli. Sgd without replacement: Sharper rates for general smooth convex functions. 2019
kxi x1k2 grows as iα2 instead of i2α2
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[4]: Prateek Jain, Dheeraj Nagaraj, and Praneeth Netrapalli. Sgd without replacement: Sharper rates for general smooth convex functions. 2019
x1
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<latexit sha1_base64="rTahWIFj4vpux9DzMKcFtQUmw=">AB8XicbZC7SgNBFIbPxluMt6ilzWAQrMKOjZYBG8sI5oLJEs5OZpMhs7PLzKwQltS+gI2FEmx9Gzvfxsml0MQfBj7+/xzmnBOmUhjr+9eYWNza3unuFva2z84PCofnzRNkmnGyRiW6HaLgUijesJK3U80xDiVvhaPbWd564tqIRD3YcqDGAdKRIKhdZjV2EokUQ92itX/Ko/F1kHuoRKbfe5PwWAeq/81e0nLIu5skyiMR3qpzbIUVvBJ+UupnhKbIRDnjHocKYmyCfTzwhF87pkyjR7ilL5u7vjhxjY8Zx6CpjtEOzms3M/7JOZqObIBcqzSxXbPFRlEliEzJbn/SF5szKsQNkWrhZCRuiRmbdkUruCHR15XVoXlWpX6X3tFLzYaEinME5XAKFa6jBHdShAQwUvMAbvHvGe/Wm3seitOAte07hj7zPHzTJkg0=</latexit><latexit sha1_base64="4+p9tVQPevYSvBbGHQTtGyWSgds=">AB8XicbZC7SgNBFIbPxlsSb1FLm8EgWEjYtdEyYGMZwVwCeHs7GwyZHZ2mZkVwpLGxhewsVDEVnwZO59GJ5dCE38Y+Pj/c5hzjp8Iro3rfjm5ldW19Y18obi5tb2zW9rb+g4VZTVaSxi1fJRM8ElqxtuBGslimHkC9b0h5eTvHnHlOaxvDGjhHUj7EsecorGWrcdib5AEva8XqnsVtypyDJ4cyhXCw/Bx/f9a1X+uwEMU0jJg0VqHXbcxPTzVAZTgUbFzupZgnSIfZ26LEiOluNp14TI6tE5AwVvZJQ6bu74MI61HkW8rIzQDvZhNzP+ydmrCi27GZIaJunsozAVxMRksj4JuGLUiJEFpIrbWQkdoEJq7JGK9gje4srL0DireG7Fu/bKVRdmysMhHMEJeHAOVbiCGtSBgoRHeIYXRztPzqvzNivNOfOeA/gj5/0HhFCTxw=</latexit><latexit sha1_base64="4+p9tVQPevYSvBbGHQTtGyWSgds=">AB8XicbZC7SgNBFIbPxlsSb1FLm8EgWEjYtdEyYGMZwVwCeHs7GwyZHZ2mZkVwpLGxhewsVDEVnwZO59GJ5dCE38Y+Pj/c5hzjp8Iro3rfjm5ldW19Y18obi5tb2zW9rb+g4VZTVaSxi1fJRM8ElqxtuBGslimHkC9b0h5eTvHnHlOaxvDGjhHUj7EsecorGWrcdib5AEva8XqnsVtypyDJ4cyhXCw/Bx/f9a1X+uwEMU0jJg0VqHXbcxPTzVAZTgUbFzupZgnSIfZ26LEiOluNp14TI6tE5AwVvZJQ6bu74MI61HkW8rIzQDvZhNzP+ydmrCi27GZIaJunsozAVxMRksj4JuGLUiJEFpIrbWQkdoEJq7JGK9gje4srL0DireG7Fu/bKVRdmysMhHMEJeHAOVbiCGtSBgoRHeIYXRztPzqvzNivNOfOeA/gj5/0HhFCTxw=</latexit><latexit sha1_base64="8eg7QxhxKCaLa/mqL4PcEPoWhA=">AB8XicbVA9SwNBEJ2LXzF+RS1tFoNgFW5tAzYWEYwH5gcYW6zlyzZ2zt294Rw5F/YWChi67+x89+4Sa7QxAcDj/dmJkXplIY6/vfXmljc2t7p7xb2ds/ODyqHp+0TZJpxlskYnuhmi4FIq3rLCSd1PNMQ4l74ST27nfeLaiEQ92GnKgxhHSkSCoXSY19hKJFEAzqo1vy6vwBZJ7QgNSjQHFS/+sOEZTFXlk0pkf91AY5aiuY5LNKPzM8RTbBEe85qjDmJsgXF8/IhVOGJEq0K2XJQv09kWNszDQOXWeMdmxWvbn4n9fLbHQT5EKlmeWKLRdFmSQ2IfP3yVBozqycOoJMC3crYWPUyKwLqeJCoKsvr5P2VZ36dXpPaw2/iKMZ3AOl0DhGhpwB01oAQMFz/AKb57xXrx372PZWvKmVP4A+/zB70vkDk=</latexit>rf1
