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Our results demonstrate that sources with high similarity to the target are", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 261, 506, 279 ], "spans": [ { "bbox": [ 104, 261, 506, 279 ], "score": 1.0, "content": "more effective at reducing the target generalization error. 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(2019); Blitzer et al. (2007); Azizzadenesheli et al. (2018);", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 516, 506, 530 ], "spans": [ { "bbox": [ 106, 516, 506, 530 ], "score": 1.0, "content": "Long et al. (2016); Shen et al. (2018). Most of this literature assume that source and target share a", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "common labeling rule but there is a shift in the marginal distributions. There are many upper bounds", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 540, 505, 551 ], "spans": [ { "bbox": [ 106, 540, 505, 551 ], "score": 1.0, "content": "for the target generalization error in this setting this setting. For instance, Ben-David et al. (2007;", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 548, 506, 564 ], "spans": [ { "bbox": [ 105, 548, 506, 564 ], "score": 1.0, "content": "2010) gives an upper bound for the target generalization error in terms of source generalization error", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "and a divergence measure between the domains that can be estimated by finitely many unlabeled", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 570, 505, 585 ], "spans": [ { "bbox": [ 105, 570, 505, 585 ], "score": 1.0, "content": "data from the source and target. In another work Mansour et al. (2009) introduces a new discrepancy", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 582, 505, 595 ], "spans": [ { "bbox": [ 105, 582, 505, 595 ], "score": 1.0, "content": "distance and generalizes the results of Ben-David et al. (2007) for a wide family of loss functions", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 594, 505, 606 ], "spans": [ { "bbox": [ 106, 594, 505, 606 ], "score": 1.0, "content": "using Rademacher complexity. Similar to this setting, but for multiple source domain adaption", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 603, 506, 618 ], "spans": [ { "bbox": [ 105, 603, 506, 618 ], "score": 1.0, "content": "scheme, Mansour et al. (2021) proposes a family of algorithms based on the idea of model selection", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 616, 505, 627 ], "spans": [ { "bbox": [ 106, 616, 505, 627 ], "score": 1.0, "content": "under the assumption that target distribution is close to some convex combination of sources. A more", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 626, 505, 639 ], "spans": [ { "bbox": [ 105, 626, 505, 639 ], "score": 1.0, "content": "recent work Lei et al. 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(2019); Blitzer et al. (2007); Azizzadenesheli et al. (2018);", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 516, 506, 530 ], "spans": [ { "bbox": [ 106, 516, 506, 530 ], "score": 1.0, "content": "Long et al. (2016); Shen et al. (2018). Most of this literature assume that source and target share a", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 528, 505, 540 ], "spans": [ { "bbox": [ 106, 528, 505, 540 ], "score": 1.0, "content": "common labeling rule but there is a shift in the marginal distributions. There are many upper bounds", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 540, 505, 551 ], "spans": [ { "bbox": [ 106, 540, 505, 551 ], "score": 1.0, "content": "for the target generalization error in this setting this setting. For instance, Ben-David et al. (2007;", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 548, 506, 564 ], "spans": [ { "bbox": [ 105, 548, 506, 564 ], "score": 1.0, "content": "2010) gives an upper bound for the target generalization error in terms of source generalization error", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 561, 505, 573 ], "spans": [ { "bbox": [ 106, 561, 505, 573 ], "score": 1.0, "content": "and a divergence measure between the domains that can be estimated by finitely many unlabeled", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 570, 505, 585 ], "spans": [ { "bbox": [ 105, 570, 505, 585 ], "score": 1.0, "content": "data from the source and target. In another work Mansour et al. (2009) introduces a new discrepancy", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 582, 505, 595 ], "spans": [ { "bbox": [ 105, 582, 505, 595 ], "score": 1.0, "content": "distance and generalizes the results of Ben-David et al. (2007) for a wide family of loss functions", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 594, 505, 606 ], "spans": [ { "bbox": [ 106, 594, 505, 606 ], "score": 1.0, "content": "using Rademacher complexity. Similar to this setting, but for multiple source domain adaption", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 603, 506, 618 ], "spans": [ { "bbox": [ 105, 603, 506, 618 ], "score": 1.0, "content": "scheme, Mansour et al. (2021) proposes a family of algorithms based on the idea of model selection", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 616, 505, 627 ], "spans": [ { "bbox": [ 106, 616, 505, 627 ], "score": 1.0, "content": "under the assumption that target distribution is close to some convex combination of sources. A more", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 626, 505, 639 ], "spans": [ { "bbox": [ 105, 626, 505, 639 ], "score": 1.0, "content": "recent work Lei et al. (2021) studies linear regression under shift distribution including covariate", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 637, 505, 651 ], "spans": [ { "bbox": [ 105, 637, 505, 651 ], "score": 1.0, "content": "shift (i.e. conditional distributions of source and target are the same) as well as model shift (i.e. only", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 648, 505, 662 ], "spans": [ { "bbox": [ 106, 648, 505, 662 ], "score": 1.0, "content": "distributions of the features of the source and target are the same) and develops algorithms achieving", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 659, 273, 673 ], "spans": [ { "bbox": [ 105, 659, 273, 673 ], "score": 1.0, "content": "near optimal minimax risk in this setting.", "type": "text" } ], "index": 47 } ], "index": 39, "bbox_fs": [ 105, 483, 506, 673 ] }, { "type": "text", "bbox": [ 107, 677, 504, 732 ], "lines": [ { "bbox": [ 106, 677, 506, 689 ], "spans": [ { "bbox": [ 106, 677, 506, 689 ], "score": 1.0, "content": "In addition to upper bounds, there are also a few results which provide lower bounds for target", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 686, 506, 700 ], "spans": [ { "bbox": [ 105, 686, 506, 700 ], "score": 1.0, "content": "generalization error. David et al. (2010) provides impossibility results under the assumption of", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "covariate shift and small discrepancy of unlabeled distributions. Mousavi Kalan et al. (2020) studies", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 709, 505, 721 ], "spans": [ { "bbox": [ 106, 709, 505, 721 ], "score": 1.0, "content": "transfer learning with one hidden layer neural networks for regression problems. This result defines a", "type": "text" } ], "index": 51 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 505, 733 ], "score": 1.0, "content": "notion of similarity between the source and target tasks based on a distance between the ground truth", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "parameters of the source and target networks. Using this distance this paper develops a statistical", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 94, 506, 106 ], "spans": [ { "bbox": [ 106, 94, 506, 106 ], "score": 1.0, "content": "minimax lower bound for the target generalization error in terms of the number of source and target", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 106, 104, 505, 117 ], "spans": [ { "bbox": [ 106, 104, 505, 117 ], "score": 1.0, "content": "samples as well as the defined similarity of the source and target under the distribution shift with the", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 106, 115, 505, 128 ], "spans": [ { "bbox": [ 106, 115, 505, 128 ], "score": 1.0, "content": "assumption that the features are generated by Gaussian distributions. Compared to Mousavi Kalan", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 105, 127, 506, 139 ], "spans": [ { "bbox": [ 105, 127, 506, 139 ], "score": 1.0, "content": "et al. (2020) our result has quite a few unique advantages: (1) We do not assume that the source", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 106, 138, 506, 150 ], "spans": [ { "bbox": [ 106, 138, 506, 150 ], "score": 1.0, "content": "and target data are generated according to a planted (teacher) network and our results now even", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 105, 149, 505, 161 ], "spans": [ { "bbox": [ 105, 149, 505, 161 ], "score": 1.0, "content": "hold in the agnostic setting. (2) Mousavi Kalan et al. (2020) applies to regression problems but this", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 105, 159, 505, 172 ], "spans": [ { "bbox": [ 105, 159, 505, 172 ], "score": 1.0, "content": "result covers classification (3) Mousavi Kalan et al. (2020) only considered one-hidden layer neural", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 105, 169, 505, 184 ], "spans": [ { "bbox": [ 105, 169, 505, 184 ], "score": 1.0, "content": "networks for predicting the labels of extracted features. In this result we can handle arbitrary deep", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 181, 505, 194 ], "spans": [ { "bbox": [ 105, 181, 505, 194 ], "score": 1.0, "content": "neural networks. (4) Our notion of similarity between the source and target distributions can be much", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 193, 505, 204 ], "spans": [ { "bbox": [ 106, 193, 505, 204 ], "score": 1.0, "content": "more easily estimated by using only a few target data without the need for estimating the ground truth", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 105, 204, 345, 215 ], "spans": [ { "bbox": [ 105, 204, 345, 215 ], "score": 1.0, "content": "target parameters which requires lots of labeled target data.", "type": "text", "cross_page": true } ], "index": 11 } ], "index": 50, "bbox_fs": [ 105, 677, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 214 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "parameters of the source and target networks. Using this distance this paper develops a statistical", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 506, 106 ], "spans": [ { "bbox": [ 106, 94, 506, 106 ], "score": 1.0, "content": "minimax lower bound for the target generalization error in terms of the number of source and target", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 117 ], "spans": [ { "bbox": [ 106, 104, 505, 117 ], "score": 1.0, "content": "samples as well as the defined similarity of the source and target under the distribution shift with the", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 115, 505, 128 ], "spans": [ { "bbox": [ 106, 115, 505, 128 ], "score": 1.0, "content": "assumption that the features are generated by Gaussian distributions. Compared to Mousavi Kalan", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 127, 506, 139 ], "spans": [ { "bbox": [ 105, 127, 506, 139 ], "score": 1.0, "content": "et al. (2020) our result has quite a few unique advantages: (1) We do not assume that the source", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 138, 506, 150 ], "spans": [ { "bbox": [ 106, 138, 506, 150 ], "score": 1.0, "content": "and target data are generated according to a planted (teacher) network and our results now even", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 149, 505, 161 ], "spans": [ { "bbox": [ 105, 149, 505, 161 ], "score": 1.0, "content": "hold in the agnostic setting. (2) Mousavi Kalan et al. (2020) applies to regression problems but this", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 159, 505, 172 ], "spans": [ { "bbox": [ 105, 159, 505, 172 ], "score": 1.0, "content": "result covers classification (3) Mousavi Kalan et al. (2020) only considered one-hidden layer neural", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 169, 505, 184 ], "spans": [ { "bbox": [ 105, 169, 505, 184 ], "score": 1.0, "content": "networks for predicting the labels of extracted features. In this result we can handle arbitrary deep", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 181, 505, 194 ], "spans": [ { "bbox": [ 105, 181, 505, 194 ], "score": 1.0, "content": "neural networks. (4) Our notion of similarity between the source and target distributions can be much", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 193, 505, 204 ], "spans": [ { "bbox": [ 106, 193, 505, 204 ], "score": 1.0, "content": "more easily estimated by using only a few target data without the need for estimating the ground truth", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 204, 345, 215 ], "spans": [ { "bbox": [ 105, 204, 345, 215 ], "score": 1.0, "content": "target parameters which requires lots of labeled target data.", "type": "text" } ], "index": 11 } ], "index": 5.5 }, { "type": "text", "bbox": [ 107, 219, 505, 319 ], "lines": [ { "bbox": [ 106, 220, 505, 232 ], "spans": [ { "bbox": [ 106, 220, 505, 232 ], "score": 1.0, "content": "More closely related to this work Hanneke & Kpotufe (2019) derives a minimax lower bound for", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 506, 243 ], "spans": [ { "bbox": [ 105, 231, 506, 243 ], "score": 1.0, "content": "target generalization error in binary classification under the assumption of a relaxed version of", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 243, 505, 254 ], "spans": [ { "bbox": [ 105, 243, 505, 254 ], "score": 1.0, "content": "covariate shift and small transfer exponent parameter which is defined to measure the discrepancy of", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 251, 505, 267 ], "spans": [ { "bbox": [ 105, 251, 505, 267 ], "score": 1.0, "content": "the source and target distributions. Our work differs from this previous work as except for assuming", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 263, 505, 276 ], "spans": [ { "bbox": [ 106, 263, 505, 276 ], "score": 1.0, "content": "the VC dimension of the model is finite we do not make any further assumptions. This makes our", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 274, 506, 288 ], "spans": [ { "bbox": [ 105, 274, 506, 288 ], "score": 1.0, "content": "results applicable in a much broader set of classifications or decision making problems. Furthermore,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 285, 505, 299 ], "spans": [ { "bbox": [ 105, 285, 505, 299 ], "score": 1.0, "content": "our lower bound can be evaluated on real data sets and serve as a guideline to practitioners helping", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 296, 505, 310 ], "spans": [ { "bbox": [ 105, 296, 505, 310 ], "score": 1.0, "content": "them decide when utilizing additional knowledge from a source domain is useful for a given target", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 307, 128, 320 ], "spans": [ { "bbox": [ 105, 307, 128, 320 ], "score": 1.0, "content": "task.", "type": "text" } ], "index": 20 } ], "index": 16 }, { "type": "text", "bbox": [ 106, 325, 505, 446 ], "lines": [ { "bbox": [ 105, 323, 506, 338 ], "spans": [ { "bbox": [ 105, 323, 506, 338 ], "score": 1.0, "content": "Most of the literature in transfer learning try to provide sufficiency and necessity results by deriving", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 506, 348 ], "spans": [ { "bbox": [ 105, 335, 506, 348 ], "score": 1.0, "content": "upper and lower bounds for target generalization error in a relatively general setting. However, these", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 346, 505, 358 ], "spans": [ { "bbox": [ 105, 346, 505, 358 ], "score": 1.0, "content": "papers often require a variety of assumptions to find the optimal classifier in a target domain in closed", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 357, 505, 370 ], "spans": [ { "bbox": [ 106, 357, 505, 370 ], "score": 1.0, "content": "form. For instance, Karbalayghareh et al. (2019; 2018) defines a joint prior distribution of source", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 369, 505, 380 ], "spans": [ { "bbox": [ 106, 369, 505, 380 ], "score": 1.0, "content": "and target domains using a Wishart distribution which relate the source and target tasks and then", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 380, 505, 391 ], "spans": [ { "bbox": [ 106, 380, 505, 391 ], "score": 1.0, "content": "makes it possible to study and understand the transferability between domains. Furthermore, in this", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 391, 505, 402 ], "spans": [ { "bbox": [ 106, 391, 505, 402 ], "score": 1.0, "content": "setting, the authors develop a closed form optimal Bayesian transfer learning and demonstrate its", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 401, 506, 414 ], "spans": [ { "bbox": [ 105, 401, 506, 414 ], "score": 1.0, "content": "advantage over a classifier obtained by only target data. Related to this setting but for regressions,", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 411, 505, 425 ], "spans": [ { "bbox": [ 105, 411, 505, 425 ], "score": 1.0, "content": "Karbalayghareh et al. (2018) obtains the optimal Bayesian transfer learning under setting of joint", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 423, 505, 436 ], "spans": [ { "bbox": [ 106, 423, 505, 436 ], "score": 1.0, "content": "Gaussian feature/label distribution. In contrast with the above in our paper we do not make any", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 434, 293, 446 ], "spans": [ { "bbox": [ 105, 434, 293, 446 ], "score": 1.0, "content": "assumptions about the distribution of the data.", "type": "text" } ], "index": 31 } ], "index": 26 }, { "type": "title", "bbox": [ 108, 461, 255, 473 ], "lines": [ { "bbox": [ 105, 459, 257, 476 ], "spans": [ { "bbox": [ 105, 459, 257, 476 ], "score": 1.0, "content": "3 PROBLEM FORMULATION", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 484, 505, 574 ], "lines": [ { "bbox": [ 105, 485, 505, 498 ], "spans": [ { "bbox": [ 105, 485, 505, 498 ], "score": 1.0, "content": "We consider a transfer learning problem where there are some labeled training data from a source", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 496, 506, 509 ], "spans": [ { "bbox": [ 106, 496, 506, 509 ], "score": 1.0, "content": "task and a target task with the goal of inferring a hypothesis function with small generalization error", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 508, 505, 520 ], "spans": [ { "bbox": [ 106, 508, 315, 520 ], "score": 1.0, "content": "in the target task. More specifically, we assume have", "type": "text" }, { "bbox": [ 315, 509, 328, 518 ], "score": 0.86, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 328, 508, 345, 520 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 345, 509, 359, 518 ], "score": 0.84, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 360, 508, 505, 520 ], "score": 1.0, "content": "source and target labeled data where", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 519, 505, 531 ], "spans": [ { "bbox": [ 106, 519, 505, 531 ], "score": 1.0, "content": "each training data consists of an input/feature as well as an output/label. We denote the source and", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 529, 505, 542 ], "spans": [ { "bbox": [ 105, 529, 172, 542 ], "score": 1.0, "content": "training data by", "type": "text" }, { "bbox": [ 172, 529, 228, 541 ], "score": 0.93, "content": "( \\pmb { x } _ { S } , y _ { S } ) \\sim \\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 228, 529, 246, 542 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 246, 529, 304, 541 ], "score": 0.93, "content": "\\bar { \\mathbf { \\Omega } } ( \\mathbf { x } _ { T } , y _ { T } ) \\sim \\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 305, 529, 387, 542 ], "score": 1.0, "content": ", respectively, where", "type": "text" }, { "bbox": [ 387, 529, 451, 541 ], "score": 0.93, "content": "y _ { S } , y _ { T } \\in \\{ 0 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 452, 529, 469, 542 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 469, 529, 489, 541 ], "score": 0.88, "content": "\\mathbb { P } , \\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 490, 529, 505, 542 ], "score": 1.0, "content": "are", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 540, 505, 554 ], "spans": [ { "bbox": [ 105, 540, 505, 554 ], "score": 1.0, "content": "the joint feature-label distributions of source and target data. Additionally, we assume that source", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 551, 506, 564 ], "spans": [ { "bbox": [ 105, 551, 295, 564 ], "score": 1.0, "content": "and target features/inputs share a same domain,", "type": "text" }, { "bbox": [ 295, 552, 344, 563 ], "score": 0.91, "content": "{ \\pmb x } _ { S } , { \\pmb x } _ { T } \\in { \\chi }", "type": "inline_equation" }, { "bbox": [ 345, 551, 365, 564 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 365, 551, 398, 561 ], "score": 0.9, "content": "\\mathcal { H } \\subset 2 ^ { \\chi }", "type": "inline_equation" }, { "bbox": [ 398, 551, 506, 564 ], "score": 1.0, "content": "denotes a fixed hypothesis", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 562, 226, 575 ], "spans": [ { "bbox": [ 106, 562, 148, 575 ], "score": 1.0, "content": "class with", "type": "text" }, { "bbox": [ 149, 563, 162, 574 ], "score": 0.88, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 163, 562, 226, 575 ], "score": 1.0, "content": "VC-dimension.", "type": "text" } ], "index": 40 } ], "index": 36.5 }, { "type": "text", "bbox": [ 105, 578, 504, 602 ], "lines": [ { "bbox": [ 105, 578, 506, 592 ], "spans": [ { "bbox": [ 105, 578, 338, 592 ], "score": 1.0, "content": "In transfer learning the goal is to find a hypothesis from", "type": "text" }, { "bbox": [ 338, 579, 348, 589 ], "score": 0.77, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 348, 578, 506, 592 ], "score": 1.0, "content": "that minimizing the target excess risk", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 590, 364, 602 ], "spans": [ { "bbox": [ 106, 590, 364, 602 ], "score": 1.0, "content": "defined below based on a combination of source and target data.", "type": "text" } ], "index": 42 } ], "index": 41.5 }, { "type": "text", "bbox": [ 106, 609, 505, 643 ], "lines": [ { "bbox": [ 106, 610, 505, 622 ], "spans": [ { "bbox": [ 106, 610, 315, 622 ], "score": 1.0, "content": "Definition 1 (Excess risk) For a hypothesis function", "type": "text" }, { "bbox": [ 316, 611, 343, 621 ], "score": 0.9, "content": "h \\in \\mathcal H", "type": "inline_equation" }, { "bbox": [ 344, 610, 505, 622 ], "score": 1.0, "content": "and source and target label-feature data", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 621, 505, 634 ], "spans": [ { "bbox": [ 105, 621, 259, 634 ], "score": 1.0, "content": "generated according to distributions", "type": "text" }, { "bbox": [ 259, 622, 267, 631 ], "score": 0.81, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 267, 621, 286, 634 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 287, 622, 297, 632 ], "score": 0.72, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 298, 621, 360, 633 ], "score": 0.87, "content": "( ( \\pmb { x } _ { S } , y _ { S } ) \\sim \\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 360, 621, 380, 634 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 380, 621, 442, 633 ], "score": 0.91, "content": "( \\pmb { x } _ { T } , \\pmb { y } _ { T } ) \\sim \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 443, 621, 505, 634 ], "score": 1.0, "content": ", we define the", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 632, 271, 645 ], "spans": [ { "bbox": [ 105, 632, 271, 645 ], "score": 1.0, "content": "source and target excess risks as follows", "type": "text" } ], "index": 45 } ], "index": 44 }, { "type": "interline_equation", "bbox": [ 211, 645, 399, 659 ], "lines": [ { "bbox": [ 211, 645, 399, 659 ], "spans": [ { "bbox": [ 211, 645, 399, 659 ], "score": 0.88, "content": "\\mathcal { E } _ { T } ( h ) = \\mathbb { Q } [ h ( \\mathbf { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { * } ( \\mathbf { x } _ { T } ) \\neq y _ { T } ]", "type": "interline_equation", "image_path": "8bed66d7e12cbd52b0964c90a062dd1167fdce9b26cd8510b60777279d9a3ef8.jpg" } ] } ], "index": 46, "virtual_lines": [ { "bbox": [ 211, 645, 399, 659 ], "spans": [], "index": 46 } ] }, { "type": "text", "bbox": [ 106, 661, 124, 672 ], "lines": [ { "bbox": [ 105, 660, 125, 672 ], "spans": [ { "bbox": [ 105, 660, 125, 672 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 47 } ], "index": 47 }, { "type": "interline_equation", "bbox": [ 106, 673, 434, 704 ], "lines": [ { "bbox": [ 132, 673, 434, 704 ], "spans": [ { "bbox": [ 132, 673, 434, 704 ], "score": 0.8, "content": "\\begin{array} { c } { \\displaystyle \\varepsilon _ { S } ( h ) = \\mathbb { P } [ h ( \\pmb { x } _ { S } ) \\neq y _ { S } ] - \\mathbb { P } [ h _ { S } ^ { \\ast } ( \\pmb { x } _ { S } ) \\neq y _ { S } ] } \\\\ { \\displaystyle h _ { T } ^ { \\ast } = \\arg \\operatorname* { m i n } _ { h \\in \\mathcal { H } } \\mathbb { Q } [ h ( \\pmb { x } _ { T } ) \\neq y _ { T } ] a n d h _ { S } ^ { \\ast } = \\arg \\operatorname* { m i n } _ { h \\in \\mathcal { H } } \\mathbb { P } [ h ( \\pmb { x } _ { S } ) \\neq y _ { S } ] } \\end{array}", "type": "interline_equation", "image_path": "367fe765182f3a8464385a5370987108764d33d5fb90ef1fac069d81b0bd5955.jpg" } ] } ], "index": 49, "virtual_lines": [ { "bbox": [ 106, 673, 434, 683.3333333333334 ], "spans": [], "index": 48 }, { "bbox": [ 106, 683.3333333333334, 434, 693.6666666666667 ], "spans": [], "index": 49 }, { "bbox": [ 106, 693.6666666666667, 434, 704.0000000000001 ], "spans": [], "index": 50 } ] }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "Next, we need to define an appropriate notion of distance between the source and target. In the", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "literature of domain adaptation, where the conditional expectation remains unchanged and there is", "type": "text" } ], "index": 52 } ], "index": 51.5 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 214 ], "lines": [], "index": 5.5, "bbox_fs": [ 105, 82, 506, 215 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 219, 505, 319 ], "lines": [ { "bbox": [ 106, 220, 505, 232 ], "spans": [ { "bbox": [ 106, 220, 505, 232 ], "score": 1.0, "content": "More closely related to this work Hanneke & Kpotufe (2019) derives a minimax lower bound for", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 506, 243 ], "spans": [ { "bbox": [ 105, 231, 506, 243 ], "score": 1.0, "content": "target generalization error in binary classification under the assumption of a relaxed version of", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 243, 505, 254 ], "spans": [ { "bbox": [ 105, 243, 505, 254 ], "score": 1.0, "content": "covariate shift and small transfer exponent parameter which is defined to measure the discrepancy of", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 251, 505, 267 ], "spans": [ { "bbox": [ 105, 251, 505, 267 ], "score": 1.0, "content": "the source and target distributions. Our work differs from this previous work as except for assuming", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 263, 505, 276 ], "spans": [ { "bbox": [ 106, 263, 505, 276 ], "score": 1.0, "content": "the VC dimension of the model is finite we do not make any further assumptions. This makes our", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 274, 506, 288 ], "spans": [ { "bbox": [ 105, 274, 506, 288 ], "score": 1.0, "content": "results applicable in a much broader set of classifications or decision making problems. Furthermore,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 285, 505, 299 ], "spans": [ { "bbox": [ 105, 285, 505, 299 ], "score": 1.0, "content": "our lower bound can be evaluated on real data sets and serve as a guideline to practitioners helping", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 296, 505, 310 ], "spans": [ { "bbox": [ 105, 296, 505, 310 ], "score": 1.0, "content": "them decide when utilizing additional knowledge from a source domain is useful for a given target", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 307, 128, 320 ], "spans": [ { "bbox": [ 105, 307, 128, 320 ], "score": 1.0, "content": "task.", "type": "text" } ], "index": 20 } ], "index": 16, "bbox_fs": [ 105, 220, 506, 320 ] }, { "type": "text", "bbox": [ 106, 325, 505, 446 ], "lines": [ { "bbox": [ 105, 323, 506, 338 ], "spans": [ { "bbox": [ 105, 323, 506, 338 ], "score": 1.0, "content": "Most of the literature in transfer learning try to provide sufficiency and necessity results by deriving", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 506, 348 ], "spans": [ { "bbox": [ 105, 335, 506, 348 ], "score": 1.0, "content": "upper and lower bounds for target generalization error in a relatively general setting. However, these", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 346, 505, 358 ], "spans": [ { "bbox": [ 105, 346, 505, 358 ], "score": 1.0, "content": "papers often require a variety of assumptions to find the optimal classifier in a target domain in closed", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 357, 505, 370 ], "spans": [ { "bbox": [ 106, 357, 505, 370 ], "score": 1.0, "content": "form. For instance, Karbalayghareh et al. (2019; 2018) defines a joint prior distribution of source", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 369, 505, 380 ], "spans": [ { "bbox": [ 106, 369, 505, 380 ], "score": 1.0, "content": "and target domains using a Wishart distribution which relate the source and target tasks and then", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 380, 505, 391 ], "spans": [ { "bbox": [ 106, 380, 505, 391 ], "score": 1.0, "content": "makes it possible to study and understand the transferability between domains. Furthermore, in this", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 391, 505, 402 ], "spans": [ { "bbox": [ 106, 391, 505, 402 ], "score": 1.0, "content": "setting, the authors develop a closed form optimal Bayesian transfer learning and demonstrate its", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 401, 506, 414 ], "spans": [ { "bbox": [ 105, 401, 506, 414 ], "score": 1.0, "content": "advantage over a classifier obtained by only target data. Related to this setting but for regressions,", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 411, 505, 425 ], "spans": [ { "bbox": [ 105, 411, 505, 425 ], "score": 1.0, "content": "Karbalayghareh et al. (2018) obtains the optimal Bayesian transfer learning under setting of joint", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 423, 505, 436 ], "spans": [ { "bbox": [ 106, 423, 505, 436 ], "score": 1.0, "content": "Gaussian feature/label distribution. In contrast with the above in our paper we do not make any", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 434, 293, 446 ], "spans": [ { "bbox": [ 105, 434, 293, 446 ], "score": 1.0, "content": "assumptions about the distribution of the data.", "type": "text" } ], "index": 31 } ], "index": 26, "bbox_fs": [ 105, 323, 506, 446 ] }, { "type": "title", "bbox": [ 108, 461, 255, 473 ], "lines": [ { "bbox": [ 105, 459, 257, 476 ], "spans": [ { "bbox": [ 105, 459, 257, 476 ], "score": 1.0, "content": "3 PROBLEM FORMULATION", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 484, 505, 574 ], "lines": [ { "bbox": [ 105, 485, 505, 498 ], "spans": [ { "bbox": [ 105, 485, 505, 498 ], "score": 1.0, "content": "We consider a transfer learning problem where there are some labeled training data from a source", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 496, 506, 509 ], "spans": [ { "bbox": [ 106, 496, 506, 509 ], "score": 1.0, "content": "task and a target task with the goal of inferring a hypothesis function with small generalization error", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 508, 505, 520 ], "spans": [ { "bbox": [ 106, 508, 315, 520 ], "score": 1.0, "content": "in the target task. More specifically, we assume have", "type": "text" }, { "bbox": [ 315, 509, 328, 518 ], "score": 0.86, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 328, 508, 345, 520 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 345, 509, 359, 518 ], "score": 0.84, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 360, 508, 505, 520 ], "score": 1.0, "content": "source and target labeled data where", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 519, 505, 531 ], "spans": [ { "bbox": [ 106, 519, 505, 531 ], "score": 1.0, "content": "each training data consists of an input/feature as well as an output/label. We denote the source and", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 529, 505, 542 ], "spans": [ { "bbox": [ 105, 529, 172, 542 ], "score": 1.0, "content": "training data by", "type": "text" }, { "bbox": [ 172, 529, 228, 541 ], "score": 0.93, "content": "( \\pmb { x } _ { S } , y _ { S } ) \\sim \\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 228, 529, 246, 542 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 246, 529, 304, 541 ], "score": 0.93, "content": "\\bar { \\mathbf { \\Omega } } ( \\mathbf { x } _ { T } , y _ { T } ) \\sim \\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 305, 529, 387, 542 ], "score": 1.0, "content": ", respectively, where", "type": "text" }, { "bbox": [ 387, 529, 451, 541 ], "score": 0.93, "content": "y _ { S } , y _ { T } \\in \\{ 0 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 452, 529, 469, 542 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 469, 529, 489, 541 ], "score": 0.88, "content": "\\mathbb { P } , \\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 490, 529, 505, 542 ], "score": 1.0, "content": "are", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 540, 505, 554 ], "spans": [ { "bbox": [ 105, 540, 505, 554 ], "score": 1.0, "content": "the joint feature-label distributions of source and target data. Additionally, we assume that source", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 551, 506, 564 ], "spans": [ { "bbox": [ 105, 551, 295, 564 ], "score": 1.0, "content": "and target features/inputs share a same domain,", "type": "text" }, { "bbox": [ 295, 552, 344, 563 ], "score": 0.91, "content": "{ \\pmb x } _ { S } , { \\pmb x } _ { T } \\in { \\chi }", "type": "inline_equation" }, { "bbox": [ 345, 551, 365, 564 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 365, 551, 398, 561 ], "score": 0.9, "content": "\\mathcal { H } \\subset 2 ^ { \\chi }", "type": "inline_equation" }, { "bbox": [ 398, 551, 506, 564 ], "score": 1.0, "content": "denotes a fixed hypothesis", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 562, 226, 575 ], "spans": [ { "bbox": [ 106, 562, 148, 575 ], "score": 1.0, "content": "class with", "type": "text" }, { "bbox": [ 149, 563, 162, 574 ], "score": 0.88, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 163, 562, 226, 575 ], "score": 1.0, "content": "VC-dimension.", "type": "text" } ], "index": 40 } ], "index": 36.5, "bbox_fs": [ 105, 485, 506, 575 ] }, { "type": "text", "bbox": [ 105, 578, 504, 602 ], "lines": [ { "bbox": [ 105, 578, 506, 592 ], "spans": [ { "bbox": [ 105, 578, 338, 592 ], "score": 1.0, "content": "In transfer learning the goal is to find a hypothesis from", "type": "text" }, { "bbox": [ 338, 579, 348, 589 ], "score": 0.77, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 348, 578, 506, 592 ], "score": 1.0, "content": "that minimizing the target excess risk", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 590, 364, 602 ], "spans": [ { "bbox": [ 106, 590, 364, 602 ], "score": 1.0, "content": "defined below based on a combination of source and target data.", "type": "text" } ], "index": 42 } ], "index": 41.5, "bbox_fs": [ 105, 578, 506, 602 ] }, { "type": "text", "bbox": [ 106, 609, 505, 643 ], "lines": [ { "bbox": [ 106, 610, 505, 622 ], "spans": [ { "bbox": [ 106, 610, 315, 622 ], "score": 1.0, "content": "Definition 1 (Excess risk) For a hypothesis function", "type": "text" }, { "bbox": [ 316, 611, 343, 621 ], "score": 0.9, "content": "h \\in \\mathcal H", "type": "inline_equation" }, { "bbox": [ 344, 610, 505, 622 ], "score": 1.0, "content": "and source and target label-feature data", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 621, 505, 634 ], "spans": [ { "bbox": [ 105, 621, 259, 634 ], "score": 1.0, "content": "generated according to distributions", "type": "text" }, { "bbox": [ 259, 622, 267, 631 ], "score": 0.81, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 267, 621, 286, 634 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 287, 622, 297, 632 ], "score": 0.72, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 298, 621, 360, 633 ], "score": 0.87, "content": "( ( \\pmb { x } _ { S } , y _ { S } ) \\sim \\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 360, 621, 380, 634 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 380, 621, 442, 633 ], "score": 0.91, "content": "( \\pmb { x } _ { T } , \\pmb { y } _ { T } ) \\sim \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 443, 621, 505, 634 ], "score": 1.0, "content": ", we define the", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 632, 271, 645 ], "spans": [ { "bbox": [ 105, 632, 271, 645 ], "score": 1.0, "content": "source and target excess risks as follows", "type": "text" } ], "index": 45 } ], "index": 44, "bbox_fs": [ 105, 610, 505, 645 ] }, { "type": "interline_equation", "bbox": [ 211, 645, 399, 659 ], "lines": [ { "bbox": [ 211, 645, 399, 659 ], "spans": [ { "bbox": [ 211, 645, 399, 659 ], "score": 0.88, "content": "\\mathcal { E } _ { T } ( h ) = \\mathbb { Q } [ h ( \\mathbf { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { * } ( \\mathbf { x } _ { T } ) \\neq y _ { T } ]", "type": "interline_equation", "image_path": "8bed66d7e12cbd52b0964c90a062dd1167fdce9b26cd8510b60777279d9a3ef8.jpg" } ] } ], "index": 46, "virtual_lines": [ { "bbox": [ 211, 645, 399, 659 ], "spans": [], "index": 46 } ] }, { "type": "text", "bbox": [ 106, 661, 124, 672 ], "lines": [ { "bbox": [ 105, 660, 125, 672 ], "spans": [ { "bbox": [ 105, 660, 125, 672 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 47 } ], "index": 47, "bbox_fs": [ 105, 660, 125, 672 ] }, { "type": "interline_equation", "bbox": [ 106, 673, 434, 704 ], "lines": [ { "bbox": [ 132, 673, 434, 704 ], "spans": [ { "bbox": [ 132, 673, 434, 704 ], "score": 0.8, "content": "\\begin{array} { c } { \\displaystyle \\varepsilon _ { S } ( h ) = \\mathbb { P } [ h ( \\pmb { x } _ { S } ) \\neq y _ { S } ] - \\mathbb { P } [ h _ { S } ^ { \\ast } ( \\pmb { x } _ { S } ) \\neq y _ { S } ] } \\\\ { \\displaystyle h _ { T } ^ { \\ast } = \\arg \\operatorname* { m i n } _ { h \\in \\mathcal { H } } \\mathbb { Q } [ h ( \\pmb { x } _ { T } ) \\neq y _ { T } ] a n d h _ { S } ^ { \\ast } = \\arg \\operatorname* { m i n } _ { h \\in \\mathcal { H } } \\mathbb { P } [ h ( \\pmb { x } _ { S } ) \\neq y _ { S } ] } \\end{array}", "type": "interline_equation", "image_path": "367fe765182f3a8464385a5370987108764d33d5fb90ef1fac069d81b0bd5955.jpg" } ] } ], "index": 49, "virtual_lines": [ { "bbox": [ 106, 673, 434, 683.3333333333334 ], "spans": [], "index": 48 }, { "bbox": [ 106, 683.3333333333334, 434, 693.6666666666667 ], "spans": [], "index": 49 }, { "bbox": [ 106, 693.6666666666667, 434, 704.0000000000001 ], "spans": [], "index": 50 } ] }, { "type": "text", "bbox": [ 106, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "Next, we need to define an appropriate notion of distance between the source and target. In the", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "literature of domain adaptation, where the conditional expectation remains unchanged and there is", "type": "text" } ], "index": 52 } ], "index": 51.5, "bbox_fs": [ 105, 710, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [ { "bbox": [ 106, 82, 506, 95 ], "spans": [ { "bbox": [ 106, 82, 506, 95 ], "score": 1.0, "content": "only a shift in input distributions, it is common to define the distance as the error of performance of", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "the best source hypothesis in the target task. We also define the distance between source and target as", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 312, 117 ], "spans": [ { "bbox": [ 105, 104, 312, 117 ], "score": 1.0, "content": "the target excess risk of the best source hypothesis.