ichigaku.takigawa@riken.jp
2022 2 24
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
if x2 ≤ θ1 then
if x1 ≤ θ2 then
return Blue
else
if x2 ≤ θ4 then
return Red
else
if x1 ≤ θ5 then
return Red
else
if x1 ≤ θ6 then
return Blue
else
return Red
else
if x1 ≤ θ3 then
if x2 ≤ θ7 then
return Blue
else
return Blue
else
return Red
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue <latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit>
X2  ✓4
yes no
Red <latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit>
X1  ✓5
yes no
Red
<latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit>
X1  ✓6
yes no
Red
Blue
X1  ✓3
yes no
Red
X1  ✓7
yes no
Blue Blue
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">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</latexit>
✓4
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="e77Z2aqe8KZFIqh8R+ZYHDOKN+I=">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</latexit>
✓5
<latexit sha1_base64="eu7hkk7xInDVK5lVQXXLPV+MjYw=">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</latexit>
✓6
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="VJpvibITWQ0L0zaDACUdk6SsPiA=">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</latexit>
✓7
<latexit sha1_base64="JcXOTtKAO5x/t4uZH9BQrQClbjc=">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</latexit>
Pi := Q1 ^ Q2 ^ Q3 ^ . . .
<latexit sha1_base64="zBiZIFwyoN/DGVfflNKDwKhdwmg=">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</latexit>
Qj :=
(
Xk  ✓l
Xk > ✓l
Blue
<latexit sha1_base64="p+ll+ob/bB8deuXgcYGzmiYX5Zc=">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</latexit>
T := P1 _ P2 _ . . .
<latexit sha1_base64="+WntnxsFSxJvfdpwMjPE/ws7ZhE=">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</latexit>
(
T = 1
T = 0
Path
Query
Not Blue (= Red)
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
https://www.kaggle.com/kaggle-survey-2021
State of Data Science and
Machine Learning 2021
The 2021 Kaggle DS & ML Survey received 25,973 usable responses from participants in 171 different countries and territories.
Q17. Which of the following ML algorithms do you use on a regular basis? (Select all that apply)
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Object recognition
Game play
J’aime la
musique I love music
Speech recognition
Machine translation
Super resolution
p1 p2 p3 p4
Random Forest
Gaussian Process
Logistic Regression
Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001.
https://doi.org/10.1214/ss/1009213726
Rashomon
Occam
Bellman
Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001.
https://doi.org/10.1214/ss/1009213726
Rashomon
Occam
Bellman
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
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x1
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x2 <latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCJTg2JcEd245CGPBAlp64gNpW3aQkTiD5i4lYUrTVwYP8APcOMPuOATjEtM3LjwUpoYJeJtpnPmzD13zsyVTU21Hca6PmFsfGJyyj8dmJmdmw+GFhbzttGwFJ5TDM2wirJkc03Vec5RHY0XTYtLdVnjBbm2198vNLllq4Z+4LRMXq5LVV09VhXJISp7WhEroQiLMTfCw0D0QARepIzQIw5xBAMKGqiDQ4dDWIMEm74SRDCYxJXRJs4ipLr7HOcIkLZBWZwyJGJr9K/SquSxOq37NW1XrdApGg2LlGFE2Qu7Zz32zB7YK/v8s1bbrdH30qJZHmi5WQleLGc//lXVaXZw8q0a6dnBMbZdryp5N12mfwtloG+edXrZnUy0vcZu2Rv5v2Fd9kQ30Jvvyl2aZ65H+JHJC70YNUj83Y5hkN+IiVuxeDoeSe56rfJjBatYp34kkMQ+UshR/SoucYWO4BdiwqaQGKQKPk+zhB8hJL8AVLKQnA==</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCIDQTGuiG5c8pBHgoS0dcCG0jZtISLxB0zcysKVJi6MH+AHuPEHXPAJxiUmblx4KU2MEvE20zlz5p47Z+ZKhqpYNmM9jzAxOTU94531zc0vLPoDS8t5S2+aMs/JuqqbRUm0uKpoPGcrtsqLhsnFhqTyglTfH+wXWty0FF07tNsGLzfEmqZUFVm0icqeVmKVQIhFmBPBURB1QQhupPTAI45wDB0ymmiAQ4NNWIUIi74SomAwiCujQ5xJSHH2Oc7hI22TsjhliMTW6V+jVcllNVoPalqOWqZTVBomKYMIsxd2z/rsmT2wV/b5Z62OU2PgpU2zNNRyo+K/WM1+/Ktq0Gzj5Fs11rONKnYcrwp5NxxmcAt5qG+ddfvZ3Uy4s8Fu2Rv5v2E99kQ30Frv8l2aZ67H+JHIC70YNSj6ux2jIB+LRLcj8XQ8lNxzW+XFGtaxSf1IIIkDpJCj+jVc4gpdwStEhC0hMUwVPK5mBT9CSH4BVtKQnQ==</latexit>
x2 <latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2 <latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCJTg2JcEd245CGPBAlp64gNpW3aQkTiD5i4lYUrTVwYP8APcOMPuOATjEtM3LjwUpoYJeJtpnPmzD13zsyVTU21Hca6PmFsfGJyyj8dmJmdmw+GFhbzttGwFJ5TDM2wirJkc03Vec5RHY0XTYtLdVnjBbm2198vNLllq4Z+4LRMXq5LVV09VhXJISp7WhEroQiLMTfCw0D0QARepIzQIw5xBAMKGqiDQ4dDWIMEm74SRDCYxJXRJs4ipLr7HOcIkLZBWZwyJGJr9K/SquSxOq37NW1XrdApGg2LlGFE2Qu7Zz32zB7YK/v8s1bbrdH30qJZHmi5WQleLGc//lXVaXZw8q0a6dnBMbZdryp5N12mfwtloG+edXrZnUy0vcZu2Rv5v2Fd9kQ30Jvvyl2aZ65H+JHJC70YNUj83Y5hkN+IiVuxeDoeSe56rfJjBatYp34kkMQ+UshR/SoucYWO4BdiwqaQGKQKPk+zhB8hJL8AVLKQnA==</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2 <latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCJTg2JcEd245CGPBAlp64gNpW3aQkTiD5i4lYUrTVwYP8APcOMPuOATjEtM3LjwUpoYJeJtpnPmzD13zsyVTU21Hca6PmFsfGJyyj8dmJmdmw+GFhbzttGwFJ5TDM2wirJkc03Vec5RHY0XTYtLdVnjBbm2198vNLllq4Z+4LRMXq5LVV09VhXJISp7WhEroQiLMTfCw0D0QARepIzQIw5xBAMKGqiDQ4dDWIMEm74SRDCYxJXRJs4ipLr7HOcIkLZBWZwyJGJr9K/SquSxOq37NW1XrdApGg2LlGFE2Qu7Zz32zB7YK/v8s1bbrdH30qJZHmi5WQleLGc//lXVaXZw8q0a6dnBMbZdryp5N12mfwtloG+edXrZnUy0vcZu2Rv5v2Fd9kQ30Jvvyl2aZ65H+JHJC70YNUj83Y5hkN+IiVuxeDoeSe56rfJjBatYp34kkMQ+UshR/SoucYWO4BdiwqaQGKQKPk+zhB8hJL8AVLKQnA==</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
N = 32 = 9 N = 52 = 25 N = 102 = 100
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">AAAChHichVHLTsJAFD3UF+ID1I2JGyLBuDBkUHzEhSG6cclDHgkS0tYBG0rbtIUEiT+gW40LV5q4MH6AH+DGH3DBJxiXmLhx4aU0MUrE20znzJl77pyZKxmqYtmMtT3C0PDI6Jh33DcxOTXtD8zMZi29bso8I+uqbuYl0eKqovGMrdgqzxsmF2uSynNSda+7n2tw01J07cBuGrxYEyuaUlZk0SYq2SwFQizCnAj2g6gLQnAjoQcecYgj6JBRRw0cGmzCKkRY9BUQBYNBXBEt4kxCirPPcQofaeuUxSlDJLZK/wqtCi6r0bpb03LUMp2i0jBJGUSYvbB71mHP7IG9ss8/a7WcGl0vTZqlnpYbJf/ZfPrjX1WNZhvH36qBnm2UseV4Vci74TDdW8g9fePkqpPeToVbS+yWvZH/G9ZmT3QDrfEu3yV56nqAH4m80ItRg6K/29EPsquR6EYkloyF4rtuq7xYwCKWqR+biGMfCWSoPsc5LnApjAorwpqw3ksVPK5mDj9C2PkC2GOP+Q==</latexit>
y
<latexit sha1_base64="WQjRG4bPVEcrBhk0IPMU7zTCz9E=">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</latexit>
f
<latexit sha1_base64="WQjRG4bPVEcrBhk0IPMU7zTCz9E=">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</latexit>
f
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y
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x1
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x2
<latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y <latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y <latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y
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y <latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y <latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y
Bronstein MM, Bruna J, Cohen T, Veličković P.
Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges.
arXiv [cs.LG]. 2021. http://arxiv.org/abs/2104.13478
for all
<latexit sha1_base64="NqBYbofjRLhsRuy5gs3o5FfAjVo=">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</latexit>
f : Rd
! R
<latexit sha1_base64="bLp7Jqe5Ke+bRuLIs0xt/zwFMHo=">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</latexit>
x, x0
2 Rd
-Lipschitz
<latexit sha1_base64="VAwN9BEfNuoROOZ0f78ZUuUItlA=">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</latexit>
|f(x) f(x0
)| 6 Lkx x0
k
<latexit sha1_base64="+ZiSkRyCDmMX1dfLuFjrVp2j6hw=">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</latexit>
L
Donoho DL,
High-dimensional data analysis: The curses and blessings of dimensionality.
Plenary Lecture, AMS National Meeting on Mathematical Challenges of the 21st Century. 2000.
<latexit sha1_base64="mchf2gvio17p/Fz9iF84xhnMH3c=">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</latexit>
V (d) =
⇡
d
2
d
2 · d
2
! 0 (d ! 1)
https://www.math.ucdavis.edu/~strohmer/courses/180BigData/180lecture1.pdf
<latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit>
1
<latexit sha1_base64="z3Ba/qsiF6tEgbjc9DFSGYR9udI=">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</latexit>
0.5
p
d
<latexit sha1_base64="rzd8+rC/ag8/gv3jUp4Y6gUrSiI=">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</latexit>
0.5
<latexit sha1_base64="Nn/M5/H5iD9Z/hOTHxuHUqAZH+I=">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</latexit>
d = 2
<latexit sha1_base64="LVcqe4i4MA2A5XBAmTvFXnqLayw=">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</latexit>
d = 4
<latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit>
1
<latexit sha1_base64="UR9ymFZxc/Q/TCzOORzighhjAlM=">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</latexit>
d > 4
<latexit sha1_base64="nHSiJ7YlA7Y9nChVeD/tJhyz2yo=">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</latexit>
0.5
p
2
<latexit sha1_base64="z3Ba/qsiF6tEgbjc9DFSGYR9udI=">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</latexit>
0.5
p
d
!?
Unit Cube in
Unit Ball
<latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit>
1
<latexit sha1_base64="PtefnFPcLJximPXvkayt2U9xPRk=">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</latexit>
> 1
!?
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Balestriero R, Pesenti J, LeCun Y.
Learning in High Dimension Always Amounts to Extrapolation.
arXiv [cs.LG]. 2021. http://arxiv.org/abs/2110.09485 https://youtu.be/86ib0sfdFtw
• "on any high-dimensional (>100) dataset,
interpolation almost surely never happens."
• "Those results challenge the validity of our
current interpolation/extrapolation definition as
an indicator of generalization performances. "
!
