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It is also sig-", "type": "text" } ], "index": 11 }, { "bbox": [ 54, 211, 290, 223 ], "spans": [ { "bbox": [ 54, 211, 290, 223 ], "score": 1.0, "content": "nificantly better than other known implementations of AD-", "type": "text" } ], "index": 12 }, { "bbox": [ 54, 222, 290, 236 ], "spans": [ { "bbox": [ 54, 222, 290, 236 ], "score": 1.0, "content": "ABOOST.MH (Zhu et al., 2009; Mukherjee & Schapire,", "type": "text" } ], "index": 13 }, { "bbox": [ 53, 234, 83, 247 ], "spans": [ { "bbox": [ 53, 234, 83, 247 ], "score": 1.0, "content": "2013).", "type": "text" } ], "index": 14 } ], "index": 7 }, { "type": "text", "bbox": [ 54, 253, 289, 324 ], "lines": [ { "bbox": [ 53, 253, 290, 265 ], "spans": [ { "bbox": [ 53, 253, 290, 265 ], "score": 1.0, "content": "The paper is organized as follows. In Section 2 we give", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 264, 290, 276 ], "spans": [ { "bbox": [ 53, 264, 290, 276 ], "score": 1.0, "content": "the formal multi-class setup used in the paper and AD-", "type": "text" } ], "index": 16 }, { "bbox": [ 54, 276, 290, 288 ], "spans": [ { "bbox": [ 54, 276, 290, 288 ], "score": 1.0, "content": "ABOOST.MH, and show how to train factorized base learn-", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 289, 290, 300 ], "spans": [ { "bbox": [ 53, 289, 290, 300 ], "score": 1.0, "content": "ers in general. The algorithm to build Hamming trees is", "type": "text" } ], "index": 18 }, { "bbox": [ 54, 301, 290, 312 ], "spans": [ { "bbox": [ 54, 301, 290, 312 ], "score": 1.0, "content": "described in Section 3. Experiments are described in Sec-", "type": "text" } ], "index": 19 }, { "bbox": [ 54, 313, 234, 324 ], "spans": [ { "bbox": [ 54, 313, 234, 324 ], "score": 1.0, "content": "tion 4 before a brief conclusion in Section 5.", "type": "text" } ], "index": 20 } ], "index": 17.5 }, { "type": "title", "bbox": [ 55, 339, 153, 353 ], "lines": [ { "bbox": [ 52, 339, 154, 354 ], "spans": [ { "bbox": [ 52, 339, 154, 354 ], "score": 1.0, "content": "2. ADABOOST.MH", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 55, 361, 289, 468 ], "lines": [ { "bbox": [ 54, 361, 290, 372 ], "spans": [ { "bbox": [ 54, 361, 290, 372 ], "score": 1.0, "content": "In this section we first introduce the general multi-", "type": "text" } ], "index": 22 }, { "bbox": [ 53, 372, 289, 384 ], "spans": [ { "bbox": [ 53, 372, 289, 384 ], "score": 1.0, "content": "class learning setup (Section 2.1), then we describe AD-", "type": "text" } ], "index": 23 }, { "bbox": [ 53, 384, 290, 397 ], "spans": [ { "bbox": [ 53, 384, 290, 397 ], "score": 1.0, "content": "ABOOST.MH in detail (Section 2.2). 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(Sec-", "type": "text" } ], "index": 29 }, { "bbox": [ 53, 456, 93, 468 ], "spans": [ { "bbox": [ 53, 456, 93, 468 ], "score": 1.0, "content": "tion 2.4).", "type": "text" } ], "index": 30 } ], "index": 26 }, { "type": "title", "bbox": [ 54, 481, 236, 505 ], "lines": [ { "bbox": [ 54, 480, 236, 494 ], "spans": [ { "bbox": [ 54, 480, 236, 494 ], "score": 1.0, "content": "2.1. The multi-class setup: single-label and", "type": "text" } ], "index": 37 }, { "bbox": [ 72, 494, 167, 505 ], "spans": [ { "bbox": [ 72, 494, 167, 505 ], "score": 1.0, "content": "multi-label/multi-task", "type": "text" } ], "index": 34 } ], "index": 35.5 }, { "type": "text", "bbox": [ 54, 511, 289, 667 ], "lines": [ { "bbox": [ 52, 510, 290, 524 ], "spans": [ { "bbox": [ 52, 510, 290, 524 ], "score": 1.0, "content": "For the formal description of ADABOOST.MH, let the", "type": "text" } ], "index": 35 }, { "bbox": [ 52, 523, 291, 537 ], "spans": [ { "bbox": [ 52, 523, 124, 537 ], "score": 1.0, "content": "training data be", "type": "text" }, { "bbox": [ 125, 523, 258, 537 ], "score": 0.92, "content": "\\mathcal { D } \\ = \\ \\big \\{ ( \\mathbf { x } _ { 1 } , \\mathbf { y } _ { 1 } ) , \\dotsc , ( \\mathbf { x } _ { n } , \\mathbf { y } _ { n } ) \\big \\} _ { . }", "type": "inline_equation" }, { "bbox": [ 258, 523, 291, 537 ], "score": 1.0, "content": ", where", "type": "text" } ], "index": 38 }, { "bbox": [ 54, 535, 291, 549 ], "spans": [ { "bbox": [ 54, 535, 90, 547 ], "score": 0.92, "content": "\\mathbf { x } _ { i } \\in \\mathbb { R } ^ { d }", "type": "inline_equation" }, { "bbox": [ 90, 535, 206, 549 ], "score": 1.0, "content": "are observation vectors, and", "type": "text" }, { "bbox": [ 207, 536, 260, 548 ], "score": 0.89, "content": "\\mathbf { y } _ { i } \\in \\{ \\pm 1 \\} ^ { \\mathrm { \\tilde { K } } }", "type": "inline_equation" }, { "bbox": [ 261, 535, 291, 549 ], "score": 1.0, "content": "are la-", "type": "text" } ], "index": 39 }, { "bbox": [ 53, 548, 289, 560 ], "spans": [ { "bbox": [ 53, 548, 264, 560 ], "score": 1.0, "content": "bel vectors. 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Without loss of gen-", "type": "text" } ], "index": 45 }, { "bbox": [ 53, 608, 290, 620 ], "spans": [ { "bbox": [ 53, 608, 169, 619 ], "score": 1.0, "content": "erality, we will suppose that", "type": "text" }, { "bbox": [ 169, 608, 254, 620 ], "score": 0.93, "content": "\\ell \\in \\mathcal { L } = \\{ 1 , \\ldots , K \\}", "type": "inline_equation" }, { "bbox": [ 255, 608, 290, 619 ], "score": 1.0, "content": ". 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193 ], "score": 1.0, "content": "The goal of learning is to infer a vector-valued multi-class", "type": "text" } ], "index": 56 }, { "bbox": [ 304, 192, 543, 206 ], "spans": [ { "bbox": [ 304, 193, 404, 205 ], "score": 1.0, "content": "discriminant function f", "type": "text" }, { "bbox": [ 404, 192, 461, 204 ], "score": 0.86, "content": ": \\mathcal { X } \\mathbb { R } ^ { K }", "type": "inline_equation" }, { "bbox": [ 461, 193, 471, 205 ], "score": 1.0, "content": ". 2", "type": "text" }, { "bbox": [ 472, 192, 543, 206 ], "score": 1.0, "content": "The single-label", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 205, 542, 218 ], "spans": [ { "bbox": [ 305, 205, 435, 218 ], "score": 1.0, "content": "output of the algorithm is then", "type": "text" }, { "bbox": [ 436, 205, 539, 217 ], "score": 0.91, "content": "\\ell _ { \\mathbf { f } } ( \\mathbf { x } ) = \\arg \\operatorname* { m a x } _ { \\ell } f _ { \\ell } ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 539, 205, 542, 218 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 58 }, { "bbox": [ 305, 216, 542, 229 ], "spans": [ { "bbox": [ 305, 216, 542, 229 ], "score": 1.0, "content": "The classical measure of the performance of the multi-", "type": "text" } ], "index": 59 }, { "bbox": [ 306, 229, 542, 240 ], "spans": [ { "bbox": [ 306, 229, 542, 240 ], "score": 1.0, "content": "class discriminant function f is the single-label one-loss", "type": "text" } ], "index": 60 }, { "bbox": [ 306, 240, 542, 254 ], "spans": [ { "bbox": [ 306, 240, 433, 254 ], "score": 0.91, "content": "L _ { \\mathbb { I } } ( \\mathbf { f } , ( \\mathbf { x } , \\boldsymbol { \\ell } ) ) = \\mathbb { I } \\left\\{ \\boldsymbol { \\ell } \\neq \\ell _ { \\mathbf { f } } ( \\mathbf { x } _ { i } ) \\right\\}", "type": "inline_equation" }, { "bbox": [ 433, 240, 542, 253 ], "score": 1.0, "content": ", which defines the single-", "type": "text" } ], "index": 61 }, { "bbox": [ 305, 252, 384, 265 ], "spans": [ 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Although this goal", "type": "text" } ], "index": 37 }, { "bbox": [ 305, 248, 543, 260 ], "spans": [ { "bbox": [ 305, 248, 543, 260 ], "score": 1.0, "content": "is clearly defined in (Schapire & Singer, 1999), efficient", "type": "text" } ], "index": 38 }, { "bbox": [ 305, 261, 542, 272 ], "spans": [ { "bbox": [ 305, 261, 542, 272 ], "score": 1.0, "content": "base learning algorithms have never been described in de-", "type": "text" } ], "index": 39 }, { "bbox": [ 305, 272, 543, 285 ], "spans": [ { "bbox": [ 305, 272, 543, 285 ], "score": 1.0, "content": "tail. In most recent papers (Zhu et al., 2009; Mukherjee", "type": "text" } ], "index": 40 }, { "bbox": [ 306, 284, 543, 296 ], "spans": [ { "bbox": [ 306, 284, 543, 296 ], "score": 1.0, "content": "& Schapire, 2013) where ADABOOST.MH is used as base-", "type": "text" } ], "index": 41 }, { "bbox": [ 305, 296, 543, 308 ], "spans": [ { "bbox": [ 305, 296, 543, 308 ], "score": 1.0, "content": "line, the base learner is a classical single-label decision tree", "type": "text" } ], "index": 42 }, { "bbox": [ 306, 308, 542, 320 ], "spans": [ { "bbox": [ 306, 308, 542, 320 ], "score": 1.0, "content": "which has to be grown rather large to satisfy the weak-", "type": "text" } ], "index": 43 }, { "bbox": [ 306, 321, 542, 332 ], "spans": [ { "bbox": [ 306, 321, 542, 332 ], "score": 1.0, "content": "learning condition, and, when boosted, yields suboptimal", "type": "text" } ], "index": 44 }, { "bbox": [ 305, 332, 542, 345 ], "spans": [ { "bbox": [ 305, 332, 542, 345 ], "score": 1.0, "content": "results (Section 4). The reason why methods for learning", "type": "text" } ], "index": 45 }, { "bbox": [ 306, 343, 542, 356 ], "spans": [ { "bbox": [ 306, 344, 352, 356 ], "score": 1.0, "content": "multi-class", "type": "text" }, { "bbox": [ 352, 343, 383, 356 ], "score": 0.93, "content": "\\{ \\pm 1 \\} ^ { K }", "type": "inline_equation" }, { "bbox": [ 383, 344, 542, 356 ], "score": 1.0, "content": "-valued base classifiers had not been de-", "type": "text" } ], "index": 46 }, { "bbox": [ 306, 356, 542, 368 ], "spans": [ { "bbox": [ 306, 356, 542, 368 ], "score": 1.0, "content": "veloped before is because they have to be boosted: since", "type": "text" } ], "index": 47 }, { "bbox": [ 306, 369, 543, 380 ], "spans": [ { "bbox": [ 306, 369, 543, 380 ], "score": 1.0, "content": "they do not select a single label, they cannot be used as", "type": "text" } ], "index": 48 }, { "bbox": [ 306, 381, 445, 392 ], "spans": [ { "bbox": [ 306, 381, 445, 392 ], "score": 1.0, "content": "stand-alone multi-class classifiers.", "type": "text" } ], "index": 49 } ], "index": 42.5, "bbox_fs": [ 305, 224, 543, 392 ] }, { "type": "text", "bbox": [ 306, 398, 542, 505 ], "lines": [ { "bbox": [ 305, 397, 543, 410 ], "spans": [ { "bbox": [ 305, 397, 543, 410 ], "score": 1.0, "content": "Although it is not described in detail, it seems that the", "type": "text" } ], "index": 50 }, { "bbox": [ 305, 409, 543, 422 ], "spans": [ { "bbox": [ 305, 409, 543, 422 ], "score": 1.0, "content": "base classifier used in the original paper of Schapire &", "type": "text" } ], "index": 51 }, { "bbox": [ 305, 421, 543, 434 ], "spans": [ { "bbox": [ 305, 421, 415, 434 ], "score": 1.0, "content": "Singer (1999) is a vector of", "type": "text" }, { "bbox": [ 415, 422, 425, 432 ], "score": 0.81, "content": "K", "type": "inline_equation" }, { "bbox": [ 425, 421, 543, 434 ], "score": 1.0, "content": "independent decision stumps", "type": "text" } ], "index": 52 }, { "bbox": [ 306, 432, 543, 447 ], "spans": [ { "bbox": [ 306, 433, 429, 447 ], "score": 0.92, "content": "\\mathbf { h } ( \\mathbf { x } ) \\ = \\ \\left( h _ { 1 } ( \\mathbf { x } ) , \\ldots , h _ { K } ( \\mathbf { x } ) \\right)", "type": "inline_equation" }, { "bbox": [ 429, 432, 543, 447 ], "score": 1.0, "content": ". 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Both vari-", "type": "text" } ], "index": 68 }, { "bbox": [ 306, 656, 542, 667 ], "spans": [ { "bbox": [ 306, 656, 542, 667 ], "score": 1.0, "content": "ants are known under the name of real ADABOOST.MH.", "type": "text" } ], "index": 69 }, { "bbox": [ 306, 668, 542, 680 ], "spans": [ { "bbox": [ 306, 668, 542, 680 ], "score": 1.0, "content": "Although there might be slight differences in the practical", "type": "text" } ], "index": 70 }, { "bbox": [ 305, 680, 543, 691 ], "spans": [ { "bbox": [ 305, 680, 543, 691 ], "score": 1.0, "content": "performance of real and discrete ADABOOST.MH, here we", "type": "text" } ], "index": 71 }, { "bbox": [ 305, 690, 542, 704 ], "spans": [ { "bbox": [ 305, 690, 542, 704 ], "score": 1.0, "content": "decided to stick to the discrete case for the sake of simplic-", "type": "text" } ], "index": 72 }, { "bbox": [ 304, 703, 322, 717 ], "spans": [ { "bbox": [ 304, 703, 322, 717 ], "score": 1.0, "content": "ity.", "type": "text" } ], "index": 73 } ], "index": 66.5, "bbox_fs": [ 304, 546, 543, 717 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 54, 68, 145, 79 ], "lines": [ { "bbox": [ 53, 67, 147, 81 ], "spans": [ { "bbox": [ 53, 67, 147, 81 ], "score": 1.0, "content": "2.4. 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In multi-", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 158, 290, 170 ], "spans": [ { "bbox": [ 53, 158, 204, 170 ], "score": 1.0, "content": "class classification it is possible that", "type": "text" }, { "bbox": [ 205, 158, 226, 170 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 226, 158, 290, 170 ], "score": 1.0, "content": "correlates with", "type": "text" } ], "index": 7 }, { "bbox": [ 52, 170, 290, 182 ], "spans": [ { "bbox": [ 52, 170, 158, 181 ], "score": 1.0, "content": "some of the class labels", "type": "text" }, { "bbox": [ 158, 172, 168, 182 ], "score": 0.74, "content": "y _ { \\ell }", "type": "inline_equation" }, { "bbox": [ 168, 170, 290, 181 ], "score": 1.0, "content": "and anti-correlates with oth-", "type": "text" } ], "index": 8 }, { "bbox": [ 52, 181, 291, 194 ], "spans": [ { "bbox": [ 52, 181, 291, 194 ], "score": 1.0, "content": "ers. This free choice is expressed by the binary “votes”", "type": "text" } ], "index": 9 }, { "bbox": [ 54, 194, 289, 207 ], "spans": [ { "bbox": [ 54, 194, 99, 207 ], "score": 0.93, "content": "v _ { \\ell } \\in \\{ \\pm 1 \\}", "type": "inline_equation" }, { "bbox": [ 100, 194, 154, 207 ], "score": 1.0, "content": ". We say that", "type": "text" }, { "bbox": [ 154, 194, 175, 206 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 175, 194, 236, 207 ], "score": 1.0, "content": "votes for class", "type": "text" }, { "bbox": [ 236, 194, 243, 204 ], "score": 0.75, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 243, 194, 252, 207 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 253, 194, 289, 205 ], "score": 0.91, "content": "v _ { \\ell } = + 1", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 53, 206, 290, 218 ], "spans": [ { "bbox": [ 53, 206, 157, 218 ], "score": 1.0, "content": "and it votes against class", "type": "text" }, { "bbox": [ 158, 207, 163, 216 ], "score": 0.79, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 164, 206, 173, 218 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 174, 207, 211, 217 ], "score": 0.91, "content": "v _ { \\ell } = - 1", "type": "inline_equation" }, { "bbox": [ 211, 206, 290, 218 ], "score": 1.0, "content": ". As in binary clas-", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 217, 291, 231 ], "spans": [ { "bbox": [ 53, 217, 96, 231 ], "score": 1.0, "content": "sification,", "type": "text" }, { "bbox": [ 97, 220, 105, 228 ], "score": 0.76, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 105, 217, 291, 231 ], "score": 1.0, "content": "expresses the overall quality of the classifier", "type": "text" } ], "index": 12 }, { "bbox": [ 54, 230, 290, 242 ], "spans": [ { "bbox": [ 54, 230, 81, 242 ], "score": 0.9, "content": "\\mathbf { v } \\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 82, 230, 87, 242 ], "score": 1.0, "content": ":", "type": "text" }, { "bbox": [ 87, 231, 95, 240 ], "score": 0.73, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 96, 230, 290, 242 ], "score": 1.0, "content": "is monotonically decreasing with respect to the", "type": "text" } ], "index": 13 }, { "bbox": [ 54, 241, 157, 254 ], "spans": [ { "bbox": [ 54, 242, 126, 254 ], "score": 1.0, "content": "weighted error of", "type": "text" }, { "bbox": [ 126, 241, 153, 254 ], "score": 0.92, "content": "\\mathbf { v } \\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 153, 242, 157, 254 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 14 } ], "index": 7.5 }, { "type": "text", "bbox": [ 54, 259, 290, 307 ], "lines": [ { "bbox": [ 53, 259, 290, 272 ], "spans": [ { "bbox": [ 53, 259, 290, 272 ], "score": 1.0, "content": "The advantage of the setup is that, given the binary classi-", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 271, 290, 284 ], "spans": [ { "bbox": [ 53, 271, 70, 284 ], "score": 1.0, "content": "fier", "type": "text" }, { "bbox": [ 70, 272, 92, 284 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 92, 271, 194, 284 ], "score": 1.0, "content": ", the optimal vote vector", "type": "text" }, { "bbox": [ 194, 274, 202, 282 ], "score": 0.53, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 202, 271, 281, 284 ], "score": 1.0, "content": "and the coefficient", "type": "text" }, { "bbox": [ 281, 274, 290, 282 ], "score": 0.75, "content": "\\alpha", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 53, 283, 290, 295 ], "spans": [ { "bbox": [ 53, 283, 290, 295 ], "score": 1.0, "content": "can be set in an efficient way. To see this, first let us define", "type": "text" } ], "index": 17 }, { "bbox": [ 54, 296, 187, 308 ], "spans": [ { "bbox": [ 54, 296, 187, 308 ], "score": 1.0, "content": "the weighted per-class error rate", "type": "text" } ], "index": 18 } ], "index": 16.5 }, { "type": "interline_equation", "bbox": [ 105, 314, 238, 347 ], "lines": [ { "bbox": [ 105, 314, 238, 347 ], "spans": [ { "bbox": [ 105, 314, 238, 347 ], "score": 0.94, "content": "\\mu _ { \\ell - } = \\sum _ { i = 1 } ^ { n } w _ { i , \\ell } \\mathbb { I } \\left\\{ \\varphi ( \\mathbf { x } _ { i } ) \\neq y _ { i , \\ell } \\right\\} ,", "type": "interline_equation", "image_path": "bbd8c1f44044fd745356ab187d882a63a4b468f2e24b4071c76c094a2e6e8d09.jpg" } ] } ], "index": 19.5, "virtual_lines": [ { "bbox": [ 105, 314, 238, 330.5 ], "spans": [], "index": 19 }, { "bbox": [ 105, 330.5, 238, 347.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 54, 353, 267, 365 ], "lines": [ { "bbox": [ 53, 353, 267, 366 ], "spans": [ { "bbox": [ 53, 353, 267, 366 ], "score": 1.0, "content": "and the weighted per-class correct classification rate", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "interline_equation", "bbox": [ 106, 372, 237, 405 ], "lines": [ { "bbox": [ 106, 372, 237, 405 ], "spans": [ { "bbox": [ 106, 372, 237, 405 ], "score": 0.94, "content": "\\mu _ { \\ell + } = \\sum _ { i = 1 } ^ { n } w _ { i , \\ell } \\mathbb { I } \\left\\{ \\varphi ( \\mathbf { x } _ { i } ) = y _ { i , \\ell } \\right\\}", "type": "interline_equation", "image_path": "3444c65b60e624987b2884bf31ba1baba263b40af689327939b5e73c4e18982a.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 106, 372, 237, 388.5 ], "spans": [], "index": 22 }, { "bbox": [ 106, 388.5, 237, 405.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 54, 413, 289, 437 ], "lines": [ { "bbox": [ 52, 411, 289, 427 ], "spans": [ { "bbox": [ 52, 411, 111, 427 ], "score": 1.0, "content": "for each class", "type": "text" }, { "bbox": [ 111, 413, 167, 425 ], "score": 0.93, "content": "\\ell = 1 , \\ldots , K", "type": "inline_equation" }, { "bbox": [ 167, 411, 248, 427 ], "score": 1.0, "content": ". 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The full multi-class", "type": "text" } ], "index": 31 }, { "bbox": [ 54, 560, 167, 571 ], "spans": [ { "bbox": [ 54, 560, 167, 571 ], "score": 1.0, "content": "edge of the classifier is then", "type": "text" } ], "index": 32 } ], "index": 31.5 }, { "type": "interline_equation", "bbox": [ 63, 577, 262, 649 ], "lines": [ { "bbox": [ 63, 577, 262, 649 ], "spans": [ { "bbox": [ 63, 577, 262, 649 ], "score": 0.94, "content": "\\begin{array} { l } { { \\displaystyle \\gamma = \\gamma ( \\mathbf { v } , \\varphi , \\mathbf { W } ) = \\sum _ { \\ell = 1 } ^ { K } \\gamma _ { \\ell } = \\sum _ { \\ell = 1 } ^ { K } v _ { \\ell } \\big ( \\mu _ { \\ell + } - \\mu _ { \\ell - } \\big ) } } \\\\ { { \\displaystyle \\ = \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } w _ { i , \\ell } v _ { \\ell } \\varphi ( \\mathbf { x } _ { i } ) y _ { i , \\ell } } . } \\end{array}", "type": "interline_equation", "image_path": "ddbff5ed176fe6c144c817d1749c36443bd920698609c485f54289bafc8c6450.jpg" } ] } ], "index": 35.5, "virtual_lines": [ { "bbox": [ 63, 577, 262, 589.0 ], "spans": [], "index": 33 }, { "bbox": [ 63, 589.0, 262, 601.0 ], "spans": [], "index": 34 }, { "bbox": [ 63, 601.0, 262, 613.0 ], "spans": [], "index": 35 }, { "bbox": [ 63, 613.0, 262, 625.0 ], "spans": [], "index": 36 }, { "bbox": [ 63, 625.0, 262, 637.0 ], "spans": [], "index": 37 }, { "bbox": [ 63, 637.0, 262, 649.0 ], "spans": [], "index": 38 } ] }, { "type": "text", "bbox": [ 54, 655, 290, 690 ], "lines": [ { "bbox": [ 53, 653, 290, 667 ], "spans": [ { "bbox": [ 53, 653, 290, 667 ], "score": 1.0, "content": "With this notation, the classical (Freund & Schapire, 1997)", "type": "text" } ], "index": 39 }, { "bbox": [ 54, 667, 290, 678 ], "spans": [ { "bbox": [ 54, 667, 126, 678 ], "score": 1.0, "content": "binary coefficient", "type": "text" }, { "bbox": [ 126, 669, 133, 677 ], "score": 0.78, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 134, 667, 290, 678 ], "score": 1.0, "content": "is recovered: it is easy to see that (9) is", "type": "text" } ], "index": 40 }, { "bbox": [ 54, 679, 123, 690 ], "spans": [ { "bbox": [ 54, 679, 123, 690 ], "score": 1.0, "content": "minimized when", "type": "text" } ], "index": 41 } ], "index": 40 }, { "type": "interline_equation", "bbox": [ 135, 695, 209, 721 ], "lines": [ { "bbox": [ 135, 695, 209, 721 ], "spans": [ { "bbox": [ 135, 695, 209, 721 ], "score": 0.94, "content": "\\alpha = \\frac { 1 } { 2 } \\log \\frac { 1 + \\gamma } { 1 - \\gamma } .", "type": "interline_equation", "image_path": "fe05a16d39699acdf8c48c1d4b9b91f6f817e255c58393d12c4b7e69f19eccd5.jpg" } ] } ], "index": 42.5, "virtual_lines": [ { "bbox": [ 135, 695, 209, 708.0 ], "spans": [], "index": 42 }, { "bbox": [ 135, 708.0, 209, 721.0 ], "spans": [], "index": 43 } ] }, { "type": "text", "bbox": [ 306, 67, 542, 117 ], "lines": [ { "bbox": [ 305, 66, 542, 80 ], "spans": [ { "bbox": [ 305, 66, 488, 80 ], "score": 1.0, "content": "With this optimal coefficient, (9) becomes", "type": "text" }, { "bbox": [ 488, 67, 542, 80 ], "score": 0.9, "content": "Z ( \\mathbf { h } , \\mathbf { W } ) \\ =", "type": "inline_equation" } ], "index": 44 }, { "bbox": [ 307, 79, 543, 93 ], "spans": [ { "bbox": [ 307, 79, 344, 93 ], "score": 0.9, "content": "\\sqrt { 1 - \\gamma ^ { 2 } }", "type": "inline_equation" }, { "bbox": [ 345, 79, 359, 93 ], "score": 1.0, "content": ", so", "type": "text" }, { "bbox": [ 359, 80, 398, 93 ], "score": 0.91, "content": "Z ( \\mathbf { h } , \\mathbf { W } )", "type": "inline_equation" }, { "bbox": [ 398, 79, 476, 93 ], "score": 1.0, "content": "is minimized when", "type": "text" }, { "bbox": [ 477, 82, 484, 92 ], "score": 0.81, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 484, 79, 543, 93 ], "score": 1.0, "content": "is maximized.", "type": "text" } ], "index": 45 }, { "bbox": [ 305, 92, 541, 105 ], "spans": [ { "bbox": [ 305, 92, 427, 105 ], "score": 1.0, "content": "From (11) it then follows that", "type": "text" }, { "bbox": [ 428, 92, 466, 104 ], "score": 0.91, "content": "Z ( \\mathbf { h } , \\mathbf { W } )", "type": "inline_equation" }, { "bbox": [ 467, 92, 531, 105 ], "score": 1.0, "content": "is minimized if", "type": "text" }, { "bbox": [ 531, 94, 541, 104 ], "score": 0.83, "content": "v _ { \\ell }", "type": "inline_equation" } ], "index": 46 }, { "bbox": [ 305, 104, 487, 118 ], "spans": [ { "bbox": [ 305, 105, 399, 117 ], "score": 1.0, "content": "agrees with the sign of", "type": "text" }, { "bbox": [ 400, 104, 452, 118 ], "score": 0.92, "content": "\\left( \\mu _ { \\ell + } - 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Indeed, if", "type": "text" }, { "bbox": [ 426, 204, 447, 217 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 447, 204, 543, 216 ], "score": 1.0, "content": "is slightly better then a", "type": "text" } ], "index": 53 }, { "bbox": [ 305, 216, 543, 228 ], "spans": [ { "bbox": [ 305, 216, 349, 228 ], "score": 1.0, "content": "coin toss,", "type": "text" }, { "bbox": [ 349, 218, 357, 228 ], "score": 0.8, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 357, 216, 543, 228 ], "score": 1.0, "content": "will be positive. 