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 105, 124, 504, 147 ], "lines": [ { "bbox": [ 105, 124, 505, 138 ], "spans": [ { "bbox": [ 105, 124, 505, 138 ], "score": 1.0, "content": "Definition 2 (Transfer distance) We define the transfer distance between a source and a target with", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 136, 238, 148 ], "spans": [ { "bbox": [ 106, 136, 159, 148 ], "score": 1.0, "content": "distributions", "type": "text" }, { "bbox": [ 159, 137, 167, 146 ], "score": 0.81, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 167, 136, 185, 148 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 137, 195, 147 ], "score": 0.84, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 195, 136, 238, 148 ], "score": 1.0, "content": "as follows", "type": "text" } ], "index": 4 } ], "index": 3.5 }, { "type": "interline_equation", "bbox": [ 204, 150, 407, 165 ], "lines": [ { "bbox": [ 204, 150, 407, 165 ], "spans": [ { "bbox": [ 204, 150, 407, 165 ], "score": 0.89, "content": "\\rho ( \\mathbb { P } , \\mathbb { Q } ) : = \\mathbb { Q } [ h _ { S } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ]", "type": "interline_equation", "image_path": "b11f0b3b4f4065ef6b13ffb2a276be6c050dfed880b72c8b55b59abf57fb1f3d.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 204, 150, 407, 165 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 173, 505, 218 ], "lines": [ { "bbox": [ 106, 173, 505, 186 ], "spans": [ { "bbox": [ 106, 173, 505, 186 ], "score": 1.0, "content": "Since we aim to derive a minimax lower bound for transfer learning in binary classifications, we", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 185, 505, 197 ], "spans": [ { "bbox": [ 106, 185, 464, 197 ], "score": 1.0, "content": "consider the class of pairs of distributions whose transfer distance is within a fixed number", "type": "text" }, { "bbox": [ 464, 185, 473, 195 ], "score": 0.81, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 474, 185, 505, 197 ], "score": 1.0, "content": ". As we", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 195, 505, 208 ], "spans": [ { "bbox": [ 105, 195, 505, 208 ], "score": 1.0, "content": "will elaborate further in Remark 7 below this notion of distance can be easily estimated/computed in", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 206, 144, 219 ], "spans": [ { "bbox": [ 104, 206, 144, 219 ], "score": 1.0, "content": "practice.", "type": "text" } ], "index": 9 } ], "index": 7.5 }, { "type": "title", "bbox": [ 108, 233, 206, 246 ], "lines": [ { "bbox": [ 105, 232, 207, 249 ], "spans": [ { "bbox": [ 105, 232, 207, 249 ], "score": 1.0, "content": "4 MAIN RESULTS", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 106, 258, 504, 281 ], "lines": [ { "bbox": [ 105, 256, 506, 272 ], "spans": [ { "bbox": [ 105, 256, 506, 272 ], "score": 1.0, "content": "In this section we characterize the fundamental limits of transfer learning in binary classifications by", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 269, 388, 282 ], "spans": [ { "bbox": [ 105, 269, 388, 282 ], "score": 1.0, "content": "deriving a minimax lower bound via information-theoretic arguments.", "type": "text" } ], "index": 12 } ], "index": 11.5 }, { "type": "text", "bbox": [ 106, 289, 506, 363 ], "lines": [ { "bbox": [ 106, 289, 505, 302 ], "spans": [ { "bbox": [ 106, 289, 375, 302 ], "score": 1.0, "content": "Theorem 1 Consider a transfer learning problem where there are", "type": "text" }, { "bbox": [ 375, 292, 388, 301 ], "score": 0.84, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 388, 289, 406, 302 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 407, 291, 420, 301 ], "score": 0.82, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 420, 289, 505, 302 ], "score": 1.0, "content": "number of source as", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 301, 506, 313 ], "spans": [ { "bbox": [ 106, 301, 278, 313 ], "score": 1.0, "content": "well as target data and the hypothesis class", "type": "text" }, { "bbox": [ 278, 302, 288, 311 ], "score": 0.73, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 288, 301, 305, 313 ], "score": 1.0, "content": "has", "type": "text" }, { "bbox": [ 305, 302, 319, 311 ], "score": 0.33, "content": "V C", "type": "inline_equation" }, { "bbox": [ 319, 301, 362, 313 ], "score": 1.0, "content": "dimension", "type": "text" }, { "bbox": [ 362, 302, 376, 312 ], "score": 0.88, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 376, 301, 410, 313 ], "score": 1.0, "content": "obeying", "type": "text" }, { "bbox": [ 410, 302, 447, 312 ], "score": 0.9, "content": "d _ { \\mathcal { H } } \\geq 1 0", "type": "inline_equation" }, { "bbox": [ 447, 301, 506, 313 ], "score": 1.0, "content": ". Furthermore,", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 312, 506, 326 ], "spans": [ { "bbox": [ 105, 313, 160, 326 ], "score": 1.0, "content": "suppose that", "type": "text" }, { "bbox": [ 160, 312, 221, 326 ], "score": 0.93, "content": "\\hat { h } = \\hat { h } ( S _ { \\mathbb { P } } , S _ { \\mathbb { Q } } )", "type": "inline_equation" }, { "bbox": [ 222, 313, 506, 326 ], "score": 1.0, "content": "is an estimated hypothesis for the target task using source and target", "type": "text" } ], "index": 15 }, { "bbox": [ 101, 319, 503, 348 ], "spans": [ { "bbox": [ 101, 319, 162, 348 ], "score": 1.0, "content": "data in which", "type": "text" }, { "bbox": [ 162, 328, 174, 339 ], "score": 0.85, "content": "S _ { \\mathbb { P } }", "type": "inline_equation" }, { "bbox": [ 174, 319, 192, 348 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 192, 328, 205, 340 ], "score": 0.87, "content": "S _ { \\mathbb { Q } }", "type": "inline_equation" }, { "bbox": [ 206, 319, 334, 348 ], "score": 1.0, "content": "denote i.i.d. feature-label data p", "type": "text" }, { "bbox": [ 342, 319, 351, 348 ], "score": 1.0, "content": "rs", "type": "text" }, { "bbox": [ 351, 325, 417, 340 ], "score": 0.94, "content": "\\{ ( \\pmb { x } _ { S } ^ { ( i ) } , \\pmb { y } _ { S } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { S } }", "type": "inline_equation" }, { "bbox": [ 417, 319, 436, 348 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 437, 325, 503, 340 ], "score": 0.91, "content": "\\{ ( \\pmb { x } _ { T } ^ { ( i ) } , \\pmb { y } _ { T } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { T } }", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 105, 339, 507, 351 ], "spans": [ { "bbox": [ 105, 339, 334, 351 ], "score": 1.0, "content": "generated according to the source and target distributions", "type": "text" }, { "bbox": [ 334, 339, 342, 348 ], "score": 0.74, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 342, 339, 359, 351 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 360, 340, 368, 350 ], "score": 0.82, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 369, 339, 462, 351 ], "score": 1.0, "content": ". Fix a transfer distance", "type": "text" }, { "bbox": [ 462, 340, 502, 349 ], "score": 0.8, "content": "\\Delta < 0 . 9 9", "type": "inline_equation" }, { "bbox": [ 502, 339, 507, 351 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 349, 466, 364 ], "spans": [ { "bbox": [ 105, 350, 160, 364 ], "score": 1.0, "content": "Then for any", "type": "text" }, { "bbox": [ 160, 349, 167, 361 ], "score": 0.81, "content": "\\hat { h }", "type": "inline_equation" }, { "bbox": [ 167, 350, 216, 364 ], "score": 1.0, "content": "there exists", "type": "text" }, { "bbox": [ 216, 351, 242, 363 ], "score": 0.91, "content": "( \\mathbb { P } , \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 243, 350, 263, 364 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 264, 351, 317, 363 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } , \\mathbb { Q } ) \\leq \\Delta", "type": "inline_equation" }, { "bbox": [ 318, 350, 419, 364 ], "score": 1.0, "content": "and a universal constant", "type": "text" }, { "bbox": [ 420, 354, 425, 361 ], "score": 0.3, "content": "c", "type": "inline_equation" }, { "bbox": [ 425, 350, 466, 364 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 18 } ], "index": 15.5 }, { "type": "interline_equation", "bbox": [ 199, 366, 410, 399 ], "lines": [ { "bbox": [ 199, 366, 410, 399 ], "spans": [ { "bbox": [ 199, 366, 410, 399 ], "score": 0.93, "content": "P _ { \\mathit { P } , \\mathit { S } _ { \\mathbb { Q } } } \\bigg ( \\mathcal { E } _ { T } ( \\hat { h } ) > c \\cdot \\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta ) \\bigg ) \\geq \\frac { 3 - 2 \\sqrt { 2 } } { 8 } ,", "type": "interline_equation", "image_path": "cc624f66a7391cce80c9448fc7b10d7d402550f1a54082f148fe7cdabb631719.jpg" } ] } ], "index": 19.5, "virtual_lines": [ { "bbox": [ 199, 366, 410, 382.5 ], "spans": [], "index": 19 }, { "bbox": [ 199, 382.5, 410, 399.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 106, 402, 132, 413 ], "lines": [ { "bbox": [ 105, 401, 135, 414 ], "spans": [ { "bbox": [ 105, 401, 135, 414 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "interline_equation", "bbox": [ 225, 415, 387, 448 ], "lines": [ { "bbox": [ 225, 415, 387, 448 ], "spans": [ { "bbox": [ 225, 415, 387, 448 ], "score": 0.95, "content": "\\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta ) = \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { n _ { S } } { d _ { \\mathcal { H } } + n _ { S } \\Delta } } } .", "type": "interline_equation", "image_path": "5cbce3d81817022bfe6e631e97ecef77737700e7973dce15397373e9c68e6deb.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 225, 415, 387, 431.5 ], "spans": [], "index": 22 }, { "bbox": [ 225, 431.5, 387, 448.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 451, 194, 462 ], "lines": [ { "bbox": [ 106, 450, 195, 463 ], "spans": [ { "bbox": [ 106, 450, 195, 463 ], "score": 1.0, "content": "This also implies that", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 202, 465, 408, 491 ], "lines": [ { "bbox": [ 202, 465, 408, 491 ], "spans": [ { "bbox": [ 202, 465, 408, 491 ], "score": 0.92, "content": "\\operatorname* { i n f } _ { \\hat { h } } \\operatorname* { s u p } _ { \\rho ( \\mathbb { P } , \\mathbb { Q } ) \\leq \\Delta } \\operatorname* { \\mathbb { E } } _ { S _ { \\mathbb { P } } , S _ { \\mathbb { Q } } } \\Big [ \\mathcal E _ { T } ( \\hat { h } ) \\Big ] \\geq c \\cdot \\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal H } , \\Delta ) .", "type": "interline_equation", "image_path": "94c6c6aa8ccfe81f2219cddc939bdacaa65c7fce6edd89d1c902de4202d7da87.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 202, 465, 408, 491 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 499, 505, 544 ], "lines": [ { "bbox": [ 105, 498, 505, 513 ], "spans": [ { "bbox": [ 105, 498, 505, 513 ], "score": 1.0, "content": "Remark 1 The bound above characterizes the fundamental limits of transfer learning by providing", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 510, 505, 523 ], "spans": [ { "bbox": [ 105, 510, 505, 523 ], "score": 1.0, "content": "a lower bound on the excess risk of any algorithm (regardless of computational tractability) as a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 522, 505, 533 ], "spans": [ { "bbox": [ 105, 522, 505, 533 ], "score": 1.0, "content": "function of the number of source and target training data, the similarity/distance between the source", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 533, 362, 545 ], "spans": [ { "bbox": [ 105, 533, 362, 545 ], "score": 1.0, "content": "and target tasks and the dimension of the hypothesis class used.", "type": "text" } ], "index": 29 } ], "index": 27.5 }, { "type": "text", "bbox": [ 105, 552, 504, 576 ], "lines": [ { "bbox": [ 106, 552, 505, 565 ], "spans": [ { "bbox": [ 106, 552, 219, 565 ], "score": 1.0, "content": "Remark 2 The assumption", "type": "text" }, { "bbox": [ 219, 553, 259, 564 ], "score": 0.87, "content": "\\Delta < 0 . 9 9", "type": "inline_equation" }, { "bbox": [ 260, 552, 505, 565 ], "score": 1.0, "content": "in the statement of Theorem 1 is just made for simplifying the", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 564, 473, 577 ], "spans": [ { "bbox": [ 106, 564, 447, 577 ], "score": 1.0, "content": "analysis and the upper bound of 0.99 can be replaced by any constant in the interval", "type": "text" }, { "bbox": [ 447, 565, 469, 576 ], "score": 0.91, "content": "( 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 470, 564, 473, 577 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 31 } ], "index": 30.5 }, { "type": "text", "bbox": [ 103, 585, 467, 601 ], "lines": [ { "bbox": [ 103, 582, 471, 603 ], "spans": [ { "bbox": [ 103, 582, 471, 603 ], "score": 1.0, "content": "Remark 3 One can show that the numerical constant c in equation 4.1 obeys c > 3−2 248 .", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 607, 506, 661 ], "lines": [ { "bbox": [ 106, 608, 505, 621 ], "spans": [ { "bbox": [ 106, 608, 505, 621 ], "score": 1.0, "content": "Remark 4 (Connection to PAC learning) We note that the well-known agnostic PAC learning result", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 619, 507, 640 ], "spans": [ { "bbox": [ 104, 619, 275, 640 ], "score": 1.0, "content": "for a single task gives a lower bound of", "type": "text" }, { "bbox": [ 276, 619, 312, 640 ], "score": 0.93, "content": "c \\cdot \\sqrt { \\frac { d _ { \\mathscr { H } } } { n } }", "type": "inline_equation" }, { "bbox": [ 312, 619, 342, 640 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 342, 626, 349, 634 ], "score": 0.56, "content": "n", "type": "inline_equation" }, { "bbox": [ 350, 619, 507, 640 ], "score": 1.0, "content": "is the number of samples of the task.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 637, 506, 650 ], "spans": [ { "bbox": [ 106, 637, 402, 650 ], "score": 1.0, "content": "Theorem 1 recovers this result when there is not any source task, namely", "type": "text" }, { "bbox": [ 403, 638, 434, 649 ], "score": 0.89, "content": "n _ { S } = 0", "type": "inline_equation" }, { "bbox": [ 434, 637, 506, 650 ], "score": 1.0, "content": ", and the transfer", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 648, 468, 662 ], "spans": [ { "bbox": [ 105, 648, 468, 662 ], "score": 1.0, "content": "learning problem reduces to learning a task without any prior knowledge from the source.", "type": "text" } ], "index": 36 } ], "index": 34.5 }, { "type": "text", "bbox": [ 106, 668, 506, 732 ], "lines": [ { "bbox": [ 106, 669, 505, 682 ], "spans": [ { "bbox": [ 106, 669, 505, 682 ], "score": 1.0, "content": "Remark 5 (Identical source and target) When the source and target tasks are identical, then the", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 681, 505, 693 ], "spans": [ { "bbox": [ 106, 681, 363, 693 ], "score": 1.0, "content": "transfer learning problem reduces to learning a single task with", "type": "text" }, { "bbox": [ 363, 681, 400, 692 ], "score": 0.88, "content": "n _ { S } + n _ { T }", "type": "inline_equation" }, { "bbox": [ 400, 681, 505, 693 ], "score": 1.0, "content": "training data. Theorem 1,", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 692, 505, 703 ], "spans": [ { "bbox": [ 106, 692, 505, 703 ], "score": 1.0, "content": "also leads to the same conclusion in this special case as when the source and target data are identical", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 703, 506, 722 ], "spans": [ { "bbox": [ 105, 705, 203, 720 ], "score": 1.0, "content": "∆ = 0 and thus \u000f = q", "type": "text" }, { "bbox": [ 174, 703, 232, 722 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\sqrt { \\frac { d _ { \\mathcal { H } } } { n _ { S } + n _ { T } } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 233, 704, 506, 722 ], "score": 1.0, "content": "which states that the lower bound is proportional to reciprocal of", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 720, 325, 732 ], "spans": [ { "bbox": [ 106, 720, 325, 732 ], "score": 1.0, "content": "combination of source and target samples as expected.", "type": "text" } ], "index": 41 } ], "index": 39 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 309, 39 ], "spans": [ { "bbox": [ 106, 25, 309, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 9 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [ { "bbox": [ 106, 82, 506, 95 ], "spans": [ { "bbox": [ 106, 82, 506, 95 ], "score": 1.0, "content": "only a shift in input distributions, it is common to define the distance as the error of performance of", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "the best source hypothesis in the target task. We also define the distance between source and target as", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 312, 117 ], "spans": [ { "bbox": [ 105, 104, 312, 117 ], "score": 1.0, "content": "the target excess risk of the best source hypothesis.", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 105, 82, 506, 117 ] }, { "type": "text", "bbox": [ 105, 124, 504, 147 ], "lines": [ { "bbox": [ 105, 124, 505, 138 ], "spans": [ { "bbox": [ 105, 124, 505, 138 ], "score": 1.0, "content": "Definition 2 (Transfer distance) We define the transfer distance between a source and a target with", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 136, 238, 148 ], "spans": [ { "bbox": [ 106, 136, 159, 148 ], "score": 1.0, "content": "distributions", "type": "text" }, { "bbox": [ 159, 137, 167, 146 ], "score": 0.81, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 167, 136, 185, 148 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 185, 137, 195, 147 ], "score": 0.84, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 195, 136, 238, 148 ], "score": 1.0, "content": "as follows", "type": "text" } ], "index": 4 } ], "index": 3.5, "bbox_fs": [ 105, 124, 505, 148 ] }, { "type": "interline_equation", "bbox": [ 204, 150, 407, 165 ], "lines": [ { "bbox": [ 204, 150, 407, 165 ], "spans": [ { "bbox": [ 204, 150, 407, 165 ], "score": 0.89, "content": "\\rho ( \\mathbb { P } , \\mathbb { Q } ) : = \\mathbb { Q } [ h _ { S } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ]", "type": "interline_equation", "image_path": "b11f0b3b4f4065ef6b13ffb2a276be6c050dfed880b72c8b55b59abf57fb1f3d.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 204, 150, 407, 165 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 173, 505, 218 ], "lines": [ { "bbox": [ 106, 173, 505, 186 ], "spans": [ { "bbox": [ 106, 173, 505, 186 ], "score": 1.0, "content": "Since we aim to derive a minimax lower bound for transfer learning in binary classifications, we", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 185, 505, 197 ], "spans": [ { "bbox": [ 106, 185, 464, 197 ], "score": 1.0, "content": "consider the class of pairs of distributions whose transfer distance is within a fixed number", "type": "text" }, { "bbox": [ 464, 185, 473, 195 ], "score": 0.81, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 474, 185, 505, 197 ], "score": 1.0, "content": ". As we", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 195, 505, 208 ], "spans": [ { "bbox": [ 105, 195, 505, 208 ], "score": 1.0, "content": "will elaborate further in Remark 7 below this notion of distance can be easily estimated/computed in", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 206, 144, 219 ], "spans": [ { "bbox": [ 104, 206, 144, 219 ], "score": 1.0, "content": "practice.", "type": "text" } ], "index": 9 } ], "index": 7.5, "bbox_fs": [ 104, 173, 505, 219 ] }, { "type": "title", "bbox": [ 108, 233, 206, 246 ], "lines": [ { "bbox": [ 105, 232, 207, 249 ], "spans": [ { "bbox": [ 105, 232, 207, 249 ], "score": 1.0, "content": "4 MAIN RESULTS", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 106, 258, 504, 281 ], "lines": [ { "bbox": [ 105, 256, 506, 272 ], "spans": [ { "bbox": [ 105, 256, 506, 272 ], "score": 1.0, "content": "In this section we characterize the fundamental limits of transfer learning in binary classifications by", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 269, 388, 282 ], "spans": [ { "bbox": [ 105, 269, 388, 282 ], "score": 1.0, "content": "deriving a minimax lower bound via information-theoretic arguments.", "type": "text" } ], "index": 12 } ], "index": 11.5, "bbox_fs": [ 105, 256, 506, 282 ] }, { "type": "text", "bbox": [ 106, 289, 506, 363 ], "lines": [ { "bbox": [ 106, 289, 505, 302 ], "spans": [ { "bbox": [ 106, 289, 375, 302 ], "score": 1.0, "content": "Theorem 1 Consider a transfer learning problem where there are", "type": "text" }, { "bbox": [ 375, 292, 388, 301 ], "score": 0.84, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 388, 289, 406, 302 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 407, 291, 420, 301 ], "score": 0.82, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 420, 289, 505, 302 ], "score": 1.0, "content": "number of source as", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 301, 506, 313 ], "spans": [ { "bbox": [ 106, 301, 278, 313 ], "score": 1.0, "content": "well as target data and the hypothesis class", "type": "text" }, { "bbox": [ 278, 302, 288, 311 ], "score": 0.73, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 288, 301, 305, 313 ], "score": 1.0, "content": "has", "type": "text" }, { "bbox": [ 305, 302, 319, 311 ], "score": 0.33, "content": "V C", "type": "inline_equation" }, { "bbox": [ 319, 301, 362, 313 ], "score": 1.0, "content": "dimension", "type": "text" }, { "bbox": [ 362, 302, 376, 312 ], "score": 0.88, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 376, 301, 410, 313 ], "score": 1.0, "content": "obeying", "type": "text" }, { "bbox": [ 410, 302, 447, 312 ], "score": 0.9, "content": "d _ { \\mathcal { H } } \\geq 1 0", "type": "inline_equation" }, { "bbox": [ 447, 301, 506, 313 ], "score": 1.0, "content": ". Furthermore,", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 312, 506, 326 ], "spans": [ { "bbox": [ 105, 313, 160, 326 ], "score": 1.0, "content": "suppose that", "type": "text" }, { "bbox": [ 160, 312, 221, 326 ], "score": 0.93, "content": "\\hat { h } = \\hat { h } ( S _ { \\mathbb { P } } , S _ { \\mathbb { Q } } )", "type": "inline_equation" }, { "bbox": [ 222, 313, 506, 326 ], "score": 1.0, "content": "is an estimated hypothesis for the target task using source and target", "type": "text" } ], "index": 15 }, { "bbox": [ 101, 319, 503, 348 ], "spans": [ { "bbox": [ 101, 319, 162, 348 ], "score": 1.0, "content": "data in which", "type": "text" }, { "bbox": [ 162, 328, 174, 339 ], "score": 0.85, "content": "S _ { \\mathbb { P } }", "type": "inline_equation" }, { "bbox": [ 174, 319, 192, 348 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 192, 328, 205, 340 ], "score": 0.87, "content": "S _ { \\mathbb { Q } }", "type": "inline_equation" }, { "bbox": [ 206, 319, 334, 348 ], "score": 1.0, "content": "denote i.i.d. feature-label data p", "type": "text" }, { "bbox": [ 342, 319, 351, 348 ], "score": 1.0, "content": "rs", "type": "text" }, { "bbox": [ 351, 325, 417, 340 ], "score": 0.94, "content": "\\{ ( \\pmb { x } _ { S } ^ { ( i ) } , \\pmb { y } _ { S } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { S } }", "type": "inline_equation" }, { "bbox": [ 417, 319, 436, 348 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 437, 325, 503, 340 ], "score": 0.91, "content": "\\{ ( \\pmb { x } _ { T } ^ { ( i ) } , \\pmb { y } _ { T } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { T } }", "type": "inline_equation" } ], "index": 16 }, { "bbox": [ 105, 339, 507, 351 ], "spans": [ { "bbox": [ 105, 339, 334, 351 ], "score": 1.0, "content": "generated according to the source and target distributions", "type": "text" }, { "bbox": [ 334, 339, 342, 348 ], "score": 0.74, "content": "\\mathbb { P }", "type": "inline_equation" }, { "bbox": [ 342, 339, 359, 351 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 360, 340, 368, 350 ], "score": 0.82, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 369, 339, 462, 351 ], "score": 1.0, "content": ". Fix a transfer distance", "type": "text" }, { "bbox": [ 462, 340, 502, 349 ], "score": 0.8, "content": "\\Delta < 0 . 9 9", "type": "inline_equation" }, { "bbox": [ 502, 339, 507, 351 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 349, 466, 364 ], "spans": [ { "bbox": [ 105, 350, 160, 364 ], "score": 1.0, "content": "Then for any", "type": "text" }, { "bbox": [ 160, 349, 167, 361 ], "score": 0.81, "content": "\\hat { h }", "type": "inline_equation" }, { "bbox": [ 167, 350, 216, 364 ], "score": 1.0, "content": "there exists", "type": "text" }, { "bbox": [ 216, 351, 242, 363 ], "score": 0.91, "content": "( \\mathbb { P } , \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 243, 350, 263, 364 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 264, 351, 317, 363 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } , \\mathbb { Q } ) \\leq \\Delta", "type": "inline_equation" }, { "bbox": [ 318, 350, 419, 364 ], "score": 1.0, "content": "and a universal constant", "type": "text" }, { "bbox": [ 420, 354, 425, 361 ], "score": 0.3, "content": "c", "type": "inline_equation" }, { "bbox": [ 425, 350, 466, 364 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 18 } ], "index": 15.5, "bbox_fs": [ 101, 289, 507, 364 ] }, { "type": "interline_equation", "bbox": [ 199, 366, 410, 399 ], "lines": [ { "bbox": [ 199, 366, 410, 399 ], "spans": [ { "bbox": [ 199, 366, 410, 399 ], "score": 0.93, "content": "P _ { \\mathit { P } , \\mathit { S } _ { \\mathbb { Q } } } \\bigg ( \\mathcal { E } _ { T } ( \\hat { h } ) > c \\cdot \\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta ) \\bigg ) \\geq \\frac { 3 - 2 \\sqrt { 2 } } { 8 } ,", "type": "interline_equation", "image_path": "cc624f66a7391cce80c9448fc7b10d7d402550f1a54082f148fe7cdabb631719.jpg" } ] } ], "index": 19.5, "virtual_lines": [ { "bbox": [ 199, 366, 410, 382.5 ], "spans": [], "index": 19 }, { "bbox": [ 199, 382.5, 410, 399.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 106, 402, 132, 413 ], "lines": [ { "bbox": [ 105, 401, 135, 414 ], "spans": [ { "bbox": [ 105, 401, 135, 414 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 105, 401, 135, 414 ] }, { "type": "interline_equation", "bbox": [ 225, 415, 387, 448 ], "lines": [ { "bbox": [ 225, 415, 387, 448 ], "spans": [ { "bbox": [ 225, 415, 387, 448 ], "score": 0.95, "content": "\\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta ) = \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { n _ { S } } { d _ { \\mathcal { H } } + n _ { S } \\Delta } } } .", "type": "interline_equation", "image_path": "5cbce3d81817022bfe6e631e97ecef77737700e7973dce15397373e9c68e6deb.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 225, 415, 387, 431.5 ], "spans": [], "index": 22 }, { "bbox": [ 225, 431.5, 387, 448.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 451, 194, 462 ], "lines": [ { "bbox": [ 106, 450, 195, 463 ], "spans": [ { "bbox": [ 106, 450, 195, 463 ], "score": 1.0, "content": "This also implies that", "type": "text" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 106, 450, 195, 463 ] }, { "type": "interline_equation", "bbox": [ 202, 465, 408, 491 ], "lines": [ { "bbox": [ 202, 465, 408, 491 ], "spans": [ { "bbox": [ 202, 465, 408, 491 ], "score": 0.92, "content": "\\operatorname* { i n f } _ { \\hat { h } } \\operatorname* { s u p } _ { \\rho ( \\mathbb { P } , \\mathbb { Q } ) \\leq \\Delta } \\operatorname* { \\mathbb { E } } _ { S _ { \\mathbb { P } } , S _ { \\mathbb { Q } } } \\Big [ \\mathcal E _ { T } ( \\hat { h } ) \\Big ] \\geq c \\cdot \\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal H } , \\Delta ) .", "type": "interline_equation", "image_path": "94c6c6aa8ccfe81f2219cddc939bdacaa65c7fce6edd89d1c902de4202d7da87.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 202, 465, 408, 491 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 499, 505, 544 ], "lines": [ { "bbox": [ 105, 498, 505, 513 ], "spans": [ { "bbox": [ 105, 498, 505, 513 ], "score": 1.0, "content": "Remark 1 The bound above characterizes the fundamental limits of transfer learning by providing", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 510, 505, 523 ], "spans": [ { "bbox": [ 105, 510, 505, 523 ], "score": 1.0, "content": "a lower bound on the excess risk of any algorithm (regardless of computational tractability) as a", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 522, 505, 533 ], "spans": [ { "bbox": [ 105, 522, 505, 533 ], "score": 1.0, "content": "function of the number of source and target training data, the similarity/distance between the source", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 533, 362, 545 ], "spans": [ { "bbox": [ 105, 533, 362, 545 ], "score": 1.0, "content": "and target tasks and the dimension of the hypothesis class used.", "type": "text" } ], "index": 29 } ], "index": 27.5, "bbox_fs": [ 105, 498, 505, 545 ] }, { "type": "text", "bbox": [ 105, 552, 504, 576 ], "lines": [ { "bbox": [ 106, 552, 505, 565 ], "spans": [ { "bbox": [ 106, 552, 219, 565 ], "score": 1.0, "content": "Remark 2 The assumption", "type": "text" }, { "bbox": [ 219, 553, 259, 564 ], "score": 0.87, "content": "\\Delta < 0 . 9 9", "type": "inline_equation" }, { "bbox": [ 260, 552, 505, 565 ], "score": 1.0, "content": "in the statement of Theorem 1 is just made for simplifying the", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 564, 473, 577 ], "spans": [ { "bbox": [ 106, 564, 447, 577 ], "score": 1.0, "content": "analysis and the upper bound of 0.99 can be replaced by any constant in the interval", "type": "text" }, { "bbox": [ 447, 565, 469, 576 ], "score": 0.91, "content": "( 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 470, 564, 473, 577 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 106, 552, 505, 577 ] }, { "type": "text", "bbox": [ 103, 585, 467, 601 ], "lines": [ { "bbox": [ 103, 582, 471, 603 ], "spans": [ { "bbox": [ 103, 582, 471, 603 ], "score": 1.0, "content": "Remark 3 One can show that the numerical constant c in equation 4.1 obeys c > 3−2 248 .", "type": "text" } ], "index": 32 } ], "index": 32, "bbox_fs": [ 103, 582, 471, 603 ] }, { "type": "text", "bbox": [ 106, 607, 506, 661 ], "lines": [ { "bbox": [ 106, 608, 505, 621 ], "spans": [ { "bbox": [ 106, 608, 505, 621 ], "score": 1.0, "content": "Remark 4 (Connection to PAC learning) We note that the well-known agnostic PAC learning result", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 619, 507, 640 ], "spans": [ { "bbox": [ 104, 619, 275, 640 ], "score": 1.0, "content": "for a single task gives a lower bound of", "type": "text" }, { "bbox": [ 276, 619, 312, 640 ], "score": 0.93, "content": "c \\cdot \\sqrt { \\frac { d _ { \\mathscr { H } } } { n } }", "type": "inline_equation" }, { "bbox": [ 312, 619, 342, 640 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 342, 626, 349, 634 ], "score": 0.56, "content": "n", "type": "inline_equation" }, { "bbox": [ 350, 619, 507, 640 ], "score": 1.0, "content": "is the number of samples of the task.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 637, 506, 650 ], "spans": [ { "bbox": [ 106, 637, 402, 650 ], "score": 1.0, "content": "Theorem 1 recovers this result when there is not any source task, namely", "type": "text" }, { "bbox": [ 403, 638, 434, 649 ], "score": 0.89, "content": "n _ { S } = 0", "type": "inline_equation" }, { "bbox": [ 434, 637, 506, 650 ], "score": 1.0, "content": ", and the transfer", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 648, 468, 662 ], "spans": [ { "bbox": [ 105, 648, 468, 662 ], "score": 1.0, "content": "learning problem reduces to learning a task without any prior knowledge from the source.", "type": "text" } ], "index": 36 } ], "index": 34.5, "bbox_fs": [ 104, 608, 507, 662 ] }, { "type": "text", "bbox": [ 106, 668, 506, 732 ], "lines": [ { "bbox": [ 106, 669, 505, 682 ], "spans": [ { "bbox": [ 106, 669, 505, 682 ], "score": 1.0, "content": "Remark 5 (Identical source and target) When the source and target tasks are identical, then the", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 681, 505, 693 ], "spans": [ { "bbox": [ 106, 681, 363, 693 ], "score": 1.0, "content": "transfer learning problem reduces to learning a single task with", "type": "text" }, { "bbox": [ 363, 681, 400, 692 ], "score": 0.88, "content": "n _ { S } + n _ { T }", "type": "inline_equation" }, { "bbox": [ 400, 681, 505, 693 ], "score": 1.0, "content": "training data. Theorem 1,", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 692, 505, 703 ], "spans": [ { "bbox": [ 106, 692, 505, 703 ], "score": 1.0, "content": "also leads to the same conclusion in this special case as when the source and target data are identical", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 703, 506, 722 ], "spans": [ { "bbox": [ 105, 705, 203, 720 ], "score": 1.0, "content": "∆ = 0 and thus \u000f = q", "type": "text" }, { "bbox": [ 174, 703, 232, 722 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\sqrt { \\frac { d _ { \\mathcal { H } } } { n _ { S } + n _ { T } } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 233, 704, 506, 722 ], "score": 1.0, "content": "which states that the lower bound is proportional to reciprocal of", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 720, 325, 732 ], "spans": [ { "bbox": [ 106, 720, 325, 732 ], "score": 1.0, "content": "combination of source and target samples as expected.", "type": "text" } ], "index": 41 } ], "index": 39, "bbox_fs": [ 105, 669, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 134 ], "lines": [ { "bbox": [ 106, 82, 506, 95 ], "spans": [ { "bbox": [ 106, 82, 506, 95 ], "score": 1.0, "content": "Remark 6 (Sharpness in a special case) We note that the above lower bound is known to be tight", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 107 ], "spans": [ { "bbox": [ 105, 93, 432, 107 ], "score": 1.0, "content": "in special cases. For instance when there is a small amount of source data and", "type": "text" }, { "bbox": [ 432, 95, 442, 104 ], "score": 0.43, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 442, 93, 506, 107 ], "score": 1.0, "content": "is rather large,", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 506, 124 ], "spans": [ { "bbox": [ 105, 108, 216, 122 ], "score": 1.0, "content": "the lower bound reduces to", "type": "text" }, { "bbox": [ 223, 105, 506, 124 ], "score": 1.0, "content": "dHn which is known to be tight based on known agnostic PAC learning", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 122, 140, 135 ], "spans": [ { "bbox": [ 105, 122, 140, 135 ], "score": 1.0, "content": "bounds.