<latexit sha1_base64="rTahWIFj4vpux9DzMKcFtQUmw=">AB8XicbZC7SgNBFIbPxluMt6ilzWAQrMKOjZYBG8sI5oLJEs5OZpMhs7PLzKwQltS+gI2FEmx9Gzvfxsml0MQfBj7+/xzmnBOmUhjr+9eYWNza3unuFva2z84PCofnzRNkmnGyRiW6HaLgUijesJK3U80xDiVvhaPbWd564tqIRD3YcqDGAdKRIKhdZjV2EokUQ92itX/Ko/F1kHuoRKbfe5PwWAeq/81e0nLIu5skyiMR3qpzbIUVvBJ+UupnhKbIRDnjHocKYmyCfTzwhF87pkyjR7ilL5u7vjhxjY8Zx6CpjtEOzms3M/7JOZqObIBcqzSxXbPFRlEliEzJbn/SF5szKsQNkWrhZCRuiRmbdkUruCHR15XVoXlWpX6X3tFLzYaEinME5XAKFa6jBHdShAQwUvMAbvHvGe/Wm3seitOAte07hj7zPHzTJkg0=</latexit><latexit sha1_base64="4+p9tVQPevYSvBbGHQTtGyWSgds=">AB8XicbZC7SgNBFIbPxlsSb1FLm8EgWEjYtdEyYGMZwVwCeHs7GwyZHZ2mZkVwpLGxhewsVDEVnwZO59GJ5dCE38Y+Pj/c5hzjp8Iro3rfjm5ldW19Y18obi5tb2zW9rb+g4VZTVaSxi1fJRM8ElqxtuBGslimHkC9b0h5eTvHnHlOaxvDGjhHUj7EsecorGWrcdib5AEva8XqnsVtypyDJ4cyhXCw/Bx/f9a1X+uwEMU0jJg0VqHXbcxPTzVAZTgUbFzupZgnSIfZ26LEiOluNp14TI6tE5AwVvZJQ6bu74MI61HkW8rIzQDvZhNzP+ydmrCi27GZIaJunsozAVxMRksj4JuGLUiJEFpIrbWQkdoEJq7JGK9gje4srL0DireG7Fu/bKVRdmysMhHMEJeHAOVbiCGtSBgoRHeIYXRztPzqvzNivNOfOeA/gj5/0HhFCTxw=</latexit><latexit sha1_base64="4+p9tVQPevYSvBbGHQTtGyWSgds=">AB8XicbZC7SgNBFIbPxlsSb1FLm8EgWEjYtdEyYGMZwVwCeHs7GwyZHZ2mZkVwpLGxhewsVDEVnwZO59GJ5dCE38Y+Pj/c5hzjp8Iro3rfjm5ldW19Y18obi5tb2zW9rb+g4VZTVaSxi1fJRM8ElqxtuBGslimHkC9b0h5eTvHnHlOaxvDGjhHUj7EsecorGWrcdib5AEva8XqnsVtypyDJ4cyhXCw/Bx/f9a1X+uwEMU0jJg0VqHXbcxPTzVAZTgUbFzupZgnSIfZ26LEiOluNp14TI6tE5AwVvZJQ6bu74MI61HkW8rIzQDvZhNzP+ydmrCi27GZIaJunsozAVxMRksj4JuGLUiJEFpIrbWQkdoEJq7JGK9gje4srL0DireG7Fu/bKVRdmysMhHMEJeHAOVbiCGtSBgoRHeIYXRztPzqvzNivNOfOeA/gj5/0HhFCTxw=</latexit><latexit sha1_base64="8eg7QxhxKCaLa/mqL4PcEPoWhA=">AB8XicbVA9SwNBEJ2LXzF+RS1tFoNgFW5tAzYWEYwH5gcYW6zlyzZ2zt294Rw5F/YWChi67+x89+4Sa7QxAcDj/dmJkXplIY6/vfXmljc2t7p7xb2ds/ODyqHp+0TZJpxlskYnuhmi4FIq3rLCSd1PNMQ4l74ST27nfeLaiEQ92GnKgxhHSkSCoXSY19hKJFEAzqo1vy6vwBZJ7QgNSjQHFS/+sOEZTFXlk0pkf91AY5aiuY5LNKPzM8RTbBEe85qjDmJsgXF8/IhVOGJEq0K2XJQv09kWNszDQOXWeMdmxWvbn4n9fLbHQT5EKlmeWKLRdFmSQ2IfP3yVBozqycOoJMC3crYWPUyKwLqeJCoKsvr5P2VZ36dXpPaw2/iKMZ3AOl0DhGhpwB01oAQMFz/AKb57xXrx372PZWvKmVP4A+/zB70vkDk=</latexit>F x = 1 n &
!"# $ %
f& x + &
!"# $ %
