Entropy and mutual information in models of deep neural networks - - PowerPoint PPT Presentation

entropy and mutual information in models of deep neural
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Entropy and mutual information in models of deep neural networks - - PowerPoint PPT Presentation

Entropy and mutual information in models of deep neural networks NeurIPS 2018 - Thursday Dec 06th - Spotlight Poster @ Room 210 & 230 AB #110 Marylou Gabri (LPS ENS), Andre Manoel (INRIA Saclay, Owkin), Clment Luneau, Jean Barbier,


slide-1
SLIDE 1

Entropy and mutual information in models of deep neural networks

NeurIPS 2018 - Thursday Dec 06th - Spotlight Poster @ Room 210 & 230 AB #110 Marylou Gabrié (LPS ENS), Andre Manoel (INRIA Saclay, Owkin), Clément Luneau, Jean Barbier, Nicolas Macris (EPFL), Lenka Zdeborová (CEA Saclay), Florent Krzakala (LPS ENS)

slide-2
SLIDE 2

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

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Less and less reliable as networks get large

è

Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

slide-3
SLIDE 3

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

è

Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

t1

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slide-4
SLIDE 4

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

è

Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

t1

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ˆ y

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t`

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t0

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I(t0; t`)

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slide-5
SLIDE 5

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

è

Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

t1

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ˆ y

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t`

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t0

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I(t0; t`)

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I(t0; ˆ y)

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slide-6
SLIDE 6

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

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Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

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t`

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t0

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I(t0; t`)

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I(t0; ˆ y)

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p(t`) = Z dt0dt1 · · · dt`−1 p(t`|t`−1) · · · p(t1|t0)p(t0)

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slide-7
SLIDE 7

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

è

Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

H(t`) = − Z dt` p(t`)log p(t`)

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t1

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ˆ y

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t`

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t0

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I(t0; t`)

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I(t0; ˆ y)

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p(t`) = Z dt0dt1 · · · dt`−1 p(t`|t`−1) · · · p(t1|t0)p(t0)

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slide-8
SLIDE 8

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

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Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leveverage statistical physics replica method

H(t`) = − Z dt` p(t`)log p(t`)

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t1

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ˆ y

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t`

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t0

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I(t0; t`)

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I(t0; ˆ y)

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p(t`) = Z dt0dt1 · · · dt`−1 p(t`|t`−1) · · · p(t1|t0)p(t0)

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slide-9
SLIDE 9

Motivations

Information theoretic arguments to deep learning

  • Theory of generalization: Shwartz-Ziv et al. 2017, Saxe et al. 2018, Goldfeld et al. 2018 etc.
  • New regularizer: Chalk et al. 2016, Alemi et al. 2017, Kolchinsky et al. 2017, Belghazi et al. 2017, Zhao et al. 2018,

Achille et al. 2018, Hjelm et al. 2018, etc.

Computing mutual informations in neural networks … is tricky

  • Intractable
  • Computing or yet more involved

Sampled based estimations (non-parametric, variational methods)

Kraskov et al. 2014, Kolchinsky et al. 2017, Belghazi et al. 2017, Goldfeld et al. 2018 etc.

è

Less and less reliable as networks get large

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Need for benchmark controlled case

Here 1) Restrict to models of DNNS, 2) Leverage statistical physics replica method

H(t`) = − Z dt` p(t`)log p(t`)

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t1

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ˆ y

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t`

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t0

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I(t0; t`)

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I(t0; ˆ y)

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p(t`) = Z dt0dt1 · · · dt`−1 p(t`|t`−1) · · · p(t1|t0)p(t0)

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slide-10
SLIDE 10

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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slide-11
SLIDE 11

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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slide-12
SLIDE 12

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

P0(t0) =

N0

Y

i=1

P0(t0i)

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t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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slide-13
SLIDE 13

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

P0(t0) =

N0

Y

i=1

P0(t0i)

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t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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slide-14
SLIDE 14

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

P0(t0) =

N0

Y

i=1

P0(t0i)

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W (`)

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t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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i.i.d Gaussian

slide-15
SLIDE 15

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

P0(t0) =

N0

Y

i=1

P0(t0i)

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= × ×

W (`)

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U (`)

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S(`)

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V (`)|

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i.i.d. from Haar

t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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i.i.d Gaussian è Orthogonally invariant

slide-16
SLIDE 16

Model: fully connected feed forward neural network with stochastic activations

With : 1) known factorized input distribution 2) weight matrices from class of random matrix class

P0(t0) =

N0

Y

i=1

P0(t0i)

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= × ×

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S(`)

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V (`)|

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i.i.d. from Haar arbitrary diagonal

t` = f✏(W(`)f✏(W(`−1) · · · f✏(W(1)t0) · · · ))

