generative adversarial nets - courses.cs.washington.edu

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Discussion Board: Available on Ed Zoom Link: Available on Canvas Instructor Contact: [email protected] Course Webpage: https://courses.cs.washington.edu/courses/cse599i/20au/ CSE 599I: Generative Models University of Washington Lecture 10 Instructor: John Thickstun Generative Adversarial Nets

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Page 1: Generative Adversarial Nets - courses.cs.washington.edu

Discussion Board: Available on Ed

Zoom Link: Available on Canvas

Instructor Contact: [email protected]

Course Webpage: https://courses.cs.washington.edu/courses/cse599i/20au/

CSE 599I: Generative Models University of WashingtonLecture 10

Instructor: John Thickstun

Generative Adversarial Nets

Page 2: Generative Adversarial Nets - courses.cs.washington.edu

Latent Variable Models

CSE 599I: Generative Models University of WashingtonLecture 10

• Given finite samples , and unlimited samples

• Generative latent variable model:

• Learn the marginal defined by

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z ⇠ r.<latexit sha1_base64="7ZpJ5/yCkLJQNnCm+DkAGYFaCtU=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBjwYt4qmBbpV1KNk3b0CS7JLNCXforvHhQxKs/x5v/xrTdg7a+EHh4Z4bMvFEihUXf//YKK6tr6xvFzdLW9s7uXnn/oGnj1DDeYLGMzX1ELZdC8wYKlPw+MZyqSPJWNLqa1luP3FgR6zscJzxUdKBFXzCKznp4Ih0rFDHVbrniV/2ZyDIEOVQgV71b/ur0YpYqrpFJam078BMMM2pQMMknpU5qeULZiA5426Gmitswmy08ISfO6ZF+bNzTSGbu74mMKmvHKnKdiuLQLtam5n+1dor9yzATOkmRazb/qJ9KgjGZXk96wnCGcuyAMiPcroQNqaEMXUYlF0KwePIyNM+qgePb80rtJo+jCEdwDKcQwAXU4Brq0AAGCp7hFd484714797HvLXg5TOH8Efe5w/xlo/b</latexit><latexit sha1_base64="7ZpJ5/yCkLJQNnCm+DkAGYFaCtU=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBjwYt4qmBbpV1KNk3b0CS7JLNCXforvHhQxKs/x5v/xrTdg7a+EHh4Z4bMvFEihUXf//YKK6tr6xvFzdLW9s7uXnn/oGnj1DDeYLGMzX1ELZdC8wYKlPw+MZyqSPJWNLqa1luP3FgR6zscJzxUdKBFXzCKznp4Ih0rFDHVbrniV/2ZyDIEOVQgV71b/ur0YpYqrpFJam078BMMM2pQMMknpU5qeULZiA5426Gmitswmy08ISfO6ZF+bNzTSGbu74mMKmvHKnKdiuLQLtam5n+1dor9yzATOkmRazb/qJ9KgjGZXk96wnCGcuyAMiPcroQNqaEMXUYlF0KwePIyNM+qgePb80rtJo+jCEdwDKcQwAXU4Brq0AAGCp7hFd484714797HvLXg5TOH8Efe5w/xlo/b</latexit><latexit sha1_base64="7ZpJ5/yCkLJQNnCm+DkAGYFaCtU=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBjwYt4qmBbpV1KNk3b0CS7JLNCXforvHhQxKs/x5v/xrTdg7a+EHh4Z4bMvFEihUXf//YKK6tr6xvFzdLW9s7uXnn/oGnj1DDeYLGMzX1ELZdC8wYKlPw+MZyqSPJWNLqa1luP3FgR6zscJzxUdKBFXzCKznp4Ih0rFDHVbrniV/2ZyDIEOVQgV71b/ur0YpYqrpFJam078BMMM2pQMMknpU5qeULZiA5426Gmitswmy08ISfO6ZF+bNzTSGbu74mMKmvHKnKdiuLQLtam5n+1dor9yzATOkmRazb/qJ9KgjGZXk96wnCGcuyAMiPcroQNqaEMXUYlF0KwePIyNM+qgePb80rtJo+jCEdwDKcQwAXU4Brq0AAGCp7hFd484714797HvLXg5TOH8Efe5w/xlo/b</latexit><latexit sha1_base64="7ZpJ5/yCkLJQNnCm+DkAGYFaCtU=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBjwYt4qmBbpV1KNk3b0CS7JLNCXforvHhQxKs/x5v/xrTdg7a+EHh4Z4bMvFEihUXf//YKK6tr6xvFzdLW9s7uXnn/oGnj1DDeYLGMzX1ELZdC8wYKlPw+MZyqSPJWNLqa1luP3FgR6zscJzxUdKBFXzCKznp4Ih0rFDHVbrniV/2ZyDIEOVQgV71b/ur0YpYqrpFJam078BMMM2pQMMknpU5qeULZiA5426Gmitswmy08ISfO6ZF+bNzTSGbu74mMKmvHKnKdiuLQLtam5n+1dor9yzATOkmRazb/qJ9KgjGZXk96wnCGcuyAMiPcroQNqaEMXUYlF0KwePIyNM+qgePb80rtJo+jCEdwDKcQwAXU4Brq0AAGCp7hFd484714797HvLXg5TOH8Efe5w/xlo/b</latexit>