f' x where, f& x = )x% + x x ≥ 0 Rx% + x x < 0 and f' x = )x% − x x ≥ 0 Rx% − x x < 0
[3]: Jeffery Z HaoChen and Suvrit Sra. Random shuffling beats sgd after finite epochs. 2018
F(x) = 1 n @
n/2
X
i=1
fa(x) +
n/2
X
i=1
fb(x) 1 A
<latexit sha1_base64="uXsAmFQl76d40yKOkgBCm8Fgl04=">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</latexit><latexit sha1_base64="uXsAmFQl76d40yKOkgBCm8Fgl04=">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</latexit><latexit sha1_base64="uXsAmFQl76d40yKOkgBCm8Fgl04=">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</latexit><latexit sha1_base64="uXsAmFQl76d40yKOkgBCm8Fgl04=">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</latexit>where, fa(x) = ( x2 + x x ≥ 0 Rx2 + x x < 0
<latexit sha1_base64="MQKihgtUtfMQ4JQV24eKnKeSYw=">ACMXicbVBNS+wFE3V50fU0dugkOTxRlaEXQhYLgxqWKo8J0HNLM7UwTWtyqx3K/CU3/hNx40IRt/4JM2MXfh0InJxz703uCVMpDHreozMyOvZnfGJyv37b3pmtjI3f2qSTHOo80Qm+jxkBqRQUEeBEs5TDSwOJZyFl/sD/+watBGJOsFeCs2YdZSIBGdopVblIEDIsbjpgoZ12o9abCVf3Q1C6AhVcDvY9N38YmMtX86Dlx5QeAel/cdzw1AtcuqVqXq1bwh6E/il6RKShy2KvdBO+FZDAq5ZMY0fC/FZsE0Ci6h7waZgZTxS9aBhqWKxWCaxXDjPv1vlTaNEm2PQjpUP3cULDamF4e2MmbYNd+9gfib18gw2m4WQqUZguIfD0WZpJjQXy0LTRwlD1LGNfC/pXyLtOMow3ZtSH431f+SU43ar5X8482q3t+GckWSRLZIX4ZIvskQNySOqEk1vyQJ7Is3PnPDovzutH6YhT9iyQL3De3gFugqkH</latexit><latexit sha1_base64="MQKihgtUtfMQ4JQV24eKnKeSYw=">ACMXicbVBNS+wFE3V50fU0dugkOTxRlaEXQhYLgxqWKo8J0HNLM7UwTWtyqx3K/CU3/hNx40IRt/4JM2MXfh0InJxz703uCVMpDHreozMyOvZnfGJyv37b3pmtjI3f2qSTHOo80Qm+jxkBqRQUEeBEs5TDSwOJZyFl/sD/+watBGJOsFeCs2YdZSIBGdopVblIEDIsbjpgoZ12o9abCVf3Q1C6AhVcDvY9N38YmMtX86Dlx5QeAel/cdzw1AtcuqVqXq1bwh6E/il6RKShy2KvdBO+FZDAq5ZMY0fC/FZsE0Ci6h7waZgZTxS9aBhqWKxWCaxXDjPv1vlTaNEm2PQjpUP3cULDamF4e2MmbYNd+9gfib18gw2m4WQqUZguIfD0WZpJjQXy0LTRwlD1LGNfC/pXyLtOMow3ZtSH431f+SU43ar5X8482q3t+GckWSRLZIX4ZIvskQNySOqEk1vyQJ7Is3PnPDovzutH6YhT9iyQL3De3gFugqkH</latexit><latexit sha1_base64="MQKihgtUtfMQ4JQV24eKnKeSYw=">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</latexit><latexit sha1_base64="MQKihgtUtfMQ4JQV24eKnKeSYw=">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</latexit>and, fb(x) = ( x2 − x x ≥ 0 Rx2 − x x < 0
