Monge blunts Bayes: Hardness Results for Adversarial Training Zac - - PowerPoint PPT Presentation

monge blunts bayes hardness results for adversarial
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Monge blunts Bayes: Hardness Results for Adversarial Training Zac - - PowerPoint PPT Presentation

Monge blunts Bayes: Hardness Results for Adversarial Training Zac Cranko Aditya Krishna Menon Richard Nock Cheng Soon Ong Zhan Shi Christian Walder Overview Hardness results on adversarial training. Key result applicable to a learner:


slide-1
SLIDE 1

Monge blunts Bayes: Hardness Results for Adversarial Training

Zac Cranko Aditya Krishna Menon Cheng Soon Ong Richard Nock Zhan Shi Christian Walder

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SLIDE 2

data shot

2 blunts — Cranko et al.

Overview

  • Hardness results on adversarial training. Key result applicable to a learner:
  • optimising any loss satisfying a mild statistical requirement, and
  • learning a classifier from any class satisfying a mild continuity assumption
  • Implementation disentangles adversarial training:
  • 1. generate adversarial data (Key result solves the compression of an OT plan)
  • 2. training as usual
  • Toy experiments against “weakly activated” adversarial data


reveal generalisation improves on clean data as well

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SLIDE 3

( in inf)

`(H, A, D) ≥ (1 − ") · `

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3 blunts — Cranko et al.

Key players: Bayes

1- Classifiers 2- Adversaries 3- (differentiable) Loss 4- general adversarial loss

H ⊆ RX

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sha1_base64="R2oIf9g7NZKmruE3mt1vPpkxCxE=">ACw3icbVHLbhMxFHWGVxleKSzZWKSVWFTRTDdlWVFV6rIg0kZKQmR7kys+DH4AUTW/AZfw7b9B/4GzRCnZYr2To6915f3NoLbh1WfZnkDx4+Ojxk52n6bPnL16+Gu6+vrDaGwYTpoU2U0osCK5g4rgTMK0NEkFXNL1SZu/A7Gcq2+uE0NC0kqxUvOiIvUcpjtzSVxK0ZEOGvw3HpqwcE3LGUhs/N1/CvYto0e8vhKBtnXeD7IN+CEdrG+XJ3EOaFZl6CckwQa2d5VrtFIMZxJqBJ595CTdiaVDCLUBEJdhG61Rq8H5kCl9rEoxzu2NsdgUhrN5LGyvaX9m6uJf+Xm3lXflgErmrvQLGbQaUX2Gnc6oQLboA5sYmAMPjXzFbEUOYi2qmaW+OrysDsD4AH1XqLRQqQ+oVZz/jACy1gXgVYFS6f7soql7ypu+GCGqE62D8kB64bjRP/pKtHqRYF7tbD/eZOvzTald815z64OBzn2Tj/dDg6/rg1bge9Re/Qe5SjI3SMztA5miCGfqHf6ApdJ6fJOjGJuylNBtueN6gXSfMXGafmA=</latexit>

domain sign = class

A ⊆ XX

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domain

` : {−1, 1} × [0, 1] → R

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(conditional) Bayes risk proper

L(⇡)

.

= infc EY∼π`(Y, c)

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π

<latexit sha1_base64="L9+ZPceyhzbahWj3wpGT6nBa0c0=">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</latexit><latexit sha1_base64="L9+ZPceyhzbahWj3wpGT6nBa0c0=">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</latexit><latexit sha1_base64="L9+ZPceyhzbahWj3wpGT6nBa0c0=">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</latexit><latexit sha1_base64="L9+ZPceyhzbahWj3wpGT6nBa0c0=">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</latexit>

`

ψ = `

<latexit sha1_base64="0y6L3Odxy4eIhTLGlhpNI0rSkzQ=">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</latexit><latexit sha1_base64="0y6L3Odxy4eIhTLGlhpNI0rSkzQ=">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</latexit><latexit sha1_base64="0y6L3Odxy4eIhTLGlhpNI0rSkzQ=">AEjHicbVPdahNBFJ62UWvU2uqlN4NtQaTEpAgKUihKRcGLivYHumYnT2bTDszO85P2zDsY/g03upD+DaeTSLtbjqQcOZ83/n7Zk9qpHC+2/27sLjUunP3vL9oOHj1Yer649OXRFsBwOeCELe5wyB1JoOPDCSzg2FphKJRyl5x8q/OgCrBOF/u7HBvqKDbXIBWceXYPVxsJSDmIMZnkihayMibGibIsTxMuLKc7tKJMLxuD1fVupzs5dN7ozYx1Mjv7g7WlvSQreFCgPZfMuZNe1/h+ZNYLqFsJ8GBYfycDeETc0UuH6cdFPSTfRkNC8s/rSnE+/NiMiUc2OVIlMxP3JNrHLehp0En7/tR6FN8KD5tFAeJPUFrWSimbDAvRyjwbgV2CvlI2YZ9yhmu12rE8zQApxvQeBM1gaKQ8vMSPArLEBVYQH/MrC6vXmT5EKai2GwdTEiqoMvB/mWCtILW1zW4VRVSdNCZnQyHq0HT/TDPjVc8kIprOYeDZGrwumrOxQ1lFjCwNWFs4hjC/egHlq0R809l9bPFL2UyfQ4/kOM8NmFBxqRqzKrYKcu4UzbywZVBhcspKU3jXpNwaxgmsM15bBJMfYa3G+CzrgK9nDlHY94k3Phjl8TqktzIpeIeGbGCo2iKo8jUkKnm3Rs8HLuWr4HjWquo2jb8s3V1bgkv4fG+X7rJCBi9drtm8cbjd6XU7va/b67vZyu4TJ6R5+QF6ZE3ZJd8IvkgHDyk/wiv8mf1krdetda2dKXVyYxTwltdP6+A9G5dR</latexit><latexit sha1_base64="0y6L3Odxy4eIhTLGlhpNI0rSkzQ=">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</latexit>

composite loss link ψ : [0, 1] → R

<latexit sha1_base64="eUXmsClgur7BYC+oFqx8qgUHjlQ=">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</latexit><latexit sha1_base64="eUXmsClgur7BYC+oFqx8qgUHjlQ=">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</latexit><latexit sha1_base64="eUXmsClgur7BYC+oFqx8qgUHjlQ=">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</latexit><latexit sha1_base64="eUXmsClgur7BYC+oFqx8qgUHjlQ=">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</latexit>

`ψ(y, v)

.