Fan J, Han F, Liu H.
Challenges of Big Data Analysis. Natl Sci Rev. 2014;1: 293–314.
<latexit sha1_base64="hsRBbjUZq2fjH2VINZUiaiPtLiw=">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</latexit>
(X1, . . . , Xd) ⇠ Nd(0, I)
<latexit sha1_base64="Ay3UPGyVpdQRyf/zf3IimzcaXzc=">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</latexit>
r̂ = max
j>2
|corr(X1, Xj)|
<latexit sha1_base64="DII9mhRSnTvcUJHdfuik1ohKbRg=">AAAC8nichVHNSiNBEK6Muv6srlEvgpfBoLggoUeCiiCIXjxqstGAI0PP2Ek6zh89naBO8gK+gKAnlT0su3cfYC97Xzz4CMseXRDBgzU/ICqrNfRU1df1VX/dZfo2DyQhNxmlq7vnQ29f/8DHwaFPw9mR0a3AawqLlS3P9kTFpAGzucvKkkubVXzBqGPabNvcX4v2t1tMBNxzv8hDn+06tObyKreoRMjIOnqdyrDYUZdV3aEHRtgutZcLnTTRTSap0cDUZlXZjlBZF05oeUJ0YmymYmizqh40HSNs6NxVS1icsNSK0dAFr9XlZzXxbSObI3kSm/o60NIgB6lteNkr0GEPPLCgCQ4wcEFibAOFAL8d0ICAj9guhIgJjHi8z6ADA8htYhXDCoroPv5rmO2kqIt51DOI2RaeYuMSyFRhilyTb+SW/CLfyR/y8N9eYdwj0nKI3ky4zDeGj8dLd++yHPQS6k+sNzVLqMJirJWjdj9GoltYCb91dHJbWipOhdPkgvxF/efkhvzEG7itf9bXTVY8e0OPiVrwxXBA2stxvA625vLafL6wWcitrKaj6oMJmIQZnMcCrMA6bEAZ+/+G+4yS6VKkcqqcK5dJqZJJOWPwzJQfj/5suoY=</latexit>
R̂ = max
|S|=4
max
j
corr
0
@X1,
X
j2S
jXj
1
A
0.3 0.4 0.5 0.6 0.5 0.6 0.7 0.8
<latexit sha1_base64="dsksfNj+oQDjs49biWmppWLfS4s=">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</latexit>
R̂
<latexit sha1_base64="XcYeasdspQvBninjyfi4DGqRhKY=">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</latexit>
r̂
K. Beyer+, When Is “Nearest Neighbor” Meaningful? ICDT’99
V. Pestov, On the geometry of similarity search: dimensionality curse and concentration of measure,
Information Processing Letters, 1999.
Inductive Bias
Decision
Tree
Random
Forest GBDT
Nearest
Neighbor
Logistic
Regression
SVM
Gaussian
Process
Neural
Network
Journal of the American Statistical Association , Jun., 2006, Vol. 101, No. 474 (Jun., 2006), pp. 578-590
https://www.jstor.org/stable/27590719
"We introduce a concept of potential nearest neighbors (k-PNNs) and show that random
forests can be viewed as adaptively weighted k-PNN methods. "
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
TheBibleor"TheYellowTerror"
byLuc,Laci,Gábor
TheBibleor"TheYellowTerror"
byLuc,Laci,Gábor
<latexit sha1_base64="jdzy7O9ZT5s5Y1wkxQhz0FOJMwM=">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</latexit>
N(µ = 1.2, = 2.5)
Given , a real i.i.d. sequence, estimate
TheBibleor"TheYellowTerror"
byLuc,Laci,Gábor
NeurIPS 2021 Invited Talk (Breiman Lecture)
Do we know how to estimate the mean?
<latexit sha1_base64="3NbZouAktoxRn4pfXQgJR8+DKAA=">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</latexit>
X1, . . . , Xn
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µ = EX1
<latexit sha1_base64="ve2fKPQE43VaTVIusM8fwhBZiOg=">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</latexit>
(
Pn
i=1 Xi)/n
Journal of Machine Learning Research, 9, 2015-2033, 2008 The Annals of Statistics, 43(4), 1716-1741, 2015.
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001.
https://doi.org/10.1214/ss/1009213726
Rashomon
Occam
Bellman
Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001.
https://doi.org/10.1214/ss/1009213726
• "What I call the Rashomon Effect is that there is often a multitude of different descriptions
[equations f︎(x)] in a class of functions giving about the same minimum error rate. The most
easily understood example is subset selection in linear regression."
• "The Rashomon Effect also occurs with decision trees and neural nets."
• "This effect is closely connected to what I call instability (Breiman,1996a) that occurs when
there are many different models crowded together that have about the same training or test
set error."
Rashomon Effect:
Kernel Ridge (RBF)
Neural Network (MLP)
Gradient Boosted Trees
SVR (RBF) Gaussian Process (RBF)
Random Forest
Nearest Neighbors Decision Tree
Kernel Ridge (RBF)
Neural Network (MLP)
Gradient Boosted Trees
SVR (RBF) Gaussian Process (RBF)
Random Forest
Nearest Neighbors Decision Tree
https://arxiv.org/abs/2011.03395
https://ai.googleblog.com/2021/10/
how-underspecification-presents.html
"While ML models are validated on held-out data, this validation is often insufficient to
guarantee that the models will have well-defined behavior when they are used in a new setting. "
Kernel Ridge (RBF)
Neural Network (MLP)
Gradient Boosted Trees
SVR (RBF) Gaussian Process (RBF)
Random Forest
Nearest Neighbors Decision Tree
Kernel Ridge (RBF)
Neural Network (MLP)
Gradient Boosted Trees
SVR (RBF) Gaussian Process (RBF)
Random Forest
Nearest Neighbors Decision Tree
Benign?
Benign? Benign?
Malignant Malignant
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Hutter et al, Algorithm Runtime Prediction: Methods & Evaluation (2014) https://arxiv.org/abs/1211.0906
https://scikit-optimize.github.io/stable/_modules/skopt/learning/forest.html
def _return_std(X, trees, predictions, min_variance):
# This derives std(y | x) as described in 4.3.2 of arXiv:1211.0906
std = np.zeros(len(X))
for tree in trees:
var_tree = tree.tree_.impurity[tree.apply(X)]
# This rounding off is done in accordance with the
# adjustment done in section 4.3.3
# of http://arxiv.org/pdf/1211.0906v2.pdf to account
# for cases such as leaves with 1 sample in which there
# is zero variance.
var_tree[var_tree < min_variance] = min_variance
mean_tree = tree.predict(X)
std += var_tree + mean_tree ** 2
std /= len(trees)
std -= predictions ** 2.0
std[std < 0.0] = 0.0
std = std ** 0.5
return std
Hutter et al, Algorithm Runtime Prediction: Methods & Evaluation (2014) https://arxiv.org/abs/1211.0906
RandomForestRegressor
(n_estimators=10, max_leaf_nodes=12)
ExtraTreesRegressor
(n_estimators=10, max_leaf_nodes=32)
GradientBoostingRegressor
(n_estimators=30, learning_rate=0.1)
GradientBoostingRegressor(loss='quantile', alpha=α)
GradientBoostingRegressor(
(n_estimators=30, learning_rate=0.1)
LGBMRegressor (n_estimators=30, learning_rate=0.1,
max_leaf_nodes=8, min_child_samples=5)
aka "Pinball loss"
Rando Forest ExtraTrees (w/o bootstrap)
ExtraTrees (w/ bootstrap)
Gradient Boosted Trees
Rando Forest ExtraTrees (w/o bootstrap)
ExtraTrees (w/ bootstrap)
Gradient Boosted Trees
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Gaussian Process Gaussian Process
MLP
(Quantile Regression)
MLP
(Quantile Regression)
MLP
(MC Dropout)
MLP
(MC Dropout)
arch:
(Linear(1, 100), ReLU,
Dropout(p),
Linear(100, 100), ReLU,
Dropout(p),
Linear(100, 1))
p=0.25
#sampling=30
p=0.25
#sampling=30
p=0.0
p=0.0
MLP
(30 Ensemble)
MLP
(30 Ensemble)
p=0.0
p=0.0
MLP
(30 Ensemble)
MLP
(30 Ensemble)
p=0.25
p=0.25
arch:
(Linear(1, 100), ReLU,
Dropout(p),
Linear(100, 100), ReLU,
Dropout(p),
Linear(100, 1))
• Tolstikhin IO, Houlsby N, Kolesnikov A, Beyer L, Zhai X, Unterthiner T, et al.
MLP-Mixer: An all-MLP Architecture for Vision. NeurIPS 2021.
• Kadra A, Lindauer M, Hutter F, Grabocka J.
Well-Tuned Simple Nets Excel on Tabular Datasets. NeurIPS 2021.
• Liu H, Dai Z, So D, Le Q.
Pay Attention to MLPs. NeurIPS 2021.
• Melas-Kyriazi L.
Do You Even Need Attention? A Stack of Feed-Forward Layers Does Surprisingly
Well on ImageNet. 2021. http://arxiv.org/abs/2105.02723
Kernel Ridge (RBF)
Neural Network (MLP)
Gradient Boosted Trees
SVR (RBF) Gaussian Process (RBF)
Random Forest
Nearest Neighbors Decision Tree
Benign?
Benign? Benign?
Malignant Malignant
PolyReg(1)
RMSE 0.299
PolyReg(3)
RMSE 0.28
PolyReg(5)
RMSE 0.225
PolyReg(7)
RMSE 0.113
PolyReg(10)
RMSE 0.0189
PolyReg(15)
RMSE 0.00737
PolyReg(20)
RMSE 0.000
PolyReg(30)
RMSE 0.000
ExtraTrees (no bootstrap)
RMSE 0.000
ExtraTrees (bootstrap)
RMSE 0.0121
Random Forest
RMSE 0.012
LGBM
RMSE 0.0508
95%-CI 95%-CI 95%-CI 95%-CI
Problematic overfitting by polynomial regression of order k
clearly overfitted but harmless (still informative)
also we can assess
the uncertainty
ExtraTrees (no bootstrap) ExtraTrees (no bootstrap) ExtraTrees (no bootstrap) ExtraTrees (no bootstrap)
Gradient Boosted Trees Gradient Boosted Trees Gradient Boosted Trees Gradient Boosted Trees
Nearest Neighbor (k=1) Nearest Neighbor (k=1) Nearest Neighbor (k=1) Nearest Neighbor (k=1)
Decision Tree Decision Tree Decision Tree Decision Tree
Random Forest Random Forest Random Forest Random Forest
Nearest Neighbor (k=3) Nearest Neighbor (k=3) Nearest Neighbor (k=3) Nearest Neighbor (k=3)
"our experiments establish that state-of-the-art convolutional networks for image classification trained with
stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively
unaffected by explicit regularization and occurs even if we replace the true images by completely unstructured
random noise. "
Berner J, Grohs P, Kutyniok G, Petersen P.
The Modern Mathematics of Deep Learning. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2105.04026
Zhang C, Bengio S, Hardt M, Recht B, Vinyals O.
Understanding Deep Learning (Still) Requires Rethinking Generalization. Commun ACM. 2021;64: 107–115.
Berner J, Grohs P, Kutyniok G, Petersen P.