<latexit sha1_base64="SmzCZGqQS5jn9RbDU3jM39gZj0I=">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</latexit><latexit sha1_base64="SmzCZGqQS5jn9RbDU3jM39gZj0I=">ACL3icbVDLatAFB2leSovp1l2M9QkJAYyQSRQuBQOgyLXUcsBwzGl3Zg0cjZeYqyAj/UTb5lWxCaSjd9i86trXI68DAmXPuvTP3hJkUBj3vlzP3YX5hcWl5xV1dW9/YrG19vDRprjm0eCpTfRUyA1IoaKFACVeZBpaEtrh8Gzit29BG5GqnzjKoJuwvhKx4Ayt1KudBwgFlkxFB3Qc98K9Yv9rEJfqJLbsWbsFtfNw2K3CPpw4wWB+6O6f/HcAFRUVfVqda/hTUHfEr8idVLhold7CKU5wko5JIZ0/G9DLsl0yi4hLEb5AYyxoesDx1LFUvAdMvpvmO6Y5WIxqm2RyGdqs87SpYM0pCW5kwHJjX3kR8z+vkGJ90S6GyHEHx2UNxLimdBIejYQGjnJkCeNa2L9SPmCacbQRuzYE/XKb8ls+F7Df/7Uf3Ur+JYJp/IZ7JHfHJMTsk3ckFahJM78kB+kyfn3nl0/jh/Z6VzTtWzTV7A+fcfjDeoEA=</latexit><latexit sha1_base64="SmzCZGqQS5jn9RbDU3jM39gZj0I=">ACL3icbVDLatAFB2leSovp1l2M9QkJAYyQSRQuBQOgyLXUcsBwzGl3Zg0cjZeYqyAj/UTb5lWxCaSjd9i86trXI68DAmXPuvTP3hJkUBj3vlzP3YX5hcWl5xV1dW9/YrG19vDRprjm0eCpTfRUyA1IoaKFACVeZBpaEtrh8Gzit29BG5GqnzjKoJuwvhKx4Ayt1KudBwgFlkxFB3Qc98K9Yv9rEJfqJLbsWbsFtfNw2K3CPpw4wWB+6O6f/HcAFRUVfVqda/hTUHfEr8idVLhold7CKU5wko5JIZ0/G9DLsl0yi4hLEb5AYyxoesDx1LFUvAdMvpvmO6Y5WIxqm2RyGdqs87SpYM0pCW5kwHJjX3kR8z+vkGJ90S6GyHEHx2UNxLimdBIejYQGjnJkCeNa2L9SPmCacbQRuzYE/XKb8ls+F7Df/7Uf3Ur+JYJp/IZ7JHfHJMTsk3ckFahJM78kB+kyfn3nl0/jh/Z6VzTtWzTV7A+fcfjDeoEA=</latexit><latexit sha1_base64="SmzCZGqQS5jn9RbDU3jM39gZj0I=">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</latexit>If F has Lipschitz Hessian, Error = O ⇣
n3 T 3 + 1 T 2
⌘ [3]
<latexit sha1_base64="Tn6dgNK7Db1Lz7nIlJu0MYwInY4=">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</latexit><latexit sha1_base64="Tn6dgNK7Db1Lz7nIlJu0MYwInY4=">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</latexit><latexit sha1_base64="Tn6dgNK7Db1Lz7nIlJu0MYwInY4=">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</latexit><latexit sha1_base64="Tn6dgNK7Db1Lz7nIlJu0MYwInY4=">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</latexit>rfa(x) = ( x 0 x < 0
<latexit sha1_base64="Od1Qas2bnXEvIblYUXsgfzitCUY=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARh3Yuds98gPscpkz+6K+L/m9Xgbh6N6+87vYc31/jD10Sz7KsLB0yhW36g6L/gVeASqkqHqn/OSHCctilIYJ0Lrlualp56AMZwLtCJnGFNgNdLFloYQYdTsf7n1PtywT0ihR9khDh+xPRw6x1v04sJ0xmGs9rg3I/7RWZqKDds5lmhmUbPRlAlqEjoIkYZcITOibwEwxe2slF2DAmZs1CUbgje+8l/QrFU9t+qd71WOa0Uc2SDbJId4pF9ckxOSZ0CMP5IW8kXfn0Xl1PpzPUeuEU3jWya9yvr4B1Gysdg=</latexit><latexit