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RN`

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RN0

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i.i.d Gaussian è Orthogonally invariant

slide-17
SLIDE 17

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

<latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit>
slide-18
SLIDE 18

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

<latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit><latexit sha1_base64="/BmQwne6mgHNYv6cj+E+eWlLRhE=">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</latexit>
slide-19
SLIDE 19

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

φ`(A, V, ˜ A, ˜ V) =I t0; t0 + ξ0 p ˜ A1 ! − 1 2

`

X

k=1

˜ αk−1 ⇥ ˜ AkVk + αkAk ˜ Vk − FWk(AkVk) ⇤ +

`−1

X

k=1

˜ αk h H(tk|ξk; ˜ Ak+1, ˜ Vk, ˜ ρk) − 1 2 log(2πe ˜ A−1

k+1)

i + ˜ α`H(t`|ξ`; ˜ V`, ˜ ρ`)

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lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

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slide-20
SLIDE 20

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

φ`(A, V, ˜ A, ˜ V) =I t0; t0 + ξ0 p ˜ A1 ! − 1 2

`

X

k=1

˜ αk−1 ⇥ ˜ AkVk + αkAk ˜ Vk − FWk(AkVk) ⇤ +

`−1

X

k=1

˜ αk h H(tk|ξk; ˜ Ak+1, ˜ Vk, ˜ ρk) − 1 2 log(2πe ˜ A−1

k+1)

i + ˜ α`H(t`|ξ`; ˜ V`, ˜ ρ`)

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lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

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slide-21
SLIDE 21

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

φ`(A, V, ˜ A, ˜ V) =I t0; t0 + ξ0 p ˜ A1 ! − 1 2

`

X

k=1

˜ αk−1 ⇥ ˜ AkVk + αkAk ˜ Vk − FWk(AkVk) ⇤ +

`−1

X

k=1

˜ αk h H(tk|ξk; ˜ Ak+1, ˜ Vk, ˜ ρk) − 1 2 log(2πe ˜ A−1

k+1)

i + ˜ α`H(t`|ξ`; ˜ V`, ˜ ρ`)

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lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

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slide-22
SLIDE 22

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

φ`(A, V, ˜ A, ˜ V) =I t0; t0 + ξ0 p ˜ A1 ! − 1 2

`

X

k=1

˜ αk−1 ⇥ ˜ AkVk + αkAk ˜ Vk − FWk(AkVk) ⇤ +

`−1

X

k=1

˜ αk h H(tk|ξk; ˜ Ak+1, ˜ Vk, ˜ ρk) − 1 2 log(2πe ˜ A−1

k+1)

i + ˜ α`H(t`|ξ`; ˜ V`, ˜ ρ`)

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lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

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slide-23
SLIDE 23

Main results

1) Formula:

potential: efficient implementation of extremization:

2) Theorem:

2-layer Gaussian random weight matrices

φ`(A, V, ˜ A, ˜ V) =I t0; t0 + ξ0 p ˜ A1 ! − 1 2

`

X

k=1

˜ αk−1 ⇥ ˜ AkVk + αkAk ˜ Vk − FWk(AkVk) ⇤ +

`−1

X

k=1

˜ αk h H(tk|ξk; ˜ Ak+1, ˜ Vk, ˜ ρk) − 1 2 log(2πe ˜ A−1

k+1)

i + ˜ α`H(t`|ξ`; ˜ V`, ˜ ρ`)

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lim

N0→∞

1 N0 H(t`) = min extr

A,V, ˜ A, ˜ V

φ`(A, V, ˜ A, ˜ V)

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replica prediction exact in limit of large networks ⇒

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t2 = f✏(W(2)f✏(W(1)t0))

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SLIDE 24

APPLICATION: Follow mutual information during SGD learning

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SLIDE 25

1) Synthetic data sets for student 2) Orthogonally invariant weight matrices during learning for the student

Applying the replica formula to learning experiments

Keras implementation

x

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˜ t1

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˜ tM

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generative model z

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i.i.d. latent non trivial distribution

slide-26
SLIDE 26

1) Synthetic data sets for student 2) Orthogonally invariant weight matrices during learning for the student

Applying the replica formula to learning experiments

y

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teacher

Keras implementation

x

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˜ t1

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˜ tM

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generative model z

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i.i.d. latent non trivial distribution

slide-27
SLIDE 27

1) Synthetic data sets for student 2) Orthogonally invariant weight matrices during learning for the student

Applying the replica formula to learning experiments

student t1

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tL

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ˆ y

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W (1)

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W (2)

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W (L)

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y

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teacher

Keras implementation

x

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˜ t1

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˜ tM

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generative model z

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i.i.d. latent non trivial distribution

slide-28
SLIDE 28

1) Synthetic data sets for student 2) Orthogonally invariant weight matrices during learning for the student

