Pairwise Comparisons with Flexible Time-Dynamics Lucas Maystre , Victor Kristof, Matthias Grossglauser KDD Research Track 2 — August 6 th , 2019
Pairwise comparison data Association football example: Questions we might want to ask: Team 1 Team 2 Score “How can we quantify the skill of France?” France Portugal 2-5 Luxembourg Greece 0-3 “How likely is South Korea ... ... ... to beat Germany?” Turkey Slovakia 1-1 Bulgaria Kosovo 0-1 → we need pairwise comparison models. 2
<latexit sha1_base64="UP3JbDQ68thUsUruLlvz9QvaE80=">ACAXicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZcFRVxWsA+YDiWTpm1oMhmSjFCGrvwGt7p2J279Epf+iZl2Frb1QOBwzr3ckxPGnGnjut9OYW19Y3OruF3a2d3bPygfHrW0TBShTSK5VJ0Qa8pZRJuGU47saJYhJy2w/FN5refqNJMRo9mEtNA4GHEBoxgYyW/K7AZEczT2mvXHGr7gxolXg5qUCORq/80+1LkgaGcKx1r7nxiZIsTKMcDotdRNY0zGeEh9SyMsqA7SWeQpOrNKHw2ksi8yaKb+3Uix0HoiQjuZRdTLXib+5/mJGVwHKYvixNCIzA8NEo6MRNn/UZ8pSgyfWIKJYjYrIiOsMDG2pYUrocg68ZYbWCWtWtW7qNYeLiv1u7ydIpzAKZyDB1dQh3toQBMISHiBV3hznp1358P5nI8WnHznGBbgfP0CWKuX+A=</latexit> <latexit sha1_base64="HG+BMlUV/2K3oGVTANntbUCUhw=">ACKnicbVDLSsNAFJ34rPUVdelmsAgVsSRV0I1QcONKtgHJCFMpN27OTBzEQsIf/hR/gNbnXtrgS/BAnbRa29cAwh3PO5c4cL2ZUSMYa0vLK6tr6WN8ubW9s6uvrfFlHCMWnhiEW86yFBGA1JS1LJSDfmBAUeIx1veJP7nSfCBY3CBzmKiROgfkh9ipFUkqvX4yqFtkgwho8n8BraPkc4NbPUhKfQJs+xBc9gVbhUXcJVESdz9YpRMyaAi8QsSAUaLr6t92LcBKQUGKGhLBMI5ZOirikmJGsbCeCxAgPUZ9YioYoIMJ3/L4LFSetCPuDqhBP170SKAiFGgaeSAZIDMe/l4n+elUj/yklpGCeShHi6yE8YlBHMi4I9ygmWbKQIwpyqt0I8QKodqeqc2eIFeSfmfAOLpF2vme1+v1FpXFXtFMCh+AIVIEJLkED3ImaAEMXsAbeAcf2qv2qY21r2l0StmDsAMtJ9f16ykeA=</latexit> <latexit sha1_base64="iBqJEKBf+8hcXui8sJPkCiQKLc4=">ACG3icbVDLSgMxFM34rPVdekmWIQKUmaqoMuCG1elgn1AZyZNOGJpkhyQhlmA/wI/wGt7p2J25duPRPzLSzsK0HAodzuXeHD9iVGnb/rZWVtfWNzYLW8Xtnd29/dLBYVuFscSkhUMWyq6PFGFUkJampFuJAniPiMdf3yT+Z1HIhUNxb2eRMTjaChoQDHSRuqXyqpPoasohxC6HOkRixpBX7PFOHD3UzkzKrtpTwGXi5KQMcjT7pR93EOKYE6ExQ0r1HDvSXoKkpiRtOjGikQIj9GQ9AwViBPlJdPpPDUKAMYhNI8oeFU/TuRIK7UhPsmd2rFr1M/M/rxTq49hIqolgTgWeLgphBHcKsGTigkmDNJoYgLKm5FeIRkghr09/cFp+nphNnsYFl0q5VnYtq7e6yXG/k7RTAMTgBFeCAK1AHt6AJWgCDJ/ACXsGb9Wy9Wx/W5y6YuUzR2AO1tcvXbGgVQ=</latexit> <latexit sha1_base64="Ix18cYdMx7NrDq0ECn91v7Keh+Q=">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</latexit> [Zermelo, 1928] Latent skill model [Thurstone, 1927] s i ∼ N (0 , σ 2 ) Germany 1.18 latent skill of i Brazil 1.09 France 0.68 1 p ( i � j ) = 1 + exp[ � ( s i � s j )] 0 “ i wins over j ” Switzerland -0.74 Given data , the posterior distribution is D Iceland -1.22 Y p ( s | D ) / p ( s ) p ( i � j ) ( i,j ) ∈ D 3
Actual setting Data come with a timestamp . Questions we might want to ask: Date Team 1 Team 2 Score “How strong was France in 1972 ? In 2018 ?” 1923-09-15 France Portugal 2-5 1923-09-16 Luxembourg Greece 0-3 “How likely is South Korea ... ... ... ... to beat Germany today ?” 2018-06-21 Turkey Slovakia 1-1 2018-06-21 Bulgaria Kosovo 0-1 → we need dynamic models. 4
<latexit sha1_base64="LuQOmJp/62XYXmAp4JLOqHRlnj8=">ACHicbVDLSgNBEJyNrxhfUY9eBqOYQAi7UdBjwIOeJIJ5QLKE2clsMmRmd5npFcKSH/Aj/AavevYmXgWP/omTx8EkFjQUVd10d3mR4Bps+9tKrayurW+kNzNb2zu7e9n9g7oOY0VZjYiVE2PaCZ4wGrAQbBmpBiRnmANb3A9huPTGkeBg8wjJgrS/gPqcEjNTJnugOz0MBtzWXuC0J9JVMbqjl3EgzwUMZwV3E42Z5fsCfAycWYkh2aodrI/7W5IY8kCoIJo3XLsCNyEKOBUsFGmHWsWETogPdYyNCSaTeZfDPCp0bpYj9UpgLAE/XvREKk1kPpmc7xvXrRG4v/ea0Y/Cs34UEUAwvodJEfCwhHkeDu1wxCmJoCKGKm1sx7RNFKJgA57Z4cmQycRYTWCb1csk5L5XvL3KVu1k6aXSEjlEeOegSVdAtqIaougJvaBX9GY9W+/Wh/U5bU1Zs5lDNAfr6xeNXqBd</latexit> <latexit sha1_base64="WbOUPImugAWxZeTwfab5BbunGSI=">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</latexit> This work: kickscore Skill becomes a (latent) stochastic process s i ( t ) ∼ GP[0 , k ( t, t 0 )] covariance function, defines time dynamics Obs. model is conditionally parametric : p ( i � j | t ) = 1 1 + exp { � [ s i ( t ) � s j ( t )] } 5
Covariance functions Brownian motion Mean-reverting, stationary dynamics Smooth dynamics Discontinuities 6
Outline 1 2 Inference Experimental algorithm evaluation 7
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