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=">AAACi3ichVG7SgNBFL1ZXzEaE7URbIIhYhVmNagEi6AIlnmYByQh7K5jMmRf7E4CMfgDljYWsVGwED/AD7DxByzyCWIZwcbCu5sF0WC8y+ycOXPPnTNzZVNlNiek7xMmJqemZ/yzgbn54EIovLhUsI2WpdC8YqiGVZIlm6pMp3nOuEpLpkUlTVZpUW4eOPvFNrVsZujHvGPSqibVdXbKFIkjVarwBuVSTayFoyRO3IiMAtEDUfAibYQfoQInYIACLdCAgg4csQoS2PiVQQQCJnJV6CJnIWLuPoVzCKC2hVkUMyRkm/iv46rssTqunZq2q1bwFBWHhcoIxMgLuScD8kweyCv5/LNW163heOngLA+11KyFLlZyH/+qNJw5NL5VYz1zOIVd1ytD76bLOLdQhvr22dUgl8zGuuvklryh/xvSJ094A739rtxlaLY3xo+MXvDFsEHi73aMgsJmXNyOJzKJaGrfa5UfVmENNrAfO5CCI0hD3u3DJfTgWggKW0JS2BumCj5Psww/Qjj8Ak3rksg=</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 寸㹀加כ锷椚䒭ה׃ג鼅鎉垥彊䕎 琎ㄤ䕎 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)
Ԩ ءٝفٕ٥鍑ꅸ׃װְׅ Ԩ ⚛⻉׃װְׅ 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)
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 x0k <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. ׃⡂Ⰵ⸂חכ⡂⳿⸂הְֲ箢ְ鸬竲䚍 -JQTDIJU[鸬竲䚍 ׃ַ铬ׁזְ㜥さחכ♧圫ז ė鵚⡂ח ė Eؔ٦تך؟ٝفٕ侧ָ䗳銲חזկ4PCPMFWDMBTTהַח׃ג葺ֻזזְկ ⢽E דչ挿ְֻպך礵䏝実זE דכ ⭙挿䗳銲דꬊ植㹋涸˘
ך⡤琎 <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 !?
High Dimension Always Amounts to Extrapolation. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2110.09485 넝如⯋דכⰻ䯏זג饯ֿ然桦כئٗהְֲطة 넝如⯋ E דכչⰻ䯏הכ⡦ַպչⰻ䯏ַ㢩䯏ַպ הְֲ㛇劤涸陽锷ׅ㹋כ噰גꬊ荈僇 https://youtu.be/86ib0sfdFtw 儗⟃♳ח床衼罏㼎锑!.-4USFFU5BML 嗚叨挿 鋅劤挿꧊さ ך⳻⺪ • "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. " ⰻ䯏 嗚叨挿ָ鎮箺ر٦ة挿ך⳻⺪ח衅הֹח ワך挿ך⦼ַךZ⦼寸ֿה
؛ٌ؎ٝؿؓ歲דכꬊ䌢ח〢ַֻ濼גְ،٦ثؿ؋ؙز 5PQMJTT 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=">AAACvnichVFNSxtBGH6y9atp1VgvBS9Lg0VBwqxIbQtCaC89amw0YGSZXSfJ6n51dhK0m/yB/oEeeigVeij+AH+ACJ7aUw/+BPFooZce+u5mQVTUd5mdZ555n3eemdcKXSdSjJ3mtAcDg0PDIw/zjx6Pjo0XJp6sRUFb2qJqB24gaxaPhOv4oqoc5YpaKAX3LFesWztvk/31jpCRE/jv1V4oNj3e9J2GY3NFlFl4VW9xFcuevqTXPb5rxtv1pvgQudxX+nxP7yasakkvtgMpezM105jTa+b2rN41C0VWYmnoN4GRgSKyWA4Kh6hjCwFstOFBwIci7IIjom8DBhhC4jYREycJOem+QA950rYpS1AGJ3aH/k1abWSsT+ukZpSqbTrFpSFJqWOa/WY/2AU7YQfsjP27tVac1ki87NFs9bUiNMc/PV39e6/Ko1mhdam607NCAy9Trw55D1MmuYXd13c+fr5YfV2Zjp+zfXZO/r+xU3ZEN/A7f+zvK6Ly5Q4/FnmhF6MGGdfbcROszZeMF6WFlYVi+U3WqhFM4RlmqB+LKOMdllGl+l9xjJ/4pZW1huZpQT9Vy2WaSVwJbfc/j5mmYg==</latexit> ˆ r = max j>2 |corr(X1, Xj)| <latexit sha1_base64="DII9mhRSnTvcUJHdfuik1ohKbRg=">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</latexit> ˆ R = 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> ˆ R <latexit sha1_base64="XcYeasdspQvBninjyfi4DGqRhKY=">AAACinichVHLTsJAFL3UFyIK6sbEDZFgXJEpIT43RF245CGPBAhp6wANfaUdSLDhB9y5MpGVJi6MH+AHuPEHXPAJxiUmblx4W5oYJeJtpnPmzD13zswVDUW2GCEDHzc1PTM7558PLAQXl0Lh5ZWCpbdNieYlXdHNkihYVJE1mmcyU2jJMKmgigotiq0jZ7/YoaYl69op6xq0qgoNTa7LksCQKlaaArPNXi0cJXHiRmQc8B6IghdpPfwIFTgDHSRogwoUNGCIFRDAwq8MPBAwkKuCjZyJSHb3KfQggNo2ZlHMEJBt4b+Bq7LHarh2alquWsJTFBwmKiMQIy/kngzJM3kgr+Tzz1q2W8Px0sVZHGmpUQtdrOU+/lWpODNofqsmemZQh13Xq4zeDZdxbiGN9J3zq2FuPxuzN8kteUP/N2RAnvAGWuddusvQbH+CHxG94Ithg/jf7RgHhUSc344nM8lo6tBrlR/WYQO2sB87kIITSEPerX8J19DnglyC2+MORqmcz9Oswo/gjr8AJ6KSvw==</latexit> ˆ r O 㔐דءىُٖ٦ز
⦐ךE如⯋挿 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.