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To understand the signif-", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 264, 542, 275 ], "spans": [ { "bbox": [ 305, 264, 542, 275 ], "score": 1.0, "content": "icance of this, consider a classical single-label base clas-", "type": "text" } ], "index": 58 }, { "bbox": [ 305, 275, 543, 288 ], "spans": [ { "bbox": [ 305, 275, 330, 288 ], "score": 1.0, "content": "sifier", "type": "text" }, { "bbox": [ 331, 276, 462, 288 ], "score": 0.9, "content": "h \\ : \\ \\mathcal { X } \\ \\to \\ \\mathcal { L } \\ = \\ \\{ 1 , . . . , K \\}", "type": "inline_equation" }, { "bbox": [ 462, 275, 543, 288 ], "score": 1.0, "content": ", required by AD-", "type": "text" } ], "index": 59 }, { "bbox": [ 305, 288, 542, 300 ], "spans": [ { "bbox": [ 305, 288, 401, 300 ], "score": 1.0, "content": "ABOOST.M1. Now if", "type": "text" }, { "bbox": [ 401, 288, 421, 300 ], "score": 0.91, "content": "h ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 422, 288, 542, 300 ], "score": 1.0, "content": "is slightly better than a coin", "type": "text" } ], "index": 60 }, { "bbox": [ 305, 300, 542, 312 ], "spans": [ { "bbox": [ 305, 300, 542, 312 ], "score": 1.0, "content": "toss, all one can hope for is an error rate slightly lower than", "type": "text" } ], "index": 61 }, { "bbox": [ 307, 309, 547, 349 ], "spans": [ { "bbox": [ 307, 311, 327, 325 ], "score": 0.88, "content": "\\textstyle { \\frac { K - 1 } { K } }", "type": "inline_equation" }, { "bbox": [ 329, 309, 432, 349 ], "score": 1.0, "content": "(which is equivalent to a. To achieve the error of", "type": "text" }, { "bbox": [ 439, 309, 547, 349 ], "score": 1.0, "content": "edge slightly higher than(zero edge), required for", "type": "text" } ], "index": 63 }, { "bbox": [ 307, 324, 439, 339 ], "spans": [ { "bbox": [ 307, 325, 329, 339 ], "score": 0.88, "content": "\\textstyle { \\frac { 2 - K } { K } } )", "type": "inline_equation" }, { "bbox": [ 432, 324, 439, 338 ], "score": 0.84, "content": "\\frac { 1 } { 2 }", "type": "inline_equation" } ], "index": 62 }, { "bbox": [ 305, 337, 543, 349 ], "spans": [ { "bbox": [ 305, 337, 543, 349 ], "score": 1.0, "content": "continuing boosting, one has to come up with a base learner", "type": "text" } ], "index": 64 }, { "bbox": [ 306, 350, 486, 361 ], "spans": [ { "bbox": [ 306, 350, 486, 361 ], "score": 1.0, "content": "which is significantly better than a coin toss.", "type": "text" } ], "index": 65 } ], "index": 58 }, { "type": "text", "bbox": [ 306, 366, 542, 510 ], "lines": [ { "bbox": [ 306, 367, 542, 379 ], "spans": [ { "bbox": [ 306, 367, 542, 379 ], "score": 1.0, "content": "There is a long line of research on output codes similar", "type": "text" } ], "index": 66 }, { "bbox": [ 305, 380, 542, 390 ], "spans": [ { "bbox": [ 305, 380, 542, 390 ], "score": 1.0, "content": "in spirit to our setup. The boosting engine in these works", "type": "text" } ], "index": 67 }, { "bbox": [ 304, 391, 543, 402 ], "spans": [ { "bbox": [ 304, 391, 543, 402 ], "score": 1.0, "content": "is usually slightly different from ADABOOST.MH since it", "type": "text" } ], "index": 68 }, { "bbox": [ 305, 403, 542, 414 ], "spans": [ { "bbox": [ 305, 403, 542, 414 ], "score": 1.0, "content": "attempts to optimize the multi-class hinge loss, but the fac-", "type": "text" } ], "index": 69 }, { "bbox": [ 306, 415, 542, 426 ], "spans": [ { "bbox": [ 306, 415, 542, 426 ], "score": 1.0, "content": "torization of the multi-class base classifier is similar to (6).", "type": "text" } ], "index": 70 }, { "bbox": [ 305, 426, 542, 438 ], "spans": [ { "bbox": [ 305, 426, 542, 438 ], "score": 1.0, "content": "Formally, the vote vector v in this framework is one column", "type": "text" } ], "index": 71 }, { "bbox": [ 305, 439, 542, 451 ], "spans": [ { "bbox": [ 305, 439, 542, 451 ], "score": 1.0, "content": "in an output code matrix. In the simplest setup this matrix", "type": "text" } ], "index": 72 }, { "bbox": [ 305, 450, 542, 463 ], "spans": [ { "bbox": [ 305, 450, 542, 463 ], "score": 1.0, "content": "is fixed beforehand by maximizing the error correcting ca-", "type": "text" } ], "index": 73 }, { "bbox": [ 305, 463, 542, 474 ], "spans": [ { "bbox": [ 305, 463, 542, 474 ], "score": 1.0, "content": "pacity of the matrix (Dietterich & Bakiri, 1995; Allwein", "type": "text" } ], "index": 74 }, { "bbox": [ 305, 474, 543, 486 ], "spans": [ { "bbox": [ 305, 474, 543, 486 ], "score": 1.0, "content": "et al., 2001). A slightly better solution (Schapire, 1997;", "type": "text" } ], "index": 75 }, { "bbox": [ 305, 486, 542, 498 ], "spans": [ { "bbox": [ 305, 486, 542, 498 ], "score": 1.0, "content": "Guruswami & Sahai, 1999; Sun et al., 2005) is to wait until", "type": "text" } ], "index": 76 }, { "bbox": [ 306, 498, 482, 511 ], "spans": [ { "bbox": [ 306, 498, 410, 511 ], "score": 1.0, "content": "the given iteration to pick", "type": "text" }, { "bbox": [ 410, 500, 418, 509 ], "score": 0.38, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 419, 498, 482, 511 ], "score": 1.0, "content": "by maximizing", "type": "text" } ], "index": 77 } ], "index": 71.5 }, { "type": "interline_equation", "bbox": [ 335, 516, 511, 551 ], "lines": [ { "bbox": [ 335, 516, 511, 551 ], "spans": [ { "bbox": [ 335, 516, 511, 551 ], "score": 0.94, "content": "\\mathbf { v } ^ { * } = \\underset { \\mathbf { v } } { \\arg \\operatorname* { m a x } } \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } w _ { i , \\ell } \\mathbb { I } \\left\\{ v _ { \\ell } \\neq v _ { \\ell ( \\mathbf { x } _ { i } ) } \\right\\} ,", "type": "interline_equation", "image_path": "d26af1d5ff7ef05698b2350d6e4c660b784bcd48e67d28f1a2b95413dc228ed4.jpg" } ] } ], "index": 78.5, "virtual_lines": [ { "bbox": [ 335, 516, 511, 533.5 ], "spans": [], "index": 78 }, { "bbox": [ 335, 533.5, 511, 551.0 ], "spans": [], "index": 79 } ] }, { "type": "text", "bbox": [ 306, 556, 542, 640 ], "lines": [ { "bbox": [ 306, 556, 542, 568 ], "spans": [ { "bbox": [ 306, 556, 496, 568 ], "score": 1.0, "content": "and then to choose the optimal binary classifier", "type": "text" }, { "bbox": [ 496, 558, 505, 568 ], "score": 0.83, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 505, 556, 542, 568 ], "score": 1.0, "content": "with this", "type": "text" } ], "index": 80 }, { "bbox": [ 306, 568, 542, 580 ], "spans": [ { "bbox": [ 306, 568, 411, 580 ], "score": 1.0, "content": "fixed vote (or code) vector", "type": "text" }, { "bbox": [ 412, 569, 423, 578 ], "score": 0.85, "content": "\\mathbf { v } ^ { * }", "type": "inline_equation" }, { "bbox": [ 424, 568, 542, 580 ], "score": 1.0, "content": "(although in practice it seems", "type": "text" } ], "index": 81 }, { "bbox": [ 305, 580, 542, 592 ], "spans": [ { "bbox": [ 305, 580, 379, 592 ], "score": 1.0, "content": "to be better to fix", "type": "text" }, { "bbox": [ 380, 582, 388, 591 ], "score": 0.53, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 388, 580, 542, 592 ], "score": 1.0, "content": "to a random binary vector; Sun et al.", "type": "text" } ], "index": 82 }, { "bbox": [ 306, 592, 542, 604 ], "spans": [ { "bbox": [ 306, 592, 542, 604 ], "score": 1.0, "content": "2005). The state of the art in this line of research is to iterate", "type": "text" } ], "index": 83 }, { "bbox": [ 306, 604, 542, 616 ], "spans": [ { "bbox": [ 306, 604, 388, 615 ], "score": 1.0, "content": "between optimizing", "type": "text" }, { "bbox": [ 388, 606, 396, 616 ], "score": 0.82, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 397, 604, 448, 615 ], "score": 1.0, "content": "with a fixed", "type": "text" }, { "bbox": [ 448, 606, 456, 614 ], "score": 0.47, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 456, 604, 542, 615 ], "score": 1.0, "content": "and then picking the", "type": "text" } ], "index": 84 }, { "bbox": [ 305, 615, 542, 628 ], "spans": [ { "bbox": [ 305, 615, 380, 628 ], "score": 1.0, "content": "best v with a fixed", "type": "text" }, { "bbox": [ 381, 617, 389, 628 ], "score": 0.8, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 389, 615, 542, 628 ], "score": 1.0, "content": "(Li, 2006; Kegl & Busa-Fekete ´ , 2009;", "type": "text" } ], "index": 85 }, { "bbox": [ 305, 627, 394, 640 ], "spans": [ { "bbox": [ 305, 627, 394, 640 ], "score": 1.0, "content": "Gao & Koller, 2011).", "type": "text" } ], "index": 86 } ], "index": 83 }, { "type": "text", "bbox": [ 306, 645, 542, 717 ], "lines": [ { "bbox": [ 305, 645, 542, 658 ], "spans": [ { "bbox": [ 305, 645, 379, 658 ], "score": 1.0, "content": "It turns out that if", "type": "text" }, { "bbox": [ 379, 648, 387, 658 ], "score": 0.83, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 387, 645, 542, 658 ], "score": 1.0, "content": "is a decision stump, exhaustive search", "type": "text" } ], "index": 87 }, { "bbox": [ 305, 657, 543, 669 ], "spans": [ { "bbox": [ 305, 657, 543, 669 ], "score": 1.0, "content": "for both the best binary cut (threshold) and the best vote", "type": "text" } ], "index": 88 }, { "bbox": [ 304, 669, 541, 682 ], "spans": [ { "bbox": [ 304, 669, 509, 682 ], "score": 1.0, "content": "vector can be carried out using one single sweep in", "type": "text" }, { "bbox": [ 510, 669, 541, 682 ], "score": 0.92, "content": "\\Theta ( n K )", "type": "inline_equation" } ], "index": 89 }, { "bbox": [ 305, 682, 542, 693 ], "spans": [ { "bbox": [ 305, 682, 542, 693 ], "score": 1.0, "content": "time. The algorithm is a simple extension of the classi-", "type": "text" } ], "index": 90 }, { "bbox": [ 305, 694, 542, 705 ], "spans": [ { "bbox": [ 305, 694, 542, 705 ], "score": 1.0, "content": "cal binary decision stump learner; for the sake of com-", "type": "text" } ], "index": 91 }, { "bbox": [ 305, 706, 542, 717 ], "spans": [ { "bbox": [ 305, 706, 542, 717 ], "score": 1.0, "content": "pleteness, we provide the pseudocode in Appendix B. The", "type": "text" } ], "index": 92 } ], "index": 89.5 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 246, 46, 350, 55 ], "lines": [ { "bbox": [ 244, 44, 351, 57 ], "spans": [ { "bbox": [ 244, 44, 351, 57 ], "score": 1.0, "content": "Multi-class Hamming trees", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 54, 68, 145, 79 ], "lines": [ { "bbox": [ 53, 67, 147, 81 ], "spans": [ { "bbox": [ 53, 67, 147, 81 ], "score": 1.0, "content": "2.4. Casting the votes", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 54, 86, 289, 253 ], "lines": [ { "bbox": [ 54, 86, 290, 98 ], "spans": [ { "bbox": [ 54, 86, 167, 98 ], "score": 1.0, "content": "To start, we show how to set", "type": "text" }, { "bbox": [ 167, 88, 175, 96 ], "score": 0.79, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 176, 86, 192, 98 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 192, 88, 200, 97 ], "score": 0.55, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 200, 86, 290, 98 ], "score": 1.0, "content": "in general if the scalar", "type": "text" } ], "index": 1 }, { "bbox": [ 53, 98, 290, 110 ], "spans": [ { "bbox": [ 53, 98, 114, 110 ], "score": 1.0, "content": "base classifier", "type": "text" }, { "bbox": [ 114, 100, 123, 110 ], "score": 0.81, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 123, 98, 290, 110 ], "score": 1.0, "content": "is given. The intuitive semantics of (6)", "type": "text" } ], "index": 2 }, { "bbox": [ 53, 110, 290, 122 ], "spans": [ { "bbox": [ 53, 110, 216, 122 ], "score": 1.0, "content": "is the following. The binary classifier", "type": "text" }, { "bbox": [ 217, 110, 238, 122 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 239, 110, 290, 122 ], "score": 1.0, "content": "cuts the in-", "type": "text" } ], "index": 3 }, { "bbox": [ 53, 123, 290, 135 ], "spans": [ { "bbox": [ 53, 123, 290, 135 ], "score": 1.0, "content": "put space into a positive and a negative region. In binary", "type": "text" } ], "index": 4 }, { "bbox": [ 52, 134, 290, 146 ], "spans": [ { "bbox": [ 52, 134, 257, 146 ], "score": 1.0, "content": "classification this is the end of the story: we need", "type": "text" }, { "bbox": [ 257, 134, 279, 146 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 279, 134, 290, 146 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 5 }, { "bbox": [ 52, 145, 290, 158 ], "spans": [ { "bbox": [ 52, 145, 240, 158 ], "score": 1.0, "content": "be well-correlated with the binary class labels", "type": "text" }, { "bbox": [ 241, 148, 248, 158 ], "score": 0.74, "content": "y", "type": "inline_equation" }, { "bbox": [ 248, 145, 290, 158 ], "score": 1.0, "content": ". In multi-", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 158, 290, 170 ], "spans": [ { "bbox": [ 53, 158, 204, 170 ], "score": 1.0, "content": "class classification it is possible that", "type": "text" }, { "bbox": [ 205, 158, 226, 170 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 226, 158, 290, 170 ], "score": 1.0, "content": "correlates with", "type": "text" } ], "index": 7 }, { "bbox": [ 52, 170, 290, 182 ], "spans": [ { "bbox": [ 52, 170, 158, 181 ], "score": 1.0, "content": "some of the class labels", "type": "text" }, { "bbox": [ 158, 172, 168, 182 ], "score": 0.74, "content": "y _ { \\ell }", "type": "inline_equation" }, { "bbox": [ 168, 170, 290, 181 ], "score": 1.0, "content": "and anti-correlates with oth-", "type": "text" } ], "index": 8 }, { "bbox": [ 52, 181, 291, 194 ], "spans": [ { "bbox": [ 52, 181, 291, 194 ], "score": 1.0, "content": "ers. This free choice is expressed by the binary “votes”", "type": "text" } ], "index": 9 }, { "bbox": [ 54, 194, 289, 207 ], "spans": [ { "bbox": [ 54, 194, 99, 207 ], "score": 0.93, "content": "v _ { \\ell } \\in \\{ \\pm 1 \\}", "type": "inline_equation" }, { "bbox": [ 100, 194, 154, 207 ], "score": 1.0, "content": ". We say that", "type": "text" }, { "bbox": [ 154, 194, 175, 206 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 175, 194, 236, 207 ], "score": 1.0, "content": "votes for class", "type": "text" }, { "bbox": [ 236, 194, 243, 204 ], "score": 0.75, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 243, 194, 252, 207 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 253, 194, 289, 205 ], "score": 0.91, "content": "v _ { \\ell } = + 1", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 53, 206, 290, 218 ], "spans": [ { "bbox": [ 53, 206, 157, 218 ], "score": 1.0, "content": "and it votes against class", "type": "text" }, { "bbox": [ 158, 207, 163, 216 ], "score": 0.79, "content": "\\ell", "type": "inline_equation" }, { "bbox": [ 164, 206, 173, 218 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 174, 207, 211, 217 ], "score": 0.91, "content": "v _ { \\ell } = - 1", "type": "inline_equation" }, { "bbox": [ 211, 206, 290, 218 ], "score": 1.0, "content": ". As in binary clas-", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 217, 291, 231 ], "spans": [ { "bbox": [ 53, 217, 96, 231 ], "score": 1.0, "content": "sification,", "type": "text" }, { "bbox": [ 97, 220, 105, 228 ], "score": 0.76, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 105, 217, 291, 231 ], "score": 1.0, "content": "expresses the overall quality of the classifier", "type": "text" } ], "index": 12 }, { "bbox": [ 54, 230, 290, 242 ], "spans": [ { "bbox": [ 54, 230, 81, 242 ], "score": 0.9, "content": "\\mathbf { v } \\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 82, 230, 87, 242 ], "score": 1.0, "content": ":", "type": "text" }, { "bbox": [ 87, 231, 95, 240 ], "score": 0.73, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 96, 230, 290, 242 ], "score": 1.0, "content": "is monotonically decreasing with respect to the", "type": "text" } ], "index": 13 }, { "bbox": [ 54, 241, 157, 254 ], "spans": [ { "bbox": [ 54, 242, 126, 254 ], "score": 1.0, "content": "weighted error of", "type": "text" }, { "bbox": [ 126, 241, 153, 254 ], "score": 0.92, "content": "\\mathbf { v } \\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 153, 242, 157, 254 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 14 } ], "index": 7.5, "bbox_fs": [ 52, 86, 291, 254 ] }, { "type": "text", "bbox": [ 54, 259, 290, 307 ], "lines": [ { "bbox": [ 53, 259, 290, 272 ], "spans": [ { "bbox": [ 53, 259, 290, 272 ], "score": 1.0, "content": "The advantage of the setup is that, given the binary classi-", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 271, 290, 284 ], "spans": [ { "bbox": [ 53, 271, 70, 284 ], "score": 1.0, "content": "fier", "type": "text" }, { "bbox": [ 70, 272, 92, 284 ], "score": 0.92, "content": "\\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 92, 271, 194, 284 ], "score": 1.0, "content": ", the optimal vote vector", "type": "text" }, { "bbox": [ 194, 274, 202, 282 ], "score": 0.53, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 202, 271, 281, 284 ], "score": 1.0, "content": "and the coefficient", "type": "text" }, { "bbox": [ 281, 274, 290, 282 ], "score": 0.75, "content": "\\alpha", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 53, 283, 290, 295 ], "spans": [ { "bbox": [ 53, 283, 290, 295 ], "score": 1.0, "content": "can be set in an efficient way. To see this, first let us define", "type": "text" } ], "index": 17 }, { "bbox": [ 54, 296, 187, 308 ], "spans": [ { "bbox": [ 54, 296, 187, 308 ], "score": 1.0, "content": "the weighted per-class error rate", "type": "text" } ], "index": 18 } ], "index": 16.5, "bbox_fs": [ 53, 259, 290, 308 ] }, { "type": "interline_equation", "bbox": [ 105, 314, 238, 347 ], "lines": [ { "bbox": [ 105, 314, 238, 347 ], "spans": [ { "bbox": [ 105, 314, 238, 347 ], "score": 0.94, "content": "\\mu _ { \\ell - } = \\sum _ { i = 1 } ^ { n } w _ { i , \\ell } \\mathbb { I } \\left\\{ \\varphi ( \\mathbf { x } _ { i } ) \\neq y _ { i , \\ell } \\right\\} ,", "type": "interline_equation", "image_path": "bbd8c1f44044fd745356ab187d882a63a4b468f2e24b4071c76c094a2e6e8d09.jpg" } ] } ], "index": 19.5, "virtual_lines": [ { "bbox": [ 105, 314, 238, 330.5 ], "spans": [], "index": 19 }, { "bbox": [ 105, 330.5, 238, 347.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 54, 353, 267, 365 ], "lines": [ { "bbox": [ 53, 353, 267, 366 ], "spans": [ { "bbox": [ 53, 353, 267, 366 ], "score": 1.0, "content": "and the weighted per-class correct classification rate", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 53, 353, 267, 366 ] }, { "type": "interline_equation", "bbox": [ 106, 372, 237, 405 ], "lines": [ { "bbox": [ 106, 372, 237, 405 ], "spans": [ { "bbox": [ 106, 372, 237, 405 ], "score": 0.94, "content": "\\mu _ { \\ell + } = \\sum _ { i = 1 } ^ { n } w _ { i , \\ell } \\mathbb { I } \\left\\{ \\varphi ( \\mathbf { x } _ { i } ) = y _ { i , \\ell } \\right\\}", "type": "interline_equation", "image_path": "3444c65b60e624987b2884bf31ba1baba263b40af689327939b5e73c4e18982a.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 106, 372, 237, 388.5 ], "spans": [], "index": 22 }, { "bbox": [ 106, 388.5, 237, 405.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 54, 413, 289, 437 ], "lines": [ { "bbox": [ 52, 411, 289, 427 ], "spans": [ { "bbox": [ 52, 411, 111, 427 ], "score": 1.0, "content": "for each class", "type": "text" }, { "bbox": [ 111, 413, 167, 425 ], "score": 0.93, "content": "\\ell = 1 , \\ldots , K", "type": "inline_equation" }, { "bbox": [ 167, 411, 248, 427 ], "score": 1.0, "content": ". 