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "text", "bbox": [ 106, 143, 505, 375 ], "lines": [ { "bbox": [ 106, 144, 506, 157 ], "spans": [ { "bbox": [ 106, 144, 506, 157 ], "score": 1.0, "content": "Remark 7 (How to apply Theorem 1 in practical settings.) In this remark we explain how Theorem", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 155, 506, 168 ], "spans": [ { "bbox": [ 106, 155, 506, 168 ], "score": 1.0, "content": "1 can be applied when using contemporary machine learning models involving artificial neural", "type": "text" } ], "index": 5 }, { "bbox": [ 104, 165, 506, 180 ], "spans": [ { "bbox": [ 104, 165, 506, 180 ], "score": 1.0, "content": "networks. In this case, the hypothesis class corresponds to all neural networks with a fixed architecture", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 176, 506, 190 ], "spans": [ { "bbox": [ 105, 176, 506, 190 ], "score": 1.0, "content": "but different parameters. It is known that the class of neural networks with a fixed architecture has", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 187, 506, 201 ], "spans": [ { "bbox": [ 104, 187, 506, 201 ], "score": 1.0, "content": "finite VC dimension and Harvey et al. (2017) gives upper and lower bounds for VC dimension", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 199, 506, 212 ], "spans": [ { "bbox": [ 105, 199, 506, 212 ], "score": 1.0, "content": "of neural networks with ReLU activation functions. Thus, to apply Theorem 1, one only needs to", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 210, 505, 222 ], "spans": [ { "bbox": [ 106, 210, 505, 222 ], "score": 1.0, "content": "have an estimate of the transfer distance per Definition 2. We note that the transfer distance 3.1", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 221, 506, 234 ], "spans": [ { "bbox": [ 105, 221, 363, 234 ], "score": 1.0, "content": "consists of two terms: To estimate the first term, we note that", "type": "text" }, { "bbox": [ 363, 221, 376, 233 ], "score": 0.86, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 376, 221, 506, 234 ], "score": 1.0, "content": "can be easily estimated due to", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 231, 506, 244 ], "spans": [ { "bbox": [ 105, 231, 422, 244 ], "score": 1.0, "content": "the abundance of source data in most applications. Also with an estimate of", "type": "text" }, { "bbox": [ 422, 232, 434, 244 ], "score": 0.86, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 435, 231, 506, 244 ], "score": 1.0, "content": "in hand one can", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 243, 506, 255 ], "spans": [ { "bbox": [ 105, 243, 142, 255 ], "score": 1.0, "content": "estimate", "type": "text" }, { "bbox": [ 142, 243, 213, 255 ], "score": 0.91, "content": "\\mathbb { Q } [ h _ { S } ^ { * } ( { \\pmb x } _ { T } ) \\neq { \\ - { \\boldsymbol y } _ { T } } ]", "type": "inline_equation" }, { "bbox": [ 213, 243, 506, 255 ], "score": 1.0, "content": "rather accurately using a simple empirical average with a few target test", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 253, 505, 266 ], "spans": [ { "bbox": [ 106, 253, 505, 266 ], "score": 1.0, "content": "data as well-known concentration of bounded functions imply that this empirical average is well", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 264, 506, 279 ], "spans": [ { "bbox": [ 105, 264, 192, 279 ], "score": 1.0, "content": "concentrated around", "type": "text" }, { "bbox": [ 193, 265, 263, 277 ], "score": 0.91, "content": "\\mathbb { Q } [ h _ { S } ^ { * } ( { \\pmb x } _ { T } ) \\neq { \\ - { \\boldsymbol y } _ { T } } ]", "type": "inline_equation" }, { "bbox": [ 263, 264, 506, 279 ], "score": 1.0, "content": ". Up on first glance it seems that estimating the second term", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "which corresponds to the lowest possible error in the target domain among the hypothesis class,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 286, 507, 301 ], "spans": [ { "bbox": [ 105, 286, 507, 301 ], "score": 1.0, "content": "requires a large amount of labeled target data which is not available in a practical problem. However,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "in an overparametrized setting, it is typical to assume that there exists a network which achieves", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 321 ], "spans": [ { "bbox": [ 106, 309, 505, 321 ], "score": 1.0, "content": "very small target generalization error so we can ignore the second term in most practical problems.", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 320, 506, 333 ], "spans": [ { "bbox": [ 106, 320, 506, 333 ], "score": 1.0, "content": "Finally we note that as stated earlier the lower bound on the target excess risk gives an estimate of", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 330, 505, 343 ], "spans": [ { "bbox": [ 106, 330, 505, 343 ], "score": 1.0, "content": "what generalization performance we can expect with a certain number of source and target samples.", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 341, 505, 354 ], "spans": [ { "bbox": [ 106, 341, 505, 354 ], "score": 1.0, "content": "Furthermore, by comparing the estimated transfer distance of different pairs of tasks, we can find the", "type": "text" } ], "index": 22 }, { "bbox": [ 104, 352, 505, 366 ], "spans": [ { "bbox": [ 104, 352, 505, 366 ], "score": 1.0, "content": "pairs that are more suitable for transfer learning. This knowledge can in turn significantly reduce the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 364, 367, 377 ], "spans": [ { "bbox": [ 106, 364, 367, 377 ], "score": 1.0, "content": "required number of target samples to achieve a certain accuracy.", "type": "text" } ], "index": 24 } ], "index": 14 }, { "type": "text", "bbox": [ 107, 384, 505, 418 ], "lines": [ { "bbox": [ 106, 384, 505, 397 ], "spans": [ { "bbox": [ 106, 384, 505, 397 ], "score": 1.0, "content": "Next, we extend our result to a multiple source transfer learning setup where instead of only one", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 396, 505, 408 ], "spans": [ { "bbox": [ 106, 396, 505, 408 ], "score": 1.0, "content": "source task there are several source tasks available and the goal is to transfer knowledge from multiple", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 406, 405, 420 ], "spans": [ { "bbox": [ 105, 406, 405, 420 ], "score": 1.0, "content": "sources to a given target task to achieve a small target generalization error.", "type": "text" } ], "index": 27 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 426, 506, 517 ], "lines": [ { "bbox": [ 105, 426, 506, 440 ], "spans": [ { "bbox": [ 105, 426, 246, 440 ], "score": 1.0, "content": "Theorem 2 Suppose that there are", "type": "text" }, { "bbox": [ 246, 429, 317, 439 ], "score": 0.87, "content": "n _ { S _ { 1 } } , n _ { S _ { 2 } } , . . . , n _ { S _ { N } }", "type": "inline_equation" }, { "bbox": [ 317, 426, 414, 440 ], "score": 1.0, "content": "number of samples from", "type": "text" }, { "bbox": [ 415, 428, 425, 437 ], "score": 0.72, "content": "N", "type": "inline_equation" }, { "bbox": [ 425, 426, 506, 440 ], "score": 1.0, "content": "source tasks as well", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 437, 506, 451 ], "spans": [ { "bbox": [ 104, 437, 117, 451 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 118, 440, 131, 449 ], "score": 0.39, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 131, 437, 374, 451 ], "score": 1.0, "content": "number of samples from a target task and the hypothesis class", "type": "text" }, { "bbox": [ 374, 439, 383, 448 ], "score": 0.78, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 384, 437, 457, 451 ], "score": 1.0, "content": "has VC dimension", "type": "text" }, { "bbox": [ 457, 438, 471, 449 ], "score": 0.87, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 471, 437, 506, 451 ], "score": 1.0, "content": "obeying", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 449, 506, 464 ], "spans": [ { "bbox": [ 107, 450, 210, 462 ], "score": 0.89, "content": "d _ { \\mathcal { H } } \\ge \\operatorname* { m a x } { ( N + 9 , N / 2 ) }", "type": "inline_equation" }, { "bbox": [ 210, 449, 323, 464 ], "score": 1.0, "content": ". Furthermore, suppose that", "type": "text" }, { "bbox": [ 323, 449, 442, 462 ], "score": 0.91, "content": "\\hat { h } = \\hat { h } ( S _ { \\mathbb { P } _ { 1 } } , S _ { \\mathbb { P } _ { 2 } } , . . . , S _ { \\mathbb { P } _ { N } } , S _ { \\mathbb { Q } } )", "type": "inline_equation" }, { "bbox": [ 442, 449, 506, 464 ], "score": 1.0, "content": "is an estimated", "type": "text" } ], "index": 30 }, { "bbox": [ 174, 461, 511, 500 ], "spans": [ { "bbox": [ 174, 461, 194, 500 ], "score": 1.0, "content": "e tarand", "type": "text" }, { "bbox": [ 250, 462, 260, 471 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 260, 461, 385, 500 ], "score": 1.0, "content": "sources and target data where generated according to souce a", "type": "text" }, { "bbox": [ 385, 462, 400, 475 ], "score": 0.89, "content": "S _ { \\mathbb { P } _ { j } }", "type": "inline_equation" }, { "bbox": [ 400, 461, 419, 500 ], "score": 1.0, "content": "and targe", "type": "text" }, { "bbox": [ 420, 462, 433, 474 ], "score": 0.86, "content": "S _ { \\mathbb { Q } }", "type": "inline_equation" }, { "bbox": [ 433, 461, 449, 500 ], "score": 1.0, "content": "denstrib", "type": "text" }, { "bbox": [ 457, 461, 475, 500 ], "score": 1.0, "content": "e i.i.ions", "type": "text" }, { "bbox": [ 487, 461, 511, 500 ], "score": 1.0, "content": "dataand", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 473, 486, 494 ], "spans": [ { "bbox": [ 105, 475, 178, 494 ], "score": 1.0, "content": "{(x(i)Sj , y(i)Sj )} ji=1", "type": "text" }, { "bbox": [ 194, 473, 260, 489 ], "score": 0.93, "content": "\\{ ( \\pmb { x } _ { T } ^ { ( i ) } , \\pmb { y } _ { T } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { T } }", "type": "inline_equation" }, { "bbox": [ 475, 477, 486, 489 ], "score": 0.89, "content": "\\mathbb { P } _ { j }", "type": "inline_equation" } ], "index": 31 }, { "bbox": [ 107, 490, 506, 506 ], "spans": [ { "bbox": [ 107, 492, 115, 503 ], "score": 0.77, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 116, 490, 131, 506 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 131, 492, 181, 503 ], "score": 0.88, "content": "j = 1 , . . . , N", "type": "inline_equation" }, { "bbox": [ 181, 490, 275, 506 ], "score": 1.0, "content": ". Fix transfer distances", "type": "text" }, { "bbox": [ 276, 491, 313, 505 ], "score": 0.92, "content": "\\{ \\Delta _ { j } \\} _ { j = 1 } ^ { N }", "type": "inline_equation" }, { "bbox": [ 313, 490, 341, 506 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 341, 492, 391, 504 ], "score": 0.91, "content": "0 \\leq \\Delta _ { j } \\leq 1", "type": "inline_equation" }, { "bbox": [ 392, 490, 449, 506 ], "score": 1.0, "content": ". Then for any", "type": "text" }, { "bbox": [ 450, 490, 456, 502 ], "score": 0.8, "content": "\\hat { h }", "type": "inline_equation" }, { "bbox": [ 457, 490, 506, 506 ], "score": 1.0, "content": "there exists", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 504, 401, 517 ], "spans": [ { "bbox": [ 106, 504, 169, 516 ], "score": 0.92, "content": "( \\mathbb { P } _ { 1 } , . . . , \\mathbb { P } _ { M } , \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 169, 504, 190, 517 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 191, 504, 252, 516 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } _ { j } , \\mathbb { Q } ) \\leq \\Delta _ { j }", "type": "inline_equation" }, { "bbox": [ 252, 504, 401, 517 ], "score": 1.0, "content": "and a universal constant c such that", "type": "text" } ], "index": 34 } ], "index": 31 }, { "type": "interline_equation", "bbox": [ 146, 520, 464, 554 ], "lines": [ { "bbox": [ 146, 520, 464, 554 ], "spans": [ { "bbox": [ 146, 520, 464, 554 ], "score": 0.92, "content": "\\operatorname* { P r o b } _ { S _ { \\mathrm { P } _ { 1 } } , \\dots , S _ { \\mathrm { P } _ { N } } , S _ { \\mathrm { Q } } } \\Bigg ( \\mathcal { E } _ { T } ( \\hat { h } ) > c \\cdot \\epsilon ( n _ { S _ { 1 } } , \\dots , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) \\Bigg ) \\geq \\frac { 3 - 2 \\sqrt { 2 } } { 8 } ,", "type": "interline_equation", "image_path": "c2e69984897141ec6f1e6db13ccb05326412dda14439c09d31d50ad1a5fd15f5.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 146, 520, 464, 531.3333333333334 ], "spans": [], "index": 35 }, { "bbox": [ 146, 531.3333333333334, 464, 542.6666666666667 ], "spans": [], "index": 36 }, { "bbox": [ 146, 542.6666666666667, 464, 554.0000000000001 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 106, 558, 132, 568 ], "lines": [ { "bbox": [ 105, 555, 135, 570 ], "spans": [ { "bbox": [ 105, 555, 135, 570 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "interline_equation", "bbox": [ 143, 571, 468, 605 ], "lines": [ { "bbox": [ 143, 571, 468, 605 ], "spans": [ { "bbox": [ 143, 571, 468, 605 ], "score": 0.92, "content": "\\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) = \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { n _ { S _ { 1 } } } { d _ { \\mathcal { H } } + n _ { S _ { 1 } } \\Delta _ { 1 } } + . . . + \\frac { n _ { S _ { N } } } { d _ { \\mathcal { H } } + n _ { S _ { N } } \\Delta _ { N } } } } .", "type": "interline_equation", "image_path": "10714a1e3a3059f2e066753346bf4b545b9f0b507d8b408c32f15e2c37308954.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 143, 571, 468, 582.3333333333334 ], "spans": [], "index": 39 }, { "bbox": [ 143, 582.3333333333334, 468, 593.6666666666667 ], "spans": [], "index": 40 }, { "bbox": [ 143, 593.6666666666667, 468, 605.0000000000001 ], "spans": [], "index": 41 } ] }, { "type": "text", "bbox": [ 107, 608, 205, 619 ], "lines": [ { "bbox": [ 106, 606, 205, 621 ], "spans": [ { "bbox": [ 106, 606, 205, 621 ], "score": 1.0, "content": "This in turn implies that", "type": "text" } ], "index": 42 } ], "index": 42 }, { "type": "interline_equation", "bbox": [ 147, 623, 463, 657 ], "lines": [ { "bbox": [ 147, 623, 463, 657 ], "spans": [ { "bbox": [ 147, 623, 463, 657 ], "score": 0.92, "content": "\\operatorname* { i n f } _ { \\hat { h } } \\operatorname* { s u p } _ { \\rho ( \\mathbb { P } _ { j } , \\mathbb { Q } ) \\leq \\Delta _ { j } } S _ { \\mathbb { P } _ { 1 } , \\ldots , \\mathbb { P } _ { N } , S _ { \\mathbb { Q } } } \\Big [ \\mathcal { E } _ { T } ( \\hat { h } ) \\Big ] \\geq c \\cdot \\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) .", "type": "interline_equation", "image_path": "8c1ae3d2063bbfe004744586a39a15eeb61c6a8eba192bc3c6ce4cef655eb056.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 147, 623, 463, 634.3333333333334 ], "spans": [], "index": 43 }, { "bbox": [ 147, 634.3333333333334, 463, 645.6666666666667 ], "spans": [], "index": 44 }, { "bbox": [ 147, 645.6666666666667, 463, 657.0000000000001 ], "spans": [], "index": 45 } ] }, { "type": "text", "bbox": [ 106, 665, 505, 732 ], "lines": [ { "bbox": [ 105, 665, 506, 678 ], "spans": [ { "bbox": [ 105, 665, 506, 678 ], "score": 1.0, "content": "Remark 8 Similar to the previous theorem, Theorem 2 provides a minimax lower bound for target", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 677, 506, 690 ], "spans": [ { "bbox": [ 105, 677, 506, 690 ], "score": 1.0, "content": "excess risk with the key distinction that now it applies in the setting where there are multiple source", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "with different transfer distances to the target. This theorem characterizes the exces risk achievable by", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "any algorithm as a function of these transfer distances as well as the number of samples from the", "type": "text" } ], "index": 49 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "different sources and the target data. Theorem 2 indicates that the more sources we have, the better", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 721, 506, 733 ], "spans": [ { "bbox": [ 105, 721, 506, 733 ], "score": 1.0, "content": "performance we can achieve in the target domain. However, this performance gain maybe marginal", "type": "text" } ], "index": 51 } ], "index": 48.5 } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 134 ], "lines": [ { "bbox": [ 106, 82, 506, 95 ], "spans": [ { "bbox": [ 106, 82, 506, 95 ], "score": 1.0, "content": "Remark 6 (Sharpness in a special case) We note that the above lower bound is known to be tight", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 107 ], "spans": [ { "bbox": [ 105, 93, 432, 107 ], "score": 1.0, "content": "in special cases. For instance when there is a small amount of source data and", "type": "text" }, { "bbox": [ 432, 95, 442, 104 ], "score": 0.43, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 442, 93, 506, 107 ], "score": 1.0, "content": "is rather large,", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 506, 124 ], "spans": [ { "bbox": [ 105, 108, 216, 122 ], "score": 1.0, "content": "the lower bound reduces to", "type": "text" }, { "bbox": [ 223, 105, 506, 124 ], "score": 1.0, "content": "dHn which is known to be tight based on known agnostic PAC learning", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 122, 140, 135 ], "spans": [ { "bbox": [ 105, 122, 140, 135 ], "score": 1.0, "content": "bounds.", "type": "text" } ], "index": 3 } ], "index": 1.5, "bbox_fs": [ 105, 82, 506, 135 ] }, { "type": "text", "bbox": [ 106, 143, 505, 375 ], "lines": [ { "bbox": [ 106, 144, 506, 157 ], "spans": [ { "bbox": [ 106, 144, 506, 157 ], "score": 1.0, "content": "Remark 7 (How to apply Theorem 1 in practical settings.) In this remark we explain how Theorem", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 155, 506, 168 ], "spans": [ { "bbox": [ 106, 155, 506, 168 ], "score": 1.0, "content": "1 can be applied when using contemporary machine learning models involving artificial neural", "type": "text" } ], "index": 5 }, { "bbox": [ 104, 165, 506, 180 ], "spans": [ { "bbox": [ 104, 165, 506, 180 ], "score": 1.0, "content": "networks. In this case, the hypothesis class corresponds to all neural networks with a fixed architecture", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 176, 506, 190 ], "spans": [ { "bbox": [ 105, 176, 506, 190 ], "score": 1.0, "content": "but different parameters. It is known that the class of neural networks with a fixed architecture has", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 187, 506, 201 ], "spans": [ { "bbox": [ 104, 187, 506, 201 ], "score": 1.0, "content": "finite VC dimension and Harvey et al. (2017) gives upper and lower bounds for VC dimension", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 199, 506, 212 ], "spans": [ { "bbox": [ 105, 199, 506, 212 ], "score": 1.0, "content": "of neural networks with ReLU activation functions. Thus, to apply Theorem 1, one only needs to", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 210, 505, 222 ], "spans": [ { "bbox": [ 106, 210, 505, 222 ], "score": 1.0, "content": "have an estimate of the transfer distance per Definition 2. We note that the transfer distance 3.1", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 221, 506, 234 ], "spans": [ { "bbox": [ 105, 221, 363, 234 ], "score": 1.0, "content": "consists of two terms: To estimate the first term, we note that", "type": "text" }, { "bbox": [ 363, 221, 376, 233 ], "score": 0.86, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 376, 221, 506, 234 ], "score": 1.0, "content": "can be easily estimated due to", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 231, 506, 244 ], "spans": [ { "bbox": [ 105, 231, 422, 244 ], "score": 1.0, "content": "the abundance of source data in most applications. Also with an estimate of", "type": "text" }, { "bbox": [ 422, 232, 434, 244 ], "score": 0.86, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 435, 231, 506, 244 ], "score": 1.0, "content": "in hand one can", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 243, 506, 255 ], "spans": [ { "bbox": [ 105, 243, 142, 255 ], "score": 1.0, "content": "estimate", "type": "text" }, { "bbox": [ 142, 243, 213, 255 ], "score": 0.91, "content": "\\mathbb { Q } [ h _ { S } ^ { * } ( { \\pmb x } _ { T } ) \\neq { \\ - { \\boldsymbol y } _ { T } } ]", "type": "inline_equation" }, { "bbox": [ 213, 243, 506, 255 ], "score": 1.0, "content": "rather accurately using a simple empirical average with a few target test", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 253, 505, 266 ], "spans": [ { "bbox": [ 106, 253, 505, 266 ], "score": 1.0, "content": "data as well-known concentration of bounded functions imply that this empirical average is well", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 264, 506, 279 ], "spans": [ { "bbox": [ 105, 264, 192, 279 ], "score": 1.0, "content": "concentrated around", "type": "text" }, { "bbox": [ 193, 265, 263, 277 ], "score": 0.91, "content": "\\mathbb { Q } [ h _ { S } ^ { * } ( { \\pmb x } _ { T } ) \\neq { \\ - { \\boldsymbol y } _ { T } } ]", "type": "inline_equation" }, { "bbox": [ 263, 264, 506, 279 ], "score": 1.0, "content": ". Up on first glance it seems that estimating the second term", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "which corresponds to the lowest possible error in the target domain among the hypothesis class,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 286, 507, 301 ], "spans": [ { "bbox": [ 105, 286, 507, 301 ], "score": 1.0, "content": "requires a large amount of labeled target data which is not available in a practical problem. However,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "in an overparametrized setting, it is typical to assume that there exists a network which achieves", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 321 ], "spans": [ { "bbox": [ 106, 309, 505, 321 ], "score": 1.0, "content": "very small target generalization error so we can ignore the second term in most practical problems.", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 320, 506, 333 ], "spans": [ { "bbox": [ 106, 320, 506, 333 ], "score": 1.0, "content": "Finally we note that as stated earlier the lower bound on the target excess risk gives an estimate of", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 330, 505, 343 ], "spans": [ { "bbox": [ 106, 330, 505, 343 ], "score": 1.0, "content": "what generalization performance we can expect with a certain number of source and target samples.", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 341, 505, 354 ], "spans": [ { "bbox": [ 106, 341, 505, 354 ], "score": 1.0, "content": "Furthermore, by comparing the estimated transfer distance of different pairs of tasks, we can find the", "type": "text" } ], "index": 22 }, { "bbox": [ 104, 352, 505, 366 ], "spans": [ { "bbox": [ 104, 352, 505, 366 ], "score": 1.0, "content": "pairs that are more suitable for transfer learning. This knowledge can in turn significantly reduce the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 364, 367, 377 ], "spans": [ { "bbox": [ 106, 364, 367, 377 ], "score": 1.0, "content": "required number of target samples to achieve a certain accuracy.", "type": "text" } ], "index": 24 } ], "index": 14, "bbox_fs": [ 104, 144, 507, 377 ] }, { "type": "text", "bbox": [ 107, 384, 505, 418 ], "lines": [ { "bbox": [ 106, 384, 505, 397 ], "spans": [ { "bbox": [ 106, 384, 505, 397 ], "score": 1.0, "content": "Next, we extend our result to a multiple source transfer learning setup where instead of only one", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 396, 505, 408 ], "spans": [ { "bbox": [ 106, 396, 505, 408 ], "score": 1.0, "content": "source task there are several source tasks available and the goal is to transfer knowledge from multiple", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 406, 405, 420 ], "spans": [ { "bbox": [ 105, 406, 405, 420 ], "score": 1.0, "content": "sources to a given target task to achieve a small target generalization error.", "type": "text" } ], "index": 27 } ], "index": 26, "bbox_fs": [ 105, 384, 505, 420 ] }, { "type": "text", "bbox": [ 106, 426, 506, 517 ], "lines": [ { "bbox": [ 105, 426, 506, 440 ], "spans": [ { "bbox": [ 105, 426, 246, 440 ], "score": 1.0, "content": "Theorem 2 Suppose that there are", "type": "text" }, { "bbox": [ 246, 429, 317, 439 ], "score": 0.87, "content": "n _ { S _ { 1 } } , n _ { S _ { 2 } } , . . . , n _ { S _ { N } }", "type": "inline_equation" }, { "bbox": [ 317, 426, 414, 440 ], "score": 1.0, "content": "number of samples from", "type": "text" }, { "bbox": [ 415, 428, 425, 437 ], "score": 0.72, "content": "N", "type": "inline_equation" }, { "bbox": [ 425, 426, 506, 440 ], "score": 1.0, "content": "source tasks as well", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 437, 506, 451 ], "spans": [ { "bbox": [ 104, 437, 117, 451 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 118, 440, 131, 449 ], "score": 0.39, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 131, 437, 374, 451 ], "score": 1.0, "content": "number of samples from a target task and the hypothesis class", "type": "text" }, { "bbox": [ 374, 439, 383, 448 ], "score": 0.78, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 384, 437, 457, 451 ], "score": 1.0, "content": "has VC dimension", "type": "text" }, { "bbox": [ 457, 438, 471, 449 ], "score": 0.87, "content": "d _ { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 471, 437, 506, 451 ], "score": 1.0, "content": "obeying", "type": "text" } ], "index": 29 }, { "bbox": [ 107, 449, 506, 464 ], "spans": [ { "bbox": [ 107, 450, 210, 462 ], "score": 0.89, "content": "d _ { \\mathcal { H } } \\ge \\operatorname* { m a x } { ( N + 9 , N / 2 ) }", "type": "inline_equation" }, { "bbox": [ 210, 449, 323, 464 ], "score": 1.0, "content": ". Furthermore, suppose that", "type": "text" }, { "bbox": [ 323, 449, 442, 462 ], "score": 0.91, "content": "\\hat { h } = \\hat { h } ( S _ { \\mathbb { P } _ { 1 } } , S _ { \\mathbb { P } _ { 2 } } , . . . , S _ { \\mathbb { P } _ { N } } , S _ { \\mathbb { Q } } )", "type": "inline_equation" }, { "bbox": [ 442, 449, 506, 464 ], "score": 1.0, "content": "is an estimated", "type": "text" } ], "index": 30 }, { "bbox": [ 174, 461, 511, 500 ], "spans": [ { "bbox": [ 174, 461, 194, 500 ], "score": 1.0, "content": "e tarand", "type": "text" }, { "bbox": [ 250, 462, 260, 471 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 260, 461, 385, 500 ], "score": 1.0, "content": "sources and target data where generated according to souce a", "type": "text" }, { "bbox": [ 385, 462, 400, 475 ], "score": 0.89, "content": "S _ { \\mathbb { P } _ { j } }", "type": "inline_equation" }, { "bbox": [ 400, 461, 419, 500 ], "score": 1.0, "content": "and targe", "type": "text" }, { "bbox": [ 420, 462, 433, 474 ], "score": 0.86, "content": "S _ { \\mathbb { Q } }", "type": "inline_equation" }, { "bbox": [ 433, 461, 449, 500 ], "score": 1.0, "content": "denstrib", "type": "text" }, { "bbox": [ 457, 461, 475, 500 ], "score": 1.0, "content": "e i.i.ions", "type": "text" }, { "bbox": [ 487, 461, 511, 500 ], "score": 1.0, "content": "dataand", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 473, 486, 494 ], "spans": [ { "bbox": [ 105, 475, 178, 494 ], "score": 1.0, "content": "{(x(i)Sj , y(i)Sj )} ji=1", "type": "text" }, { "bbox": [ 194, 473, 260, 489 ], "score": 0.93, "content": "\\{ ( \\pmb { x } _ { T } ^ { ( i ) } , \\pmb { y } _ { T } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { T } }", "type": "inline_equation" }, { "bbox": [ 475, 477, 486, 489 ], "score": 0.89, "content": "\\mathbb { P } _ { j }", "type": "inline_equation" } ], "index": 31 }, { "bbox": [ 107, 490, 506, 506 ], "spans": [ { "bbox": [ 107, 492, 115, 503 ], "score": 0.77, "content": "\\mathbb { Q }", "type": "inline_equation" }, { "bbox": [ 116, 490, 131, 506 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 131, 492, 181, 503 ], "score": 0.88, "content": "j = 1 , . . . , N", "type": "inline_equation" }, { "bbox": [ 181, 490, 275, 506 ], "score": 1.0, "content": ". Fix transfer distances", "type": "text" }, { "bbox": [ 276, 491, 313, 505 ], "score": 0.92, "content": "\\{ \\Delta _ { j } \\} _ { j = 1 } ^ { N }", "type": "inline_equation" }, { "bbox": [ 313, 490, 341, 506 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 341, 492, 391, 504 ], "score": 0.91, "content": "0 \\leq \\Delta _ { j } \\leq 1", "type": "inline_equation" }, { "bbox": [ 392, 490, 449, 506 ], "score": 1.0, "content": ". Then for any", "type": "text" }, { "bbox": [ 450, 490, 456, 502 ], "score": 0.8, "content": "\\hat { h }", "type": "inline_equation" }, { "bbox": [ 457, 490, 506, 506 ], "score": 1.0, "content": "there exists", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 504, 401, 517 ], "spans": [ { "bbox": [ 106, 504, 169, 516 ], "score": 0.92, "content": "( \\mathbb { P } _ { 1 } , . . . , \\mathbb { P } _ { M } , \\mathbb { Q } )", "type": "inline_equation" }, { "bbox": [ 169, 504, 190, 517 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 191, 504, 252, 516 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } _ { j } , \\mathbb { Q } ) \\leq \\Delta _ { j }", "type": "inline_equation" }, { "bbox": [ 252, 504, 401, 517 ], "score": 1.0, "content": "and a universal constant c such that", "type": "text" } ], "index": 34 } ], "index": 31, "bbox_fs": [ 104, 426, 511, 517 ] }, { "type": "interline_equation", "bbox": [ 146, 520, 464, 554 ], "lines": [ { "bbox": [ 146, 520, 464, 554 ], "spans": [ { "bbox": [ 146, 520, 464, 554 ], "score": 0.92, "content": "\\operatorname* { P r o b } _ { S _ { \\mathrm { P } _ { 1 } } , \\dots , S _ { \\mathrm { P } _ { N } } , S _ { \\mathrm { Q } } } \\Bigg ( \\mathcal { E } _ { T } ( \\hat { h } ) > c \\cdot \\epsilon ( n _ { S _ { 1 } } , \\dots , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) \\Bigg ) \\geq \\frac { 3 - 2 \\sqrt { 2 } } { 8 } ,", "type": "interline_equation", "image_path": "c2e69984897141ec6f1e6db13ccb05326412dda14439c09d31d50ad1a5fd15f5.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 146, 520, 464, 531.3333333333334 ], "spans": [], "index": 35 }, { "bbox": [ 146, 531.3333333333334, 464, 542.6666666666667 ], "spans": [], "index": 36 }, { "bbox": [ 146, 542.6666666666667, 464, 554.0000000000001 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 106, 558, 132, 568 ], "lines": [ { "bbox": [ 105, 555, 135, 570 ], "spans": [ { "bbox": [ 105, 555, 135, 570 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 38 } ], "index": 38, "bbox_fs": [ 105, 555, 135, 570 ] }, { "type": "interline_equation", "bbox": [ 143, 571, 468, 605 ], "lines": [ { "bbox": [ 143, 571, 468, 605 ], "spans": [ { "bbox": [ 143, 571, 468, 605 ], "score": 0.92, "content": "\\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) = \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { n _ { S _ { 1 } } } { d _ { \\mathcal { H } } + n _ { S _ { 1 } } \\Delta _ { 1 } } + . . . + \\frac { n _ { S _ { N } } } { d _ { \\mathcal { H } } + n _ { S _ { N } } \\Delta _ { N } } } } .", "type": "interline_equation", "image_path": "10714a1e3a3059f2e066753346bf4b545b9f0b507d8b408c32f15e2c37308954.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 143, 571, 468, 582.3333333333334 ], "spans": [], "index": 39 }, { "bbox": [ 143, 582.3333333333334, 468, 593.6666666666667 ], "spans": [], "index": 40 }, { "bbox": [ 143, 593.6666666666667, 468, 605.0000000000001 ], "spans": [], "index": 41 } ] }, { "type": "text", "bbox": [ 107, 608, 205, 619 ], "lines": [ { "bbox": [ 106, 606, 205, 621 ], "spans": [ { "bbox": [ 106, 606, 205, 621 ], "score": 1.0, "content": "This in turn implies that", "type": "text" } ], "index": 42 } ], "index": 42, "bbox_fs": [ 106, 606, 205, 621 ] }, { "type": "interline_equation", "bbox": [ 147, 623, 463, 657 ], "lines": [ { "bbox": [ 147, 623, 463, 657 ], "spans": [ { "bbox": [ 147, 623, 463, 657 ], "score": 0.92, "content": "\\operatorname* { i n f } _ { \\hat { h } } \\operatorname* { s u p } _ { \\rho ( \\mathbb { P } _ { j } , \\mathbb { Q } ) \\leq \\Delta _ { j } } S _ { \\mathbb { P } _ { 1 } , \\ldots , \\mathbb { P } _ { N } , S _ { \\mathbb { Q } } } \\Big [ \\mathcal { E } _ { T } ( \\hat { h } ) \\Big ] \\geq c \\cdot \\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } ) .", "type": "interline_equation", "image_path": "8c1ae3d2063bbfe004744586a39a15eeb61c6a8eba192bc3c6ce4cef655eb056.jpg" } ] } ], "index": 44, "virtual_lines": [ { "bbox": [ 147, 623, 463, 634.3333333333334 ], "spans": [], "index": 43 }, { "bbox": [ 147, 634.3333333333334, 463, 645.6666666666667 ], "spans": [], "index": 44 }, { "bbox": [ 147, 645.6666666666667, 463, 657.0000000000001 ], "spans": [], "index": 45 } ] }, { "type": "text", "bbox": [ 106, 665, 505, 732 ], "lines": [ { "bbox": [ 105, 665, 506, 678 ], "spans": [ { "bbox": [ 105, 665, 506, 678 ], "score": 1.0, "content": "Remark 8 Similar to the previous theorem, Theorem 2 provides a minimax lower bound for target", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 677, 506, 690 ], "spans": [ { "bbox": [ 105, 677, 506, 690 ], "score": 1.0, "content": "excess risk with the key distinction that now it applies in the setting where there are multiple source", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "with different transfer distances to the target. This theorem characterizes the exces risk achievable by", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "any algorithm as a function of these transfer distances as well as the number of samples from the", "type": "text" } ], "index": 49 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "different sources and the target data. Theorem 2 indicates that the more sources we have, the better", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 721, 506, 733 ], "spans": [ { "bbox": [ 105, 721, 506, 733 ], "score": 1.0, "content": "performance we can achieve in the target domain. However, this performance gain maybe marginal", "type": "text" } ], "index": 51 }, { "bbox": [ 104, 81, 505, 96 ], "spans": [ { "bbox": [ 104, 81, 505, 96 ], "score": 1.0, "content": "for source tasks that have a large transfer distance to the target or where there are very few training", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "data. In these cases of course it may be more computationally efficient to discard these sources given", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 104, 459, 117 ], "spans": [ { "bbox": [ 105, 104, 459, 117 ], "score": 1.0, "content": "the marginal improvement in the generalization performance suggested by this theorem.", "type": "text", "cross_page": true } ], "index": 2 } ], "index": 48.5, "bbox_fs": [ 105, 665, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [ { "bbox": [ 104, 81, 505, 96 ], "spans": [ { "bbox": [ 104, 81, 505, 96 ], "score": 1.0, "content": "for source tasks that have a large transfer distance to the target or where there are very few training", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "data. In these cases of course it may be more computationally efficient to discard these sources given", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 459, 117 ], "spans": [ { "bbox": [ 105, 104, 459, 117 ], "score": 1.0, "content": "the marginal improvement in the generalization performance suggested by this theorem.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 106, 124, 505, 177 ], "lines": [ { "bbox": [ 106, 124, 506, 137 ], "spans": [ { "bbox": [ 106, 124, 506, 137 ], "score": 1.0, "content": "Remark 9 (Identical sources) if all the source tasks are identical, then there are effec-", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 135, 507, 150 ], "spans": [ { "bbox": [ 105, 135, 134, 150 ], "score": 1.0, "content": "tively", "type": "text" }, { "bbox": [ 134, 137, 212, 148 ], "score": 0.74, "content": "n _ { S _ { 1 } } ~ + ~ . . . ~ + ~ n _ { S _ { N } }", "type": "inline_equation" }, { "bbox": [ 212, 135, 507, 150 ], "score": 1.0, "content": "number of source samples and by Theorem 1 the lower bound", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 149, 505, 171 ], "spans": [ { "bbox": [ 104, 149, 150, 162 ], "score": 1.0, "content": "would be", "type": "text" }, { "bbox": [ 188, 149, 228, 171 ], "score": 1.0, "content": "1PNj=1 nSj", "type": "text" }, { "bbox": [ 240, 150, 505, 163 ], "score": 1.0, "content": ". Theorem 2 also gives the same order wise lower bound as", "type": "text" } ], "index": 5 }, { "bbox": [ 162, 159, 239, 182 ], "spans": [ { "bbox": [ 162, 159, 181, 171 ], "score": 1.0, "content": "nT +", "type": "text" }, { "bbox": [ 162, 166, 174, 176 ], "score": 1.0, "content": "dH", "type": "text" }, { "bbox": [ 177, 162, 239, 182 ], "score": 1.0, "content": "dH+∆ PNj=1 nSj", "type": "text" } ], "index": 6 } ], "index": 4.5 }, { "type": "interline_equation", "bbox": [ 107, 177, 416, 212 ], "lines": [ { "bbox": [ 107, 177, 416, 212 ], "spans": [ { "bbox": [ 107, 177, 416, 212 ], "score": 0.78, "content": "\\begin{array} { r } { \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\sum _ { j = 1 } ^ { N } \\frac { n _ { j } } { d _ { \\mathcal { H } } + \\Delta n _ { j } } } } \\le \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } } { d _ { \\mathcal { H } } + \\Delta \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } } } } \\le \\sqrt { N } \\cdot \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\sum _ { j = 1 } ^ { N } \\frac { n _ { j } } { d _ { \\mathcal { H } } + \\Delta n _ { j } } } } } \\end{array}", "type": "interline_equation", "image_path": "57a113d83402013b437fcb88ca432b9a348d35f06a962b20191fccb31df3858f.