= `(y, −1(v))

<latexit sha1_base64="FQWi2BI0lPpPhRICTC9xur37Y=">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</latexit><latexit sha1_base64="FQWi2BI0lPpPhRICTC9xur37Y=">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</latexit><latexit sha1_base64="FQWi2BI0lPpPhRICTC9xur37Y=">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</latexit><latexit sha1_base64="FQWi2BI0lPpPhRICTC9xur37Y=">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</latexit>

canonical loss

ψ

.

= −L0

<latexit sha1_base64="l6hu/y32eQZrbtosN/wuz7uocw=">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</latexit><latexit sha1_base64="l6hu/y32eQZrbtosN/wuz7uocw=">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</latexit><latexit sha1_base64="l6hu/y32eQZrbtosN/wuz7uocw=">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</latexit><latexit sha1_base64="l6hu/y32eQZrbtosN/wuz7uocw=">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</latexit>

blunt predictor

h

.

= ψ 1

2

  • <latexit sha1_base64="gptT/ebG3IuSyk0t7WX4heiafE=">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</latexit><latexit sha1_base64="gptT/ebG3IuSyk0t7WX4heiafE=">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</latexit><latexit sha1_base64="gptT/ebG3IuSyk0t7WX4heiafE=">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</latexit><latexit sha1_base64="gptT/ebG3IuSyk0t7WX4heiafE=">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</latexit>

= 0

<latexit sha1_base64="KdhJD46RQujLwUPIEn2mTEhbZlA=">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</latexit><latexit sha1_base64="KdhJD46RQujLwUPIEn2mTEhbZlA=">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</latexit><latexit sha1_base64="KdhJD46RQujLwUPIEn2mTEhbZlA=">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</latexit><latexit sha1_base64="KdhJD46RQujLwUPIEn2mTEhbZlA=">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</latexit>

(often) corresponding loss:

`(H, A, D)

.

= min

h∈H E(X,Y)∼D

 max

a∈A `(Y, h a(X))

  • <latexit sha1_base64="7JboIx635P7dUEXfb0sb2QEebM=">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</latexit><latexit sha1_base64="7JboIx635P7dUEXfb0sb2QEebM=">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</latexit><latexit sha1_base64="7JboIx635P7dUEXfb0sb2QEebM=">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</latexit><latexit sha1_base64="7JboIx635P7dUEXfb0sb2QEebM=">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</latexit>

( “hidden”)

ψ

<latexit sha1_base64="iWga5ZXxIOqjq9iJGKhQVoq+qk=">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</latexit><latexit sha1_base64="iWga5ZXxIOqjq9iJGKhQVoq+qk=">ACqHicbVHLbhMxFHWGVxkeTUGs2FiklRCqoplu6LKCDcsgkaYoiSLbc2dixS/5AY2s+Ri28EX8Dc40i07LlWwd3XMf9jnUCO58UfwdZA8ePnr85OBp/uz5i5eHw6NXl04Hy2DKtND2ihIHgiuYeu4FXBkLRFIBM7r5vONnP8A6rtU3vzWwlKRvOaM+JRaDd8cx0U3JVqo2rgwjrft8Wo4KsZF/g+KPdghPYxWR0N4qLSLEhQngni3LwsjF9GYj1nAtp8ERwYwjakgXmCikhwy9htbvFJylS41jYd5XGXvd0RiXRuK2mqlMSv3V1ul/wfNw+Pl9GrkzwoNjNojoI7DXeiYErboF5sU2AMvTWzFbE0uYT5LleW9PMI0F2JxCYET0PhQbS8yas+u0AEtIV0VWJWf3C5ygda8CbYvRkzqJH+gPpVBeG71z5N5W4o1aLC3fdwv7nTL092lXfNuQ8uz8ZlMS6/no0uPu2NO0Bv0Tv0HpXoI7pAX9AETRFDEf1Cv9Gf7EM2yWbZ95vSbLDveY16kdF/mMfUYg=</latexit><latexit sha1_base64="iWga5ZXxIOqjq9iJGKhQVoq+qk=">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</latexit><latexit sha1_base64="iWga5ZXxIOqjq9iJGKhQVoq+qk=">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</latexit>

kδk  δ∗

<latexit sha1_base64="N801/Fgd9SeWocT49UR5Je4XdcI=">AD6HicbZNLb9NAEMe3DY8SXikckZBFWgmhKrJ7gQtSBaoEocikTZVHKL1epws2fW6+0iJlr3xAbghrhw5Al+Gb8PaiWjtMJKt2fn9d3dmPE4KRpUOwz8bm60rV69d37rRvnr9p27ne17x0oYSaBPBNykGAFjObQ1QzGBQSME8YnCSzlyU/mYNUVOTv9KAEceTnGaUYO1D487DnfhTPAcbp8A0dn7B4CxYrt4/2Rl3umEvrCxYd6KV0UrOxpvb/6MU0EMh1wThpUaRmGhRxZLTQkD146NgKTGZ7A0Ls5qBGtirEBbs+kgaZkP7JdVBFL+wmCu14IlXcqynqsnK4P/Y0Ojs2cjSvDAacrK8KDMs0CIouxKkVALRbOEdTCT1uQZkiUm2veu3a7dY4qJBJjtgSGY1QqyE4mLKSUf/QUBFxL8KwWZt3cvi5RJMjoxst4M67vjPxRke9wTaU4r+OEl4cmgqVBV5Q31z1rxYqVJlyvnsczgngnOcpzbWeOG1yhSu9I2r0KAiQTSnkMjDUwSaSPm9xXVU6cfeMaglOPq/wye9pkgws2aLDC3bYZClkcOa50r40CWyplNz2nLPXbPAObhNLJxwm03WqPU/wDg/h3xmnuFn/KoOdPrzvF+Lwp70dv97sGL1bxvoQfoEXqMIvQUHaBX6Aj1EUGf0Q/0C/1ufWh9aX1tfVtKNzdWe+6jmrW+/wU5JlXS</latexit><latexit sha1_base64="N801/Fgd9SeWocT49UR5Je4XdcI=">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</latexit><latexit sha1_base64="N801/Fgd9SeWocT49UR5Je4XdcI=">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</latexit><latexit sha1_base64="N801/Fgd9SeWocT49UR5Je4XdcI=">AD6HicbZNLb9NAEMe3DY8SXikckZBFWgmhKrJ7gQtSBaoEocikTZVHKL1epws2fW6+0iJlr3xAbghrhw5Al+Gb8PaiWjtMJKt2fn9d3dmPE4KRpUOwz8bm60rV69d37rRvnr9p27ne17x0oYSaBPBNykGAFjObQ1QzGBQSME8YnCSzlyU/mYNUVOTv9KAEceTnGaUYO1D487DnfhTPAcbp8A0dn7B4CxYrt4/2Rl3umEvrCxYd6KV0UrOxpvb/6MU0EMh1wThpUaRmGhRxZLTQkD146NgKTGZ7A0Ls5qBGtirEBbs+kgaZkP7JdVBFL+wmCu14IlXcqynqsnK4P/Y0Ojs2cjSvDAacrK8KDMs0CIouxKkVALRbOEdTCT1uQZkiUm2veu3a7dY4qJBJjtgSGY1QqyE4mLKSUf/QUBFxL8KwWZt3cvi5RJMjoxst4M67vjPxRke9wTaU4r+OEl4cmgqVBV5Q31z1rxYqVJlyvnsczgngnOcpzbWeOG1yhSu9I2r0KAiQTSnkMjDUwSaSPm9xXVU6cfeMaglOPq/wye9pkgws2aLDC3bYZClkcOa50r40CWyplNz2nLPXbPAObhNLJxwm03WqPU/wDg/h3xmnuFn/KoOdPrzvF+Lwp70dv97sGL1bxvoQfoEXqMIvQUHaBX6Aj1EUGf0Q/0C/1ufWh9aX1tfVtKNzdWe+6jmrW+/wU5JlXS</latexit>