Applying the replica formula to learning experiments

= × ×

fixed fixed

updated by backprop

student t1

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tL

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ˆ y

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W (1)

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W (2)

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W (L)

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W (`)

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U (`)

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S(`)

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V (`)

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y

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teacher

Keras implementation

x

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˜ t1

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˜ tM

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generative model z

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i.i.d. latent non trivial distribution

slide-29
SLIDE 29

1) Synthetic data sets for student 2) Orthogonally invariant weight matrices during learning for the student

Applying the replica formula to learning experiments

= × ×

fixed fixed

updated by backprop

student t1

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tL

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ˆ y

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W (1)

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W (2)

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W (L)

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W (`)

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U (`)

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S(`)

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V (`)

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y

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teacher

Keras implementation

x

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˜ t1

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˜ tM

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generative model z

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i.i.d. latent non trivial distribution

slide-30
SLIDE 30

Exploratory experiment: A binary classification example

x

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t1

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ˆ y

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W1

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W2

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t2

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t3

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W3

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W4

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W5

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t4

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y

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z

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y = sign(z0)

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z ∼

N0

Y

i=0

N(zi; 0, 1)

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slide-31
SLIDE 31

200 400 600 800 1000

epochs

2.40 2.45 2.50 2.55 2.60 2.65

I(X; T4) layer 4 - relu

Exploratory experiment: A binary classification example

x

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t1

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ˆ y

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W1

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W2

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t2

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t3

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W3

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W4

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W5

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t4

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y

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z

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y = sign(z0)

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z ∼

N0

Y

i=0

N(zi; 0, 1)

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I(x; t4)

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σ2

init = 1/N

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slide-32
SLIDE 32

200 400 600 800 1000

epochs

2.40 2.45 2.50 2.55 2.60 2.65

I(X; T4) layer 4 - relu

500 1000 1500 2000 2500

epochs

3.8 3.9 4.0 4.1 4.2

I(X; T4) layer 4 - hardtanh

Exploratory experiment: A binary classification example

Different non-linearities

x

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t1

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ˆ y

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W1

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W2

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t2

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t3

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W3

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W4

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W5

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t4

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y

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z

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y = sign(z0)

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z ∼

N0

Y

i=0

N(zi; 0, 1)

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I(x; t4)

<latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit>

σ2

init = 1/N

<latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit>

σ2

init = 1/N

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slide-33
SLIDE 33

200 400 600 800 1000

epochs

2.40 2.45 2.50 2.55 2.60 2.65

I(X; T4) layer 4 - relu

500 1000 1500 2000 2500

epochs

3.8 3.9 4.0 4.1 4.2

I(X; T4) layer 4 - hardtanh

500 1000 1500 2000 2500

epochs

2.4 2.6 2.8 3.0 3.2

I(X; T4) layer 4 - hardtanh

500 1000 1500 2000 2500

epochs

2.85 2.90 2.95 3.00 3.05 3.10 3.15

I(X; T4) layer 4 - hardtanh

Exploratory experiment: A binary classification example

Different non-linearities

x

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t1

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ˆ y

<latexit sha1_base64="4u0L09J3REqFw4Fk2PhpQjE57w4=">AB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoN4KXjxWMFpoQtlsN+3SzSbsbgoh5J948aDi1Z/izX/jps1BWwcWhpn3eLMTpwp7TjfVmNtfWNzq7nd2tnd2z+wD48eVZJQj2S8ET2Q6woZ4J6mlO+6mkOA45fQqnt5X/NKNSsUQ86DylQYzHgkWMYG2koW37MdaTMCr8CdZFXpZDu+10nDnQKnFr0oYavaH95Y8SksVUaMKxUgPXSXVQYKkZ4bRs+ZmiKSZTPKYDQwWOqQqKefISnRlhKJEmic0mqu/NwocK5XHoZmscqplrxL/8waZjq6Dgok01SQxaEo40gnqKoBjZikRPcEwkM1kRmWCJiTZltUwJ7vKXV4l30bnpuPeX7a5Tt9GEziFc3DhCrpwBz3wgMAMnuEV3qzCerHerY/FaMOqd47hD6zPH7m/k90=</latexit><latexit sha1_base64="4u0L09J3REqFw4Fk2PhpQjE57w4=">AB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoN4KXjxWMFpoQtlsN+3SzSbsbgoh5J948aDi1Z/izX/jps1BWwcWhpn3eLMTpwp7TjfVmNtfWNzq7nd2tnd2z+wD48eVZJQj2S8ET2Q6woZ4J6mlO+6mkOA45fQqnt5X/NKNSsUQ86DylQYzHgkWMYG2koW37MdaTMCr8CdZFXpZDu+10nDnQKnFr0oYavaH95Y8SksVUaMKxUgPXSXVQYKkZ4bRs+ZmiKSZTPKYDQwWOqQqKefISnRlhKJEmic0mqu/NwocK5XHoZmscqplrxL/8waZjq6Dgok01SQxaEo40gnqKoBjZikRPcEwkM1kRmWCJiTZltUwJ7vKXV4l30bnpuPeX7a5Tt9GEziFc3DhCrpwBz3wgMAMnuEV3qzCerHerY/FaMOqd47hD6zPH7m/k90=</latexit><latexit sha1_base64="4u0L09J3REqFw4Fk2PhpQjE57w4=">AB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoN4KXjxWMFpoQtlsN+3SzSbsbgoh5J948aDi1Z/izX/jps1BWwcWhpn3eLMTpwp7TjfVmNtfWNzq7nd2tnd2z+wD48eVZJQj2S8ET2Q6woZ4J6mlO+6mkOA45fQqnt5X/NKNSsUQ86DylQYzHgkWMYG2koW37MdaTMCr8CdZFXpZDu+10nDnQKnFr0oYavaH95Y8SksVUaMKxUgPXSXVQYKkZ4bRs+ZmiKSZTPKYDQwWOqQqKefISnRlhKJEmic0mqu/NwocK5XHoZmscqplrxL/8waZjq6Dgok01SQxaEo40gnqKoBjZikRPcEwkM1kRmWCJiTZltUwJ7vKXV4l30bnpuPeX7a5Tt9GEziFc3DhCrpwBz3wgMAMnuEV3qzCerHerY/FaMOqd47hD6zPH7m/k90=</latexit><latexit sha1_base64="4u0L09J3REqFw4Fk2PhpQjE57w4=">AB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoN4KXjxWMFpoQtlsN+3SzSbsbgoh5J948aDi1Z/izX/jps1BWwcWhpn3eLMTpwp7TjfVmNtfWNzq7nd2tnd2z+wD48eVZJQj2S8ET2Q6woZ4J6mlO+6mkOA45fQqnt5X/NKNSsUQ86DylQYzHgkWMYG2koW37MdaTMCr8CdZFXpZDu+10nDnQKnFr0oYavaH95Y8SksVUaMKxUgPXSXVQYKkZ4bRs+ZmiKSZTPKYDQwWOqQqKefISnRlhKJEmic0mqu/NwocK5XHoZmscqplrxL/8waZjq6Dgok01SQxaEo40gnqKoBjZikRPcEwkM1kRmWCJiTZltUwJ7vKXV4l30bnpuPeX7a5Tt9GEziFc3DhCrpwBz3wgMAMnuEV3qzCerHerY/FaMOqd47hD6zPH7m/k90=</latexit>