sha1_base64="Od1Qas2bnXEvIblYUXsgfzitCUY=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARh3Yuds98gPscpkz+6K+L/m9Xgbh6N6+87vYc31/jD10Sz7KsLB0yhW36g6L/gVeASqkqHqn/OSHCctilIYJ0Lrlualp56AMZwLtCJnGFNgNdLFloYQYdTsf7n1PtywT0ihR9khDh+xPRw6x1v04sJ0xmGs9rg3I/7RWZqKDds5lmhmUbPRlAlqEjoIkYZcITOibwEwxe2slF2DAmZs1CUbgje+8l/QrFU9t+qd71WOa0Uc2SDbJId4pF9ckxOSZ0CMP5IW8kXfn0Xl1PpzPUeuEU3jWya9yvr4B1Gysdg=</latexit><latexit sha1_base64="Od1Qas2bnXEvIblYUXsgfzitCUY=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARh3Yuds98gPscpkz+6K+L/m9Xgbh6N6+87vYc31/jD10Sz7KsLB0yhW36g6L/gVeASqkqHqn/OSHCctilIYJ0Lrlualp56AMZwLtCJnGFNgNdLFloYQYdTsf7n1PtywT0ihR9khDh+xPRw6x1v04sJ0xmGs9rg3I/7RWZqKDds5lmhmUbPRlAlqEjoIkYZcITOibwEwxe2slF2DAmZs1CUbgje+8l/QrFU9t+qd71WOa0Uc2SDbJId4pF9ckxOSZ0CMP5IW8kXfn0Xl1PpzPUeuEU3jWya9yvr4B1Gysdg=</latexit><latexit sha1_base64="Od1Qas2bnXEvIblYUXsgfzitCUY=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARh3Yuds98gPscpkz+6K+L/m9Xgbh6N6+87vYc31/jD10Sz7KsLB0yhW36g6L/gVeASqkqHqn/OSHCctilIYJ0Lrlualp56AMZwLtCJnGFNgNdLFloYQYdTsf7n1PtywT0ihR9khDh+xPRw6x1v04sJ0xmGs9rg3I/7RWZqKDds5lmhmUbPRlAlqEjoIkYZcITOibwEwxe2slF2DAmZs1CUbgje+8l/QrFU9t+qd71WOa0Uc2SDbJId4pF9ckxOSZ0CMP5IW8kXfn0Xl1PpzPUeuEU3jWya9yvr4B1Gysdg=</latexit>2Rx + 1 2x − 1 2Rx − 1 rfb(x) = ( x 0 x < 0
<latexit sha1_base64="4i2tqHSPLVcd2qDNVRyeEp41e5Q=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARp1g5273yA+wy2XO7Iv6vuT3ehmEo3v7zu9iz/X9MfbQLfkow8LSKVfcqjs+hd4BaiQouqd8pMfJiyLURomQOuW56amnYMynAm0I2QaU2A30MWhRJi1O18uPc93bJMSKNE2SMNHbI/HTnEWvfjwHbGYK71uDYg/9NamYkO2jmXaWZQstFHUSaoSegRBpyhcyIvgXAFLezUnYNCpixUZdsCN74yn9Bs1b13Kp3vlc5rhVxzJENskl2iEf2yTE5JXSIw8kBfyRt6dR+fV+XA+R60TuFZJ7/K+foG1jSsdw=</latexit><latexit sha1_base64="4i2tqHSPLVcd2qDNVRyeEp41e5Q=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARp1g5273yA+wy2XO7Iv6vuT3ehmEo3v7zu9iz/X9MfbQLfkow8LSKVfcqjs+hd4BaiQouqd8pMfJiyLURomQOuW56amnYMynAm0I2QaU2A30MWhRJi1O18uPc93bJMSKNE2SMNHbI/HTnEWvfjwHbGYK71uDYg/9NamYkO2jmXaWZQstFHUSaoSegRBpyhcyIvgXAFLezUnYNCpixUZdsCN74yn9Bs1b13Kp3vlc5rhVxzJENskl2iEf2yTE5JXSIw8kBfyRt6dR+fV+XA+R60TuFZJ7/K+foG1jSsdw=</latexit><latexit