The Modern Mathematics of Deep Learning. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2105.04026
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
PNAS (2020) Ann. Statist. (2020)
NeurIPS (2018) arXiv (2019)
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
PolyReg(10)
Train RMSE 0.0189
PolyReg(15)
Train RMSE 0.00737
KernelRidge RBF(γ=1)
Train RMSE 0.103
KernelRidge RBF(γ=10)
Train RMSE 0.0139
KernelRidge RBF(γ=100)
Train RMSE 0.00726
KernelRidge RBF(γ=1000)
Train RMSE 0.00726
ExtraTrees (#trees=1, #leaves=8,
bootstrap=off), Train RMSE 0.0189
ExtraTrees (#trees=10, #leaves=8,
bootstrap=off), Train RMSE 0.0274
ExtraTrees (#trees=102, #leaves=8,
bootstrap=off), Train RMSE 0.0279
ExtraTrees (#trees=103, #leaves=8,
bootstrap=off), Train RMSE 0.0243
GBDT w/ 1 ExtraTree (#leaves=8)
Train RMSE 0.29
GBDT w/ 10 ExtraTree (#leaves=8)
Train RMSE 0.17
GBDT w/ 102, ExtraTree (#leaves=8)
Train RMSE 0.00597
GBDT w/ 103, ExtraTree (#leaves=8)
Train RMSE 0.000997
Random Forest (#trees=100)
Train RMSE 0.0109
Random Forest (#trees=1000)
Train RMSE 0.0274
ExtraTrees (#trees=100, bootstrap)
Train RMSE 0.0109
ExtraTrees (#trees=1000, bootstrap)
Train RMSE 0.0243
Random Forest (#trees=100)
Train RMSE 0.0112
Random Forest (#trees=100)
Train RMSE 0.0114
ExtraTrees (#trees=100, bootstrap)
Train RMSE 0.0099
ExtraTrees (#trees=1000, bootstrap)
Train RMSE 0.00977
With Random Fourier Features (100-dim Random Kitchen Sinks)
1-1-1 ReLU MLP
Train RMSE 0.287
1-5-1 ReLU MLP
Train RMSE 0.287
1-10-1 ReLU MLP
Train RMSE 0.0191
1-100-1 ReLU MLP
Train RMSE 0.00816
Train RMSE 0.0139 Train RMSE 0.0066 Train RMSE 0.0121 Train RMSE 0.0139
…
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k
Optimized With L-BFGS
arch:
(Linear(1, k),
ReLU,
Linear(k, 1))
1-1-1 ReLU MLP
Train RMSE 0.289
1-5-1 ReLU MLP
Train RMSE 0.297
1-10-1 ReLU MLP
Train RMSE 0.0237
1-100-1 ReLU MLP
Train RMSE 0.0243
…
<latexit sha1_base64="OtDdB3Z2MZrUEFM+jHf2FtPg0CM=">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</latexit>
k
Optimized With Adam
arch:
(Linear(1, k),
ReLU,
Linear(k, 1))
k=1 ReLU MLP
Train RMSE 0.305
k=5 ReLU MLP
Train RMSE 0.303
k=10 ReLU MLP
Train RMSE 0.0131
k=100 ReLU MLP
Train RMSE 0.00979
Optimized With Adam
arch:
(Linear(1, k),
ReLU,
Linear(k, k),
ReLU,
Linear(k, 1))
x 9
Train RMSE 0.0443 Train RMSE 0.0124 Train RMSE 0.0811 Train RMSE 0.00864
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks.
Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z
Ingrid Daubechies Ronald DeVore
Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks.
Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z
Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks.
Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z
1-1-1 ReLU MLP
Train RMSE 0.289
1-5-1 ReLU MLP
Train RMSE 0.297
1-10-1 ReLU MLP
Train RMSE 0.0237
1-100-1 ReLU MLP
Train RMSE 0.0243
Gradient Boosted Trees
Random Forest
Nearest Neighbors Decision Tree
1-1-1 ReLU MLP
Train RMSE 0.289
1-5-1 ReLU MLP
Train RMSE 0.297
1-10-1 ReLU MLP
Train RMSE 0.0237
1-100-1 ReLU MLP
Train RMSE 0.0243
ExtraTrees (#trees=103, #leaves=8,
bootstrap=off), Train RMSE 0.0243
ExtraTrees (#trees=1000, bootstrap)
Train RMSE 0.0243
Geurts, P., Ernst, D. & Wehenkel, L. Extremely randomized trees. Mach Learn 63, 3–42 (2006). https://doi.org/10.1007/s10994-006-6226-1
Decision
Tree
Random
Forest GBDT
Nearest
Neighbor
Logistic
Regression
SVM
Gaussian
Process
Neural
Network
NeurIPS 2019
ICML 2018
"A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
"A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
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!1
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!2
<latexit sha1_base64="FFDeu0DUpj47hM4y6m8eBm4BZtQ=">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</latexit>
!3
<latexit sha1_base64="sVWa2gbxZQIcX+Uuo5Pa/ZykHGo=">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</latexit>
R = 3
<latexit sha1_base64="OjCfKqSNh49UObVnmIiT+2dVPoU=">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</latexit>
a1x + b1
<latexit sha1_base64="ypt4OullvQj05vXVFfrmSxA2QBQ=">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</latexit>
a2x + b2
<latexit sha1_base64="KJcELS5QlR94EdZgQoFs9xFpLGc=">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</latexit>
a3x + b3
<latexit sha1_base64="cMrtMzvn4NG+7k/rWFifrZke5Q4=">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</latexit>
K
<latexit sha1_base64="SX+cUk0bt832iGox/GTAFgu/TOQ=">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</latexit>
x 2 RD
"A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
<latexit sha1_base64="9hP3p02sWXHHDzXWLshlHDq6Jj0=">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</latexit>
f⇥(x) =
⇣
f
(L)
✓(L) · · · f
(1)
✓(1)
⌘
(x)
<latexit sha1_base64="up8ecpPgYfR7pYjRae+QJ3x7AHo=">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</latexit>
⇥ =
n
✓(1)
, . . . , ✓(L)
o
<latexit sha1_base64="CrSrBbNDDRBNHbWah4z26wv6p/I=">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</latexit>
f
(i)
✓(i)
• fully-connected
• convolution
• activation
• ReLU
• leaky ReLU
• absolute value
• pooling (max, average, channel, etc)
• recurrent
• skip connection
MASO
DNs are signal-dependent affine
transformations. The particular affine
mapping applied to x depends on which
parrtition of the spline it falls in Rd.
"A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
ReLU
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">AAAChHichVHLTsJAFD3UF+ID1I2JGyLBuDBkUHzEhSG6cclDHgkS0tYBG0rbtIUEiT+gW40LV5q4MH6AH+DGH3DBJxiXmLhx4aU0MUrE20znzJl77pyZKxmqYtmMtT3C0PDI6Jh33DcxOTXtD8zMZi29bso8I+uqbuYl0eKqovGMrdgqzxsmF2uSynNSda+7n2tw01J07cBuGrxYEyuaUlZk0SYq2SwFQizCnAj2g6gLQnAjoQcecYgj6JBRRw0cGmzCKkRY9BUQBYNBXBEt4kxCirPPcQofaeuUxSlDJLZK/wqtCi6r0bpb03LUMp2i0jBJGUSYvbB71mHP7IG9ss8/a7WcGl0vTZqlnpYbJf/ZfPrjX1WNZhvH36qBnm2UseV4Vci74TDdW8g9fePkqpPeToVbS+yWvZH/G9ZmT3QDrfEu3yV56nqAH4m80ItRg6K/29EPsquR6EYkloyF4rtuq7xYwCKWqR+biGMfCWSoPsc5LnApjAorwpqw3ksVPK5mDj9C2PkC2GOP+Q==</latexit>
y
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="wThlqV4N/f6TJkASaopeeXy6hGk=">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</latexit>
z1
<latexit sha1_base64="eRVZ9lBHtuomnSZ9U9icTCKNkZM=">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</latexit>
z2
<latexit sha1_base64="ujDIq6S6SKYxF3DBHU4kFLGtayo=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCKDohhXRDcuecgjQULaOmBDaZu2kADxB0zcysKVJi6MH+AHuPEHXPAJxiUmblx4KU2MEvE20zlz5p47Z+ZKhqpYNmM9jzAxOTU94531zc0vLPoDS8s5S2+YMs/KuqqbBUm0uKpoPGsrtsoLhsnFuqTyvFQ7HOznm9y0FF07tlsGL9XFqqZUFFm0icq0y9vlQIhFmBPBURB1QQhuJPXAI05wCh0yGqiDQ4NNWIUIi74iomAwiCuhQ5xJSHH2Oc7hI22DsjhliMTW6F+lVdFlNVoPalqOWqZTVBomKYMIsxd2z/rsmT2wV/b5Z62OU2PgpUWzNNRyo+y/WM18/Kuq02zj7Fs11rONCvYcrwp5NxxmcAt5qG+2u/3Mfjrc2WC37I3837Aee6IbaM13+S7F09dj/EjkhV6MGhT93Y5RkNuKRHcjsVQslDhwW+XFGtaxSf2II4EjJJGl+lVc4gpdwStEhB0hPkwVPK5mBT9CSHwBXTaQoA==</latexit>
z3
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z4
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z5
<latexit sha1_base64="wThlqV4N/f6TJkASaopeeXy6hGk=">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</latexit>
z1
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z2
<latexit sha1_base64="ujDIq6S6SKYxF3DBHU4kFLGtayo=">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</latexit>
z3
<latexit sha1_base64="RfN7iAoKUyccE78XRExQP3n1Z8Y=">AAAChnichVG7TgJBFD2sL8QHqI2JDZFgrMhgUIwV0caShzwSJGR3HXHDvrK7kADxB0xspbDSxML4AX6AjT9gwScYS0xsLLwsmxgl4t3Mzpkz99w5M1cyVcV2GOv5hInJqekZ/2xgbn5hMRhaWi7YRsOSeV42VMMqSaLNVUXneUdxVF4yLS5qksqLUv1gsF9scstWDP3IaZm8ook1XTlVZNEhKteuJqqhCIsxN8KjIO6BCLxIG6FHHOMEBmQ0oIFDh0NYhQibvjLiYDCJq6BDnEVIcfc5zhEgbYOyOGWIxNbpX6NV2WN1Wg9q2q5aplNUGhYpw4iyF3bP+uyZPbBX9vlnrY5bY+ClRbM01HKzGrxYzX38q9JodnD2rRrr2cEpdl2vCnk3XWZwC3mob7a7/dxeNtrZYLfsjfzfsB57ohvozXf5LsOz12P8SOSFXowaFP/djlFQ2IrFd2KJTCKS2vda5cca1rFJ/UgihUOkkaf6NVziCl3BL8SEbSE5TBV8nmYFP0JIfQFfVpCh</latexit>
z4
<latexit sha1_base64="jGmtKxPXGnV1kZ3bfATsRPXurLo=">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</latexit>
z5
<latexit sha1_base64="wThlqV4N/f6TJkASaopeeXy6hGk=">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</latexit>
z1
<latexit sha1_base64="eRVZ9lBHtuomnSZ9U9icTCKNkZM=">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</latexit>
z2
<latexit sha1_base64="ujDIq6S6SKYxF3DBHU4kFLGtayo=">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</latexit>
z3
<latexit sha1_base64="RfN7iAoKUyccE78XRExQP3n1Z8Y=">AAAChnichVG7TgJBFD2sL8QHqI2JDZFgrMhgUIwV0caShzwSJGR3HXHDvrK7kADxB0xspbDSxML4AX6AjT9gwScYS0xsLLwsmxgl4t3Mzpkz99w5M1cyVcV2GOv5hInJqekZ/2xgbn5hMRhaWi7YRsOSeV42VMMqSaLNVUXneUdxVF4yLS5qksqLUv1gsF9scstWDP3IaZm8ook1XTlVZNEhKteuJqqhCIsxN8KjIO6BCLxIG6FHHOMEBmQ0oIFDh0NYhQibvjLiYDCJq6BDnEVIcfc5zhEgbYOyOGWIxNbpX6NV2WN1Wg9q2q5aplNUGhYpw4iyF3bP+uyZPbBX9vlnrY5bY+ClRbM01HKzGrxYzX38q9JodnD2rRrr2cEpdl2vCnk3XWZwC3mob7a7/dxeNtrZYLfsjfzfsB57ohvozXf5LsOz12P8SOSFXowaFP/djlFQ2IrFd2KJTCKS2vda5cca1rFJ/UgihUOkkaf6NVziCl3BL8SEbSE5TBV8nmYFP0JIfQFfVpCh</latexit>
z4
<latexit sha1_base64="jGmtKxPXGnV1kZ3bfATsRPXurLo=">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</latexit>
z5
-12.43 15.48 2.04 2.05 -2.48
arch:
(Linear(2, 5),
ReLU,
Linear(5, 1))
ReLU
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit>
y
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
<latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">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</latexit>
x1
<latexit sha1_base64="C1ohMOVeKO2tnLKPo8dfEeHHC5o=">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</latexit>
x2
Power diagram (PD)
aka Laguerre–Voronoi diagram
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Balestriero, Randall. "Max-Affine Splines Insights Into Deep Learning." (2021) Diss., Rice University. https://hdl.handle.net/1911/110439.