W1

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W2

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t2

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t3

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W3

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W4

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W5

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t4

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y

<latexit sha1_base64="Z8nNPeM+ejMQyUlsq4q2PO/v68=">AB8HicbVBNS8NAFHzxs9avqkcvi0XwVBIR1FvBi8cKxhbUDbTbt0swm7L0IJ/RdePKh49ed489+4aXPQ1oGFYeY9dt6EqRQGXfbWVldW9/YrGxVt3d29/ZrB4cPJsk04z5LZKI7ITVcCsV9FCh5J9WcxqHk7XB8U/jtJ6NSNQ9TlIexHSoRCQYRSs9mKozDKJ9N+re423BnIMvFKUocSrX7tqzdIWBZzhUxSY7qem2KQU42CST6t9jLDU8rGdMi7lioacxPks8RTcmqVAYkSbZ9CMlN/b+Q0NmYSh3aySGgWvUL8z+tmGF0FuVBphlyx+UdRJgkmpDifDITmDOXEsq0sFkJG1FNGdqSqrYEb/HkZeKfN64b3t1FvemWbVTgGE7gDy4hCbcQgt8YKDgGV7hzTHOi/PufMxHV5xy5wj+wPn8AWPqkN8=</latexit><latexit sha1_base64="Z8nNPeM+ejMQyUlsq4q2PO/v68=">AB8HicbVBNS8NAFHzxs9avqkcvi0XwVBIR1FvBi8cKxhbUDbTbt0swm7L0IJ/RdePKh49ed489+4aXPQ1oGFYeY9dt6EqRQGXfbWVldW9/YrGxVt3d29/ZrB4cPJsk04z5LZKI7ITVcCsV9FCh5J9WcxqHk7XB8U/jtJ6NSNQ9TlIexHSoRCQYRSs9mKozDKJ9N+re423BnIMvFKUocSrX7tqzdIWBZzhUxSY7qem2KQU42CST6t9jLDU8rGdMi7lioacxPks8RTcmqVAYkSbZ9CMlN/b+Q0NmYSh3aySGgWvUL8z+tmGF0FuVBphlyx+UdRJgkmpDifDITmDOXEsq0sFkJG1FNGdqSqrYEb/HkZeKfN64b3t1FvemWbVTgGE7gDy4hCbcQgt8YKDgGV7hzTHOi/PufMxHV5xy5wj+wPn8AWPqkN8=</latexit><latexit sha1_base64="Z8nNPeM+ejMQyUlsq4q2PO/v68=">AB8HicbVBNS8NAFHzxs9avqkcvi0XwVBIR1FvBi8cKxhbUDbTbt0swm7L0IJ/RdePKh49ed489+4aXPQ1oGFYeY9dt6EqRQGXfbWVldW9/YrGxVt3d29/ZrB4cPJsk04z5LZKI7ITVcCsV9FCh5J9WcxqHk7XB8U/jtJ6NSNQ9TlIexHSoRCQYRSs9mKozDKJ9N+re423BnIMvFKUocSrX7tqzdIWBZzhUxSY7qem2KQU42CST6t9jLDU8rGdMi7lioacxPks8RTcmqVAYkSbZ9CMlN/b+Q0NmYSh3aySGgWvUL8z+tmGF0FuVBphlyx+UdRJgkmpDifDITmDOXEsq0sFkJG1FNGdqSqrYEb/HkZeKfN64b3t1FvemWbVTgGE7gDy4hCbcQgt8YKDgGV7hzTHOi/PufMxHV5xy5wj+wPn8AWPqkN8=</latexit><latexit sha1_base64="Z8nNPeM+ejMQyUlsq4q2PO/v68=">AB8HicbVBNS8NAFHzxs9avqkcvi0XwVBIR1FvBi8cKxhbUDbTbt0swm7L0IJ/RdePKh49ed489+4aXPQ1oGFYeY9dt6EqRQGXfbWVldW9/YrGxVt3d29/ZrB4cPJsk04z5LZKI7ITVcCsV9FCh5J9WcxqHk7XB8U/jtJ6NSNQ9TlIexHSoRCQYRSs9mKozDKJ9N+re423BnIMvFKUocSrX7tqzdIWBZzhUxSY7qem2KQU42CST6t9jLDU8rGdMi7lioacxPks8RTcmqVAYkSbZ9CMlN/b+Q0NmYSh3aySGgWvUL8z+tmGF0FuVBphlyx+UdRJgkmpDifDITmDOXEsq0sFkJG1FNGdqSqrYEb/HkZeKfN64b3t1FvemWbVTgGE7gDy4hCbcQgt8YKDgGV7hzTHOi/PufMxHV5xy5wj+wPn8AWPqkN8=</latexit>

z

<latexit sha1_base64="wFWaygzE2cXN9wVruiaSGnuoJPo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LgxmUF+8A2lMl0g6dTMLMjVBD/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTm9zvPHJtRKzucZpwP6IjJULBKFrpoR9RHAdh9jQbVGtu3Z2DrBKvIDUo0BxUv/rDmKURV8gkNabnuQn6GdUomOSzSj81PKFsQke8Z6miETd+Nk8I2dWGZIw1vYpJHP190ZGI2OmUWAn84Rm2cvF/7xeiuG1nwmVpMgVW3wUpJgTPLzyVBozlBOLaFMC5uVsDHVlKEtqWJL8JZPXiXti7pn+d1lreEWdZThBE7hHDy4gbcQhNawEDBM7zCm2OcF+fd+ViMlpxi5xj+wPn8AfrbkQw=</latexit><latexit sha1_base64="wFWaygzE2cXN9wVruiaSGnuoJPo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LgxmUF+8A2lMl0g6dTMLMjVBD/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTm9zvPHJtRKzucZpwP6IjJULBKFrpoR9RHAdh9jQbVGtu3Z2DrBKvIDUo0BxUv/rDmKURV8gkNabnuQn6GdUomOSzSj81PKFsQke8Z6miETd+Nk8I2dWGZIw1vYpJHP190ZGI2OmUWAn84Rm2cvF/7xeiuG1nwmVpMgVW3wUpJgTPLzyVBozlBOLaFMC5uVsDHVlKEtqWJL8JZPXiXti7pn+d1lreEWdZThBE7hHDy4gbcQhNawEDBM7zCm2OcF+fd+ViMlpxi5xj+wPn8AfrbkQw=</latexit><latexit sha1_base64="wFWaygzE2cXN9wVruiaSGnuoJPo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LgxmUF+8A2lMl0g6dTMLMjVBD/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTm9zvPHJtRKzucZpwP6IjJULBKFrpoR9RHAdh9jQbVGtu3Z2DrBKvIDUo0BxUv/rDmKURV8gkNabnuQn6GdUomOSzSj81PKFsQke8Z6miETd+Nk8I2dWGZIw1vYpJHP190ZGI2OmUWAn84Rm2cvF/7xeiuG1nwmVpMgVW3wUpJgTPLzyVBozlBOLaFMC5uVsDHVlKEtqWJL8JZPXiXti7pn+d1lreEWdZThBE7hHDy4gbcQhNawEDBM7zCm2OcF+fd+ViMlpxi5xj+wPn8AfrbkQw=</latexit><latexit sha1_base64="wFWaygzE2cXN9wVruiaSGnuoJPo=">AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LgxmUF+8A2lMl0g6dTMLMjVBD/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTm9zvPHJtRKzucZpwP6IjJULBKFrpoR9RHAdh9jQbVGtu3Z2DrBKvIDUo0BxUv/rDmKURV8gkNabnuQn6GdUomOSzSj81PKFsQke8Z6miETd+Nk8I2dWGZIw1vYpJHP190ZGI2OmUWAn84Rm2cvF/7xeiuG1nwmVpMgVW3wUpJgTPLzyVBozlBOLaFMC5uVsDHVlKEtqWJL8JZPXiXti7pn+d1lreEWdZThBE7hHDy4gbcQhNawEDBM7zCm2OcF+fd+ViMlpxi5xj+wPn8AfrbkQw=</latexit>