particular case, Madry et al.’18: s.t.

x

a

→ x + δ

<latexit sha1_base64="0UjQ21UiAexwvrn/dqj7IxXdD58=">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</latexit><latexit sha1_base64="0UjQ21UiAexwvrn/dqj7IxXdD58=">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</latexit><latexit sha1_base64="0UjQ21UiAexwvrn/dqj7IxXdD58=">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</latexit><latexit sha1_base64="0UjQ21UiAexwvrn/dqj7IxXdD58=">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</latexit>

ε

<latexit sha1_base64="rld6jd05QhufWNRJ69d1Vyob7g=">AB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMF+wFpKJvtpF262Q27m0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelHKmjed9O6WNza3tnfJuZW/4PCoenzS1jJTFtUcqm6EdHImcCWYZjN1VIkohjJxrfz/3OBJVmUjyZaYphQoaCxYwSY6WgNyEKU824FP1qzat7C7jrxC9IDQo0+9Wv3kDSLEFhKCdaB76XmjAnyjDKcVbpZRpTQsdkiIGlgiSow3x8sy9sMrAjaWyJYy7UH9P5CTRepEtjMhZqRXvbn4nxdkJr4NcybSzKCgy0Vxl0j3fn/7oApIZPLSFUMXurS0dEWpsShUbgr/68jpX9V9r+4/Xtcad0UcZTiDc7gEH26gAQ/QhBZQkPAMr/DmGOfFeXc+lq0lp5g5hT9wPn8Auc+Rhw=</latexit><latexit sha1_base64="rld6jd05QhufWNRJ69d1Vyob7g=">AB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMF+wFpKJvtpF262Q27m0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelHKmjed9O6WNza3tnfJuZW/4PCoenzS1jJTFtUcqm6EdHImcCWYZjN1VIkohjJxrfz/3OBJVmUjyZaYphQoaCxYwSY6WgNyEKU824FP1qzat7C7jrxC9IDQo0+9Wv3kDSLEFhKCdaB76XmjAnyjDKcVbpZRpTQsdkiIGlgiSow3x8sy9sMrAjaWyJYy7UH9P5CTRepEtjMhZqRXvbn4nxdkJr4NcybSzKCgy0Vxl0j3fn/7oApIZPLSFUMXurS0dEWpsShUbgr/68jpX9V9r+4/Xtcad0UcZTiDc7gEH26gAQ/QhBZQkPAMr/DmGOfFeXc+lq0lp5g5hT9wPn8Auc+Rhw=</latexit><latexit sha1_base64="rld6jd05QhufWNRJ69d1Vyob7g=">AB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMF+wFpKJvtpF262Q27m0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelHKmjed9O6WNza3tnfJuZW/4PCoenzS1jJTFtUcqm6EdHImcCWYZjN1VIkohjJxrfz/3OBJVmUjyZaYphQoaCxYwSY6WgNyEKU824FP1qzat7C7jrxC9IDQo0+9Wv3kDSLEFhKCdaB76XmjAnyjDKcVbpZRpTQsdkiIGlgiSow3x8sy9sMrAjaWyJYy7UH9P5CTRepEtjMhZqRXvbn4nxdkJr4NcybSzKCgy0Vxl0j3fn/7oApIZPLSFUMXurS0dEWpsShUbgr/68jpX9V9r+4/Xtcad0UcZTiDc7gEH26gAQ/QhBZQkPAMr/DmGOfFeXc+lq0lp5g5hT9wPn8Auc+Rhw=</latexit><latexit sha1_base64="rld6jd05QhufWNRJ69d1Vyob7g=">AB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMF+wFpKJvtpF262Q27m0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelHKmjed9O6WNza3tnfJuZW/4PCoenzS1jJTFtUcqm6EdHImcCWYZjN1VIkohjJxrfz/3OBJVmUjyZaYphQoaCxYwSY6WgNyEKU824FP1qzat7C7jrxC9IDQo0+9Wv3kDSLEFhKCdaB76XmjAnyjDKcVbpZRpTQsdkiIGlgiSow3x8sy9sMrAjaWyJYy7UH9P5CTRepEtjMhZqRXvbn4nxdkJr4NcybSzKCgy0Vxl0j3fn/7oApIZPLSFUMXurS0dEWpsShUbgr/68jpX9V9r+4/Xtcad0UcZTiDc7gEH26gAQ/QhBZQkPAMr/DmGOfFeXc+lq0lp5g5hT9wPn8Auc+Rhw=</latexit> -defeated by on iff

H

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A

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sha1_base64="m+PVbZjTx2L2th7tHcKc1S7OuyA=">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</latexit> ε → 0 <latexit 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sha1_base64="G4O3LJd+M3Pl5BO1E+eczfAZJ5M=">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</latexit><latexit sha1_base64="G4O3LJd+M3Pl5BO1E+eczfAZJ5M=">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</latexit><latexit sha1_base64="G4O3LJd+M3Pl5BO1E+eczfAZJ5M=">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</latexit>
slide-4
SLIDE 4
  • For any proper composite loss , classifiers , adversaries (+integrability

assumptions),

4 blunts — Cranko et al.

Main negative result

`

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H

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A

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`(H, A, D) ≥ ✓ ` − 1 2 · min

a2A a

+

<latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit>

βa

.