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSTayFoyRO3IiMAtEDUfAibYQfoQInYIACLdCAgg4csQoS2PiVQQQCJnJV6CJnIWLuPoVzCKC2hVkUMyRkm/iv46rssTqunZq2q1bwFBWHhcoIxMgLuScD8kweyCv5/LNW163heOngLA+11KyFLlZyH/+qNJw5NL5VYz1zOIVd1ytD76bLOLdQhvr22dUgl8zGuuvklryh/xvSJ094A739rtxlaLY3xo+MXvDFsEHi73aMgsJmXNyOJzKJaGrfa5UfVmENNrAfO5CCI0hD3u3DJfTgWggKW0JS2BumCj5Psww/Qjj8Ak3rksg=</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSTayFoyRO3IiMAtEDUfAibYQfoQInYIACLdCAgg4csQoS2PiVQQQCJnJV6CJnIWLuPoVzCKC2hVkUMyRkm/iv46rssTqunZq2q1bwFBWHhcoIxMgLuScD8kweyCv5/LNW163heOngLA+11KyFLlZyH/+qNJw5NL5VYz1zOIVd1ytD76bLOLdQhvr22dUgl8zGuuvklryh/xvSJ094A739rtxlaLY3xo+MXvDFsEHi73aMgsJmXNyOJzKJaGrfa5UfVmENNrAfO5CCI0hD3u3DJfTgWggKW0JS2BumCj5Psww/Qjj8Ak3rksg=</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit>
X1  ✓3
yes no
Red
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSTayFoyRO3IiMAtEDUfAibYQfoQInYIACLdCAgg4csQoS2PiVQQQCJnJV6CJnIWLuPoVzCKC2hVkUMyRkm/iv46rssTqunZq2q1bwFBWHhcoIxMgLuScD8kweyCv5/LNW163heOngLA+11KyFLlZyH/+qNJw5NL5VYz1zOIVd1ytD76bLOLdQhvr22dUgl8zGuuvklryh/xvSJ094A739rtxlaLY3xo+MXvDFsEHi73aMgsJmXNyOJzKJaGrfa5UfVmENNrAfO5CCI0hD3u3DJfTgWggKW0JS2BumCj5Psww/Qjj8Ak3rksg=</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">AAACi3ichVG7TgJBFL2sL0QR1MbEhkgwVmRAooZYEI2JJQ95JEDI7jrAhH1ldyBB4g9Y2lhgo4mF8QP8ABt/wIJPMJaY2Fh4d9nEKBHvZnbOnLnnzpm5kqEwixMy8AhT0zOzc95538KifykQXF4pWHrblGle1hXdLEmiRRWm0TxnXKElw6SiKim0KLUO7f1ih5oW07UT3jVoVRUbGqszWeRIlSq8SblYi9eCYRIlToTGQcwFYXAjrQcfoQKnoIMMbVCBggYcsQIiWPiVIQYEDOSq0EPORMScfQrn4ENtG7MoZojItvDfwFXZZTVc2zUtRy3jKQoOE5UhiJAXck+G5Jk8kFfy+WetnlPD9tLFWRppqVELXKzlPv5VqThzaH6rJnrmUIc9xytD74bD2LeQR/rO2dUwl8xGepvklryh/xsyIE94A63zLt9laLY/wY+EXvDFsEGx3+0YB4V4NLYTTWQS4dSB2yovrMMGbGE/diEFx5CGvNOHS+jDteAXtoWksD9KFTyuZhV+hHD0BVALksk=</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">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</latexit>
✓4
<latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit>
X2  ✓4
yes no
Red
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit>
X1  ✓3
yes no
Red
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSLVELR0mcuBEZBaIHouBF2gg/QgVOwAAFWqABBR04YhUksPErgwgETOSq0EXOQsTcfQrnEEBtC7MoZkjINvFfx1XZY3VcOzVtV63gKSoOC5URiJEXck8G5Jk8kFfy+WetrlvD8dLBWR5qqVkLXazkPv5VaThzaHyrxnrmcAq7rleG3k2XcW6hDPXts6tBLpmNddfJLXlD/zekT57wBnr7XbnL0GxvjB8ZveCLYYPE3+0YBYXNuLgdT2QS0dS+1yo/rMIabGA/diAFR5CGvNuHS+jBtRAUtoSksDdMFXyeZhl+hHD4BVRLkss=</latexit>
✓4
<latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit>
X2  ✓4
yes no
Red
<latexit sha1_base64="e77Z2aqe8KZFIqh8R+ZYHDOKN+I=">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</latexit>
✓5
<latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit>
X1  ✓5
yes no
Red
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit>
X1  ✓3
yes no
Red
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">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</latexit>
✓4
<latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit>
X2  ✓4
yes no
Red
<latexit sha1_base64="e77Z2aqe8KZFIqh8R+ZYHDOKN+I=">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</latexit>
✓5
<latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit>
X1  ✓5
yes no
Red
<latexit sha1_base64="eu7hkk7xInDVK5lVQXXLPV+MjYw=">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</latexit>
✓6
<latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit>
X1  ✓6
yes no
Red
Blue
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit>
X1  ✓3
yes no
Red
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit>
X2  ✓1
yes no
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit>
X1  ✓2
yes no
Blue
<latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">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</latexit>
✓4
<latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit>
X2  ✓4
yes no
Red
<latexit sha1_base64="e77Z2aqe8KZFIqh8R+ZYHDOKN+I=">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</latexit>
✓5
<latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit>
X1  ✓5
yes no
Red
<latexit sha1_base64="eu7hkk7xInDVK5lVQXXLPV+MjYw=">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</latexit>
✓6
<latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit>
X1  ✓6
yes no
Red
Blue
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit>
X1  ✓3
yes no
Red
<latexit sha1_base64="VJpvibITWQ0L0zaDACUdk6SsPiA=">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</latexit>
✓7
<latexit sha1_base64="bG79VND5DJn1e15l/47tj0nzmbk=">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</latexit>
X1  ✓7
yes no
Blue Blue
19
8
35 2
predict_proba(x)
red blue
8/27 = 0.296
19/27 = 0.704
0.0
1.0
0.704
0.296
Piecewise "constant"
Piecewise "constant"
Piecewise "constant"
Piecewise "constant"
Piecewise "constant"
= + +
+ + +
RandomForestClassifier
(n_estimators=6,
max_leaf_nodes=4)
6 × DecisionTreeClassifier(max_leaf_nodes=4)
Piecewise "constant"
RandomForestRegressor
(n_estimators=6,
max_leaf_nodes=8)
= + +
+ + +
6 × DecisionTreeRegressor(max_leaf_nodes=8)
Piecewise "constant"
Random forests ntree=10
Extra trees ntree=10
Piecewise "constant"
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
<latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit>
✓2
<latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit>
✓3
<latexit sha1_base64="qjiSJoxcwS4tq1ZQQKyk/d3Jx4c=">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</latexit>
X1
<latexit sha1_base64="8MqRReVFvhIT3zNdIBs0YcQ9SnU=">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</latexit>
X2
<latexit sha1_base64="q73WpNp2doZjySRNvFreZWUKlsk=">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</latexit>
<latexit sha1_base64="ZfCGc5FghO0FNxwpLzVn2qoZjcM=">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</latexit>
<latexit sha1_base64="fJBMCHp13koMDCQnuypJ5B3z94E=">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</latexit>
<latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit>
✓1
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✓2
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X1
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X2
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Random forests ntree=10
Extra trees ntree=10
x1
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x2
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x1
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x2
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n0
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dn0
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x1
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x2
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n0
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dn0
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n1
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dn1
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x1
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x2
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n0
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dn0
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`5
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`6
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n4
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dn4
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n1
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dn1
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x1
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x2
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n0
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dn0
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`5
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`6
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n4
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n1
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dn1
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`1
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`2
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n2
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dn2
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x1
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x2
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n0
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`5
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`6
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dn4
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n1
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dn1
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`1
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`2
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n2
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dn2
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`3
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`4
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n3
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dn3
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x1
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x2
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n0
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dn0
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`5
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n4
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dn4
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n1
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dn1
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`1
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`2
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n2
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dn2
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`3
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`4
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n3
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dn3
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<latexit sha1_base64="nSO8qFZKcyDEg6yWfnpy67LEjMY=">AAAB/nicdVDLSsNAFJ34rPVVdelmsAiuwiRtsd0V3bisYB/QhjCZTNqhk0mYmQglFPwFt7p3J279Fbd+idOHoEUPXDiccy/33hOknCmN0Ie1tr6xubVd2Cnu7u0fHJaOjjsqySShbZLwRPYCrChngrY105z2UklxHHDaDcbXM797T6ViibjTk5R6MR4KFjGCtZG6oZ8LvzL1S2VkNxqoWq1BZNeQ67p1Q1DFrTcc6NhojjJYouWXPgdhQrKYCk04VqrvoFR7OZaaEU6nxUGmaIrJGA9p31CBY6q8fH7uFJ4bJYRRIk0JDefqz4kcx0pN4sB0xliP1Ko3E//0gnhls47qXs5EmmkqyGJxlHGoEzjLAoZMUqL5xBBMJDO3QzLCEhNtEiuaUL4/h/+Tjms7yHZuq+Xm1TKeAjgFZ+ACOOASNMENaIE2IGAMHsETeLYerBfr1XpbtK5Zy5kT8AvW+xcFHJZq</latexit>
<latexit sha1_base64="nSO8qFZKcyDEg6yWfnpy67LEjMY=">AAAB/nicdVDLSsNAFJ34rPVVdelmsAiuwiRtsd0V3bisYB/QhjCZTNqhk0mYmQglFPwFt7p3J279Fbd+idOHoEUPXDiccy/33hOknCmN0Ie1tr6xubVd2Cnu7u0fHJaOjjsqySShbZLwRPYCrChngrY105z2UklxHHDaDcbXM797T6ViibjTk5R6MR4KFjGCtZG6oZ8LvzL1S2VkNxqoWq1BZNeQ67p1Q1DFrTcc6NhojjJYouWXPgdhQrKYCk04VqrvoFR7OZaaEU6nxUGmaIrJGA9p31CBY6q8fH7uFJ4bJYRRIk0JDefqz4kcx0pN4sB0xliP1Ko3E//0gnhls47qXs5EmmkqyGJxlHGoEzjLAoZMUqL5xBBMJDO3QzLCEhNtEiuaUL4/h/+Tjms7yHZuq+Xm1TKeAjgFZ+ACOOASNMENaIE2IGAMHsETeLYerBfr1XpbtK5Zy5kT8AvW+xcFHJZq</latexit>
<latexit sha1_base64="Z3hqgm++IGzADOmBhR7v6UmjU3Y=">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</latexit>
■ OC1/Oblique Tree (Murthy+ 1994)
■ Perceptron Tree (Utgoff , 1988)
■ Large Margin Tree (Wu+ 1999; Bennett+ 2000)
■ Margin Tree (Tibshirani & Hastie 2007)
■ Geometric Decision Tree (Manwani & Sastry,
2012)
■ HHCART (Wickramarachchi+, 2016)
■ M5/M5’ (Quinlan, 1992; Wang & Witten 1997)
■ RETIS (Karalic & Cestnik 1991)
■ SECRET (Dobra & Gehrke, 2002)
■ SMOTI (Malerba+ 2004)
■ MAUVE (Vens & Blockeel, 2006)
■ LLRT (Vogel, Asparouhov, Scheffer, 2007)
■ Cubist (Quinlan, 2011)
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Regression Trees M5 MARS
Classification Trees OC1 Multivariate CART
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
y=0.89 + -0.36 max(0,x-0.99)
+ -0.88 max(0,0.99-x)
y=1.23 + -0.59 max(0,x-0.99)
+ -1.32 max(0,0.99-x)
+ 1.41 max(0,x-4.96)
y=0.98 + -0.23 max(0,x-0.99)
+ -1.0 max(0,0.99-x)
+ 1.69 max(0,x-4.96)
+ -0.51 max(0,x-2.31)
y=0.95 + -0.12 max(0,x-0.99)
+ -0.96 max(0,0.99-x)
+ 0.77 max(0,x-4.96)
+ -0.79 max(0,x-2.31)
+ 0.87 max(0,x-2.31)
y=0.66 + -0.29 max(0,x-0.99)
Friedman JH. Multivariate Adaptive Regression Splines. The Annals of Statistics, 1991;19: 1–67.