p✓(x) =

Z

Zp✓(x|z)r(z) dz.

<latexit sha1_base64="Pl7U9wQw9Eqal+eA8JgMp1zDB5s=">AAACInicbVDLSgMxFM34rPVVdekmWIQWpMyIoC6EghtxVcE+sFOGTJq2oZnMkNwR27Hf4sZfceNCUVeCH2P6WNTWA4HDOeeSe48fCa7Btr+thcWl5ZXV1Fp6fWNzazuzs1vRYawoK9NQhKrmE80El6wMHASrRYqRwBes6ncvh371ninNQ3kLvYg1AtKWvMUpASN5mXMceS50GJDcQx5fYJdL8BI3INChRCR3g8FU4LGfV7l+HrtHzX7By2Ttgj0CnifOhGTRBCUv8+k2QxoHTAIVROu6Y0fQSIgCTgUbpN1Ys4jQLmmzuqGSBEw3ktGJA3xolCZuhco8CXikTk8kJNC6F/gmOdxdz3pD8T+vHkPrrJFwGcXAJB1/1IoFhhAP+8JNrhgF0TOEUMXNrph2iCIUTKtpU4Ize/I8qRwXHMNvTrLF60kdKbSPDlAOOegUFdEVKqEyougJvaA39G49W6/Wh/U1ji5Yk5k99AfWzy+6GaMr</latexit><latexit sha1_base64="Pl7U9wQw9Eqal+eA8JgMp1zDB5s=">AAACInicbVDLSgMxFM34rPVVdekmWIQWpMyIoC6EghtxVcE+sFOGTJq2oZnMkNwR27Hf4sZfceNCUVeCH2P6WNTWA4HDOeeSe48fCa7Btr+thcWl5ZXV1Fp6fWNzazuzs1vRYawoK9NQhKrmE80El6wMHASrRYqRwBes6ncvh371ninNQ3kLvYg1AtKWvMUpASN5mXMceS50GJDcQx5fYJdL8BI3INChRCR3g8FU4LGfV7l+HrtHzX7By2Ttgj0CnifOhGTRBCUv8+k2QxoHTAIVROu6Y0fQSIgCTgUbpN1Ys4jQLmmzuqGSBEw3ktGJA3xolCZuhco8CXikTk8kJNC6F/gmOdxdz3pD8T+vHkPrrJFwGcXAJB1/1IoFhhAP+8JNrhgF0TOEUMXNrph2iCIUTKtpU4Ize/I8qRwXHMNvTrLF60kdKbSPDlAOOegUFdEVKqEyougJvaA39G49W6/Wh/U1ji5Yk5k99AfWzy+6GaMr</latexit><latexit sha1_base64="Pl7U9wQw9Eqal+eA8JgMp1zDB5s=">AAACInicbVDLSgMxFM34rPVVdekmWIQWpMyIoC6EghtxVcE+sFOGTJq2oZnMkNwR27Hf4sZfceNCUVeCH2P6WNTWA4HDOeeSe48fCa7Btr+thcWl5ZXV1Fp6fWNzazuzs1vRYawoK9NQhKrmE80El6wMHASrRYqRwBes6ncvh371ninNQ3kLvYg1AtKWvMUpASN5mXMceS50GJDcQx5fYJdL8BI3INChRCR3g8FU4LGfV7l+HrtHzX7By2Ttgj0CnifOhGTRBCUv8+k2QxoHTAIVROu6Y0fQSIgCTgUbpN1Ys4jQLmmzuqGSBEw3ktGJA3xolCZuhco8CXikTk8kJNC6F/gmOdxdz3pD8T+vHkPrrJFwGcXAJB1/1IoFhhAP+8JNrhgF0TOEUMXNrph2iCIUTKtpU4Ize/I8qRwXHMNvTrLF60kdKbSPDlAOOegUFdEVKqEyougJvaA39G49W6/Wh/U1ji5Yk5k99AfWzy+6GaMr</latexit><latexit sha1_base64="Pl7U9wQw9Eqal+eA8JgMp1zDB5s=">AAACInicbVDLSgMxFM34rPVVdekmWIQWpMyIoC6EghtxVcE+sFOGTJq2oZnMkNwR27Hf4sZfceNCUVeCH2P6WNTWA4HDOeeSe48fCa7Btr+thcWl5ZXV1Fp6fWNzazuzs1vRYawoK9NQhKrmE80El6wMHASrRYqRwBes6ncvh371ninNQ3kLvYg1AtKWvMUpASN5mXMceS50GJDcQx5fYJdL8BI3INChRCR3g8FU4LGfV7l+HrtHzX7By2Ttgj0CnifOhGTRBCUv8+k2QxoHTAIVROu6Y0fQSIgCTgUbpN1Ys4jQLmmzuqGSBEw3ktGJA3xolCZuhco8CXikTk8kJNC6F/gmOdxdz3pD8T+vHkPrrJFwGcXAJB1/1IoFhhAP+8JNrhgF0TOEUMXNrph2iCIUTKtpU4Ize/I8qRwXHMNvTrLF60kdKbSPDlAOOegUFdEVKqEyougJvaA39G49W6/Wh/U1ji5Yk5k99AfWzy+6GaMr</latexit>