= max

h∈H{ϕ(P, f, π, 2L(1)) − ϕ(N, f, 1 − π, −2L(0))}

<latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">AEVnicbZPfatswFMadNl27E/T7WawG62h0Iw02L3ZbgZlo9DBGB2sbUocgqwcJ6KS7Epy2yD8BnuaPc32GnuAsWMnW2t3ApvD+X3y0Tn6HKWCG+v7Pxsrq821B+sbD1uPHj95utnenZqkwzOGJSPQgogYEV3BiuRUwSDVQGQk4iy4+FPzsCrThifpq5ymMJ0qHnNGLabG7W9hBFOuHFwqjWdv85bmLF0TEk4gRguCQklvRm7GQm5KmI7Y1S4ozwnoQuvqE5nfPe4R+IeCVPeI/shi/Ru0O2SPfIXfy5xsFcI9hYCv9sNsRSoyZ3S43bH7/vlIveDYBl0vOU6Hm+tfA8nCcskKMsENWY+KkdOaotZwKwQGYgpeyCTmGIoaISzMiVY8vJDmYmJE40PsqSMnt3h6PSmLmMUFm0beqsSP6PDTMbvx05rtLMgmKLQnEmiE1IcQdkwjUwK+YUKY5npWwGdWUWbypVqtSJ0unGuCiBxmOvdKQm2qK02U3WIDIRAO+JqBVa+euyGRzKeZrg7D4XTQFhD3ZCYs18l1FUey+GiUiAkp2yPVzeX8KqlCpU1scjy9gmuWSEnxbkNL56g1WZoXcZXaqTFLRIjEMQtQwGgXzmcKuCn+7T3lNcI64PF/szutscMsGdXZ4yw7rHQ9cmOxNQ1iodTS9fPcvcvrDV5BPgxGLoyk6wT3KMfDfJ/n/goUYEuD+qevh+c7vcDvx982e8cvF/6fcN76W17u17gvfEOvCPv2DvxmPer8aLxqrG9+mP1d3Otub6QrjSWe57ldVs/wHKTXf/</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">AEVnicbZPfatswFMadNl27E/T7WawG62h0Iw02L3ZbgZlo9DBGB2sbUocgqwcJ6KS7Epy2yD8BnuaPc32GnuAsWMnW2t3ApvD+X3y0Tn6HKWCG+v7Pxsrq821B+sbD1uPHj95utnenZqkwzOGJSPQgogYEV3BiuRUwSDVQGQk4iy4+FPzsCrThifpq5ymMJ0qHnNGLabG7W9hBFOuHFwqjWdv85bmLF0TEk4gRguCQklvRm7GQm5KmI7Y1S4ozwnoQuvqE5nfPe4R+IeCVPeI/shi/Ru0O2SPfIXfy5xsFcI9hYCv9sNsRSoyZ3S43bH7/vlIveDYBl0vOU6Hm+tfA8nCcskKMsENWY+KkdOaotZwKwQGYgpeyCTmGIoaISzMiVY8vJDmYmJE40PsqSMnt3h6PSmLmMUFm0beqsSP6PDTMbvx05rtLMgmKLQnEmiE1IcQdkwjUwK+YUKY5npWwGdWUWbypVqtSJ0unGuCiBxmOvdKQm2qK02U3WIDIRAO+JqBVa+euyGRzKeZrg7D4XTQFhD3ZCYs18l1FUey+GiUiAkp2yPVzeX8KqlCpU1scjy9gmuWSEnxbkNL56g1WZoXcZXaqTFLRIjEMQtQwGgXzmcKuCn+7T3lNcI64PF/szutscMsGdXZ4yw7rHQ9cmOxNQ1iodTS9fPcvcvrDV5BPgxGLoyk6wT3KMfDfJ/n/goUYEuD+qevh+c7vcDvx982e8cvF/6fcN76W17u17gvfEOvCPv2DvxmPer8aLxqrG9+mP1d3Otub6QrjSWe57ldVs/wHKTXf/</latexit>

f

.

= (L0) ψ1 h a

<latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit>

with and

ϕ(Q, f, u, v)

.

= Z

X

u · (f(x) + v)dQ(x)

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“+” ex. “-” ex.

Example: if and , then is Integral Probability Metric for class L(0) = L(1)

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π = 1/2

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βa

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{(L0) ψ1 h a : h 2 H}

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(more in paper / poster)

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slide-5
SLIDE 5

Example: if and , then is Integral Probability Metric for class

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  • For any proper composite loss , classifiers , adversaries (+integrability

assumptions),

5 blunts — Cranko et al.

Main negative result — consequence #1

`

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H

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A

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`(H, A, D) ≥ ✓ ` − 1 2 · min

a2A a

+

<latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit>

βa

.

= max

h∈H{ϕ(P, f, π, 2L(1)) − ϕ(N, f, 1 − π, −2L(0))}

<latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit>

f

.

= (L0) ψ1 h a

<latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">AD/nicbZPNbtNAEMe3NR8lfKVw5LIirShSG8UVElyQKlAlkDgUibSp4hCt1+NklV2v2Y+WaGWJ5+ABuCEkThy5whvwNqydiNYOI9kez+/uzPjcZxzpk2v92dtPbhy9dr1jRutm7du37nb3rx3rKVFPpUcqkGMdHAWQZ9wyHQa6AiJjDSTx7WfKTM1CayeydmecwEmSsZRYnxo3H6yleIogRQ+4J29iMbq0WMcUaYojnLN3ru9sFi+T6tHC2OyNW53et1eZXjVCZdOBy3taLy5/j1KJLUCMkM50XoY9nIzckQZRjkUrchqyAmdkQkMvZsRAXrkqvIKvO0jCU6l8ldmcBW9vMIRofVcxF4piJnqJiuD/2NDa9JnI8ey3BrI6OKg1HJsJC57hROmgBo+9w6hivlcMZ0SRajxHW21aufYfKIAZrtgKeG1gtxEkXzK6Ed/ABZSgb8loLW9mWRtnHKJlbVm+F8d/zng3RXWG6Ykud1HIty01jyBFfl4friqn+1UKlSOtWFz6DcyqFIFniIkPmXqtXpS+Leo0VzIHxaXWHgPnDezHxsdt5qsq59C9KRqCU4+r/FJ32mSDCzZosMLdthk1dB6ro0vTQFfKJVw3aJwz4tmgWdQDMORi2LhOuEKZf63gOLfFq+FV/gpD5szveoc73fDXjd8u985eLGc9w30AD1EOyhET9EBeoWOUB9R9Bn9RL/Q7+BT8CX4GnxbSNfXlmvuo5oFP/4CPxhb4g=</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit>

with and

ϕ(Q, f, u, v)

.