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
STOC 2008 NIPS 2010
Google scholar citations: 419 Google scholar citations: 10
RP決定木の領域分割
10 RP vectors
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Popov S, Morozov S, Babenko A. Neural Oblivious Decision Ensembles for Deep Learning on
Tabular Data. ICLR 2020, http://arxiv.org/abs/1909.06312
Kontschieder P, Fiterau M, Criminisi A, Bulo SR. Deep Neural Decision Forests. IJCAI 2016,
https://www.ijcai.org/Proceedings/16/Papers/628.pdf
Lee G-H, Jaakkola TS. Oblique Decision Trees from Derivatives of ReLU Networks.
ICLR 2020, https://openreview.net/pdf?id=Bke8UR4FPB
Lay N, Harrison AP, Schreiber S, Dawer G, Barbu A. Random Hinge Forest for Differentiable
Learning. ICML 2018, http://arxiv.org/abs/1802.03882
Zantedeschi V, Kusner MJ, Niculae V. Learning Binary Decision Trees by Argmin Differentiation.
ICML 2021, http://arxiv.org/abs/2010.04627
Hazimeh H, Ponomareva N, Mol P, Tan Z, Mazumder R. The Tree Ensemble Layer: Differentiability
meets Conditional Computation. ICML 2020, https://arxiv.org/abs/2002.07772
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Thanks to
https://stanfordmlgroup.github.io/projects/ngboost/
https://towardsdatascience.com/interpreting-the-probabilistic-predictions-from-ngboost-868d6f3770b2
NGBRegressor
決定森回帰の信頼区間推定, Benign Overfitting, 多変量木とReLUネットの入力空間分割
Uncertainty Quantification (UQ)
https://en.wikipedia.org/wiki/Uncertainty_quantification
https://en.wikipedia.org/wiki/Uncertainty_quantification#Aleatoric_and_epistemic
• Accuracy alone does not suffice for reliable, consequential decision-making; we also need
uncertainty.
• Distribution-free UQ gives finite-sample statistical guarantees for any predictive model, no
matter how bad/misspecified, and any data distribution, even if unknown.
• DF techniques such as conformal prediction represent a new, principled approach to UQ for
complex prediction systems, such as deep learning.
• conformal prediction
• tolerance regions
• risk-controlling prediction sets
• calibration by binning
and more
https://en.wikipedia.org/wiki/Conformal_prediction
"Conformity"
Non-conformity measure
https://github.com/valeman/awesome-conformal-prediction
Awesome Conformal Prediction
https://github.com/valeman/awesome-conformal-prediction
Awesome Conformal Prediction
https://www.stat.cmu.edu/~aramdas/conformal.html
https://github.com/scikit-learn-contrib/MAPIE
With MAPIE, uncertainties are back in machine learning
https://towardsdatascience.com/with-mapie-uncertainties-are-back-in-machine-learning-882d5c17fdc3
https://github.com/scikit-learn-contrib/MAPIE
regressor = ExtraTreesRegressor(max_leaf_nodes=32, bootstrap=True)
MapieRegressor(regressor,
method="plus", cv=-1)
MapieRegressor(regressor,
method="plus",
cv=Subsample(n_resampling
s=50))
MapieRegressor(regressor,
method="plus", cv=-1)
MapieRegressor(regressor,
method="plus",
cv=Subsample(n_resampling
s=50))
95% prediction intervals
95% prediction intervals
90% prediction intervals
90% prediction intervals
Jacknife+
Jacknife+ after bootstrap
Jacknife+
Jacknife+ after bootstrap

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  • 6. if x2 ≤ θ1 then if x1 ≤ θ2 then return Blue else if x2 ≤ θ4 then return Red else if x1 ≤ θ5 then return Red else if x1 ≤ θ6 then return Blue else return Red else if x1 ≤ θ3 then if x2 ≤ θ7 then return Blue else return Blue else return Red <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no <latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit> X1  ✓2 yes no Blue <latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit> X2  ✓4 yes no Red <latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit> X1  ✓5 yes no Red <latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit> X1  ✓6 yes no Red Blue X1  ✓3 yes no Red X1  ✓7 yes no Blue Blue <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit 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sha1_base64="zBiZIFwyoN/DGVfflNKDwKhdwmg=">AAACynichVHLTttAFD24DyB9kJYNUjdWIxCraIIQIKRWUatKLFgQIBAJI8s2lzBkYrv2JFJqZceqP9BFV0ViUfEB/YBuumXBgvIFiCWVuumCG8cSAgS91njOnLnnzpm5bqhkrIU4GTAePHz0eHBoOPfk6bPnI/kXL9fioBV5VPUCFUQ114lJSZ+qWmpFtTAip+kqWncb73v7622KYhn4q7oT0mbTqftyW3qOZsrOf6jYu+b8G9NyqS79xONScTdXsxumpeijaekd0o6tTMtKybdXTM4ifysT2PmCKIo0zNuglIECslgK8j9gYQsBPLTQBMGHZqzgIOZvAyUIhMxtImEuYiTTfUIXOda2OIs4w2G2wf86rzYy1ud1r2acqj0+RfGIWGliXByL7+JC/BKH4kz8u7NWktboeenw7Pa1FNojn8dW/v5X1eRZY+dKda9njW3MpV4lew9TpncLr69vf/pysTK/PJ5MiH1xzv6/iRPxk2/gt/94BxVa/nqPH5e98Itxg0o323EbrE0VSzPF6cp0ofwua9UQXuE1JrkfsyhjAUuocv1DHOE3To1FIzI6RtJPNQYyzSiuhbF3CZqVqlw=</latexit> Qj := ( Xk  ✓l Xk > ✓l Blue <latexit sha1_base64="p+ll+ob/bB8deuXgcYGzmiYX5Zc=">AAACoXichVHLSsNAFD3G97NVN4IugqXiqkyKqAiC6EZ3tbZaaEtJ0rGGpklIpgUtblz6Ay5cKQiKK934AW78ARd+gris4MaFt2lAtFhvmMyZM/fcOTNXc0zDE4y9dEndPb19/QODQ8Mjo2Oh8PjErmdXXZ2nddu03Yymetw0LJ4WhjB5xnG5WtFMvqeVN5r7ezXueoZtpcShw/MVtWQZ+4auCqIK4ZmUvLIqJwqKnKtxTiDeArmiLbxCOMJizA+5HSgBiCCIhB1+QA5F2NBRRQUcFgRhEyo8+rJQwOAQl0edOJeQ4e9zHGOItFXK4pShElumf4lW2YC1aN2s6flqnU4xabiklBFlz+yGNdgTu2Wv7PPPWnW/RtPLIc1aS8udQuh0aufjX1WFZoGDb1VHzwL7WPa9GuTd8ZnmLfSWvnZ01thZSUbrc+ySvZH/C/bCHukGVu1dv9rmyfMOfjTyQi9GDVJ+t6Md7MZjymJsYXshsrYetGoA05jFPPVjCWvYRAJpqn+Ca9zhXopIW1JCSrZSpa5AM4kfIWW/AAtRmVk=</latexit> T := P1 _ P2 _ . . . <latexit sha1_base64="+WntnxsFSxJvfdpwMjPE/ws7ZhE=">AAACqXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoxCSlpmfmVScDzSiu5QqxNYyJUQBSBlwxqXkpUOF4AWUDPQMwUMBkGEIZygxQEJAvsJ0hhiGFIZ8hmaGUIZchlSGPoQTIzmFIZCgGwmgGQwYDhgKgWCxDNVCsCMjKBMunMtQycAH1lgJVpQJVJAJFs4FkOpAXDRXNA/JBZhaDdScDbckB4iKgTgUGVYOrBisNPhucMFht8NLgD06zqsFmgNxSCaSTIHpTC+L5uySCvxPUlQukSxgyELrwurmEIY3BAuzWTKDbC8AiIF8kQ/SXVU3/HGwVpFqtZrDI4DXQ/QsNbhocBvogr+xL8tLA1KDZeNyTBHQLMMSAEWSIHh2YjDAjPUMzPZNAE2UHJ2hUcTBIMygxaADjw5zBgcGDIYAhFGh+F8MGhp0Mu5i0mQKZIpiiIEqZGKF6hBlQAFMyAOxdnYo=</latexit> ( T = 1 T = 0 Path Query Not Blue (= Red)
  • 8. https://www.kaggle.com/kaggle-survey-2021 State of Data Science and Machine Learning 2021 The 2021 Kaggle DS & ML Survey received 25,973 usable responses from participants in 171 different countries and territories. Q17. Which of the following ML algorithms do you use on a regular basis? (Select all that apply)
  • 10. Object recognition Game play J’aime la musique I love music Speech recognition Machine translation Super resolution
  • 11. p1 p2 p3 p4 Random Forest Gaussian Process Logistic Regression
  • 12. Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001. https://doi.org/10.1214/ss/1009213726 Rashomon Occam Bellman
  • 13. Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001. https://doi.org/10.1214/ss/1009213726 Rashomon Occam Bellman
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sha1_base64="ms/uPGAGeTg0lt5op+vO01P7rjg=">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</latexit> y
  • 18. Bronstein MM, Bruna J, Cohen T, Veličković P. Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2104.13478 for all <latexit sha1_base64="NqBYbofjRLhsRuy5gs3o5FfAjVo=">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</latexit> f : Rd ! R <latexit sha1_base64="bLp7Jqe5Ke+bRuLIs0xt/zwFMHo=">AAACmnichVHLLgRBFD3T3u9BJBIWExOPhUxqRBCrCRtiM8a8EsOkuxUq+pXumgk68wN+wMKKxAIf4ANs/ICFTxBLEhsLd3o6EQS3U12nTt1z61RdzTGEJxl7jChNzS2tbe0dnV3dPb190f6BvGdXXJ3ndNuw3aKmetwQFs9JIQ1edFyumprBC9rBcn2/UOWuJ2wrK48cvmWqe5bYFboqiSpHhw6nDydjJWHFSqYq9zXNz9S2d8rROEuwIGI/QTIEcYSRtqO3KGEHNnRUYILDgiRsQIVH3yaSYHCI24JPnEtIBPscNXSStkJZnDJUYg/ov0erzZC1aF2v6QVqnU4xaLikjGGcPbAr9sLu2Q17Yu+/1vKDGnUvRzRrDS13yn0nwxtv/6pMmiX2P1V/epbYxULgVZB3J2Dqt9Ab+urx6cvGYmbcn2AX7Jn8n7NHdkc3sKqv+uU6z5z94UcjL/Ri1KDk93b8BPmZRHIuMbs+G08tha1qxwjGMEX9mEcKK0gjR/V9nOMaN8qosqSsKmuNVCUSagbxJZTsB6MZmAU=</latexit> x, x0 2 Rd -Lipschitz <latexit sha1_base64="VAwN9BEfNuoROOZ0f78ZUuUItlA=">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</latexit> |f(x) f(x0 )| 6 Lkx x0 k <latexit sha1_base64="+ZiSkRyCDmMX1dfLuFjrVp2j6hw=">AAAChHichVG7SgNBFD1ZNcZ31EawCQbFQsJE4wMLCdpYWJjERCGK7K6TOLjZXXYngRj8AW0VCysFC/ED/AAbf8AinyCWEWwsvNksiAb1LrNz5sw9d87M1WxDuJKxekDp6OwKdod6evv6BwaHwsMjOdcqOzrP6pZhOTua6nJDmDwrhTT4ju1wtaQZfFs7Wmvub1e44wrL3JJVm++V1KIpCkJXJVGpjf1wlMWYF5F2EPdBFH5sWuEH7OIAFnSUUQKHCUnYgAqXvjziYLCJ20ONOIeQ8PY5TtBL2jJlccpQiT2if5FWeZ81ad2s6XpqnU4xaDikjGCSPbM71mBP7J69sI9fa9W8Gk0vVZq1lpbb+0OnY5n3f1UlmiUOv1R/epYoYMnzKsi77THNW+gtfeX4spFZTk/WptgNeyX/16zOHukGZuVNv03x9NUffjTyQi9GDYr/bEc7yM3G4guxRCoRTa76rQphHBOYpn4sIol1bCJL9TnOcI4LJajMKHPKfCtVCfiaUXwLZeUTeMOPzA==</latexit> L Donoho DL, High-dimensional data analysis: The curses and blessings of dimensionality. Plenary Lecture, AMS National Meeting on Mathematical Challenges of the 21st Century. 2000.