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The boosting engine in these works", "type": "text" } ], "index": 67 }, { "bbox": [ 304, 391, 543, 402 ], "spans": [ { "bbox": [ 304, 391, 543, 402 ], "score": 1.0, "content": "is usually slightly different from ADABOOST.MH since it", "type": "text" } ], "index": 68 }, { "bbox": [ 305, 403, 542, 414 ], "spans": [ { "bbox": [ 305, 403, 542, 414 ], "score": 1.0, "content": "attempts to optimize the multi-class hinge loss, but the fac-", "type": "text" } ], "index": 69 }, { "bbox": [ 306, 415, 542, 426 ], "spans": [ { "bbox": [ 306, 415, 542, 426 ], "score": 1.0, "content": "torization of the multi-class base classifier is similar to (6).", "type": "text" } ], "index": 70 }, { "bbox": [ 305, 426, 542, 438 ], "spans": [ { "bbox": [ 305, 426, 542, 438 ], "score": 1.0, "content": "Formally, the vote vector v in this framework is one column", "type": "text" } ], "index": 71 }, { "bbox": [ 305, 439, 542, 451 ], "spans": [ { "bbox": [ 305, 439, 542, 451 ], "score": 1.0, "content": "in an output code matrix. In the simplest setup this matrix", "type": "text" } ], "index": 72 }, { "bbox": [ 305, 450, 542, 463 ], "spans": [ { "bbox": [ 305, 450, 542, 463 ], "score": 1.0, "content": "is fixed beforehand by maximizing the error correcting ca-", "type": "text" } ], "index": 73 }, { "bbox": [ 305, 463, 542, 474 ], "spans": [ { "bbox": [ 305, 463, 542, 474 ], "score": 1.0, "content": "pacity of the matrix (Dietterich & Bakiri, 1995; Allwein", "type": "text" } ], "index": 74 }, { "bbox": [ 305, 474, 543, 486 ], "spans": [ { "bbox": [ 305, 474, 543, 486 ], "score": 1.0, "content": "et al., 2001). A slightly better solution (Schapire, 1997;", "type": "text" } ], "index": 75 }, { "bbox": [ 305, 486, 542, 498 ], "spans": [ { "bbox": [ 305, 486, 542, 498 ], "score": 1.0, "content": "Guruswami & Sahai, 1999; Sun et al., 2005) is to wait until", "type": "text" } ], "index": 76 }, { "bbox": [ 306, 498, 482, 511 ], "spans": [ { "bbox": [ 306, 498, 410, 511 ], "score": 1.0, "content": "the given iteration to pick", "type": "text" }, { "bbox": [ 410, 500, 418, 509 ], "score": 0.38, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 419, 498, 482, 511 ], "score": 1.0, "content": "by maximizing", "type": "text" } ], "index": 77 } ], "index": 71.5, "bbox_fs": [ 304, 367, 543, 511 ] }, { "type": "interline_equation", "bbox": [ 335, 516, 511, 551 ], "lines": [ { "bbox": [ 335, 516, 511, 551 ], "spans": [ { "bbox": [ 335, 516, 511, 551 ], "score": 0.94, "content": "\\mathbf { v } ^ { * } = \\underset { \\mathbf { v } } { \\arg \\operatorname* { m a x } } \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } w _ { i , \\ell } \\mathbb { I } \\left\\{ v _ { \\ell } \\neq v _ { \\ell ( \\mathbf { x } _ { i } ) } \\right\\} ,", "type": "interline_equation", "image_path": "d26af1d5ff7ef05698b2350d6e4c660b784bcd48e67d28f1a2b95413dc228ed4.jpg" } ] } ], "index": 78.5, "virtual_lines": [ { "bbox": [ 335, 516, 511, 533.5 ], "spans": [], "index": 78 }, { "bbox": [ 335, 533.5, 511, 551.0 ], "spans": [], "index": 79 } ] }, { "type": "text", "bbox": [ 306, 556, 542, 640 ], "lines": [ { "bbox": [ 306, 556, 542, 568 ], "spans": [ { "bbox": [ 306, 556, 496, 568 ], "score": 1.0, "content": "and then to choose the optimal binary classifier", "type": "text" }, { "bbox": [ 496, 558, 505, 568 ], "score": 0.83, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 505, 556, 542, 568 ], "score": 1.0, "content": "with this", "type": "text" } ], "index": 80 }, { "bbox": [ 306, 568, 542, 580 ], "spans": [ { "bbox": [ 306, 568, 411, 580 ], "score": 1.0, "content": "fixed vote (or code) vector", "type": "text" }, { "bbox": [ 412, 569, 423, 578 ], "score": 0.85, "content": "\\mathbf { v } ^ { * }", "type": "inline_equation" }, { "bbox": [ 424, 568, 542, 580 ], "score": 1.0, "content": "(although in practice it seems", "type": "text" } ], "index": 81 }, { "bbox": [ 305, 580, 542, 592 ], "spans": [ { "bbox": [ 305, 580, 379, 592 ], "score": 1.0, "content": "to be better to fix", "type": "text" }, { "bbox": [ 380, 582, 388, 591 ], "score": 0.53, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 388, 580, 542, 592 ], "score": 1.0, "content": "to a random binary vector; Sun et al.", "type": "text" } ], "index": 82 }, { "bbox": [ 306, 592, 542, 604 ], "spans": [ { "bbox": [ 306, 592, 542, 604 ], "score": 1.0, "content": "2005). The state of the art in this line of research is to iterate", "type": "text" } ], "index": 83 }, { "bbox": [ 306, 604, 542, 616 ], "spans": [ { "bbox": [ 306, 604, 388, 615 ], "score": 1.0, "content": "between optimizing", "type": "text" }, { "bbox": [ 388, 606, 396, 616 ], "score": 0.82, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 397, 604, 448, 615 ], "score": 1.0, "content": "with a fixed", "type": "text" }, { "bbox": [ 448, 606, 456, 614 ], "score": 0.47, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 456, 604, 542, 615 ], "score": 1.0, "content": "and then picking the", "type": "text" } ], "index": 84 }, { "bbox": [ 305, 615, 542, 628 ], "spans": [ { "bbox": [ 305, 615, 380, 628 ], "score": 1.0, "content": "best v with a fixed", "type": "text" }, { "bbox": [ 381, 617, 389, 628 ], "score": 0.8, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 389, 615, 542, 628 ], "score": 1.0, "content": "(Li, 2006; Kegl & Busa-Fekete ´ , 2009;", "type": "text" } ], "index": 85 }, { "bbox": [ 305, 627, 394, 640 ], "spans": [ { "bbox": [ 305, 627, 394, 640 ], "score": 1.0, "content": "Gao & Koller, 2011).", "type": "text" } ], "index": 86 } ], "index": 83, "bbox_fs": [ 305, 556, 542, 640 ] }, { "type": "text", "bbox": [ 306, 645, 542, 717 ], "lines": [ { "bbox": [ 305, 645, 542, 658 ], "spans": [ { "bbox": [ 305, 645, 379, 658 ], "score": 1.0, "content": "It turns out that if", "type": "text" }, { "bbox": [ 379, 648, 387, 658 ], "score": 0.83, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 387, 645, 542, 658 ], "score": 1.0, "content": "is a decision stump, exhaustive search", "type": "text" } ], "index": 87 }, { "bbox": [ 305, 657, 543, 669 ], "spans": [ { "bbox": [ 305, 657, 543, 669 ], "score": 1.0, "content": "for both the best binary cut (threshold) and the best vote", "type": "text" } ], "index": 88 }, { "bbox": [ 304, 669, 541, 682 ], "spans": [ { "bbox": [ 304, 669, 509, 682 ], "score": 1.0, "content": "vector can be carried out using one single sweep in", "type": "text" }, { "bbox": [ 510, 669, 541, 682 ], "score": 0.92, "content": "\\Theta ( n K )", "type": "inline_equation" } ], "index": 89 }, { "bbox": [ 305, 682, 542, 693 ], "spans": [ { "bbox": [ 305, 682, 542, 693 ], "score": 1.0, "content": "time. The algorithm is a simple extension of the classi-", "type": "text" } ], "index": 90 }, { "bbox": [ 305, 694, 542, 705 ], "spans": [ { "bbox": [ 305, 694, 542, 705 ], "score": 1.0, "content": "cal binary decision stump learner; for the sake of com-", "type": "text" } ], "index": 91 }, { "bbox": [ 305, 706, 542, 717 ], "spans": [ { "bbox": [ 305, 706, 542, 717 ], "score": 1.0, "content": "pleteness, we provide the pseudocode in Appendix B. The", "type": "text" } ], "index": 92 }, { "bbox": [ 54, 67, 290, 80 ], "spans": [ { "bbox": [ 54, 67, 290, 80 ], "score": 1.0, "content": "computational efficiency of this learning algorithm com-", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 53, 79, 290, 91 ], "spans": [ { "bbox": [ 53, 79, 290, 91 ], "score": 1.0, "content": "bined with the factorized form (6) of the classifier allows", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 53, 91, 290, 104 ], "spans": [ { "bbox": [ 53, 91, 290, 104 ], "score": 1.0, "content": "us to build multiclass Hamming trees in an efficient man-", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 53, 104, 291, 115 ], "spans": [ { "bbox": [ 53, 104, 291, 115 ], "score": 1.0, "content": "ner, circumventing the problem of global maximization of", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 53, 115, 188, 128 ], "spans": [ { "bbox": [ 53, 115, 151, 128 ], "score": 1.0, "content": "the edge with respect to", "type": "text", "cross_page": true }, { "bbox": [ 151, 117, 159, 127 ], "score": 0.82, "content": "\\varphi", "type": "inline_equation", "cross_page": true }, { "bbox": [ 159, 115, 177, 128 ], "score": 1.0, "content": "and", "type": "text", "cross_page": true }, { "bbox": [ 177, 117, 184, 126 ], "score": 0.36, "content": "\\mathbf { v }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 185, 115, 188, 128 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 4 } ], "index": 89.5, "bbox_fs": [ 304, 645, 543, 717 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 55, 68, 290, 127 ], "lines": [ { "bbox": [ 54, 67, 290, 80 ], "spans": [ { "bbox": [ 54, 67, 290, 80 ], "score": 1.0, "content": "computational efficiency of this learning algorithm com-", "type": "text" } ], "index": 0 }, { "bbox": [ 53, 79, 290, 91 ], "spans": [ { "bbox": [ 53, 79, 290, 91 ], "score": 1.0, "content": "bined with the factorized form (6) of the classifier allows", "type": "text" } ], "index": 1 }, { "bbox": [ 53, 91, 290, 104 ], "spans": [ { "bbox": [ 53, 91, 290, 104 ], "score": 1.0, "content": "us to build multiclass Hamming trees in an efficient man-", "type": "text" } ], "index": 2 }, { "bbox": [ 53, 104, 291, 115 ], "spans": [ { "bbox": [ 53, 104, 291, 115 ], "score": 1.0, "content": "ner, circumventing the problem of global maximization of", "type": "text" } ], "index": 3 }, { "bbox": [ 53, 115, 188, 128 ], "spans": [ { "bbox": [ 53, 115, 151, 128 ], "score": 1.0, "content": "the edge with respect to", "type": "text" }, { "bbox": [ 151, 117, 159, 127 ], "score": 0.82, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 159, 115, 177, 128 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 177, 117, 184, 126 ], "score": 0.36, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 185, 115, 188, 128 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "title", "bbox": [ 54, 142, 146, 156 ], "lines": [ { "bbox": [ 52, 141, 148, 159 ], "spans": [ { "bbox": [ 52, 141, 148, 159 ], "score": 1.0, "content": "3. Hamming trees", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 54, 163, 289, 342 ], "lines": [ { "bbox": [ 54, 163, 290, 175 ], "spans": [ { "bbox": [ 54, 163, 290, 175 ], "score": 1.0, "content": "Classification trees (Quinlan, 1986) have been widely used", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 174, 290, 187 ], "spans": [ { "bbox": [ 53, 174, 290, 187 ], "score": 1.0, "content": "for multivariate classification since the 80s. They are", "type": "text" } ], "index": 7 }, { "bbox": [ 52, 187, 291, 199 ], "spans": [ { "bbox": [ 52, 187, 291, 199 ], "score": 1.0, "content": "especially efficient when used as base learners in AD-", "type": "text" } ], "index": 8 }, { "bbox": [ 53, 199, 290, 212 ], "spans": [ { "bbox": [ 53, 199, 290, 212 ], "score": 1.0, "content": "ABOOST (Caruana & Niculescu-Mizil, 2006; Quinlan,", "type": "text" } ], "index": 9 }, { "bbox": [ 54, 211, 290, 223 ], "spans": [ { "bbox": [ 54, 211, 290, 223 ], "score": 1.0, "content": "1996). Their main disadvantage is their variance with re-", "type": "text" } ], "index": 10 }, { "bbox": [ 53, 223, 290, 235 ], "spans": [ { "bbox": [ 53, 223, 262, 235 ], "score": 1.0, "content": "spect to the training data, but when averaged over", "type": "text" }, { "bbox": [ 263, 223, 272, 233 ], "score": 0.75, "content": "T", "type": "inline_equation" }, { "bbox": [ 272, 223, 290, 235 ], "score": 1.0, "content": "dif-", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 235, 291, 247 ], "spans": [ { "bbox": [ 53, 235, 291, 247 ], "score": 1.0, "content": "ferent runs, this problem largely disappears. The most", "type": "text" } ], "index": 12 }, { "bbox": [ 52, 247, 290, 258 ], "spans": [ { "bbox": [ 52, 247, 290, 258 ], "score": 1.0, "content": "commonly used tree learner is C4.5 of Quinlan (1993).", "type": "text" } ], "index": 13 }, { "bbox": [ 53, 259, 291, 271 ], "spans": [ { "bbox": [ 53, 259, 291, 271 ], "score": 1.0, "content": "Whereas this tree implementation is a perfect choice for", "type": "text" } ], "index": 14 }, { "bbox": [ 53, 271, 290, 282 ], "spans": [ { "bbox": [ 53, 271, 290, 282 ], "score": 1.0, "content": "binary ADABOOST, it is suboptimal for ADABOOST.MH", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 283, 290, 295 ], "spans": [ { "bbox": [ 53, 283, 290, 295 ], "score": 1.0, "content": "since it outputs a single-label classifier with no guarantee", "type": "text" } ], "index": 16 }, { "bbox": [ 53, 294, 291, 308 ], "spans": [ { "bbox": [ 53, 294, 291, 308 ], "score": 1.0, "content": "of a positive multi-class edge (11). Although this problem", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 307, 290, 319 ], "spans": [ { "bbox": [ 53, 307, 290, 319 ], "score": 1.0, "content": "can be solved in practice by building large trees, it seems", "type": "text" } ], "index": 18 }, { "bbox": [ 54, 319, 290, 331 ], "spans": [ { "bbox": [ 54, 319, 290, 331 ], "score": 1.0, "content": "that using these large single-class trees is suboptimal (Sec-", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 330, 86, 343 ], "spans": [ { "bbox": [ 53, 330, 86, 343 ], "score": 1.0, "content": "tion 4).", "type": "text" } ], "index": 20 } ], "index": 13 }, { "type": "text", "bbox": [ 54, 348, 289, 551 ], "lines": [ { "bbox": [ 54, 349, 291, 361 ], "spans": [ { "bbox": [ 54, 349, 291, 361 ], "score": 1.0, "content": "The main technical difficulty of building trees out of", "type": "text" } ], "index": 21 }, { "bbox": [ 52, 360, 291, 373 ], "spans": [ { "bbox": [ 52, 360, 86, 373 ], "score": 1.0, "content": "generic", "type": "text" }, { "bbox": [ 87, 360, 118, 373 ], "score": 0.93, "content": "\\{ \\pm 1 \\} ^ { K }", "type": "inline_equation" }, { "bbox": [ 118, 360, 239, 373 ], "score": 1.0, "content": "-valued multi-class classifiers", "type": "text" }, { "bbox": [ 239, 361, 261, 372 ], "score": 0.9, "content": "\\mathbf { h } ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 261, 360, 291, 373 ], "score": 1.0, "content": "is that", "type": "text" } ], "index": 22 }, { "bbox": [ 54, 372, 290, 385 ], "spans": [ { "bbox": [ 54, 373, 241, 385 ], "score": 1.0, "content": "they do not necessarily implement a binary cut", "type": "text" }, { "bbox": [ 241, 372, 286, 385 ], "score": 0.93, "content": "{ \\bf x } \\mapsto \\{ \\pm 1 \\}", "type": "inline_equation" }, { "bbox": [ 287, 373, 290, 385 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 52, 384, 291, 396 ], "spans": [ { "bbox": [ 52, 384, 231, 396 ], "score": 1.0, "content": "and partitioning the data into all the possibly", "type": "text" }, { "bbox": [ 231, 384, 245, 395 ], "score": 0.87, "content": "2 ^ { K }", "type": "inline_equation" }, { "bbox": [ 245, 384, 291, 396 ], "score": 1.0, "content": "children at", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 397, 290, 409 ], "spans": [ { "bbox": [ 53, 397, 290, 409 ], "score": 1.0, "content": "a tree node leads to rapid overfitting. Factorizing the multi-", "type": "text" } ], "index": 25 }, { "bbox": [ 53, 408, 290, 421 ], "spans": [ { "bbox": [ 53, 408, 114, 421 ], "score": 1.0, "content": "class classifier", "type": "text" }, { "bbox": [ 114, 408, 136, 421 ], "score": 0.89, "content": "\\mathbf { h } ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 136, 408, 290, 421 ], "score": 1.0, "content": "into an input-independent vote vector", "type": "text" } ], "index": 26 }, { "bbox": [ 54, 420, 290, 433 ], "spans": [ { "bbox": [ 54, 422, 62, 431 ], "score": 0.38, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 62, 420, 230, 433 ], "score": 1.0, "content": "and a label-independent binary classifier", "type": "text" }, { "bbox": [ 231, 420, 252, 433 ], "score": 0.92, "content": "\\varphi ( x )", "type": "inline_equation" }, { "bbox": [ 252, 420, 290, 433 ], "score": 1.0, "content": "as in (6)", "type": "text" } ], "index": 27 }, { "bbox": [ 54, 433, 291, 444 ], "spans": [ { "bbox": [ 54, 433, 291, 444 ], "score": 1.0, "content": "solves this problem. Base classifiers are trained as usual at", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 444, 290, 455 ], "spans": [ { "bbox": [ 53, 444, 290, 455 ], "score": 1.0, "content": "each new tree leaf. 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The most", "type": "text" } ], "index": 12 }, { "bbox": [ 52, 247, 290, 258 ], "spans": [ { "bbox": [ 52, 247, 290, 258 ], "score": 1.0, "content": "commonly used tree learner is C4.5 of Quinlan (1993).", "type": "text" } ], "index": 13 }, { "bbox": [ 53, 259, 291, 271 ], "spans": [ { "bbox": [ 53, 259, 291, 271 ], "score": 1.0, "content": "Whereas this tree implementation is a perfect choice for", "type": "text" } ], "index": 14 }, { "bbox": [ 53, 271, 290, 282 ], "spans": [ { "bbox": [ 53, 271, 290, 282 ], "score": 1.0, "content": "binary ADABOOST, it is suboptimal for ADABOOST.MH", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 283, 290, 295 ], "spans": [ { "bbox": [ 53, 283, 290, 295 ], "score": 1.0, "content": "since it outputs a single-label classifier with no guarantee", "type": "text" } ], "index": 16 }, { "bbox": [ 53, 294, 291, 308 ], "spans": [ { "bbox": [ 53, 294, 291, 308 ], "score": 1.0, "content": "of a positive multi-class edge (11). Although this problem", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 307, 290, 319 ], "spans": [ { "bbox": [ 53, 307, 290, 319 ], "score": 1.0, "content": "can be solved in practice by building large trees, it seems", "type": "text" } ], "index": 18 }, { "bbox": [ 54, 319, 290, 331 ], "spans": [ { "bbox": [ 54, 319, 290, 331 ], "score": 1.0, "content": "that using these large single-class trees is suboptimal (Sec-", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 330, 86, 343 ], "spans": [ { "bbox": [ 53, 330, 86, 343 ], "score": 1.0, "content": "tion 4).", "type": "text" } ], "index": 20 } ], "index": 13, "bbox_fs": [ 52, 163, 291, 343 ] }, { "type": "text", "bbox": [ 54, 348, 289, 551 ], "lines": [ { "bbox": [ 54, 349, 291, 361 ], "spans": [ { "bbox": [ 54, 349, 291, 361 ], "score": 1.0, "content": "The main technical difficulty of building trees out of", "type": "text" } ], "index": 21 }, { "bbox": [ 52, 360, 291, 373 ], "spans": [ { "bbox": [ 52, 360, 86, 373 ], "score": 1.0, "content": "generic", "type": "text" }, { "bbox": [ 87, 360, 118, 373 ], "score": 0.93, "content": "\\{ \\pm 1 \\} ^ { K }", "type": "inline_equation" }, { "bbox": [ 118, 360, 239, 373 ], "score": 1.0, "content": "-valued multi-class classifiers", "type": "text" }, { "bbox": [ 239, 361, 261, 372 ], "score": 0.9, "content": "\\mathbf { h } ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 261, 360, 291, 373 ], "score": 1.0, "content": "is that", "type": "text" } ], "index": 22 }, { "bbox": [ 54, 372, 290, 385 ], "spans": [ { "bbox": [ 54, 373, 241, 385 ], "score": 1.0, "content": "they do not necessarily implement a binary cut", "type": "text" }, { "bbox": [ 241, 372, 286, 385 ], "score": 0.93, "content": "{ \\bf x } \\mapsto \\{ \\pm 1 \\}", "type": "inline_equation" }, { "bbox": [ 287, 373, 290, 385 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 52, 384, 291, 396 ], "spans": [ { "bbox": [ 52, 384, 231, 396 ], "score": 1.0, "content": "and partitioning the data into all the possibly", "type": "text" }, { "bbox": [ 231, 384, 245, 395 ], "score": 0.87, "content": "2 ^ { K }", "type": "inline_equation" }, { "bbox": [ 245, 384, 291, 396 ], "score": 1.0, "content": "children at", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 397, 290, 409 ], "spans": [ { "bbox": [ 53, 397, 290, 409 ], "score": 1.0, "content": "a tree node leads to rapid overfitting. Factorizing the multi-", "type": "text" } ], "index": 25 }, { "bbox": [ 53, 408, 290, 421 ], "spans": [ { "bbox": [ 53, 408, 114, 421 ], "score": 1.0, "content": "class classifier", "type": "text" }, { "bbox": [ 114, 408, 136, 421 ], "score": 0.89, "content": "\\mathbf { h } ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 136, 408, 290, 421 ], "score": 1.0, "content": "into an input-independent vote vector", "type": "text" } ], "index": 26 }, { "bbox": [ 54, 420, 290, 433 ], "spans": [ { "bbox": [ 54, 422, 62, 431 ], "score": 0.38, "content": "\\mathbf { v }", "type": "inline_equation" }, { "bbox": [ 62, 420, 230, 433 ], "score": 1.0, "content": "and a label-independent binary classifier", "type": "text" }, { "bbox": [ 231, 420, 252, 433 ], "score": 0.92, "content": "\\varphi ( x )", "type": "inline_equation" }, { "bbox": [ 252, 420, 290, 433 ], "score": 1.0, "content": "as in (6)", "type": "text" } ], "index": 27 }, { "bbox": [ 54, 433, 291, 444 ], "spans": [ { "bbox": [ 54, 433, 291, 444 ], "score": 1.0, "content": "solves this problem. Base classifiers are trained as usual at", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 444, 290, 455 ], "spans": [ { "bbox": [ 53, 444, 290, 455 ], "score": 1.0, "content": "each new tree leaf. 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Since", "type": "text" } ], "index": 77 }, { "bbox": [ 306, 517, 543, 529 ], "spans": [ { "bbox": [ 306, 517, 354, 529 ], "score": 1.0, "content": "the data set", "type": "text" }, { "bbox": [ 354, 517, 385, 529 ], "score": 0.92, "content": "( \\mathbf { X } , \\mathbf { Y } )", "type": "inline_equation" }, { "bbox": [ 385, 517, 543, 529 ], "score": 1.0, "content": "is different at each node, we include it", "type": "text" } ], "index": 78 }, { "bbox": [ 305, 529, 520, 542 ], "spans": [ { "bbox": [ 305, 529, 520, 542 ], "score": 1.0, "content": "explicitly in the argument of the full multi-class edge", "type": "text" } ], "index": 79 } ], "index": 72, "bbox_fs": [ 305, 362, 543, 542 ] }, { "type": "interline_equation", "bbox": [ 307, 549, 540, 585 ], "lines": [ { "bbox": [ 307, 549, 540, 585 ], "spans": [ { "bbox": [ 307, 549, 540, 585 ], "score": 0.93, "content": "\\gamma ( \\mathbf { v } , \\varphi , \\mathbf { X } , \\mathbf { Y } , \\mathbf { W } ) = \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } \\mathbb { I } \\left\\{ x _ { i } \\in \\mathbf { X } \\right\\} w _ { i , \\ell } v _ { \\ell } \\varphi ( \\mathbf { x } _ { i } ) y _ { i , \\ell } .", "type": "interline_equation", "image_path": "4be1759aeb131b39ae7d4209eedba8bd51d57cbf366037b0d941f0769edc0a74.jpg" } ] } ], "index": 81, "virtual_lines": [ { "bbox": [ 307, 549, 540, 561.0 ], "spans": [], "index": 80 }, { "bbox": [ 307, 561.0, 540, 573.0 ], "spans": [], "index": 81 }, { "bbox": [ 307, 573.0, 540, 585.0 ], "spans": [], "index": 82 } ] }, { "type": "text", "bbox": [ 307, 592, 542, 689 ], "lines": [ { "bbox": [ 304, 592, 542, 605 ], "spans": [ { "bbox": [ 304, 592, 542, 605 ], "score": 1.0, "content": "Note that in this definition we do not require that the", "type": "text" } ], "index": 83 }, { "bbox": [ 305, 605, 542, 617 ], "spans": [ { "bbox": [ 305, 605, 542, 617 ], "score": 1.0, "content": "weights of the selected points add up to 1. 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This is a crucial property: it assures that the edge of", "type": "text" } ], "index": 89 }, { "bbox": [ 306, 677, 542, 689 ], "spans": [ { "bbox": [ 306, 677, 542, 689 ], "score": 1.0, "content": "the tree is always positive as long as the local edges in the", "type": "text" } ], "index": 90 }, { "bbox": [ 53, 66, 289, 81 ], "spans": [ { "bbox": [ 53, 66, 268, 81 ], "score": 1.0, "content": "inner nodes are positive, so any weak binary classifier", "type": "text", "cross_page": true }, { "bbox": [ 268, 68, 289, 80 ], "score": 0.89, "content": "\\phi ( \\mathbf { x } )", "type": "inline_equation", "cross_page": true } ], "index": 0 }, { "bbox": [ 53, 79, 257, 92 ], "spans": [ { "bbox": [ 53, 79, 257, 92 ], "score": 1.0, "content": "can be used to define the inner cuts and the leaves.", "type": "text", "cross_page": true } ], "index": 1 } ], "index": 86.5, "bbox_fs": [ 304, 592, 543, 689 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 54, 68, 289, 91 ], "lines": [ { "bbox": [ 53, 66, 289, 81 ], "spans": [ { "bbox": [ 53, 66, 268, 81 ], "score": 1.0, "content": "inner nodes are positive, so any weak binary classifier", "type": "text" }, { "bbox": [ 268, 68, 289, 80 ], "score": 0.89, "content": "\\phi ( \\mathbf { x } )", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 53, 79, 257, 92 ], "spans": [ { "bbox": [ 53, 79, 257, 92 ], "score": 1.0, "content": "can be used to define the inner cuts and the leaves.