y = sign(z0)

<latexit sha1_base64="yj9/AQBl/Ff3zjnmVRtfTUJ60W4=">ACAXicbZDLSsNAFIYn9VbrLepGcDNYhApSEhF0IxTcuKxgL9CGMJlO2qEzkzAzEWqIG1/FjQtF3PoW7nwbJ20W2vrDwMd/zmHO+YOYUaUd59sqLS2vrK6V1ysbm1vbO/buXltFicSkhSMWyW6AFGFUkJampFuLAniASOdYHyd1zv3RCoaiTs9iYnH0VDQkGKkjeXbBxN4Bfsc6ZHkqaJDkdUe/NTJTny76tSdqeAiuAVUQaGmb3/1BxFOBEaM6RUz3Vi7aVIaoZySr9RJEY4TEakp5BgThRXjq9IPHxhnAMJLmCQ2n7u+JFHGlJjwnfmuar6Wm/VeokOL72UijRODZR2HCoI5gHgcUEmwZhMDCEtqdoV4hCTC2oRWMSG48ycvQvus7hq+Pa82Tos4yuAQHIEacMEFaIAb0AQtgMEjeAav4M16sl6sd+tj1lqyipl98EfW5w+95Zb</latexit><latexit sha1_base64="yj9/AQBl/Ff3zjnmVRtfTUJ60W4=">ACAXicbZDLSsNAFIYn9VbrLepGcDNYhApSEhF0IxTcuKxgL9CGMJlO2qEzkzAzEWqIG1/FjQtF3PoW7nwbJ20W2vrDwMd/zmHO+YOYUaUd59sqLS2vrK6V1ysbm1vbO/buXltFicSkhSMWyW6AFGFUkJampFuLAniASOdYHyd1zv3RCoaiTs9iYnH0VDQkGKkjeXbBxN4Bfsc6ZHkqaJDkdUe/NTJTny76tSdqeAiuAVUQaGmb3/1BxFOBEaM6RUz3Vi7aVIaoZySr9RJEY4TEakp5BgThRXjq9IPHxhnAMJLmCQ2n7u+JFHGlJjwnfmuar6Wm/VeokOL72UijRODZR2HCoI5gHgcUEmwZhMDCEtqdoV4hCTC2oRWMSG48ycvQvus7hq+Pa82Tos4yuAQHIEacMEFaIAb0AQtgMEjeAav4M16sl6sd+tj1lqyipl98EfW5w+95Zb</latexit><latexit sha1_base64="yj9/AQBl/Ff3zjnmVRtfTUJ60W4=">ACAXicbZDLSsNAFIYn9VbrLepGcDNYhApSEhF0IxTcuKxgL9CGMJlO2qEzkzAzEWqIG1/FjQtF3PoW7nwbJ20W2vrDwMd/zmHO+YOYUaUd59sqLS2vrK6V1ysbm1vbO/buXltFicSkhSMWyW6AFGFUkJampFuLAniASOdYHyd1zv3RCoaiTs9iYnH0VDQkGKkjeXbBxN4Bfsc6ZHkqaJDkdUe/NTJTny76tSdqeAiuAVUQaGmb3/1BxFOBEaM6RUz3Vi7aVIaoZySr9RJEY4TEakp5BgThRXjq9IPHxhnAMJLmCQ2n7u+JFHGlJjwnfmuar6Wm/VeokOL72UijRODZR2HCoI5gHgcUEmwZhMDCEtqdoV4hCTC2oRWMSG48ycvQvus7hq+Pa82Tos4yuAQHIEacMEFaIAb0AQtgMEjeAav4M16sl6sd+tj1lqyipl98EfW5w+95Zb</latexit><latexit sha1_base64="yj9/AQBl/Ff3zjnmVRtfTUJ60W4=">ACAXicbZDLSsNAFIYn9VbrLepGcDNYhApSEhF0IxTcuKxgL9CGMJlO2qEzkzAzEWqIG1/FjQtF3PoW7nwbJ20W2vrDwMd/zmHO+YOYUaUd59sqLS2vrK6V1ysbm1vbO/buXltFicSkhSMWyW6AFGFUkJampFuLAniASOdYHyd1zv3RCoaiTs9iYnH0VDQkGKkjeXbBxN4Bfsc6ZHkqaJDkdUe/NTJTny76tSdqeAiuAVUQaGmb3/1BxFOBEaM6RUz3Vi7aVIaoZySr9RJEY4TEakp5BgThRXjq9IPHxhnAMJLmCQ2n7u+JFHGlJjwnfmuar6Wm/VeokOL72UijRODZR2HCoI5gHgcUEmwZhMDCEtqdoV4hCTC2oRWMSG48ycvQvus7hq+Pa82Tos4yuAQHIEacMEFaIAb0AQtgMEjeAav4M16sl6sd+tj1lqyipl98EfW5w+95Zb</latexit>