Variational Autoencoder Generative Adversarial Net

• Generative latent variable model

• Learn the pushforward defined by:

1. z ⇠ r,

2. x = g✓(z) ⇠ p✓(x).<latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit>

1. z ⇠ r,

2. x ⇠ p✓(·|z).<latexit sha1_base64="5E0G1GS7GiUm6xMxb4xc5z95ems=">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</latexit><latexit sha1_base64="5E0G1GS7GiUm6xMxb4xc5z95ems=">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</latexit><latexit sha1_base64="5E0G1GS7GiUm6xMxb4xc5z95ems=">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</latexit><latexit sha1_base64="5E0G1GS7GiUm6xMxb4xc5z95ems=">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</latexit>

p✓(x) = r(g�1✓ (x))|rxg

�1✓ (x)|.

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Page 3: Generative Adversarial Nets - courses.cs.washington.edu

Maximize the Likelihood?

CSE 599I: Generative Models University of WashingtonLecture 10

• Generative latent variable model:

• The maximum likelihood estimator:

• The log-likelihood of a sample:

1. z ⇠ r,

2. x = g✓(z) ⇠ p✓(x).<latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit><latexit sha1_base64="9CaYLQfTr+Jh74ik7SsrRgPDGlg=">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</latexit>

log p✓(x) = log r(g�1✓ (x)) + logdet

�rxg

�1✓ (x)

�.