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 54, 97, 289, 169 ], "lines": [ { "bbox": [ 53, 97, 290, 109 ], "spans": [ { "bbox": [ 53, 97, 290, 109 ], "score": 1.0, "content": "The basic operation when adding a tree node with a scalar", "type": "text" } ], "index": 2 }, { "bbox": [ 54, 109, 290, 121 ], "spans": [ { "bbox": [ 54, 110, 141, 121 ], "score": 1.0, "content": "binary classifier (cut)", "type": "text" }, { "bbox": [ 141, 111, 150, 121 ], "score": 0.82, "content": "\\varphi", "type": "inline_equation" }, { "bbox": [ 150, 110, 277, 121 ], "score": 1.0, "content": "is to separate the data matrices", "type": "text" }, { "bbox": [ 277, 109, 286, 120 ], "score": 0.3, "content": "\\mathbf { X }", "type": "inline_equation" }, { "bbox": [ 287, 110, 290, 121 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 3 }, { "bbox": [ 54, 120, 290, 134 ], "spans": [ { "bbox": [ 54, 121, 64, 132 ], "score": 0.42, "content": "\\mathbf { Y }", "type": "inline_equation" }, { "bbox": [ 64, 120, 250, 134 ], "score": 1.0, "content": ", and W according to the sign of classification", "type": "text" }, { "bbox": [ 250, 121, 275, 133 ], "score": 0.93, "content": "\\varphi ( \\mathbf { x } _ { i } )", "type": "inline_equation" }, { "bbox": [ 275, 120, 290, 134 ], "score": 1.0, "content": "for", "type": "text" } ], "index": 4 }, { "bbox": [ 54, 133, 290, 145 ], "spans": [ { "bbox": [ 54, 134, 66, 145 ], "score": 1.0, "content": "all", "type": "text" }, { "bbox": [ 66, 133, 98, 145 ], "score": 0.91, "content": "\\mathbf { x } _ { i } \\in \\mathbf { X }", "type": "inline_equation" }, { "bbox": [ 99, 134, 290, 145 ], "score": 1.0, "content": ". The pseudocode is straightforward, but for the", "type": "text" } ], "index": 5 }, { "bbox": [ 53, 145, 290, 159 ], "spans": [ { "bbox": [ 53, 145, 290, 159 ], "score": 1.0, "content": "sake of completeness, we include it in the supplementary", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 157, 153, 170 ], "spans": [ { "bbox": [ 53, 157, 153, 170 ], "score": 1.0, "content": "(Appendix C, Figure 5).", "type": "text" } ], "index": 7 } ], "index": 4.5 }, { "type": "text", "bbox": [ 54, 174, 289, 414 ], "lines": [ { "bbox": [ 54, 175, 290, 188 ], "spans": [ { "bbox": [ 54, 175, 290, 188 ], "score": 1.0, "content": "Building a tree is usually described in a recursive way but", "type": "text" } ], "index": 8 }, { "bbox": [ 53, 187, 291, 199 ], "spans": [ { "bbox": [ 53, 187, 291, 199 ], "score": 1.0, "content": "we find the iterative procedure easier to explain, so our", "type": "text" } ], "index": 9 }, { "bbox": [ 52, 199, 291, 211 ], "spans": [ { "bbox": [ 52, 199, 291, 211 ], "score": 1.0, "content": "pseudocode in Figure 2 contains this version. The main", "type": "text" } ], "index": 10 }, { "bbox": [ 53, 211, 290, 223 ], "spans": [ { "bbox": [ 53, 211, 290, 223 ], "score": 1.0, "content": "idea is to maintain a priority queue, a data structure that", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 223, 290, 235 ], "spans": [ { "bbox": [ 53, 223, 290, 235 ], "score": 1.0, "content": "allows inserting objects with numerical keys into a set, and", "type": "text" } ], "index": 12 }, { "bbox": [ 53, 235, 290, 248 ], "spans": [ { "bbox": [ 53, 235, 290, 248 ], "score": 1.0, "content": "extracting the object with the maximum key (Cormen et al.,", "type": "text" } ], "index": 13 }, { "bbox": [ 53, 246, 290, 260 ], "spans": [ { "bbox": [ 53, 246, 290, 260 ], "score": 1.0, "content": "2009). The key will represent the improvement of the edge", "type": "text" } ], "index": 14 }, { "bbox": [ 54, 259, 290, 270 ], "spans": [ { "bbox": [ 54, 259, 290, 270 ], "score": 1.0, "content": "when cutting a leaf. We first call the base learner on the", "type": "text" } ], "index": 15 }, { "bbox": [ 53, 271, 290, 283 ], "spans": [ { "bbox": [ 53, 271, 290, 283 ], "score": 1.0, "content": "full data set (line 1) and insert it into the priority queue", "type": "text" } ], "index": 16 }, { "bbox": [ 53, 282, 291, 295 ], "spans": [ { "bbox": [ 53, 283, 109, 295 ], "score": 1.0, "content": "with its edge", "type": "text" }, { "bbox": [ 110, 282, 184, 295 ], "score": 0.92, "content": "\\gamma ( \\mathbf { v } , \\varphi , \\mathbf { X } , \\mathbf { Y } , \\mathbf { W } )", "type": "inline_equation" }, { "bbox": [ 184, 283, 291, 295 ], "score": 1.0, "content": "(line 3) as the key. Then", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 295, 291, 307 ], "spans": [ { "bbox": [ 53, 295, 291, 307 ], "score": 1.0, "content": "in each iteration, we extract the leaf that would provide the", "type": "text" } ], "index": 18 }, { "bbox": [ 53, 307, 290, 319 ], "spans": [ { "bbox": [ 53, 307, 290, 319 ], "score": 1.0, "content": "best edge improvement among all the leaves in the priority", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 319, 290, 330 ], "spans": [ { "bbox": [ 53, 319, 290, 330 ], "score": 1.0, "content": "queue (line 7), we partition the data set (line 11), call the", "type": "text" } ], "index": 20 }, { "bbox": [ 53, 330, 290, 343 ], "spans": [ { "bbox": [ 53, 330, 290, 343 ], "score": 1.0, "content": "base learners on the two new leaves (line 12), and insert", "type": "text" } ], "index": 21 }, { "bbox": [ 53, 342, 291, 356 ], "spans": [ { "bbox": [ 53, 342, 291, 356 ], "score": 1.0, "content": "them into the priority queue using the difference between", "type": "text" } ], "index": 22 }, { "bbox": [ 53, 354, 290, 367 ], "spans": [ { "bbox": [ 53, 354, 290, 367 ], "score": 1.0, "content": "the old edge on the partitioned data sets and the new edges", "type": "text" } ], "index": 23 }, { "bbox": [ 54, 366, 290, 379 ], "spans": [ { "bbox": [ 54, 366, 290, 379 ], "score": 1.0, "content": "of the base classifiers in the two new leaves (line 13). When", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 379, 290, 391 ], "spans": [ { "bbox": [ 53, 379, 290, 391 ], "score": 1.0, "content": "inserting a leaf into the queue, we also save the sign of the", "type": "text" } ], "index": 25 }, { "bbox": [ 53, 391, 290, 403 ], "spans": [ { "bbox": [ 53, 391, 290, 403 ], "score": 1.0, "content": "cut (left or right child) and the index of the parent, so the", "type": "text" } ], "index": 26 }, { "bbox": [ 53, 402, 254, 414 ], "spans": [ { "bbox": [ 53, 402, 132, 414 ], "score": 1.0, "content": "index vectors l and", "type": "text" }, { "bbox": [ 132, 404, 138, 412 ], "score": 0.3, "content": "\\mathfrak { r }", "type": "inline_equation" }, { "bbox": [ 139, 402, 254, 414 ], "score": 1.0, "content": "can be set properly in line 8.", "type": "text" } ], "index": 27 } ], "index": 17.5 }, { "type": "text", "bbox": [ 54, 420, 289, 552 ], "lines": [ { "bbox": [ 53, 419, 290, 433 ], "spans": [ { "bbox": [ 53, 419, 290, 433 ], "score": 1.0, "content": "When the priority queue is implemented as a heap, both", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 432, 290, 444 ], "spans": [ { "bbox": [ 53, 432, 290, 444 ], "score": 1.0, "content": "the insertion and the extraction of the maximum takes", "type": "text" } ], "index": 29 }, { "bbox": [ 54, 443, 291, 457 ], "spans": [ { "bbox": [ 54, 444, 94, 456 ], "score": 0.92, "content": "{ \\cal O } ( \\log N )", "type": "inline_equation" }, { "bbox": [ 95, 443, 291, 457 ], "score": 1.0, "content": "time (Cormen et al., 2009), so the total running", "type": "text" } ], "index": 30 }, { "bbox": [ 53, 456, 290, 469 ], "spans": [ { "bbox": [ 53, 456, 150, 468 ], "score": 1.0, "content": "time of the procedure is", "type": "text" }, { "bbox": [ 151, 456, 259, 469 ], "score": 0.89, "content": "O \\big ( N ( T _ { \\mathrm { B a s E } } + n + \\log N ) \\big )", "type": "inline_equation" }, { "bbox": [ 259, 456, 290, 468 ], "score": 1.0, "content": ", where", "type": "text" } ], "index": 31 }, { "bbox": [ 54, 468, 291, 480 ], "spans": [ { "bbox": [ 54, 468, 78, 480 ], "score": 0.91, "content": "T _ { \\mathrm { B A S E } }", "type": "inline_equation" }, { "bbox": [ 78, 468, 259, 480 ], "score": 1.0, "content": "is the running time of the base learner. Since", "type": "text" }, { "bbox": [ 259, 469, 270, 478 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 270, 468, 291, 480 ], "score": 1.0, "content": "can-", "type": "text" } ], "index": 32 }, { "bbox": [ 52, 479, 290, 493 ], "spans": [ { "bbox": [ 52, 479, 123, 493 ], "score": 1.0, "content": "not be more than", "type": "text" }, { "bbox": [ 123, 482, 131, 491 ], "score": 0.73, "content": "n", "type": "inline_equation" }, { "bbox": [ 131, 479, 212, 493 ], "score": 1.0, "content": ", the running time is", "type": "text" }, { "bbox": [ 212, 479, 286, 493 ], "score": 0.92, "content": "O \\big ( N ( T _ { \\mathrm { B a S E } } + n ) \\big )", "type": "inline_equation" }, { "bbox": [ 286, 479, 290, 493 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 33 }, { "bbox": [ 52, 491, 290, 505 ], "spans": [ { "bbox": [ 52, 491, 290, 505 ], "score": 1.0, "content": "If the base learners cutting the leaves are decision stumps,", "type": "text" } ], "index": 34 }, { "bbox": [ 52, 503, 291, 517 ], "spans": [ { "bbox": [ 52, 503, 154, 517 ], "score": 1.0, "content": "the total running time is", "type": "text" }, { "bbox": [ 155, 504, 200, 516 ], "score": 0.93, "content": "O ( n K d N )", "type": "inline_equation" }, { "bbox": [ 201, 503, 291, 517 ], "score": 1.0, "content": ". In the procedure we", "type": "text" } ], "index": 35 }, { "bbox": [ 52, 515, 291, 528 ], "spans": [ { "bbox": [ 52, 515, 291, 528 ], "score": 1.0, "content": "have no explicit control over the shape of the tree, but if", "type": "text" } ], "index": 36 }, { "bbox": [ 52, 527, 291, 540 ], "spans": [ { "bbox": [ 52, 527, 291, 540 ], "score": 1.0, "content": "it happens to be balanced, the running time can further be", "type": "text" } ], "index": 37 }, { "bbox": [ 53, 540, 171, 553 ], "spans": [ { "bbox": [ 53, 540, 105, 553 ], "score": 1.0, "content": "improved to", "type": "text" }, { "bbox": [ 105, 540, 167, 552 ], "score": 0.92, "content": "O ( n K d \\log N )", "type": "inline_equation" }, { "bbox": [ 167, 540, 171, 553 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 38 } ], "index": 33 }, { "type": "title", "bbox": [ 55, 567, 132, 580 ], "lines": [ { "bbox": [ 52, 565, 135, 583 ], "spans": [ { "bbox": [ 52, 565, 135, 583 ], "score": 1.0, "content": "4. Experiments", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 55, 587, 289, 671 ], "lines": [ { "bbox": [ 53, 587, 290, 600 ], "spans": [ { "bbox": [ 53, 587, 290, 600 ], "score": 1.0, "content": "Full reproducibility was one of the key motivations when", "type": "text" } ], "index": 40 }, { "bbox": [ 53, 600, 290, 612 ], "spans": [ { "bbox": [ 53, 600, 290, 612 ], "score": 1.0, "content": "we designed our experimental setup. All experiments were", "type": "text" } ], "index": 41 }, { "bbox": [ 54, 612, 290, 623 ], "spans": [ { "bbox": [ 54, 612, 290, 623 ], "score": 1.0, "content": "done using the open source multiboost software of Ben-", "type": "text" } ], "index": 42 }, { "bbox": [ 54, 624, 290, 635 ], "spans": [ { "bbox": [ 54, 624, 290, 635 ], "score": 1.0, "content": "bouzid et al. (2012), version 1.2. In addition, we will make", "type": "text" } ], "index": 43 }, { "bbox": [ 53, 636, 291, 648 ], "spans": [ { "bbox": [ 53, 636, 291, 648 ], "score": 1.0, "content": "public all the configuration files, train/test/validation cuts,", "type": "text" } ], "index": 44 }, { "bbox": [ 53, 648, 290, 660 ], "spans": [ { "bbox": [ 53, 648, 290, 660 ], "score": 1.0, "content": "and the scripts that we used to set up the hyperparameter", "type": "text" } ], "index": 45 }, { "bbox": [ 54, 660, 98, 671 ], "spans": [ { "bbox": [ 54, 660, 98, 671 ], "score": 1.0, "content": "validation.", "type": "text" } ], "index": 46 } ], "index": 43 }, { "type": "text", "bbox": [ 54, 677, 289, 713 ], "lines": [ { "bbox": [ 53, 676, 290, 690 ], "spans": [ { "bbox": [ 53, 676, 290, 690 ], "score": 1.0, "content": "We carried out experiments on five mid-sized (isolet,", "type": "text" } ], "index": 47 }, { "bbox": [ 53, 689, 290, 702 ], "spans": [ { "bbox": [ 53, 689, 290, 702 ], "score": 1.0, "content": "letter, optdigits, pendigits, and USPS) and nine small", "type": "text" } ], "index": 48 }, { "bbox": [ 53, 700, 291, 714 ], "spans": [ { "bbox": [ 53, 700, 291, 714 ], "score": 1.0, "content": "(balance, blood, wdbc, breast, ecoli, iris, pima, sonar,", "type": "text" } ], "index": 49 } ], "index": 48 }, { "type": "text", "bbox": [ 307, 68, 541, 222 ], "lines": [ { "bbox": [ 305, 66, 543, 80 ], "spans": [ { "bbox": [ 305, 66, 514, 80 ], "score": 1.0, "content": "and wine) data sets from the UCI repository.", "type": "text" }, { "bbox": [ 519, 66, 543, 79 ], "score": 1.0, "content": "The", "type": "text" } ], "index": 50 }, { "bbox": [ 305, 79, 543, 91 ], "spans": [ { "bbox": [ 305, 79, 543, 91 ], "score": 1.0, "content": "five sets were chosen to overlap with the selections of", "type": "text" } ], "index": 51 }, { "bbox": [ 304, 90, 543, 104 ], "spans": [ { "bbox": [ 304, 90, 543, 104 ], "score": 1.0, "content": "most of the recent multi-class boosting papers (Kegl & ´", "type": "text" } ], "index": 52 }, { "bbox": [ 305, 102, 542, 116 ], "spans": [ { "bbox": [ 305, 102, 542, 116 ], "score": 1.0, "content": "Busa-Fekete, 2009; Li, 2009a;b; Zhu et al., 2009; Sun", "type": "text" } ], "index": 53 }, { "bbox": [ 305, 115, 542, 128 ], "spans": [ { "bbox": [ 305, 115, 542, 128 ], "score": 1.0, "content": "et al., 2012; Mukherjee & Schapire, 2013), The small", "type": "text" } ], "index": 54 }, { "bbox": [ 305, 127, 542, 140 ], "spans": [ { "bbox": [ 305, 127, 542, 140 ], "score": 1.0, "content": "data sets were selected for comparing ADABOOST.MH", "type": "text" } ], "index": 55 }, { "bbox": [ 306, 139, 542, 150 ], "spans": [ { "bbox": [ 306, 139, 542, 150 ], "score": 1.0, "content": "with SVMs using Gaussian kernels, taking the results", "type": "text" } ], "index": 56 }, { "bbox": [ 306, 151, 542, 162 ], "spans": [ { "bbox": [ 306, 151, 542, 162 ], "score": 1.0, "content": "of a recent paper (Duch et al., 2012) whose experimen-", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 163, 542, 174 ], "spans": [ { "bbox": [ 305, 163, 542, 174 ], "score": 1.0, "content": "tal setup we adopted. All numerical results (multi-class", "type": "text" } ], "index": 58 }, { "bbox": [ 305, 173, 542, 187 ], "spans": [ { "bbox": [ 305, 174, 351, 187 ], "score": 1.0, "content": "test errors", "type": "text" }, { "bbox": [ 351, 173, 376, 187 ], "score": 0.84, "content": "\\widehat { R } _ { \\mathbb { I } } ( \\mathbf { f } )", "type": "inline_equation" }, { "bbox": [ 376, 174, 542, 187 ], "score": 1.0, "content": "(1) and test learning curves) are avail-", "type": "text" } ], "index": 59 }, { "bbox": [ 305, 186, 541, 199 ], "spans": [ { "bbox": [ 305, 186, 541, 199 ], "score": 1.0, "content": "able at https://www.lri.fr/˜kegl/research/", "type": "text" } ], "index": 60 }, { "bbox": [ 305, 199, 542, 212 ], "spans": [ { "bbox": [ 305, 199, 542, 212 ], "score": 1.0, "content": "multiboostResults.pdf, one experiment per page", "type": "text" } ], "index": 61 }, { "bbox": [ 305, 210, 542, 223 ], "spans": [ { "bbox": [ 305, 210, 542, 223 ], "score": 1.0, "content": "for clarity. Tables 1 and 2 contain summaries of the results.", "type": "text" } ], "index": 62 } ], "index": 56 }, { "type": "table", "bbox": [ 349, 234, 498, 337 ], "blocks": [ { "type": "table_body", "bbox": [ 349, 234, 498, 337 ], "group_id": 0, "lines": [ { "bbox": [ 349, 234, 498, 337 ], "spans": [ { "bbox": [ 349, 234, 498, 337 ], "score": 0.956, "html": "
AB.MHSVM
balance6.0±4.010.0±2.0
blood22.0±4.021.0± 5.0
wdbc3.0± 2.02.0±3.0
breast34.0 ±13.037.0±8.0
ecoli15.0±6.016.0±6.0
iris7.0±6.05.0±6.0
pima24.0±5.024.0±4.0
sonar13.0±10.014.0±8.0
wine2.0±3.03.0±4.0
", "type": "table", "image_path": "6c320c18c62cb527aa52b7174e1c627728143aa9d5cf94cbe968cb791b3259ae.jpg" } ] } ], "index": 63.5, "virtual_lines": [ { "bbox": [ 349, 234, 498, 285.5 ], "spans": [], "index": 63 }, { "bbox": [ 349, 285.5, 498, 337.0 ], "spans": [], "index": 64 } ] }, { "type": "table_caption", "bbox": [ 307, 345, 528, 356 ], "group_id": 0, "lines": [ { "bbox": [ 306, 345, 528, 357 ], "spans": [ { "bbox": [ 306, 345, 528, 357 ], "score": 1.0, "content": "Table 2. Test error percentages on small benchmark data sets.", "type": "text" } ], "index": 65 } ], "index": 65 } ], "index": 64.25 }, { "type": "text", "bbox": [ 307, 369, 541, 477 ], "lines": [ { "bbox": [ 305, 368, 543, 383 ], "spans": [ { "bbox": [ 305, 368, 543, 383 ], "score": 1.0, "content": "Hyperparameter optimization is largely swept under the rug", "type": "text" } ], "index": 66 }, { "bbox": [ 306, 381, 541, 393 ], "spans": [ { "bbox": [ 306, 381, 541, 393 ], "score": 1.0, "content": "in papers describing alternative multi-class boosting meth-", "type": "text" } ], "index": 67 }, { "bbox": [ 306, 393, 542, 405 ], "spans": [ { "bbox": [ 306, 393, 542, 405 ], "score": 1.0, "content": "ods. Some report results with fixed hyperparameters (Zhu", "type": "text" } ], "index": 68 }, { "bbox": [ 306, 406, 542, 416 ], "spans": [ { "bbox": [ 306, 406, 542, 416 ], "score": 1.0, "content": "et al., 2009; Sun et al., 2012) and others give the full table", "type": "text" } ], "index": 69 }, { "bbox": [ 305, 417, 542, 429 ], "spans": [ { "bbox": [ 305, 417, 542, 429 ], "score": 1.0, "content": "of test errors for a grid of hyperparameters (Kegl & Busa- ´", "type": "text" } ], "index": 70 }, { "bbox": [ 306, 429, 542, 441 ], "spans": [ { "bbox": [ 306, 429, 378, 441 ], "score": 1.0, "content": "Fekete, 2009; Li,", "type": "text" }, { "bbox": [ 378, 430, 412, 441 ], "score": 0.28, "content": "{ 2 0 0 9 \\mathrm { a } ; \\mathrm { b } }", "type": "inline_equation" }, { "bbox": [ 412, 429, 542, 441 ], "score": 1.0, "content": "; Mukherjee & Schapire, 2013).", "type": "text" } ], "index": 71 }, { "bbox": [ 306, 441, 542, 453 ], "spans": [ { "bbox": [ 306, 441, 542, 453 ], "score": 1.0, "content": "Although the following procedure is rather old, we feel the", "type": "text" } ], "index": 72 }, { "bbox": [ 305, 453, 542, 465 ], "spans": [ { "bbox": [ 305, 453, 542, 465 ], "score": 1.0, "content": "need to detail it for promoting a more scrupulous compari-", "type": "text" } ], "index": 73 }, { "bbox": [ 306, 466, 380, 477 ], "spans": [ { "bbox": [ 306, 466, 380, 477 ], "score": 1.0, "content": "son across papers.", "type": "text" } ], "index": 74 } ], "index": 70 }, { "type": "text", "bbox": [ 307, 483, 542, 627 ], "lines": [ { "bbox": [ 305, 482, 542, 495 ], "spans": [ { "bbox": [ 305, 482, 423, 495 ], "score": 1.0, "content": "For the small data sets we ran", "type": "text" }, { "bbox": [ 424, 483, 453, 494 ], "score": 0.89, "content": "1 0 \\times 1 0", "type": "inline_equation" }, { "bbox": [ 454, 482, 542, 495 ], "score": 1.0, "content": "cross-validation (CV)", "type": "text" } ], "index": 75 }, { "bbox": [ 305, 496, 542, 507 ], "spans": [ { "bbox": [ 305, 496, 542, 507 ], "score": 1.0, "content": "to optimize the hyperparameters and the estimate the gen-", "type": "text" } ], "index": 76 }, { "bbox": [ 306, 507, 543, 519 ], "spans": [ { "bbox": [ 306, 507, 543, 519 ], "score": 1.0, "content": "eralization error. For the number of inner nodes we do a", "type": "text" } ], "index": 77 }, { "bbox": [ 306, 519, 542, 531 ], "spans": [ { "bbox": [ 306, 519, 542, 531 ], "score": 1.0, "content": "grid search (we also considered using the “one sigma” rule", "type": "text" } ], "index": 78 }, { "bbox": [ 306, 532, 542, 542 ], "spans": [ { "bbox": [ 306, 532, 542, 542 ], "score": 1.0, "content": "for biasing the selection towards smaller trees, but the sim-", "type": "text" } ], "index": 79 }, { "bbox": [ 305, 543, 543, 555 ], "spans": [ { "bbox": [ 305, 543, 543, 555 ], "score": 1.0, "content": "ple minimization proved to be better). For robustly esti-", "type": "text" } ], "index": 80 }, { "bbox": [ 305, 555, 543, 567 ], "spans": [ { "bbox": [ 305, 555, 543, 567 ], "score": 1.0, "content": "mating the optimal stopping time we use a smoothed test", "type": "text" } ], "index": 81 }, { "bbox": [ 304, 565, 544, 580 ], "spans": [ { "bbox": [ 304, 565, 460, 580 ], "score": 1.0, "content": "error. For the formal description, let", "type": "text" }, { "bbox": [ 460, 565, 478, 577 ], "score": 0.9, "content": "\\widehat { R } ^ { ( t ) }", "type": "inline_equation" }, { "bbox": [ 479, 565, 544, 580 ], "score": 1.0, "content": "be the average", "type": "text" } ], "index": 82 }, { "bbox": [ 305, 579, 542, 590 ], "spans": [ { "bbox": [ 305, 579, 478, 590 ], "score": 1.0, "content": "test error (1) of the ten validation runs after", "type": "text" }, { "bbox": [ 478, 580, 484, 589 ], "score": 0.75, "content": "t", "type": "inline_equation" }, { "bbox": [ 484, 579, 542, 590 ], "score": 1.0, "content": "iterations. We", "type": "text" } ], "index": 83 }, { "bbox": [ 305, 591, 543, 603 ], "spans": [ { "bbox": [ 305, 591, 408, 603 ], "score": 1.0, "content": "run ADABOOST.MH for", "type": "text" }, { "bbox": [ 408, 591, 430, 602 ], "score": 0.91, "content": "T _ { \\mathrm { m a x } }", "type": "inline_equation" }, { "bbox": [ 430, 591, 543, 603 ], "score": 1.0, "content": "iterations, and compute the", "type": "text" } ], "index": 84 }, { "bbox": [ 306, 603, 542, 614 ], "spans": [ { "bbox": [ 306, 603, 542, 614 ], "score": 1.0, "content": "optimal stopping time using the minimum of the smoothed", "type": "text" } ], "index": 85 }, { "bbox": [ 306, 615, 538, 627 ], "spans": [ { "bbox": [ 306, 615, 538, 627 ], "score": 1.0, "content": "test error using a linearly growing sliding window, that is,", "type": "text" } ], "index": 86 } ], "index": 80.5 }, { "type": "interline_equation", "bbox": [ 317, 635, 514, 673 ], "lines": [ { "bbox": [ 317, 635, 514, 673 ], "spans": [ { "bbox": [ 317, 635, 514, 673 ], "score": 0.92, "content": "T ^ { * } = \\operatorname * { a r g m i n } _ { T : T _ { \\mathrm { m i n } } < T \\leq T _ { \\mathrm { m a x } } } \\frac { 1 } { T - \\lfloor 0 . 8 T \\rfloor } \\sum _ { t = \\lfloor 0 . 8 T \\rfloor } ^ { T } \\widehat { R } ^ { ( t ) } ,", "type": "interline_equation", "image_path": "562d41023d6bb36672a40809eea86f872f135f21bc0cade503027d368602d153.jpg" } ] } ], "index": 88, "virtual_lines": [ { "bbox": [ 317, 635, 514, 647.6666666666666 ], "spans": [], "index": 87 }, { "bbox": [ 317, 647.6666666666666, 514, 660.3333333333333 ], "spans": [], "index": 88 }, { "bbox": [ 317, 660.3333333333333, 514, 672.9999999999999 ], "spans": [], "index": 89 } ] }, { "type": "text", "bbox": [ 306, 682, 542, 717 ], "lines": [ { "bbox": [ 305, 681, 543, 695 ], "spans": [ { "bbox": [ 305, 681, 333, 695 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 333, 682, 354, 693 ], "score": 0.9, "content": "T _ { \\mathrm { m i n } }", "type": "inline_equation" }, { "bbox": [ 354, 681, 543, 695 ], "score": 1.0, "content": "was set to a constant 50 to avoid stopping too", "type": "text" } ], "index": 90 }, { "bbox": [ 306, 694, 542, 705 ], "spans": [ { "bbox": [ 306, 694, 542, 705 ], "score": 1.0, "content": "early due to fluctuations. 