= Z

X

u · (f(x) + v)dQ(x)

<latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">AEVHicbVTdatRAFM62VeuqtVXwxgsHS2Gry7Lpjd4IRSkoeNGCbdslmUyOdkOnZmk87PtMswL+Dg+jOBb+ABeDbd2iZ1IOHwfd/MmXPOl6Sl4Mb2+79aS8sr9+4/WH3YfvT4ydrT9Y1nR6ZwmsEhK0ShByk1ILiCQ8utgEGpgcpUwHF69mnOH09BG16ob3ZWwkjSieI5Z9QiNF7/nqQw4crDuaJa09mb0N5qQKGdTKkuT3noEvyLnFdMt0mSQY5nJOEKzv2iaT2lFHhByEQl7CsJ28k0zBX4Zt8hb1lUJLn4WDa7ydgMpuJR6vb/Z7/WqRu0G8CDajxdofbyz9SLKCOQnKMkGNGcb90o481ZYzAXhvZ6Ck7IxOYIihohLMyFdNC2QLkYzkhcZHWVKht3d4Ko2ZyRSV87ubJjcH/8cNnc3fjzxXpbOg2FWi3AliCzKfAMm4BmbFDAPKNMe7EnZKNWUW59Ru1/K4cqIBzrgsLu1gvxEU5wJu8QERBYa8JWBVji+WyLj0pxPnK43w2N30BSQd6UTluviok6ncn5oWoiMVOWR+uaqfzWoGq7JTcDbK7hghZQUZ5tYOkOtcWYxy7U2VIXJWhRGIM0CNGgWaoRdwqrmrvbfw0NwUm4Mp7J/UmTG9xwgya3d8PtNbnK1cgbi6VpEP7auL0Q/IfQLHAKYRiPfJKvxnfYTl+bBD+HfFogJdHjc9fTc42unF/V58sLO5+3Hh9XoZfQ6kRx9C7ajT5H+9FhxKLfrRetVy2y/HP5zwr+Ja6kS63FnudRba2s/QWY3tr</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit>

“+” ex.

L(0) = L(1)

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π = 1/2

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βa

<latexit sha1_base64="3HAsMguSPk3GQ/+xe9TMts0uQ=">AD0XicbZJLb9NAEMe3NY8SXi0cuViklThUkd1LuSBVoEogcSiCtImSKFqvx8kq+2IfLdHKEuLKkSt8Cb4N34a1E9HaYaW1RvP7z87DkylGjU2SP1vb0a3bd+7u3Ovcf/Dw0ePdvSfnRjpNoE8k3qQYQOMCuhbahkMlAbMwYX2eJNxS8uQRsqxSe7VDheCZoQm2wTXcH2dg8RTvT3e7S+pT7xpGuji9bnbLq3/XucS+I4CEsYNmaUJspOPNaWEgZlZ+wMKEwWeAajYArMwUx8XEZHwRPHhdShytsXHtvRnjMjVnyLCg5tnPTZpXzf2zkbPFy4qlQzoIgq0SFY7GVcdV+nFMNxLJlMDRNQakznWmNgwpE6nkcepmQZYHIjmDUa8jON1ZySLyFBzKWG8MlBi87BTZFxWUFnTjeH4cN0wh+B4pA7ZqmWV02c8erRTLI8rtuLm8H1/BquSqVNYcpQvYArIjnHIvdji5dBa5wqK9uVTaq0VKCZNCZgYKyFSaD34nQVbVa/n3ZEgwDrusr/LDNBtds0Gan1+y0zXIo4HPgxobWNLCVUnPfK0v/qmw3eAnlKJ34cZ9N92gNGw6lP+eMeDImx52t7pTeP8qJcmvfTDUfk9Xrfd9Az9By9QCk6RifoLTpDfUQRz/QT/Qr+hgto6/Rt5V0e2sd8xQ1TvT9L8wTJY=</latexit><latexit sha1_base64="3HAsMguSPk3GQ/+xe9TMts0uQ=">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</latexit><latexit sha1_base64="3HAsMguSPk3GQ/+xe9TMts0uQ=">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</latexit><latexit sha1_base64="3HAsMguSPk3GQ/+xe9TMts0uQ=">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</latexit>

{(L0) ψ1 h a : h 2 H}

<latexit sha1_base64="tcN9wKvJZoNTUg2dml+F0LfCtA=">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</latexit><latexit sha1_base64="tcN9wKvJZoNTUg2dml+F0LfCtA=">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</latexit><latexit sha1_base64="tcN9wKvJZoNTUg2dml+F0LfCtA=">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</latexit><latexit sha1_base64="tcN9wKvJZoNTUg2dml+F0LfCtA=">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</latexit>