<latexit sha1_base64="zB3hp8sfUYhu1D5u/Fw9qKlNMvY=">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</latexit><latexit sha1_base64="zB3hp8sfUYhu1D5u/Fw9qKlNMvY=">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</latexit><latexit sha1_base64="zB3hp8sfUYhu1D5u/Fw9qKlNMvY=">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</latexit><latexit sha1_base64="zB3hp8sfUYhu1D5u/Fw9qKlNMvY=">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</latexit>

✓̂mle ⌘ argmax✓

Ex⇠p

log p✓(x) ⇡ argmax✓

1

n

nX

i=1

log p✓(xi), where xi ⇠ p.<latexit sha1_base64="3bYNzSKeYC5ne6HFFoR+leCMWYs=">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</latexit><latexit sha1_base64="3bYNzSKeYC5ne6HFFoR+leCMWYs=">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</latexit><latexit sha1_base64="3bYNzSKeYC5ne6HFFoR+leCMWYs=">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</latexit><latexit sha1_base64="3bYNzSKeYC5ne6HFFoR+leCMWYs=">AAAETXicfVJta9RAEM71qq3nW6sf/bJ4LbQSyqUICkUoWKUI0hPsC3TPsNlM7pZms3F3r01d94/5J/zsD/GbiJO7aHutupDMw8w8M/PsTlLmwthe71trrj1/4+bC4q3O7Tt3791fWn5wYNRYc9jnKlf6KGEGclHAvhU2h6NSA5NJDofJycs6fngK2ghVvLfnJQwkGxYiE5xZdMVLX+iIWWpHYFlMLVTWyRw8ofBxLE4JZXooWRVPEwiVzI5U6SY2Sdwr72NXEWqEJCWScjUkZZO8Vq0jvSy1qq6VyTTjLvKu8NSMZezEi8h/KK7yY7EekulM5GwEGsgK+ppuK34jXur2NnqTQ66DqAHdoDn9eHnO0VTxsYTC8pwZcxz1SjtwTFvBUXWHjg2UjJ+wIRwjLJgEM3CTS/ZkFT0pyZTGr7Bk4r3McEya+l5CtOZcJrW1I1mbmmDCnKEWjIRJIsPKIA3CoWblSPAqzHLFbFjzrVK58WR2GJs9HzhRlGMLBZ/Oko1zYhWpH5WkQgO3+TkCxrVAOYSPGN6yxafvdOgOoF4Nb7H8XgmaWaWfuMmriMKj/iENa/S/RFb9TkT010RXv18K1rvGYuMUMgqlcfVP5KpuoaGAM66kZEXqKKr3jp4yXYOLSfsMFaU7uNZSoIYjR0WRpeK08sebA7fm3bp3qx1CuhHdwiaYZsQnoJ/pVndztYNdLveYUr3b+VNkNt5E4+wi/q9BHC2UljhxfgAalVI9sZ3Zglwrg+uD8d2Lkriu0dXlvA4ONjcixO+edrffNIu7GDwKHgdrQRQ8C7aD3aAf7Ae8RVqvW3utfvtr+3v7R/vnNHWu1XAeBjNnfuEX9Sl3Vg==</latexit>

Page 4: Generative Adversarial Nets - courses.cs.washington.edu

What are our options?

CSE 599I: Generative Models University of WashingtonLecture 10

• The maximum likelihood estimator:

• Write down parameterized families with simple inverses and Jacobians?

• Work hard and compute the inverses and Jacobians?

• Generative adversarial nets: Give up on the MLE and try something else.

argmax✓

1

n

nX

i=1

log r(g�1✓ (xi)) + logdet

�rxg

�1✓ (xi)

�.

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Page 5: Generative Adversarial Nets - courses.cs.washington.edu

Revisiting the MLE

CSE 599I: Generative Models University of WashingtonLecture 10

• The MLE minimizes KL divergence:

• Where KL divergence is given by

• What’s so special about KL anyway?

D(p k q) =

Z

Xp(x) log

p(x)

q(x).

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argmin✓

D(p k p✓) = argmin✓

H(p) +D(p k p✓)

= argmin✓

Ex⇠p

� logp✓(x)

p(x)p(x)

= argmax✓

Ex⇠p

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Page 6: Generative Adversarial Nets - courses.cs.washington.edu

What’s so special about MLE?