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Then", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 295, 291, 307 ], "spans": [ { "bbox": [ 53, 295, 291, 307 ], "score": 1.0, "content": "in each iteration, we extract the leaf that would provide the", "type": "text" } ], "index": 18 }, { "bbox": [ 53, 307, 290, 319 ], "spans": [ { "bbox": [ 53, 307, 290, 319 ], "score": 1.0, "content": "best edge improvement among all the leaves in the priority", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 319, 290, 330 ], "spans": [ { "bbox": [ 53, 319, 290, 330 ], "score": 1.0, "content": "queue (line 7), we partition the data set (line 11), call the", "type": "text" } ], "index": 20 }, { "bbox": [ 53, 330, 290, 343 ], "spans": [ { "bbox": [ 53, 330, 290, 343 ], "score": 1.0, "content": "base learners on the two new leaves (line 12), and insert", "type": "text" } ], "index": 21 }, { "bbox": [ 53, 342, 291, 356 ], "spans": [ { "bbox": [ 53, 342, 291, 356 ], "score": 1.0, "content": "them into the priority queue using the difference between", "type": "text" } ], "index": 22 }, { "bbox": [ 53, 354, 290, 367 ], "spans": [ { "bbox": [ 53, 354, 290, 367 ], "score": 1.0, "content": "the old edge on the partitioned data sets and the new edges", "type": "text" } ], "index": 23 }, { "bbox": [ 54, 366, 290, 379 ], "spans": [ { "bbox": [ 54, 366, 290, 379 ], "score": 1.0, "content": "of the base classifiers in the two new leaves (line 13). When", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 379, 290, 391 ], "spans": [ { "bbox": [ 53, 379, 290, 391 ], "score": 1.0, "content": "inserting a leaf into the queue, we also save the sign of the", "type": "text" } ], "index": 25 }, { "bbox": [ 53, 391, 290, 403 ], "spans": [ { "bbox": [ 53, 391, 290, 403 ], "score": 1.0, "content": "cut (left or right child) and the index of the parent, so the", "type": "text" } ], "index": 26 }, { "bbox": [ 53, 402, 254, 414 ], "spans": [ { "bbox": [ 53, 402, 132, 414 ], "score": 1.0, "content": "index vectors l and", "type": "text" }, { "bbox": [ 132, 404, 138, 412 ], "score": 0.3, "content": "\\mathfrak { r }", "type": "inline_equation" }, { "bbox": [ 139, 402, 254, 414 ], "score": 1.0, "content": "can be set properly in line 8.", "type": "text" } ], "index": 27 } ], "index": 17.5, "bbox_fs": [ 52, 175, 291, 414 ] }, { "type": "text", "bbox": [ 54, 420, 289, 552 ], "lines": [ { "bbox": [ 53, 419, 290, 433 ], "spans": [ { "bbox": [ 53, 419, 290, 433 ], "score": 1.0, "content": "When the priority queue is implemented as a heap, both", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 432, 290, 444 ], "spans": [ { "bbox": [ 53, 432, 290, 444 ], "score": 1.0, "content": "the insertion and the extraction of the maximum takes", "type": "text" } ], "index": 29 }, { "bbox": [ 54, 443, 291, 457 ], "spans": [ { "bbox": [ 54, 444, 94, 456 ], "score": 0.92, "content": "{ \\cal O } ( \\log N )", "type": "inline_equation" }, { "bbox": [ 95, 443, 291, 457 ], "score": 1.0, "content": "time (Cormen et al., 2009), so the total running", "type": "text" } ], "index": 30 }, { "bbox": [ 53, 456, 290, 469 ], "spans": [ { "bbox": [ 53, 456, 150, 468 ], "score": 1.0, "content": "time of the procedure is", "type": "text" }, { "bbox": [ 151, 456, 259, 469 ], "score": 0.89, "content": "O \\big ( N ( T _ { \\mathrm { B a s E } } + n + \\log N ) \\big )", "type": "inline_equation" }, { "bbox": [ 259, 456, 290, 468 ], "score": 1.0, "content": ", where", "type": "text" } ], "index": 31 }, { "bbox": [ 54, 468, 291, 480 ], "spans": [ { "bbox": [ 54, 468, 78, 480 ], "score": 0.91, "content": "T _ { \\mathrm { B A S E } }", "type": "inline_equation" }, { "bbox": [ 78, 468, 259, 480 ], "score": 1.0, "content": "is the running time of the base learner. Since", "type": "text" }, { "bbox": [ 259, 469, 270, 478 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 270, 468, 291, 480 ], "score": 1.0, "content": "can-", "type": "text" } ], "index": 32 }, { "bbox": [ 52, 479, 290, 493 ], "spans": [ { "bbox": [ 52, 479, 123, 493 ], "score": 1.0, "content": "not be more than", "type": "text" }, { "bbox": [ 123, 482, 131, 491 ], "score": 0.73, "content": "n", "type": "inline_equation" }, { "bbox": [ 131, 479, 212, 493 ], "score": 1.0, "content": ", the running time is", "type": "text" }, { "bbox": [ 212, 479, 286, 493 ], "score": 0.92, "content": "O \\big ( N ( T _ { \\mathrm { B a S E } } + n ) \\big )", "type": "inline_equation" }, { "bbox": [ 286, 479, 290, 493 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 33 }, { "bbox": [ 52, 491, 290, 505 ], "spans": [ { "bbox": [ 52, 491, 290, 505 ], "score": 1.0, "content": "If the base learners cutting the leaves are decision stumps,", "type": "text" } ], "index": 34 }, { "bbox": [ 52, 503, 291, 517 ], "spans": [ { "bbox": [ 52, 503, 154, 517 ], "score": 1.0, "content": "the total running time is", "type": "text" }, { "bbox": [ 155, 504, 200, 516 ], "score": 0.93, "content": "O ( n K d N )", "type": "inline_equation" }, { "bbox": [ 201, 503, 291, 517 ], "score": 1.0, "content": ". In the procedure we", "type": "text" } ], "index": 35 }, { "bbox": [ 52, 515, 291, 528 ], "spans": [ { "bbox": [ 52, 515, 291, 528 ], "score": 1.0, "content": "have no explicit control over the shape of the tree, but if", "type": "text" } ], "index": 36 }, { "bbox": [ 52, 527, 291, 540 ], "spans": [ { "bbox": [ 52, 527, 291, 540 ], "score": 1.0, "content": "it happens to be balanced, the running time can further be", "type": "text" } ], "index": 37 }, { "bbox": [ 53, 540, 171, 553 ], "spans": [ { "bbox": [ 53, 540, 105, 553 ], "score": 1.0, "content": "improved to", "type": "text" }, { "bbox": [ 105, 540, 167, 552 ], "score": 0.92, "content": "O ( n K d \\log N )", "type": "inline_equation" }, { "bbox": [ 167, 540, 171, 553 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 38 } ], "index": 33, "bbox_fs": [ 52, 419, 291, 553 ] }, { "type": "title", "bbox": [ 55, 567, 132, 580 ], "lines": [ { "bbox": [ 52, 565, 135, 583 ], "spans": [ { "bbox": [ 52, 565, 135, 583 ], "score": 1.0, "content": "4. Experiments", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 55, 587, 289, 671 ], "lines": [ { "bbox": [ 53, 587, 290, 600 ], "spans": [ { "bbox": [ 53, 587, 290, 600 ], "score": 1.0, "content": "Full reproducibility was one of the key motivations when", "type": "text" } ], "index": 40 }, { "bbox": [ 53, 600, 290, 612 ], "spans": [ { "bbox": [ 53, 600, 290, 612 ], "score": 1.0, "content": "we designed our experimental setup. All experiments were", "type": "text" } ], "index": 41 }, { "bbox": [ 54, 612, 290, 623 ], "spans": [ { "bbox": [ 54, 612, 290, 623 ], "score": 1.0, "content": "done using the open source multiboost software of Ben-", "type": "text" } ], "index": 42 }, { "bbox": [ 54, 624, 290, 635 ], "spans": [ { "bbox": [ 54, 624, 290, 635 ], "score": 1.0, "content": "bouzid et al. (2012), version 1.2. In addition, we will make", "type": "text" } ], "index": 43 }, { "bbox": [ 53, 636, 291, 648 ], "spans": [ { "bbox": [ 53, 636, 291, 648 ], "score": 1.0, "content": "public all the configuration files, train/test/validation cuts,", "type": "text" } ], "index": 44 }, { "bbox": [ 53, 648, 290, 660 ], "spans": [ { "bbox": [ 53, 648, 290, 660 ], "score": 1.0, "content": "and the scripts that we used to set up the hyperparameter", "type": "text" } ], "index": 45 }, { "bbox": [ 54, 660, 98, 671 ], "spans": [ { "bbox": [ 54, 660, 98, 671 ], "score": 1.0, "content": "validation.", "type": "text" } ], "index": 46 } ], "index": 43, "bbox_fs": [ 53, 587, 291, 671 ] }, { "type": "text", "bbox": [ 54, 677, 289, 713 ], "lines": [ { "bbox": [ 53, 676, 290, 690 ], "spans": [ { "bbox": [ 53, 676, 290, 690 ], "score": 1.0, "content": "We carried out experiments on five mid-sized (isolet,", "type": "text" } ], "index": 47 }, { "bbox": [ 53, 689, 290, 702 ], "spans": [ { "bbox": [ 53, 689, 290, 702 ], "score": 1.0, "content": "letter, optdigits, pendigits, and USPS) and nine small", "type": "text" } ], "index": 48 }, { "bbox": [ 53, 700, 291, 714 ], "spans": [ { "bbox": [ 53, 700, 291, 714 ], "score": 1.0, "content": "(balance, blood, wdbc, breast, ecoli, iris, pima, sonar,", "type": "text" } ], "index": 49 }, { "bbox": [ 305, 66, 543, 80 ], "spans": [ { "bbox": [ 305, 66, 514, 80 ], "score": 1.0, "content": "and wine) data sets from the UCI repository.", "type": "text" }, { "bbox": [ 519, 66, 543, 79 ], "score": 1.0, "content": "The", "type": "text" } ], "index": 50 }, { "bbox": [ 305, 79, 543, 91 ], "spans": [ { "bbox": [ 305, 79, 543, 91 ], "score": 1.0, "content": "five sets were chosen to overlap with the selections of", "type": "text" } ], "index": 51 }, { "bbox": [ 304, 90, 543, 104 ], "spans": [ { "bbox": [ 304, 90, 543, 104 ], "score": 1.0, "content": "most of the recent multi-class boosting papers (Kegl & ´", "type": "text" } ], "index": 52 }, { "bbox": [ 305, 102, 542, 116 ], "spans": [ { "bbox": [ 305, 102, 542, 116 ], "score": 1.0, "content": "Busa-Fekete, 2009; Li, 2009a;b; Zhu et al., 2009; Sun", "type": "text" } ], "index": 53 }, { "bbox": [ 305, 115, 542, 128 ], "spans": [ { "bbox": [ 305, 115, 542, 128 ], "score": 1.0, "content": "et al., 2012; Mukherjee & Schapire, 2013), The small", "type": "text" } ], "index": 54 }, { "bbox": [ 305, 127, 542, 140 ], "spans": [ { "bbox": [ 305, 127, 542, 140 ], "score": 1.0, "content": "data sets were selected for comparing ADABOOST.MH", "type": "text" } ], "index": 55 }, { "bbox": [ 306, 139, 542, 150 ], "spans": [ { "bbox": [ 306, 139, 542, 150 ], "score": 1.0, "content": "with SVMs using Gaussian kernels, taking the results", "type": "text" } ], "index": 56 }, { "bbox": [ 306, 151, 542, 162 ], "spans": [ { "bbox": [ 306, 151, 542, 162 ], "score": 1.0, "content": "of a recent paper (Duch et al., 2012) whose experimen-", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 163, 542, 174 ], "spans": [ { "bbox": [ 305, 163, 542, 174 ], "score": 1.0, "content": "tal setup we adopted. 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Tables 1 and 2 contain summaries of the results.", "type": "text" } ], "index": 62 } ], "index": 48, "bbox_fs": [ 53, 676, 291, 714 ] }, { "type": "text", "bbox": [ 307, 68, 541, 222 ], "lines": [], "index": 56, "bbox_fs": [ 304, 66, 543, 223 ], "lines_deleted": true }, { "type": "table", "bbox": [ 349, 234, 498, 337 ], "blocks": [ { "type": "table_body", "bbox": [ 349, 234, 498, 337 ], "group_id": 0, "lines": [ { "bbox": [ 349, 234, 498, 337 ], "spans": [ { "bbox": [ 349, 234, 498, 337 ], "score": 0.956, "html": "
AB.MHSVM
balance6.0±4.010.0±2.0
blood22.0±4.021.0± 5.0
wdbc3.0± 2.02.0±3.0
breast34.0 ±13.037.0±8.0
ecoli15.0±6.016.0±6.0
iris7.0±6.05.0±6.0
pima24.0±5.024.0±4.0
sonar13.0±10.014.0±8.0
wine2.0±3.03.0±4.0
", "type": "table", "image_path": "6c320c18c62cb527aa52b7174e1c627728143aa9d5cf94cbe968cb791b3259ae.jpg" } ] } ], "index": 63.5, "virtual_lines": [ { "bbox": [ 349, 234, 498, 285.5 ], "spans": [], "index": 63 }, { "bbox": [ 349, 285.5, 498, 337.0 ], "spans": [], "index": 64 } ] }, { "type": "table_caption", "bbox": [ 307, 345, 528, 356 ], "group_id": 0, "lines": [ { "bbox": [ 306, 345, 528, 357 ], "spans": [ { "bbox": [ 306, 345, 528, 357 ], "score": 1.0, "content": "Table 2. Test error percentages on small benchmark data sets.", "type": "text" } ], "index": 65 } ], "index": 65 } ], "index": 64.25 }, { "type": "text", "bbox": [ 307, 369, 541, 477 ], "lines": [ { "bbox": [ 305, 368, 543, 383 ], "spans": [ { "bbox": [ 305, 368, 543, 383 ], "score": 1.0, "content": "Hyperparameter optimization is largely swept under the rug", "type": "text" } ], "index": 66 }, { "bbox": [ 306, 381, 541, 393 ], "spans": [ { "bbox": [ 306, 381, 541, 393 ], "score": 1.0, "content": "in papers describing alternative multi-class boosting meth-", "type": "text" } ], "index": 67 }, { "bbox": [ 306, 393, 542, 405 ], "spans": [ { "bbox": [ 306, 393, 542, 405 ], "score": 1.0, "content": "ods. Some report results with fixed hyperparameters (Zhu", "type": "text" } ], "index": 68 }, { "bbox": [ 306, 406, 542, 416 ], "spans": [ { "bbox": [ 306, 406, 542, 416 ], "score": 1.0, "content": "et al., 2009; Sun et al., 2012) and others give the full table", "type": "text" } ], "index": 69 }, { "bbox": [ 305, 417, 542, 429 ], "spans": [ { "bbox": [ 305, 417, 542, 429 ], "score": 1.0, "content": "of test errors for a grid of hyperparameters (Kegl & Busa- ´", "type": "text" } ], "index": 70 }, { "bbox": [ 306, 429, 542, 441 ], "spans": [ { "bbox": [ 306, 429, 378, 441 ], "score": 1.0, "content": "Fekete, 2009; Li,", "type": "text" }, { "bbox": [ 378, 430, 412, 441 ], "score": 0.28, "content": "{ 2 0 0 9 \\mathrm { a } ; \\mathrm { b } }", "type": "inline_equation" }, { "bbox": [ 412, 429, 542, 441 ], "score": 1.0, "content": "; Mukherjee & Schapire, 2013).", "type": "text" } ], "index": 71 }, { "bbox": [ 306, 441, 542, 453 ], "spans": [ { "bbox": [ 306, 441, 542, 453 ], "score": 1.0, "content": "Although the following procedure is rather old, we feel the", "type": "text" } ], "index": 72 }, { "bbox": [ 305, 453, 542, 465 ], "spans": [ { "bbox": [ 305, 453, 542, 465 ], "score": 1.0, "content": "need to detail it for promoting a more scrupulous compari-", "type": "text" } ], "index": 73 }, { "bbox": [ 306, 466, 380, 477 ], "spans": [ { "bbox": [ 306, 466, 380, 477 ], "score": 1.0, "content": "son across papers.", "type": "text" } ], "index": 74 } ], "index": 70, "bbox_fs": [ 305, 368, 543, 477 ] }, { "type": "text", "bbox": [ 307, 483, 542, 627 ], "lines": [ { "bbox": [ 305, 482, 542, 495 ], "spans": [ { "bbox": [ 305, 482, 423, 495 ], "score": 1.0, "content": "For the small data sets we ran", "type": "text" }, { "bbox": [ 424, 483, 453, 494 ], "score": 0.89, "content": "1 0 \\times 1 0", "type": "inline_equation" }, { "bbox": [ 454, 482, 542, 495 ], "score": 1.0, "content": "cross-validation (CV)", "type": "text" } ], "index": 75 }, { "bbox": [ 305, 496, 542, 507 ], "spans": [ { "bbox": [ 305, 496, 542, 507 ], "score": 1.0, "content": "to optimize the hyperparameters and the estimate the gen-", "type": "text" } ], "index": 76 }, { "bbox": [ 306, 507, 543, 519 ], "spans": [ { "bbox": [ 306, 507, 543, 519 ], "score": 1.0, "content": "eralization error. 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"inline_equation" }, { "bbox": [ 299, 184, 405, 199 ], "score": 1.0, "content": ". child index of parent", "type": "text" } ], "index": 8 }, { "bbox": [ 62, 199, 388, 212 ], "spans": [ { "bbox": [ 62, 199, 73, 212 ], "score": 1.0, "content": "9", "type": "text" }, { "bbox": [ 98, 201, 388, 211 ], "score": 0.34, "content": "\\begin{array} { r l } { \\mathfrak { H } \\mathrm { A P P E N D } \\big ( \\mathfrak { H } , \\mathbf { v } _ { j } \\varphi _ { j } ( \\cdot ) \\big ) } & { { } \\triangleright a d d i n g \\mathbf { h } _ { j } ( \\cdot ) = \\mathbf { v } _ { j } \\varphi _ { j } ( \\cdot ) t o \\tilde { \\mathfrak { H } } } \\end{array}", "type": "inline_equation", "image_path": "dab80b8596069c43067f7f1bca45b9d1cc4fcbae3a72e78b1b8a83ce0c43c53c.jpg" } ], "index": 9 }, { "bbox": [ 57, 212, 384, 227 ], "spans": [ { "bbox": [ 57, 212, 73, 226 ], "score": 1.0, "content": "10", "type": "text" }, { "bbox": [ 96, 214, 384, 227 ], "score": 0.58, "content": "( \\mathbf { X } _ { - } , \\mathbf { Y } _ { - } , \\mathbf { W } _ { - } , \\mathbf { X } _ { + } , \\mathbf { Y } _ { + } , \\mathbf { W } _ { + } ) \\xleftarrow - \\mathrm { C U T D A T A S E T } \\big ( \\mathbf { X } _ { j } , \\mathbf { Y } _ { j } , \\mathbf { W } , \\varphi _ { j } ( \\cdot ) \\big )", "type": "inline_equation", "image_path": "f521652042f20977e7a434d348dc01919783ee6c9690076ccb93794cdd4bb3c2.jpg" } ], "index": 10 }, { "bbox": [ 58, 227, 321, 240 ], "spans": [ { "bbox": [ 58, 227, 72, 240 ], "score": 1.0, "content": "11", "type": "text" }, { "bbox": [ 97, 227, 102, 240 ], "score": 1.0, "content": "f", "type": "text" }, { "bbox": [ 102, 227, 163, 240 ], "score": 0.43, "content": "\\mathbf { \\dot { o } r } \\bullet \\in \\{ - , + \\}", "type": "inline_equation" }, { "bbox": [ 163, 227, 321, 240 ], "score": 1.0, "content": ". insert children into priority queue", "type": "text" } ], "index": 11 }, { "bbox": [ 57, 240, 272, 255 ], "spans": [ { "bbox": [ 57, 240, 73, 254 ], "score": 1.0, "content": "12", "type": "text" }, { "bbox": [ 112, 241, 272, 255 ], "score": 0.63, "content": "( \\alpha _ { \\bullet } , \\mathbf { v _ { \\bullet } } , \\varphi _ { \\bullet } ( \\cdot ) ) \\mathrm { B A S E } ( \\mathbf { X _ { \\bullet } } , \\mathbf { Y _ { \\bullet } } , \\mathbf { W _ { \\bullet } } )", "type": "inline_equation", "image_path": "d60416122754cbd4a0bed5020beeefc6f7269fc13f294b5d10497e084a304508.jpg" } ], "index": 12 }, { "bbox": [ 57, 255, 453, 270 ], "spans": [ { "bbox": [ 57, 255, 73, 270 ], "score": 1.0, "content": "13", "type": "text" }, { "bbox": [ 105, 255, 453, 269 ], "score": 0.28, "content": "\\mathrm { I N S E R T } \\big ( S , \\big ( \\mathbf { v _ { * } } , \\varphi _ { \\bullet } ( \\cdot ) , \\mathbf { X _ { \\bullet } } , \\mathbf { Y _ { \\bullet } } , \\bullet , j \\big ) , \\gamma ( \\mathbf { v _ { * } } , \\varphi _ { \\bullet } , \\mathbf { X _ { \\bullet } } , \\mathbf { Y _ { \\bullet } } , \\mathbf { W _ { \\bullet } } ) - \\gamma ( \\mathbf { v } _ { j } , \\varphi _ { j } , \\mathbf { X _ { \\bullet } } , \\mathbf { Y _ { \\bullet } } , \\mathbf { W _ { \\bullet } } ) \\big )", "type": "inline_equation", "image_path": "afb56addab40422c667c94442aa7411b4adfe71577c698dadcffe6a9aff1811c.jpg" } ], "index": 13 }, { "bbox": [ 121, 266, 302, 280 ], "spans": [ { "bbox": [ 121, 266, 302, 280 ], "score": 1.0, "content": ". key = edge improvement over parent edge", "type": "text" } ], "index": 14 }, { "bbox": [ 107, 276, 191, 293 ], "spans": [ { "bbox": [ 107, 281, 114, 290 ], "score": 1.0, "content": "1", "type": "text" }, { "bbox": [ 129, 276, 191, 293 ], "score": 1.0, "content": "1 + γ(h1, W)", "type": "text" } ], "index": 15 }, { "bbox": [ 57, 285, 421, 307 ], "spans": [ { "bbox": [ 57, 285, 73, 299 ], "score": 1.0, "content": "14", "type": "text" }, { "bbox": [ 85, 288, 106, 298 ], "score": 1.0, "content": "α =", "type": "text" }, { "bbox": [ 110, 286, 191, 307 ], "score": 1.0, "content": "2 log 1 − γ(h1, W)", "type": "text" }, { "bbox": [ 203, 286, 389, 298 ], "score": 1.0, "content": ". standard coefficient of the full tree classifier", "type": "text" }, { "bbox": [ 390, 286, 402, 298 ], "score": 0.63, "content": "{ \\mathfrak { h } } _ { 1 }", "type": "inline_equation" }, { "bbox": [ 402, 286, 421, 298 ], "score": 1.0, "content": "(14)", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 293, 115, 304 ], "spans": [ { "bbox": [ 106, 293, 115, 304 ], "score": 0.999, "content": "2", "type": "text" } ], "index": 17 }, { "bbox": [ 59, 304, 164, 321 ], "spans": [ { "bbox": [ 59, 306, 73, 319 ], "score": 1.0, "content": "15", "type": "text" }, { "bbox": [ 84, 304, 164, 321 ], "score": 1.0, "content": "return \u0000α, H, l, r\u0001", "type": "text" } ], "index": 18 } ], "index": 9.5 }, { "type": "table", "bbox": [ 54, 363, 547, 479 ], "blocks": [ { "type": "table_caption", "bbox": [ 51, 331, 540, 353 ], "group_id": 0, "lines": [ { "bbox": [ 53, 331, 542, 344 ], "spans": [ { "bbox": [ 53, 331, 282, 344 ], "score": 1.0, "content": "Figure 2. The pseudocode of the Hamming tree base learner.", "type": "text" }, { "bbox": [ 283, 332, 293, 341 ], "score": 0.8, "content": "N", "type": "inline_equation" }, { "bbox": [ 293, 331, 542, 344 ], "score": 1.0, "content": "is the number of inner nodes. The algorithm returns a list of base", "type": "text" } ], "index": 19 }, { "bbox": [ 52, 341, 436, 355 ], "spans": [ { "bbox": [ 52, 341, 91, 355 ], "score": 1.0, "content": "classifiers", "type": "text" }, { "bbox": [ 91, 343, 100, 352 ], "score": 0.69, "content": "{ \\mathfrak H }", "type": "inline_equation" }, { "bbox": [ 100, 341, 178, 355 ], "score": 1.0, "content": ", two index lists l and", "type": "text" }, { "bbox": [ 178, 344, 184, 352 ], "score": 0.31, "content": "\\mathfrak { r }", "type": "inline_equation" }, { "bbox": [ 184, 341, 273, 355 ], "score": 1.0, "content": ", and the base coefficient", "type": "text" }, { "bbox": [ 273, 344, 281, 352 ], "score": 0.74, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 281, 341, 436, 355 ], "score": 1.0, "content": ". The tree classifier is then defined by (14).", "type": "text" } ], "index": 20 } ], "index": 19.5 }, { "type": "table_body", "bbox": [ 54, 363, 547, 479 ], "group_id": 0, "lines": [ { "bbox": [ 54, 363, 547, 479 ], "spans": [ { "bbox": [ 54, 363, 547, 479 ], "score": 0.974, "html": "
MethodisoletletteroptdigitspendigitsUSPS
ADABoosT.MH w Hamming trees3.5±0.52.1±0.22.0±0.32.1±0.34.5±0.5
ADABoosT.MH w Hamming prod. (Kégl & Busa-Fekete, 2009)4.2±0.52.5±0.22.1 ± 0.42.1±0.24.4±0.5
AOSOLOGITBoOST J= 20,v = 0.1 (Sun et al.,2012)3.5 ± 0.52.3±0.22.1 ± 0.32.4± 0.34.9±0.5
ABCLOGITB0OSTJ= 20,v= 0.1 (Li,2009b)4.2±0.52.2±0.23.1± 0.42.9±0.34.9±0.5
ABCMARTJ= 20,v=0.1(Li,2009a)5.0±0.62.5±0.22.6±0.43.0± 0.35.2±0.5
LOGITB0OST J= 20,v = 0.1 (Li,2009b)4.7± 0.52.8±0.33.6±0.43.1± 0.35.8±0.5
SAMME w single-label trees (Zhu etal., 2009)2.3±0.22.5 ± 0.3
ADABoosT.MH w single-label trees (Zhu et al., 2009)2.6± 0.32.8±0.3
ADABoosT.MM (Mukherjee & Schapire,2013)2.5± 0.22.7 ± 0.3
ADABoosT.MH w single-label trees (Mukherjee & Schapire,2013)9.0±0.57.0± 0.4
", "type": "table", "image_path": "b2605a0500719a817c760fece751ea1bfbd28f4775fca649517681ca01ce2278.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 54, 363, 547, 401.6666666666667 ], "spans": [], "index": 21 }, { "bbox": [ 54, 401.6666666666667, 547, 440.33333333333337 ], "spans": [], "index": 22 }, { "bbox": [ 54, 440.33333333333337, 547, 479.00000000000006 ], "spans": [], "index": 23 } ] }, { "type": "table_caption", "bbox": [ 179, 488, 416, 498 ], "group_id": 0, "lines": [ { "bbox": [ 178, 487, 418, 500 ], "spans": [ { "bbox": [ 178, 487, 418, 500 ], "score": 1.0, "content": "Table 1. Test error percentages on mid-sized benchmark data sets.", "type": "text" } ], "index": 24 } ], "index": 24 } ], "index": 22 }, { "type": "text", "bbox": [ 54, 519, 165, 530 ], "lines": [ { "bbox": [ 53, 519, 165, 531 ], "spans": [ { "bbox": [ 53, 519, 165, 531 ], "score": 1.0, "content": "error over a predefined grid", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 125, 537, 218, 558 ], "lines": [ { "bbox": [ 125, 537, 218, 558 ], "spans": [ { "bbox": [ 125, 537, 218, 558 ], "score": 0.92, "content": "N ^ { * } = \\operatorname* { m i n } _ { N \\in \\mathcal { N } } \\widehat { R } ^ { ( T _ { N } ^ { * } ) } ( N )", "type": "interline_equation", "image_path": "3021175b108f1d0092cdde2dcb99eb301de4d842fee80598f218bf1c4e15165c.