sha1_base64="4i2tqHSPLVcd2qDNVRyeEp41e5Q=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARp1g5273yA+wy2XO7Iv6vuT3ehmEo3v7zu9iz/X9MfbQLfkow8LSKVfcqjs+hd4BaiQouqd8pMfJiyLURomQOuW56amnYMynAm0I2QaU2A30MWhRJi1O18uPc93bJMSKNE2SMNHbI/HTnEWvfjwHbGYK71uDYg/9NamYkO2jmXaWZQstFHUSaoSegRBpyhcyIvgXAFLezUnYNCpixUZdsCN74yn9Bs1b13Kp3vlc5rhVxzJENskl2iEf2yTE5JXSIw8kBfyRt6dR+fV+XA+R60TuFZJ7/K+foG1jSsdw=</latexit><latexit sha1_base64="4i2tqHSPLVcd2qDNVRyeEp41e5Q=">ACN3icbVBNSwMxEM36bf2qevQSLIpeym4R9KAgePEkFWwVuqXMZmdrMJvdJlmxLP4rL/4Nb3rxoIhX/4FpuwetDiQ83puXzLwgFVwb1312Jianpmdm5+ZLC4tLyvl1bWmTjLFsMESkairADQKLrFhuBF4lSqEOB4GdycDPTLW1SaJ/LC9FNsx9CVPOIMjKU65TNfQiCARp1g5273yA+wy2XO7Iv6vuT3ehmEo3v7zu9iz/X9MfbQLfkow8LSKVfcqjs+hd4BaiQouqd8pMfJiyLURomQOuW56amnYMynAm0I2QaU2A30MWhRJi1O18uPc93bJMSKNE2SMNHbI/HTnEWvfjwHbGYK71uDYg/9NamYkO2jmXaWZQstFHUSaoSegRBpyhcyIvgXAFLezUnYNCpixUZdsCN74yn9Bs1b13Kp3vlc5rhVxzJENskl2iEf2yTE5JXSIw8kBfyRt6dR+fV+XA+R60TuFZJ7/K+foG1jSsdw=</latexit>4 , |xi| ≥ Cα
rfa(x) = ( x 0 x < 0
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sha1_base64="vVRZ8mRNWV13Ovzo42xIJqc1ajo=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzMFDwIvHCOYB2WZnUySITOzyzEsOQ3vHhQxKs/482/cZLsQRMLGoqbrq7kowzbXz/21tb39jc2i7tlHf39g8OK0fHbZ1aRWiLpDxV3QRrypmkLcMp91MUSwSTjvJ+Hbmdx6p0iyVD2aS0UjgoWQDRrBxUhqK+L8KWY3/jSuVP2aPwdaJUFBqlCgGVe+wn5KrKDSEI617gV+ZqIcK8MIp9NyaDXNMBnjIe05KrGgOsrnN0/RuVP6aJAqV9Kgufp7IsdC64lIXKfAZqSXvZn4n9ezZlCPciYza6gki0UDy5FJ0SwA1GeKEsMnjmCimLsVkRFWmBgXU9mFECy/vEral7XArwX3V9VGvYijBKdwBhcQwDU04A6a0AICGTzDK7x51nvx3r2PReuaV8ycwB94nz8AEZGc</latexit><latexit sha1_base64="vVRZ8mRNWV13Ovzo42xIJqc1ajo=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzMFDwIvHCOYB2WZnUySITOzyzEsOQ3vHhQxKs/482/cZLsQRMLGoqbrq7kowzbXz/21tb39jc2i7tlHf39g8OK0fHbZ1aRWiLpDxV3QRrypmkLcMp91MUSwSTjvJ+Hbmdx6p0iyVD2aS0UjgoWQDRrBxUhqK+L8KWY3/jSuVP2aPwdaJUFBqlCgGVe+wn5KrKDSEI617gV+ZqIcK8MIp9NyaDXNMBnjIe05KrGgOsrnN0/RuVP6aJAqV9Kgufp7IsdC64lIXKfAZqSXvZn4n9ezZlCPciYza6gki0UDy5FJ0SwA1GeKEsMnjmCimLsVkRFWmBgXU9mFECy/vEral7XArwX3V9VGvYijBKdwBhcQwDU04A6a0AICGTzDK7x51nvx3r2PReuaV8ycwB94nz8AEZGc</latexit><latexit sha1_base64="vVRZ8mRNWV13Ovzo42xIJqc1ajo=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzMFDwIvHCOYB2WZnUySITOzyzEsOQ3vHhQxKs/482/cZLsQRMLGoqbrq7kowzbXz/21tb39jc2i7tlHf39g8OK0fHbZ1aRWiLpDxV3QRrypmkLcMp91MUSwSTjvJ+Hbmdx6p0iyVD2aS0UjgoWQDRrBxUhqK+L8KWY3/jSuVP2aPwdaJUFBqlCgGVe+wn5KrKDSEI617gV+ZqIcK8MIp9NyaDXNMBnjIe05KrGgOsrnN0/RuVP6aJAqV9Kgufp7IsdC64lIXKfAZqSXvZn4n9ezZlCPciYza6gki0UDy5FJ0SwA1GeKEsMnjmCimLsVkRFWmBgXU9mFECy/vEral7XArwX3V9VGvYijBKdwBhcQwDU04A6a0AICGTzDK7x51nvx3r2PReuaV8ycwB94nz8AEZGc</latexit>xi≥0