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 107, 177, 416, 188.66666666666666 ], "spans": [], "index": 7 }, { "bbox": [ 107, 188.66666666666666, 416, 200.33333333333331 ], "spans": [], "index": 8 }, { "bbox": [ 107, 200.33333333333331, 416, 211.99999999999997 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 106, 220, 505, 262 ], "lines": [ { "bbox": [ 102, 216, 506, 241 ], "spans": [ { "bbox": [ 102, 216, 506, 241 ], "score": 1.0, "content": "Remark 10 (Infinitely many source samples) When ∆i > 0 and nSi → ∞, the fraction nSidH+nS ∆i", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 237, 505, 253 ], "spans": [ { "bbox": [ 106, 237, 156, 252 ], "score": 1.0, "content": "saturates at", "type": "text" }, { "bbox": [ 156, 237, 169, 253 ], "score": 0.89, "content": "\\frac { 1 } { \\Delta _ { i } }", "type": "inline_equation" }, { "bbox": [ 169, 237, 505, 252 ], "score": 1.0, "content": "which shows that when the source and target have positive distance, the source can", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 251, 271, 263 ], "spans": [ { "bbox": [ 106, 251, 271, 263 ], "score": 1.0, "content": "never compensate for the target samples.", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "text", "bbox": [ 106, 271, 504, 305 ], "lines": [ { "bbox": [ 105, 271, 506, 284 ], "spans": [ { "bbox": [ 105, 271, 310, 284 ], "score": 1.0, "content": "Remark 11 In the lower bound, the product terms", "type": "text" }, { "bbox": [ 311, 272, 337, 284 ], "score": 0.91, "content": "\\Delta _ { i } n _ { S _ { i } }", "type": "inline_equation" }, { "bbox": [ 337, 271, 506, 284 ], "score": 1.0, "content": "appear which indicate that a source with", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 283, 505, 294 ], "spans": [ { "bbox": [ 106, 283, 505, 294 ], "score": 1.0, "content": "large transfer distance can sometimes be as useful as a source with small transfer distance when", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 293, 375, 306 ], "spans": [ { "bbox": [ 105, 293, 375, 306 ], "score": 1.0, "content": "there is a large amount of training data available from that source.", "type": "text" } ], "index": 15 } ], "index": 14 }, { "type": "title", "bbox": [ 108, 320, 257, 333 ], "lines": [ { "bbox": [ 105, 320, 258, 334 ], "spans": [ { "bbox": [ 105, 320, 258, 334 ], "score": 1.0, "content": "5 EXPERIMENTAL RESULTS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 344, 505, 401 ], "lines": [ { "bbox": [ 105, 343, 505, 358 ], "spans": [ { "bbox": [ 105, 343, 505, 358 ], "score": 1.0, "content": "In this section we evaluate our theoretical results on real data sets for action recognition and image", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 355, 506, 368 ], "spans": [ { "bbox": [ 106, 355, 506, 368 ], "score": 1.0, "content": "classification tasks. By estimating the parameters appearing in Theorem 1 for different pairs of tasks,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 367, 506, 380 ], "spans": [ { "bbox": [ 105, 367, 506, 380 ], "score": 1.0, "content": "we first plot the lower bounds and then by running weighted empirical risk minimization investigate", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 377, 505, 390 ], "spans": [ { "bbox": [ 105, 377, 505, 390 ], "score": 1.0, "content": "the sharpness of the bounds. We also investigate the effectiveness of different source tasks with", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 388, 350, 402 ], "spans": [ { "bbox": [ 105, 388, 350, 402 ], "score": 1.0, "content": "different transfer distances on the target generalization error.", "type": "text" } ], "index": 21 } ], "index": 19 }, { "type": "title", "bbox": [ 108, 413, 230, 424 ], "lines": [ { "bbox": [ 106, 413, 231, 426 ], "spans": [ { "bbox": [ 106, 413, 231, 426 ], "score": 1.0, "content": "5.1 ACTION RECOGNITION", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 106, 433, 505, 511 ], "lines": [ { "bbox": [ 106, 433, 507, 447 ], "spans": [ { "bbox": [ 106, 433, 507, 447 ], "score": 1.0, "content": "Experimental setup. We first perform experiments on the UCF101 action recognition data set.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 444, 506, 457 ], "spans": [ { "bbox": [ 105, 444, 506, 457 ], "score": 1.0, "content": "We pick CricketBowling and TableTennis videos from UCF101 as the target task as well as three", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 455, 505, 468 ], "spans": [ { "bbox": [ 106, 455, 505, 468 ], "score": 1.0, "content": "different pairs of classes as the source tasks: 1- CricketBowling and BaseballPitch, 2- Cricketshot and", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 466, 505, 479 ], "spans": [ { "bbox": [ 105, 466, 505, 479 ], "score": 1.0, "content": "Archery, 3- BasketballDunk and Basketball. We pass the videos through an i3d network pretrained on", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 477, 506, 490 ], "spans": [ { "bbox": [ 105, 477, 506, 490 ], "score": 1.0, "content": "kinetics400 Carreira & Zisserman (2017) with the fully connected top classifier removed and extract", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 489, 505, 500 ], "spans": [ { "bbox": [ 106, 489, 505, 500 ], "score": 1.0, "content": "the corresponding features of dimension 2048 from the raw videos. We then work with the extracted", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 500, 245, 511 ], "spans": [ { "bbox": [ 106, 500, 245, 511 ], "score": 1.0, "content": "features instead of the raw videos.", "type": "text" } ], "index": 29 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 515, 505, 594 ], "lines": [ { "bbox": [ 106, 516, 505, 528 ], "spans": [ { "bbox": [ 106, 516, 505, 528 ], "score": 1.0, "content": "Training. We train a one hidden layer neural network with 15 number of hidden units and ReLU", "type": "text" } ], "index": 30 }, { "bbox": [ 104, 526, 506, 541 ], "spans": [ { "bbox": [ 104, 526, 506, 541 ], "score": 1.0, "content": "activation functions for each pair of data sets. Table 3 consists of test accuracy on CricketBowling", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 539, 505, 550 ], "spans": [ { "bbox": [ 106, 539, 505, 550 ], "score": 1.0, "content": "vs. TableTennis, when using the network trained on each source task. We use these accuracies for", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 550, 505, 562 ], "spans": [ { "bbox": [ 106, 550, 505, 562 ], "score": 1.0, "content": "deriving the corresponding lower bounds. Furthermore, we run weighted empirical risk minimization", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 560, 506, 573 ], "spans": [ { "bbox": [ 105, 560, 506, 573 ], "score": 1.0, "content": "as a simple transfer learning approach to find some upper bounds on the target generalization error.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 572, 505, 584 ], "spans": [ { "bbox": [ 106, 572, 132, 584 ], "score": 1.0, "content": "Given", "type": "text" }, { "bbox": [ 133, 573, 146, 583 ], "score": 0.84, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 146, 572, 163, 584 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 164, 573, 177, 582 ], "score": 0.85, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 177, 572, 505, 584 ], "score": 1.0, "content": "number of source and target samples, for estimating the corresponding one hidden", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 582, 444, 595 ], "spans": [ { "bbox": [ 105, 582, 444, 595 ], "score": 1.0, "content": "layer neural network parameters we minimize the following weighted empirical risk", "type": "text" } ], "index": 36 } ], "index": 33 }, { "type": "interline_equation", "bbox": [ 111, 596, 480, 629 ], "lines": [ { "bbox": [ 111, 596, 480, 629 ], "spans": [ { "bbox": [ 111, 596, 480, 629 ], "score": 0.94, "content": "\\operatorname* { m i n } _ { W _ { 1 } , W _ { 2 } } \\frac { 1 - \\lambda } { n _ { T } } \\sum _ { i = 1 } ^ { n _ { T } } \\mathbf { C o s t } ( W _ { 2 } \\mathbf { R e L U } ( W _ { 1 } \\pmb { x } _ { T } ^ { ( i ) } ) , y _ { T } ^ { ( i ) } ) + \\frac { \\lambda } { n _ { S } } \\sum _ { i = 1 } ^ { n _ { S } } \\mathbf { C o s t } ( W _ { 2 } \\mathbf { R e L U } ( W _ { 1 } \\pmb { x } _ { S } ^ { ( i ) } ) , y _ { S } ^ { ( i ) } )", "type": "interline_equation", "image_path": "3c19850b2377a8366c3c35efac700649b14797e228f9403eabe5cee32aa0f04d.jpg" } ] } ], "index": 38, "virtual_lines": [ { "bbox": [ 111, 596, 480, 607.0 ], "spans": [], "index": 37 }, { "bbox": [ 111, 607.0, 480, 618.0 ], "spans": [], "index": 38 }, { "bbox": [ 111, 618.0, 480, 629.0 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 105, 637, 505, 660 ], "lines": [ { "bbox": [ 106, 637, 505, 650 ], "spans": [ { "bbox": [ 106, 637, 370, 650 ], "score": 1.0, "content": "where the function Cost denotes the logistic regression cost and", "type": "text" }, { "bbox": [ 370, 637, 483, 650 ], "score": 0.91, "content": "\\lambda \\in \\{ 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 484, 637, 505, 650 ], "score": 1.0, "content": ". We", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 648, 343, 662 ], "spans": [ { "bbox": [ 105, 648, 343, 662 ], "score": 1.0, "content": "then pick the lambda which minimizes the target test error.", "type": "text" } ], "index": 41 } ], "index": 40.5 }, { "type": "text", "bbox": [ 106, 665, 505, 732 ], "lines": [ { "bbox": [ 105, 664, 505, 679 ], "spans": [ { "bbox": [ 105, 664, 505, 679 ], "score": 1.0, "content": "Results. First we calculate the transfer distance by Definition 2 for each source/target pairs using", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 676, 506, 689 ], "spans": [ { "bbox": [ 105, 676, 506, 689 ], "score": 1.0, "content": "Table 1. To this end, we assume that best target generalization error is zero and using the Table 1 we", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 687, 506, 700 ], "spans": [ { "bbox": [ 106, 687, 506, 700 ], "score": 1.0, "content": "obtain the transfer distance for each pair which is demonstrated in Table 2. As it can be observed", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 699, 506, 712 ], "spans": [ { "bbox": [ 106, 699, 506, 712 ], "score": 1.0, "content": "by Table 2, the pair of Source1 and Target has the lowest transfer distance among other pairs since", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "both of the source and target tasks share a same class which is CricketBowling. Furthermore, Table 2", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 720, 477, 733 ], "spans": [ { "bbox": [ 105, 720, 477, 733 ], "score": 1.0, "content": "determines which pairs are more suitable for transferring the source knowledge to the target.", "type": "text" } ], "index": 47 } ], "index": 44.5 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 310, 761 ], "spans": [ { "bbox": [ 302, 750, 310, 761 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [], "index": 1, "bbox_fs": [ 104, 81, 505, 117 ], "lines_deleted": true }, { "type": "text", "bbox": [ 106, 124, 505, 177 ], "lines": [ { "bbox": [ 106, 124, 506, 137 ], "spans": [ { "bbox": [ 106, 124, 506, 137 ], "score": 1.0, "content": "Remark 9 (Identical sources) if all the source tasks are identical, then there are effec-", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 135, 507, 150 ], "spans": [ { "bbox": [ 105, 135, 134, 150 ], "score": 1.0, "content": "tively", "type": "text" }, { "bbox": [ 134, 137, 212, 148 ], "score": 0.74, "content": "n _ { S _ { 1 } } ~ + ~ . . . ~ + ~ n _ { S _ { N } }", "type": "inline_equation" }, { "bbox": [ 212, 135, 507, 150 ], "score": 1.0, "content": "number of source samples and by Theorem 1 the lower bound", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 149, 505, 171 ], "spans": [ { "bbox": [ 104, 149, 150, 162 ], "score": 1.0, "content": "would be", "type": "text" }, { "bbox": [ 188, 149, 228, 171 ], "score": 1.0, "content": "1PNj=1 nSj", "type": "text" }, { "bbox": [ 240, 150, 505, 163 ], "score": 1.0, "content": ". Theorem 2 also gives the same order wise lower bound as", "type": "text" } ], "index": 5 }, { "bbox": [ 162, 159, 239, 182 ], "spans": [ { "bbox": [ 162, 159, 181, 171 ], "score": 1.0, "content": "nT +", "type": "text" }, { "bbox": [ 162, 166, 174, 176 ], "score": 1.0, "content": "dH", "type": "text" }, { "bbox": [ 177, 162, 239, 182 ], "score": 1.0, "content": "dH+∆ PNj=1 nSj", "type": "text" } ], "index": 6 } ], "index": 4.5, "bbox_fs": [ 104, 124, 507, 182 ] }, { "type": "interline_equation", "bbox": [ 107, 177, 416, 212 ], "lines": [ { "bbox": [ 107, 177, 416, 212 ], "spans": [ { "bbox": [ 107, 177, 416, 212 ], "score": 0.78, "content": "\\begin{array} { r } { \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\sum _ { j = 1 } ^ { N } \\frac { n _ { j } } { d _ { \\mathcal { H } } + \\Delta n _ { j } } } } \\le \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\frac { \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } } { d _ { \\mathcal { H } } + \\Delta \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } } } } \\le \\sqrt { N } \\cdot \\sqrt { \\frac { 1 } { \\frac { n _ { T } } { d _ { \\mathcal { H } } } + \\sum _ { j = 1 } ^ { N } \\frac { n _ { j } } { d _ { \\mathcal { H } } + \\Delta n _ { j } } } } } \\end{array}", "type": "interline_equation", "image_path": "57a113d83402013b437fcb88ca432b9a348d35f06a962b20191fccb31df3858f.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 107, 177, 416, 188.66666666666666 ], "spans": [], "index": 7 }, { "bbox": [ 107, 188.66666666666666, 416, 200.33333333333331 ], "spans": [], "index": 8 }, { "bbox": [ 107, 200.33333333333331, 416, 211.99999999999997 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 106, 220, 505, 262 ], "lines": [ { "bbox": [ 102, 216, 506, 241 ], "spans": [ { "bbox": [ 102, 216, 506, 241 ], "score": 1.0, "content": "Remark 10 (Infinitely many source samples) When ∆i > 0 and nSi → ∞, the fraction nSidH+nS ∆i", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 237, 505, 253 ], "spans": [ { "bbox": [ 106, 237, 156, 252 ], "score": 1.0, "content": "saturates at", "type": "text" }, { "bbox": [ 156, 237, 169, 253 ], "score": 0.89, "content": "\\frac { 1 } { \\Delta _ { i } }", "type": "inline_equation" }, { "bbox": [ 169, 237, 505, 252 ], "score": 1.0, "content": "which shows that when the source and target have positive distance, the source can", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 251, 271, 263 ], "spans": [ { "bbox": [ 106, 251, 271, 263 ], "score": 1.0, "content": "never compensate for the target samples.", "type": "text" } ], "index": 12 } ], "index": 11, "bbox_fs": [ 102, 216, 506, 263 ] }, { "type": "text", "bbox": [ 106, 271, 504, 305 ], "lines": [ { "bbox": [ 105, 271, 506, 284 ], "spans": [ { "bbox": [ 105, 271, 310, 284 ], "score": 1.0, "content": "Remark 11 In the lower bound, the product terms", "type": "text" }, { "bbox": [ 311, 272, 337, 284 ], "score": 0.91, "content": "\\Delta _ { i } n _ { S _ { i } }", "type": "inline_equation" }, { "bbox": [ 337, 271, 506, 284 ], "score": 1.0, "content": "appear which indicate that a source with", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 283, 505, 294 ], "spans": [ { "bbox": [ 106, 283, 505, 294 ], "score": 1.0, "content": "large transfer distance can sometimes be as useful as a source with small transfer distance when", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 293, 375, 306 ], "spans": [ { "bbox": [ 105, 293, 375, 306 ], "score": 1.0, "content": "there is a large amount of training data available from that source.", "type": "text" } ], "index": 15 } ], "index": 14, "bbox_fs": [ 105, 271, 506, 306 ] }, { "type": "title", "bbox": [ 108, 320, 257, 333 ], "lines": [ { "bbox": [ 105, 320, 258, 334 ], "spans": [ { "bbox": [ 105, 320, 258, 334 ], "score": 1.0, "content": "5 EXPERIMENTAL RESULTS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 344, 505, 401 ], "lines": [ { "bbox": [ 105, 343, 505, 358 ], "spans": [ { "bbox": [ 105, 343, 505, 358 ], "score": 1.0, "content": "In this section we evaluate our theoretical results on real data sets for action recognition and image", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 355, 506, 368 ], "spans": [ { "bbox": [ 106, 355, 506, 368 ], "score": 1.0, "content": "classification tasks. By estimating the parameters appearing in Theorem 1 for different pairs of tasks,", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 367, 506, 380 ], "spans": [ { "bbox": [ 105, 367, 506, 380 ], "score": 1.0, "content": "we first plot the lower bounds and then by running weighted empirical risk minimization investigate", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 377, 505, 390 ], "spans": [ { "bbox": [ 105, 377, 505, 390 ], "score": 1.0, "content": "the sharpness of the bounds. We also investigate the effectiveness of different source tasks with", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 388, 350, 402 ], "spans": [ { "bbox": [ 105, 388, 350, 402 ], "score": 1.0, "content": "different transfer distances on the target generalization error.", "type": "text" } ], "index": 21 } ], "index": 19, "bbox_fs": [ 105, 343, 506, 402 ] }, { "type": "title", "bbox": [ 108, 413, 230, 424 ], "lines": [ { "bbox": [ 106, 413, 231, 426 ], "spans": [ { "bbox": [ 106, 413, 231, 426 ], "score": 1.0, "content": "5.1 ACTION RECOGNITION", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 106, 433, 505, 511 ], "lines": [ { "bbox": [ 106, 433, 507, 447 ], "spans": [ { "bbox": [ 106, 433, 507, 447 ], "score": 1.0, "content": "Experimental setup. We first perform experiments on the UCF101 action recognition data set.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 444, 506, 457 ], "spans": [ { "bbox": [ 105, 444, 506, 457 ], "score": 1.0, "content": "We pick CricketBowling and TableTennis videos from UCF101 as the target task as well as three", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 455, 505, 468 ], "spans": [ { "bbox": [ 106, 455, 505, 468 ], "score": 1.0, "content": "different pairs of classes as the source tasks: 1- CricketBowling and BaseballPitch, 2- Cricketshot and", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 466, 505, 479 ], "spans": [ { "bbox": [ 105, 466, 505, 479 ], "score": 1.0, "content": "Archery, 3- BasketballDunk and Basketball. We pass the videos through an i3d network pretrained on", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 477, 506, 490 ], "spans": [ { "bbox": [ 105, 477, 506, 490 ], "score": 1.0, "content": "kinetics400 Carreira & Zisserman (2017) with the fully connected top classifier removed and extract", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 489, 505, 500 ], "spans": [ { "bbox": [ 106, 489, 505, 500 ], "score": 1.0, "content": "the corresponding features of dimension 2048 from the raw videos. We then work with the extracted", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 500, 245, 511 ], "spans": [ { "bbox": [ 106, 500, 245, 511 ], "score": 1.0, "content": "features instead of the raw videos.", "type": "text" } ], "index": 29 } ], "index": 26, "bbox_fs": [ 105, 433, 507, 511 ] }, { "type": "text", "bbox": [ 106, 515, 505, 594 ], "lines": [ { "bbox": [ 106, 516, 505, 528 ], "spans": [ { "bbox": [ 106, 516, 505, 528 ], "score": 1.0, "content": "Training. We train a one hidden layer neural network with 15 number of hidden units and ReLU", "type": "text" } ], "index": 30 }, { "bbox": [ 104, 526, 506, 541 ], "spans": [ { "bbox": [ 104, 526, 506, 541 ], "score": 1.0, "content": "activation functions for each pair of data sets. Table 3 consists of test accuracy on CricketBowling", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 539, 505, 550 ], "spans": [ { "bbox": [ 106, 539, 505, 550 ], "score": 1.0, "content": "vs. TableTennis, when using the network trained on each source task. We use these accuracies for", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 550, 505, 562 ], "spans": [ { "bbox": [ 106, 550, 505, 562 ], "score": 1.0, "content": "deriving the corresponding lower bounds. Furthermore, we run weighted empirical risk minimization", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 560, 506, 573 ], "spans": [ { "bbox": [ 105, 560, 506, 573 ], "score": 1.0, "content": "as a simple transfer learning approach to find some upper bounds on the target generalization error.", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 572, 505, 584 ], "spans": [ { "bbox": [ 106, 572, 132, 584 ], "score": 1.0, "content": "Given", "type": "text" }, { "bbox": [ 133, 573, 146, 583 ], "score": 0.84, "content": "n _ { S }", "type": "inline_equation" }, { "bbox": [ 146, 572, 163, 584 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 164, 573, 177, 582 ], "score": 0.85, "content": "n _ { T }", "type": "inline_equation" }, { "bbox": [ 177, 572, 505, 584 ], "score": 1.0, "content": "number of source and target samples, for estimating the corresponding one hidden", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 582, 444, 595 ], "spans": [ { "bbox": [ 105, 582, 444, 595 ], "score": 1.0, "content": "layer neural network parameters we minimize the following weighted empirical risk", "type": "text" } ], "index": 36 } ], "index": 33, "bbox_fs": [ 104, 516, 506, 595 ] }, { "type": "interline_equation", "bbox": [ 111, 596, 480, 629 ], "lines": [ { "bbox": [ 111, 596, 480, 629 ], "spans": [ { "bbox": [ 111, 596, 480, 629 ], "score": 0.94, "content": "\\operatorname* { m i n } _ { W _ { 1 } , W _ { 2 } } \\frac { 1 - \\lambda } { n _ { T } } \\sum _ { i = 1 } ^ { n _ { T } } \\mathbf { C o s t } ( W _ { 2 } \\mathbf { R e L U } ( W _ { 1 } \\pmb { x } _ { T } ^ { ( i ) } ) , y _ { T } ^ { ( i ) } ) + \\frac { \\lambda } { n _ { S } } \\sum _ { i = 1 } ^ { n _ { S } } \\mathbf { C o s t } ( W _ { 2 } \\mathbf { R e L U } ( W _ { 1 } \\pmb { x } _ { S } ^ { ( i ) } ) , y _ { S } ^ { ( i ) } )", "type": "interline_equation", "image_path": "3c19850b2377a8366c3c35efac700649b14797e228f9403eabe5cee32aa0f04d.jpg" } ] } ], "index": 38, "virtual_lines": [ { "bbox": [ 111, 596, 480, 607.0 ], "spans": [], "index": 37 }, { "bbox": [ 111, 607.0, 480, 618.0 ], "spans": [], "index": 38 }, { "bbox": [ 111, 618.0, 480, 629.0 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 105, 637, 505, 660 ], "lines": [ { "bbox": [ 106, 637, 505, 650 ], "spans": [ { "bbox": [ 106, 637, 370, 650 ], "score": 1.0, "content": "where the function Cost denotes the logistic regression cost and", "type": "text" }, { "bbox": [ 370, 637, 483, 650 ], "score": 0.91, "content": "\\lambda \\in \\{ 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 484, 637, 505, 650 ], "score": 1.0, "content": ". We", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 648, 343, 662 ], "spans": [ { "bbox": [ 105, 648, 343, 662 ], "score": 1.0, "content": "then pick the lambda which minimizes the target test error.", "type": "text" } ], "index": 41 } ], "index": 40.5, "bbox_fs": [ 105, 637, 505, 662 ] }, { "type": "text", "bbox": [ 106, 665, 505, 732 ], "lines": [ { "bbox": [ 105, 664, 505, 679 ], "spans": [ { "bbox": [ 105, 664, 505, 679 ], "score": 1.0, "content": "Results. First we calculate the transfer distance by Definition 2 for each source/target pairs using", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 676, 506, 689 ], "spans": [ { "bbox": [ 105, 676, 506, 689 ], "score": 1.0, "content": "Table 1. To this end, we assume that best target generalization error is zero and using the Table 1 we", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 687, 506, 700 ], "spans": [ { "bbox": [ 106, 687, 506, 700 ], "score": 1.0, "content": "obtain the transfer distance for each pair which is demonstrated in Table 2. As it can be observed", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 699, 506, 712 ], "spans": [ { "bbox": [ 106, 699, 506, 712 ], "score": 1.0, "content": "by Table 2, the pair of Source1 and Target has the lowest transfer distance among other pairs since", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "both of the source and target tasks share a same class which is CricketBowling. Furthermore, Table 2", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 720, 477, 733 ], "spans": [ { "bbox": [ 105, 720, 477, 733 ], "score": 1.0, "content": "determines which pairs are more suitable for transferring the source knowledge to the target.", "type": "text" } ], "index": 47 } ], "index": 44.5, "bbox_fs": [ 105, 664, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 160, 81, 450, 149 ], "blocks": [ { "type": "table_body", "bbox": [ 160, 81, 450, 149 ], "group_id": 0, "lines": [ { "bbox": [ 160, 81, 450, 149 ], "spans": [ { "bbox": [ 160, 81, 450, 149 ], "score": 0.982, "html": "
TaskTest accuracy of Target us- ing the source network
Target:CricketBowlingvs.TableTennis Source1: CricketBowling vs.Baseball Pitch Source2: Cricketshot vs.Archery Source3:BasketballDunk vs.Basketball1 0.946 0.61 0.52
", "type": "table", "image_path": "4bb8800ad63456739c3720d9a7e28ad40a8a471860367c5e46a894b0ed7212fd.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 160, 81, 450, 103.66666666666667 ], "spans": [], "index": 0 }, { "bbox": [ 160, 103.66666666666667, 450, 126.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 160, 126.33333333333334, 450, 149.0 ], "spans": [], "index": 2 } ] }, { "type": "table_caption", "bbox": [ 290, 157, 321, 168 ], "group_id": 0, "lines": [ { "bbox": [ 289, 156, 322, 169 ], "spans": [ { "bbox": [ 289, 156, 322, 169 ], "score": 1.0, "content": "Table 1", "type": "text" } ], "index": 3 } ], "index": 3 } ], "index": 2.0 }, { "type": "table", "bbox": [ 213, 180, 397, 227 ], "blocks": [ { "type": "table_body", "bbox": [ 213, 180, 397, 227 ], "group_id": 1, "lines": [ { "bbox": [ 213, 180, 397, 227 ], "spans": [ { "bbox": [ 213, 180, 397, 227 ], "score": 0.974, "html": "
pair of tasksp(Source,Target)
(Source1, Target)0.053
(Source2, Target)0.39
(Source3,Target)0.48
", "type": "table", "image_path": "015860d91bcd3a380f0d20a130395c8ef195c05a1b440386bad5605cdabff171.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 213, 180, 397, 195.66666666666666 ], "spans": [], "index": 4 }, { "bbox": [ 213, 195.66666666666666, 397, 211.33333333333331 ], "spans": [], "index": 5 }, { "bbox": [ 213, 211.33333333333331, 397, 226.99999999999997 ], "spans": [], "index": 6 } ] } ], "index": 5 }, { "type": "text", "bbox": [ 131, 235, 477, 247 ], "lines": [ { "bbox": [ 131, 234, 479, 248 ], "spans": [ { "bbox": [ 131, 234, 479, 248 ], "score": 1.0, "content": "Table 2: Transfer distance of pairs of source and target on UCF101 action recognition.", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "image", "bbox": [ 115, 259, 477, 407 ], "blocks": [ { "type": "image_body", "bbox": [ 115, 259, 477, 407 ], "group_id": 0, "lines": [ { "bbox": [ 115, 259, 477, 407 ], "spans": [ { "bbox": [ 115, 259, 477, 407 ], "score": 0.97, "type": "image", "image_path": "2c241193221490f096d20c1db5bb3ebe4553c3e1b29a5f9a4c43fa4d3551c9e2.jpg" } ] } ], "index": 9, "virtual_lines": [ { "bbox": [ 115, 259, 477, 308.3333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 115, 308.3333333333333, 477, 357.66666666666663 ], "spans": [], "index": 9 }, { "bbox": [ 115, 357.66666666666663, 477, 406.99999999999994 ], "spans": [], "index": 10 } ] }, { "type": "image_caption", "bbox": [ 106, 416, 506, 449 ], "group_id": 0, "lines": [ { "bbox": [ 106, 416, 506, 428 ], "spans": [ { "bbox": [ 106, 416, 506, 428 ], "score": 1.0, "content": "Figure 1: (a) depicts our lower bounds for three pairs of source and target tasks on action classification.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 427, 505, 439 ], "spans": [ { "bbox": [ 106, 427, 505, 439 ], "score": 1.0, "content": "(b) depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 438, 164, 449 ], "spans": [ { "bbox": [ 105, 438, 164, 449 ], "score": 1.0, "content": "minimization.", "type": "text" } ], "index": 13 } ], "index": 12 } ], "index": 10.5 }, { "type": "text", "bbox": [ 107, 471, 505, 581 ], "lines": [ { "bbox": [ 106, 471, 506, 483 ], "spans": [ { "bbox": [ 106, 471, 506, 483 ], "score": 1.0, "content": "Next, we draw the lower bound curves for each pair in Fig 1a. To this end, we need to find the", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 483, 506, 494 ], "spans": [ { "bbox": [ 106, 483, 506, 494 ], "score": 1.0, "content": "VC dimension of the hypothesis class which consists of neural networks with the architecture of", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 492, 505, 506 ], "spans": [ { "bbox": [ 106, 493, 160, 504 ], "score": 0.91, "content": "2 0 4 8 * 1 5 * 1", "type": "inline_equation" }, { "bbox": [ 160, 492, 505, 506 ], "score": 1.0, "content": "with ReLU activation functions. Theorem 1 in Harvey et al. (2017) gives a lower bound", "type": "text" } ], "index": 16 }, { "bbox": [ 103, 500, 504, 522 ], "spans": [ { "bbox": [ 103, 500, 399, 522 ], "score": 1.0, "content": "for VC dimension of neural networks with ReLU activation functions by", "type": "text" }, { "bbox": [ 399, 504, 463, 517 ], "score": 0.93, "content": "\\begin{array} { r } { \\frac { 1 } { 6 4 0 } \\dot { W } \\dot { L } \\log _ { 2 } \\frac { W } { L } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 464, 500, 492, 522 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 492, 504, 504, 515 ], "score": 0.57, "content": "W", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 106, 515, 506, 528 ], "spans": [ { "bbox": [ 106, 515, 123, 528 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 132, 515, 506, 528 ], "score": 1.0, "content": "are the number of parameters and layers, respectively. Then in Figure 1b we plot the lower", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 525, 505, 538 ], "spans": [ { "bbox": [ 106, 525, 505, 538 ], "score": 1.0, "content": "bounds along with the upper bounds obtained via Formula 5.1 for three different pairs of source and", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 537, 505, 550 ], "spans": [ { "bbox": [ 106, 537, 505, 550 ], "score": 1.0, "content": "target as well as using only target samples. We obtained these upper bounds by running Formula 5.1", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 547, 505, 560 ], "spans": [ { "bbox": [ 106, 547, 505, 560 ], "score": 1.0, "content": "five times and then averaging the results. Fig 1b shows that when the distance of a source from the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 558, 506, 572 ], "spans": [ { "bbox": [ 105, 558, 506, 572 ], "score": 1.0, "content": "target is small it would be more effective in achieving small target generalization error. We would", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 569, 507, 583 ], "spans": [ { "bbox": [ 105, 569, 507, 583 ], "score": 1.0, "content": "like to mention that in all of these plots we choose the same number of source samples for each pair.", "type": "text" } ], "index": 23 } ], "index": 18.5 }, { "type": "text", "bbox": [ 107, 587, 505, 642 ], "lines": [ { "bbox": [ 105, 586, 506, 600 ], "spans": [ { "bbox": [ 105, 586, 223, 600 ], "score": 1.0, "content": "Figure 2 shows the average", "type": "text" }, { "bbox": [ 223, 587, 230, 597 ], "score": 0.62, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 231, 586, 506, 600 ], "score": 1.0, "content": ", the weight appearing in Formula 5.1, when the number of target", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 598, 505, 610 ], "spans": [ { "bbox": [ 106, 598, 437, 610 ], "score": 1.0, "content": "samples is 100 to 150. It shows that in the pair Source1 and Target the average", "type": "text" }, { "bbox": [ 438, 599, 445, 608 ], "score": 0.77, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 446, 598, 505, 610 ], "score": 1.0, "content": "is high which", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 609, 505, 621 ], "spans": [ { "bbox": [ 106, 609, 471, 621 ], "score": 1.0, "content": "demonstrate the usefulness of the source in the target task. Furthermore, the small value of", "type": "text" }, { "bbox": [ 472, 609, 479, 619 ], "score": 0.8, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 479, 609, 505, 621 ], "score": 1.0, "content": "in the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 619, 506, 634 ], "spans": [ { "bbox": [ 105, 619, 506, 634 ], "score": 1.0, "content": "pair Source3 and Target suggests that when the transfer distance is high, source samples are no longer", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 631, 138, 642 ], "spans": [ { "bbox": [ 105, 631, 138, 642 ], "score": 1.0, "content": "usefull.", "type": "text" } ], "index": 28 } ], "index": 26 }, { "type": "title", "bbox": [ 108, 656, 235, 667 ], "lines": [ { "bbox": [ 105, 655, 236, 669 ], "spans": [ { "bbox": [ 105, 655, 236, 669 ], "score": 1.0, "content": "5.2 IMAGE CLASSIFICATION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 106, 676, 506, 690 ], "spans": [ { "bbox": [ 106, 676, 506, 690 ], "score": 1.0, "content": "Experimental setup. In this section we focus on image classification tasks and utilize Theorem 1 to", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "recognize appropriate pairs of tasks that are suitable for transfer learning. We choose some classes of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "the DomainNet data set Peng et al. (2019) as source and target tasks. We pick Clock and Ambulance", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "from DomainNet Clipart for the target task and three different pairs of classes as the source tasks:", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 719, 505, 733 ], "spans": [ { "bbox": [ 105, 719, 505, 733 ], "score": 1.0, "content": "1- Clock and Ambulance, 2- Cricketshot and TableTennis, 3- TableTennis and FrontCraw. Here we", "type": "text" } ], "index": 34 } ], "index": 32 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 160, 81, 450, 149 ], "blocks": [ { "type": "table_body", "bbox": [ 160, 81, 450, 149 ], "group_id": 0, "lines": [ { "bbox": [ 160, 81, 450, 149 ], "spans": [ { "bbox": [ 160, 81, 450, 149 ], "score": 0.982, "html": "
TaskTest accuracy of Target us- ing the source network
Target:CricketBowlingvs.TableTennis Source1: CricketBowling vs.Baseball Pitch Source2: Cricketshot vs.Archery Source3:BasketballDunk vs.Basketball1 0.946 0.61 0.52
", "type": "table", "image_path": "4bb8800ad63456739c3720d9a7e28ad40a8a471860367c5e46a894b0ed7212fd.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 160, 81, 450, 103.66666666666667 ], "spans": [], "index": 0 }, { "bbox": [ 160, 103.66666666666667, 450, 126.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 160, 126.33333333333334, 450, 149.0 ], "spans": [], "index": 2 } ] }, { "type": "table_caption", "bbox": [ 290, 157, 321, 168 ], "group_id": 0, "lines": [ { "bbox": [ 289, 156, 322, 169 ], "spans": [ { "bbox": [ 289, 156, 322, 169 ], "score": 1.0, "content": "Table 1", "type": "text" } ], "index": 3 } ], "index": 3 } ], "index": 2.0 }, { "type": "table", "bbox": [ 213, 180, 397, 227 ], "blocks": [ { "type": "table_body", "bbox": [ 213, 180, 397, 227 ], "group_id": 1, "lines": [ { "bbox": [ 213, 180, 397, 227 ], "spans": [ { "bbox": [ 213, 180, 397, 227 ], "score": 0.974, "html": "
pair of tasksp(Source,Target)
(Source1, Target)0.053
(Source2, Target)0.39
(Source3,Target)0.48