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 125, 537, 218, 558 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 55, 568, 289, 718 ], "lines": [ { "bbox": [ 52, 565, 292, 584 ], "spans": [ { "bbox": [ 52, 565, 81, 584 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 81, 570, 95, 582 ], "score": 0.9, "content": "T _ { N } ^ { * }", "type": "inline_equation" }, { "bbox": [ 96, 565, 114, 584 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 114, 567, 149, 582 ], "score": 0.94, "content": "\\widehat { R } ^ { ( t ) } ( N )", "type": "inline_equation" }, { "bbox": [ 150, 565, 292, 584 ], "score": 1.0, "content": "are the optimal stopping time (15)", "type": "text" } ], "index": 27 }, { "bbox": [ 53, 581, 291, 593 ], "spans": [ { "bbox": [ 53, 581, 254, 593 ], "score": 1.0, "content": "and the test error, respectively, in the run with", "type": "text" }, { "bbox": [ 254, 582, 265, 591 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 265, 581, 291, 593 ], "score": 1.0, "content": "inner", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 593, 291, 606 ], "spans": [ { "bbox": [ 53, 593, 100, 606 ], "score": 1.0, "content": "nodes, and", "type": "text" }, { "bbox": [ 100, 594, 111, 604 ], "score": 0.8, "content": "\\mathcal { N }", "type": "inline_equation" }, { "bbox": [ 111, 593, 291, 606 ], "score": 1.0, "content": "is the set of inner nodes participating in the", "type": "text" } ], "index": 29 }, { "bbox": [ 53, 605, 291, 618 ], "spans": [ { "bbox": [ 53, 605, 291, 618 ], "score": 1.0, "content": "grid search. Then we re-run ADABOOST.MH on the joined", "type": "text" } ], "index": 30 }, { "bbox": [ 54, 617, 291, 630 ], "spans": [ { "bbox": [ 54, 617, 291, 630 ], "score": 1.0, "content": "training/validation set using the selected hyperparameters", "type": "text" } ], "index": 31 }, { "bbox": [ 54, 627, 291, 642 ], "spans": [ { "bbox": [ 54, 629, 69, 640 ], "score": 0.86, "content": "N ^ { * }", "type": "inline_equation" }, { "bbox": [ 69, 629, 88, 642 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 89, 630, 107, 642 ], "score": 0.91, "content": "T _ { N ^ { * } } ^ { * }", "type": "inline_equation" }, { "bbox": [ 107, 629, 155, 642 ], "score": 1.0, "content": ". The error", "type": "text" }, { "bbox": [ 156, 627, 168, 640 ], "score": 0.9, "content": "\\widetilde { R _ { i } }", "type": "inline_equation" }, { "bbox": [ 168, 629, 291, 642 ], "score": 1.0, "content": "in the ith training/test fold is", "type": "text" } ], "index": 32 }, { "bbox": [ 53, 641, 291, 653 ], "spans": [ { "bbox": [ 53, 641, 291, 653 ], "score": 1.0, "content": "then computed on the held-out test set. In the tables we", "type": "text" } ], "index": 33 }, { "bbox": [ 53, 654, 291, 665 ], "spans": [ { "bbox": [ 53, 654, 291, 665 ], "score": 1.0, "content": "report the mean error and the standard deviation. On the", "type": "text" } ], "index": 34 }, { "bbox": [ 53, 665, 290, 677 ], "spans": [ { "bbox": [ 53, 665, 180, 677 ], "score": 1.0, "content": "medium-size data sets we ran", "type": "text" }, { "bbox": [ 181, 665, 207, 676 ], "score": 0.56, "content": "1 \\times 5", "type": "inline_equation" }, { "bbox": [ 207, 665, 290, 677 ], "score": 1.0, "content": "CV (using the des-", "type": "text" } ], "index": 35 }, { "bbox": [ 53, 677, 291, 690 ], "spans": [ { "bbox": [ 53, 677, 291, 690 ], "score": 1.0, "content": "ignated test sets where available) following the same pro-", "type": "text" } ], "index": 36 }, { "bbox": [ 53, 688, 291, 701 ], "spans": [ { "bbox": [ 53, 688, 291, 701 ], "score": 1.0, "content": "cedure. In this case the report the binomial standard devia-", "type": "text" } ], "index": 37 }, { "bbox": [ 52, 700, 291, 720 ], "spans": [ { "bbox": [ 52, 702, 72, 719 ], "score": 1.0, "content": "tion", "type": "text" }, { "bbox": [ 72, 700, 134, 720 ], "score": 0.92, "content": "\\sqrt { \\widehat { R } ( 1 - \\widehat { R } ) / n }", "type": "inline_equation" }, { "bbox": [ 134, 702, 291, 719 ], "score": 1.0, "content": ". Further details and the description and", "type": "text" } ], "index": 38 } ], "index": 32.5 }, { "type": "text", "bbox": [ 306, 519, 541, 554 ], "lines": [ { "bbox": [ 305, 518, 542, 531 ], "spans": [ { "bbox": [ 305, 518, 542, 531 ], "score": 1.0, "content": "explanation of some slight variations of this experimental", "type": "text" } ], "index": 39 }, { "bbox": [ 306, 531, 542, 543 ], "spans": [ { "bbox": [ 306, 531, 542, 543 ], "score": 1.0, "content": "setup are available at https://www.lri.fr/˜kegl/", "type": "text" } ], "index": 40 }, { "bbox": [ 306, 543, 490, 555 ], "spans": [ { "bbox": [ 306, 543, 490, 555 ], "score": 1.0, "content": "research/multiboostResults.pdf.", "type": "text" } ], "index": 41 } ], "index": 40 }, { "type": "text", "bbox": [ 307, 561, 542, 703 ], "lines": [ { "bbox": [ 306, 560, 543, 573 ], "spans": [ { "bbox": [ 306, 560, 543, 573 ], "score": 1.0, "content": "On the small data sets, Duch et al. (2012) used the exact", "type": "text" } ], "index": 42 }, { "bbox": [ 305, 573, 543, 585 ], "spans": [ { "bbox": [ 305, 573, 543, 585 ], "score": 1.0, "content": "same protocol, so, although the folds are not the same, the", "type": "text" } ], "index": 43 }, { "bbox": [ 305, 584, 543, 597 ], "spans": [ { "bbox": [ 305, 584, 543, 597 ], "score": 1.0, "content": "results are directly comparable. The error bars represent", "type": "text" } ], "index": 44 }, { "bbox": [ 306, 596, 543, 608 ], "spans": [ { "bbox": [ 306, 596, 543, 608 ], "score": 1.0, "content": "the standard deviation of the test errors over the ten test", "type": "text" } ], "index": 45 }, { "bbox": [ 305, 607, 543, 621 ], "spans": [ { "bbox": [ 305, 608, 393, 621 ], "score": 1.0, "content": "folds not divided by", "type": "text" }, { "bbox": [ 394, 607, 413, 620 ], "score": 0.91, "content": "\\sqrt { 1 0 }", "type": "inline_equation" }, { "bbox": [ 414, 608, 543, 621 ], "score": 1.0, "content": ", contrary to common practice,", "type": "text" } ], "index": 46 }, { "bbox": [ 306, 621, 542, 632 ], "spans": [ { "bbox": [ 306, 621, 542, 632 ], "score": 1.0, "content": "since the training set of the folds are highly correlated. The", "type": "text" } ], "index": 47 }, { "bbox": [ 304, 632, 543, 644 ], "spans": [ { "bbox": [ 304, 632, 543, 644 ], "score": 1.0, "content": "large error bars are the consequence of the small size and", "type": "text" } ], "index": 48 }, { "bbox": [ 306, 644, 542, 656 ], "spans": [ { "bbox": [ 306, 644, 542, 656 ], "score": 1.0, "content": "the noisiness of these sets. They make it difficult to es-", "type": "text" } ], "index": 49 }, { "bbox": [ 306, 657, 542, 668 ], "spans": [ { "bbox": [ 306, 657, 542, 668 ], "score": 1.0, "content": "tablish any significant trends. We can safely state that AD-", "type": "text" } ], "index": 50 }, { "bbox": [ 305, 667, 543, 682 ], "spans": [ { "bbox": [ 305, 667, 543, 682 ], "score": 1.0, "content": "ABOOST.MH is on par with SVM (it is certainly not worse,", "type": "text" } ], "index": 51 }, { "bbox": [ 306, 680, 543, 693 ], "spans": [ { "bbox": [ 306, 680, 543, 693 ], "score": 1.0, "content": "“winning” on six of the nine sets), widely considered one", "type": "text" } ], "index": 52 }, { "bbox": [ 306, 692, 533, 704 ], "spans": [ { "bbox": [ 306, 692, 533, 704 ], "score": 1.0, "content": "of the the best classification methods for small data sets.", "type": "text" } ], "index": 53 } ], "index": 47.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 246, 46, 350, 55 ], "lines": [ { "bbox": [ 244, 44, 351, 57 ], "spans": [ { "bbox": [ 244, 44, 351, 57 ], "score": 1.0, "content": "Multi-class Hamming trees", "type": "text" } ] } ] } ], "para_blocks": [ { "type": 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The pseudocode of the Hamming tree base learner.", "type": "text" }, { "bbox": [ 283, 332, 293, 341 ], "score": 0.8, "content": "N", "type": "inline_equation" }, { "bbox": [ 293, 331, 542, 344 ], "score": 1.0, "content": "is the number of inner nodes. The algorithm returns a list of base", "type": "text" } ], "index": 19 }, { "bbox": [ 52, 341, 436, 355 ], "spans": [ { "bbox": [ 52, 341, 91, 355 ], "score": 1.0, "content": "classifiers", "type": "text" }, { "bbox": [ 91, 343, 100, 352 ], "score": 0.69, "content": "{ \\mathfrak H }", "type": "inline_equation" }, { "bbox": [ 100, 341, 178, 355 ], "score": 1.0, "content": ", two index lists l and", "type": "text" }, { "bbox": [ 178, 344, 184, 352 ], "score": 0.31, "content": "\\mathfrak { r }", "type": "inline_equation" }, { "bbox": [ 184, 341, 273, 355 ], "score": 1.0, "content": ", and the base coefficient", "type": "text" }, { "bbox": [ 273, 344, 281, 352 ], "score": 0.74, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 281, 341, 436, 355 ], "score": 1.0, "content": ". The tree classifier is then defined by (14).", "type": "text" } ], "index": 20 } ], "index": 19.5 }, { "type": "table_body", "bbox": [ 54, 363, 547, 479 ], "group_id": 0, "lines": [ { "bbox": [ 54, 363, 547, 479 ], "spans": [ { "bbox": [ 54, 363, 547, 479 ], "score": 0.974, "html": "
MethodisoletletteroptdigitspendigitsUSPS
ADABoosT.MH w Hamming trees3.5±0.52.1±0.22.0±0.32.1±0.34.5±0.5
ADABoosT.MH w Hamming prod. (Kégl & Busa-Fekete, 2009)4.2±0.52.5±0.22.1 ± 0.42.1±0.24.4±0.5
AOSOLOGITBoOST J= 20,v = 0.1 (Sun et al.,2012)3.5 ± 0.52.3±0.22.1 ± 0.32.4± 0.34.9±0.5
ABCLOGITB0OSTJ= 20,v= 0.1 (Li,2009b)4.2±0.52.2±0.23.1± 0.42.9±0.34.9±0.5
ABCMARTJ= 20,v=0.1(Li,2009a)5.0±0.62.5±0.22.6±0.43.0± 0.35.2±0.5
LOGITB0OST J= 20,v = 0.1 (Li,2009b)4.7± 0.52.8±0.33.6±0.43.1± 0.35.8±0.5
SAMME w single-label trees (Zhu etal., 2009)2.3±0.22.5 ± 0.3
ADABoosT.MH w single-label trees (Zhu et al., 2009)2.6± 0.32.8±0.3
ADABoosT.MM (Mukherjee & Schapire,2013)2.5± 0.22.7 ± 0.3
ADABoosT.MH w single-label trees (Mukherjee & Schapire,2013)9.0±0.57.0± 0.4
", "type": "table", "image_path": "b2605a0500719a817c760fece751ea1bfbd28f4775fca649517681ca01ce2278.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 54, 363, 547, 401.6666666666667 ], "spans": [], "index": 21 }, { "bbox": [ 54, 401.6666666666667, 547, 440.33333333333337 ], "spans": [], "index": 22 }, { "bbox": [ 54, 440.33333333333337, 547, 479.00000000000006 ], "spans": [], "index": 23 } ] }, { "type": "table_caption", "bbox": [ 179, 488, 416, 498 ], "group_id": 0, "lines": [ { "bbox": [ 178, 487, 418, 500 ], "spans": [ { "bbox": [ 178, 487, 418, 500 ], "score": 1.0, "content": "Table 1. Test error percentages on mid-sized benchmark data sets.", "type": "text" } ], "index": 24 } ], "index": 24 } ], "index": 22 }, { "type": "text", "bbox": [ 54, 519, 165, 530 ], "lines": [ { "bbox": [ 53, 519, 165, 531 ], "spans": [ { "bbox": [ 53, 519, 165, 531 ], "score": 1.0, "content": "error over a predefined grid", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 53, 519, 165, 531 ] }, { "type": "interline_equation", "bbox": [ 125, 537, 218, 558 ], "lines": [ { "bbox": [ 125, 537, 218, 558 ], "spans": [ { "bbox": [ 125, 537, 218, 558 ], "score": 0.92, "content": "N ^ { * } = \\operatorname* { m i n } _ { N \\in \\mathcal { N } } \\widehat { R } ^ { ( T _ { N } ^ { * } ) } ( N )", "type": "interline_equation", "image_path": "3021175b108f1d0092cdde2dcb99eb301de4d842fee80598f218bf1c4e15165c.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 125, 537, 218, 558 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 55, 568, 289, 718 ], "lines": [ { "bbox": [ 52, 565, 292, 584 ], "spans": [ { "bbox": [ 52, 565, 81, 584 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 81, 570, 95, 582 ], "score": 0.9, "content": "T _ { N } ^ { * }", "type": "inline_equation" }, { "bbox": [ 96, 565, 114, 584 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 114, 567, 149, 582 ], "score": 0.94, "content": "\\widehat { R } ^ { ( t ) } ( N )", "type": "inline_equation" }, { "bbox": [ 150, 565, 292, 584 ], "score": 1.0, "content": "are the optimal stopping time (15)", "type": "text" } ], "index": 27 }, { "bbox": [ 53, 581, 291, 593 ], "spans": [ { "bbox": [ 53, 581, 254, 593 ], "score": 1.0, "content": "and the test error, respectively, in the run with", "type": "text" }, { "bbox": [ 254, 582, 265, 591 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 265, 581, 291, 593 ], "score": 1.0, "content": "inner", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 593, 291, 606 ], "spans": [ { "bbox": [ 53, 593, 100, 606 ], "score": 1.0, "content": "nodes, and", "type": "text" }, { "bbox": [ 100, 594, 111, 604 ], "score": 0.8, "content": "\\mathcal { N }", "type": "inline_equation" }, { "bbox": [ 111, 593, 291, 606 ], "score": 1.0, "content": "is the set of inner nodes participating in the", "type": "text" } ], "index": 29 }, { "bbox": [ 53, 605, 291, 618 ], "spans": [ { "bbox": [ 53, 605, 291, 618 ], "score": 1.0, "content": "grid search. Then we re-run ADABOOST.MH on the joined", "type": "text" } ], "index": 30 }, { "bbox": [ 54, 617, 291, 630 ], "spans": [ { "bbox": [ 54, 617, 291, 630 ], "score": 1.0, "content": "training/validation set using the selected hyperparameters", "type": "text" } ], "index": 31 }, { "bbox": [ 54, 627, 291, 642 ], "spans": [ { "bbox": [ 54, 629, 69, 640 ], "score": 0.86, "content": "N ^ { * }", "type": "inline_equation" }, { "bbox": [ 69, 629, 88, 642 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 89, 630, 107, 642 ], "score": 0.91, "content": "T _ { N ^ { * } } ^ { * }", "type": "inline_equation" }, { "bbox": [ 107, 629, 155, 642 ], "score": 1.0, "content": ". The error", "type": "text" }, { "bbox": [ 156, 627, 168, 640 ], "score": 0.9, "content": "\\widetilde { R _ { i } }", "type": "inline_equation" }, { "bbox": [ 168, 629, 291, 642 ], "score": 1.0, "content": "in the ith training/test fold is", "type": "text" } ], "index": 32 }, { "bbox": [ 53, 641, 291, 653 ], "spans": [ { "bbox": [ 53, 641, 291, 653 ], "score": 1.0, "content": "then computed on the held-out test set. In the tables we", "type": "text" } ], "index": 33 }, { "bbox": [ 53, 654, 291, 665 ], "spans": [ { "bbox": [ 53, 654, 291, 665 ], "score": 1.0, "content": "report the mean error and the standard deviation. On the", "type": "text" } ], "index": 34 }, { "bbox": [ 53, 665, 290, 677 ], "spans": [ { "bbox": [ 53, 665, 180, 677 ], "score": 1.0, "content": "medium-size data sets we ran", "type": "text" }, { "bbox": [ 181, 665, 207, 676 ], "score": 0.56, "content": "1 \\times 5", "type": "inline_equation" }, { "bbox": [ 207, 665, 290, 677 ], "score": 1.0, "content": "CV (using the des-", "type": "text" } ], "index": 35 }, { "bbox": [ 53, 677, 291, 690 ], "spans": [ { "bbox": [ 53, 677, 291, 690 ], "score": 1.0, "content": "ignated test sets where available) following the same pro-", "type": "text" } ], "index": 36 }, { "bbox": [ 53, 688, 291, 701 ], "spans": [ { "bbox": [ 53, 688, 291, 701 ], "score": 1.0, "content": "cedure. In this case the report the binomial standard devia-", "type": "text" } ], "index": 37 }, { "bbox": [ 52, 700, 291, 720 ], "spans": [ { "bbox": [ 52, 702, 72, 719 ], "score": 1.0, "content": "tion", "type": "text" }, { "bbox": [ 72, 700, 134, 720 ], "score": 0.92, "content": "\\sqrt { \\widehat { R } ( 1 - \\widehat { R } ) / n }", "type": "inline_equation" }, { "bbox": [ 134, 702, 291, 719 ], "score": 1.0, "content": ". Further details and the description and", "type": "text" } ], "index": 38 }, { "bbox": [ 305, 518, 542, 531 ], "spans": [ { "bbox": [ 305, 518, 542, 531 ], "score": 1.0, "content": "explanation of some slight variations of this experimental", "type": "text" } ], "index": 39 }, { "bbox": [ 306, 531, 542, 543 ], "spans": [ { "bbox": [ 306, 531, 542, 543 ], "score": 1.0, "content": "setup are available at https://www.lri.fr/˜kegl/", "type": "text" } ], "index": 40 }, { "bbox": [ 306, 543, 490, 555 ], "spans": [ { "bbox": [ 306, 543, 490, 555 ], "score": 1.0, "content": "research/multiboostResults.pdf.", "type": "text" } ], "index": 41 } ], "index": 32.5, "bbox_fs": [ 52, 565, 292, 720 ] }, { "type": "text", "bbox": [ 306, 519, 541, 554 ], "lines": [], "index": 40, "bbox_fs": [ 305, 518, 542, 555 ], "lines_deleted": true }, { "type": "text", "bbox": [ 307, 561, 542, 703 ], "lines": [ { "bbox": [ 306, 560, 543, 573 ], "spans": [ { "bbox": [ 306, 560, 543, 573 ], "score": 1.0, "content": "On the small data sets, Duch et al. (2012) used the exact", "type": "text" } ], "index": 42 }, { "bbox": [ 305, 573, 543, 585 ], "spans": [ { "bbox": [ 305, 573, 543, 585 ], "score": 1.0, "content": "same protocol, so, although the folds are not the same, the", "type": "text" } ], "index": 43 }, { "bbox": [ 305, 584, 543, 597 ], "spans": [ { "bbox": [ 305, 584, 543, 597 ], "score": 1.0, "content": "results are directly comparable. The error bars represent", "type": "text" } ], "index": 44 }, { "bbox": [ 306, 596, 543, 608 ], "spans": [ { "bbox": [ 306, 596, 543, 608 ], "score": 1.0, "content": "the standard deviation of the test errors over the ten test", "type": "text" } ], "index": 45 }, { "bbox": [ 305, 607, 543, 621 ], "spans": [ { "bbox": [ 305, 608, 393, 621 ], "score": 1.0, "content": "folds not divided by", "type": "text" }, { "bbox": [ 394, 607, 413, 620 ], "score": 0.91, "content": "\\sqrt { 1 0 }", "type": "inline_equation" }, { "bbox": [ 414, 608, 543, 621 ], "score": 1.0, "content": ", contrary to common practice,", "type": "text" } ], "index": 46 }, { "bbox": [ 306, 621, 542, 632 ], "spans": [ { "bbox": [ 306, 621, 542, 632 ], "score": 1.0, "content": "since the training set of the folds are highly correlated. The", "type": "text" } ], "index": 47 }, { "bbox": [ 304, 632, 543, 644 ], "spans": [ { "bbox": [ 304, 632, 543, 644 ], "score": 1.0, "content": "large error bars are the consequence of the small size and", "type": "text" } ], "index": 48 }, { "bbox": [ 306, 644, 542, 656 ], "spans": [ { "bbox": [ 306, 644, 542, 656 ], "score": 1.0, "content": "the noisiness of these sets. They make it difficult to es-", "type": "text" } ], "index": 49 }, { "bbox": [ 306, 657, 542, 668 ], "spans": [ { "bbox": [ 306, 657, 542, 668 ], "score": 1.0, "content": "tablish any significant trends. We can safely state that AD-", "type": "text" } ], "index": 50 }, { "bbox": [ 305, 667, 543, 682 ], "spans": [ { "bbox": [ 305, 667, 543, 682 ], "score": 1.0, "content": "ABOOST.MH is on par with SVM (it is certainly not worse,", "type": "text" } ], "index": 51 }, { "bbox": [ 306, 680, 543, 693 ], "spans": [ { "bbox": [ 306, 680, 543, 693 ], "score": 1.0, "content": "“winning” on six of the nine sets), widely considered one", "type": "text" } ], "index": 52 }, { "bbox": [ 306, 692, 533, 704 ], "spans": [ { "bbox": [ 306, 692, 533, 704 ], "score": 1.0, "content": "of the the best classification methods for small data sets.", "type": "text" } ], "index": 53 } ], "index": 47.5, "bbox_fs": [ 304, 560, 543, 704 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 55, 69, 289, 389 ], "lines": [ { "bbox": [ 53, 67, 291, 79 ], "spans": [ { "bbox": [ 53, 67, 291, 79 ], "score": 1.0, "content": "Even though on the mid-sized data sets there are dedicated", "type": "text" } ], "index": 0 }, { "bbox": [ 53, 78, 291, 93 ], "spans": [ { "bbox": [ 53, 78, 291, 93 ], "score": 1.0, "content": "test sets used by most of the experimenters, comparing", "type": "text" } ], "index": 1 }, { "bbox": [ 53, 91, 290, 103 ], "spans": [ { "bbox": [ 53, 91, 290, 103 ], "score": 1.0, "content": "ADABOOST.MH to alternative multi-class boosting tech-", "type": "text" } ], "index": 2 }, { "bbox": [ 53, 103, 291, 116 ], "spans": [ { "bbox": [ 53, 103, 291, 116 ], "score": 1.0, "content": "niques is somewhat more difficult since none of the papers", "type": "text" } ], "index": 3 }, { "bbox": [ 54, 116, 290, 127 ], "spans": [ { "bbox": [ 54, 116, 290, 127 ], "score": 1.0, "content": "do proper hyperparameter tuning. Most of the papers re-", "type": "text" } ], "index": 4 }, { "bbox": [ 52, 127, 291, 141 ], "spans": [ { "bbox": [ 52, 127, 291, 141 ], "score": 1.0, "content": "port results with a table of errors given for a set of hy-", "type": "text" } ], "index": 5 }, { "bbox": [ 52, 138, 291, 153 ], "spans": [ { "bbox": [ 52, 138, 291, 153 ], "score": 1.0, "content": "perparameter choices, without specifying which hyperpa-", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 151, 290, 162 ], "spans": [ { "bbox": [ 53, 151, 290, 162 ], "score": 1.0, "content": "rameter choice would be picked by proper validation. For", "type": "text" } ], "index": 7 }, { "bbox": [ 53, 163, 290, 174 ], "spans": [ { "bbox": [ 53, 163, 290, 174 ], "score": 1.0, "content": "methods that are non-competitive with ADABOOST.MH", "type": "text" } ], "index": 8 }, { "bbox": [ 54, 174, 291, 187 ], "spans": [ { "bbox": [ 54, 174, 291, 187 ], "score": 1.0, "content": "(SAMME of Zhu et al. (2009) and ADABOOST.MM of", "type": "text" } ], "index": 9 }, { "bbox": [ 53, 187, 291, 200 ], "spans": [ { "bbox": [ 53, 187, 291, 200 ], "score": 1.0, "content": "Mukherjee & Schapire (2013)) we report the post-validated", "type": "text" } ], "index": 10 }, { "bbox": [ 54, 199, 290, 210 ], "spans": [ { "bbox": [ 54, 199, 290, 210 ], "score": 1.0, "content": "best error which may be significantly lower than the er-", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 210, 290, 223 ], "spans": [ { "bbox": [ 53, 210, 290, 223 ], "score": 1.0, "content": "ror corresponding to the hyperparameter choice selected by", "type": "text" } ], "index": 12 }, { "bbox": [ 52, 223, 291, 235 ], "spans": [ { "bbox": [ 52, 223, 291, 235 ], "score": 1.0, "content": "proper validation. For methods where this choice would", "type": "text" } ], "index": 13 }, { "bbox": [ 54, 234, 291, 247 ], "spans": [ { "bbox": [ 54, 234, 291, 247 ], "score": 1.0, "content": "unfairly bias the comparison (AOSOLOGITBOOST (Sun", "type": "text" } ], "index": 14 }, { "bbox": [ 53, 247, 290, 258 ], "spans": [ { "bbox": [ 53, 247, 290, 258 ], "score": 1.0, "content": "et al., 2012), ABCLOGITBOOST, LOGITBOOST, and", "type": "text" } ], "index": 15 }, { "bbox": [ 54, 258, 291, 271 ], "spans": [ { "bbox": [ 54, 258, 291, 271 ], "score": 1.0, "content": "ABCMART (Li, 2009a;b)), we chose the best overall", "type": "text" } ], "index": 16 }, { "bbox": [ 52, 270, 290, 284 ], "spans": [ { "bbox": [ 52, 270, 121, 284 ], "score": 1.0, "content": "hyperparameter", "type": "text" }, { "bbox": [ 121, 271, 162, 282 ], "score": 0.9, "content": "\\textit { J } = \\ 2 0", "type": "inline_equation" }, { "bbox": [ 163, 270, 184, 284 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 271, 227, 282 ], "score": 0.88, "content": "\\nu ~ = ~ 0 . 1", "type": "inline_equation" }, { "bbox": [ 227, 270, 290, 284 ], "score": 1.0, "content": ", suggested by", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 282, 289, 295 ], "spans": [ { "bbox": [ 53, 282, 82, 295 ], "score": 1.0, "content": "the Li", "type": "text" }, { "bbox": [ 82, 283, 119, 294 ], "score": 0.32, "content": "( 2 0 0 9 \\mathrm { a } ; \\mathrm { b } )", "type": "inline_equation" }, { "bbox": [ 120, 282, 289, 295 ], "score": 1.0, "content": ". At https://www.lri.fr/˜kegl/", "type": "text" } ], "index": 18 }, { "bbox": [ 53, 295, 290, 306 ], "spans": [ { "bbox": [ 53, 295, 290, 306 ], "score": 1.0, "content": "research/multiboostResults.pdf (but not in", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 305, 290, 319 ], "spans": [ { "bbox": [ 53, 305, 290, 319 ], "score": 1.0, "content": "Table 1) we give both errors for some of the methods.", "type": "text" } ], "index": 20 }, { "bbox": [ 53, 318, 291, 331 ], "spans": [ { "bbox": [ 53, 318, 291, 331 ], "score": 1.0, "content": "Proper hyperparameter-validation should put the correct", "type": "text" } ], "index": 21 }, { "bbox": [ 53, 330, 291, 343 ], "spans": [ { "bbox": [ 53, 330, 291, 343 ], "score": 1.0, "content": "test error estimates between those two limits. Since", "type": "text" } ], "index": 22 }, { "bbox": [ 54, 341, 290, 355 ], "spans": [ { "bbox": [ 54, 341, 290, 355 ], "score": 1.0, "content": "ADABOOST.MH with decision products (Kegl & Busa- ´", "type": "text" } ], "index": 23 }, { "bbox": [ 53, 353, 290, 366 ], "spans": [ { "bbox": [ 53, 353, 290, 366 ], "score": 1.0, "content": "Fekete, 2009) is also implemented in multiboost (Ben-", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 366, 290, 378 ], "spans": [ { "bbox": [ 53, 366, 290, 378 ], "score": 1.0, "content": "bouzid et al., 2012), for this method we re-ran experiments", "type": "text" } ], "index": 25 }, { "bbox": [ 54, 379, 194, 390 ], "spans": [ { "bbox": [ 54, 379, 194, 390 ], "score": 1.0, "content": "with the protocol described above.", "type": "text" } ], "index": 26 } ], "index": 13 }, { "type": "text", "bbox": [ 55, 397, 289, 599 ], "lines": [ { "bbox": [ 53, 395, 291, 408 ], "spans": [ { "bbox": [ 53, 395, 291, 408 ], "score": 1.0, "content": "The overall conclusion is that AOSOLOGITBOOST (Sun", "type": "text" } ], "index": 27 }, { "bbox": [ 53, 408, 291, 420 ], "spans": [ { "bbox": [ 53, 408, 291, 420 ], "score": 1.0, "content": "et al., 2012) and ADABOOST.MH with Hamming trees", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 420, 290, 433 ], "spans": [ { "bbox": [ 53, 420, 290, 433 ], "score": 1.0, "content": "are the best algorithms (ADABOOST.MH winning on all", "type": "text" } ], "index": 29 }, { "bbox": [ 53, 431, 291, 444 ], "spans": [ { "bbox": [ 53, 431, 291, 444 ], "score": 1.0, "content": "the five data sets but within one standard deviation).", "type": "text" } ], "index": 30 }, { "bbox": [ 53, 443, 291, 456 ], "spans": [ { "bbox": [ 53, 443, 291, 456 ], "score": 1.0, "content": "ADABOOST.MH with decision products (Kegl & Busa- ´", "type": "text" } ], "index": 31 }, { "bbox": [ 53, 455, 290, 468 ], "spans": [ { "bbox": [ 53, 455, 290, 468 ], "score": 1.0, "content": "Fekete, 2009) and ABCLOGITBOOST are slightly weaker,", "type": "text" } ], "index": 32 }, { "bbox": [ 52, 468, 291, 480 ], "spans": [ { "bbox": [ 52, 468, 291, 480 ], "score": 1.0, "content": "as also noted by (Sun et al., 2012). SAMME (Zhu", "type": "text" } ], "index": 33 }, { "bbox": [ 52, 480, 291, 493 ], "spans": [ { "bbox": [ 52, 480, 291, 493 ], "score": 1.0, "content": "et al., 2009) and ADABOOST.MM (Mukherjee & Schapire,", "type": "text" } ], "index": 34 }, { "bbox": [ 52, 492, 291, 504 ], "spans": [ { "bbox": [ 52, 492, 291, 504 ], "score": 1.0, "content": "2013) perform below the rest of the methods on the", "type": "text" } ], "index": 35 }, { "bbox": [ 53, 504, 290, 516 ], "spans": [ { "bbox": [ 53, 504, 290, 516 ], "score": 1.0, "content": "two data sets shared among all the papers (even though", "type": "text" } ], "index": 36 }, { "bbox": [ 54, 516, 290, 528 ], "spans": [ { "bbox": [ 54, 516, 290, 528 ], "score": 1.0, "content": "we give post-validated results). Another important con-", "type": "text" } ], "index": 37 }, { "bbox": [ 54, 528, 290, 539 ], "spans": [ { "bbox": [ 54, 528, 290, 539 ], "score": 1.0, "content": "clusion is that ADABOOST.MH with Hamming trees is", "type": "text" } ], "index": 38 }, { "bbox": [ 53, 540, 290, 552 ], "spans": [ { "bbox": [ 53, 540, 290, 552 ], "score": 1.0, "content": "significantly better then other implementations of AD-", "type": "text" } ], "index": 39 }, { "bbox": [ 53, 551, 290, 564 ], "spans": [ { "bbox": [ 53, 551, 290, 564 ], "score": 1.0, "content": "ABOOST.MH in (Zhu et al., 2009; Mukherjee & Schapire,", "type": "text" } ], "index": 40 }, { "bbox": [ 52, 562, 291, 577 ], "spans": [ { "bbox": [ 52, 562, 291, 577 ], "score": 1.0, "content": "2013), assumably implemented using single-label trees (the", "type": "text" } ], "index": 41 }, { "bbox": [ 53, 576, 290, 588 ], "spans": [ { "bbox": [ 53, 576, 290, 588 ], "score": 1.0, "content": "errors reported by Mukherjee & Schapire (2013) are espe-", "type": "text" } ], "index": 42 }, { "bbox": [ 54, 588, 136, 600 ], "spans": [ { "bbox": [ 54, 588, 136, 600 ], "score": 1.0, "content": "cially conspicuous).", "type": "text" } ], "index": 43 } ], "index": 35 }, { "type": "text", "bbox": [ 55, 605, 289, 712 ], "lines": [ { "bbox": [ 54, 605, 291, 618 ], "spans": [ { "bbox": [ 54, 605, 291, 618 ], "score": 1.0, "content": "ADABOOST.MH with Hamming trees also achieves good", "type": "text" } ], "index": 44 }, { "bbox": [ 53, 618, 290, 629 ], "spans": [ { "bbox": [ 53, 618, 290, 629 ], "score": 1.0, "content": "results on image recognition problems. On MNIST, boost-", "type": "text" } ], "index": 45 }, { "bbox": [ 53, 630, 291, 641 ], "spans": [ { "bbox": [ 53, 630, 291, 641 ], "score": 1.0, "content": "ing trees of stumps over pixels with eight inner nodes and", "type": "text" } ], "index": 46 }, { "bbox": [ 53, 640, 291, 654 ], "spans": [ { "bbox": [ 53, 640, 226, 654 ], "score": 1.0, "content": "about 50000 iterations has a test error of", "type": "text" }, { "bbox": [ 226, 641, 253, 653 ], "score": 0.88, "content": "1 . 