z ∼

N0

Y

i=0

N(zi; 0, 1)

<latexit sha1_base64="Adyvm6AfK+mrFtiok2pvdDuinc=">ACI3icbVDLSsNAFJ3UV62vqks3g0WoUEoigqIBTeuSgX7gCaGyWTSDp08mJkIbci/uPFX3LhQihsX/ouTtAtPTBwOdc5t7jRIwKqetfWmFldW19o7hZ2tre2d0r7x90RBhzTNo4ZCHvOUgQRgPSlQy0os4Qb7DSNcZ3WZ+94lwQcPgQY4jYvloEFCPYiSVZJevTB/JoeMlkxSagvrQjHjo2gm90dPHpGnraR7AiCXNtAonNr2GOqxBA57a5Ype13PAZWLMSQXM0bLU9MNceyTQGKGhOgbeiStBHFJMSNpyYwFiRAeoQHpKxognwgryW9M4YlSXOiFXL1Awlz9PZEgX4ix76hktrBY9DLxP68fS+/SmgQxZIEePaRFzMoQ5gVBl3KCZsrAjCnKpdIR4ijrBUtZUCcbiycukc1Y3FL8/rzRq8zqK4AgcgyowAVogDvQAm2AwTN4Be/gQ3vR3rSp9jmLFrT5zCH4A+37B8O0ouY=</latexit><latexit sha1_base64="Adyvm6AfK+mrFtiok2pvdDuinc=">ACI3icbVDLSsNAFJ3UV62vqks3g0WoUEoigqIBTeuSgX7gCaGyWTSDp08mJkIbci/uPFX3LhQihsX/ouTtAtPTBwOdc5t7jRIwKqetfWmFldW19o7hZ2tre2d0r7x90RBhzTNo4ZCHvOUgQRgPSlQy0os4Qb7DSNcZ3WZ+94lwQcPgQY4jYvloEFCPYiSVZJevTB/JoeMlkxSagvrQjHjo2gm90dPHpGnraR7AiCXNtAonNr2GOqxBA57a5Ype13PAZWLMSQXM0bLU9MNceyTQGKGhOgbeiStBHFJMSNpyYwFiRAeoQHpKxognwgryW9M4YlSXOiFXL1Awlz9PZEgX4ix76hktrBY9DLxP68fS+/SmgQxZIEePaRFzMoQ5gVBl3KCZsrAjCnKpdIR4ijrBUtZUCcbiycukc1Y3FL8/rzRq8zqK4AgcgyowAVogDvQAm2AwTN4Be/gQ3vR3rSp9jmLFrT5zCH4A+37B8O0ouY=</latexit><latexit sha1_base64="Adyvm6AfK+mrFtiok2pvdDuinc=">ACI3icbVDLSsNAFJ3UV62vqks3g0WoUEoigqIBTeuSgX7gCaGyWTSDp08mJkIbci/uPFX3LhQihsX/ouTtAtPTBwOdc5t7jRIwKqetfWmFldW19o7hZ2tre2d0r7x90RBhzTNo4ZCHvOUgQRgPSlQy0os4Qb7DSNcZ3WZ+94lwQcPgQY4jYvloEFCPYiSVZJevTB/JoeMlkxSagvrQjHjo2gm90dPHpGnraR7AiCXNtAonNr2GOqxBA57a5Ype13PAZWLMSQXM0bLU9MNceyTQGKGhOgbeiStBHFJMSNpyYwFiRAeoQHpKxognwgryW9M4YlSXOiFXL1Awlz9PZEgX4ix76hktrBY9DLxP68fS+/SmgQxZIEePaRFzMoQ5gVBl3KCZsrAjCnKpdIR4ijrBUtZUCcbiycukc1Y3FL8/rzRq8zqK4AgcgyowAVogDvQAm2AwTN4Be/gQ3vR3rSp9jmLFrT5zCH4A+37B8O0ouY=</latexit><latexit sha1_base64="Adyvm6AfK+mrFtiok2pvdDuinc=">ACI3icbVDLSsNAFJ3UV62vqks3g0WoUEoigqIBTeuSgX7gCaGyWTSDp08mJkIbci/uPFX3LhQihsX/ouTtAtPTBwOdc5t7jRIwKqetfWmFldW19o7hZ2tre2d0r7x90RBhzTNo4ZCHvOUgQRgPSlQy0os4Qb7DSNcZ3WZ+94lwQcPgQY4jYvloEFCPYiSVZJevTB/JoeMlkxSagvrQjHjo2gm90dPHpGnraR7AiCXNtAonNr2GOqxBA57a5Ype13PAZWLMSQXM0bLU9MNceyTQGKGhOgbeiStBHFJMSNpyYwFiRAeoQHpKxognwgryW9M4YlSXOiFXL1Awlz9PZEgX4ix76hktrBY9DLxP68fS+/SmgQxZIEePaRFzMoQ5gVBl3KCZsrAjCnKpdIR4ijrBUtZUCcbiycukc1Y3FL8/rzRq8zqK4AgcgyowAVogDvQAm2AwTN4Be/gQ3vR3rSp9jmLFrT5zCH4A+37B8O0ouY=</latexit>