<latexit sha1_base64="50mNsqgKCkaZ+04s7PjI0dVaZw=">AB+XicbVBNS8NAEN3Ur1q/oh69LBZBEoigj0WvHisYFuhDWGznbRLdzdxd1Msof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMi1LOtPG8b6e0tr6xuVXeruzs7u0fuIdHbZ1kikKLJjxRDxHRwJmElmGw0OqgIiIQyca3cz8zhiUZom8N5MUAkEGksWMEmOl0HUvejoTYf4Ust4AHr1p6Fa9mjcHXiV+QaqoQDN0v3r9hGYCpKGcaN31vdQEOVGUQ7TSi/TkBI6IgPoWiqJAB3k8un+MwqfRwnypY0eK7+nsiJ0HoiItspiBnqZW8m/ud1MxPXg5zJNDMg6WJRnHFsEjyLAfeZAmr4xBJCFbO3YjokilBjw6rYEPzl1dJ+7LmezX/7qraqBdxlNEJOkXnyEfXqIFuURO1EVj9Ixe0ZuTOy/Ou/OxaC05xcwx+gPn8wdxDZN9</latexit><latexit sha1_base64="50mNsqgKCkaZ+04s7PjI0dVaZw=">AB+XicbVBNS8NAEN3Ur1q/oh69LBZBEoigj0WvHisYFuhDWGznbRLdzdxd1Msof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMi1LOtPG8b6e0tr6xuVXeruzs7u0fuIdHbZ1kikKLJjxRDxHRwJmElmGw0OqgIiIQyca3cz8zhiUZom8N5MUAkEGksWMEmOl0HUvejoTYf4Ust4AHr1p6Fa9mjcHXiV+QaqoQDN0v3r9hGYCpKGcaN31vdQEOVGUQ7TSi/TkBI6IgPoWiqJAB3k8un+MwqfRwnypY0eK7+nsiJ0HoiItspiBnqZW8m/ud1MxPXg5zJNDMg6WJRnHFsEjyLAfeZAmr4xBJCFbO3YjokilBjw6rYEPzl1dJ+7LmezX/7qraqBdxlNEJOkXnyEfXqIFuURO1EVj9Ixe0ZuTOy/Ou/OxaC05xcwx+gPn8wdxDZN9</latexit><latexit sha1_base64="50mNsqgKCkaZ+04s7PjI0dVaZw=">AB+XicbVBNS8NAEN3Ur1q/oh69LBZBEoigj0WvHisYFuhDWGznbRLdzdxd1Msof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMi1LOtPG8b6e0tr6xuVXeruzs7u0fuIdHbZ1kikKLJjxRDxHRwJmElmGw0OqgIiIQyca3cz8zhiUZom8N5MUAkEGksWMEmOl0HUvejoTYf4Ust4AHr1p6Fa9mjcHXiV+QaqoQDN0v3r9hGYCpKGcaN31vdQEOVGUQ7TSi/TkBI6IgPoWiqJAB3k8un+MwqfRwnypY0eK7+nsiJ0HoiItspiBnqZW8m/ud1MxPXg5zJNDMg6WJRnHFsEjyLAfeZAmr4xBJCFbO3YjokilBjw6rYEPzl1dJ+7LmezX/7qraqBdxlNEJOkXnyEfXqIFuURO1EVj9Ixe0ZuTOy/Ou/OxaC05xcwx+gPn8wdxDZN9</latexit><latexit sha1_base64="50mNsqgKCkaZ+04s7PjI0dVaZw=">AB+XicbVBNS8NAEN3Ur1q/oh69LBZBEoigj0WvHisYFuhDWGznbRLdzdxd1Msof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMi1LOtPG8b6e0tr6xuVXeruzs7u0fuIdHbZ1kikKLJjxRDxHRwJmElmGw0OqgIiIQyca3cz8zhiUZom8N5MUAkEGksWMEmOl0HUvejoTYf4Ust4AHr1p6Fa9mjcHXiV+QaqoQDN0v3r9hGYCpKGcaN31vdQEOVGUQ7TSi/TkBI6IgPoWiqJAB3k8un+MwqfRwnypY0eK7+nsiJ0HoiItspiBnqZW8m/ud1MxPXg5zJNDMg6WJRnHFsEjyLAfeZAmr4xBJCFbO3YjokilBjw6rYEPzl1dJ+7LmezX/7qraqBdxlNEJOkXnyEfXqIFuURO1EVj9Ixe0ZuTOy/Ou/OxaC05xcwx+gPn8wdxDZN9</latexit>xi≥0