2 5 \\%", "type": "inline_equation" }, { "bbox": [ 253, 640, 291, 654 ], "score": 1.0, "content": ", making", "type": "text" } ], "index": 47 }, { "bbox": [ 54, 654, 290, 664 ], "spans": [ { "bbox": [ 54, 654, 290, 664 ], "score": 1.0, "content": "it one of the best no-domain-knowledge “shallow” classi-", "type": "text" } ], "index": 48 }, { "bbox": [ 53, 665, 290, 677 ], "spans": [ { "bbox": [ 53, 665, 290, 677 ], "score": 1.0, "content": "fiers. Using stumps over Haar filters (Viola & Jones, 2004),", "type": "text" } ], "index": 49 }, { "bbox": [ 53, 677, 290, 689 ], "spans": [ { "bbox": [ 53, 677, 290, 689 ], "score": 1.0, "content": "boosted trees with four inner nodes and 10000 iterations", "type": "text" } ], "index": 50 }, { "bbox": [ 54, 689, 290, 701 ], "spans": [ { "bbox": [ 54, 689, 145, 701 ], "score": 1.0, "content": "achieves a test error of", "type": "text" }, { "bbox": [ 145, 689, 172, 700 ], "score": 0.89, "content": "0 . 8 5 \\%", "type": "inline_equation" }, { "bbox": [ 172, 689, 290, 701 ], "score": 1.0, "content": ", comparable to classical con-", "type": "text" } ], "index": 51 }, { "bbox": [ 54, 701, 200, 713 ], "spans": [ { "bbox": [ 54, 701, 200, 713 ], "score": 1.0, "content": "volutional nets (LeCun et al., 1998).", "type": "text" } ], "index": 52 } ], "index": 48 }, { "type": "text", "bbox": [ 307, 68, 541, 222 ], "lines": [ { "bbox": [ 306, 68, 542, 79 ], "spans": [ { "bbox": [ 306, 68, 542, 79 ], "score": 1.0, "content": "ADABOOST.MH with Hamming trees, usually combined", "type": "text" } ], "index": 53 }, { "bbox": [ 305, 78, 542, 92 ], "spans": [ { "bbox": [ 305, 78, 542, 92 ], "score": 1.0, "content": "with calibration (Platt, 2000; Niculescu-Mizil & Caruana,", "type": "text" } ], "index": 54 }, { "bbox": [ 306, 92, 542, 103 ], "spans": [ { "bbox": [ 306, 92, 542, 103 ], "score": 1.0, "content": "2005) and model averaging, has been also successful in re-", "type": "text" } ], "index": 55 }, { "bbox": [ 305, 104, 542, 115 ], "spans": [ { "bbox": [ 305, 104, 542, 115 ], "score": 1.0, "content": "cent data challenges. On the Kaggle emotions data chal-", "type": "text" } ], "index": 56 }, { "bbox": [ 305, 116, 542, 127 ], "spans": [ { "bbox": [ 305, 116, 542, 127 ], "score": 1.0, "content": "lenge, although not competitive with deep learning tech-", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 127, 542, 140 ], "spans": [ { "bbox": [ 305, 127, 542, 140 ], "score": 1.0, "content": "niques, out-of-the-box ADABOOST.MH with Hamming", "type": "text" } ], "index": 58 }, { "bbox": [ 306, 139, 543, 150 ], "spans": [ { "bbox": [ 306, 139, 543, 150 ], "score": 1.0, "content": "trees over Haar filters finished 17th place with a test error of", "type": "text" } ], "index": 59 }, { "bbox": [ 306, 150, 542, 163 ], "spans": [ { "bbox": [ 306, 151, 325, 162 ], "score": 0.87, "content": "5 7 \\%", "type": "inline_equation" }, { "bbox": [ 326, 150, 542, 163 ], "score": 1.0, "content": ". In the Yahoo! Learning-to-Rank Challenge (Chapelle", "type": "text" } ], "index": 60 }, { "bbox": [ 305, 163, 542, 174 ], "spans": [ { "bbox": [ 305, 163, 542, 174 ], "score": 1.0, "content": "et al., 2011) it achieved top ten performances with results", "type": "text" } ], "index": 61 }, { "bbox": [ 305, 174, 542, 188 ], "spans": [ { "bbox": [ 305, 174, 542, 188 ], "score": 1.0, "content": "not significantly different from the winning scores. Finally,", "type": "text" } ], "index": 62 }, { "bbox": [ 305, 187, 542, 199 ], "spans": [ { "bbox": [ 305, 187, 542, 199 ], "score": 1.0, "content": "in the recent INTERSPEECH Challenge it won the Emo-", "type": "text" } ], "index": 63 }, { "bbox": [ 305, 198, 541, 212 ], "spans": [ { "bbox": [ 305, 198, 541, 212 ], "score": 1.0, "content": "tion sub-challenge and it was runner up in the Social Sig-", "type": "text" } ], "index": 64 }, { "bbox": [ 305, 210, 385, 223 ], "spans": [ { "bbox": [ 305, 210, 385, 223 ], "score": 1.0, "content": "nals sub-challenge.", "type": "text" } ], "index": 65 } ], "index": 59 }, { "type": "title", "bbox": [ 306, 238, 376, 252 ], "lines": [ { "bbox": [ 304, 236, 378, 253 ], "spans": [ { "bbox": [ 304, 236, 378, 253 ], "score": 1.0, "content": "5. Conclusion", "type": "text" } ], "index": 66 } ], "index": 66 }, { "type": "text", "bbox": [ 307, 259, 541, 366 ], "lines": [ { "bbox": [ 305, 258, 542, 271 ], "spans": [ { "bbox": [ 305, 258, 542, 271 ], "score": 1.0, "content": "In this paper we introduced Hamming trees that optimize", "type": "text" } ], "index": 67 }, { "bbox": [ 306, 271, 541, 282 ], "spans": [ { "bbox": [ 306, 271, 541, 282 ], "score": 1.0, "content": "the multi-class edge prescribed by ADABOOST.MH with-", "type": "text" } ], "index": 68 }, { "bbox": [ 306, 283, 542, 295 ], "spans": [ { "bbox": [ 306, 283, 478, 295 ], "score": 1.0, "content": "out reducing the multi-class problem to", "type": "text" }, { "bbox": [ 478, 283, 489, 293 ], "score": 0.79, "content": "K", "type": "inline_equation" }, { "bbox": [ 489, 283, 542, 295 ], "score": 1.0, "content": "binary one-", "type": "text" } ], "index": 69 }, { "bbox": [ 305, 295, 542, 306 ], "spans": [ { "bbox": [ 305, 295, 542, 306 ], "score": 1.0, "content": "against-all classifications. We showed that without this", "type": "text" } ], "index": 70 }, { "bbox": [ 306, 307, 542, 318 ], "spans": [ { "bbox": [ 306, 307, 542, 318 ], "score": 1.0, "content": "restriction, often considered mandatory, ADABOOST.MH", "type": "text" } ], "index": 71 }, { "bbox": [ 305, 319, 542, 330 ], "spans": [ { "bbox": [ 305, 319, 542, 330 ], "score": 1.0, "content": "is one of the best off-the-shelf multi-class classification", "type": "text" } ], "index": 72 }, { "bbox": [ 305, 331, 542, 343 ], "spans": [ { "bbox": [ 305, 331, 542, 343 ], "score": 1.0, "content": "algorithms. The algorithm retains the conceptual ele-", "type": "text" } ], "index": 73 }, { "bbox": [ 305, 343, 542, 355 ], "spans": [ { "bbox": [ 305, 343, 542, 355 ], "score": 1.0, "content": "gance, power, and computational efficiency of binary AD-", "type": "text" } ], "index": 74 }, { "bbox": [ 306, 355, 347, 366 ], "spans": [ { "bbox": [ 306, 355, 347, 366 ], "score": 1.0, "content": "ABOOST.", "type": "text" } ], "index": 75 } ], "index": 71 }, { "type": "text", "bbox": [ 307, 373, 541, 503 ], "lines": [ { "bbox": [ 306, 373, 542, 384 ], "spans": [ { "bbox": [ 306, 373, 542, 384 ], "score": 1.0, "content": "Using decision stumps at the inner nodes and at the leaves", "type": "text" } ], "index": 76 }, { "bbox": [ 306, 384, 543, 396 ], "spans": [ { "bbox": [ 306, 384, 543, 396 ], "score": 1.0, "content": "of the tree is a natural choice due to the efficiency of", "type": "text" } ], "index": 77 }, { "bbox": [ 306, 397, 542, 408 ], "spans": [ { "bbox": [ 306, 397, 542, 408 ], "score": 1.0, "content": "the learning algorithm, nevertheless, the general setup de-", "type": "text" } ], "index": 78 }, { "bbox": [ 306, 409, 542, 420 ], "spans": [ { "bbox": [ 306, 409, 542, 420 ], "score": 1.0, "content": "scribed in this paper allows for using any binary classifier.", "type": "text" } ], "index": 79 }, { "bbox": [ 305, 419, 542, 433 ], "spans": [ { "bbox": [ 305, 419, 542, 433 ], "score": 1.0, "content": "One of the avenues investigated for future work is to try", "type": "text" } ], "index": 80 }, { "bbox": [ 305, 432, 542, 443 ], "spans": [ { "bbox": [ 305, 432, 542, 443 ], "score": 1.0, "content": "stronger classifiers, such as SVMs, as binary cuts. The for-", "type": "text" } ], "index": 81 }, { "bbox": [ 305, 444, 542, 455 ], "spans": [ { "bbox": [ 305, 444, 542, 455 ], "score": 1.0, "content": "mal setup described in Section 2.1 does not restrict the al-", "type": "text" } ], "index": 82 }, { "bbox": [ 305, 457, 542, 468 ], "spans": [ { "bbox": [ 305, 457, 542, 468 ], "score": 1.0, "content": "gorithm to single-label problems; another direction for fu-", "type": "text" } ], "index": 83 }, { "bbox": [ 306, 468, 542, 479 ], "spans": [ { "bbox": [ 306, 468, 542, 479 ], "score": 1.0, "content": "ture work is to benchmark it on standard multi-label and", "type": "text" } ], "index": 84 }, { "bbox": [ 305, 481, 542, 491 ], "spans": [ { "bbox": [ 305, 481, 542, 491 ], "score": 1.0, "content": "sequence-to-sequence classification problems (Dietterich", "type": "text" } ], "index": 85 }, { "bbox": [ 305, 492, 358, 503 ], "spans": [ { "bbox": [ 305, 492, 358, 503 ], "score": 1.0, "content": "et al., 2008).", "type": "text" } ], "index": 86 } ], "index": 81 }, { "type": "title", "bbox": [ 307, 520, 363, 532 ], "lines": [ { "bbox": [ 305, 518, 365, 534 ], "spans": [ { "bbox": [ 305, 518, 365, 534 ], "score": 1.0, "content": "References", "type": "text" } ], "index": 87 } ], "index": 87 }, { "type": "text", "bbox": [ 306, 539, 542, 717 ], "lines": [ { "bbox": [ 306, 538, 542, 551 ], "spans": [ { "bbox": [ 306, 538, 542, 551 ], "score": 1.0, "content": "Allwein, E. L., Schapire, R. E., and Singer, Y. Reduc-", "type": "text" } ], "index": 88 }, { "bbox": [ 316, 551, 542, 563 ], "spans": [ { "bbox": [ 316, 551, 542, 563 ], "score": 1.0, "content": "ing multiclass to binary: a unifying approach for margin", "type": "text" } ], "index": 89 }, { "bbox": [ 315, 562, 542, 575 ], "spans": [ { "bbox": [ 315, 562, 542, 575 ], "score": 1.0, "content": "classifiers. Journal of Machine Learning Research, 1:", "type": "text" } ], "index": 90 }, { "bbox": [ 316, 575, 380, 585 ], "spans": [ { "bbox": [ 316, 575, 380, 585 ], "score": 1.0, "content": "113–141, 2001.", "type": "text" } ], "index": 91 }, { "bbox": [ 305, 593, 542, 608 ], "spans": [ { "bbox": [ 305, 593, 542, 608 ], "score": 1.0, "content": "Benbouzid, D., Busa-Fekete, R., Casagrande, N., Collin,", "type": "text" } ], "index": 92 }, { "bbox": [ 315, 606, 542, 619 ], "spans": [ { "bbox": [ 315, 606, 542, 619 ], "score": 1.0, "content": "F.-D., and Kegl, B. MultiBoost: a multi-purpose boost- ´", "type": "text" } ], "index": 93 }, { "bbox": [ 315, 618, 542, 631 ], "spans": [ { "bbox": [ 315, 618, 542, 631 ], "score": 1.0, "content": "ing package. Journal of Machine Learning Research, 13:", "type": "text" } ], "index": 94 }, { "bbox": [ 315, 629, 381, 642 ], "spans": [ { "bbox": [ 315, 629, 381, 642 ], "score": 1.0, "content": "549–553, 2012.", "type": "text" } ], "index": 95 }, { "bbox": [ 305, 650, 542, 663 ], "spans": [ { "bbox": [ 305, 650, 542, 663 ], "score": 1.0, "content": "Boser, B., Guyon, I., and Vapnik, V. A training algorithm", "type": "text" } ], "index": 96 }, { "bbox": [ 316, 662, 542, 675 ], "spans": [ { "bbox": [ 316, 662, 542, 675 ], "score": 1.0, "content": "for optimal margin classifiers. In Fifth Annual Workshop", "type": "text" } ], "index": 97 }, { "bbox": [ 315, 674, 542, 686 ], "spans": [ { "bbox": [ 315, 674, 542, 686 ], "score": 1.0, "content": "on Computational Learning Theory, pp. 144–152, 1992.", "type": "text" } ], "index": 98 }, { "bbox": [ 306, 693, 542, 706 ], "spans": [ { "bbox": [ 306, 693, 542, 706 ], "score": 1.0, "content": "Caruana, R. and Niculescu-Mizil, A. An empirical compar-", "type": "text" } ], "index": 99 }, { "bbox": [ 315, 705, 542, 718 ], "spans": [ { "bbox": [ 315, 705, 542, 718 ], "score": 1.0, "content": "ison of supervised learning algorithms. In Proceedings", "type": "text" } ], "index": 100 } ], "index": 94 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 246, 46, 350, 55 ], "lines": [ { "bbox": [ 244, 44, 352, 57 ], "spans": [ { "bbox": [ 244, 44, 352, 57 ], "score": 1.0, "content": "Multi-class Hamming trees", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 55, 69, 289, 389 ], "lines": [ { "bbox": [ 53, 67, 291, 79 ], "spans": [ { "bbox": [ 53, 67, 291, 79 ], "score": 1.0, "content": "Even though on the mid-sized data sets there are dedicated", "type": "text" } ], "index": 0 }, { "bbox": [ 53, 78, 291, 93 ], "spans": [ { "bbox": [ 53, 78, 291, 93 ], "score": 1.0, "content": "test sets used by most of the experimenters, comparing", "type": "text" } ], "index": 1 }, { "bbox": [ 53, 91, 290, 103 ], "spans": [ { "bbox": [ 53, 91, 290, 103 ], "score": 1.0, "content": "ADABOOST.MH to alternative multi-class boosting tech-", "type": "text" } ], "index": 2 }, { "bbox": [ 53, 103, 291, 116 ], "spans": [ { "bbox": [ 53, 103, 291, 116 ], "score": 1.0, "content": "niques is somewhat more difficult since none of the papers", "type": "text" } ], "index": 3 }, { "bbox": [ 54, 116, 290, 127 ], "spans": [ { "bbox": [ 54, 116, 290, 127 ], "score": 1.0, "content": "do proper hyperparameter tuning. Most of the papers re-", "type": "text" } ], "index": 4 }, { "bbox": [ 52, 127, 291, 141 ], "spans": [ { "bbox": [ 52, 127, 291, 141 ], "score": 1.0, "content": "port results with a table of errors given for a set of hy-", "type": "text" } ], "index": 5 }, { "bbox": [ 52, 138, 291, 153 ], "spans": [ { "bbox": [ 52, 138, 291, 153 ], "score": 1.0, "content": "perparameter choices, without specifying which hyperpa-", "type": "text" } ], "index": 6 }, { "bbox": [ 53, 151, 290, 162 ], "spans": [ { "bbox": [ 53, 151, 290, 162 ], "score": 1.0, "content": "rameter choice would be picked by proper validation. For", "type": "text" } ], "index": 7 }, { "bbox": [ 53, 163, 290, 174 ], "spans": [ { "bbox": [ 53, 163, 290, 174 ], "score": 1.0, "content": "methods that are non-competitive with ADABOOST.MH", "type": "text" } ], "index": 8 }, { "bbox": [ 54, 174, 291, 187 ], "spans": [ { "bbox": [ 54, 174, 291, 187 ], "score": 1.0, "content": "(SAMME of Zhu et al. (2009) and ADABOOST.MM of", "type": "text" } ], "index": 9 }, { "bbox": [ 53, 187, 291, 200 ], "spans": [ { "bbox": [ 53, 187, 291, 200 ], "score": 1.0, "content": "Mukherjee & Schapire (2013)) we report the post-validated", "type": "text" } ], "index": 10 }, { "bbox": [ 54, 199, 290, 210 ], "spans": [ { "bbox": [ 54, 199, 290, 210 ], "score": 1.0, "content": "best error which may be significantly lower than the er-", "type": "text" } ], "index": 11 }, { "bbox": [ 53, 210, 290, 223 ], "spans": [ { "bbox": [ 53, 210, 290, 223 ], "score": 1.0, "content": "ror corresponding to the hyperparameter choice selected by", "type": "text" } ], "index": 12 }, { "bbox": [ 52, 223, 291, 235 ], "spans": [ { "bbox": [ 52, 223, 291, 235 ], "score": 1.0, "content": "proper validation. For methods where this choice would", "type": "text" } ], "index": 13 }, { "bbox": [ 54, 234, 291, 247 ], "spans": [ { "bbox": [ 54, 234, 291, 247 ], "score": 1.0, "content": "unfairly bias the comparison (AOSOLOGITBOOST (Sun", "type": "text" } ], "index": 14 }, { "bbox": [ 53, 247, 290, 258 ], "spans": [ { "bbox": [ 53, 247, 290, 258 ], "score": 1.0, "content": "et al., 2012), ABCLOGITBOOST, LOGITBOOST, and", "type": "text" } ], "index": 15 }, { "bbox": [ 54, 258, 291, 271 ], "spans": [ { "bbox": [ 54, 258, 291, 271 ], "score": 1.0, "content": "ABCMART (Li, 2009a;b)), we chose the best overall", "type": "text" } ], "index": 16 }, { "bbox": [ 52, 270, 290, 284 ], "spans": [ { "bbox": [ 52, 270, 121, 284 ], "score": 1.0, "content": "hyperparameter", "type": "text" }, { "bbox": [ 121, 271, 162, 282 ], "score": 0.9, "content": "\\textit { J } = \\ 2 0", "type": "inline_equation" }, { "bbox": [ 163, 270, 184, 284 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 271, 227, 282 ], "score": 0.88, "content": "\\nu ~ = ~ 0 . 1", "type": "inline_equation" }, { "bbox": [ 227, 270, 290, 284 ], "score": 1.0, "content": ", suggested by", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 282, 289, 295 ], "spans": [ { "bbox": [ 53, 282, 82, 295 ], "score": 1.0, "content": "the Li", "type": "text" }, { "bbox": [ 82, 283, 119, 294 ], "score": 0.32, "content": "( 2 0 0 9 \\mathrm { a } ; \\mathrm { b } )", "type": "inline_equation" }, { "bbox": [ 120, 282, 289, 295 ], "score": 1.0, "content": ". At https://www.lri.fr/˜kegl/", "type": "text" } ], "index": 18 }, { "bbox": [ 53, 295, 290, 306 ], "spans": [ { "bbox": [ 53, 295, 290, 306 ], "score": 1.0, "content": "research/multiboostResults.pdf (but not in", "type": "text" } ], "index": 19 }, { "bbox": [ 53, 305, 290, 319 ], "spans": [ { "bbox": [ 53, 305, 290, 319 ], "score": 1.0, "content": "Table 1) we give both errors for some of the methods.", "type": "text" } ], "index": 20 }, { "bbox": [ 53, 318, 291, 331 ], "spans": [ { "bbox": [ 53, 318, 291, 331 ], "score": 1.0, "content": "Proper hyperparameter-validation should put the correct", "type": "text" } ], "index": 21 }, { "bbox": [ 53, 330, 291, 343 ], "spans": [ { "bbox": [ 53, 330, 291, 343 ], "score": 1.0, "content": "test error estimates between those two limits. Since", "type": "text" } ], "index": 22 }, { "bbox": [ 54, 341, 290, 355 ], "spans": [ { "bbox": [ 54, 341, 290, 355 ], "score": 1.0, "content": "ADABOOST.MH with decision products (Kegl & Busa- ´", "type": "text" } ], "index": 23 }, { "bbox": [ 53, 353, 290, 366 ], "spans": [ { "bbox": [ 53, 353, 290, 366 ], "score": 1.0, "content": "Fekete, 2009) is also implemented in multiboost (Ben-", "type": "text" } ], "index": 24 }, { "bbox": [ 53, 366, 290, 378 ], "spans": [ { "bbox": [ 53, 366, 290, 378 ], "score": 1.0, "content": "bouzid et al., 2012), for this method we re-ran experiments", "type": "text" } ], "index": 25 }, { "bbox": [ 54, 379, 194, 390 ], "spans": [ { "bbox": [ 54, 379, 194, 390 ], "score": 1.0, "content": "with the protocol described above.", "type": "text" } ], "index": 26 } ], "index": 13, "bbox_fs": [ 52, 67, 291, 390 ] }, { "type": "text", "bbox": [ 55, 397, 289, 599 ], "lines": [ { "bbox": [ 53, 395, 291, 408 ], "spans": [ { "bbox": [ 53, 395, 291, 408 ], "score": 1.0, "content": "The overall conclusion is that AOSOLOGITBOOST (Sun", "type": "text" } ], "index": 27 }, { "bbox": [ 53, 408, 291, 420 ], "spans": [ { "bbox": [ 53, 408, 291, 420 ], "score": 1.0, "content": "et al., 2012) and ADABOOST.MH with Hamming trees", "type": "text" } ], "index": 28 }, { "bbox": [ 53, 420, 290, 433 ], "spans": [ { "bbox": [ 53, 420, 290, 433 ], "score": 1.0, "content": "are the best algorithms (ADABOOST.MH winning on all", "type": "text" } ], "index": 29 }, { "bbox": [ 53, 431, 291, 444 ], "spans": [ { "bbox": [ 53, 431, 291, 444 ], "score": 1.0, "content": "the five data sets but within one standard deviation).", "type": "text" } ], "index": 30 }, { "bbox": [ 53, 443, 291, 456 ], "spans": [ { "bbox": [ 53, 443, 291, 456 ], "score": 1.0, "content": "ADABOOST.MH with decision products (Kegl & Busa- ´", "type": "text" } ], "index": 31 }, { "bbox": [ 53, 455, 290, 468 ], "spans": [ { "bbox": [ 53, 455, 290, 468 ], "score": 1.0, "content": "Fekete, 2009) and ABCLOGITBOOST are slightly weaker,", "type": "text" } ], "index": 32 }, { "bbox": [ 52, 468, 291, 480 ], "spans": [ { "bbox": [ 52, 468, 291, 480 ], "score": 1.0, "content": "as also noted by (Sun et al., 2012). SAMME (Zhu", "type": "text" } ], "index": 33 }, { "bbox": [ 52, 480, 291, 493 ], "spans": [ { "bbox": [ 52, 480, 291, 493 ], "score": 1.0, "content": "et al., 2009) and ADABOOST.MM (Mukherjee & Schapire,", "type": "text" } ], "index": 34 }, { "bbox": [ 52, 492, 291, 504 ], "spans": [ { "bbox": [ 52, 492, 291, 504 ], "score": 1.0, "content": "2013) perform below the rest of the methods on the", "type": "text" } ], "index": 35 }, { "bbox": [ 53, 504, 290, 516 ], "spans": [ { "bbox": [ 53, 504, 290, 516 ], "score": 1.0, "content": "two data sets shared among all the papers (even though", "type": "text" } ], "index": 36 }, { "bbox": [ 54, 516, 290, 528 ], "spans": [ { "bbox": [ 54, 516, 290, 528 ], "score": 1.0, "content": "we give post-validated results). Another important con-", "type": "text" } ], "index": 37 }, { "bbox": [ 54, 528, 290, 539 ], "spans": [ { "bbox": [ 54, 528, 290, 539 ], "score": 1.0, "content": "clusion is that ADABOOST.MH with Hamming trees is", "type": "text" } ], "index": 38 }, { "bbox": [ 53, 540, 290, 552 ], "spans": [ { "bbox": [ 53, 540, 290, 552 ], "score": 1.0, "content": "significantly better then other implementations of AD-", "type": "text" } ], "index": 39 }, { "bbox": [ 53, 551, 290, 564 ], "spans": [ { "bbox": [ 53, 551, 290, 564 ], "score": 1.0, "content": "ABOOST.MH in (Zhu et al., 2009; Mukherjee & Schapire,", "type": "text" } ], "index": 40 }, { "bbox": [ 52, 562, 291, 577 ], "spans": [ { "bbox": [ 52, 562, 291, 577 ], "score": 1.0, "content": "2013), assumably implemented using single-label trees (the", "type": "text" } ], "index": 41 }, { "bbox": [ 53, 576, 290, 588 ], "spans": [ { "bbox": [ 53, 576, 290, 588 ], "score": 1.0, "content": "errors reported by Mukherjee & Schapire (2013) are espe-", "type": "text" } ], "index": 42 }, { "bbox": [ 54, 588, 136, 600 ], "spans": [ { "bbox": [ 54, 588, 136, 600 ], "score": 1.0, "content": "cially conspicuous).", "type": "text" } ], "index": 43 } ], "index": 35, "bbox_fs": [ 52, 395, 291, 600 ] }, { "type": "text", "bbox": [ 55, 605, 289, 712 ], "lines": [ { "bbox": [ 54, 605, 291, 618 ], "spans": [ { "bbox": [ 54, 605, 291, 618 ], "score": 1.0, "content": "ADABOOST.MH with Hamming trees also achieves good", "type": "text" } ], "index": 44 }, { "bbox": [ 53, 618, 290, 629 ], "spans": [ { "bbox": [ 53, 618, 290, 629 ], "score": 1.0, "content": "results on image recognition problems. On MNIST, boost-", "type": "text" } ], "index": 45 }, { "bbox": [ 53, 630, 291, 641 ], "spans": [ { "bbox": [ 53, 630, 291, 641 ], "score": 1.0, "content": "ing trees of stumps over pixels with eight inner nodes and", "type": "text" } ], "index": 46 }, { "bbox": [ 53, 640, 291, 654 ], "spans": [ { "bbox": [ 53, 640, 226, 654 ], "score": 1.0, "content": "about 50000 iterations has a test error of", "type": "text" }, { "bbox": [ 226, 641, 253, 653 ], "score": 0.88, "content": "1 . 