  • 19. <latexit sha1_base64="mchf2gvio17p/Fz9iF84xhnMH3c=">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</latexit> V (d) = ⇡ d 2 d 2 · d 2 ! 0 (d ! 1) https://www.math.ucdavis.edu/~strohmer/courses/180BigData/180lecture1.pdf <latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit> 1 <latexit sha1_base64="z3Ba/qsiF6tEgbjc9DFSGYR9udI=">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</latexit> 0.5 p d <latexit sha1_base64="rzd8+rC/ag8/gv3jUp4Y6gUrSiI=">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</latexit> 0.5 <latexit sha1_base64="Nn/M5/H5iD9Z/hOTHxuHUqAZH+I=">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</latexit> d = 2 <latexit sha1_base64="LVcqe4i4MA2A5XBAmTvFXnqLayw=">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</latexit> d = 4 <latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit> 1 <latexit sha1_base64="UR9ymFZxc/Q/TCzOORzighhjAlM=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCKDQTEuDNGNSx7ySJCQtgzYUNqmLSRI/AETt7JwpYkL4wf4AW78ARd8gnGJiRsXXkoTo0S8zXTOnLnnzpm5kqEqls1YzyNMTE5Nz3hnfXPzC4v+wNJyztKbpsyzsq7qZkESLa4qGs/aiq3ygmFysSGpPC/VDwf7+RY3LUXXju22wUsNsaYpVUUWbaIylf1YORBiEeZEcBREXRCCG0k98IgTVKBDRhMNcGiwCasQYdFXRBQMBnEldIgzCSnOPsc5fKRtUhanDJHYOv1rtCq6rEbrQU3LUct0ikrDJGUQYfbC7lmfPbMH9so+/6zVcWoMvLRploZabpT9F6uZj39VDZptnH6rxnq2UcWu41Uh74bDDG4hD/Wts24/s5cOdzbYLXsj/zesx57oBlrrXb5L8fT1GD8SeaEXowZFf7djFOS2ItGdSCwVCyUO3FZ5sYZ1bFI/4kjgCElkqX4Nl7hCV/AKEWFbiA9TBY+rWcGPEBJf6hqQag==</latexit> d > 4 <latexit sha1_base64="nHSiJ7YlA7Y9nChVeD/tJhyz2yo=">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</latexit> 0.5 p 2 <latexit sha1_base64="z3Ba/qsiF6tEgbjc9DFSGYR9udI=">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</latexit> 0.5 p d !? Unit Cube in Unit Ball <latexit sha1_base64="eWTwm+uSU1d/PxikW7VncIaHMRY=">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</latexit> 1 <latexit sha1_base64="PtefnFPcLJximPXvkayt2U9xPRk=">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</latexit> > 1 !?
  • 21. Balestriero R, Pesenti J, LeCun Y. Learning in High Dimension Always Amounts to Extrapolation. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2110.09485 https://youtu.be/86ib0sfdFtw • "on any high-dimensional (>100) dataset, interpolation almost surely never happens." • "Those results challenge the validity of our current interpolation/extrapolation definition as an indicator of generalization performances. "
  • 22. ! Fan J, Han F, Liu H. Challenges of Big Data Analysis. Natl Sci Rev. 2014;1: 293–314. <latexit sha1_base64="hsRBbjUZq2fjH2VINZUiaiPtLiw=">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</latexit> (X1, . . . , Xd) ⇠ Nd(0, I) <latexit sha1_base64="Ay3UPGyVpdQRyf/zf3IimzcaXzc=">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</latexit> r̂ = max j>2 |corr(X1, Xj)| <latexit sha1_base64="DII9mhRSnTvcUJHdfuik1ohKbRg=">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</latexit> R̂ = max |S|=4 max j corr 0 @X1, X j2S jXj 1 A 0.3 0.4 0.5 0.6 0.5 0.6 0.7 0.8 <latexit sha1_base64="dsksfNj+oQDjs49biWmppWLfS4s=">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</latexit> R̂ <latexit sha1_base64="XcYeasdspQvBninjyfi4DGqRhKY=">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</latexit> r̂
  • 23. K. Beyer+, When Is “Nearest Neighbor” Meaningful? ICDT’99 V. Pestov, On the geometry of similarity search: dimensionality curse and concentration of measure, Information Processing Letters, 1999.
  • 26. Journal of the American Statistical Association , Jun., 2006, Vol. 101, No. 474 (Jun., 2006), pp. 578-590 https://www.jstor.org/stable/27590719 "We introduce a concept of potential nearest neighbors (k-PNNs) and show that random forests can be viewed as adaptively weighted k-PNN methods. "
  • 30. Given , a real i.i.d. sequence, estimate TheBibleor"TheYellowTerror" byLuc,Laci,Gábor NeurIPS 2021 Invited Talk (Breiman Lecture) Do we know how to estimate the mean? <latexit sha1_base64="3NbZouAktoxRn4pfXQgJR8+DKAA=">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</latexit> X1, . . . , Xn <latexit sha1_base64="luqROJ+77OljeLhhMh6h7Ajj7RY=">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</latexit> µ = EX1 <latexit sha1_base64="ve2fKPQE43VaTVIusM8fwhBZiOg=">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</latexit> ( Pn i=1 Xi)/n
  • 31. Journal of Machine Learning Research, 9, 2015-2033, 2008 The Annals of Statistics, 43(4), 1716-1741, 2015.
  • 33. Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001. https://doi.org/10.1214/ss/1009213726 Rashomon Occam Bellman
  • 34. Breiman L, Statistical Modeling: The Two Cultures. Statist. Sci. 16(3): 199-231, 2001. https://doi.org/10.1214/ss/1009213726 • "What I call the Rashomon Effect is that there is often a multitude of different descriptions [equations f︎(x)] in a class of functions giving about the same minimum error rate. The most easily understood example is subset selection in linear regression." • "The Rashomon Effect also occurs with decision trees and neural nets." • "This effect is closely connected to what I call instability (Breiman,1996a) that occurs when there are many different models crowded together that have about the same training or test set error." Rashomon Effect:
  • 35. Kernel Ridge (RBF) Neural Network (MLP) Gradient Boosted Trees SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree
  • 36. Kernel Ridge (RBF) Neural Network (MLP) Gradient Boosted Trees SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree
  • 37. https://arxiv.org/abs/2011.03395 https://ai.googleblog.com/2021/10/ how-underspecification-presents.html "While ML models are validated on held-out data, this validation is often insufficient to guarantee that the models will have well-defined behavior when they are used in a new setting. "
  • 38. Kernel Ridge (RBF) Neural Network (MLP) Gradient Boosted Trees SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree
  • 39. Kernel Ridge (RBF) Neural Network (MLP) Gradient Boosted Trees SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree Benign? Benign? Benign? Malignant Malignant
  • 41. Hutter et al, Algorithm Runtime Prediction: Methods & Evaluation (2014) https://arxiv.org/abs/1211.0906
  • 42. https://scikit-optimize.github.io/stable/_modules/skopt/learning/forest.html def _return_std(X, trees, predictions, min_variance): # This derives std(y | x) as described in 4.3.2 of arXiv:1211.0906 std = np.zeros(len(X)) for tree in trees: var_tree = tree.tree_.impurity[tree.apply(X)] # This rounding off is done in accordance with the # adjustment done in section 4.3.3 # of http://arxiv.org/pdf/1211.0906v2.pdf to account # for cases such as leaves with 1 sample in which there # is zero variance. var_tree[var_tree < min_variance] = min_variance mean_tree = tree.predict(X) std += var_tree + mean_tree ** 2 std /= len(trees) std -= predictions ** 2.0 std[std < 0.0] = 0.0 std = std ** 0.5 return std Hutter et al, Algorithm Runtime Prediction: Methods & Evaluation (2014) https://arxiv.org/abs/1211.0906
  • 45. GradientBoostingRegressor( (n_estimators=30, learning_rate=0.1) LGBMRegressor (n_estimators=30, learning_rate=0.1, max_leaf_nodes=8, min_child_samples=5) aka "Pinball loss"
  • 46. Rando Forest ExtraTrees (w/o bootstrap) ExtraTrees (w/ bootstrap) Gradient Boosted Trees Rando Forest ExtraTrees (w/o bootstrap) ExtraTrees (w/ bootstrap) Gradient Boosted Trees
  • 50. MLP (Quantile Regression) MLP (Quantile Regression) MLP (MC Dropout) MLP (MC Dropout) arch: (Linear(1, 100), ReLU, Dropout(p), Linear(100, 100), ReLU, Dropout(p), Linear(100, 1)) p=0.25 #sampling=30 p=0.25 #sampling=30 p=0.0 p=0.0
  • 51. MLP (30 Ensemble) MLP (30 Ensemble) p=0.0 p=0.0 MLP (30 Ensemble) MLP (30 Ensemble) p=0.25 p=0.25 arch: (Linear(1, 100), ReLU, Dropout(p), Linear(100, 100), ReLU, Dropout(p), Linear(100, 1))
  • 52. • Tolstikhin IO, Houlsby N, Kolesnikov A, Beyer L, Zhai X, Unterthiner T, et al. MLP-Mixer: An all-MLP Architecture for Vision. NeurIPS 2021. • Kadra A, Lindauer M, Hutter F, Grabocka J. Well-Tuned Simple Nets Excel on Tabular Datasets. NeurIPS 2021. • Liu H, Dai Z, So D, Le Q. Pay Attention to MLPs. NeurIPS 2021. • Melas-Kyriazi L. Do You Even Need Attention? A Stack of Feed-Forward Layers Does Surprisingly Well on ImageNet. 2021. http://arxiv.org/abs/2105.02723
  • 53. Kernel Ridge (RBF) Neural Network (MLP) Gradient Boosted Trees SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree Benign? Benign? Benign? Malignant Malignant
  • 54. PolyReg(1) RMSE 0.299 PolyReg(3) RMSE 0.28 PolyReg(5) RMSE 0.225 PolyReg(7) RMSE 0.113 PolyReg(10) RMSE 0.0189 PolyReg(15) RMSE 0.00737 PolyReg(20) RMSE 0.000 PolyReg(30) RMSE 0.000 ExtraTrees (no bootstrap) RMSE 0.000 ExtraTrees (bootstrap) RMSE 0.0121 Random Forest RMSE 0.012 LGBM RMSE 0.0508 95%-CI 95%-CI 95%-CI 95%-CI Problematic overfitting by polynomial regression of order k clearly overfitted but harmless (still informative) also we can assess the uncertainty
  • 55. ExtraTrees (no bootstrap) ExtraTrees (no bootstrap) ExtraTrees (no bootstrap) ExtraTrees (no bootstrap) Gradient Boosted Trees Gradient Boosted Trees Gradient Boosted Trees Gradient Boosted Trees
  • 56. Nearest Neighbor (k=1) Nearest Neighbor (k=1) Nearest Neighbor (k=1) Nearest Neighbor (k=1) Decision Tree Decision Tree Decision Tree Decision Tree
  • 57. Random Forest Random Forest Random Forest Random Forest Nearest Neighbor (k=3) Nearest Neighbor (k=3) Nearest Neighbor (k=3) Nearest Neighbor (k=3)
  • 58. "our experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization and occurs even if we replace the true images by completely unstructured random noise. " Berner J, Grohs P, Kutyniok G, Petersen P. The Modern Mathematics of Deep Learning. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2105.04026 Zhang C, Bengio S, Hardt M, Recht B, Vinyals O. Understanding Deep Learning (Still) Requires Rethinking Generalization. Commun ACM. 2021;64: 107–115.