", "type": "table", "image_path": "015860d91bcd3a380f0d20a130395c8ef195c05a1b440386bad5605cdabff171.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 213, 180, 397, 195.66666666666666 ], "spans": [], "index": 4 }, { "bbox": [ 213, 195.66666666666666, 397, 211.33333333333331 ], "spans": [], "index": 5 }, { "bbox": [ 213, 211.33333333333331, 397, 226.99999999999997 ], "spans": [], "index": 6 } ] } ], "index": 5 }, { "type": "text", "bbox": [ 131, 235, 477, 247 ], "lines": [ { "bbox": [ 131, 234, 479, 248 ], "spans": [ { "bbox": [ 131, 234, 479, 248 ], "score": 1.0, "content": "Table 2: Transfer distance of pairs of source and target on UCF101 action recognition.", "type": "text" } ], "index": 7 } ], "index": 7, "bbox_fs": [ 131, 234, 479, 248 ] }, { "type": "image", "bbox": [ 115, 259, 477, 407 ], "blocks": [ { "type": "image_body", "bbox": [ 115, 259, 477, 407 ], "group_id": 0, "lines": [ { "bbox": [ 115, 259, 477, 407 ], "spans": [ { "bbox": [ 115, 259, 477, 407 ], "score": 0.97, "type": "image", "image_path": "2c241193221490f096d20c1db5bb3ebe4553c3e1b29a5f9a4c43fa4d3551c9e2.jpg" } ] } ], "index": 9, "virtual_lines": [ { "bbox": [ 115, 259, 477, 308.3333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 115, 308.3333333333333, 477, 357.66666666666663 ], "spans": [], "index": 9 }, { "bbox": [ 115, 357.66666666666663, 477, 406.99999999999994 ], "spans": [], "index": 10 } ] }, { "type": "image_caption", "bbox": [ 106, 416, 506, 449 ], "group_id": 0, "lines": [ { "bbox": [ 106, 416, 506, 428 ], "spans": [ { "bbox": [ 106, 416, 506, 428 ], "score": 1.0, "content": "Figure 1: (a) depicts our lower bounds for three pairs of source and target tasks on action classification.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 427, 505, 439 ], "spans": [ { "bbox": [ 106, 427, 505, 439 ], "score": 1.0, "content": "(b) depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 438, 164, 449 ], "spans": [ { "bbox": [ 105, 438, 164, 449 ], "score": 1.0, "content": "minimization.", "type": "text" } ], "index": 13 } ], "index": 12 } ], "index": 10.5 }, { "type": "text", "bbox": [ 107, 471, 505, 581 ], "lines": [ { "bbox": [ 106, 471, 506, 483 ], "spans": [ { "bbox": [ 106, 471, 506, 483 ], "score": 1.0, "content": "Next, we draw the lower bound curves for each pair in Fig 1a. To this end, we need to find the", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 483, 506, 494 ], "spans": [ { "bbox": [ 106, 483, 506, 494 ], "score": 1.0, "content": "VC dimension of the hypothesis class which consists of neural networks with the architecture of", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 492, 505, 506 ], "spans": [ { "bbox": [ 106, 493, 160, 504 ], "score": 0.91, "content": "2 0 4 8 * 1 5 * 1", "type": "inline_equation" }, { "bbox": [ 160, 492, 505, 506 ], "score": 1.0, "content": "with ReLU activation functions. Theorem 1 in Harvey et al. (2017) gives a lower bound", "type": "text" } ], "index": 16 }, { "bbox": [ 103, 500, 504, 522 ], "spans": [ { "bbox": [ 103, 500, 399, 522 ], "score": 1.0, "content": "for VC dimension of neural networks with ReLU activation functions by", "type": "text" }, { "bbox": [ 399, 504, 463, 517 ], "score": 0.93, "content": "\\begin{array} { r } { \\frac { 1 } { 6 4 0 } \\dot { W } \\dot { L } \\log _ { 2 } \\frac { W } { L } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 464, 500, 492, 522 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 492, 504, 504, 515 ], "score": 0.57, "content": "W", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 106, 515, 506, 528 ], "spans": [ { "bbox": [ 106, 515, 123, 528 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 132, 515, 506, 528 ], "score": 1.0, "content": "are the number of parameters and layers, respectively. Then in Figure 1b we plot the lower", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 525, 505, 538 ], "spans": [ { "bbox": [ 106, 525, 505, 538 ], "score": 1.0, "content": "bounds along with the upper bounds obtained via Formula 5.1 for three different pairs of source and", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 537, 505, 550 ], "spans": [ { "bbox": [ 106, 537, 505, 550 ], "score": 1.0, "content": "target as well as using only target samples. We obtained these upper bounds by running Formula 5.1", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 547, 505, 560 ], "spans": [ { "bbox": [ 106, 547, 505, 560 ], "score": 1.0, "content": "five times and then averaging the results. Fig 1b shows that when the distance of a source from the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 558, 506, 572 ], "spans": [ { "bbox": [ 105, 558, 506, 572 ], "score": 1.0, "content": "target is small it would be more effective in achieving small target generalization error. We would", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 569, 507, 583 ], "spans": [ { "bbox": [ 105, 569, 507, 583 ], "score": 1.0, "content": "like to mention that in all of these plots we choose the same number of source samples for each pair.", "type": "text" } ], "index": 23 } ], "index": 18.5, "bbox_fs": [ 103, 471, 507, 583 ] }, { "type": "text", "bbox": [ 107, 587, 505, 642 ], "lines": [ { "bbox": [ 105, 586, 506, 600 ], "spans": [ { "bbox": [ 105, 586, 223, 600 ], "score": 1.0, "content": "Figure 2 shows the average", "type": "text" }, { "bbox": [ 223, 587, 230, 597 ], "score": 0.62, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 231, 586, 506, 600 ], "score": 1.0, "content": ", the weight appearing in Formula 5.1, when the number of target", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 598, 505, 610 ], "spans": [ { "bbox": [ 106, 598, 437, 610 ], "score": 1.0, "content": "samples is 100 to 150. It shows that in the pair Source1 and Target the average", "type": "text" }, { "bbox": [ 438, 599, 445, 608 ], "score": 0.77, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 446, 598, 505, 610 ], "score": 1.0, "content": "is high which", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 609, 505, 621 ], "spans": [ { "bbox": [ 106, 609, 471, 621 ], "score": 1.0, "content": "demonstrate the usefulness of the source in the target task. Furthermore, the small value of", "type": "text" }, { "bbox": [ 472, 609, 479, 619 ], "score": 0.8, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 479, 609, 505, 621 ], "score": 1.0, "content": "in the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 619, 506, 634 ], "spans": [ { "bbox": [ 105, 619, 506, 634 ], "score": 1.0, "content": "pair Source3 and Target suggests that when the transfer distance is high, source samples are no longer", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 631, 138, 642 ], "spans": [ { "bbox": [ 105, 631, 138, 642 ], "score": 1.0, "content": "usefull.", "type": "text" } ], "index": 28 } ], "index": 26, "bbox_fs": [ 105, 586, 506, 642 ] }, { "type": "title", "bbox": [ 108, 656, 235, 667 ], "lines": [ { "bbox": [ 105, 655, 236, 669 ], "spans": [ { "bbox": [ 105, 655, 236, 669 ], "score": 1.0, "content": "5.2 IMAGE CLASSIFICATION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 106, 676, 506, 690 ], "spans": [ { "bbox": [ 106, 676, 506, 690 ], "score": 1.0, "content": "Experimental setup. In this section we focus on image classification tasks and utilize Theorem 1 to", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "recognize appropriate pairs of tasks that are suitable for transfer learning. We choose some classes of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "the DomainNet data set Peng et al. (2019) as source and target tasks. We pick Clock and Ambulance", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "from DomainNet Clipart for the target task and three different pairs of classes as the source tasks:", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 719, 505, 733 ], "spans": [ { "bbox": [ 105, 719, 505, 733 ], "score": 1.0, "content": "1- Clock and Ambulance, 2- Cricketshot and TableTennis, 3- TableTennis and FrontCraw. Here we", "type": "text" } ], "index": 34 } ], "index": 32, "bbox_fs": [ 105, 676, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 220, 78, 392, 178 ], "blocks": [ { "type": "image_body", "bbox": [ 220, 78, 392, 178 ], "group_id": 0, "lines": [ { "bbox": [ 220, 78, 392, 178 ], "spans": [ { "bbox": [ 220, 78, 392, 178 ], "score": 0.946, "type": "image", "image_path": "1df4f1137b5f43be56ef563df21cd603dfe5643d8bffa4d7a5b2e8b91d2fa116.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 220, 78, 392, 92.28571428571429 ], "spans": [], "index": 0 }, { "bbox": [ 220, 92.28571428571429, 392, 106.57142857142858 ], "spans": [], "index": 1 }, { "bbox": [ 220, 106.57142857142858, 392, 120.85714285714288 ], "spans": [], "index": 2 }, { "bbox": [ 220, 120.85714285714288, 392, 135.14285714285717 ], "spans": [], "index": 3 }, { "bbox": [ 220, 135.14285714285717, 392, 149.42857142857144 ], "spans": [], "index": 4 }, { "bbox": [ 220, 149.42857142857144, 392, 163.71428571428572 ], "spans": [], "index": 5 }, { "bbox": [ 220, 163.71428571428572, 392, 178.0 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 106, 186, 505, 209 ], "group_id": 0, "lines": [ { "bbox": [ 106, 186, 505, 199 ], "spans": [ { "bbox": [ 106, 186, 181, 199 ], "score": 1.0, "content": "Figure 2: Average", "type": "text" }, { "bbox": [ 182, 187, 189, 196 ], "score": 0.71, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 189, 186, 505, 199 ], "score": 1.0, "content": "in weighted empirical risk minimization for three different pairs of source and", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 197, 245, 210 ], "spans": [ { "bbox": [ 106, 197, 245, 210 ], "score": 1.0, "content": "target tasks for action recognition.", "type": "text" } ], "index": 8 } ], "index": 7.5 } ], "index": 5.25 }, { "type": "table", "bbox": [ 160, 223, 450, 292 ], "blocks": [ { "type": "table_body", "bbox": [ 160, 223, 450, 292 ], "group_id": 0, "lines": [ { "bbox": [ 160, 223, 450, 292 ], "spans": [ { "bbox": [ 160, 223, 450, 292 ], "score": 0.963, "html": "
TaskTest Accuracy of Target using the source network
Target: Clock vs. Ambulance (Clipart) Source1: Clock vs.Ambulance (Sketch) Source2: Clock vs. Crow(Sketch)0.916 0.697
", "type": "table", "image_path": "e82dd66dddbb4945be0d81c77adaaa3d5b80a765efe387e96e7043766fe30a4b.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 160, 223, 450, 246.0 ], "spans": [], "index": 9 }, { "bbox": [ 160, 246.0, 450, 269.0 ], "spans": [], "index": 10 }, { "bbox": [ 160, 269.0, 450, 292.0 ], "spans": [], "index": 11 } ] }, { "type": "table_caption", "bbox": [ 290, 300, 321, 310 ], "group_id": 0, "lines": [ { "bbox": [ 289, 298, 322, 312 ], "spans": [ { "bbox": [ 289, 298, 322, 312 ], "score": 1.0, "content": "Table 3", "type": "text" } ], "index": 12 } ], "index": 12 } ], "index": 11.0 }, { "type": "image", "bbox": [ 120, 330, 478, 474 ], "blocks": [ { "type": "image_body", "bbox": [ 120, 330, 478, 474 ], "group_id": 1, "lines": [ { "bbox": [ 120, 330, 478, 474 ], "spans": [ { "bbox": [ 120, 330, 478, 474 ], "score": 0.971, "type": "image", "image_path": "13e028571da16adec8995d8d638a4e0ce066f5cf2190ce602d12a912311c3bd1.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 120, 330, 478, 378.0 ], "spans": [], "index": 13 }, { "bbox": [ 120, 378.0, 478, 426.0 ], "spans": [], "index": 14 }, { "bbox": [ 120, 426.0, 478, 474.0 ], "spans": [], "index": 15 } ] }, { "type": "image_caption", "bbox": [ 106, 483, 506, 516 ], "group_id": 1, "lines": [ { "bbox": [ 105, 482, 506, 496 ], "spans": [ { "bbox": [ 105, 482, 506, 496 ], "score": 1.0, "content": "Figure 3: (a) depicts our lower bounds for three pairs of source and target tasks on image classification.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 494, 505, 506 ], "spans": [ { "bbox": [ 106, 494, 505, 506 ], "score": 1.0, "content": "(b) depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 505, 164, 516 ], "spans": [ { "bbox": [ 105, 505, 164, 516 ], "score": 1.0, "content": "minimization.", "type": "text" } ], "index": 18 } ], "index": 17 } ], "index": 15.5 }, { "type": "table", "bbox": [ 213, 530, 397, 577 ], "blocks": [ { "type": "table_body", "bbox": [ 213, 530, 397, 577 ], "group_id": 1, "lines": [ { "bbox": [ 213, 530, 397, 577 ], "spans": [ { "bbox": [ 213, 530, 397, 577 ], "score": 0.972, "html": "
pair of tasksp(Source, Target)
(Source1, Target)0.083
(Source2, Target)0.3
(Source3, Target)0.35
", "type": "table", "image_path": "33c339905dc6a07bdd4a9080faf43de277ba01cee2fa67e11a373a258ecd5764.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 213, 530, 397, 545.6666666666666 ], "spans": [], "index": 19 }, { "bbox": [ 213, 545.6666666666666, 397, 561.3333333333333 ], "spans": [], "index": 20 }, { "bbox": [ 213, 561.3333333333333, 397, 576.9999999999999 ], "spans": [], "index": 21 } ] } ], "index": 20 }, { "type": "text", "bbox": [ 117, 585, 491, 597 ], "lines": [ { "bbox": [ 117, 583, 492, 599 ], "spans": [ { "bbox": [ 117, 583, 492, 599 ], "score": 1.0, "content": "Table 4: Transfer distance of pairs of source and target on DomainNet image classifications].", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 621, 503, 643 ], "lines": [ { "bbox": [ 106, 621, 505, 633 ], "spans": [ { "bbox": [ 106, 621, 505, 633 ], "score": 1.0, "content": "use ResNet50 network pretrained on Imagenet for extracting features of dimension 2048 and in the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 631, 405, 645 ], "spans": [ { "bbox": [ 105, 631, 405, 645 ], "score": 1.0, "content": "sequel we work with the extracted features rather than the raw image data.", "type": "text" } ], "index": 24 } ], "index": 23.5 }, { "type": "text", "bbox": [ 107, 648, 505, 704 ], "lines": [ { "bbox": [ 106, 649, 504, 660 ], "spans": [ { "bbox": [ 106, 649, 504, 660 ], "score": 1.0, "content": "Training. We train a one hidden layer neural network with 15 number of hidden units and ReLU", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 659, 505, 672 ], "spans": [ { "bbox": [ 105, 659, 505, 672 ], "score": 1.0, "content": "activation functions for each of pairs of the tasks. Table 3 includes the test accuracy on the target task", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 669, 506, 685 ], "spans": [ { "bbox": [ 105, 669, 506, 685 ], "score": 1.0, "content": "when using the networks trained on different sources, which is necessary for estimating/calculating", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 680, 505, 695 ], "spans": [ { "bbox": [ 105, 680, 505, 695 ], "score": 1.0, "content": "the transfer distance as demonstrated in Table 4. Similar to the subsection 5.1, we also run weighted", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 693, 491, 705 ], "spans": [ { "bbox": [ 106, 693, 491, 705 ], "score": 1.0, "content": "empirical risk minimization for finding upper bounds for the pairs of the source and target tasks.", "type": "text" } ], "index": 29 } ], "index": 27 }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Results. Similar to the previous section on action recognition, using Table 3 we can obtain the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 505, 733 ], "score": 1.0, "content": "transfer distances and based on this distance we can identify suitable pairs of source and target tasks", "type": "text" } ], "index": 31 } ], "index": 30.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 309, 38 ], "spans": [ { "bbox": [ 106, 25, 309, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 300, 750, 309, 761 ], "spans": [ { "bbox": [ 300, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 220, 78, 392, 178 ], "blocks": [ { "type": "image_body", "bbox": [ 220, 78, 392, 178 ], "group_id": 0, "lines": [ { "bbox": [ 220, 78, 392, 178 ], "spans": [ { "bbox": [ 220, 78, 392, 178 ], "score": 0.946, "type": "image", "image_path": "1df4f1137b5f43be56ef563df21cd603dfe5643d8bffa4d7a5b2e8b91d2fa116.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 220, 78, 392, 92.28571428571429 ], "spans": [], "index": 0 }, { "bbox": [ 220, 92.28571428571429, 392, 106.57142857142858 ], "spans": [], "index": 1 }, { "bbox": [ 220, 106.57142857142858, 392, 120.85714285714288 ], "spans": [], "index": 2 }, { "bbox": [ 220, 120.85714285714288, 392, 135.14285714285717 ], "spans": [], "index": 3 }, { "bbox": [ 220, 135.14285714285717, 392, 149.42857142857144 ], "spans": [], "index": 4 }, { "bbox": [ 220, 149.42857142857144, 392, 163.71428571428572 ], "spans": [], "index": 5 }, { "bbox": [ 220, 163.71428571428572, 392, 178.0 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 106, 186, 505, 209 ], "group_id": 0, "lines": [ { "bbox": [ 106, 186, 505, 199 ], "spans": [ { "bbox": [ 106, 186, 181, 199 ], "score": 1.0, "content": "Figure 2: Average", "type": "text" }, { "bbox": [ 182, 187, 189, 196 ], "score": 0.71, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 189, 186, 505, 199 ], "score": 1.0, "content": "in weighted empirical risk minimization for three different pairs of source and", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 197, 245, 210 ], "spans": [ { "bbox": [ 106, 197, 245, 210 ], "score": 1.0, "content": "target tasks for action recognition.", "type": "text" } ], "index": 8 } ], "index": 7.5 } ], "index": 5.25 }, { "type": "table", "bbox": [ 160, 223, 450, 292 ], "blocks": [ { "type": "table_body", "bbox": [ 160, 223, 450, 292 ], "group_id": 0, "lines": [ { "bbox": [ 160, 223, 450, 292 ], "spans": [ { "bbox": [ 160, 223, 450, 292 ], "score": 0.963, "html": "
TaskTest Accuracy of Target using the source network
Target: Clock vs. Ambulance (Clipart) Source1: Clock vs.Ambulance (Sketch) Source2: Clock vs. Crow(Sketch)0.916 0.697
", "type": "table", "image_path": "e82dd66dddbb4945be0d81c77adaaa3d5b80a765efe387e96e7043766fe30a4b.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 160, 223, 450, 246.0 ], "spans": [], "index": 9 }, { "bbox": [ 160, 246.0, 450, 269.0 ], "spans": [], "index": 10 }, { "bbox": [ 160, 269.0, 450, 292.0 ], "spans": [], "index": 11 } ] }, { "type": "table_caption", "bbox": [ 290, 300, 321, 310 ], "group_id": 0, "lines": [ { "bbox": [ 289, 298, 322, 312 ], "spans": [ { "bbox": [ 289, 298, 322, 312 ], "score": 1.0, "content": "Table 3", "type": "text" } ], "index": 12 } ], "index": 12 } ], "index": 11.0 }, { "type": "image", "bbox": [ 120, 330, 478, 474 ], "blocks": [ { "type": "image_body", "bbox": [ 120, 330, 478, 474 ], "group_id": 1, "lines": [ { "bbox": [ 120, 330, 478, 474 ], "spans": [ { "bbox": [ 120, 330, 478, 474 ], "score": 0.971, "type": "image", "image_path": "13e028571da16adec8995d8d638a4e0ce066f5cf2190ce602d12a912311c3bd1.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 120, 330, 478, 378.0 ], "spans": [], "index": 13 }, { "bbox": [ 120, 378.0, 478, 426.0 ], "spans": [], "index": 14 }, { "bbox": [ 120, 426.0, 478, 474.0 ], "spans": [], "index": 15 } ] }, { "type": "image_caption", "bbox": [ 106, 483, 506, 516 ], "group_id": 1, "lines": [ { "bbox": [ 105, 482, 506, 496 ], "spans": [ { "bbox": [ 105, 482, 506, 496 ], "score": 1.0, "content": "Figure 3: (a) depicts our lower bounds for three pairs of source and target tasks on image classification.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 494, 505, 506 ], "spans": [ { "bbox": [ 106, 494, 505, 506 ], "score": 1.0, "content": "(b) depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 505, 164, 516 ], "spans": [ { "bbox": [ 105, 505, 164, 516 ], "score": 1.0, "content": "minimization.", "type": "text" } ], "index": 18 } ], "index": 17 } ], "index": 15.5 }, { "type": "table", "bbox": [ 213, 530, 397, 577 ], "blocks": [ { "type": "table_body", "bbox": [ 213, 530, 397, 577 ], "group_id": 1, "lines": [ { "bbox": [ 213, 530, 397, 577 ], "spans": [ { "bbox": [ 213, 530, 397, 577 ], "score": 0.972, "html": "
pair of tasksp(Source, Target)
(Source1, Target)0.083
(Source2, Target)0.3
(Source3, Target)0.35
", "type": "table", "image_path": "33c339905dc6a07bdd4a9080faf43de277ba01cee2fa67e11a373a258ecd5764.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 213, 530, 397, 545.6666666666666 ], "spans": [], "index": 19 }, { "bbox": [ 213, 545.6666666666666, 397, 561.3333333333333 ], "spans": [], "index": 20 }, { "bbox": [ 213, 561.3333333333333, 397, 576.9999999999999 ], "spans": [], "index": 21 } ] } ], "index": 20 }, { "type": "text", "bbox": [ 117, 585, 491, 597 ], "lines": [ { "bbox": [ 117, 583, 492, 599 ], "spans": [ { "bbox": [ 117, 583, 492, 599 ], "score": 1.0, "content": "Table 4: Transfer distance of pairs of source and target on DomainNet image classifications].", "type": "text" } ], "index": 22 } ], "index": 22, "bbox_fs": [ 117, 583, 492, 599 ] }, { "type": "text", "bbox": [ 107, 621, 503, 643 ], "lines": [ { "bbox": [ 106, 621, 505, 633 ], "spans": [ { "bbox": [ 106, 621, 505, 633 ], "score": 1.0, "content": "use ResNet50 network pretrained on Imagenet for extracting features of dimension 2048 and in the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 631, 405, 645 ], "spans": [ { "bbox": [ 105, 631, 405, 645 ], "score": 1.0, "content": "sequel we work with the extracted features rather than the raw image data.", "type": "text" } ], "index": 24 } ], "index": 23.5, "bbox_fs": [ 105, 621, 505, 645 ] }, { "type": "text", "bbox": [ 107, 648, 505, 704 ], "lines": [ { "bbox": [ 106, 649, 504, 660 ], "spans": [ { "bbox": [ 106, 649, 504, 660 ], "score": 1.0, "content": "Training. We train a one hidden layer neural network with 15 number of hidden units and ReLU", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 659, 505, 672 ], "spans": [ { "bbox": [ 105, 659, 505, 672 ], "score": 1.0, "content": "activation functions for each of pairs of the tasks. Table 3 includes the test accuracy on the target task", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 669, 506, 685 ], "spans": [ { "bbox": [ 105, 669, 506, 685 ], "score": 1.0, "content": "when using the networks trained on different sources, which is necessary for estimating/calculating", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 680, 505, 695 ], "spans": [ { "bbox": [ 105, 680, 505, 695 ], "score": 1.0, "content": "the transfer distance as demonstrated in Table 4. Similar to the subsection 5.1, we also run weighted", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 693, 491, 705 ], "spans": [ { "bbox": [ 106, 693, 491, 705 ], "score": 1.0, "content": "empirical risk minimization for finding upper bounds for the pairs of the source and target tasks.", "type": "text" } ], "index": 29 } ], "index": 27, "bbox_fs": [ 105, 649, 506, 705 ] }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Results. Similar to the previous section on action recognition, using Table 3 we can obtain the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 720, 505, 733 ], "spans": [ { "bbox": [ 105, 720, 505, 733 ], "score": 1.0, "content": "transfer distances and based on this distance we can identify suitable pairs of source and target tasks", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 231, 505, 242 ], "spans": [ { "bbox": [ 106, 231, 505, 242 ], "score": 1.0, "content": "for transfer learning. In the pair1 Source and target tasks share the same objects which are Clock and", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 241, 505, 253 ], "spans": [ { "bbox": [ 106, 241, 505, 253 ], "score": 1.0, "content": "Ambulance which results in low transfer distance. In pair2, still one of the objects which is Clock is", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 106, 253, 506, 264 ], "spans": [ { "bbox": [ 106, 253, 506, 264 ], "score": 1.0, "content": "the same in the source and target and we can see that the transfer distance for pair2 is lower than that", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 105, 262, 505, 276 ], "spans": [ { "bbox": [ 105, 262, 505, 276 ], "score": 1.0, "content": "for pair3. Then we plot the lower bounds in Fig 3a and the corresponding upper bounds obtained by", "type": "text", "cross_page": true } ], "index": 12 }, { "bbox": [ 106, 274, 506, 287 ], "spans": [ { "bbox": [ 106, 274, 506, 287 ], "score": 1.0, "content": "weighted empirical risk minimization in Fig 3b. One can see that sources that are closer to the target", "type": "text", "cross_page": true } ], "index": 13 }, { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 506, 298 ], "score": 1.0, "content": "according to our notion of distance are more effective in achieving small target generalization error.", "type": "text", "cross_page": true } ], "index": 14 } ], "index": 30.5, "bbox_fs": [ 105, 709, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 221, 78, 392, 178 ], "blocks": [ { "type": "image_body", "bbox": [ 221, 78, 392, 178 ], "group_id": 0, "lines": [ { "bbox": [ 221, 78, 392, 178 ], "spans": [ { "bbox": [ 221, 78, 392, 178 ], "score": 0.963, "type": "image", "image_path": "7a303c3706559f7a90ce3dfe420c8b7330324be99f6ba51d8e74ec5e3805fcc7.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 221, 78, 392, 92.28571428571429 ], "spans": [], "index": 0 }, { "bbox": [ 221, 92.28571428571429, 392, 106.57142857142858 ], "spans": [], "index": 1 }, { "bbox": [ 221, 106.57142857142858, 392, 120.85714285714288 ], "spans": [], "index": 2 }, { "bbox": [ 221, 120.85714285714288, 392, 135.14285714285717 ], "spans": [], "index": 3 }, { "bbox": [ 221, 135.14285714285717, 392, 149.42857142857144 ], "spans": [], "index": 4 }, { "bbox": [ 221, 149.42857142857144, 392, 163.71428571428572 ], "spans": [], "index": 5 }, { "bbox": [ 221, 163.71428571428572, 392, 178.0 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 106, 186, 504, 209 ], "group_id": 0, "lines": [ { "bbox": [ 105, 185, 505, 199 ], "spans": [ { "bbox": [ 105, 185, 181, 199 ], "score": 1.0, "content": "Figure 4: Average", "type": "text" }, { "bbox": [ 182, 187, 189, 196 ], "score": 0.71, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 189, 185, 505, 199 ], "score": 1.0, "content": "in weighted empirical risk minimization for three different pairs of source and", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 198, 252, 210 ], "spans": [ { "bbox": [ 105, 198, 252, 210 ], "score": 1.0, "content": "target tasks for image classification.", "type": "text" } ], "index": 8 } ], "index": 7.5 } ], "index": 5.25 }, { "type": "text", "bbox": [ 106, 230, 505, 297 ], "lines": [ { "bbox": [ 106, 231, 505, 242 ], "spans": [ { "bbox": [ 106, 231, 505, 242 ], "score": 1.0, "content": "for transfer learning. In the pair1 Source and target tasks share the same objects which are Clock and", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 241, 505, 253 ], "spans": [ { "bbox": [ 106, 241, 505, 253 ], "score": 1.0, "content": "Ambulance which results in low transfer distance. In pair2, still one of the objects which is Clock is", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 253, 506, 264 ], "spans": [ { "bbox": [ 106, 253, 506, 264 ], "score": 1.0, "content": "the same in the source and target and we can see that the transfer distance for pair2 is lower than that", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 505, 276 ], "spans": [ { "bbox": [ 105, 262, 505, 276 ], "score": 1.0, "content": "for pair3. Then we plot the lower bounds in Fig 3a and the corresponding upper bounds obtained by", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 274, 506, 287 ], "spans": [ { "bbox": [ 106, 274, 506, 287 ], "score": 1.0, "content": "weighted empirical risk minimization in Fig 3b. One can see that sources that are closer to the target", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 285, 506, 298 ], "spans": [ { "bbox": [ 105, 285, 506, 298 ], "score": 1.0, "content": "according to our notion of distance are more effective in achieving small target generalization error.", "type": "text" } ], "index": 14 } ], "index": 11.5 }, { "type": "text", "bbox": [ 107, 302, 505, 346 ], "lines": [ { "bbox": [ 106, 302, 505, 314 ], "spans": [ { "bbox": [ 106, 302, 505, 314 ], "score": 1.0, "content": "CricketBowling is common both in the source and Target1. This suggests that these tasks are", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 312, 506, 326 ], "spans": [ { "bbox": [ 106, 312, 506, 326 ], "score": 1.0, "content": "similar to each other and the estimated transfer distance conforms with this intuition. Furthermore,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 324, 505, 336 ], "spans": [ { "bbox": [ 105, 324, 505, 336 ], "score": 1.0, "content": "CricketBowling and Cricketshot are intuitively similar to one another and this is also reflected in the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 335, 315, 347 ], "spans": [ { "bbox": [ 105, 335, 315, 347 ], "score": 1.0, "content": "lower transfer distance between source and Target2.", "type": "text" } ], "index": 18 } ], "index": 16.5 }, { "type": "text", "bbox": [ 107, 352, 505, 397 ], "lines": [ { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 227, 365 ], "score": 1.0, "content": "In Fig 4 we plot the average", "type": "text" }, { "bbox": [ 227, 353, 234, 362 ], "score": 0.74, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 235, 352, 505, 365 ], "score": 1.0, "content": ", the weight appearing in Formula 5.1 when the number of target", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 362, 506, 375 ], "spans": [ { "bbox": [ 105, 362, 506, 375 ], "score": 1.0, "content": "samples varies from 150 to 200. 4 demonstrates that when a source is close to the target the weight of", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 374, 505, 386 ], "spans": [ { "bbox": [ 105, 374, 505, 386 ], "score": 1.0, "content": "source risk in weighted empirical risk becomes high which shows the effectiveness of source samples", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 386, 290, 397 ], "spans": [ { "bbox": [ 106, 386, 290, 397 ], "score": 1.0, "content": "in achieving small target generalization error.", "type": "text" } ], "index": 22 } ], "index": 20.5 }, { "type": "title", "bbox": [ 107, 412, 210, 425 ], "lines": [ { "bbox": [ 105, 411, 211, 427 ], "spans": [ { "bbox": [ 105, 411, 211, 427 ], "score": 1.0, "content": "6 PROOF OUTLINE", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 437, 460, 449 ], "lines": [ { "bbox": [ 105, 437, 461, 451 ], "spans": [ { "bbox": [ 105, 437, 461, 451 ], "score": 1.0, "content": "The main idea of proof is based on the following proposition proved in Tsybakov (2009)", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 457, 506, 492 ], "lines": [ { "bbox": [ 106, 457, 506, 471 ], "spans": [ { "bbox": [ 106, 457, 362, 471 ], "score": 1.0, "content": "Proposition 1 [Theorem 2.5 of Tsybakov (2009)] Assume that", "type": "text" }, { "bbox": [ 362, 459, 393, 469 ], "score": 0.88, "content": "M \\geq 2", "type": "inline_equation" }, { "bbox": [ 393, 457, 462, 471 ], "score": 1.0, "content": "and the function", "type": "text" }, { "bbox": [ 463, 458, 487, 470 ], "score": 0.91, "content": "d ( \\cdot , \\cdot )", "type": "inline_equation" }, { "bbox": [ 487, 457, 506, 471 ], "score": 1.0, "content": "is a", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 468, 506, 483 ], "spans": [ { "bbox": [ 105, 468, 243, 483 ], "score": 1.0, "content": "semi-distance. Also suppose that", "type": "text" }, { "bbox": [ 243, 469, 287, 482 ], "score": 0.93, "content": "\\{ P _ { \\theta _ { j } } \\} _ { \\theta _ { j } \\in \\Theta }", "type": "inline_equation" }, { "bbox": [ 288, 468, 506, 483 ], "score": 1.0, "content": "is a family of distributions indexed over a parameter", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 480, 346, 493 ], "spans": [ { "bbox": [ 105, 480, 134, 493 ], "score": 1.0, "content": "space,", "type": "text" }, { "bbox": [ 134, 481, 143, 490 ], "score": 0.73, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 143, 480, 164, 493 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 165, 481, 173, 490 ], "score": 0.75, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 174, 480, 248, 493 ], "score": 1.0, "content": "contains elements", "type": "text" }, { "bbox": [ 249, 481, 302, 492 ], "score": 0.89, "content": "\\theta _ { 0 } , \\bar { \\theta } _ { 1 } , . . . , \\theta _ { M }", "type": "inline_equation" }, { "bbox": [ 303, 480, 346, 493 ], "score": 1.0, "content": "such that:", "type": "text" } ], "index": 27 } ], "index": 26 }, { "type": "interline_equation", "bbox": [ 141, 501, 303, 514 ], "lines": [ { "bbox": [ 141, 501, 303, 514 ], "spans": [ { "bbox": [ 141, 501, 303, 514 ], "score": 0.75, "content": "d ( \\theta _ { i } , \\theta _ { j } ) \\geq 2 s > 0 , \\ \\forall 0 \\leq j < k \\leq M", "type": "interline_equation", "image_path": "7d8273f42d18c53cb998b4153a3dd2ae520f7b101cbf282a2be487792abd913b.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 141, 501, 303, 514 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 126, 519, 273, 532 ], "lines": [ { "bbox": [ 124, 518, 273, 533 ], "spans": [ { "bbox": [ 124, 518, 142, 533 ], "score": 1.0, "content": "(ii)", "type": "text" }, { "bbox": [ 142, 519, 249, 533 ], "score": 0.83, "content": "P _ { j } \\ll P _ { 0 } , \\ \\forall \\ j = 1 , . . . , M .", "type": "inline_equation" }, { "bbox": [ 250, 518, 273, 533 ], "score": 1.0, "content": ", and", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "interline_equation", "bbox": [ 259, 538, 387, 573 ], "lines": [ { "bbox": [ 259, 538, 387, 573 ], "spans": [ { "bbox": [ 259, 538, 387, 573 ], "score": 0.94, "content": "\\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } { \\mathcal { D } } _ { k l } ( P _ { j } | P _ { 0 } ) \\leq \\alpha \\log M", "type": "interline_equation", "image_path": "e2c49d38ef7ce39bd19178d1c98fbe4a52106f16bc3c92bafbfa854aa4e2f3fb.jpg" } ] } ], "index": 30.5, "virtual_lines": [ { "bbox": [ 259, 538, 387, 555.5 ], "spans": [], "index": 30 }, { "bbox": [ 259, 555.5, 387, 573.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 142, 578, 503, 591 ], "lines": [ { "bbox": [ 141, 578, 505, 592 ], "spans": [ { "bbox": [ 141, 578, 161, 592 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 162, 578, 216, 591 ], "score": 0.93, "content": "0 < \\alpha < 1 / 8", "type": "inline_equation" }, { "bbox": [ 216, 578, 234, 592 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 234, 578, 272, 592 ], "score": 0.78, "content": "P _ { j } = P _ { \\theta _ { j } }", "type": "inline_equation" }, { "bbox": [ 273, 578, 276, 592 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 277, 578, 338, 591 ], "score": 0.79, "content": "j = 0 , 1 , . . . , M", "type": "inline_equation" }, { "bbox": [ 339, 578, 356, 592 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 357, 579, 372, 590 ], "score": 0.9, "content": "\\mathcal { D } _ { k l }", "type": "inline_equation" }, { "bbox": [ 373, 578, 505, 592 ], "score": 1.0, "content": "denotes the KL-divergence. Then", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 204, 596, 442, 626 ], "lines": [ { "bbox": [ 204, 596, 442, 626 ], "spans": [ { "bbox": [ 204, 596, 442, 626 ], "score": 0.93, "content": "\\operatorname* { i n f } _ { \\hat { \\theta } } \\operatorname* { s u p } _ { \\theta \\in \\Theta } P _ { \\theta } ( d ( \\hat { \\theta } , \\theta ) \\geq s ) \\geq \\frac { \\sqrt { M } } { 1 + \\sqrt { M } } \\big ( 1 - 2 \\alpha - \\sqrt { \\frac { 2 \\alpha } { \\log M } } \\big )", "type": "interline_equation", "image_path": "c4c241467485be3eafb8654a97f572e1d7abb12b5fd076d296ec66f7449e8de3.jpg" } ] } ], "index": 33.5, "virtual_lines": [ { "bbox": [ 204, 596, 442, 611.0 ], "spans": [], "index": 33 }, { "bbox": [ 204, 611.0, 442, 626.0 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 106, 636, 505, 725 ], "lines": [ { "bbox": [ 105, 636, 505, 650 ], "spans": [ { "bbox": [ 105, 636, 505, 650 ], "score": 1.0, "content": "Based on Proposition 1 we construct a family of pairs of distributions, namely source and target", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 647, 505, 660 ], "spans": [ { "bbox": [ 105, 647, 305, 660 ], "score": 1.0, "content": "distributions, whose transfer distances satisfy the", "type": "text" }, { "bbox": [ 305, 648, 315, 658 ], "score": 0.83, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 315, 647, 505, 660 ], "score": 1.0, "content": "-constraint. To do so we pick some points from", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 658, 506, 672 ], "spans": [ { "bbox": [ 105, 658, 155, 672 ], "score": 1.0, "content": "the domain", "type": "text" }, { "bbox": [ 155, 660, 163, 670 ], "score": 0.82, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 164, 658, 506, 672 ], "score": 1.0, "content": "shattered by the hypothesis class and define appropriate distributions on this set of", "type": "text" } ], "index": 37 }, { "bbox": [ 104, 669, 506, 682 ], "spans": [ { "bbox": [ 104, 669, 411, 682 ], "score": 1.0, "content": "points. Furthermore, this family of distributions are indexed in the space of", "type": "text" }, { "bbox": [ 411, 669, 448, 682 ], "score": 0.93, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 449, 669, 506, 682 ], "score": 1.0, "content": "which can be", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 680, 505, 693 ], "spans": [ { "bbox": [ 105, 680, 505, 693 ], "score": 1.0, "content": "a metric space using Hamming distance. In order to satisfy the condition (i) in Proposition 1, the", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 691, 505, 704 ], "spans": [ { "bbox": [ 105, 691, 505, 704 ], "score": 1.0, "content": "indexes have to be well separated which can be achieved using the well-known Gilbert-Varshamov’s", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 702, 505, 715 ], "spans": [ { "bbox": [ 105, 702, 505, 715 ], "score": 1.0, "content": "bound. Finally we show that estimating a parameter with small hamming distance is equivalent to", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 714, 369, 726 ], "spans": [ { "bbox": [ 105, 714, 369, 726 ], "score": 1.0, "content": "estimating an appropriate hypothesis with small excess risk error.", "type": "text" } ], "index": 42 } ], "index": 38.5 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "9", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 106, 27, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 309, 38 ], "spans": [ { "bbox": [ 106, 25, 309, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 221, 78, 392, 178 ], "blocks": [ { "type": "image_body", "bbox": [ 221, 78, 392, 178 ], "group_id": 0, "lines": [ { "bbox": [ 221, 78, 392, 178 ], "spans": [ { "bbox": [ 221, 78, 392, 178 ], "score": 0.963, "type": "image", "image_path": "7a303c3706559f7a90ce3dfe420c8b7330324be99f6ba51d8e74ec5e3805fcc7.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 221, 78, 392, 92.28571428571429 ], "spans": [], "index": 0 }, { "bbox": [ 221, 92.28571428571429, 392, 106.57142857142858 ], "spans": [], "index": 1 }, { "bbox": [ 221, 106.57142857142858, 392, 120.85714285714288 ], "spans": [], "index": 2 }, { "bbox": [ 221, 120.85714285714288, 392, 135.14285714285717 ], "spans": [], "index": 3 }, { "bbox": [ 221, 135.14285714285717, 392, 149.42857142857144 ], "spans": [], "index": 4 }, { "bbox": [ 221, 149.42857142857144, 392, 163.71428571428572 ], "spans": [], "index": 5 }, { "bbox": [ 221, 163.71428571428572, 392, 178.0 ], "spans": [], "index": 6 } ] }, { "type": "image_caption", "bbox": [ 106, 186, 504, 209 ], "group_id": 0, "lines": [ { "bbox": [ 105, 185, 505, 199 ], "spans": [ { "bbox": [ 105, 185, 181, 199 ], "score": 1.0, "content": "Figure 4: Average", "type": "text" }, { "bbox": [ 182, 187, 189, 196 ], "score": 0.71, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 189, 185, 505, 199 ], "score": 1.0, "content": "in weighted empirical risk minimization for three different pairs of source and", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 198, 252, 210 ], "spans": [ { "bbox": [ 105, 198, 252, 210 ], "score": 1.0, "content": "target tasks for image classification.", "type": "text" } ], "index": 8 } ], "index": 7.5 } ], "index": 5.25 }, { "type": "text", "bbox": [ 106, 230, 505, 297 ], "lines": [], "index": 11.5, "bbox_fs": [ 105, 231, 506, 298 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 302, 505, 346 ], "lines": [ { "bbox": [ 106, 302, 505, 314 ], "spans": [ { "bbox": [ 106, 302, 505, 314 ], "score": 1.0, "content": "CricketBowling is common both in the source and Target1. This suggests that these tasks are", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 312, 506, 326 ], "spans": [ { "bbox": [ 106, 312, 506, 326 ], "score": 1.0, "content": "similar to each other and the estimated transfer distance conforms with this intuition. Furthermore,", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 324, 505, 336 ], "spans": [ { "bbox": [ 105, 324, 505, 336 ], "score": 1.0, "content": "CricketBowling and Cricketshot are intuitively similar to one another and this is also reflected in the", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 335, 315, 347 ], "spans": [ { "bbox": [ 105, 335, 315, 347 ], "score": 1.0, "content": "lower transfer distance between source and Target2.", "type": "text" } ], "index": 18 } ], "index": 16.5, "bbox_fs": [ 105, 302, 506, 347 ] }, { "type": "text", "bbox": [ 107, 352, 505, 397 ], "lines": [ { "bbox": [ 105, 352, 505, 365 ], "spans": [ { "bbox": [ 105, 352, 227, 365 ], "score": 1.0, "content": "In Fig 4 we plot the average", "type": "text" }, { "bbox": [ 227, 353, 234, 362 ], "score": 0.74, "content": "\\lambda", "type": "inline_equation" }, { "bbox": [ 235, 352, 505, 365 ], "score": 1.0, "content": ", the weight appearing in Formula 5.1 when the number of target", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 362, 506, 375 ], "spans": [ { "bbox": [ 105, 362, 506, 375 ], "score": 1.0, "content": "samples varies from 150 to 200. 