Hence, if such that then a ≤ 2"`

< l a t e x i t s h a 1 _ b a s e 6 4 = " G W N N d R z z K b O G 4 p j p x x + / K T a P 3 B c = " > A A A D 9 3 i c b Z P L b t N A F I a n N Z c S L m 1 h y W Z E W
  • l
F i e J u Y I N U g S q B x K J I p E V h 2 g 8 P k 5 H n Y s 7 l 5 R
  • 5
K f g A d g h t r B j C / B 2 z B 2 I l K 7 j G T r 6 H z / m X P x c V p w Z m y / / 2 d t P b p x 8 9 b t j T u d u / f u P 9 j c 2 n 5 4 b J T T F A Z U c a W H K T H A m Y S B Z Z b D s N B A R M r h J D 1 / X f G T G W j D l P x g 5 w W M B Z l K l j N K b H B N t p 7 t J C l Y M i E 4 4 X C B 9 5 M Z V A Y x p X E S a F V A Z
  • r
Y z 4 m l G m 6 M 9 n q 9 n v 9 + u D r R r w u m h 5 j i b b 6 z + S T F E n Q F r K i T G j u F / Y s S f a M s q h 7 C T O Q E H
  • O
Z n C K J i S C D B j X / d V 4 t 3 g y X C u d H i k x b X 3 a
  • Q
n w p i 5 S I N S E H t m 2 q x y /
  • +
N n M 1 f j D 2 T h b M g 6 S J R 7 j i 2 C l d D w h n T Q C 2 f B 4 N Q z U K t m J 4 R T a g N
  • +
x G n l c M d U A 5 3 v g K O G N h v x U k + K M U 8 h A R Z K Q 3 h l
  • G
V n 9 6 r I u D R n U 6 e b w / B h O u G 7 Q b 4 n H L d M q 8 s m T k V 1 a a p 4 h u v 2 c D O 4 n l / D V a m y U Z q p d w S Z U Q R G Y + s W Q e t M Y V Z W W 7 s k l X G x A w c N 7 C N N X B 7 2 T
  • q
l p A / 6 5 s C U 4 D r u v L / W m b D V d s 2 G a H K 3 b Y Z h n k c B G 4 s a E 1 D X y h 1 M L 3 y t K / L N s N z q A c x W O f p M J 3 4 2 u U h f 8 B y n 9 X v B V B E b Y 8 b u / d e N 4 v x f 3 e / H 7 / e 7 B q + W + b 6 D H 6 A l 6 i m L H B 2 g N + g I D R B F n 9 F P 9 A v 9 j u b R l + h r 9 G h X V 9 b x j x C j R N 9 / w t x 3 1 w O < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G W N N d R z z K b O G 4 p j p x x + / K T a P 3 B c = " > A A A D 9 3 i c b Z P L b t N A F I a n N Z c S L m 1 h y W Z E W
  • l
F i e J u Y I N U g S q B x K J I p E V h 2 g 8 P k 5 H n Y s 7 l 5 R
  • 5
K f g A d g h t r B j C / B 2 z B 2 I l K 7 j G T r 6 H z / m X P x c V p w Z m y / / 2 d t P b p x 8 9 b t j T u d u / f u P 9 j c 2 n 5 4 b J T T F A Z U c a W H K T H A m Y S B Z Z b D s N B A R M r h J D 1 / X f G T G W j D l P x g 5 w W M B Z l K l j N K b H B N t p 7 t J C l Y M i E 4 4 X C B 9 5 M Z V A Y x p X E S a F V A Z
  • r
Y z 4 m l G m 6 M 9 n q 9 n v 9 + u D r R r w u m h 5 j i b b 6 z + S T F E n Q F r K i T G j u F / Y s S f a M s q h 7 C T O Q E H
  • O
Z n C K J i S C D B j X / d V 4 t 3 g y X C u d H i k x b X 3 a
  • Q
n w p i 5 S I N S E H t m 2 q x y /
  • +
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A

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`

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∃ a ∈ A

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is -defeated by on

slide-6
SLIDE 6

Example: if and , then is Integral Probability Metric for class

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  • For any proper composite loss , classifiers , adversaries (+integrability

assumptions),

6 blunts — Cranko et al.

Main negative result — consequence #2

`

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H

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A

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`(H, A, D) ≥ ✓ ` − 1 2 · min

a2A a

+

<latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit><latexit sha1_base64="g6/w4GLNiRwPs1Lm9mephVxyjfk=">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</latexit>

βa

.

= max

h∈H{ϕ(P, f, π, 2L(1)) − ϕ(N, f, 1 − π, −2L(0))}

<latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit><latexit sha1_base64="KnIpapXZGpTXBwu5h4ly+97T0n8=">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</latexit>

f

.

= (L0) ψ1 h a

<latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit><latexit sha1_base64="yEn3Y1VioexjzPuKb/E3szQW1OM=">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</latexit>

with and

ϕ(Q, f, u, v)

.

= Z

X

u · (f(x) + v)dQ(x)

<latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit><latexit sha1_base64="ZqaZKvj2E2tnsDZGZTQtTFOX1SY=">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</latexit>

“+” ex.

L(0) = L(1)

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π = 1/2

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βa

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{(L0) ψ1 h a : h 2 H}

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RHS: roles permuted — adversary in outer optimisation suggests 2-stages optimisation to train classifier: (i) (build adversary) craft adversarial training data
 (ii) train from adversarial data

min

h∈H E(X,Y)∼D

 max

a∈A `(Y, h a(X))

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=

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slide-7
SLIDE 7
  • Direct link with Maximum Mean Discrepancy (MMD)
  • Let be the unit ball of a RKHS w/ reproducing kernel .

Adversarial mean embedding of on (Adversarial) MMD between and

mmd[P, N|a]

.

= kµa,P µa,NkH

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7 blunts — Cranko et al.

Adversaries 1/3: MMD

H

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κ

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µa,Q

.

= Z

X

κ(a(x), .)dQ(x)

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a

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Q

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P

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is -defeated by on if L(0) = L(1)

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π = 1/2

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ε

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H

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A

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`

<latexit sha1_base64="oFXi7J9YjZLxf17eZQBPYxeGBOw=">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</latexit><latexit sha1_base64="oFXi7J9YjZLxf17eZQBPYxeGBOw=">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</latexit><latexit sha1_base64="oFXi7J9YjZLxf17eZQBPYxeGBOw=">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</latexit><latexit sha1_base64="oFXi7J9YjZLxf17eZQBPYxeGBOw=">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</latexit>

βa = 1 4 · mmd[P, N|a]