  • 59. Berner J, Grohs P, Kutyniok G, Petersen P. The Modern Mathematics of Deep Learning. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2105.04026
  • 61. PNAS (2020) Ann. Statist. (2020) NeurIPS (2018) arXiv (2019)
  • 66. PolyReg(10) Train RMSE 0.0189 PolyReg(15) Train RMSE 0.00737 KernelRidge RBF(γ=1) Train RMSE 0.103 KernelRidge RBF(γ=10) Train RMSE 0.0139 KernelRidge RBF(γ=100) Train RMSE 0.00726 KernelRidge RBF(γ=1000) Train RMSE 0.00726
  • 67. ExtraTrees (#trees=1, #leaves=8, bootstrap=off), Train RMSE 0.0189 ExtraTrees (#trees=10, #leaves=8, bootstrap=off), Train RMSE 0.0274 ExtraTrees (#trees=102, #leaves=8, bootstrap=off), Train RMSE 0.0279 ExtraTrees (#trees=103, #leaves=8, bootstrap=off), Train RMSE 0.0243 GBDT w/ 1 ExtraTree (#leaves=8) Train RMSE 0.29 GBDT w/ 10 ExtraTree (#leaves=8) Train RMSE 0.17 GBDT w/ 102, ExtraTree (#leaves=8) Train RMSE 0.00597 GBDT w/ 103, ExtraTree (#leaves=8) Train RMSE 0.000997
  • 68. Random Forest (#trees=100) Train RMSE 0.0109 Random Forest (#trees=1000) Train RMSE 0.0274 ExtraTrees (#trees=100, bootstrap) Train RMSE 0.0109 ExtraTrees (#trees=1000, bootstrap) Train RMSE 0.0243 Random Forest (#trees=100) Train RMSE 0.0112 Random Forest (#trees=100) Train RMSE 0.0114 ExtraTrees (#trees=100, bootstrap) Train RMSE 0.0099 ExtraTrees (#trees=1000, bootstrap) Train RMSE 0.00977 With Random Fourier Features (100-dim Random Kitchen Sinks)
  • 69. 1-1-1 ReLU MLP Train RMSE 0.287 1-5-1 ReLU MLP Train RMSE 0.287 1-10-1 ReLU MLP Train RMSE 0.0191 1-100-1 ReLU MLP Train RMSE 0.00816 Train RMSE 0.0139 Train RMSE 0.0066 Train RMSE 0.0121 Train RMSE 0.0139 … <latexit sha1_base64="OtDdB3Z2MZrUEFM+jHf2FtPg0CM=">AAAChHichVHLTsJAFD3UF+ID1I2JGyLBuDBkUHzEhSG6cclDHgkS0tYBG0rbtIUEiT+gW40LV5q4MH6AH+DGH3DBJxiXmLhx4aU0MUrE20znzJl77pyZKxmqYtmMtT3C0PDI6Jh33DcxOTXtD8zMZi29bso8I+uqbuYl0eKqovGMrdgqzxsmF2uSynNSda+7n2tw01J07cBuGrxYEyuaUlZk0SYqWS0FQizCnAj2g6gLQnAjoQcecYgj6JBRRw0cGmzCKkRY9BUQBYNBXBEt4kxCirPPcQofaeuUxSlDJLZK/wqtCi6r0bpb03LUMp2i0jBJGUSYvbB71mHP7IG9ss8/a7WcGl0vTZqlnpYbJf/ZfPrjX1WNZhvH36qBnm2UseV4Vci74TDdW8g9fePkqpPeToVbS+yWvZH/G9ZmT3QDrfEu3yV56nqAH4m80ItRg6K/29EPsquR6EYkloyF4rtuq7xYwCKWqR+biGMfCWSoPsc5LnApjAorwpqw3ksVPK5mDj9C2PkCuqOP6w==</latexit> k Optimized With L-BFGS arch: (Linear(1, k), ReLU, Linear(k, 1))
  • 70. 1-1-1 ReLU MLP Train RMSE 0.289 1-5-1 ReLU MLP Train RMSE 0.297 1-10-1 ReLU MLP Train RMSE 0.0237 1-100-1 ReLU MLP Train RMSE 0.0243 … <latexit sha1_base64="OtDdB3Z2MZrUEFM+jHf2FtPg0CM=">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</latexit> k Optimized With Adam arch: (Linear(1, k), ReLU, Linear(k, 1))
  • 71. k=1 ReLU MLP Train RMSE 0.305 k=5 ReLU MLP Train RMSE 0.303 k=10 ReLU MLP Train RMSE 0.0131 k=100 ReLU MLP Train RMSE 0.00979 Optimized With Adam arch: (Linear(1, k), ReLU, Linear(k, k), ReLU, Linear(k, 1)) x 9 Train RMSE 0.0443 Train RMSE 0.0124 Train RMSE 0.0811 Train RMSE 0.00864
  • 73. Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks. Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z Ingrid Daubechies Ronald DeVore
  • 74. Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks. Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z
  • 75. Daubechies, I., DeVore, R., Foucart, S. et al. Nonlinear Approximation and (Deep) ReLU Networks. Constr Approx 55, 127–172 (2022). https://doi.org/10.1007/s00365-021-09548-z
  • 76. 1-1-1 ReLU MLP Train RMSE 0.289 1-5-1 ReLU MLP Train RMSE 0.297 1-10-1 ReLU MLP Train RMSE 0.0237 1-100-1 ReLU MLP Train RMSE 0.0243 Gradient Boosted Trees Random Forest Nearest Neighbors Decision Tree
  • 77. 1-1-1 ReLU MLP Train RMSE 0.289 1-5-1 ReLU MLP Train RMSE 0.297 1-10-1 ReLU MLP Train RMSE 0.0237 1-100-1 ReLU MLP Train RMSE 0.0243 ExtraTrees (#trees=103, #leaves=8, bootstrap=off), Train RMSE 0.0243 ExtraTrees (#trees=1000, bootstrap) Train RMSE 0.0243
  • 78. Geurts, P., Ernst, D. & Wehenkel, L. Extremely randomized trees. Mach Learn 63, 3–42 (2006). https://doi.org/10.1007/s10994-006-6226-1
  • 81. "A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
  • 82. "A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
  • 83. <latexit sha1_base64="hTsNc7Yxq+3zKV9wCSgyaPQAP+M=">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</latexit> !1 <latexit sha1_base64="Tgm67LizZXft8YrsZzmIczqCO8g=">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</latexit> !2 <latexit sha1_base64="FFDeu0DUpj47hM4y6m8eBm4BZtQ=">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</latexit> !3 <latexit sha1_base64="sVWa2gbxZQIcX+Uuo5Pa/ZykHGo=">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</latexit> R = 3 <latexit sha1_base64="OjCfKqSNh49UObVnmIiT+2dVPoU=">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</latexit> a1x + b1 <latexit sha1_base64="ypt4OullvQj05vXVFfrmSxA2QBQ=">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</latexit> a2x + b2 <latexit sha1_base64="KJcELS5QlR94EdZgQoFs9xFpLGc=">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</latexit> a3x + b3
  • 84. <latexit sha1_base64="cMrtMzvn4NG+7k/rWFifrZke5Q4=">AAAChHichVG7SgNBFD1ZNcZ31EawCQbFQsJE4wMLCdoINiYxUYgiu+skDm52l91JIAZ/QFvFwkrBQvwAP8DGH7DIJ4hlBBsLbzYLokG9y+ycOXPPnTNzNdsQrmSsHlA6OruC3aGe3r7+gcGh8PBIzrXKjs6zumVYzo6mutwQJs9KIQ2+YztcLWkG39aO1pr72xXuuMIyt2TV5nsltWiKgtBVSVRqYz8cZTHmRaQdxH0QhR+bVvgBuziABR1llMBhQhI2oMKlL484GGzi9lAjziEkvH2OE/SStkxZnDJUYo/oX6RV3mdNWjdrup5ap1MMGg4pI5hkz+yONdgTu2cv7OPXWjWvRtNLlWatpeX2/tDpWOb9X1WJZonDL9WfniUKWPK8CvJue0zzFnpLXzm+bGSW05O1KXbDXsn/NauzR7qBWXnTb1M8ffWHH4280ItRg+I/29EOcrOx+EIskUpEk6t+q0IYxwSmqR+LSGIdm8hSfY4znONCCSozypwy30pVAr5mFN9CWfkEdqOPyw==</latexit> K <latexit sha1_base64="SX+cUk0bt832iGox/GTAFgu/TOQ=">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</latexit> x 2 RD "A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
  • 85. <latexit sha1_base64="9hP3p02sWXHHDzXWLshlHDq6Jj0=">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</latexit> f⇥(x) = ⇣ f (L) ✓(L) · · · f (1) ✓(1) ⌘ (x) <latexit sha1_base64="up8ecpPgYfR7pYjRae+QJ3x7AHo=">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</latexit> ⇥ = n ✓(1) , . . . , ✓(L) o <latexit sha1_base64="CrSrBbNDDRBNHbWah4z26wv6p/I=">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</latexit> f (i) ✓(i) • fully-connected • convolution • activation • ReLU • leaky ReLU • absolute value • pooling (max, average, channel, etc) • recurrent • skip connection MASO DNs are signal-dependent affine transformations. The particular affine mapping applied to x depends on which parrtition of the spline it falls in Rd. "A large class of DNs can be written as a composition of max-affine spline operators (MASOs)"
  • 86. ReLU <latexit sha1_base64="mpuRLmJhhTRKlPt+QsNIZJRw5HU=">AAAChnichVHLTsJAFD3UF+ID1I2JGyLBuCJTg2JcEd245CGPBAlp64gNpW3aQkTiD5i4lYUrTVwYP8APcOMPuOATjEtM3LjwUpoYJeJtpnPmzD13zsyVTU21Hca6PmFsfGJyyj8dmJmdmw+GFhbzttGwFJ5TDM2wirJkc03Vec5RHY0XTYtLdVnjBbm2198vNLllq4Z+4LRMXq5LVV09VhXJISp7WhEroQiLMTfCw0D0QARepIzQIw5xBAMKGqiDQ4dDWIMEm74SRDCYxJXRJs4ipLr7HOcIkLZBWZwyJGJr9K/SquSxOq37NW1XrdApGg2LlGFE2Qu7Zz32zB7YK/v8s1bbrdH30qJZHmi5WQleLGc//lXVaXZw8q0a6dnBMbZdryp5N12mfwtloG+edXrZnUy0vcZu2Rv5v2Fd9kQ30Jvvyl2aZ65H+JHJC70YNUj83Y5hkN+IiVuxeDoeSe56rfJjBatYp34kkMQ+UshR/SoucYWO4BdiwqaQGKQKPk+zhB8hJL8AVLKQnA==</latexit> x1 <latexit 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  • 89. Power diagram (PD) aka Laguerre–Voronoi diagram
  • 91. Balestriero, Randall. "Max-Affine Splines Insights Into Deep Learning." (2021) Diss., Rice University. https://hdl.handle.net/1911/110439.