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. "
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: 葺ְ堣唒㷕统ٌرٕך㢳ꅾ䚍 ꬊ♧䠐䚍 堣唒㷕统؝ٝلדכ♧אךر٦ةإحزח㼎׃ג㢳圫זٌرָٕ䲿⳿ׁկ ✮庠礵䏝כ♳⡘כ׀朐䡾חז㹋欽♳כקרず瘝ח葺ְٌرٕהזׇկ ⢽
data, this validation is often insufficient to guarantee that the models will have well-defined behavior when they are used in a new setting. " 6OEFSTQFDJGJDBUJPOכؽحؚر٦ة✲⢽ד饯ֿגְ չ葺ְ侄䌌ٓكָٕאֽ㣐鋉垷ر٦ةך㹋⢽דպ饯ֿגְךַ
SVR (RBF) Gaussian Process (RBF) Random Forest Nearest Neighbors Decision Tree չ寸㹀加،ٝ؟ٝـٕדכ㔐䌓刼简תד㔐䌓⦼ָעװְׅպכوؤְ暴䚍ַ ̔ההך؟ٝفٕ⦼ָ溪ך⫘ぢַٓٝتي㢌⹛׃גְךַֿך玎䏝✮庠⦼ָ䮶 խךכ鷞ח⨳Ⰻדכչ✮庠ⴓ侔պ鎘皾׃ג➰♷ׅל׃㹋欽欽鷿ד(PPEזכ׆
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
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 ׯה׃ر٦ةד㷕统黝ⴖח؝ٝزٗ٦ׁٕגְל$//5SBOTGPSNFS "UUFOUJPOׅ銲זְַ˘ .-1ֲתֻ⢪ִל״ְֽזךַ
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 #FOJHO0WFSGJUUJOHظ؎ؤ֮ر٦ةד鎮箺铎䊴ד搀㹱
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.
Modern Mathematics of Deep Learning. arXiv [cs.LG]. 2021. http://arxiv.org/abs/2105.04026 穗꿀涸✲㹋ה׃גծ㹋ꥷך帾㾴㷕统ך植㜥דכ侄䌌ٓكٕחظ؎ؤֲָָ֮זֲַָծ ذأز铎䊴ָ㼭ְׁٌرٕכչ鎮箺铎䊴㼭ְׁ קרئٗ铎䊴 պז㜥さָהג㢳ְկ
of DNs can be written as a composition of max-affine spline operators (MASOs)" %JTKPJOUז걄㚖ךQBSUJUJPOָ֮ ぐ걄㚖׀החꟼ侧 סאֲ㢳갪䒭 BQQMZֽׅוչ㞮歲ד鸬竲חז ״ֲחպזגְךָأفٓ؎ٝ
· 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)"
no <latexit sha1_base64="OBBuUzKAHrHcbt2xZlrMzfgelZg=">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</latexit> ✓2 <latexit sha1_base64="fUwPPBtoir3sq20H215QULGdmUg=">AAAClnichVHLSsNAFD3Gd3201Y3gplgUV2UiRcWFFEV02YfVgpWQxLENpklMphUt/oA/4EJcKKiIH+AHuPEHXPQTxGUFNy68TQOiot4wmTNn7rlzZq7mmIYnGGt0SJ1d3T29ff2hgcGh4XAkOrLh2VVX53ndNm23oKkeNw2L54UhTF5wXK5WNJNvanvLrf3NGnc9w7bWxaHDtytqyTJ2DV0VRCmRaEGRY0WT78eKosyFqswokThLMD9iP4EcgDiCSNuRexSxAxs6qqiAw4IgbEKFR98WZDA4xG2jTpxLyPD3OY4RIm2VsjhlqMTu0b9Eq62AtWjdqun5ap1OMWm4pIxhkj2xW9Zkj+yOPbP3X2vV/RotL4c0a20td5TwyVju7V9VhWaB8qfqT88Cu5j3vRrk3fGZ1i30tr52dNrMLWQn61Pskr2Q/wvWYA90A6v2ql9lePbsDz8aeaEXowbJ39vxE2zMJOTZRDKTjKeWglb1YRwTmKZ+zCGFNaSRp/oHOMc1bqQxaVFakVbbqVJHoBnFl5DSHzOelho=</latexit> X1 ✓2 yes no Blue
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
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=">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</latexit> X1 ✓3 yes no Red