Different initializations I(x; t4)

<latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit>

σ2

init = 1/N

<latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit>

σ2

init = 1/N

<latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit>

σ2

init = 0.25/N

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σ2

init = 4/N

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slide-34
SLIDE 34

200 400 600 800 1000

epochs

2.40 2.45 2.50 2.55 2.60 2.65

I(X; T4) layer 4 - relu

500 1000 1500 2000 2500

epochs

3.8 3.9 4.0 4.1 4.2

I(X; T4) layer 4 - hardtanh

500 1000 1500 2000 2500

epochs

2.4 2.6 2.8 3.0 3.2

I(X; T4) layer 4 - hardtanh

500 1000 1500 2000 2500

epochs

2.85 2.90 2.95 3.00 3.05 3.10 3.15

I(X; T4) layer 4 - hardtanh

Exploratory experiment: A binary classification example

Different non-linearities

x

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t1

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ˆ y

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W1

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W2

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t2

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t3

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W3

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W4

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W5

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t4

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y

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z

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y = sign(z0)

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z ∼

N0

Y

i=0

N(zi; 0, 1)

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Different initializations Poster @ Room 210 & 230 AB #110 I(x; t4)

<latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit><latexit sha1_base64="/Oz5leZPKrPXAU8th9Z52wD3FE=">ACBXicbZDLSsNAFIYnXmu9RV3qYrAIdVOSUlBwU3Cjuwr2Am0Ik+mkHTq5MHMiltCNG1/FjQtF3PoO7nwbJ20Ebf1h4OM/5zDn/F4suAL+jKWldW19YLG8XNre2dXNv6WiRFLWpJGIZMcjigkesiZwEKwTS0YCT7C2N7rM6u07JhWPwlsYx8wJyCDkPqcEtOWaR9flXkBg6Pnp/eQC/zBM3Nopds2SVbGmwotg51BCuRqu+dnrRzQJWAhUEKW6thWDkxIJnAo2KfYSxWJCR2TAuhpDEjDlpNMrJvhEO3sR1K/EPDU/T2RkCpceDpzmxLNV/LzP9q3QT8cyflYZwAC+nsIz8RGCKcRYL7XDIKYqyBUMn1rpgOiSQUdHBFHYI9f/IitKoVW/NrVSv5nEU0CE6RmVkozNUR1eogZqIogf0hF7Qq/FoPBtvxvusdcnIZw7QHxkf3EZl9M=</latexit>

σ2

init = 1/N

<latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit>

σ2

init = 1/N

<latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit><latexit sha1_base64="4JmNw1tioRuSTP1/GguJ3sBmRzA=">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</latexit>

σ2

init = 0.25/N

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σ2

init = 4/N

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