  • 94. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no
  • 95. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no <latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit> ✓2 <latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit> X1  ✓2 yes no Blue
  • 96. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no <latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit> ✓2 <latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit> X1  ✓2 yes no Blue <latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit> ✓3 <latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit> X1  ✓3 yes no Red
  • 97. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no <latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit> ✓2 <latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">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</latexit> X1  ✓2 yes no Blue <latexit sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">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</latexit> ✓4 <latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit> X2  ✓4 yes no Red <latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit> ✓3 <latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">AAAClnichVHLSsNAFD3GV3221Y3gplgUV2WiRcWFFEV0Wa3VgpWQxKmGpklMphUt/oA/4EJcKKiIH+AHuPEHXPQTxKWCGxfepAFRUW+YzJkz99w5M1dzTMMTjDVapNa29o7OSFd3T29ffzQWH1j37Kqr87xum7Zb0FSPm4bF88IQJi84Llcrmsk3tPKCv79R465n2NaaOHD4VkXdsYySoauCKCUWLyhyomjyvURR7HKhKpNKLMlSLIjETyCHIIkwsnbsDkVsw4aOKirgsCAIm1Dh0bcJGQwOcVuoE+cSMoJ9jiN0k7ZKWZwyVGLL9N+h1WbIWrT2a3qBWqdTTBouKRMYZY/shr2wB3bLntj7r7XqQQ3fywHNWlPLHSV6PJR7+1dVoVlg91P1p2eBEmYCrwZ5dwLGv4Xe1NcOT15ys6uj9TF2wZ7J/zlrsHu6gVV71S9X+OrpH3408kIvRg2Sv7fjJ1ifSMlTqfRKOpmZD1sVwTBGME79mEYGy8giT/X3cYYrXEtD0py0KC01U6WWUDOILyFlPwA1vpYb</latexit> X1  ✓3 yes no Red
  • 98. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit sha1_base64="KLK3uKDnOLsAzd1uz4imbactEN0=">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</latexit> X2  ✓1 yes no <latexit 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  • 99. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit 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sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">AAAClnichVHLSsNAFD3Gd321uhHcFIviqkykVHEhRRFd9mG1YCUkcWyDaRKTaUWLP+APuBAXCiriB/gBbvwBF/0EcVnBjQtv04CoqDdM5syZe+6cmas5puEJxhodUmdXd09vX39oYHBoeCQcGd3w7Kqr87xum7Zb0FSPm4bF88IQJi84Llcrmsk3tb3l1v5mjbueYVvr4tDh2xW1ZBm7hq4KopRwpKDI0aLJ96NFUeZCVZJKOMbizI/oTyAHIIYg0nb4HkXswIaOKirgsCAIm1Dh0bcFGQwOcduoE+cSMvx9jmOESFulLE4ZKrF79C/RaitgLVq3anq+WqdTTBouKaOYYk/sljXZI7tjz+z911p1v0bLyyHNWlvLHWXkZDz39q+qQrNA+VP1p2eBXcz7Xg3y7vhM6xZ6W187Om3mFrJT9Wl2yV7I/wVrsAe6gVV71a8yPHv2hx+NvNCLUYPk7+34CTZm43IynsgkYqmloFV9mMAkZqgfc0hhDWnkqf4BznGNG2lcWpRWpNV2qtQRaMbwJaT0Bzwelh4=</latexit> X1  ✓6 yes no Red Blue <latexit 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  • 100. <latexit sha1_base64="FVULVmCNg47QKtvg46uxkYeCQJE=">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</latexit> ✓1 <latexit 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sha1_base64="587jdLEUJKtkMM/L+3DOcHwPLKw=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSLVELR0mcuBEZBaIHouBF2gg/QgVOwAAFWqABBR04YhUksPErgwgETOSq0EXOQsTcfQrnEEBtC7MoZkjINvFfx1XZY3VcOzVtV63gKSoOC5URiJEXck8G5Jk8kFfy+WetrlvD8dLBWR5qqVkLXazkPv5VaThzaHyrxnrmcAq7rleG3k2XcW6hDPXts6tBLpmNddfJLXlD/zekT57wBnr7XbnL0GxvjB8ZveCLYYPE3+0YBYXNuLgdT2QS0dS+1yo/rMIabGA/diAFR5CGvNuHS+jBtRAUtoSksDdMFXyeZhl+hHD4BVRLkss=</latexit> ✓4 <latexit sha1_base64="TQ4GrszeMgM294G1rFaPjchY/V8=">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</latexit> X2  ✓4 yes no Red <latexit sha1_base64="e77Z2aqe8KZFIqh8R+ZYHDOKN+I=">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</latexit> ✓5 <latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">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</latexit> X1  ✓5 yes no Red <latexit sha1_base64="eu7hkk7xInDVK5lVQXXLPV+MjYw=">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</latexit> ✓6 <latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit> X1  ✓6 yes no Red Blue <latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">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</latexit> ✓3 <latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit> X1  ✓3 yes no Red <latexit sha1_base64="VJpvibITWQ0L0zaDACUdk6SsPiA=">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</latexit> ✓7 <latexit sha1_base64="bG79VND5DJn1e15l/47tj0nzmbk=">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</latexit> X1  ✓7 yes no Blue Blue
  • 101. 19 8 35 2 predict_proba(x) red blue 8/27 = 0.296 19/27 = 0.704 0.0 1.0 0.704 0.296 Piecewise "constant"
  • 106. = + + + + + RandomForestClassifier (n_estimators=6, max_leaf_nodes=4) 6 × DecisionTreeClassifier(max_leaf_nodes=4) Piecewise "constant"
  • 107. RandomForestRegressor (n_estimators=6, max_leaf_nodes=8) = + + + + + 6 × DecisionTreeRegressor(max_leaf_nodes=8) Piecewise "constant"
  • 108. Random forests ntree=10 Extra trees ntree=10 Piecewise "constant"
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  • 117. ■ OC1/Oblique Tree (Murthy+ 1994) ■ Perceptron Tree (Utgoff , 1988) ■ Large Margin Tree (Wu+ 1999; Bennett+ 2000) ■ Margin Tree (Tibshirani & Hastie 2007) ■ Geometric Decision Tree (Manwani & Sastry, 2012) ■ HHCART (Wickramarachchi+, 2016) ■ M5/M5’ (Quinlan, 1992; Wang & Witten 1997) ■ RETIS (Karalic & Cestnik 1991) ■ SECRET (Dobra & Gehrke, 2002) ■ SMOTI (Malerba+ 2004) ■ MAUVE (Vens & Blockeel, 2006) ■ LLRT (Vogel, Asparouhov, Scheffer, 2007) ■ Cubist (Quinlan, 2011) <latexit 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  • 118. Regression Trees M5 MARS Classification Trees OC1 Multivariate CART
  • 120. y=0.89 + -0.36 max(0,x-0.99) + -0.88 max(0,0.99-x) y=1.23 + -0.59 max(0,x-0.99) + -1.32 max(0,0.99-x) + 1.41 max(0,x-4.96) y=0.98 + -0.23 max(0,x-0.99) + -1.0 max(0,0.99-x) + 1.69 max(0,x-4.96) + -0.51 max(0,x-2.31) y=0.95 + -0.12 max(0,x-0.99) + -0.96 max(0,0.99-x) + 0.77 max(0,x-4.96) + -0.79 max(0,x-2.31) + 0.87 max(0,x-2.31) y=0.66 + -0.29 max(0,x-0.99) Friedman JH. Multivariate Adaptive Regression Splines. The Annals of Statistics, 1991;19: 1–67.
  • 122. STOC 2008 NIPS 2010 Google scholar citations: 419 Google scholar citations: 10
  • 125. Popov S, Morozov S, Babenko A. Neural Oblivious Decision Ensembles for Deep Learning on Tabular Data. ICLR 2020, http://arxiv.org/abs/1909.06312 Kontschieder P, Fiterau M, Criminisi A, Bulo SR. Deep Neural Decision Forests. IJCAI 2016, https://www.ijcai.org/Proceedings/16/Papers/628.pdf Lee G-H, Jaakkola TS. Oblique Decision Trees from Derivatives of ReLU Networks. ICLR 2020, https://openreview.net/pdf?id=Bke8UR4FPB Lay N, Harrison AP, Schreiber S, Dawer G, Barbu A. Random Hinge Forest for Differentiable Learning. ICML 2018, http://arxiv.org/abs/1802.03882 Zantedeschi V, Kusner MJ, Niculae V. Learning Binary Decision Trees by Argmin Differentiation. ICML 2021, http://arxiv.org/abs/2010.04627 Hazimeh H, Ponomareva N, Mol P, Tan Z, Mazumder R. The Tree Ensemble Layer: Differentiability meets Conditional Computation. ICML 2020, https://arxiv.org/abs/2002.07772
  • 133. • Accuracy alone does not suffice for reliable, consequential decision-making; we also need uncertainty. • Distribution-free UQ gives finite-sample statistical guarantees for any predictive model, no matter how bad/misspecified, and any data distribution, even if unknown. • DF techniques such as conformal prediction represent a new, principled approach to UQ for complex prediction systems, such as deep learning.
  • 134. • conformal prediction • tolerance regions • risk-controlling prediction sets • calibration by binning and more
  • 141. With MAPIE, uncertainties are back in machine learning https://towardsdatascience.com/with-mapie-uncertainties-are-back-in-machine-learning-882d5c17fdc3
  • 142. https://github.com/scikit-learn-contrib/MAPIE regressor = ExtraTreesRegressor(max_leaf_nodes=32, bootstrap=True) MapieRegressor(regressor, method="plus", cv=-1) MapieRegressor(regressor, method="plus", cv=Subsample(n_resampling s=50)) MapieRegressor(regressor, method="plus", cv=-1) MapieRegressor(regressor, method="plus", cv=Subsample(n_resampling s=50)) 95% prediction intervals 95% prediction intervals 90% prediction intervals 90% prediction intervals Jacknife+ Jacknife+ after bootstrap Jacknife+ Jacknife+ after bootstrap