CSE 599I: Generative Models University of WashingtonLecture 10

• The MLE is consistent:

• That’s nice, but the same argument applies to the reverse KL

• Or to the symmetrized KL

1. D(p k p✓) � 0 and D(p k p✓) = 0 i↵ p = p✓.

2. argmax✓

Ex⇠p

log p✓(x) = argmin✓

D(p k p✓).

3. limn!1

1

n

nX

i=1

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Page 7: Generative Adversarial Nets - courses.cs.washington.edu

Monte Carlo Estimation

CSE 599I: Generative Models University of WashingtonLecture 10

• We can construct a Monte Carlo estimate of the MLE:

• How would we do this for reverse-KL?

argmin✓

D(p✓ k p) = argmin✓

Ex⇠p✓

logp✓(x)

p(x)

⇡ argmin✓

1

n

nX

i=1

logp✓(x)

p(x), where xi ⇠ p✓.

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argmin✓

D(p k p✓) = argmax✓

Ex⇠p

log p✓(x)

⇡ argmax✓

1

n

nX

i=1

log p✓(xi), where xi ⇠ p.<latexit sha1_base64="nyvx/uNVoOv1osWzuyN/gk6/JN8=">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</latexit><latexit sha1_base64="nyvx/uNVoOv1osWzuyN/gk6/JN8=">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</latexit><latexit sha1_base64="nyvx/uNVoOv1osWzuyN/gk6/JN8=">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</latexit><latexit sha1_base64="nyvx/uNVoOv1osWzuyN/gk6/JN8=">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</latexit>

Page 8: Generative Adversarial Nets - courses.cs.washington.edu

Information Divergences

CSE 599I: Generative Models University of WashingtonLecture 10

• We can’t compute so the Monte Carlo estimator isn’t that useful.

• This frees us to think about other information divergences. E.g.

• We can construct lower bounds on an f-divergence.

• Does the choice of information divergence make a difference? Unclear!

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Df (p k q) ⌘Z

Xq(x)f

✓p(x)

q(x)

◆dx.

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Page 9: Generative Adversarial Nets - courses.cs.washington.edu

f-Divergences

CSE 599I: Generative Models University of WashingtonLecture 10

f(x) = x log x<latexit sha1_base64="JaY0pjjSFhmdveql+OBG6EKZEuM=">AAADbnicfVJtaxNBEN4mvtT40lbBLyIuhkAqR8kFQaEIBfNBCmIE0wZyMeztzV2W7suxu6kX1/sfftV/5b/wJ7iXXqq11oFjHmaeZ3ZmbuKcM2N7vR8bjea16zdubt5q3b5z997W9s79I6MWmsKIKq70OCYGOJMwssxyGOcaiIg5HMcnr6v88Slow5T8YJc5TAXJJEsZJdaHPqbdYhe/wkXEVYaL2Xa7t9dbGb4Mwhq0UW3D2U7jMEoUXQiQlnJizCTs5XbqiLaMcihb0cJATugJyWDioSQCzNSt2i5xx0cSnCrtP2nxKvqnwhFhBLHzwHuzFHHl7VxUrhKYgBMLhc8EcSyCwngZBJkm+ZzRIki5Ijao9FYpboJYlPhiPzZ9OXVM5gsLkp61ky44tgpXm8IJ00AtX3pAqGZ+IkznRBNq/T5brWgAfmQNb/0L73LQxCr9zEVEZ4LJ0q8gi4IK/Y9IijXRo38SXfVnErClq/1V5YxK7areGlxRzrBMKJZ4Xg1+TzIkfuJk4G9JMD/j2EVMpgk7LcpJf+q6pdstXaeFcTuM9qOkohn2GaIv0X6732n5ByV8okoIIpO1tHSD8yIX83V2lp7n/fWFf9/aZXDU3ws9fv+8fXBY3+EmeoSeoi4K0Qt0gN6gIRohijT6ir6h742fzYfNx80nZ9TGRq15gC5Ys/sLhbUgxg==</latexit><latexit sha1_base64="JaY0pjjSFhmdveql+OBG6EKZEuM=">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</latexit><latexit sha1_base64="JaY0pjjSFhmdveql+OBG6EKZEuM=">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</latexit><latexit sha1_base64="JaY0pjjSFhmdveql+OBG6EKZEuM=">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</latexit>