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∃a ∈ A

< l a t e x i t s h a 1 _ b a s e 6 4 = " V u j G Z 9 L d L l O 7 W r f p R K
  • p
j e w x a 2 4 = " > A A A D 5 X i c b Z J L T x R B E M c b 1 g e u L 9 A j l 4 4 L i Q e y 2 e G C F x P U k G j i A R O B J T s b t N T s 3 S 2 H 2 M / g E 1 n D n 4 A b 8 a r R 6 / q t / H b W L N s h B n s p C e V + v 2 r 6 z G V l V I 4 P x j 8 W V r u 3 L p 9 5 + 7 K v e 7 9 B w 8 f P V 5 d e 3 L
  • T
L A c D r i R x g 4 z 5 k A K D Q d e e A n D g J T m Y S j b P q m 5 k d n Y J w + q O f l T B W b K J F I T j z 6 D p Z X d 9 I 4 Q L T O M p
  • K
j R N F f O n n M n 4 q t
  • 4
W e N +
  • P
5
  • T
e N Z G H y O L s n 6 w t / 5 z w 4 M C 7 b l k z
  • 2
S Q e n H k V k v u I S q m w Y H J e N T N
  • E
R m p
  • p
c O M 4 7 6 K i m + j J a W E s X u 3 p 3 H s 9 I j L l 3 E x l q K y L d G 1 W O / / H R s E X L 8 Z R 6 D J 4 P w y U R E k 9 Y b W I 6 G 5 s M C 9 n K H B u B V Y K + W n z D L u c X D d b i N P K C c W Y L
  • F
A Y f U a C h O L C t P B b / A B F Q Z C / j J w e r u 5 n W R C 1 k h J s E 2 h x F x O v i X
  • N
h S Q X p h z X k T Z 6 p + N D M y p / P 2 a D N 4 P r + G q 1 Z Z V 7 g K q 9 d w z
  • 1
S T O c x 9 W y G W h f K q r Z D 1 a S l N S V Y a Z x D D F K 2 M M 8 s +
  • P
G r u p 1 i + + r l u A Y 8 b y + I h 6 3 2 f C K D d t s 7 4 r t t V k O B X x C 7 j y 2 Z k F e K q 2 K / a q K L 6 t 2 g 2 d Q j Z J x T D M V e 8 k N K n D 7
  • f
r 3 x D u F C t z y p L 3 T N 4 3 D 7 X 4 y 6 C c f t n u 7 r x f 7 v k L W y T P y n C R k h + y S t 2 S f H B B O P p M f 5 C f 5 1 Z l v n S + d r 5 d S p e X F j F P S e N v v 8 F Q m 1 U I g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " V u j G Z 9 L d L l O 7 W r f p R K
  • p
j e w x a 2 4 = " > A A A D 5 X i c b Z J L T x R B E M c b 1 g e u L 9 A j l 4 4 L i Q e y 2 e G C F x P U k G j i A R O B J T s b t N T s 3 S 2 H 2 M / g E 1 n D n 4 A b 8 a r R 6 / q t / H b W L N s h B n s p C e V + v 2 r 6 z G V l V I 4 P x j 8 W V r u 3 L p 9 5 + 7 K v e 7 9 B w 8 f P V 5 d e 3 L
  • T
L A c D r i R x g 4 z 5 k A K D Q d e e A n D g J T m Y S j b P q m 5 k d n Y J w + q O f l T B W b K J F I T j z 6 D p Z X d 9 I 4 Q L T O M p
  • K
j R N F f O n n M n 4 q t
  • 4
W e N +
  • P
5
  • T
e N Z G H y O L s n 6 w t / 5 z w 4 M C 7 b l k z
  • 2
S Q e n H k V k v u I S q m w Y H J e N T N
  • E
R m p
  • p
c O M 4 7 6 K i m + j J a W E s X u 3 p 3 H s 9 I j L l 3 E x l q K y L d G 1 W O / / H R s E X L 8 Z R 6 D J 4 P w y U R E k 9 Y b W I 6 G 5 s M C 9 n K H B u B V Y K + W n z D L u c X D d b i N P K C c W Y L
  • F
A Y f U a C h O L C t P B b / A B F Q Z C / j J w e r u 5 n W R C 1 k h J s E 2 h x F x O v i X
  • N
h S Q X p h z X k T Z 6 p + N D M y p / P 2 a D N 4 P r + G q 1 Z Z V 7 g K q 9 d w z
  • 1
S T O c x 9 W y G W h f K q r Z D 1 a S l N S V Y a Z x D D F K 2 M M 8 s +
  • P
G r u p 1 i + + r l u A Y 8 b y + I h 6 3 2 f C K D d t s 7 4 r t t V k O B X x C 7 j y 2 Z k F e K q 2 K / a q K L 6 t 2 g 2 d Q j Z J x T D M V e 8 k N K n D 7
  • f
r 3 x D u F C t z y p L 3 T N 4 3 D 7 X 4 y 6 C c f t n u 7 r x f 7 v k L W y T P y n C R k h + y S t 2 S f H B B O P p M f 5 C f 5 1 Z l v n S + d r 5 d S p e X F j F P S e N v v 8 F Q m 1 U I g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " V u j G Z 9 L d L l O 7 W r f p R K
  • p
j e w x a 2 4 = " > A A A D 5 X i c b Z J L T x R B E M c b 1 g e u L 9 A j l 4 4 L i Q e y 2 e G C F x P U k G j i A R O B J T s b t N T s 3 S 2 H 2 M / g E 1 n D n 4 A b 8 a r R 6 / q t / H b W L N s h B n s p C e V + v 2 r 6 z G V l V I 4 P x j 8 W V r u 3 L p 9 5 + 7 K v e 7 9 B w 8 f P V 5 d e 3 L
  • T
L A c D r i R x g 4 z 5 k A K D Q d e e A n D g J T m Y S j b P q m 5 k d n Y J w + q O f l T B W b K J F I T j z 6 D p Z X d 9 I 4 Q L T O M p
  • K
j R N F f O n n M n 4 q t
  • 4
W e N +
  • P
5
  • T
e N Z G H y O L s n 6 w t / 5 z w 4 M C 7 b l k z
  • 2
S Q e n H k V k v u I S q m w Y H J e N T N
  • E
R m p
  • p
c O M 4 7 6 K i m + j J a W E s X u 3 p 3 H s 9 I j L l 3 E x l q K y L d G 1 W O / / H R s E X L 8 Z R 6 D J 4 P w y U R E k 9 Y b W I 6 G 5 s M C 9 n K H B u B V Y K + W n z D L u c X D d b i N P K C c W Y L
  • F
A Y f U a C h O L C t P B b / A B F Q Z C / j J w e r u 5 n W R C 1 k h J s E 2 h x F x O v i X
  • N
h S Q X p h z X k T Z 6 p + N D M y p / P 2 a D N 4 P r + G q 1 Z Z V 7 g K q 9 d w z
  • 1
S T O c x 9 W y G W h f K q r Z D 1 a S l N S V Y a Z x D D F K 2 M M 8 s +
  • P
G r u p 1 i + + r l u A Y 8 b y + I h 6 3 2 f C K D d t s 7 4 r t t V k O B X x C 7 j y 2 Z k F e K q 2 K / a q K L 6 t 2 g 2 d Q j Z J x T D M V e 8 k N K n D 7
  • f
r 3 x D u F C t z y p L 3 T N 4 3 D 7 X 4 y 6 C c f t n u 7 r x f 7 v k L W y T P y n C R k h + y S t 2 S f H B B O P p M f 5 C f 5 1 Z l v n S + d r 5 d S p e X F j F P S e N v v 8 F Q m 1 U I g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " V u j G Z 9 L d L l O 7 W r f p R K
  • p
j e w x a 2 4 = " > A A A D 5 X i c b Z J L T x R B E M c b 1 g e u L 9 A j l 4 4 L i Q e y 2 e G C F x P U k G j i A R O B J T s b t N T s 3 S 2 H 2 M / g E 1 n D n 4 A b 8 a r R 6 / q t / H b W L N s h B n s p C e V + v 2 r 6 z G V l V I 4 P x j 8 W V r u 3 L p 9 5 + 7 K v e 7 9 B w 8 f P V 5 d e 3 L
  • T
L A c D r i R x g 4 z 5 k A K D Q d e e A n D g J T m Y S j b P q m 5 k d n Y J w + q O f l T B W b K J F I T j z 6 D p Z X d 9 I 4 Q L T O M p
  • K
j R N F f O n n M n 4 q t
  • 4
W e N +
  • P
5
  • T
e N Z G H y O L s n 6 w t / 5 z w 4 M C 7 b l k z
  • 2
S Q e n H k V k v u I S q m w Y H J e N T N
  • E
R m p
  • p
c O M 4 7 6 K i m + j J a W E s X u 3 p 3 H s 9 I j L l 3 E x l q K y L d G 1 W O / / H R s E X L 8 Z R 6 D J 4 P w y U R E k 9 Y b W I 6 G 5 s M C 9 n K H B u B V Y K + W n z D L u c X D d b i N P K C c W Y L
  • F
A Y f U a C h O L C t P B b / A B F Q Z C / j J w e r u 5 n W R C 1 k h J s E 2 h x F x O v i X
  • N
h S Q X p h z X k T Z 6 p + N D M y p / P 2 a D N 4 P r + G q 1 Z Z V 7 g K q 9 d w z
  • 1
S T O c x 9 W y G W h f K q r Z D 1 a S l N S V Y a Z x D D F K 2 M M 8 s +
  • P
G r u p 1 i + + r l u A Y 8 b y + I h 6 3 2 f C K D d t s 7 4 r t t V k O B X x C 7 j y 2 Z k F e K q 2 K / a q K L 6 t 2 g 2 d Q j Z J x T D M V e 8 k N K n D 7
  • f
r 3 x D u F C t z y p L 3 T N 4 3 D 7 X 4 y 6 C c f t n u 7 r x f 7 v k L W y T P y n C R k h + y S t 2 S f H B B O P p M f 5 C f 5 1 Z l v n S + d r 5 d S p e X F j F P S e N v v 8 F Q m 1 U I g = = < / l a t e x i t >