4 demonstrates that when a source is close to the target the weight of", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 374, 505, 386 ], "spans": [ { "bbox": [ 105, 374, 505, 386 ], "score": 1.0, "content": "source risk in weighted empirical risk becomes high which shows the effectiveness of source samples", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 386, 290, 397 ], "spans": [ { "bbox": [ 106, 386, 290, 397 ], "score": 1.0, "content": "in achieving small target generalization error.", "type": "text" } ], "index": 22 } ], "index": 20.5, "bbox_fs": [ 105, 352, 506, 397 ] }, { "type": "title", "bbox": [ 107, 412, 210, 425 ], "lines": [ { "bbox": [ 105, 411, 211, 427 ], "spans": [ { "bbox": [ 105, 411, 211, 427 ], "score": 1.0, "content": "6 PROOF OUTLINE", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 437, 460, 449 ], "lines": [ { "bbox": [ 105, 437, 461, 451 ], "spans": [ { "bbox": [ 105, 437, 461, 451 ], "score": 1.0, "content": "The main idea of proof is based on the following proposition proved in Tsybakov (2009)", "type": "text" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 105, 437, 461, 451 ] }, { "type": "text", "bbox": [ 107, 457, 506, 492 ], "lines": [ { "bbox": [ 106, 457, 506, 471 ], "spans": [ { "bbox": [ 106, 457, 362, 471 ], "score": 1.0, "content": "Proposition 1 [Theorem 2.5 of Tsybakov (2009)] Assume that", "type": "text" }, { "bbox": [ 362, 459, 393, 469 ], "score": 0.88, "content": "M \\geq 2", "type": "inline_equation" }, { "bbox": [ 393, 457, 462, 471 ], "score": 1.0, "content": "and the function", "type": "text" }, { "bbox": [ 463, 458, 487, 470 ], "score": 0.91, "content": "d ( \\cdot , \\cdot )", "type": "inline_equation" }, { "bbox": [ 487, 457, 506, 471 ], "score": 1.0, "content": "is a", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 468, 506, 483 ], "spans": [ { "bbox": [ 105, 468, 243, 483 ], "score": 1.0, "content": "semi-distance. Also suppose that", "type": "text" }, { "bbox": [ 243, 469, 287, 482 ], "score": 0.93, "content": "\\{ P _ { \\theta _ { j } } \\} _ { \\theta _ { j } \\in \\Theta }", "type": "inline_equation" }, { "bbox": [ 288, 468, 506, 483 ], "score": 1.0, "content": "is a family of distributions indexed over a parameter", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 480, 346, 493 ], "spans": [ { "bbox": [ 105, 480, 134, 493 ], "score": 1.0, "content": "space,", "type": "text" }, { "bbox": [ 134, 481, 143, 490 ], "score": 0.73, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 143, 480, 164, 493 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 165, 481, 173, 490 ], "score": 0.75, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 174, 480, 248, 493 ], "score": 1.0, "content": "contains elements", "type": "text" }, { "bbox": [ 249, 481, 302, 492 ], "score": 0.89, "content": "\\theta _ { 0 } , \\bar { \\theta } _ { 1 } , . . . , \\theta _ { M }", "type": "inline_equation" }, { "bbox": [ 303, 480, 346, 493 ], "score": 1.0, "content": "such that:", "type": "text" } ], "index": 27 } ], "index": 26, "bbox_fs": [ 105, 457, 506, 493 ] }, { "type": "interline_equation", "bbox": [ 141, 501, 303, 514 ], "lines": [ { "bbox": [ 141, 501, 303, 514 ], "spans": [ { "bbox": [ 141, 501, 303, 514 ], "score": 0.75, "content": "d ( \\theta _ { i } , \\theta _ { j } ) \\geq 2 s > 0 , \\ \\forall 0 \\leq j < k \\leq M", "type": "interline_equation", "image_path": "7d8273f42d18c53cb998b4153a3dd2ae520f7b101cbf282a2be487792abd913b.jpg" } ] } ], "index": 28, "virtual_lines": [ { "bbox": [ 141, 501, 303, 514 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 126, 519, 273, 532 ], "lines": [ { "bbox": [ 124, 518, 273, 533 ], "spans": [ { "bbox": [ 124, 518, 142, 533 ], "score": 1.0, "content": "(ii)", "type": "text" }, { "bbox": [ 142, 519, 249, 533 ], "score": 0.83, "content": "P _ { j } \\ll P _ { 0 } , \\ \\forall \\ j = 1 , . . . , M .", "type": "inline_equation" }, { "bbox": [ 250, 518, 273, 533 ], "score": 1.0, "content": ", and", "type": "text" } ], "index": 29 } ], "index": 29, "bbox_fs": [ 124, 518, 273, 533 ] }, { "type": "interline_equation", "bbox": [ 259, 538, 387, 573 ], "lines": [ { "bbox": [ 259, 538, 387, 573 ], "spans": [ { "bbox": [ 259, 538, 387, 573 ], "score": 0.94, "content": "\\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } { \\mathcal { D } } _ { k l } ( P _ { j } | P _ { 0 } ) \\leq \\alpha \\log M", "type": "interline_equation", "image_path": "e2c49d38ef7ce39bd19178d1c98fbe4a52106f16bc3c92bafbfa854aa4e2f3fb.jpg" } ] } ], "index": 30.5, "virtual_lines": [ { "bbox": [ 259, 538, 387, 555.5 ], "spans": [], "index": 30 }, { "bbox": [ 259, 555.5, 387, 573.0 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 142, 578, 503, 591 ], "lines": [ { "bbox": [ 141, 578, 505, 592 ], "spans": [ { "bbox": [ 141, 578, 161, 592 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 162, 578, 216, 591 ], "score": 0.93, "content": "0 < \\alpha < 1 / 8", "type": "inline_equation" }, { "bbox": [ 216, 578, 234, 592 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 234, 578, 272, 592 ], "score": 0.78, "content": "P _ { j } = P _ { \\theta _ { j } }", "type": "inline_equation" }, { "bbox": [ 273, 578, 276, 592 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 277, 578, 338, 591 ], "score": 0.79, "content": "j = 0 , 1 , . . . , M", "type": "inline_equation" }, { "bbox": [ 339, 578, 356, 592 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 357, 579, 372, 590 ], "score": 0.9, "content": "\\mathcal { D } _ { k l }", "type": "inline_equation" }, { "bbox": [ 373, 578, 505, 592 ], "score": 1.0, "content": "denotes the KL-divergence. Then", "type": "text" } ], "index": 32 } ], "index": 32, "bbox_fs": [ 141, 578, 505, 592 ] }, { "type": "interline_equation", "bbox": [ 204, 596, 442, 626 ], "lines": [ { "bbox": [ 204, 596, 442, 626 ], "spans": [ { "bbox": [ 204, 596, 442, 626 ], "score": 0.93, "content": "\\operatorname* { i n f } _ { \\hat { \\theta } } \\operatorname* { s u p } _ { \\theta \\in \\Theta } P _ { \\theta } ( d ( \\hat { \\theta } , \\theta ) \\geq s ) \\geq \\frac { \\sqrt { M } } { 1 + \\sqrt { M } } \\big ( 1 - 2 \\alpha - \\sqrt { \\frac { 2 \\alpha } { \\log M } } \\big )", "type": "interline_equation", "image_path": "c4c241467485be3eafb8654a97f572e1d7abb12b5fd076d296ec66f7449e8de3.jpg" } ] } ], "index": 33.5, "virtual_lines": [ { "bbox": [ 204, 596, 442, 611.0 ], "spans": [], "index": 33 }, { "bbox": [ 204, 611.0, 442, 626.0 ], "spans": [], "index": 34 } ] }, { "type": "text", "bbox": [ 106, 636, 505, 725 ], "lines": [ { "bbox": [ 105, 636, 505, 650 ], "spans": [ { "bbox": [ 105, 636, 505, 650 ], "score": 1.0, "content": "Based on Proposition 1 we construct a family of pairs of distributions, namely source and target", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 647, 505, 660 ], "spans": [ { "bbox": [ 105, 647, 305, 660 ], "score": 1.0, "content": "distributions, whose transfer distances satisfy the", "type": "text" }, { "bbox": [ 305, 648, 315, 658 ], "score": 0.83, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 315, 647, 505, 660 ], "score": 1.0, "content": "-constraint. To do so we pick some points from", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 658, 506, 672 ], "spans": [ { "bbox": [ 105, 658, 155, 672 ], "score": 1.0, "content": "the domain", "type": "text" }, { "bbox": [ 155, 660, 163, 670 ], "score": 0.82, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 164, 658, 506, 672 ], "score": 1.0, "content": "shattered by the hypothesis class and define appropriate distributions on this set of", "type": "text" } ], "index": 37 }, { "bbox": [ 104, 669, 506, 682 ], "spans": [ { "bbox": [ 104, 669, 411, 682 ], "score": 1.0, "content": "points. Furthermore, this family of distributions are indexed in the space of", "type": "text" }, { "bbox": [ 411, 669, 448, 682 ], "score": 0.93, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 449, 669, 506, 682 ], "score": 1.0, "content": "which can be", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 680, 505, 693 ], "spans": [ { "bbox": [ 105, 680, 505, 693 ], "score": 1.0, "content": "a metric space using Hamming distance. In order to satisfy the condition (i) in Proposition 1, the", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 691, 505, 704 ], "spans": [ { "bbox": [ 105, 691, 505, 704 ], "score": 1.0, "content": "indexes have to be well separated which can be achieved using the well-known Gilbert-Varshamov’s", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 702, 505, 715 ], "spans": [ { "bbox": [ 105, 702, 505, 715 ], "score": 1.0, "content": "bound. Finally we show that estimating a parameter with small hamming distance is equivalent to", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 714, 369, 726 ], "spans": [ { "bbox": [ 105, 714, 369, 726 ], "score": 1.0, "content": "estimating an appropriate hypothesis with small excess risk error.", "type": "text" } ], "index": 42 } ], "index": 38.5, "bbox_fs": [ 104, 636, 506, 726 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 176, 93 ], "lines": [ { "bbox": [ 106, 82, 176, 94 ], "spans": [ { "bbox": [ 106, 82, 176, 94 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 100, 506, 133 ], "lines": [ { "bbox": [ 105, 99, 505, 114 ], "spans": [ { "bbox": [ 105, 99, 505, 114 ], "score": 1.0, "content": "Kamyar Azizzadenesheli, Anqi Liu, Fanny Yang, and Animashree Anandkumar. 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Then there exists a subset", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 338, 366, 354 ], "spans": [ { "bbox": [ 107, 340, 175, 353 ], "score": 0.92, "content": "\\{ w ^ { ( 0 ) } , . . . , w ^ { ( M ) } \\}", "type": "inline_equation" }, { "bbox": [ 176, 338, 187, 354 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 187, 340, 245, 353 ], "score": 0.93, "content": "\\Omega = \\{ - 1 , \\mathrm { \\bar { 1 } } \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 246, 338, 285, 354 ], "score": 1.0, "content": "such that", "type": "text" }, { "bbox": [ 286, 340, 361, 353 ], "score": 0.91, "content": "w ^ { ( 0 ) } = ( 1 , 1 , . . . , 1 )", "type": "inline_equation" }, { "bbox": [ 362, 338, 366, 354 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "interline_equation", "bbox": [ 188, 356, 421, 380 ], "lines": [ { "bbox": [ 188, 356, 421, 380 ], "spans": [ { "bbox": [ 188, 356, 421, 380 ], "score": 0.91, "content": "d i s t ( w ^ { ( j ) } , w ^ { ( k ) } ) \\geq \\frac { d } { 8 } , \\ \\forall 0 \\leq j < k \\leq M a n d M \\geq 2 ^ { d / 8 } ,", "type": "interline_equation", "image_path": "f6b493e691595152c71338845cdff381f598128497689d2bf6d2a7d054288642.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 188, 356, 421, 380 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 384, 504, 410 ], "lines": [ { "bbox": [ 101, 376, 506, 406 ], "spans": [ { "bbox": [ 101, 376, 134, 406 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 383, 281, 397 ], "score": 0.9, "content": "\\begin{array} { r } { d i s t ( w , w ^ { \\prime } ) = \\sum _ { k = 1 } ^ { d } I ( w _ { k } \\ne w _ { k } ^ { \\prime } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 282, 376, 506, 406 ], "score": 1.0, "content": "is the Hamming distance between binary sequences", "type": "text" } ], "index": 18 }, { "bbox": [ 107, 393, 273, 412 ], "spans": [ { "bbox": [ 107, 397, 177, 409 ], "score": 0.91, "content": "w = ( w _ { 1 } , . . . , w _ { d } )", "type": "inline_equation" }, { "bbox": [ 177, 393, 195, 412 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 196, 397, 267, 409 ], "score": 0.82, "content": "\\boldsymbol { w } ^ { \\prime } = \\bar { ( } w _ { 1 } ^ { \\prime } , . . . , w _ { d } ^ { \\prime } )", "type": "inline_equation" }, { "bbox": [ 268, 393, 273, 412 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 19 } ], "index": 18.5 }, { "type": "text", "bbox": [ 106, 417, 505, 462 ], "lines": [ { "bbox": [ 104, 416, 506, 430 ], "spans": [ { "bbox": [ 104, 416, 506, 430 ], "score": 1.0, "content": "We will also use the following lemma proved in Hanneke & Kpotufe (2019). We would like to", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 427, 506, 441 ], "spans": [ { "bbox": [ 104, 427, 506, 441 ], "score": 1.0, "content": "mention that some ideas of the proof are similar to those in Hanneke & Kpotufe (2019). However, as", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 439, 506, 452 ], "spans": [ { "bbox": [ 106, 439, 506, 452 ], "score": 1.0, "content": "discussed in section 2, the problem setting of Hanneke & Kpotufe (2019) is different from that of this", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 451, 369, 462 ], "spans": [ { "bbox": [ 105, 451, 369, 462 ], "score": 1.0, "content": "work which results in constructing a different set of distributions.", "type": "text" } ], "index": 23 } ], "index": 21.5 }, { "type": "text", "bbox": [ 106, 470, 315, 483 ], "lines": [ { "bbox": [ 105, 469, 315, 484 ], "spans": [ { "bbox": [ 105, 469, 167, 484 ], "score": 1.0, "content": "Lemma 1 Let", "type": "text" }, { "bbox": [ 167, 470, 219, 483 ], "score": 0.91, "content": "0 < \\epsilon < 1 / 2", "type": "inline_equation" }, { "bbox": [ 219, 469, 237, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 237, 470, 288, 483 ], "score": 0.92, "content": "z \\in \\{ - 1 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 288, 469, 315, 484 ], "score": 1.0, "content": ". Then", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 116, 486, 451, 513 ], "lines": [ { "bbox": [ 116, 486, 451, 513 ], "spans": [ { "bbox": [ 116, 486, 451, 513 ], "score": 0.59, "content": "\\mathcal { D } _ { k l } \\bigg ( B e r \\big ( 1 / 2 + ( z / 2 ) \\cdot \\epsilon \\big ) , B e r \\big ( 1 / 2 - ( z / 2 ) \\cdot \\epsilon \\big ) \\bigg ) \\le c _ { 0 } \\cdot \\epsilon ^ { 2 } f o r s o m e c _ { 0 } \\le 4 i n d e p ,", "type": "interline_equation", "image_path": "4828ff1f7aec6729fb1b7348e4bd530420929c0b4f9981033fd4676f44bee708.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 116, 486, 451, 513 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 108, 522, 503, 545 ], "lines": [ { "bbox": [ 105, 519, 504, 537 ], "spans": [ { "bbox": [ 105, 519, 353, 537 ], "score": 1.0, "content": "Now we are in place to provide the proof of Theorem 1. Let", "type": "text" }, { "bbox": [ 353, 523, 403, 534 ], "score": 0.91, "content": "d = d _ { \\mathcal { H } } - 2", "type": "inline_equation" }, { "bbox": [ 403, 519, 441, 537 ], "score": 1.0, "content": "and pick", "type": "text" }, { "bbox": [ 442, 524, 504, 534 ], "score": 0.88, "content": "\\pmb { x } _ { - 1 } , \\pmb { x } _ { 0 } , . . . , \\pmb { x } _ { d }", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 106, 533, 202, 546 ], "spans": [ { "bbox": [ 106, 533, 128, 546 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 128, 535, 136, 545 ], "score": 0.83, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 137, 533, 188, 546 ], "score": 1.0, "content": "shattered by", "type": "text" }, { "bbox": [ 188, 534, 198, 543 ], "score": 0.81, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 198, 533, 202, 546 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 27 } ], "index": 26.5 }, { "type": "text", "bbox": [ 106, 549, 506, 594 ], "lines": [ { "bbox": [ 104, 549, 506, 563 ], "spans": [ { "bbox": [ 104, 549, 325, 563 ], "score": 1.0, "content": "Next, we construct a family of pairs of distributions", "type": "text" }, { "bbox": [ 326, 550, 365, 562 ], "score": 0.92, "content": "\\left( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } \\right)", "type": "inline_equation" }, { "bbox": [ 365, 549, 415, 563 ], "score": 1.0, "content": "indexed by", "type": "text" }, { "bbox": [ 416, 549, 476, 562 ], "score": 0.94, "content": "w \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 476, 549, 506, 563 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 560, 505, 574 ], "spans": [ { "bbox": [ 107, 560, 144, 573 ], "score": 0.91, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 145, 560, 316, 574 ], "score": 1.0, "content": "is the parameter space playing the role of", "type": "text" }, { "bbox": [ 317, 562, 326, 571 ], "score": 0.83, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 326, 560, 487, 574 ], "score": 1.0, "content": "in Proposition 1. For the following, fix", "type": "text" }, { "bbox": [ 488, 563, 505, 572 ], "score": 0.84, "content": "\\epsilon =", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 107, 571, 504, 586 ], "spans": [ { "bbox": [ 107, 572, 212, 585 ], "score": 0.89, "content": "\\begin{array} { r } { \\dot { c } _ { 1 } \\cdot \\epsilon ( \\dot { n _ { S } } , n _ { T } , d _ { \\mathcal { H } } ^ { \\cdot } , \\Delta ) \\leq \\frac { 1 } { 2 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 213, 571, 285, 586 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 285, 574, 295, 583 ], "score": 0.84, "content": "c _ { 1 }", "type": "inline_equation" }, { "bbox": [ 295, 571, 434, 586 ], "score": 1.0, "content": "to be determined later in proof and", "type": "text" }, { "bbox": [ 434, 572, 504, 584 ], "score": 0.93, "content": "\\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta )", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 583, 207, 595 ], "spans": [ { "bbox": [ 105, 583, 207, 595 ], "score": 1.0, "content": "is defined in Theorem 1.", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 105, 599, 503, 623 ], "lines": [ { "bbox": [ 105, 597, 505, 614 ], "spans": [ { "bbox": [ 105, 597, 161, 614 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 600, 199, 611 ], "score": 0.5, "content": "\\mathbb { Q } _ { w } \\colon \\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 199, 597, 476, 614 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 476, 600, 505, 612 ], "score": 0.86, "content": "\\mathbb { Q } _ { w } =", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 106, 609, 351, 626 ], "spans": [ { "bbox": [ 106, 611, 154, 626 ], "score": 0.93, "content": "\\mathbb { Q } _ { x } ^ { w } \\times \\mathbb { Q } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 154, 609, 351, 626 ], "score": 1.0, "content": ". We define the marginaldistributions as follows:", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "interline_equation", "bbox": [ 228, 627, 383, 683 ], "lines": [ { "bbox": [ 228, 627, 383, 683 ], "spans": [ { "bbox": [ 228, 627, 383, 683 ], "score": 0.92, "content": "\\begin{array} { l l l } { \\mathbb { Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - 1 } ) = \\Delta } \\\\ { \\mathbb { Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { 0 } ) = 0 . 9 9 - \\Delta } \\\\ { \\mathbb { Q } _ { \\pmb x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { i } ) = \\displaystyle \\frac { 1 } { 1 0 0 d } \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "d84c28e37c0bc3d3f125e90b0113f50db790c891892b54f012c8b7b7aa220881.jpg" } ] } ], "index": 35.5, "virtual_lines": [ { "bbox": [ 228, 627, 383, 641.0 ], "spans": [], "index": 34 }, { "bbox": [ 228, 641.0, 383, 655.0 ], "spans": [], "index": 35 }, { "bbox": [ 228, 655.0, 383, 669.0 ], "spans": [], "index": 36 }, { "bbox": [ 228, 669.0, 383, 683.0 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 106, 684, 239, 695 ], "lines": [ { "bbox": [ 106, 682, 239, 696 ], "spans": [ { "bbox": [ 106, 682, 239, 696 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "interline_equation", "bbox": [ 199, 698, 412, 731 ], "lines": [ { "bbox": [ 199, 698, 412, 731 ], "spans": [ { "bbox": [ 199, 698, 412, 731 ], "score": 0.9, "content": "\\begin{array} { r l } & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - 1 } ) = \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "efca9a2b8b8b954f76e0452c62405345dc616295d7a7dd534be10dab25eadf6e.jpg" } ] } ], "index": 39.5, "virtual_lines": [ { "bbox": [ 199, 698, 412, 714.5 ], "spans": [], "index": 39 }, { "bbox": [ 199, 714.5, 412, 731.0 ], "spans": [], "index": 40 } ] } ], "page_idx": 10, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 26, 308, 38 ], "lines": [ { "bbox": [ 107, 25, 308, 39 ], "spans": [ { "bbox": [ 107, 25, 308, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 765 ], "spans": [ { "bbox": [ 299, 750, 312, 765 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 82, 504, 105 ], "lines": [ { "bbox": [ 105, 81, 505, 96 ], "spans": [ { "bbox": [ 105, 81, 505, 96 ], "score": 1.0, "content": "Sinno Jialin Pan and Qiang Yang. 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In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.", "type": "text" } ], "index": 6 } ], "index": 5.5, "bbox_fs": [ 106, 152, 506, 175 ] }, { "type": "text", "bbox": [ 106, 181, 504, 203 ], "lines": [ { "bbox": [ 106, 180, 506, 194 ], "spans": [ { "bbox": [ 106, 180, 506, 194 ], "score": 1.0, "content": "Alexandre B Tsybakov. Introduction to Nonparametric Estimation. Springer series in statistics.", "type": "text" } ], "index": 7 }, { "bbox": [ 115, 192, 315, 203 ], "spans": [ { "bbox": [ 115, 192, 315, 203 ], "score": 1.0, "content": "Springer, Dordrecht, 2009. doi: 10.1007/b13794.", "type": "text" } ], "index": 8 } ], "index": 7.5, "bbox_fs": [ 106, 180, 506, 203 ] }, { "type": "text", "bbox": [ 109, 210, 505, 233 ], "lines": [ { "bbox": [ 106, 209, 506, 223 ], "spans": [ { "bbox": [ 106, 209, 506, 223 ], "score": 1.0, "content": "Karl Weiss, Taghi M Khoshgoftaar, and DingDing Wang. A survey of transfer learning. Journal of", "type": "text" } ], "index": 9 }, { "bbox": [ 115, 221, 225, 233 ], "spans": [ { "bbox": [ 115, 221, 225, 233 ], "score": 1.0, "content": "Big data, 3(1):1–40, 2016.", "type": "text" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 106, 209, 506, 233 ] }, { "type": "title", "bbox": [ 107, 252, 180, 265 ], "lines": [ { "bbox": [ 105, 250, 182, 268 ], "spans": [ { "bbox": [ 105, 250, 182, 268 ], "score": 1.0, "content": "7 APPENDIX", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "title", "bbox": [ 107, 276, 228, 288 ], "lines": [ { "bbox": [ 106, 276, 228, 289 ], "spans": [ { "bbox": [ 106, 276, 228, 289 ], "score": 1.0, "content": "7.1 PROOF OF THEOREM 1", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 297, 505, 321 ], "lines": [ { "bbox": [ 106, 297, 505, 310 ], "spans": [ { "bbox": [ 106, 297, 505, 310 ], "score": 1.0, "content": "We also use the following famous result in information theory known as Gilbert-Varhsamov’s bound", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 308, 197, 321 ], "spans": [ { "bbox": [ 106, 308, 197, 321 ], "score": 1.0, "content": "for packing argument.", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 106, 297, 505, 321 ] }, { "type": "text", "bbox": [ 105, 328, 504, 353 ], "lines": [ { "bbox": [ 105, 327, 506, 342 ], "spans": [ { "bbox": [ 105, 327, 336, 342 ], "score": 1.0, "content": "Proposition 2 (Lemma 2.9 of Tsybakov (2009)) Let", "type": "text" }, { "bbox": [ 337, 329, 373, 340 ], "score": 0.87, "content": "d \\_ 8", "type": "inline_equation" }, { "bbox": [ 373, 327, 506, 342 ], "score": 1.0, "content": ". Then there exists a subset", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 338, 366, 354 ], "spans": [ { "bbox": [ 107, 340, 175, 353 ], "score": 0.92, "content": "\\{ w ^ { ( 0 ) } , . . . , w ^ { ( M ) } \\}", "type": "inline_equation" }, { "bbox": [ 176, 338, 187, 354 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 187, 340, 245, 353 ], "score": 0.93, "content": "\\Omega = \\{ - 1 , \\mathrm { \\bar { 1 } } \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 246, 338, 285, 354 ], "score": 1.0, "content": "such that", "type": "text" }, { "bbox": [ 286, 340, 361, 353 ], "score": 0.91, "content": "w ^ { ( 0 ) } = ( 1 , 1 , . . . , 1 )", "type": "inline_equation" }, { "bbox": [ 362, 338, 366, 354 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 105, 327, 506, 354 ] }, { "type": "interline_equation", "bbox": [ 188, 356, 421, 380 ], "lines": [ { "bbox": [ 188, 356, 421, 380 ], "spans": [ { "bbox": [ 188, 356, 421, 380 ], "score": 0.91, "content": "d i s t ( w ^ { ( j ) } , w ^ { ( k ) } ) \\geq \\frac { d } { 8 } , \\ \\forall 0 \\leq j < k \\leq M a n d M \\geq 2 ^ { d / 8 } ,", "type": "interline_equation", "image_path": "f6b493e691595152c71338845cdff381f598128497689d2bf6d2a7d054288642.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 188, 356, 421, 380 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 384, 504, 410 ], "lines": [ { "bbox": [ 101, 376, 506, 406 ], "spans": [ { "bbox": [ 101, 376, 134, 406 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 383, 281, 397 ], "score": 0.9, "content": "\\begin{array} { r } { d i s t ( w , w ^ { \\prime } ) = \\sum _ { k = 1 } ^ { d } I ( w _ { k } \\ne w _ { k } ^ { \\prime } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 282, 376, 506, 406 ], "score": 1.0, "content": "is the Hamming distance between binary sequences", "type": "text" } ], "index": 18 }, { "bbox": [ 107, 393, 273, 412 ], "spans": [ { "bbox": [ 107, 397, 177, 409 ], "score": 0.91, "content": "w = ( w _ { 1 } , . . . , w _ { d } )", "type": "inline_equation" }, { "bbox": [ 177, 393, 195, 412 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 196, 397, 267, 409 ], "score": 0.82, "content": "\\boldsymbol { w } ^ { \\prime } = \\bar { ( } w _ { 1 } ^ { \\prime } , . . . , w _ { d } ^ { \\prime } )", "type": "inline_equation" }, { "bbox": [ 268, 393, 273, 412 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 19 } ], "index": 18.5, "bbox_fs": [ 101, 376, 506, 412 ] }, { "type": "text", "bbox": [ 106, 417, 505, 462 ], "lines": [ { "bbox": [ 104, 416, 506, 430 ], "spans": [ { "bbox": [ 104, 416, 506, 430 ], "score": 1.0, "content": "We will also use the following lemma proved in Hanneke & Kpotufe (2019). We would like to", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 427, 506, 441 ], "spans": [ { "bbox": [ 104, 427, 506, 441 ], "score": 1.0, "content": "mention that some ideas of the proof are similar to those in Hanneke & Kpotufe (2019). However, as", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 439, 506, 452 ], "spans": [ { "bbox": [ 106, 439, 506, 452 ], "score": 1.0, "content": "discussed in section 2, the problem setting of Hanneke & Kpotufe (2019) is different from that of this", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 451, 369, 462 ], "spans": [ { "bbox": [ 105, 451, 369, 462 ], "score": 1.0, "content": "work which results in constructing a different set of distributions.", "type": "text" } ], "index": 23 } ], "index": 21.5, "bbox_fs": [ 104, 416, 506, 462 ] }, { "type": "text", "bbox": [ 106, 470, 315, 483 ], "lines": [ { "bbox": [ 105, 469, 315, 484 ], "spans": [ { "bbox": [ 105, 469, 167, 484 ], "score": 1.0, "content": "Lemma 1 Let", "type": "text" }, { "bbox": [ 167, 470, 219, 483 ], "score": 0.91, "content": "0 < \\epsilon < 1 / 2", "type": "inline_equation" }, { "bbox": [ 219, 469, 237, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 237, 470, 288, 483 ], "score": 0.92, "content": "z \\in \\{ - 1 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 288, 469, 315, 484 ], "score": 1.0, "content": ". Then", "type": "text" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 105, 469, 315, 484 ] }, { "type": "interline_equation", "bbox": [ 116, 486, 451, 513 ], "lines": [ { "bbox": [ 116, 486, 451, 513 ], "spans": [ { "bbox": [ 116, 486, 451, 513 ], "score": 0.59, "content": "\\mathcal { D } _ { k l } \\bigg ( B e r \\big ( 1 / 2 + ( z / 2 ) \\cdot \\epsilon \\big ) , B e r \\big ( 1 / 2 - ( z / 2 ) \\cdot \\epsilon \\big ) \\bigg ) \\le c _ { 0 } \\cdot \\epsilon ^ { 2 } f o r s o m e c _ { 0 } \\le 4 i n d e p ,", "type": "interline_equation", "image_path": "4828ff1f7aec6729fb1b7348e4bd530420929c0b4f9981033fd4676f44bee708.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 116, 486, 451, 513 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 108, 522, 503, 545 ], "lines": [ { "bbox": [ 105, 519, 504, 537 ], "spans": [ { "bbox": [ 105, 519, 353, 537 ], "score": 1.0, "content": "Now we are in place to provide the proof of Theorem 1. Let", "type": "text" }, { "bbox": [ 353, 523, 403, 534 ], "score": 0.91, "content": "d = d _ { \\mathcal { H } } - 2", "type": "inline_equation" }, { "bbox": [ 403, 519, 441, 537 ], "score": 1.0, "content": "and pick", "type": "text" }, { "bbox": [ 442, 524, 504, 534 ], "score": 0.88, "content": "\\pmb { x } _ { - 1 } , \\pmb { x } _ { 0 } , . . . , \\pmb { x } _ { d }", "type": "inline_equation" } ], "index": 26 }, { "bbox": [ 106, 533, 202, 546 ], "spans": [ { "bbox": [ 106, 533, 128, 546 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 128, 535, 136, 545 ], "score": 0.83, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 137, 533, 188, 546 ], "score": 1.0, "content": "shattered by", "type": "text" }, { "bbox": [ 188, 534, 198, 543 ], "score": 0.81, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 198, 533, 202, 546 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 27 } ], "index": 26.5, "bbox_fs": [ 105, 519, 504, 546 ] }, { "type": "text", "bbox": [ 106, 549, 506, 594 ], "lines": [ { "bbox": [ 104, 549, 506, 563 ], "spans": [ { "bbox": [ 104, 549, 325, 563 ], "score": 1.0, "content": "Next, we construct a family of pairs of distributions", "type": "text" }, { "bbox": [ 326, 550, 365, 562 ], "score": 0.92, "content": "\\left( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } \\right)", "type": "inline_equation" }, { "bbox": [ 365, 549, 415, 563 ], "score": 1.0, "content": "indexed by", "type": "text" }, { "bbox": [ 416, 549, 476, 562 ], "score": 0.94, "content": "w \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 476, 549, 506, 563 ], "score": 1.0, "content": "where", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 560, 505, 574 ], "spans": [ { "bbox": [ 107, 560, 144, 573 ], "score": 0.91, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 145, 560, 316, 574 ], "score": 1.0, "content": "is the parameter space playing the role of", "type": "text" }, { "bbox": [ 317, 562, 326, 571 ], "score": 0.83, "content": "\\Theta", "type": "inline_equation" }, { "bbox": [ 326, 560, 487, 574 ], "score": 1.0, "content": "in Proposition 1. For the following, fix", "type": "text" }, { "bbox": [ 488, 563, 505, 572 ], "score": 0.84, "content": "\\epsilon =", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 107, 571, 504, 586 ], "spans": [ { "bbox": [ 107, 572, 212, 585 ], "score": 0.89, "content": "\\begin{array} { r } { \\dot { c } _ { 1 } \\cdot \\epsilon ( \\dot { n _ { S } } , n _ { T } , d _ { \\mathcal { H } } ^ { \\cdot } , \\Delta ) \\leq \\frac { 1 } { 2 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 213, 571, 285, 586 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 285, 574, 295, 583 ], "score": 0.84, "content": "c _ { 1 }", "type": "inline_equation" }, { "bbox": [ 295, 571, 434, 586 ], "score": 1.0, "content": "to be determined later in proof and", "type": "text" }, { "bbox": [ 434, 572, 504, 584 ], "score": 0.93, "content": "\\epsilon ( n _ { S } , n _ { T } , d _ { \\mathcal { H } } , \\Delta )", "type": "inline_equation" } ], "index": 30 }, { "bbox": [ 105, 583, 207, 595 ], "spans": [ { "bbox": [ 105, 583, 207, 595 ], "score": 1.0, "content": "is defined in Theorem 1.", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 104, 549, 506, 595 ] }, { "type": "text", "bbox": [ 105, 599, 503, 623 ], "lines": [ { "bbox": [ 105, 597, 505, 614 ], "spans": [ { "bbox": [ 105, 597, 161, 614 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 600, 199, 611 ], "score": 0.5, "content": "\\mathbb { Q } _ { w } \\colon \\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 199, 597, 476, 614 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 476, 600, 505, 612 ], "score": 0.86, "content": "\\mathbb { Q } _ { w } =", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 106, 609, 351, 626 ], "spans": [ { "bbox": [ 106, 611, 154, 626 ], "score": 0.93, "content": "\\mathbb { Q } _ { x } ^ { w } \\times \\mathbb { Q } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 154, 609, 351, 626 ], "score": 1.0, "content": ". We define the marginaldistributions as follows:", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 105, 597, 505, 626 ] }, { "type": "interline_equation", "bbox": [ 228, 627, 383, 683 ], "lines": [ { "bbox": [ 228, 627, 383, 683 ], "spans": [ { "bbox": [ 228, 627, 383, 683 ], "score": 0.92, "content": "\\begin{array} { l l l } { \\mathbb { Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - 1 } ) = \\Delta } \\\\ { \\mathbb { Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { 0 } ) = 0 . 9 9 - \\Delta } \\\\ { \\mathbb { Q } _ { \\pmb x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { i } ) = \\displaystyle \\frac { 1 } { 1 0 0 d } \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "d84c28e37c0bc3d3f125e90b0113f50db790c891892b54f012c8b7b7aa220881.jpg" } ] } ], "index": 35.5, "virtual_lines": [ { "bbox": [ 228, 627, 383, 641.0 ], "spans": [], "index": 34 }, { "bbox": [ 228, 641.0, 383, 655.0 ], "spans": [], "index": 35 }, { "bbox": [ 228, 655.0, 383, 669.0 ], "spans": [], "index": 36 }, { "bbox": [ 228, 669.0, 383, 683.0 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 106, 684, 239, 695 ], "lines": [ { "bbox": [ 106, 682, 239, 696 ], "spans": [ { "bbox": [ 106, 682, 239, 696 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 38 } ], "index": 38, "bbox_fs": [ 106, 682, 239, 696 ] }, { "type": "interline_equation", "bbox": [ 199, 698, 412, 731 ], "lines": [ { "bbox": [ 199, 698, 412, 731 ], "spans": [ { "bbox": [ 199, 698, 412, 731 ], "score": 0.9, "content": "\\begin{array} { r l } & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - 1 } ) = \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "efca9a2b8b8b954f76e0452c62405345dc616295d7a7dd534be10dab25eadf6e.jpg" } ] } ], "index": 39.5, "virtual_lines": [ { "bbox": [ 199, 698, 412, 714.5 ], "spans": [], "index": 39 }, { "bbox": [ 199, 714.5, 412, 731.0 ], "spans": [], "index": 40 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 103, 82, 504, 105 ], "lines": [ { "bbox": [ 105, 81, 505, 97 ], "spans": [ { "bbox": [ 105, 81, 161, 97 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 83, 175, 93 ], "score": 0.84, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 176, 81, 182, 97 ], "score": 1.0, "content": ":", "type": "text" }, { "bbox": [ 182, 83, 196, 93 ], "score": 0.79, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 197, 81, 477, 97 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 477, 83, 505, 94 ], "score": 0.86, "content": "\\mathbb { P } _ { w } =", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 106, 88, 352, 112 ], "spans": [ { "bbox": [ 106, 93, 151, 108 ], "score": 0.92, "content": "\\mathbb { P } _ { x } ^ { w } \\times \\mathbb { P } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 151, 88, 352, 112 ], "score": 1.0, "content": ". We define the marginal distributions as follows:", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 195, 109, 416, 161 ], "lines": [ { "bbox": [ 195, 109, 416, 161 ], "spans": [ { "bbox": [ 195, 109, 416, 161 ], "score": 0.87, "content": "\\begin{array} { l l l } { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { - 1 } ) = \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 / 2 \\big ( 1 - \\frac { d } { d + n _ { S } \\Delta } \\big ) } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { i } ) = \\frac { 1 } { d + n _ { S } \\Delta } \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "d3d0b43c36674461ec8ef939913b82325f74ec1e7a6a9fc1e6e33d321456acd6.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 195, 109, 416, 126.33333333333333 ], "spans": [], "index": 2 }, { "bbox": [ 195, 126.33333333333333, 416, 143.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 195, 143.66666666666666, 416, 161.0 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 161, 239, 172 ], "lines": [ { "bbox": [ 106, 160, 239, 173 ], "spans": [ { "bbox": [ 106, 160, 239, 173 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 198, 174, 411, 222 ], "lines": [ { "bbox": [ 198, 174, 411, 222 ], "spans": [ { "bbox": [ 198, 174, 411, 222 ], "score": 0.91, "content": "\\begin{array} { r l } & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - 1 } ) = 0 } \\\\ & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "4be167508deaf5ee6508ebbd0cf2a08748ac083628dd34d91d1166864799b860.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 198, 174, 411, 190.0 ], "spans": [], "index": 6 }, { "bbox": [ 198, 190.0, 411, 206.0 ], "spans": [], "index": 7 }, { "bbox": [ 198, 206.0, 411, 222.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 105, 223, 465, 236 ], "lines": [ { "bbox": [ 106, 223, 466, 237 ], "spans": [ { "bbox": [ 106, 223, 148, 237 ], "score": 1.0, "content": "Verifying", "type": "text" }, { "bbox": [ 148, 223, 215, 236 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } ) \\leq \\Delta", "type": "inline_equation" }, { "bbox": [ 215, 223, 395, 237 ], "score": 1.0, "content": ": Bayes classifier of the domain generated by", "type": "text" }, { "bbox": [ 396, 224, 409, 235 ], "score": 0.89, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 410, 223, 466, 237 ], "score": 1.0, "content": "is as follows:", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "interline_equation", "bbox": [ 174, 237, 436, 279 ], "lines": [ { "bbox": [ 174, 237, 436, 279 ], "spans": [ { "bbox": [ 174, 237, 436, 279 ], "score": 0.81, "content": "\\begin{array} { r l } & { h _ { S } ^ { * } ( { \\pmb x } _ { - 1 } ) = 0 } \\\\ & { h _ { S } ^ { * } ( { \\pmb x } _ { 0 } ) = 1 } \\\\ & { h _ { S } ^ { * } ( { \\pmb x } _ { i } ) = 1 \\mathrm { i f } w _ { i } = 1 \\mathrm { , , o t h e r w i s e } h _ { S } ^ { * } ( { \\pmb x } _ { i } ) = 0 \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "5aaaa6e33dd5bbc8e456624989a8b5fbadf6ba07307a406c27ffb7a6fc4c15dd.