Df (p k q) = D(p k q).<latexit sha1_base64="73vz3N5Yd1D6G8i4Bu/6cJDUu6U=">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</latexit><latexit sha1_base64="73vz3N5Yd1D6G8i4Bu/6cJDUu6U=">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</latexit><latexit sha1_base64="73vz3N5Yd1D6G8i4Bu/6cJDUu6U=">AAADf3icfVJdaxNBFJ10/ajrR1N9FGEwBKssYRMElSIUzIMUxAimDWRCmJ2d3Qydj3VmUhPHffLX+Kq/xn/jbLppjbVeWO7Ze8+5M2e4ScGZsXH8q7EVXLt+4+b2rfD2nbv3dpq794+MmmtCh0RxpUcJNpQzSYeWWU5HhaZYJJweJydvqv7xKdWGKfnRLgs6ETiXLGMEW1+aNh+FKEvZqStK96mEryFi8uK/M2224k68CngZdGvQAnUMprtbhyhVZC6otIRjY8bduLATh7VlhNMyRHNDC0xOcE7HHkosqJm4lY8Stn0lhZnS/pMWrqp/KhwWRmA7i3w2S5FU2c5ElSqBiTi2dOE7UZKIaGG8jEa5xsWMkUWUcYVtVOmtUtxEiSjh5n1s9nLimCzmlkpydp1szqFVsHo6mDJNieVLDzDRzDuCZIY1JtY/cBiiPvWWNX3nT3hfUI2t0s8cwjoXTJb+CXIUVeh/RLxYEz36J9EhrvKU2tLV+apxRmV2NW8NrhhnWC4USz2vBhdOBtg7Tvt+uQTzHkeu3o5FOe5N3F7pnpauHULY6qJ9lFY0w75Q9BXtt3rt0B8o6WeihMAyXUtL1z8fstmvu9PsvO+3r/v3rl0GR71O1+MPz1sHh/UeboOH4DHYA13wAhyAt2AAhoCAb+A7+AF+Bo3gSdAJ4jPqVqPWPAAbEbz6DUwGJ2w=</latexit><latexit sha1_base64="73vz3N5Yd1D6G8i4Bu/6cJDUu6U=">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</latexit>

f(x) = � log x<latexit sha1_base64="DKqWKPTWk5cNtU6a3EnNpA7liOk=">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</latexit><latexit sha1_base64="DKqWKPTWk5cNtU6a3EnNpA7liOk=">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</latexit><latexit sha1_base64="DKqWKPTWk5cNtU6a3EnNpA7liOk=">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</latexit><latexit sha1_base64="DKqWKPTWk5cNtU6a3EnNpA7liOk=">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</latexit>

Df (p k q) = D(q k p).<latexit sha1_base64="Qigid8tRVpyrP94CNe6NIgG3YZw=">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</latexit><latexit sha1_base64="Qigid8tRVpyrP94CNe6NIgG3YZw=">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</latexit><latexit sha1_base64="Qigid8tRVpyrP94CNe6NIgG3YZw=">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</latexit><latexit sha1_base64="Qigid8tRVpyrP94CNe6NIgG3YZw=">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</latexit>

Df (p k q) = Ex⇠q

f

✓p(x)

q(x)

◆�� f

✓E

x⇠q

p(x)

q(x)

�◆= f(1) = 0.

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• For example, suppose . Then

• If then

• Non-negative:

Page 10: Generative Adversarial Nets - courses.cs.washington.edu

A Divergence Lower Bound

CSE 599I: Generative Models University of WashingtonLecture 10

• For any function

• The function is the convex conjugate of , defined by:

• Furthermore [Nguyen, Wainwright, and Jordan 2010]:

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Df (p k q) = supT :X!R

E

x⇠pT (x)� E

x⇠qf⇤(T (x))

�.