<latexit sha1_base64="OmUN7o15gBwrSbQz2UYW17u/s0s=">ACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIthlwY3LCvYBTQiT6aQdOo84M5GW0I2/4saFIm79DHf+jdM2C209cOFwzr3ce0+cMqN5307pbX1jc2t8nZlZ3dv/8A9PGprmSlMWlgyqbox0oRQVqGka6qSKIx4x04tHNzO8EqWpFPdmkpKQo4GgCcXIWClyTwKUpkqOYaAzHuXjiAYD8uBNI7fq1bw54CrxC1IFBZqR+xX0Jc4EQYzpHXP91IT5kgZihmZVoJMkxThERqQnqUCcaLDfP7AFJ5bpQ8TqWwJA+fq74kca0nPLadHJmhXvZm4n9eLzNJPcypSDNDBF4sSjIGjYSzNGCfKoINm1iCsKL2VoiHSCFsbGYVG4K/PIqaV/WfK/m31VG/UijI4BWfgAvjgGjTALWiCFsBgCp7BK3hznpwX5935WLSWnGLmGPyB8/kDGM+Wrg=</latexit><latexit sha1_base64="OmUN7o15gBwrSbQz2UYW17u/s0s=">ACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIthlwY3LCvYBTQiT6aQdOo84M5GW0I2/4saFIm79DHf+jdM2C209cOFwzr3ce0+cMqN5307pbX1jc2t8nZlZ3dv/8A9PGprmSlMWlgyqbox0oRQVqGka6qSKIx4x04tHNzO8EqWpFPdmkpKQo4GgCcXIWClyTwKUpkqOYaAzHuXjiAYD8uBNI7fq1bw54CrxC1IFBZqR+xX0Jc4EQYzpHXP91IT5kgZihmZVoJMkxThERqQnqUCcaLDfP7AFJ5bpQ8TqWwJA+fq74kca0nPLadHJmhXvZm4n9eLzNJPcypSDNDBF4sSjIGjYSzNGCfKoINm1iCsKL2VoiHSCFsbGYVG4K/PIqaV/WfK/m31VG/UijI4BWfgAvjgGjTALWiCFsBgCp7BK3hznpwX5935WLSWnGLmGPyB8/kDGM+Wrg=</latexit><latexit sha1_base64="OmUN7o15gBwrSbQz2UYW17u/s0s=">ACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIthlwY3LCvYBTQiT6aQdOo84M5GW0I2/4saFIm79DHf+jdM2C209cOFwzr3ce0+cMqN5307pbX1jc2t8nZlZ3dv/8A9PGprmSlMWlgyqbox0oRQVqGka6qSKIx4x04tHNzO8EqWpFPdmkpKQo4GgCcXIWClyTwKUpkqOYaAzHuXjiAYD8uBNI7fq1bw54CrxC1IFBZqR+xX0Jc4EQYzpHXP91IT5kgZihmZVoJMkxThERqQnqUCcaLDfP7AFJ5bpQ8TqWwJA+fq74kca0nPLadHJmhXvZm4n9eLzNJPcypSDNDBF4sSjIGjYSzNGCfKoINm1iCsKL2VoiHSCFsbGYVG4K/PIqaV/WfK/m31VG/UijI4BWfgAvjgGjTALWiCFsBgCp7BK3hznpwX5935WLSWnGLmGPyB8/kDGM+Wrg=</latexit><latexit sha1_base64="OmUN7o15gBwrSbQz2UYW17u/s0s=">ACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIthlwY3LCvYBTQiT6aQdOo84M5GW0I2/4saFIm79DHf+jdM2C209cOFwzr3ce0+cMqN5307pbX1jc2t8nZlZ3dv/8A9PGprmSlMWlgyqbox0oRQVqGka6qSKIx4x04tHNzO8EqWpFPdmkpKQo4GgCcXIWClyTwKUpkqOYaAzHuXjiAYD8uBNI7fq1bw54CrxC1IFBZqR+xX0Jc4EQYzpHXP91IT5kgZihmZVoJMkxThERqQnqUCcaLDfP7AFJ5bpQ8TqWwJA+fq74kca0nPLadHJmhXvZm4n9eLzNJPcypSDNDBF4sSjIGjYSzNGCfKoINm1iCsKL2VoiHSCFsbGYVG4K/PIqaV/WfK/m31VG/UijI4BWfgAvjgGjTALWiCFsBgCp7BK3hznpwX5935WLSWnGLmGPyB8/kDGM+Wrg=</latexit>(assume R >> 1)