s.t.mmd[P, N|a] ≤ 8"`

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(more in paper / poster)

N

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(if and )

slide-8
SLIDE 8
  • Allows to build efficient adversaries when classifiers are Lipschitz
  • Solve the compression of an optimal transport plan
  • Monge efficient for on , iff

8 blunts — Cranko et al.

Adversaries 2/3: Monge

C(a, P, N)

.

= inf

µ2Π(P,N)

Z c(a(x), a(x0))dµ(x, x0)

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(Adversarial) OT plan between and

P

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N

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Monge efficiency (for cost )

c : X × X → R

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(more general statement in paper / poster)

δ

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A

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c

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P

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N

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∃a ∈ A : C(a, P, N) ≤ δ

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Suppose is -Lipschitz with respect to , is -Monge efficient for on , . Suppose , .

H

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c

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A

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δ

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c

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P

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N

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L(0) = L(1)

<latexit sha1_base64="pN93QRCEQ14U6yZNRgS9Hl3cXY=">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</latexit><latexit sha1_base64="pN93QRCEQ14U6yZNRgS9Hl3cXY=">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</latexit><latexit sha1_base64="pN93QRCEQ14U6yZNRgS9Hl3cXY=">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</latexit><latexit sha1_base64="pN93QRCEQ14U6yZNRgS9Hl3cXY=">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</latexit>

π = 1/2

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K

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≤ 8"`/K

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If , then is -defeated by on

H

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ε

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A

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`

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vs

slide-9
SLIDE 9
  • A. It is possible to -defeat simultaneously on a whole set of symmetric losses



 Simple way to defeat strategies learning/tuning the loss 
 — important case because common losses fit in (log, square, Matsushita, etc.)


  • B. It is possible to craft very strong adversaries from very weak ones



 RKHS example — suppose there exists a weakly contractive adversary in a feature map of the RKHS:
 Then ,composing just adversaries yields -Monge efficiency

9 blunts — Cranko et al.

Adversaries 3/3: Boosting

(complete statements in paper / poster)

ε

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H

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a

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L

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Φ

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kΦ a(x) Φ a(x0)kH  (1 η) · kΦ(x) Φ(x0)kH, 8x, x0 2 X

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δ

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∀δ > 0

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W Φ

1

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= 1-Wasserstein distance between and in P

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N

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Φ

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(1/η) · log(W Φ

1 /δ)

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slide-10
SLIDE 10
  • A. Replace adversarial training by training from adversarial data

  • B. If loss in specific classes, incl. popular losses, adversary can be loss agnostic

  • C. If learner’s is Lipschitz, use Lipschitz cost in an OT compression problem

  • D. Adversarial boosting: craft strong adversaries from weak adversaries

10 blunts — Cranko et al.

Take home theoretical messages

vs

H

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slide-11
SLIDE 11
  • A. Mixup adversaries (named after Zhang, Cissé, Dauphin & Lopez-Paz ’18)



 
 general transformation: 
 


  • B. Monge adversary — for a tight control on Monge efficiency, focus on



 
 min Wasserstein-OT s.t.

a(x)

.

= (1 − λ) · x + λ · x0

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11 blunts — Cranko et al.

Some Monge efficient adversaries

neighbour in other class cluster / class centroid sample centroid, etc.

(more in paper / poster)

d(a(x), x) ≤ α, ∀x

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a

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“budget”

slide-12
SLIDE 12

blunts — Cranko et al.

Toy experiments 1/2 — data & transformations

  • A. Mixup-to-sample-centroid



 
 
 


  • B. Monge adversary for Wasserstein = and (cvx) 



 


12

x0 = ED[X]

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W 2

2

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USPS data

1D normal classes classes 1 & 3

λ

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α W1

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d = k.k1

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slide-13
SLIDE 13

blunts — Cranko et al.

Toy experiments 2/2 — findings

  • A. Mixup-to-sample-centroid



 
 
 


  • B. Monge adversary for Wasserstein = and (cvx) 



 


13

x0 = ED[X]

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W 2

2

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USPS data

1D normal classes classes 1 & 3

λ

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α W1

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λ

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logistic loss

train test

c = clean a = adversarial

worst trains on c ! best trains on (weak) a (even tested on c) !

d = k.k1

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slide-14
SLIDE 14

blunts — Cranko et al.

Conclusion

  • Replacement of adversarial training by training from adversarial data
  • Adversaries that can be effective against wide ranges of 

  • Adversarial strategy against Lipschitz classifiers: compression of OT plans

(between class marginals)


  • Toy experiments reveal that sufficiently weak adversarial data can improve

generalisation on clean data
 
 
 


  • Next step: explain such a “vaccination phenomenon”

14

(H, `)

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data shot

slide-15
SLIDE 15

blunts — Cranko et al.

Thank you

(get your data shot at poster # 191)