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Df (p k q) � Ex⇠p

T (x)� Ex⇠q

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f⇤ : R ! R<latexit sha1_base64="jxTxoMxTzCjZipLuDSp8FWqaxlI=">AAADgXicfVLbbtNAEN3GXEq4pfDIAyuiSG1lVXaEBCIvlcgDqoQIiLSRsiFar9fOqnuxdjclYfEjX8MrfAx/wzp1SkMpI1lzPHPO7MxokoIzY6Po11YjuHHz1u3tO8279+4/eNjaeXRs1FwTOiSKKz1KsKGcSTq0zHI6KjTFIuH0JDl9XeVPzqg2TMmPdlnQicC5ZBkj2PrQtPU0+7QPX0EksJ0liftQQmTVpd9pqx0dRCuDV0FcgzaobTDdaRyhVJG5oNISjo0Zx1FhJw5rywinZRPNDS0wOcU5HXsosaBm4laTlLDjIynMlPaftHAVvaxwWJiqt9B7sxRJ5e1MVK4SmJBjSxc+EyaJCBfGy2iYa1zMGFmEGVfYhpXeKsVNmIgSbvZjs5cTx2Qxt1SS83ayOYd+JdXyYMo0JZYvPcBEMz8RJDOsMbF+xc0m6lM/sqZv/QvvCqqxVXrfIaxzwWTpV5CjsEL/I+LFmujRP4kOcZWn1Jau9teVMyqzq3prcE05w3KhWOp5NfgzyQD7idO+Py/B/Iwjh5jMUna2KMfdidst3V7pOk0I2zHqobSiGfaFoq+o1+52mv5BST8TJQSW6Vpauv5Fkc18nZ1mF3l/ffHft3YVHHcPYo/fP28fHtV3uA2egGdgF8TgBTgEb8AADAEB38B38AP8DIJgL4iC7jm1sVVrHoMNC3q/ASHPJ1c=</latexit><latexit sha1_base64="jxTxoMxTzCjZipLuDSp8FWqaxlI=">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</latexit><latexit sha1_base64="jxTxoMxTzCjZipLuDSp8FWqaxlI=">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</latexit><latexit sha1_base64="jxTxoMxTzCjZipLuDSp8FWqaxlI=">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</latexit>

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f⇤(t) ⌘ supx{tx� f(x)}.

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Page 11: Generative Adversarial Nets - courses.cs.washington.edu

• Solve a saddle-point problem:

• Use an expressive parameterized family of functions

• Adversarial: optimize to minimize the objective, and to maximize it.

• The objective only requires samples from :

Generative Adversarial Nets

CSE 599I: Generative Models University of WashingtonLecture 10

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Page 12: Generative Adversarial Nets - courses.cs.washington.edu

5-Minute Break

CSE 599I: Generative Models University of WashingtonLecture 10

Page 13: Generative Adversarial Nets - courses.cs.washington.edu

• Solve a saddle-point problem:

• Using the fact that:

• Where does this fact come from?

Generative Adversarial Nets

CSE 599I: Generative Models University of WashingtonLecture 10

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Df (p k q) = supT :X!R

E

x⇠pT (x)� E

x⇠qf⇤(T (x))

�.

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Page 14: Generative Adversarial Nets - courses.cs.washington.edu

• The function is the convex conjugate of , defined by:

• For a convex, lower-semicontinuous function : (Fenchel duality).

• A variational representation of :

• Lift a variational representation of to a variational representation of

Fenchel Duality

CSE 599I: Generative Models University of WashingtonLecture 10

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f⇤(t) ⌘ supx{tx� f(x)}.

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Df (p k q) =

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q(x)

◆dx.

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f(t) = f⇤⇤(t) ⌘ supt{tx� f⇤(t)}.