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 174, 237, 436, 251.0 ], "spans": [], "index": 10 }, { "bbox": [ 174, 251.0, 436, 265.0 ], "spans": [], "index": 11 }, { "bbox": [ 174, 265.0, 436, 279.0 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 107, 280, 315, 291 ], "lines": [ { "bbox": [ 106, 279, 316, 293 ], "spans": [ { "bbox": [ 106, 279, 261, 293 ], "score": 1.0, "content": "Similarly for the domain generated by", "type": "text" }, { "bbox": [ 261, 280, 276, 291 ], "score": 0.89, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 276, 279, 316, 293 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "interline_equation", "bbox": [ 174, 293, 437, 321 ], "lines": [ { "bbox": [ 174, 293, 437, 321 ], "spans": [ { "bbox": [ 174, 293, 437, 321 ], "score": 0.81, "content": "\\begin{array} { r l } & { h _ { T } ^ { * } ( { \\pmb x } _ { - 1 } ) = h _ { T } ^ { * } ( { \\pmb x } _ { 0 } ) = 1 } \\\\ & { h _ { T } ^ { * } ( { \\pmb x } _ { i } ) = 1 \\mathrm { i f } w _ { i } = 1 \\mathrm { , o t h e r w i s e } h _ { T } ^ { * } ( { \\pmb x } _ { i } ) = 0 \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "4fc579e2f99bd0ce951af4d71aa968812fd6c6bca0f05ea836841c704b154552.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 174, 293, 437, 321 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 322, 330, 334 ], "lines": [ { "bbox": [ 105, 321, 330, 336 ], "spans": [ { "bbox": [ 105, 321, 119, 336 ], "score": 1.0, "content": "So", "type": "text" }, { "bbox": [ 119, 322, 132, 334 ], "score": 0.87, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 132, 321, 150, 336 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 151, 323, 164, 334 ], "score": 0.9, "content": "h _ { T } ^ { * }", "type": "inline_equation" }, { "bbox": [ 164, 321, 233, 336 ], "score": 1.0, "content": "disagree only on", "type": "text" }, { "bbox": [ 233, 324, 252, 334 ], "score": 0.87, "content": "{ \\pmb x } _ { - 1 }", "type": "inline_equation" }, { "bbox": [ 252, 321, 330, 336 ], "score": 1.0, "content": "which implies that", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "interline_equation", "bbox": [ 188, 335, 423, 349 ], "lines": [ { "bbox": [ 188, 335, 423, 349 ], "spans": [ { "bbox": [ 188, 335, 423, 349 ], "score": 0.85, "content": "\\rho ( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } ) = \\mathbb { Q } [ h _ { S } ^ { \\ast } ( { \\pmb x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( { \\pmb x } _ { T } ) \\neq y _ { T } ] = \\Delta", "type": "interline_equation", "image_path": "eb6e1bf0578ba67796515fc16f4e0a5ea117df9513a8d38a08383fcebc1a6084.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 188, 335, 423, 349 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 356, 506, 415 ], "lines": [ { "bbox": [ 106, 356, 505, 368 ], "spans": [ { "bbox": [ 106, 356, 505, 368 ], "score": 1.0, "content": "Since we want to derive a lower bound for the minimax risk stated in Theorem 1, among the", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 367, 506, 379 ], "spans": [ { "bbox": [ 106, 367, 234, 379 ], "score": 1.0, "content": "hypotheses that they agree on", "type": "text" }, { "bbox": [ 234, 369, 245, 379 ], "score": 0.86, "content": "\\mathbf { \\boldsymbol { x } } _ { i }", "type": "inline_equation" }, { "bbox": [ 245, 367, 263, 379 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 263, 367, 312, 379 ], "score": 0.92, "content": "i = 1 , . . . , d", "type": "inline_equation" }, { "bbox": [ 312, 367, 432, 379 ], "score": 1.0, "content": ", the hypothesis that outputs", "type": "text" }, { "bbox": [ 432, 369, 451, 379 ], "score": 0.89, "content": "{ \\pmb x } _ { - 1 }", "type": "inline_equation" }, { "bbox": [ 451, 367, 471, 379 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 471, 369, 483, 379 ], "score": 0.86, "content": "\\scriptstyle { \\pmb x } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 484, 367, 506, 379 ], "score": 1.0, "content": "as 1", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 378, 506, 392 ], "spans": [ { "bbox": [ 105, 380, 385, 392 ], "score": 1.0, "content": "results in a smaller target error. Hence, we can restrict ourselves to", "type": "text" }, { "bbox": [ 386, 378, 396, 390 ], "score": 0.86, "content": "\\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 396, 380, 506, 392 ], "score": 1.0, "content": "which is the projection of", "type": "text" } ], "index": 19 }, { "bbox": [ 107, 390, 505, 405 ], "spans": [ { "bbox": [ 107, 392, 116, 402 ], "score": 0.81, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 117, 390, 138, 405 ], "score": 1.0, "content": "onto", "type": "text" }, { "bbox": [ 139, 391, 176, 403 ], "score": 0.91, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 176, 390, 273, 405 ], "score": 1.0, "content": "with the constraint that", "type": "text" }, { "bbox": [ 274, 392, 362, 404 ], "score": 0.94, "content": "h ( \\pmb { x } _ { - 1 } ) = h ( \\pmb { x } _ { 0 } ) = 1", "type": "inline_equation" }, { "bbox": [ 362, 390, 390, 405 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 390, 390, 418, 402 ], "score": 0.91, "content": "h \\in \\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 419, 390, 505, 405 ], "score": 1.0, "content": ". Furthermor, for any", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 402, 216, 415 ], "spans": [ { "bbox": [ 107, 403, 178, 415 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 179, 402, 216, 415 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 21 } ], "index": 19 }, { "type": "interline_equation", "bbox": [ 226, 416, 385, 441 ], "lines": [ { "bbox": [ 226, 416, 385, 441 ], "spans": [ { "bbox": [ 226, 416, 385, 441 ], "score": 0.94, "content": "\\mathcal { E } _ { T } ( h _ { w ^ { \\prime } } ) = \\frac { \\mathrm { d i s t } ( w , w ^ { \\prime } ) } { 1 0 0 d } \\cdot \\epsilon , ~ \\forall ~ h _ { w ^ { \\prime } } \\in \\tilde { \\mathcal { H } }", "type": "interline_equation", "image_path": "df150745bf6e2f61646ee55386173930870964749e9bc2236e0c4a5b24aeaace.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 226, 416, 385, 441 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 442, 280, 454 ], "lines": [ { "bbox": [ 105, 439, 280, 457 ], "spans": [ { "bbox": [ 105, 439, 265, 457 ], "score": 1.0, "content": "when the target domain is generated by", "type": "text" }, { "bbox": [ 265, 443, 280, 453 ], "score": 0.88, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 106, 459, 505, 493 ], "lines": [ { "bbox": [ 105, 458, 504, 472 ], "spans": [ { "bbox": [ 105, 458, 376, 472 ], "score": 1.0, "content": "Reduction to a packing: By using Proposition 2, we can get a subset", "type": "text" }, { "bbox": [ 376, 460, 385, 469 ], "score": 0.83, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 385, 458, 396, 472 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 396, 458, 433, 471 ], "score": 0.93, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 434, 458, 504, 472 ], "score": 1.0, "content": "whose cardinality", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 468, 506, 486 ], "spans": [ { "bbox": [ 104, 468, 116, 486 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 117, 470, 160, 482 ], "score": 0.93, "content": "M \\geq 2 ^ { d / 8 }", "type": "inline_equation" }, { "bbox": [ 160, 468, 211, 486 ], "score": 1.0, "content": "and for any", "type": "text" }, { "bbox": [ 212, 471, 235, 483 ], "score": 0.91, "content": "w , w ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 235, 468, 290, 486 ], "score": 1.0, "content": "belonging to", "type": "text" }, { "bbox": [ 291, 471, 299, 481 ], "score": 0.81, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 299, 468, 340, 486 ], "score": 1.0, "content": "we have", "type": "text" }, { "bbox": [ 340, 471, 412, 483 ], "score": 0.91, "content": "\\operatorname* { l i s t } ( w , w ^ { \\prime } ) \\geq d / 8", "type": "inline_equation" }, { "bbox": [ 412, 468, 506, 486 ], "score": 1.0, "content": ". Furthermore, for any", "type": "text" } ], "index": 25 }, { "bbox": [ 107, 482, 186, 494 ], "spans": [ { "bbox": [ 107, 483, 149, 493 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\Sigma", "type": "inline_equation" }, { "bbox": [ 149, 482, 186, 494 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 246, 495, 365, 519 ], "lines": [ { "bbox": [ 246, 495, 365, 519 ], "spans": [ { "bbox": [ 246, 495, 365, 519 ], "score": 0.95, "content": "\\mathcal { E } _ { T } ( h _ { w ^ { \\prime } } ) \\geq \\frac { d } { 8 } \\cdot \\frac { \\epsilon } { 1 0 0 d } = \\frac { \\epsilon } { 8 0 0 }", "type": "interline_equation", "image_path": "d806dc2a8e1a45316cfe4e4686cf3c1524d0e441e81b314604beb1822dc609f3.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 246, 495, 365, 519 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 522, 506, 569 ], "lines": [ { "bbox": [ 106, 521, 506, 535 ], "spans": [ { "bbox": [ 106, 522, 316, 535 ], "score": 1.0, "content": "On the other hand, there is a bijective map between", "type": "text" }, { "bbox": [ 316, 522, 354, 534 ], "score": 0.94, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 354, 522, 421, 535 ], "score": 1.0, "content": "and elements of", "type": "text" }, { "bbox": [ 422, 521, 432, 533 ], "score": 0.87, "content": "\\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 432, 522, 506, 535 ], "score": 1.0, "content": "and any classifier", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 533, 506, 549 ], "spans": [ { "bbox": [ 107, 534, 181, 547 ], "score": 0.92, "content": "\\hat { h } : \\{ { \\pmb x } _ { i } \\} \\{ 0 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 182, 533, 203, 549 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 203, 533, 290, 547 ], "score": 0.94, "content": "\\hat { h } ( { \\pmb x } _ { - 1 } ) = \\hat { h } ( { \\pmb x } _ { 0 } ) = 1", "type": "inline_equation" }, { "bbox": [ 291, 533, 371, 549 ], "score": 1.0, "content": "can be reduced to a", "type": "text" }, { "bbox": [ 371, 534, 428, 547 ], "score": 0.93, "content": "w \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 428, 533, 506, 549 ], "score": 1.0, "content": ". So we can choose", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 545, 505, 559 ], "spans": [ { "bbox": [ 106, 547, 115, 556 ], "score": 0.79, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 115, 545, 505, 559 ], "score": 1.0, "content": "as the set of indices in Proposition 1 with Hamming distance as the semi-metric and the expression", "type": "text" } ], "index": 30 }, { "bbox": [ 107, 556, 344, 570 ], "spans": [ { "bbox": [ 107, 557, 196, 570 ], "score": 0.9, "content": "P _ { w } ( \\mathrm { d i s t } ( \\hat { w } , w ) > d / 8 )", "type": "inline_equation" }, { "bbox": [ 197, 556, 258, 570 ], "score": 1.0, "content": "translates into", "type": "text" }, { "bbox": [ 258, 557, 339, 570 ], "score": 0.91, "content": "P _ { w } ( \\mathcal { E } _ { T } ( h _ { \\hat { w } } ) > c \\cdot \\epsilon )", "type": "inline_equation" }, { "bbox": [ 340, 556, 344, 570 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 106, 573, 504, 596 ], "lines": [ { "bbox": [ 104, 570, 504, 591 ], "spans": [ { "bbox": [ 104, 570, 347, 591 ], "score": 1.0, "content": "KL divergence bound (part (ii) of Proposition 1): Define", "type": "text" }, { "bbox": [ 347, 574, 423, 586 ], "score": 0.92, "content": "P _ { w } = \\mathbb { P } _ { w } ^ { n _ { S } } \\times \\mathbb { Q } _ { w } ^ { n _ { T } }", "type": "inline_equation" }, { "bbox": [ 423, 570, 461, 591 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 461, 574, 504, 586 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\Sigma", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 105, 583, 143, 598 ], "spans": [ { "bbox": [ 105, 583, 143, 598 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "interline_equation", "bbox": [ 127, 597, 483, 713 ], "lines": [ { "bbox": [ 127, 597, 483, 713 ], "spans": [ { "bbox": [ 127, 597, 483, 713 ], "score": 0.95, "content": "\\begin{array} { l } { \\mathcal { D } _ { k l } ( P _ { w } | P _ { w ^ { \\prime } } ) = n _ { S } \\cdot \\mathcal { D } _ { k l } ( \\mathbb { P } _ { w } | \\mathbb { P } _ { w } ^ { \\prime } ) + n _ { T } \\cdot \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { w } | \\mathbb { Q } _ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad = n _ { S } \\cdot \\frac { \\mathbb { E } } { \\mathbb { P } _ { \\alpha } } \\mathcal { D } _ { k l } ( \\mathbb { P } _ { y | \\alpha } ^ { w } | \\mathbb { P } _ { y | \\alpha } ^ { w ^ { \\prime } } ) + n _ { T } \\cdot \\mathbb { E } \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { y | x } ^ { w } | \\mathbb { Q } _ { y | x } ^ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad = n _ { S } \\cdot \\sum _ { i = 1 } ^ { d } \\frac { 1 } { d + n _ { S } \\Delta } \\mathcal { D } _ { k l } ( \\mathbb { P } _ { y | x _ { i } } ^ { w } | \\mathbb { P } _ { y | x _ { i } } ^ { w ^ { \\prime } } ) + n _ { T } \\cdot \\sum _ { i = 1 } ^ { d } \\frac { 1 } { 1 0 0 d } \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { y | x _ { i } } ^ { w } | \\mathbb { Q } _ { y | x _ { i } } ^ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad \\leq n _ { S } \\cdot \\frac { d } { d + n _ { S } \\Delta } c _ { 0 } \\epsilon ^ { 2 } + n _ { T } \\cdot \\frac { 1 } { 1 0 0 } c _ { 0 } \\epsilon ^ { 2 } } \\\\ { \\displaystyle \\quad \\leq c _ { 0 } c _ { 1 } ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "dda90f027f9f50b7da746aaa2a608e769472d1d58d089d2a94eb3f2f3f026f0f.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 127, 597, 483, 635.6666666666666 ], "spans": [], "index": 34 }, { "bbox": [ 127, 635.6666666666666, 483, 674.3333333333333 ], "spans": [], "index": 35 }, { "bbox": [ 127, 674.3333333333333, 483, 712.9999999999999 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 106, 719, 335, 734 ], "lines": [ { "bbox": [ 104, 717, 336, 736 ], "spans": [ { "bbox": [ 104, 717, 115, 736 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 115, 720, 145, 734 ], "score": 0.93, "content": "\\begin{array} { r } { c _ { 1 } < \\frac { 1 } { 6 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 145, 717, 165, 736 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 165, 719, 204, 734 ], "score": 0.94, "content": "c _ { 0 } c _ { 1 } ^ { 2 } < \\frac { 1 } { 8 }", "type": "inline_equation" }, { "bbox": [ 204, 717, 336, 736 ], "score": 1.0, "content": "and we can apply Proposition 1.", "type": "text" } ], "index": 37 } ], "index": 37 } ], "page_idx": 11, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 765 ], "spans": [ { "bbox": [ 299, 750, 312, 765 ], "score": 1.0, "content": "12", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 103, 82, 504, 105 ], "lines": [ { "bbox": [ 105, 81, 505, 97 ], "spans": [ { "bbox": [ 105, 81, 161, 97 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 83, 175, 93 ], "score": 0.84, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 176, 81, 182, 97 ], "score": 1.0, "content": ":", "type": "text" }, { "bbox": [ 182, 83, 196, 93 ], "score": 0.79, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 197, 81, 477, 97 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 477, 83, 505, 94 ], "score": 0.86, "content": "\\mathbb { P } _ { w } =", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 106, 88, 352, 112 ], "spans": [ { "bbox": [ 106, 93, 151, 108 ], "score": 0.92, "content": "\\mathbb { P } _ { x } ^ { w } \\times \\mathbb { P } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 151, 88, 352, 112 ], "score": 1.0, "content": ". We define the marginal distributions as follows:", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 81, 505, 112 ] }, { "type": "interline_equation", "bbox": [ 195, 109, 416, 161 ], "lines": [ { "bbox": [ 195, 109, 416, 161 ], "spans": [ { "bbox": [ 195, 109, 416, 161 ], "score": 0.87, "content": "\\begin{array} { l l l } { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { - 1 } ) = \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 / 2 \\big ( 1 - \\frac { d } { d + n _ { S } \\Delta } \\big ) } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w } ( \\pmb { x } = \\pmb { x } _ { i } ) = \\frac { 1 } { d + n _ { S } \\Delta } \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "d3d0b43c36674461ec8ef939913b82325f74ec1e7a6a9fc1e6e33d321456acd6.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 195, 109, 416, 126.33333333333333 ], "spans": [], "index": 2 }, { "bbox": [ 195, 126.33333333333333, 416, 143.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 195, 143.66666666666666, 416, 161.0 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 106, 161, 239, 172 ], "lines": [ { "bbox": [ 106, 160, 239, 173 ], "spans": [ { "bbox": [ 106, 160, 239, 173 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 5 } ], "index": 5, "bbox_fs": [ 106, 160, 239, 173 ] }, { "type": "interline_equation", "bbox": [ 198, 174, 411, 222 ], "lines": [ { "bbox": [ 198, 174, 411, 222 ], "spans": [ { "bbox": [ 198, 174, 411, 222 ], "score": 0.91, "content": "\\begin{array} { r l } & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - 1 } ) = 0 } \\\\ & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { P } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "4be167508deaf5ee6508ebbd0cf2a08748ac083628dd34d91d1166864799b860.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 198, 174, 411, 190.0 ], "spans": [], "index": 6 }, { "bbox": [ 198, 190.0, 411, 206.0 ], "spans": [], "index": 7 }, { "bbox": [ 198, 206.0, 411, 222.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 105, 223, 465, 236 ], "lines": [ { "bbox": [ 106, 223, 466, 237 ], "spans": [ { "bbox": [ 106, 223, 148, 237 ], "score": 1.0, "content": "Verifying", "type": "text" }, { "bbox": [ 148, 223, 215, 236 ], "score": 0.92, "content": "\\rho ( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } ) \\leq \\Delta", "type": "inline_equation" }, { "bbox": [ 215, 223, 395, 237 ], "score": 1.0, "content": ": Bayes classifier of the domain generated by", "type": "text" }, { "bbox": [ 396, 224, 409, 235 ], "score": 0.89, "content": "\\mathbb { P } _ { w }", "type": "inline_equation" }, { "bbox": [ 410, 223, 466, 237 ], "score": 1.0, "content": "is as follows:", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 106, 223, 466, 237 ] }, { "type": "interline_equation", "bbox": [ 174, 237, 436, 279 ], "lines": [ { "bbox": [ 174, 237, 436, 279 ], "spans": [ { "bbox": [ 174, 237, 436, 279 ], "score": 0.81, "content": "\\begin{array} { r l } & { h _ { S } ^ { * } ( { \\pmb x } _ { - 1 } ) = 0 } \\\\ & { h _ { S } ^ { * } ( { \\pmb x } _ { 0 } ) = 1 } \\\\ & { h _ { S } ^ { * } ( { \\pmb x } _ { i } ) = 1 \\mathrm { i f } w _ { i } = 1 \\mathrm { , , o t h e r w i s e } h _ { S } ^ { * } ( { \\pmb x } _ { i } ) = 0 \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "5aaaa6e33dd5bbc8e456624989a8b5fbadf6ba07307a406c27ffb7a6fc4c15dd.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 174, 237, 436, 251.0 ], "spans": [], "index": 10 }, { "bbox": [ 174, 251.0, 436, 265.0 ], "spans": [], "index": 11 }, { "bbox": [ 174, 265.0, 436, 279.0 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 107, 280, 315, 291 ], "lines": [ { "bbox": [ 106, 279, 316, 293 ], "spans": [ { "bbox": [ 106, 279, 261, 293 ], "score": 1.0, "content": "Similarly for the domain generated by", "type": "text" }, { "bbox": [ 261, 280, 276, 291 ], "score": 0.89, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 276, 279, 316, 293 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 13 } ], "index": 13, "bbox_fs": [ 106, 279, 316, 293 ] }, { "type": "interline_equation", "bbox": [ 174, 293, 437, 321 ], "lines": [ { "bbox": [ 174, 293, 437, 321 ], "spans": [ { "bbox": [ 174, 293, 437, 321 ], "score": 0.81, "content": "\\begin{array} { r l } & { h _ { T } ^ { * } ( { \\pmb x } _ { - 1 } ) = h _ { T } ^ { * } ( { \\pmb x } _ { 0 } ) = 1 } \\\\ & { h _ { T } ^ { * } ( { \\pmb x } _ { i } ) = 1 \\mathrm { i f } w _ { i } = 1 \\mathrm { , o t h e r w i s e } h _ { T } ^ { * } ( { \\pmb x } _ { i } ) = 0 \\mathrm { f o r } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "4fc579e2f99bd0ce951af4d71aa968812fd6c6bca0f05ea836841c704b154552.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 174, 293, 437, 321 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 322, 330, 334 ], "lines": [ { "bbox": [ 105, 321, 330, 336 ], "spans": [ { "bbox": [ 105, 321, 119, 336 ], "score": 1.0, "content": "So", "type": "text" }, { "bbox": [ 119, 322, 132, 334 ], "score": 0.87, "content": "h _ { S } ^ { * }", "type": "inline_equation" }, { "bbox": [ 132, 321, 150, 336 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 151, 323, 164, 334 ], "score": 0.9, "content": "h _ { T } ^ { * }", "type": "inline_equation" }, { "bbox": [ 164, 321, 233, 336 ], "score": 1.0, "content": "disagree only on", "type": "text" }, { "bbox": [ 233, 324, 252, 334 ], "score": 0.87, "content": "{ \\pmb x } _ { - 1 }", "type": "inline_equation" }, { "bbox": [ 252, 321, 330, 336 ], "score": 1.0, "content": "which implies that", "type": "text" } ], "index": 15 } ], "index": 15, "bbox_fs": [ 105, 321, 330, 336 ] }, { "type": "interline_equation", "bbox": [ 188, 335, 423, 349 ], "lines": [ { "bbox": [ 188, 335, 423, 349 ], "spans": [ { "bbox": [ 188, 335, 423, 349 ], "score": 0.85, "content": "\\rho ( \\mathbb { P } _ { w } , \\mathbb { Q } _ { w } ) = \\mathbb { Q } [ h _ { S } ^ { \\ast } ( { \\pmb x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( { \\pmb x } _ { T } ) \\neq y _ { T } ] = \\Delta", "type": "interline_equation", "image_path": "eb6e1bf0578ba67796515fc16f4e0a5ea117df9513a8d38a08383fcebc1a6084.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 188, 335, 423, 349 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 356, 506, 415 ], "lines": [ { "bbox": [ 106, 356, 505, 368 ], "spans": [ { "bbox": [ 106, 356, 505, 368 ], "score": 1.0, "content": "Since we want to derive a lower bound for the minimax risk stated in Theorem 1, among the", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 367, 506, 379 ], "spans": [ { "bbox": [ 106, 367, 234, 379 ], "score": 1.0, "content": "hypotheses that they agree on", "type": "text" }, { "bbox": [ 234, 369, 245, 379 ], "score": 0.86, "content": "\\mathbf { \\boldsymbol { x } } _ { i }", "type": "inline_equation" }, { "bbox": [ 245, 367, 263, 379 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 263, 367, 312, 379 ], "score": 0.92, "content": "i = 1 , . . . , d", "type": "inline_equation" }, { "bbox": [ 312, 367, 432, 379 ], "score": 1.0, "content": ", the hypothesis that outputs", "type": "text" }, { "bbox": [ 432, 369, 451, 379 ], "score": 0.89, "content": "{ \\pmb x } _ { - 1 }", "type": "inline_equation" }, { "bbox": [ 451, 367, 471, 379 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 471, 369, 483, 379 ], "score": 0.86, "content": "\\scriptstyle { \\pmb x } _ { 0 }", "type": "inline_equation" }, { "bbox": [ 484, 367, 506, 379 ], "score": 1.0, "content": "as 1", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 378, 506, 392 ], "spans": [ { "bbox": [ 105, 380, 385, 392 ], "score": 1.0, "content": "results in a smaller target error. Hence, we can restrict ourselves to", "type": "text" }, { "bbox": [ 386, 378, 396, 390 ], "score": 0.86, "content": "\\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 396, 380, 506, 392 ], "score": 1.0, "content": "which is the projection of", "type": "text" } ], "index": 19 }, { "bbox": [ 107, 390, 505, 405 ], "spans": [ { "bbox": [ 107, 392, 116, 402 ], "score": 0.81, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 117, 390, 138, 405 ], "score": 1.0, "content": "onto", "type": "text" }, { "bbox": [ 139, 391, 176, 403 ], "score": 0.91, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 176, 390, 273, 405 ], "score": 1.0, "content": "with the constraint that", "type": "text" }, { "bbox": [ 274, 392, 362, 404 ], "score": 0.94, "content": "h ( \\pmb { x } _ { - 1 } ) = h ( \\pmb { x } _ { 0 } ) = 1", "type": "inline_equation" }, { "bbox": [ 362, 390, 390, 405 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 390, 390, 418, 402 ], "score": 0.91, "content": "h \\in \\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 419, 390, 505, 405 ], "score": 1.0, "content": ". Furthermor, for any", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 402, 216, 415 ], "spans": [ { "bbox": [ 107, 403, 178, 415 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 179, 402, 216, 415 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 21 } ], "index": 19, "bbox_fs": [ 105, 356, 506, 415 ] }, { "type": "interline_equation", "bbox": [ 226, 416, 385, 441 ], "lines": [ { "bbox": [ 226, 416, 385, 441 ], "spans": [ { "bbox": [ 226, 416, 385, 441 ], "score": 0.94, "content": "\\mathcal { E } _ { T } ( h _ { w ^ { \\prime } } ) = \\frac { \\mathrm { d i s t } ( w , w ^ { \\prime } ) } { 1 0 0 d } \\cdot \\epsilon , ~ \\forall ~ h _ { w ^ { \\prime } } \\in \\tilde { \\mathcal { H } }", "type": "interline_equation", "image_path": "df150745bf6e2f61646ee55386173930870964749e9bc2236e0c4a5b24aeaace.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 226, 416, 385, 441 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 442, 280, 454 ], "lines": [ { "bbox": [ 105, 439, 280, 457 ], "spans": [ { "bbox": [ 105, 439, 265, 457 ], "score": 1.0, "content": "when the target domain is generated by", "type": "text" }, { "bbox": [ 265, 443, 280, 453 ], "score": 0.88, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 105, 439, 280, 457 ] }, { "type": "text", "bbox": [ 106, 459, 505, 493 ], "lines": [ { "bbox": [ 105, 458, 504, 472 ], "spans": [ { "bbox": [ 105, 458, 376, 472 ], "score": 1.0, "content": "Reduction to a packing: By using Proposition 2, we can get a subset", "type": "text" }, { "bbox": [ 376, 460, 385, 469 ], "score": 0.83, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 385, 458, 396, 472 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 396, 458, 433, 471 ], "score": 0.93, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 434, 458, 504, 472 ], "score": 1.0, "content": "whose cardinality", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 468, 506, 486 ], "spans": [ { "bbox": [ 104, 468, 116, 486 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 117, 470, 160, 482 ], "score": 0.93, "content": "M \\geq 2 ^ { d / 8 }", "type": "inline_equation" }, { "bbox": [ 160, 468, 211, 486 ], "score": 1.0, "content": "and for any", "type": "text" }, { "bbox": [ 212, 471, 235, 483 ], "score": 0.91, "content": "w , w ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 235, 468, 290, 486 ], "score": 1.0, "content": "belonging to", "type": "text" }, { "bbox": [ 291, 471, 299, 481 ], "score": 0.81, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 299, 468, 340, 486 ], "score": 1.0, "content": "we have", "type": "text" }, { "bbox": [ 340, 471, 412, 483 ], "score": 0.91, "content": "\\operatorname* { l i s t } ( w , w ^ { \\prime } ) \\geq d / 8", "type": "inline_equation" }, { "bbox": [ 412, 468, 506, 486 ], "score": 1.0, "content": ". Furthermore, for any", "type": "text" } ], "index": 25 }, { "bbox": [ 107, 482, 186, 494 ], "spans": [ { "bbox": [ 107, 483, 149, 493 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\Sigma", "type": "inline_equation" }, { "bbox": [ 149, 482, 186, 494 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 104, 458, 506, 494 ] }, { "type": "interline_equation", "bbox": [ 246, 495, 365, 519 ], "lines": [ { "bbox": [ 246, 495, 365, 519 ], "spans": [ { "bbox": [ 246, 495, 365, 519 ], "score": 0.95, "content": "\\mathcal { E } _ { T } ( h _ { w ^ { \\prime } } ) \\geq \\frac { d } { 8 } \\cdot \\frac { \\epsilon } { 1 0 0 d } = \\frac { \\epsilon } { 8 0 0 }", "type": "interline_equation", "image_path": "d806dc2a8e1a45316cfe4e4686cf3c1524d0e441e81b314604beb1822dc609f3.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 246, 495, 365, 519 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 522, 506, 569 ], "lines": [ { "bbox": [ 106, 521, 506, 535 ], "spans": [ { "bbox": [ 106, 522, 316, 535 ], "score": 1.0, "content": "On the other hand, there is a bijective map between", "type": "text" }, { "bbox": [ 316, 522, 354, 534 ], "score": 0.94, "content": "\\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 354, 522, 421, 535 ], "score": 1.0, "content": "and elements of", "type": "text" }, { "bbox": [ 422, 521, 432, 533 ], "score": 0.87, "content": "\\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 432, 522, 506, 535 ], "score": 1.0, "content": "and any classifier", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 533, 506, 549 ], "spans": [ { "bbox": [ 107, 534, 181, 547 ], "score": 0.92, "content": "\\hat { h } : \\{ { \\pmb x } _ { i } \\} \\{ 0 , 1 \\}", "type": "inline_equation" }, { "bbox": [ 182, 533, 203, 549 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 203, 533, 290, 547 ], "score": 0.94, "content": "\\hat { h } ( { \\pmb x } _ { - 1 } ) = \\hat { h } ( { \\pmb x } _ { 0 } ) = 1", "type": "inline_equation" }, { "bbox": [ 291, 533, 371, 549 ], "score": 1.0, "content": "can be reduced to a", "type": "text" }, { "bbox": [ 371, 534, 428, 547 ], "score": 0.93, "content": "w \\in \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 428, 533, 506, 549 ], "score": 1.0, "content": ". So we can choose", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 545, 505, 559 ], "spans": [ { "bbox": [ 106, 547, 115, 556 ], "score": 0.79, "content": "\\Sigma", "type": "inline_equation" }, { "bbox": [ 115, 545, 505, 559 ], "score": 1.0, "content": "as the set of indices in Proposition 1 with Hamming distance as the semi-metric and the expression", "type": "text" } ], "index": 30 }, { "bbox": [ 107, 556, 344, 570 ], "spans": [ { "bbox": [ 107, 557, 196, 570 ], "score": 0.9, "content": "P _ { w } ( \\mathrm { d i s t } ( \\hat { w } , w ) > d / 8 )", "type": "inline_equation" }, { "bbox": [ 197, 556, 258, 570 ], "score": 1.0, "content": "translates into", "type": "text" }, { "bbox": [ 258, 557, 339, 570 ], "score": 0.91, "content": "P _ { w } ( \\mathcal { E } _ { T } ( h _ { \\hat { w } } ) > c \\cdot \\epsilon )", "type": "inline_equation" }, { "bbox": [ 340, 556, 344, 570 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 106, 521, 506, 570 ] }, { "type": "text", "bbox": [ 106, 573, 504, 596 ], "lines": [ { "bbox": [ 104, 570, 504, 591 ], "spans": [ { "bbox": [ 104, 570, 347, 591 ], "score": 1.0, "content": "KL divergence bound (part (ii) of Proposition 1): Define", "type": "text" }, { "bbox": [ 347, 574, 423, 586 ], "score": 0.92, "content": "P _ { w } = \\mathbb { P } _ { w } ^ { n _ { S } } \\times \\mathbb { Q } _ { w } ^ { n _ { T } }", "type": "inline_equation" }, { "bbox": [ 423, 570, 461, 591 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 461, 574, 504, 586 ], "score": 0.91, "content": "w , w ^ { \\prime } \\in \\Sigma", "type": "inline_equation" } ], "index": 32 }, { "bbox": [ 105, 583, 143, 598 ], "spans": [ { "bbox": [ 105, 583, 143, 598 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 104, 570, 504, 598 ] }, { "type": "interline_equation", "bbox": [ 127, 597, 483, 713 ], "lines": [ { "bbox": [ 127, 597, 483, 713 ], "spans": [ { "bbox": [ 127, 597, 483, 713 ], "score": 0.95, "content": "\\begin{array} { l } { \\mathcal { D } _ { k l } ( P _ { w } | P _ { w ^ { \\prime } } ) = n _ { S } \\cdot \\mathcal { D } _ { k l } ( \\mathbb { P } _ { w } | \\mathbb { P } _ { w } ^ { \\prime } ) + n _ { T } \\cdot \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { w } | \\mathbb { Q } _ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad = n _ { S } \\cdot \\frac { \\mathbb { E } } { \\mathbb { P } _ { \\alpha } } \\mathcal { D } _ { k l } ( \\mathbb { P } _ { y | \\alpha } ^ { w } | \\mathbb { P } _ { y | \\alpha } ^ { w ^ { \\prime } } ) + n _ { T } \\cdot \\mathbb { E } \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { y | x } ^ { w } | \\mathbb { Q } _ { y | x } ^ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad = n _ { S } \\cdot \\sum _ { i = 1 } ^ { d } \\frac { 1 } { d + n _ { S } \\Delta } \\mathcal { D } _ { k l } ( \\mathbb { P } _ { y | x _ { i } } ^ { w } | \\mathbb { P } _ { y | x _ { i } } ^ { w ^ { \\prime } } ) + n _ { T } \\cdot \\sum _ { i = 1 } ^ { d } \\frac { 1 } { 1 0 0 d } \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { y | x _ { i } } ^ { w } | \\mathbb { Q } _ { y | x _ { i } } ^ { w ^ { \\prime } } ) } \\\\ { \\displaystyle \\quad \\leq n _ { S } \\cdot \\frac { d } { d + n _ { S } \\Delta } c _ { 0 } \\epsilon ^ { 2 } + n _ { T } \\cdot \\frac { 1 } { 1 0 0 } c _ { 0 } \\epsilon ^ { 2 } } \\\\ { \\displaystyle \\quad \\leq c _ { 0 } c _ { 1 } ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "dda90f027f9f50b7da746aaa2a608e769472d1d58d089d2a94eb3f2f3f026f0f.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 127, 597, 483, 635.6666666666666 ], "spans": [], "index": 34 }, { "bbox": [ 127, 635.6666666666666, 483, 674.3333333333333 ], "spans": [], "index": 35 }, { "bbox": [ 127, 674.3333333333333, 483, 712.9999999999999 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 106, 719, 335, 734 ], "lines": [ { "bbox": [ 104, 717, 336, 736 ], "spans": [ { "bbox": [ 104, 717, 115, 736 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 115, 720, 145, 734 ], "score": 0.93, "content": "\\begin{array} { r } { c _ { 1 } < \\frac { 1 } { 6 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 145, 717, 165, 736 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 165, 719, 204, 734 ], "score": 0.94, "content": "c _ { 0 } c _ { 1 } ^ { 2 } < \\frac { 1 } { 8 }", "type": "inline_equation" }, { "bbox": [ 204, 717, 336, 736 ], "score": 1.0, "content": "and we can apply Proposition 1.", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 104, 717, 336, 736 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 102, 505, 126 ], "lines": [ { "bbox": [ 105, 102, 505, 116 ], "spans": [ { "bbox": [ 105, 102, 505, 116 ], "score": 1.0, "content": "Proof of Theorem 2 is similar to that of Theorem 1. However, we construct different target and source", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 114, 208, 127 ], "spans": [ { "bbox": [ 105, 114, 208, 127 ], "score": 1.0, "content": "probability distributions.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 106, 131, 506, 179 ], "lines": [ { "bbox": [ 105, 130, 506, 145 ], "spans": [ { "bbox": [ 105, 130, 125, 145 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 125, 131, 211, 142 ], "score": 0.91, "content": "d \\ = \\ d _ { \\mathcal { H } } \\ - \\ N - \\ 1", "type": "inline_equation" }, { "bbox": [ 212, 130, 256, 145 ], "score": 1.0, "content": "and pick", "type": "text" }, { "bbox": [ 257, 133, 349, 142 ], "score": 0.88, "content": "x _ { - M } , . . . , x _ { 0 } , x _ { 1 } , . . . , x _ { d }", "type": "inline_equation" }, { "bbox": [ 349, 130, 376, 145 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 376, 133, 384, 143 ], "score": 0.82, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 385, 130, 443, 145 ], "score": 1.0, "content": "shattered by", "type": "text" }, { "bbox": [ 444, 132, 453, 141 ], "score": 0.79, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 454, 130, 506, 145 ], "score": 1.0, "content": ". Then we", "type": "text" } ], "index": 2 }, { "bbox": [ 101, 139, 505, 163 ], "spans": [ { "bbox": [ 101, 139, 256, 163 ], "score": 1.0, "content": "construct a family of distributions", "type": "text" }, { "bbox": [ 257, 142, 337, 156 ], "score": 0.92, "content": "( \\mathbb { P } _ { w } ^ { ( 1 ) } , . . . , \\mathbb { P } _ { w } ^ { ( N ) } , \\mathbb { Q } _ { w } )", "type": "inline_equation" }, { "bbox": [ 338, 139, 391, 163 ], "score": 1.0, "content": "indexed by", "type": "text" }, { "bbox": [ 392, 144, 456, 157 ], "score": 0.93, "content": "w ~ \\in ~ \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 456, 139, 484, 163 ], "score": 1.0, "content": ". Let", "type": "text" }, { "bbox": [ 484, 146, 505, 156 ], "score": 0.83, "content": "\\epsilon =", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 107, 156, 506, 168 ], "spans": [ { "bbox": [ 107, 156, 270, 168 ], "score": 0.91, "content": "c _ { 1 } \\cdot \\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } )", "type": "inline_equation" }, { "bbox": [ 270, 156, 346, 168 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 347, 156, 376, 167 ], "score": 0.89, "content": "c _ { 1 } < 1", "type": "inline_equation" }, { "bbox": [ 376, 156, 506, 168 ], "score": 1.0, "content": "to be determined later in proof.", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 165, 446, 180 ], "spans": [ { "bbox": [ 104, 165, 315, 180 ], "score": 1.0, "content": "Furthermore, without loss of generality assume that", "type": "text" }, { "bbox": [ 315, 167, 442, 178 ], "score": 0.9, "content": "1 \\ge \\Delta _ { 1 } \\ge \\Delta _ { 2 } \\ge . . . \\ge \\Delta _ { N } \\ge 0", "type": "inline_equation" }, { "bbox": [ 443, 165, 446, 180 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 5 } ], "index": 3.5 }, { "type": "text", "bbox": [ 105, 183, 505, 207 ], "lines": [ { "bbox": [ 105, 182, 505, 198 ], "spans": [ { "bbox": [ 105, 182, 161, 198 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 184, 199, 195 ], "score": 0.42, "content": "\\mathbb { Q } _ { w } \\colon \\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 199, 182, 476, 198 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 476, 184, 505, 196 ], "score": 0.86, "content": "\\mathbb { Q } _ { w } =", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 106, 192, 353, 211 ], "spans": [ { "bbox": [ 106, 195, 154, 209 ], "score": 0.93, "content": "\\mathbb { Q } _ { x } ^ { w } \\times \\mathbb { Q } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 154, 192, 353, 211 ], "score": 1.0, "content": ". We define the marginal distributions as follows:", "type": "text" } ], "index": 7 } ], "index": 6.5 }, { "type": "interline_equation", "bbox": [ 145, 213, 466, 269 ], "lines": [ { "bbox": [ 145, 213, 466, 269 ], "spans": [ { "bbox": [ 145, 213, 466, 269 ], "score": 0.85, "content": "\\begin{array} { l } { { \\mathbb Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - i } ) = \\Delta _ { i } - \\Delta _ { i + 1 } \\mathrm { ~ f o r ~ } i = 1 , . . . , N - 1 \\mathrm { ~ a n d ~ } \\mathbb Q _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - N } ) = \\Delta _ { N } } \\\\ { { \\mathbb Q } _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { 0 } ) = 0 . 