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=">AAAClnichVHLSsNAFD3GV62vVjeCm2JRXJWJ+MKFFEV0WVurBSshiVMNTZOYTCta/AF/wIW4UFARP8APcOMPuOgniEsFNy68SQOiot4wmTNn7rlzZq7mmIYnGGu0SK1t7R2dka5od09vX38sPrDu2VVX53ndNm23oKkeNw2L54UhTF5wXK5WNJNvaOVFf3+jxl3PsK01ceDwrYq6YxklQ1cFUUosXlDkRNHke4mi2OVCVaaUWJKlWBCJn0AOQRJhZOzYHYrYhg0dVVTAYUEQNqHCo28TMhgc4rZQJ84lZAT7HEeIkrZKWZwyVGLL9N+h1WbIWrT2a3qBWqdTTBouKRMYZY/shr2wB3bLntj7r7XqQQ3fywHNWlPLHaX/eCj39q+qQrPA7qfqT88CJcwGXg3y7gSMfwu9qa8dnrzk5rKj9TF2wZ7J/zlrsHu6gVV71S9Xefb0Dz8aeaEXowbJ39vxE6xPpOTp1OTqZDK9ELYqgmGMYJz6MYM0VpBBnurv4wxXuJaGpHlpSVpupkotoWYQX0LKfAA5/pYd</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
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=">AAACi3ichVG7TgJBFL2sL0QR1MbEhkgwVmRQgoZYEI2JJQ95JEDI7jrAhH1ldyBB4g9Y2lhgo4mF8QP8ABt/wIJPMJaY2Fh4d9nEKBHvZnbOnLnnzpm5kqEwixMy8AhT0zOzc95538KifykQXF4pWHrblGle1hXdLEmiRRWm0TxnXKElw6SiKim0KLUO7f1ih5oW07UT3jVoVRUbGqszWeRIlSq8SblYS9SCYRIlToTGQcwFYXAjrQcfoQKnoIMMbVCBggYcsQIiWPiVIQYEDOSq0EPORMScfQrn4ENtG7MoZojItvDfwFXZZTVc2zUtRy3jKQoOE5UhiJAXck+G5Jk8kFfy+WetnlPD9tLFWRppqVELXKzlPv5VqThzaH6rJnrmUIc9xytD74bD2LeQR/rO2dUwl8xGepvklryh/xsyIE94A63zLt9laLY/wY+EXvDFsEGx3+0YB4XtaCwRjWfi4dSB2yovrMMGbGE/diEFx5CGvNOHS+jDteAXdoSksD9KFTyuZhV+hHD0BViLks0=</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
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=">AAACi3ichVG7TgJBFL2sL0QR1MbEhkgwVmRQfIRYEI2JJQ95JEDI7jrAhH1ldyBB4g9Y2lhgo4mF8QP8ABt/wIJPMJaY2Fh4d9nEKBHvZnbOnLnnzpm5kqEwixPS9wgTk1PTM95Z39y8fyEQXFzKW3rLlGlO1hXdLEqiRRWm0RxnXKFFw6SiKim0IDUP7f1Cm5oW07UT3jFoRRXrGqsxWeRIFcu8QblY3a4GwyRKnAiNgpgLwuBGSg8+QhlOQQcZWqACBQ04YgVEsPArQQwIGMhVoIuciYg5+xTOwYfaFmZRzBCRbeK/jquSy2q4tmtajlrGUxQcJipDECEv5J4MyDN5IK/k889aXaeG7aWDszTUUqMauFjJfvyrUnHm0PhWjfXMoQZ7jleG3g2HsW8hD/Xts6tBNpGJdNfJLXlD/zekT57wBlr7Xb5L00xvjB8JveCLYYNiv9sxCvKb0dhONJ6Oh5MHbqu8sAprsIH92IUkHEMKck4fLqEH14Jf2BISwv4wVfC4mmX4EcLRF1Zrksw=</latexit> ✓5 <latexit