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Page 15: Generative Adversarial Nets - courses.cs.washington.edu

Proposition [Nguyen, Wainwright, and Jordan 2010]:

Proof:

Proof of the Lower Bound

CSE 599I: Generative Models University of WashingtonLecture 10

Df (p k q) =

Z

Xq(x)f

✓p(x)

q(x)

◆dx =

Z

Xq(x) sup

t

tp(x)

q(x)� f⇤(t)

�dx

=

Z

Xsupt

[tp(x)� f⇤(t)q(x)] dx = supT :X!R

Z

X(T (x)p(x)� f⇤(T (x))q(x)) dx

= supT :X!R

E

x⇠pT (x)� E

x⇠qf⇤(T (x))

�.

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Df (p k q) = supT :X!R

E

x⇠pT (x)� E

x⇠qf⇤(T (x))

�.

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Page 16: Generative Adversarial Nets - courses.cs.washington.edu

• Choose , resulting in

• The Jensen-Shannon Divergence is given by

• The convex conjugate of is .

• Parameterize . Then

The Goodfellow GAN

CSE 599I: Generative Models University of WashingtonLecture 10

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f<latexit sha1_base64="/Ay5zcjyvSyNEmiTMlPRD1udfc8=">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</latexit><latexit sha1_base64="/Ay5zcjyvSyNEmiTMlPRD1udfc8=">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</latexit><latexit sha1_base64="/Ay5zcjyvSyNEmiTMlPRD1udfc8=">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</latexit><latexit sha1_base64="/Ay5zcjyvSyNEmiTMlPRD1udfc8=">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</latexit>

f⇤(t) = � log(1� et)<latexit sha1_base64="4vWPv5YqB/xfuvLFNvaB1wsn3tk=">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</latexit><latexit sha1_base64="4vWPv5YqB/xfuvLFNvaB1wsn3tk=">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</latexit><latexit sha1_base64="4vWPv5YqB/xfuvLFNvaB1wsn3tk=">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</latexit><latexit sha1_base64="4vWPv5YqB/xfuvLFNvaB1wsn3tk=">AAADhHicfVLfaxNBEN42Wuv5q9UHH3xZDIW0XEsuKgpFKdiHIogVTFvopmFvby5Zunt77G5i4np/ja/6B/nfOJdG21h04G4+Zuab+WaZtFTS+Xb759Jy48bNlVurt6M7d+/df7C2/vDImZEV0BVGGXuScgdKFtD10is4KS1wnSo4Ts/f1vnjMVgnTfHJT0voaT4oZC4F9xjqrz3Oz7ZafpO+pttMmUErodsUzvxmf63Z3mnPjF4HyRw0ydwO++vLXZYZMdJQeKG4c6dJu/S9wK2XQkEVsZGDkotzPoBThAXX4HphtkFFNzCS0dxY/ApPZ9GrjMC109wPY/RuqtPa+6GuXU1wseIeJpiJ01THE4c0iAeWl0MpJnGuDPdxzffGKFfRRTE+f9ULsihHHgpxoSUfKeoNrV+MZtKC8GqKgAsrcR0qhtxy4fFdo4jtA+5r4T22/1CC5d7YrcC4HWhZVLj/gMU1+l8hn/wuRIQtM8gZlC7UP6lMTbZQwGdhtOZFFhjuVQU25rYGlxoOOWrN9vEatER1J4HJIs/keFKddnqhVYXNKmxElDYTtotDsMzJL8C+st1mZyPCKVdnXFCrsP+nyWJ+nu3nl/l/CQmsMFajYnUE1qO3Mx8tNhTWODwMzB9ctsRDTP4+u+vgqLOTIP74vLn3bn6Sq+QJeUpaJCEvyR45IIekSwSpyDfynfxorDTixrPGi4vS5aU55xFZsMabX6FaJgY=</latexit>

T'(x) = log(d'(x))<latexit sha1_base64="XxiWXkIl4TzQlgGzELGJNjg6oMo=">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</latexit><latexit sha1_base64="XxiWXkIl4TzQlgGzELGJNjg6oMo=">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</latexit><latexit sha1_base64="XxiWXkIl4TzQlgGzELGJNjg6oMo=">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</latexit><latexit sha1_base64="XxiWXkIl4TzQlgGzELGJNjg6oMo=">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</latexit>

✓f = argmin✓

sup'

E

x⇠plog d'(x) + E

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