9 9 - \\Delta _ { 1 } } \\\\ { { \\mathbb Q } _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { i } ) = \\displaystyle \\frac 1 { 1 0 0 d } \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "e9e5b8904808a8fed2ebb930113e1db6e92d7d487cfd8fdc010449713b33850e.jpg" } ] } ], "index": 9, "virtual_lines": [ { "bbox": [ 145, 213, 466, 231.66666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 145, 231.66666666666666, 466, 250.33333333333331 ], "spans": [], "index": 9 }, { "bbox": [ 145, 250.33333333333331, 466, 269.0 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 106, 272, 239, 284 ], "lines": [ { "bbox": [ 106, 271, 239, 285 ], "spans": [ { "bbox": [ 106, 271, 239, 285 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 197, 288, 414, 338 ], "lines": [ { "bbox": [ 197, 288, 414, 338 ], "spans": [ { "bbox": [ 197, 288, 414, 338 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - i } ) = 1 \\mathrm { f o r } i = 1 , . . . , N } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { f o r } i = 1 , . . . , d } \\end{array}", "type": "interline_equation", "image_path": "1854d41c2c0260a4149d15784617ba953546d885b71c6fb61a5ec23085fd47af.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 197, 288, 414, 304.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 197, 304.6666666666667, 414, 321.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 197, 321.33333333333337, 414, 338.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 105, 344, 504, 371 ], "lines": [ { "bbox": [ 102, 338, 505, 365 ], "spans": [ { "bbox": [ 102, 338, 161, 365 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 343, 201, 357 ], "score": 0.34, "content": "\\mathbb { P } _ { w } ^ { ( i ) } \\colon \\mathbb { P } _ { w } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 201, 338, 475, 365 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 475, 343, 505, 358 ], "score": 0.89, "content": "\\mathbb { P } _ { w } ^ { ( i ) } =", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 106, 355, 369, 374 ], "spans": [ { "bbox": [ 106, 357, 170, 374 ], "score": 0.92, "content": "\\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } \\times \\mathbb { P } _ { \\pmb { y } | \\pmb { x } } ^ { w ( i ) }", "type": "inline_equation" }, { "bbox": [ 155, 355, 369, 374 ], "score": 1.0, "content": "x(i). We define the marginal distributions as follows:", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "interline_equation", "bbox": [ 177, 378, 434, 460 ], "lines": [ { "bbox": [ 177, 378, 434, 460 ], "spans": [ { "bbox": [ 177, 378, 434, 460 ], "score": 0.94, "content": "\\begin{array} { l l } { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { - j } ) = \\frac { 1 } { N + 1 } \\big ( 1 - \\frac { d } { d + n _ { S _ { i } } \\Delta _ { i } } \\big ) \\mathrm { ~ f o r ~ } j = 1 , . . . , N } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { 0 } ) = \\frac { 1 } { N + 1 } \\big ( 1 - \\frac { d } { d + n _ { S _ { i } } \\Delta _ { i } } \\big ) } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { j } ) = \\frac { 1 } { d + n _ { S _ { i } } \\Delta _ { i } } \\mathrm { ~ f o r ~ } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "493b53dd2330737e9c4713e5429764f738cfac2afb9fbe73404e0ffd63720a8a.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 177, 378, 434, 405.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 177, 405.3333333333333, 434, 432.66666666666663 ], "spans": [], "index": 18 }, { "bbox": [ 177, 432.66666666666663, 434, 459.99999999999994 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 464, 239, 475 ], "lines": [ { "bbox": [ 106, 464, 239, 476 ], "spans": [ { "bbox": [ 106, 464, 239, 476 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 115, 480, 478, 529 ], "lines": [ { "bbox": [ 115, 480, 478, 529 ], "spans": [ { "bbox": [ 115, 480, 478, 529 ], "score": 0.73, "content": "\\begin{array} { r l } & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { - j } ) = 0 \\mathrm { i f } j \\geq i , \\mathrm { o t h e r w i s e } \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . } \\\\ & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { 0 } ) = 1 } \\\\ & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { j } ) = 1 / 2 + ( w _ { j } ) \\epsilon \\mathrm { f o r } j = 1 , . . . , d } \\end{array}", "type": "interline_equation", "image_path": "be7ca788c3d35cefcb8072d3c144774d69b545dd45d64dcfad3eece9fddf6940.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 115, 480, 478, 496.3333333333333 ], "spans": [], "index": 21 }, { "bbox": [ 115, 496.3333333333333, 478, 512.6666666666666 ], "spans": [], "index": 22 }, { "bbox": [ 115, 512.6666666666666, 478, 529.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 542, 224, 556 ], "lines": [ { "bbox": [ 105, 540, 222, 559 ], "spans": [ { "bbox": [ 105, 540, 148, 559 ], "score": 1.0, "content": "Verifying", "type": "text" }, { "bbox": [ 148, 541, 222, 556 ], "score": 0.91, "content": "\\rho ( \\mathbb { P } _ { w } ^ { ( i ) } , \\mathbb { Q } _ { w } ) \\leq \\Delta _ { i }", "type": "inline_equation" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 106, 562, 355, 576 ], "lines": [ { "bbox": [ 104, 559, 358, 579 ], "spans": [ { "bbox": [ 104, 559, 282, 579 ], "score": 1.0, "content": "Bayes classifier of the domain generated by", "type": "text" }, { "bbox": [ 283, 561, 299, 574 ], "score": 0.92, "content": "\\mathbb { P } _ { w } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 299, 559, 358, 579 ], "score": 1.0, "content": "is as follows:", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 166, 580, 442, 627 ], "lines": [ { "bbox": [ 166, 580, 442, 627 ], "spans": [ { "bbox": [ 166, 580, 442, 627 ], "score": 0.83, "content": "\\begin{array} { r l } & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { - j } ) = 0 \\mathrm { i f } j \\geq i , \\mathrm { o t h e r w i s e } h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . , N } \\\\ & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { 0 } ) = 1 } \\\\ & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { j } ) = 1 \\mathrm { i f } w _ { j } = 1 , \\mathrm { o t h e r w i s e } h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { j } ) = 0 \\mathrm { f o r } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "7c117f4ca1d1aa05f502e08c66a6e3c6901262d5c93c8754db2e5bd5f5257d5f.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 166, 580, 442, 595.6666666666666 ], "spans": [], "index": 26 }, { "bbox": [ 166, 595.6666666666666, 442, 611.3333333333333 ], "spans": [], "index": 27 }, { "bbox": [ 166, 611.3333333333333, 442, 626.9999999999999 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 107, 630, 315, 643 ], "lines": [ { "bbox": [ 106, 630, 316, 644 ], "spans": [ { "bbox": [ 106, 630, 261, 644 ], "score": 1.0, "content": "Similarly for the domain generated by", "type": "text" }, { "bbox": [ 261, 631, 276, 642 ], "score": 0.9, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 276, 630, 316, 644 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "interline_equation", "bbox": [ 172, 647, 438, 691 ], "lines": [ { "bbox": [ 172, 647, 438, 691 ], "spans": [ { "bbox": [ 172, 647, 438, 691 ], "score": 0.84, "content": "\\begin{array} { r l } & { h _ { T } ^ { * } ( \\pmb { x } _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . , N } \\\\ & { h _ { T } ^ { * } ( \\pmb { x } _ { 0 } ) = 1 } \\\\ & { h _ { T } ^ { * } ( \\pmb { x } _ { j } ) = 1 \\mathrm { i f } w _ { j } = 1 , \\mathrm { o t h e r w i s e } h _ { T } ^ { * } ( \\pmb { x } _ { j } ) = 0 \\mathrm { f o r } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "1bdfa6b731e98014c985ce7720c50207b19c2b6d8941e6d5712b7a8a6dd524d5.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 172, 647, 438, 661.6666666666666 ], "spans": [], "index": 30 }, { "bbox": [ 172, 661.6666666666666, 438, 676.3333333333333 ], "spans": [], "index": 31 }, { "bbox": [ 172, 676.3333333333333, 438, 690.9999999999999 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 106, 695, 345, 708 ], "lines": [ { "bbox": [ 105, 694, 347, 711 ], "spans": [ { "bbox": [ 105, 694, 119, 711 ], "score": 1.0, "content": "So", "type": "text" }, { "bbox": [ 119, 696, 135, 708 ], "score": 0.89, "content": "h _ { S _ { i } } ^ { * }", "type": "inline_equation" }, { "bbox": [ 135, 694, 153, 711 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 153, 696, 166, 708 ], "score": 0.9, "content": "h _ { T } ^ { * }", "type": "inline_equation" }, { "bbox": [ 167, 694, 216, 711 ], "score": 1.0, "content": "disagree on", "type": "text" }, { "bbox": [ 216, 698, 268, 707 ], "score": 0.91, "content": "\\pmb { x } _ { - i } , . . , \\pmb { x } _ { - N }", "type": "inline_equation" }, { "bbox": [ 268, 694, 347, 711 ], "score": 1.0, "content": "which implies that", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "interline_equation", "bbox": [ 184, 714, 426, 730 ], "lines": [ { "bbox": [ 184, 714, 426, 730 ], "spans": [ { "bbox": [ 184, 714, 426, 730 ], "score": 0.88, "content": "\\rho ( \\mathbb { P } _ { w } ^ { ( i ) } , \\mathbb { Q } _ { w } ) = \\mathbb { Q } [ h _ { S _ { i } } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] = \\Delta _ { i }", "type": "interline_equation", "image_path": "1fa467d600b617b573ad168363ac52e464d481ff42e1afb14c3bcad5da0931b2.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 184, 714, 426, 730 ], "spans": [], "index": 34 } ] } ], "page_idx": 12, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2022", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "13", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 107, 82, 228, 93 ], "lines": [ { "bbox": [ 106, 81, 230, 95 ], "spans": [ { "bbox": [ 106, 81, 230, 95 ], "score": 1.0, "content": "7.2 PROOF OF THEOREM 2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 102, 505, 126 ], "lines": [ { "bbox": [ 105, 102, 505, 116 ], "spans": [ { "bbox": [ 105, 102, 505, 116 ], "score": 1.0, "content": "Proof of Theorem 2 is similar to that of Theorem 1. However, we construct different target and source", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 114, 208, 127 ], "spans": [ { "bbox": [ 105, 114, 208, 127 ], "score": 1.0, "content": "probability distributions.", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 102, 505, 127 ] }, { "type": "text", "bbox": [ 106, 131, 506, 179 ], "lines": [ { "bbox": [ 105, 130, 506, 145 ], "spans": [ { "bbox": [ 105, 130, 125, 145 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 125, 131, 211, 142 ], "score": 0.91, "content": "d \\ = \\ d _ { \\mathcal { H } } \\ - \\ N - \\ 1", "type": "inline_equation" }, { "bbox": [ 212, 130, 256, 145 ], "score": 1.0, "content": "and pick", "type": "text" }, { "bbox": [ 257, 133, 349, 142 ], "score": 0.88, "content": "x _ { - M } , . . . , x _ { 0 } , x _ { 1 } , . . . , x _ { d }", "type": "inline_equation" }, { "bbox": [ 349, 130, 376, 145 ], "score": 1.0, "content": "from", "type": "text" }, { "bbox": [ 376, 133, 384, 143 ], "score": 0.82, "content": "\\chi", "type": "inline_equation" }, { "bbox": [ 385, 130, 443, 145 ], "score": 1.0, "content": "shattered by", "type": "text" }, { "bbox": [ 444, 132, 453, 141 ], "score": 0.79, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 454, 130, 506, 145 ], "score": 1.0, "content": ". Then we", "type": "text" } ], "index": 2 }, { "bbox": [ 101, 139, 505, 163 ], "spans": [ { "bbox": [ 101, 139, 256, 163 ], "score": 1.0, "content": "construct a family of distributions", "type": "text" }, { "bbox": [ 257, 142, 337, 156 ], "score": 0.92, "content": "( \\mathbb { P } _ { w } ^ { ( 1 ) } , . . . , \\mathbb { P } _ { w } ^ { ( N ) } , \\mathbb { Q } _ { w } )", "type": "inline_equation" }, { "bbox": [ 338, 139, 391, 163 ], "score": 1.0, "content": "indexed by", "type": "text" }, { "bbox": [ 392, 144, 456, 157 ], "score": 0.93, "content": "w ~ \\in ~ \\{ - 1 , 1 \\} ^ { d }", "type": "inline_equation" }, { "bbox": [ 456, 139, 484, 163 ], "score": 1.0, "content": ". Let", "type": "text" }, { "bbox": [ 484, 146, 505, 156 ], "score": 0.83, "content": "\\epsilon =", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 107, 156, 506, 168 ], "spans": [ { "bbox": [ 107, 156, 270, 168 ], "score": 0.91, "content": "c _ { 1 } \\cdot \\epsilon ( n _ { S _ { 1 } } , . . . , n _ { S _ { N } } , n _ { T } , d _ { \\mathcal { H } } , \\Delta _ { 1 } , . . . , \\Delta _ { N } )", "type": "inline_equation" }, { "bbox": [ 270, 156, 346, 168 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 347, 156, 376, 167 ], "score": 0.89, "content": "c _ { 1 } < 1", "type": "inline_equation" }, { "bbox": [ 376, 156, 506, 168 ], "score": 1.0, "content": "to be determined later in proof.", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 165, 446, 180 ], "spans": [ { "bbox": [ 104, 165, 315, 180 ], "score": 1.0, "content": "Furthermore, without loss of generality assume that", "type": "text" }, { "bbox": [ 315, 167, 442, 178 ], "score": 0.9, "content": "1 \\ge \\Delta _ { 1 } \\ge \\Delta _ { 2 } \\ge . . . \\ge \\Delta _ { N } \\ge 0", "type": "inline_equation" }, { "bbox": [ 443, 165, 446, 180 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 101, 130, 506, 180 ] }, { "type": "text", "bbox": [ 105, 183, 505, 207 ], "lines": [ { "bbox": [ 105, 182, 505, 198 ], "spans": [ { "bbox": [ 105, 182, 161, 198 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 184, 199, 195 ], "score": 0.42, "content": "\\mathbb { Q } _ { w } \\colon \\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 199, 182, 476, 198 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 476, 184, 505, 196 ], "score": 0.86, "content": "\\mathbb { Q } _ { w } =", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 106, 192, 353, 211 ], "spans": [ { "bbox": [ 106, 195, 154, 209 ], "score": 0.93, "content": "\\mathbb { Q } _ { x } ^ { w } \\times \\mathbb { Q } _ { y | x } ^ { w }", "type": "inline_equation" }, { "bbox": [ 154, 192, 353, 211 ], "score": 1.0, "content": ". We define the marginal distributions as follows:", "type": "text" } ], "index": 7 } ], "index": 6.5, "bbox_fs": [ 105, 182, 505, 211 ] }, { "type": "interline_equation", "bbox": [ 145, 213, 466, 269 ], "lines": [ { "bbox": [ 145, 213, 466, 269 ], "spans": [ { "bbox": [ 145, 213, 466, 269 ], "score": 0.85, "content": "\\begin{array} { l } { { \\mathbb Q } _ { x } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - i } ) = \\Delta _ { i } - \\Delta _ { i + 1 } \\mathrm { ~ f o r ~ } i = 1 , . . . , N - 1 \\mathrm { ~ a n d ~ } \\mathbb Q _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { - N } ) = \\Delta _ { N } } \\\\ { { \\mathbb Q } _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { 0 } ) = 0 . 9 9 - \\Delta _ { 1 } } \\\\ { { \\mathbb Q } _ { { \\pmb x } } ^ { w } ( { \\pmb x } = { \\pmb x } _ { i } ) = \\displaystyle \\frac 1 { 1 0 0 d } \\mathrm { ~ f o r ~ } i = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "e9e5b8904808a8fed2ebb930113e1db6e92d7d487cfd8fdc010449713b33850e.jpg" } ] } ], "index": 9, "virtual_lines": [ { "bbox": [ 145, 213, 466, 231.66666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 145, 231.66666666666666, 466, 250.33333333333331 ], "spans": [], "index": 9 }, { "bbox": [ 145, 250.33333333333331, 466, 269.0 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 106, 272, 239, 284 ], "lines": [ { "bbox": [ 106, 271, 239, 285 ], "spans": [ { "bbox": [ 106, 271, 239, 285 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 11 } ], "index": 11, "bbox_fs": [ 106, 271, 239, 285 ] }, { "type": "interline_equation", "bbox": [ 197, 288, 414, 338 ], "lines": [ { "bbox": [ 197, 288, 414, 338 ], "spans": [ { "bbox": [ 197, 288, 414, 338 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { - i } ) = 1 \\mathrm { f o r } i = 1 , . . . , N } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { 0 } ) = 1 } \\\\ & { \\mathbb { Q } _ { y | \\pmb { x } } ^ { w } ( y = 1 | \\pmb { x } = \\pmb { x } _ { i } ) = 1 / 2 + ( w _ { i } ) \\epsilon \\mathrm { f o r } i = 1 , . . . , d } \\end{array}", "type": "interline_equation", "image_path": "1854d41c2c0260a4149d15784617ba953546d885b71c6fb61a5ec23085fd47af.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 197, 288, 414, 304.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 197, 304.6666666666667, 414, 321.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 197, 321.33333333333337, 414, 338.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 105, 344, 504, 371 ], "lines": [ { "bbox": [ 102, 338, 505, 365 ], "spans": [ { "bbox": [ 102, 338, 161, 365 ], "score": 1.0, "content": "Distribution", "type": "text" }, { "bbox": [ 162, 343, 201, 357 ], "score": 0.34, "content": "\\mathbb { P } _ { w } ^ { ( i ) } \\colon \\mathbb { P } _ { w } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 201, 338, 475, 365 ], "score": 1.0, "content": "is composed of a marginal and a conditional distribution, namely", "type": "text" }, { "bbox": [ 475, 343, 505, 358 ], "score": 0.89, "content": "\\mathbb { P } _ { w } ^ { ( i ) } =", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 106, 355, 369, 374 ], "spans": [ { "bbox": [ 106, 357, 170, 374 ], "score": 0.92, "content": "\\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } \\times \\mathbb { P } _ { \\pmb { y } | \\pmb { x } } ^ { w ( i ) }", "type": "inline_equation" }, { "bbox": [ 155, 355, 369, 374 ], "score": 1.0, "content": "x(i). We define the marginal distributions as follows:", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 102, 338, 505, 374 ] }, { "type": "interline_equation", "bbox": [ 177, 378, 434, 460 ], "lines": [ { "bbox": [ 177, 378, 434, 460 ], "spans": [ { "bbox": [ 177, 378, 434, 460 ], "score": 0.94, "content": "\\begin{array} { l l } { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { - j } ) = \\frac { 1 } { N + 1 } \\big ( 1 - \\frac { d } { d + n _ { S _ { i } } \\Delta _ { i } } \\big ) \\mathrm { ~ f o r ~ } j = 1 , . . . , N } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { 0 } ) = \\frac { 1 } { N + 1 } \\big ( 1 - \\frac { d } { d + n _ { S _ { i } } \\Delta _ { i } } \\big ) } \\\\ { \\displaystyle \\mathbb { P } _ { \\pmb { x } } ^ { w ( i ) } ( \\pmb { x } = \\pmb { x } _ { j } ) = \\frac { 1 } { d + n _ { S _ { i } } \\Delta _ { i } } \\mathrm { ~ f o r ~ } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "493b53dd2330737e9c4713e5429764f738cfac2afb9fbe73404e0ffd63720a8a.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 177, 378, 434, 405.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 177, 405.3333333333333, 434, 432.66666666666663 ], "spans": [], "index": 18 }, { "bbox": [ 177, 432.66666666666663, 434, 459.99999999999994 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 464, 239, 475 ], "lines": [ { "bbox": [ 106, 464, 239, 476 ], "spans": [ { "bbox": [ 106, 464, 239, 476 ], "score": 1.0, "content": "For the conditional distributions:", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 106, 464, 239, 476 ] }, { "type": "interline_equation", "bbox": [ 115, 480, 478, 529 ], "lines": [ { "bbox": [ 115, 480, 478, 529 ], "spans": [ { "bbox": [ 115, 480, 478, 529 ], "score": 0.73, "content": "\\begin{array} { r l } & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { - j } ) = 0 \\mathrm { i f } j \\geq i , \\mathrm { o t h e r w i s e } \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . } \\\\ & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { 0 } ) = 1 } \\\\ & { \\mathbb { P } _ { y | x } ^ { w } ( y = 1 | x = x _ { j } ) = 1 / 2 + ( w _ { j } ) \\epsilon \\mathrm { f o r } j = 1 , . . . , d } \\end{array}", "type": "interline_equation", "image_path": "be7ca788c3d35cefcb8072d3c144774d69b545dd45d64dcfad3eece9fddf6940.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 115, 480, 478, 496.3333333333333 ], "spans": [], "index": 21 }, { "bbox": [ 115, 496.3333333333333, 478, 512.6666666666666 ], "spans": [], "index": 22 }, { "bbox": [ 115, 512.6666666666666, 478, 529.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 542, 224, 556 ], "lines": [ { "bbox": [ 105, 540, 222, 559 ], "spans": [ { "bbox": [ 105, 540, 148, 559 ], "score": 1.0, "content": "Verifying", "type": "text" }, { "bbox": [ 148, 541, 222, 556 ], "score": 0.91, "content": "\\rho ( \\mathbb { P } _ { w } ^ { ( i ) } , \\mathbb { Q } _ { w } ) \\leq \\Delta _ { i }", "type": "inline_equation" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 105, 540, 222, 559 ] }, { "type": "text", "bbox": [ 106, 562, 355, 576 ], "lines": [ { "bbox": [ 104, 559, 358, 579 ], "spans": [ { "bbox": [ 104, 559, 282, 579 ], "score": 1.0, "content": "Bayes classifier of the domain generated by", "type": "text" }, { "bbox": [ 283, 561, 299, 574 ], "score": 0.92, "content": "\\mathbb { P } _ { w } ^ { ( i ) }", "type": "inline_equation" }, { "bbox": [ 299, 559, 358, 579 ], "score": 1.0, "content": "is as follows:", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 104, 559, 358, 579 ] }, { "type": "interline_equation", "bbox": [ 166, 580, 442, 627 ], "lines": [ { "bbox": [ 166, 580, 442, 627 ], "spans": [ { "bbox": [ 166, 580, 442, 627 ], "score": 0.83, "content": "\\begin{array} { r l } & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { - j } ) = 0 \\mathrm { i f } j \\geq i , \\mathrm { o t h e r w i s e } h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . , N } \\\\ & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { 0 } ) = 1 } \\\\ & { h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { j } ) = 1 \\mathrm { i f } w _ { j } = 1 , \\mathrm { o t h e r w i s e } h _ { S _ { i } } ^ { * } ( \\boldsymbol { x } _ { j } ) = 0 \\mathrm { f o r } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "7c117f4ca1d1aa05f502e08c66a6e3c6901262d5c93c8754db2e5bd5f5257d5f.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 166, 580, 442, 595.6666666666666 ], "spans": [], "index": 26 }, { "bbox": [ 166, 595.6666666666666, 442, 611.3333333333333 ], "spans": [], "index": 27 }, { "bbox": [ 166, 611.3333333333333, 442, 626.9999999999999 ], "spans": [], "index": 28 } ] }, { "type": "text", "bbox": [ 107, 630, 315, 643 ], "lines": [ { "bbox": [ 106, 630, 316, 644 ], "spans": [ { "bbox": [ 106, 630, 261, 644 ], "score": 1.0, "content": "Similarly for the domain generated by", "type": "text" }, { "bbox": [ 261, 631, 276, 642 ], "score": 0.9, "content": "\\mathbb { Q } _ { w }", "type": "inline_equation" }, { "bbox": [ 276, 630, 316, 644 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 29 } ], "index": 29, "bbox_fs": [ 106, 630, 316, 644 ] }, { "type": "interline_equation", "bbox": [ 172, 647, 438, 691 ], "lines": [ { "bbox": [ 172, 647, 438, 691 ], "spans": [ { "bbox": [ 172, 647, 438, 691 ], "score": 0.84, "content": "\\begin{array} { r l } & { h _ { T } ^ { * } ( \\pmb { x } _ { - j } ) = 1 \\mathrm { f o r } j = 1 , . . . , N } \\\\ & { h _ { T } ^ { * } ( \\pmb { x } _ { 0 } ) = 1 } \\\\ & { h _ { T } ^ { * } ( \\pmb { x } _ { j } ) = 1 \\mathrm { i f } w _ { j } = 1 , \\mathrm { o t h e r w i s e } h _ { T } ^ { * } ( \\pmb { x } _ { j } ) = 0 \\mathrm { f o r } j = 1 , . . , d } \\end{array}", "type": "interline_equation", "image_path": "1bdfa6b731e98014c985ce7720c50207b19c2b6d8941e6d5712b7a8a6dd524d5.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 172, 647, 438, 661.6666666666666 ], "spans": [], "index": 30 }, { "bbox": [ 172, 661.6666666666666, 438, 676.3333333333333 ], "spans": [], "index": 31 }, { "bbox": [ 172, 676.3333333333333, 438, 690.9999999999999 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 106, 695, 345, 708 ], "lines": [ { "bbox": [ 105, 694, 347, 711 ], "spans": [ { "bbox": [ 105, 694, 119, 711 ], "score": 1.0, "content": "So", "type": "text" }, { "bbox": [ 119, 696, 135, 708 ], "score": 0.89, "content": "h _ { S _ { i } } ^ { * }", "type": "inline_equation" }, { "bbox": [ 135, 694, 153, 711 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 153, 696, 166, 708 ], "score": 0.9, "content": "h _ { T } ^ { * }", "type": "inline_equation" }, { "bbox": [ 167, 694, 216, 711 ], "score": 1.0, "content": "disagree on", "type": "text" }, { "bbox": [ 216, 698, 268, 707 ], "score": 0.91, "content": "\\pmb { x } _ { - i } , . . , \\pmb { x } _ { - N }", "type": "inline_equation" }, { "bbox": [ 268, 694, 347, 711 ], "score": 1.0, "content": "which implies that", "type": "text" } ], "index": 33 } ], "index": 33, "bbox_fs": [ 105, 694, 347, 711 ] }, { "type": "interline_equation", "bbox": [ 184, 714, 426, 730 ], "lines": [ { "bbox": [ 184, 714, 426, 730 ], "spans": [ { "bbox": [ 184, 714, 426, 730 ], "score": 0.88, "content": "\\rho ( \\mathbb { P } _ { w } ^ { ( i ) } , \\mathbb { Q } _ { w } ) = \\mathbb { Q } [ h _ { S _ { i } } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] - \\mathbb { Q } [ h _ { T } ^ { \\ast } ( \\pmb { x } _ { T } ) \\neq y _ { T } ] = \\Delta _ { i }", "type": "interline_equation", "image_path": "1fa467d600b617b573ad168363ac52e464d481ff42e1afb14c3bcad5da0931b2.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 184, 714, 426, 730 ], "spans": [], "index": 34 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 82, 505, 107 ], "lines": [ { "bbox": [ 105, 81, 506, 94 ], "spans": [ { "bbox": [ 105, 81, 457, 94 ], "score": 1.0, "content": "With the same argument we used in the proof of Theorem 1 we can restrict ourselves to", "type": "text" }, { "bbox": [ 458, 81, 468, 93 ], "score": 0.86, "content": "\\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 468, 81, 506, 94 ], "score": 1.0, "content": "which is", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 496, 107 ], "spans": [ { "bbox": [ 105, 94, 174, 107 ], "score": 1.0, "content": "the projection of", "type": "text" }, { "bbox": [ 175, 95, 185, 105 ], "score": 0.81, "content": "\\mathcal { H }", "type": "inline_equation" }, { "bbox": [ 185, 94, 280, 107 ], "score": 1.0, "content": "with the constraint that", "type": "text" }, { "bbox": [ 280, 94, 437, 107 ], "score": 0.92, "content": "h ( \\pmb { x } _ { - N } ) = \\ldots = h ( \\pmb { x } _ { - 1 } ) = h ( \\pmb { x } _ { 0 } ) = 1", "type": "inline_equation" }, { "bbox": [ 437, 94, 464, 107 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 465, 93, 492, 105 ], "score": 0.91, "content": "h \\in \\tilde { \\mathcal { H } }", "type": "inline_equation" }, { "bbox": [ 492, 94, 496, 107 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 108, 111, 475, 123 ], "lines": [ { "bbox": [ 106, 110, 476, 125 ], "spans": [ { "bbox": [ 106, 110, 476, 125 ], "score": 1.0, "content": "The rest of the proof is exactly the same except the part regarding the KL divergence bound.", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "text", "bbox": [ 106, 128, 397, 144 ], "lines": [ { "bbox": [ 101, 124, 394, 150 ], "spans": [ { "bbox": [ 101, 126, 234, 150 ], "score": 1.0, "content": "KL divergence bound: Define", "type": "text" }, { "bbox": [ 234, 128, 394, 144 ], "score": 0.9, "content": "P _ { w } = \\mathbb { P } _ { w } ^ { ( 1 ) ^ { n _ { S _ { 1 } } } } \\times \\ldots \\times \\mathbb { P } _ { w } ^ { ( N ) ^ { n _ { S _ { N } } } } \\times \\mathbb { Q } _ { w } ^ { n _ { T } } .", "type": "inline_equation" }, { "bbox": [ 289, 124, 369, 149 ], "score": 1.0, "content": "1 × ... × P(N )w nSN ×", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "interline_equation", "bbox": [ 111, 163, 511, 325 ], "lines": [ { "bbox": [ 111, 163, 511, 325 ], "spans": [ { "bbox": [ 111, 163, 511, 325 ], "score": 0.93, "content": "\\begin{array} { r l } { { \\operatorname* { P } _ { k l } ( P _ { w } | P _ { w ^ { \\prime } } ) = \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } \\cdot P _ { k l } ( \\mathbb { P } _ { w } ^ { ( j ) } | \\mathbb { P } _ { w ^ { \\prime } } ^ { ( j ) } ) + n _ { I } \\cdot \\mathcal { P } _ { k l } ( \\mathbb { Q } _ { w } | \\mathbb { Q } _ { w ^ { \\prime } } ) } } \\\\ & { = \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } \\cdot \\mathbb { E } _ { \\mathbb { P } } \\mathbb { P } _ { k l } ( \\mathbb { P } _ { y | z } ^ { w } ) | \\mathbb { P } _ { y | z } ^ { w ^ { \\prime } ( j ) } ) + n _ { T } \\cdot \\mathbb { E } _ { \\mathbb { P } } \\mathcal { P } _ { k l } ( \\mathbb { Q } _ { y | z } ^ { w } | \\mathbb { Q } _ { y | z } ^ { n ^ { \\prime } } ) } \\\\ & { = \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } \\cdot \\displaystyle \\sum _ { \\mathrm { i } = 1 } ^ { G } \\frac { 1 } { d + n _ { S _ { j } } \\Delta _ { j } } \\mathcal { D } _ { k l } ( | \\mathbb { P } _ { y | z } ^ { w } ( \\cdot ) ( i ) | \\mathbb { P } _ { y | z _ { h } } ^ { n ^ { \\prime } } ( \\cdot ) + n _ { T } \\cdot \\displaystyle \\sum _ { \\mathrm { i } = 1 } ^ { d } \\frac { 1 } { 1 0 0 d } \\mathcal { D } _ { k l } ( \\mathbb { Q } _ { y | z _ { h } } ^ { w } | \\mathbb { Q } _ { y | z _ { h } } ^ { n ^ { \\prime } } ) } \\\\ & { \\leq \\displaystyle \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } \\cdot \\frac { d } { d + n _ { S _ { j } } \\Delta _ { j } } c _ { 0 } e ^ { 2 } + n _ { T } \\cdot \\frac { 1 } { 1 0 0 } c _ { 0 } e ^ { 2 } } \\\\ & { \\leq \\displaystyle \\sum _ { j = 1 } ^ { N } n _ { S _ { j } } \\cdot \\frac { d } { d + n _ { S _ { j } } \\Delta _ { j } } c _ { 0 } e ^ { 2 } + n _ { T } \\cdot \\frac { 1 } { 1 0 0 } c _ { 0 } e ^ { 2 } } \\\\ & { \\leq c _ { 0 } c _ { j } ^ { 2 } . d } \\end{array}", "type": "interline_equation", "image_path": "f5a15e30f96dbbc1332084bb2f85d24c7e1aec9e9abe8d7511d0e8910c21ac0e.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 111, 163, 511, 217.0 ], "spans": [], "index": 4 }, { "bbox": [ 111, 217.0, 511, 271.0 ], "spans": [], "index": 5 }, { "bbox": [ 111, 271.0, 511, 325.0 ], "spans": [], "index": 6 } ] }, { "type": "text", "bbox": [ 107, 334, 299, 347 ], "lines": [ { "bbox": [ 106, 333, 301, 348 ], "spans": [ { "bbox": [ 106, 333, 176, 348 ], "score": 1.0, "content": "for small enough", "type": "text" }, { "bbox": [ 177, 336, 186, 345 ], "score": 0.85, "content": "c _ { 1 }", "type": "inline_equation" }, { "bbox": [ 187, 333, 301, 348 ], "score": 1.0, "content": "we can apply Proposition 1.", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "title", "bbox": [ 108, 360, 299, 371 ], "lines": [ { "bbox": [ 106, 359, 300, 372 ], "spans": [ { "bbox": [ 106, 359, 300, 372 ], "score": 1.0, "content": "7.3 ADDITIONAL EXPERIMENTAL RESULTS", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 106, 380, 506, 468 ], "lines": [ { "bbox": [ 105, 379, 505, 392 ], "spans": [ { "bbox": [ 105, 379, 505, 392 ], "score": 1.0, "content": "In section 5 we fix number of source samples and vary the number of target samples. Here in order to", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 391, 505, 403 ], "spans": [ { "bbox": [ 106, 391, 505, 403 ], "score": 1.0, "content": "investigate the effect of source samples on the target generalization error, we fix the number of target", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 402, 505, 414 ], "spans": [ { "bbox": [ 105, 402, 150, 414 ], "score": 1.0, "content": "samples at", "type": "text" }, { "bbox": [ 150, 404, 181, 413 ], "score": 0.85, "content": "\\cdot", "type": "inline_equation" }, { "bbox": [ 182, 402, 505, 414 ], "score": 1.0, "content": "and vary the number of source samples. Fig 5 depicts the theoretical lower bounds", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 413, 505, 425 ], "spans": [ { "bbox": [ 106, 413, 505, 425 ], "score": 1.0, "content": "along with the upper bounds obtained by empirical risk minimization for image classifications. We", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 424, 505, 435 ], "spans": [ { "bbox": [ 106, 424, 505, 435 ], "score": 1.0, "content": "use the same source/target pairs as used in section 5.2. Fig 5 demonstrates that Source1 is more", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 433, 506, 447 ], "spans": [ { "bbox": [ 105, 433, 506, 447 ], "score": 1.0, "content": "helpful in reducing the target generalization error because it has a low distance from the target.", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 446, 505, 457 ], "spans": [ { "bbox": [ 106, 446, 505, 457 ], "score": 1.0, "content": "Furthermore, it shows that increasing the number of source samples is useful up to a point and beyond", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 456, 450, 468 ], "spans": [ { "bbox": [ 105, 456, 450, 468 ], "score": 1.0, "content": "that point the error saturates and does not decrease further as discussed in Remark 10.", "type": "text" } ], "index": 16 } ], "index": 12.5 }, { "type": "image", "bbox": [ 201, 485, 403, 642 ], "blocks": [ { "type": "image_body", "bbox": [ 201, 485, 403, 642 ], "group_id": 0, "lines": [ { "bbox": [ 201, 485, 403, 642 ], "spans": [ { "bbox": [ 201, 485, 403, 642 ], "score": 0.97, "type": "image", "image_path": "68afa0a6eca3c2060650667d2fbccf678290c955da93e7e8eaee731207bf9f06.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 201, 485, 403, 498.0833333333333 ], "spans": [], "index": 17 }, { "bbox": [ 201, 498.0833333333333, 403, 511.16666666666663 ], "spans": [], "index": 18 }, { "bbox": [ 201, 511.16666666666663, 403, 524.25 ], "spans": [], "index": 19 }, { "bbox": [ 201, 524.25, 403, 537.3333333333334 ], "spans": [], "index": 20 }, { "bbox": [ 201, 537.3333333333334, 403, 550.4166666666667 ], "spans": [], "index": 21 }, { "bbox": [ 201, 550.4166666666667, 403, 563.5000000000001 ], "spans": [], "index": 22 }, { "bbox": [ 201, 563.5000000000001, 403, 576.5833333333335 ], "spans": [], "index": 23 }, { "bbox": [ 201, 576.5833333333335, 403, 589.6666666666669 ], "spans": [], "index": 24 }, { "bbox": [ 201, 589.6666666666669, 403, 602.7500000000002 ], "spans": [], "index": 25 }, { "bbox": [ 201, 602.7500000000002, 403, 615.8333333333336 ], "spans": [], "index": 26 }, { "bbox": [ 201, 615.8333333333336, 403, 628.916666666667 ], "spans": [], "index": 27 }, { "bbox": [ 201, 628.916666666667, 403, 642.0000000000003 ], "spans": [], "index": 28 } ] }, { "type": "image_caption", "bbox": [ 106, 652, 505, 675 ], "group_id": 0, "lines": [ { "bbox": [ 106, 652, 505, 663 ], "spans": [ { "bbox": [ 106, 652, 505, 663 ], "score": 1.0, "content": "Figure 5: Depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 662, 413, 675 ], "spans": [ { "bbox": [ 105, 662, 380, 675 ], "score": 1.0, "content": "minimization. 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Here in order to", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 391, 505, 403 ], "spans": [ { "bbox": [ 106, 391, 505, 403 ], "score": 1.0, "content": "investigate the effect of source samples on the target generalization error, we fix the number of target", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 402, 505, 414 ], "spans": [ { "bbox": [ 105, 402, 150, 414 ], "score": 1.0, "content": "samples at", "type": "text" }, { "bbox": [ 150, 404, 181, 413 ], "score": 0.85, "content": "\\cdot", "type": "inline_equation" }, { "bbox": [ 182, 402, 505, 414 ], "score": 1.0, "content": "and vary the number of source samples. Fig 5 depicts the theoretical lower bounds", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 413, 505, 425 ], "spans": [ { "bbox": [ 106, 413, 505, 425 ], "score": 1.0, "content": "along with the upper bounds obtained by empirical risk minimization for image classifications. 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Fig 5 demonstrates that Source1 is more", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 433, 506, 447 ], "spans": [ { "bbox": [ 105, 433, 506, 447 ], "score": 1.0, "content": "helpful in reducing the target generalization error because it has a low distance from the target.", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 446, 505, 457 ], "spans": [ { "bbox": [ 106, 446, 505, 457 ], "score": 1.0, "content": "Furthermore, it shows that increasing the number of source samples is useful up to a point and beyond", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 456, 450, 468 ], "spans": [ { "bbox": [ 105, 456, 450, 468 ], "score": 1.0, "content": "that point the error saturates and does not decrease further as discussed in Remark 10.", "type": "text" } ], "index": 16 } ], "index": 12.5, "bbox_fs": [ 105, 379, 506, 468 ] }, { "type": "image", "bbox": [ 201, 485, 403, 642 ], "blocks": [ { "type": "image_body", "bbox": [ 201, 485, 403, 642 ], "group_id": 0, "lines": [ { "bbox": [ 201, 485, 403, 642 ], "spans": [ { "bbox": [ 201, 485, 403, 642 ], "score": 0.97, "type": "image", "image_path": "68afa0a6eca3c2060650667d2fbccf678290c955da93e7e8eaee731207bf9f06.jpg" } ] } ], "index": 22.5, "virtual_lines": [ { "bbox": [ 201, 485, 403, 498.0833333333333 ], "spans": [], "index": 17 }, { "bbox": [ 201, 498.0833333333333, 403, 511.16666666666663 ], "spans": [], "index": 18 }, { "bbox": [ 201, 511.16666666666663, 403, 524.25 ], "spans": [], "index": 19 }, { "bbox": [ 201, 524.25, 403, 537.3333333333334 ], "spans": [], "index": 20 }, { "bbox": [ 201, 537.3333333333334, 403, 550.4166666666667 ], "spans": [], "index": 21 }, { "bbox": [ 201, 550.4166666666667, 403, 563.5000000000001 ], "spans": [], "index": 22 }, { "bbox": [ 201, 563.5000000000001, 403, 576.5833333333335 ], "spans": [], "index": 23 }, { "bbox": [ 201, 576.5833333333335, 403, 589.6666666666669 ], "spans": [], "index": 24 }, { "bbox": [ 201, 589.6666666666669, 403, 602.7500000000002 ], "spans": [], "index": 25 }, { "bbox": [ 201, 602.7500000000002, 403, 615.8333333333336 ], "spans": [], "index": 26 }, { "bbox": [ 201, 615.8333333333336, 403, 628.916666666667 ], "spans": [], "index": 27 }, { "bbox": [ 201, 628.916666666667, 403, 642.0000000000003 ], "spans": [], "index": 28 } ] }, { "type": "image_caption", "bbox": [ 106, 652, 505, 675 ], "group_id": 0, "lines": [ { "bbox": [ 106, 652, 505, 663 ], "spans": [ { "bbox": [ 106, 652, 505, 663 ], "score": 1.0, "content": "Figure 5: Depicts the lower bounds along with the upper bounds obtained via weighted empirical risk", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 662, 413, 675 ], "spans": [ { "bbox": [ 105, 662, 380, 675 ], "score": 1.0, "content": "minimization. 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