2 5 \\%", "type": "inline_equation" }, { "bbox": [ 253, 640, 291, 654 ], "score": 1.0, "content": ", making", "type": "text" } ], "index": 47 }, { "bbox": [ 54, 654, 290, 664 ], "spans": [ { "bbox": [ 54, 654, 290, 664 ], "score": 1.0, "content": "it one of the best no-domain-knowledge “shallow” classi-", "type": "text" } ], "index": 48 }, { "bbox": [ 53, 665, 290, 677 ], "spans": [ { "bbox": [ 53, 665, 290, 677 ], "score": 1.0, "content": "fiers. Using stumps over Haar filters (Viola & Jones, 2004),", "type": "text" } ], "index": 49 }, { "bbox": [ 53, 677, 290, 689 ], "spans": [ { "bbox": [ 53, 677, 290, 689 ], "score": 1.0, "content": "boosted trees with four inner nodes and 10000 iterations", "type": "text" } ], "index": 50 }, { "bbox": [ 54, 689, 290, 701 ], "spans": [ { "bbox": [ 54, 689, 145, 701 ], "score": 1.0, "content": "achieves a test error of", "type": "text" }, { "bbox": [ 145, 689, 172, 700 ], "score": 0.89, "content": "0 . 8 5 \\%", "type": "inline_equation" }, { "bbox": [ 172, 689, 290, 701 ], "score": 1.0, "content": ", comparable to classical con-", "type": "text" } ], "index": 51 }, { "bbox": [ 54, 701, 200, 713 ], "spans": [ { "bbox": [ 54, 701, 200, 713 ], "score": 1.0, "content": "volutional nets (LeCun et al., 1998).", "type": "text" } ], "index": 52 } ], "index": 48, "bbox_fs": [ 53, 605, 291, 713 ] }, { "type": "text", "bbox": [ 307, 68, 541, 222 ], "lines": [ { "bbox": [ 306, 68, 542, 79 ], "spans": [ { "bbox": [ 306, 68, 542, 79 ], "score": 1.0, "content": "ADABOOST.MH with Hamming trees, usually combined", "type": "text" } ], "index": 53 }, { "bbox": [ 305, 78, 542, 92 ], "spans": [ { "bbox": [ 305, 78, 542, 92 ], "score": 1.0, "content": "with calibration (Platt, 2000; Niculescu-Mizil & Caruana,", "type": "text" } ], "index": 54 }, { "bbox": [ 306, 92, 542, 103 ], "spans": [ { "bbox": [ 306, 92, 542, 103 ], "score": 1.0, "content": "2005) and model averaging, has been also successful in re-", "type": "text" } ], "index": 55 }, { "bbox": [ 305, 104, 542, 115 ], "spans": [ { "bbox": [ 305, 104, 542, 115 ], "score": 1.0, "content": "cent data challenges. On the Kaggle emotions data chal-", "type": "text" } ], "index": 56 }, { "bbox": [ 305, 116, 542, 127 ], "spans": [ { "bbox": [ 305, 116, 542, 127 ], "score": 1.0, "content": "lenge, although not competitive with deep learning tech-", "type": "text" } ], "index": 57 }, { "bbox": [ 305, 127, 542, 140 ], "spans": [ { "bbox": [ 305, 127, 542, 140 ], "score": 1.0, "content": "niques, out-of-the-box ADABOOST.MH with Hamming", "type": "text" } ], "index": 58 }, { "bbox": [ 306, 139, 543, 150 ], "spans": [ { "bbox": [ 306, 139, 543, 150 ], "score": 1.0, "content": "trees over Haar filters finished 17th place with a test error of", "type": "text" } ], "index": 59 }, { "bbox": [ 306, 150, 542, 163 ], "spans": [ { "bbox": [ 306, 151, 325, 162 ], "score": 0.87, "content": "5 7 \\%", "type": "inline_equation" }, { "bbox": [ 326, 150, 542, 163 ], "score": 1.0, "content": ". In the Yahoo! 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Showing (9)", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 114, 105, 480, 391 ], "lines": [ { "bbox": [ 114, 105, 480, 391 ], "spans": [ { "bbox": [ 114, 105, 480, 391 ], "score": 0.93, "content": "\\begin{array} { r l } { Z _ { 1 } \\mathbf { h } _ { N } \\boldsymbol { u } } & { = \\underbrace { \\frac { 1 } { \\sqrt { 2 } } \\frac { \\epsilon _ { \\mathrm { N } } } { \\epsilon _ { \\mathrm { N } } } \\boldsymbol { u } _ { 1 } \\cdot \\boldsymbol { u } _ { N } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { \\mathrm { o l d } } ^ { \\mathrm { o p } } ( x , y ) \\boldsymbol { \\Lambda } _ { 1 } ^ { \\mathrm { o r } } \\geq \\frac { \\epsilon _ { \\mathrm { N } } ^ { \\mathrm { o r } } \\gamma _ { 1 } } { \\omega _ { 1 } ^ { 2 } } , \\alpha \\epsilon _ { \\mathrm { N } } \\gamma _ { 1 } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { \\mathrm { o l d } } ^ { \\mathrm { o p } } ( x , y ) \\boldsymbol { \\Lambda } _ { 1 } ^ { \\mathrm { o r } } \\} , } \\\\ & { = \\quad \\sum _ { i = 1 } ^ { \\infty } \\Bigg \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , \\Bigg . } \\\\ & { = \\quad \\frac { 1 } { \\sqrt { 2 } } \\omega _ { 1 } } ( \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) - \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , } \\\\ & { \\quad - \\sum _ { i = 1 } ^ { \\infty } \\{ \\beta _ { 1 } ^ { \\mathrm { o p } } \\boldsymbol { u } _ { i } ^ { \\mathrm { o p } } ( x , y ) + \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , } \\\\ & = \\quad \\sum _ { i = 1 } ^ { \\infty } \\{ \\alpha _ { i } ^ \\end{array}", "type": "interline_equation", "image_path": "a47054a0d130dcbcae99dc68ca9dd789192e2afd3b0714c261c91560c134de60.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 114, 105, 480, 200.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 114, 200.33333333333331, 480, 295.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 114, 295.66666666666663, 480, 390.99999999999994 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 54, 391, 541, 415 ], "lines": [ { "bbox": [ 51, 388, 543, 407 ], "spans": [ { "bbox": [ 51, 388, 204, 407 ], "score": 1.0, "content": "(16) comes from the definition (6) of", "type": "text" }, { "bbox": [ 204, 393, 212, 402 ], "score": 0.72, "content": "\\mathbf { h }", "type": "inline_equation" }, { "bbox": [ 212, 388, 420, 407 ], "score": 1.0, "content": "and (17) follows from the definitions (7) and (8) of", "type": "text" }, { "bbox": [ 420, 394, 437, 404 ], "score": 0.86, "content": "\\mu _ { \\ell - }", "type": "inline_equation" }, { "bbox": [ 437, 388, 456, 407 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 456, 394, 473, 404 ], "score": 0.87, "content": "\\mu _ { \\ell + }", "type": "inline_equation" }, { "bbox": [ 473, 388, 543, 407 ], "score": 1.0, "content": ". In the final step", "type": "text" } ], "index": 4 }, { "bbox": [ 53, 403, 158, 416 ], "spans": [ { "bbox": [ 53, 403, 158, 416 ], "score": 1.0, "content": "(18) we used the fact that", "type": "text" } ], "index": 5 } ], "index": 4.5 }, { "type": "interline_equation", "bbox": [ 219, 412, 375, 448 ], "lines": [ { "bbox": [ 219, 412, 375, 448 ], "spans": [ { "bbox": [ 219, 412, 375, 448 ], "score": 0.94, "content": "\\sum _ { \\ell = 1 } ^ { K } \\left( \\mu _ { \\ell + } + \\mu _ { \\ell - } \\right) = \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } w _ { i , \\ell } = 1 .", "type": "interline_equation", "image_path": "3ab4138ff2abc6cae8a014285a97e71c3ddf6f125dac894db89eac0726ac511e.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 219, 412, 375, 430.0 ], "spans": [], "index": 6 }, { "bbox": [ 219, 430.0, 375, 448.0 ], "spans": [], "index": 7 } ] }, { "type": "title", "bbox": [ 55, 460, 210, 473 ], "lines": [ { "bbox": [ 53, 459, 210, 476 ], "spans": [ { "bbox": [ 53, 459, 210, 476 ], "score": 1.0, "content": "B. Multi-class decision stumps", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 57, 480, 541, 503 ], "lines": [ { "bbox": [ 55, 479, 543, 495 ], "spans": [ { "bbox": [ 55, 479, 543, 495 ], "score": 1.0, "content": "The simplest scalar base learner used in practice on numerical features is the decision stump, a one-decision two-leaf", "type": "text" } ], "index": 9 }, { "bbox": [ 55, 492, 155, 506 ], "spans": [ { "bbox": [ 55, 492, 155, 506 ], "score": 1.0, "content": "decision tree of the form", "type": "text" } ], "index": 10 } ], "index": 9.5 }, { "type": "interline_equation", "bbox": [ 235, 500, 359, 534 ], "lines": [ { "bbox": [ 235, 500, 359, 534 ], "spans": [ { "bbox": [ 235, 500, 359, 534 ], "score": 0.95, "content": "\\varphi _ { j , b } ( \\mathbf { x } ) = \\left\\{ { \\begin{array} { r l } { 1 } & { { } { \\mathrm { ~ i f ~ } } x ^ { ( j ) } \\geq b , } \\\\ { - 1 } & { { } { \\mathrm { ~ o t h e r w i s e } } , } \\end{array} } \\right.", "type": "interline_equation", "image_path": "33c7ae9b34dc3cab52430ca81bc175baacd8b78ede3aec78206d1ca1158dc3ef.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 235, 500, 359, 517.0 ], "spans": [], "index": 11 }, { "bbox": [ 235, 517.0, 359, 534.0 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 54, 540, 543, 577 ], "lines": [ { "bbox": [ 53, 534, 542, 558 ], "spans": [ { "bbox": [ 53, 534, 81, 558 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 81, 542, 87, 553 ], "score": 0.82, "content": "j", "type": "inline_equation" }, { "bbox": [ 87, 534, 247, 558 ], "score": 1.0, "content": "is the index of the selected feature and", "type": "text" }, { "bbox": [ 248, 542, 253, 551 ], "score": 0.78, "content": "b", "type": "inline_equation" }, { "bbox": [ 254, 534, 443, 558 ], "score": 1.0, "content": "is the decision threshold. If the feature values", "type": "text" }, { "bbox": [ 443, 538, 507, 554 ], "score": 0.92, "content": "\\big ( x _ { 1 } ^ { ( j ) } , \\dots , x _ { n } ^ { ( j ) } \\big )", "type": "inline_equation" }, { "bbox": [ 507, 540, 542, 555 ], "score": 1.0, "content": "are pre-", "type": "text" } ], "index": 13 }, { "bbox": [ 52, 552, 543, 567 ], "spans": [ { "bbox": [ 52, 552, 500, 567 ], "score": 1.0, "content": "ordered before the first boosting iteration, a decision stump maximizing the edge (11) (or minimizing the energy", "type": "text" }, { "bbox": [ 500, 553, 522, 564 ], "score": 0.86, "content": "( 1 6 ) ^ { 5 }", "type": "inline_equation" }, { "bbox": [ 522, 552, 543, 567 ], "score": 1.0, "content": ") can", "type": "text" } ], "index": 14 }, { "bbox": [ 52, 563, 226, 578 ], "spans": [ { "bbox": [ 52, 563, 164, 578 ], "score": 1.0, "content": "be found very efficiently in", "type": "text" }, { "bbox": [ 165, 565, 201, 577 ], "score": 0.91, "content": "\\Theta ( n d K )", "type": "inline_equation" }, { "bbox": [ 201, 563, 226, 578 ], "score": 1.0, "content": "time.", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "text", "bbox": [ 54, 582, 543, 678 ], "lines": [ { "bbox": [ 53, 581, 544, 596 ], "spans": [ { "bbox": [ 53, 581, 459, 596 ], "score": 1.0, "content": "The pseudocode of the algorithm is given in Figure 3. STUMPBASE first calculates the edge vector", "type": "text" }, { "bbox": [ 460, 582, 478, 595 ], "score": 0.91, "content": "\\gamma ^ { ( 0 ) }", "type": "inline_equation" }, { "bbox": [ 478, 581, 544, 596 ], "score": 1.0, "content": "of the constant", "type": "text" } ], "index": 16 }, { "bbox": [ 52, 593, 544, 608 ], "spans": [ { "bbox": [ 52, 593, 92, 608 ], "score": 1.0, "content": "classifier", "type": "text" }, { "bbox": [ 92, 594, 144, 607 ], "score": 0.93, "content": "{ \\bf h } ^ { ( 0 ) } ( { \\bf x } ) \\equiv { \\bf 1 }", "type": "inline_equation" }, { "bbox": [ 144, 593, 544, 608 ], "score": 1.0, "content": "which will serve as the initial edge vector for each featurewise edge-maximizer. Then it loops over", "type": "text" } ], "index": 17 }, { "bbox": [ 53, 606, 543, 619 ], "spans": [ { "bbox": [ 53, 606, 543, 619 ], "score": 1.0, "content": "the features, calls BESTSTUMP to return the best featurewise stump, and then selects the best of the best by minimizing the", "type": "text" } ], "index": 18 }, { "bbox": [ 52, 618, 544, 632 ], "spans": [ { "bbox": [ 52, 618, 306, 632 ], "score": 1.0, "content": "energy (16). BESTSTUMP loops over all (sorted) feature values", "type": "text" }, { "bbox": [ 306, 620, 358, 631 ], "score": 0.9, "content": "s _ { 1 } , \\ldots , s _ { n - 1 }", "type": "inline_equation" }, { "bbox": [ 359, 618, 466, 632 ], "score": 1.0, "content": ". 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Showing (9)", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 52, 63, 133, 83 ] }, { "type": "interline_equation", "bbox": [ 114, 105, 480, 391 ], "lines": [ { "bbox": [ 114, 105, 480, 391 ], "spans": [ { "bbox": [ 114, 105, 480, 391 ], "score": 0.93, "content": "\\begin{array} { r l } { Z _ { 1 } \\mathbf { h } _ { N } \\boldsymbol { u } } & { = \\underbrace { \\frac { 1 } { \\sqrt { 2 } } \\frac { \\epsilon _ { \\mathrm { N } } } { \\epsilon _ { \\mathrm { N } } } \\boldsymbol { u } _ { 1 } \\cdot \\boldsymbol { u } _ { N } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { \\mathrm { o l d } } ^ { \\mathrm { o p } } ( x , y ) \\boldsymbol { \\Lambda } _ { 1 } ^ { \\mathrm { o r } } \\geq \\frac { \\epsilon _ { \\mathrm { N } } ^ { \\mathrm { o r } } \\gamma _ { 1 } } { \\omega _ { 1 } ^ { 2 } } , \\alpha \\epsilon _ { \\mathrm { N } } \\gamma _ { 1 } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { \\mathrm { o l d } } ^ { \\mathrm { o p } } ( x , y ) \\boldsymbol { \\Lambda } _ { 1 } ^ { \\mathrm { o r } } \\} , } \\\\ & { = \\quad \\sum _ { i = 1 } ^ { \\infty } \\Bigg \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , \\Bigg . } \\\\ & { = \\quad \\frac { 1 } { \\sqrt { 2 } } \\omega _ { 1 } } ( \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) - \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , } \\\\ & { \\quad - \\sum _ { i = 1 } ^ { \\infty } \\{ \\beta _ { 1 } ^ { \\mathrm { o p } } \\boldsymbol { u } _ { i } ^ { \\mathrm { o p } } ( x , y ) + \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\{ \\alpha \\mathbf { h } _ { i } ^ { \\mathrm { o r } } ( x , y ) \\} , } \\\\ & = \\quad \\sum _ { i = 1 } ^ { \\infty } \\{ \\alpha _ { i } ^ \\end{array}", "type": "interline_equation", "image_path": "a47054a0d130dcbcae99dc68ca9dd789192e2afd3b0714c261c91560c134de60.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 114, 105, 480, 200.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 114, 200.33333333333331, 480, 295.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 114, 295.66666666666663, 480, 390.99999999999994 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 54, 391, 541, 415 ], "lines": [ { "bbox": [ 51, 388, 543, 407 ], "spans": [ { "bbox": [ 51, 388, 204, 407 ], "score": 1.0, "content": "(16) comes from the definition (6) of", "type": "text" }, { "bbox": [ 204, 393, 212, 402 ], "score": 0.72, "content": "\\mathbf { h }", "type": "inline_equation" }, { "bbox": [ 212, 388, 420, 407 ], "score": 1.0, "content": "and (17) follows from the definitions (7) and (8) of", "type": "text" }, { "bbox": [ 420, 394, 437, 404 ], "score": 0.86, "content": "\\mu _ { \\ell - }", "type": "inline_equation" }, { "bbox": [ 437, 388, 456, 407 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 456, 394, 473, 404 ], "score": 0.87, "content": "\\mu _ { \\ell + }", "type": "inline_equation" }, { "bbox": [ 473, 388, 543, 407 ], "score": 1.0, "content": ". In the final step", "type": "text" } ], "index": 4 }, { "bbox": [ 53, 403, 158, 416 ], "spans": [ { "bbox": [ 53, 403, 158, 416 ], "score": 1.0, "content": "(18) we used the fact that", "type": "text" } ], "index": 5 } ], "index": 4.5, "bbox_fs": [ 51, 388, 543, 416 ] }, { "type": "interline_equation", "bbox": [ 219, 412, 375, 448 ], "lines": [ { "bbox": [ 219, 412, 375, 448 ], "spans": [ { "bbox": [ 219, 412, 375, 448 ], "score": 0.94, "content": "\\sum _ { \\ell = 1 } ^ { K } \\left( \\mu _ { \\ell + } + \\mu _ { \\ell - } \\right) = \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { K } w _ { i , \\ell } = 1 .", "type": "interline_equation", "image_path": "3ab4138ff2abc6cae8a014285a97e71c3ddf6f125dac894db89eac0726ac511e.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 219, 412, 375, 430.0 ], "spans": [], "index": 6 }, { "bbox": [ 219, 430.0, 375, 448.0 ], "spans": [], "index": 7 } ] }, { "type": "title", "bbox": [ 55, 460, 210, 473 ], "lines": [ { "bbox": [ 53, 459, 210, 476 ], "spans": [ { "bbox": [ 53, 459, 210, 476 ], "score": 1.0, "content": "B. Multi-class decision stumps", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 57, 480, 541, 503 ], "lines": [ { "bbox": [ 55, 479, 543, 495 ], "spans": [ { "bbox": [ 55, 479, 543, 495 ], "score": 1.0, "content": "The simplest scalar base learner used in practice on numerical features is the decision stump, a one-decision two-leaf", "type": "text" } ], "index": 9 }, { "bbox": [ 55, 492, 155, 506 ], "spans": [ { "bbox": [ 55, 492, 155, 506 ], "score": 1.0, "content": "decision tree of the form", "type": "text" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 55, 479, 543, 506 ] }, { "type": "interline_equation", "bbox": [ 235, 500, 359, 534 ], "lines": [ { "bbox": [ 235, 500, 359, 534 ], "spans": [ { "bbox": [ 235, 500, 359, 534 ], "score": 0.95, "content": "\\varphi _ { j , b } ( \\mathbf { x } ) = \\left\\{ { \\begin{array} { r l } { 1 } & { { } { \\mathrm { ~ i f ~ } } x ^ { ( j ) } \\geq b , } \\\\ { - 1 } & { { } { \\mathrm { ~ o t h e r w i s e } } , } \\end{array} } \\right.", "type": "interline_equation", "image_path": "33c7ae9b34dc3cab52430ca81bc175baacd8b78ede3aec78206d1ca1158dc3ef.jpg" } ] } ], "index": 11.5, "virtual_lines": [ { "bbox": [ 235, 500, 359, 517.0 ], "spans": [], "index": 11 }, { "bbox": [ 235, 517.0, 359, 534.0 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 54, 540, 543, 577 ], "lines": [ { "bbox": [ 53, 534, 542, 558 ], "spans": [ { "bbox": [ 53, 534, 81, 558 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 81, 542, 87, 553 ], "score": 0.82, "content": "j", "type": "inline_equation" }, { "bbox": [ 87, 534, 247, 558 ], "score": 1.0, "content": "is the index of the selected feature and", "type": "text" }, { "bbox": [ 248, 542, 253, 551 ], "score": 0.78, "content": "b", "type": "inline_equation" }, { "bbox": [ 254, 534, 443, 558 ], "score": 1.0, "content": "is the decision threshold. If the feature values", "type": "text" }, { "bbox": [ 443, 538, 507, 554 ], "score": 0.92, "content": "\\big ( x _ { 1 } ^ { ( j ) } , \\dots , x _ { n } ^ { ( j ) } \\big )", "type": "inline_equation" }, { "bbox": [ 507, 540, 542, 555 ], "score": 1.0, "content": "are pre-", "type": "text" } ], "index": 13 }, { "bbox": [ 52, 552, 543, 567 ], "spans": [ { "bbox": [ 52, 552, 500, 567 ], "score": 1.0, "content": "ordered before the first boosting iteration, a decision stump maximizing the edge (11) (or minimizing the energy", "type": "text" }, { "bbox": [ 500, 553, 522, 564 ], "score": 0.86, "content": "( 1 6 ) ^ { 5 }", "type": "inline_equation" }, { "bbox": [ 522, 552, 543, 567 ], "score": 1.0, "content": ") can", "type": "text" } ], "index": 14 }, { "bbox": [ 52, 563, 226, 578 ], "spans": [ { "bbox": [ 52, 563, 164, 578 ], "score": 1.0, "content": "be found very efficiently in", "type": "text" }, { "bbox": [ 165, 565, 201, 577 ], "score": 0.91, "content": "\\Theta ( n d K )", "type": "inline_equation" }, { "bbox": [ 201, 563, 226, 578 ], "score": 1.0, "content": "time.", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 52, 534, 543, 578 ] }, { "type": "text", "bbox": [ 54, 582, 543, 678 ], "lines": [ { "bbox": [ 53, 581, 544, 596 ], "spans": [ { "bbox": [ 53, 581, 459, 596 ], "score": 1.0, "content": "The pseudocode of the algorithm is given in Figure 3. STUMPBASE first calculates the edge vector", "type": "text" }, { "bbox": [ 460, 582, 478, 595 ], "score": 0.91, "content": "\\gamma ^ { ( 0 ) }", "type": "inline_equation" }, { "bbox": [ 478, 581, 544, 596 ], "score": 1.0, "content": "of the constant", "type": "text" } ], "index": 16 }, { "bbox": [ 52, 593, 544, 608 ], "spans": [ { "bbox": [ 52, 593, 92, 608 ], "score": 1.0, "content": "classifier", "type": "text" }, { "bbox": [ 92, 594, 144, 607 ], "score": 0.93, "content": "{ \\bf h } ^ { ( 0 ) } ( { \\bf x } ) \\equiv { \\bf 1 }", "type": "inline_equation" }, { "bbox": [ 144, 593, 544, 608 ], "score": 1.0, "content": "which will serve as the initial edge vector for each featurewise edge-maximizer. 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The total edge", "type": "text" } ], "index": 22 }, { "bbox": [ 52, 666, 506, 680 ], "spans": [ { "bbox": [ 52, 666, 64, 680 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 65, 667, 92, 678 ], "score": 0.92, "content": "\\mathbf { v } \\varphi ( \\mathbf { x } )", "type": "inline_equation" }, { "bbox": [ 92, 666, 442, 680 ], "score": 1.0, "content": "with optimal votes (13) is then the sum of the absolute values of the classwise edges of", "type": "text" }, { "bbox": [ 442, 666, 469, 678 ], "score": 0.91, "content": "{ \\bf 1 } \\varphi ( { \\bf x } )", "type": "inline_equation" }, { "bbox": [ 470, 666, 506, 680 ], "score": 1.0, "content": "(line 7).", "type": "text" } ], "index": 23 } ], "index": 19.5, "bbox_fs": [ 50, 581, 545, 680 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 50, 65, 545, 481 ], "blocks": [ { "type": "table_body", "bbox": [ 50, 65, 545, 481 ], "group_id": 0, "lines": [ { "bbox": [ 50, 65, 545, 481 ], "spans": [ { "bbox": [ 50, 65, 545, 481 ], "score": 0.801, "html": "
STUMPBASE(X, Y,W)
1 forl←1 to K>for all classes n > 2 20↑ Wi,lyi,l> classwise edges (1O) of constant classifier h(O)(x) = 1
3 for j←1 to di=1 D all (numerical) features
4s ← SORT(𝑥x(),..,x)） sort the jth column of X
5(vj,bj,γj) ← BEsTSTUMP(s,Y,W,γ(0))D best stump per feature
6 αj←1+Yj log base coefficient (12) 2 1-Yj
7 j* ← arg min Z(αjVjj,b,W)> best stump across features
J return (αj*,Vj*,j*,b*(·))
BESTSTUMP(s,Y,W, γ(0))
1γ*← γ(0）best edge vector
2γ ← γ(o）initial edge vector
3for i← 1to n-1 >for all points in order s1 ≤...≤ sn-1
4for l ← 1 to K>for all classes
5
Ye ← Ye - 2wi,lyi,l > update classwise edges of stump with v = 1
6if si Si+1 then> no threshold if identical coordinates Si = Si+1
7if∑=1 lel >∑ε-1l|thenfound beter stump
8γ* ←γupdate best edge vector
96*←Si+si+1 update best threshold 2
10forl ←1 to Kfor all classes
11U ← sign(γe) D set vote vector according to (13)
12ifγ*=γ(0） did not beat the constant classifier
13return (v*,-O,llγ*lli）D constant classifier with optimal votes
14else
15
return (v*,b*,llγ*ll1）)>best stump
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STUMPBASE(X, Y,W)
1 forl←1 to K>for all classes n > 2 20↑ Wi,lyi,l> classwise edges (1O) of constant classifier h(O)(x) = 1
3 for j←1 to di=1 D all (numerical) features
4s ← SORT(𝑥x(),..,x)） sort the jth column of X
5(vj,bj,γj) ← BEsTSTUMP(s,Y,W,γ(0))D best stump per feature
6 αj←1+Yj log base coefficient (12) 2 1-Yj
7 j* ← arg min Z(αjVjj,b,W)> best stump across features
J return (αj*,Vj*,j*,b*(·))
BESTSTUMP(s,Y,W, γ(0))
1γ*← γ(0）best edge vector
2γ ← γ(o）initial edge vector
3for i← 1to n-1 >for all points in order s1 ≤...≤ sn-1
4for l ← 1 to K>for all classes
5
Ye ← Ye - 2wi,lyi,l > update classwise edges of stump with v = 1
6if si Si+1 then> no threshold if identical coordinates Si = Si+1
7if∑=1 lel >∑ε-1l|thenfound beter stump
8γ* ←γupdate best edge vector
96*←Si+si+1 update best threshold 2
10forl ←1 to Kfor all classes
11U ← sign(γe) D set vote vector according to (13)
12ifγ*=γ(0） did not beat the constant classifier
13return (v*,-O,llγ*lli）D constant classifier with optimal votes
14else
15
return (v*,b*,llγ*ll1）)>best stump
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