sha1_base64="FaDvqDDJ/DHLyBEhkovQS4NqCn0=">AAAClnichVHLSsNAFD3GV62vVjeCm2JRXJWJ+MKFFEV0WVurBSshiVMNTZOYTCta/AF/wIW4UFARP8APcOMPuOgniEsFNy68SQOiot4wmTNn7rlzZq7mmIYnGGu0SK1t7R2dka5od09vX38sPrDu2VVX53ndNm23oKkeNw2L54UhTF5wXK5WNJNvaOVFf3+jxl3PsK01ceDwrYq6YxklQ1cFUUosXlDkRNHke4mi2OVCVaaUWJKlWBCJn0AOQRJhZOzYHYrYhg0dVVTAYUEQNqHCo28TMhgc4rZQJ84lZAT7HEeIkrZKWZwyVGLL9N+h1WbIWrT2a3qBWqdTTBouKRMYZY/shr2wB3bLntj7r7XqQQ3fywHNWlPLHaX/eCj39q+qQrPA7qfqT88CJcwGXg3y7gSMfwu9qa8dnrzk5rKj9TF2wZ7J/zlrsHu6gVV71S9Xefb0Dz8aeaEXowbJ39vxE6xPpOTp1OTqZDK9ELYqgmGMYJz6MYM0VpBBnurv4wxXuJaGpHlpSVpupkotoWYQX0LKfAA5/pYd</latexit> X1 ✓5 yes no Red <latexit sha1_base64="eu7hkk7xInDVK5lVQXXLPV+MjYw=">AAACi3ichVG7TgJBFL2sL0QR1MbEhkgwVmRQgoZYEI2JJQ95JEDI7jrAhH1ldyBB4g9Y2lhgo4mF8QP8ABt/wIJPMJaY2Fh4d9nEKBHvZnbOnLnnzpm5kqEwixMy8AhT0zOzc95538KifykQXF4pWHrblGle1hXdLEmiRRWm0TxnXKElw6SiKim0KLUO7f1ih5oW07UT3jVoVRUbGqszWeRIlSq8SblYS9SCYRIlToTGQcwFYXAjrQcfoQKnoIMMbVCBggYcsQIiWPiVIQYEDOSq0EPORMScfQrn4ENtG7MoZojItvDfwFXZZTVc2zUtRy3jKQoOE5UhiJAXck+G5Jk8kFfy+WetnlPD9tLFWRppqVELXKzlPv5VqThzaH6rJnrmUIc9xytD74bD2LeQR/rO2dUwl8xGepvklryh/xsyIE94A63zLt9laLY/wY+EXvDFsEGx3+0YB4XtaCwRjWfi4dSB2yovrMMGbGE/diEFx5CGvNOHS+jDteAXdoSksD9KFTyuZhV+hHD0BViLks0=</latexit> ✓6 <latexit sha1_base64="j6aRObEml8aKBOmlM9//dP7CY5I=">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</latexit> X1 ✓6 yes no Red Blue <latexit sha1_base64="YU9dEL0zbRAMaEFNhHkGnXZKQpA=">AAACi3ichVG7TgJBFL2sL0QR1MbEhkgwVmQQooZYEI2JJQ95JEDI7jrAhH1ldyBB4g9Y2lhgo4mF8QP8ABt/wIJPMJaY2Fh4d9nEKBHvZnbOnLnnzpm5kqEwixMy8AhT0zOzc95538KifykQXF4pWHrblGle1hXdLEmiRRWm0TxnXKElw6SiKim0KLUO7f1ih5oW07UT3jVoVRUbGqszWeRIlSq8SblYi9eCYRIlToTGQcwFYXAjrQcfoQKnoIMMbVCBggYcsQIiWPiVIQYEDOSq0EPORMScfQrn4ENtG7MoZojItvDfwFXZZTVc2zUtRy3jKQoOE5UhiJAXck+G5Jk8kFfy+WetnlPD9tLFWRppqVELXKzlPv5VqThzaH6rJnrmUIc9xytD74bD2LeQR/rO2dUwl8xGepvklryh/xsyIE94A63zLt9laLY/wY+EXvDFsEGx3+0YB4XtaGwnmsgkwqkDt1VeWIcN2MJ+7EIKjiENeacPl9CHa8EvxIWksD9KFTyuZhV+hHD0BVIrkso=</latexit> ✓3 <latexit sha1_base64="3d4/BR7YXLLt+AHkvejt6OKrM6I=">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</latexit> X1 ✓3 yes no Red <latexit 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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 ˖ %JFSFOUJBCMFז寸㹀加/FVSBM/FUXPSLTַך寸㹀加圓眠 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
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.
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